Listening to the Universe through Indirect Detection
LListening to the Universe through Indirect Detection by Nicholas Llewellyn Rodd
M.Sc., University of Melbourne (2012)B.Sc. and LL.B., University of Melbourne (2010)Submitted to the Department of Physicsin partial fulfillment of the requirements for the degree ofDoctor of Philosophyat theMASSACHUSETTS INSTITUTE OF TECHNOLOGYJune 2018c ○ Massachusetts Institute of Technology 2018. All rights reserved.Author . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Department of PhysicsApril 20, 2018Certified by . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Tracy R. SlatyerAssistant Professor of PhysicsThesis SupervisorAccepted by . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Scott A. HughesInterim Associate Department Head of Physics a r X i v : . [ h e p - ph ] M a y istening to the Universe through Indirect Detection byNicholas Llewellyn Rodd Submitted to the Department of Physicson April 20, 2018, in partial fulfillment of therequirements for the degree ofDoctor of Philosophy
Abstract
Indirect detection is the search for the particle nature of dark matter with astro-physical probes. Manifestly, it exists right at the intersection of particle physics andastrophysics, and the discovery potential for dark matter can be greatly extendedusing insights from both disciplines. This thesis provides an exploration of this phi-losophy. On the one hand, I will show how astrophysical observations of dark matter,through its gravitational interaction, can be exploited to determine the most promis-ing locations on the sky to observe a particle dark matter signal. On the other, Idemonstrate that refined theoretical calculations of the expected dark matter inter-actions can be used disentangle signals from astrophysical backgrounds. Both of theseapproaches will be discussed in the context of general searches, but also applied to thecase of an excess of photons observed at the center of the Milky Way. This galacticcenter excess represents both the challenges and joys of indirect detection. Initiallythought to be a signal of annihilating dark matter at the center of our own galaxy,it now appears more likely to be associated with a population of millisecond pulsars.Yet these pulsars were completely unanticipated, and highlight that indirect detectioncan lead to many new insights about the universe, hopefully one day including theparticle nature of dark matter.Thesis Supervisor: Tracy R. SlatyerTitle: Assistant Professor of Physics 3 cknowledgments
Life and science are fundamentally collaborative. In this sense I feel that only havingmy name listed as an author for this thesis is misleading. The work I undertookin grad school, culminating in this thesis, would not have been possible without thepeople around me. For this reason, here I want to take the time to thank the followingpeople for far more than just useful discussions and comments, but for helping megrow as a physicist and a person over the last five years.My best decision and piece of good fortune during grad school was to have TracySlatyer as my advisor. Tracy had me working on research almost immediately afterwe met at the MIT open house. Although I knew little about dark matter andnothing about the
Fermi telescope, she involved me in a project that turned into thegalactic center analysis presented in this thesis. This was undoubtedly one of themost exciting projects I’ve ever worked on, thinking that we may actually be lookingat a signal emanating from dark matter (as you will see in the thesis, we were not).During this period we would meet several times a week, and Tracy with near infinitepatience brought me to a level where I could contribute to that paper. Over timeTracy slowly encouraged me to become more independent, but in a manner where Ihardly noticed it was happening, leaving me feeling confident about life after gradschool. Beyond this Tracy is a truly incredible academic, teacher, and person. Whilethis may be common knowledge, to have such a mentor over the last five years hasbeen a privilege, and I cannot thank her enough. So if you are reading this thanksagain Tracy!After Tracy, the person who had the most influence on my academic experience atMIT was Ben Safdi. Ben, who arrived at MIT as a postdoc in my second year, quicklywent about setting us straight on the galactic center excess, demonstrating that it wasmost likely coming from unresolved point sources, not dark matter. This work wasextremely impressive, and it was clear in my mind Ben was someone I wanted to workwith. So shortly after we started working on a project. After several twists and turnsthis became the galaxy group analysis presented in this thesis, and was just the first5f many projects we undertook. Throughout this time Ben acted almost as a secondadvisor to me, teaching me many new ways to look at problems, and emphasisingagain and again the importance of Monte Carlo! In his role as pseudo advisor, Benhas been an incredible mentor. But beyond this he is a remarkable academic, and hisability to generate interesting research questions and directions is an inspiration. Ourcollaboration was one of my absolute highlights of grad school, and I look forward toour ongoing work in the future.Although Tracy and Ben played the largest role, my academic experience at gradschool has been shaped by a much larger number of people. MIT, and in particularthe CTP, has been the ideal grad school environment. I have had the good fortune towork with all of the MIT phenomenology faculty, which in addition to Tracy includesIain Stewart and Jesse Thaler. Jesse and Iain are giants of the field. Being ableto see how they approach problems and how they dragged me through my own hastaught me a lot. They were incredibly generous with their time both when it came toresearch, but also more generally in discussing career advice. The CTP itself was alsoincredibly welcoming. In my early years, older students were often looking out forme, and really made me feel welcome. Other highlights have included morning coffeediscussions with Yotam and the resource provided by Tracy’s other grad studentsHongwan, Chih-Liang, and Patrick. Non-academic aspects of life in the CTP werealways smooth, and as I know this does not just happen on its own, I have a lot ofappreciation for the hard work of Scott Morley, Joyce Berggren, Charles Suggs, andCathy Modica. My research has enormously benefited from the many collaboratorsI’ve had both at MIT and elsewhere. I have learnt from each and every one of them,but in particular I want to thank long time collaborator Sid Mishra-Sharma for savingour 2mass2furious project by initiating a full rewrite of our code, Grigory Ovanesyanfor dragging me through my first box integral, and Josh Foster for patiently remindingme many times of basic probability. Finally, I also owe a det of gratitude to my physicsadvisors during my masters in Australia. Archil Kobakhidze, Elisabetta Barberio, andin particular Ray Volkas set me off on the path I am now on and have continued toprovide useful advice at times during grad school.6’ve been fortunate enough to have a fantastic experience in the US over the lastfive years, an experience which has been due to more than just science, and is largelythanks to the number of great friends I’ve made during this time. In particular Iwant to thank my next door office mate Nikhil for endless amusing discussions andpointing out the many logical flaws in Journey’s “Don’t Stop Believin’ ” (who arethe streetlight people?), Erik for always motivating me to go the gym and take fulladvantage of nights out, and my other housemates for the good times shared at401, especially Darius and Ryan. Beyond this I am fortunate enough to have stayedconnected with so many friends back in Australia, which is one of the reasons I alwaysenjoy heading home.Finally I want to thank the people who have played the greatest role in getting meto my present position: my family Annabelle, Robyn, David, and Helen. In particularI want to single out my mother. From a young age, her encouragement and supporthave always driven me forwards, and no one in my life has played a larger part inmy development than her. But every member of my family has played an importantrole in supporting me, even though I’m now living on the other side of the world, andI’m endlessly grateful for it. If any one of them had been missing I cannot imagineachieving half of what I have. For the providing this same support I am also extremelygrateful to my partner Felicia. Meeting you has been the best thing to happen to meduring this period. You provide me strength and support, whilst making every daybrighter. I’ve cherished our time together so far and cannot wait for the next chapterto unfold in California. 7 ontents
Planck . . . . . . . . . . . . . . . . . . . . . . . . 1446.6 Dwarf Limits from
Fermi . . . . . . . . . . . . . . . . . . . . . . . . . 1466.7 Positron bounds from AMS-02 . . . . . . . . . . . . . . . . . . . . . . 1516.8 General Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1566.9 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
A.1 Stability Under Modifications to the Analysis . . . . . . . . . . . . . 185A.1.1 Changing the Region of Interest . . . . . . . . . . . . . . . . . 185A.1.2 Varying the Event Selection . . . . . . . . . . . . . . . . . . . 187A.1.3 A Simplified Test of Elongation . . . . . . . . . . . . . . . . . 188A.1.4 Sensitivity of the Spectral Shape to the Assumed Morphology 191A.2 Modeling of Background Diffuse Emission in the Inner Galaxy . . . . 192A.2.1 The
Fermi
Bubbles . . . . . . . . . . . . . . . . . . . . . . . . 192A.2.2 The Choice of Diffuse Model . . . . . . . . . . . . . . . . . . . 193A.2.3 Variation in the 𝜋 Contribution to the Galactic Diffuse Emission194A.2.4 Modulating the 𝜋 Contribution . . . . . . . . . . . . . . . . . 196A.3 Modifications to the Point Source Modeling and Masking for the InnerGalaxy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200A.4 Shifting the Dark Matter Contribution Along the Plane . . . . . . . . 201A.5 Variations to the Galactic Center Analysis . . . . . . . . . . . . . . . 20311
Dark Matter in Galaxy Groups 205
B.1 Extended results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205B.2 Variations on the Analysis . . . . . . . . . . . . . . . . . . . . . . . . 218
C Dark Matter Decay 229
C.1 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229C.1.1 Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229C.1.2 Data analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 236C.2 Likelihood Profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238C.3 Systematics Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242C.4 Extended Theory Interpretation . . . . . . . . . . . . . . . . . . . . . 249C.4.1 Additional Final States . . . . . . . . . . . . . . . . . . . . . . 249C.4.2 Extending
Fermi limits beyond 10 PeV . . . . . . . . . . . . 251C.4.3 Additional Models . . . . . . . . . . . . . . . . . . . . . . . . 254
D Cascade Spectra and the GCE 257
D.1 0-step Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257D.1.1 Annihilations to 𝑒 + 𝑒 − . . . . . . . . . . . . . . . . . . . . . . 258D.1.2 Annihilations to 𝜇 + 𝜇 − . . . . . . . . . . . . . . . . . . . . . . 258D.1.3 Annihilations to 𝜏 + 𝜏 − . . . . . . . . . . . . . . . . . . . . . . 259D.1.4 Annihilations to 𝑏 ¯ 𝑏 . . . . . . . . . . . . . . . . . . . . . . . . 259D.2 Kinematics of a Multi-step Cascade . . . . . . . . . . . . . . . . . . . 259D.3 Model-Building Considerations . . . . . . . . . . . . . . . . . . . . . 262D.3.1 A Simple Model . . . . . . . . . . . . . . . . . . . . . . . . . . 262D.3.2 The Sommerfeld Enhancement . . . . . . . . . . . . . . . . . . 263 E Limits on Cascade Spectra 267
E.1 Details of 𝑛 -body Cascades . . . . . . . . . . . . . . . . . . . . . . . . 267E.2 Details of the CMB Results . . . . . . . . . . . . . . . . . . . . . . . 272E.3 Pass 7 versus Pass 8 for the Dwarfs . . . . . . . . . . . . . . . . . . . 273E.4 Description of Cascade Spectra Files . . . . . . . . . . . . . . . . . . 27412 Dark Matter Annihilation at One-Loop 279
F.1 One-loop Calculation of 𝜒 𝑎 𝜒 𝑏 → 𝑊 𝑐 𝑊 𝑑 in the Full Theory . . . . . . 279F.2 Consistency Check on the High-Scale Matching . . . . . . . . . . . . 299F.3 Low-Scale Matching Calculation . . . . . . . . . . . . . . . . . . . . . 300F.4 Consistency Check on the Low-Scale Matching . . . . . . . . . . . . . 318F.5 Analytic Form of Π . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321134 hapter 1Introduction There is an enormous body of evidence pointing to the existence of dark matter allaround us, and this evidence is entirely consistent with dark matter being a new fun-damental particle. Yet we are almost completely in the dark as to the basic propertiesof this particle if it exists. For example, the mass of the dark matter particle andwhether it experiences interactions with itself or the standard model beyond gravityare completely unknown, although limits exist. Answering these questions, beyondresolving the question of what makes up 85% of the mass in our universe, wouldhave profound implications for both particle physics and astrophysics. Famously, thestandard model of particle physics does not contain a dark matter candidate. Inthis sense, any insight as to the particle nature of dark matter would open a windowinto physics beyond the standard model. In addition, insights into dark matter self-interactions, as well as the interactions between dark matter and the standard modelcould prove important ingredients towards a deeper understanding of how structure The evidence for dark matter is due to a body of theoretical, numerical, and experimental workconducted over decades by large fractions of the physics and astronomy community. The range ofscales over which dark matter’s influence has been observed is staggering. The effects of dark matterstretch from its influence on our local region in the Milky Way, to the role it played in creatingstructures in the early universe, the imprints of which are left in the cosmic microwave background.A recent review of the history of dark matter and the different threads of evidence pointing to itsexistence can be found in [1]. The neutrino, being electrically neutral and having only feeble interactions with the rest of thestandard model, is the only possible candidate. Yet as neutrinos obey Fermi-Dirac statistics, there isa limit on the number density that can be packed into a given structure like a dark matter halo [2].This, combined with existing constraints on the smallness of the neutrino mass, forbid neutrinosfrom making up an 𝒪 (1) fraction of the observed dark matter. One strategy is tolook for the production of dark matter at a collider. At the Large Hadron Collider,for example, dark matter could be produced in a proton-proton collision, which wedepict schematically as 𝑝𝑝 → DM DM . Of course, dark matter is famously hard todetect, so this would not result in an event we could actually see at the experiment.If, however, one of the initial state protons emitted an observable particle such as ajet, weak boson, or photon, that we collectively denote 𝑋 , then the process wouldbecome 𝑝𝑝 → DM DM + 𝑋 . By looking for this single 𝑋 particle, and a largeamount of missing energy associated with the fact we cannot see the dark matterparticles, one can effectively search for various dark matter candidates. This strategyis generically referred to as a mono- 𝑋 search, and for a recent review of the colliderapproach, see e.g. [4]. The second strategy is to search for the signatures of a darkmatter particle scattering with the standard model, through an interaction of theform SM DM → SM DM . Such a scattering would cause the standard model particleto recoil, and if such an effect were detected it would be a direct indication of theinfluence of dark matter particles. This approach, referred to as direct detection, hasdeveloped into a small industry, setting incredibly strong limits on the rate at whichsuch scattering can occur. A review of this approach can be found in, e.g. [5].The final paradigm, which represents the focus of this thesis, is referred to asindirect detection, and will be introduced in the following section. Before proceeding,we note that often when referring to dark matter as a particle in this thesis, therewill be an implicit assumption that it is a particle with a mass not too differentto the particles in the standard model, the canonical example being an electroweakscale, 𝒪 (100 GeV) , supersymmetric weakly interactive massive particle (WIMP), seee.g. [6,7]. Nevertheless, we mention in passing that it is possible the dark matter could Another possibility is to search for dark matter self-interactions, which could leave fingerprints onstructures throughout the universe, as many of these are dark matter dominated. A comprehensivediscussion of this approach can be found in [3].
16e an extremely light boson, potentially as light as − eV [8]. This mass rangeincludes theoretically well motivated particles such as the QCD axion [9–12], which inaddition to solving the strong CP problem is a viable dark matter candidate [13–15].For such particles it is often more useful to think of the dark matter as a coherent field,rather than individual particles, just as how it is convenient to move from photonsto waves when describing the electromagnetic field at lower energies. This leads to amodification for the search strategies. In recent years there has been a resurgence ofefforts to search for the axion, and during grad school I contributed to this effort byintroducing an analysis framework for the direct detection of axions [16], although itwill not be discussed further here. If dark matter is a particle that has a fundamental interaction with the standardmodel, then it is possible that it could annihilate or decay into standard model finalstates. This possibility, first suggested in 1978 [17, 18], is the dark matter analogueof familiar processes in the standard model, such as electron-positron annihilation tophotons 𝑒 + 𝑒 − → 𝛾𝛾 or muon decay 𝜇 − → 𝑒 − ¯ 𝜈 𝑒 𝜈 𝜇 . We can represent the dark sectoranalogues schematically as DM DM −→ SM particles , DM −→ SM particles . (1.1)In both cases the identity of the standard model (SM) particles in the final statedepends on the model and the mass of the dark matter (DM), as if it is too lightcertain states become kinematically inaccessible. For annihilations, the standardWIMP picture is that the dark matter is its own antiparticle, allowing this processto occur, although if this is not the case, then the process above represents a darkmatter particle antiparticle annihilation. A schematic depiction of the annihilationcase is presented in Fig. 1-1.Such interactions appear generically in a large class of dark matter models. The17igure 1-1: A schematic depiction of dark matter annihilation to standard model finalstates.same interactions that can give rise to dark matter annihilations can play an importantrole in the early universe, as at temperatures well above the dark matter mass theycan keep the dark matter and standard model in thermal equilibrium. Then, usingour detailed understanding of the thermal history of the universe, this process leadsto a prediction for the resultant dark matter mass fraction in the universe, whichis a well measured observable. Famously, if the dark matter mass and cross sectionboth occur at the electroweak scale, we can exactly explain the observed dark matterdensity, a phenomena known as the WIMP miracle. In fact for a wide range of masses,an electroweak cross section of ⟨ 𝜎𝑣 ⟩ ≈ × − cm / s is required in order to obtainthe observed abundance of dark matter. We refer elsewhere for a detailed review ofthese points, see e.g. [19], however for our purposes this indicates a particular crosssection value as an important target for indirect detection searches. For the case ofdark matter decay, for this particle to make up the dark matter of our universe, weexpect its lifetime to be much larger than the age of the universe, which is ∼ s .Considering the type of particle interactions that could induce such a decay, valuesfor the dark matter lifetime of s or larger are well motivated, although we putthe details aside for now as this will be discussed in detail in Chapter 4 of this thesis.The important point at this stage, is that these values provide a benchmark forexperimental searches for these effects. 18he central idea of indirect detection is that if these processes are occurringthroughout the universe, then the standard model final states could be detectable.As a simple example, if the final states are photons, then the universe should beilluminated by these processes in regions of high dark matter density. Already atthis stage, the two challenges of indirect detection can be identified. On the one handastrophysical inputs are required to determine what these regions of high dark matterdensity are, essentially identifying where we should look on the sky. On the other,particle physics dictates the result of the processes in (1.1); in particular it sets whattypes of final states we should see in a detector, and at what energies they will beobserved. In the next section we will make this precise and derive some basic resultsfor indirect detection that will be used throughout this thesis. The goal of this section is to calculate carefully what flux of particles from dark mat-ter annihilations and decays would be predicted to arrive at a detector on Earth.To simplify the calculation we will imagine that the standard model final states arephotons, which will effectively free stream directly from the site of their productionto the detector. This should be contrasted with charged final states, such as electronsand positrons, that due to the magnetic fields that permeate the Milky Way and uni-verse more generally take a much more complicated path to a detector. Although thisdiffusion of charged final states can be approximately accounted for, within this thesiswe will focus almost exclusively on photon detectors and so the current discussionwill suffice.To begin with let us consider the case of dark matter annihilation to photons.The ultimate goal is to derive the flux of photons deposited on a detector at Earthdue to these annihilations, but as a starting point we will derive the rate at whichannihilations are occurring in some arbitrary volume in the universe. To this end,imagine we had the configuration shown in Fig. 1-2: a box of volume 𝑉 uniformlyfilled with a large number 𝑁 of identical dark matter particles, which are their own19igure 1-2: A cartoon depiction of the framework used to derive the expected flux atan experiment due to dark matter annihilation. The starting point to this argumentis to consider 𝑁 dark matter particles in a box of volume 𝑉 , and consider the rate atwhich annihilations occur in this box. Here we are considering the dark matter as itsown antiparticle, and thus any of these particles can annihilate with any other. Thebackground image is of the Andromeda galaxy, and is taken from [20].antiparticle. This last condition means any particle can annihilate with any other, andis chosen as it is a common feature in dark matter models, although the case wherethe particle and antiparticle are distinguishable is a straightforward generalisation.Now consider one of these particles and let us move to a frame where it is at rest. Wewill consider this particle to be a target for the interaction leading to the annihilation.The size of the target is set by the cross sectional area, 𝜎 . In this frame the remaining 𝑁 − particles will form an incident flux, and if one intersects the target’s cross sectionit will initiate the annihilation. The number density of the particles contributing tothe flux is 𝑛 DM ≡ ( 𝑁 − /𝑉 ≈ 𝑁/𝑉 , (1.2)20here we assumed the number of dark matter particles is large. In terms of thisthe incident flux density of particles is 𝑣𝑛 DM , where 𝑣 represents the relative velocitybetween the particles as we are in the target’s rest frame. For the time being weimagine this velocity is fixed for all particles, but we will consider the more realisticcase where it is drawn from a distribution shortly. Combining this with the targetarea, 𝜎 , the rate at which an annihilation with this one particle would occur is simplythe target area combined with the incident flux, or 𝜎𝑣𝑛 DM . To determine the rate ofannihilations in the whole box, we repeat this exercise but letting each particle takea turn as the target, which enhances the rate by 𝑁 . But this enhancement includesa double counting. To see this, if two of the particles were labelled 𝑎 and 𝑏 , we havecounted the case where 𝑎 is a target hit by 𝑏 , and where 𝑏 is a target hit by 𝑎 . Moregenerally the number of pairs we can make from 𝑁 particles is 𝑁 ( 𝑁 − / , not 𝑁 ( 𝑁 − as our naive initial counting suggested. Consequently the rate at whichannihilations occur in this box is given by 𝑁 𝜎𝑣𝑛 DM . Then to remove any referenceto the box, which was just a calculational tool, we instead re-express this as therate of annihilations per unit volume, 𝜎𝑣𝑛 . As a final step, we address the factthat realistically the relative velocities will be drawn from a distribution which weshould average over. This then allows us to write the number of annihilations perunit volume per unit time as 𝑑𝑁 ann . 𝑑𝑉 𝑑𝑡 = 12 ⟨ 𝜎𝑣 ⟩ 𝑛 . (1.3)In this expression the ⟨ · ⟩ indicates an averaging over the velocity distribution, andwe have also accounted for the fact that the cross section can in general depend onthe relative velocity.The above expression contains the intuitive fact that the higher the number den-sity of dark matter particles, the more often the particles will find each other, andtherefore the larger the annihilation rate will be. Nevertheless, the rate depends onthe number density of dark matter particles at a specific location in the universe,which is not something we can experimentally observe at present. Instead our best21ool is gravitational probes of dark matter, and gravitational effects are sensitive tothe mass density 𝜌 DM = 𝑚 DM 𝑛 DM . Here the dark matter mass, 𝑚 DM , has to be viewedas an input from the particle physics side. To account for this, it is convenient torewrite the above expression as 𝑑𝑁 ann . 𝑑𝑉 𝑑𝑡 = ⟨ 𝜎𝑣 ⟩ 𝑚 𝜌 . (1.4)The expression in (1.4) achieves our goal of giving the rate of annihilations pervolume at some point in the universe. Next we want to determine the incident fluxand spectrum of photons resulting from these annihilations. For this we need aparticle physics input, which is the spectrum of photons per annihilation describingthe schematic process in (1.1), denoted 𝑑𝑁 𝛾 /𝑑𝐸 . The spectrum is a function ofenergy itself, and can be defined as 𝑑𝑁 𝛾 /𝑑𝐸 ( 𝐸 ) giving the number of photons in theenergy range [ 𝐸, 𝐸 + 𝑑𝐸 ] . Further, the total number of dark matter particles expectedfrom an annihilation can be readily determined from the spectrum as 𝑁 𝛾 = ∫︁ 𝐸 max 𝑑𝐸 𝑑𝑁 𝛾 𝑑𝐸 , (1.5)where 𝐸 max is the maximum photon energy allowed by the kinematics of the process,so 𝐸 max = 𝑚 DM for annihilation and 𝑚 DM / for decay. Note 𝑁 𝛾 need not be aninteger, as the annihilation process is dictated by quantum mechanics and hence isintrinsically probabilistic. Instead the actual number of particles emerging from agiven annihilation will be a draw from a Poisson distribution with mean 𝑁 𝛾 . Theshape of the spectrum is dictated by at what energies it is most probable to emit aphoton, and this probability is determined by quantum field theory. Accordingly, it We note another common convention is to consider the annihilation spectrum per dark matterparticle rather than per annihilation, which will reduce spectra by a factor of two compared to thosepresented in this thesis. 𝑁 𝛾 𝑑𝑁 𝛾 𝑑𝐸 = 1 ⟨ 𝜎𝑣 ⟩ 𝑑 ⟨ 𝜎𝑣 ⟩ 𝑑𝐸 . (1.6)The spectrum is highly model dependent and will be discussed extensively in thisthesis, but to provide a concrete example, consider the particularly simple case wheredark matter annihilates to two photons, DM DM → 𝛾𝛾 . In the center of mass framefor the collision, this spectrum takes the form 𝑑𝑁 𝛾 𝑑𝐸 = 2 𝛿 ( 𝐸 − 𝑚 DM ) , (1.7)such that there are two photons produced, and their energy is fixed to be the darkmatter mass by the simple → kinematics. Returning to the case of a generalspectrum, combining this with the annihilation rate in (1.4), the number of photonsper unit volume and per unit energy produced by annihilations is then 𝑑𝑁 𝛾 𝑑𝐸𝑑𝑉 𝑑𝑡 = ⟨ 𝜎𝑣 ⟩ 𝑚 𝑑𝑁 𝛾 𝑑𝐸 𝜌 . (1.8)At this stage we just know the rate at which photons are being injected into theuniverse. If we want to detect this effect, the quantity of interest is the number ofthese photons incident on a detector at Earth. On average, the photons produced willdisperse isotropically out over a sphere. If the proper distance between the volumeelement under consideration and the telescope is 𝑠 , then by the time the photons reachthe Earth they are spread over an area 𝜋𝑠 . Imagining that we have a detector witha differential effective area 𝑑𝐴 , then only 𝑑𝐴/ (4 𝜋𝑠 ) of the photons produced will A similar result for decays holds if the cross section is replaced by the decay rate, ⟨ 𝜎𝑣 ⟩ → Γ .This point is expanded upon in App. D. We note that detection at Earth is not the only way to determine the impact of these processes.If they have been occurring throughout the history of the universe, then their impact can be seenelsewhere, for example through perturbations to the cosmic microwave background. This is a pow-erful probe and can be used to constrain annihilation [21, 22], decay [23], and even contributions toprocesses such as reionization [24]. The effective area is an efficiency corrected notion of detector area. The larger the collection areaof the telescope, the more photons will be detected following the argument in the text. Neverthelessany realistic experiment will not perfectly detect every incident photon, and instead will only do so
23e detected, and we have to downweight the number of photons produced as givenin (1.8) by this factor. Doing so, we arrive at 𝑑𝑁 𝛾 𝑑𝐸𝑑𝑉 𝑑𝐴𝑑𝑡 = ⟨ 𝜎𝑣 ⟩ 𝜋𝑚 𝑑𝑁 𝛾 𝑑𝐸 𝜌 𝑠 . (1.9)It is convenient at this point to define the notion of differential photon flux incidenton the detector, as 𝑑 Φ 𝛾 ≡ 𝑑𝑁 𝛾 𝑑𝐴𝑑𝑡 , (1.10)which has units of photons per effective area per time. It includes all the experimentalquantities, such as telescope size, efficiency, and observation time; increasing any ofthese leads to more collected photons. This definition allows us to rewrite (1.9) as 𝑑 Φ 𝛾 𝑑𝐸 = ⟨ 𝜎𝑣 ⟩ 𝜋𝑚 𝑑𝑁 𝛾 𝑑𝐸 𝜌 𝑠 𝑑𝑉 . (1.11)This last expression represents the differential energy flux produced by dark matterannihilations from a differential volume element 𝑑𝑉 a distance 𝑠 away. But of coursethere is a lot of dark matter out there in the universe, all of which can contributephotons at the detector. To account for this we will want to integrate over somevolume of dark matter, accounting for the fact that dark matter is not distributed with some efficiency. The effective area is a way of quantifying this, and operates such that if youhave a telescope of area 2 m with a 50% efficiency, then the effective area is 1 m . In general theefficiency and hence effective area will vary with energy, and additional details such as the incidentangle of the photon on the detector and where it hit, although we will put these complications asidefor the present discussion. By writing the same energy on both sides of this expression we have implicitly assumed theemitted photon energy is equal to the detected value. In general this will not be true. For oneexample, a photon emitted a cosmological distance from the detector will be redshifted during itspropagation. Another example would be if the center of mass frame of the annihilation differs fromthe detector rest frame, the energy will be shifted. This effect would cause an initial 𝛿 function lineto be smeared out by the velocity dispersion of the dark matter. We put these caveats aside for now,but where relevant will address them in the main body of this thesis. There are two common variants of flux used in indirect detection: 1. particles per unit effectivearea per unit time; and 2. particles per unit effective area per unit time per unit solid angle on thesky. Our definition here corresponds to the former, and note that the difference between them isthat the first definition is the integrated version of the second over the full sky. We will not addressthis issue further, although we emphasize caution is required as the two naively differ by a factor 𝒪 (4 𝜋 ) , and errors due to confusing the two exist in the literature. A careful description of the unitsappeared in one of the works I completed during grad school, see App. A of [25]. 𝜌 DM position dependent. It is then convenient to perform thisintegral in a spherical coordinate system centered on the Earth, so that we can write 𝑑𝑉 = 𝑠 𝑑𝑠𝑑 Ω . From here note that by observing different regions of the celestialsphere, we can restrict the patch of solid angle we look at, but we cannot isolate aspecific radial scale in general. Incident photons could have come from a dark matterannihilation 1 mm or 1 Gpc from the detector and we could not distinguish them.As such we need to integrate over all distances. With this in mind, let us say weobserve a region of solid angle Σ , which could be the full sky or a one degree circlearound the galactic center for example, then the total detected energy flux is simplythe integrated version of (1.11), and is given by 𝑑 Φ 𝛾 𝑑𝐸 = ⟨ 𝜎𝑣 ⟩ 𝜋𝑚 𝑑𝑁 𝛾 𝑑𝐸 ∫︁ ∞ 𝑑𝑠 ∫︁ Σ 𝑑 Ω 𝜌 ( 𝑠, Ω) . (1.12)This expression achieves our goal of expressing the photon flux arriving at anexperiment on Earth due to dark matter annihilations. Integrated over the energyrange of the telescope, we can determine the expected flux, which we can turn into anexpected number of observed photons when combined with the experimental param-eters, specified via the detector effective area and observation time. Yet predictingthis flux is entirely dependent upon our ability to determine the various quantitiesappearing on the right hand side of (1.12). This is the central challenge of indirectdetection and the focus of the work in this thesis.Observe that the various terms appearing in (1.12) have factorized into quantitiesdictated by particle physics—the cross section, mass, and spectrum—and those fixedby astrophysics—the dark matter density. Motivated by this, it is common to In some models of dark matter, this factorization is not always exact. For example, if the crosssection has a large velocity dependence, which can happen in models with Sommerfeld enhancementof the annihilation [26–30], then the result is dependent on the velocity distribution of dark matterwithin Σ , which is also determined by astrophysics. We will not consider such cases in this thesis,although see [31, 32], for some recent work in this direction. 𝑑 Φ 𝛾 𝑑𝐸 = 𝑑 Φ PP 𝛾 𝑑𝐸 × 𝐽 Σ ,𝑑 Φ PP 𝛾 𝑑𝐸 ≡ ⟨ 𝜎𝑣 ⟩ 𝜋𝑚 𝑑𝑁 𝛾 𝑑𝐸 ,𝐽 Σ ≡ ∫︁ ∞ 𝑑𝑠 ∫︁ Σ 𝑑 Ω 𝜌 ( 𝑠, Ω) . (1.13)The quantities on the final two lines are referred to as the particle physics factor, andthe 𝐽 -factor respectively.For the case of decaying dark matter, an analogous argument holds, which wesketch out below. Consider again 𝑁 particles in a box of volume 𝑉 , and assumenow these particles have a lifetime 𝜏 . The probability that one of these particleshas decayed after a time 𝑡 is given by the cumulative distribution function − 𝑒 − 𝑡/𝜏 .Accordingly if we have 𝑁 particles undergoing the same process, then the expectednumber remaining after a time 𝑡 is 𝑁 (1 − 𝑒 − 𝑡/𝜏 ) , and rate of these decays is then thetime derivative of this quantity. Thus the rate of decays per unit volume is given by 𝑑𝑁 dec . 𝑑𝑉 𝑑𝑡 = 𝑒 − 𝑡/𝜏 𝜏 𝑛 DM . (1.14)Recall for the dark matter to still be around, we require 𝜏 ≫ 𝑡 universe , and indeedmany models predict the lifetime to be many orders of magnitude longer than the ageof the universe. As such, 𝑒 − 𝑡/𝜏 ≈ is a very good approximation, and again movingto the more convenient mass density, we have 𝑑𝑁 dec . 𝑑𝑉 𝑑𝑡 = 1 𝑚 DM 𝜏 𝜌 DM . (1.15)From here the argument is completely analogous to the annihilation case, and we26rrive at the factorized expression 𝑑 Φ 𝛾 𝑑𝐸 = 𝑑 Φ PP 𝛾 𝑑𝐸 × 𝐷 Σ ,𝑑 Φ PP 𝛾 𝑑𝐸 ≡ 𝜋𝑚 DM 𝜏 𝑑𝑁 𝛾 𝑑𝐸 ,𝐷 Σ ≡ ∫︁ ∞ 𝑑𝑠 ∫︁ Σ 𝑑 Ω 𝜌 DM ( 𝑠, Ω) . (1.16)Both the particle physics and astrophysical factors are different in this case, witharguably the most striking distinction occurring in the astrophysical dependence. Forannihilation we have 𝐽 , which depends on 𝜌 , whereas 𝐷 is related to 𝜌 DM . Due tothis, at first order the 𝐷 -factor is just sensitive to the total dark matter mass of theobject being observed, whereas the 𝐽 -factor has a more complicated dependence onthe substructure of the dark matter within an object, and in particular contributionsfrom locally overdense regions can be strongly enhanced due to the 𝜌 scaling. Thisimplies immediately that the optimal targets of observation could well differ betweenannihilation and decay, and this is a topic explored in this thesis. In the main work presented in this thesis we will consider very explicit forms of thevarious indirect detection quantities derived in the previous section. Before going intothis however, here we consider simplified versions of these expressions and considertheir scaling with the various parameters. Further, we will consider how to use thesescalings to estimate the reach of indirect detection experiments.To facilitate this estimate, consider the particularly simple case of annihilation ordecay to two photons, as then we can use the simplified form of the spectrum providedin (1.7) to simplify the results in (1.13) and (1.16). Doing so, we have 𝑑 Φ ann .𝛾 𝑑𝐸 = ⟨ 𝜎𝑣 ⟩ 𝐽 Σ 𝜋𝑚 𝛿 ( 𝐸 − 𝑚 DM ) ,𝑑 Φ dec .𝛾 𝑑𝐸 = 𝐷 Σ 𝜋𝑚 DM 𝜏 𝛿 ( 𝐸 − 𝑚 DM ) . (1.17)27urther, if we assume that the detector effective area ℰ and the observation time 𝑇 are independent of energy and Σ , we can determine the expected number of photonsfrom the above as 𝑁 ann .𝛾 ⃒⃒ 𝐸 = 𝑚 DM = ⟨ 𝜎𝑣 ⟩ 𝐽 Σ ℰ 𝑇 𝜋𝑚 ,𝑁 dec .𝛾 ⃒⃒ 𝐸 = 𝑚 DM = 𝐷 Σ ℰ 𝑇 𝜋𝑚 DM 𝜏 , (1.18)where we have noted that this only holds for 𝐸 = 𝑚 DM , otherwise the detector willsee zero photons.At this point we can already estimate what type of experiment we might needto observe these effects. To this end, consider the case of a 100 GeV dark mattercandidate, which annihilates with the canonical thermal relic cross section ⟨ 𝜎𝑣 ⟩ =3 × − cm /s. For our observation, we take the Andromeda galaxy, which isexpected to be the brightest extragalactic source of dark matter annihilation, with 𝐽 Σ ≈ × GeV / cm · sr [33]. To detect this, we clearly need to see at least onephoton in our detector, and to have a better chance imagine we wanted to observe40. Then inverting the relation in (1.18), we obtain ℰ 𝑇 ≈ 𝜋 × cm · s . (1.19)The photons produced from this annihilation will be exactly at 100 GeV. This isa particularly challenging energy to observe gamma-rays, as they interact stronglywith the Earth’s atmosphere, initiating a shower of particles. At TeV and higherenergies these showers become large enough that they can be observed on the planet’ssurface. Yet at 100 GeV this is challenging, and to obtain a significant flux a satellitebased experiment is required. Given the costs and challenges associated with gettingexperiments into orbit, a telescope with a collection area of ∼ , or cm , isclose to as large as we could expect the effective area of such an instrument to be.Such an experiment would then have to observe Andromeda for 𝑇 ≈ 𝜋 × s ≈
10 years . (1.20)28igure 1-3: The gamma-ray sky as observed by the Fermi
Space Telescope. Muchof the work in this thesis is devoted to searching for the imprints of dark matterin this dataset. In detail, the data represents photons collected from 200 MeV – 2TeV between August 4, 2008 through July 7, 2016. In addition we are only showingthe photons with the highest quality angular reconstruction. The image represents aMollweide projection of the celestial sphere in galactic coordinates, with the centerof the Milky Way in the middle of the image.This is a significant, although not unimaginable time scale. Further, by combiningobservations of different targets with comparable values for 𝐽 Σ , one could hope tobuild up a similar amount of dark matter flux. Excitingly, the Fermi gamma-rayspace telescope exactly fits the criteria described above. Launched on June 11, 2008,it has almost exactly 10 years of data, and has a collection area of approximately1 m . This indicates that the ideal dataset for detecting electroweak scale darkmatter annihilation has already been collected, and explains why data from the Fermi satellite will feature heavily in this thesis. The dataset collected by
Fermi is shownin Fig. 1-3, which is a picture of the sky in gamma-rays.Returning to our estimate for the scalings of indirect detection, we need to addressthe reality that the universe emits photons at almost all energies due to non-dark mat-ter related phenomena. The challenge then is to find a hint for a dark matter signalon top of this background, and for this we need an estimate for its contribution. Thebackground contribution will vary with energy, however generically follows a power29aw ( 𝑑𝑁/𝑑𝐸 ∼ 𝐸 − 𝑛 ), with various breaks corresponding to different physical phenom-ena. Depending on the energy range of interest, the physical processes responsiblefor generating these photons varies. In this thesis we will be primarily focussed ongamma-rays, which are photons with energy higher than ∼ MeV. At these energies,astrophysical photons emerge from non-thermal processes. The dominant contribu-tion arises from cosmic-ray proton collisions with interstellar hydrogen, which leadsto a 𝑝 − 𝑝 collision, much like the Large Hadron Collider. In such processes, neutralpions are produced copiously, which leads to photon production through their decay 𝜋 → 𝛾𝛾 . Generically, we expect the initial cosmic-ray protons to have an energyspectrum which is a power law, scaling as 𝐸 − , as a result of Fermi shock acceler-ation [34], one of the dominant astrophysical mechanisms for accelerating chargedparticles to high energies. The photons produced from the subsequent 𝑝 − 𝑝 collisonswill be expected to have a softer spectrum than the initial protons, but as a first orderestimate, we can take the gamma-ray spectrum to also have a generic 𝐸 − scaling. Accordingly around these energies we expect the approximate scaling Φ bkg .𝛾 𝑑𝐸 = 𝐴 (︂ 𝐸 )︂ − , (1.21)where 𝐴 is an energy independent constant that determines the flux received at 1 GeV.Assuming we can look away from the plane of the Milky Way, then a large contributionto the background comes from the position independent isotropic emission, and Fermi measurements [35] estimate this to have amplitude 𝐴 ≈ Ω Σ − photons / cm / s / GeV . (1.22)The amount of flux arriving from isotropic emission is of course dependent upon the More realistically, the spectra associated with various astrophysical sources are not perfectlydescribed by a power law, and where they are the index can deviate from − . For example, the Fermi gamma-ray telescope has estimated the isotropic gamma-ray spectrum to scale as ∼ 𝐸 − . ,but with an exponential cutoff on the spectrum at several hundred GeV [35]. Further, the dominant 𝑝 − 𝑝 galactic contribution, has an even softer spectrum of ∼ 𝐸 − . , although sources with harderspectra exist, such as the Fermi bubbles [36]. As such, the background model used here is only arough approximation, but is sufficient for the simple scaling arguments presented. Ω Σ in the above expression. Now tocompare to our line search, we are interested in the background flux near a particulardark matter mass. For the sake of estimating the sensitivity to GeV scale dark matter,if we approximate the energy resolution as ≈ 𝑁 bkg .𝛾 = 10 − Ω Σ ℰ 𝑇𝑚 . (1.23)where we now require 𝑚 DM is measured in GeV. Of course we emphasize that thisis a crude estimate of the actual gamma-ray background, for at least two reasons.Firstly, as suggested above the background emission is often softer than 𝐸 − , althoughnot dramatically. Secondly, generically the highly non-isotropic gamma-ray emissionassociated with sources within the Milky Way has a significant contribution, so thisusually must be accounted for. This second point is evident in Fig. 1-3, where the Fermi dataset is brightest along the plane of the Milky Way. Nevertheless thesecomplications should not significantly impact the simple estimates we are seekinghere.So now from (1.18) and (1.23) we have a model for the expected signal and back-ground contributions to the photon flux at an experiment. We want to combine thetwo of these, which we will refer to as the signal counts 𝑆 and background counts 𝐵 , to determine the scaling of the indirect detection sensitivity. To this end we needa statistical, and more specifically a likelihood, framework. Focussing on higher en-ergies, say X-ray and above, the experiments are effectively counting the number ofincident photons. This implies the correct likelihood framework is the Poisson like-lihood, where we have a predicted mean 𝑆 + 𝐵 . When performing analyses in theremainder of this thesis, this is the approach we will use, but for the following simpleestimate we will assume we have a large enough number of photons that the Gaus-sian likelihood with mean 𝑆 + 𝐵 and standard deviation √ 𝑆 + 𝐵 provides a goodapproximation. In detail, if we observe 𝑑 photons, then we can write the likelihoodas ℒ ( 𝑑 | 𝑆, 𝐵 ) = 1 √︀ 𝜋 ( 𝑆 + 𝐵 ) exp [︃ − ( 𝑑 − 𝑆 − 𝐵 ) 𝑆 + 𝐵 ) ]︃ . (1.24)31o test for the discovery of dark matter, a convenient test statistic to define is twicethe log ratio of a hypothesis with and without dark matter, specifically TS = 2 [ln ℒ ( 𝑑 | 𝑆, 𝐵 ) − ln ℒ ( 𝑑 | , 𝐵 )] . (1.25)Substituting in the Gaussian form of the likelihood, we have TS = − ( 𝑑 − 𝑆 − 𝐵 ) 𝑆 + 𝐵 + ( 𝑑 − 𝐵 ) 𝐵 − ln (︂ 𝑆𝐵 )︂ , (1.26)which is of course again a function of the signal and background models, as well asthe data. To calibrate our expectations, imagine that the data is actually perfectlydescribed by a model with the background and signal, i.e. 𝑑 is a Poisson drawfrom 𝑆 + 𝐵 . To determine our expected reach in this scenario, we can use theAsimov analysis framework [37], where we obtain the asymptotic expectation for TSunder many experimental realisations by using 𝑑 = 𝑆 + 𝐵 . If so, then denoting theasymptotic TS as ̃︁ TS , we have ̃︁ TS = 𝑆 𝐵 − ln (︂ 𝑆𝐵 )︂ ≈ 𝑆 ( 𝑆 − 𝐵 ≈ 𝑆 𝐵 . (1.27)In the second step above, we assumed that 𝑆 + 𝐵 ≈ 𝐵 , namely that the backgroundwill always be much larger than the signal. Given that we have not discovered darkmatter using these techniques, this is often a good approximation. In the thirdstep, we used the fact that even though 𝐵 ≫ 𝑆 , we still want significantly more thanone photons from dark matter to have a chance of discovering it, so 𝑆 ≫ . Now inthe case where we can ignore the look elsewhere effect, we can relate the TS to thelocal significance for discovery, 𝜎 , according to √︀ ̃︁ TS = 𝜎 . Requiring a 5 𝜎 significancediscovery then fixes 𝑆 = 5 √ 𝐵 . (1.28) An exception to this can occur for line searches, where the entire dark matter flux is verylocalised in energy, whereas the background is not. As such, if the energy resolution of the detectoris good enough one can potentially achieve
𝑆 > 𝐵 . This point will not impact the thrust of ourmain scaling arguments, however, so we put it aside. ⟨ 𝜎𝑣 ⟩ ∼ 𝑚 DM 𝐽 Σ √︂ Ω Σ ℰ 𝑇 ,𝜏 − ∼ 𝐷 Σ √︂ Ω Σ ℰ 𝑇 . (1.29)A number of basic scalings for indirect detection can be seen in these results. Aswould be expected, increasing the 𝐽 or 𝐷 -factor, or similarly the effective area orobservation time, allow us to reach smaller cross sections and inverse lifetimes, asdoes reducing the background. Interestingly, we see that finding better targets forobservation, namely finding objects with better astrophysics factors, has a largerimpact than improvements on the other parameters. Note also that sensitivity to theannihilation cross section degrades with increasing dark matter mass, as highlightedby the presence of 𝑚 DM in the above expression, whilst the lifetime sensitivity ismass independent. This basic scaling is common in indirect detection results, andis usually described as originating from the following heuristic argument. Due togravitational probes, we can determine the amount of dark matter mass in an object.As we increase the mass of the individual dark matter particles, we must reducetheir number density 𝑛 DM . As annihilation is dependent upon 𝑛 and decay 𝑛 DM ,this alone leads to a reduction of sensitivity in the two cases as /𝑚 and /𝑚 DM respectively. Yet in both cases, the higher mass increases the power injected per eventby 𝑚 DM . The combination of the two effects reproduces the scaling in (1.29). Yetfrom the derivation of that result, we can see that alternative assumptions about theshape of the signal spectrum or the background can lead to variations in the scalingwith mass.We have reached the limit of what we can achieve with the rough scaling argumentspresented above. Within this thesis we will not only refine such arguments, but moreimportantly go beyond them to extend the discovery potential for dark matter inindirect detection through novel analysis strategies and refined theoretical predictions.33 .4 Organization of this Thesis As we have seen, the indirect detection flux factorizes into a contribution from astro-physics and particle physics. In the same fashion, this thesis and much of my workduring grad school approximately factorize down the same line. The first half of thethesis, Chapters 2, 3, and 4, will focus on how we can search for evidence of darkmatter, or set limits in its absence, by considering promising astrophysical targets.The second half, Chapters 5, 6, and 7, will turn to refining the particle physics pre-dictions, and demonstrating how these can enhance our understanding of how darkmatter might first appear in the sky. Note that for each of the substantive chaptersin the main text there is an associated appendix where many of the technical detailsappear.In more detail, the first half will be further subdivided into three parts. In thefirst of these, presented in Chapter 2 we provide a taste of what is considered theultimate aim of indirect detection, analysis of a putative dark matter signal. Thissignal is an excess of gamma-rays observed by the
Fermi telescope near the galacticcenter, and as such is commonly referred to as the galactic center excess (GCE). Theexcess was observed almost as soon as the
Fermi data became publicly available [38],and then was followed up in a number of studies [39–44]. That it was seen so quicklyis consistent with a dark matter interpretation, as the galactic center is the locationon the sky with the largest 𝐽 Σ . My first contribution to the GCE anomaly camein [45] and this represents the contents of Chapter 2. In that work we demonstratedthat the excess satisfies many properties you would expect for dark matter, such asbeing far more spherically symmetric than the expected background contributions.This work generated a lot of excitement, as it gave further indication that this excesswas due to dark matter. Nevertheless it was later realised, due to the application ofa novel statistical framework [46] and a wavelet based technique [47], that in fact theexcess looks to be coming from a population of point sources. The novel statisticaltechnique is known as the non-Poissonian template fit, and my work has includedapplying this method to further study the GCE and in particular its spectrum at34igh energies [48], and also in making the method into a publicly available code [49].Dark matter is not expected to have point-source-like spatial morphology, and thusthe leading hypothesis is that GCE is due to an unresolved population of point sources,which are most likely millisecond pulsars, see e.g. [50], although there is much ongoingwork to fully understand this excess, examples of which include [51–54]. Attemptshave been made to find evidence for the existence of these millisecond pulsars amongstthe resolved point sources Fermi has seen [55], although as my collaborators and Ihave demonstrated such searches are at present unable to say anything definitive [56].Another basic challenge is presented by the fact that if the excess was due to darkmatter, then we may have expected to see a signal from other regions with a large 𝐽 Σ ,such as the Milky Way dwarf spheroidal galaxies. Searches in the dwarfs, however,have not seen a similar excess [57, 58], and in [59] my collaborators and I quantifiedthe existing tension between these two measurements. As such, it is unlikely theGCE is associated with dark matter, but Chapter 2 represents a study of the sort onewould perform in the presence of an excess.In Chapter 3 we look beyond the galactic center, and indeed beyond our own MilkyWay, in search of extragalactic signals of dark matter annihilation. As mentionedabove, the largest expected regions of 𝐽 Σ beyond the galactic center are associatedwith structures within the Milky Way, in particular dwarf spheroidal galaxies. Non-observation of a dark matter signal in these objects leads to some of the strongestconstraints on the annihilation cross section [57, 58]. The question explored in Chap-ter 3, is whether extragalactic observations can compete with the dwarf searches,and indeed we will demonstrate that they can. The intuition is that even thoughextragalactic objects are much further away, they can be significantly more massive.For example, the mass of the Milky Way is ≈ 𝑀 ⊙ , whereas the Virgo galaxygroup has a substantially larger mass of ≈ × 𝑀 ⊙ . The aim is to exploit thisadditional mass, combined with the fact there are an extraordinarily large numberof galaxies and galaxy clusters outside the Milky Way, to compensate for the ad-ditional distance. This Chapter represents work published in [33], which appearedwith a companion paper [25], where my collaborators and I extensively validated our35ethods on 𝑁 -body simulations.Moving beyond annihilation, in Chapter 4 we consider how to search for darkmatter decay. In this chapter, based on [60], my collaborators and I used data fromthe Fermi telescope to set the strongest constraints on the dark matter lifetime overalmost six orders of magnitude from a GeV to almost a PeV. That we set limits isan indication that no clear signs of an excess was observed, although the methodswe introduce allow for some of the deepest searches ever performed. Further thesemethods have application beyond
Fermi , and as an example I worked with the HAWCcollaboration to apply our methods to their instrument [61].Starting in Chapter 5 we pivot to the particle physics side of indirect detection.This chapter, based on work appearing in [62], should be considered as the particlephysics side of exploring a potential excess, again in the context of the GCE. As men-tioned, the GCE was for a time considered a very promising candidate for a signalfrom annihilating dark matter, and as such generated a great deal of excitement andwork in the particle physics community. The central idea was to try and determinewhat type of particle physics interaction could give rise to the specific spectrum
Fermi was seeing, basically trying to determine exactly what was going on in Fig. 1-1. Anenormous number of proposals were put forward, and the work presented in Chapter 5focussed on trying to organize the space of models in a convenient way. In that workwe demonstrated that many complicated dark matter models, where there can be alot of structure in the blob of Fig. 1-1, can be well approximated using relativistickinematics. For example, if instead of having
DM DM → SM SM , the dark matterannihilated to an intermediate state particle 𝜑 , then the process is now a cascadeannihilation: DM DM → 𝜑𝜑 , followed by two copies of 𝜑 → SM SM . The spectrumobtained in this more complicated case can actually be derived form the earlier spec-trum by use of relativistic kinematics, and in this way starting from simple modelswe can generate the expectation for more complicated scenarios straightforwardly,even when many cascades occur in the dark sector. This formed a framework thatallowed for a broad consideration of the type of models that could explain the GCE,and represents the contents of this chapter.36hapter 6 builds off the insights of cascade annihilations derived in Chapter 5 toconsider how this general framework can be turned to set more model-independentconstraints on dark matter. In this chapter, based on work appearing in [63], wedemonstrated how the usual constraints on dark matter annihilation arising frommeasurements of photons from
Fermi , electrons, positrons, and antiprotons at AMS-02, and observations of the cosmic microwave background by
Planck , are modifiedwhen a more complicated dark sector is considered. This work significantly extendsthe use of the standard published limits, and allows theorists to more easily convertthose results into ones applicable to more complicated dark matter models.Chapter 7 represents the final substantive topic of the thesis, and is devoted toa detailed study of the physics involved in a specific dark matter annihilation. Forthis purpose we focus on a particular model for dark matter, the supersymmetricwino, and perform a full one-loop calculation for
DM DM → 𝛾𝛾 cross section in thistheory. The calculation demonstrates many of the complications that can arise whena given model is considered in detail, for example the Sommerfeld enhancement andthe resummation of large logarithms are both relevant and included. This chapterrepresents the work that appeared in [64]. This work was recently followed up bymy collaborators and I in [65], where we showed that contributions from other finalstates such as 𝑊 + 𝑊 − 𝛾 can play an important role in the photon spectrum near thedark matter mass. For any realistic instrument with imperfect energy resolution,such effects are impossible to disentangle from the pure 𝛾𝛾 contribution and thus weshowed they can significantly modify the experimental expectation.The focus of my research at grad school has been on the two faces of indirectdetection as described above. Nevertheless there are two projects I have completedthat fall outside this general program. The first of these related to exploiting noveljet algorithms on data collected at LHCb to uncover the splitting functions of heavyquarks, in particular the charm 𝑐 and bottom 𝑏 quarks [66]. The second, which wasmentioned above, related to an analysis framework for axion direct detection [16],which is a method for searching for dark matter that is much lighter than what weconsider in the remainder of this thesis. 378 hapter 2The Characterization of theGamma-Ray Signal from the CentralMilky Way: A Case for AnnihilatingDark Matter Weakly interacting massive particles (WIMPs) are a leading class of candidates forthe dark matter of our universe. If the dark matter consists of such particles, thentheir annihilations are predicted to produce potentially observable fluxes of energeticparticles, including gamma rays, cosmic rays, and neutrinos. Of particular interestare gamma rays from the region of the Galactic Center which, due to its proximityand high dark matter density, is expected to be the brightest source of dark matterannihilation products on the sky, hundreds of times brighter than the most promisingdwarf spheroidal galaxies.Over the past few years, several groups analyzing data from the
Fermi
Gamma-Ray Space Telescope have reported the detection of a gamma-ray signal from theinner few degrees around the Galactic Center (corresponding to a region several hun-39red parsecs in radius), with a spectrum and angular distribution compatible withthat anticipated from annihilating dark matter particles [38–44]. More recently, thissignal was shown to also be present throughout the larger Inner Galaxy region, ex-tending kiloparsecs from the center of the Milky Way [67, 68]. While the spectrumand morphology of the Galactic Center and Inner Galaxy signals have been shown tobe compatible with that predicted from the annihilations of an approximately 30-40GeV WIMP annihilating to quarks (or a ∼ Fermi should have resolveda much greater number of such objects. Accounting for this constraint, Ref. [70]concluded that no more than ∼ Fermi data in an effort toconstrain and characterize this signal more definitively, with the ultimate goal beingto confidently determine its origin. One way in which we expand upon previous workis by selecting photons based on the value of the
Fermi event parameter CTBCORE.Through the application of this cut, we select only those events with more reliabledirectional reconstruction, allowing us to better separate the various gamma-ray com-ponents, and to better limit the degree to which emission from the Galactic Disk leaks40nto the regions studied in our Inner Galaxy analysis. We produce a new and robustdetermination of the spectrum and morphology of the Inner Galaxy and the GalacticCenter signals. We go on to apply a number of tests to this data, and determinethat the anomalous emission in question agrees well with that predicted from theannihilations of a 36-51 GeV WIMP annihilating mostly to 𝑏 quarks (or a somewhatlower mass WIMP if its annihilations proceed to first or second generation quarks).Our results now appear to disfavor the previously considered 7-10 GeV mass windowin which the dark matter annihilates significantly to tau leptons [39,41,43,44,67] (theanalysis of Ref. [43] also disfavored this scenario). The morphology of the signal isconsistent with spherical symmetry, and strongly disfavors any significant elongationalong the Galactic Plane. The emission decreases with the distance to the GalacticCenter at a rate consistent with a dark matter halo profile which scales as 𝜌 ∝ 𝑟 − 𝛾 ,with 𝛾 ≈ . − . . The signal can be identified out to angles of ≃ ∘ from theGalactic Center, beyond which systematic uncertainties related to the Galactic dif-fuse model become significant. The annihilation cross section required to normalizethe observed signal is 𝜎𝑣 ∼ − cm /s, in good agreement with that predicted fordark matter in the form of a simple thermal relic.The remainder of this chapter is structured as follows. In the following section,we review the calculation of the spectrum and angular distribution of gamma rayspredicted from annihilating dark matter. In Sec. 2.3, we describe the event selectionused in our analysis, including the application of cuts on the Fermi event parameterCTBCORE. In Secs. 2.4 and 2.5, we describe our analyses of the Inner Galaxy andGalactic Center regions, respectively. In each of these analyses, we observe a signif-icant gamma-ray excess, with a spectrum and morphology in good agreement withthat predicted from annihilating dark matter. We further investigate the angulardistribution of this emission in Sec. 2.6, and discuss the dark matter interpretation ofthis signal in Sec. 2.7. In Sec. 2.8 we discuss the implications of these observations,and offer predictions for other upcoming observations. Finally, we summarize ourresults and conclusions in Sec. 2.9. In the associated appendix of this chapter, weinclude supplemental material intended for those interested in further details of our41igure 2-1: Left frame: The dark matter density as a function of the distance tothe Galactic Center, for several halo profiles, each normalized such that 𝜌 = 0 . GeV/cm at 𝑟 = 8 . kpc. Right frame: The line-of-sight integral of the densitysquared, as defined in Eq. 2.3, for the same set of halo profiles, as a function of theangular distance from the Galactic Center, 𝜓 .analysis. Dark matter searches using gamma-ray telescopes have a number of advantages overother indirect detection strategies. Unlike signals associated with cosmic rays (elec-trons, positrons, antiprotons, etc), gamma rays are not deflected by magnetic fields.Furthermore, gamma-ray energy losses are negligible on galactic scales. As a result,gamma-ray telescopes can potentially acquire both spectral and spatial information,unmolested by astrophysical effects.The flux of gamma rays generated by annihilating dark matter particles, as afunction of the direction observed, 𝜓 , is given by: Φ( 𝐸 𝛾 , 𝜓 ) = 𝜎𝑣 𝜋𝑚 𝑋 d 𝑁 𝛾 d 𝐸 𝛾 ∫︁ los 𝜌 ( 𝑟 ) d 𝑙, (2.1)where 𝑚 𝑋 is the mass of the dark matter particle, 𝜎𝑣 is the annihilation cross section(times the relative velocity of the particles), 𝑑𝑁 𝛾 /𝑑𝐸 𝛾 is the gamma-ray spectrumproduced per annihilation, and the integral of the density squared is performed over42igure 2-2: Left frame: The spectrum of gamma rays produced per dark matterannihilation for a 30 GeV WIMP mass and a variety of annihilation channels. Rightframe: An estimate for the bremsstrahlung emission from the electrons produced indark matter annihilations taking place near the Galactic Center, for the case of a 30GeV WIMP annihilating to 𝑏 ¯ 𝑏 . At | 𝑧 | (cid:46) . kpc ( | 𝑏 | (cid:46) ∘ ) and at energies below ∼ 𝑟 . Throughout this study, we will consider dark matterdistributions described by a generalized Navarro-Frenk-White (NFW) halo profile [74,75]: 𝜌 ( 𝑟 ) = 𝜌 ( 𝑟/𝑟 𝑠 ) − 𝛾 (1 + 𝑟/𝑟 𝑠 ) − 𝛾 . (2.2)Throughout this chapter, we adopt a scale radius of 𝑟 𝑠 = 20 kpc, and select 𝜌 such that the local dark matter density (at . from the Galactic Center) is . GeV/cm , consistent with dynamical constraints [76, 77]. Although dark matter-only simulations generally favor inner slopes near the canonical NFW value ( 𝛾 =1 ) [78, 79], baryonic effects are expected to have a non-negligible impact on the darkmatter distribution within the inner ∼
10 kiloparsecs of the Milky Way [80–90]. Themagnitude and direction of such baryonic effects, however, are currently a topic ofdebate. With this in mind, we remain agnostic as to the value of the inner slope, andtake 𝛾 to be a free parameter. 43n the left frame of Fig. 2-1, we plot the density of dark matter as a function of 𝑟 for several choices of the halo profile. Along with generalized NFW profiles usingthree values of the inner slope ( 𝛾 =1.0, 1.2, 1.4), we also show for comparison theresults for an Einasto profile (with 𝛼 = 0 . ) [91]. In the right frame, we plot thevalue of the integral in Eq. 2.1 for the same halo profiles, denoted by the quantity, 𝐽 ( 𝜓 ) : 𝐽 ( 𝜓 ) = ∫︁ los 𝜌 ( 𝑟 ) 𝑑𝑙, (2.3)where 𝜓 is the angle observed away from the Galactic Center. In the NFW case(with 𝛾 = 1 ), for example, the value of 𝐽 averaged over the inner degree around theGalactic Center exceeds that of the most promising dwarf spheroidal galaxies by afactor of ∼ [92]. If the Milky Way’s dark matter halo is contracted by baryons or isotherwise steeper than predicted by NFW, this ratio could easily be ∼ or greater.The spectrum of gamma rays produced per dark matter annihilation, 𝑑𝑁 𝛾 /𝑑𝐸 𝛾 ,depends on the mass of the dark matter particle and on the types of particles pro-duced in this process. In the left frame of Fig. 2-2, we plot 𝑑𝑁 𝛾 /𝑑𝐸 𝛾 for the caseof a 30 GeV WIMP mass, and for a variety of annihilation channels (as calculatedusing PYTHIA [93], except for the 𝑒 + 𝑒 − case, for which the final state radiation wascalculated analytically [94,95]). In each case, a distinctive bump-like feature appears,although at different energies and with different widths, depending on the final state.In addition to prompt gamma rays, dark matter annihilations can produce elec-trons and positrons which subsequently generate gamma rays via inverse Comptonand bremsstrahlung processes. For dark matter annihilations taking place near theGalactic Plane, the low-energy gamma-ray spectrum can receive a non-negligible con-tribution from bremsstrahlung. In the right frame of Fig. 2-2, we plot the gamma-ray spectrum from dark matter (per annihilation), including an estimate for thebremsstrahlung contribution. In estimating the contribution from bremsstrahlung,we neglect diffusion, but otherwise follow the calculation of Ref. [96]. In particu-lar, we consider representative values of ⟨ 𝐵 ⟩ = 10 𝜇 G for the magnetic field, and 10eV / cm for the radiation density throughout the region of the Galactic Center. For44he distribution of gas, we adopt a density of 10 particles per cm near the GalacticPlane ( 𝑧 = 0 ), with a dependence on 𝑧 given by exp( −| 𝑧 | / .
15 kpc) . Within ∼ ∘ – ∘ of the Galactic Plane, we find that bremsstrahlung could potentially contributenon-negligibly to the low energy ( (cid:46) In most analyses of
Fermi data, one makes use of all of the events within a givenclass (Transient, Source, Clean, or Ultraclean). Each of these event classes reflects adifferent trade-off between the effective area and the efficiency of cosmic-ray rejection.Higher quality event classes also allow for somewhat greater angular resolution (asquantified by the point spread function, PSF). The optimal choice of event class fora given analysis depends on the nature of the signal and background in question.The Ultraclean event class, for example, is well suited to the study of large angularregions, and to situations where the analysis is sensitive to spectral features thatmight be caused by cosmic ray backgrounds. The Transient event class, in contrast,is best suited for analyses of short duration events, with little background. Searchesfor dark matter annihilation products from the Milky Way’s halo significantly benefitfrom the high background rejection and angular resolution of the Ultraclean class andthus can potentially fall into the former category.As a part of event reconstruction, the
Fermi
Collaboration estimates the accuracyof the reconstructed direction of each event. Inefficiencies and inactive regions withinthe detector reduce the quality of the information available for certain events. Fac-tors such as whether an event is front-converting or back-converting, whether thereare multiple tracks that can be combined into a vertex, and the amount of energydeposited into the calorimeter each impact the reliability of the reconstructed direc-tion [97]. 45 I n t en s i t y ( no r m a li z ed ) E γ = 0.42 GeV E γ = 1.33 GeV E γ = 7.50 GeV Figure 2-3: The point spread function (PSF) of the
Fermi
Gamma-Ray Space Tele-scope, for front-converting, Ultraclean class events. The solid lines represent the PSFfor the full dataset, using the
Fermi
Collaboration’s default cuts on the parameterCTBCORE. The dotted and dashed lines, in contrast, denote the PSFs for the toptwo quartiles (Q2) and top quartile (Q1) of these events, respectively, as ranked byCTBCORE. See text for details.In their most recent public data releases, the
Fermi
Collaboration has begun toinclude a greater body of information about each event, including a value for the pa-rameter CTBCORE, which quantifies the reliability of the directional reconstruction.By selecting only events with a high value of CTBCORE, one can reduce the tails ofthe PSF, although at the expense of effective area [97].For this study, we have created a set of new event classes by increasing the CTB-CORE cut from the default values used by the
Fermi
Collaboration. To accomplishthis, we divided all front-converting, Ultraclean events (Pass 7, Reprocessed) intoquartiles, ranked by CTBCORE. Those events in the top quartile make up the eventclass Q1, while those in the top two quartiles make up Q2, etc. For each new eventclass, we calibrate the on-orbit PSF [98, 99] using the Geminga pulsar. Taking ad-vantage of Geminga’s pulsation, we remove the background by taking the differencebetween the on-phase and off-phase images. We fit the PSF in each energy bin bya single King function, and smooth the overall PSF with energy. We also rescale
Fermi ’s effective area according to the fraction of events that are removed by theCTBCORE cut, as a function of energy and incidence angle.These cuts on CTBCORE have a substantial impact on
Fermi ’s PSF, especiallyat low energies. In Fig. 2-3, we show the PSF for front-converting, Ultraclean events,46t three representative energies, for different cuts on CTBCORE (all events, Q2, andQ1).Such a cut can be used to mitigate the leakage of astrophysical emission from theGalactic Plane and point sources into our regions of interest. This leakage is mostproblematic at low energies, where the PSF is quite broad and where the CTBCOREcut has the greatest impact. These new event classes and their characterization arefurther detailed in [100], and accompanied by a data release of all-sky maps for eachclass, and the instrument response function files necessary for use with the
Fermi
Science Tools.Throughout the remainder of this study, we will employ the Q2 event class bydefault, corresponding to the top 50% (by CTBCORE) of
Fermi ’s front-converting,Ultraclean photons, to maximize event quality. We select Q2 rather than Q1 toimprove statistics, since as demonstrated in Fig. 2-3, the angular resolution improve-ment in moving from Q2 to Q1 is minimal. In Appendix F.2 we demonstrate thatour results are stable upon removing the CTBCORE cut (thus doubling the dataset),or expanding the dataset to include lower-quality events. In this section, we follow the procedure previously pursued in Ref. [67] (see alsoRefs. [36, 101]) to study the gamma-ray emission from the Inner Galaxy. We use theterm “Inner Galaxy” to denote the region of the sky that lies within several tens ofdegrees around the Galactic Center, excepting the Galactic Plane itself ( | 𝑏 | < ∘ ),which we mask in this portion of our analysis.Throughout our analysis, we make use of the Pass 7 (V15) reprocessed data takenbetween August 4, 2008 and December 5, 2013, using only front-converting, Ultra- An earlier version of this work found a number of apparent peculiarities in the results withoutthe CTBCORE cut that were removed on applying the cut. However, we now attribute thosepeculiarities to an incorrect smoothing of the diffuse background model. When the background modelis smoothed correctly, we find results that are much more stable to the choice of CTBCORE cut,and closely resemble the results previously obtained with Q2 events. Accordingly, the CTBCOREcut appears to be effective at separating signal from poorly-modeled background emission, but isless necessary when the background is well-modeled.
180 90 0 -90 -18000 -90-450459000 -5-4-3-2-10-5-4-3-2-10
180 90 0 -90 -18000 -90-450459000 -5-4-3-2-10-5-4-3-2-10
180 90 0 -90 -18000 -90-450459000 -5-4-3-2-10-5-4-3-2-10
Figure 2-4: The spatial templates (in galactic coordinates) for the Galactic diffusemodel (upper left), the
Fermi bubbles (upper right), and dark matter annihilationproducts (lower), as used in our Inner Galaxy analysis. The scale is logarithmic(base 10), normalized to the brightest point in each map. The diffuse model templateis shown as evaluated at 1 GeV, and the dark matter template corresponds to ageneralized NFW profile with an inner slope of 𝛾 = 1 . . Red dashed lines indicatethe boundaries of our standard Region of Interest (we also mask bright point sourcesand the region of the Galactic plane with | 𝑏 | < ∘ ).clean class events which pass the Q2 CTBCORE cut as described in Sec. 2.3. Wealso apply standard cuts to ensure data quality (zenith angle < ∘ , instrumentalrocking angle < ∘ , DATA_QUAL = 1,
LAT_CONFIG =1). Using this data set, we havegenerated a series of maps of the gamma-ray sky binned in energy, with 30 logarith-mically spaced energy bins spanning the range from 0.3-300 GeV. For the analysespresented in this chapter, by default we restrict to energies 50 GeV and lower toensure numerical stability of the fit. We apply the point source subtraction methoddescribed in Ref. [36], updated to employ the 2FGL catalogue, and masking out the300 brightest and most variable sources at a mask radius corresponding to con-tainment. We then perform a pixel-based maximum likelihood analysis on the map,fitting the data in each energy bin to a sum of spatial templates. These templatesconsist of: 1) the
Fermi
Collaboration p6v11
Galactic diffuse model (which we referto as the p6v11 diffuse model),
2) an isotropic map, intended to account for the ex- Unlike more recently released Galactic diffuse models, the p6v11 diffuse model does not implicitly
Fermi
Bubbles, as described in Ref. [36]. In addition to these three background templates,we include an additional dark matter template, motivated by the hypothesis that thepreviously reported gamma-ray excess originates from annihilating dark matter. Inparticular, our dark matter template is taken to be proportional to the line-of-sightintegral of the dark matter density squared, 𝐽 ( 𝜓 ) , for a generalized NFW densityprofile (see Eqs. 2.2–2.3). The spatial morphology of the Galactic diffuse model (asevaluated at 1 GeV), Fermi
Bubbles, and dark matter templates are each shown inFig. 2-4.We smooth the Galactic diffuse model template to match the data using the gtsrcmaps routine in the
Fermi
Science Tools, to ensure that the tails of the PSF areproperly taken into account. Because the Galactic diffuse model template is muchbrighter than the other contributions in the region of interest, relatively small errorsin its smoothing could potentially bias our results. However, the other templatesare much fainter, and so we simply perform a Gaussian smoothing, with a FWHMmatched to the FWHM of the
Fermi
PSF at the minimum energy for the bin (sincemost of the counts are close to this minimum energy). In all cases, when usingCTBCORE data, we employ the appropriate (narrower) PSF, as derived in [100].By default, we employ a Region of Interest (ROI) of | ℓ | < ∘ , ∘ < | 𝑏 | < ∘ . Anearlier version of this work used the full sky (with the plane masked at 1 degree) as thedefault ROI; we find that restricting to a smaller ROI alleviates oversubtraction in theinner Galaxy and improves the stability of our results. Thus we present “baseline”results for the smaller region, but show the impact of changing the ROI in AppendixF.2, and in selected figures in the main text. Where we refer to the “full sky” analysisthe Galactic plane is masked for | 𝑏 | < ∘ unless noted otherwise. include a component corresponding to the Fermi
Bubbles. By using this model, we are free to fitthe
Fermi
Bubbles component independently. See Appendix A.2 for a discussion of the impact ofvarying the diffuse model. We checked the impact of smoothing the diffuse model with a Gaussian and found no significantimpact on our results. This approach was in part inspired by the work presented in Ref. [102]. −
2Δ ln ℒ (referred to as TS) extracted fromthe likelihood fit, as a function of the inner slope of the dark matter halo profile, 𝛾 .All values are relative to the result for the best-fit (highest TS) template, and positivevalues thus indicate a reduction in TS. Results are shown using gamma-ray data fromthe full sky (solid line) and only the southern sky (dashed line). Unlike in the analysisof Ref. [67], we do not find any large north-south asymmetry in the preferred valueof 𝛾 .As found in previous studies [67, 68], the inclusion of the dark matter templatedramatically improves the quality of the fit to the Fermi data. For the best-fit spec-trum and halo profile, we find that the inclusion of the dark matter template improvesthe formal fit by TS ≡ −
2Δ ln
ℒ ≃ (here TS stands for “test statistic”). Thisdark matter template has 22 degrees of freedom, corresponding to its normalizationin each of the 22 energy bins below 50 GeV. A naive translation from TS to 𝑝 -valueresults in an apparent statistical preference greater than 30 𝜎 ; however, when consid-ering this enormous statistical significance, one should keep in mind that in additionto statistical errors there is a degree of unavoidable and unaccounted-for systematicerror. Neither model (with or without a dark matter component) is a “good fit” inthe sense of describing the sky to the level of Poisson noise. That being said, the datado very strongly prefer the presence of a gamma-ray component with a morphologysimilar to that predicted from annihilating dark matter (see Appendices F.2-A.5 forfurther details).As in Ref. [67], we vary the value of the inner slope of the generalized NFW profile, 𝛾 , and compare the change in the log-likelihood, Δ ln ℒ , between the resulting fits50igure 2-6: Left frame: The spectrum of the dark matter component, extracted froma fit in our standard ROI ( ∘ < | 𝑏 | < ∘ , | 𝑙 | < ∘ ) for a template corresponding to ageneralized NFW halo profile with an inner slope of 𝛾 = 1 . (normalized to the fluxat an angle of 5 ∘ from the Galactic Center). Shown for comparison (solid line) is thespectrum predicted from a 43.0 GeV dark matter particle annihilating to 𝑏 ¯ 𝑏 with across section of 𝜎𝑣 = 2 . × − cm /s × [(0 . / cm ) /𝜌 local ] . Right frame: asleft frame, but for a full-sky ROI ( | 𝑏 | > ∘ ), with 𝛾 = 1 . ; shown for comparison (solidline) is the spectrum predicted from a 36. 6 GeV dark matter particle annihilating to 𝑏 ¯ 𝑏 with a cross section of 𝜎𝑣 = 0 . × − cm /s × [(0 . / cm ) /𝜌 local ] .in order to determine the preferred range for the value of 𝛾 . The results of thisexercise are shown in Fig. 2-5. We find that our default ROI has a best-fit value of 𝛾 = 1 . , consistent with previous studies of the inner Galaxy (which did not employany additional cuts on CTBCORE) that preferred an inner slope of 𝛾 ≃ . [67].Fitting over the full sky, we find a preference for a slightly steeper value of 𝛾 ≃ . .These results are quite stable to our mask of the Galactic plane; masking the regionwith | 𝑏 | < ∘ changes the preferred value to 𝛾 = 1 . in our default ROI, and 𝛾 = 1 . over the whole sky. In contrast to Ref. [67], we find no significant difference in theslope preferred by the fit over the standard ROI, and by a fit only over the southernhalf ( 𝑏 < ) of the ROI (we also find no significant difference between the fit overthe full sky and the southern half of the full sky). This can be seen directly fromFig. 2-5, where the full-sky and southern-sky fits for the same level of masking arefound to favor quite similar values of 𝛾 (the southern sky distribution is broader thanthat for the full sky simply due to the difference in the number of photons). The Throughout, we describe the improvement in −
2Δ ln ℒ induced by inclusion of a specific templateas the “test statistic” or TS for that template. .5-1 GeV residual -20-1001020 00 -20-100102000 0510152005101520 - c oun t s / c m / s / s r -20-1001020 00 -20-100102000 -2024681012-2024681012 - c oun t s / c m / s / s r -20-1001020 00 -20-100102000 0123401234 - c oun t s / c m / s / s r -20-1001020 00 -20-100102000 - c oun t s / c m / s / s r Figure 2-7: Intensity maps (in galactic coordinates) after subtracting the point sourcemodel and best-fit Galactic diffuse model,
Fermi bubbles, and isotropic templates.Template coefficients are obtained from the fit including these three templates anda 𝛾 = 1 . DM-like template. Masked pixels are indicated in black. All maps havebeen smoothed to a common PSF of 2 degrees for display, before masking (the cor-responding masks have not been smoothed; they reflect the actual masks used inthe analysis). At energies between ∼ i.e. in the first three frames), thedark-matter-like emission is clearly visible around the Galactic Center.best-fit values for gamma, from fits in the southern half of the standard ROI and thesouthern half of the full sky, are 1.13 and 1.26 respectively.In Fig. 2-6, we show the spectrum of the emission correlated with the dark mattertemplate in the default ROI and full-sky analysis, for their respective best-fit values52f 𝛾 = 1 . and 1.28. While no significant emission is absorbed by this template atenergies above ∼
10 GeV, a bright and robust component is present at lower energies,peaking near ∼ Shown for comparison (as a solid line) is the spectrum predicted from(left panel) a 43.0 GeV dark matter particle annihilating to 𝑏 ¯ 𝑏 with a cross sectionof 𝜎𝑣 = 2 . × − cm /s × [(0 . / cm ) /𝜌 local ] , and (right panel) a 36.6 GeVdark matter particle annihilating to 𝑏 ¯ 𝑏 with a cross section of 𝜎𝑣 = 0 . × − cm /s × [(0 . / cm ) /𝜌 local ] . The spectra extracted for this component are inmoderately good agreement with the predictions of the dark matter models, yieldingfits of 𝜒 = 44 and over the 22 error bars between 0.3 and 50 GeV. We emphasizethat these uncertainties (and the resulting 𝜒 values) are purely statistical, and thereare significant systematic uncertainties which are not accounted for here (see thediscussion in the appendices). We also note that the spectral shape of the darkmatter template is quite robust to variations in 𝛾 , within the range where good fitsare obtained (see Appendix F.2).In Fig. 2-7, we plot the maps of the gamma-ray sky in four energy ranges aftersubtracting the best-fit diffuse model, Fermi
Bubbles, and isotropic templates. Inthe 0.5-1 GeV, 1-3 GeV, and 3-10 GeV maps, the dark-matter-like emission is clearlyvisible in the region surrounding the Galactic Center. Much less central emissionis visible at 10-50 GeV, where the dark matter component is absent, or at leastsignificantly less bright.We note that the p6v11 diffuse model, like all other diffuse models created by the
Fermi
Collaboration, was designed for point source subtraction rather than study ofextended sources or large-scale diffuse excesses. Accordingly, the
Fermi
Collaborationdoes not recommend any standard background model for extended diffuse analyses,stating that the approach for such studies should be determined and tested on a case- A comparison between the two ROIs with 𝛾 held constant is presented in Appendix F.2. An earlier version of this work found this improvement only in the presence of the CTBCOREcut; we now find this hardening independent of the CTBCORE cut. The model also inherits fundamental limitations from the
GALPROP code [103–105] used to compute the distribution of cosmic rays (for example, thiscode treats the Galaxy as axisymmetric). Finally, the p6v11 diffuse model was createdbased on a much earlier Fermi dataset than the one employed in this work, with adifferent event selection. It was fitted to the data assuming (a) no dark mattercomponent, and (b) a set of instrument response functions that have since beensuperseded. The p6v11 diffuse model itself is a physical model for the gamma-rayemission, and is not convolved with those original instrument response functions.However, there is no guarantee that it would still yield the best fit to our updatedand modified dataset if the same analysis to be repeated, due to both increasedstatistics, and low-level systematic errors in the instrument response functions (thatdiffer between our analysis and the original fit of the p6v11 diffuse model to thedata). More fundamentally, in the absence of an accurate model for the cosmic raydistribution in the inner Galaxy, any diffuse model we construct will have difficult-to-gauge systematic differences from the data. It is not unexpected that – as mentionedabove – none of our models provide formally good fits to the data, to the level ofPoisson noise.Acknowledging these caveats, we proceed using the p6v11 diffuse model, motivatedby the results of Ref. [67], that demonstrated that (in a study of the inner Galaxy)consistent results were obtained using the p6v11 diffuse model and a data-drivenmodel for the diffuse emission based on the 0.5-1 GeV energy band (where the excessappears to be relatively faint). By letting the model components float independentlyin each energy bin, as was done in Ref. [67], we remove any systematic effect from amismodeled energy-dependent effective area in the original construction of the p6v11 diffuse model.Our approach in this work is to test the robustness of the excess to variationsin the background modeling, rather than trying to construct the best possible back-ground model. In Appendix A.2 we explore the effects of changing to a later
Fermi http://fermi.gsfc.nasa.gov/ssc/data/analysis/LAT_caveats.html . GALPROP is publicly available at http://galprop.stanford.edu . p6v11 diffuse model. Both of these modifications can change the detailsof the extracted spectrum for the excess, but its presence and general shape (peakedat 1-3 GeV) appear fairly robust. Only when a template sculpted to be near-sphericaland sharply peaked toward the Galactic Center is added to the “background” modeldo we find substantial degeneracy with the excess at low energies, and a resultingshift in the peak of its spectrum to higher energies. We also perform the tests offitting in different sub-regions (Appendix F.2), as one would expect the systematicsdue to mis-subtraction of the diffuse background to differ over the sky. Finally, aswe will show in the next section, the features of the excess discussed in this work arealso reproduced in the Galactic Center, where we would again expect the systematicerrors due to mismodeled diffuse emission to be different from those present at higherlatitudes.
In this section, we describe our analysis of the
Fermi data from the region of theGalactic Center, defined as | 𝑏 | < ∘ , | 𝑙 | < ∘ . We make use of the same Pass 7 dataset, with Q2 cuts on CTBCORE, as described in the previous section. We performeda binned likelihood analysis to this data set using the Fermi tool gtlike , dividingthe region into 200 ×
200 spatial bins (each . ∘ × . ∘ ), and 12 logarithmically-spaced energy bins between 0.316-10.0 GeV. Included in the fit is a model for theGalactic diffuse emission, supplemented by a model spatially tracing the observed 20cm emission [106], a model for the isotropic gamma-ray background, and all gamma-ray sources listed in the 2FGL catalog [107], as well as the two additional point sourcesdescribed in Ref. [108]. We allow the flux and spectral shape of all high-significance After the original appearance of this work, an independent study demonstrated that consideringa wide range of
GALPROP -based models also does not significantly alter the qualitative features ofthe excess [102]. 𝛾 = 1 . (left) or 1.3 (right), normalized to the flux at an angle of 5 ∘ fromthe Galactic Center. We caution that significant and difficult to estimate system-atic uncertainties exist in this determination, especially at energies below ∼ 𝑏 ¯ 𝑏 with a cross section of 𝜎𝑣 = 1 . × − cm /s × [(0 . / cm ) /𝜌 local ] (left) or 𝜎𝑣 = 0 . × − cm /s × [(0 . / cm ) /𝜌 local ] (right). The dot-dash and dotted curves include an estimated contribution frombremsstrahlung, as shown in the right frame of Fig. 2-2.( √ TS > ) 2FGL sources located within ∘ of the Galactic Center to vary. Forsomewhat more distant or lower significance sources ( 𝜓 = 7 ∘ − ∘ and √ TS > , 𝜓 = 2 ∘ − ∘ and √ TS = 10 − , or 𝜓 < ∘ and any TS), we adopt the best-fit spectralshape as presented in the 2FGL catalog, but allow the overall normalization to float.We additionally allow the spectrum and normalization of the two new sources fromRef. [108], the 20 cm template, and the extended sources W28 and W30 [107] to float.We fix the emission from all other sources to the best-fit 2FGL values.For the Galactic diffuse emission, we adopt the model gal_2yearp7v6_v0 . Al-though an updated Galactic diffuse model has recently been released by the Fermi
Collaboration, that model includes additional empirically fitted features at scalesgreater than 2 ∘ , and therefore is not recommended for studies of extended gamma-ray emission. We use this “ p7v6 model” in preference to the p6v11 diffuse model forthe Galactic Center analysis because of its finer binning and improved treatment ofthe Galactic plane and point sources. These factors are much more important in http://fermi.gsfc.nasa.gov/ssc/data/access/lat/Model_details/Pass7_galactic. 𝛾 , as found in our Galactic Center likelihoodanalysis. All values are relative to the result for the best-fit (highest TS) template,and positive values thus indicate a reduction in TS. The best-fit value is very similarto that found in our analysis of the larger Inner Galaxy region (in the default ROI),favoring 𝛾 ∼ . (compared to 𝛾 ≃ . in the Inner Galaxy analysis).this region than in the inner Galaxy. The disadvantage of the p7v6 model for diffuseemission analyses, relative to its p6v11 counterpart, is its inclusion of fixed, non-physical templates for large-scale residuals. However, the impact of these templatesis small in the bright Galactic Center region.For the isotropic component, we adopt the model of Ref. [109]. We allow theoverall normalization of the Galactic diffuse and isotropic emission to freely vary.In our fits, we found that the isotropic component prefers a normalization that isconsiderably brighter than the extragalactic gamma-ray background. In order toaccount for this additional isotropic emission in our region of interest, we attemptedsimulations in which we allowed the spectrum of the isotropic component to vary, butfound this to have a negligible impact on the fit.In addition to these astrophysical components, we include a spatially extendedmodel in our fits motivated by the possibility of annihilating dark matter. Themorphology of this component is again taken to follow the line-of-sight integralof the square of the dark matter density, as described in Sec. 2.2. We adopt ageneralized NFW profile centered around the location of Sgr A * ( 𝑏 = − . ∘ , html /s/sr. The right frames clearly contain asignificant central and spatially extended excess, peaking at ∼ ∘ Gaussian. 𝑙 = − . ∘ [110]), and allow the inner slope ( 𝛾 ) and overall normalization (set bythe annihilation cross section) to freely float.In Figs. 2-8 and 2-9, we show the main results of our Galactic Center likelihoodanalysis. In Fig. 2-9, we plot the change of the log-likelihood of our fit as a function58f the inner slope of the halo profile, 𝛾 . For our best-fit value of 𝛾 = 1 . , theinclusion of the dark matter component (with two degrees of freedom correspondingto the normalization of spectrum based on the best-fit dark matter mass) can improvethe overall fit with TS ≃ , corresponding to a statistical preference for such acomponent at the level of ∼ 𝜎 . In Fig. 2-8, we show the spectrum of the dark-matter-like component, for values of 𝛾 = 1 . (left frame) and 𝛾 = 1 . (right frame).Shown for comparison is the spectrum predicted from a 35.25 GeV WIMP annihilatingto 𝑏 ¯ 𝑏 . The solid line represents the contribution from prompt emission, whereasthe dot-dashed and dotted lines also include an estimate for the contribution frombremsstrahlung (for the 𝑧 = 0 . and 0.3 kpc cases, as shown in the right frame ofFig. 2-2, respectively). The normalizations of the Galactic Center and Inner Galaxysignals are compatible (see Figs. 2-6 and 2-8), although the details of this comparisondepend on the precise morphology that is adopted.We note that the Fermi tool gtlike determines the quality of the fit assuming agiven spectral shape for the dark matter template, but does not generally provide amodel-independent spectrum for this or other components. In order to make a model-independent determination of the dark matter component’s spectrum, we adopt thefollowing procedure. First, assuming a seed spectrum for the dark matter component,the normalization and spectral shape of the various astrophysical components are eachvaried and set to their best-fit values. Then, the fit is performed again, allowing thespectrum of the dark matter component to vary in each energy bin. The resultantdark matter spectrum is then taken to be the new seed, and this procedure is repeatediteratively until convergence is reached.In Fig. 2-10, we plot the gamma-ray count maps of the Galactic Center region. Inthe left frames, we show the raw maps, while in the right frames we have subtractedthe best-fit contributions from each component in the fit except for that correspond-ing to the dark matter template (the Galactic diffuse model, 20 cm template, pointsources, and isotropic template). In each frame, the map has been smoothed bya 0.25 ∘ Gaussian (0.59 ∘ full-width-half-maximum). The excess emission is clearlypresent in the right frames, and most evidently in the 1.0-3.16 GeV range, where the59ignal is most significant.The slope favored by our Galactic Center analysis ( 𝛾 ≃ . –1.24) is very similar tothat found in the Inner Galaxy analysis ( 𝛾 ≃ | 𝑏 | < . ∘ , | 𝑙 | < . ∘ ), and found a preference for 𝛾 ≃ . ± . . Wediscuss this question further in Sec. 2.6.As mentioned above, in addition to the Galactic diffuse model, we include a spatialtemplate in our Galactic Center fit with a morphology tracing the 20 cm (1.5 GHz)map of Ref. [106]. This map is dominated by synchrotron emission, and thus tracesa convolution of the distribution of cosmic-ray electrons and magnetic fields in theregion. As cosmic-ray electrons also generate gamma rays via bremsstrahlung andinverse Compton processes, the inclusion of the 20 cm template in our fit is intendedto better account for these sources of gamma rays. And although the Galactic dif-fuse model already includes contributions from bremsstrahlung and inverse Comptonemission, the inclusion of this additional template allows for more flexibility in thefit. In actuality, however, we find that this template has only a marginal impact onthe results of our fit, absorbing some of the low energy emission that (without the 20cm template) would have been associated with our dark matter template. In the previous two sections, we showed that the gamma-ray emission observed fromthe regions of the Inner Galaxy and Galactic Center is significantly better fit whenwe include an additional component with an angular distribution that follows thatpredicted from annihilating dark matter. In particular, our fits favor a morphologyfor this component that follows the square of a generalized NFW halo profile withan inner slope of 𝛾 ≃ . − . . Implicit in those fits, however, was the assumptionthat the angular distribution of the anomalous emission is spherically symmetric with60igure 2-11: The variation in TS for the dark matter template, as performed inSec. 2.4’s Inner Galaxy analysis (left frame) and Sec. 2.5’s Galactic Center analysis(right frame), when breaking our assumption of spherical symmetry for the dark mat-ter template. All values shown are relative to the choice of axis ratio with the highestTS; positive values thus indicate a reduction in TS. The axis ratio is defined such thatvalues less than one are elongated along the Galactic Plane, whereas values greaterthan one are elongated with Galactic latitude. The fit strongly prefers a morphologyfor the anomalous component that is approximately spherically symmetric, with anaxis ratio near unity.respect to the dynamical center of the Milky Way. In this section, we challenge thisassumption and test whether other morphologies might provide a better fit to theobserved emission.We begin by considering templates which are elongated either along or perpen-dicular to the direction of the Galactic Plane. In Fig. 2-11, we plot the change inthe TS of the Inner Galaxy (left) and Galactic Center (right) fits with such an asym-metric template, relative to the case of spherical symmetry. The axis ratio is definedsuch that values less than unity are elongated in the direction of the Galactic Plane,while values greater than one are preferentially extended perpendicular to the plane.The profile slope averaged over all orientations is taken to be 𝛾 = 1 . in both cases.From this figure, it is clear that the gamma-ray excess in the Galactic Center prefersto be fit by an approximately spherically symmetric distribution, and disfavors anyaxis ratio which departs from unity by more than approximately 20%. In the InnerGalaxy there is a preference for a stretch perpendicular to the plane, with an axisratio of ∼ . . As we will discuss in Appendix F.2, however, there are reasons to61igure 2-12: The change in the quality of the fit in our Galactic Center analysis, fora dark matter template that is elongated along an arbitrary orientation (x-axis) andwith an arbitrary axis ratio (y-axis). As shown in Fig. 2-11, the fit worsens if thethis template is significantly stretched either along or perpendicular to the directionof the Galactic Plane (corresponding to ∘ or ∘ on the x-axis, respectively). Amild statistical preference, however, is found for a morphology with an axis ratio of ∼ ∼ ∘ clockwise from the Galactic Plane.Figure 2-13: To test whether the excess emission is centered around the dynamicalcenter of the Milky Way (Sgr A * ), we plot the change in the TS associated withthe dark matter template found in our Galactic Center analysis, as a function of thecenter of the template. Positive values correspond to a worse fit (lower TS). The fitclearly prefers this template to be centered within ∼ . ∘ of the location of Sgr A * .62elieve this may be due to the oversubtraction of the diffuse model along the plane,and this result is especially sensitive to the choice of ROI.In Fig. 2-12, we generalize this approach within our Galactic Center analysis totest morphologies that are not only elongated along or perpendicular to the GalacticPlane, but along any arbitrary orientation. Again, we find that that the quality of thefit worsens if the the template is significantly elongated either along or perpendicularto the direction of the Galactic Plane. A mild statistical preference is found, however,for a morphology with an axis ratio of ∼ ∼ ∘ clockwise from the Galactic Plane in galactic coordinates. While this may be astatistical fluctuation, or the product of imperfect background templates, it could alsopotentially reflect a degree of triaxiality in the underlying dark matter distribution.We have also tested whether the excess emission is, in fact, centered around thedynamical center of the Milky Way (Sgr A * ), as we have thus far assumed. In Fig. 2-13, we plot the change in TS of the dark-matter-motivated template, as found in ourGalactic Center analysis, when we vary the center of the template. The fit clearlyprefers this template to be centered within ∼ . ∘ of the location of Sgr A * .We smooth the ring templates to a ∘ Gaussian (full-width-half-max), and fit thenormalization of each ring template independently. Instead of allowing the spectrumof the ring templates to each vary freely (which would have introduced an untenablenumber of free parameters), we fix their spectral shape to that found for the darkmatter component in the single template fit. We also break the template associatedwith the
Fermi
Bubbles into five templates, in 10 ∘ latitude slices (each with the samespectrum, but with independent normalizations).The dark-matter-like emission is clearly and consistently present in each ring tem-plate out to ∼ ∘ , beyond which systematic and statistical limitations make such de-terminations difficult. For comparison, we also show the predictions for a generalizedNFW profile with 𝛾 = 1 . (after appropriate smoothing). While this value for theprofile slope is steeper that that found in Secs 2.4 and 2.5, we caution that systematicuncertainties associated with the diffuse model template may be biasing this fit to- We define a “clockwise” rotation such that a 90 ∘ rotation turns +l into +b. ∼ ∘ , beyond which systematic and statistical limitations makesuch determinations difficult. For comparison, we also show the predictions for ageneralized NFW profile with 𝛾 = 1 . . The spectrum of the rings is held fixed at thatof Fig.2-6, and the fluxes displayed in the plot correspond to an energy of 2.67 GeV.ward somewhat steeper values of 𝛾 (we discuss this question further in Appendix F.2,in the context of the increased values of 𝛾 found for larger ROIs). It is also plausiblethat the dark matter slope could vary with distance from the Galactic Center, forexample as exhibited by an Einasto profile [91].An important question to address is to what degree the gamma-ray excess isspatially extended, and over what range of angles from the Galactic Center can it bedetected? To address this issue, we have repeated our Inner Galaxy analysis, replacingthe dark matter template with 8 concentric, rotationally symmetric ring templates,each 1 ∘ wide, and centered around the Galactic Center. However instead of allowingthe spectrum of the ring templates to each vary freely (which would have introducedan untenable number of free parameters), we fix their spectral shape between 0.3GeV - 30 GeV to that found for the dark matter component in the single templatefit. By floating the ring coefficients with a fixed spectral dependence, we obtainanother handle on the spatial extent and morphology of the excess. In order to be64elf-consistent we inherit the background modeling and ROI from the Inner Galaxyanalysis (except that we mask the plane for | 𝑏 | < ∘ rather than | 𝑏 | < ∘ ) and fixthe spectra of all the other templates to the best fit values from the Inner Galaxyfit. We also break the template associated with the Fermi
Bubbles into two sub-templates, in 10 ∘ latitude slices (each with the same spectrum, but with independentnormalizations). We smooth the templates to the Fermi
PSF.The results of this fit are shown in Fig. 2-14. The dark-matter-like emission isclearly and consistently present in each ring template out to ∼ ∘ , beyond whichsystematic and statistical limitations make such determinations difficult. In order tocompare the radial dependence with that expected from a generalized NFW profile,we weight the properly smoothed NFW squared/projected template with each ringto obtain ring coefficients expected from an ideal NFW distribution. We then per-form a minimum 𝜒 fit on the data-driven ring coefficients taking as the templatethe coefficients obtained from an NFW profile with 𝛾 = 1 . . We exclude the twooutermost outlier ring coefficients from this fit in order to avoid systematic bias onthe preferred 𝛾 value. Since the ring templates spatially overlap upon smoothing, wetake into account the correlated errors of the maximum likelihood fit, which add tothe spectral errors in quadrature. We show an interpolation of the best fit NFW ringcoefficients with the solid line on the same figure.We caution that systematic uncertainties associated with the diffuse model tem-plate may be biasing this fit toward somewhat steeper values of 𝛾 (we discuss thisquestion further in Appendix F.2, in the context of the increased values of 𝛾 found forlarger ROIs). It is also plausible that the dark matter slope could vary with distancefrom the Galactic Center, for example as exhibited by an Einasto profile [91].To address the same question within the context of our Galactic Center analysis,we have re-performed our fit using dark matter templates which are based on densityprofiles which are set to zero beyond a given radius. We find that templates corre-sponding to density profiles set to zero outside of 800 pc (600 pc, 400 pc) provide afit that is worse relative to that found using an untruncated template at the level of Δ TS=10.7 (57.6, 108, respectively). 65e have also tested our Galactic Center fit to see if a cored dark matter profilecould also provide a good fit to the data. We find, however, that the inclusion ofeven a fairly small core is disfavored. Marginalizing over the inner slope of the darkmatter profile, we find that flattening the density profile within a radius of 10 pc(30 pc, 50 pc, 70 pc, 90 pc) worsens the overall fit by Δ TS=3.6 (12.2, 22.4, 30.6,39.2, respectively). The fit thus strongly disfavors any dark matter profile with a corelarger than a few tens of parsecs.Lastly, we confirm that the morphology of the anomalous emission does not signif-icantly vary with energy. If we fit the inner slope of the dark matter template in ourInner Galaxy analysis one energy bin at a time, we find a similar value of 𝛾 ∼ ∼ . GeV and lower, the fit preferssomewhat steeper slopes ( 𝛾 ∼ . or higher) and a corresponding spectrum with avery soft spectral index, probably reflecting contamination from the Galactic Plane.At energies above ∼ GeV, the fit again tends to prefers a steeper profile.The results of this section indicate that the gamma-ray excess exhibits a mor-phology which is both approximately spherically symmetric and steeply falling (yetdetectable) over two orders of magnitude in galactocentric distance (between ∼ ∼ In this section, we use the results of the previous sections to constrain the charac-teristics of the dark matter particle species potentially responsible for the observedgamma-ray excess.We begin by fitting various dark matter models to the spectrum of the gamma-rayexcess as found in our Inner Galaxy analysis (as shown in the left frame of Fig. 2-6).In Fig. 2-15, we plot the quality of this fit ( 𝜒 ) as a function of the WIMP mass,for a number of dark matter annihilation channels (or combination of channels),66igure 2-15: The quality of the fit ( 𝜒 , over 25-1 degrees-of-freedom) for various an-nihilating dark matter models to the spectrum of the anomalous gamma-ray emissionfrom the Inner Galaxy (as shown in the left frame of Fig. 2-6) as a function of mass,and marginalized over the value of the annihilation cross section. In the left frame,we show results for dark matter particles which annihilate uniquely to 𝑏 ¯ 𝑏 , 𝑐 ¯ 𝑐 , 𝑠 ¯ 𝑠 , lightquarks ( 𝑢 ¯ 𝑢 and/or 𝑑 ¯ 𝑑 ), or 𝜏 + 𝜏 − . In the right frame, we consider models in which thedark matter annihilates to a combination of channels, with cross sections proportionalto the square of the mass of the final state particles, the square of the charge of thefinal state particles, democratically to all kinematically accessible Standard Modelfermions, or 80% to 𝜏 + 𝜏 − and 20% to 𝑏 ¯ 𝑏 . The best fits are found for dark matterparticles with masses in the range of ∼ 𝑝 -value of 0.05 (95%CL) corresponds to a 𝜒 of approximately 36.8. Given the systematic uncertaintiesassociated with the choice of background templates, we take any value of 𝜒 (cid:46) to constitute a reasonably “good fit” to the Inner Galaxy spectrum. Good fits arefound for dark matter that annihilates to bottom, strange, or charm quarks. The fitsare slightly worse for annihilations to light quarks, or to combinations of fermionsproportional to the square of the mass of the final state, the square of the charge ofthe final state, or equally to all fermionic degrees of freedom (democratic). In the lightmass region ( 𝑚 𝑋 ∼ 𝜏 + 𝜏 − , with anadditional small fraction to quarks, such as 𝑏 ¯ 𝑏 . Even this scenario, however, providesa somewhat poor fit, significantly worse that that found for heavier ( 𝑚 𝑋 ∼ − GeV) dark matter particles annihilating mostly to quarks.67igure 2-16: The range of the dark matter mass and annihilation cross section requiredto fit the gamma-ray spectrum observed from the Inner Galaxy, for a variety ofannihilation channels or combination of channels (see Fig. 2-15). We show results forour standard ROI (black) and as fit over the full sky (blue). The observed gamma-rayspectrum is generally best fit by dark matter particles with a mass of ∼ 𝜎𝑣 ∼ − cm /s. Note thatthe cross-section for each model is computed for the best-fit slope 𝛾 in that ROI andthe assumed dark matter densities at 5 ∘ from the Galactic Center (where the signal isnormalized) are different for different values of 𝛾 . This is responsible for roughly halfof the variation between the best-fit cross-sections. Figures A-1 and A-2 show theimpact of changing the ROI when holding the assumed DM density profile constant.In Fig. 2-16, we show the regions of the dark matter mass-annihilation crosssection plane that are best fit by the gamma-ray spectrum shown in Fig. 2-6. Foreach annihilation channel (or combination of channels), the 1, 2 and 3 𝜎 contoursare shown around the best-fit point (corresponding to Δ 𝜒 = 2 . , 6.18, and 11.83,respectively). Again, in the left frame we show results for dark matter particles whichannihilate entirely to a single final state, while the right frame considers insteadcombinations of final states. Generally speaking, the best-fit models are those inwhich the dark matter annihilates to second or third generation quarks with a crosssection of 𝜎𝑣 ∼ − cm /s. This range of values favored for the dark matter’s annihilation cross section is The cross sections shown in Fig. 2-16 were normalized assuming a local dark matter densityof 0.4 GeV/cm . Although this value is near the center of the range preferred by the combinationof dynamical and microlensing data (for 𝛾 = 1 . ), there are non-negligible uncertainties in thisquantity. The analysis of Ref. [76], for example, finds a range of 𝜌 local = 0 . − . GeV/cm atthe 2 𝜎 level. This range of densities corresponds to a potential rescaling of the y-axis of Fig. 2-16by up to a factor of 0.7-2.4. 𝛾 = 𝑏 ¯ 𝑏 .quite interesting from the perspective of early universe cosmology. For the massrange being considered here, a WIMP with an annihilation cross section of 𝜎𝑣 ≃ . × − cm /s (as evaluated at the temperature of freeze-out) will freeze-outin the early universe with a relic abundance equal to the measured cosmologicaldark matter density (assuming the standard thermal history) [117]. The dark matterannihilation cross section evaluated in the low-velocity limit (as is relevant for indirectsearches), however, is slightly lower than the value at freeze-out in many models. Fora generic 𝑠 -wave annihilation process, for example, one generally expects dark matterin the form of a thermal relic to annihilate at low-velocities with a cross section near 𝜎𝑣 𝑣 =0 ≃ (1 − × − cm /s, in good agreement with the range of values favoredby the observed gamma-ray excess.Thus far in this section, we have fit the predictions of various dark matter modelsto the gamma-ray spectrum derived from our Inner Galaxy analysis. In Fig. 2-17,we compare the mass range best fit to the Inner Galaxy spectrum to that favoredby our Galactic Center analysis. Overall, these two analyses favor a similar range ofdark matter masses and annihilation channels, although the Galactic Center spectrumdoes appear to be slightly softer, and thus prefers WIMP masses that are a few GeVlower than favored by the Inner Galaxy analysis. This could, however, be the result69f bremsstrahlung, which can soften the gamma-ray spectrum from dark matter inregions near the Galactic Plane (see Fig. 2-8 and the right frame of Fig. 2-2). Suchemission could plausibly cause a ∼ ∼ In this chapter (and in previous studies [38, 39, 41–44, 67, 68]), it has been shownthat the gamma-ray excess observed from the Inner Galaxy and Galactic Centeris compatible with that anticipated from annihilating dark matter particles. Thisis not, however, the first time that an observational anomaly has been attributed todark matter. Signals observed by numerous experiments, including
INTEGRAL [118],
PAMELA [119],
ATIC [120],
Fermi [121,122],
WMAP [123,124],
DAMA/LIBRA [115,116],
CoGeNT [111, 112],
CDMS [113], and
CRESST [114], among others, have re-ceived a great deal of attention as possible detections of dark matter particles. Most,if not all, of these signals, have nothing to do with dark matter, but instead result fromsome combination of astrophysical, environmental, and instrumental backgrounds (seee.g. [125–132]). Given the frequency of such false alarms, we would be wise to applya very high standard before concluding that any new signal is, in fact, the result ofannihilating dark matter.There are significant reasons to conclude, however, that the gamma-ray signaldescribed in this chapter is more likely to be a detection of dark matter than anyof the previously reported anomalies. Firstly, this signal consists of a very largenumber of events, and has been detected with overwhelming statistical significance.The excess consists of ∼ gamma rays per square meter per year above 1 GeV (fromwithin 10 ∘ of the Galactic Center). Not only does this large number of events enableus to conclude with confidence that the signal is present, but it also allows us todetermine its spectrum and morphology in some detail. And as shown, the measuredspectrum, angular distribution, and normalization of this emission does indeed match70ell with that expected from annihilating dark matter particles.It is possible that a systematic mismodeling of the background could bias theextracted properties of the signal. However, if this mismodeling is not itself fairlysymmetric about the Galactic Center, it would be peculiar that the properties ofthe signal seem fairly consistent in different sub-regions of the ROI, and between theGalactic Center and the more extended inner Galaxy region. Further, a mismodelingthat is symmetric about the Galactic Center would be unexpected, as neither thedata nor the background model possess this symmetry. While it is possible that thetrue “source of the excess” and a background mismodeling could combine to yieldan apparently spherically symmetric excess with a spectrum that does not appearto change significantly with position, even if neither component has such propertieson its own, it would require a coincidence. Accordingly, it seems likely that theseobserved properties reflect the actual nature of the signal. (However, it is certainlypossible that background mismodeling could induce subtle changes in the extractedspectrum or morphology or the signal.)Secondly, the gamma-ray signal from annihilating dark matter can be calculatedstraightforwardly, and generally depends on only a few unknown parameters. Themorphology of this signal, in particular, depends only on the distribution of darkmatter in the Inner Galaxy (as parameterized in our study by the inner slope, 𝛾 ).The spectral shape of the signal depends only on the mass of the dark matter particleand on what Standard Model particles are produced in its annihilations. The Galac-tic gamma-ray signal from dark matter can thus be predicted relatively simply, incontrast to, e.g ., dark matter searches using cosmic rays, where putative signals areaffected by poorly constrained diffusion and energy-loss processes. In other words,for the gamma-ray signal at hand, there are relatively few “knobs to turn”, makingit less likely that one would be able to mistakenly fit a well-measured astrophysicalsignal with that of an annihilating dark matter model. (However, it is true that thespectrum of gamma-rays from measured millisecond pulsars closely resembles thatarising from light DM annihilating into simple final states; we discuss this possibleexplanation below.) 71hirdly, we once again note that the signal described in this study can be ex-plained by a very simple dark matter candidate, without any baroque or otherwiseunexpected features. After accounting for uncertainties in the overall mass of theMilky Way’s dark matter halo profile [76], our results favor dark matter particleswith an annihilation cross section of 𝜎𝑣 = (0 . − . × − cm /s (for annihilationsto 𝑏 ¯ 𝑏 , see Fig. 2-16). This range covers the long predicted value that is required ofa thermal relic that freezes-out in the early universe with an abundance equal tothe measured cosmological dark matter density ( . × − cm /s). No substruc-ture boost factors, Sommerfeld enhancements, or non-thermal histories are required.Furthermore, it is not difficult to construct simple models in which a ∼ i.e. radiation, gas, dust, star formation, etc.). Anenergetic event at the Galactic Center might conceivably give rise to a sphericallysymmetric flux of cosmic rays, but the targets on which they scatter to producegamma rays (gas, starlight) would not share this symmetry. Furthermore, the lack ofany indication for a change in the spectrum with spatial position may argue againstmodels where the signal photons are produced by scattering of electrons and positrons(whether originating from DM or astrophysical sources), as electrons lose energyrapidly as they diffuse through the Galactic halo. In contrast, both the sphericalsymmetry and uniform spectrum of the signal are natural in the context of photonsarising directly from DM annihilation, as the dark matter halo is inferred to be muchmore spherical than the disk (see e.g. [147, 148] for discussions of general DM halos,72nd e.g. [149, 150] for studies of the Milky Way halo specifically).The astrophysical interpretation most often discussed within the context of thissignal is that it might originate from a large population of unresolved millisecondpulsars. The millisecond pulsars observed within the Milky Way are largely locatedeither within globular clusters or in or around the Galactic Disk (with an exponentialscale height of 𝑧 𝑠 ∼ ∼ ∘ of the Galactic Center, this model predicts thatmillisecond pulsars should account for ∼
1% of the observed diffuse emission, and lessthan ∼ 𝑛 MSP ∝ 𝑟 − . ), those sourcescould plausibly account for much of the gamma-ray excess observed within the inner ∼ ∘ around the Galactic Center [39, 41–44, 69]. However, it is more challenging toimagine that such a concentrated population could account for the more extendedcomponent of this excess, which we have shown to be present out to at least ∼ ∘ from the Galactic Center.If the number of faint millisecond pulsars required to generate the signal arepresent ∼ ∘ ( ∼ Fermi and appeared within the2FGL catalog [70, 107, 151]. The lack of such resolved sources strongly limits the73igure 2-18: A comparison of the spectral shape of the gamma-ray excess described inthis chapter (error bars) to that measured from a number of high-significance globularclusters (NGC 6266, 47 Tuc, and Terzan 5), and from the sum of all millisecond pulsarsdetected as individual point sources by
Fermi . The gamma-ray spectrum measuredfrom millisecond pulsars and from globular clusters (whose emission is believed tobe dominated by millisecond pulsars) is consistently softer than that of the observedexcess at energies below ∼ Furthermore, the shape of the gamma-ray spectrum observed from resolved mil-lisecond pulsars and from globular clusters (whose emission is believed to be domi-nated by millisecond pulsars) appears to be somewhat softer than that of the gamma-ray excess observed from the Inner Galaxy. n Fig. 2-18, we compare the spectral shapeof the gamma-ray excess to that measured from a number of globular clusters, andfrom the sum of all resolved millisecond pulsars. Here, we have selected the threehighest significance globular clusters (NGC 6266, 47 Tuc, and Terzan 5), and plottedtheir best fit spectra as reported by the
Fermi
Collaboration [152]. For the emissionfrom resolved millisecond pulsars, we include the 37 sources as described in Ref. [70].Although each of these spectral shapes provides a reasonably good fit to the high-energy spectrum, they also each significantly exceed the amount of emission that is Since this work was posted to the arXiv, a toy model for a spherically symmetric pulsar popula-tion extending to high latitudes has been put forward [50]; some evidence has also been presented infavor of a point source origin for the excess [46,47]. Our arguments here cannot rule out a point sourcepopulation with a different luminosity function and spatial distribution than the known/predictedpulsar populations we initially considered. ∼ Fermi ’s sensitivityfrom observations of dwarf spheroidal galaxies. In fact, the
Fermi
Collaboration hasreported a modestly statistically significant excess ( ∼ 𝜎 ) in their search for anni-hilating dark matter particles in dwarf galaxies. If interpreted as a detection of darkmatter, this observation would imply a similar mass and cross section to that favoredby our analysis [92]. A similar ( ∼ 𝜎 ) excess has also been reported from the directionof the Virgo Cluster [153,154]. With the full dataset anticipated from Fermi ’s 10 yearmission, it may be possible to make statistically significant detections of dark mat-ter annihilation products from a few of the brightest dwarf galaxies, galaxy clusters,and perhaps nearby dark matter subhalos [155]. Anticipated measurements of thecosmic-ray antiproton-to-proton ratio by
AMS may also be sensitive to annihilatingdark matter with the characteristics implied by our analysis [156, 157].
In this study, we have revisited and scrutinized the gamma-ray emission from thecentral regions of the Milky Way, as measured by the
Fermi
Gamma-Ray Space Tele-scope. In doing so, we have confirmed a robust and highly statistically significantexcess, with a spectrum and angular distribution that is in excellent agreement withthat expected from annihilating dark matter. The signal is distributed with approx-75mate spherical symmetry around the Galactic Center, with a flux that falls off as 𝐹 𝛾 ∝ 𝑟 − (2 . − . , implying a dark matter distribution of 𝜌 ∝ 𝑟 − 𝛾 , with 𝛾 ≃ . − . .The spectrum of the excess peaks at ∼ 𝑏 ¯ 𝑏 . The annihilation cross section required to nor-malize this signal is 𝜎𝑣 = (1 . − . × − cm /s (for a local dark matter densityof 0.4 GeV/cm ), in good agreement with the value predicted for a simple thermalrelic. Consequently, a dark matter particle with this cross section will freeze-out ofthermal equilibrium in the early universe to yield an abundance approximately equalto the measured cosmological dark matter density (for the range of masses and crosssections favored for other annihilation channels, see Sec. 2.7).In addition to carrying out two different analyses (as described in Secs. 2.4 and 2.5),subject to different systematic uncertainties, we have applied a number of tests toour results in order to more stringently determine whether the characteristics of theobserved excess are in fact robust and consistent with the signal predicted from an-nihilating dark matter. These tests uniformly confirm that the signal is presentthroughout the Galactic Center and Inner Galaxy (extending out to angles of at least ∘ from the Galactic Center), without discernible spectral variation or significantdepartures from spherical symmetry.At present, no known or proposed astrophysical diffuse emission mechanism nat-urally gives rise to these properties of the excess. A population of several thousandmillisecond pulsars could have plausibly been responsible for much of the anomalousemission observed from within the innermost ∼ ∘ − ∘ around the Galactic Center,but the extension of this signal into regions well beyond the confines of the centralstellar cluster disfavors such objects as the primary source of this signal, unless theinner Galaxy hosts a new dense and approximately spherical pulsar population withan intrinsically fainter luminosity function than observed elsewhere in the Galaxy.In light of these considerations, we consider annihilating dark matter particles tobe the leading explanation for the origin of this signal, with potentially profoundimplications for cosmology and particle physics. Note added: Since the completion of the work in this chapter, several analyses ave demonstrated that the photon statistics of the excess are more consistent with apoint source like population than smooth emission [46, 47]. This poses a significantchallenge for dark matter explanations of the excess, and the leading hypothesis at thetime of writing is that the excess originates from a population of unresolved millisecondpulsars. It remains true that such a population of pulsars would need to take onseveral surprising characteristics, although examples of how to generate these havebeen proposed, see e.g. [50]. hapter 3A Search for Dark MatterAnnihilation in Galaxy Groups Weakly-interacting massive particles, which acquire their cosmological abundancethrough thermal freeze-out in the early universe, are leading candidates for darkmatter (DM). Such particles can annihilate into Standard Model states in the lateuniverse, leading to striking gamma-ray signatures that can be detected with obser-vatories such as the
Fermi
Large Area Telescope. Some of the strongest limits on theannihilation cross section have been set by searching for excess gamma-rays in theMilky Way’s dwarf spheroidal satellite galaxies (dSphs) [57, 58]. In this chapter, wepresent competitive constraints that are obtained using hundreds of galaxy groupswithin 𝑧 (cid:46) . .This work is complemented by a companion publication in which we describe theprocedure for utilizing galaxy group catalogs in searches for extragalactic DM [25].Previous attempts to search for DM outside the Local Group were broad in scope, butyielded weaker constraints than the dSph studies. For example, limits on the annihila-tion rate were set by requiring that the DM-induced flux not overproduce the isotropicgamma-ray background [158]. These bounds could be improved by further resolvingthe contribution of sub-threshold point sources to the isotropic background [159,160],79r by looking at the auto-correlation spectrum [161,161–163]. A separate approach in-volves cross-correlating [164–170] the Fermi data with galaxy-count maps constructedfrom, e.g. , the Two Micron All-Sky Survey (2MASS) [171, 172]. A positive cross-correlation was detected with 2MASS galaxy counts [167], which could arise fromannihilating DM with mass ∼ – GeV and a near-thermal annihilation rate [168].However, other source classes, such as misaligned Active Galactic Nuclei, could alsoexplain the signal [169].An alternative to studying the full-sky imprint of extragalactic DM annihilation isto use individual galaxy clusters [173–182]. Previous analyses along these lines havelooked at a small number of ∼ – M ⊙ X-ray–selected clusters. Like the dSphsearches, the cluster studies have the advantage that the expected signal is localizedin the sky, which reduces the systematic uncertainties associated with modeling theforegrounds and unresolved extragalactic sources. As we will show, however, the sen-sitivity to DM annihilation is enhanced—and is more robust—when a larger numberof targets are included compared to previous studies.Our work aims to combine the best attributes of the cross-correlation and clusterstudies to improve the search for extragalactic DM annihilation. We use the galaxygroup catalogs in Refs. [183] and [184] (hereby T15 and T17, respectively), whichcontain accurate mass estimates for halos with mass greater than ∼ M ⊙ and 𝑧 (cid:46) . , to systematically determine the galaxy groups that are expected to yieldthe best limits on the annihilation rate. The T15 catalog provides reliable redshiftestimates in the range . (cid:46) 𝑧 (cid:46) . , while the T17 catalog provides measureddistances for nearby galaxies, 𝑧 (cid:46) . , based on Ref. [185]. The T15 catalog waspreviously used for a gamma-ray line search [181], but our focus here is on the broader,and more challenging, class of continuum signatures. We search for gamma-ray fluxfrom these galaxy groups and interpret the null results as bounds on the annihilationcross section. 80 .2 Galaxy Group Selection The observed gamma-ray flux from DM annihilation in an extragalactic halo is pro-portional to both the particle physics properties of the DM, as well as its astrophysicaldistribution: 𝑑 Φ 𝑑𝐸 𝛾 = 𝐽 × ⟨ 𝜎𝑣 ⟩ 𝜋𝑚 𝜒 ∑︁ 𝑖 Br 𝑖 𝑑𝑁 𝑖 𝑑𝐸 ′ 𝛾 ⃒⃒⃒⃒⃒ 𝐸 ′ 𝛾 =(1+ 𝑧 ) 𝐸 𝛾 , (3.1)with units of [counts cm − s − GeV − ] . Here, 𝐸 𝛾 is the gamma-ray energy, ⟨ 𝜎𝑣 ⟩ isthe annihilation cross section, 𝑚 𝜒 is the DM mass, Br 𝑖 is the branching fraction tothe 𝑖 th annihilation channel, and 𝑧 is the cosmological redshift. The energy spec-trum for each channel is described the function 𝑑𝑁 𝑖 /𝑑𝐸 𝛾 , which is modeled usingPPPC4DMID [186]. The 𝐽 -factor that appears in Eq. 3.1 encodes the astrophysicalproperties of the halo. It is proportional to the line-of-sight integral of the squaredDM density distribution, 𝜌 DM , and is written in full as 𝐽 = (1 + 𝑏 sh [ 𝑀 vir ]) ∫︁ 𝑑𝑠 𝑑 Ω 𝜌 DM ( 𝑠, Ω) , (3.2)where 𝑏 sh [ 𝑀 vir ] is the boost factor, which accounts for the enhancement due to sub-structure. For an extragalactic halo, where the angular diameter distance 𝑑 𝐴 [ 𝑧 ] ismuch greater than the virial radius 𝑟 vir , the integral in Eq. 5.3 scales as 𝑀 vir 𝑐 𝜌 𝑐 /𝑑 𝐴 [ 𝑧 ] for the Navarro-Frenk-White (NFW) density profile [75]. Here, 𝑀 vir is the virial mass, 𝜌 𝑐 is the critical density, and 𝑐 vir = 𝑟 vir /𝑟 𝑠 is the virial concentration, with 𝑟 𝑠 the scaleradius. We infer 𝑐 vir using the concentration-mass relation from Ref. [187], whichwe update with the Planck 2015 cosmology [188]. For a given mass and redshift,the concentration is modeled as a log-normal distribution with mean given by theconcentration-mass relation. We estimate the dispersion by matching to that observedin the DarkSky-400 simulation for an equivalent 𝑀 vir [189]. Typical dispersions rangefrom ∼ . – . over the halo masses considered.The halo mass and redshift also determine the boost factor enhancement thatarises from annihilation in DM substructure. Accurately modeling the boost factor is81hallenging as it involves extrapolating the halo-mass function and concentration tomasses smaller than can be resolved with current simulations. Some previous analysesof extragalactic DM annihilation have estimated boost factors ∼ – for cluster-size halos (see, for example, Ref. [190]) based on phenomenological extrapolations ofthe subhalo mass and concentration relations. However, more recent studies indi-cate that the concentration-mass relation likely flattens at low masses [187, 191, 192],suppressing the enhancement. We use the model of Ref. [193]—specifically, the “self-consistent” model with 𝑀 min = 10 − M ⊙ —which accounts for tidal stripping of boundsubhalos and yields a modest boost ∼ for ∼ M ⊙ halos. Additionally, we modelthe boost factor as a multiplicative enhancement to the rate in our main analysis,though we consider the effect of possible spatial extension from the subhalo annihila-tion in the Supplementary Material. In particular, we find that modeling the boostcomponent of the signal as tracing a subhalo population distributed as 𝜌 NFW ratherthan 𝜌 NFW degrades the upper limits obtained by almost an order of magnitude athigher masses 𝑚 𝜒 (cid:38) GeV while strengthening the limit by a small 𝒪 (1) factor atlower masses 𝑚 𝜒 (cid:46) GeV.The halo masses and redshifts are taken from the galaxy group catalog T15 [183],which is based on the 2MASS Redshift Survey (2MRS) [194], and T17 [184], whichcompiles an inventory of nearby galaxies and distances from several sources. Thecatalogs provide group associations for these galaxies as well as mass estimates anduncertainties of the host halos, constructed from a luminosity-to-mass relation. Themass distribution is assumed to follow a log-normal distribution with uncertaintyfixed at 1% in log-space [25], which translates to typical absolute uncertainties of 25-40%. This is conservative compared to the 20% uncertainty estimate given in T15due to their inference procedure. The halo centers are assumed to coincide with thelocations of the brightest galaxy in the group. We infer the 𝐽 -factor using Eq. 5.3 andcalculate its uncertainty by propagating the errors on 𝑀 vir and 𝑐 vir , which we take tobe uncorrelated. Note that we neglect the distance uncertainties, which are expected To translate, approximately, between log- and linear-space uncertainties for the mass, we maywrite 𝑥 = log 𝑀 vir , which implies that the linear-space fractional uncertainties are 𝛿𝑀 vir /𝑀 vir ∼ ( 𝛿𝑥/𝑥 ) log 𝑀 vir . ame log 𝐽 log 𝑀 vir 𝑧 × ℓ 𝑏 log 𝑐 vir 𝜃 s 𝑏 sh [GeV cm − sr] [ 𝑀 ⊙ ] [deg] [deg] [deg]NGC4472 19.11 ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± Table 3.1: The top five halos included in the analysis, as ranked by inferred 𝐽 -factor, including the boost factor. For each group, we show the brightest centralgalaxy, as well as the virial mass, cosmological redshift, Galactic longitude ℓ , Galacticlatitude 𝑏 , inferred virial concentration [187], angular extent, and boost factor [193].The angular extent is defined as 𝜃 s ≡ tan − ( 𝑟 s /𝑑 𝐴 [ 𝑧 ]) , where 𝑑 𝐴 [ 𝑧 ] is the angulardiameter distance and 𝑟 s is the NFW scale radius. Common names for NGC4472and NGC4696 are Virgo and Centaurus, respectively. A complete table of the galaxygroups used in this analysis, as well as their associated properties, are provided at https://github.com/bsafdi/DMCat .to be ∼
5% [184, 185], as they are subdominant compared to the uncertainties onmass and concentration. We compile an initial list of nearby targets using the T17catalog, supplementing these with the T15 catalog. We exclude from T15 all groupswith Local Sheet velocity 𝑉 LS < km s − ( 𝑧 (cid:46) . ) and 𝑉 LS > , kms − ( 𝑧 (cid:38) . ), the former because of peculiar velocity contamination and the latterbecause of large uncertainties in halo mass estimation due to less luminous satellites.When groups overlap between the two catalogs, we preferentially choose distance andmass measurements from T17.The galaxy groups are ranked by their inferred 𝐽 -factors, excluding any groupsthat lie within | 𝑏 | ≤ ∘ to mitigate contamination from Galactic diffuse emission. Werequire that halos do not overlap to within ∘ of each other, which is approximatelythe scale radius of the largest halos. The exclusion procedure is applied sequentiallystarting with a halo list ranked by 𝐽 -factor. We manually exclude Andromeda, thebrightest halo in the catalog, because its large angular size is not ideally suited to ouranalysis pipeline and requires careful individual study [195]. As discussed later in thischapter, halos are also excluded if they show large residuals that are inconsistent withDM annihilation in the other groups in the sample. Starting with the top 1000 halos,83e end up with 495 halos that pass all these requirements. Of the excluded halos, 276are removed because they fall too close to the Galactic plane, 134 are removed by the ∘ proximity requirement, and 95 are removed because of the cut on large residuals.Table 3.1 lists the top five galaxy groups included in the analysis, labeled bytheir central galaxy or common name, if one exists. We provide the inferred 𝐽 -factorincluding the boost factor, the halo mass, redshift, position in Galactic coordinates,inferred concentration, and boost factor. Additionally, we show 𝜃 s ≡ tan − ( 𝑟 s /𝑑 𝐴 [ 𝑧 ]) to indicate the spatial extension of the halo. We find that 𝜃 s is typically betweenthe 68% and 95% containment radius for emission associated with annihilation inthe halos, without accounting for spread from the point-spread function (PSF). Forreference, Andromeda has 𝜃 s ∼ . ∘ . We analyze 413 weeks of Pass 8
Fermi data in the UltracleanVeto event class, fromAugust 4, 2008 through July 7, 2016. The data is binned in 26 logarithmically-spaced energy bins between 502 MeV and 251 GeV and spatially with a HEALPixpixelation [196] with nside =128. The recommended set of quality cuts are ap-plied to the data corresponding to zenith angle less than ∘ , LAT_CONFIG = 1 , and
DATA_QUAL > . We also mask known large-scale structures [25].The template analysis that we perform using
NPTFit [49] is similar to that ofprevious dSph studies [57, 58] and is detailed in our companion paper [25]. We sum-marize the relevant points here. Each region-of-interest (ROI), defined as the ∘ area surrounding each halo center, has its own likelihood. In each energy bin, thislikelihood is the product, over all pixels, of the Poisson probability for the observedphoton counts per pixel. This probability depends on the mean expected counts perpixel, which depends on contributions from known astrophysical emission as well as Our energy binning is constructed by taking 40 log-spaced bins between 200 MeV and 2 TeVand then removing the lowest four and highest ten bins, for reasons discussed in the companionpaper [25]. https://fermi.gsfc.nasa.gov/ssc/data/analysis/documentation/Cicerone/Cicerone_Data_Exploration/Data_preparation.html .
84 potential DM signal. Note that the likelihood is also multiplied by the appropriatelog-normal distribution for 𝐽 , which we treat as a single nuisance parameter for eachhalo and account for through the profile likelihood method.To model the expected counts per pixel, we include several templates in the anal-ysis that trace the emission associated with: (i) the projected NFW-squared pro-file modeling the putative DM signal, (ii) the diffuse background, as described bythe Fermi gll_iem_v06 (p8r2) model, (iii) isotropic emission, (iv) the
Fermi bub-bles [36], (v) 3FGL sources within ∘ to ∘ of the halo center, floated togetherafter fixing their individual fluxes to the values predicted by the 3FGL catalog [197],and (vi) all individual 3FGL point sources within ∘ of the halo center. Note thatwe do not model the contributions from annihilation in the smooth Milky Way halobecause the brightest groups have peak flux significantly (approximately an orderof magnitude for the groups in Tab. 3.1) over the foreground emission from Galac-tic annihilation and because we expect Galactic annihilation to be subsumed by theisotropic component.We assume that the best-fit normalizations ( i.e. , profiled values) of the astrophysi-cal components, which we treat as nuisance parameters, do not vary appreciably withDM template normalization. This allows us to obtain the likelihood profile in a givenROI and energy bin by profiling over them in the presence of the DM template, thenfixing the normalizations of the background components to the best-fit values andscanning over the DM intensity. We then obtain the total likelihood by taking theproduct of the individual likelihoods from each energy bin. In order to avoid degen-eracies at low energies due to the large PSF, we only include the DM template whenobtaining the best-fit background normalizations at energies above ∼ GeV. At theend of this procedure, the likelihood is only a function of the DM template intensity,which can then be mapped onto a mass and cross section for a given annihilationchannel. We emphasize that the assumptions described above have been thoroughlyvetted in our companion paper [25], where we show that this procedure is robust inthe presence of a potential signal.The final step of the analysis involves stacking the likelihoods from each ROI. The85tacked log-likelihood, log ℒ , is simply the sum of the log-likelihoods for each ROI. Itfollows that the test statistic for data 𝑑 is defined as TS( ℳ , ⟨ 𝜎𝑣 ⟩ , 𝑚 𝜒 ) ≡ ℒ ( 𝑑 |ℳ , ⟨ 𝜎𝑣 ⟩ , 𝑚 𝜒 ) − log ℒ ( 𝑑 |ℳ , ̂︂ ⟨ 𝜎𝑣 ⟩ , 𝑚 𝜒 ) ]︁ , (3.3)where ̂︂ ⟨ 𝜎𝑣 ⟩ is the cross section that maximizes the likelihood for DM model ℳ . The95% upper limit on the annihilation cross section is given by the value of ⟨ 𝜎𝑣 ⟩ > ̂︂ ⟨ 𝜎𝑣 ⟩ where TS = − . .Galaxy groups are expected to emit gamma-rays from standard cosmic-ray pro-cesses. Using group catalogs to study gamma-ray emission from cosmic rays in theseobjects is an interesting study in its own right (see, e.g. , Ref. [176, 178, 198, 199]),which we leave to future work. For the purpose of the present analysis, however, wewould like a way to remove groups with large residuals, likely arising from standardastrophysical processes in the clusters, to maintain maximum sensitivity to DM an-nihilation. This requires care, however, as we must guarantee that the procedure forremoving halos does not remove a real signal, if one were present.We adopt the following algorithm to remove halos with large residuals that areinconsistent with DM annihilation in the other groups in the sample. A group isexcluded if it meets two conditions. First, to ensure it is a statistically significantexcess, we require twice the difference between the maximum log likelihood and the loglikelihood with ⟨ 𝜎𝑣 ⟩ = 0 to be greater than 9 at any DM mass. This selects sourceswith large residuals at a given DM mass. Second, the residuals must be stronglyinconsistent with limits set by other galaxy groups. Specifically, the halo must satisfy ⟨ 𝜎𝑣 ⟩ best > × ⟨ 𝜎𝑣 ⟩ * lim , where ⟨ 𝜎𝑣 ⟩ best is the halo’s best-fit cross section at any massand ⟨ 𝜎𝑣 ⟩ * lim is the strongest limit out of all halos at the specified 𝑚 𝜒 . These conditionsare designed to exclude galaxy groups where the gamma-ray emission is inconsistentwith a DM origin. This prescription has been extensively tested on mock data and,crucially, does not exclude injected signals [25].86 m χ [GeV]10 − − − − − − h σ v i [ c m s − ] Stacked Galaxy Groups
Fermi -LAT Pass 8 Data, b ¯ b Thermal relic cross section
Galaxy groups (this work)68 /
95% containmentGalaxy groups, no boost
Fermi dwarfs (2016) N H (halo number)10 − − − h σ v i [ c m s − ] Cumulative Galaxy Groups b ¯ b, m χ = 100 GeV Figure 3-1: (Left) The solid black line shows the 95% confidence limit on the DMannihilation cross section, ⟨ 𝜎𝑣 ⟩ , as a function of the DM mass, 𝑚 𝜒 , for the 𝑏 ¯ 𝑏 finalstate, assuming the fiducial boost factor [193]. The containment regions are computedby performing the data analysis multiple times for random sky locations of the halos.For comparison, the dashed black line shows the limit assuming no boost factor. The Fermi dwarf limit is also shown, as well as the 𝜎 regions where DM may contributeto the Galactic Center Excess (see text for details). The thermal relic cross sectionfor a generic weakly interacting massive particle [117] is indicated by the thin dottedline. (Right) The change in the limit for 𝑚 𝜒 = 100 GeV as a function of the number ofhalos that are included in the analysis, which are ranked in order of largest 𝐽 -factor.The result is compared to the expectation from random sky locations; the 68 and95% expectations from 200 random sky locations are indicated by the red bands. The left panel of Fig. 3-1 illustrates the main results of the stacked analysis. Thesolid black line represents the limit obtained for DM annihilating to a 𝑏 ¯ 𝑏 final stateusing the fiducial boost factor model [193], while the dashed line shows the limitwithout the boost factor enhancement. To estimate the expected limit under the nullhypothesis, we repeat the analysis by randomizing the locations of the halos on thesky 200 times, though still requiring they pass the selection cuts described above. Thecolored bands indicate the 68 and 95% containment regions for the expected limit.The limit is consistent with the expectation under the null hypothesis.The right panel of Fig. 3-1 illustrates how the limits evolve for the 𝑏 ¯ 𝑏 final statewith 𝑚 𝜒 = 100 GeV as an increasing number of halos are stacked. We also show theexpected 68% and 95% containment regions, which are obtained from the randomsky locations. As can be seen, no single halo dominates the bounds. For example,87emoving Virgo, the brightest halo in the catalog, from the stacking has no significanteffect on the limit. Indeed, the inclusion of all 495 halos buys one an additional orderof magnitude in the sensitivity reach.The limit derived in this work is complementary to the published dSph bound [57,58], shown as the solid gray line in the left panel of Fig. 3-1. Given the large systematicuncertainties associated with the dwarf analyses (see e.g. , Ref. [200]), we stress theimportance of using complementary targets and detection strategies to probe thesame region of parameter space. Our limit also probes the parameter space thatmay explain the Galactic Center excess (GCE); the best-fit models are marked bythe orange cross [44], blue [43], red [45], and orange [102] 𝜎 regions. The GCEis a spherically symmetric excess of ∼ GeV gamma-rays observed to arise from thecenter of the Milky Way [38, 39, 201, 202]. The GCE has received a considerableamount of attention because it can be explained by annihilating DM. However, itcan also be explained by more standard astrophysical sources; indeed, recent analyseshave shown that the distribution of photons in this region of sky is more consistentwith a population of unresolved point sources, such as millisecond pulsars, comparedto smooth emission from DM [46–48, 55]. Because systematic uncertainties can besignificant and hard to quantify in indirect searches for DM, it is crucial to haveindependent probes of the parameter space where DM can explain the GCE. Whileour null findings do not exclude the DM interpretation of the GCE, their consistencywith the dwarf bounds put it further in tension. This does not, however, account forthe fact that the systematics on the modeling of the Milky Way’s density distributioncan potentially alleviate the tension by changing the best-fit cross section for theGCE.
This chapter presents the results of the first systematic search for annihilating DM innearby galaxy groups. We introduced and validated a prescription to infer propertiesof DM halos associated with these groups, thereby allowing us to build a map of DM88nnihilation in the local universe. Using this map, we performed a stacked analysisof several hundred galaxy groups and obtained bounds that exclude thermal crosssections for DM annihilating to 𝑏 ¯ 𝑏 with mass below ∼ GeV, assuming a conservativeboost factor model. These limits are competitive with those obtained from the
Fermi dSph analyses and are in tension with the range of parameter space that can explainthe GCE. Moving forward, we plan to investigate the objects with gamma-ray excessesto see if they can be interpreted in the context of astrophysical emission. In so doing,we can also develop more refined metrics for selecting the optimal galaxy groups forDM studies. 890 hapter 4Gamma-ray Constraints on DecayingDark Matter and Implications forIceCube
A primary goal of the particle physics program is to discover the connection betweendark matter (DM) and the Standard Model (SM). While the DM is known to be stableover cosmological timescales, rare DM decays may give rise to observable signals inthe spectrum of high-energy cosmic rays. Such decays would be induced throughoperators involving both the dark sector and the SM. In this chapter, we derive someof the strongest constraints to date on decaying DM for masses from ∼
400 MeV to ∼ GeV by performing a dedicated analysis of
Fermi gamma-ray data from 200 MeVto 2 TeV.The solid red line in Fig. 4-1 gives an example of our constraint on the DM ( 𝜒 )lifetime, 𝜏 , as a function of its mass, 𝑚 𝜒 , assuming the DM decays exclusively toa pair of bottom quarks. Our analysis includes three contributions to the photonspectrum: (1) prompt emission, (2) gamma-rays that are up-scattered by primaryelectrons/positrons through inverse Compton (IC) within the Galaxy, and (3) extra-91 m χ [GeV]10 τ [ s ] DM → b ¯ b Fermi (this work)IceCube (this work)IceCube 3 σ (Comb.)IceCube 3 σ (MESE) Figure 4-1: Limits on DM decays to 𝑏 ¯ 𝑏 , as compared to previously computed lim-its using data from Fermi (2,3,5), AMS-02 (1,4), and PAO/KASCADE/CASA-MIA(6). The hashed green (blue) region suggests parameter space where DM decay mayprovide a ∼ 𝜎 improvement to the description of the combined maximum likelihood(MESE) IceCube neutrino flux. The best-fit points, marked as stars, are in strongtension with our gamma-ray results. The red dotted line provides a limit if we as-sume a combination of DM decay and astrophysical sources are responsible for thespectrum.galactic contributions.In addition to deriving some of the strongest limits on the DM lifetime acrossmany DM decay channels, our results provide the first dedicated constraints on DMusing the latest Fermi data for 𝑚 𝜒 (cid:38) TeV. To emphasize this point, we providea comparison with other limits in Fig. 4-1. The dashed red curve indicates ournew estimate of the limits set by high-energy neutrino observations at the IceCubeexperiment [203–206]. Our IceCube constraint dominates in the range from ∼ to GeV.Constraints from previous studies are plotted as solid grey lines labeled from 1-6.Curve 6 shows that for masses above ∼ GeV, limits from null observations of ultra-high-energy gamma-rays at air shower experiments [207], such as the Pierre Auger Ob-servatory (PAO) [208], KASCADE [209], and CASA-MIA [210], surpass our IceCubelimits. Curves 2, 5, and 3 are from previous analyses of the extragalactic [211,212] and92alactic [213]
Fermi gamma-ray flux (for related work see [214–216]). Our results areless sensitive to astrophysical modeling than [211], which makes assumptions aboutthe classes of sources and their spectra that contribute to the unresolved component ofthe extragalactic gamma-ray background. We improve and extend beyond [212, 213]in a number of ways: by including state-of-the-art modeling for cosmic-ray-inducedgamma-ray emission in the Milky Way, a larger and cleaner data set, and a novelanalysis technique that allows us to search for a combination of Galactic and extra-galactic flux arising from DM decay. The limits labeled 1 and 4 in Fig. 4-1 are fromthe AMS-02 antiproton [217, 218] and positron [219, 220] measurements, respectively;these constraints are subject to considerable astrophysical uncertainties, due to thepropagation of charged cosmic rays from their source to Earth.An additional motivation for this chapter is the measurement of the so far un-explained high-energy neutrinos observed by the IceCube experiment [203–206]. Ifthe DM has both a mass 𝑚 𝜒 ∼ PeV and a long lifetime 𝜏 ∼ seconds, its decayscould contribute to the upper end of the IceCube spectrum. These DM candidateswould produce correlated cosmic-ray signals, yielding a broad spectrum of gammarays with energies extending well into Fermi ’s energy range. Taking this correlationbetween neutrino and photon spectra into account enables us to constrain the DMinterpretation of these neutrinos using the
Fermi data.Figure 4-1 illustrates regions of parameter space where we fit a decaying DMspectrum to the high-energy neutrino flux at IceCube in hashed green. The corre-sponding region for the analysis of Ref. [221] using lower-energy neutrinos is shownin blue. Clearly, much of the parameter space relevant for IceCube is disfavored bythe gamma-ray limits; the best fit points (indicated by stars) are in strong tensionwith the
Fermi observations. We conclude that models where decaying DM couldaccount for the entire astrophysical neutrino flux observed by IceCube are disfavored.Furthermore, models where the neutrino flux results from a mix of decaying DM andastrophysical sources are strongly constrained.The rest of this chapter is organized as follows. First, we discuss the variouscontributions to the gamma-ray flux resulting from DM decay. Then, we give an93verview of the data set and analysis techniques used in this chapter. Next, weprovide context for these limits by interpreting them as constraints on a concretemodel (glueball DM), before concluding.
Decaying DM contributes both a Galactic and extragalactic flux. The Galactic con-tribution results primarily from prompt gamma-ray emission due to the decay itself,which is simulated with
Pythia e.g. [225–235]).These effects can be the only source of photons for channels such as 𝜒 → 𝜈 ¯ 𝜈 .In addition, the electrons and positrons from these decays IC scatter off of cosmicbackground radiation (CBR), producing gamma-rays (see e.g. [236,237]). The promptcontribution follows the spatial morphology obtained from the line-of-sight (LOS)integral of the DM density, which we model with a Navarro-Frenk-White (NFW)profile [74, 75], setting the local DM density 𝜌 = 0 . GeV / cm , and the scale ra-dius 𝑟 𝑠 = 20 kpc (variations to the profile lead to similar results, see the associ-ated appendix). We only consider IC scattering off of the cosmic microwave back-ground (CMB), as scattering from integrated stellar radiation and the infrared back-ground is expected to be sub-dominant, see the associated appendix. For scatteringoff of the CMB, the resulting gamma-ray morphology also follows the LOS integralof the DM density. Importantly, as scattering off of the other radiation fields onlyincreases the gamma-ray flux, neglecting these effects is conservative. In the samespirit, we conservatively assume that the electrons and positrons lose energy due tosynchrotron emission in a rather strong, uniform 𝐵 = 2 . 𝜇 G magnetic field (see e.g. [238–240]) and show variations in the associated appendix.In addition to the Galactic fluxes, there is an essentially isotropic extragalacticcontribution, arising from DM decays throughout the broader universe [241]. The ex-tragalactic flux receives three important contributions: (1) attenuated prompt emis-sion; (2) attenuated emission from IC of primary electrons and positrons; and (3)94mission from gamma-ray cascades. The cascade emission arises when an electron-positron pair is created by high-energy gamma rays scattering off of the CBR, in-ducing IC emission along with adiabatic energy loss. We account for these effectsfollowing [212, 236].
We assess how well predicted Galactic (NFW-correlated) and extragalactic (isotropic)fluxes describe the data using the profile-likelihood method (see e.g. [242]), describedin more detail in the associated appendix. To this end, we perform a template fit-ting analysis (using
NPTFit [49]) with 413 weeks of
Fermi
Pass 8 data collected fromAugust 4, 2008 to July 7, 2016. We restrict to the UltracleanVeto event class; further-more, we only use the top quartile of events as ranked by point-spread function (PSF).The UltracleanVeto event class is used to minimize contamination from cosmic rays,while the PSF cut is imposed to mitigate effects from mis-modeling bright regions.We bin the data in 40 logarithmically-spaced energy bins between
MeV and TeV, and we apply the recommended quality cuts
DATA_QUAL>0 && LAT_CONFIG==1 and zenith angle less than ∘ . The data is binned spatially using a HEALPix [196]pixelation with nside =128.We constrain this data to a region of interest (ROI) defined by Galactic latitude | 𝑏 | ≥ ∘ within ∘ of the Galactic Center (GC). The Galactic plane is masked inorder to avoid issues related to mismodeling of diffuse emission in that region. Simi-larly, we do not extend our region out further from the GC to avoid over-subtractionissues that may arise when fitting diffuse templates over large regions of the sky (see e.g. [45,48,243]). Finally we mask all point sources (PSs) in the 3FGL PS catalog [197]at their 95% containment radius.Using this restricted dataset, we then independently fit templates in each energybin in order to construct a likelihood profile as a function of the extragalactic andGalactic flux. We separate our model parameters into those of interest 𝜓 and the See http://fermi.gsfc.nasa.gov/ssc/data/analysis/documentation/Cicerone . 𝜆 . The 𝜓 include parameters for an isotropic template to accountfor the extragalactic emission, along with a template following a LOS-integrated NFWprofile to model the Galactic emission. Note that both the prompt and IC contributeto the same template, see the associated appendix for justification. The 𝜆 includeparameters for the flux from diffuse emission within the Milky Way, flux from the Fermi bubbles, flux from isotropic emission that does not arise from DM decay ( e.g. emission from blazars and other extragalactic sources, along with misidentified cosmicrays), and flux from PSs, both Galactic and extragalactic, in the 3FGL PS catalog.Importantly, each spatial template is given a separate, uncorrelated degree of freedomin the northern and southern hemispheres, further alleviating over-subtraction.In our main analysis, we use the Pass 7 diffuse model gal_2yearp7v6_v0 ( p7v6 )to account for diffuse emission in the Milky Way, coming from gas-correlated emission(mostly pion decay and bremsstrahlung from high-energy electrons), IC emission, andemission from large-scale structures such as the Fermi bubbles [36] and Loop 1 [244].Additionally, even though the
Fermi bubbles are included to some extent in the p7v6 model, we add an additional degree of freedom for the bubbles, following the uniformspatial template given in [36]. We add a single template for all 3FGL PSs basedon the spectra in [197], though we emphasize again that all PSs are masked at 95%containment. See the associated appendix for variations of these choices.Given the templates described above, we are able to construct 2-d log-likelihoodprofiles log 𝑝 𝑖 ( 𝑑 𝑖 |{ 𝐼 𝑖 iso , 𝐼 𝑖 NFW } ) as functions of the isotropic and NFW-correlated DM-induced emission 𝐼 𝑖 iso and 𝐼 𝑖 NFW , respectively, in each of the energy bins 𝑖 . Here, 𝑑 𝑖 isthe data in that energy bin, which simply consists of the number of counts in eachpixel. The likelihood profiles are given by maximizing the Poisson likelihood functionsover the 𝜆 parameters.Any decaying DM model may be constrained from the set of likelihood profiles ineach energy bin, which are provided as Supplementary Data [245]. Concretely, givena DM model ℳ , the total log-likelihood log 𝑝 ( 𝑑 |ℳ , { 𝜏, 𝑚 𝜒 } ) is simply the sum ofthe log 𝑝 𝑖 , where the intensities in each energy bin are functions of the DM mass andlifetime. The test-statistics (TS) used to constrain the model is twice the difference96etween the log-likelihood at a given 𝜏 and the value at 𝜏 = ∞ , where the DMcontributes no flux. The 95% limit is given by TS = − . .In order to compare our gamma-ray results to potential signals from IceCube, wedetermine the region of parameter space where DM may contribute to the observedhigh-energy neutrino flux. We use the recent high-energy astrophysical neutrino spec-trum measurement by the IceCube collaboration [205]. In that work, neutrino fluxmeasurements from a combination of muon-track and shower data are given in 9logarithmically-spaced energy bins between 10 TeV and 10 PeV, under the assump-tion of equal flavor ratios and an isotropic flux. We assume that DM decays arethe only source of high-energy neutrino flux. In Fig. 4-1 (assuming the DM decaysexclusively to 𝑏 ¯ 𝑏 ) we show the region where the DM model provides at least a 3 𝜎 improvement over the null hypothesis of no high-energy flux at all. The best-fit pointis marked with a star. The blue region in Fig. 4-1 is the best-fit region [221] forexplaining an apparent excess in the 2-year medium energy starting event (MESE)IceCube data, which extends down to energies ∼ 𝜏 and 𝑚 𝜒 . We emphasize that we allow the spectral index to float,as opposed to the analysis of [221], which fixes the index equal to two. In Fig. 4-1, we show our total constraint on the DM lifetime for a model where 𝜒 → 𝑏 ¯ 𝑏 . This result demonstrates tension in models where decaying DM explainsor contributes to the astrophysical neutrino flux observed by IceCube. PeV-scale Constraints at high masses may be improved by incorporating recent results from [246], whichfocused on neutrino events with energies greater than 10 PeV. e.g. [207, 236, 248–268]).In particular, while conventional astrophysical models such as those involving star-forming galaxies and galaxy clusters provide viable explanations for the neutrino dataabove 100 TeV (see [269] for a summary of recent ideas), the MESE data have beendifficult to explain with conventional models [270, 271]. Moreover, it is natural toexpect heavy DM to slowly decay to the SM in a wide class of scenarios where, forexample, the DM is stabilized through global symmetries in a hidden sector that areexpected to be violated at the Planck scale or perhaps the scale of grand unification(the GUT scale).From a purely data-driven point of view it is worthwhile to ask whether any setof SM final states may contribute significantly to or explain the IceCube data whilebeing consistent with the gamma-ray constraints. In the associated appendix weprovide limits on a variety of two-body SM final states.It is also important to interpret the bounds as constraints on the parameter spaceof UV models or gauge-invariant effective field theory (EFT) realizations. If the decayis mediated by irrelevant operators, and given the long lifetimes we are probing, itis natural to assume very high cut-off scales Λ , such as the GUT scale ∼ GeVor the Planck scale 𝑚 Pl ≃ . × GeV. We expect all gauge invariant operatorsconnecting the dark sector to the SM to appear in the EFT suppressed by a scale 𝑚 Pl or less (assuming no accidentally small coefficients and, perhaps, discrete globalsymmetries).It is also interesting to consider models that could yield signals relevant for thisanalysis. Many cases are explored in the associated appendix, and here we highlightone simple option: a hidden sector that consists of a confining gauge theory, at scale Λ 𝐷 [272], without additional light matter. Hidden gauge sectors that decouple fromthe SM at high scales appear to be generic in many string constructions (see [273] for arecent discussion). Denoting the hidden-sector field strength as 𝐺 𝐷𝜇𝜈 , then the lowestdimensional operator connecting the hidden sector to the SM appears at dimension-6:
ℒ ⊃ 𝜆 𝐷 𝐺 𝐷𝜇𝜈 𝐺 𝜇𝜈𝐷 | 𝐻 | / Λ , where 𝜆 𝐷 is a dimensionless coupling constant, Λ is the scale where this operator is generated, and 𝐻 the SM Higgs doublet. The98 m χ [GeV]10 τ [ s ] Glueball DM
EG onlyIC onlyPrompt only
Fermi combinedIceCube m χ [GeV]0 . . . B r a n c h i n g r a t i o s hhb ¯ bττWW + ZZ Figure 4-2: Limits on decaying glueball DM (see text for detals). We show limitsobtained from prompt, IC, and EG emission only, along with the 95% confidencewindow for the expectation of each limit from MC simulations. Furthermore, theparameter space where the IceCube data may be interpreted as a ∼ 𝜎 hint for DMis shown in shaded green, with the best fit point represented by the star. (inset) Thedominant glueball DM branching ratios.lightest ++ glueball state in the hidden gauge theory is a simple DM candidate 𝜒 ,with 𝑚 𝜒 ∼ Λ 𝐷 , though heavier, long-lived states may also play important roles (see e.g. [274]). The lowest dimension EFT operator connecting 𝜒 to the SM is then ∼ 𝜒 | 𝐻 | Λ 𝐷 / Λ . Furthermore, Λ 𝐷 (cid:38) MeV in order to avoid constraints on DMself-interactions [275].At masses comparable to and lower than the electroweak scale, the glueball decaysprimary to 𝑏 quarks through mixing with the SM Higgs, while at high masses theglueball decays predominantly to 𝑊 ± , 𝑍 , and Higgs boson pairs (see the inset ofFig. 4-2 for the dominant branching ratios). In the high-mass limit, the lifetime isapproximately 𝜏 ≃ · s (︂ 𝑁 𝐷 𝜋𝜆 𝐷 )︂ (︂ Λ 𝑚 Pl )︂ (︂ . PeV Λ 𝐷 )︂ , (4.1)with 𝑁 𝐷 the number of colors. This is roughly the right lifetime to be relevant for99he IceCube neutrino flux.In Fig. 4-2, we show our constraint on this glueball model. Using Eq. (4.1), theseresults suggest that models with Λ 𝐷 (cid:38) . PeV, 𝜆 𝐷 (cid:38) / (4 𝜋 ) , and Λ = 𝑚 Pl areexcluded. As in Fig. 4-1, the shaded green is the region of parameter space where themodel may contribute significantly to IceCube, and the dashed red line provides thelimit we obtain from IceCube allowing for an astrophysical contribution to the flux.As in the case of the 𝑏 ¯ 𝑏 final state, the gamma-ray limits derived in this chapter arein tension with the decaying-DM origin of the signal.Figure 4-2 also illustrates the relative contribution of prompt, IC and extragalacticemissions to the total limit. The 95% confidence interval is shown for each source,assuming background templates only, where the normalizations are fit to the data.Across almost all of the mass range, and particularly at the highest masses, the limitsobtained on the real data align with the expectations from MC. In the statistics-dominated regime, we would expect the real-data limits to be consistent with thosefrom MC, while in the systematics dominated regime the limits on real data maydiffer from those obtained from MC. This is because the real data can have residualscoming from mis-modeling the background templates, and the overall goodness of fitmay increase with flux from the NFW-correlated template, for example, even in theabsence of DM. Alternatively, the background templates may overpredict the fluxat certain regions of the sky, leading to over-subtraction issues that could make thelimits artificially strong. In this chapter, we presented some of the strongest limits to date on decaying DMfrom a dedicated analysis of
Fermi gamma-ray data incorporating spectral and spatialinformation, along with up-to-date modeling of diffuse emission in the Milky Way.Our results disfavor a decaying DM explanation of the IceCube high-energy neutrinodata.There are several ways that our analysis could be expanded upon. We have not at-100empted to characterize the spectral composition of the astrophysical contributions tothe isotropic emission, which may strengthen our limits. On the other hand, ideally,for a given, fixed decaying DM flux in the profile likelihood, we should marginal-ize not just over the normalization of the diffuse template but also over all of theindividual components that go into making this template, such as IC emission andbremsstrahlung.A variety of strategies beyond those described here have been used to constrainDM lifetimes (see e.g. [276] for a review). These include gamma-ray line searches,such as those performed in [129,277–279], which are complementary to the constraintson broader energy emission given in this chapter. Limits from direct decay into neu-trinos have also been considered [280]. Less competitive limits have been set on DMdecays resulting in broad energy deposition and nearby galaxies and galaxy clus-ters [182, 281], large scale Galactic and extragalactic emission [213, 282–285], MilkyWay Dwarfs [286, 287], and the CMB [23]. The upcoming Cherenkov Telescope Ar-ray (CTA) experiment [288] may have similar sensitivity as our results to DM masses ∼
10 TeV [289]. However, more work needs to be done in order to assess the potentialfor CTA to constrain or detect heavier, ∼ PeV decaying DM. On the other hand, theHigh-Altitude Walter Cherenkov Observatory (HAWC) [290] and air-shower experi-ments such as Tibet AS+MD [291] will provide meaningful constraints on the Galacticdiffuse gamma-ray emission. The constraints on DM lifetimes might be as stringentas − s for PeV masses and hadronic channels, assuming no astrophysicalemission is seen [236, 237, 292].Finally, we mention that our results also have implications for possible decayingDM interpretations (see e.g. [293]) of the positron [219,294] and antiproton fluxes [217]measured by AMS-02. Recent measurements of the positron flux appear to exhibit abreak at high masses that could indicate evidence for decaying DM to, for example, 𝑒 + 𝑒 − with 𝑚 𝜒 ∼ 𝜏 ∼ s. However, our results appear to rule out thedecaying DM interpretation of the positron flux for this and other final states. Forexample, in the 𝑒 + 𝑒 − case our limit for 𝑚 𝜒 ∼ 𝜏 (cid:38) × s.10102 hapter 5Multi-Step Cascade Annihilations ofDark Matter and the Galactic CenterExcess Over the past five years, numerous independent studies have confirmed a flux offew-GeV gamma rays from the inner Milky Way, steeply peaked toward the GalacticCenter, that is not captured by models for the known diffuse backgrounds [38–45, 67,68, 102]. This “Galactic Center excess” (GCE), detected using public data from the
Fermi
Gamma-Ray Space Telescope, has a spatial morphology well described by thesquare of a generalized Navarro-Frenk-White (NFW) profile, projected along the lineof sight. Furthermore, it is highly spherically symmetric, centered on the GalacticCenter (GC), and extends at least 10 degrees from the GC [45]; these conclusionsremain unchanged when accounting for systematic uncertainties in the modeling ofthe diffuse backgrounds [102]. These spatial properties suggest the excess emissioncould arise from the annihilation of dark matter (DM) with an NFW-like densityprofile. Competing interpretations include a transient event at the GC producing This analysis exploited improvements to the
Fermi point spread function as described in [100]. The presence of an intermediate step between DM annihilation and the productionof SM particles broadens the spectrum of SM particles produced, and consequentlyalso broadens the resulting gamma-ray spectrum, unless the mediator is degeneratein mass with either the DM or the total mass of the SM decay products. The gamma-ray multiplicity is increased by a factor of two, if each mediator decays into two SMparticles, and the typical energy of the gamma-rays is reduced accordingly. Thuscascade models for the excess generically tend to accommodate: ∙ Higher DM masses, ∙ Decays of the mediator to SM final states whose decays produce a more sharplypeaked gamma-ray spectrum than favored by direct annihilation.In general, there may be more than one decay step within the dark sector; thedominant annihilation of the DM need not be to the lightest dark sector particle Annihilation into the dark sector can also lead to a novel spatial distribution for the signal [310],but the GCE favors a cuspy morphology, so in this chapter we assume all decays are prompt. 𝜒𝜒 → 𝜑 𝑛 𝜑 𝑛 → × 𝜑 𝑛 − 𝜑 𝑛 − → ... → 𝑛 − × 𝜑 𝜑 → 𝑛 × 𝑓 ¯ 𝑓 . (5.1)Here 𝑓 ¯ 𝑓 are SM lepton or quark pairs, which can subsequently decay; the decaysshown above may also produce photons in the final step via final state radiation(FSR). By fitting the resulting photon spectrum to the GCE, we determine the allowedvalues of cross-section and DM mass for cascades with one to six steps, for a varietyof SM final states. Provided that the masses of the particles at each step in thecascade are not near-degenerate, the final spectrum of gamma-rays becomes nearlyindependent of the exact masses at each step. This assumption is not limiting, as105esults for the quality of fit for the more general case of non-hierarchical cascades(with nearly-degenerate steps) can be simply extracted from results derived assuminga large hierarchy.In Sec. 5.2 we outline the determination of the photon spectrum for an 𝑛 -stepcascade with specified SM final state, and discuss the procedure used to comparesuch a spectrum to the GCE. We present sample results of these fits in Sec. 5.3 undercertain assumptions. Section 5.4 extends our results for general cascades, and containsour complete fit results. In Sec. 5.5 we outline the existing experimental constraintsa complete model for the GCE via cascade decays would need to satisfy. We presentour conclusions in Sec. 7.5. In the appendices we provide additional details of ourmethodology and discuss some further model-dependent considerations. The photon flux generated by the annihilations of self-conjugate DM as a functionof the direction observed in the sky, is given by: Φ ( 𝐸 𝛾 , 𝑙, 𝑏 ) = ⟨ 𝜎𝑣 ⟩ 𝜋𝑚 𝜒 𝑑𝑁 𝛾 𝑑𝐸 𝛾 𝐽 ( 𝑙, 𝑏 ) , (5.2)where ⟨ 𝜎𝑣 ⟩ is the thermally averaged annihilation cross-section, 𝑚 𝜒 is the DM mass,and 𝑑𝑁 𝛾 /𝑑𝐸 𝛾 is the photon spectrum per DM annihilation, which has contributionsfrom FSR and from the decay of the leptons or quarks and their subsequent hadroniza-tion products. The 𝐽 -factor, the integral of DM density squared along the line-of-sight, is a function of the observed direction in the sky expressed in terms of Galacticcoordinates 𝑙 and 𝑏 : 𝐽 ( 𝑙, 𝑏 ) = ∫︁ ∞ 𝜌 (︂√︁ 𝑠 + 2 𝑟 ⊙ 𝑠 cos 𝑙 cos 𝑏 + 𝑟 ⊙ )︂ 𝑑𝑠 , (5.3) As discussed in Appendix D.1, our results can be readily translated to the case of decays,although the steeply peaked morphology of the GCE disfavors this interpretation. 𝑟 ⊙ ≈ . kpc is the distance from the Sun to the Galactic Center, and 𝑠 parametrizes the integral along the line-of-sight. We parameterize the DM density bya generalized NFW halo profile [74, 75]: 𝜌 ( 𝑟, 𝛾 ) = 𝜌 ( 𝑟/𝑟 𝑠 ) − 𝛾 (1 + 𝑟/𝑟 𝑠 ) − 𝛾 . (5.4)Here we use 𝑟 𝑠 = 20 kpc, 𝜌 = 0 . GeV/cm and 𝛾 = 1 . , following [102], as we willcompare our models to the data using the spectrum and covariance matrix determinedby that work.We focus on 𝑛 -step cascades ending in 𝜑 → 𝑓 ¯ 𝑓 , where 𝑓 ¯ 𝑓 is a pair of electrons,muons, taus or 𝑏 -quarks. Other SM final states are possible, of course, but these casesspan the range from steeply peaked photon spectra close to the DM mass throughto the lower-energy and broader spectra characteristic of annihilation to hadrons. Inorder to generate the cascade spectrum, we first start with the result from direct DMannihilation, which is equivalent to the spectrum from 𝜑 decay (in the 𝜑 rest frame)if the DM mass is half the 𝜑 mass. For the case of electrons or muons we determinethis spectrum analytically using the results of [313], whilst for taus and 𝑏 -quarks theresults are simulated in Pythia8 [222]. We have relegated the details of calculatingthese spectra to Appendix D.1.We denote the spectrum obtained at this “0th step” by 𝑑𝑁 𝛾 /𝑑𝑥 , where 𝑥 =2 𝐸 /𝑚 , 𝑚 is the mass of 𝜑 and 𝐸 is the energy of the photon in the 𝜑 rest frame.The shape of the photon spectrum is determined by the identity of the final stateparticle 𝑓 and also the ratio 𝜖 𝑓 = 2 𝑚 𝑓 /𝑚 . In the limit where the decay of 𝜑 isdominated by a two-body final state (at least for the purposes of photon production),the photon spectrum converges to a constant shape (as a function of 𝑥 ) as 𝜖 𝑓 → and the 𝑓 ¯ 𝑓 become highly relativistic. However, final state radiation (FSR) andhadronization depend on the energy of the 𝑓 ¯ 𝑓 products of the 𝜑 decay in the 𝜑 rest frame, so in cases where these effects dominate, the dependence of the photonspectrum on 𝜖 𝑓 is more complex.In Fig. 5-1 we show 𝑑𝑁 𝛾 /𝑑𝑥 per annihilation for the four different final states107 - - x x d N γ / dx � - ���� ������� � × �� μ × �� τ � ϵ � = ��� ϵ � = ��� Figure 5-1: 𝑑𝑁 𝛾 /𝑑𝑥 for 𝜑 decaying to ( 𝑒, 𝜇, 𝜏, 𝑏 ) in (blue, red, green, orange). Solid curves correspond to 𝜖 𝑓 = 0 . , and dashed to 𝜖 𝑓 = 0 . . Theelectron and muon spectra have been magnified by a factor of ten to appear comparable to the tausand 𝑏 s. we considered, for 𝜖 𝑓 = 0 . and 𝜖 𝑓 = 0 . . The photon spectra from electron andmuon production are dominated by FSR, whereas for 𝑏 -quarks fragmentation andhadronization are important. In the photon spectrum from taus, these effects aresubdominant and so the impact of varying 𝜖 𝑓 is minimal. Note that the spectrum for 𝑏 -quarks is peaked at a significantly lower 𝑥 , highlighting why models with this finalstate tend to accommodate higher DM masses.Given the 0-step spectrum, determining the photon spectrum from an 𝑛 -stepcascade is particularly simple in the case of scalar mediators, where the calculationessentially reduces to Lorentz-boosting the photon spectrum up the ladder of particlesappearing in the cascade. We review this calculation in Appendix D.2. As observedin [313], in the case of large mass hierarchies between the steps in the cascade, thefinal photon spectrum can be simplified even further, as we now outline.Consider the 𝑖 th step in the cascade, where the decay is 𝜑 𝑖 +1 → 𝜑 𝑖 𝜑 𝑖 . Let usdefine 𝜖 𝑖 = 2 𝑚 𝑖 /𝑚 𝑖 +1 , and assume 𝜖 𝑖 ≪ . Suppose the photon spectrum from decayof a single 𝜑 𝑖 (and the subsequent cascade), in the rest frame of the 𝜑 𝑖 particle, is We discuss the case of vector mediators in Sec. 5.4. Note that the earlier-defined 𝜖 𝑓 parameter does not function in exactly the same way as these 𝜖 𝑖 parameters: 𝜖 𝑓 fully parameterizes the photon spectrum associated with production and decay ofthe SM particles, whereas the 𝜖 𝑖 only describe Lorentz boosts. - - - x x d N γ / dx ������� ������� � = τ � ϵ τ = ��� ����� ������� Figure 5-2:
An example photon spectrum from direct annihilation to taus (grey) and hierarchicalcascades with 𝑛 = (1,2,3,4,5,6) steps, corresponding to (purple, blue, green, pink, orange, red) curves.The presence of each additional step in the cascade acts to broaden and soften the spectrum, andshift the peak to lower masses. All spectra are per annihilation. known and denoted by 𝑑𝑁 𝛾 /𝑑𝑥 𝑖 − . Then, in the presence of a large mass hierarchy,the decay of 𝜑 𝑖 +1 produces two highly relativistic 𝜑 𝑖 particles, each (in the rest frameof the 𝜑 𝑖 +1 ) carrying energy equal to 𝑚 𝑖 +1 / 𝑚 𝑖 /𝜖 𝑖 . The photon spectrum in therest frame of the 𝜑 𝑖 +1 is then given by a Lorentz boost (see Appendix D.2), and inthe limit 𝜖 𝑖 ≪ takes the simple form [313]: 𝑑𝑁 𝛾 𝑑𝑥 𝑖 = 2 ∫︁ 𝑥 𝑖 𝑑𝑥 𝑖 − 𝑥 𝑖 − 𝑑𝑁 𝛾 𝑑𝑥 𝑖 − + 𝒪 ( 𝜖 𝑖 ) . (5.5)Here we have introduced the dimensionless variable 𝑥 𝑖 = 2 𝐸 𝑖 /𝑚 𝑖 +1 , where 𝐸 𝑖 isthe photon energy in the 𝜑 𝑖 +1 rest frame. Following this, once we know the 0-stepspectrum we can iteratively derive the 𝑛 -step result. The error introduced by thisassumption is 𝒪 ( 𝜖 𝑖 ) , as we quantify in Appendix D.2.Beyond simplifying calculations, the large hierarchy approximation is also conve-nient for the following two reasons. Firstly in this limit, we can specify the shape ofthe spectrum simply by the identity of the final state 𝑓 , the value of 𝜖 𝑓 , and finallythe number of steps 𝑛 . This is in contrast to the many possible parameters that couldbe present in a generic cascade. Secondly, as we will elaborate further in Sec. 5.4, it isalso possible to read off the results for a generic hierarchy once we know the small 𝜖 𝑖 𝜖 𝑖 → ), the 𝜑 𝑖 ’s are producedat rest. When they subsequently decay, there is no boost to the 𝜑 𝑖 +1 rest frame, andso an 𝑛 -step cascade effectively reduces to a hierarchical ( 𝑛 − -step cascade, exceptfor the additional final state multiplicity.The Galactic frame is approximately the rest frame of the (cold) DM; consequently,to determine the measured photon spectrum, we need to calculate the photon spec-trum in the rest frame of the original DM particles. For an 𝑛 -step cascade, thiswill involve 𝑛 such convolutions, starting from the 𝑑𝑁 𝛾 /𝑑𝑥 𝑚 𝑖 = 𝑛 = 2 𝑚 𝜒 . Thus 𝑥 𝑖 = 𝑛 = 𝐸 𝑛 /𝑚 𝜒 ,and the Galactic-frame photon spectrum will be 𝑑𝑁 𝛾 /𝑑𝑥 𝑛 = 𝑚 𝜒 𝑑𝑁 𝛾 /𝑑𝐸 𝑛 . Fig. 5-2shows the resulting spectrum for a 0-6 step cascade in the case of final state tauswith 𝜖 𝜏 = 0 . . Each step in the cascade broadens out and softens the spectrum, andsimilar behaviour is seen for other final states.In order to determine the favored parameter space, for a given choice of 𝑓 , 𝜖 𝑓 , andnumber of steps in the cascade 𝑛 , we vary 𝑚 𝜒 and an overall normalization parameter 𝜂 (proportional to ⟨ 𝜎𝑣 ⟩ /𝑚 𝜒 , as we will see below) and compare the model to the datausing the spectrum and covariance matrix of [102]. In detail we calculate 𝜒 accordingto: 𝜒 = ∑︁ 𝑖𝑗 ( 𝒩 𝑖, model − 𝒩 𝑖, data ) 𝐶 − 𝑖𝑗 ( 𝒩 𝑗, model − 𝒩 𝑗, data ) , (5.6)where 𝒩 𝑖, model = (︂ 𝜂𝑚 𝜒 𝐸 𝑛 𝑑𝑁𝑑𝑥 𝑛 )︂ 𝑖, model (5.7) 𝒩 𝑖, data = (︂ 𝐸 𝑑𝑁𝑑𝐸 )︂ 𝑖, data (5.8)and both model and data are expressed in units of GeV/cm /s/sr averaged overthe region of interest. Here the 𝐶 − 𝑖𝑗 are elements of the inverse covariance matrix,which together with the data points are taken from [102]. By Eq. 5.2, the fitted110ormalization 𝜂 is related to the DM mass and the J-factor by: ⟨ 𝜎𝑣 ⟩ = 8 𝜋𝑚 𝜒 𝜂𝐽 norm . (5.9)For consistency with the spectrum normalization of [102] the J-factor is averaged overthe ROI | 𝑙 | ≤ ∘ and ∘ ≤ | 𝑏 | ≤ ∘ , so that: 𝐽 norm = ∫︁ ROI 𝑑 Ω 𝐽 ( 𝑙, 𝑏 ) / ∫︁ ROI 𝑑 Ω ∼ . × GeV cm − . (5.10)(Note that 𝑑 Ω = 𝑑𝑙𝑑 sin 𝑏 , not 𝑑𝑙𝑑 cos 𝑏 , since 𝑏 measures the angle from the Galacticequator, not the north pole.) Self-Consistency Requirements:
The procedure outlined above treats 𝑚 𝜒 as a freeparameter that can be adjusted to modify the 0-step spectrum; the fit only uses theshape of the spectrum provided by the 0-step result and the boost of Eq. 5.5. However,there is an additional condition required for a cascade scenario to be physically self-consistent: the mass hierarchy between the DM mass and the particles produced inthe final state must be sufficiently large to accommodate the specified number ofsteps. Equivalently, there is a hard upper limit on the number of steps allowed, for agiven DM mass and final state.Recall that for an 𝑛 -step cascade ending in a final state 𝑓 , we defined 𝜖 𝑓 = 2 𝑚 𝑓 /𝑚 , 𝜖 = 2 𝑚 /𝑚 , 𝜖 = 2 𝑚 /𝑚 all the way up to 𝜖 𝑛 = 𝑚 𝑛 /𝑚 𝜒 . Combining these, theDM mass is given in terms of 𝑚 𝑓 and the 𝜖 factors by: 𝑚 𝜒 = 2 𝑛 𝑚 𝑓 𝜖 𝑓 𝜖 𝜖 ...𝜖 𝑛 , (5.11)If the 𝜖 𝑖 factors are allowed to float, we can still say that < 𝜖 𝑖 ≤ in all cases (sinceeach decaying particle must have enough mass to provide the rest masses of the decayproducts), setting a strict lower bound on the DM mass of: 𝑚 𝜒 ≥ 𝑛 𝑚 𝑓 /𝜖 𝑓 . (5.12)111n the remainder of this article we refer to this bound as a “self-consistency” conditionor defining “kinematically allowed” masses. For consistency with the assumption ofhierarchical decays (i.e. 𝜖 𝑖 ≪ ), the true bound on 𝑚 𝜒 will in general be somewhatstronger than this conservative estimate (although as we will discuss in Sec. 5.4, 𝜖 𝑖 can become quite close to 1 before significantly modifying the fit relative to the 𝜖 𝑖 → case). Here we present the results from the fits performed using the procedure outlinedin the previous section. Assuming hierarchical cascades, we perform fits for fourdifferent final states – electrons, muons, taus, and 𝑏 -quarks – and fit over the photonenergy range . GeV ≤ 𝐸 𝛾 ≤ GeV. Later in this section we discuss the effectsof cutting out high energy data points, and how the fits would change if we onlyconsidered statistical uncertainties.In Fig. 5-3 we show a sample result, in which we plot Δ 𝜒
1, 2 and 3 𝜎 contours in ( 𝑚 𝜒 , ⟨ 𝜎𝑣 ⟩ ) space for 1-6 step cascades ending in muons with 𝜖 𝜇 = 0 . . The trend inthe best fit point for each step is as expected. Recall the generic behavior illustratedin Fig. 5-2; each progressive step in the cascade acts to reduce the height of the peakand shift it to lower masses. Therefore higher steps in the cascades will be better fit bylarger DM mass and cross-section as is indeed the case in Fig. 5-3. The larger cross-section results from an interplay of effects as can be seen from Eq. 5.9: an increasedDM mass leads to a lower number density and hence a higher cross-section (scalingas 𝑚 𝜒 ), but the increased power per annihilation implies a lower 𝜂 (adding a factorof 𝑚 − 𝜒 ), and finally the reduced height of the peak in the dimensionless spectrum forhigher steps (as shown in Fig. 5-2) requires a larger 𝜂 . By default, we omit the low energy data points with . GeV ≤ 𝐸 𝛾 ≤ . GeV, as in thisregion the spectrum suffers larger uncertainties under variations of the background modeling, andthe preferred value of the NFW 𝛾 parameter is not robust [45]. We have confirmed that includingthese low-energy data points has little impact on our results. - - - - - m χ [ GeV ] L og σ v [ c m / s ec ] μ �������� ���� ��� �������� �� �� � σ �������� ϵ μ = ��� ����� ������ Figure 5-3:
Contours of Δ 𝜒 from the best-fit point (for a given step number 𝑛 ) corre-sponding to 1, 2 and 3 𝜎 for final state 𝜇 ’s,with 𝜖 𝜇 = 0 . . The purple, blue, green, pink,orange and red colors correspond to 𝑛 = 𝜇 ’s. Here we have fixed 𝜖 𝜇 = 0 . and fitover the range 0.5 GeV ≤ 𝐸 𝛾 ≤
300 GeV.
1. 1.5 2. 2.5 3. - - - - - - - - - Log m χ [ GeV ] L og σ v [ c m / s ec ] ���� ��� �� �� � σ �������� ������ ������� ��� ������������������ ������� ϵ � = ��� ��������� ��������� � - ������ Figure 5-4:
Contours of Δ 𝜒 correspond-ing to 1, 2 and 3 𝜎 for 𝑛 = 1 − steps for 𝑒 , 𝜇 , 𝜏 and 𝑏 final states with 𝜖 𝑓 = 0 . . The fitis performed over the range 0.5 GeV ≤ 𝐸 𝛾 ≤
300 GeV. The best fit point of each step forall four final states follows a power law rela-tion between 𝑚 𝜒 and ⟨ 𝜎𝑣 ⟩ , with index ∼ . .Only the darker regions are kinematically al-lowed. See text for details. In Fig. 5-4 we show the corresponding Δ 𝜒 contours for electron, muon, tau, and 𝑏 -quark final states, again fixing 𝜖 𝑓 = 0 . . The best-fit mass and cross-section foreach of the final states are empirically found to follow an approximate power lawwith ⟨ 𝜎𝑣 ⟩ ∝ 𝑚 . 𝜒 . As discussed above we would expect ⟨ 𝜎𝑣 ⟩ ∝ 𝑚 𝜒 if the spectrumdid not change in shape (simply being rescaled proportionally to 𝑚 𝜒 to ensure energyconservation); the additional 𝑚 . 𝜒 scaling factor reflects the change in shape of thespectrum.As discussed above, for a given DM mass and final-state fermion with mass 𝑚 𝑓 ,there is an absolute upper limit on the number of steps allowed in a cascade, sinceevery step corresponds to a change in mass scale of at least a factor of 2. In Fig.5-4, we show the contours if the limitation of Eq. 6.5 is ignored , since this conveysinformation on the mass scale and number of steps at which the broadness of thespectrum best matches the data; however, the mass values that violate this conditionand so do not represent a self-consistent physical scenario are shown in lighter shading.This issue is relevant for the heavier final-state fermions, taus and 𝑏 -quarks, andparticularly acute for taus. Finally note that the irregular shape of the contours for113 n Δ χ ��������� ϵ � = ��� ϵ � = ��� ϵ � = ���� - σ - σ - σ > σ n Δ χ ����� ϵ μ = ��� ϵ μ = ��� - σ - σ - σ > σ n Δ χ ���� ϵ τ = ��� ϵ τ = ��� �������� - σ - σ - σ > σ n Δ χ � - ������ ϵ � = ��� ϵ � = ��� �������� - σ - σ - σ > σ Figure 5-5:
Clockwise panels show the overall best fit for DM annihilating through an 𝑛 -stepcascade to electron, muon, 𝑏 -quark and tau final states. The grey solid, dashed (and dotted) linescorrespond to the Δ 𝜒 between the best fit at that step, and the best fit for all 𝑛 , for 𝜖 𝑓 = 0 . , . (and . ) respectively. In the case of tau and 𝑏 -quark final states, the blue dotted curves, de-noted ‘physical,’ correspond to the case where only kinematically allowed (self-consistent) massesare considered as per the discussion in Sec. 5.2 (we set 𝜖 𝑓 = 0 . for these curves). Note that inthe case of taus, the “physical” best-fit points for 0 and 1 steps have the same 𝜒 as the best-fitpoints when “unphysical” scenarios are allowed, but as the overall best fit is different (with higher 𝜒 ) their Δ 𝜒 is lower. The shaded bands correspond to the quality of fit. 0-step results are notincluded for electrons and muons, as these fits are poor and have Δ 𝜒 values well above the plotted 𝑦 -axis. Electrons, muons and taus prefer longer 3-5 step cascades, whilst annihilations to 𝑏 -quarksprefer shorter 0-2 step cascades. This is not surprising, since as has been already pointed out in theliterature, 𝑏 -quark final states are preferred for direct annihilations. Non-integer values of 𝑛 can beassociated with cascades containing steps with one or more large 𝜖 𝑖 , as discussed in Sec. 5.4. - Log E γ E γ d N γ / d E γ ( × - ) ���� ��� ������� � μτ � ϵ � = ��� Figure 5-6:
The blue, red, green and orange curves correspond to the overall best fit spectrumfor e, 𝜇 , 𝜏 and 𝑏 -quarks as determined from Fig. 5-5. Overlaid are the data points and systematicerrors from [102]. Note that due to correlations between energies, the best fit curves are not whatwould be naively expected if only statistical errors were present. 𝑛 -step 𝑚 𝜒 (GeV) 𝜎𝑣 (cm / sec) 𝜒 e 5 67.2 . × − 𝜇 . × − 𝜏 unphysical . × − 𝜏 physical . × − 𝑏 . × − 𝜖 𝑓 = 0 . . For thecase of taus we show a best fit point if we include kinematically disallowed masses(unphysical) and also if we restrict ourselves to physical masses as discussed in Sec. 5.2.Fits were performed over 20 degrees of freedom.Final State 𝑛 -step 𝑚 𝜒 (GeV) 𝜎𝑣 (cm / sec)e 3-6 28-107 − . - − . 𝜇 − . - − . 𝜏 unphysical − . - − . 𝜏 physical − . 𝑏 − . - − . Table 5.2: Range of parameters within 1 𝜎 of the best fit step for 𝜖 𝑓 = 0 . for electrons,muons, taus and 𝑏 -quarks. As in Table 5.1 we show both physical and unphysical tauresults.the one-step electrons and muons can be traced to the fact the 0-step FSR spectrumis both sharply peaked and has a kinematic edge, leading to a poor fit.In Fig. 5-5 we show the Δ 𝜒 values between the best fit at a given step number 𝑛 and the best fit overall, for each final state. We show results for both 𝜖 𝑓 = 0 . and . in all cases, and also include 𝜖 𝑓 = 0 . for electrons. As expected the results donot depend strongly on 𝜖 𝑓 , especially in the case of taus, which is in accord with theresults of Fig. 5-1. Note that the nominal overall best fit for the taus ( 𝑛 = 4 ) fallsinto the kinematically disallowed (inconsistent) region; 𝑛 = 4 cannot be physicallyaccommodated within 3 𝜎 of its preferred DM mass. For this reason the results fortaus and 𝑏 -quarks were rerun allowing only self-consistent scenarios (in the sense ofEq. 6.5); in these cases we obtain the results shown by the blue dotted curves inFig. 5-5. We summarize the best fit results for 𝜖 𝑓 = 0 . in Table 5.1 and the 1 𝜎 rangeas determined from Fig. 5-5 on these parameters in Table 5.2.115 - - x x d N γ / dx � - ���� �� γγ ������� ϵ � = ���� ������� ���� ����������� �� ��������������� ������� ���� � = � - �� μτ � δ - �� Figure 5-7:
The 0-step spectra for e, 𝜇 , 𝜏 and 𝑏 -quarks with 𝜖 𝑓 = 0 . are shown as the blue, red,green and orange curves. The dashed curves show the spectrum of a hierarchical 𝑛 -step cascade thatends in 𝜑 → 𝛾𝛾 (a 𝛿 -function in the 𝜑 rest frame) for 𝑛 = 1 − , with lighter curves correspondingto progressively longer cascades. In order to compare the shape of the spectra we have magnifiedthe 0-step spectra by a factor of , , . and . for e, 𝜇 , 𝜏 and 𝑏 -quarks respectively. We seethe electron spectrum is closest to a 2-3 step cascade ending in a 𝛿 -function, muons and taus areclosest to a 3-4 step cascade, whilst 𝑏 -quarks most resemble 6-7. In Fig. 5-6 we show the overall best fit spectrum for electron, muons, taus, and 𝑏 -quarks with 𝜖 𝑓 = 0 . . Although the spectra for direct annihilation to these finalstates are quite different, after introducing the freedom to have multi-step cascades, asimilar best fit spectrum is picked out in each case. To expand on this, we can comparethe various 0-step spectra - as displayed in Fig. 5-1 - to the result of a hierarchical 𝑛 -step cascade that ends in 𝜑 → 𝛾𝛾 . This comparison is shown in Fig. 5-7. Thespectrum of photons from this process is just a 𝛿 -function in the 𝜑 rest frame, and isin a sense the simplest possible photon spectrum. We find that the photon spectrumfrom direct annihilation to electrons is similar to that obtained by a 2-3 step cascadeterminating in 𝜑 → 𝛾𝛾 ; for muons and taus the closest match is a 3-4 step cascade;and for 𝑏 -quarks 6-7. Of course these correspondences are not exact – for example,the 𝑏 -quark spectrum is more complex than just applying Eq. 5.5 to a 𝛿 -function – butthey allow us to regard these 0-step spectra as arising approximately from a common( 𝛿 -function) spectrum convolved with differing numbers of cascade steps. We canthen intuit how many additional steps are required in each case, to bring the spectrato a similar shape. Combining these numbers with the preferred number of steps seenin Table 5.1, we find the GCE prefers a spectrum that can be roughly modeled as a116 -function occurring at the endpoint of 7-9 cascade decays. In this sense it seems fitsto the GCE prefer a cascade with a large number of steps, and that these can occurin the SM or dark sector.Likewise, this general picture can approximately describe showers in the darksector [309]. Such showers will effectively contain decay cascades of different lengths,but we find that the spectrum of [309] can be well described by a 𝛿 -function 𝜑 → 𝛾𝛾 broadened by ∼ decay steps. The best-fit scenario found in that paper correspondsto a DM mass of ∼ GeV; this is consistent with the preferred mass for our 1-stepelectron case, which also corresponds to a 𝛿 -function at the endpoint of a ∼ -stepcascade. A better fit to the data might therefore be obtained by combining such darkshowering with a short dark-sector cascade. In Sec. 5.4 we will return to this point,and discuss the sense in which our results may be used to estimate the parameterspace for dark shower models. A few comments about the various final states are in order.
Electrons:
The photon spectrum from direct annihilations 𝜒𝜒 → 𝑒 + 𝑒 − is sharplypeaked. This tends to produce a worse fit to the GCE. As such we need severalsteps in the cascade in order to broaden the spectrum sufficiently to allow for aparameter space where a significantly improved fit is possible, and this is shown bythe substantial decrease in the quality of fit at low 𝑛 in Fig. 5-5. It should be notedthat any model for the GCE with direct annihilation into electrons will likely be insevere tension with the data from AMS [314]. This tension is likely to persist for atleast the 𝑛 = 1 cascade, and possibly higher steps as well. As we go to higher-stepcascades the spectrum broadens and the AMS bounds are expected to weaken, butthe exact bounds should be worked out for any cascade scenario with a branchingfraction to electrons. For the purposes of this chapter, we use the electron case asan example of a sharply peaked photon spectrum to demonstrate the impact of thespectral broadening, not necessarily as a realistic explanation for the excess. Similarly, Private communication, Wei Xue. 𝑏 and tau regions largely unconstrained. The figure of meritfor CMB constraints is ⟨ 𝜎𝑣 ⟩ /𝑚 𝜒 [315, 316], up to an 𝒪 (1) factor which is channel-and spectrum-dependent [317, 318]. As discussed above, for the best-fit regions (forhierarchical decays), this quantity scales as ∼ 𝑚 . 𝜒 as the number of steps increases;thus, we expect the constraint to become slightly stronger for longer cascades. Muons:
In Fig. 5-5 we see that the muon final state spectrum has the samequalitative behavior as the electrons, and will be subject to similar constraints. Thisis unsurprising as the muon spectrum is quite similar to that from electrons, albeitwith a less pronounced peak (see Fig. 5-1).
Taus:
As with other leptonic final states, taus also prefer multi-step cascades forthe best fit. Note that the best fit point at 4 steps is in fact kinematically disallowed(inconsistent) as can be seen in Fig. 5-4 and as discussed in Sec. 5.2. However, thebest fit point after imposing the consistency condition, at 2 steps, is still a better fitthan the high-step cases with electron and muon final states. 𝑏 -quarks: DM annihilation to 𝑏 -quarks is the preferred channel for direct annihi-lation identified in [45, 102], where it already provides a good fit. Accordingly thereis no need to broaden the spectrum with a large number of cascades – however, aswe will discuss in Sec. 5.5, even a short cascade can greatly alleviate constraints fromcolliders and direct searches (see also [306, 307] and references therein). A cascadewith several steps can still give an equally good or slightly better fit, and of courseaccommodates higher masses than for the case of direct annihilation. However, sincethe spectrum is already fairly broad, adding too many additional steps makes the fitworse, as shown in Fig. 5-5. Accordingly, the DM mass cannot be pushed far above100 GeV without significantly worsening the fit, at least in the context of hierarchicalcascades. 118 . 1.5 2. - - - - - Log m χ [ GeV ] L og σ v [ c m / s ec ] ���� ��� � σ �������� � = �� ϵ � = ��� ����������� � γ ≤ ��� �������������� � γ ≤ �� ��������������� � γ ≤ ��� ��� Figure 5-8:
The 3 𝜎 contours for 1-6 step cascade annihilations to final state electrons with 𝜖 𝑒 = 0 . .Red contours correspond to fitting over the entire energy range . GeV ≤ 𝐸 𝛾 ≤ GeV with thefull covariance matrix of [102]. Orange contours correspond to fitting with a cut on high energies 𝐸 𝛾 ≤ GeV. Green contours correspond to a fit over the full energy range but with only thestatistical errors of [102].
In the results presented above we have fit the data of [102] to the photon spectrumfrom DM annihilations through multi-step cascades to various final states. The fit wasperformed over the energy range . GeV ≤ 𝐸 𝛾 ≤ GeV. There is some evidencethat the emission detected above 10 GeV may not share the same spatial profile asthe main excess, suggesting a possible independent origin (for example, these high-energy data appear to prefer a morphology centered at negative ℓ and with a shallowspatial slope [102]), so we also test the impact of omitting the data above 10 GeV.Finally, we explore the impact of including only the statistical uncertainties of [102],omitting systematic errors, to test the degree to which the constraints could improvewith reduction in the systematic uncertainties.We display the results of this study in Fig. 5-8-5-9, for the case of 𝑛 -step cascadeannihilations to final state electrons with 𝜖 𝑒 = 0 . . Annihilations to other final statesgenerically display the same behavior as the energy range and error estimates arevaried. Cutting out the high energy data points generically shifts the fit to prefer lower119 n Δ χ ��������� ����� ���� ������ � = �� ϵ � = ��� � γ ≤ ��� ���� γ ≤ �� ��� n Δ χ ��������� ����� ����� ������ � = �� ϵ � = ��� � γ ≤ ��� ���� γ ≤ �� ��� - Log E γ E γ d N γ / d E γ ( × - ) ���� ��� ������� � = �� ϵ � = ���� ���� ������ � γ ≤ ��� ���� γ ≤ �� ��� - Log E γ E γ d N γ / d E γ ( × - ) ���� ��� ������� � = �� ϵ � = ���� ����� ������ � γ ≤ ��� ���� γ ≤ �� ��� Figure 5-9:
Top Panels: We show the impact on the preferred number of steps when changing theenergy range and error types considered. Each curve is for final state electrons with 𝜖 𝑒 = 0 . . The leftfigure shows the use of systematic errors over the full and a restricted energy range ( 𝐸 𝛾 ≤ GeV)in red and orange respectively. The right figure is the equivalent for statistical errors, with the fullenergy range shown in green and the restricted in blue. Bottom panels: Here the best fit curves asdetermined from the top panels are shown with the appropriate data and errors from [102] overlaid,for the example case of the electron final state. The left panel shows the results for systematic errors,where the best fit point was 𝑛 = 5 for the full range (red curve) and 𝑛 = 3 for the restricted range(orange curve). The right panel shows the equivalent for statistical errors, where for the full rangethe 𝑛 = 6 curve is shown in green and for the restricted range the 𝑛 = 2 curve is in blue. masses and narrower spectra, and therefore corresponds to cascades with fewer steps– resembling a 𝛿 -function at the endpoint of a 5-7 step cascade, rather than a 7-9 stepcascade. At a fixed number of steps, the main impact of omitting the high-energy datapoints is to raise the preferred cross-section and shrink the contours. Understandingthe high-energy data will thus be important in distinguishing quantitative models forthe GeV excess.Fitting over statistical errors increases the actual 𝜒 values, and the rate at which 𝜒 increases away from its minimum (as expected), as demonstrated by the shrinkinggreen contours of Fig. 5-8. The overall preferred step in the cascade however is notdramatically affected, only changing by 0-1 steps, as shown in the top panels of Fig. 5-9 - we display the corresponding best fit spectra in the bottom panels. At a fixed120umber of steps, the preferred cross-section increases, becoming more similar to whatwe find when omitting the high energy points. The results displayed in the previous section were obtained assuming large masshierarchies between each cascade step. It is possible to recast these results to gaininsight into the case of general 𝜖 𝑖 values. To see this, consider the decay 𝜑 𝑖 +1 → 𝜑 𝑖 𝜑 𝑖 .As previously discussed, in the limit when two mass scales become degenerate ( 𝜖 𝑖 → ), an 𝑛 -step cascade effectively reduces to an ( 𝑛 − -step cascade, except for theadditional final state multiplicity. Thus adding a degenerate step to a cascade is muchsimpler than adding one with a large hierarchy: we need only multiply the spectrumby two to account for the increased multiplicity, and halve the photon energy scaleto account for the initial energy being spread between twice as many particles. (Forcompleteness, we check analytically that the limit of 𝜖 𝑖 → has this behavior inAppendix D.2.)In light of this, an 𝑛 -step cascade with one degenerate step and an ( 𝑛 − -step hierarchical cascade must provide equally good fits to the GCE, with the formerpreferring twice the annihilation cross-section and DM mass relative to the latter. Theincreased DM mass results from the halving of the energy scale, whilst to understandthe cross-section we look back to Eq. 5.9: adding the degenerate step doubles thephoton multiplicity, which halves 𝜂 to compensate, but the doubling of the DM massmeans overall the cross-section is increased by a factor of two. As such the results inFig. 5-4 can be readily extended for additional degenerate steps. For each additionaldegenerate step on top of an initial hierarchical cascade (the degenerate step mayoccur anywhere in the cascade), the shape of the 𝜒 contours remains the same, butshifted upward by a factor of two in mass and cross-section. With a sufficiently largenumber of degenerate decays, the DM mass required to fit the GCE could be made121 . 0.2 0.4 0.60.0.050.1 x x d N γ / dx ������� ��� ������� ϵ � = �� � = τ � ϵ τ = ��� ϵ = � ϵ = �� < ϵ < � ������ ����� ���� �������� ��� ��� ����� ��� ϵ ={ �������������������� } Figure 5-10:
The transition of the spectra between 𝜖 = 0 and 𝜖 = 1 , calculated using Eq. D.9.The example case is a 2-step cascade with final state taus and 𝜖 𝜏 = 0 . . The dark blue is for 𝜖 = 0 and is what would result from the large hierarchies approximation. The 𝜖 = 1 case shown inlight blue corresponds to a completely degenerate spectrum, and as such is equivalent to a shifted1-step curve. In between these two, we show intermediate 𝜖 values as the dashed curves, specifically 𝜖 = { . , . , . , . , . } . Note the rate of transition between the two cases is in keeping with theerror in the large hierarchies case being of order 𝒪 ( 𝜖 𝑖 ) . arbitrarily high, although this would seem to require considerable fine-tuning. (Anatural scenario in which one degenerate step arises due to a symmetry is discussedin [319].)Cascades with general values of 𝜖 𝑖 in turn interpolate between the two simpler casesalready considered, with small and large 𝜖 𝑖 . We give the general convolution formulain Appendix D.2, and an example of how spectra evolve as a single 𝜖 𝑖 shifts from 0to 1 is shown in Fig. 5-10. This interpolation provides an alternate interpretationfor Fig. 5-5: the 𝑛 on the 𝑥 -axis of these plots can be thought of as representing thenumber of steps with a large hierarchy, rather than the total number of steps. If oneof these steps becomes degenerate (while holding the total number of steps fixed), aspreviously discussed, we will move from 𝑛 to 𝑛 − steps in terms of the spectral shapeand hence quality of fit. Intermediate 𝜖 𝑖 values will interpolate smoothly between thesetwo cases. Thus for any arbitrary collection of hierarchical and degenerate steps, thequality of the fit and the location of the best-fit region in 𝑚 𝜒 − ⟨ 𝜎𝑣 ⟩ parameter spacecan already be estimated from Figs. 5-4-5-5. A concrete example of the transitionin preferred DM mass and cross-section is shown in Fig. 5-11, which corresponds to122 - - - m χ [ GeV ] L og σ v [ c m / s ec ] ���� ��� ������� ϵ � = �� � = τ � ϵ τ = ��� ϵ = � ϵ = �� < ϵ < � ������ ����� ���� ������� ��� �������� ��� ϵ ={ �������������������� } ���� �� ������ ���� σ �������� Figure 5-11:
The transition of the best fit point and 1 𝜎 contours between 𝜖 = 0 and 𝜖 = 1 ,calculated using Eq. D.9. The example case is a 2-step cascade with final state taus and 𝜖 𝜏 = 0 . .The transition is between the 𝜖 = 0 case in dark blue and 𝜖 = 1 in light blue. The dashed curvesmap out the transition with intermediate values, specifically 𝜖 = { . , . , . , . , . } . the variation of the spectrum shown in Fig. 5-10. The curve plotted out by the bestfit point for intermediate values of 𝜖 is not a straight line between the two extremevalues, but does not deviate far from this. Similar behavior was seen for other finalstates and choice of degenerate step.At a fixed DM mass, the perturbation to the 𝜖 𝑖 = 0 photon spectrum evolvesroughly as 𝜖 𝑖 as 𝜖 𝑖 varies from 0 to 1 (as discussed in Appendix D.2); this behavioris shown in Fig. 5-10, where the 𝜖 = 0 . spectrum is almost indistinguishable fromthe 𝜖 = 0 spectrum, and 𝜖 = 0 . , 𝜖 = 0 . and 𝜖 = 0 . give spectra intermediatebetween the 𝜖 = 0 and 𝜖 = 1 cases. The perturbation to the best-fit 𝜒 will tend toincrease even more slowly than 𝜖 𝑖 , in the case where 𝜖 𝑖 = 0 is a better fit than 𝜖 𝑖 = 1 ,since the DM mass and cross-section can float to absorb changes in the spectrum andreduce the increase in 𝜒 . In all examples tested the best-fit 𝜒 remains essentiallyunchanged from the 𝜖 𝑖 = 0 case out to 𝜖 𝑖 = 0 . .In general a cascade with 𝑛 total steps, 𝑛 𝑑 of which are degenerate ( 𝑛 𝑑 values of 𝜖 𝑖 → ) will have the same spectrum as a cascade with ( 𝑛 − 𝑛 𝑑 ) hierarchical steps witha factor of 𝑛 𝑑 enhancement in mass and cross-section. This is illustrated in Fig. 5-12 for the case of decays to final state 𝜏 ’s with 1-6 total cascade steps. Relaxing123 .5 2. 2.5 - - - - Log m χ [ GeV ] L og σ v [ c m / s ec ] ������ �� ���������� ����� ���� ��� ������� � = τ � ϵ τ = ��� ����� ������ ���� ����� ������� ��� ���� �� ����������� �� ������������ ����� ��� ���� χ � ����� ����� ������������� ������� Figure 5-12:
The purple, blue, green, pink, orange and red points correspond to the best fit ( 𝑚 𝜒 , 𝜎𝑣 ) point for a total number of cascade steps (degenerate + hierarchical) 𝑛 = 1, 2, 3, 4, 5, 6respectively; for annihilations to final state taus with 𝜖 𝜏 = 0 . . Points living on the same line have thesame number of hierarchical steps and therefore result in equally good fits to the data. Points of thesame color, but with progressively greater values of ( 𝑚 𝜒 , 𝜎𝑣 ) , correspond to successively replacinghierarchical steps with degenerate steps, holding the number of total steps fixed. For the above caseof taus only the one and two step hierarchical cascades are kinematically allowed as indicated inFig. 5-4 (note that the kinematic constraint applies to lines as a whole, not individual points; seetext), thus only points living on the solid lines are allowed as these lines correspond to cascades withone and two hierarchical steps respectively. the assumption of large hierarchies therefore results in a preferred triangular slice ofparameter space, bounded by curves with ⟨ 𝜎𝑣 ⟩ ∝ 𝑚 𝜒 and ⟨ 𝜎𝑣 ⟩ ∝ 𝑚 . 𝜒 . We can nowunderstand the results of Fig. 5-5 as mapping out the variation in 𝜒 when movingbetween classes of scenarios, each defined by a fixed number of hierarchical stepsbut containing scenarios with varying numbers of degenerate steps (each of theseclasses is represented by a line in Fig. 5-12). Note also that the kinematic constraintEq. 5.11 acts on classes rather than individual scenarios (since adding a degeneratestep doubles the DM mass but increases the number of steps by 1, strengthening theconstraint on DM mass by a factor of 2); if one scenario is disallowed the entire classis disallowed.Fig. 5-13 summarizes our combined results. There, the top panels display theregions mapped out in the ⟨ 𝜎𝑣 ⟩ − 𝑚 𝜒 plane by the best fit points involving 1-6 steps(either hierarchical or degenerate) cascades to final state electrons, muons, taus and124 ▼ ▼▼ ▼▼ ▼▼ ▼▼ ▼▼▼▼ ▼▼ ▼▼ ▼▼ ▼▼▼▼ ▼▼ ▼▼ ▼▼▼▼ ▼▼ ▼▼▼▼ ▼▼▼▼■■ ■■ ■■ ■■ ■■ ■■■■ ■■ ■■ ■■ ■■■■ ■■ ■■ ■■■■ ■■ ■■■■ ■■ ■■ ●● ●● ●● ●● ●● ●●●● ●● ●● ●● ●●●● ●● ●● ●●●● ●● ●●●● ●● ●● ×× ×× ×× ×× ×× ×××× ×× ×× ×× ×××× ×× ×× ×××× ×× ×××× ×× ××
1. 1.5 2. 2.5 3. - - - - Log m χ [ GeV ] L og σ v [ c m / s ec ] ���� ��� ������ ϵ � = ��� ����� ����� ������������� ���������� ���� ���� ������ ����� ������ ������ ▼ � ■ μ ● τ × � - σ - σ - σ > σ ▼▼ ▼▼ ▼▼ ▼▼ ▼▼ ▼▼▼▼ ▼▼ ▼▼ ▼▼ ▼▼▼▼ ▼▼ ▼▼ ▼▼▼▼ ▼▼ ▼▼▼▼ ▼▼ ▼▼■■ ■■ ■■ ■■ ■■ ■■■■ ■■ ■■ ■■ ■■■■ ■■ ■■ ■■■■ ■■ ■■■■ ■■ ■■ ●● ●● ●● ●● ●● ●●●● ●● ●● ●● ●●●● ●● ●● ●●●● ●● ●●●● ●● ●● ×× ×× ×× ×× ×× ×××× ×× ×× ×× ×××× ×× ×× ×××× ×× ×××× ×× ××
1. 1.5 2. 2.5 3. - - - - Log m χ [ GeV ] L og σ v [ c m / s ec ] ���� ��� ������ ϵ � = ��� ����� ����� ������������� ���������� ����������� � γ ≤ �� ��� ������ ������ ▼ � ■ μ ● τ × � - σ - σ - σ > σ ▼ ▼ ▼ ▼ ▼ ▼■ ■ ■ ■ ■ ■ ● ● ● ● ● ● × × × × × × n Δ χ ��� ���� ���� ������ ����� ▼ � ■ μ ● τ × � ϵ � = ��� - σ - σ - σ > σ ▼ ▼ ▼ ▼ ▼ ▼■ ■ ■ ■ ■ ■ ● ● ● ● ● ● × × × × × × n Δ χ ��� ����������� � γ ≤ �� ��� ▼ � ■ μ ● τ × � ϵ � = ��� - σ - σ - σ > σ Figure 5-13:
Combined results of fits with 𝜖 𝑓 = 0 . over the full energy range (left) or witha restriction 𝐸 𝛾 ≤ GeV (right). Top panels: Best fit ( 𝑚 𝜒 , 𝜎𝑣 ) for a cascade with 1-6 total(degenerate + hierarchical) steps ending in electrons, muons, taus of 𝑏 -quarks. Points on the sameline have the same number of hierarchical steps and therefore result in equally good fits to the data,following the discussion in Sec. 5.4. Points of the same color, but with sequentially greater values of ( 𝑚 𝜒 , 𝜎𝑣 ) , correspond to progressively replacing hierarchical steps with degenerate steps, holding thetotal number of steps fixed. The color of the lines indicate goodness of fit and only solid lines arekinematically allowed (as explained in see Sec. 5.2). Bottom panels: Show the overall best fit for DMannihilation through an 𝑛 -step hierarchical cascade to electron, muon, tau and 𝑏 -quark final states.The curves show the Δ 𝜒 of the best fit at that step and final state, as compared with best fit overall steps and final states. No restriction to physical kinematics is imposed, but where restrictionswould apply can be inferred from the top panels. The shaded bands correspond to the quality of fit.For fits over the full energy range a fairly short cascade terminating in decay to 𝑏 -quarks gives thepreferred spectrum, whilst over the restricted energy range each final state can potentially provideapproximately equally good fits. 𝑏 -quarks. In the bottom panels, we indicate which hierarchical step and final stateyield the best fit, and the comparative quality of fit for other combinations. Weshow all these results for fits over the full (left panels) and restricted (right panels)energy ranges. Additionally as shown in the top panels, electrons (taus) and muons( 𝑏 -quarks) have some degree of overlap, especially once degenerate steps are included.The overlap of these regions is reduced when the high energy data points are excluded,as is clear by comparing the right and left panels.The positions of the triangular regions in Fig. 5-13 largely reflect the differing125 ▼ ▼▼ ▼▼ ▼▼ ▼▼ ▼▼▼▼ ▼▼ ▼▼ ▼▼ ▼▼▼▼ ▼▼ ▼▼ ▼▼▼▼ ▼▼ ▼▼▼▼ ▼▼▼▼■■ ■■ ■■ ■■ ■■ ■■■■ ■■ ■■ ■■ ■■■■ ■■ ■■ ■■■■ ■■ ■■■■ ■■ ■■ ●● ●● ●● ●● ●● ●●●● ●● ●● ●● ●●●● ●● ●● ●●●● ●● ●●●● ●● ●● ×× ×× ×× ×× ×× ×××× ×× ×× ×× ×××× ×× ×× ×××× ×× ×××× ×× ××
1. 1.5 2. 2.5 3. - - - Log m χ [ GeV ] L og k σ v [ c m / s ec ] ���� ��� ������ ������ �� ������ ����� ϵ � = ��� ��� ���� ���� ������ ����� ������ ������ ▼ � ■ μ ● τ × � Figure 5-14:
Colored points indicate the best fits for different numbers of hierarchical and de-generate cascade steps, and different final states, as in Fig. 5-13. However, here we rescale thecross-section by the fraction of power into photons 𝑘 for each final state ( . × − , . × − , . and . for electrons, muons, taus and 𝑏 -quarks respectively). All final states then pick out thesame region of ( 𝑚 𝜒 , 𝑘𝜎𝑣 ) parameter space. The dashed lines indicate curves with 𝑘 ⟨ 𝜎𝑣 ⟩ ∝ 𝑚 𝜒 and 𝑘 ⟨ 𝜎𝑣 ⟩ ∝ 𝑚 . 𝜒 , chosen to originate from the lowest-mass point studied; these curves approximatelybound the full parameter space of interest (see text). branching ratios to photons (rather than other stable SM particles) for the differentfinal states. For each of the direct annihilation (0-step) spectra, we can compute afactor 𝑘 , defined as the total energy in photons (per annihilation) as a fraction of 𝑚 = 2 𝑚 𝜒 . For example, direct annihilation/decay to 𝛾𝛾 would have 𝑘 = 1 . Forthe final states we consider, we find 𝑘 = 3 . × − , . × − , . and . forelectrons, muons, taus and 𝑏 -quarks respectively. Final states with smaller 𝑘 willnaturally require higher cross-sections in order to fit the signal. In Fig. 5-14 we showthe results of Fig. 5-13 replotted in terms of 𝑘 ⟨ 𝜎𝑣 ⟩ and 𝑚 𝜒 : we see that once thisfactor is taken into account, all channels pick out essentially the same triangularregion of parameter space, bounded by curves with 𝑘 ⟨ 𝜎𝑣 ⟩ ∝ 𝑚 𝜒 and 𝑘 ⟨ 𝜎𝑣 ⟩ ∝ 𝑚 . 𝜒 . Incorporating dark showers:
This concordance between the different final statessuggests that dark shower models may be expected to also inhabit this region. Forinstance, the authors of [309] find a preferred cross-section of × − cm /s for their 𝑆𝑈 (2) 𝑉 model, with a roughly branching ratio into stable dark sector baryons(with other decay channels ending in photons), and a preferred mass of ∼ GeV. At126rst glance this suggests a somewhat higher value for 𝑘 ⟨ 𝜎𝑣 ⟩ than the lower tip of thetriangular region identified in Fig. 5-14. However, [309] fits to a different spectrumfor the GCE excess (taken from [45]), without a systematic uncertainty estimate,and assumes a lower local DM density (0.3 GeV/cm rather than 0.4 GeV/cm ). Inour analysis, omitting systematic errors (or removing high-energy data points) raisesthe preferred cross-section by a factor of ∼ (Fig. 5-8), and likewise lowering thelocal DM density from 0.4 to 0.3 GeV/cm would raise the required cross-section bya factor of ∼ ; the lower tip of our triangular region would then reside at 𝑚 𝜒 ∼ GeV and 𝑘 ⟨ 𝜎𝑣 ⟩ ∼ × − cm /s, which seems roughly consistent with [309]. Thus far we have considered models of multi-step cascades through scalar mediators.However models in which the hidden sector mediators include vector, fermion orpseudo-scalar particles are at least as equally well motivated (e.g. [305] or the darkshower example discussed above [309]). In the case of vector or fermionic mediatorsthe simple recursion formula Eq. 5.5 will in general no longer hold, since the photonspectrum from the decay of mediators with spin need not in general be isotropic. Thestandard recursion formula will also break down if a decay is more than two-body,or if the decay is two-body but the decay products have different masses (althoughif the decay is strongly hierarchical the impact will be tiny), since these possibilitiesmodify the Lorentz boost from the 𝜑 𝑖 frame to the 𝜑 𝑖 +1 frame. Note this is differentto having several possible decay chains with different branching ratios; in this caseour analysis does apply, and the final spectrum will simply be a linear combinationof the spectra produced by the different decay chains.Anisotropy of the photon spectrum is not in itself a sufficient condition for therecursion formula to break down. To modify the recursion, for some step 𝑖 , thedifferential decay rate of 𝜑 𝑖 must be a function of the angle 𝜃 between (1) the momentaof the decay products in the 𝜑 𝑖 rest frame and (2) the boost direction from the 𝜑 𝑖 rest frame to the 𝜑 𝑖 − rest frame. (Here we use 𝜑 𝑖 to denote arbitrary mediators, Private communication, Dean Robinson. 𝜑 𝑖 rest frame do not “know” aboutthe 𝜑 𝑖 +1 frame, this sort of correlation is only possible if (1) the direction of thespin/polarization vector of the 𝜑 𝑖 in its rest frame depends on the momentum withwhich it was produced in the 𝜑 𝑖 − rest frame, and (2) the spectrum of the decayproducts of 𝜑 𝑖 is a function of the angle between their momentum and the rest-framespin/polarization vector of 𝜑 𝑖 . If only one of the two applies, averaging over thespin/polarization of 𝜑 𝑖 will leave no 𝜃 -dependence. However, both these propertieswill generically hold if 𝜑 𝑖 is a vector: typically the decay of 𝜑 𝑖 − will prefer eitherlongitudinally or transversely polarized vectors 𝜑 𝑖 , which will in turn decay withdifferent angular distributions.Let us consider the potential impact of such a 𝜃 -dependence. For illustrativepurposes, let us suppose that the photons produced in the decays of 𝜑 (whetherdirectly or by subsequent decays of the fermions) have essentially the same energy spectrum as in the pure-scalar case, in the rest frame of the 𝜑 . This assumptionmight fail if the spin of 𝜑 affects the correlations (if any) between the fermion spins,fermion momenta and photon momenta, but by making it we can isolate the impactof angular dependence in a single step of the cascade.Consider a one step cascade 𝜒𝜒 → 𝜑 𝜑 , 𝜑 → 𝑓 ¯ 𝑓 , where 𝜑 is a vector boson.Suppose the full spectrum of photons in the 𝜑 rest frame can be written as 𝑑𝑁𝑑𝑥 = 𝑓 ( 𝑦 ) 𝑑𝑁/𝑑𝑥 , where 𝑦 = cos 𝜃 and 𝑑𝑁/𝑑𝑥 is the spectrum for the scalar mediatorcase 𝑓 = 1 . Then the now familiar formula for the energy spectrum in the 𝜒𝜒 centerof mass frame is: 𝑑𝑁 𝛾 𝑑𝑥 = 2 ∫︁ − 𝑑𝑦 ∫︁ 𝑑𝑥 𝑓 ( 𝑦 ) 𝑑𝑁 𝛾 𝑑𝑥 𝛿 (︂ 𝑥 − 𝑥 − 𝑦 𝑥 √︁ − 𝜖 )︂ = 2 ∫︁ 𝑥 𝑑𝑥 𝑥 𝑓 (︂ 𝑥 𝑥 − )︂ 𝑑𝑁 𝛾 𝑑𝑥 + 𝒪 ( 𝜖 ) . (5.13)where we calculated the 𝑦 integral assuming 𝜖 ≪ . Again we could extend thisexpression to an 𝑛 -step cascade using the same formalism as in Appendix D.2. The128 - - x x d N γ / dx ������� ���� � ������ ���� � = τ � ϵ τ = ��� ����� ���� �� ��� � θ �� ( � + ��� � θ ) � ( θ ) � Figure 5-15:
Spectrum for a 1-3 step cascade with a vector mediator in the final step of thecascade 𝜑 → 𝑉 𝑉 , 𝑉 → 𝑓 ¯ 𝑓 . We consider three separate cases: 𝑓 ( 𝜃 ) = 1 , (3 / 𝜃 ) , and (3 /
2) sin 𝜃 . The first of these is equivalent to a cascade with only intermediate scalars (and henceisotropic decays), the others correspond to common angular dependences (see text). angular dependence at each step will in general be different depending on the model;we can parameterize this by specifying different functions 𝑓 𝑖 ( 𝑦 𝑖 ) at each step. In thelimit of small 𝜖 𝑖 we find: 𝑑𝑁 𝛾 𝑑𝑥 𝑖 = 2 ∫︁ 𝑥 𝑖 𝑑𝑥 𝑖 − 𝑥 𝑖 − 𝑓 𝑖 − (︂ 𝑥 𝑖 𝑥 𝑖 − − )︂ 𝑑𝑁𝑑𝑥 𝑖 − + 𝒪 ( 𝜖 𝑖 ) . (5.14)A detailed study of the impact of vector or fermionic mediators is beyond thescope of this chapter; we leave it to future work. However, we can work out an explicitexample motivated by the case where at the end of the cascade, a scalar/pseudoscalarresonance decays to two vectors which subsequently each decay into two fermions.This scenario has been studied in the context of Higgs decays [320], furnishing resultsfor a general resonance 𝑋 decaying to two identical vectors 𝑉 𝑉 , which each in turnsubsequently decay to 𝑓 ¯ 𝑓 . (In our notation, the 𝑉 here would correspond to 𝜑 and 𝑋 to 𝜑 .) The differential decay rate to fermions in this case is a linear combinationof terms proportional to sin 𝜃 , 𝜃 and cos 𝜃 (where 𝜃 is the angle defined aboveand in Appendix D.2), with coefficients depending on the axial and vector couplingsof the fermions to the 𝑉 , and the parity of the initial state 𝑋 [320]. In hierarchicaldecays of a scalar or pseudoscalar resonance to 𝑉 𝑉 , where 𝑉 has vector (rather thanaxial vector) couplings to 𝑓 ¯ 𝑓 , the dominant angular dependence is either 𝜃 or sin 𝜃 . For these specific (but common) angular dependences in the 𝜑 decay, we show129he resulting changes to the photon spectrum in Fig. 5-15. The impact is modest,and so we expect our qualitative results should hold for more general cascades. While we have remained agnostic regarding the choice of an actual model, we pointout that any model with new light states in a dark sector that explains the GCE mustalso be consistent with the following experimental constraints: ∙ Direct Detection: The coupling controlling 𝜎 𝐷𝐷 must not be so large as to bein conflict with bounds from DM direct detection experiments [306]. ∙ Big Bang Nucleosynthesis (BBN): New light states must decay fast enough suchthat they do not spoil the predictions of BBN. ∙ Collider constraints.These experimental constraints on a multi-step cascade will be very similar to thoseon a one-step cascade, with the key parameter being the coupling of the dark sectorto the SM in both cases.The simplest models that explain the GCE by direct DM annihilations to SMstates are generally in conflict with direct detection bounds: the same coupling thatmust be small enough to avoid the LUX bound [132], must also be large enoughto explain the GCE with a thermal WIMP (note however that this conclusion is notinevitable; there are effective DM-SM couplings and simplified models that genericallyevade the bounds, e.g. [300, 301]). As pointed out in [305–307], the addition of a darksector with a single mediator allows for an explanation of the GCE while alleviatingdirect detection constraints. The reason is straightforward: any direct detection signalwill be controlled by the coupling of the mediator to the SM, whereas the annihilation rate is independent of this quantity, so the two can be tuned largely independently.We make this point more explicit in Appendix D.3. Exactly the same property holdsin models with expanded cascades, where the direct detection signal is controlled130y the coupling between the dark sector and the SM; indeed, the direct detectionsignal may be suppressed even further if the coupling between the DM and the SMrequires multiple mediators. If the couplings within the dark sector are not highlysuppressed, decays within the dark sector should in general proceed promptly (ontimescales ≪ s), and so the constraint from BBN will primarily limit the couplingof the final mediator in the cascade to the SM. Accordingly, since it has been shownthat for one-step cascades the constraint from BBN can be consistent with a nullsignal in direct detection experiments [306], the same should hold true for multi-stepcascades (since in the multi-step case, the final step controls the coupling to the SMand hence provides the only relevant parameter for both BBN and direct detection).Collider bounds and limits from invisible decays of SM particles are also controlledby this final coupling, so can accordingly be dialled down in the same way as for one-step cascades, consistent with BBN bounds on the final coupling [306]. A complexdark sector with multiple mediators could potentially give rise to interesting collidersignatures (e.g. [311, 321, 322]), but a detailed discussion is beyond the scope of thischapter. We have laid out a general framework for characterizing the photon spectrum frommulti-step decays within a secluded dark sector terminating in a decay to SM particles,and explored the ability of such a framework to produce the GeV gamma-ray excessobserved in the central Milky Way.For any given SM final state, allowing multi-step decays expands the preferredregion of 𝑚 𝜒 −⟨ 𝜎𝑣 ⟩ to a triangular region of parameter space, probed by cascades withdifferent numbers of degenerate and hierarchical decays (where the decay products areslow-moving or relativistic, respectively), and bounded by curves with ⟨ 𝜎𝑣 ⟩ ∝ 𝑚 𝜒 and ⟨ 𝜎𝑣 ⟩ ∝ 𝑚 . 𝜒 . Decays to different Standard model final steps correspond to differenttriangular regions in parameters space as shown in Fig. 5-13. Large numbers ofdegenerate decays can raise the mass scale for the DM without bound, albeit at the131ost of requiring a cross-section much higher than the thermal relic value and somedegree of fine-tuning. Hierarchical decays broaden the photon spectrum, permittinga better fit to the data for SM final states that produce a sharply peaked photonspectrum; however, more than 4-5 hierarchical decays begin to reduce the qualityof the fit even if the initial spectrum is very sharply peaked. In the absence ofdegenerate decays, the preferred mass range for the DM can then be constrained, andis consistently ∼ − GeV across all channels; the corresponding cross-sectionsare close to the thermal relic value for tau and 𝑏 -quark final states, and 1-2 ordersof magnitude higher for 𝑒 and 𝜇 final states. Regardless of the final state, with theadditional freedom of hierarchical decays the preferred spectrum tends to a similarshape, which can be approximated as the result of a cascade of 7-9 hierarchical decaysterminating in a two-body 𝛾𝛾 decay. We find that the best overall fits are still attainedby DM annihilating to 𝑏 -quarks (or other hadronic channels) with 0-2 hierarchicalsteps.Our preferred ⟨ 𝜎𝑣 ⟩ − 𝑚 𝜒 regions are fairly insensitive to the details of the uncer-tainty analysis or the range of data points included. However, omitting high-energydata (above 10 GeV) substantially reduces the preferred number of hierarchical decaysteps (from 4-5 to 2) for channels where the photon spectrum from direct annihilationis sharply peaked. There is currently disagreement between different analyses as tothe high-energy photon spectrum associated with the excess; we do not take a posi-tion on this question, but note that its resolution may affect the range of dark-sectormodels that can provide viable explanations of the excess.In this chapter we assumed that the directions of decay products in the rest frameof their progenitor are uncorrelated with the direction of the Lorentz boost to therest frame of the previous progenitor particle in the sequence. Whilst always truefor scalars, this may not hold for vector and fermionic mediators. We leave a moredetailed discussion of concrete multi-step cascade models exploring these issues forfuture work. 132 hapter 6Model-Independent IndirectDetection Constraints on HiddenSector Dark Matter Indirect searches provide one of the best ways to probe the nature of dark matter(DM) beyond gravitational interactions. Through the observation of gamma rays, cos-mic rays, and the anisotropies of the Cosmic Microwave Background (CMB), we mayfind a hint of DM annihilations to Standard Model (SM) particles. Many models havebeen proposed in which DM annihilates directly to a pair of SM particles through, forexample, a Higgs [323, 324], gauge boson [325], axion [312], or neutrino [326]. Goingbeyond these simple models, we can consider scenarios in which DM is secluded inits own rich dark sector; such a setup is well motivated from top-down considera-tions (e.g. [327] and references therein). In such scenarios, the DM does not coupledirectly to SM particles (or such couplings are highly suppressed), but instead an-nihilates to unstable dark sector particles. These states may decay to SM particlesor to other dark sector states, but eventually mediator particles that couple to theSM are produced. The mediators subsequently decay into SM particles, which in133 ¯ n n n n n n ...... ... SMSM
SMSM } n ⇥ S M S M ¯ ... SMSM
SMSM } n ⇥ S M S M Figure 6-1:
Left: Schematic diagram of a generic hidden sector cascade. The DM, secluded inits own hidden sector, first annihilates to a pair of hidden sector particles. These 𝜑 𝑛 mediatorssubsequently decay to lighter particles in the hidden sector and finally to SM particles. Here weconsider SM = { 𝛾, 𝑒, 𝜇, 𝜏, 𝑏, 𝐻, 𝑊, 𝑔 } and 𝑛 = 𝑛 = 𝑛 -body state in the hidden sectorwhich then decays to SM particles. turn decay to stable and detectable photons, neutrinos, electrons, positrons, protonsand/or antiprotons. We refer to this pattern as a “cascade annihilation” or simply“cascade”, with a number of steps given by the number of distinct on-shell dark-sectorstates between the initial DM annihilation and the production of SM particles. Weillustrate this setup schematically in Fig. 6-1.Hidden Sector DM scenarios encompass a broad class of models. For instancemodels containing one light dark photon mediator [29, 305, 328], generically give riseto one-step cascades decays; DM annihilates to two dark photons which decay to SMparticles. Multi-step cascades can occur naturally in hidden valley models [329, 330].In such models, production of the DM at terrestrial colliders and scattering in directdetection experiments can be generically suppressed by the small coupling betweenthe dark and visible sectors. In contrast, indirect detection signals depend primarilyon the annihilation rate of the dark matter to particles within the dark sector; thesmall coupling between the sectors only suppresses the decay rate of the mediators toSM particles, which does not affect indirect searches provided the decay rate is smallon astrophysical timescales (as in e.g. [310]). Thus cascade annihilations scenarios areoften invoked to explain anomalies and suggest new DM signals. For instance in [62,306–308, 331–335] multi-step cascades were used to explain the apparent excess GeVgamma-rays identified in the central Milky Way [38–45, 67, 68, 102, 201], while evading134ounds from dark matter direct detection experiments. In general the injectionof photons and other high energy secondary particles produced is constrained by anumber of indirect searches. In particular we focus on: ∙ Measurements of the CMB by
Planck [188] ∙ Bounds set by
Fermi from DM searches in the Dwarf Spheroidal Galaxies of theMilky Way [57] ∙ Measurements of the 𝑒 + flux by AMS-02 [336, 337]Constraints from the above three experiments can be parametrized model-independentlyfor the case of direct DM annihilations (see for instance [338]), by classifying annihi-lations to all possible two-body SM final states, DM + DM → SM + SM. For a givenDM mass and final state, the spectra of secondary particles, is fixed independently ofthe form of the DM interaction and spin. Therefore constraints on DM annihilationrates are usually quoted in terms of the parameters relevant to the direct annihilationscenario, and do not encompass DM models embedded in a hidden sector. Given thebroad space of Hidden Sector DM models, it is essential to provide model-independentmethods that cover the majority of model space.In the present chapter, we present DM mass dependent bounds on the DM crosssection from the above three indirect detection experiments for DM annihilations via0-6 step cascades to eight SM final states: 𝛾𝛾 , 𝑒 + 𝑒 − , 𝜇 + 𝜇 − , 𝜏 + 𝜏 − , 𝑏 ¯ 𝑏 , 𝑔𝑔 , 𝑊 + 𝑊 − ,and ℎ ¯ ℎ . We remain agnostic about the details of the hidden sector, thus making ourstatements robust and model-independent. Limits from the Fermi dwarfs and AMS-02 generally provide the strongest constraints on channels that are rich in photonsand those that are not, respectively (although at sufficiently high masses, limits fromH.E.S.S. [340] and VERITAS [341] overtake those from
Fermi ). While there maybe arguably stronger photon bounds from the Galactic Center (e.g. [342]) or galaxyclusters (e.g. [343]), these limits depend strongly on the assumed dark matter density There is recent evidence this excess may originate from a population of point-like objects, ratherthan DM [46, 47]. Signals and constraints for a class of 1-step hidden sector models were studied in [339]. ∙ The
Planck
CMB bounds are robust and nearly model-independent varying byat most a factor of 1.5 over cascades with up to 6 steps for all final states. ∙ For photon-rich final states (all states considered except electrons and muons),we find the dwarf limits yield the most sensitive robust constraint, and can beweakened or strengthened by about an order of magnitude or more as comparedto the direct annihilation case. For high (low) DM masses and small (large) stepnumber the dwarf bounds can be overtaken by the robust CMB bounds as themost limiting constraints. ∙ For final states with few photons (electrons and muons), constraints from AMS-02 generally dominate the limits for low number of cascade steps. The limits canchange by several orders of magnitude as compared to the direct case. As theseweaken for higher DM masses and larger number of steps, CMB constraintsbecome more important. ∙ Taking the above three points into account we find that for a fixed DM massand final state, the presence of a hidden sector can change the overall crosssection constraints by up to an order of magnitude in either direction (althoughthe effect can be much smaller).In addition to these constraints we also discuss how the bounds from multi-stepcascades can be generalized to include the case of decays to 𝑛 -body states in thedark sector. Finally as a supplement to this chapter we release code to generate thecascade spectrum. 136n Sec. 6.2 we review the procedure used in [62] to calculate the photon, electronand positron spectra from a multi-step cascade. Section 6.3 contains a descriptionof the SM final state spectra used. Then in Sec. 6.4 we describe how results formulti-body decays can be estimated from our cascade results. Our main results arepresented in Sec. 6.5-6.7 where we show the model-independent bounds extractedfrom the CMB, dwarfs, and AMS-02 respectively. We discuss the interplay of thevarious experimental limit in Sec. 6.8 and conclude in Sec. 7.5. In the Appendices wedescribe the contents of the publicly available code, as well as additional details andcross-checks. The multi-step cascade annihilation scenario is illustrated schematically in the leftpanel of Fig. 6-1. In this setup the DM pair annihilates into two scalar mediators (thecase of non-scalar mediators was discussed in [62] where the conclusions proved tobe relatively insensitive to choice of vector or scalar mediator ) which subsequentlydecay through a (possibly) multi-step cascade in the dark sector, eventually producinga dark-sector state (with high multiplicity) that decays to the SM. Schematically wehave: 𝜒𝜒 → 𝜑 𝑛 𝜑 𝑛 → × 𝜑 𝑛 − 𝜑 𝑛 − → ... → 𝑛 − × 𝜑 𝜑 → 𝑛 × (SM final state) . (6.1)Here 𝑛 is the number of steps as defined above.A variation on this picture occurs when any of the heavy hidden sector mediatorsgoes off-shell and can therefore be integrated out yielding an effective vertex, nowwith a multi-body decay in the hidden sector of the form 𝜑 𝑛 → 𝑚𝜑 𝑛 − , with 𝑚 > .This possibility is illustrated schematically in the right panel of Fig. 6-1 and for a1-step cascade the analogue to Eq. 6.1 would be: 𝜒𝜒 → 𝑛 × 𝜑 → 𝑛 × (SM final state) , (6.2) A thorough investigation of possible exceptions to this result is left to future work. 𝑛 -step cascade results. This again highlights the point emphasized in [62] thatthe simple framework of 𝑛 -step 2-body scalar cascades can describe a wide class ofmodels and in this sense provide a relatively model-independent framework.In Eq. 6.1 and Eq. 6.2 “(SM final state)” denotes the SM particles produced bya single decay of 𝜑 , which in turn will (in general) subsequently decay to produceobservable photons, neutrinos and charged stable particles. For example a SM fi-nal state may produce additional photons due to final state radiation (FSR) or thedecay of neutral pions produced during hadronization. The mass ratio between 𝜑 and the sum of the masses of the SM particles in this state, which we denote 𝜖 𝑓 ( 𝜖 𝑓 ≡ ( ∑︀ 𝑚 SM ) /𝑚 ) , controls the level of FSR and hadronization, and so is a usefulparameters for describing these decays; the details are discussed in [62]. When theSM particles are massless, the relevant parameter is instead just the mass of the 𝜑 ,which we denote interchangeably as 𝑚 or 𝑚 𝜑 .The spectrum of particles in an intermediate step of a cascade may be obtainedusing the method discussed in [62], which we briefly review in this section. Considerthe “ith step” decay 𝜑 𝑖 +1 → 𝜑 𝑖 𝜑 𝑖 . In the rest frame of 𝜑 𝑖 +1 we will denote the spectrumof the subsequent photons, electrons or positrons as 𝑑𝑁/𝑑𝑥 𝑖 , where 𝑥 𝑖 = 2 𝐸 𝑖 /𝑚 𝑖 +1 , 𝑚 𝑖 +1 is the mass of 𝜑 𝑖 +1 and 𝐸 𝑖 is the energy of the photon, electron or positron inthe 𝜑 𝑖 +1 rest frame. We define 𝜖 𝑖 = 2 𝑚 𝑖 /𝑚 𝑖 +1 , and will (by default) assume a largemass hierarchy between cascades steps such that 𝜖 𝑖 ≪ . Assume that the spectrumin the rest frame of the 𝜑 𝑖 particle is known and denoted by 𝑑𝑁/𝑑𝑥 𝑖 − . In the limit ofa large mass hierarchy the decay of 𝜑 𝑖 +1 produces two highly relativistic 𝜑 𝑖 particles,each (in the rest frame of the 𝜑 𝑖 +1 ) carrying energy equal to 𝑚 𝑖 +1 / 𝑚 𝑖 /𝜖 𝑖 . Thephoton, electron, or positron spectrum per annihilation in the rest frame of the 𝜑 𝑖 +1 𝑑𝑁𝑑𝑥 𝑖 = 2 ∫︁ 𝑥 𝑖 𝑑𝑥 𝑖 − 𝑥 𝑖 − 𝑑𝑁𝑑𝑥 𝑖 − + 𝒪 ( 𝜖 𝑖 ) . (6.3)In this way, we can begin with a direct spectrum of 𝑑𝑁/𝑑𝑥 from 𝜑 → SM final state – the details of which are described in the next section – and generate a cascade spec-trum inductively. By repeated application of this formula we can see that the presenceof each additional step in a cascade acts to broaden and soften the spectrum, andshift the peak to lower masses. Importantly the shapes of these cascade spectra arevery simple, being characterized by just three pieces of information: the number ofsteps 𝑛 , the SM final state (often denoted 𝑓 ), and the value of 𝜖 𝑓 . Such cascades areindependent of the details of each of the intermediate steps, within the large-hierarchy( 𝜖 𝑖 ≪ ) approximation, and as such are independent of the various 𝜖 𝑖 . As pointed out in [62], although the large-hierarchy approximation seems to dis-card information, the more general case can be recovered quite easily. To see this,consider the opposite limit where 𝜖 𝑖 → , so that 𝑚 𝑖 ≈ 𝑚 𝑖 +1 . In this case, the restframes of the 𝜑 𝑖 +1 and 𝜑 𝑖 are the same, so no boost needs to be applied. As such,in this “degenerate limit”, the final spectrum of annihilation products is the same asthat for a hierarchical cascade with one fewer step, with half the initial dark mattermass and half the annihilation cross-section. The intermediate regime, where neither 𝜖 𝑖 nor − 𝜖 𝑖 are particularly small, smoothly interpolates between these two cases.Thus by studying the parameter space of ( 𝑚 𝜒 , ⟨ 𝜎𝑣 ⟩ , no. of steps) in the hierarchicallimit, it is possible to quickly estimate results for a general cascade.Again this framework is more general than it might initially appear. For exam-ple, simple extensions where a 𝜑 𝑖 decays to two 𝜑 𝑖 − with different masses will notchange our results in the large-hierarchy limit, as those results are independent ofthe intermediate masses. Additionally, as pointed out in [62], for larger 𝑛 our cas-cade scenarios can approximate models with hadronization in the dark sector (seee.g. [309, 344]), and additionally as we will show in Sec. 6.4, multi-body decays can The order of the error in the large-hierarchy approximation is 𝜖 𝑖 ; see [62] for more details. 𝑛 -step cascade ending in a final state consisting of two particles eachwith mass 𝑚 𝑓 , we defined 𝜖 𝑓 = 2 𝑚 𝑓 /𝑚 , 𝜖 = 2 𝑚 /𝑚 , 𝜖 = 2 𝑚 /𝑚 and so onuntil 𝜖 𝑛 = 𝑚 𝑛 /𝑚 𝜒 . Combining these, the DM mass is given in terms of 𝑚 𝑓 and the 𝜖 factors by: 𝑚 𝜒 = 2 𝑛 𝑚 𝑓 𝜖 𝑓 𝜖 𝜖 ...𝜖 𝑛 . (6.4)If the 𝜖 𝑖 factors are allowed to float, we can still say that < 𝜖 𝑖 ≤ in all cases (sinceeach decaying particle must have enough mass to provide the rest masses of the decayproducts), setting a strict lower bound on the DM mass of: 𝑚 𝜒 ≥ 𝑛 𝑚 𝑓 /𝜖 𝑓 . (6.5)Where this limit is not satisfied, the spectra should not be thought of as potentiallyrepresenting a physical dark-sector scenario, but only as a parameterization for gen-eral spectral broadening. For the massless final states considered here (photons andgluons) 𝑚 𝑓 = 0 , but we can still derive a condition from the value of 𝑚 𝜑 , specifically: 𝑚 𝜒 ≥ 𝑛 − 𝑚 𝜑 . (6.6) Using the formalism outlined in the previous section, from a given “direct” spectrumwe can straightforwardly generate an 𝑛 -step cascade spectrum, to compare with var-ious indirect searches. We outline the different SM final states considered for thedirect (0-step) spectra in this section. To obtain limits using bounds from the dwarfs,CMB and AMS-02 we need the spectrum of photons, electrons and positrons, and140 x x d N / dx [ × - ] � - ���� ������ ������� � × �� μ × �� τ ���� ϵ � = ���� ϕ = �� ��� x x d N / dx [ × - ] � - ���� �������� ������� μ ÷ � τ ���� ϵ � = ���� ϕ = �� ��� x x d N / dx [ × - ] � - ���� ���������� ������� ���� ϵ � = ���� ϕ = �� ��� Figure 6-2:
The 0-step or direct photon (left), positron (center) or antiproton (right) spectrumfor the various final states considered in this chapter. We have left out the 𝛾𝛾 spectrum in thephoton case and the electron spectrum in the positron case as both of these are 𝛿 -functions. Whereapplicable spectra are plotted with 𝜖 𝑓 = 0 . or 𝑚 𝜑 = 20 GeV in the case of gluons. so we determine the spectrum for these particles arising from the boosted decays ofthe following eight SM states: 𝛾𝛾 , 𝑒 + 𝑒 − , 𝜇 + 𝜇 − , 𝜏 + 𝜏 − , ¯ 𝑏𝑏 , 𝑊 + 𝑊 − , ℎ ¯ ℎ , and 𝑔𝑔 . Wechoose these states as a representative sample of possible spectra. For example decaysof light quarks generally give signals similar to those of 𝑏 -quarks and the 𝑍𝑍 finalstate is similar to 𝑊 + 𝑊 − .As discussed in the previous section, many of the cascade spectra depend onthe parameter 𝜖 𝑓 = ∑︀ 𝑚 SM /𝑚 = 2 𝑚 𝑓 /𝑚 (the final equality holds for all theprocesses we consider here). In the context of generating the direct (0-step) spec-trum, we can imagine two analogous processes: either the direct annihilation 𝜒𝜒 → SM final state , in which case 𝜖 𝑓 = 𝑚 𝑓 /𝑚 𝜒 , or the final step in a cascade annihilation, 𝜑 → SM final state , so that 𝜖 𝑓 = 2 𝑚 𝑓 /𝑚 as stated. If the (SM final state) is aphoton or a gluon, then clearly 𝜖 𝑓 is no longer a useful parameter; instead 𝑚 𝜑 = 𝑚 (equivalent to 𝑚 𝜒 in the case of direct annihilation) plays this role. For many spec-tra no such parameter is needed. For example the 𝛾𝛾 photon spectrum, as well asthe positron spectra from 𝑒 + 𝑒 − or 𝜇 + 𝜇 − final states, are independent of any suchparameter, since they are either just 𝛿 -functions or arise from decay rather than FSRor hadronization.In all but five cases, we use the results of the PPPC4DMID package [186] toproduce the direct spectra (hereafter referred to simply as PPPC). The exceptions tothis are: In our publicly released code we also provide the antiproton spectrum for 𝑏 -quarks, 𝑊 -bosons,Higgs and gluons. the 𝛾𝛾 photon and 𝑒 + 𝑒 − electron or positron spectra, which are just 𝛿 -functions,to a good approximation (we neglect the effect of FSR on the 𝑒 + 𝑒 − spectra inthe case of annihilation/decay to 𝑒 + 𝑒 − ), ∙ the spectra of photons produced in conjunction with the 𝑒 + 𝑒 − and 𝜇 + 𝜇 − fi-nal states, for which we use the analytic results of [62, 313] and [62, 313, 345]respectively, ∙ the spectrum of electrons or positrons from muon decay, where we use theanalytic Michel spectrum [313, 346].Finally we briefly comment on the 𝜖 𝑓 or 𝑚 𝜑 dependence of the various directspectra as it is often useful in interpreting results, noting that [62] has a more detaileddiscussion of several cases for photon spectra. For photons produced from 𝑒 + 𝑒 − and 𝜇 + 𝜇 − final states, the spectra arise entirely from FSR and so are strongly dependenton 𝜖 𝑓 , increasing in flux and becoming more sharply peaked near the maximum energyas 𝜖 𝑓 decreases. Similarly the photon spectrum produced from the 𝑊 -boson finalstate, in addition to a broad continuum peaked at low 𝑥 , acquires a sharp spikeat high 𝑥 due to FSR when 𝜖 𝑓 becomes small. The photon spectrum from the 𝑏 -quarks final state broadens and moves its peak to smaller 𝑥 as 𝜖 𝑓 decreases; the gluonspectrum behaves similarly as 𝑚 𝜑 increases. Finally the photon spectra from 𝜏 + 𝜏 − and ¯ ℎℎ final states are largely independent of 𝜖 𝑓 .The positron spectra produced from the Higgs and tau final states again show noreal variation with 𝜖 𝑓 . For positrons the spectrum from the 𝑊 + 𝑊 − final state is alsoquite independent of 𝜖 𝑓 , whilst the 𝑏 -quark and gluon spectra behave much as theydid in the photon case. Lastly, for antiprotons, once more the spectra from Higgs and 𝑊 -boson final states are independent of 𝜖 𝑓 , whilst now for decreasing 𝜖 𝑓 (increasing 𝑚 𝜑 ) the 𝑏 -quark (gluon) spectrum increases in height without substantially changingthe position of its peak. 142 .4 Multi-Body Cascades So far we have focused on cascades comprised of 2-body scalar decays. In this sectionwe discuss the extension of this framework to the case of 𝑛 -body cascades, schemat-ically illustrated on the right of Fig. 6-1. As we will see, in the large hierarchiesregime the 𝑛 -body decays can be understood in terms of our existing 2-body results,again emphasizing the model-independence of our results. The explicit calculationsand examples to help build intuition are provided in App. E.1.As explained in the introduction, a multi-body decay can arise if there is a heavymediator in the cascade that has been integrated out. This can happen anywherein a cascade, but here we restrict to a 1-step cascade of the form 𝜒𝜒 → 𝑛 × 𝜑 → 𝑛 × (SM final state) (c.f. Eq. 6.1). From here the extension to higher step cascadesis intuitively clear, and in practice can be calculated using Eq. 6.3. As shown in theappendix, an analogue of this equation can be derived for the multi-body case: 𝑑𝑁𝑑𝑥 = 𝑛 ( 𝑛 − 𝑛 − ∫︁ 𝑑𝜉 (1 − 𝜉 ) 𝑛 − ∫︁ 𝑥 /𝜉 𝑑𝑥 𝑥 𝑑𝑁𝑑𝑥 + 𝒪 ( 𝜖 ) (6.7)where again 𝑑𝑁/𝑑𝑥 represents the direct spectrum. Intuitively, the 𝑑𝑥 integralaccounts for the boosting of the decay products, just as in Eq. 6.3, whilst the 𝜉 integral samples from the 𝑛 -body phase space to give the correct degree of boosting.At first glance it appears that this formula could produce marked differences toour standard cascade framework, but as we show in Fig. 6-3, this is not the case.There we show the 1-step spectrum for an 𝑛 -body cascade ending in annihilation intothe SM state 𝑏 ¯ 𝑏 with 𝜖 𝑏 = 0 . , for 𝑛 = 𝑚 -step 2-body cascadefor 𝑚 = 𝑛 = 𝑚 = 𝑛 and increasing 𝑚 perturb the spectra in quite similar ways (albeitto different degrees, as expected since the multiplicities of final-state particles are notequal for 𝑚 = 𝑛 with 𝑛 > ), and so we expect the observational signatures of multi-body decays to lie within the space mapped out by the simple cascade annihilations.An example of the constraints on multi-body decays, and how they lie within theband of cascade constraints, is given in Fig. E-3, in App. E.1.143 .10.010.001012 x x d N / dx [ × - ] � - ���� ������� � = �� ϵ � = ���� ������� ���� ���������� ������ ��� � � - ���� � - ���� ������ ��� � = � - �� ���� � - �������� ����� ��� Figure 6-3:
The 1-step spectrum for an 𝑛 -body cascade from a direct annihilation to 𝑏 ¯ 𝑏 with 𝜖 𝑏 = 0 . is shown as the dashed gray curves for 𝑛 = 𝑛 . In purple, green and orange we show a 2-body 1-step, 3-step and 5-step cascade spectrumrespectively, for the same direct spectrum. These three curves outline the 𝑛 -body results and showthat the result of 1-step multi-body spectra should be encapsulated in the multi-step 2-body results. Planck
Dark matter annihilation during the cosmic dark ages can inject ionizing particlesinto the universe, modify the ionization history of the hydrogen and helium gas,and consequently perturb the anisotropies of the CMB. Sensitive measurements ofthe CMB by
Planck [188] (and previously WMAP and other experiments) can placequite model-independent limits on such energy injections, which when applied to darkmatter annihilation are competitive with other indirect searches.The figure of merit for CMB limits on dark matter annihilation is the parameter 𝑝 ann = 𝑓 eff ⟨ 𝜎𝑣 ⟩ /𝑚 𝜒 , where 𝑓 eff is an efficiency factor that depends on the spectrum ofinjected electrons and photons, and the other factors describe the total power injectedby dark matter annihilation per unit time. In principle, different dark matter modelscould give rise to different patterns of anisotropies in the CMB – but for WIMP modelsof dark matter that annihilate through 𝑠 -wave processes, it has been shown [347] thatthe impact on the CMB is identical at the sub-percent level up to an overall rescalingby 𝑝 ann (related studies [348–350] independently found that the signal was largelycontrolled by a single parameter). In [21]- [22] this result was generalized to includeany class of DM models for which ⟨ 𝜎𝑣 ⟩ can be treated as a constant during the144osmic dark ages, which is generically true for the models considered in the presentchapter. We use the results of [21] to compute 𝑓 eff as a function of dark matter massand annihilation channel. In particular, we compute positron and photon spectra forthe case of direct annihilations using PPPC, and then convolve to find the resultingspectrum for an 𝑛 -step cascade as discussed above. The spectrum of electrons isequal to that of positrons by the assumption of charge symmetry. We then integratethe resulting spectra over the 𝑓 eff ( 𝐸 ) curves provided in [21] to obtain the weighted 𝑓 eff ( 𝑚 𝜒 ) for 𝑛 = 0 -6 step cascades for dark matter annihilations to various final states: 𝑓 eff ( 𝑚 𝜒 ) = ∫︀ 𝑚 𝜒 𝐸𝑑𝐸 [︁ 𝑓 𝑒 + 𝑒 − eff (︀ 𝑑𝑁𝑑𝐸 )︀ 𝑒 + + 𝑓 𝛾 eff (︀ 𝑑𝑁𝑑𝐸 )︀ 𝛾 ]︁ 𝑚 𝜒 . (6.8)We neglect the contribution from protons and antiprotons, as for all the channels weconsider, the fraction of power proceeding into these channels is rather small, andconsequently including them should only slightly increase 𝑓 eff [351]. The constraintswe present are therefore somewhat conservative (they could be strengthened slightlyby a careful treatment of protons and antiprotons). As discussed in [22], we usethe best-estimate curves suited for the “3 keV” baseline prescription, which are mostappropriate for applying constraints derived by Planck .The bound set on the annihilation parameter, 𝑝 ann , from Planck temperature andpolarization data is taken to be [188]: 𝑓 eff ⟨ 𝜎𝑣 ⟩ 𝑚 𝜒 < . × − cm / s / GeV . (6.9)In Fig. 6-5 we present our results for the bound on DM cross-section as a functionof 𝑚 𝜒 for various numbers of cascade steps and SM final states. We note that thenumber of steps does not affect the total power deposited by dark matter annihilationper unit time (at least in the simple scenario where all that power eventually goesinto SM particles). Each additional step reduces the average energy of the final-state photons/positrons/electrons by a factor of 2, but simultaneously increases their145ultiplicity by a factor of 2. Thus the only possible impact on the constraints comesfrom the energy dependence of 𝑓 eff , combined with the softening and broadening ofthe spectrum.In accordance with our expectations, we find that the effect of the spectral broad-ening and softening is rather mild, typically changing the constraints by no morethen 0.1-0.15 decades (corresponding to a factor of ∼ . ). There is no general trend,in that constraints on these high-multiplicity final states may be either weaker orstronger than those pertaining to direct annihilation; this arises from the fact that 𝑓 eff is not a monotonic function of energy, so lowering the average energy of the in-jected particles may either increase or decrease the deposition efficiency. In general, 𝑓 eff and hence the upper bound on the ratio ⟨ 𝜎𝑣 ⟩ /𝑚 𝜒 varies less as a function of massfor higher-multiplicity final states (as expected, from the broader resulting spectrum),but this effect is very small. The choice of 𝜖 parameters, again, does not perturb theconstraints outside this ∼ . -decade band. We refer the reader to the App. E.2 foradditional details regarding the behavior of 𝑓 eff . Fermi
The dwarf spheroidal galaxies of the Milky Way are expected to produce some of thebrightest signals of DM annihilation on the sky. Whilst less intense than the emis-sion expected from the galactic center, the dwarfs have the distinct advantage of anenormous reduction in the expected astrophysical background. These features makethem ideal candidates for analysis with the data available from the
Fermi
Gamma-Ray Space Telescope. Indeed the
Fermi
Collaboration has set stringent limits onthe dark matter annihilation cross-section using the dwarfs [57], and together withthe DES Collaboration have used 8 newly discovered dwarf satellites [352, 353] to setindependent limits [354]. We note in passing that several groups have pointed outan apparent gamma-ray excess in the direction of one of the new dwarfs, ReticulumII [354–356], albeit with considerable variation as to its significance (with estimatesranging from ∼ 𝜎 to completely insignificant). We will not discuss this tentative146xcess here, other than to note as it appears roughly consistent with the emissioncoming from the GCE, the implications for dark sector cascades will be analogous tothose discussed in [62].Here we focus on understanding how the presence of cascade annihilations canmodify the limits obtained from these dwarf galaxies. In order to do this we use thepublicly released bin-by-bin likelihoods provided for each of the dwarfs consideredin [57]. This analysis made use of 6 years of Pass 8 data and found no evidence foran excess over the expected background. Note the
Fermi collaboration produced anearlier analysis of the same dwarfs using 4 years of Pass 7 data in [92]. In App. E.3we show that the results are similar between the two, but that the limits set usingthe newer analysis are usually about half an order of magnitude stronger.Although [57] considered 25 dwarf galaxies, when setting limits they restricted thisto 15, choosing a non-overlapping subset of dwarfs with kinematically determined 𝐽 -factors. Specifically the 15 dwarfs considered were: Bootes I, Canes Venatici II,Carina, Coma Berenices, Draco, Fornax, Hercules, Leo II, Leo IV, Sculptor, Segue 1,Sextans, Ursa Major II, Ursa Minor, and Willman 1.For a given dwarf Fermi provides the likelihood curves as a function of the inte-grated energy flux in each of the energy bins considered in their analysis, covering theenergy range from 500 MeV to 500 GeV. Thus to obtain the likelihood curves for ourcascade models we need to firstly determine the integrated energy flux per bin. Thiswill be a function of the DM mass 𝑚 𝜒 , annihilation cross-section ⟨ 𝜎𝑣 ⟩ , and shape ofthe cascade spectrum 𝑑𝑁/𝑑𝑥 – which itself depends on the number of cascade steps,the identity of the final state particle and possibly either 𝜖 𝑓 or 𝑚 𝜑 . For an energy binrunning from 𝐸 min to 𝐸 max , the energy flux in GeV/cm /s is: Φ 𝐸 = ⟨ 𝜎𝑣 ⟩ 𝜋𝑚 𝜒 [︂∫︁ 𝐸 max 𝐸 min 𝐸 𝑑𝑁𝑑𝐸 𝑑𝐸 ]︂ 𝐽 𝑖 , (6.10)where 𝐽 𝑖 is the 𝐽 -factor appropriate for the individual dwarf 𝑖 . Treating the energybins as independent, we can simply multiply the likelihoods for the various bins to 𝑖 : ℒ 𝑖 ( 𝜇 |𝒟 𝑖 ) , which is a function of both themodel parameters 𝜇 and the data 𝒟 𝑖 . At a given mass and for a given channel (finalstate and number of cascade steps), 𝜇 just describes the annihilation cross-section ⟨ 𝜎𝑣 ⟩ . There is, however, one additional source of error that should be accountedfor: the uncertainty in the 𝐽 -factor. Following [57] we incorporate this as a nuisanceparameter on the global likelihood, modifying the likelihood as follows: ˜ ℒ 𝑖 ( 𝜇 , 𝐽 𝑖 |𝒟 𝑖 ) = ℒ 𝑖 ( 𝜇 |𝒟 𝑖 ) × 𝐽 𝑖 √ 𝜋𝜎 𝑖 𝑒 − ( log ( 𝐽 𝑖 ) − log ( 𝐽 𝑖 ) ) / 𝜎 𝑖 , (6.11)where for log ( 𝐽 𝑖 ) and 𝜎 𝑖 we use the values provided in [57] for a Navarro-Frenk-White profile [75]. This approach allows us to account for the 𝐽 -factor uncertaintiesusing the profile likelihood method [242]. We obtain the full likelihood function bymultiplying the likelihoods for each of the 15 dwarfs together.Using this likelihood function, for a given DM mass and cascade spectrum we canthen determine the 95% confidence bound on the annihilation cross-section. We followthis procedure for cascade annihilations with 0-6 steps, for final state electrons, muons,taus, 𝑏 -quarks, 𝑊 -bosons, Higgses, photons and gluons, considering two differentvalues of 𝜖 𝑓 or 𝑚 𝜑 where appropriate.Results are shown in Fig. 6-6. For the final states considered in [57], our direct/0-step results are in agreement. Recall that there is a physical limitation on realizinga given cascade scenario set by 𝑚 𝜒 ≥ 𝑛 𝑚 𝑓 /𝜖 𝑓 , as mentioned in Sec. 6.2. The con-straints corresponding to scenarios that satisfy this condition are indicated by darkerlines, but we also show the limits for cases that do not satisfy this condition (and socannot be physically realized as a cascade annihilation of the type we have consid-ered), to demonstrate the effect of spectral broadening.Before discussing results for each final state independently, there are a few genericfeatures worth pointing out. Recall that higher-step cascades have a spectrum peakedat lower 𝑥 = 𝐸 𝛾 /𝑚 𝜒 . Thus in order to produce emission at an equivalent energy,higher-step cascades require a larger DM mass, which in turn requires a larger cross-148ection to inject the same amount of power (as the DM number density scales inverselywith the mass). Equivalently, at a fixed mass and cross-section, larger numbers ofcascade steps will tend to produce a larger number of lower-energy photons; at lowmasses, some of these photons may lie outside the energy range of the Fermi analysis,and the astrophysical backgrounds will also generally be larger at low energies. Thesefactors tend to weaken the constraints, and indeed we see a systematic trend forweaker bounds with increasing 𝑛 for low-mass dark matter, for all channels.Nevertheless this conclusion is not inevitable. Specific energy bins may allowstronger constraints than neighboring bins, purely due to statistical accidents; addingcascade steps smooths out such effects. The total number of emitted photons isincreased with larger 𝑛 (albeit while preserving the total injected power).Most generically, if the DM mass is large, much of the spectrum may be above the500 GeV cutoff of this analysis in the case of direct annihilation. In this case, addingcascade steps can strengthen the constraints by moving the photons into the rangeof sensitivity for the search. This effect is most pronounced, and occurs at the lowestDM masses, for final states with spectra peaked at large 𝑥 (electrons, muons, taus andphotons): for softer direct-annihilation spectra, even at the heaviest masses tested,the peak of the spectrum does not move past 500 GeV. Inclusion of higher-energydata, e.g. from studies of the dwarf galaxies with VERITAS [341], would potentiallystrengthen the constraints at high DM masses, but for this reason we expect theimprovement to be smaller for higher-step cascades.Thus in general we see a weakening in the cascade constraints relative to thedirect-annihilation case at low DM masses, and a strengthening at high DM masses,with the crossover point and the width of the band varying based on the SM finalstate. For some final states, the cascade constraints can be weaker or stronger thanthose for the direct-annihilation case by more than an order of magnitude. Let usnow discuss the detailed results for each SM final state (shown in Fig. 6-6) separately: Electrons: the generic behaviors discussed above are clearly demonstrated in theseresults. There is also a striking difference between the results for direct and cascadeannihilations. The photon spectrum in the direct case originates from FSR and is149ery sharply peaked (especially for small 𝜖 𝑓 ); even a single cascade step will smoothout the spectrum and considerably change its shape. Further, the bounds are stronglydependent on the value of 𝜖 𝑓 , as the FSR photon spectrum diverges as 𝜖 𝑓 → . Assuch, for smaller 𝜖 𝑓 we expect stronger limits, and this is exactly what we observe.Nonetheless note that the position of the peak of the spectrum in 𝑥 is not stronglydependent on 𝜖 𝑓 , so we should expect the crossover behavior between different spectramentioned above should happen at a similar location for different 𝜖 𝑓 values and thisis exactly what we observe. Finally note that the bumps in the direct spectrum are aresult of the sharply peaked 0-step spectrum moving between energy bins. The widthof these bumps is exactly the width of the energy bins in the data. As we move tocascade scenarios, the spectrum is smoothed out and the majority of the emission isno longer in a single bin, meaning these bumps vanish. 𝛾𝛾 : the most noticeable feature here is the jagged direct spectrum. As the directspectrum of 𝛾𝛾 is just a 𝛿 -function at the mass considered, these jumps are an extremerealization of the issue mentioned for the 0-step electron limits: we get a jump asthe emission moves from one of the energy bins considered to the next. Of coursephysically the Fermi instrument has a finite energy resolution, which will act tosmooth out such a sharp feature. To approximate this we smooth the 0-step by aGaussian with a width set to 10% of the energy value, yet this ultimately had littleimpact on the extracted limit. Note also that once the spike moves beyond 500 GeV,which occurs at roughly log 𝑚 𝜒 = 2 . , the Fermi data can no longer constrain thisscenario so the limit completely drops off.
Muons: the photon spectrum for the muon final state is very similar to that forthe electron final state, except that it is slightly less dependent on 𝜖 𝑓 . The resultshere are accordingly very similar to those for the electron final state, except that thevariations with 𝜖 𝑓 are less pronounced. Taus: the fact that the tau spectrum is only weakly dependent on 𝜖 𝑓 is clearlyvisible; otherwise only the generic behavior is apparent. 𝑏 - quarks: There is a modest dependence on 𝜖 𝑓 , which does not change the qual-itative results. The crossover where the direct constraints become weaker than the150ascade constraints occurs at a DM mass around 100 GeV. Due to the kinematicbounds, over the physically allowed region the variation in the band width is fairlymodest varying by at most 0.4 decades. Gluons: the gluon spectrum behaves very similarly to the 𝑏 -quark spectrum, ifwe swap decreasing 𝜖 𝑓 for increasing 𝑚 𝜑 . As such the results are similar to those for 𝑏 -quarks. 𝑊 - bosons: firstly note that the kinematic edge in these results appears from thethreshold requirement to have enough energy to create on-shell 𝑊 ’s. Other than thiswe see that the limits are somewhat stronger for smaller values of 𝜖 𝑓 , which is becausethe 𝑊 spectrum includes a small FSR component which is larger for smaller 𝜖 𝑓 . Thewidth of the band of possible results is at most 0.7 decades. Again we also see acrossover where the direct constraints become weaker than the cascade constraints,here at roughly 500 GeV. Higgs: as with the 𝑊 -bosons, our results again have a kinematic edge. Further-more, like final state taus, the Higgs spectrum is only weakly dependent on 𝜖 𝑓 andthus so are the results. As for the 𝑊 case, the width again has a maximum around0.7 decades, whilst this time the direct crossover first happens at about a TeV. AMS-02 has recently released a precise measurement of cosmic ray electrons andpositrons in the energy range of ∼ GeV to ∼ GeV [336, 337]. The measuredpositron ratio exceeds the prediction of the standard cosmic ray propagation modelsat energies larger than ∼ GeV. There are many possible explanations for this risein the positron ratio, including DM physics (although the annihilation scenario seemschallenged by a range of other null results, e.g. [188]), nearby pulsars [125, 126] orsupernovae [357].The presence of an apparent large positron excess of unknown origin makes itchallenging to set stringent limits on general DM annihilation scenarios. The situationis further complicated by the effects of solar modulation at energies below ∼ 𝑒 + divided by the flux of 𝑒 + + 𝑒 − ) and the fluxes ofcosmic ray electrons and positrons are fairly smooth; there is no clear structure inthe spectrum within the energy resolution of AMS-02. Accordingly, it is possible toset quite strong constraints on DM models that predict a sharp spectral feature inthe positron spectrum (e.g. [220, 314, 361]).As discussed in the previous sections, DM annihilation through multi-step cascadesusually gives rise to a softer and broader spectrum than direct annihilation to the SMstates, generally leading to weaker bounds from AMS-02. In this section, we studythis effect quantitatively. We note that our goal here is to study the impact of thesespectral changes, not to explore possible explanations for the rise in the positronfraction or systematic uncertainties in the modeling of the background or signal.To set bounds on annihilating DM, we first need to parametrize or model thebackgrounds. Here the backgrounds that we refer to are the astrophysical cosmic rayflux, plus some new smooth ingredient to account for the observed rise in the positronflux. Since we do not know the source of the new ingredient, polynomial functionsof degree 6 are introduced to fit the AMS-02 positron flux data (the 6 degrees areemployed to obtain a good 𝜒 fit to the data). To derive the limits, we float the6 parameters from the polynomial functions within 30 % of the best fit values fromthe fit without DM, together with the DM annihilation cross-section. We check thatincreasing the range of allowed values for the background parameters does not weakenthe constraints.We derive limits from only the positron flux, as both the positron and electronbackgrounds are required to float in the fit to the positron ratio. Such an analysiswould require many additional free parameters, and is beyond the scope of the currentchapter. As a cross-check we attempted a simplified fit to the positron ratio data(using AMS-02 measurements of the total 𝑒 + + 𝑒 − spectrum) and found constraintsof comparable strength to those we present here.The positron flux from DM annihilation is obtained by propagating the injected152ositron spectrum using the public code DRAGON [362, 363]. There are substantialsystematic uncertainties in the propagation of electrons and positrons in the galaxy,affecting diffusion, energy loss, convection, and solar modulation. In particular, ac-counting for uncertainties in the modeling of energy loss and solar modulation cansignificantly weaken the constraints on DM annihilation.Once electrons and positrons are injected into the halo, they will diffuse and loseenergy. Their number density 𝑁 𝑖 evolves according to the following diffusion equation, 𝜕𝑁 𝑖 𝜕𝑡 = ⃗ ∇ · (︁ 𝐷 ⃗ ∇ − ⃗𝑣 𝑐 )︁ 𝑁 𝑖 + 𝜕𝜕𝑝 (︁ ˙ 𝑝 − 𝑝 ⃗ ∇ · ⃗𝑣 𝑐 )︁ 𝑁 𝑖 + 𝜕𝜕𝑝 𝑝 𝐷 𝑝𝑝 𝜕𝜕𝑝 𝑁 𝑖 𝑝 + 𝑄 𝑖 ( 𝑝, 𝑟, 𝑧 )+ ∑︁ 𝑗>𝑖 𝛽𝑛 gas ( 𝑟, 𝑧 ) 𝜎 𝑗𝑖 𝑁 𝑗 − 𝛽𝑛 gas 𝜎 𝑖𝑛𝑖 ( 𝐸 𝑘 ) 𝑁 𝑖 , (6.12)where 𝐷 is the spatial diffusion coefficient, depending on the spatial position andenergy. It is parametrized by the following form 𝐷 ( 𝜌, 𝑟, 𝑧 ) = 𝐷 e | 𝑧 | /𝑧 𝑡 (︂ 𝜌𝜌 )︂ 𝛿 , (6.13)where we assume the diffusion zone is axisymmetric, and use the cylindrical coordinatesystem ( 𝑟, 𝑧 ) . Most of the electrons and positrons are trapped in the diffusion zonewith thickness 𝑧 𝑡 . Here 𝜌 = 𝑝/ ( 𝑍𝑒 ) is the rigidity of the charged particle with 𝑍 = 1 for electron and positron. 𝐷 normalizes the diffusion at the rigidity 𝜌 = 4 GV. InEq. 6.12, 𝑣 𝑐 is the velocity of the convection winds; ˙ 𝑝 accounts for the energy loss; 𝑄 𝑖 is the source of the cosmic ray, where DM is one kind of the source; 𝐷 𝑝𝑝 accountsfor the diffusion in the momentum space; the last two terms in Eq. 6.12 parameterizehow the nuclei inelastic scattering with the gas to affects the number density of thecosmic rays. Although there are many parameters in the diffusion equation, we donot simulate the backgrounds (instead modeling them with a polynomial function),which decreases the systematic uncertainties of the limit substantially.We use a specific model to propagate the electrons and positrons injected by DMannihilation [364]. In this model, 𝐷 = 2 . × cm /s, 𝑧 𝑡 = 4 kpc, 𝛿 = 0 . and153 - - - ��� ������ ������ γ � μτ ���� ������ - - - - ��� - �� ����������� ��� γ � �� � �������������� - �� Log σ v [ c m / s e c ] Log m χ [ GeV ] Figure 6-4:
Constraints on ⟨ 𝜎𝑣 ⟩ for the case of direct annihilations to photons, electrons, muons,taus, 𝑏 -quarks, gluons, 𝑊 ’s and Higgs final states derived from CMB (top left), dwarfs (top right)and AMS-02 (bottom left). In the bottom right panel we overlay the constraints from all threeexperiments for the case of direct annihilations to final state photons, electrons and 𝑏 -quarks. we take the local density of DM to be 0.4 GeV/cm . We set the convection anddiffusion in momentum space equal to zero, since they do not change the spectrumsignificantly in the energy range of interest [105]. There are other propagation modelswith different diffusion terms or diffusion zone heights that can be employed here.However, since the energy loss effect is dominant for the propagation of high energyleptons, we choose only one propagation model to derive the limits. While theremay be remaining systematic effects due to the choice of propagation model, wereiterate that the purpose of this analysis is not to explore all the uncertainties inthese constraints.Cosmic-ray propagation is affected by the magnetic field, which determines bothhow the cosmic rays diffuse and their energy losses due to synchrotron radiation.The magnetic field is modeled by two components, one regular and one turbulent(irregular) [365, 366]. [367] gives the constraints on these components. To be conser-vative, here we set the value of the magnetic field at the Sun to 𝐵 ⊙ ∼ . 𝜇 G. With154his magnetic field, the local radiation field and magnetic field energy density is 3.1eV/cm , which is close to (but somewhat higher than) the 2.6 eV/cm value used forconservative constraints in [314]. For this reason, the constraint we obtain for thedirect annihilation is slightly weaker than even the conservative case studied in [314],as the energy loss rate for the positrons is higher. The main effect of changing thelocal energy density is to rescale all the constraint curves, with lesser effects on thevariation of the constraint with DM mass and number of cascade steps.For cosmic rays with energy smaller than GeV, although there are many otherparameters in the propagation model, we only consider the uncertainty from the solarmodulation, which is modeled by the modulation potential. The modulation potential 𝜑 in the range of (0 , GeV is fixed by minimizing the 𝜒 to fit the AMS-02 data.In summary, we derive the limits on DM annihilation by using AMS-02 positronflux starting from . The background is parametrized by a polynomial functionof 6 degree, and to derive the bounds we let the 6 parameters float within 30 % of theirbest fit values. The diffusion model is employed here to propagate the DM positronflux, and the solar modulation potential is allowed to float in the range (0 ,
1) GeV when minimizing the likelihood function. The limits are summarized in Fig. 6-7.In general, similar to the dwarf galaxies, the constraints on cascade models canbe substantially weaker than those on the direct-annihilation case for low DM masses(below ∼ GeV), by up to several orders of magnitude depending on the channel.This weakening likely arises from a combination of (a) positrons falling below theminimum energy of the search, and (b) broadening of the spectrum making it easierfor the background model to compensate for a DM component. The effect can be upto two orders of magnitude in most channels at sufficiently low masses (the exceptionsare the 𝑊 , Higgs and 𝑏 -quark final states where low DM masses are kinematicallyforbidden). At high masses, the bands of possible constraints are narrower, of orderhalf an order of magnitude or less; for the electron, muon, tau and gamma final statesthe direct-annihilation constraints are systematically weaker than those for cascadescenarios. This is likely due to the cascade scenarios producing greater numbers ofpositrons in the energy range of the search, but may also be due to the hardening of155he positron flux at high energies mimicking a hard signal from DM annihilation. We summarize our main results in Fig. 6-8, where we overlay the combined constraintsfrom the three experiments as a function of DM mass for an 𝑛 = Fermi limits for large numbers of cascade steps at low masses, orsmall numbers of cascade steps at high masses. We summarize the results the variousSM final states below.
Electrons:
The spectrum of positron and photon spectrum is very sharply peakedin the case of direct DM annihilations to 𝑒 + 𝑒 − . Thus AMS-02 places the most con-straining bound for 𝑛 = 𝑚 𝜒 (cid:46) . As thenumber of steps increases, 𝑛 > , the spectrum smooths and broadens thereby weak-ening the AMS-02 bound so that the CMB bound becomes the most constraining.The CMB bounds are generically stronger at high DM masses, above a few hundredGeV. The dwarf limits are, in all cases, 1-2 orders of magnitude less constrainingthan the AMS-02 and CMB bounds. This is unsurprising given the dwarfs are onlysensitive to the photon spectrum from the final state electrons, which represents onlya small fraction of the available power per annihilation. 𝛾𝛾 : The strongest constraints almost always arise from the Fermi dwarfs, althoughat high DM masses and for small numbers of steps, the CMB bounds may be more156tringent. However, in this case VERITAS or H.E.S.S dwarf searches may actuallyprovide a stronger limit. For AMS-02 the positron spectrum is similar in shape tothat of the electron channel; the photon generates a hard electron spectrum via Drell-Yan. Nonetheless this process is suppressed by a factor of 𝛼 𝑒 as well as phase space.Combining these, approximately two order magnitude suppression relative to electroncase would be expected and is in fact observed. Muons:
Recall that the spectrum of positrons and photons from DM annihilationsto muons is similar to the corresponding spectra for the electron final state, exceptthe photon spectrum is less dependent on 𝜖 𝑓 and the positron spectrum is somewhatbroader. For 0-2 cascade steps, the most stringent constraints are from AMS-02 atlow masses, below a few hundred GeV. At higher step numbers (for all masses) andhigher masses (for all cascade scenarios), the CMB limit becomes more restrictive. Taus:
The tau final state is richer in photons than the other leptonic final states,and yields smoother and broader photon and positron spectra even in case of directannihilation. Thus the bound from the dwarfs is more sensitive and constraining thanAMS-02, and generally also stronger than the CMB limits. The exceptions are at lowmass and large number of steps, or inversely high mass and a small number of steps,as in both cases the CMB bounds dominate the constraint. 𝑏 - quarks: The direct spectrum for DM annihilations to 𝑏 ¯ 𝑏 is much softer then thepreviously discussed channels. So the Fermi dwarf limits almost always provide thestrongest constraint; for low masses and 𝑛 = Gluons:
As previously discussed the gluon spectrum behaves very similarly to the 𝑏 -quark spectrum, if we swap the decreasing 𝜖 𝑓 for increasing 𝑚 𝜑 . As such the resultsare similar to those for 𝑏 ¯ 𝑏 . 𝑊 - bosons: For annihilation to 𝑊 final states the bounds are quite robust, withthe dwarfs always setting the strongest limits. Higgs:
Annihilations to final state Higgses is similar to the 𝑊 case; the resultsare almost identical aside from the difference in the kinematic edge between the H157nd 𝑊 mass. We have shown that results from current DM indirect searches can be extended toconstrain a broad space of dark sector models. We summarize our main points below: ∙ Photon rich final states are generally most constrained by bounds from the
Fermi dwarfs. ∙ Electron and muon final states are generally most constrained by AMS-02 (albeitsubject to uncertainties in the propagation and background modeling). ∙ The CMB bounds from
Planck are robust and insensitive to number of darksector steps. As a result the CMB bounds may become the most limiting incertain cases where the AMS-02 or dwarf bounds weaken as a result of large 𝑚 𝜒 or increasing number of dark sector steps. ∙ We find that for a fixed DM mass and final state, the presence of a hidden sectorcan change the overall cross section constraints by up to an order of magnitudein either direction (although the effect can be much smaller).For hadronic SM final states ( 𝑏 -quarks, gluons, gauge bosons, Higgses), constraintsfrom gamma-ray studies of dwarf galaxies generally remain the most limiting, and –within the kinematically allowed region – are generally fairly robust, although theycan weaken at low masses and strengthen at high masses. More specifically for smallbut kinematically allowed masses the bound for final state gauge bosons, Higgses and 𝑏 -quarks can weaken by about 0.1 decades. For the gluon final state, where very lowDM masses are in principle possible, this bound can weaken by up to 1.1 decade;however a careful consideration of this regime would require taking into account themass of the mediators, which may be comparable to Λ QCD . At high masses the boundswill strengthen by about 0.3-0.5 decades for the hadronic final states.158he photon-rich tau and photon final states behave similarly, with the dwarflimits dominating the constraints except perhaps at very high masses (where it maybe important to take constraints from VERITAS and H.E.S.S. into account). Addingextra cascade steps has little effect on the dwarf constraints on the photon final stateat low masses (after the addition of the first cascade step, which weakens the limit byup to 0.8 decades), whereas for the tau final state it can weaken the bound by about0.1 decades within the kinematically allowed regime.For leptonic final states with few photons (electrons and muons), constraints fromAMS-02 often appear to dominate the limits, but are quite sensitive to the numberof cascade steps (as well as assumptions on the cosmic-ray propagation and localmagnetic field; our limits are more conservative than others in the literature). Atlow masses (below a few hundred GeV), increasing the number of cascade steps canweaken the constraints by up to 2 orders of magnitude, at which point bounds fromthe CMB become more constraining. Above a few hundred GeV, however, addingmore cascade steps tends to strengthen the constraints, so using results quoted fordirect annihilation gives conservative bounds; the CMB limits are also genericallystronger than the AMS-02 limits in this mass range.If a quick estimate of constraints is needed, the CMB limits almost always appearto be within an order of magnitude of the strongest limit, for the cases we have tested,and vary by at most a factor of 1.3 to 1.5 over cascades with up to 6 steps.The details of our code for 𝑛 -step cascades which were used to produced ourresults are described in App. E.4 and are available at: http://web.mit.edu/lns/research/CascadeSpectra.html . 159 - - - - ������� � + � - ��� ������ ϵ � = ��� ϵ � = ���������� ������� ���������������� ������� γγ - - - - μ + μ - τ + τ - - - - - �� �� � ϕ = �� ���� ϕ = � ��� - - - - � + � - �� Log σ v [ c m / s e c ] Log m χ [ GeV ] Figure 6-5:
The bound on DM cross-section Eq. 6.9 for 𝑛 = 𝜖 𝑓 = 0 . (solid) and 𝜖 𝑓 = 0 . (dashed). The shaded out portions of the plot correspondto values of 𝑚 𝜒 that are kinematically forbidden. As discussed above the number of steps does notaffect the total power deposited by the DM annihilation per unit time. Therefore the constraintsare insensitive to the number of steps as the only impact comes from the energy dependence of 𝑓 eff and the broadening of the spectrum. - - - - - - ������� � + � - ����� ������ ϵ � = ��� ϵ � = ���������� ������� ���������������� ������� γγ - - - - - - - μ + μ - τ + τ - - - - - - - - �� �� � ϕ = �� ���� ϕ = � ��� - - - - - - - � + � - �� Log σ v [ c m / s e c ] Log m χ [ GeV ] Figure 6-6:
95% confidence limits on dark matter cross-section for cascade models using the datafrom 15 dwarf spheroidal galaxies. Results are shown for the photon spectrum obtained from eightdifferent final states: electrons, photons, muons, taus, 𝑏 -quarks, gluons, 𝑊 -bosons, and Higgs. Ineach case we show the results of a 0 (direct), or 1-6 step cascade. Additionally where it makes sensewe show results for two different 𝜖 𝑓 values, solid lines representing . and dashed . . Note the 𝛾𝛾 spectrum is independent of 𝜖 𝑓 , so we only show one set of limits there, and for the gluon spectrumthe relevant parameter is instead 𝑚 𝜑 and we show results for GeV in solid and TeV as dashed.Only the darker regions are kinematically allowed. See text for a discussion of the results. - - - - - - ������� � + � - ��� - �� ������ ������ ������� ���������������� ������� γγ � ϕ = �� ��� - - - - - - - μ + μ - τ + τ - - - - - - - - �� ϵ � = ��� ϵ � = ���� �� � ϕ = �� ���� ϕ = � ��� - - - - - - - � + � - �� Log σ v [ c m / s e c ] Log m χ [ GeV ] Figure 6-7:
95% confidence limits on DM cross-section for cascade models. Details are similar tothe previous two plots. The limits obtained are strongest for electron and muon final states, andgenerically we find that the addition of cascades steps can change the limits by up to several ordersof magnitude. - - - - - - �������������� - �� � + � - ����������� ϵ � = ���������� �� ������� � = � - � ����� γγ � ϕ = �� ��� ( ��� ) - - - - - - - μ + μ - τ + τ - - - - - - - - �� �� � ϕ = �� ��� - - - - - - - � + � - �� Log σ v [ c m / s e c ] Log m χ [ GeV ] Figure 6-8:
Overlaid constraints from the CMB (green), AMS-02 (red) and the
Fermi
Dwarfs(blue) for 𝑛 = hapter 7The One-Loop Correction to HeavyDark Matter Annihilation It is now well established that if dark matter (DM) is composed of TeV scale WeaklyInteracting Massive Particles (WIMPs) then its present day annihilation rate is poorlydescribed by the tree-level amplitude. Correcting this shortcoming is important fordetermining accurate theoretical predictions for existing and future indirect detectionexperiments focussing on the TeV mass range, such as H.E.S.S [368, 369], HAWC[370–372], CTA [288], VERITAS [373–375], and MAGIC [376, 377].The origin of the breakdown in the lowest order approximation can be traced totwo independent effects. The first of these is the so called Sommerfeld enhancement:the large enhancement in the annihilation cross section when the initial states aresubject to a long-range potential. In the case of WIMPs this potential is due tothe exchange of electroweak gauge bosons and photons. This effect has been widelystudied (see for example [26–30]) and can alter the cross section by as much as severalorders of magnitude. The Sommerfeld enhancement is particularly important whenthe relative velocity of the annihilating DM particles is low, as it is thought to be inthe present day Milky Way halo.The second effect is due to large electroweak Sudakov logarithms of the heavy165M mass, 𝑚 𝜒 , over the electroweak scale, which enhance loop-level diagrams andcause a breakdown in the usual perturbative expansion. The origin of these largecorrections can be traced to the fact that the initial state in the annihilation is notan electroweak gauge singlet, and that a particular 𝛾 or 𝑍 final state is selected,implying that the KLN theorem does not apply [378–381]. While the importanceof this effect for indirect detection has only been appreciated more recently (seefor example [382–387]), it must be accounted for, as it can induce 𝒪 (1) changesto the cross section. Hryczuk and Iengo [382] (hereafter HI) calculated the one-loop correction to the annihilation rate of heavy winos to 𝛾𝛾 and 𝛾𝑍 , and foundlarge corrections to the tree-level result, even after including a prescription for theSommerfeld enhancement. These large corrections are symptomatic of the presenceof large logarithms ln(2 𝑚 𝜒 /𝑚 𝑍 ) and ln(2 𝑚 𝜒 /𝑚 𝑊 ) , which can generally be resummedusing effective field theory (EFT) techniques. This observation has been made bya number of authors who introduced EFTs to study a variety of models and finalstates. The list includes the case of exclusive annihilation into 𝛾 or 𝑍 final states forthe standard fermionic wino [385] and also a scalar version of the wino [384], as well assemi-inclusive annihilation into 𝛾 + 𝑋 for the wino [383, 386, 387] and higgsino [387].In principle the EFT calculations are systematically improvable to higher orderand in a manner where the perturbative expansion is now under control. In orderto fully demonstrate perturbative control has been regained, however, it is importantto extend these works to higher order. To this end, in this chapter we extend thecalculation of exclusive annihilation of the wino, which has already been calculated tonext-to-leading logarithmic (NLL) accuracy [385]. Doing so includes determining theone-loop correction in the full theory, as already considered in HI. Nonetheless theresults in that reference were calculated numerically and are not in the form needed toextend the EFT calculation to higher order. As such, here we revisit that calculationand analytically determine high or DM-scale one-loop matching coefficients. Wefurther calculate the low or electroweak-scale matching at one loop, thereby includingthe effects of finite gauge boson masses. Taken together these two effects extend thecalculation to NLL ′ = NLL + 𝒪 ( 𝛼 ) one-loop corrections, where 𝛼 = 𝑔 / 𝜋 and166 is the SU(2) L coupling. We estimate that our result reduces the perturbativeuncertainty from Sudakov effects to 𝒪 (1%) , improving on the NLL result where theuncertainty was 𝒪 (5%) . Our calculation is complementary to the NLL ′ calculationfor the scalar wino considered in [384], and where relevant we have cross checked ourwork against that reference. In Sec. 7.2 we outline the EFT setup and review the NLLcalculation. Then in Sec. 7.3 we state the main results of this chapter, the one-loophigh and low-scale matching, leaving the details of their calculation to App. F.1 andApp. F.3 respectively. Detailed cross checks on the results are provided in App. F.2and App. F.4, whilst lengthy formulae are delayed till App. F.5. We compare ouranalytic results to the numerical ones of HI in Sec. 7.4 and then conclude in Sec. 7.5. We begin by outlining the EFT framework for our calculation, and in doing so re-view the calculation of heavy DM annihilation to NLL, focussing on the treatment ofthe large logarithms that were partly responsible for the breakdown in the tree-levelapproximation. We choose the concrete model of pure wino DM – the same as usedin HI and [385] – to study these effects. Nevertheless we emphasize the point thatthe central aim is to quantify the effect of large logarithms which can occur in manymodels of heavy DM, rather than to better understand this particular model. Ulti-mately it would be satisfying to extend these results to DM with arbitrary chargesunder a general gauge group to make the analysis less model specific. This is possiblefor GeV scale DM indirect detection where the tree-level approximation is generallyaccurate (see for example [62, 63]). However, understanding the effects in a simplemodel is an important step towards this goal.The model considered takes the DM to be a wino: an SU(2) L triplet of Majoranafermions. As already highlighted, this is a simple example where both the Sommer-feld enhancement and large logarithms are important. Furthermore this model is ofinterest in its own right. Neutralino DM is generic in supersymmetric theories [6, 7];models of “split supersymmetry” naturally accommodate wino-like DM close to the167eak scale, while the scalar superpartners can be much heavier [388–390]. DM trans-forming as an SU(2) L triplet has been studied extensively in the literature, bothwithin split-SUSY scenarios [391–393] and more generally [28, 394, 395]. The modelaugments the Standard Model (SM) Lagrangian with ℒ DM = 12 Tr ¯ 𝜒 (︀ 𝑖 /𝐷 − 𝑀 𝜒 )︀ 𝜒 . (7.1)We take 𝑀 𝜒 = 𝑚 𝜒 I , such that in the unbroken theory all the DM fermions have thesame mass. After electroweak symmetry breaking, the three states 𝜒 , , break into aMajorana fermion 𝜒 and a Dirac fermion 𝜒 + . A small mass difference, 𝛿𝑚 , betweenthese states is then generated radiatively, ensuring that 𝜒 makes up the observedstable DM. Note, however, that both the charged and neutral states will be includedin the EFT.An effective field theory for this model, NRDM-SCET, was introduced in [385]and used to calculate the rates for the annihilation processes 𝜒𝜒 → 𝑍𝑍, 𝑍𝛾, 𝛾𝛾 .Specifically the EFT generalizes soft-collinear effective theory (SCET) [396–399] toinclude non-relativistic dark matter (NRDM) in the initial state. Schematically thecalculation involves several steps. Firstly the full theory has to be matched onto therelevant NRDM-SCET EW operators at the high scale of 𝜇 ≃ 𝑚 𝜒 . The qualifier EWindicates that this is a theory where electroweak degrees of freedom – the 𝑊 and 𝑍 bosons, top quark, and the Higgs – are dynamical, as introduced in [381, 400–403].These operators then need to be run down to the electroweak scale, 𝜇 ≃ 𝑚 𝑍 . Atthis low scale, we then match NRDM-SCET EW onto a theory where the electroweakdegrees of freedom are no longer dynamical, NRDM-SCET 𝛾 . This matching accountsfor the effects of electroweak symmetry breaking, such as the finite gauge bosonmasses. At this stage we can now calculate the low scale matrix elements whichprovide the Sommerfeld enhancement. We now briefly review each of these steps.The first requirement is to match NRDM-SCET EW and the full theory at the highscale 𝜇 𝑚 𝜒 . The relevant operators in the EFT to describe DM annihilation have the168ollowing form: 𝑂 𝑟 = 12 (︀ 𝜒 𝑎𝑇𝑣 𝑖𝜎 𝜒 𝑏𝑣 )︀ (︁ 𝑆 𝑎𝑏𝑐𝑑𝑟 ℬ 𝑖𝑐𝑛 ⊥ ℬ 𝑗𝑑 ¯ 𝑛 ⊥ )︁ 𝑖𝜖 𝑖𝑗𝑘 ( 𝑛 − ¯ 𝑛 ) 𝑘 , (7.2)which is written in terms of the basic building blocks of the effective theory, and in thecenter of momentum frame we can define 𝑣 = (1 , , , , 𝑛 = (1 , ^ 𝑛 ) , and ¯ 𝑛 = (1 , − ^ 𝑛 ) where ^ 𝑛 is the direction of an outgoing gauge boson. In more detail 𝜒 𝑎𝑣 is a non-relativistic two-component fermionic field of gauge index 𝑎 corresponding to the DMand ℬ ¯ 𝑛,𝑛 contain the outgoing (anti-)collinear gauge bosons 𝐴 𝜇 ¯ 𝑛,𝑛 , which can be seenas ℬ 𝜇𝑛 ⊥ = 𝐴 𝜇𝑛 ⊥ − 𝑘 𝜇 ⊥ ¯ 𝑛 · 𝑘 ¯ 𝑛 · 𝐴 𝜇𝑛 + . . . , (7.3)where the higher order terms in this expression involve two or more collinear gaugefields. For ℬ 𝜇 ¯ 𝑛 ⊥ we simply interchange 𝑛 ↔ ¯ 𝑛 . The full form of ℬ 𝜇𝑛 ⊥ can be found in[404], and is collinear gauge invariant on its own. Finally the gauge index connectionis encoded in 𝑆 𝑎𝑏𝑐𝑑𝑟 : 𝑆 𝑎𝑏𝑐𝑑 = 𝛿 𝑎𝑏 ( 𝒮 𝑐𝑒𝑛 𝒮 𝑑𝑒 ¯ 𝑛 ) ,𝑆 𝑎𝑏𝑐𝑑 = ( 𝒮 𝑎𝑒𝑣 𝒮 𝑐𝑒𝑛 )( 𝒮 𝑏𝑓𝑣 𝒮 𝑑𝑓 ¯ 𝑛 ) . (7.4)These expressions are written in terms of adjoint Wilson lines of soft gauge bosonsalong some direction 𝑛 , ¯ 𝑛 , or 𝑣 ; in position space the incoming Wilson line is 𝒮 𝑣 ( 𝑥 ) = 𝑃 exp [︂ 𝑖𝑔 ∫︁ −∞ 𝑑𝑠𝑣 · 𝐴 𝑣 ( 𝑥 + 𝑛𝑠 ) ]︂ , (7.5)where the matrix 𝐴 𝑏𝑐𝑣 = − 𝑖𝑓 𝑎𝑏𝑐 𝐴 𝑎𝑣 and for outgoing Wilson lines the integral runsfrom to ∞ .The fact there are only two possible forms of 𝑆 𝑎𝑏𝑐𝑑𝑟 means there are only tworelevant NRDM-SCET operators. An important requirement of the operators is thatthe incoming DM fields must be in an 𝑠 -wave configuration. Then being a two-particlestate of identical fermions, the initial state must be a spin singlet. If the annihilationwas 𝑝 -wave or higher, it would be suppressed by powers of the low DM velocityrelative to these operators. The Wilson coefficients associated with these operators169re determined by the matching. Calculating to NLL simply requires the tree-levelresult where 𝐶 ( 𝜇 𝑚 𝜒 ) = − 𝐶 ( 𝜇 𝑚 𝜒 ) = − 𝜋𝛼 ( 𝜇 𝑚 𝜒 ) /𝑚 𝜒 , where again 𝛼 is the SU(2) L fine structure constant. We extend this result to one loop in Sec. 7.3.After matching, the next step is to evolve these operators down to the low scale,effectively resumming the large logarithms that caused a breakdown in the perturba-tive expansion of the coupling. This is done using the anomalous dimension matrix ^ 𝛾 of the two operators (a matrix as the operators will in general mix during the run-ning). In general the matrix can be broken into a diagonal collinear piece 𝛾 𝑊 𝑇 , anda non-diagonal soft contribution ^ 𝛾 𝑆 , as ^ 𝛾 = 2 𝛾 𝑊 𝑇 I + ^ 𝛾 𝑆 . (7.6)To NLL these results are given by [385]: 𝛾 𝑊 𝑇 = 𝛼 𝜋 Γ 𝑔 ln 2 𝑚 𝜒 𝜇 − 𝛼 𝜋 𝑏 + (︁ 𝛼 𝜋 )︁ Γ 𝑔 ln 2 𝑚 𝜒 𝜇 , ^ 𝛾 𝑆 = 𝛼 𝜋 (1 − 𝑖𝜋 ) ⎛⎝ − ⎞⎠ − 𝛼 𝜋 ⎛⎝ ⎞⎠ . (7.7)Here the collinear anomalous dimension has been written in terms of the SU(2) L one-loop 𝛽 -function, 𝑏 = 19 / , as well as the cusp anomalous dimensions, Γ 𝑔 = 8 and Γ 𝑔 = 8 (︀ − 𝜋 )︀ . Below the matching scale, the spin of the DM is no longer impor-tant. As such the anomalous dimension determined in [385] for the fermionic winoshould resum the same logarithms as those that appear in the scalar case consideredin [384], and we have confirmed they agree.We can then explicitly use the full anomalous dimension to evolve the operatorsas follows: ⎡⎣ 𝐶 𝑋 ± ( { 𝑚 𝑖 } ) 𝐶 𝑋 ( { 𝑚 𝑖 } ) ⎤⎦ = 𝑒 ^ 𝐷 𝑋 ( 𝜇 𝑍 , { 𝑚 𝑖 } )) 𝑃 exp (︃∫︁ 𝜇 𝑍 𝜇 𝑚𝜒 𝑑𝜇𝜇 ^ 𝛾 ( 𝜇, 𝑚 𝜒 ) )︃ × ⎡⎣ 𝐶 ( 𝜇 𝑚 𝜒 , 𝑚 𝜒 ) 𝐶 ( 𝜇 𝑚 𝜒 , 𝑚 𝜒 ) ⎤⎦ , (7.8)170et us carefully explain the origin and dependence of each of these terms. Startingfrom the right, 𝐶 and 𝐶 are the high-scale Wilson coefficients of the operators statedin Eq. (7.2), resulting from a matching of the full theory onto NRDM-SCET EW .These only depend on the high scales, specifically 𝜇 𝑚 𝜒 and 𝑚 𝜒 . Next the anomalousdimension ^ 𝛾 is also a high scale object, and so only depends on 𝑚 𝜒 and now 𝜇 as itruns between the relevant scales. ^ 𝐷 𝑋 is a factor accounting for the low-scale matchingfrom NRDM-SCET EW onto NRDM-SCET 𝛾 – a theory where the electroweak modeshave been integrated out, see [381, 400–403]. It is a matrix as soft gauge bosonexchanges can mix the operators. Furthermore ^ 𝐷 𝑋 is labelled by 𝑋 to denote itsdependence on the specific final state considered, 𝛾𝛾 , 𝛾𝑍 or 𝑍𝑍 . This object dependson the low-scale physics and so depends on 𝜇 𝑍 and all the masses in the problem,which we denote as { 𝑚 𝑖 } . Finally on the left we have our final coefficients 𝐶 𝑋 ± and 𝐶 𝑋 ,which as explained below can be associated with the charged and neutral annihilationprocesses. In an all orders calculation of all terms in Eq. (7.8), the scale dependencewould completely cancel on the right hand side, implying that 𝐶 𝑋 ± and 𝐶 𝑋 dependonly on the mass scales in the problem and not 𝜇 𝑚 𝜒 or 𝜇 𝑍 . Nevertheless at anyfinite perturbative order, the scale dependence does not cancel completely and so aresidual dependence is induced in these coefficients. We will exploit this to estimatethe uncertainty in our results associated with missing higher order terms.As we are performing a resummed calculation, the order to which we calculate isdefined in terms of the large electroweak logarithms we can resum. In general thestructure of the logarithms can be written schematically as: ln 𝐶𝐶 tree ∼ ∞ ∑︁ 𝑘 =1 [︀ 𝛼 𝑘 ln 𝑘 +1 ⏟ ⏞ LL + 𝛼 𝑘 ln 𝑘 ⏟ ⏞ NLL + 𝛼 𝑘 ln 𝑘 − ⏟ ⏞ NNLL + . . . ]︀ , (7.9)where since Sudakov logarithms exponentiate, we have defined the counting in termsof the log of the result. Furthermore all corrections are defined with respect to thetree level result 𝐶 tree ∼ 𝒪 ( 𝛼 ) , which is a convention we will follow throughout. Withthis definition of the counting, to perform the running in Eq. (7.8) to NLL order, thereare three effects that must be accounted for: 1. high-scale matching at tree level; 2.171wo-loop cusp and one-loop non-cusp anomalous dimensions; and 3. the low-scalematching at tree level, together with the rapidity renormalization group ( [405, 406])at NLL. To extend this to NNLL all three of these need to be calculated to oneorder higher. In between these two is the NLL ′ result we present here, which involvesdetermining both the high and low-scale matching at one loop. In terms Eq. (7.9),this amounts to determining the leading 𝑘 = 1 piece of the NNLL result. To theextent that 𝒪 ( 𝛼 ) corrections are larger than those at 𝒪 ( 𝛼 ln( 𝜇 𝑚 𝜒 /𝜇 𝑍 )) , the NLL ′ result is an improvement over NLL and more important than NNLL.Before presenting the result of that calculation, however, it is worth emphasizinganother advantage gained from the effective theory. In addition to allowing us toresum the Sudakov logarithms, the effective theory also allows this problem to becleanly separated from the issue of low velocity Sommerfeld enhancement in the am-plitude – in NRDM-SCET there is a Sommerfeld-Sudakov factorization. At leadingpower the relevant SCET Lagrangian contains no interaction with the DM field. Onthe other hand NRDM does contain soft modes, which are responsible for runningthe couplings, however these modes do not couple the Sommerfeld potential to thehard interaction at leading power. Consequently matrix elements for the DM fac-torize from the elements of the states annihilated into. This allows for an all ordersfactorized formula for the DM annihilation amplitude in this theory: ℳ 𝜒 𝜒 → 𝑋 = 4 √ 𝑚 𝜒 𝑃 𝑋 [︁ 𝑠 (︀ Σ 𝑋 − Σ 𝑋 )︀ + √ 𝑠 ± Σ 𝑋 ]︁ , ℳ 𝜒 + 𝜒 − → 𝑋 = 4 𝑚 𝜒 𝑃 𝑋 [︁ 𝑠 ± (︀ Σ 𝑋 − Σ 𝑋 )︀ + √ 𝑠 ±± Σ 𝑋 ]︁ . (7.10)Here 𝑋 can be 𝛾𝛾 , 𝛾𝑍 or 𝑍𝑍 and 𝑃 𝛾𝛾 = − 𝑒 𝜖 𝑖𝑛 ⊥ 𝜖 𝑗 ¯ 𝑛 ⊥ 𝜖 𝑖𝑗𝑘 ^ 𝑛 𝑘 / (2 𝑚 𝜒 ) , whilst 𝑃 𝛾𝑍 =cot ¯ 𝜃 𝑊 𝑃 𝛾𝛾 and 𝑃 𝑍𝑍 = cot ¯ 𝜃 𝑊 𝑃 𝛾𝛾 , with ¯ 𝜃 𝑊 the MS Weinberg angle. The key physicsin this equation is that the contribution from Sommerfeld enhancement is capturedin the terms 𝑠 𝑖𝑗 , whilst the contribution from electroweak logarithms is in Σ 𝑋𝑖 ; thetwo are manifestly factorized and can be calculated independently.The focus of the present chapter is to extend the calculation of the Sudakov effects.In terms of the factorized result stated in Eq. (7.10) this amounts to a modification172f Σ 𝑋𝑖 . Explicitly, from there we can see that: ⃒⃒ Σ 𝑋 ⃒⃒ = 𝜎 (cid:8)(cid:8) SE 𝜒 + 𝜒 − → 𝑋 𝜎 tree 𝜒 + 𝜒 − → 𝑋 , ⃒⃒ Σ 𝑋 − Σ 𝑋 ⃒⃒ = 𝜎 (cid:8)(cid:8) SE 𝜒 𝜒 → 𝑋 𝜎 tree 𝜒 + 𝜒 − → 𝑋 , (7.11)where (cid:8)(cid:8) SE denotes a calculation where Sommerfeld Enhancement is intentionally leftout. To be even more explicit, we can write these Sudakov effects in terms of theWilson coefficients in Eq. (7.8). Specifically we have: Σ 𝑋 = 𝐶 𝑋 ± 𝐶 tree1 , Σ 𝑋 − Σ 𝑋 = 𝐶 𝑋 𝐶 tree1 , (7.12)where as stated above 𝐶 tree1 = − 𝜋𝛼 /𝑚 𝜒 . In terms of the formalism described in the previous section, we now state one of themain results of this chapter: the high-scale Wilson coefficients 𝐶 𝑟 calculated to oneloop as shown in Eq. (7.13). The details have been eschewed to App. F.1. In shortthis calculation involves enumerating and evaluating the 25 one-loop diagrams thatmediate 𝜒 𝑎 𝜒 𝑏 → 𝑊 𝑐 𝑊 𝑑 in the unbroken full theory and then matching this resultonto the NRDM-SCET EW operators. For example, we evaluate diagrams such asand provide the analytic expression graph by graph. In addition we account for thecounter term contribution, the change in the running of the coupling through thematching, and also ensure that the calculation maintains the Sudakov-Sommerfeld173actorization. Combining all of these we find 𝐶 ( 𝜇 ) = − 𝜋𝛼 ( 𝜇 ) 𝑚 𝜒 + 𝛼 ( 𝜇 ) 𝑚 𝜒 [︂ 𝜇 𝑚 𝜒 +2 ln 𝜇 𝑚 𝜒 + 2 𝑖𝜋 ln 𝜇 𝑚 𝜒 + 8 − 𝜋 ]︂ ,𝐶 ( 𝜇 ) = 𝜋𝛼 ( 𝜇 ) 𝑚 𝜒 − 𝛼 ( 𝜇 ) 𝑚 𝜒 [︂ ln 𝜇 𝑚 𝜒 +3 ln 𝜇 𝑚 𝜒 − 𝑖𝜋 ln 𝜇 𝑚 𝜒 − 𝜋 ]︂ , (7.13)where here and throughout this section 𝛼 ( 𝜇 ) is the coupling defined below the scaleof the DM mass, 𝑚 𝜒 . We explain this distinction carefully in App. F.1. For eachcoefficient in Eq. (7.13) the first term represents the tree-level contribution. A crosscheck on this result is provided in App. F.2, where we check that the 𝜇 dependenceof this result properly cancels with that of the NLL resummation for the 𝒪 ( 𝛼 ) corrections. The cancellation occurs between our result in Eq. (7.13) and the runninginduced by the anomalous dimension stated in Eqs. (7.6) and (7.7); this can be seenclearly in Eq. (7.8) as these are the only two objects that depend on 𝜇 𝑚 𝜒 . As theanomalous dimension is independent of the DM spin, the logarithms appearing in ourhigh-scale matching coefficients should also be, and indeed ours match those in thescalar calculation of [384]. Of course the finite terms should not be, and are not, thesame.We next state the contribution from the low-scale matching. Unsurprisingly, asthis effect accounts for electroweak symmetry breaking effects such as the gauge bosonmasses, it is in general dependent upon the identity of the final states. Again thisis a matching calculation and involves evaluating diagrams that appear in SCET EW ,but not SCET 𝛾 , and we provide three examples below. W/Z
W/Z
W/Z
174 central difficulty in the calculation is accounting for the effects of electroweaksymmetry breaking, see for example [407] for a recent discussion. In order to simplifythis we make use of the general formalism for electroweak SCET of [381, 400–403],which we have extended to include the case of non-relativistic external states. Wepostpone the details to App. F.3. The approach breaks the full low-scale matching intoa soft and collinear component, which are the labels associated with the non-diagonaland diagonal contributions respectively, rather than the effective theory modes thatgive rise to them. This distinction is explained in detail in App. F.3. In our case, ^ 𝐷 𝑋 ( 𝜇 ) in Eq. (7.8) can be specified through exp [︁ ^ 𝐷 𝑋 ( 𝜇 ) ]︁ = [︁ ^ 𝐷 𝑠 ( 𝜇 ) ]︁ [︁ 𝐷 𝜒𝑐 ( 𝜇 ) I ]︁ exp [︃∑︁ 𝑖 ∈ 𝑋 𝐷 𝑖𝑐 ( 𝜇 ) I ]︃ , (7.14)where again 𝑋 can be 𝛾𝛾 , 𝛾𝑍 or 𝑍𝑍 , ^ 𝐷 𝑠 ( 𝜇 ) is the non-diagonal soft contributionand a matrix as it mixes the operators, whilst 𝐷 𝜒𝑐 ( 𝜇 ) and 𝐷 𝑖𝑐 ( 𝜇 ) are the initial andfinal state diagonal contributions respectively. Note both ^ 𝐷 𝑠 ( 𝜇 ) and the identitymatrix I are × matrices. The terms that are not exponentiated in Eq. (7.14)are only determined to 𝒪 ( 𝛼 ) , whereas the final state collinear contribution has itslargest contribution resummed to all orders. Using this definition we find that thecomponents of the soft matrix are (see App. F.3): [ ^ 𝐷 𝑠 ] = 1 + 𝛼 ( 𝜇 )2 𝜋 [︂ ln 𝑚 𝑊 𝜇 (1 − 𝑖𝜋 ) + 𝑐 𝑊 ln 𝑚 𝑍 𝜇 ]︂ , [ ^ 𝐷 𝑠 ] = 𝛼 ( 𝜇 )2 𝜋 ln 𝑚 𝑊 𝜇 (1 − 𝑖𝜋 ) , (7.15) [ ^ 𝐷 𝑠 ] = 1 + 𝛼 ( 𝜇 )2 𝜋 ln 𝑚 𝑊 𝜇 (2 − 𝑖𝜋 ) , [ ^ 𝐷 𝑠 ] = 1 . Here and throughout we use the shorthand 𝑐 𝑊 = cos ¯ 𝜃 𝑊 and 𝑠 𝑊 = sin ¯ 𝜃 𝑊 . Further, This calculation can also be performed using the rapidity renormalization group [405, 406], butwe will not use that formalism here. 𝐷 𝜒𝑐 ( 𝜇 ) =1 − 𝛼 ( 𝜇 )2 𝜋 [︂ ln 𝑚 𝑊 𝜇 + 𝑐 𝑊 ln 𝑚 𝑍 𝜇 ]︂ ,𝐷 𝑖𝑐 ( 𝜇 ) = 𝛼 ( 𝜇 )2 𝜋 [︂ ln 𝑚 𝑊 𝜇 ln 4 𝑚 𝜒 𝜇 −
12 ln 𝑚 𝑊 𝜇 − ln 𝑚 𝑊 𝜇 + 𝑐 𝑖 ln 𝑚 𝑍 𝜇 + 𝑐 𝑖 ]︂ , (7.16)where 𝑖 = 𝑍 or 𝛾 and we have: 𝑐 𝑍 = 5 − 𝑠 𝑊 − 𝑠 𝑊 𝑐 𝑊 ,𝑐 𝛾 = 1 − 𝑠 𝑊 , (7.17)and 𝑐 𝑍 = − . − . 𝑖 ,𝑐 𝛾 = − . . (7.18)Analytic expressions for these last results are provided in App. F.3 and App. F.5, andwe give numerical values here as the expressions are lengthy.The 𝜇 dependence of the low-scale matching is demonstrated to cancel with that inour high-scale matching result when the running is turned off, the details being shownin App. F.4. We emphasize that this cross check involves not only the 𝜇 dependenceof the objects in Eq. (7.14), but also the 𝜇 dependence of the high-scale coefficientsstated in Eq. (7.13) and further the SM SU(2) L and U(1) 𝑌 𝛽 -functions. The full 𝜇 cancellation is non-trivial – it requires the interplay between each of these objects.This ultimately provides us with confidence in the results as stated. As a furthercheck, our low-scale matching result does not depend on the spin of the DM. As suchwe should be again able to compare our result to the scalar case calculated in [384].In that work they only considered the 𝛾𝛾 final state, and also neglected the impact ofSM fermions. Restricting our calculation to the same assumptions, we confirm thatthe 𝜇 dependence in our result matches theirs.Taking our results in combination, we can extend the NLL calculation to NLL ′ .176 m χ [ TeV ] | Σ ������� ������������ �� χ + χ - → ��� � γ � γγ �������� ′ m χ [ TeV ] | Σ - Σ ������� ������������ �� χ � χ � → ��� � γ � γγ �������� ′ Figure 7-1:
Here we show the NLL ′ electroweak corrections to the charged (left) and neutral(right) DM annihilations obtained by adding the one-loop high and low-scale corrections to the NLLresult. The correction is in good agreement with the NLL calculation, whilst the scale uncertaintieshave been reduced. Bands are derived by varying the high scale between 𝑚 𝜒 and 𝑚 𝜒 . m χ [ TeV ] | Σ ������� ������������ �� χ + χ - → ��� � γ � γγ �������� ′ m χ [ TeV ] | Σ - Σ ������� ������������ �� χ � χ � → ��� � γ � γγ �������� ′ Figure 7-2:
As for Fig. 7-1, but showing a variation in the low-scale matching between 𝑚 𝑍 / and 𝑚 𝑍 , rather than a variation of the high-scale matching as shown there. As can be seen the NLL ′ contribution has reduced the low scale dependence in both cases, but is again consistent with theNLL result. Of course we cannot show full NNLL results in the absence of the higher order anoma-lous dimension calculation, nevertheless the results we state here determine the crosssection with perturbative uncertainties on the Sudakov effects reduced to the percentlevel. At 𝒪 ( 𝛼 ) , our calculation accounts for all terms of the form 𝛼 ln ( 𝜇 𝑚 𝜒 /𝜇 𝑍 ) , 𝛼 ln ( 𝜇 𝑚 𝜒 /𝜇 𝑍 ) , and 𝛼 ln ( 𝜇 𝑚 𝜒 /𝜇 𝑍 ) . The first perturbative term we are missing atthis order is 𝛼 ln( 𝜇 𝑚 𝜒 /𝜇 𝑍 ) . Taking 𝜇 𝑍 = 𝑚 𝑍 and 𝑚 𝜒 anywhere from 𝑚 𝑍 to TeV, we find the absence of these terms induces an uncertainty that is less than 1%,demonstrating the claimed accuracy. Again note that all counting here is relative to the lowest order contribution, which occurs at 𝐶 tree ∼ 𝒪 ( 𝛼 ) . As such the absolute order of the terms in this sentence is 𝒪 ( 𝛼 ) . .5 1 5 1010 - - - - m χ [ TeV ] σ γγ + Z γ / v [ c m / s ] �� + ����� + ����� ′ + ��� - ���� + �� ( �� ) �� ′ γ + � ( �� ) ���� ����� ( ��� ) ��� ���������� ( ��� ) Figure 7-3:
The impact of the NLL ′ result on the full cross section, which includes the SommerfeldEnhancement (SE), is shown to be consistent with the lower orders result, suggesting the electroweakcorrections are under control. Also shown is the rate for the semi-inclusive process 𝛾 + 𝑋 calculatedto LL ′ in [387]. In addition on this plot we show current bounds from H.E.S.S. and projected onesfrom CTA, determined assuming 5 hours of observation time. See text for details. To combine the various results stated above into the cross section we take thefactorized results in Eq. (7.10), and note that as the higher order Wilson coefficientshave nothing to do with the Sommerfeld enhancement, their contribution is includedin the Σ terms as given explicitly in Eq. (7.12). We know that at tree level 𝑠 = 𝑠 ±± = 1 and 𝑠 ± = 𝑠 ± = 0 , implying that when the Sommerfeld enhancement can beignored we can associate | Σ | with the Sudakov contribution to 𝜒 + 𝜒 − annihilationand | Σ − Σ | with 𝜒 𝜒 .For this reason, in Fig. 7-1 and Fig. 7-2 we show the contributions to | Σ | and | Σ − Σ | for LL, NLL and NLL ′ . In both cases we see the addition of the one-loop corrections is completely consistent with the NLL results, suggesting that thisapproach has the Sudakov logarithms under control. In these plots we take a centralvalue of 𝜇 𝑚 𝜒 = 2 𝑚 𝜒 and 𝜇 𝑍 = 𝑚 𝑍 . In Fig. 7-1 the bands are derived from varying thehigh-scale matching between 𝑚 𝜒 and 𝑚 𝜒 . Recall that if we were able to calculatethese quantities to all orders, they would be independent of 𝜇 , and so varying thesescales estimates the impact of missing higher order terms. For the | Σ | NLL result,taking 𝜇 𝑚 𝜒 = 2 𝑚 𝜒 is a minimum in the range varied over, so we symmetrise theuncertainties in order to indicate the range of uncertainty. Similarly in Fig. 7-2 we178how the equivalent plot, but here the bands are derived by varying the low scale 𝜇 𝑍 from 𝑚 𝑍 / to 𝑚 𝑍 . Improving on the high and low-scale matching, as we have donehere, should lead to a reduction in the scale uncertainty. In all four cases shown thisis clearly visible and furthermore all results are still consistent with the NLL resultwithin the uncertainty bands.We can also take this result and determine the impact on the full DM annihilationcross section into line photons from 𝛾𝛾 and 𝛾𝑍 in this model, as we show in Fig. 7-3.We take the uncertainty on our final result to include the high and low-scale variationsadded in quadrature. For H.E.S.S. limits we use [369], whilst for the CTA projectionwe assume 5 hours of observation time and use [394, 408]. For both we assume anNFW profile with a local DM density of 0.4 GeV/cm . We see again that our partialNLL ′ results are consistent with the NLL conclusions. In this figure we also includethe LL ′ result for the semi-inclusive process 𝛾 + 𝑋 taken from Fig. 7 of [387], denotedby (BV). The semi-inclusive result is above our line photon result, except at lowDM masses. Note that this work does not show scale uncertainties, so the precisedifference is hard to quantify numerically. In addition to using our results from the previous section in conjunction with therunning due to the anomalous dimension, we can also consider the case where wetake our one-loop result in isolation. In this sense we should be able to reproduce theinitial problem of large logarithms seen by HI. We show this in Fig. 7-4, comparedto the LL and NLL result. For Σ our one-loop result is consistent with that fromNLL, indicating the importance of the 𝛼 ln ( 𝜇 𝑚 𝜒 /𝜇 𝑍 ) and 𝛼 ln( 𝜇 𝑚 𝜒 /𝜇 𝑍 ) correctionsto 𝐶 tree . For Σ − Σ , which starts at NLL, our one-loop result is only consistent withthe NLL expression in the small 𝑚 𝜒 region.For the | Σ | case we also show on that plot the equivalent curve for HI as extractedfrom Fig. 11 of their paper. From here it is clear that the qualitative shape of our A digitized version of our cross section is available with the arXiv submission or upon request. m χ [ TeV ] | Σ ������� ������������ �� χ + χ - → ��� � γ � γγ ������ - ���� ( ���� ��� ) � - ���� ( �� ) m χ [ TeV ] | Σ - Σ ������� ������������ �� χ � χ � → ��� � γ � γγ ������ - ���� ( ���� ��� ) Figure 7-4:
Similar to Fig. 7-1, but instead of the NLL ′ results we show our high and low-scale one-loop results including no running from the anomalous dimension. For the case of 𝜒 + 𝜒 − annihilationwe further show the equivalent result of HI, taken from Fig. 11 of their work (which only extends upto 3 TeV and is the origin of the cutoff). There is evidently some discrepancy between the results.Note that at low masses where the Sudakov logarithms are not too large, our result is consistentwith the NLL result as would be expected. See text for details. m χ [ TeV ] | Σ ��� - ���� ���������� � - ���� ���� ( ���� ��� ) � - ���� ���� + ��� ( ���� ��� ) � - ���� ( �� ) m χ [ TeV ] | Σ ��� - ���� ���������� � - ���� ���� ( ���� ��� ) � �������� - ���� ���� + ��� ( ���� ��� ) � �������� - ���� ( �� ) Figure 7-5:
We show the result of HI for | Σ | compared to two variations of our result. Firstlyin the left panel we show our result with the high only or high and low-scale calculations comparedto the result of HI, taken from Fig. 11 of their paper. In the right panel we take our results andshift them by a constant factor. The shifted results show that above around 1 TeV the shape of ourresult is in good agreement with HI, but the constant offset highlights there is tension. results agrees with theirs but that there is some tension. This tension is already clearin Fig. 7-3, but in Fig. 7-5 we explore this difference in more detail. In the left panelwe show the difference between their result and ours, showing our calculation withand without the low-scale matching included. Given the low-scale matching accountsfor the electroweak masses, which were included in HI, we would expect includingit to improve the agreement. This is seen, but it does not substantially relieve thetension.To further explore the difference, in the right panel of Fig. 7-5 we take our results180nd shift them down by a constant: 0.175 for the high only result and 0.137 for thehigh and low combination. Such a constant offset could originate from a difference in 𝑚 𝜒 independent terms between our result and HI. Unfortunately, however, a differencein such terms could originate from almost any of the graphs contributing to the result.Comparing our analytic expressions to the numerical results of HI we have been unableto pinpoint the exact location of the disagreement, although it is clear that we agreeon the shape of the higher order corrections.Despite the discrepancy between our result and that of HI, we emphasize that wehave confidence in our result as stated. This confidence is derived from the non-trivialcross checks we have performed on our result. In detail, these are ∙ The cancellation in the 𝒪 ( 𝛼 ) corrections of the 𝜇 𝑚 𝜒 dependence in our high-scale matching coefficients, stated in Eq. (7.13), with the high-scale dependenceentering from the anomalous dimension, as stated in Eqs. (7.6) and (7.7). Thiscancellation is demonstrated in App. F.2; ∙ In the absence of running, the cancellation in the 𝒪 ( 𝛼 ) corrections of the 𝜇 dependence between our high and low-scale results, where the latter is statedin Eqs. (7.14), (7.15), (7.16), (7.17), and (7.18). This cancellation also dependson the SM SU(2) L and U(1) 𝑌 𝛽 -functions and is shown in App. F.4; ∙ We have confirmed that the 𝜇 dependence in our low-scale result matches thatin [384], when we reduce our calculation to make the same assumptions used inthat work; ∙ The form of the dominant 𝜇 independent terms in the low-scale matching arein agreement with the results of [381, 400–403], as discussed in App. F.3; and ∙ We have confirmed that the framework used to calculate the low-scale match-ing for our non-relativistic initial state kinematics, reproduces the results of[381, 400–403] when we instead consider massless initial states as used in thosereferences. 181 .5 Conclusion
In this chapter we provide analytic expressions for the full one-loop corrections toheavy wino dark matter annihilation, allowing the systematic resummation of elec-troweak Sudakov logarithms to NLL ′ for the line cross section. We have comparedour result to earlier numerical calculations of such effects, finding results similar inbehaviour but quantitatively different. Our result is stated in a manner that canbe straightforwardly extended to higher order, with our result already reducing theperturbative uncertainty from Sudakov effects on this process to 𝒪 (1%) .182 hapter 8Conclusions In this thesis I have shown just a taste of how insights from astrophysics and particlephysics can help uncover the fingerprints of dark matter through indirect detection.Yet in a very real sense there is work left to be done, as we remain in the dark as to theparticle nature of dark matter. Many of the techniques presented in this thesis can befurther refined, and indeed in ongoing work I am already pursuing these directions.Beyond this the data from recent and upcoming experiments like IceCube, HAWC,and CTA will provide new avenues for exploration. Taken together I believe there isreason to be optimistic that a combination of improved techniques and experimentscould in the near future uncover the first hints of dark matter shining back at us, andfinally revealing its true nature. 18384 ppendix AA Gamma-Ray Signal in the CentralMilky Way
A.1 Stability Under Modifications to the Analysis
A.1.1 Changing the Region of Interest
In Fig. A-1, we compare the spectrum correlated with the dark matter template (with 𝛾 = 1 . ) for variations of the ROI. In the left panel, we study different degrees ofmasking the Galactic Plane ( | 𝑏 | > ∘ and | 𝑏 | > ∘ ), and the impact of performingthe fit only in the southern sky (where the diffuse backgrounds are somewhat fainter)rather than in the full ROI. In the right panel, we show the impact of expanding orshrinking the ROI.There is no evidence of asymmetry between the southern sky and the overall signal.Masking at ∘ gives rise to a similar spectral shape but a lower overall normalizationthan obtained with the ∘ mask, albeit with large error bars. As discussed in Sec. 2.6,this may reflect a steepening of the spatial profile at larger distances from the GalacticCenter, although the fainter emission at these larger radii is likely also more sensitiveto mismodeling of the diffuse gamma-ray background.Shrinking or expanding the size of the ROI also changes the height of the peak,while preserving a “bump”-like spectrum that rises steeply at low energies and peaks185 γ (GeV)-1•10 -6 -6 -6 -6 -6 -6 E d N / d E ( G e V / c m / s / s r) |b|>1, full region|b|>1, south|b|>4, full region|b|>4, south γ (GeV)-1•10 -6 -6 -6 -6 -6 -6 E d N / d E ( G e V / c m / s / s r) Figure A-1: The spectrum of the dark matter template found in our Inner Galaxyanalysis when performing the fit over different regions of the sky. Using our standardROI as a baseline, in the left panel we show variations of the Galactic plane maskand fits restricted to the southern sky, where backgrounds are typically somewhatlower, i.e. | 𝑏 | > ∘ , 𝑏 < − ∘ , | 𝑏 | > ∘ , and 𝑏 < − ∘ . All fits employ a single templatefor the Bubbles, the p6v11 Fermi diffuse model, and a dark matter motivated signaltemplate with an inner profile slope of 𝛾 = 1 . . In the right frame, we show theimpact of varying the region over which the fit is performed. All ROIs have | 𝑏 | > ∘ ;aside from this Galactic plane mask, the ROIs are | 𝑏 | < ∘ , | 𝑙 | < ∘ (“ × ”), | 𝑏 | < ∘ , | 𝑙 | < ∘ (“ × ”, standard ROI), | 𝑏 | < ∘ , | 𝑙 | < ∘ (“ × ”), and thefull sky.Figure A-2: A comparison of the regions of the dark matter mass-annihilation crosssection plane (for annihilations to 𝑏 ¯ 𝑏 and an inner slope of 1.18) best fit by thespectrum found in our default Inner Galaxy analysis (fit over the central ∘ × ∘ region), to that found for fits to other ROIs. See text for details.186round ∼ GeV. In general, larger ROIs give rise to lower normalizations for thesignal. This effect appears to be driven by a higher normalization of the diffusebackground model for larger ROIs; when the fit is confined to the inner Galaxy, thediffuse model prefers a lower coefficient than when fitted over the full sky, suggestingthat the p6v11 model has a tendency to overpredict the data in this region. Thismay also explain why larger ROIs prefer a somewhat steeper slope for the profile(higher 𝛾 ); subtracting a larger background will lead to a greater relative decreasein the signal at large radii, where it is fainter. We also find evidence for substantialoversubtraction of the Galactic plane in larger ROIs, consistent with this hypothesis,as we will discuss in Appendix A.1.3.In Fig. A-2, we show the regions of the dark matter mass-annihilation cross sectionplane favored by our fit, for several choices of the ROI (for annihilations to 𝑏 ¯ 𝑏 and aninner slope of 1.18). The degree of variation shown in this figure provides a measureof the systematic uncertainties involved in this determination; we see that the crosssection is always very close to the thermal relic value, but the best-fit mass canshift substantially (from ∼ − GeV). As previously, the contours are based onstatistical errors only.
A.1.2 Varying the Event Selection
By default, we employ cuts on the CTBCORE parameter to improve angular res-olution and minimize cross-leakage between the background and the signal. In anearlier version of this work, this resulted in a pronounced improvement in the con-sistency of the spectrum between different regions (in particular, in the hardness ofthe low-energy spectrum); however, this appears to have been due to a mismodelingof the background emission. We now find that when the backgrounds are treatedcorrectly, the spectrum has a consistent shape independent of the CTBCORE cut,and the significant changes in the tails of the point spread function (PSF) associated We suggested in that earlier work that the soft low-energy spectrum observed in the absence ofa CTBCORE cut was likely due to contamination by mismodeled diffuse emission from the Galacticplane; our current results support that interpretation. γ (GeV)01•10 -6 -6 -6 -6 E d N / d E ( G e V / c m / s / s r) Q2F UltracleanQ4F UltracleanQ4FB UltracleanQ4FB Clean
Figure A-3: The spectrum of emission associated with the dark matter template, cor-responding to a generalized NFW profile with an inner slope of 𝛾 = 1 . , as performedfor four different event selections. Black diamonds indicate the spectrum extractedfrom the usual fit. The blue stars, red crosses and green triangles represent the spectraextracted from repeating our analysis on datasets without a CTBCORE cut, for (re-spectively) ULTRACLEAN front-converting events, all ULTRACLEAN events, andall CLEAN events.with a CTBCORE cut do not materially affect our results. Similarly, we find that ourresults are robust to the choice of ULTRACLEAN or CLEAN event selection, and tothe inclusion or exclusion of back-converting events. We show the spectra extractedfor several different event selections in Fig. A-3. Systematics associated with thesechoices are therefore unlikely to affect the observed excess. A.1.3 A Simplified Test of Elongation
Probing the morphology of the Inner Galaxy excess is complicated by the brightemission correlated with the Galactic Plane. In Ref. [67], it proved difficult to ro-bustly determine whether any signal was present outside of the regions occupied bythe
Fermi
Bubbles, as the regions both close to the Galactic Center and outside ofthe Bubbles were dominated by the bright emission from the Galactic Plane. Theimproved analysis presented in this work mitigates this issue.In addition to the detailed study of morphology described in Sec. 2.6, we performhere a fit dividing the signal template into two independent templates, one with | 𝑙 | > | 𝑏 | and the other with | 𝑏 | > | 𝑙 | . The former template favors the Galactic Plane,while the latter contains the Fermi
Bubbles. As previously, the fit also includes188 γ (GeV)-1•10 -6 -6 -6 -6 -6 E d N / d E ( G e V / c m / s / s r) Complete templateNorth/SouthEast/West
180 90 0 -90 -18000 -90-450459000 -5-4-3-2-10-5-4-3-2-10
180 90 0 -90 -18000 -90-450459000 -5-4-3-2-10-5-4-3-2-10
Figure A-4: In the upper frame, we show the spectra of the emission associated withthe dark matter template, corresponding to a generalized NFW profile with an innerslope of 𝛾 = 1 . , as performed over three regions of the sky. Black diamonds indicatethe spectrum extracted from the usual fit, whereas the blue stars and red crossesrepresent the spectra correlated with the parts of the template in which | 𝑏 | > | 𝑙 | and | 𝑏 | < | 𝑙 | , respectively (when the two are allowed to vary independently). Thecorresponding spatial templates are shown in the lower row, in logarithmic (base 10)units, normalized to the brightest point in each map. Red dashed lines indicate theboundaries of our standard ROI. γ (GeV)-2•10 -6 -6 -6 E d N / d E ( G e V / c m / s / s r) Complete templateNorth/SouthEast/West γ (GeV)-2•10 -6 -6 -6 E d N / d E ( G e V / c m / s / s r) Complete templateNorth/SouthEast/West γ (GeV)-2•10 -6 -6 -6 E d N / d E ( G e V / c m / s / s r) Complete templateNorth/SouthEast/West γ (GeV)-2•10 -6 -6 -6 E d N / d E ( G e V / c m / s / s r) Complete templateNorth/SouthEast/West
Figure A-5: As the upper panel of Fig. A-4, but for ROIs given by (upper left frame) | 𝑏 | , | 𝑙 | < ∘ , (upper right frame) | 𝑏 | , | 𝑙 | < ∘ , (lower left frame) | 𝑏 | , | 𝑙 | < ∘ , (lowerright frame) full sky. In all cases the Galactic plane is masked for | 𝑏 | < ∘ . We at-tribute the lower emission in the East/West quadrants in the larger ROIs to oversub-traction by the Galactic diffuse model along the Galactic plane. The slope parameterfor the dark matter template is set to 𝛾 = 1 . in all cases.189 single template for the Bubbles in addition to the Fermi diffuse model and anisotropic offset. The extracted spectra of the signal templates are shown in Fig. A-4. Both regions exhibit a clear spectral feature with broadly consistent shape andnormalization, although the best-fit spectrum for the region with | 𝑙 | > | 𝑏 | is generallyslightly lower and has larger uncertainties. A lower normalization in these quadrantsis expected, from the preference for a slight stretch perpendicular to the Galacticplane noted for the inner Galaxy in Sec. 2.6.As shown in Appendix A.1.1, the impact of the choice of ROI on the overall shapeof the spectrum is modest. However, upon repeating this analysis in each of the ROIs,we find that the spectrum extracted from the quadrants lying along the Galactic plane( | 𝑙 | > | 𝑏 | ) is much more sensitive to this choice. While a spectral “bump” peaked at ∼ GeV is always present, it appears to be superimposed on a negative offset whichgrows larger as the size of the ROI is increased. As discussed above, we believe this isdue to oversubtraction along the plane by the Galactic diffuse model, which is mostacute when the diffuse model normalization is determined by regions outside the innerGalaxy. We display this progression explicitly in Fig. A-5.The relative heights of the spectra in the | 𝑙 | > | 𝑏 | and | 𝑏 | > | 𝑙 | regions are areasonable proxy for sphericity of the signal; the former will be higher if the signalis elongated along the plane, and lower if the signal has perpendicular extension.Increased oversubtraction along the plane thus induces an apparent elongation of thesignal perpendicular to the plane; we suspect this may be the origin of the apparentstretch perpendicular to the plane shown in Fig. 2-11.One might wonder whether this oversubtraction might give rise to apparent spheric-ity even if the true signal were elongated perpendicular to the plane. We argue thatthis is unlikely, as our results appear to converge to sphericity as the size of the ROI isreduced and the constraint on the normalization of the diffuse background is relaxed;the Galactic Center analysis, which includes the peak of the excess and the regionwhere the signal-to-background ratio is largest, also prefers a spherical excess.We also performed the additional test of not including any model for the pointsources in the fit, allowing their flux to be absorbed by the NFW template. Since190 γ (GeV)-2•10 -6 -6 -6 -6 E d N / d E ( G e V / c m / s / s r) NFW profile, γ =1.0NFW profile, γ =1.1NFW profile, γ =1.2NFW profile, γ =1.3NFW profile, γ =1.4NFW profile, γ =1.5NFW profile, γ =1.6 Figure A-6: The central values of the spectra of the dark matter templates for differentvalues of the dark matter profile’s inner slope, 𝛾 . To better facilitate comparison,each curve has been rescaled to match the 𝛾 = 1 . curve at 1 GeV. All fits havebeen performed with the p6v11 Fermi diffuse model, a single flat template for theBubbles, and the dark matter signal template. The region between the 𝛾 = 1 . and 𝛾 = 1 . lines, preferred by the fit, is cross-hatched. Error bars are not shownto avoid cluttering the plot. In this preferred range, the spectra are remarkablyconsistent. Allowing very high values of 𝛾 seems to pick up a much softer spectrum,likely due to contamination by the Galactic plane, but these high values of 𝛾 providecommensurately worse fits to the data.many point sources are clustered along the plane, over-subtracting them could bias theextracted morphology of the signal and hide an elongation along the plane. However,we found that even when no sources were subtracted, there was no ROI in whichthe spectrum extracted from the | 𝑙 | > | 𝑏 | quadrants exceeded that for the | 𝑏 | > | 𝑙 | quadrants. A.1.4 Sensitivity of the Spectral Shape to the Assumed Mor-phology
In our main analyses, we have derived spectra for the component associated withthe dark matter template assuming a dark matter density profile with a given innerslope, 𝛾 . One might ask, however, to what degree uncertainties in the morphology ofthe template might bias the spectral shape extracted from our analysis. In Fig. A-6,we plot the (central values of the) spectrum found for the dark matter template inour Inner Galaxy analysis, for a number of values of 𝛾 . The shapes of the spectraare quite consistent, within the range of slopes favored by our fits ( 𝛾 = 1 . − . );191 γ (GeV)01•10 -6 -6 -6 -6 E d N / d E ( G e V / c m / s / s r) Single Bubbles templateTwo Bubbles templates
Figure A-7: The spectrum of the emission correlated with a dark matter template,corresponding to a generalized NFW profile with an inner slope of 𝛾 = 1 . , obtainedby a fit containing either a single template for the Fermi
Bubbles (black diamonds)or two templates for 10-degree-wide slices in Galactic latitude through the Bubbles(blue stars). The latter allows the spectrum of the
Fermi
Bubbles to vary somewhatwith Galactic latitude (there are only two templates, in contrast to the five employedin [67], because the ROI only extends to ± degrees).the extracted spectrum is not highly sensitive to the specified signal morphology.However, for 𝛾 (cid:38) . this statement is no longer true: higher values of 𝛾 pick up amuch softer spectrum, which we ascribe to contamination from the Galactic plane atthe edge of the mask. Of course, such high values of 𝛾 also have much worse TS. A.2 Modeling of Background Diffuse Emission in theInner Galaxy
A.2.1 The
Fermi
Bubbles
The fit described in Sec. 2.4 is a simplified version of the analysis performed inRef. [67], where the spectrum of the Bubbles was allowed to vary with latitude. Fromthe results in Ref. [67], it appears that this freedom is not necessary – the spectrumand normalization of the Bubbles varies only slightly with Galactic latitude.It is straightforward to reintroduce this freedom, and we show in Fig. A-7 thespectrum correlated with the dark matter template if this is done. Above 0.5 GeV, thespectrum of the excess is not significantly altered by fixing the Bubbles to have a single192 γ (GeV)01•10 -6 -6 -6 -6 E d N / d E ( G e V / c m / s / s r) P6V11 backgroundP7V6 background
Figure A-8: The spectra of the emission correlated with a dark matter template,corresponding to a generalized NFW profile with an inner slope of 𝛾 = 1 . , withthe background modeled by the p6v11 diffuse model (black diamonds) or the p7v6 diffuse model (blue stars). In both cases, the fit also contains an isotropic offset anda template for the Fermi
Bubbles.spectrum; at low energies, reintroducing this freedom slightly raises the extractedspectrum for the dark matter template.
A.2.2 The Choice of Diffuse Model
Throughout our Inner Galaxy analysis, we employed the p6v11 diffuse model releasedby the
Fermi
Collaboration, rather than the more up-to-date p7v6 model. As notedearlier, this choice was made because the p7v6 model contains artificial templates forthe
Fermi
Bubbles and other large-scale features (with fixed spectra), making it moredifficult to interpret any residuals.Having shown that a single flat-luminosity template for the Bubbles is sufficientto capture their contribution without biasing the spectrum of the signal template,one might also employ the p7v6 model in addition to an independent template forthe Bubbles, in order to absorb any deviations between the true spectrum of theBubbles and their description in the model. Unfortunately, the template for the
Fermi
Bubbles employed in constructing the p7v6 diffuse model (which is not sepa-rately characterized from the overall Galactic diffuse emission) is different to the oneemployed in our analysis, especially in the regions close to the Galactic plane. Con-sequently, this approach gives rise to residuals correlated with the spatial differences193igure A-9: A comparison of the regions of the dark matter mass-annihilation crosssection plane (for annihilations to 𝑏 ¯ 𝑏 ) best fit by the spectrum found in our defaultInner Galaxy analysis (using the p6v11 Galactic diffuse model, and fit over the stan-dard ROI), to that found for the spectra shown in Figs. A-4 and A-8. See text fordetails.between these templates. For this reason, we employ the p6v11 diffuse model for ourprincipal analysis. However, using the p7v6 model does not quantitatively change ourresults, although the peak of the spectrum is somewhat lower (yielding results morecomparable to that obtained from the full-sky ROI with the p6v11 model). A directcomparison of these two results is shown in Fig. A-8.In Fig. A-9, we compare the regions of the dark matter mass-annihilation crosssection plane (for annihilations to 𝑏 ¯ 𝑏 ) that are best fit by the spectrum found in ourdefault Inner Galaxy analysis (using the p6v11 Galactic diffuse model, and fit over the | 𝑙 | < ∘ , ∘ > | 𝑏 | > ∘ ROI), to that found for the spectra shown in Figs. A-4 and A-8. The excess is still clearly present and consistent with a dark matter interpretation,and the qualitative results do not change with choice of diffuse model.
A.2.3 Variation in the 𝜋 Contribution to the Galactic DiffuseEmission
Although the spectrum of the observed excess does not appear to be consistent withgamma rays produced by interactions of proton cosmic rays with gas, one might won-der whether the difference between the true spectrum and the model might give riseto an artificially peaked spectrum. While we fit the spectrum of emission correlated194 γ (GeV)10 -7 -6 -5 -4 E d N / d E ( G e V / c m / s / s r) UniformWhole BubbleDiffuseSFD Dust γ (GeV)10 -7 -6 -5 -4 E d N / d E ( G e V / c m / s / s r) UniformWhole BubbleDiffuseSFD DustNFW
Figure A-10: In the left frame, we show the spectra correlated with the varioustemplates, from a fit with the usual backgrounds as well as the Schlegel-Finkbeiner-Davis (SFD) dust map, with the standard ROI. The right frame shows the results ofthe same fit, but also including a dark matter template with 𝛾 = 1 . . The spectrafor the dust map and diffuse model represent the average flux correlated with thosetemplates outside the | 𝑏 | < ∘ mask and within ∘ of the Galactic Center.with the Fermi diffuse model from the data, the model contains at least two principalemission components with quite different spectra (the gamma rays from the inverseCompton scattering of cosmic-ray electrons, and those from the interactions betweencosmic-ray protons and gas), and their ratio is essentially fixed by our choice to usea single template for the diffuse Galactic emission (although we do allow for an ar-bitrary isotropic offset). Mismodeling of the cosmic-ray spectrum or density in theinner Galaxy could also give rise to residual differences between the data and model.As a first step in exploring such issues, we consider relaxing the constraints onthe background model by adding the Schlegel-Finkbeiner-Davis (SFD) map of in-terstellar dust [409] as an additional template. This dust map has previously beenused effectively as a template for the gas-correlated gamma-ray emission [36,101]. Byallowing its spectrum to vary independently of the
Fermi diffuse model, we hope toabsorb systematic differences between the model and the data correlated with the gas.While the approximately spherical nature of the observed excess (see Sec. 2.6) makesthe dust template unlikely to absorb the majority of this signal, if the spectrum ofthe excess were to change drastically as a result of this new component, that couldindicate a systematic uncertainty associated with the background modeling.In Fig. A-10, we show the results of a template fit using the three background195emplates described in Sec. 2.4, as well as the SFD dust map. The additional templateimproves the fit markedly, and absorbs significant emission across a broad range ofenergies. However, when the dark matter template is added, the fit still stronglyprefers its presence and recovers the familiar spectrum with power peaked at ∼ A.2.4 Modulating the 𝜋 Contribution
The use of the SFD dust map as a tracer for the emission from cosmic-ray protoninteractions with gas (producing neutral pions) is predicated on the assumption thatthe distribution of cosmic-ray protons is approximately spatially uniform. In thisappendix, we demonstrate the robustness of the observed signal against the relaxationof this assumption. Specifically we consider an otherwise unmotivated modulation ofthe gas-correlated emission that seems most likely to be capable of mimicking thesignal: the proton density at energies of a few tens of GeV increasing toward theGalactic Center in such a way as to produce the spatially concentrated spectral featurefound in the data. Since the gas density is strongly correlated with the Galactic Diskwhile the signal appears to be quite spherically symmetric (see Sec. 2.6), this wouldrequire the modulation from varying the cosmic-ray proton density to be alignedperpendicular to the Galactic Plane.To this end we created additional templates of the form:
Modulation = (SFD dust map) × 𝑓 ( 𝑟 ) 𝑔 ( 𝑟 ) , (A.1)where 𝑓 ( 𝑟 ) is a projected squared NFW template and 𝑔 ( 𝑟 ) is a simple data-drivencharacterization of how the SFD dust map falls off with increasing galactic latitudeand longitude. In this sense we have factored out how the dust map itself increasestowards the Galactic Center and replaced this with a slope that matches a generalizedNFW profile. Different modulations were generated by varying 𝑓 ( 𝑟 ) , which was doneby choosing various values of the NFW inner slope, 𝛾 , from 0.5 to 2.0 in 0.1 increments.In order to determine 𝑔 ( 𝑟 ) , the dust map was binned in longitude and latitude and196
180 90 0 -90 -18000 -90-450459000 -5-4-3-2-10-5-4-3-2-10
180 90 0 -90 -18000 -90-450459000 -5-4-3-2-10-5-4-3-2-10
Figure A-11: Left frame: The Schlegel-Finkbeiner-Davis dust map, used as a tracer foremission from proton-gas interactions. Right frame: An example of a dust-modulationtemplate, created by multiplying the dust map by 𝑓 ( 𝑟 ) /𝑔 ( 𝑟 ) , in the case where 𝑓 ( 𝑟 ) is a projected squared NFW with 𝛾 = 0 . . Red dashed lines indicate the boundariesof our standard ROI. All maps are given in logarithmic (base 10) units, normalizedto the brightest point in each map. The modulated-dust template is artificially setto zero for | 𝑏 | > ∘ and | 𝑙 | > ∘ , to avoid errors due to the denominator factorbecoming small; as these bounds lie outside our ROI, they will not affect our results.See text for details.a rough functional form was chosen for each. For longitude, we analyzed the regionwith | 𝑙 | < ∘ , and fit the profile of the dust map with a Gaussian. For latitude, weconsidered | 𝑏 | < ∘ and determined a best-fit using a combination of an exponentialand linear function. These two best-fits were then multiplied to give 𝑔 ( 𝑟 ) . Each ofthe new templates were normalized such that the average value of all pixels with anangle between 4.9 and 5.1 degrees from the Galactic Center was set to unity. Thiswas done in order to aid a comparison with the projected squared NFW template,which is normalized similarly. An example of the final template is shown in Fig. A-11,which was created using an 𝑓 ( 𝑟 ) with 𝛾 = 0 . .Note that there is no particular physics motivation behind this choice of modu-lating function; we are attempting to create a dust-correlated map that mimics theobserved signal as closely as possible, even if it is not physically reasonable. Sincethe dust map is integrated along the line of sight, the modulation we have performedis also not precisely equivalent to the effect of changing the cosmic ray density inthe inner Galaxy – this analysis serves as a test of correlation with the gas, but themodulation should not be interpreted as a cosmic-ray density map.Each of the modulated-dust templates was combined with the three backgroundtemplates described in Sec. 2.4 and run through the maximum likelihood analysis.197 .5 1.0 1.5 2.0f(r) Inner Profile Slope, γ T S γ T S Figure A-12: In the left frame, we plot the improvement in TS between the templatefit performed using known backgrounds and a modulated Schlegel-Finkbeiner-Davisdust map (22 degrees of freedom, corresponding to the 22 energy bins), and the fitusing only the known backgrounds, as a function of the inner profile slope 𝛾 of the 𝑓 ( 𝑟 ) template used in constructing the modulation. In the right frame, we show theimprovement in TS when a 𝛾 = 1 . dark matter template is added to the previousfit, as a function of the inner profile slope 𝛾 of the 𝑓 ( 𝑟 ) template.The results can be seen in the left frame of Fig. A-12. Generically, the modulated-dusttemplate acquires an appreciable coefficient in a similar energy range to the observedexcess. (This should not be surprising, as the modulated-dust templates have beendesigned to absorb the excess to the greatest degree possible.) The spectrum associ-ated with the template fit using an 𝑓 ( 𝑟 ) of 𝛾 = 0 . , near where the 𝜒 was improvedmost, is shown in the left frame of Fig. A-13. Nevertheless, when a dark mattertemplate was added to the analysis, there was always a substantial improvement inquality of the fit, as shown in the right frame of Fig. A-12 for a dark matter templatewith an inner slope of 𝛾 = 1 . .When the dark matter template and modulated dust map are added to the fit to-gether, both acquire non-negligible coefficients, as shown in the right frame of Fig. A-13. The modulated dust map is correlated with a soft spectrum, similar to that ofthe diffuse model, while the dark matter template acquires power in the ∼ − GeVrange around the peak of the excess. The presence of the modulated dust map does γ (GeV)10 -7 -6 -5 -4 E d N / d E ( G e V / c m / s / s r) UniformWhole BubbleDiffuseModulated Dust γ (GeV)10 -7 -6 -5 -4 E d N / d E ( G e V / c m / s / s r) UniformWhole BubbleDiffuseModulated DustNFW
Figure A-13: The left frame shows the spectra obtained from a template fit employingthe standard backgrounds and a modulated dust template, choosing 𝑓 ( 𝑟 ) with 𝛾 = 0 . (see text), in the standard ROI. In the right frame, we plot the coefficients from thesame template fit, but with an additional 𝛾 = 1 . dark matter template included.The normalization of the spectrum for the modulated dust template is described inthe text; the normalization of the diffuse model spectrum is as in Fig. A-10. Dueto the large variation in the amplitudes of the different spectra, we use a log scale;where the central values are negative, we instead plot the 𝜎 upper limit in that bin.in this case substantially bias the extracted spectrum for the dark matter template– this is not greatly surprising, as by construction the two templates are very similarin shape.The observant reader may note that the TS of the best-fit modulated dust map isactually greater than the TS for the dark matter template. However, it appears thismay be due to the modulated dust map doing a better job of picking up unmodeledemission correlated with the dust , rather than with the few-GeV excess. If the SFDdust map is added to the fit to provide an additional degree of freedom to the diffusemodel, as described in App. A.2.3, the TS for the best fit dark matter templatebecomes 1748, compared to 1302 for the best fit modulated dust.The above conclusions were checked to be robust against the choice of ROI anddiffuse model; very similar results were found when the analysis was repeated usingthe full sky or the p7v6 model.It thus appears that a spatial modulation of the gas-correlated emission withcoincidental similarities to a dark matter signal could significantly bias the extractedspectrum, but it is difficult (at least within the tests we have performed) to absorb theexcess completely. The Galactic Center analysis also finds no evidence for correlation199etween the excess and known gas structures. Thus even if the 𝜋 background hasbeen modeled incorrectly, this deficiency seems unlikely to provide an explanation forthe observed signal. A.3 Modifications to the Point Source Modeling andMasking for the Inner Galaxy
As the point sources are concentrated along the Galactic disk and toward the Galac-tic Center, mismodeling of point sources might plausibly affect the extraction of thesignal. To study the potential impact of mismodeling, and check the validity of ourpoint source model, in the Inner Galaxy analysis, we perform the following indepen-dent tests: ∙ We allow the overall normalization of the point source model to float indepen-dently in each energy bin (the relative normalizations of different sources at thesame energy are held fixed). ∙ We halve or double the flux of all sources in the point source model, relative tothe values given in the catalog. ∙ We omit the point source model from the fit entirely. ∙ We furthermore investigate the impact of our (fairly arbitrary) choice of maskradius, which is set at the containment radius of the (energy-dependent)PSF by default.Plots showing the results of these various checks are found in Fig. A-14. Wefind that the impact on the spectrum of even quite severe errors in the point sourcemodeling (such as omitting it entirely or multiplying all source fluxes by a factor oftwo) is negligible, with the standard mask. Reducing the mask to a very small valuehas a greater effect, but is still only substantial at the lowest energies; we attribute theextra emission here to leakage from unmasked and poorly-subtracted bright sources.200 γ (GeV)01•10 -6 -6 -6 -6 E d N / d E ( G e V / c m / s / s r) Mask 1.0Mask 0.6Mask 0.2 γ (GeV)01•10 -6 -6 -6 -6 E d N / d E ( G e V / c m / s / s r) Point Source 1.0Point Source 0.5Point Source 2Point Source 0 γ (GeV)01•10 -6 -6 -6 -6 E d N / d E ( G e V / c m / s / s r) Fixed Point SourceFloating Point Source
Figure A-14: Upper panel: Here we show the impact of changing the point sourcemask radius, shrinking its original size from 1.0 to 0.6 and 0.2. We see that only atthe lower energies is there any impact. Lower panel, left frame: We show the resultof subtracting the point sources multiplied by a several values: 0, 0.5, 1 and 2. Lowerpanel, right frame: We show the difference of allowing the point source model to floatat each energy as opposed to keeping it fixed. In the floating case we only performthe fit up to 10 GeV; beyond this point it becomes numerically unstable. Our NFWtemplate has 𝛾 = 1 . for all fits. A.4 Shifting the Dark Matter Contribution Alongthe Plane
The maps of Fig. 6 show residual bright structure along the Galactic plane. Thepresence of other bright excesses with the same spectrum along the disk, not simplyin the Galactic Center, could favor astrophysical explanations for the signal. To testthis possibility, we shift the DM-annihilation-like spatial template along the Galacticplane in 30 ∘ increments; as usual, the other templates in the fit are the Fermi
Bubbles,the diffuse model and an isotropic offset. For numerical stability and consistency ofthe background modeling, we perform these fits over the full sky rather than the201 γ (GeV)-1•10 -6 -6 -6 -6 E d N / d E ( G e V / c m / s / s r) Galactic centerCentered at l=30 ° Figure A-15: Red stars indicate the Galactic Center spectrum, whereas blue diamondsindicate the spectrum correlated with a DM-annihilation-like template (correspondingto an NFW profile with an inner slope 𝛾 = 1 . ) centered at 𝑏 = 0 ∘ , 𝑙 = 30 ∘ , instead ofat the Galactic Center. The band of horizontal lines indicates the spread of the best-fit spectra correlated with DM-annihilation-like templates shifted in ∘ incrementsalong the Galactic plane: for the ten other cases sampled ( 𝑙 = 60 ∘ , ∘ , ..., ∘ ), theemission correlated with the DM-annihilation-like template was nearly an order ofmagnitude below the Galactic Center excess at its peak, with no evidence of spectralsimilarity. In this case we perform the fit over the full-sky ROI (with an appropriatebest-fit 𝛾 ), rather than our standard ROI, to ensure stability of the fit and keep thefitted normalizations of the background templates similar over the different runs.standard ROI. All templates are normalized so that their spectra reflect the fluxfive degrees from their centers. For ten of the twelve points sampled, the emissioncorrelated with this template is very small; the cross-hatched band in Fig. A-15shows the full range of the central values for these ten cases. For the point centeredat 𝑙 = 30 ∘ , there is substantial emission correlated with the template at energiesbelow 1 GeV, but its spectrum is very soft, resembling the Galactic plane more thanthe excess at the Galactic Center. The last point is the Galactic Center.We have performed the same test shifting the center of the DM-annihilation-liketemplate in ∘ increments from 𝑙 = − ∘ to 𝑙 = 30 ∘ . The templates centered at 𝑙 = ± ∘ absorb emission associated with the Galactic Center excess, albeit withlower amplitude; none of the other cases detect any excess of comparable size with asimilar spectrum. 202 .5 Variations to the Galactic Center Analysis In the default set of templates used in our Galactic Center analysis, we have employedastrophysical emission models which include several additional components that arenot included within the official
Fermi diffuse models or source catalogs. These in-clude the two point sources described in Ref. [108] and a model tracing the 20 cmsynchrotron emission. In models without a dark matter contribution, these struc-tures are extremely significant; the addition of the 20 cm template is preferred withTS=130 (when fit with a broken power-law slope with 4 d.o.f), and the inclusion ofthe additional two point sources is favored with TS=15.9 and 59.3 (when the first isfit with a broken power-law with 4 d.o.f. the second with a simple power-law with 2d.o.f).Upon including the dark matter template in the fit, however, the significance ofthese additional components is lessened substantially. In this fit, the addition of the20 cm template and the two new point sources is preferred at only TS=12.2, 21.8, and14.6, respectively. Additionally, our best-fit models attribute extremely soft spectrato each of these sources. The 20 cm component has a hard spectrum at low energiesbut breaks to a spectral index of -3.3 above 0.6 GeV. The spectral indices of the twopoint sources are -3.1 and -2.8, respectively. The total improvement in TS for theaddition of these combined sources is 47.6.In the upper frame of Fig. A-16, we compare the spectrum of the dark mattertemplate found in our default analysis to that found when the 20 cm template andtwo additional point sources are not included (for 𝛾 = 1 . . The exclusion of theseadditional components from the fit leads to a softer spectrum at energies below ∼ Fermi
Collaboration. As a test of the robustness203 x10 -7 -6 E dN/dE (GeV cm -2 s -1 ) E (cid:97) (GeV) DefaultNo 20 cm, no new PS -7 -6 E dN/dE (GeV cm -2 s -1 sr -1 ) E (cid:97) (GeV) DefaultSpectral Variations Allowed
Figure A-16: Left frame: A comparison of the spectrum of the dark matter templatefound in our default Galactic Center analysis to that found when the 20 cm templateand two additional point sources are not included in the fit (for 𝛾 = 1 . . Theexclusion of these additional components from the fit leads to a softer spectrum atenergies below ∼ ppendix BDark Matter in Galaxy Groups This appendix is organized as follows. First, we provide an extended descriptionof the main analysis results presented in chapter 3, including limits for differentannihilation channels, injected signal tests, individual bounds on the top ten galaxygroups studied, and sky maps of the extragalactic DM halos. Secondly, we show howthe results are affected by variations in the analysis procedure, focusing specificallyon the halo selection criteria, data set type, foreground models, halo density andconcentration, substructure boost, and the galaxy group catalog.
B.1 Extended results
The b¯b
Channel.
In the main Letter, the right panel of Fig. 3-1 demonstrateshow the limit on the 𝑏 ¯ 𝑏 annihilation cross section depends on the number of halosincluded in the stacking, for the case where 𝑚 𝜒 = 100 GeV. In Fig. B-1, we show thecorresponding plot for 𝑚 𝜒 = 10 GeV (left) and TeV (right). As in the 100 GeVcase, we see that no single halo dominates the bound and that stacking a large numberof halos considerably improves the sensitivity.The left panel of Fig. B-2 shows the maximum test statistic, TS max , recovered forthe stacked analysis in the 𝑏 ¯ 𝑏 channel. For a given data set 𝑑 , we define the maximumtest-statistic in preference for the DM model, relative to the null hypothesis without205 N H (halo number)10 − − − h σ v i [ c m s − ] Cumulative Galaxy Groups b ¯ b, m χ = 10 GeV N H (halo number)10 − − − h σ v i [ c m s − ] Cumulative Galaxy Groups b ¯ b, m χ = 10 TeV Figure B-1: The change in the limit on the 𝑏 ¯ 𝑏 annihilation channel as a function ofthe number of halos included in the stacking, for 𝑚 𝜒 = 10 GeV (left) and 10 TeV(right). The 68 and 95% expectations from 200 random sky locations are indicatedby the red bands.DM, as TS max ( ℳ , 𝑚 𝜒 ) ≡ [︁ log ℒ ( 𝑑 |ℳ , ̂︂ ⟨ 𝜎𝑣 ⟩ , 𝑚 𝜒 ) − log ℒ ( 𝑑 |ℳ , ⟨ 𝜎𝑣 ⟩ = 0 , 𝑚 𝜒 ) ]︁ , (B.1)where ̂︂ ⟨ 𝜎𝑣 ⟩ is the cross section that maximizes the likelihood for DM model ℳ .The observed TS max is negligible at all masses and well-within the null expectation(green/yellow bands), consistent with the conclusion that we find no evidence for DMannihilation. Other Annihilation Channels.
In general, DM may annihilate to a variety ofStandard Model final states. Figure B-2 (right) interprets the results of the analysisin terms of limits on additional final states that also lead to continuum gamma-rayemission. Final states that predominantly decay hadronically ( 𝑊 + 𝑊 − , 𝑍𝑍 , 𝑞 ¯ 𝑞 , 𝑐 ¯ 𝑐 , 𝑏 ¯ 𝑏 , 𝑡 ¯ 𝑡 ) give similar limits because their energy spectra are mostly set by boosted piondecay. The leptonic channels ( 𝑒 + 𝑒 − , 𝜇 + 𝜇 − ) give weaker limits because gamma-rayspredominantly arise from final-state radiation or, in the case of the muon, radiativedecays. The 𝜏 + 𝜏 − limit is intermediate because roughly 35% of the 𝜏 decays areleptonic, while the remaining are hadronic. Of course, the DM could annihilateinto even more complicated final states than the two-body cases considered here206 m χ [GeV]0102030405060 T S m a x Data68 /
95% containment 10 m χ [GeV]10 − − − − − − h σ v i [ c m s − ] Thermal relic cross section
Stacked Galaxy Groups
Other annihilation channels beµτq ctWZ
Figure B-2: (Left) Maximum test statistic, TS max , for the stacked analysis comparingthe model with and without DM annihilating to 𝑏 ¯ 𝑏 . The green (yellow) bands showthe 68% (95%) containment over multiple random sky locations. (Right) The 95%confidence limits on the DM annihilation cross section, as a function of the DM mass,for the Standard Model final states indicated in the legend. These limits assume thefiducial boost factor taken from Ref. [193]. Note that we neglect Inverse Comptonemission and electromagnetic cascades, which can be relevant for the leptonic decaychannels at high energies.and the results can be extended to these cases [62, 63]. Note that the limits wepresent for the leptonic final states are conservative, as they neglect Inverse Compton(IC) emission and electromagnetic cascades, which are likely important at high DMmasses—see e.g. , Ref. [212,282]. A more careful treatment of these final states requiresmodeling the magnetic field strength and energy loss mechanisms within the galaxygroups. Injected Signal.
An important consistency requirement is to ensure that the limit-setting procedure does not exclude a putative DM signal. The likelihood procedureemployed here was extensively vetted in our companion paper [25], where we demon-strated that the limit never excludes an injected signal. In Fig. B-3, we demonstratea data-driven version of this test. In detail, we inject a DM signal on top of the actualdata set used in the main analysis, focusing on the case of DM annihilation to 𝑏 ¯ 𝑏 fora variety of cross sections and masses. We then apply the analysis pipeline to thesemaps. The top panel of Fig. B-3 shows the recovered cross sections, as a functionof the injected values. The green line corresponds to the 95% cross section limit,207 − − − h σv i inj [cm s − ]10 − − − h σ v i [ c m s − ] m χ = 10 GeV h σv i TS max h σv i Lim − − − h σv i inj [cm s − ]10 − − − h σ v i [ c m s − ] m χ = 100 GeV h σv i TS max h σv i Lim − − − h σv i inj [cm s − ]10 − − − h σ v i [ c m s − ] m χ = 10 TeV h σv i TS max h σv i Lim − − − h σv i inj [cm s − ]10 − T S m a x − − − h σv i inj [cm s − ]10 − T S m a x − − − h σv i inj [cm s − ] − T S m a x Figure B-3: (Top) Recovered cross section at maxiumum test statistic, TS max , (blueline) and limit (green line) obtained for various signals injected on top of the data.(Bottom) The maximum test statistic obtained at various injected cross section values.while the blue line shows the best-fit cross section. Note that statistical uncertaintiesarising from DM annihilation photon counts are not significant here, as the dominantsource of counts arises from the data itself. The columns correspond to 10, 100, and10 GeV DM annihilating to 𝑏 ¯ 𝑏 (left, center, right, respectively). The bottom rowshows the maximum test statistic in favor of the model with DM as a function ofthe injected cross section. The best-fit cross sections are only meaningful when themaximum test statistic is (cid:38) , implying evidence for DM annihilation. We see thatacross all masses, the cross section limit (green line) is always weaker than the in-jected value. Additionally, the recovered cross section (blue line) closely approachesthat of the injected signal as the significance of the DM excess increases. Results for Individual Halos.
Here, we explore the properties of the individualgalaxy groups that are included in the stacked analysis. These galaxy groups aretaken from the catalogs in Ref. [183] and [184], which we refer to as T15 and T17,respectively. Table B.1 lists the top thirty galaxy groups, ordered by the relativebrightness of their inferred 𝐽 -factor. Not all groups in this table are included in the208tacking, as some of them satisfy one or more of the following conditions: ⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩ | 𝑏 | ≤ ∘ , overlaps another halo to within ∘ of its center , TS max > and ( 𝜎𝑣 ) best > × ( 𝜎𝑣 ) * lim . (B.2)Note that the overlap criteria is applied sequentially in order of increasing 𝐽 -factor.These selection criteria have been extensively studied on mock data in our companionpaper [25] and have been verified to not exclude a potential DM signal, even on dataas discussed above. Of the five halos with the largest 𝐽 -factors that are excluded,Andromeda is removed because of its large angular extent, and the rest fail the latitudecut.The exclusion of Andromeda is not a result of the criteria in Eq. B.2, so somemore justification is warranted. As can be seen in Table B.1, the angular extent ofAndromeda’s scale radius, 𝜃 𝑠 , is significantly larger than that of any other halo. Tojustify 𝜃 𝑠 as a proxy for angular extent of the emission, we calculate the 68% (95%)containment angle of the expected DM annihilation flux, without accounting for thePSF, and find 1.2 ∘ (4.4 ∘ ). This can be contrasted with the equivalent numbers for thenext most important halo, Virgo, where the corresponding 68% (95%) containmentangles are 0.5 ∘ (2.0 ∘ ). Because Andromeda is noticeably more extended beyond the Fermi
PSF, one must carefully model the spatial distribution of both the smoothDM component and the substructure. Such a dedicated analysis of Andromeda wasrecently performed by the
Fermi collaboration [195]. Out of an abundance of caution,we remove Andromeda from the main joint analysis, but we do show how the limitschange when Andromeda is included further below.Figure B-4 shows the individual limits on the 𝑏 ¯ 𝑏 annihilation cross section forthe top ten halos that pass the selection cuts and Fig. B-5 shows the maximum teststatistic (TS max ), as a function of 𝑚 𝜒 , for these same halos. The green and yellowbands in Fig. B-4 and B-5 represent the 68% and 95% containment regions obtainedby randomly changing the sky location of each individual halo 200 times (subject to209he selection criteria listed above). As is evident, the individual limits for the halosare consistent with expectation under the null hypothesis— i.e. , the black line fallswithin the green/yellow bands for each of these halos. Some of these groups have beenanalyzed in previous cluster studies. For example, the Fermi
Collaboration providedDM bounds for Virgo [176]; our limit is roughly consistent with theirs, and possiblya bit stronger, though an exact comparison is difficult to make due to differences inthe data set and DM model assumptions. Figure B-6 provides the 95% upper limits on the gamma-ray flux associated withthe DM template for each of the top ten halos. The upper limits are provided for26 energy bins and compared to the expectations under the null hypothesis. Theupper limits are generally consistent with the expectations under the null hypothesis,though small systematic discrepancies do exist for a few halos, such as NGC3031,at high energies. This could be due to subtle differences in the sky locations andangular extents between the objects of interest and the set of representative halosused to create the null hypothesis expectations.To demonstrate the case of a galaxy group with an excess, we show the TS max distribution and the limit for NGC6822 in Fig. B-7. This object fails the selectioncriteria because it is too close to the Galactic plane. However, it also exhibits aTS max excess and, as expected, the limit is weaker than the expectation under thenull hypothesis.
Sky maps.
Fig. B-8 shows a Mollweide projection of all the 𝐽 -factors inferred usingthe T15 and T17 catalogs, smoothed at ∘ with a Gaussian kernel. The map is shownin Galactic coordinates with the Galactic Center at the origin. Looking beyondastrophysical sources, this is how an extragalactic DM signal might show up in thesky. Although this map has no masks added to it, a clear extinction is still visiblealong the Galactic plane. This originates from the incompleteness of the catalogsalong the Galactic plane.In Fig. B-9, we show the counts map in ∘ × ∘ square regions around each ofthe top nine halos that pass the selection cuts. For each map, we show all photons Note that the 𝐽 -factor in Ref. [176] is a factor of 𝜋 too large. ∼
500 MeV, indicate all
Fermi 𝜃 𝑠 with a dashed orange circle. Given a DM signal, wewould expect to see emission extend out to 𝜃 𝑠 at the center of these images.211 ame log 𝐽 log 𝑀 vir 𝑧 × ℓ 𝑏 log 𝑐 vir 𝜃 s 𝑏 sh TS max Andromeda 19.8 ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± Table B.1: The top thirty halos included from the T15 [183] and T17 [184] catalogs,as ranked by inferred 𝐽 -factor, which includes the boost factor. For each group, weshow the brightest central galaxy and the common name, if one exists, as well as thevirial mass, cosmological redshift, Galactic coordinates, inferred concentration usingRef. [187], angular extension, boost factor using the fiducial model from Ref. [193],and the maximum test statistic (TS max ) over all 𝑚 𝜒 between the model with andwithout DM annihilating to 𝑏 ¯ 𝑏 . A complete listing of all the halos used in this studyis provided as Supplementary Data. 212 m χ [GeV]10 − − − − h σ v i [ c m s − ] Thermal relic
Object 1 / Virgolog J = 19 . m χ [GeV]10 − − − − Thermal relic
Object 3 / NGC0253log J = 18 . m χ [GeV]10 − − − − h σ v i [ c m s − ] Thermal relic
Object 6 / NGC3031log J = 18 . m χ [GeV]10 − − − − Thermal relic
Object 7 / Centauruslog J = 18 . m χ [GeV]10 − − − − h σ v i [ c m s − ] Thermal relic
Object 8 / NGC1399log J = 18 . m χ [GeV]10 − − − − Thermal relic
Object 10 / NGC4594log J = 18 . m χ [GeV]10 − − − − h σ v i [ c m s − ] Thermal relic
Object 12 / IC 1613log J = 18 . m χ [GeV]10 − − − − Thermal relic
Object 14 / NGC4736log J = 18 . m χ [GeV]10 − − − − h σ v i [ c m s − ] Thermal relic
Object 20 / NGC4565log J = 17 . m χ [GeV]10 − − − − Thermal relic
Object 21 / Hydralog J = 17 . Figure B-4: The 95% confidence limit on the DM annihilation cross section to the 𝑏 ¯ 𝑏 final state for each of the top ten halos listed in Tab. B.1 that pass the selection cuts.For each halo, we show the 68% and 95% containment regions (green and yellow,respectively), which are obtained by placing the halo at 200 random sky locations.The inferred 𝐽 -factors, assuming the fiducial boost factor model [193], are providedfor each object. 213 T S m a x Object 1 / Virgolog J = 19 . Object 3 / NGC0253log J = 18 . T S m a x Object 6 / NGC3031log J = 18 . Object 7 / Centauruslog J = 18 . T S m a x Object 8 / NGC1399log J = 18 . Object 10 / NGC4594log J = 18 . T S m a x Object 12 / IC 1613log J = 18 . Object 14 / NGC4736log J = 18 . T S m a x Object 20 / NGC4565log J = 17 . Object 21 / Hydralog J = 17 . Figure B-5: Same as Fig. B-4, except showing the maximum test statistic (TS max )for each individual halo, as a function of DM mass. These results correspond to the 𝑏 ¯ 𝑏 annihilation channel. 214 E [GeV] − − − Object 1 / Virgo E [GeV] − − − Object 3 / NGC0253 E [GeV] − − − Object 6 / NGC3031 E [GeV] − − − Object 7 / Centaurus E [GeV] − − − Object 8 / NGC1399 E [GeV] − − − Object 10 / NGC4594 E [GeV] − − − Object 12 / IC 1613 E [GeV] − − − Object 14 / NGC4736 E [GeV] − − − Object 20 / NGC4565 E [GeV] − − − Object 21 / Hydra
Figure B-6: Same as Fig. B-4, except showing the 95% upper limit on the gamma-rayflux correlated with the DM annihilation profile in each halo. We use 26 logarithmi-cally spaced energy bins between 502 MeV and 251 GeV.215 T S m a x Object 5 / NGC6822log J = 18 . m χ [GeV]10 − − − − h σ v i [ c m s − ] Thermal relic
Object 5 / NGC6822log J = 18 . Figure B-7: NGC6822 has one of the largest 𝐽 -factors of the objects in the catalog,but it fails the selection requirements because of its proximity to the Galactic plane.We show the analog of Fig. B-5 (left) and Fig. B-4 (right). We see that this objecthas a broad TS max excess over many masses and a weaker limit than expected fromrandom sky locations. Galaxy Group J -factors0 1e+15 GeV cm − sr Figure B-8: Mollweide projection of all the 𝐽 -factors inferred using the T15 and T17catalogs, smoothed at ∘ with a Gaussian kernel. If we could see beyond conventionalastrophysics to an extragalactic DM signal, this is how it would appear on the sky.216 irgo0 100 NGC02530 100 NGC30310 100Centaurus0 100 NGC13990 100 NGC45940 100IC 16130 100 NGC47360 100 NGC45650 100 Figure B-9: The
Fermi -LAT data centered on the top nine halos that are included inthe stacked sample. We show the photon counts (for the energies analyzed) within a ∘ × ∘ square centered on the region of interest. The dotted circle shows the scaleradius 𝜃 s , which is a proxy for the scale of DM annihilation, and the orange starsindicate the Fermi .2 Variations on the Analysis
We have performed a variety of systematic tests to understand the robustness of theresults presented in chapter 3. Several of these uncertainties are discussed in detailin our companion paper [25]; here, we focus specifically on how they affect the resultsof the data analysis.
Halo Selection Criteria.
Here, we demonstrate how variations on the halo selectionconditions listed above affect the baseline results of Fig. 3-1. In the left panel ofFig. B-10, the red line shows the limit that is obtained when starting with 10,000halos instead of 1000, but requiring the same selection conditions. Despite the modestimprovement in the limit, we choose to use 1000 halos in the baseline study becausesystematically testing the robustness of the analysis procedure, as done in Ref. [25],becomes computationally prohibitive otherwise. In order to calibrate the analysis forhigher halo numbers, it would be useful to use semi-analytic methods to project thesensitivity, such as those discussed in Ref. [37, 410], although we leave the details tofuture work.Virgo is the object with the highest 𝐽 -factor in the stacked sample. As madeclear in the dedicated study of this object by the Fermi
Collaboration [176], there arechallenges associated with modeling the diffuse emission in Virgo’s vicinity. However,we emphasize that the baseline limit is not highly sensitive to any one halo, includingthe brightest in the sample. For example, the dotted line in the left panel of Fig. B-10shows the impact on the limit after removing Virgo from the stacking. Critically, wesee that the limit is almost unchanged, highlighting that the stacked result is notsolely driven by the object with the largest 𝐽 -factor.The effect of including Andromeda (M31) is shown as the gray solid line. Weexclude Andromeda from the baseline analysis because of its large angular size, asdiscussed in detail above. Our analysis relies on the assumption that the DM halosare approximately point-like on the sky, which fails for Andromeda, and we thereforedeem it to fall outside the scope of the systematic studies performed here.The dashed line shows the effect of tightening the condition on overlapping halos218 m χ [GeV]10 − − − − − − h σ v i [ c m s − ] Thermal relic cross section
Stacked Galaxy Groups
Halo selection criteria
BaselineNo overlap within 5 ◦ | b | ≥ ◦ Excluding VirgoIncluding M3110k halos m χ [GeV]10 − − − − − − h σ v i [ c m s − ] Thermal relic cross section
Stacked Galaxy Groups
Effect of TS max and d h σv i cuts Baseline d h σv i > × h σv i ∗ lim ( m χ )TS max = 4No cuts Figure B-10: The same as the baseline analysis shown in the left panel of Fig. 3-1of chapter 3, except varying several assumptions made in the analysis. (Left) Weshow the effect of relaxing the overlapping halo criterion to ∘ (dashed), reducingthe latitude cut to | 𝑏 | ≥ ∘ (dot-dashed), excluding Virgo (dotted), and includingAndromeda (gray). The limit obtained when starting from an initial 10,000 halos isshown as the red line. (Right) We show the effect of strengthening the cross section(dashed) or weakening the TS max (dot-dashed) selection criteria, as well as completelyremoving the TS max and cross section cuts (dotted).from ∘ to ∘ . Predictably, the limit is slightly weakened due to the smaller poolof available targets. We also show the effect of decreasing the latitude cut to 𝑏 ≥ ∘ (dot-dashed line). In this case, the number of halos included in the stackedanalysis increases, but the limit is weaker—considerably so below 𝑚 𝜒 ∼ GeV.The weakened limits are likely due to enhanced diffuse emission along the plane aswell as contributions from unresolved point sources, both of which are difficult toaccurately model. In cases with such mismodeling, the addition of a DM templatecan generically improve the quality of the fit, which leads to excesses at low energies,in particular. The baseline latitude cut ameliorates precisely these concerns.The right panel of Fig. B-10 illustrates the effects of changing, or removing com-pletely, the cross section and TS max cuts on the halos. Specifically, the dashed blackline shows what happens when we require that a halo’s excess be even more in-consistent with the limits set by other galaxy groups; specifically, requiring that ( 𝜎𝑣 ) best > × ( 𝜎𝑣 ) * lim . The dot-dashed line shows the limit when we decrease thestatistical significance requirement to TS max > . Note that the two changes have219 m χ [GeV]10 − − − − − − h σ v i [ c m s − ] Thermal relic cross section
Stacked Galaxy Groups
No TS and h σv i cuts Figure B-11: The results of the baseline analysis with the default cuts, as shown inthe left of Fig. 3-1, compared to the corresponding result when no cuts are placed onthe TS max or cross section of the halos in the catalog. The significant offset betweenthe limit obtained with no cuts (dotted line) and the corresponding expectation fromrandom sky locations (red/blue band) demonstrates that many of the objects thatare removed by the TS max and cross section cuts are legitimately associated withastrophysical emission. See text for details.opposite effects on the limits. This is expected because more halos with excesses areincluded in the stacking procedure with the more stringent cross section requirement,which weakens the limit, whereas fewer are included if we reduce the TS max cut,strengthening the limit.The dotted line in the right panel of Fig. B-10 shows what happens when norequirement at all is placed on the TS max and cross section; in this case, the limit isdramatically weakened by several orders of magnitude. We show the same result inFig. B-11 (dotted line), but with a comparison to the null hypothesis corresponding tono TS max and cross section cuts, which is shown as the 68% (95%) red (blue) bands. In the baseline case, the limit is consistent with the random sky locations— i.e. , thesolid black line falls within the green/yellow bands. However, with no TS max and crosssection cuts, this is no longer true— i.e. , the dotted black line falls outside the red/bluebands. Clear excesses are observed above the background expectation in this case, butthey are inconsistent with a DM interpretation as they are strongly excluded by otherhalos in the stack. When deciding on the TS max and cross section requirements that We thank A. Drlica-Wagner for suggesting this test. m χ [GeV]10 − − − − − − h σ v i [ c m s − ] Thermal relic cross section
Stacked Galaxy Groups
Data set and foreground model variations
Baseline p7v6 diffuse modelTop PSF quartile m χ [GeV]10 − − − − − − h σ v i [ c m s − ] Thermal relic cross section
Stacked Galaxy Groups
Halo profile and concentration variations
Baseline ρ NFW -boosted profileBurkert profileDiemer concentration
Figure B-12: The same as the baseline analysis shown in the left panel of Fig. 3-1of chapter 3, except varying several assumptions made in the analysis. (Left) Weshow the effect of using the top PSF quartile of the UltracleanVeto data set (dot-dashed) and the p7v6 diffuse model (dashed). (Right) We show the effect of using thecored Burkert profile [411] (dot-dashed) and the Diemer and Kravtsov concentrationmodel [412] (dotted). The “ 𝜌 NFW -boosted profile” (dashed) shows what happens whenthe annihilation flux from the subhalo boost is assumed to follow the NFW profile(as opposed to a squared-NFW profile).we used for the baseline analysis in Fig. 3-1, our goal was to maximize the sensitivityreach while simultaneously ensuring that an actual DM signal would not be excluded.We verified the selection criteria thoroughly by performing injected signal tests onthe data (discussed above) as well as on mock data (discussed in Ref. [25]). Ideally,galaxy groups would be excluded from the stacking based on the specific propertiesof the astrophysical excesses that they exhibit, as opposed to the TS max and crosssection requirements used here. For example, one can imagine excluding groups thatare known to host AGN or galaxies with high amounts of star-formation activity. Weplan to study such possibilities in future work.
Data Set and Foreground Models.
In the results presented thus far, we have usedall quartiles of the UltracleanVeto event class of the
Fermi data. Alternatively, we canrestrict ourselves to the top quartile of events, as ranked by PSF. Using this subsetof data has the advantage of improved angular resolution, but the disadvantage of a ∼
75% reduction in statistics. The left panel of Fig. B-12 shows the limit (dot-dashedline) obtained by repeating the analysis with the top quartile of UltracleanVeto data;221he bounds are weaker than in the all-quartile case, as would be expected. However,the amount by which the limit weakens is not completely consistent with the decreasein statistics. Rather, it appears that when we lower the photon statistics, more halosthat were previously excluded by the cross section and TS max criteria in the baselineanalysis are allowed into the stacking and collectively weaken the limit.Another choice that we made for the baseline analysis was to use the p8r2 fore-ground model for gamma-ray emission from cosmic-ray processes in the Milky Way.In this model, the bremsstrahlung and boosted pion emission are traced with gascolumn-density maps and the IC emission is modeled using
Galprop [105]. Afterfitting the data with these three components, any ‘extended emission excesses’ areidentified and added back into the foreground model [413]. To study the dependenceof the results on the choice of foreground model, we repeat the analysis using thePass 7 gal_2yearp7v6_v0.fits ( p7v6 ) model, which includes large-scale structures likeLoop 1 and the Fermi bubbles—in addition to the bremsstrahlung, pion, and ICemission—but does not account for any data-driven excesses as is done in p8r2 . Theresults of the stacked analysis using the p7v6 model are shown in the left panel ofFig. B-12 (dashed line). The limit is somewhat weaker to that obtained using p8r2 ,though it is broadly similar to the latter. This is to be expected for stacked anal-yses, where the dependence on mismodeling of the foreground emission is reducedbecause the fits are done on small, independent regions of the sky, so that offsets inthe point-to-point normalizations of the diffuse model can have less impact. For morediscussion of this point, see Ref. [45, 48, 60, 243].
Halo Density Profile and Concentration.
Our baseline analysis makes two as-sumptions about the profiles of gamma-ray emission from the extragalactic halos.The first assumption is that the DM profile of the smooth halo is described by anNFW profile: 𝜌 NFW ( 𝑟 ) = 𝜌 𝑠 𝑟/𝑟 𝑠 (1 + 𝑟/𝑟 𝑠 ) , (B.3)where 𝜌 𝑠 is the normalization and 𝑟 𝑠 the scale radius [75]. The NFW profile suc-cessfully describes the shape of cluster-size DM halos in 𝑁 -body simulations with222nd without baryons (see, e.g. , Ref. [91, 414]). However, some evidence exists point-ing to cored density profiles on smaller scales ( e.g. , dwarf galaxies), and the densityprofiles in these systems may be better described by the phenomenological Burkertprofile [411]: 𝜌 Burkert ( 𝑟 ) = 𝜌 𝐵 (1 + 𝑟/𝑟 𝐵 )(1 + ( 𝑟/𝑟 𝐵 ) ) , (B.4)where 𝜌 𝐵 and 𝑟 𝐵 are the Burkert corollaries to the NFW 𝜌 𝑠 and 𝑟 𝑠 , but have numer-ically different values. While it appears unlikely that the Burkert profile is a gooddescription of the DM profiles of the cluster-scale halos considered here, using thisprofile provides a useful systematic variation because it predicts less annihilation fluxthan the NFW profile does. The right panel of Fig. B-12 shows the effect of using theBurkert profile to describe the halos in the T15 and T17 catalogs (dot-dashed line);the limit is slightly weaker, as expected.The second assumption we made is that the shape of the gamma-ray emissionfrom DM annihilation follows the projected integral of the DM-distribution squared.This is likely incorrect because the contribution from the boost factor, which can besubstantial, should have the spatial morphology of the distribution of DM subhalos.Neglecting tidal effects, we expect the subhalos to follow the DM distribution (insteadof the squared distribution). Including tidal effects is complicated, as subhalos closerto the halo center are more likely to be tidally stripped, which both increases theirconcentration and decreases their number density. We do not attempt to model thechange in the spatial morphology of the subhalo distribution from tidal stripping andinstead consider the limit where the annihilation flux from the subhalo boost followsthe NFW distribution. This gives a much wider angular profile for the annihilationflux for large clusters, compared to the case where the boost is simply a multiplicativefactor. The dashed line in the right panel of Fig. B-12 shows the effect on the limitof modeling the gamma-ray emission in this way (labeled “ 𝜌 NFW -boosted profile”).The extended spatial profile leads to a minimal change in the limit over most of themass range, which is to be expected given that most of the galaxy groups can bewell-approximated as point sources. 223 halo’s virial concentration is an indicator of its overall density and is defined as 𝑐 vir ≡ 𝑟 vir /𝑟 𝑠 , where 𝑟 vir is the virial radius and 𝑟 𝑠 the NFW scale radius of the halo.A variety of models exist in the literature that map from halo mass to concentration.Our fiducial case is the Correa et al. model from Ref. [187]. Here we show how thelimit (dotted line) changes when we use the model of Diemer and Kravtsov [412],updated with the Planck 2015 cosmology [188]. The change to the limit is minimal,which is perhaps a reflection of the fact that the change in the mean concentrationsbetween the concentration-mass models is small compared to the statistical spreadpredicted in these models, which is incorporated into the 𝐽 -factor uncertainties. Wehave also verified that increasing the dispersion on the concentration for the Correa etal. model to 0.24 [415], which is above the 0.14–0.19 range used in the baseline study,worsens the limit by a 𝒪 (1) factor. Substructure Boost.
Hierarchical structure formation implies that larger struc-tures can host smaller substructures, the presence of which can significantly en-hance signatures of DM annihilation in host halos. Although several models existin the literature to characterize this effect, the precise enhancement sensitively de-pends on the methods used as well as the astrophysical and particle physics proper-ties that are assumed. Phenomenological extrapolation of subhalo properties ( e.g. ,the concentration-mass relation) over many orders of magnitude down to very smallmasses 𝒪 (10 − ) M ⊙ lead to large enhancements of 𝒪 (10 ) and 𝒪 (10 ) for galaxy-and cluster-sized halos, respectively [190]. Recent numerical simulations and analyticstudies [187, 191, 192] suggest that the concentration-mass relation flattens at smallermasses, yielding boosts that are much more modest, about an order-of-magnitude be-low phenomenological extrapolations [416, 417]. In addition, the concentration-massrelation for field halos cannot simply be applied to subhalos, because the latter un-dergo tidal stripping as they fall into and orbit their host. Such effects tend to makethe subhalos more concentrated—and therefore more luminous—than their field-halocounterparts, though the number-density of such subhalos is also reduced [193].When taken together, the details of the halo formation process shape the subhalomass function 𝑑𝑛/𝑑𝑀 sh ∝ 𝑀 − 𝛼 sh , where 𝛼 ∈ [1 . , . . The mass function does not224 log ( M vir [M (cid:12) ])10 − b s h Substructure boost models
BartelsMolin´eGao α = 2 . M min = 10 − M (cid:12) α = α ( M vir ), M min = 10 M (cid:12) α = α ( M vir ), M min = 10 − M (cid:12) m χ [GeV]10 − − − − − − h σ v i [ c m s − ] Thermal relic cross section
Stacked Galaxy Groups
Boost model variations
Baseline (Bartels)Bartels M min = 10 M (cid:12) Bartels α = 2 . Figure B-13: (Left) Examples of substructure boost models commonly used in theliterature, reproduced from [25]. Our fiducial model, based on Ref. [193] using 𝑀 min =10 − M ⊙ and self-consistently computing 𝛼 , is shown as the thick green solid line.Variations on 𝑀 min and 𝛼 are shown with the dotted and dashed lines, respectively.Also plotted are the boost models of Moliné [418] (red) and Gao [190] (grey). (Right)The same as the baseline analysis shown in the left panel of Fig. 3-1 of chapter 3,except varying the boost model.follow a power-law to arbitrarily low masses, however, because the underlying particlephysics model for the DM can place a minimum cutoff on the subhalo mass, 𝑀 min .For example, DM models with longer free-streaming lengths wash out smaller-scalestructures, resulting in higher cutoffs.The left panel of Fig. B-13 shows a variety of boost models commonly used inDM studies. The fiducial boost model used here [193] is shown as the thick greensolid line and variations on 𝑀 min and 𝛼 are also plotted. The right panel of Fig. B-13shows that the expected limit when 𝑀 min = 10 M ⊙ instead of 𝑀 min = 10 − M ⊙ (dot-dashed) is weaker across all masses. While a minimum subhalo mass of M ⊙ is likely inconsistent with bounds on the kinetic decoupling temperature of thermalDM, this example illustrates the importance played by 𝑀 min in the sensitivity reach.Additionally, Fig. B-13 demonstrates the case where 𝛼 = 2 . (dashed line). Increasingthe inner slope of the subhalo mass function leads to a correspondingly stronger limit,however observations tend to favor a slope closer to 𝛼 = 1 . (which is what the mostmassive halos correspond to in our fiducial case).Ref. [417] derived a boost factor model that accounts for the flattening of the225oncentration-mass relation at low masses, but does not include the effect of tidalstripping. They assume a minimum sub-halo mass of − M ⊙ and a halo-massfunction 𝑑𝑁/𝑑𝑀 ∼ 𝑀 − . This was updated by Ref. [418] to account for the effectof tidal disruption. This updated boost factor model, which takes 𝛼 = 1 . , givesthe constraint shown in Fig. B-13 labeled “Moliné” (dotted). This model is to becontrasted with the boost factor model of Ref. [190], labeled “Gao” in Fig. B-13 (grey-dashed), which uses a phenomenological power-law extrapolation of the concentration-mass relation to low sub-halo masses. Because the annihilation rate increases withincreasing concentration parameter, the model in Ref. [190] predicts substantiallylarger boosts than other scenarios that take into account a more realistic flatteningof the concentration-mass relation at low subhalo masses. Galaxy Group Catalog.
We now explore the dependence of the results on thegroup catalog that is used to select the halos. In this way, we can better understandhow the DM bounds are affected by uncertainties on galaxy clustering algorithmsand the inference of the virial mass of the halos. The baseline limits are based onthe T15 and T17 catalogs, but here we repeat the analysis using the Lu et al. cat-alog [419], which solely relies on 2MRS observations. The group-finding algorithmused by Ref. [419] is different to that of T15 and T17 in many ways, relying on afriends-of-friends algorithm as opposed to one based on matching group propertiesat different scales to 𝑁 -body simulations. Lu et al. also use a different halo massdetermination. For these reasons, it provides a good counterpoint to T15 and T17for estimating systematic uncertainties associated with the identification of galaxygroups. While T17 includes measured distances for nearby groups, the Lu catalogcorrects for the effect of peculiar velocities following the prescription in Ref. [420]and the effect of Virgo infall as in Ref. [421]. Figure B-14 is a repeat of Fig. 3-1 inchapter 3, except using the Lu et al. catalog. Despite important differences betweenthe group catalogs used, the Lu et al. results are very similar to the baseline case.There are a variety of sources of systematic uncertainty beyond those described herethat deserve further study. For example, a systematic bias in the 𝐽 -factor determi-226 m χ [GeV]10 − − − − − − h σ v i [ c m s − ] Thermal relic cross section
Stacked Galaxy Groups
Lu group catalog
Baseline (Tully catalogs)Lu group catalog N h − − − h σ v i [ c m s − ] Stacked Galaxy Groups
Lu group catalog
Baseline (Tully catalogs)Lu group catalog
Figure B-14: The same as Fig. 3-1 of chapter 3, except using the Lu et al. galaxygroup catalog [419] (dashed) instead of the T15 and T17 catalogs in the baselineanalysis.nation due to offsets in either the mass inference or the concentration-mass relationcan be a potential source of uncertainty. A better understanding of the galaxy-haloconnection and the small-scale structure of halos is required to mitigate this. Further-more, we assumed distance uncertainties to be subdominant in our analysis. Whilethis is certainly a good assumption over the redshift range of interest—nearby groupshave measured distances, while groups further away come with spectroscopic redshiftmeasurements with small expected peculiar velocity contamination—uncertainties onthese do exist. We have also assumed that our targets consist of virialized halos andhave not accounted for possible out-of-equilibrium effects in modeling these [422].22728 ppendix CDark Matter Decay
The appendix is organized as follows. In Sec. C.1, we provide more detail regarding themethods used in chapter 4. In particular, we discuss the calculations of the gamma-ray spectra and the data analysis. In Sec. C.2, we give extended results beyond thosegiven in chapter 4. Then, in Sec. C.3, we characterize and test sources of systematicuncertainty that could affect our results. Lastly, in Sec. C.4, we describe additionaltheory considerations for our analysis, including additional final states, extendingour results to higher masses, and also additional models beyond those discussed inchapter 4.
C.1 Methods
We begin this section by detailing the calculations of the prompt and secondaryspectra from DM decay. Then, we discuss in detail the likelihood profile techniqueused in this paper.
C.1.1 Spectra
This section provides a more detailed description of the gamma-ray spectra that re-sult from heavy DM decay. There is a natural decomposition into three components:(1) prompt Galactic gamma-ray emission, (2) Galactic inverse Compton (IC) emis-229ion from high-energy electrons and positrons up-scattering background photons, and(3) extragalactic flux from DM decay outside of our Galaxy. As mentioned in chap-ter 4, when calculating the prompt spectrum (and the primary electron and positronflux) it is crucial, for certain final states, to included electroweak radiative processes,as these may be the only source of gamma-ray emission. To illustrate this point,in Tab. C.1 we show the average number of primary gamma-rays, neutrinos, andelectrons coming from DM decay to 𝑏 ¯ 𝑏 and 𝜈 ¯ 𝜈 for various DM masses. We notethat for 𝑚 𝜒 = 100 GeV there are in average 3 (0) hadrons in the final state, whilefor 𝑚 𝜒 = 1 PeV there are 77 (1) hadrons for the 𝑏 ¯ 𝑏 ( 𝜈 𝑒 ¯ 𝜈 𝑒 ) decay mode. The energyfraction of these hadrons is 13 (0) % and 16 (0.5) % for 𝑏 ¯ 𝑏 ( 𝜈 𝑒 ¯ 𝜈 𝑒 ) modes with a DMmass of 100 GeV and 1 PeV, respectively. In addition, the energy fractions of pho-tons, neutrinos and electrons are almost independent of the DM mass for the 𝑏 ¯ 𝑏 decaymode. This can be understood as the majority of these final states originate frompion decays. 𝜒 → 𝑏 ¯ 𝑏 𝜒 → 𝜈 𝑒 ¯ 𝜈 𝑒 𝑚 𝜒 𝛾 𝜈 𝑒 − /𝑒 + All 𝛾 𝜈 𝑒 − /𝑒 + All100 GeV 26 66 23 120 0 2 0 21 TeV 58 150 51 270 0.37 3 0.36 3.810 TeV 120 320 110 570 2.0 7.4 1.9 12100 TeV 250 660 230 1200 5.1 15 4.8 261 PeV 490 1300 440 2300 9.2 27 8.7 46Table C.1: Average number of final state particles for DM decay to bottom quarks orelectron neutrinos. For the neutrino case, the presence of electroweak corrections hasa large impact on the resulting spectrum for higher masses, whereas for the hadronicfinal state the effect is less important.Additionally, we show the typical number of radiated 𝑊 and 𝑍 bosons. In the 𝑏 ¯ 𝑏 case, electroweak corrections are not significant even for 1 PeV DM. However, in the230 - - - - - � [ ��� ] � � � Φ / � � [ � � � / � � � / � / � � ] �� → �� � χ = �� ���� τ = �� �� � ������ γ ������� ������� γ ������������� γ �������� + �� ν - - - - - � [ ��� ] � � � Φ / � � [ � � � / � � � / � / � � ] ��������� � χ = �� ���� τ = �� �� � Figure C-1: Gamma-ray and neutrino spectra for DM decaying to 𝑏 ¯ 𝑏 (left) and amodel of gravitino DM (right) as detailed in Sec. C.4 below, with 𝑚 𝜒 = 10 PeV and 𝜏 = 10 s. All fluxes are normalized within the ROI used in our main analysis. Fermi can detect photons in the range ∼ . − GeV. For heavy DM decays, the fluxin the
Fermi energy range is dominated by the IC and extragalactic contributions,rather than the prompt Galactic emission. 𝜈 ¯ 𝜈 case the radiated 𝑊 and 𝑍 bosons are responsible for the majority of the primaryparticles (and all of the gamma-rays and electrons) at masses above the electroweakscale. The importance of these electroweak corrections on dark matter annihila-tion/decay spectra have been previously noted (see e.g. [225–235]). For DM massesabove 10 PeV, the large number of final states implies that generation of the spectrathrough showering in Pythia is no longer practical. We discuss in Appendix C.4.2how we extend our spectra beyond these masses. As was shown in Fig. 4-2 in chapter 4, the prompt flux tends to be most importantfor lower DM masses near the
Fermi energy range, while the IC emission may play aleading role for DM masses near the PeV scale. The extragalactic flux is importantover the whole mass range, but at very high masses – well above the PeV scale –the extragalactic flux may be the only source of gamma-rays in the
Fermi energyrange. To illustrate these points, Fig. C-1 shows the gamma-ray and neutrino spectra Publicly available DM spectra, such as those in [62, 63, 186], do not extend up to these highmasses, which is why we have recalculated them. While there are certainly modeling errors associatedwith running
Pythia at these very high energies, they are expected to be subdominant to theastrophysical uncertainties inherent in this analysis. We extend the spectra above 10 PeV by rescalingthe appropriately normalized spectrum, as described and validated in this appendix. - - - - - � [ ��� ] � � � Φ / � � [ � � � / � � � / � / � � ] �� → �� � χ = ��� ���� τ = �� �� � ������ γ ������������� γ �������� + �� ν - - - - � [ ��� ] � � � Φ / � � [ � � � / � � � / � / � � ] �� → �� � χ = � ���� τ = �� �� � Figure C-2: Gamma-ray and neutrino spectra for DM decaying to 𝑏 ¯ 𝑏 for two differ-ent DM masses: GeV (left) and GeV (right). These should be comparedto the
Fermi energy range of ∼ . − GeV. For the lighter DM case, promptemission dominates, whilst at higher masses the dominant contribution is from theextragalactic flux. In neither of these cases is IC emission relevant, this only con-tributes meaningfully for intermediate 𝒪 (PeV − EeV) masses, as seen on the left ofFig. C-1.at Earth, normalized to within the ROI used in our main analysis, for 10 PeV DMdecay with 𝜏 = 10 s. We consider two final states, 𝑏 ¯ 𝑏 (left) and the gravitino model(right), which is described in more detail later in this appendix.Importantly, for DM masses (cid:38) 𝑊 ± ℓ ∓ , where ℓ ∓ are SM leptons, and 50% of the time into 𝑍 𝜈 and ℎ 𝜈 . Theselatter two final states are responsible for the sharp rise in the Galactic and extragalac-tic neutrino spectrum in the gravitino model at energies approaching the DM mass(10 PeV in this case). In both cases, however, the prompt gamma-ray spectra are seento be sub-dominant within the Fermi energy range, which extends up to ∼ Fermi energy range, the IC emission is the dominant source offlux, while the extragalactic emission extends to much lower energies.To illustrate this point further, we show in Fig. C-2 the 𝑏 ¯ 𝑏 final-state spectra for 𝑚 𝜒 = 100 GeV and ZeV (︀ = 10 GeV )︀ . In the low-mass case, the IC emissionis produced in the Thomson regime and peaks well below the Fermi energy range.Furthermore, in this case the extragalactic spectrum is generally sub-dominant tothe prompt Galactic emission. In the high-mass case, the extragalactic flux is the232nly source of emission within the
Fermi energy range. Indeed, it is well known thatthe extragalactic spectrum approaches a universal form, regardless of the primaryspectra ( e.g. see [212]; also as plotted in Fig. C-12). This can be seen by comparingthe extragalactic spectrum on the right of Fig. C-2 to that on the left on Fig. C-1, andthis is explored in more detail in Sec. C.4.2. Finally for the ZeV DM decays, the ICemission is still largely peaked in the
Fermi energy range, but has now transitionedcompletely to the Klein-Nishina regime, where the cross section is greatly reduced. Assuch its contribution is several orders of magnitude sub-dominat to the extragalacticflux. Note that in Fig. C-2, and in subsequent spectral plots, we have used a galactic 𝐽 -factor that is averaged over our ROI. In detail, if we define 𝜌 ( 𝑠, 𝑙, 𝑏 ) to be the DMdensity as a function of distance from Earth 𝑠 , as well as galactic longitude 𝑙 andlatitude 𝑏 , then we used: 𝐽 = ∫︁ ROI d Ω ∫︁ d 𝑠 𝜌 ( 𝑠, 𝑙, 𝑏 ) / ∫︁ ROI d Ω ≃ . × GeV cm − . (C.1)This is larger than the all-sky averaged value by a factor of 2.6.In chapter 4, we assumed that for the energies relevant for Fermi , the IC morphol-ogy will be effectively identical to that of the prompt DM decay flux. This justifiedthe combination of the prompt and IC flux into a single spatial template which fol-lowed the above 𝐽 -factor. In principle there are at least three places additional spatialdependence could enter, beyond the prompt 𝑒 ± spatial distribution injected by DMdecays: 1. the distribution of the seed photon fields; 2. the distribution of the mag-netic fields under which the electrons cool; and 3. the diffusion of the 𝑒 ± . Referringto the first of these, there are three fields available to up-scatter off: the CMB, theintegrated stellar radiation, and the infrared background due to the irradiated stellarradiation. These last two are position dependent and tend to decrease rapidly offthe plane. So as long as we look off the plane, as we do, the CMB dominates andis position independent. Importantly, neglecting the other contributions is conserva-tive, as they would only contribute additional flux. Regarding the second point, theregular and halo magnetic fields play an important role in the 𝑒 ± cooling. The former233 - - - - - � [ ��� ] � � � Φ / � � [ � � � / � � � / � / � � ] ����������� �� - - - - - � [ ��� ] � � � Φ / � � [ � � � / � � � / � / � � ] �� ���������������� �������� ��������� ( ����� ���� ) Figure C-3: A comparison between the PeV 𝑏 ¯ 𝑏 DM spectrum and that from ourbackground models, for the largest lifetime we can constrain using only either Galactic( 𝜏 = 8 . × s) or extragalactic ( 𝜏 = 1 . × s) DM flux. Spectra are averagedover the ROI used in our analysis. Left: Here we show the diffuse Galactic spectrum,compared to the smallest Galactic (prompt and IC) flux we can constrain. For thediffuse model we show the 68% confidence interval determined from the posterior ofour fit in each energy bin. Diffuse emission is responsible for the vast majority ofthe photons seen in our analysis, and it sits several orders of magnitude above theDM flux we can constrain in most energy bins. Right: The 68% confidence intervalson the spectrum of our isotropic and point source models, compared to the weakestextragalactic DM flux we can constrain. We also show in this plot the bin-by-bin 95%limit we set on extragalactic flux, homogenious across the northern and southern sub-regions. Further, we illustrate the IGRB as measured by Fermi [35], which is in goodagreement with our isotropic spectrum across most of the energy range. See text fordetails.component highly depends on the Galactic latitude and decays off the plane; it issubdominant to the halo magnetic field in our ROI, so we ignore it. Finally, for theenergies of interest, the diffusion of the 𝑒 ± can be neglected to a good approximationon the scales of interest, as discussed in [237]. The halo field is expected to be strongenough for electrons and positrons to lose their energy in the halo.Finally, Fig. C-3 shows the spectrum of the weakest Galactic and extragalactic DMfluxes we can constrain for PeV DM decaying to bottom quarks, directly comparedto the background contributions. In these figures, the three background componentsfrom our fits – diffuse, isotropic, and point source emission – are shown via a bandbetween the 16 and 84 percentiles on these parameters extracted from the posterior,234here the values are given directly in each of our 40 energy bins. Between these fig-ures we see that diffuse emission dominates over essentially the entire energy range.We also see that the value of the isotropic flux is not particularly well constrainedwithin our small ROI, especially at higher energy. It is is important to note that ourisotropic spectrum is found by averaging the spectra in the north and south, whichare fit independently. As a comparison, we also show the 95% limit on homogenousextragalactic emission, which is by definition the same in the northern and southernhemispheres. Reassuringly, our limit on extragalactic emission tends to be weakerthan the isotropic gamma-ray background (IGRB) as measured by
Fermi [35], whichis also shown in that figure. The IGRB was determined from a dedicated analysis athigh-latitudes using a data set with very low cosmic-ray contamination. Even thoughour ROI and data set are far from ideal for determining the IGRB, we see that ourisotropic spectrum is generally in very good agreement with the
Fermi
IGRB up toenergies of around a few hundred GeV; at higher energies, our isotropic spectrum ap-pears higher than the IGRB, perhaps because of cosmic ray contamination. However,this should only make our high-energy extragalactic results conservative.235 .1.2 Data analysis
In this section, we expand upon the profile-likelihood analysis technique used in thiswork (see [242] for comments on this method). The starting point for this is the dataitself, which we show in Fig. C-4. There we show our ROI in the context of the fulldataset. Recall this ROI is defined by | 𝑏 | > ∘ and 𝑟 < ∘ , with 3FGL PSs masked;this particular choice is discussed in detail in Sec. C.3. The raw Fermi data is a listof photons with associated energies and positions on the sky. We bin these photonsinto 40 energy bins, indexed by 𝑖 , that are equally log spaced from 200 MeV and 2TeV. In each energy bin we then take the resulting data 𝑑 𝑖 , and spatially bin it usinga HEALPix [196] pixelation with nside =128. This divides our ROI into 12,474 pixels(before the application of a point source mask), which we index with 𝑝 . The result ofthis energy and spatial binning reduces the raw data into a list of integers 𝑛 𝑝𝑖 for thenumber of photons in pixel 𝑝 in the 𝑖 th energy bin.To determine the allowable DM decay contribution to this data, we need to de-scribe it with a set of model parameters 𝜃 = { 𝜓, 𝜆 } . As discussed in chapter 4, 𝜓 are the parameters of interest which describe the DM flux, while 𝜆 are the set of nui-sance parameters. In detail 𝜓 accounts for the Galactic and separately extragalacticFigure C-4: The data within our Region of Interest (ROI), defined by | 𝑏 | > ∘ and 𝑟 < ∘ , where 𝑟 is the angular distance from the GC. This ROI is shown in thecontext of the full data, shown with a lower opacity, for two different energy ranges:0.6-1.6 GeV (left) and 20-63 GeV (right). In both cases the data has been smoothedto ∘ , and all 3FGL point sources within our ROI have been masked at their 95%containment radius. These are shown in blue, and are much larger on the left thanthe right as the Fermi
PSF increases with decreasing energy. In our lowest energybin (not shown), the point source mask covers most of our ROI. In both figures, redshades indicate increased photon counts, while in the left (right) orange (blue) shadesillustrate regions of low photon counts. 236M decay flux, and 𝜆 models the Galactic diffuse emission, Fermi bubbles, isotropicflux, and emission from PSs. Recall that each of the nuisance parameters is given aseparate degree of freedom in the northern and southern Galactic hemispheres.In terms of these model parameters, we can then build up a likelihood functionin terms of the binned data. In doing so, we treat each energy bin independently, sothat in the 𝑖 th bin we have: 𝑝 𝑖 (︀ 𝑑 𝑖 ⃒⃒ 𝜃 𝑖 )︀ = ∏︁ 𝑝 𝜇 𝑝𝑖 ( 𝜃 𝑖 ) 𝑛 𝑝𝑖 𝑒 − 𝜇 𝑝𝑖 ( 𝜃 𝑖 ) 𝑛 𝑝𝑖 ! , (C.2)where 𝜇 𝑝𝑖 ( 𝜃 𝑖 ) is the mean predicted number of photon counts in that pixel as a func-tion of the model parameters 𝜃 𝑖 = { 𝜓 𝑖 , 𝜆 𝑖 } . The 𝜇 𝑝𝑖 ( 𝜃 𝑖 ) are calculated from the setof templates used in the fit, which describe the spatial distribution of the variouscontributions described above. More specifically, if the 𝑗 th template in energy bin 𝑖 predicts 𝑇 𝑗,𝑝𝑖 counts in the pixel 𝑝 , then 𝜇 𝑝𝑖 ( 𝜃 𝑖 ) = ∑︀ 𝑗 𝐴 𝑗𝑖 ( 𝜃 𝑖 ) 𝑇 𝑗,𝑝𝑖 , where 𝐴 𝑗𝑖 ( 𝜃 𝑖 ) isthe normalization of the 𝑗 th template as a function of the model parameters. In ouranalysis, all of the normalization functions are linear in the model parameters, and inparticular there is a model parameter that simply rescales the normalization of eachtemplate in each energy bin.The likelihood profile in the single energy bin, as a function of the parameters ofinterest 𝜓 𝑖 , is then given by maximizing the log likelihood over the nuisance parame-ters 𝜆 𝑖 : log 𝑝 𝑖 (︀ 𝑑 𝑖 ⃒⃒ 𝜓 𝑖 )︀ = max 𝜆 𝑖 log 𝑝 𝑖 (︀ 𝑑 𝑖 ⃒⃒ 𝜃 𝑖 )︀ . (C.3)This choice to remove the nuisance parameters by taking their maximum is whatdefines the profile-likelihood method. After doing so we have reduced the likelihoodto a function of just the DM parameters, which are equivalent to the isotropic andLOS integrated NFW correlated flux coming from DM decay. As such, we can write log 𝑝 𝑖 (︀ 𝑑 𝑖 ⃒⃒ 𝜓 𝑖 )︀ = log 𝑝 𝑖 (︁ 𝑑 𝑖 ⃒⃒⃒{︁ 𝐼 𝑖 iso , 𝐼 𝑖 NFW }︁)︁ . (C.4)237or a given DM decay model, ℳ , there will be a certain set of values for { 𝐼 𝑖 iso , 𝐼 𝑖 NFW } in each energy bin. Given these, the likelihood associated with that model is givenby: log 𝑝 (︀ 𝑑 ⃒⃒ ℳ , { 𝜏, 𝑚 𝜒 } )︀ = ∑︁ 𝑖 =0 log 𝑝 𝑖 (︁ 𝑑 𝑖 ⃒⃒⃒{︁ 𝐼 𝑖 iso , 𝐼 𝑖 NFW }︁)︁ , (C.5)where we have made explicit the fact that in most models the lifetime 𝜏 and mass 𝑚 𝜒 are free parameters. We then define the test statistic (TS) used to constrain themodel ℳ by TS (︀ ℳ , { 𝜏, 𝑚 𝜒 } )︀ = 2 × [︁ log 𝑝 (︀ 𝑑 ⃒⃒ ℳ , { 𝜏, 𝑚 𝜒 } )︀ − log 𝑝 (︀ 𝑑 ⃒⃒ ℳ , { 𝜏 = ∞ , 𝑚 𝜒 } )︀]︁ . (C.6)Note that fundamentally it is the list of values { 𝐼 𝑖 iso , 𝐼 𝑖 NFW } that determine the TS.This means we can build a 2-d table of TS values in each energy bin as a functionof the extragalactic and Galactic DM flux. This table only needs to be computedonce; afterwards a given model can be mapped onto a set of flux values, which hasan associated TS in the tables. This is the approach we have followed, and we showthese DM flux versus TS functions in Sec. C.2. The table of TS values is also availableas Supplementary Data [245]. C.2 Likelihood Profiles
As described in chapter 4, our limits on specific DM final states and models areobtained from 2-d likelihood profiles, where the two dimensions encompass LOS inte-grated NFW correlated Galactic gamma-ray flux and extragalactic gamma-ray flux.In Figs. C-5 and C-6 we show slices of these log-likelihood profiles when the extragalac-tic DM-induced flux is set to zero. The bands indicate the 68% and 95% confidenceintervals for the expected profiles obtained from background-only MC simulations. Note that this TS stands in contrast to that used in [213]; in that work, the TS was similarlydefined, except that instead of using 𝜏 = ∞ as a reference the 𝜏 of maximal likelihood was used.The definition of TS used here is more conservative than that in [213], though formally, with Wilk’stheorem in mind, our limits do not have the interpretation of 95% constraints. - - ���� - ���� ��� ��������� ����� �� % ����� ����� �� % ���� - ���� ��� ���� - ���� ��� ���� - ���� ��� - - - ���� - ���� ��� ���� - ���� ��� ���� - ���� ��� ���� - ���� ��� - - - ���� - ���� ��� ���� - ���� ��� ���� - ���� ��� ���� - ���� ��� - - - ���� - ���� ��� ���� - ���� ��� ���� - ���� ��� ���� - ���� ��� - - - - - - - - - ���� - ���� ��� - - - - - ���� - ���� ��� - - - - - ���� - ���� ��� - - - - - - ���� - ���� ��� T e s t S t a t i s t i c ( T S ) � � �� ��� / �� [ ��� / �� � / � / �� ] Figure C-5: The change in log-likelihood, TS ≡ 𝑝 𝑖 ( 𝑑 𝑖 |{ 𝐼 𝑖 NFW } ) − 𝑝 𝑖 ( 𝑑 𝑖 |{ 𝐼 𝑖 NFW = 0 } ) , asa function of the intensity 𝐼 𝑖 NFW of NFW-correlated emission in the first 20 energy bins.The measurement is given by the dashed red line, and the 68% and 95% confidenceregions as derived from MC are given by the purple and pink bands respectively. Inmost energy bins, the likelihood curves from the analysis of the data is seen to agree,within statistical uncertainties, with the expectation from the background templatesonly, as indicated by the MC bands.The simulations use the set of background (“nuisance”) templates normalized to thebest-fit values obtained from a template analysis of the data in the given energy bin.In most energy bins, the results obtained on the real data are consistent with theMC expectations, showing that – for the most part – we are in a statistics-dominated239 - - ����� - ����� ��� ��������� ����� �� % ����� ����� �� % ����� - ����� ��� ����� - ����� ��� ����� - ����� ��� - - - ����� - ����� ��� ����� - ����� ��� ����� - ����� ��� ����� - ����� ��� - - - ����� - ����� ��� ����� - ����� ��� ����� - ����� ��� ����� - ����� ��� - - - ����� - ����� ��� ����� - ����� ��� ����� - ����� ��� ����� - ����� ��� - - - - - - - - - ����� - ���� ��� - - - - - ���� - ���� ��� - - - - - ���� - ���� ��� - - - - - - ���� - ���� ��� T e s t S t a t i s t i c ( T S ) � � �� ��� / �� [ ��� / �� � / � / �� ] Figure C-6: As in C-5, except for the later 20 energy bins.regime. In some energy bins, such as that from . – . GeV, the data shows a smallexcess in the TS compared to the MC expectation. While such an excess is perhapsnot surprising since we are looking at multiple independent energy bins, it could alsoarise from a systematic discrepancy between the background templates and the realdata. More of a concern are energy bins where the limits set from the real data aremore constraining than the MC expectation, such as the energy bin from . – . GeV. It is possible that this discrepancy, in part, arises from an over-subtraction ofdiffuse emission in certain regions of sky since the diffuse template is not a perfect240 - - ���� - ���� ��� ��������� ����� �� % ����� ����� �� % ���� - ���� ��� ���� - ���� ��� ���� - ���� ��� - - - - - - - - - ���� - ���� ��� - - - - - ���� - ���� ��� - - - - - ��� - ��� ��� - - - - - - ��� - ��� ��� T e s t S t a t i s t i c ( T S ) � � �� ��� / �� [ ��� / �� � / � / �� ] Figure C-7: As in C-5, except for a selection of energy bins for the extragalactic onlyflux.match for the real cosmic-ray induced emission in our Galaxy. This possibility – andthe efforts that we have taken to minimize its impact—is discussed further in Sec. C.3.In Fig. C-7, we show a selection of the log-likelihood profiles found for vanishingGalactic DM-induced gamma-ray flux and shown instead as functions of the extra-galactic DM-induced flux. It is important to remember that in the template fit wemarginalize over isotropic emission. As a result, it is impossible with our method tofind a positive change in the TS as we increase the DM-induced isotropic flux 𝐼 iso . Inwords, we remain completely agnostic towards the origin of the IGRB in our analysis.That is, we do not assume the IGRB is due to standard astrophysical emission butwe also do not assume it is due to DM decay. The 1-d likelihood profiles as functionsof 𝐼 iso instead show the limits obtained for the isotropic flux coming simply from therequirement that they do not overproduce the observed data.In some energy bins, particularly at high energies (such as the energy bin from632-796 GeV in Fig. C-7), the data is seen to be more constraining than the MCexpectation. However, we stress that the isotropic flux is not well determined, espe-cially at these high energies, in our small region. With that said, the isotropic fluxdetermined in this small region tends to be larger than the IGRB determined froma dedicated analysis at high latitudes (see Fig. C-3). As a result, our limits on the241xtragalactic flux are likely conservative.The full 2-d likelihood profiles are available as Supplementary Data [245]. Theseare given as a function of the average Galactic and extragalactic DM flux in our ROI,without including any point source mask. The absence of the point source mask ischosen to simplify the use of our flux-TS tables. C.3 Systematics Tests
We have performed a variety of systematic tests to understand the robustness of ouranalysis. Figure C-8 summarizes the results of some of the more important tests.In Fig. C-8, we show limits on the 𝑏 ¯ 𝑏 final state with a variety of different variationson the analysis method. Certain variations are shown to cause very little difference,such as not including an extra Fermi bubbles template, taking 𝐵 = 0 . 𝜇 G whencomputing the IC flux, and using the more up-to-date Pass 8 model gll_iem_v06 ( p8r2 ) diffuse model instead of the p7v6 model. As the p8r2 model identifies regionsof extended excess emission in the data and adds these back to the model, it is unclearif such a model would absorb a potential DM signal. Due to this concern, we usedthe p7v6 model as our default in the main analyses.Assuming 𝐵 = 5 . 𝜇 G when computing the IC flux leads to slightly weaker con-straints at higher masses due to the decrease in the IC contribution, as would beexpected. However, we emphasize that Faraday rotation measurements suggest that 𝐵 ≤ . 𝜇 G across most of our ROI [238], so . 𝜇 G is likely overly conservative.We also note that the limit 𝐵 → . 𝜇 G must be taken with care. Without anymagnetic field, the energy loss rate of high energy electrons and positrons from ICemission alone is not sufficient to keep the leptons confined to the halo. However,even taking 𝐵 ∼ . − nG, which is a typical value quoted for intergalactic magneticfields, the Larmor radius ∼ . 𝐸 𝑒 / TeV )(1 nG /𝐵 ) kpc, with 𝐸 𝑒 the lepton energy,is sufficiently small to confine the 𝑒 ± in our ROI. Larger values of circumgalacticmagnetic fields in the halo are more likely.An additional systematic is the assumption of the DM profile, as direct obser-242 m χ [GeV]10 τ [ s ] DM → b ¯ b IceCube 3 σ (HESE)IceCube 3 σ (MESE) Fermi (default)Burkert
E <
100 GeV
E > | b | > ◦ r < ◦ Top 300 PS mask B = 0 . µ G B = 5 . µ GNo Bubblesp8R2North = SouthAll Quartiles m χ [GeV]10 τ [ s ] DM → b ¯ b IceCube 3 σ (HESE)IceCube 3 σ (MESE) Fermi (default)Rotate ROI
Figure C-8: Left: The limit derived for DM decay to 𝑏 ¯ 𝑏 for ten systematic variationson our analysis, as compared to our default analysis. Right: A purely data drivensystematic cross check, where we have moved the position of our default ROI to fivenon-overlapping locations around the Galactic plane ( 𝑏 = 0 ) and show the band ofthe limits derived from these regions is consistent with what we found for an ROIcentered at the GC. See text for details.vations do not sufficiently constrain the profile over our ROI and we must rely onmodels. In this work, we have assumed the NFW profile. Another well-motivatedprofile is the Burkert profile [411], which is similar to the NFW at large distances buthas an inner core that results in less DM towards the center of the Galaxy. In Fig. C-8we show the limit we obtain using the Burkert profile with scale radius 𝑟 = 13 . kpc. From this analysis we conclude that the systematic uncertainty from the DMprofile is less significant than other sources of uncertainty associated with the dataanalysis.Masking the top 300 brightest and most variable PSs across the full sky, insteadof masking all PSs, and masking the Galactic plane at | 𝑏 | > ∘ , instead of ∘ , bothlead to stronger constraints at low energies. This is not surprising considering thatthe PS mask at low energies significantly reduces the ROI, and so any increase to thesize of the ROI helps strengthen the limit. Going out to distances within ∘ of theGC, on the other hand, slightly strengthens the limit at low masses, gives a similarlimit at high masses, but weakens the limit at intermediate masses. This is due tothe fact that the diffuse templates often provide poor fits to the data when fit overtoo large of regions. As a result, it becomes more probable that the added NFW-243orrelated template can provide an improved fit to the data, which is the case at afew intermediate energies. This is also the reason why the limit is found to be slightlyworse when the templates are not floated separately in the North and South, butrather floated together across the entire ROI (North=South in Fig. C-8). As a result,we find that the addition of the NFW-correlated template often slightly improves theoverall fit to the data in this case. Since it is hard to imagine a scenario where a DMsignal would show up in the North=South fit and not in the fit where the North andSouth are floated independently, and since the latter analysis provides a better fitto the data, we float the templates independently above and below the plane in ourmain analysis. Reassuringly, most of the systematics do not have significant effectsat high masses, where we are generally in the statistics dominated regime.Many of the variations discussed above are associated with minimizing the impactof over-subtraction as discussed in chapter 4. Fundamentally, we do not possess abackground model that describes the gamma-ray sky to the level of Poisson noise, andthe choice of ROI can exacerbate the issues associated with having a poor backgroundmodel. To determine our default ROI we considered a large number of possibilitiesand chose the one where we had the best agreement between data and MC, whichultimately led us to the relatively small ROI shown in Fig. C-4 used for our defaultanalysis. We emphasize that we did not choose the ROI where we obtained thestrongest limits, as is clear from Fig. C-8, and as such we do not need to impose atrials factor from considering many different limits.A further important systematic is our choice of data set. In our main analysis, weused the top quartile of events, as ranked by the PSF, from the UltracleanVeto class.Roughly four times as much data is available, within the same event class, if we takeall photons regardless of their PSF ranking. Naively using all of the available datawould strengthen our bound. However, as we show in Fig. C-8, this is not the case—infact, the limit we obtain using all photons is weaker than the limit we obtain usingthe top quartile of events. There are two reasons for this, both of which are illustratedin Fig. C-9. The first reason is simply that since we mask PSs at 95% containment,as determined by the PSF, there is less area in our ROI in the analysis that uses all244 E [ GeV ] N γ ������� �� ��� �� ������� ��� - �������� ������� - �������� ����� �������� - ��������� ������� - ��������� ����� ����� E [ GeV ] R O I A r ea [ s r ] ���� ��� �� ������� ��� - �������� ������� - ��������� ������ ����� ������ ���� - E [ GeV ] - LL ������� �� ��� �� ������� ��� - �������� ������� - �������� ����� �������� - ��������� ������� - ��������� ����� ����� - - - - - - - - - - E d Φ / dE [ GeV / cm / s / sr ] T S ����� - ����� ��� ��������� ����� �� % ����� ����� �� % Figure C-9: Top left: the number of photons in our ROI as a function of energy for thetop-quartile and all-quartiles. We show both the result in data and MC, where for theMC we indicate the 68% and 95% confidence intervals constructed from multiple MCrealizations in each energy bin. Top right: the size of the ROI, in sr, as a functionof energy. The variation with energy is due to the changing size of the PS mask.Bottom left: As in the top left plot, but here we show the quality of fit (the negativeof the log-likelihood) as a function of energy. Bottom right: At intermediate energiesthere are residuals in the all-quartile data that can be absorbed by our Galactic DMtemplate, leading to large excesses such as the one shown here. Such excesses playa role in the all-quartile limit being weaker than the top-quartile limit, as shown inFig. C-8. However, the all-quartile limits are also weaker in part due to the reducedROI at low energies.quartiles of events relative to the analysis using the top quartile. Indeed, in Fig. C-9we show the number of counts 𝑁 𝛾 in the different energy bins in our ROI for the top-quartile and all-quartile analyses. At high energies, the top-quartile analysis has fewerphotons than the all-quartile analysis, as would be expected. However, since the PSF245ecomes increasingly broad at low energies, we find that at energies less than around1 GeV, the top-quartile data has a larger 𝑁 𝛾 . Since both the IC and extragalacticemission tends to be quite soft, the data at low energies has an important impact onthe limits. We further emphasize this by showing the size of the ROI as a functionof energy in Fig. C-9.The second difference between the two data sets is that with the top-quartileonly we find that the data is generally consistent with the background models, upto statistical uncertainties, while with the full data set there are systematic differ-ences between the data and background models across almost all energies. This isillustrated in the bottom left panel of Fig. C-9, where we compare the data resultto expectations from MC (68% and 95% statistical confidence intervals) from thebackground templates only for the maximum log-likelihood. There we see that in thetop quartile case the data is consistent with MC up to energies ∼
100 GeV. In theall-quartile case, on the other hand, the data appears to systematically have a largerlog-likelihood than the MC at energies less than around 50 GeV. This difference couldagain be due to the increased PSF in the all quartile case, which smears out smallerrors in background mis-modeling onto larger scales. The addition of a GalacticDM template can then be used to improve this mis-modeling, which can lead to astrong preference for the DM decay flux in isolated energy bins, an example of whichis shown in the bottom right panel of Fig. C-9. Such excesses weaken the limit thatcan be set and ultimately play a central role in the all-quartile limit being weakerthan naively expected.We note that even in the top-quartile case there does appear to be some sys-tematic difference between the MC expectation and the data at energies greater thanaround 100 GeV. In particular, the data appears to generally have fewer photons thanexpected from MC. With that said, this is a low-statistics regime where some energybins have 𝑁 𝛾 = 0 . This difference is also not too surprising, considering that the PSmodel and diffuse model were calibrated at lower energies and simply extrapolated tosuch high energies. Part of this difference could also be due to cosmic ray contamina-tion. Thus, systematic discrepancies between data and MC at energies greater than246round 100 GeV should be expected. To illustrate the importance of this high energydata on our results, we show in Fig. C-8 the limit obtained when only including datawith photon energies less than GeV; at 10 PeV, the limit is around 5 times weakerwithout the high-energy data. We also show in that plot the impact of removing thedata below GeV, which has a large impact at lower masses but a minimal impactat higher masses.In addition to the numerous variations of our modeling discussed above, we havealso performed a purely data driven systematic cross check on our analysis shownon the right of Fig. C-8, similar to that used in [102]. In the absence of any DMdecay flux in the
Fermi data, there should be nothing particularly special about theROI near the Inner Galaxy that we have used—shown in Fig. C-4—and we shouldbe able to set similar limits in other regions of the sky. This is exactly what weconfirm in Fig. C-8, where in addition to our default limit we show the band of limitsderived from moving our ROI to five non-overlapping regions around the Galacticplane ( 𝑏 = 0 ). As shown in the figure, even allowing for this data driven variation,the best fit IceCube points always remain in tension with the limit we would derive.As a final note, we emphasize the importance of modeling non-DM contributionsto the gamma-ray data in addition to the spatial morphology of the signal. The limitson the DM lifetime would be weaker if we used a more simplistic analysis that didnot incorporate background modeling and spatial dependence into the likelihood. Forexample, we may set a conservative limit on the DM lifetime by using a likelihoodfunction log 𝑝 𝑖 ( 𝑑 | 𝜓 ) = ∑︁ 𝑖 max 𝜆 𝑖 ⎡⎢⎣ − (︁∑︀ 𝑝 𝜇 𝑝𝑖 ( 𝜓, 𝜆 𝑖 ) − ∑︀ 𝑝 𝑛 𝑝𝑖 )︁ ∑︀ 𝑝 𝜇 𝑝𝑖 ( 𝜓, 𝜆 𝑖 ) −
12 log (︃ 𝜋 ∑︁ 𝑝 𝜇 𝑝𝑖 ( 𝜓, 𝜆 𝑖 ) )︃⎤⎥⎦ . (C.7)The likelihood function depends on ∑︀ 𝑝 𝜇 𝑝𝑖 ( 𝜓, 𝜆 𝑖 ) ≡ ∑︀ 𝑝 𝜇 𝑝𝑖 ( 𝜓 ) + 𝜆 𝑖 , which is a functionof the DM model parameters 𝜓 . The 𝜆 𝑖 are nuisance parameters that allow us toadd an arbitrary (positive) amount of emission in each energy bin. These nuisance247arameters account for the fact that we are assuming no knowledge of the mechanismsthat would yield the gamma-rays recorded in this data set—the data may arise fromDM decay or from something else. As a corollary to this point, we may only determinelimits with this likelihood function; by construction, we cannot find evidence fordecaying DM. Using (C.7) within our ROI, we estimate a limit 𝜏 ≈ × s forDM decay to 𝑏 ¯ 𝑏 with 𝑚 𝜒 = 1 PeV. This should be contrasted with the limit 𝜏 ≈ × s that we obtain with the full likelihood function, as given in Eq. (C.2).This emphasizes the importance of including spatial dependence and backgroundmodeling in the likelihood analysis, as this knowledge increases the limit by aroundan order of magnitude in this example. Even more important is the inclusion of energydependence in (C.7). Were we to modify (C.7) to only use one large energy bin from MeV to TeV, then the limit would drop to ∼ s in this example. However,it is important to emphasize that the DM-induced flux is orders of magnitude largerthan the data at high energies for this lifetime.248 .4 Extended Theory Interpretation In this section, we expand upon the decaying DM interpretation of our results in thecontext of additional final states and also specific simplified models. We begin bygiving limits on a variety of two-body final states. Then, we comment on how wemay use universal scaling relations to extend our result to high DM masses, beyondwhere it is possible to generate the spectra in
Pythia . Finally, we illustrate thispoint by providing two example models. The limits on all final states and modelsconsidered in this work are provided as part of the Supplementary Data [245].
C.4.1 Additional Final States
In addition to DM decays directly to bottom quarks, the benchmark final state usedextensively in this work, we also determine the
Fermi limits on DM decay into anumber of two-body final states. In detail we consider all flavor conserving decaysto charged leptons, neutrinos, quarks, electroweak bosons, and Higgs bosons. Dueto our emphasis on modes that yield high energy neutrinos, we also include threemixed final states, 𝑍𝜈 , 𝑊 ℓ and ℎ𝜈 . For these last three cases we consider an equaladmixture of lepton and neutrino flavors. These limits are all shown in Fig. C-10.Figure C-10 has some interesting features. Channels which produce more electronsand positrons tend to have stronger limits at high masses due to the associatedGalactic IC flux. This is clear for DM decays to 𝑒 + 𝑒 − and also to 𝜈 𝑒 ¯ 𝜈 𝑒 . Most ofthe quark final states lead to nearly identical limits; these channels produce a largenumber of pions regardless of flavor yielding a similar final state spectrum. Theonly difference is for the top quark, which first decays to 𝑏𝑊 , thereby generatinga prompt spectrum which differs from the lighter quarks. Note that for the lighterquarks, the threshold is still always set at 20 GeV; Pythia does not operate belowthis energy since they do not simulate the full spectrum of QCD resonances. We leavethe extension of our results to lower masses for colored final states to future work.In addition to the limits, we also show the best fit point for a fit to the IceCubedata as a star for each channel. The best fit point is always in tension with the limits249 ★★ m χ [ GeV ] τ [ s ] � μτ ★ ★★ m χ [ GeV ] τ [ s ] ��� ★★ ★ m χ [ GeV ] τ [ s ] ��� ★★ ★ m χ [ GeV ] τ [ s ] ν � ν μ ν τ ★★★ m χ [ GeV ] τ [ s ] ��� ★★★ m χ [ GeV ] τ [ s ] � ν ��� ν Figure C-10: Limits on all final states considered in this work. For each final statewe show both the limit on the decay lifetime as a function of the DM mass, and alsothe best fit point for an interpretation of the IceCube flux with this channel as a star.Except for decay directly into neutrinos, for every other final state this best fit pointis in tension with the limit we derive from
Fermi .we derive from
Fermi , except for decays directly into neutrinos. However, as we showin the next subsection, when modeling the DM interactions in a consistent theorycontext, one must rely on a very restricted setup to manifest exclusive decays intoneutrino pairs.The quality of fit for the different stars represented in Fig. C-10 are not identical.This point is highlighted in Fig. C-11 where we show the quality of fit (for DM only)for three two-body final states, 𝑏 , 𝑒 , and 𝜈 𝑒 , as well as two models, glueball andgravitino dark matter. The quality of fit is shown with respect to the best fit powerlaw multiplied by an exponential cutoff, chosen to represent an astrophysical fit to thedata. The astrophysical model always gives the best fit, with the 𝑏 ¯ 𝑏 DM-only modela worse fit to the data by a TS ∼ . . A number of the other final states and modelsalso give a comparable quality of fit to the 𝑏 ¯ 𝑏 final state, as their neutrino spectraare all broad enough to fit the data in a number of energy bins. This is not the casefor 𝜈 𝑒 ¯ 𝜈 𝑒 —the only final state not in tension with our limits from Fermi —where the250 b e + e G l u e b a ll G r a v i t i n o T S Figure C-11: Quality of the best fit to the combined IceCube data for a selection offinal states and models. The quality of the fit is represented as a TS for the DM-onlymodel defined with respect to the best fit power law with an exponential cutoff; thissimple model is meant to represent an astrophysical fit to the data. Among the DM-only models, 𝑏 ¯ 𝑏 provides the best fit to the data, which motivated our choice to focuson it in chapter 4. Other final states and models give a comparative goodness of fit,except for the case of decay directly to neutrinos which gives a sharp spectrum andconsequently a poor fit.sharp neutrino spectrum can at most meaningfully contribute to a single energy bin. C.4.2 Extending
Fermi limits beyond 10 PeV
As discussed above, generating the prompt spectra much above 10 PeV in
Pythia isnot feasible. The issue is already clear in Tab. C.1: as the DM mass is increased, so isthe energy injected into the final state decays which leads to a large number of finalstates resulting from the showering and hadronization processes. At some point thisprocess simply takes too long to generate directly. Nevertheless, this section providesthe details of the spectrum generation for 𝑏 ¯ 𝑏 , and then how these are utilized toextend our Fermi limits up to energies ∼ GeV.The key observation is that when the prompt photon, electron/positron, or neu-trino spectra are considered in terms of 𝑑𝑁/𝑑𝑥 where 𝑥 = 𝐸/𝑚 𝜒 , for 𝑏 ¯ 𝑏 , and likelymany other channels though we have not fully characterized this for all final states,they approach a universal form independent of mass. This is shown for the case ofphotons on the left of Fig. C-12. There we show Pythia generated spectra up to 100251 - - - - - � � � / � � �� → �� ������ γ �������� ��� ����� ���� ������ ����� �� ��� ( �� ) �� ����� - - - - � [ ��� ] � � � Φ / � � [ � � � / � � � / � / � � ] �� → �� τ = �� �� � � ��� ( �� �� ) � ��� ( �� �� ) � ��� ( �� �� ) � ��� ( �� �� ) �� ����� Figure C-12: Left: The prompt photon DM decay 𝑏 ¯ 𝑏 spectrum approaches a universalform in 𝑑𝑁/𝑑𝑥 , where 𝑥 = 𝐸/𝑚 . All spectra except for the one at GeV weredetermined using
Pythia ; the spectrum at the GUT scale is taken from [207] andlabelled as KK. The prompt 𝑒 ± and neutrino spectra also approach universal forms.Taken together this allows us to determine the 𝑏 ¯ 𝑏 spectrum up to masses ∼ GeV. Right: At very high masses the Galactic flux from DM decay expected inthe
Fermi energy range is negligible. Nevertheless due to cascade processes, theextragalactic flux, shown here here for DM with 𝜏 = 10 s, approaches a universalform. This implies Fermi can set an essentially mass independent limit on very heavydark matter, as shown in Fig. C-13.PeV, and compare them to a spectrum at the GUT scale determined in [207]. Thecomputation in [207] takes the fragmentation function for bottom quarks at lower en-ergies, and then runs them to the GUT scale using the DGLAP evolution equations.This universality allows us to determine the prompt spectra for DM → 𝑏 ¯ 𝑏 with 𝑚 𝜒 well above the PeV scale.Given these spectra, the next consideration is whether a meaningful flux fromthese decays populates the Fermi energy range. For prompt and IC flux from theMilky Way the answer is no, as is evident already in Fig. C-2. The synchrotron fluxfrom electrons and positrons is expected in the Fermi range or even higher energies forDM mass of (cid:38) GeV, which can improve the lifetime limits by a factor of 2-3 [212].However, the results depend on halo magnetic fields that are uncertain. Thus, wehere consider conservative constraints without the Galactic synchrotron component.Nevertheless the situation is different for the extragalactic flux, as shown on the rightof Fig. C-12. There we see that the amount of flux approaches a universal form,252 m χ [GeV]10 τ [ s ] DM → b ¯ b EG onlyIC onlyPrompt only
Fermi combinedIceCubeIceCube 3 σ Figure C-13: Using the universal form of the spectra shown in Fig. C-12,
Fermi canset limits on DM decays up to masses well above the PeV scale. At higher massesthis limit comes only from the extragalactic contribution, such that after about GeV, the limit set becomes essentially independent of mass. Note that at these highmasses, the
Fermi limits are noticeably weaker than those obtained by direct searchesfor prompt Galactic photons from the decay of these heavy particles, as determinedin [207]. Note that the labeling is the same as in Fig. 4-2 of chapter 4.essentially independent of the DM mass. The intuition for how this is possible is asfollows. The total DM energy injected in decays is independent of mass: as we increasethe mass of each DM particle, the number density decreases as /𝑚 𝜒 , but at the sametime the power injected per decay increases as 𝑚 𝜒 , keeping the total injected powerconstant. Extragalactic cascades reprocess this power into the universal spectrumshown, and this implies that above a certain mass the extragalactic flux seen in the Fermi energy range becomes a constant.Using this, we extend our limits on the 𝑏 ¯ 𝑏 final state up to the masses ∼ GeVin Fig. C-13. There we see that above ∼ GeV, the limit becomes independentof mass and is coming only from the extragalactic contribution, exactly because ofthe universal form of the extragalactic flux. The same is not true for the neutrinospectrum—there is no significant reprocessing of the Galactic or extragalactic neutrinoflux—and as such the limits IceCube would be able to set decrease with increasingmass. Despite this, limits determined from direct searches for prompt Galactic pho-253ons, which at these high energies are not significantly attenuated, set considerablystronger limits as shown in [207]. Nevertheless, given that
Fermi cannot see photonsmuch above 2 TeV, we find it impressive that the instrument can set limits up tothese masses.We have cut Fig. C-13 off at masses ∼ GeV because at higher energies pro-cesses such as double pair-production may become important (see [423] for a reviewand references therein). The neutrino limits may also be affected by scattering offthe cosmic neutrino background at very high masses. We leave such discussions tofuture work.
C.4.3 Additional Models
In this section, we give limits on two additional DM models of interest beyond theexample of a hidden sector glueball decaying via the operator 𝜆 𝐷 𝐺 𝐷𝜇𝜈 𝐺 𝜇𝜈𝐷 | 𝐻 | / Λ discussed in chapter 4.Gravitino DM whose decay is due to the presence of bi-linear 𝑅 -parity viola-tion (via the super-potential coupling 𝑊 ⊃ 𝐻 𝑢 𝐿 ) is a well studied scenario. If thegravitino, denoted by 𝜓 / , is very heavy, it will decay via the following four chan-nels: 𝜓 / → 𝜈 𝛾, 𝜈 𝑍 , 𝜈 ℎ, ℓ ± 𝑊 ∓ [424, 425]. For 𝑚 / near the weak scale, thebranching ratios are somewhat sensitive to the details of the SUSY breaking masses.However, once 𝑚 / ≫ 𝑣 , the decay pattern quickly asymptotes to for the 𝜈 𝑍 , 𝜈 ℎ, ℓ ± 𝑊 ∓ channels respectively, as expected from the Goldstone equivalencetheorem.In Fig. C-14 we show the constraints on decaying gravitino DM assuming the abovedecay modes, with branching ratios given as functions of mass in the inset, using thebenchmark parameters of [424]. At masses below the electroweak scale, the 𝛾𝜈 finalstate dominates. This channel is best searched for using a gamma-ray line search,which is beyond the scope of this work. For this reason, we only show our constraintsfor masses above 𝑚 𝑊 . Note that the region where decaying DM could provide a ∼ 𝜎 improvement over the null hypothesis for the IceCube, the ultra-high-energyneutrino flux (green hashed region) is almost completely excluded by our gamma-ray254 m χ [GeV]10 τ [ s ] Gravitino DM
EG onlyIC onlyPrompt only
Fermi combinedIceCubeIceCube 3 σ m χ [GeV]0 . . . B r a n c h i n g r a t i o s W‘Zνhν m χ [GeV]10 τ [ s ] χ ( LH ) / Λ EG onlyIC onlyPrompt only
Fermi combinedIceCubeIceCube 3 σ m χ [GeV]0 . . . B r a n c h i n g r a t i o s Figure C-14: Constraints on decaying gravitino DM (left) and DM decaying via theoperator 𝜒 ( 𝐿 𝐻 ) (right). Notation and labeling is as in Fig. 4-2 in chapter 4.constraints. The IceCube constraints, determined using the same methods discussedin chapter 4, begin to dominate at scales above ∼ TeV.In Fig. C-14, we also show limits obtained on the lifetime of the DM 𝜒 underthe assumption that 𝜒 interacts with the SM through the operator in Eq. ( ?? )—thismodel was discussed in detail in the previous subsection, see Tab. ?? . The inset plotshows the branching ratios as a function of energy, and illustrates the transition fromtwo- to three- to four-body decays dominating as the mass is increased. In this caseas well, almost the entire range of parameter space relevant for IceCube is disfavoredby our gamma-ray limits.We use FeynRules
MadGraph5_aMC@NLO [427,428] to compute the parton-level decay interfacedwith
Pythia for the showering/hadronization of the decays 𝜒 → 𝜈 𝜈 ℎ ℎ , 𝜈 𝜈 𝑍 ℎ , 𝜈 𝜈 𝑍 𝑍 , 𝜈 𝑒 − ℎ 𝑊 + , 𝜈 𝑒 − 𝑍 𝑊 + , 𝑒 − 𝑒 − 𝑊 + 𝑊 + and 𝜒 → 𝜈 𝜈 ℎ , 𝜈 𝜈 𝑍 , 𝜈 𝑒 − 𝑊 + .25556 ppendix DCascade Spectra and the GCE D.1 0-step Spectra
In order to calculate the photon spectrum, it is more straightforward to first determinethe density of states according to: annihilations : 1 𝑁 𝛾 𝑑𝑁 𝛾 𝑑𝐸 𝛾 = 1 ⟨ 𝜎𝑣 ⟩ 𝑑 ⟨ 𝜎𝑣 ⟩ 𝑑𝐸 𝛾 decays : 1 𝑁 𝛾 𝑑𝑁 𝛾 𝑑𝐸 𝛾 = 1Γ 𝑑 Γ 𝑑𝐸 𝛾 (D.1)from which the spectrum can be easily backed out. Note that as pointed out in [429],the case of decays as a zeroth step in a cascade of 𝜒 → 𝜑 𝑛 𝜑 𝑛 will give an identicalphoton spectrum to the annihilation scenario for a DM particle with twice the massas for annihilations. This is the sense in which our results are readily transferredto the case of decaying DM. The key difference for the decaying case is the spatialmorphology of the signal will generically require a line of sight integral over theDM density, rather than density squared as appears in the 𝐽 -factor in Eq. 5.3. Theobserved spatial morphology of the GCE appears to disfavour decaying scenarios,which is why we do not mention them further here, although see [430] for a noveldecay scenario that is distributed like density squared.The result of Eq. D.1 is that in some circumstances it is possible to calculatevarious step cascades analytically. This approach is shown for several cases in [429].257et in many cases - most notably those involving hadronic processes in their finalstates - analytic calculations are not feasible. For the present analysis we used acombination of analytic and numeric results depending on the final state employed.The details for each case is outlined below. D.1.1 Annihilations to 𝑒 + 𝑒 − The only contribution to the photon spectrum arises from FSR via the decay 𝜑 → 𝑒 + 𝑒 − 𝛾 . The spectrum in this case can be calculated analytically using Eq. D.1, whichwas done in [313] for the generic case of 𝜑 → 𝑓 + 𝑓 − 𝛾 . As pointed out there, whenusing the simple convolution formula Eq. 6.3, consistency requires throwing awayterms 𝒪 ( 𝜖 𝑓 ) and higher, where 𝜖 𝑓 = 2 𝑚 𝑓 /𝑚 . Doing so they obtained the followingexpression for the spectrum that we include for completeness: 𝑑𝑁 FSR 𝛾 𝑑𝑥 = 𝛼 EM 𝜋 − 𝑥 ) 𝑥 [︃ ln (︃ − 𝑥 ) 𝜖 𝑓 )︃ − ]︃ . (D.2)Note the ln term will dominate for small 𝜖 𝑓 , and the − is simply included to en-sure consistency with the large hierarchies approximation. We confirmed that thisspectrum is in agreement with the output from Pythia8 in the case of final stateelectrons. From here, by repeated use of the convolution formula it is possible toobtain completely analytic formula for the 𝑛 -step cascade, which were used in ourfits. For example, the first two steps are shown in [313]. D.1.2 Annihilations to 𝜇 + 𝜇 − For final state muons, in addition to FSR, as pointed out in [313] the radiative decayof the muon 𝜇 → 𝑒 ¯ 𝜈 𝑒 𝜈 𝜇 𝛾 will meaningfully contribute to the photon spectrum. Thisdecay was calculated in [345], and again for completeness we include it here as it waspresented in [313]: 𝑑𝑁 𝜇 → 𝛾 𝑑𝑥 − = 𝛼 EM 𝜋 𝑥 − (︂ 𝑇 − ( 𝑥 − ) ln 1 𝑟 + 𝑈 − ( 𝑥 − ) )︂ , (D.3)258here 𝑟 = 𝑚 𝑒 /𝑚 𝜇 and 𝑇 − ( 𝑥 ) =(1 − 𝑥 )(3 − 𝑥 + 4 𝑥 − 𝑥 ) 𝑈 − ( 𝑥 ) =(1 − 𝑥 ) (︂ −
172 + 236 𝑥 − 𝑥 + 5512 𝑥 +(3 − 𝑥 + 4 𝑥 − 𝑥 ) ln(1 − 𝑥 ) )︀ (D.4)Note the subscript − here is used to remind us this is the spectrum calculated inthe rest frame of the muon. To then obtain the 0-step cascade we would have toapply Eq. 6.3 once, assuming 𝜖 𝜇 = 2 𝑚 𝜇 /𝑚 ≪ , and then combine this with theFSR spectrum in Eq. D.2. D.1.3 Annihilations to 𝜏 + 𝜏 − For the case of final state taus, FSR will now be a subdominant contribution. Insteadthe spectrum will have a much larger contribution from leptonic and semi-leptonictau decays: 𝜏 − → 𝜈 𝜏 𝑙 − ¯ 𝜈 𝑙 and 𝜈 𝜏 𝑑 ¯ 𝑢 . The quarks will then hadronize (dominantly topions) which will result in large contributions to the photon spectrum. We simulatedthis final state in Pythia8 to generate an initial spectrum, to which we could thenapply the convolution formula.
D.1.4 Annihilations to 𝑏 ¯ 𝑏 Much like for taus, in the case of final state 𝑏 -quarks FSR is a subdominant contribu-tion, and instead the spectrum is largely determined by hadronic processes. As suchwe again utilize Pythia8 to obtain the initial spectrum.
D.2 Kinematics of a Multi-step Cascade
As already emphasized the utility of the small 𝜖 𝑖 = 2 𝑚 𝑖 /𝑚 𝑖 +1 - or large hierarchies -approximation is threefold: 259. It simplifies calculations in that we can use Eq. 6.3, rather than the generalformula we display below;2. More importantly it allows us to describe a cascade using just the identity ofthe final state 𝑓 , the value of 𝜖 𝑓 , and the number of steps 𝑛 , in contrast to themany possible parameters of the generic case;3. Despite the simplifications afforded, results in this framework can be used toestimate the results even for general 𝜖 𝑖 , as described in Sec. 5.4.In this appendix we show how the kinematics of scalar cascade decays lead to anexpression for the 𝑛 -step spectrum in terms of the ( 𝑛 − -step result. In addition weoutline how Eq. 6.3 emerges in the small 𝜖 limit, with error 𝒪 ( 𝜖 𝑖 ) , as well as how thetransition to the degenerate case as 𝜖 → occurs.Our starting point is the 0-step spectrum 𝑑𝑁 𝛾 /𝑑𝑥 where 𝑥 = 2 𝐸 /𝑚 and 𝐸 isthe photon energy in the rest frame of 𝜑 . This results from the process 𝜑 → 𝛾𝑋 ,where the identity of 𝑋 depends on the final state considered. From here we wantto calculate 𝑑𝑁 𝛾 /𝑑𝑥 - the spectrum from a cascade that includes 𝜑 → 𝜑 𝜑 andso is one step longer - where 𝑥 = 2 𝐸 /𝑚 and 𝐸 is the photon energy in the 𝜑 rest frame. If we assume isotropic scalar decays, then we can obtain this by simplyintegrating the 0-step result over all allowed energies and emission angles: 𝑑𝑁 𝛾 𝑑𝑥 =2 ∫︁ − 𝑑 cos 𝜃 ∫︁ 𝑑𝑥 𝑑𝑁 𝛾 𝑑𝑥 𝛿 (︂ 𝑥 − 𝑥 − cos 𝜃𝑥 √︁ − 𝜖 )︂ , (D.5)where 𝜃 is defined as the angle between the photon momentum and the 𝜑 boostaxis as it is measured in the 𝜑 rest frame. The limits of integration ≤ 𝑥 ≤ reflect the fact that the photon energy in the 𝜑 rest frame can be arbitrarily soft onthe one side, and on the other it can have an energy at most half the mass of theinitial particle, 𝑚 / here. The 𝛿 function is simply enforcing how the photon energychanges when we move from the 𝜑 to the 𝜑 rest frame, i.e. from 𝐸 to 𝐸 . It also260ets the kinematic range for 𝑥 , which is: ≤ 𝑥 ≤ (︂ √︁ − 𝜖 )︂ . (D.6)Now if we then use the 𝛿 function to perform the angular integral, the one stepspectrum reduces to: 𝑑𝑁 𝛾 𝑑𝑥 = 2 ∫︁ 𝑡 , max 𝑡 , min 𝑑𝑥 𝑥 √︀ − 𝜖 𝑑𝑁 𝛾 𝑑𝑥 , (D.7)where we have introduced: 𝑡 , max = min [︂ , 𝑥 𝜖 (︂ √︁ − 𝜖 )︂]︂ 𝑡 , min = 2 𝑥 𝜖 (︂ − √︁ − 𝜖 )︂ (D.8)The maximum here is either set by the maximum physical value of 𝑥 , which is , oralternatively by where the 𝛿 function loses support. We can then repeat this processto recursively obtain the 𝑖 th order spectrum from the ( 𝑖 − th order result. Explicitlywe find: 𝑑𝑁 𝛾 𝑑𝑥 𝑖 = 2 ∫︁ 𝑡 𝑖, max 𝑡 𝑖, min 𝑑𝑥 𝑖 − 𝑥 𝑖 − √︀ − 𝜖 𝑖 𝑑𝑁 𝛾 𝑑𝑥 𝑖 − , (D.9)where we have defined: 𝑡 𝑖, max = min [︃ 𝑖 − 𝑖 − ∏︁ 𝑘 =1 (︂ √︁ − 𝜖 𝑘 )︂ , 𝑥 𝑖 𝜖 𝑖 (︂ √︁ − 𝜖 𝑖 )︂]︂ 𝑡 𝑖, min = 2 𝑥 𝑖 𝜖 𝑖 (︂ − √︁ − 𝜖 𝑖 )︂ (D.10)and now the kinematic range of 𝑥 𝑖 is ≤ 𝑥 𝑖 ≤ 𝑖 𝑖 ∏︁ 𝑘 =1 (︂ √︁ − 𝜖 𝑘 )︂ . (D.11)With the exact result of Eq. D.9, we can now see that in the small 𝜖 limit the result261educes to Eq. 6.3 with corrections at most of order 𝜖 , as claimed. The exact resultalso captures an additional feature that the large hierarchies result does not: theemergence of a degenerate step in the cascade as 𝜖 𝑖 → for some 𝑖 . As discussed inSec. 5.4, when this occurs, just from the kinematics we can see that the ( 𝑖 + 1) -stepresult will reduce to the 𝑖 -step spectrum, but shifted in energy and normalisation.Starting with Eq. D.9, setting − 𝜖 𝑖 ≡ 𝑧 and then taking 𝑧 → it is straightforwardto confirm that the exact result also reproduces this behaviour.As discussed in Sec. 5.4, there should be a smooth interpolation between the twoextreme cases of 𝜖 𝑖 = 0 and 𝜖 𝑖 = 1 , and using Eq. D.9 we can demonstrate that indeedthere is. This is shown in Fig. 5-10, where we take the case of a 1-step cascade for finalstate taus with 𝜖 𝜏 = 0 . . We plot the two extreme cases and show how intermediate 𝜖 transition between these by plotting five values: . , . , . , . and . . Note thatas claimed earlier, the transition is roughly quadratic in 𝜖 ; for small and intermediatevalues of 𝜖 , the result is well approximated by the 𝜖 = 0 result, again highlighting theutility of the large hierarchies approximation. D.3 Model-Building Considerations
D.3.1 A Simple Model
Let us extend the usual Higgs Portal [323,324] model to include a rich dark sector with 𝑛 scalar mediators and a set of 𝑛 Z symmetries. This will serve as an illustrativeexample of how different observable signatures depend on different model parameters,as discussed in the main text.Consider the potential: 𝑉 ( 𝜒, 𝜑 , 𝐻 ) = 𝑉 𝜒 + 𝑉 𝐻 + 𝑐 𝑘 𝜑 | 𝐻 | + 𝑛 ∑︁ 𝑖 =1 (︂ 𝜆 ,𝑖 𝜒 𝜑 𝑖 − 𝑚 𝑖 𝜑 𝑖 )︂ + 𝑛 ∑︁ 𝑖,𝑗 =1 𝜆 𝑖𝑗 𝜑 𝑖 𝜑 𝑗 , (D.12) A more complex symmetry structure could allow off-diagonal couplings between the scalarsand the Higgs, with potentially rich observational signatures. We thank Jessie Shelton for thisobservation. 𝑉 𝜒 and 𝑉 𝐻 contain the usual mass and quartic terms for the DM and Higgs fields.As discussed previously it is reasonable that the dark sector is secluded such that thedominant portal coupling is 𝑐 𝑘 𝜑 | 𝐻 | . Upon electroweak and Z symmetry breakingthe 𝜆 ,𝑖 couplings allow annihilations 𝜒𝜒 → 𝜑 𝑖 𝜑 𝑖 . We assume that DM annihilatespreferentially to the heaviest mediator through 𝜆 ,𝑛 𝜒 𝜑 𝑛 . So it is 𝜆 ,𝑛 that dominantlycontrols the thermal annihilation cross-section and therefore the DM relic abundance Ω 𝜒 ℎ ∼ . . The dark sector quartic term will generate interactions of the form 𝜆 𝑖𝑗 ⟨ 𝜑 𝑖 ⟩ 𝜑 𝑖 𝜑 𝑗 , allowing the mediators to cascade decay in the dark sector. Additionallythe Higgs Portal interaction will generate a mixing between 𝜑 and the Higgs. Theend result will be a dark cascade ending in the 𝑐 𝑘 suppressed decay 𝜑 → 𝑓 ¯ 𝑓 , with asubsequent photon spectrum that can be fit to the GCE.While the thermal relic cross-section depends on 𝜆 ,𝑛 , the direct detection cross-section will also depend on the portal coupling 𝑐 𝑘 . This additional small parametergives us the needed freedom to explain the GCE while alleviating constraints fromdirect detection. Additionally we point out that the size of the couplings 𝜆 𝑖𝑗 will needto be large enough such that decays of the new light states occur before BBN. Giventhe number of new free parameters, this setup should not be difficult to construct.Finally we point out that the Higgs Portal interaction also contains a coupling whichleads to the decay ℎ → 𝜑 𝜑 . Invisible Higgs decay is constrained by collider searcheswhich impose an upper bound of about 𝑐 𝑘 (cid:46) − [306]. D.3.2 The Sommerfeld Enhancement
We have seen that the preferred cross-section steadily increases with the number ofsteps in the cascade, moving away from the thermal relic value that is favored for thedirect case. This increased cross-section is also accompanied by an increase in thepreferred mass scale for the DM (indeed, the requirement for a larger cross-sectionis largely driven by the reduced number density of heavier DM). In the presence ofa mediator much lighter than the DM, exchange of such a mediator could enhancethe present-day annihilation cross-section via the Sommerfeld enhancement (e.g. [26,27, 29, 431, 432]), naturally leading to an apparently larger-than-thermal annihilation263ignal.However, there are some obstacles to such an interpretation, at least in the simplecase we have studied where the particles involved in the cascade are all scalars. For thecase of fermionic DM coupled to a light scalar or vector of mass 𝑚 𝜑 with coupling 𝛼 𝐷 ,the Sommerfeld enhancement at low velocity is parametrically given by 𝑚 𝜑 /𝛼 𝐷 𝑚 𝜒 .A large enhancement thus requires 𝛼 𝐷 (cid:38) 𝑚 𝜑 /𝑚 𝜒 . In order to obtain the correctrelic density, we typically require 𝛼 𝐷 to be 𝒪 (0 . , and so a significant Sommerfeldenhancement would require the first step in the cascade to involve a mass gap of twoorders of magnitude. This may be plausible for the electron and even muon channels,but is challenging for final states involving heavier particles such as taus and 𝑏 -quarks;if the mediator is heavy enough to decay to these particles, the required DM massbecomes much too large to fit the GCE even for a one-step cascade, and addingmore hierarchical steps only exacerbates the self-consistency issue (as discussed inSecs. 5.2-5.3).Furthermore, if the DM is a fermion, its annihilation into scalars is generically 𝑝 -wave suppressed, making it difficult to obtain a large enough cross-section to obtainthe GCE. If instead the DM is a heavy (singlet) scalar, the simplest way to coupleit to the light scalar to which it annihilates is an interaction of the form ℒ quartic = 𝜆 𝜒 𝜑 𝑛 . When the light scalar obtains a vacuum expectation value, this gives rise toan interaction of the form 𝜆 ⟨ 𝜑 𝑛 ⟩ 𝜑 𝑛 𝜒 , and repeated exchanges of the light scalar 𝜑 𝑛 can give rise to enhanced annihilation. However, assuming ⟨ 𝜑 𝑛 ⟩ ∼ 𝑚 𝑛 , the size ofthe coupling is suppressed by the small mass of the light scalar, even as its range isenhanced. Accordingly, a large enhancement to annihilation is not expected, at leastin this simple scenario.As discussed in Sec. 5.4, our results can be extended to cascades including parti-cles other than scalars, in which these later issues do not arise; for example, in theaxion portal [312], two-step cascades occur through 𝜒𝜒 → 𝑠𝑎 , 𝑠 → 𝑎𝑎 , 𝑎 → 𝑓 ¯ 𝑓 ,where 𝑠 is a dark scalar and 𝑎 a dark pseudoscalar. This annihilation channel is 𝑠 -wave and can be Sommerfeld-enhanced by exchange of the 𝑠 . However, the firstdifficulty described above may still apply, with the large hierarchy between the 𝜒 and264 potentially implying a DM mass too large to easily fit the GCE.26566 ppendix ELimits on Cascade Spectra E.1 Details of 𝑛 -body Cascades In this appendix we will derive Eq. 6.7 and provide some additional intuition for thiscase as well as pointing out that for a small number of steps, the cascade setup canprovide an excellent approximation (albeit with some dependence on the channel).To set up this problem, firstly recall that the key physics encapsulated in Eq. 6.3is that when we add in a cascade step we need to boost the spectrum to the newrest frame. In the case of 2-body decays this is particularly simple, because we knowexactly how much to boost by. Explicitly, if we have added in a step of the form 𝜑 𝑖 → 𝜑 𝑖 − 𝜑 𝑖 − , then in the 𝜑 𝑖 rest frame we know the 𝜑 𝑖 − particles must be emittedback to back, meaning we know their energy and hence their boost. If instead weintroduce a step via 𝜑 𝑖 → 𝜑 𝑖 − 𝜑 𝑖 − 𝜑 𝑖 − , we no longer know the boost exactly, insteadwe can only associate a probability with any boost which we can determine fromthe energy distribution for a given 𝜑 𝑖 − . Accordingly what we need to calculate isthe energy spectrum of a particular 𝜑 in the decay 𝜒𝜒 → 𝑛 × 𝜑 , and then combinethis with a version of Eq. 6.3 suitable for a general boost. Below we will firstly dothis exactly for the case of a 3-body decay, show what this becomes after applyingthe large hierarchies approximation, and then we will show the general 𝑛 -body resultassuming hierarchical decays.As discussed, our starting point is the energy spectrum of a particular 𝜑 in the267ecay 𝜒𝜒 → × 𝜑 , which can be determined from the three body phase space. Forthis purpose we make use of the analytic formula for the 𝑛 -body phase space outlinedin [433,434]. In the case where our three final state scalars have mass 𝑚 , we can writethe 3-body phase space as: Φ = (4 𝜋 ) ∫︁ ( 𝑀 − 𝑚 ) 𝑚 𝑑𝑀 √︀ 𝜆 ( 𝑀 , 𝑀 , 𝑚 )8 𝑀 × √︀ 𝜆 ( 𝑀 , 𝑚 , 𝑚 )8 𝑀 , (E.1)where 𝜆 ( 𝑥, 𝑦, 𝑧 ) = 𝑥 + 𝑦 + 𝑧 − 𝑥𝑦 − 𝑦𝑧 − 𝑧𝑥 and if we say the mass of thedark matter is 𝑚 𝜒 and the energy of one 𝜑 particle is 𝐸 , then 𝑀 = 4 𝑚 𝜒 and 𝑀 = 4 𝑚 𝜒 + 𝑚 − 𝑚 𝜒 𝐸 . Using this, the energy spectrum of the scalars is simply: 𝑑𝑁 𝜑 𝑑𝐸 ∝ 𝑑 Φ 𝑑𝐸 , (E.2)where the constant of proportionality can be determined by normalising the spectrum.Before proceeding, it is useful to introduce a set of dimensionless variables to workwith as we did in the 2-body case. As there, we firstly define 𝜖 = 𝑚/𝑚 𝜒 , but notehere that 𝜖 ∈ [0 , / , rather than [0 , as in the 2-body case. To play a similar roleto 𝑥 , we also introduce 𝜉 = 𝐸/𝑚 𝜒 ∈ [ 𝜖 , − 𝜖 / , where the limits here are fixedby Eq. E.1 and can also be seen from the kinematics. In terms of these variables, wecan use Eq. E.2 and Eq. E.1 to arrive at: 𝑑𝑁 𝜑 𝑑𝜉 = 𝐶 √︃ ( 𝜉 − 𝜖 )(4 − 𝜖 − 𝜉 )4 + 𝜖 − 𝜉 , (E.3)where 𝐶 is a constant that normalises the spectrum and can be determined numeri-cally. Note that when 𝜖 → / , this distribution approaches a 𝛿 function, as expectedwhen the particles are all produced at rest. We will return to the limit of small 𝜖 shortly.Using this result, we can then revisit the derivation of the boost formula givenin [62], a hierarchical version of which is given in Eq. 6.3, and derive the analogue268or an arbitrary boost. Doing so, if we label the spectrum of the decay of 𝜑 → × (SM final state) as 𝑑𝑁/𝑑𝑥 , we can write the spectrum of the same particle fromthe decay 𝜒𝜒 → × 𝜑 → × (SM final state) as: 𝑑𝑁𝑑𝑥 = 3 ∫︁ − (3 / 𝜖 𝜖 𝑑𝜉𝐶 √︃ ( 𝜉 − 𝜖 )(4 − 𝜖 − 𝜉 )4 + 𝜖 − 𝜉 × ∫︁ 𝑡 max 𝑡 min 𝑑𝑥 𝑥 √︀ 𝜉 − 𝜖 𝑑𝑁𝑑𝑥 , (E.4)where we have defined: 𝑡 max ≡ min [︂ , 𝑥 𝜖 (︂ 𝜉 + √︁ 𝜉 − 𝜖 )︂]︂ 𝑡 min ≡ 𝑥 𝜖 (︂ 𝜉 − √︁ 𝜉 − 𝜖 )︂ (E.5)There are two directions the above result can be generalised. For one, we could extendthis to a longer cascade of 3-body decays, although the logic here is identical to thegeneral 2-body case discussed in [62], so we will not repeat that here. Secondly wecan look to extend this to 4-body decays and higher. The difficulty with this is thatthe 𝑛 -body phase space quickly becomes analytically intractable. Nevertheless asobserved in [334], in the large hierarchies regime ( 𝜖 ≪ ) we regain analytic controlas we will now outline.Returning to Eq. E.3, taking the 𝜖 → limit we find that: 𝑑𝑁 𝜑 𝑑𝜉 = 2 𝜉 + 𝒪 ( 𝜖 ) , (E.6)where now 𝜉 ∈ [0 , . Following [334], this can then be generalised to the 𝑛 -body case,where we find: 𝑑𝑁 𝜑 𝑑𝜉 = ( 𝑛 − 𝑛 − − 𝜉 ) 𝑛 − 𝜉 + 𝒪 ( 𝜖 ) , (E.7)where again 𝜉 ∈ [0 , . Using this we can finally give the equivalent expression of269 x x d N / dx � - ���� �� �� - ���� � = γ �� ���� � - ����� � - ����� - ����� �� - ���� Figure E-1:
Spectra for a cascades contain-ing 𝑛 -step 2-body decay and a 1-step 𝑛 -bodydecay, both to 𝛾𝛾 , are shown as the solid anddashed curves respectively, for the case of 𝑛 = (1,2,3,4) in (purple, blue, green, pink). Wesee that for 𝑛 = 𝑛 = x x d N / dx [ × - ] � - ���� �� �� - ���� � = �� ϵ � = ��� �� ���� � - ����� � - ����� - ����� �� - ���� Figure E-2:
The same as Fig. E-1, but for afinal state 𝑏 ¯ 𝑏 with 𝜖 𝑏 = 0 . . Note that againwe get close agreement in the 𝑛 = Eq. 6.3 for the 𝑛 -body case: 𝑑𝑁𝑑𝑥 = 𝑛 ( 𝑛 − 𝑛 − ∫︁ 𝑑𝜉 (1 − 𝜉 ) 𝑛 − ∫︁ 𝑥 /𝜉 𝑑𝑥 𝑥 𝑑𝑁𝑑𝑥 + 𝒪 ( 𝜖 ) (E.8)thereby demonstrating Eq. 6.7.As a simple example of how this can be used, consider the decay 𝜑 → 𝛾𝛾 whichhas the spectrum 𝑑𝑁 𝛾 /𝑑𝑥 = 2 𝛿 ( 𝑥 − . If we substitute this in, we find the spectrumfor 𝜒𝜒 → 𝑛 × 𝜑 → 𝑛 × 𝛾 is just 𝑑𝑁 𝛾 𝑑𝑥 = 2 𝑛 ( 𝑛 − − 𝑥 ) 𝑛 − . (E.9)Integrating this over 𝑥 ∈ [0 , , we find 𝑁 𝛾 = 2 𝑛 , as expected.To follow on from this, consider the spectrum derived by repeated application ofthe boost formula in Eq. 6.3 to the same 𝜑 → 𝛾𝛾 spectrum, 𝑑𝑁 𝛾 /𝑑𝑥 = 2 𝛿 ( 𝑥 − .Doing so we obtain: 𝑑𝑁 𝛾 𝑑𝑥 𝑛 = ( − 𝑛 +1 ( 𝑛 − 𝑛 − 𝑥 𝑛 . (E.10)Note that if we integrate this over 𝑥 𝑛 ∈ [0 , , we find 𝑁 𝛾 = 2 𝑛 +1 . Now by definition270 - - - - - Log m (cid:1) [ GeV ] L og (cid:1) v [ c m / s ec ] (cid:1)(cid:1) (cid:1)(cid:2)(cid:3)(cid:4)(cid:5) (cid:6)(cid:7)(cid:8)(cid:7)(cid:9)(cid:10) (cid:1)(cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7) (cid:3)(cid:7)(cid:8)(cid:9) (cid:7)(cid:6)(cid:10)(cid:7)(cid:6)(cid:11)(cid:6)(cid:12)(cid:5)(cid:11) (cid:13)(cid:8)(cid:7)(cid:3)(cid:6)(cid:7) (cid:12) (cid:2)(cid:12) (cid:8)(cid:14) - (cid:11)(cid:5)(cid:6)(cid:10) (cid:12) - (cid:15)(cid:16)(cid:17)(cid:9) (cid:17)(cid:6)(cid:18)(cid:8)(cid:9)(cid:19) (cid:20)(cid:16)(cid:7) (cid:12) = (cid:21) - (cid:14)(cid:22) (cid:1)(cid:2)(cid:3)(cid:2) (cid:4) - (cid:5)(cid:6)(cid:7)(cid:8)(cid:9)(cid:10)(cid:11)(cid:12) (cid:13)(cid:11)(cid:14)(cid:15)(cid:16) (cid:1)(cid:2)(cid:3) (cid:1) - (cid:2)(cid:3)(cid:4)(cid:5) (cid:6)(cid:7) (cid:8) - (cid:9)(cid:10)(cid:11)(cid:12) Figure E-3:
Dwarf limits for n-body vs m-step cascades for the 𝛾𝛾 final state. We show themulti-body case for 𝑛 = 𝑛 . In orange, greenand purple we also show the 1, 3 and 5-step 2-body cascade for the same final state. As discussedin the text, for the multi-body case the spectrum sits in between the cascade spectra, and thus weexpect the limits to do the same. The figure makes this clear and emphasizes how the multi-bodyframework is captured within the cascade setup. Eq. E.9 with 𝑛 = 𝑛 =
1, as in this case they both representa 2-body 1-step cascade. Note also though that if we take Eq. E.9 with 𝑛 = 𝑛 =
2, then both situations have the same number of final state photonsfrom different kinematic setups. In Fig. E-1 we compare an 𝑛 -step 2-body decay anda 1-step 𝑛 -body decay for final state photons, for 𝑛 = (1,2,3,4). We see that whilstthey agree for 𝑛 = 𝑛 = 𝑏 ¯ 𝑏 with 𝜖 𝑏 = 0 . . Here we see the agreement isbetter although still beginning to break down for larger 𝑛 . We also tested this forseveral other final states, with the common theme that the spectrum of a 4-body1-step decay is often well approximated by that of a 2-step 2-body decay.Lastly let us confirm the claim from the main text that the results for multi-bodydecays sit in between our multi-step cascade results. For this purpose consider againthe photon spectrum obtained from 𝜑 decaying into 𝑏 ¯ 𝑏 with 𝜖 𝑏 = 0 . . In this casewe plot the 𝑛 -body spectrum for 𝑛 = Log m χ [ GeV ] f e ff ������� ��� � ��� ϵ � = ��� �������� ���� ������ ������� ���������������� ������� Figure E-4: 𝑓 eff for 𝑛 = 𝜖 𝑓 = 0 . . Note that the difference in pat-tern between 𝑓 eff for direct and single stepelectrons and higher step cascades can be un-derstood by recalling that the direct electronFSR spectrum is sharply peaked. Each sub-sequent cascade step smooths out this spec-trum thus changing the shape significantly. - - - - - Log m χ [ GeV ] L og σ v [ c m / s ec ] ������� ��� ������ ϵ � = ��� ���� �������� ������ ������� ���������������� ������� Figure E-5:
The bound on ⟨ 𝜎𝑣 ⟩ for 𝑛 = 𝜖 𝑓 = 0 . . main text. In the figure we also plot the case of a 1-step, 3-step and 5-step 2-bodycascade. Clearly the 𝑛 -body results sit in between these multi step cascade cases,which indicates that the constraints on an 𝑛 -body 1-step cascade will be largelycontained within the limits on a 2-body 𝑛 -step cascade. To demonstrate this, inFig. E-3 we show that for the case of the 𝛾𝛾 limits extracted from the dwarfs, the 𝑛 = E.2 Details of the CMB Results
In this appendix we present additional results from the CMB analysis. For many of thefinal states considered in the main text, the kinematic threshold on their productionmeans that it is not sensible to go to lower masses than we presented. This is not thecase, however, for electrons and muons, where we can take our results to much lowermasses.In Fig. E-4 we present the value of 𝑓 eff for cascades ending in final state electronsand muons with 𝜖 𝑓 = 0 . . Here we consider DM with mass as low 𝒪 (keV) which is272elevant to various CMB studies, which should be compared with our general resultsfor 𝑓 eff in Fig. E-7 for DM annihilations into the eight final states considered in themain text. As expected 𝑓 eff is largest for annihilations to final state electrons.The corresponding bound ⟨ 𝜎𝑣 ⟩ for a given 𝑚 𝜒 for light DM annihilating to finalstate electrons and muons is displayed in Fig E-5, and more generally in Fig 6-5. Asthis bound is fairly insensitive to the final state and number of steps it is interestingto examine the rescaled bound on ⟨ 𝜎𝑣 ⟩ /𝑚 𝜒 which we display in Fig. E-8 for 𝑛 = 𝜖 𝑓 values, and cascade step falls within the narrow range ⟨ 𝜎𝑣 ⟩ /𝑚 𝜒 = 10 − . − − . cm / s / GeV.
E.3 Pass 7 versus Pass 8 for the Dwarfs
As discussed in the main body, the limits displayed in Fig. 6-6 were derived using 6years of Pass 8 data collected using the
Fermi
Gamma-Ray Space Telescope; morespecifically using the publicly available results of [57] made from analysing this data.This work was an updated version of the analysis that appeared in [92], which setlimits using the same 15 dwarf spheroidal galaxies, but only with 4 years of Pass7 data. These results are also publicly available, meaning we can cross check howmuch our results change when between datasets. We did this for each of the finalstates considered in Fig. 6-6 and found generically the shape of the limit curveswere unchanged, but that the limits themselves improved by roughly half an orderof magnitude when using the updated analysis. We show an example of this forthe case of electrons in Fig. E-6, and we see that the generic features of the limitsare unchanged but the results strengthen as we move from the Pass 7 to the Pass 8dataset. - - - - - L og σ v [ c m / s ec ] ������� � + � - ����� ������ ϵ � = ��� ϵ � = ���� ���� � � + � - ���� � Log m χ [ GeV ] Figure E-6:
Here we recreate the results shown in Fig. 6-6 for the case of final state 𝑒 + 𝑒 − , for thecase of the 4 years of Pass 7 data analysed in [92] (left) and for the 6 years of Pass 8 data consideredin [57] (right). We see that the updated dataset essentially just strengthens the limits by roughlyhalf an order of magnitude, without noticeably changing other basic features. E.4 Description of Cascade Spectra Files
All of the spectra used in this work are publicly available in .dat format at:http://web.mit.edu/lns/research/CascadeSpectra.html.The details of how these spectra were generated from the direct spectra mentioned inSec. 6.3 is discussed in Sec. 6.2 and more comprehensively in [62]. The format of thespectra files has been modeled after those made available by [186], in the hope thatanyone who has used the results of that paper should have no difficulty using ours.In addition to the files themselves we have also included two example files showinghow to load the spectra in Mathematica and Python.There are four basic file types included, which we describe briefly in turn. ∙ AtProduction_{gammas,positrons,antiprotons}.dat: these are the files providedby [186] and contain the 0-step or direct annihilation spectrum of {photons,positrons, antiprotons} for various final states; ∙ Cascade_{Gam,E,Mu,Tau,B,W,H,G}_gammas.dat: photon spectrum from fi-nal state {photons, electrons, muons, taus, 𝑏 -quarks, Ws, Higgs, gluons}; ∙ Cascade_{Gam,E,Mu,Tau,B,W,H,G}_positrons.dat - positron spectrum fromfinal state {photons, electrons, muons, taus, 𝑏 -quarks, 𝑊 s, Higgs, gluons}; and ∙ Cascade_{B,W,H,G}_antiprotons.dat - antiproton spectrum from final state274 𝑏 -quarks, 𝑊 s, Higgs, gluons}.Again we emphasize that the AtProduction files were created by the authors of [186],we only include them in our results as it is convenient to store the 0-step spectra inseparate files from the cascade results, yet having them in the same place is useful.The contents of the three AtProduction_{gammas,positrons,antiprotons}.dat havethe following format: ∙ Each file has 30 columns and 11099 rows, where the first row contains columnlabels and all others contain numerical values. ∙ The first column contains the dark matter mass in GeV, running from 5 GeVup to 100 TeV. Note using these direct spectra below 5 GeV is not advised asthe extrapolation is often unreliable. ∙ The second column contains log ( 𝑥 ) values, where 𝑥 = 𝐸/𝑚 𝜒 . This rangesfrom -8.9 to 0 in steps of 0.05. ∙ Finally the columns 3-30 contain the value of the spectrum in 𝑑𝑁/𝑑 log ( 𝑥 ) =ln(10) 𝑥𝑑𝑁/𝑑𝑥 of the spectrum at that value of 𝑚 𝜒 and 𝑥 . The columns ofrelevance for us are 5 (electrons), 8 (muons), 11 (taus), 14 ( 𝑏 -quarks), 18 ( 𝑊 -bosons), 22 (gluons), 23 (photons) and 24 (Higgs).The contents of the 19 Cascade_{Final State}_{Spectrum Type}.dat has beenmodeled on these files. To be explicit we have: ∙ Each file has 8 columns and 1612 rows, where the first row contains columnlabels and all others contain numerical values. ∙ The first column contains the value of 𝜖 𝑓 . We include the spectra for the values0.01, 0.03, 0.05, 0.07, 0.1, 0.2, 0.3, 0.4 and 0.5. The only exception to this is forgluons or the positron spectrum from photons, where the first column contains 𝑚 𝜑 values instead, and we include values of 10, 20, 40, 50, 80, 100, 500, 1000and 2000 GeV. Within these parameter ranges the interpolation is quite reliable,but outside these ranges linear interpolation is recommended. Note that several275pectra, such as the 𝛾𝛾 photons spectrum or the electron positron spectrum haveno dependence on 𝜖 𝑓 or 𝑚 𝜑 . Nevertheless we still include an 𝜖 𝑓 column in thosefiles for consistency, and note picking any value of this parameter will result inan identical spectrum. ∙ The second column contains log ( 𝑥 ) values, where 𝑥 = 𝐸/𝑚 𝜒 . This rangesfrom -8.9 to 0 in steps of 0.05. ∙ Finally the columns 3-8 contain the value of the spectrum in 𝑑𝑁/𝑑 log ( 𝑥 ) =ln(10) 𝑥𝑑𝑁/𝑑𝑥 of the spectrum at that value of 𝑚 𝜒 and 𝜖 𝑓 or 𝑚 𝜑 . The columnsrepresent an 𝑛 = 𝑛 = .20.40.60.81. ������� � + � - ��� � ��� ϵ � = ��� ϵ � = ���������� ������� ���������������� ������� γγ μ + μ - τ + τ - �� �� � ϕ = �� ���� ϕ = � ��� � + � - �� f e ff Log m χ [ GeV ] Figure E-7: 𝑓 eff for 𝑛 = 𝜖 𝑓 = 0 . (solid) and 𝜖 𝑓 = 0 . (dashed). The shaded out portions of the plot correspond to values of 𝑚 𝜒 that are kinematicallyforbidden. For the case of direct annihilation (gray line) only the spectrum for 𝑚 𝜒 > GeV isdisplayed, since for lower values of 𝑚 𝜒 the PPPC is unreliable. For direct annihilations to photonsthe spectrum is simply a delta function so in this case we plot 𝑓 eff down to lower masses as well. - - - ���� ��� � + � - ��� �������� ϵ � = ��� ϵ � = ���������� ������� ���������������� ������� γγ - - - μ + μ - τ + τ - - - - �� �� � ϕ = �� ���� ϕ = � ��� - - - � + � - �� Log σ v / m χ [ c m / s e c / G e V ] Log m χ [ GeV ] Figure E-8:
Values of the bound on ⟨ 𝜎𝑣 ⟩ /𝑚 𝜒 for various final states. The bound in very robustand we find of order roughly − . − − . cm / s / GeV, independent of final state (although thebound is slightly higher for electrons and photons), number of steps, or 𝜖 𝑓 . ppendix FDark Matter Annihilation atOne-Loop F.1 One-loop Calculation of 𝜒 𝑎 𝜒 𝑏 → 𝑊 𝑐 𝑊 𝑑 in the FullTheory In this appendix we outline the details of the high-scale matching calculation, whichgives rise to the Wilson coefficients stated in Eq. (7.13). These coefficients are deter-mined solely by the ultraviolet (UV) physics, allowing us to simplify the calculationby working in the unbroken theory with 𝑚 𝑊 = 𝑚 𝑍 = 𝛿𝑚 = 0 . Combining this withthe heavy Majorana fermion DM being non-relativistic, there are only two possibleDirac structures that can appear in the result: ℳ 𝐴 = 𝜖 * 𝜇 ( 𝑝 ) 𝜖 * 𝜈 ( 𝑝 ) 𝜖 𝜎𝜇𝜈𝛼 𝑝 𝛼 𝑖 ¯ 𝑣 ( 𝑝 ) 𝛾 𝜎 𝛾 𝑢 ( 𝑝 ) , ℳ 𝐵 = 𝜖 * 𝜇 ( 𝑝 ) 𝜖 * 𝜈 ( 𝑝 ) 𝑔 𝜇𝜈 ¯ 𝑣 ( 𝑝 ) /𝑝 𝑢 ( 𝑝 ) , (F.1)where 𝑝 and 𝑝 are the momenta of the incoming fermions, whilst 𝑝 and 𝑝 corre-spond to the outgoing bosons. The symmetry properties of these structures underthe interchange of initial and final state particles allow us to write our full amplitude279s: ℳ 𝑎𝑏𝑐𝑑 = 4 𝜋𝛼 𝑚 𝜒 { [ 𝐵 𝛿 𝑎𝑏 𝛿 𝑐𝑑 + 𝐵 ( 𝛿 𝑎𝑐 𝛿 𝑏𝑑 + 𝛿 𝑎𝑑 𝛿 𝑏𝑐 )] ℳ 𝐴 + 𝐵 ( 𝛿 𝑎𝑐 𝛿 𝑏𝑑 − 𝛿 𝑎𝑑 𝛿 𝑏𝑐 ) ℳ 𝐵 } . (F.2)The above equation serves to define the Wilson coefficients 𝐵 𝑟 in a convenient form.These coefficients are related to the EFT coefficients of the operators defined inEq. (7.2) and (7.4) via: 𝐶 = ( − 𝜋𝛼 /𝑚 𝜒 ) 𝐵 , 𝐶 = ( − 𝜋𝛼 /𝑚 𝜒 ) 𝐵 . (F.3)For NLL accuracy we only need the tree-level value of these coefficients, which receivea contribution from 𝑠 , 𝑡 and 𝑢 -channel type graphs and were calculated in [385]. Forcompleteness we state their values here: 𝐵 (0)1 = 1 , 𝐵 (0)2 = − , 𝐵 (0)3 = 0 . (F.4)Combining these with Eq. (F.3), we see that the first terms in Eq. (7.13) are indeedthe tree-level contributions as claimed.The operator associated with 𝐵 was not discussed in the earlier work of [385] as itcannot contribute to the high-scale matching calculation at any order, as we will nowargue. Firstly note that the 𝐵 operator is skew under the interchange 𝑎 ↔ 𝑏 . Due tothe mass splitting between the neutral and charged states, present day annihilation isinitiated purely by 𝜒 𝜒 = 𝜒 𝜒 , a symmetric state that cannot overlap with 𝐵 . Onemay worry that exchange of one or more weak bosons between the initial states – thehallmark of the Sommerfeld enhancement – may nullify this argument. But it canbe checked that if the initial states to such an exchange have identical gauge indices,then so will the final states. As such 𝐵 is not relevant for calculating high-scalematching. Diagrams where a soft gauge boson is exchanged between an initial and final state particlewould in principle allow 𝐵 to contribute. Such a contributions would however be to the low-scalematching, which we discuss in App. F.3. As discussed there, 𝐵 contributions to present day DM 𝐵 here. From a practicalpoint of view 𝐵 gives us an additional handle on the consistency of our result,which we check in App. F.2. Given that many graphs that generate 𝐵 and 𝐵 alsocontribute to 𝐵 , the consistency of 𝐵 provides greater confidence in the results forthe operators we are interested in. Further, from a physics point of view, although 𝐵 is not relevant for high-scale matching when considering present day indirectdetection experiments, it could be relevant for calculating the annihilation rate in theearly universe, where all states in the DM triplet were present, to the extent that thenon-relativistic approximation is still relevant. For this reasons we state it in case itis of interest for future work, such as expanding on calculations of the relic densityat one loop (see for example [435–437]). Determining Matching Coefficients
Let us briefly review how matching coefficients are calculated at one loop. To beginwith we can write the general structure of the UV and infrared (IR) divergences ofthe one-loop result in the full theory as: ℳ fullbare = 𝐾𝜖 + 𝐿𝜖 IR + 𝑀𝜖 UV + 𝑁 (︂ 𝜖 UV − 𝜖 IR )︂ + 𝐶 , (F.5)where 𝑁 is the coefficient associated with the various scaleless integrals, and 𝐶 is thefinite contribution. Now the full theory is a renormalizable gauge theory, so we knowthe counter-term must be of the form: 𝛿 full = − 𝑀 + 𝑁𝜖 UV + 𝐷 + 𝐸𝜖 + 𝐹𝜖 IR , (F.6)where the values of 𝐷 , 𝐸 and 𝐹 are scheme dependent. Nonetheless when calculatingmatching coefficients it is easiest to work in the on-shell scheme for the wave-functionrenormalization factors, so below to denote this we add an “os” subscript to 𝐷 , 𝐸 and 𝐹 . The reason this scheme is the most straightforward, is that in any other scheme annihilation are power suppressed, and therefore do not contribute at any order in the leading powereffective theory. ℳ full onto the 𝑆 -matrix ele-ments we want via the LSZ reduction, there will be non-trivial residues correspond-ing to the external particles. When using the on-shell scheme for the wave-functionrenormalization factors, however, these residues are just unity, which simplifies thecalculation as we can then ignore them. We emphasize that whatever scheme oneuses, the final result for the Wilson coefficients in MS will be the same.With this in mind, if we then combine 𝛿 full with the bare results we obtain a UVfinite answer: ℳ fullren . = 𝐾 + 𝐸 os 𝜖 + 𝐿 − 𝑁 + 𝐹 os 𝜖 IR + 𝐶 + 𝐷 os . (F.7)In our calculation we will use dimensional regularization to regulate both UV and IRdivergences, which effectively sets 𝜖 UV = 𝜖 IR , causing all scaleless integrals to vanish.Naively this seems to change the above argument, but as long as we still use thecorrect counter-term in Eq. (F.6) we find: ℳ fullren . = 𝐾𝜖 + 𝐿𝜖 + 𝑀𝜖 + 𝐶 − 𝑀 + 𝑁𝜖 + 𝐷 os + 𝐸 os 𝜖 + 𝐹 os 𝜖 = 𝐾 + 𝐸 os 𝜖 + 𝐿 − 𝑁 + 𝐹 os 𝜖 + 𝐶 + 𝐷 os . (F.8)Comparing this with Eq. (F.7), we see that if we interpret all of the divergences inthe final result as IR, then this method is equivalent to carefully distinguishing 𝜖 UV and 𝜖 IR throughout.In the EFT, with the above choice of zero masses and working on-shell withdimensional regularization, all graphs are scaleless. At one loop they have the generalform: ℳ EFTbare = 𝑂 (︂ 𝜖 − 𝜖 )︂ + 𝑃 (︂ 𝜖 UV − 𝜖 IR )︂ . (F.9)Importantly if we have the correct EFT description of the full theory, then the twotheories must have the same IR divergences. Comparing Eq. (F.9) to Eq. (F.7), we see One may worry there could also be scaleless integrals of the form (︀ 𝜖 − − 𝜖 − )︀ , but the use ofthe zero-bin subtraction [438] ensures such contributions cannot appear. 𝑂 = − 𝐾 − 𝐸 os and 𝑃 = 𝑁 − 𝐿 − 𝐹 os . The EFT is again a renormalizabletheory, so we can cancel the UV divergences using 𝛿 EFT = ( 𝐾 + 𝐸 os ) 𝜖 − + ( 𝐿 + 𝐹 os − 𝑁 ) 𝜖 − . Note as all EFT graphs are scaleless there are no finite contributions thatcould be absorbed into the counter-term, so in any scheme there is no finite correctionto 𝛿 EFT . Using this counter-term, we conclude: ℳ EFTren . = 𝐾 + 𝐸 os 𝜖 + 𝐿 − 𝑁 + 𝐹 os 𝜖 IR . (F.10)Again note that for a similar argument to that in the full theory, if we had set 𝜖 UV = 𝜖 IR at the outset, then as long as we still used the correct counter-term wewould arrive at the same result.The matching coefficient is then obtained from subtracting the renormalized EFTfrom the renormalized full theory result, so taking the appropriate results above weconclude: ℳ fullren . − ℳ EFTren . = 𝐶 + 𝐷 os . (F.11)Comparing this with Eq. (F.7), we see that provided we have the correct EFT, thenthe matching coefficient is just the finite contribution to the renormalized full theoryamplitude in the on-shell scheme. Even though this result makes explicit referenceto a scheme in 𝐷 on − shell , it is in fact scheme independent. The reason for this is thatif we worked in a different scheme, although 𝐷 would change, we would also have toaccount for the now non-trivial external particle residues that enter via LSZ. Theircontribution is what ensures Eq. (F.11) is scheme independent. Results of the Calculation
As outlined above, in order to obtain the matching coefficients we need the finitecontribution to the renormalized full theory amplitude. Now to compute this in theparticular theory we consider in this work, we need to calculate the 25 diagrams thatcontribute to the one-loop correction to 𝜒 𝑎 𝜒 𝑏 → 𝑊 𝑐 𝑊 𝑑 . The diagrams are identicalto those considered in [439], where they defined a numbering scheme for the diagrams,283rouping them by topology and labelling them as 𝑇 𝑖 for various 𝑖 . We follow thatnumbering scheme here, but cannot use their results as they considered massless initialstate fermions whilst ours are massive and non-relativistic. In general we calculatethe diagrams using dimensional regularization with 𝑑 = 4 − 𝜖 to regulate the UVand IR, and work in ‘t Hooft-Feynman gauge. Loop integrals are determined usingPassarino-Veltman reduction [440], and we further make use of the results in [441–444]as well as FeynCalc [445, 446] and Package-X [447].In the EFT description of the full theory outlined in Sec. 7.2, the factorization ofthe matrix elements ensured a separation between the Sommerfeld and Sudakov con-tributions. Yet for the full theory no clear separation exists and there will be graphsthat contribute to both effects – in particular the graph 𝑇 𝑐 considered below. Thepurpose of the Wilson coefficients we are calculating here is to provide corrections tothe Sudakov contribution – we do not want to spoil the EFT distinction by includingSommerfeld effects in these coefficients. In order to cleanly separate the contribu-tions we take the relative velocity of our non-relativistic initial states to be zero. Thisensures that any contribution of the form /𝑣 , characteristic of Sommerfeld enhance-ment, become power divergences and therefore vanish in dimensional regularization.This is different to the treatment in HI, where they calculated the diagram withoutsending 𝑣 → and subtracted the Sommerfeld contribution by hand.In our calculation the DM is a Majorana fermion. It turns out that for almost allthe graphs below the result is identical regardless of whether we think of the fermion asMajorana or Dirac – a result that is also true at tree-level. The additional symmetryfactors in the Majorana case are exactly cancelled by the factors of / enteringfrom the Majorana Lagrangian. The exceptions to this are for graphs containing aclosed loop of fermions, specifically 𝑇 𝑑 and 𝑇 𝑑 below, as well as closed fermion loopcontributions to the counter-terms.Using the approach outlined above we now state the contribution to 𝐵 𝑟 as definedin Eq. (F.2) graph by graph. Throughout we define 𝐿 ≡ ln 𝜇/ 𝑚 𝜒 .284 𝑎 The result for this graph and its cross term is: 𝐵 [1 𝑎 ]1 = 𝛼 𝜋 [︂ − 𝜖 − 𝜖 (4 𝐿 + 2 𝑖𝜋 + 2) − 𝐿 − 𝐿 − 𝑖𝜋𝐿 − 𝜋 ]︂ ,𝐵 [1 𝑎 ]2 = 12 𝐵 [1 𝑎 ]1 ,𝐵 [1 𝑎 ]3 = 𝛼 𝜋 [︂ 𝜖 + 14 𝜖 (2 𝐿 − 𝑖𝜋 −
2) + 12 𝐿 − 𝐿 − 𝑖𝜋𝐿 + 17 𝜋 −
16 (2 + 7 𝑖𝜋 − ]︂ . (F.12)In calculating this graph in the non-relativistic limit via Passarino-Veltman reduc-tion there are additional spurious divergences that must be regulated. The origin ofthese divergences is that Passarino-Veltman assumes the momenta appearing in theintegrals to be linearly independent. But in the center of momentum frame if we take 𝑣 = 0 , then 𝑝 and 𝑝 are identical and this assumption breaks down, leading to thedivergences of the form ( 𝑠 − 𝑚 𝜒 ) − , where 𝑠 = ( 𝑝 + 𝑝 ) . A simple way to regulatethem is to give the initial states a small relative velocity. This does not lead to aviolation of our separation of Sommerfeld and Sudakov effects as this graph does notcontribute to the Sommerfeld enhancement. As such this procedure introduces no /𝑣 contributions to the final result and the regulator can be safely removed at theend. This is the only diagram where this issue appears – if it occurred in a graph thatdid contribute to the Sommerfeld effect we would need to use a different regulator.285 𝑏 This graph has a single crossed term and combining the two yields: 𝐵 [1 𝑏 ]1 = 𝐵 [1 𝑏 ]3 = 0 ,𝐵 [1 𝑏 ]2 = 𝛼 𝜋 [︂ 𝜖 + 4 𝐿 + 2 𝜖 + 4 𝐿 ( 𝐿 + 1) − 𝜋 − ]︂ . (F.13) 𝑇 𝑐 The combination of this graph and its crossed term is: 𝐵 [1 𝑐 ]1 = 𝛼 𝜋 [︂ 𝜖 − 𝐿 + 4 ln 2 ]︂ ,𝐵 [1 𝑐 ]2 = 12 𝐵 [1 𝑐 ]1 ,𝐵 [1 𝑐 ]3 = 𝛼 𝜋 [︂ 𝜖 − 𝐿 + 𝜋 − ]︂ . (F.14)Formally this graph also gives a contribution to the Sommerfeld enhancement in thefull theory. Nevertheless as we take 𝑣 = 0 at the outset, the contribution here ispurely to the Sudakov terms. 286 𝑑 The contribution from this diagram vanishes in the non-relativistic limit, i.e. 𝐵 [1 𝑑 ]1 = 𝐵 [1 𝑑 ]2 = 𝐵 [1 𝑑 ]3 = 0 . (F.15) 𝑇 𝑎 For the case of ghosts running in the loop of the above graph we have its contributionand the crossed term giving 𝐵 [2 𝑎 ]1 = 𝐵 [2 𝑎 ]2 = 0 ,𝐵 [2 𝑎 ]3 = 𝛼 𝜋 [︂ 𝜖 + 2 𝐿 + 𝑖𝜋
24 + 1172 ]︂ . (F.16) 𝑇 𝑏 For a scalar Higgs in the loop, the graph and its cross term contribute: 𝐵 [2 𝑏 ]1 = 𝐵 [2 𝑏 ]2 = 0 ,𝐵 [2 𝑏 ]3 = 𝛼 𝜋 [︂ 𝜖 + 2 𝐿 + 𝑖𝜋
12 + 1136 ]︂ . (F.17)287 𝑐 There is no crossed graph associated with the graph above as the gauge bosons runningin the loop are real fields. As such taking just this graph gives 𝐵 [2 𝑐 ]1 = 𝐵 [2 𝑐 ]2 = 0 ,𝐵 [2 𝑐 ]3 = 𝛼 𝜋 [︂ 𝜖 + 1 𝜖 (︂
34 (2 𝐿 + 𝑖𝜋 ) + 178 )︂ + 38 (2 𝐿 + 𝑖𝜋 ) + 178 (2 𝐿 + 𝑖𝜋 ) + 9524 − 𝜋 ]︂ . (F.18) 𝑇 𝑑 There are two types of fermions that can run in the loop: the Majorana triplet fermionthat make up our DM or left-handed SM doublets. As with the gauge bosons theseSM fermions are taken to be massless and for generality we say there are 𝑛 𝐷 of them. For the SM doublets there is a crossed graph, whilst for the Majorana DM field thereis not, so that: 𝐵 [2 𝑑 ]1 = 𝐵 [2 𝑑 ]2 = 0 ,𝐵 [2 𝑑 ]3 = 𝛼 𝜋 [︂ − (︂ 𝜖 + 43 𝐿 + 43 ln 2 −
59 + 𝜋 )︂ − 𝑛 𝐷 (︂ 𝜖 + 16 (2 𝐿 + 𝑖𝜋 ) + 736 )︂]︂ . (F.19) For the SM well above the electroweak scale 𝑛 𝐷 = 12 . In detail, for each generation thereare four doublets: the lepton doublet and due to color, three quark doublets. As such for threegenerations we have twelve left-handed SM doublets. 𝐵 [2 𝑑 ]3 gets multiplied by 2.The factor of / we find in the last line of 𝐵 [2 𝑑 ]3 is consistent with the expressionfound for this graph, but with different kinematics, in [439], but disagrees with [448]. 𝑇 𝑒 − ℎ The four graphs shown above do not contribute to our one-loop result; the graphs onthe top row do not generate either ℳ 𝐴 or ℳ 𝐵 , whilst the loops on the second lineare both scaleless and so vanish in dimensional regularization. As such we have: 𝐵 [2 𝑒 − 𝑓 ]1 = 𝐵 [2 𝑒 − 𝑓 ]2 = 𝐵 [2 𝑒 − 𝑓 ]3 = 0 . (F.20) 𝑇 𝑎 and 𝑇 𝑎 The two graphs shown above have identical amplitudes. For each graph indepen-289ently, the sum of it and its crossed graph is: 𝐵 [3 𝑎/ 𝑎 ]1 = 𝛼 𝜋 [︂ − 𝜖 + 2 − 𝐿𝜖 − 𝐿 +4 𝐿 − 𝜋 ]︂ ,𝐵 [3 𝑎/ 𝑎 ]2 = − 𝐵 [3 𝑎/ 𝑎 ]1 ,𝐵 [3 𝑎/ 𝑎 ]3 = 12 𝐵 [3 𝑎/ 𝑎 ]1 . (F.21) 𝑇 𝑏 and 𝑇 𝑏 As for 𝑇 𝑎 and 𝑇 𝑎 , these two graphs also have equal amplitudes. Again we providethe combination of each with its crossed graph: 𝐵 [3 𝑏/ 𝑏 ]1 = 𝛼 𝜋 [︂ 𝜖 + 2 𝐿 − 𝜋 ]︂ ,𝐵 [3 𝑏/ 𝑏 ]2 = − 𝐵 [3 𝑏/ 𝑏 ]1 ,𝐵 [3 𝑏/ 𝑏 ]3 = 12 𝐵 [3 𝑏/ 𝑏 ]1 . (F.22) 𝑇 𝑎 Whether the above graph has a crossed graph associated with interchanging the initialstates depends on the identity of the initial state fermions. For Majorana fermionsthere is such a crossing, whilst for Dirac there is not. Despite this, in either case thecombination of the graph and its crossing (where it exists) is the same in both cases290nd is simply: 𝐵 [5 𝑎 ]1 = 𝐵 [5 𝑎 ]2 = 0 ,𝐵 [5 𝑎 ]3 = 𝛼 𝜋 [︂ − 𝜖 − 𝐿 −
133 ln 2 −
83 + 23 𝑖𝜋 ]︂ . (F.23) 𝑇 𝑏 As for 𝑇 𝑎 the existence of a crossed graph depends on the nature of the DM. Re-gardless again the result is the same if we take it to be Dirac or Majorana, whichis: 𝐵 [5 𝑏 ]1 = 𝐵 [5 𝑏 ]2 = 0 ,𝐵 [5 𝑏 ]3 = 𝛼 𝜋 [︂ 𝜖 + 3 𝐿 + 3 ln 2 − ]︂ . (F.24) 𝑇 𝑎 For a gauge boson in the loop we have: 𝐵 [6 𝑎 ]1 = 𝐵 [6 𝑎 ]2 = 0 ,𝐵 [6 𝑎 ]3 = 𝛼 𝜋 [︂ − 𝜖 − 𝐿 − − 𝑖𝜋 ]︂ . (F.25)Note this graph and the remaining 𝑇 type topologies have no crossed graphs. 𝑇 𝑏 𝐵 [6 𝑏 ]1 = 𝐵 [6 𝑏 ]2 = 0 ,𝐵 [6 𝑏 ]3 = 𝛼 𝜋 [︂ − 𝜖 − 𝐿 − − 𝑖𝜋 ]︂ . (F.26) 𝑇 𝑐 For a scalar Higgs we have an identical contribution to 𝑇 𝑏 : 𝐵 [6 𝑐 ]1 = 𝐵 [6 𝑐 ]2 = 0 ,𝐵 [6 𝑐 ]3 = 𝛼 𝜋 [︂ − 𝜖 − 𝐿 − − 𝑖𝜋 ]︂ . (F.27) 𝑇 𝑑 As for 𝑇 𝑑 the fermion in the loop could again be either DM or SM. Allowing thereto be 𝑛 𝐷 left-handed SM doublets we have 𝐵 [6 𝑑 ]1 = 𝐵 [6 𝑑 ]2 = 0 ,𝐵 [6 𝑑 ]3 = 𝛼 𝜋 [︂(︂ 𝜖 + 43 𝐿 + 43 ln 2 + 169 )︂ + 𝑛 𝐷 (︂ 𝜖 + 13 𝐿 + 518 + 16 𝑖𝜋 )︂]︂ . (F.28)Here there is a symmetry factor of / for the loop in the case of the Majorana DMfield. If the DM was a Dirac fermion instead, the first line of 𝐵 [6 𝑑 ]3 would get multipliedby as this symmetry factor would not be present.292 𝑒 and 𝑇 𝑓 Both of these integrals are scaleless and vanish in dimensional regularization, so: 𝐵 [6 𝑒 − 𝑓 ]1 = 𝐵 [6 𝑒 − 𝑓 ]2 = 𝐵 [6 𝑒 − 𝑓 ]3 = 0 . (F.29) 𝑇 For the final graph we again have a crossed contribution, and combining the two gives: 𝐵 [7]1 = 𝛼 𝜋 [︂ − 𝜖 − 𝐿 − ]︂ ,𝐵 [7]2 = − 𝐵 [7]1 ,𝐵 [7]3 = 12 𝐵 [7]1 . (F.30)293 ounter-terms To begin with, as 𝐵 vanishes at tree level there are no counter-term corrections toits value at one loop. Instead we only need to consider graphs that would contributeto 𝐵 and 𝐵 , of which there are three:The graph on the left corresponds to the wave-function and mass renormalizationof the DM – denoted as 𝑍 𝜒 and 𝑍 𝑚 – whilst the remaining two graphs accountfor the renormalization of the DM and electroweak gauge boson interaction vertex 𝑔 ¯ 𝜒 /𝑊 𝜒 – here 𝑍 . Now if we calculate the above three graphs, we find a contributionproportional to the tree-level amplitude ℳ tree , as well as a term that would contributeto 𝐵 . As we know this latter term must be cancelled by other graphs given 𝐵 (0)3 = 0 ,we keep only the former term which gives: (2 𝛿 − 𝛿 𝜒 − 𝛿 𝑚 ) ℳ tree , (F.31)where we have used 𝑍 𝑖 = 1 + 𝛿 𝑖 .Next, when determining the 𝛿 𝑖 we need to pick a scheme. As explained above,when calculating matching coefficients it is easiest to work in the on-shell scheme toensure we do not have to worry about residues from the LSZ reduction. The meaningof the on-shell values of 𝛿 𝜒 and 𝛿 𝑚 is clear, whereas the interpretation of the on-shell 𝛿 is ambiguous in a non-abelian theory. Here we treat this counter-term as follows.By definition we know 𝛿 = 𝛿 𝑔 + 𝛿 𝑊 + 𝛿 𝜒 , where 𝛿 𝑔 and 𝛿 𝑊 are the counter-termsfor the coupling and gauge boson wave-functions respectively. For the gauge-bosonwave-function we use the on-shell scheme as usual. For the coupling counter-term,however, we define it to be purely UV, in the MS scheme and as our full theory isdefined with the DM a propagating degree of freedom, this coupling is defined abovethe 𝑚 𝜒 . In the EFT the DM is integrated out, so the appropriate coupling for the294atching is one defined below 𝑚 𝜒 . We put this issue aside for now and return to itin the next section.The above choices then define our scheme for 𝛿 in a manner that ensures allresidues are still 1. With this scheme, we can then calculate the relevant counter-terms and find: 𝛿 𝜒 = − 𝛼 𝜋 [︂ 𝜖 UV + 4 𝐿 + 4 ln 2 + 4 ]︂ , (F.32) 𝛿 𝑚 = − 𝛼 𝜋 [︂ 𝜖 UV + 12 𝐿 + 12 ln 2 + 8 ]︂ ,𝛿 𝑊 = − 𝛼 𝜋 [︂ 𝑛 𝐷 − 𝜖 UV + 19 − 𝑛 𝑓 𝜖 IR + 163 𝐿 + 163 ln 2 ]︂ ,𝛿 𝑔 = − 𝛼 𝜋 [︂ − 𝑛 𝐷 𝜖 UV ]︂ ,𝛿 = − 𝛼 𝜋 [︂ 𝜖 UV + 19 − 𝑛 𝐷 𝜖 IR + 203 𝐿 + 203 ln 2 + 4 ]︂ , where 𝑛 𝐷 is again the number of left-handed SM doublets. Recall that in determiningthe counter-terms we cannot neglect scaleless integrals as we did for the main calcu-lation, so their contribution has been included here and we explicitly distinguish 𝜖 UV from 𝜖 IR . Subbing these results into Eq. (F.31), we find the crossed contribution is: 𝐵 [CT]1 = 𝛼 𝜋 [︂ 𝑛 𝐷 − 𝜖 IR + 83 𝐿 + 83 ln 2 + 4 ]︂ ,𝐵 [CT]2 = 𝛼 𝜋 [︂ − 𝑛 𝐷 𝜖 IR − 𝐿 −
86 ln 2 − ]︂ ,𝐵 [CT]3 = 0 . (F.33)Interestingly the counter-term contribution is UV finite. This implies that the sum ofall one-loop graphs before adding in counter-terms must be UV finite. Given that weused dimensional regularization to regulate both UV and IR divergences this cannotbe immediately read off from our results, but going back to the integrals and keepingtrack of the UV divergences we confirmed that the sum is indeed UV finite. Thiscancellation appears to be purely accidental.Note if our DM field had instead been a Dirac fermion, there would be several295odifications to the above. Firstly the 𝐿 and ln 2 dependence in 𝛿 𝑊 and 𝛿 wouldbe modified, whilst the 𝜖 UV dependence in 𝛿 𝑊 and 𝛿 𝑔 would also change. In thecombination stated in Eq. (F.33) this only changes the 𝐿 and ln 2 dependence, but ina way that is exactly cancelled when we account for the scale of the coupling in thenext section. Scale of the Coupling
Throughout the above calculation we have treated the DM as a propagating degree offreedom and included its effects in loop diagrams. This implies that the coupling usedso far throughout this appendix implicitly depends on 𝑛 𝐷 + 1 flavors – 𝑛 𝐷 left-handedSM doublets and one Majorana DM fermion – i.e. we have used 𝛼 = 𝛼 ( 𝑛 𝐷 +1)2 ( 𝜇 ) .In the EFT however, the DM is no longer a propagating field and so the appropriatecoupling is 𝛼 ( 𝑛 𝐷 )2 ( 𝜇 ) . At order 𝛼 , which we are working to at one loop, the distinctionwill lead to a finite contribution because of the matching at the scale 𝜇 = 𝑚 𝜒 , whichwe calculate in this section.Let us start by reviewing the treatment of the running coupling in general. Thisrunning is captured by the 𝛽 -function, which is defined by 𝛽 ( 𝛼 ) = 𝜇𝑑𝛼 /𝑑𝜇 , wherehere 𝛼 is the renormalized coupling; the bare coupling is independent of 𝜇 . In generalthe 𝛽 -function can be written as: 𝛽 ( 𝛼 ) = − 𝛼 [︃ 𝜖 + ∞ ∑︁ 𝑛 =1 (︁ 𝛼 𝜋 )︁ 𝑛 𝑏 𝑛 − ]︃ = − 𝜖𝛼 − 𝑏 𝜋 𝛼 + . . . , (F.34)where we have expanded it to the order we work to in the second equality. At thisorder we can solve for the running of the coupling as: 𝛼 ( 𝜇 ) = 𝛼 ( 𝜇 )1 + 𝛼 ( 𝜇 ) 𝑏 𝜋 ln 𝜇𝜇 + . . . . (F.35)Now the above is completely general, so let us focus in to the specific problem wehave. At this order it suffices to demand that the couplings match at the scale 𝑚 𝜒 ,296nd at one-loop this is captured by a difference in 𝑏 . For our problem we define 𝑏 ( 𝑛 𝐷 +1)0 to be the value above 𝑚 𝜒 and 𝑏 ( 𝑛 𝐷 )0 the value below. Then using Eq. (F.35)to define 𝛼 ( 𝑛 𝐷 +1)2 ( 𝜇 ) and 𝛼 ( 𝑛 𝐷 )2 ( 𝜇 ) , we demand they match at a scale 𝑚 𝜒 , which gives: 𝛼 ( 𝑛 𝐷 +1)2 ( 𝜇 ) = 𝛼 ( 𝑛 𝐷 )2 ( 𝜇 ) [︂ 𝛼 ( 𝑛 𝐷 )2 ( 𝜇 )2 𝜋 (︁ 𝑏 ( 𝑛 𝐷 +1)0 − 𝑏 ( 𝑛 𝐷 )0 )︁ ln 𝜇𝑚 𝜒 + . . . ]︃ . (F.36)So now we just need to determine 𝑏 ( 𝑛 𝐷 +1)0 − 𝑏 ( 𝑛 𝐷 )0 . In general for a theory containingjust gauge bosons, Weyl fermions (WF), Majorana fermions (MF) and charged scalars(CS), we can write: 𝑏 = 113 𝐶 𝐴 − ∑︁ 𝑖 ∈ WF 𝐶 ( 𝑅 𝑖 ) − ∑︁ 𝑖 ∈ MF 𝐶 ( 𝑅 𝑖 ) − ∑︁ 𝑖 ∈ CS 𝐶 ( 𝑅 𝑖 ) . (F.37)Our calculation has all four of these ingredients: electroweak gauge bosons, the left-handed SM fermions (which are Weyl because only one chirality couples to the gaugebosons), the Majorana DM fermion and the Higgs. Then using 𝐶 𝐴 = 2 , 𝐶 ( 𝑅 ) = 1 / for the SM left-handed fermions and the Higgs, and 𝐶 ( 𝑅 ) = 2 for the adjoint Wino,we conclude: 𝑏 ( 𝑛 𝐷 )0 = 43 − 𝑛 𝐷 ,𝑏 ( 𝑛 𝐷 +1)0 = 35 − 𝑛 𝐷 . (F.38)From this Eq. (F.36) tells us that to the order we are working: 𝛼 ( 𝑛 𝐷 +1)2 ( 𝜇 ) = 𝛼 ( 𝑛 𝐷 )2 ( 𝜇 ) [︃ − 𝛼 ( 𝑛 𝐷 )2 ( 𝜇 )4 𝜋 (︂ 𝐿 + 83 ln 2 )︂]︃ . (F.39)Now as there is only a difference between the couplings at next to leading order, thisonly corrects the tree level result stated in Eq. (F.4). As such the impact of changingto the coupling defined below 𝑚 𝜒 , which is relevant for the matching, is to add the297ollowing contribution: 𝐵 [Matching]1 = 𝛼 𝜋 [︂ − 𝐿 −
83 ln 2 ]︂ ,𝐵 [Matching]2 = − 𝐵 [Matching]1 ,𝐵 [Matching]3 = 0 , (F.40)where now here and in all earlier one-loop results we can take 𝛼 = 𝛼 ( 𝑛 𝐷 )2 . Asalluded to above, this result is modified for a Dirac DM fermion, but in a way exactlycompensated by a change in the counter-term contribution. Combination
Combining the 25 graphs above with the counter-terms and the matching contribu-tions, we arrive at the following result: 𝐵 (1)1 = 𝛼 𝜋 [︂ − 𝜖 − 𝐿 + 12 𝑖𝜋 + 31 − 𝑛 𝐷 𝜖 − 𝐿 − 𝐿 − 𝑖𝜋𝐿 − 𝜋 ]︂ ,𝐵 (1)2 = 𝛼 𝜋 [︂ 𝜖 + 48 𝐿 − 𝑖𝜋 + 55 − 𝑛 𝐷 𝜖 + 4 𝐿 + 6 𝐿 − 𝑖𝜋𝐿 − 𝜋 ]︂ ,𝐵 (1)3 = 𝛼 𝜋 [︂ 𝑛 𝐷 −
72 ln 2 −
71 + 3 𝜋 ]︂ , (F.41)where recall 𝐿 = ln 𝜇/ 𝑚 𝜒 , 𝑛 𝐷 is the number of SM left-handed doublets and now all 𝜖 = 𝜖 IR .As explained in detail at the outset of the calculation, the one-loop contributionto the matching coefficient is just the finite part of this result. Combining this withthe tree-level term in Eq. (F.4) and mapping back to 𝐶 𝑟 using Eq. (F.3) then givesus the Wilson coefficients in Eq. (7.13), which we set out to justify.If instead we had a Dirac DM triplet rather than a Majorana, then the only impacton the above would be for 𝐵 (1)3 , and we would instead have 𝐵 (1)3 = 𝛼 𝜋 [︂ 𝑛 𝐷 −
72 ln 2 − ]︂ . (F.42)298 .2 Consistency Check on the High-Scale Matching As a non-trivial check on our high-scale calculation, we can calculate the ln 𝜇 , or 𝐿 in our case, pieces of Eq. (F.41) independently using the NLL results. To begin with,if we define 𝐶 ≡ ( 𝐶 𝐶 𝐶 ) 𝑇 , then from the definition of the anomalous dimensionwe have: 𝜇 𝑑𝑑𝜇 𝐶 ( 𝜇 ) = ^ 𝛾 ( 𝜇 ) 𝐶 ( 𝜇 ) . (F.43)Next we expand the coefficients as a series in 𝛼 : 𝐶 ( 𝜇 ) = 𝐶 (0) ( 𝜇 ) + 𝐶 (1) ( 𝜇 ) + ... , where 𝐶 (0) ( 𝜇 ) is the tree-level contribution and 𝐶 (1) ( 𝜇 ) the one-loop result. Now we want across check on the one-loop contribution, so we evaluate Eq. (F.43) at 𝒪 ( 𝛼 ) , giving 𝜇 𝑑𝛼 𝑑𝜇 𝜕𝐶 (0) 𝜕𝛼 + 𝜇 𝜕𝐶 (1) ( 𝜇 ) 𝜕𝜇 = ^ 𝛾 − loop ( 𝜇 ) 𝐶 (0) ( 𝜇 ) , (F.44)and rearranging we arrive at: 𝜇 𝜕𝐶 (1) ( 𝜇 ) 𝜕𝜇 = ^ 𝛾 − loop ( 𝜇 ) 𝐶 (0) ( 𝜇 ) − 𝜇 𝑑𝛼 𝑑𝜇 𝜕𝐶 (0) 𝜕𝛼 . (F.45)This equation shows that we can derive the 𝜇 and hence 𝐿 dependence of the one-loop Wilson coefficient from the one-loop anomalous dimension and tree-level Wilsoncoefficient, both of which are known from the NLL result. To be more explicit, wecan write the bare Wilson coefficient as 𝐶 bare = 𝜇 𝜖 (︂ 𝑎𝜖 + 𝑏𝜖 + 𝜇 − independent )︂ = 𝑎𝜖 + 𝑏 + 2 𝑎𝐿𝜖 + 2 𝑎𝐿 + 2 𝑏𝐿 + 𝜇 − independent , (F.46)where in the second equality we swapped form ln 𝜇 to 𝐿 and absorbed the additional ln 2 factors into the 𝜇 -independent term. From here we can write the renormalizedWilson coefficient as 𝐶 ren . = 2 𝑎𝐿 + 2 𝑏𝐿 + 𝜇 − independent , (F.47)299hich we can then substitute into the left-hand side of Eq. (F.45) to derive 𝑎 and 𝑏 for each Wilson coefficient. Doing this and then mapping back to 𝐵 𝑟 using Eq. (F.3),we find 𝐵 (1)1 = 𝛼 𝜋 [︂ − 𝐿𝜖 − 𝐿 − 𝐿 − 𝑖𝜋𝐿 + 𝜇 − ind . ]︂ ,𝐵 (1)2 = 𝛼 𝜋 [︂ 𝐿𝜖 + 4 𝐿 + 6 𝐿 − 𝑖𝜋𝐿 + 𝜇 − ind . ]︂ ,𝐵 (1)3 = 𝛼 𝜋 [0 + 𝜇 − ind . ] , (F.48)in exact agreement with Eq. (F.41). In particular, as 𝐵 (0)3 = 0 , we needed 𝐵 (1)3 to beindependent of 𝐿 , as we found. F.3 Low-Scale Matching Calculation
The focus of this appendix is to derive the low-scale matching conditions stated inEqs. (7.14), (7.15), (7.16), (7.17), and (7.18). At this scale, the matching is from aneffective theory where the 𝑊 , 𝑍 , top and Higgs are dynamical degrees of freedom –NRDM-SCET EW – onto a theory where these electroweak modes have been integratedout – NRDM-SCET 𝛾 .In order to perform the calculation we will make use of the formalism of elec-troweak SCET developed in [381, 400–403]. As we are working in SCET, there areboth collinear and soft gauge boson diagrams that will appear in the one-loop match-ing. In [381] it was proven that at one-loop the total low-scale matching contributionfrom these soft and collinear SCET modes can always be decomposed into a contri-bution that is diagonal, in that it leads to no operator mixing, and another that isnon-diagonal, as it does induce mixing. In their works, they then refer to the diagonalparts as collinear and non-diagonal ones as soft , a notation we follow. At one loop the matching amounts to evaluating the diagrams that appear in This explains the notation used in the main text, and caution is required to distinguish the two.As an example, our initial heavy DM states can have a soft wave-function type one-loop correction.But as this cannot mix operators, we call it a collinear contribution and label it as such in Eq. (7.14). EW but not NRDM-SCET 𝛾 . These diagrams can be broken into threeclasses:1. Wave-function diagrams correcting our initial non-relativistic states;2. Diagrams where a soft gauge boson is exchanged between two different externalstates; and3. Final state wave-function diagrams, which are now corrections to collinearstates.Each class will be discussed separately below. Before doing so, however, we firstdefine our operators and outline how the low-scale matching proceeds at tree level.Unlike for the high-scale matching, here we only consider the two operators thatmatch onto ℳ 𝐴 in Eq. (F.1), as opposed to the third operator coming from ℳ 𝐵 .The reason for this is the additional operator does not contribute to the low-scalematching calculation for present day DM annihilation at any order in leading powerNRDM-SCET. To understand this note that the operators coming from ℳ 𝐴 and ℳ 𝐵 have different spin structures. In order to mix these structures we need to transferangular momentum between the states. The only low-scale graphs we can write downto do this are soft gauge boson exchanges. The spin structure of the coupling of asoft exchange to an 𝑛 -collinear gauge boson is /𝑛 and the corresponding coupling toour non-relativistic DM field is /𝑣 . Neither coupling allows for a transfer of angularmomentum, demonstrating that these operators cannot mix. Unlike for the high-scalematching, we will not make use of the operator corresponding to ℳ 𝐵 for our low-scaleconsistency check, so we drop it from consideration at the outset.301 perator Definition and Tree-level Matching Prior to electroweak symmetry breaking, the two relevant operators in NRDM-SCET EW can be written schematically as: 𝒪 = 12 𝛿 𝑎𝑏 𝛿 𝑐𝑑 𝜒 𝑎 𝜒 𝑏 𝑊 𝑐 𝑊 𝑑 , 𝒪 = 14 ( 𝛿 𝑎𝑐 𝛿 𝑏𝑑 + 𝛿 𝑎𝑑 𝛿 𝑏𝑐 ) 𝜒 𝑎 𝜒 𝑏 𝑊 𝑐 𝑊 𝑑 . (F.49)Our notation here is schematic in the sense that we have suppressed the Lorentzstructure and soft Wilson lines. The form of these is written out explicitly in Eq. (7.2)and is left out for convenience as it appears in every operator written down in thisappendix. Further, in this equation the factor of / is introduced for convenience;as 𝜒 is a Majorana field this factor ensures the Feynman rule associated with theseoperators has no additional numerical factor. Note also that the gauge bosons arelabelled as they are associated with a collinear direction. At tree-level the low-scalematching is effected simply by mapping the fields in these operators onto their brokenform. Explicitly we have: 𝜒 = 1 √ (︀ 𝜒 + + 𝜒 − )︀ , 𝜒 = 𝑖 √ (︀ 𝜒 + − 𝜒 − )︀ , 𝜒 = 𝜒 ,𝑊 = 1 √ (︀ 𝑊 + + 𝑊 − )︀ , 𝑊 = 𝑖 √ (︀ 𝑊 + − 𝑊 − )︀ , 𝑊 = 𝑠 𝑊 𝐴 + 𝑐 𝑊 𝑍 . (F.50)Substituting these into Eq. (F.49) yields 22 operators in the broken theory. Of these,14 involve a 𝑊 ± in the final state, so we will not consider them further. We definethe remaining 8 as: ^ 𝒪 = 12 𝜒 𝜒 𝐴 𝐴 , ^ 𝒪 = 12 𝜒 𝜒 𝑍 𝐴 , ^ 𝒪 = 12 𝜒 𝜒 𝐴 𝑍 , ^ 𝒪 = 12 𝜒 𝜒 𝑍 𝑍 , ^ 𝒪 = 𝜒 + 𝜒 − 𝐴 𝐴 , ^ 𝒪 = 𝜒 + 𝜒 − 𝑍 𝐴 , ^ 𝒪 = 𝜒 + 𝜒 − 𝐴 𝑍 , ^ 𝒪 = 𝜒 + 𝜒 − 𝑍 𝑍 , (F.51)302here again we have used the schematic notation of Eq. (F.49), as we will for alloperators in this appendix. At tree level, the operators in Eq. (F.49) and (F.51) arerelated simply by the change of variables in Eq. (F.50). This mapping is performedby a × matrix, but again we only state the part of this matrix we are interestedin: ^ 𝐷 (0) 𝑠, − = ⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣ 𝑠 𝑊 𝑠 𝑊 𝑠 𝑊 𝑐 𝑊 𝑠 𝑊 𝑐 𝑊 𝑠 𝑊 𝑐 𝑊 𝑠 𝑊 𝑐 𝑊 𝑐 𝑊 𝑐 𝑊 𝑠 𝑊 𝑠 𝑊 𝑐 𝑊 𝑠 𝑊 𝑐 𝑊 𝑐 𝑊 ⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ . (F.52)In terms of the calculation presented in the main text, what we actually want is themapping onto the Sudakov factors Σ , defined in Eq. (7.10), not the broken operatorsin Eq. (F.51). As given there, the 𝑠 𝑊 and 𝑐 𝑊 factors are absorbed into 𝑃 𝑋 , andso will not contribute to the Σ factors. Then ^ 𝒪 − represent the contributions toneutral annihilation 𝜒 𝜒 → 𝑋 , represented by Σ − Σ , and ^ 𝒪 − the contributionsto charged annihilation 𝜒 + 𝜒 − → 𝑋 , represented by Σ . Accordingly we have: ^ 𝐷 (0) = ln ⎡⎣ ⎤⎦ . (F.53)This provides the tree-level result we could use in Eq. (7.8), where the ln is usedto remove the exponential in that equation. Next we turn to calculating this one-loop low-scale matching in full, considering the three classes of diagrams that cancontribute in turn. 303 nitial State Wave-function Graphs There are two graphs that fall under the category of initial state wave-function cor-rections, and these are shown below.Note here we follow the standard SCET conventions of drawing collinear fields asgluons with a solid line through them, whereas soft fields are represented simply bygluon lines. In these graphs, the soft gauge field can be either a 𝑊 or 𝑍 boson. Ineither case the integral to be calculated is: − 𝑔 ∫︁ ¯ 𝑑 𝑑 𝑘 𝜇 𝜖 [ 𝑘 − 𝑚 ] 𝑣 · ( 𝑘 + 𝑝 ) , (F.54)where 𝑔 is the coupling – 𝑔 for a 𝑊 boson, 𝑐 𝑊 𝑔 for a 𝑍 boson, 𝑝 is the externalmomentum, 𝑘 is the loop momentum, 𝑚 the gauge boson mass, and 𝑣 is the velocityassociated with the non-relativistic 𝜒 field. Given our initial state is heavy, this isunsurprisingly exactly the heavy quark effective theory wave-function renormalizationgraph. The analytic solution can be found in e.g. [449, 450], and using this we find: = − 𝑖𝑣 · 𝑝 𝛼 𝜋 [︂ 𝜖 + ln 𝜇 𝑚 ]︂ , (F.55)where 𝛼 = 𝑔 / 𝜋 . Now in addition to the one-loop graphs we drew above, at thisorder there will also be a counter-term of the form 𝑖𝑣 · 𝑝 ( 𝑍 𝜒 − . Again working inthe on-shell scheme so that we do not need to consider the residues, we conclude: 𝑍 𝜒 = 1 + 𝛼 ( 𝜇 )2 𝜋 [︂ 𝜖 − ln 𝑚 𝑊 𝜇 − 𝑐 𝑊 ln 𝑚 𝑍 𝜇 ]︂ . (F.56)304ow each of our initial states will contribute 𝑍 / 𝜒 , implying that the contribution to ^ 𝐷 ( 𝜇 ) given in Eq. (7.14) is ln 𝐷 𝜒𝑐 ( 𝜇 ) I , where 𝐷 𝜒𝑐 ( 𝜇 ) = 1 − 𝛼 ( 𝜇 )2 𝜋 [︂ ln 𝑚 𝑊 𝜇 + 𝑐 𝑊 ln 𝑚 𝑍 𝜇 ]︂ , (F.57)and the subscript 𝑐 indicates this is a collinear contribution in the sense that it leadsto no operator mixing. This is exactly as in Eq. (7.16) and justifies this part of thelow-scale matching. Soft Gauge Boson Exchange Graphs
In this section we calculate the contribution from the exchange of a soft 𝑊 or 𝑍 gaugeboson between different external final states. As these gauge bosons carry SU(2) L gauge indices, unsurprisingly these graphs will lead to operator mixing. Consequently,in terms of the notation introduced above these graphs will lead to non-diagonal orsoft contributions. Nonetheless they will also induce diagonal or collinear terms, andwe will carefully separate the two below.Once separated, we will group the collinear contribution with those we get fromthe final state wave-function graphs we consider in the next subsection. The reasonfor this is that these collinear contributions for photon and 𝑍 final states, as we have,were already evaluated in [403], and we will not fully recompute them here. In thatwork, however, the collinear contribution was only stated in full. The breakdowninto the soft boson exchange and final state wave-function graphs was not provided.This raises a potential issue because in that work all external states were taken tobe collinear, not non-relativistic. As such, in this section we will explicitly calculatethe soft gauge boson exchange graphs for both kinematics and demonstrate that thediagonal contribution is identical in the two cases.Before calculating the graphs, we first introduce some useful notation. At one loopthe gauge bosons will have two couplings to the four external states. Each of thesecouplings will have an associated gauge index structure, and in order to deal withthis it is convenient to introduce gauge index or color operators T . This notation was305rst introduced in [451, 452], and it allows the gauge index structure to be organizedgenerally rather than case by case. Examples can be found in the original papers andalso in e.g. [381, 403, 453]. An example relevant for our purposes is the action of T onan SU(2) L adjoint, which is the representation of both our initial and final states: T 𝜒 𝑎 = ( 𝑇 𝑐𝐴 ) 𝑎𝑎 ′ 𝜒 𝑎 ′ = − 𝑖𝜖 𝑐𝑎𝑎 ′ 𝜒 𝑎 ′ , T 𝑊 𝑎 = ( 𝑇 𝑐𝐴 ) 𝑎𝑎 ′ 𝑊 𝑎 ′ = − 𝑖𝜖 𝑐𝑎𝑎 ′ 𝑊 𝑎 ′ . (F.58)In terms of this notation then, we can write the gauge index structure of all relevantone-loop low-scale matching graphs as T 𝑖 · T 𝑗 , where 𝑖, 𝑗 label any of the four externallegs. Because of this we label the result from these soft exchange diagrams as 𝑆 𝑖𝑗 forthe case of our kinematics – non-relativistic initial states and collinear final states– and we use 𝑆 ′ 𝑖𝑗 to denote the kinematics of [403] – all external states collinear.Following [381,403], we take all external momenta to be incoming and further rapiditydivergences will be regulated with the Δ -regulator [454]. Now let us turn to the graphsone by one. 𝑆 ( ′ )12 In this graph the soft gauge boson exchanged between the initial state can be a 𝑊 or 𝑍 boson. In either case, the value of this graph is: 𝑆 = 𝛼 𝜋 T · T [︂ 𝜖 − ln 𝑚 𝜇 ]︂ , (F.59) 𝑆 ′ = 𝛼 𝜋 T · T [︂ 𝜖 − 𝜖 (︂ ln 𝛿 𝛿 𝜇 + 𝑖𝜋 )︂ −
12 ln 𝑚 𝜇 + 𝑖𝜋 ln 𝑚 𝜇 + ln 𝑚 𝜇 ln 𝛿 𝛿 𝜇 − 𝜋 ]︂ , 𝛼 = 𝑔 / 𝜋 and the identity 𝑔 and 𝑚 depend on whether this is for a 𝑊 or 𝑍 . In 𝑆 ′ , 𝛿 / are the Δ -regulators and unsurprisingly these only appear forthe collinear kinematics. 𝑆 ( ′ )13 , 𝑆 ( ′ )14 , 𝑆 ( ′ )23 , and 𝑆 ( ′ )24 Again the exchanged soft boson can be a 𝑊 or 𝑍 . These four graphs are groupedtogether as they have a common form, for example: 𝑆 = 𝛼 𝜋 T · T [︂ 𝜖 − 𝜖 ln 𝛿 𝜇 −
14 ln 𝑚 𝜇 (F.60) + 12 ln 𝛿 𝜇 ln 𝑚 𝜇 − 𝜋 ]︂ ,𝑆 ′ = 𝛼 𝜋 T · T [︂ 𝜖 − 𝜖 ln (︂ − 𝛿 𝛿 𝜇 𝑤 )︂ −
12 ln 𝑚 𝜇 + ln 𝑚 𝜇 ln (︂ − 𝛿 𝛿 𝜇 𝑤 )︂ − 𝜋 ]︂ . Then 𝑆 ( ′ )14 is given by the same expressions but with → , whilst 𝑆 ( ′ )23 and 𝑆 ( ′ )24 aregiven by similar replacements. For the all collinear case we have defined the followingfunctions of the kinematics: 𝑤 = 𝑤 ≡ 𝑛 · 𝑛 = 12 𝑛 · 𝑛 = 𝑡𝑠 ,𝑤 = 𝑤 ≡ 𝑛 · 𝑛 = 12 𝑛 · 𝑛 = 𝑢𝑠 , (F.61)where 𝑠 , 𝑡 , and 𝑢 are the Mandelstam variables relevant for all incoming momenta.307he signs inside the logs in Eq. (F.60) can be understood by noting that as 𝑡 < , 𝑢 < , whilst 𝑠 > , we have 𝑤 𝑖𝑗 < . 𝑆 ( ′ )34 Finally we have the graph above, which yields: 𝑆 = 𝛼 𝜋 T · T [︂ 𝜖 − 𝜖 (︂ ln 𝛿 𝛿 𝜇 + 𝑖𝜋 )︂ −
12 ln 𝑚 𝜇 + 𝑖𝜋 ln 𝑚 𝜇 + ln 𝑚 𝜇 ln 𝛿 𝛿 𝜇 − 𝜋 ]︂ ,𝑆 ′ = 𝑆 . (F.62)This completes the list of graphs to evaluate. As written it appears that all graphsare non-diagonal from their gauge index structure. However as we will now show, thecombinations of all graphs can be reduced to a diagonal and non-diagonal piece.Firstly for the case of all collinear external states we have: 𝑆 ′ + 𝑆 ′ + 𝑆 ′ + 𝑆 ′ + 𝑆 ′ + 𝑆 ′ ≡ ∑︁ ⟨ 𝑖𝑗 ⟩ 𝑆 ′ 𝑖𝑗 , (F.63)which serves to define ⟨ 𝑖𝑗 ⟩ . The part of this sum that involving the rapidity regulatorscan be written as 𝛼 𝜋 ln 𝑚 𝜇 ∑︁ ⟨ 𝑖𝑗 ⟩ T 𝑖 · T 𝑗 (︂ ln 𝛿 𝑖 𝜇 + ln 𝛿 𝑗 𝜇 )︂ . (F.64)This can be simplified using the following identity: ∑︁ ⟨ 𝑖𝑗 ⟩ ( 𝑓 𝑖 + 𝑓 𝑗 ) T 𝑖 · T 𝑗 = − ∑︁ 𝑖 𝑓 𝑖 T 𝑖 · T 𝑖 . (F.65) This and the gauge index identity stated below in Eq. (F.69) follow simply from the fact ∑︀ 𝑖 T 𝑖 =0 when it acts on gauge index singlet operators, see for example [381]. 𝑓 𝑖 = ln 𝛿 𝑖 /𝜇 , then Eq. (F.64) becomes: = − 𝛼 𝜋 ln 𝑚 𝜇 ∑︁ ⟨ 𝑖𝑗 ⟩ T 𝑖 · T 𝑖 ln 𝛿 𝑖 𝜇 , (F.66)which is now diagonal in the gauge indices. For the remaining terms that are inde-pendent of 𝛿 , we organize them as follows: ∑︁ ⟨ 𝑖𝑗 ⟩ 𝑆 ′ 𝑖𝑗 = 12 [ 𝑆 ′ + 𝑆 ′ + 𝑆 ′ ]+ 12 [ 𝑆 ′ + 𝑆 ′ + 𝑆 ′ ]+ 12 [ 𝑆 ′ + 𝑆 ′ + 𝑆 ′ ]+ 12 [ 𝑆 ′ + 𝑆 ′ + 𝑆 ′ ] , (F.67)where we used the fact 𝑆 ′ 𝑖𝑗 = 𝑆 ′ 𝑗𝑖 . Each of these groups can now be simplified. Forexample, the first group can be written as: 𝑆 ′ + 𝑆 ′ + 𝑆 ′ = 𝛼 𝜋 ( T · T + T · T + T · T ) × [︂ −
12 ln 𝑚 𝜇 − 𝜋 ]︂ + 𝛼 𝜋 T · T [︂ 𝑖𝜋 ln 𝑚 𝜇 ]︂ (F.68) − 𝛼 𝜋 T · T [︂ ln (︂ − 𝑡𝑠 )︂ ln 𝑚 𝜇 ]︂ − 𝛼 𝜋 T · T [︂ ln (︁ − 𝑢𝑠 )︁ ln 𝑚 𝜇 ]︂ , If we then use ∑︁ 𝑗,𝑗 ̸ = 𝑖 T 𝑖 · T 𝑗 = − T 𝑖 · T 𝑖 , (F.69)309q. (F.68) can be rewritten as: = 𝛼 𝜋 T · T [︂
12 ln 𝑚 𝜇 + 𝜋 ]︂ + 𝛼 𝜋 T · T [︂ 𝑖𝜋 ln 𝑚 𝜇 ]︂ (F.70) − 𝛼 𝜋 T · T [︂ ln (︂ − 𝑡𝑠 )︂ ln 𝑚 𝜇 ]︂ − 𝛼 𝜋 T · T [︂ ln (︁ − 𝑢𝑠 )︁ ln 𝑚 𝜇 ]︂ . Repeating this for the remaining three terms in Eq. (F.67) and reinserting the 𝛿 contributions, we can rewrite the combination of all terms as: ∑︁ ⟨ 𝑖𝑗 ⟩ 𝑆 ′ 𝑖𝑗 ≡ ∑︁ ⟨ 𝑖𝑗 ⟩ ^ 𝑆 ′ 𝑖𝑗 + ∑︁ 𝑖 𝐶 𝑖 , (F.71)where we have defined: ^ 𝑆 ′ 𝑖𝑗 ≡ − 𝛼 𝜋 ln 𝑚 𝜇 T 𝑖 · T 𝑗 𝑈 ′ 𝑖𝑗 , (F.72) 𝐶 𝑖 ≡ 𝛼 𝜋 T 𝑖 · T 𝑖 [︂
14 ln 𝑚 𝜇 + 𝜋 −
12 ln 𝑚 𝜇 ln 𝛿 𝑖 𝜇 ]︂ , and from the above we can see that: 𝑈 ′ = 𝑈 ′ = − 𝑖𝜋 ,𝑈 ′ = 𝑈 ′ = ln (︂ − 𝑡𝑠 )︂ ,𝑈 ′ = 𝑈 ′ = ln (︁ − 𝑢𝑠 )︁ . (F.73)Thus as claimed, we have reduced ∑︀ ⟨ 𝑖𝑗 ⟩ 𝑆 ′ 𝑖𝑗 in Eq. (F.71) into a diagonal and non-diagonal piece. Importantly we have explicitly isolated the collinear contribution 𝐶 𝑖 , and as we will now show we get exactly the same diagonal contribution for thekinematics of interest in this work.Before doing so, however, note that the irreducibly non-diagonal contributiongiven in Eq. (F.72) and Eq. (F.73) agrees with Eq. (150) in [381], where they gave310he general form of 𝑈 ′ 𝑖𝑗 for the case of all external collinear particles: 𝑈 ′ 𝑖𝑗 = ln − 𝑛 𝑖 · 𝑛 𝑗 − 𝑖 + . (F.74)Next we repeat this procedure for ∑︀ ⟨ 𝑖𝑗 ⟩ 𝑆 𝑖𝑗 , where we have non-relativistic fields inthe initial state. As before we consider the contribution from the rapidity regulatorsat the outset, which for 𝛿 yield: 𝛼 𝜋 ( T · T + T · T + T · T ) [︂
12 ln 𝑚 𝜇 ln 𝛿 𝜇 ]︂ = − 𝛼 𝜋 T · T [︂
12 ln 𝑚 𝜇 ln 𝛿 𝜇 ]︂ , (F.75)where we again used Eq. (F.69). An identical relation will hold for 𝛿 , and this timethere is no 𝛿 / as the non-relativistic fields do not lead to rapidity divergences. Forthe remaining terms, we now organize them as follows: ∑︁ ⟨ 𝑖𝑗 ⟩ 𝑆 𝑖𝑗 = 𝑆 + [︂ 𝑆 + 𝑆 + 12 𝑆 ]︂ + [︂ 𝑆 + 𝑆 + 12 𝑆 ]︂ . (F.76)Evaluating each of the terms in square brackets and simplifying the gauge indexstructure as before, we arrive at the following: ∑︁ ⟨ 𝑖𝑗 ⟩ 𝑆 𝑖𝑗 ≡ ∑︁ ⟨ 𝑖𝑗 ⟩ ^ 𝑆 𝑖𝑗 + 𝐶 + 𝐶 , (F.77)where we again have: ^ 𝑆 𝑖𝑗 ≡ − 𝛼 𝜋 ln 𝑚 𝜇 T 𝑖 · T 𝑗 𝑈 𝑖𝑗 , (F.78) 𝐶 𝑖 ≡ 𝛼 𝜋 T 𝑖 · T 𝑖 [︂
14 ln 𝑚 𝜇 + 𝜋 −
12 ln 𝑚 𝜇 ln 𝛿 𝑖 𝜇 ]︂ , 𝑈 = 1 ,𝑈 = − 𝑖𝜋 ,𝑈 = 𝑈 = 𝑈 = 𝑈 = 0 . (F.79)Critically, although the non-diagonal contribution is different to the case of all collinearkinematics, we see that the collinear function defined in Eq. (F.78) is identical to thatin Eq. (F.72). This justifies the claim made earlier that the diagonal part of this cal-culation is the same for both kinematics. As such we put the 𝐶 𝑖 terms aside for themoment, and return to them when we consider the final state wave-function graphs.What remains here then is to evaluate the irreducibly non-diagonal contribution: ∑︀ ⟨ 𝑖𝑗 ⟩ ^ 𝑆 𝑖𝑗 . This essentially amounts to calculating the gauge index structure, whichthe use of gauge index operators has allowed us to put off until now. In additionwe need to recall that we have a contribution to each graph from a 𝑊 and 𝑍 bosonexchange. As above we closely follow the approach in [381, 403], except accountingfor the differences in our kinematics. To this end, we begin by observing that afterelectroweak symmetry breaking the unbroken SU(2) L and U(1) 𝑌 generators, t and 𝑌 ,become 𝛼 t · t + 𝛼 𝑌 · 𝑌 → 𝛼 𝑊 ( 𝑡 + 𝑡 − + 𝑡 − 𝑡 + )+ 𝛼 𝑍 𝑡 𝑍 · 𝑡 𝑍 + 𝛼 em 𝑄 · 𝑄 , (F.80)where 𝛼 = 𝛼 em /𝑠 𝑊 , 𝛼 = 𝛼 em /𝑐 𝑊 , 𝛼 𝑊 = 𝛼 , 𝛼 𝑍 = 𝛼 /𝑐 𝑊 , and 𝑡 𝑍 = 𝑡 − 𝑠 𝑊 𝑄 . Thisimplies that we can write the full contribution as: ^ 𝐷 (1) 𝑠 = 𝛼 𝑊 ( 𝜇 )2 𝜋 ln 𝑚 𝑊 𝜇 ⎡⎣ − ∑︁ ⟨ 𝑖𝑗 ⟩
12 ( 𝑡 + 𝑡 − + 𝑡 − 𝑡 + ) 𝑈 𝑖𝑗 ⎤⎦ + 𝛼 𝑍 ( 𝜇 )2 𝜋 ln 𝑚 𝑍 𝜇 ⎡⎣ − ∑︁ ⟨ 𝑖𝑗 ⟩ 𝑡 𝑍𝑖 𝑡 𝑍𝑗 𝑈 𝑖𝑗 ⎤⎦ . (F.81)Now the contribution on the first line is more complicated, because ( 𝑡 + 𝑡 − + 𝑡 − 𝑡 + ) 𝑈 𝑖𝑗 is a non-diagonal × matrix, whereas as we will see 𝑡 𝑍𝑖 𝑡 𝑍𝑗 𝑈 𝑖𝑗 is diagonal. Never-312heless we can simplify the non-diagonal part by using the following relation:
12 ( 𝑡 + 𝑡 − + 𝑡 − 𝑡 + ) = t · t − 𝑡 · 𝑡 . (F.82)Here 𝑡 · 𝑡 is again diagonal, and whilst t · t is non-diagonal, it is written in terms ofthe unbroken operators so that we can calculate it in the unbroken theory where weonly have 2 operators not 22. Thus it is now a × matrix. In terms of this we cannow write the non-diagonal contribution to the low-scale matching as: ^ 𝐷 𝑠 = ^ 𝐷 (0) 𝑠 + ^ 𝐷 (1) 𝑠,𝑊 + ^ 𝐷 (1) 𝑠,𝑍 , ^ 𝐷 (1) 𝑠,𝑊 = 𝛼 𝑊 ( 𝜇 )2 𝜋 ln 𝑚 𝑊 𝜇 [︁ ^ 𝐷 (0) 𝑠 S + D 𝑊 ^ 𝐷 (0) 𝑠 ]︁ , ^ 𝐷 (1) 𝑠,𝑍 = 𝛼 𝑍 ( 𝜇 )2 𝜋 ln 𝑚 𝑍 𝜇 [︁ D 𝑍 ^ 𝐷 (0) 𝑠 ]︁ , (F.83)where ^ 𝐷 (0) 𝑠 is given in Eq. (F.52) and as we will now demonstrate ^ 𝐷 𝑠 is effectivelythe matrix given in Eq. (7.15) that we set out to justify. In order to do this we haveto evaluate the remaining terms: S ≡ − ∑︁ ⟨ 𝑖𝑗 ⟩ t 𝑖 · t 𝑗 𝑈 𝑖𝑗 , D 𝑊 ≡ ∑︁ ⟨ 𝑖𝑗 ⟩ t 𝑖 · t 𝑗 𝑈 𝑖𝑗 , D 𝑍 ≡ − ∑︁ ⟨ 𝑖𝑗 ⟩ t 𝑍𝑖 · t 𝑍𝑗 𝑈 𝑖𝑗 . (F.84)The form of each of these matrices can be evaluated by acting with them on theoperators – the unbroken operators in Eq. (F.49) for S and the broken operatorsin Eq. (F.51) for D 𝑊/𝑍 – where the action of the gauge index operators is given byEq. (F.58). Doing this, we find: S = ⎡⎣ − 𝑖𝜋 − 𝑖𝜋 𝑖𝜋 − ⎤⎦ , (F.85)313hilst D 𝑊, − = diag (0 , , , , − , − , − , − , D 𝑍 = − 𝑐 𝑊 D 𝑊 . (F.86)Substituting these results into Eq. (F.83), we find: ^ 𝐷 𝑠, − = ⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣ 𝑠 𝑊 [1 + 𝐺 ( 𝜇 )] 𝑠 𝑊 𝑠 𝑊 𝑐 𝑊 [1 + 𝐺 ( 𝜇 )] 𝑠 𝑊 𝑐 𝑊 𝑠 𝑊 𝑐 𝑊 [1 + 𝐺 ( 𝜇 )] 𝑠 𝑊 𝑐 𝑊 𝑐 𝑊 [1 + 𝐺 ( 𝜇 )] 𝑐 𝑊 𝑠 𝑊 [1 + 𝐻 ( 𝜇 )] 𝑠 𝑊 𝐼 ( 𝜇 ) 𝑠 𝑊 𝑐 𝑊 [1 + 𝐻 ( 𝜇 )] 𝑠 𝑊 𝑐 𝑊 𝐼 ( 𝜇 ) 𝑠 𝑊 𝑐 𝑊 [1 + 𝐻 ( 𝜇 )] 𝑠 𝑊 𝑐 𝑊 𝐼 ( 𝜇 ) 𝑐 𝑊 [1 + 𝐻 ( 𝜇 )] 𝑐 𝑊 𝐼 ( 𝜇 ) ⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ , (F.87)where we have defined: 𝐺 ( 𝜇 ) ≡ 𝛼 𝑊 ( 𝜇 )2 𝜋 ln 𝑚 𝑊 𝜇 (2 − 𝑖𝜋 ) ,𝐻 ( 𝜇 ) ≡ 𝛼 𝑊 ( 𝜇 )2 𝜋 ln 𝑚 𝑊 𝜇 (1 − 𝑖𝜋 )+ 𝑐 𝑊 𝛼 𝑍 ( 𝜇 )2 𝜋 ln 𝑚 𝑍 𝜇 ,𝐼 ( 𝜇 ) ≡ 𝛼 𝑊 𝜋 ln 𝑚 𝑊 𝜇 (1 − 𝑖𝜋 ) . (F.88)From the form of ^ 𝐷 𝑠 given in Eq. (F.87), we can again reduce this to a × matrixwhich maps onto Σ and Σ − Σ , exactly as we did for the tree-level low-scalematching. Doing this, the × matrix we obtain is exactly Eq. (7.15), which we setout to justify. Final State Wave-function Graphs
Finally we have the last contribution, which is the combination of final state wave-function graphs as well as 𝐶 + 𝐶 , as defined in Eq. (F.78). As mentioned in the314revious subsection, this calculation has already been performed in [403], and giventhat the form of 𝐶 𝑖 is the same for our kinematics as it is for theirs, we take theresult from their work. In that paper they calculated this collinear contribution forall possible weak bosons. For our calculation we are only interested in a final statephoton or 𝑍 , for which they give: 𝐷 𝑍𝑐 = 𝛼 𝜋 [︂ 𝐹 𝑊 + 𝑓 𝑆 (︂ 𝑚 𝑍 𝑚 𝑊 , )︂]︂ + 12 𝛿 R 𝑍 + tan ¯ 𝜃 𝑊 R 𝛾 → 𝑍 ,𝐷 𝛾𝑐 = 𝛼 𝜋 [ 𝐹 𝑊 + 𝑓 𝑆 (0 , 𝛿 R 𝛾 + cot ¯ 𝜃 𝑊 R 𝑍 → 𝛾 . (F.89)The various terms in these equations are outlined below. Nonetheless, once the fullexpressions are written out the analytic result for the terms in Eq. (7.18) can beextracted as the terms independent of ln 𝜇 .To begin with we have: 𝐹 𝑊 ≡ ln 𝑚 𝑊 𝜇 ln 𝑠𝜇 −
12 ln 𝑚 𝑊 𝜇 − ln 𝑚 𝑊 𝜇 − 𝜋
12 + 1 , (F.90)where note for our calculation 𝑠 = 4 𝑚 𝜒 . Next 𝑓 𝑆 ( 𝑤, 𝑧 ) is defined as: 𝑓 𝑆 ( 𝑤, 𝑧 ) ≡ ∫︁ 𝑑𝑥 (2 − 𝑥 ) 𝑥 ln 1 − 𝑥 + 𝑧𝑥 − 𝑤𝑥 (1 − 𝑥 )1 − 𝑥 , (F.91)such that an explicit calculation gives us 𝑓 𝑆 (︂ 𝑚 𝑍 𝑚 𝑊 , )︂ = 1 . ,𝑓 𝑆 (0 ,
1) = 𝜋 − . (F.92)315inally the R contributions are defined by: 𝛿 R 𝑍 ≡ Π ′ 𝑍𝑍 ( 𝑚 𝑍 ) ,𝛿 R 𝛾 ≡ Π ′ 𝛾𝛾 (0) , R 𝛾 → 𝑍 ≡ 𝑚 𝑍 Π 𝑍𝛾 ( 𝑚 𝑍 ) , R 𝑍 → 𝛾 ≡ − 𝑚 𝑍 Π 𝛾𝑍 (0) , (F.93)where Π ′ ≡ 𝜕 Π( 𝑘 ) /𝜕𝑘 and the various Π functions are defined via the inverse of thetransverse gauge boson propagator − 𝑖 (︂ 𝑔 𝜇𝜈 − 𝑘 𝜇 𝑘 𝜈 𝑘 )︂ ⎡⎣ 𝑘 − 𝑚 𝑍 − Π 𝑍𝑍 ( 𝑘 ) − Π 𝑍𝛾 ( 𝑘 ) − Π 𝛾𝑍 ( 𝑘 ) 𝑘 − Π 𝛾𝛾 ( 𝑘 ) ⎤⎦ . (F.94)The form of the Π functions is not given explicitly in [403], but can be determinedfrom the results of e.g. [448, 455]. When doing so, there are two factors that must beaccounted for. Firstly the Π functions must be calculated in MS . This is because [403]accounts for the residues explicitly in (F.89). If we used the on-shell scheme forexternal particles, as we did for the high-scale matching, we would double countthe contribution from the residues. Secondly we need to respect that the low-scalematching is performed above and below the electroweak scale, which means the Π functions for the photon and 𝑍 must be treated differently. Above the matching scalethe 𝑊 , 𝑍 , top and Higgs are dynamical degrees of freedom, but below it they are not.Light degrees of freedom like the photon, bottom quark or electron are dynamicalabove and below. This means for the 𝑍 contributions, we need to include all degreesof freedom – heavy and light – in the loops, as the 𝑍 itself does not propagate belowthe matching. For the photon contributions, however, only the heavy degrees of Note there is a typo in Eq. B2 of [403], where R 𝛾 → 𝑍 and R 𝑍 → 𝛾 involved Π ′ rather than Π . Theexpressions stated here are the correct ones, and we thank Aneesh Manohar for confirming this andfor providing a numerical cross check on our results for these terms. 𝛿 R 𝑍 = 𝛼 𝜋 [︂ − 𝑠 𝑊 + 46 𝑠 𝑊 𝑐 𝑊 ln 𝑚 𝑍 𝜇 𝑍 +1 . − . 𝑖 ]︂ ,𝛿 R 𝛾 = 𝛼 𝜋 [︂ − 𝑠 𝑊 ln 𝑚 𝑍 𝜇 𝑍 + 0 . ]︂ , R 𝛾 → 𝑍 = 𝛼 𝜋 [︂ − 𝑠 𝑊 + 34 𝑠 𝑊 𝑐 𝑊 tan ¯ 𝜃 𝑊 ln 𝑚 𝑍 𝜇 𝑍 +0 . − . 𝑖 ]︂ , R 𝑍 → 𝛾 = 𝛼 𝜋 [︂ 𝑠 𝑊 𝑐 𝑊 ln 𝑚 𝑍 𝜇 𝑍 − . ]︂ . (F.95)Analytic forms for the Π functions are provided in App. F.5, we do not provide thefull expressions here as they are lengthy. In order to determine the numerical valuesabove we have used the following: 𝑚 𝑍 = 91 . ,𝑚 𝑊 = 80 .
385 GeV ,𝑚 𝑡 = 173 .
21 GeV ,𝑚 𝐻 = 125 GeV ,𝑚 𝑏 = 4 .
18 GeV ,𝑚 𝑐 = 1 .
275 GeV ,𝑚 𝜏 = 1 . ,𝑚 𝑠 = 𝑚 𝑑 = 𝑚 𝑢 = 𝑚 𝜇 = 𝑚 𝑒 = 0 GeV ,𝑐 𝑊 = 𝑚 𝑊 /𝑚 𝑍 . (F.96)This completes the list of ingredients for Eq. (F.89). Substituting them into thatequation gives exactly the relevant terms in Eqs. (7.16), (7.17), and (7.18), justifyingthe collinear part of the low-scale matching.We have now justified each of the pieces making up the low-scale one-loop match-317ng. All that remains is to cross check this result, which we turn to in the nextappendix. F.4 Consistency Check on the Low-Scale Matching
In this appendix we provide a cross check on the low-scale one-loop matching calcula-tion, much as we did for the high-scale result in App. F.2. Given that we already crosschecked the high-scale result, we here make use of that to determine whether the ln 𝜇 contributions at the low scale are correct. In order to do this, we take Eq. (7.8) andturn off the running, which amounts to setting 𝜇 𝑚 𝜒 = 𝜇 𝑍 ≡ 𝜇 . In detail we obtain: ⎡⎣ 𝐶 𝑋 ± 𝐶 𝑋 ⎤⎦ = 𝑒 ^ 𝐷 𝑋 ( 𝜇 ) ⎡⎣ 𝐶 ( 𝜇 ) 𝐶 ( 𝜇 ) ⎤⎦ . (F.97)Now as we have the full one-loop result, the ln 𝜇 dependence between these two termsmust cancel at 𝒪 ( 𝛼 ) for any 𝑋 , which we will now demonstrate.Before doing this in general, we first consider the simpler case where electroweaksymmetry remains unbroken and we just have a 𝑊 𝑊 final state. In this case, asin general, to capture all 𝜇 dependence at 𝒪 ( 𝛼 ) we also need to account for the 𝛽 -function. If SU(2) L remains unbroken, however, this is just simply captured in: 𝛼 ( 𝜇 ) = 𝛼 ( 𝑚 𝑍 ) + 𝛼 ( 𝑚 𝑍 ) 𝑏 𝜋 ln 𝑚 𝑍 𝜇 , (F.98)where 𝑏 = (43 − 𝑛 𝐷 ) / , with 𝑛 𝐷 the number of SM doublets. This follows directlyfrom Eq. (F.35). In the unbroken theory we can simply set 𝑐 𝑊 = 1 and 𝑠 𝑊 = 0 , soif we do this and substitute our results from Eqs. (7.13), (7.14), (7.15), (7.16), (7.17)into Eq. (F.97), then we find: 𝐶 𝑊 ± = 1 𝑚 𝜒 (︂ 𝑏 𝑐 𝑊 − )︂ ln 𝜇 + 𝜇 − ind . ,𝐶 𝑊 = 𝜇 − ind . , (F.99)318ow we can calculate that 𝑐 𝑊 = (2 𝑛 𝐷 − / , which taking 𝑛 𝐷 = 12 exactlyagrees with 𝑐 𝑍 in Eq. (7.17) when 𝑐 𝑊 = 1 and 𝑠 𝑊 = 0 as it must. Then recalling 𝑏 from above we see that both coefficients are then 𝜇 independent at this order,demonstrating the required consistency.We now consider the same cross check in the full broken theory. The addedcomplication here is that for our different final states, 𝛾𝛾 , 𝛾𝑍 , and 𝑍𝑍 , the couplingis actually 𝑠 𝑊 𝛼 , 𝑠 𝑊 𝑐 𝑊 𝛼 , and 𝑐 𝑊 𝛼 respectively. As we work in MS , we need toaccount for the fact that 𝑠 𝑊 and 𝑐 𝑊 are functions 𝜇 . Explicit calculation demonstratesthat the running is only relevant for the consistency of 𝐶 𝑋 ± – the cancellation in 𝐶 𝑋 is independent of the 𝛽 -function at this order – and in fact we find: 𝐶 𝑋 ± = 1 𝑚 𝜒 (︃ 𝑏 ( 𝑋 )0 ∑︁ 𝑖 ∈ 𝑋 𝑐 𝑖 − )︃ ln 𝜇 + 𝜇 − ind . . (F.100)To derive this we simply used Eq. (F.98), with 𝑏 → 𝑏 ( 𝑋 )0 , leaving us to derive theappropriate for of 𝑏 ( 𝑋 )0 for 𝑋 = 𝛾𝛾 , 𝛾𝑍 , 𝑍𝑍 . Firstly note that 𝑠 𝑊 ( 𝜇 ) = 𝛼 ( 𝜇 ) 𝛼 ( 𝜇 ) + 𝛼 ( 𝜇 ) ,𝑐 𝑊 ( 𝜇 ) = 𝛼 ( 𝜇 ) 𝛼 ( 𝜇 ) + 𝛼 ( 𝜇 ) , (F.101)where 𝛼 is the U(1) 𝑌 coupling. We can write a similar expression to Eq. (F.98) for 𝛼 , but this time we have 𝑏 (1)0 = − / . To avoid confusion we also now refer to theSU(2) L 𝑏 as 𝑏 (2)0 = 19 / .Now for the case of two 𝑍 bosons in the final state, the appropriate 𝛽 -function is: 𝛽 𝑍𝑍 = 𝜇 𝑑𝑑𝜇 [︀ 𝑐 𝑊 𝛼 ]︀ . (F.102)319ombining this with Eq. (F.101), we conclude that: 𝑏 ( 𝑍𝑍 )0 = (︀ 𝑠 𝑊 + 1 )︀ 𝑏 (2)0 − 𝑠 𝑊 𝑐 𝑊 𝑏 (1)0 = 19 + 22 𝑠 𝑊 𝑐 𝑊 . (F.103)There is an additional factor of 𝑐 𝑊 in this expression than if we were just calculatingthe 𝛽 -function for 𝛼 𝑍 . The reason for this is that 𝑏 ( 𝑍𝑍 )0 is the appropriate replacementfor 𝑏 in Eq. (F.98), which represents the correction to 𝛼 = 𝑐 𝑊 𝛼 𝑍 not 𝛼 𝑍 . Substitut-ing this into Eq. (F.100) along with the definition of 𝑐 𝑍 from Eq. (7.17) demonstratesconsistency for the 𝑍𝑍 case.The case of two final state photons has to be treated differently, because of thefact our low-scale matching integrated out the electroweak degrees of freedom, whichdid not include the photon. This means we need to use a modified version of theSU(2) L and U(1) 𝑌 couplings that only include the running due to the modes beingremoved. This amounts to accounting for the running from the Higgs, 𝑊 and 𝑍 bosons, and the top quark, which we treat as an SU(2) L singlet Dirac fermion toensure it is entirely removed through the matching. Doing so, the SM 𝛽 -functionsnow evaluate to 𝑏 (2) ′ = 43 / and 𝑏 (1) ′ = − / . Repeating the same calculation aswe used to determine 𝑏 ( 𝑍𝑍 )0 , we find that: 𝑏 ( 𝛾𝛾 )0 = (︁ 𝑏 (1) ′ + 𝑏 (2) ′ )︁ 𝑠 𝑊 = 479 𝑠 𝑊 . (F.104)Again, substituting this into Eq. (F.100) shows that the two photon case is also con-sistent. The final case 𝛾𝑍 , but it is straightforward to see that in this case Eq. (F.100)breaks into two conditions that are satisfied if the 𝑍𝑍 and 𝛾𝛾 cases are, so this is notan independent cross check.As such, in the absence of running, all the 𝜇 dependence in our calculation vanishesat 𝒪 ( 𝛼 ) , as it must. But we emphasize that this is a non-trivial cross check, thatinvolves all aspects of the calculation in the full broken theory.320 .5 Analytic Form of Π Here we state the analytic expressions for the MS electroweak Π functions for photonand 𝑍 boson, appropriate for the matching from SCET EW to SCET 𝛾 . These resultscan be determined using standard references, such as [448, 455]. As the photon is adynamical degree of freedom above and below the matching, we only need to con-sider loop diagrams involving electroweak modes that are integrated out through thematching. This simplifies the evaluation, and we have the following two functions: Π ′ 𝛾𝛾 (0) = 𝛼 𝑠 𝑊 𝜋 {︂ −
169 ln 𝜇 𝑚 𝑡 + 3 ln 𝜇 𝑚 𝑊 + 23 }︂ , Π 𝛾𝑍 (0) = 𝛼 𝑠 𝑊 𝜋 {︂ 𝑚 𝑊 𝑠 𝑊 𝑐 𝑊 ln 𝜇 𝑚 𝑊 }︂ . (F.105)As the 𝑍 itself is being integrated out, we need to include all relevant loops whencalculating Π 𝑍𝛾 and Π ′ 𝑍𝑍 . In order to simplify the expressions, we firstly introducethe following expressions: 𝛽 ≡ √︂ 𝑚 𝑠 − , 𝜉 ≡ √︂ − 𝑚 𝑠 , (F.106) 𝜆 ± ≡ 𝑠 (︂ 𝑠 − 𝑚 + 𝑚 ± √︁ ( 𝑠 − 𝑚 + 𝑚 ) − 𝑠 ( 𝑚 − 𝑖𝜖 ) )︂ . In terms of these we then define: 𝑎 ( 𝑚 , 𝑚 ) ≡ 𝑚 𝑚 − 𝑚 ln 𝑚 𝑚 , (F.107) 𝑏 ( 𝑠, 𝑚 ) ≡ 𝑖𝛽 ln (︂ 𝛽 + 𝑖𝛽 − 𝑖 )︂ , 𝑏 ( 𝑠, 𝑚 ) ≡ − 𝜉 ln 1 + 𝜉 − 𝜉 + 𝑖𝜋𝜉 ,𝑐 ( 𝑠, 𝑚 ) ≡ − 𝑚 𝑠 𝛽 (︂ 𝛽 𝛽 + 𝑖 ln 𝛽 + 𝑖𝛽 − 𝑖 )︂ , 𝑐 ( 𝑠, 𝑚 ) ≡ 𝑚 𝑠 𝜉 (︂ 𝜉𝜉 − − ln 1 + 𝜉 − 𝜉 )︂ ,𝑑 ( 𝑠, 𝑚 , 𝑚 ) ≡ 𝜆 + ln (︂ 𝜆 + − 𝜆 + )︂ − ln ( 𝜆 + −
1) + 𝜆 − ln (︂ 𝜆 − − 𝜆 − )︂ − ln ( 𝜆 − − ,𝑒 ( 𝑠, 𝑚 , 𝑚 ) ≡ − 𝑠 + ln (︂ 𝜆 + − 𝜆 + )︂ 𝜕𝜆 + 𝜕𝑠 + ln (︂ 𝜆 − − 𝜆 − )︂ 𝜕𝜆 − 𝜕𝑠 . Π 𝑍𝛾 ( 𝑚 𝑍 ) = 𝛼 𝑠 𝑊 𝜋 {︂ − 𝑠 𝑊 𝑐 𝑊 𝑠 𝑊 [︂ 𝑚 𝑍 − 𝑚 𝑍 ln 𝜇 𝑚 𝑡 − ( 𝑚 𝑍 + 2 𝑚 𝑡 ) 𝑏 ( 𝑚 𝑍 , 𝑚 𝑡 ) ]︂ + 3 − 𝑠 𝑊 𝑐 𝑊 𝑠 𝑊 [︂ 𝑚 𝑍 − 𝑚 𝑍 ln 𝜇 𝑚 𝑏 − ( 𝑚 𝑍 + 2 𝑚 𝑏 ) 𝑏 ( 𝑚 𝑍 , 𝑚 𝑏 ) ]︂ + 6 − 𝑠 𝑊 𝑐 𝑊 𝑠 𝑊 [︂ 𝑚 𝑍 − 𝑚 𝑍 ln 𝜇 𝑚 𝑐 − ( 𝑚 𝑍 + 2 𝑚 𝑐 ) 𝑏 ( 𝑚 𝑍 , 𝑚 𝑐 ) ]︂ + 1 − 𝑠 𝑊 𝑐 𝑊 𝑠 𝑊 [︂ 𝑚 𝑍 − 𝑚 𝑍 ln 𝜇 𝑚 𝜏 − ( 𝑚 𝑍 + 2 𝑚 𝜏 ) 𝑏 ( 𝑚 𝑍 , 𝑚 𝜏 ) ]︂ + 𝑚 𝑍 𝑠 𝑊 − 𝑐 𝑊 𝑠 𝑊 [︂
53 + 𝑖𝜋 + ln 𝜇 𝑚 𝑍 ]︂ + 13 𝑠 𝑊 𝑐 𝑊 {︂[︂(︂ 𝑐 𝑊 + 12 )︂ 𝑚 𝑍 + (︀ 𝑐 𝑊 + 4 )︀ 𝑚 𝑊 ]︂ × (︂ ln 𝜇 𝑚 𝑊 + 𝑏 ( 𝑚 𝑍 , 𝑚 𝑊 ) )︂ − (12 𝑐 𝑊 − 𝑚 𝑊 ln 𝜇 𝑚 𝑊 + 13 𝑚 𝑍 }︂}︂ , (F.108)322nd finally Π ′ 𝑍𝑍 ( 𝑚 𝑍 ) = 𝛼 𝑠 𝑊 𝜋 {︂ {︂ − 𝑠 𝑊 + 32 𝑠 𝑊 𝑐 𝑊 𝑠 𝑊 [︂ − ln 𝜇 𝑚 𝑡 − 𝑏 ( 𝑚 𝑍 , 𝑚 𝑡 ) − ( 𝑚 𝑍 + 2 𝑚 𝑡 ) 𝑐 ( 𝑚 𝑍 , 𝑚 𝑡 ) + 13 ]︂ + 34 𝑠 𝑊 𝑐 𝑊 𝑚 𝑡 𝑐 ( 𝑚 𝑍 , 𝑚 𝑡 ) }︂ + 2 {︂ − 𝑠 𝑊 + 8 𝑠 𝑊 𝑐 𝑊 𝑠 𝑊 [︂ − ln 𝜇 𝑚 𝑏 − 𝑏 ( 𝑚 𝑍 , 𝑚 𝑏 ) − ( 𝑚 𝑍 + 2 𝑚 𝑏 ) 𝑐 ( 𝑚 𝑍 , 𝑚 𝑏 ) + 13 ]︂ + 34 𝑠 𝑊 𝑐 𝑊 𝑚 𝑏 𝑐 ( 𝑚 𝑍 , 𝑚 𝑏 ) }︂ + 2 {︂ − 𝑠 𝑊 + 32 𝑠 𝑊 𝑐 𝑊 𝑠 𝑊 [︂ − ln 𝜇 𝑚 𝑐 − 𝑏 ( 𝑚 𝑍 , 𝑚 𝑐 ) − ( 𝑚 𝑍 + 2 𝑚 𝑐 ) 𝑐 ( 𝑚 𝑍 , 𝑚 𝑐 ) + 13 ]︂ + 34 𝑠 𝑊 𝑐 𝑊 𝑚 𝑐 𝑐 ( 𝑚 𝑍 , 𝑚 𝑐 ) }︂ + 23 {︂ − 𝑠 𝑊 + 8 𝑠 𝑊 𝑐 𝑊 𝑠 𝑊 [︂ − ln 𝜇 𝑚 𝜏 − 𝑏 ( 𝑚 𝑍 , 𝑚 𝜏 ) − ( 𝑚 𝑍 + 2 𝑚 𝜏 ) 𝑐 ( 𝑚 𝑍 , 𝑚 𝜏 ) + 13 ]︂ + 34 𝑠 𝑊 𝑐 𝑊 𝑚 𝜏 𝑐 ( 𝑚 𝑍 , 𝑚 𝜏 ) }︂ + 7 − 𝑠 𝑊 + 16 𝑠 𝑊 𝑠 𝑊 𝑐 𝑊 [︂ − − ln 𝜇 𝑚 𝑍 − 𝑖𝜋 ]︂ + 16 𝑠 𝑊 𝑐 𝑊 {︂(︂ 𝑐 𝑊 + 2 𝑐 𝑊 − )︂ (︂ ln 𝜇 𝑚 𝑊 + 𝑏 ( 𝑚 𝑍 , 𝑚 𝑊 ) )︂ + 13 (︀ 𝑐 𝑊 − )︀ + [︂(︂ 𝑐 𝑊 + 2 𝑐 𝑊 − )︂ 𝑚 𝑍 + (︀ 𝑐 𝑊 + 16 𝑐 𝑊 − )︀ 𝑚 𝑊 ]︂ 𝑐 ( 𝑚 𝑍 , 𝑚 𝑊 ) }︂ + 112 𝑠 𝑊 𝑐 𝑊 {︂ − (︂ ln 𝜇 𝑚 𝑍 + 𝑑 ( 𝑚 𝑍 , 𝑚 𝑍 , 𝑚 𝐻 ) )︂ + (︀ 𝑚 𝐻 − 𝑚 𝑍 )︀ 𝑒 ( 𝑚 𝑍 , 𝑚 𝑍 , 𝑚 𝐻 ) − ( 𝑚 𝑍 − 𝑚 𝐻 ) 𝑚 𝑍 𝑒 ( 𝑚 𝑍 , 𝑚 𝑍 , 𝑚 𝐻 ) −
23+ ( 𝑚 𝑍 − 𝑚 𝐻 ) 𝑚 𝑍 (︂ ln 𝑚 𝐻 𝑚 𝑍 + 𝑑 ( 𝑚 𝑍 , 𝑚 𝑍 , 𝑚 𝐻 ) − 𝑎 ( 𝑚 𝑍 , 𝑚 𝐻 ) )︂}︂}︂ . (F.109)32324 ibliography [1] G. Bertone and D. Hooper, A History of Dark Matter , Submitted to: Rev.Mod. Phys. (2016) [ arXiv:1605.04909 ].[2] S. Tremaine and J. E. Gunn,
Dynamical Role of Light Neutral Leptons inCosmology , Phys. Rev. Lett. (1979) 407–410.[3] M. R. Buckley and A. H. G. Peter, Gravitational probes of dark matterphysics , arXiv:1712.06615 .[4] F. Kahlhoefer, Review of LHC Dark Matter Searches , Int. J. Mod. Phys.
A32 (2017), no. 13 1730006, [ arXiv:1702.02430 ].[5] T. Marrodán Undagoitia and L. Rauch,
Dark matter direct-detectionexperiments , J. Phys.
G43 (2016), no. 1 013001, [ arXiv:1509.08767 ].[6] G. F. Giudice, M. A. Luty, H. Murayama, and R. Rattazzi,
Gaugino masswithout singlets , JHEP (1998) 027, [ hep-ph/9810442 ].[7] L. Randall and R. Sundrum, Out of this world supersymmetry breaking , Nucl.Phys.
B557 (1999) 79–118, [ hep-th/9810155 ].[8] L. Hui, J. P. Ostriker, S. Tremaine, and E. Witten,
Ultralight scalars ascosmological dark matter , Phys. Rev.
D95 (2017), no. 4 043541,[ arXiv:1610.08297 ].[9] R. D. Peccei and H. R. Quinn,
CP Conservation in the Presence ofInstantons , Phys. Rev. Lett. (1977) 1440–1443.32510] R. D. Peccei and H. R. Quinn, Constraints Imposed by CP Conservation inthe Presence of Instantons , Phys. Rev.
D16 (1977) 1791–1797.[11] S. Weinberg,
A New Light Boson? , Phys. Rev. Lett. (1978) 223–226.[12] F. Wilczek, Problem of Strong p and t Invariance in the Presence ofInstantons , Phys. Rev. Lett. (1978) 279–282.[13] J. Preskill, M. B. Wise, and F. Wilczek, Cosmology of the Invisible Axion , Phys. Lett. (1983) 127–132.[14] L. F. Abbott and P. Sikivie,
A Cosmological Bound on the Invisible Axion , Phys. Lett. (1983) 133–136.[15] M. Dine and W. Fischler,
The Not So Harmless Axion , Phys. Lett. (1983) 137–141.[16] J. W. Foster, N. L. Rodd, and B. R. Safdi,
Revealing the Dark Matter Halowith Axion Direct Detection , arXiv:1711.10489 .[17] J. E. Gunn, B. W. Lee, I. Lerche, D. N. Schramm, and G. Steigman, SomeAstrophysical Consequences of the Existence of a Heavy Stable Neutral Lepton , Astrophys. J. (1978) 1015–1031.[18] F. W. Stecker,
The Cosmic Gamma-Ray Background from the Annihilation ofPrimordial Stable Neutral Heavy Leptons , Astrophys. J. (1978) 1032–1036.[19] T. R. Slatyer,
TASI Lectures on Indirect Detection of Dark Matter , in
Theoretical Advanced Study Institute in Elementary Particle Physics:Anticipating the Next Discoveries in Particle Physics (TASI 2016) Boulder,CO, USA, June 6-July 1, 2016 , 2017. arXiv:1710.05137 .[20] A. Evans, “M31, the andromeda galaxy (now with h-alpha).” . Accessed:2018-03-14. 32621] T. R. Slatyer,
Indirect dark matter signatures in the cosmic dark ages. I.Generalizing the bound on s-wave dark matter annihilation from Planckresults , Phys. Rev.
D93 (2016), no. 2 023527, [ arXiv:1506.03811 ].[22] T. R. Slatyer,
Indirect Dark Matter Signatures in the Cosmic Dark Ages II.Ionization, Heating and Photon Production from Arbitrary Energy Injections , Phys. Rev.
D93 (2016), no. 2 023521, [ arXiv:1506.03812 ].[23] T. R. Slatyer and C.-L. Wu,
General Constraints on Dark Matter Decay fromthe Cosmic Microwave Background , Phys. Rev.
D95 (2017), no. 2 023010,[ arXiv:1610.06933 ].[24] H. Liu, T. R. Slatyer, and J. Zavala,
Contributions to cosmic reionization fromdark matter annihilation and decay , Phys. Rev.
D94 (2016), no. 6 063507,[ arXiv:1604.02457 ].[25] M. Lisanti, S. Mishra-Sharma, N. L. Rodd, B. R. Safdi, and R. H. Wechsler,
Mapping Extragalactic Dark Matter Annihilation with Galaxy Surveys: ASystematic Study of Stacked Group Searches , arXiv:1709.00416 .[26] J. Hisano, S. Matsumoto, and M. M. Nojiri, Explosive dark matterannihilation , Phys. Rev. Lett. (2004) 031303, [ hep-ph/0307216 ].[27] J. Hisano, S. Matsumoto, M. M. Nojiri, and O. Saito, Non-perturbative effecton dark matter annihilation and gamma ray signature from galactic center , Phys. Rev.
D71 (2005) 063528, [ hep-ph/0412403 ].[28] M. Cirelli, A. Strumia, and M. Tamburini,
Cosmology and Astrophysics ofMinimal Dark Matter , Nucl. Phys.
B787 (2007) 152–175, [ arXiv:0706.4071 ].[29] N. Arkani-Hamed, D. P. Finkbeiner, T. R. Slatyer, and N. Weiner,
A Theoryof Dark Matter , Phys. Rev.
D79 (2009) 015014, [ arXiv:0810.0713 ].[30] K. Blum, R. Sato, and T. R. Slatyer,
Self-consistent Calculation of theSommerfeld Enhancement , JCAP (2016), no. 06 021,[ arXiv:1603.01383 ]. 32731] K. K. Boddy, J. Kumar, L. E. Strigari, and M.-Y. Wang,
Sommerfeld-Enhanced 𝐽 -Factors For Dwarf Spheroidal Galaxies , Phys. Rev.
D95 (2017), no. 12 123008, [ arXiv:1702.00408 ].[32] K. Boddy, J. Kumar, D. Marfatia, and P. Sandick,
Model-independentconstraints on dark matter annihilation in dwarf spheroidal galaxies , arXiv:1802.03826 .[33] M. Lisanti, S. Mishra-Sharma, N. L. Rodd, and B. R. Safdi, A Search forDark Matter Annihilation in Galaxy Groups , arXiv:1708.09385 .[34] E. Fermi, On the Origin of the Cosmic Radiation , Phys. Rev. (1949)1169–1174.[35] Fermi-LAT
Collaboration, M. Ackermann et al.,
The spectrum of isotropicdiffuse gamma-ray emission between 100 MeV and 820 GeV , Astrophys. J. (2015) 86, [ arXiv:1410.3696 ].[36] M. Su, T. R. Slatyer, and D. P. Finkbeiner,
Giant Gamma-ray Bubbles fromFermi-LAT: AGN Activity or Bipolar Galactic Wind? , Astrophys. J. (2010) 1044–1082, [ arXiv:1005.5480 ].[37] G. Cowan, K. Cranmer, E. Gross, and O. Vitells,
Asymptotic formulae forlikelihood-based tests of new physics , Eur. Phys. J.
C71 (2011) 1554,[ arXiv:1007.1727 ]. [Erratum: Eur. Phys. J.C73,2501(2013)].[38] L. Goodenough and D. Hooper,
Possible Evidence For Dark MatterAnnihilation In The Inner Milky Way From The Fermi Gamma Ray SpaceTelescope , arXiv:0910.2998 .[39] D. Hooper and L. Goodenough, Dark Matter Annihilation in The GalacticCenter As Seen by the Fermi Gamma Ray Space Telescope , Phys. Lett.
B697 (2011) 412–428, [ arXiv:1010.2752 ].32840] A. Boyarsky, D. Malyshev, and O. Ruchayskiy,
A comment on the emissionfrom the Galactic Center as seen by the Fermi telescope , Phys. Lett.
B705 (2011) 165–169, [ arXiv:1012.5839 ].[41] D. Hooper and T. Linden,
On The Origin Of The Gamma Rays From TheGalactic Center , Phys. Rev.
D84 (2011) 123005, [ arXiv:1110.0006 ].[42] K. N. Abazajian and M. Kaplinghat,
Detection of a Gamma-Ray Source in theGalactic Center Consistent with Extended Emission from Dark MatterAnnihilation and Concentrated Astrophysical Emission , Phys. Rev.
D86 (2012) 083511, [ arXiv:1207.6047 ]. [Erratum: Phys. Rev.D87,129902(2013)].[43] C. Gordon and O. Macias,
Dark Matter and Pulsar Model Constraints fromGalactic Center Fermi-LAT Gamma Ray Observations , Phys. Rev.
D88 (2013), no. 8 083521, [ arXiv:1306.5725 ]. [Erratum: Phys.Rev.D89,no.4,049901(2014)].[44] K. N. Abazajian, N. Canac, S. Horiuchi, and M. Kaplinghat,
Astrophysicaland Dark Matter Interpretations of Extended Gamma-Ray Emission from theGalactic Center , Phys. Rev.
D90 (2014), no. 2 023526, [ arXiv:1402.4090 ].[45] T. Daylan, D. P. Finkbeiner, D. Hooper, T. Linden, S. K. N. Portillo, N. L.Rodd, and T. R. Slatyer,
The characterization of the gamma-ray signal fromthe central Milky Way: A case for annihilating dark matter , Phys. Dark Univ. (2016) 1–23, [ arXiv:1402.6703 ].[46] S. K. Lee, M. Lisanti, B. R. Safdi, T. R. Slatyer, and W. Xue, Evidence forUnresolved 𝛾 -Ray Point Sources in the Inner Galaxy , Phys. Rev. Lett. (2016), no. 5 051103, [ arXiv:1506.05124 ].[47] R. Bartels, S. Krishnamurthy, and C. Weniger,
Strong support for themillisecond pulsar origin of the Galactic center GeV excess , Phys. Rev. Lett. (2016), no. 5 051102, [ arXiv:1506.05104 ].32948] T. Linden, N. L. Rodd, B. R. Safdi, and T. R. Slatyer,
High-energy tail of theGalactic Center gamma-ray excess , Phys. Rev.
D94 (2016), no. 10 103013,[ arXiv:1604.01026 ].[49] S. Mishra-Sharma, N. L. Rodd, and B. R. Safdi,
NPTFit: A code package forNon-Poissonian Template Fitting , Astron. J. (2017), no. 6 253,[ arXiv:1612.03173 ].[50] T. D. Brandt and B. Kocsis,
Disrupted Globular Clusters Can Explain theGalactic Center Gamma Ray Excess , Astrophys. J. (2015), no. 1 15,[ arXiv:1507.05616 ].[51] O. Macias, C. Gordon, R. M. Crocker, B. Coleman, D. Paterson, S. Horiuchi,and M. Pohl,
X-Shaped Bulge Preferred Over Dark Matter for the GalacticCenter Gamma-Ray Excess , arXiv:1611.06644 .[52] R. Bartels, E. Storm, C. Weniger, and F. Calore, The Fermi-LAT GeV ExcessTraces Stellar Mass in the Galactic Bulge , arXiv:1711.04778 .[53] B. Balaji, I. Cholis, P. J. Fox, and S. D. McDermott, Analyzing theGamma-ray Sky with Wavelets , arXiv:1803.01952 .[54] R. Bartels, F. Calore, E. Storm, and C. Weniger, Galactic Binaries CanExplain the Fermi Galactic Center Excess and 511 keV Emission , arXiv:1803.04370 .[55] Fermi-LAT
Collaboration, M. Ajello et al.,
Characterizing the population ofpulsars in the inner Galaxy with the Fermi Large Area Telescope , Submittedto: Astrophys. J. (2017) [ arXiv:1705.00009 ].[56] R. Bartels, D. Hooper, T. Linden, S. Mishra-Sharma, N. L. Rodd, B. R. Safdi,and T. R. Slatyer,
Comment on "Characterizing the population of pulsars inthe Galactic bulge with the Fermi Large Area Telescope" [arXiv:1705.00009v1] , arXiv:1710.10266 . 33057] Fermi-LAT
Collaboration, M. Ackermann et al.,
Searching for Dark MatterAnnihilation from Milky Way Dwarf Spheroidal Galaxies with Six Years ofFermi Large Area Telescope Data , Phys. Rev. Lett. (2015), no. 23 231301,[ arXiv:1503.02641 ].[58]
DES, Fermi-LAT
Collaboration, A. Albert et al.,
Searching for Dark MatterAnnihilation in Recently Discovered Milky Way Satellites with Fermi-LAT , Astrophys. J. (2017), no. 2 110, [ arXiv:1611.03184 ].[59] R. Keeley, K. Abazajian, A. Kwa, N. Rodd, and B. Safdi,
What the MilkyWay’s Dwarfs tell us about the Galactic Center extended excess , arXiv:1710.03215 .[60] T. Cohen, K. Murase, N. L. Rodd, B. R. Safdi, and Y. Soreq, Gamma-rayConstraints on Decaying Dark Matter and Implications for IceCube , Phys.Rev. Lett. (2017), no. 2 021102, [ arXiv:1612.05638 ].[61]
HAWC
Collaboration, A. U. Abeysekara et al.,
A Search for Dark Matter inthe Galactic Halo with HAWC , JCAP (2018), no. 02 049,[ arXiv:1710.10288 ].[62] G. Elor, N. L. Rodd, and T. R. Slatyer,
Multistep cascade annihilations ofdark matter and the Galactic Center excess , Phys. Rev.
D91 (2015) 103531,[ arXiv:1503.01773 ].[63] G. Elor, N. L. Rodd, T. R. Slatyer, and W. Xue,
Model-Independent IndirectDetection Constraints on Hidden Sector Dark Matter , JCAP (2016),no. 06 024, [ arXiv:1511.08787 ].[64] G. Ovanesyan, N. L. Rodd, T. R. Slatyer, and I. W. Stewart,
One-loopcorrection to heavy dark matter annihilation , Phys. Rev.
D95 (2017), no. 5055001, [ arXiv:1612.04814 ]. 33165] M. Baumgart, T. Cohen, I. Moult, N. L. Rodd, T. R. Slatyer, M. P. Solon,I. W. Stewart, and V. Vaidya,
Resummed Photon Spectra for WIMPAnnihilation , arXiv:1712.07656 .[66] P. Ilten, N. L. Rodd, J. Thaler, and M. Williams, Disentangling Heavy Flavorat Colliders , Phys. Rev.
D96 (2017), no. 5 054019, [ arXiv:1702.02947 ].[67] D. Hooper and T. R. Slatyer,
Two Emission Mechanisms in the FermiBubbles: A Possible Signal of Annihilating Dark Matter , Phys. Dark Univ. (2013) 118–138, [ arXiv:1302.6589 ].[68] W.-C. Huang, A. Urbano, and W. Xue, Fermi Bubbles under Dark MatterScrutiny. Part I: Astrophysical Analysis , arXiv:1307.6862 .[69] K. N. Abazajian, The Consistency of Fermi-LAT Observations of the GalacticCenter with a Millisecond Pulsar Population in the Central Stellar Cluster , JCAP (2011) 010, [ arXiv:1011.4275 ].[70] D. Hooper, I. Cholis, T. Linden, J. Siegal-Gaskins, and T. Slatyer,
PulsarsCannot Account for the Inner Galaxy’s GeV Excess , Phys. Rev.
D88 (2013)083009, [ arXiv:1305.0830 ].[71] T. Linden, E. Lovegrove, and S. Profumo,
The Morphology of HadronicEmission Models for the Gamma-Ray Source at the Galactic Center , Astrophys. J. (2012) 41, [ arXiv:1203.3539 ].[72] O. Macias and C. Gordon,
Contribution of cosmic rays interacting withmolecular clouds to the Galactic Center gamma-ray excess , Phys. Rev.
D89 (2014), no. 6 063515, [ arXiv:1312.6671 ].[73] M. Kuhlen, J. Diemand, and P. Madau,
The shapes, orientation, andalignment of Galactic dark matter subhalos , Astrophys. J. (2007) 1135,[ arXiv:0705.2037 ]. 33274] J. F. Navarro, C. S. Frenk, and S. D. M. White,
The Structure of cold darkmatter halos , Astrophys. J. (1996) 563–575, [ astro-ph/9508025 ].[75] J. F. Navarro, C. S. Frenk, and S. D. M. White,
A Universal density profilefrom hierarchical clustering , Astrophys. J. (1997) 493–508,[ astro-ph/9611107 ].[76] F. Iocco, M. Pato, G. Bertone, and P. Jetzer,
Dark Matter distribution in theMilky Way: microlensing and dynamical constraints , JCAP (2011) 029,[ arXiv:1107.5810 ].[77] R. Catena and P. Ullio,
A novel determination of the local dark matterdensity , JCAP (2010) 004, [ arXiv:0907.0018 ].[78] J. F. Navarro, A. Ludlow, V. Springel, J. Wang, M. Vogelsberger, S. D. M.White, A. Jenkins, C. S. Frenk, and A. Helmi,
The Diversity and Similarity ofCold Dark Matter Halos , Mon. Not. Roy. Astron. Soc. (2010) 21,[ arXiv:0810.1522 ].[79] J. Diemand, M. Kuhlen, P. Madau, M. Zemp, B. Moore, D. Potter, andJ. Stadel,
Clumps and streams in the local dark matter distribution , Nature (2008) 735–738, [ arXiv:0805.1244 ].[80] J. N. Fry,
Statistics of Voids in Hierarchical Universes , Phys. Lett. (1985) 331–335.[81] B. S. Ryden and J. E. Gunn,
Galaxy formation by gravitational collapse , Astrophys. J. (1987) 15.[82] O. Y. Gnedin, D. Ceverino, N. Y. Gnedin, A. A. Klypin, A. V. Kravtsov,R. Levine, D. Nagai, and G. Yepes,
Halo Contraction Effect in HydrodynamicSimulations of Galaxy Formation , arXiv:1108.5736 .[83] O. Y. Gnedin, A. V. Kravtsov, A. A. Klypin, and D. Nagai, Response of darkmatter halos to condensation of baryons: Cosmological simulations and mproved adiabatic contraction model , Astrophys. J. (2004) 16–26,[ astro-ph/0406247 ].[84] F. Governato, A. Zolotov, A. Pontzen, C. Christensen, S. H. Oh, A. M.Brooks, T. Quinn, S. Shen, and J. Wadsley,
Cuspy No More: How OutflowsAffect the Central Dark Matter and Baryon Distribution in Lambda CDMGalaxies , Mon. Not. Roy. Astron. Soc. (2012) 1231–1240,[ arXiv:1202.0554 ].[85] M. Kuhlen, J. Guedes, A. Pillepich, P. Madau, and L. Mayer,
An Off-centerDensity Peak in the Milky Way’s Dark Matter Halo? , Astrophys. J. (2013) 10, [ arXiv:1208.4844 ].[86] M. D. Weinberg and N. Katz,
Bar-driven dark halo evolution: a resolution ofthe cusp-core controversy , Astrophys. J. (2002) 627–633,[ astro-ph/0110632 ].[87] M. D. Weinberg and N. Katz,
The bar-halo interaction. 2. secular evolutionand the religion of n-body simulations , Mon. Not. Roy. Astron. Soc. (2007) 460–476, [ astro-ph/0601138 ].[88] J. A. Sellwood,
Bars and dark matter halo cores , Astrophys. J. (2003)638–648, [ astro-ph/0210079 ].[89] O. Valenzuela and A. Klypin,
Secular bar formation in galaxies withsignificant amount of dark matter , Mon. Not. Roy. Astron. Soc. (2003)406, [ astro-ph/0204028 ].[90] P. Colin, O. Valenzuela, and A. Klypin,
Bars and cold dark matter halos , Astrophys. J. (2006) 687–700, [ astro-ph/0506627 ].[91] V. Springel, J. Wang, M. Vogelsberger, A. Ludlow, A. Jenkins, A. Helmi, J. F.Navarro, C. S. Frenk, and S. D. M. White,
The Aquarius Project: the subhalosof galactic halos , Mon. Not. Roy. Astron. Soc. (2008) 1685–1711,[ arXiv:0809.0898 ]. 33492]
Fermi-LAT
Collaboration, M. Ackermann et al.,
Dark matter constraintsfrom observations of 25 Milky Way satellite galaxies with the Fermi LargeArea Telescope , Phys. Rev.
D89 (2014) 042001, [ arXiv:1310.0828 ].[93] T. Sjostrand, S. Mrenna, and P. Z. Skands,
PYTHIA 6.4 Physics and Manual , JHEP (2006) 026, [ hep-ph/0603175 ].[94] L. Bergstrom, T. Bringmann, M. Eriksson, and M. Gustafsson, Gamma raysfrom Kaluza-Klein dark matter , Phys. Rev. Lett. (2005) 131301,[ astro-ph/0410359 ].[95] A. Birkedal, K. T. Matchev, M. Perelstein, and A. Spray, Robust gamma raysignature of WIMP dark matter , hep-ph/0507194 .[96] M. Cirelli, P. D. Serpico, and G. Zaharijas, Bremsstrahlung gamma rays fromlight Dark Matter , JCAP (2013) 035, [ arXiv:1307.7152 ].[97]
Fermi-LAT
Collaboration, W. B. Atwood et al.,
The Large Area Telescopeon the Fermi Gamma-ray Space Telescope Mission , Astrophys. J. (2009)1071–1102, [ arXiv:0902.1089 ].[98]
Fermi-LAT
Collaboration, M. Ackermann et al.,
The Fermi Large AreaTelescope On Orbit: Event Classification, Instrument Response Functions, andCalibration , Astrophys. J. Suppl. (2012) 4, [ arXiv:1206.1896 ].[99]
Fermi-LAT
Collaboration, M. Ackermann et al.,
Determination of thePoint-Spread Function for the Fermi Large Area Telescope from On-orbit Dataand Limits on Pair Halos of Active Galactic Nuclei , Astrophys. J. (2013),no. 1 54, [ arXiv:1309.5416 ].[100] S. K. N. Portillo and D. P. Finkbeiner,
Sharper Fermi LAT Images:instrument response functions for an improved event selection , Astrophys. J. (2014), no. 1 54, [ arXiv:1406.0507 ].335101] G. Dobler, D. P. Finkbeiner, I. Cholis, T. R. Slatyer, and N. Weiner,
TheFermi Haze: A Gamma-Ray Counterpart to the Microwave Haze , Astrophys.J. (2010) 825–842, [ arXiv:0910.4583 ].[102] F. Calore, I. Cholis, and C. Weniger,
Background Model Systematics for theFermi GeV Excess , JCAP (2015) 038, [ arXiv:1409.0042 ].[103] A. W. Strong and I. V. Moskalenko,
Propagation of cosmic-ray nucleons in thegalaxy , Astrophys. J. (1998) 212–228, [ astro-ph/9807150 ].[104] A. W. Strong and I. V. Moskalenko,
The galprop program for cosmic raypropagation: new developments , in
Proceedings, 26th International Cosmic RayConference (ICRC), August 17-25, 1999, Salt Lake City: Invited, Rapporteur,and Highlight Papers , p. 255, 1999. astro-ph/9906228 . [4,255(1999)].[105] A. W. Strong, I. V. Moskalenko, and V. S. Ptuskin,
Cosmic-ray propagationand interactions in the Galaxy , Ann. Rev. Nucl. Part. Sci. (2007) 285–327,[ astro-ph/0701517 ].[106] C. J. Law, F. Yusef-Zadeh, W. D. Cotton, and R. J. Maddalena, GBTMultiwavelength Survey of the Galactic Center Region , Astrophys. J. Suppl. (2008) 255, [ arXiv:0801.4294 ].[107]
Fermi-LAT
Collaboration, P. Nolan et al.,
Fermi Large Area TelescopeSecond Source Catalog , Astrophys. J. Suppl. (2012) 31,[ arXiv:1108.1435 ].[108] F. Yusef-Zadeh et al.,
Interacting Cosmic Rays with Molecular Clouds: ABremsstrahlung Origin of Diffuse High Energy Emission from the Inner 2degby 1deg of the Galactic Center , Astrophys. J. (2013) 33,[ arXiv:1206.6882 ].[109]
Fermi-LAT
Collaboration, A. A. Abdo et al.,
The Spectrum of the IsotropicDiffuse Gamma-Ray Emission Derived From First-Year Fermi Large AreaTelescope Data , Phys. Rev. Lett. (2010) 101101, [ arXiv:1002.3603 ].336110] F. Yusef-Zadeh, D. Choate, and W. Cotton,
The position of sgr 𝑎 * at thegalactic center , Astrophys. J. (1999) L33, [ astro-ph/9904142 ].[111]
CoGeNT
Collaboration, C. E. Aalseth et al.,
Results from a Search forLight-Mass Dark Matter with a P-type Point Contact Germanium Detector , Phys. Rev. Lett. (2011) 131301, [ arXiv:1002.4703 ].[112] C. E. Aalseth et al.,
Search for an Annual Modulation in a P-type PointContact Germanium Dark Matter Detector , Phys. Rev. Lett. (2011)141301, [ arXiv:1106.0650 ].[113]
CDMS
Collaboration, R. Agnese et al.,
Silicon Detector Dark Matter Resultsfrom the Final Exposure of CDMS II , Phys. Rev. Lett. (2013), no. 25251301, [ arXiv:1304.4279 ].[114] G. Angloher et al.,
Results from 730 kg days of the CRESST-II Dark MatterSearch , Eur. Phys. J.
C72 (2012) 1971, [ arXiv:1109.0702 ].[115]
DAMA
Collaboration, R. Bernabei et al.,
First results from DAMA/LIBRAand the combined results with DAMA/NaI , Eur. Phys. J.
C56 (2008) 333–355,[ arXiv:0804.2741 ].[116]
DAMA, LIBRA
Collaboration, R. Bernabei et al.,
New results fromDAMA/LIBRA , Eur. Phys. J.
C67 (2010) 39–49, [ arXiv:1002.1028 ].[117] G. Steigman, B. Dasgupta, and J. F. Beacom,
Precise Relic WIMP Abundanceand its Impact on Searches for Dark Matter Annihilation , Phys. Rev.
D86 (2012) 023506, [ arXiv:1204.3622 ].[118] C. Boehm, D. Hooper, J. Silk, M. Casse, and J. Paul,
MeV dark matter: Hasit been detected? , Phys. Rev. Lett. (2004) 101301, [ astro-ph/0309686 ].[119] PAMELA
Collaboration, O. Adriani et al.,
An anomalous positronabundance in cosmic rays with energies 1.5-100 GeV , Nature (2009)607–609, [ arXiv:0810.4995 ]. 337120] J. Chang et al.,
An excess of cosmic ray electrons at energies of 300-800 GeV , Nature (2008) 362–365.[121] C. Weniger,
A Tentative Gamma-Ray Line from Dark Matter Annihilation atthe Fermi Large Area Telescope , JCAP (2012) 007, [ arXiv:1204.2797 ].[122] M. Su and D. P. Finkbeiner,
Strong Evidence for Gamma-ray Line Emissionfrom the Inner Galaxy , arXiv:1206.1616 .[123] D. P. Finkbeiner, WMAP microwave emission interpreted as dark matterannihilation in the inner galaxy , astro-ph/0409027 .[124] D. Hooper, D. P. Finkbeiner, and G. Dobler, Possible evidence for dark matterannihilations from the excess microwave emission around the center of theGalaxy seen by the Wilkinson Microwave Anisotropy Probe , Phys. Rev.
D76 (2007) 083012, [ arXiv:0705.3655 ].[125] D. Hooper, P. Blasi, and P. D. Serpico,
Pulsars as the Sources of High EnergyCosmic Ray Positrons , JCAP (2009) 025, [ arXiv:0810.1527 ].[126] S. Profumo,
Dissecting cosmic-ray electron-positron data with Occam’s Razor:the role of known Pulsars , Central Eur. J. Phys. (2011) 1–31,[ arXiv:0812.4457 ].[127] G. Dobler, A Last Look at the Microwave Haze/Bubbles with WMAP , Astrophys. J. (2012) 17, [ arXiv:1109.4418 ].[128] G. Dobler,
Identifying the Radio Bubble Nature of the Microwave Haze , Astrophys. J. (2012) L8, [ arXiv:1208.2690 ].[129]
Fermi-LAT
Collaboration, M. Ackermann et al.,
Search for Gamma-raySpectral Lines with the Fermi Large Area Telescope and Dark MatterImplications , Phys. Rev.
D88 (2013) 082002, [ arXiv:1305.5597 ].338130]
CoGeNT
Collaboration, C. E. Aalseth et al.,
CoGeNT: A Search forLow-Mass Dark Matter using p-type Point Contact Germanium Detectors , Phys. Rev.
D88 (2013) 012002, [ arXiv:1208.5737 ].[131]
XENON100
Collaboration, E. Aprile et al.,
Dark Matter Results from 225Live Days of XENON100 Data , Phys. Rev. Lett. (2012) 181301,[ arXiv:1207.5988 ].[132]
LUX
Collaboration, D. S. Akerib et al.,
First results from the LUX darkmatter experiment at the Sanford Underground Research Facility , Phys. Rev.Lett. (2014) 091303, [ arXiv:1310.8214 ].[133] C. Boehm, M. J. Dolan, C. McCabe, M. Spannowsky, and C. J. Wallace,
Extended gamma-ray emission from Coy Dark Matter , JCAP (2014)009, [ arXiv:1401.6458 ].[134] E. Hardy, R. Lasenby, and J. Unwin,
Annihilation Signals from AsymmetricDark Matter , JHEP (2014) 049, [ arXiv:1402.4500 ].[135] K. P. Modak, D. Majumdar, and S. Rakshit, A Possible Explanation of LowEnergy 𝛾 -ray Excess from Galactic Centre and Fermi Bubble by a Dark MatterModel with Two Real Scalars , JCAP (2015) 011, [ arXiv:1312.7488 ].[136] W.-C. Huang, A. Urbano, and W. Xue,
Fermi Bubbles under Dark MatterScrutiny Part II: Particle Physics Analysis , JCAP (2014) 020,[ arXiv:1310.7609 ].[137] N. Okada and O. Seto,
Gamma ray emission in Fermi bubbles and Higgsportal dark matter , Phys. Rev.
D89 (2014), no. 4 043525, [ arXiv:1310.5991 ].[138] K. Hagiwara, S. Mukhopadhyay, and J. Nakamura,
10 GeV neutralino darkmatter and light stau in the MSSM , Phys. Rev.
D89 (2014), no. 1 015023,[ arXiv:1308.6738 ]. 339139] M. R. Buckley, D. Hooper, and J. Kumar,
Phenomenology of Dirac NeutralinoDark Matter , Phys. Rev.
D88 (2013) 063532, [ arXiv:1307.3561 ].[140] L. A. Anchordoqui and B. J. Vlcek,
W-WIMP Annihilation as a Source of theFermi Bubbles , Phys. Rev.
D88 (2013) 043513, [ arXiv:1305.4625 ].[141] M. R. Buckley, D. Hooper, and J. L. Rosner,
A Leptophobic Z’ And DarkMatter From Grand Unification , Phys. Lett.
B703 (2011) 343–347,[ arXiv:1106.3583 ].[142] M. S. Boucenna and S. Profumo,
Direct and Indirect Singlet Scalar DarkMatter Detection in the Lepton-Specific two-Higgs-doublet Model , Phys. Rev.
D84 (2011) 055011, [ arXiv:1106.3368 ].[143] G. Marshall and R. Primulando,
The Galactic Center Region Gamma RayExcess from A Supersymmetric Leptophilic Higgs Model , JHEP (2011) 026,[ arXiv:1102.0492 ].[144] G. Zhu, WIMPless dark matter and the excess gamma rays from the Galacticcenter , Phys. Rev.
D83 (2011) 076011, [ arXiv:1101.4387 ].[145] M. R. Buckley, D. Hooper, and T. M. P. Tait,
Particle Physics Implicationsfor CoGeNT, DAMA, and Fermi , Phys. Lett.
B702 (2011) 216–219,[ arXiv:1011.1499 ].[146] H. E. Logan,
Dark matter annihilation through a lepton-specific Higgs boson , Phys. Rev.
D83 (2011) 035022, [ arXiv:1010.4214 ].[147] B. Allgood, R. A. Flores, J. R. Primack, A. V. Kravtsov, R. H. Wechsler,A. Faltenbacher, and J. S. Bullock,
The shape of dark matter halos:dependence on mass, redshift, radius, and formation , Mon. Not. Roy. Astron.Soc. (2006) 1781–1796, [ astro-ph/0508497 ].[148] N. Bernal, J. E. Forero-Romero, R. Garani, and S. Palomares-Ruiz,
Systematic uncertainties from halo asphericity in dark matter searches , JCAP (2014) 004, [ arXiv:1405.6240 ].340149] D. R. Law, S. R. Majewski, and K. V. Johnston,
Evidence for a Triaxial MilkyWay Dark Matter Halo from the Sagittarius Stellar Tidal Stream , Astrophys.J. (2009) L67–L71, [ arXiv:0908.3187 ].[150] D. R. Law and S. R. Majewski,
The Sagittarius Dwarf Galaxy: a Model forEvolution in a Triaxial Milky Way Halo , Astrophys. J. (2010) 229–254,[ arXiv:1003.1132 ].[151] T. Grégoire and J. Knödlseder,
Constraining the Galactic millisecond pulsarpopulation using Fermi Large Area Telescope , Astron. Astrophys. (2013)A62, [ arXiv:1305.1584 ].[152]
Fermi-LAT
Collaboration, A. Abdo et al.,
A population of gamma-rayemitting globular clusters seen with the Fermi Large Area Telescope , Astron.Astrophys. (2010) A75, [ arXiv:1003.3588 ].[153] J. Han, C. S. Frenk, V. R. Eke, L. Gao, S. D. M. White, A. Boyarsky,D. Malyshev, and O. Ruchayskiy,
Constraining Extended Gamma-rayEmission from Galaxy Clusters , Mon. Not. Roy. Astron. Soc. (2012)1651–1665, [ arXiv:1207.6749 ].[154] O. Macias-Ramirez, C. Gordon, A. M. Brown, and J. Adams,
Evaluating theGamma-Ray Evidence for Self-Annihilating Dark Matter from the VirgoCluster , Phys. Rev.
D86 (2012) 076004, [ arXiv:1207.6257 ].[155] A. Berlin and D. Hooper,
Stringent Constraints on the Dark MatterAnnihilation Cross Section From Subhalo Searches with the Fermi Gamma-RaySpace Telescope , Phys. Rev.
D89 (2014), no. 1 016014, [ arXiv:1309.0525 ].[156] M. Cirelli and G. Giesen,
Antiprotons from Dark Matter: Current constraintsand future sensitivities , JCAP (2013) 015, [ arXiv:1301.7079 ].[157] N. Fornengo, L. Maccione, and A. Vittino,
Constraints on particle dark matterfrom cosmic-ray antiprotons , JCAP (2014), no. 04 003,[ arXiv:1312.3579 ]. 341158]
Fermi-LAT
Collaboration, M. Ackermann et al.,
Limits on Dark MatterAnnihilation Signals from the Fermi LAT 4-year Measurement of the IsotropicGamma-Ray Background , JCAP (2015), no. 09 008,[ arXiv:1501.05464 ].[159] H.-S. Zechlin, A. Cuoco, F. Donato, N. Fornengo, and M. Regis,
StatisticalMeasurement of the Gamma-ray Source-count Distribution as a Function ofEnergy , Astrophys. J. (2016), no. 2 L31, [ arXiv:1605.04256 ].[160] M. Lisanti, S. Mishra-Sharma, L. Necib, and B. R. Safdi,
DecipheringContributions to the Extragalactic Gamma-Ray Background from 2 GeV to 2TeV , Astrophys. J. (2016), no. 2 117, [ arXiv:1606.04101 ].[161]
Fermi-LAT
Collaboration, M. Ackermann et al.,
Anisotropies in the diffusegamma-ray background measured by the Fermi LAT , Phys. Rev.
D85 (2012)083007, [ arXiv:1202.2856 ].[162] S. Ando, E. Komatsu, T. Narumoto, and T. Totani,
Dark matter annihilationor unresolved astrophysical sources? Anisotropy probe of the origin of cosmicgamma-ray background , Phys. Rev.
D75 (2007) 063519, [ astro-ph/0612467 ].[163] S. Ando and E. Komatsu,
Constraints on the annihilation cross section ofdark matter particles from anisotropies in the diffuse gamma-ray backgroundmeasured with Fermi-LAT , Phys. Rev.
D87 (2013), no. 12 123539,[ arXiv:1301.5901 ].[164] J.-Q. Xia, A. Cuoco, E. Branchini, M. Fornasa, and M. Viel,
Across-correlation study of the Fermi-LAT 𝛾 -ray diffuse extragalactic signal , Mon. Not. Roy. Astron. Soc. (2011) 2247–2264, [ arXiv:1103.4861 ].[165] S. Ando,
Power spectrum tomography of dark matter annihilation with localgalaxy distribution , JCAP (2014), no. 10 061, [ arXiv:1407.8502 ].342166] S. Ando, A. Benoit-Lévy, and E. Komatsu,
Mapping dark matter in thegamma-ray sky with galaxy catalogs , Phys. Rev.
D90 (2014), no. 2 023514,[ arXiv:1312.4403 ].[167] J.-Q. Xia, A. Cuoco, E. Branchini, and M. Viel,
Tomography of the Fermi-lat 𝛾 -ray Diffuse Extragalactic Signal via Cross Correlations With GalaxyCatalogs , Astrophys. J. Suppl. (2015), no. 1 15, [ arXiv:1503.05918 ].[168] M. Regis, J.-Q. Xia, A. Cuoco, E. Branchini, N. Fornengo, and M. Viel,
Particle dark matter searches outside the Local Group , Phys. Rev. Lett. (2015), no. 24 241301, [ arXiv:1503.05922 ].[169] A. Cuoco, J.-Q. Xia, M. Regis, E. Branchini, N. Fornengo, and M. Viel,
DarkMatter Searches in the Gamma-ray Extragalactic Background viaCross-correlations With Galaxy Catalogs , Astrophys. J. Suppl. (2015),no. 2 29, [ arXiv:1506.01030 ].[170] S. Ando and K. Ishiwata,
Constraining particle dark matter using local galaxydistribution , JCAP (2016), no. 06 045, [ arXiv:1604.02263 ].[171] T. H. Jarrett, T. Chester, R. Cutri, S. Schneider, M. Skrutskie, and J. P.Huchra, , Astron. J. (2000) 2498–2531, [ astro-ph/0004318 ].[172] M. Bilicki, T. H. Jarrett, J. A. Peacock, M. E. Cluver, and L. Steward, , arXiv:1311.5246 . [Astrophys. J. Suppl.210,9(2014)].[173] M. Ackermann et al., Constraints on Dark Matter Annihilation in Clusters ofGalaxies with the Fermi Large Area Telescope , JCAP (2010) 025,[ arXiv:1002.2239 ].[174] S. Ando and D. Nagai,
Fermi-LAT constraints on dark matter annihilationcross section from observations of the Fornax cluster , JCAP (2012) 017,[ arXiv:1201.0753 ]. 343175]
Fermi-LAT
Collaboration, M. Ackermann et al.,
Search for cosmic-rayinduced gamma-ray emission in Galaxy Clusters , Astrophys. J. (2014) 18,[ arXiv:1308.5654 ].[176]
Fermi-LAT
Collaboration, M. Ackermann et al.,
Search for extendedgamma-ray emission from the Virgo galaxy cluster with Fermi-LAT , Astrophys. J. (2015), no. 2 159, [ arXiv:1510.00004 ].[177] B. Anderson, S. Zimmer, J. Conrad, M. Gustafsson, M. Sánchez-Conde, andR. Caputo,
Search for Gamma-Ray Lines towards Galaxy Clusters with theFermi-LAT , JCAP (2016), no. 02 026, [ arXiv:1511.00014 ].[178]
Fermi-LAT
Collaboration, M. Ackermann et al.,
Search for gamma-rayemission from the Coma Cluster with six years of Fermi-LAT data , Astrophys.J. (2016), no. 2 149, [ arXiv:1507.08995 ].[179]
MAGIC
Collaboration, M. L. Ahnen et al.,
Deep onservation of the NGC1275 region with MAGIC: search of diffuse 𝛾 -ray emission from cosmic rays inthe Perseus cluster , Astron. Astrophys. (2016) A33, [ arXiv:1602.03099 ].[180] Y.-F. Liang, Z.-Q. Shen, X. Li, Y.-Z. Fan, X. Huang, S.-J. Lei, L. Feng, E.-W.Liang, and J. Chang,
Search for a gamma-ray line feature from a group ofnearby galaxy clusters with Fermi LAT Pass 8 data , Phys. Rev.
D93 (2016),no. 10 103525, [ arXiv:1602.06527 ].[181] D. Q. Adams, L. Bergstrom, and D. Spolyar,
Improved Constraints on DarkMatter Annihilation to a Line using Fermi-LAT observations of GalaxyClusters , arXiv:1606.09642 .[182] X. Huang, G. Vertongen, and C. Weniger, Probing Dark Matter Decay andAnnihilation with Fermi LAT Observations of Nearby Galaxy Clusters , JCAP (2012) 042, [ arXiv:1110.1529 ].[183] R. B. Tully,
Galaxy Groups: A 2MASS Catalog , Astron. J. (2015) 171,[ arXiv:1503.03134 ]. 344184] E. Kourkchi and R. B. Tully,
Galaxy Groups Within 3500 km s − , Ap. J. (2017) 16, [ arXiv:1705.08068 ].[185] R. B. Tully, H. M. Courtois, and J. G. Sorce,
Cosmicflows-3 , Astron. J. (2016) 50, [ arXiv:1605.01765 ].[186] M. Cirelli, G. Corcella, A. Hektor, G. Hutsi, M. Kadastik, P. Panci,M. Raidal, F. Sala, and A. Strumia,
PPPC 4 DM ID: A Poor ParticlePhysicist Cookbook for Dark Matter Indirect Detection , JCAP (2011)051, [ arXiv:1012.4515 ]. [Erratum: JCAP1210,E01(2012)].[187] C. A. Correa, J. S. B. Wyithe, J. Schaye, and A. R. Duffy,
The accretionhistory of dark matter haloes – III. A physical model for theconcentration–mass relation , Mon. Not. Roy. Astron. Soc. (2015), no. 21217–1232, [ arXiv:1502.00391 ].[188]
Planck
Collaboration, P. A. R. Ade et al.,
Planck 2015 results. XIII.Cosmological parameters , Astron. Astrophys. (2016) A13,[ arXiv:1502.01589 ].[189] B. V. Lehmann, Y.-Y. Mao, M. R. Becker, S. W. Skillman, and R. H.Wechsler,
The Concentration Dependence of the Galaxy-Halo Connection:Modeling Assembly Bias with Abundance Matching , Astrophys. J. (2017),no. 1 37, [ arXiv:1510.05651 ].[190] L. Gao, C. S. Frenk, A. Jenkins, V. Springel, and S. D. M. White,
Where willsupersymmetric dark matter first be seen? , Mon. Not. Roy. Astron. Soc. (2012) 1721, [ arXiv:1107.1916 ].[191] D. Anderhalden and J. Diemand,
Density Profiles of CDM Microhalos andtheir Implications for Annihilation Boost Factors , JCAP (2013) 009,[ arXiv:1302.0003 ]. [Erratum: JCAP1308,E02(2013)].[192] A. D. Ludlow, J. F. Navarro, R. E. Angulo, M. Boylan-Kolchin, V. Springel,C. Frenk, and S. D. M. White,
The mass–concentration–redshift relation of old dark matter haloes , Mon. Not. Roy. Astron. Soc. (2014), no. 1378–388, [ arXiv:1312.0945 ].[193] R. Bartels and S. Ando,
Boosting the annihilation boost: Tidal effects on darkmatter subhalos and consistent luminosity modeling , Phys. Rev.
D92 (2015),no. 12 123508, [ arXiv:1507.08656 ].[194] A. C. Crook, J. P. Huchra, N. Martimbeau, K. L. Masters, T. Jarrett, andL. M. Macri,
Groups of Galaxies in the Two Micron All-Sky Redshift Survey , Astrophys. J. (2007) 790–813, [ astro-ph/0610732 ].[195]
Fermi-LAT
Collaboration, M. Ackermann et al.,
Observations of M31 andM33 with the Fermi Large Area Telescope: A Galactic Center Excess inAndromeda? , Astrophys. J. (2017), no. 2 208, [ arXiv:1702.08602 ].[196] K. M. Gorski, E. Hivon, A. J. Banday, B. D. Wandelt, F. K. Hansen,M. Reinecke, and M. Bartelman,
HEALPix - A Framework for high resolutiondiscretization, and fast analysis of data distributed on the sphere , Astrophys.J. (2005) 759–771, [ astro-ph/0409513 ].[197]
Fermi-LAT
Collaboration, F. Acero et al.,
Fermi Large Area Telescope ThirdSource Catalog , Astrophys. J. Suppl. (2015), no. 2 23,[ arXiv:1501.02003 ].[198] T. E. Jeltema, J. Kehayias, and S. Profumo,
Gamma Rays from Clusters andGroups of Galaxies: Cosmic Rays versus Dark Matter , Phys. Rev.
D80 (2009)023005, [ arXiv:0812.0597 ].[199] B. Huber, C. Tchernin, D. Eckert, C. Farnier, A. Manalaysay, U. Straumann,and R. Walter,
Probing the cosmic-ray content of galaxy clusters by stackingFermi-LAT count maps , Astron. Astrophys. (2013) A64,[ arXiv:1308.6278 ]. 346200] A. Geringer-Sameth, S. M. Koushiappas, and M. G. Walker,
Comprehensivesearch for dark matter annihilation in dwarf galaxies , Phys. Rev.
D91 (2015),no. 8 083535, [ arXiv:1410.2242 ].[201]
Fermi-LAT
Collaboration, M. Ajello et al.,
Fermi-LAT Observations ofHigh-Energy 𝛾 -Ray Emission Toward the Galactic Center , Astrophys. J. (2016), no. 1 44, [ arXiv:1511.02938 ].[202] C. Karwin, S. Murgia, T. M. P. Tait, T. A. Porter, and P. Tanedo,
DarkMatter Interpretation of the Fermi-LAT Observation Toward the GalacticCenter , Phys. Rev.
D95 (2017), no. 10 103005, [ arXiv:1612.05687 ].[203]
IceCube
Collaboration, M. G. Aartsen et al.,
First observation ofPeV-energy neutrinos with IceCube , Phys. Rev. Lett. (2013) 021103,[ arXiv:1304.5356 ].[204]
IceCube
Collaboration, M. G. Aartsen et al.,
Evidence for High-EnergyExtraterrestrial Neutrinos at the IceCube Detector , Science (2013)1242856, [ arXiv:1311.5238 ].[205]
IceCube
Collaboration, M. G. Aartsen et al.,
A combined maximum-likelihoodanalysis of the high-energy astrophysical neutrino flux measured with IceCube , Astrophys. J. (2015), no. 1 98, [ arXiv:1507.03991 ].[206]
IceCube
Collaboration, M. G. Aartsen et al.,
Evidence for AstrophysicalMuon Neutrinos from the Northern Sky with IceCube , Phys. Rev. Lett. (2015), no. 8 081102, [ arXiv:1507.04005 ].[207] O. K. Kalashev and M. Yu. Kuznetsov,
Constraining heavy decaying darkmatter with the high energy gamma-ray limits , Phys. Rev.
D94 (2016), no. 6063535, [ arXiv:1606.07354 ].[208]
The Pierre Auger Observatory: Contributions to the 34th InternationalCosmic Ray Conference (ICRC 2015) , 2015.347209] D. Kang et al.,
A limit on the diffuse gamma-rays measured withKASCADE-Grande , J. Phys. Conf. Ser. (2015), no. 1 012013.[210]
CASA-MIA
Collaboration, M. C. Chantell et al.,
Limits on the isotropicdiffuse flux of ultrahigh-energy gamma radiation , Phys. Rev. Lett. (1997)1805–1808, [ astro-ph/9705246 ].[211] S. Ando and K. Ishiwata, Constraints on decaying dark matter from theextragalactic gamma-ray background , JCAP (2015), no. 05 024,[ arXiv:1502.02007 ].[212] K. Murase and J. F. Beacom,
Constraining Very Heavy Dark Matter UsingDiffuse Backgrounds of Neutrinos and Cascaded Gamma Rays , JCAP (2012) 043, [ arXiv:1206.2595 ].[213]
Fermi-LAT
Collaboration, M. Ackermann et al.,
Constraints on the GalacticHalo Dark Matter from Fermi-LAT Diffuse Measurements , Astrophys. J. (2012) 91, [ arXiv:1205.6474 ].[214] G. Hutsi, A. Hektor, and M. Raidal,
Implications of the Fermi-LAT diffusegamma-ray measurements on annihilating or decaying Dark Matter , JCAP (2010) 008, [ arXiv:1004.2036 ].[215] M. Cirelli, E. Moulin, P. Panci, P. D. Serpico, and A. Viana,
Gamma rayconstraints on Decaying Dark Matter , Phys. Rev.
D86 (2012) 083506,[ arXiv:1205.5283 ].[216] O. Kalashev,
Constraining Dark Matter and Ultra-High Energy Cosmic RaySources with Fermi-LAT Diffuse Gamma Ray Background , EPJ Web Conf. (2016) 02012, [ arXiv:1608.07530 ].[217] Ams-02 Collaboration, talks at the ‘AMS Days at CERN’, 2015, 15-17 april.[218] G. Giesen, M. Boudaud, Y. Génolini, V. Poulin, M. Cirelli, P. Salati, andP. D. Serpico,
AMS-02 antiprotons, at last! Secondary astrophysical omponent and immediate implications for Dark Matter , JCAP (2015),no. 09 023, [ arXiv:1504.04276 ].[219]
AMS
Collaboration, M. Aguilar et al.,
First Result from the Alpha MagneticSpectrometer on the International Space Station: Precision Measurement ofthe Positron Fraction in Primary Cosmic Rays of 0.5–350 GeV , Phys. Rev.Lett. (2013) 141102.[220] A. Ibarra, A. S. Lamperstorfer, and J. Silk,
Dark matter annihilations anddecays after the AMS-02 positron measurements , Phys. Rev.
D89 (2014), no. 6063539, [ arXiv:1309.2570 ].[221] M. Chianese, G. Miele, and S. Morisi,
Dark Matter interpretation of low energyIceCube MESE excess , JCAP (2017), no. 01 007, [ arXiv:1610.04612 ].[222] T. Sjostrand, S. Mrenna, and P. Z. Skands,
A Brief Introduction to PYTHIA8.1 , Comput. Phys. Commun. (2008) 852–867, [ arXiv:0710.3820 ].[223] T. Sjöstrand, S. Ask, J. R. Christiansen, R. Corke, N. Desai, P. Ilten,S. Mrenna, S. Prestel, C. O. Rasmussen, and P. Z. Skands,
An Introduction toPYTHIA 8.2 , Comput. Phys. Commun. (2015) 159–177,[ arXiv:1410.3012 ].[224] J. R. Christiansen and T. Sjöstrand,
Weak Gauge Boson Radiation in PartonShowers , JHEP (2014) 115, [ arXiv:1401.5238 ].[225] M. Kachelriess and P. D. Serpico, Model-independent dark matter annihilationbound from the diffuse 𝛾 ray flux , Phys. Rev.
D76 (2007) 063516,[ arXiv:0707.0209 ].[226] M. Regis and P. Ullio,
Multi-wavelength signals of dark matter annihilations atthe Galactic center , Phys. Rev.
D78 (2008) 043505, [ arXiv:0802.0234 ].[227] G. D. Mack, T. D. Jacques, J. F. Beacom, N. F. Bell, and H. Yuksel,
Conservative Constraints on Dark Matter Annihilation into Gamma Rays , Phys. Rev.
D78 (2008) 063542, [ arXiv:0803.0157 ].349228] N. F. Bell, J. B. Dent, T. D. Jacques, and T. J. Weiler,
ElectroweakBremsstrahlung in Dark Matter Annihilation , Phys. Rev.
D78 (2008) 083540,[ arXiv:0805.3423 ].[229] J. B. Dent, R. J. Scherrer, and T. J. Weiler,
Toward a Minimum BranchingFraction for Dark Matter Annihilation into Electromagnetic Final States , Phys. Rev.
D78 (2008) 063509, [ arXiv:0806.0370 ].[230] E. Borriello, A. Cuoco, and G. Miele,
Radio constraints on dark matterannihilation in the galactic halo and its substructures , Phys. Rev.
D79 (2009)023518, [ arXiv:0809.2990 ].[231] G. Bertone, M. Cirelli, A. Strumia, and M. Taoso,
Gamma-ray and radio testsof the e+e- excess from DM annihilations , JCAP (2009) 009,[ arXiv:0811.3744 ].[232] N. F. Bell and T. D. Jacques,
Gamma-ray Constraints on Dark MatterAnnihilation into Charged Particles , Phys. Rev.
D79 (2009) 043507,[ arXiv:0811.0821 ].[233] M. Cirelli and P. Panci,
Inverse Compton constraints on the Dark Matter e+e-excesses , Nucl. Phys.
B821 (2009) 399–416, [ arXiv:0904.3830 ].[234] M. Kachelriess, P. D. Serpico, and M. A. Solberg,
On the role of electroweakbremsstrahlung for indirect dark matter signatures , Phys. Rev.
D80 (2009)123533, [ arXiv:0911.0001 ].[235] P. Ciafaloni, D. Comelli, A. Riotto, F. Sala, A. Strumia, and A. Urbano,
WeakCorrections are Relevant for Dark Matter Indirect Detection , JCAP (2011) 019, [ arXiv:1009.0224 ].[236] K. Murase, R. Laha, S. Ando, and M. Ahlers,
Testing the Dark MatterScenario for PeV Neutrinos Observed in IceCube , Phys. Rev. Lett. (2015),no. 7 071301, [ arXiv:1503.04663 ].350237] A. Esmaili and P. D. Serpico,
Gamma-ray bounds from EAS detectors andheavy decaying dark matter constraints , JCAP (2015), no. 10 014,[ arXiv:1505.06486 ].[238] S. A. Mao, N. M. McClure-Griffiths, B. M. Gaensler, J. C. Brown, C. L. vanEck, M. Haverkorn, P. P. Kronberg, J. M. Stil, A. Shukurov, and A. R. Taylor,
New Constraints on the Galactic Halo Magnetic Field using RotationMeasures of Extragalactic Sources Towards the Outer Galaxy , Astrophys. J. (2012) 21, [ arXiv:1206.3314 ].[239] M. Haverkorn,
Magnetic Fields in the Milky Way , arXiv:1406.0283 .[240] M. C. Beck, A. M. Beck, R. Beck, K. Dolag, A. W. Strong, and P. Nielaba, New constraints on modelling the random magnetic field of the MW , JCAP (2016), no. 05 056, [ arXiv:1409.5120 ].[241] G. D. Kribs and I. Z. Rothstein,
Bounds on longlived relics from diffusegamma-ray observations , Phys. Rev.
D55 (1997) 4435–4449,[ hep-ph/9610468 ]. [Erratum: Phys. Rev.D56,1822(1997)].[242] W. A. Rolke, A. M. Lopez, and J. Conrad,
Limits and confidence intervals inthe presence of nuisance parameters , Nucl. Instrum. Meth.
A551 (2005)493–503, [ physics/0403059 ].[243] S. A. Narayanan and T. R. Slatyer,
A Latitude-Dependent Analysis of theLeptonic Hypothesis for the Fermi Bubbles , Mon. Not. Roy. Astron. Soc. (2017), no. 3 3051–3070, [ arXiv:1603.06582 ].[244]
Fermi-LAT
Collaboration, J.-M. Casandjian and I. Grenier,
High EnergyGamma-Ray Emission from the Loop I region , arXiv:0912.3478 .[245] Supplementary Data for “Gamma-ray Constraints on Decaying Dark Matterand Implications for IceCube". 351246] IceCube
Collaboration, M. G. Aartsen et al.,
Constraints onUltrahigh-Energy Cosmic-Ray Sources from a Search for Neutrinos above 10PeV with IceCube , Phys. Rev. Lett. (2016), no. 24 241101,[ arXiv:1607.05886 ]. [Erratum: Phys. Rev. Lett.119,no.25,259902(2017)].[247]
IceCube
Collaboration, M. G. Aartsen et al.,
Atmospheric and astrophysicalneutrinos above 1 TeV interacting in IceCube , Phys. Rev.
D91 (2015), no. 2022001, [ arXiv:1410.1749 ].[248] A. Esmaili and P. D. Serpico,
Are IceCube neutrinos unveiling PeV-scaledecaying dark matter? , JCAP (2013) 054, [ arXiv:1308.1105 ].[249] B. Feldstein, A. Kusenko, S. Matsumoto, and T. T. Yanagida,
Neutrinos atIceCube from Heavy Decaying Dark Matter , Phys. Rev.
D88 (2013), no. 1015004, [ arXiv:1303.7320 ].[250] Y. Ema, R. Jinno, and T. Moroi,
Cosmic-Ray Neutrinos from the Decay ofLong-Lived Particle and the Recent IceCube Result , Phys. Lett.
B733 (2014)120–125, [ arXiv:1312.3501 ].[251] J. Zavala,
Galactic PeV neutrinos from dark matter annihilation , Phys. Rev.
D89 (2014), no. 12 123516, [ arXiv:1404.2932 ].[252] A. Bhattacharya, M. H. Reno, and I. Sarcevic,
Reconciling neutrino flux fromheavy dark matter decay and recent events at IceCube , JHEP (2014) 110,[ arXiv:1403.1862 ].[253] T. Higaki, R. Kitano, and R. Sato, Neutrinoful Universe , JHEP (2014)044, [ arXiv:1405.0013 ].[254] C. Rott, K. Kohri, and S. C. Park, Superheavy dark matter and IceCubeneutrino signals: Bounds on decaying dark matter , Phys. Rev.
D92 (2015),no. 2 023529, [ arXiv:1408.4575 ]. 352255] C. S. Fong, H. Minakata, B. Panes, and R. Zukanovich Funchal,
PossibleInterpretations of IceCube High-Energy Neutrino Events , JHEP (2015)189, [ arXiv:1411.5318 ].[256] E. Dudas, Y. Mambrini, and K. A. Olive, Monochromatic neutrinos generatedby dark matter and the seesaw mechanism , Phys. Rev.
D91 (2015) 075001,[ arXiv:1412.3459 ].[257] Y. Ema, R. Jinno, and T. Moroi,
Cosmological Implications of High-EnergyNeutrino Emission from the Decay of Long-Lived Particle , JHEP (2014)150, [ arXiv:1408.1745 ].[258] A. Esmaili, S. K. Kang, and P. D. Serpico, IceCube events and decaying darkmatter: hints and constraints , JCAP (2014), no. 12 054,[ arXiv:1410.5979 ].[259] L. A. Anchordoqui, V. Barger, H. Goldberg, X. Huang, D. Marfatia, L. H. M.da Silva, and T. J. Weiler,
IceCube neutrinos, decaying dark matter, and theHubble constant , Phys. Rev.
D92 (2015), no. 6 061301, [ arXiv:1506.08788 ].[Erratum: Phys. Rev.D94,no.6,069901(2016)].[260] S. M. Boucenna, M. Chianese, G. Mangano, G. Miele, S. Morisi, O. Pisanti,and E. Vitagliano,
Decaying Leptophilic Dark Matter at IceCube , JCAP (2015), no. 12 055, [ arXiv:1507.01000 ].[261] P. Ko and Y. Tang,
IceCube Events from Heavy DM decays through theRight-handed Neutrino Portal , Phys. Lett.
B751 (2015) 81–88,[ arXiv:1508.02500 ].[262] C. El Aisati, M. Gustafsson, T. Hambye, and T. Scarna,
Dark Matter Decayto a Photon and a Neutrino: the Double Monochromatic Smoking GunScenario , Phys. Rev.
D93 (2016), no. 4 043535, [ arXiv:1510.05008 ].[263] M. D. Kistler,
On TeV Gamma Rays and the Search for Galactic Neutrinos , arXiv:1511.05199 . 353264] M. Chianese, G. Miele, S. Morisi, and E. Vitagliano, Low energy IceCube dataand a possible Dark Matter related excess , Phys. Lett.
B757 (2016) 251–256,[ arXiv:1601.02934 ].[265] M. Re Fiorentin, V. Niro, and N. Fornengo,
A consistent model forleptogenesis, dark matter and the IceCube signal , JHEP (2016) 022,[ arXiv:1606.04445 ].[266] P. S. B. Dev, D. Kazanas, R. N. Mohapatra, V. L. Teplitz, and Y. Zhang, Heavy right-handed neutrino dark matter and PeV neutrinos at IceCube , JCAP (2016), no. 08 034, [ arXiv:1606.04517 ].[267] P. Di Bari, P. O. Ludl, and S. Palomares-Ruiz,
Unifying leptogenesis, darkmatter and high-energy neutrinos with right-handed neutrino mixing via Higgsportal , JCAP (2016), no. 11 044, [ arXiv:1606.06238 ].[268] M. Chianese and A. Merle,
A Consistent Theory of Decaying Dark MatterConnecting IceCube to the Sesame Street , JCAP (2017), no. 04 017,[ arXiv:1607.05283 ].[269] K. Murase and E. Waxman,
Constraining High-Energy Cosmic NeutrinoSources: Implications and Prospects , Phys. Rev.
D94 (2016), no. 10 103006,[ arXiv:1607.01601 ].[270] K. Murase, D. Guetta, and M. Ahlers,
Hidden Cosmic-Ray Accelerators as anOrigin of TeV-PeV Cosmic Neutrinos , Phys. Rev. Lett. (2016), no. 7071101, [ arXiv:1509.00805 ].[271] A. Palladino, M. Spurio, and F. Vissani,
On the IceCube spectral anomaly , JCAP (2016), no. 12 045, [ arXiv:1610.07015 ].[272] A. E. Faraggi and M. Pospelov,
Selfinteracting dark matter from the hiddenheterotic string sector , Astropart. Phys. (2002) 451–461, [ hep-ph/0008223 ].354273] J. Halverson, B. D. Nelson, and F. Ruehle, String Theory and the DarkGlueball Problem , Phys. Rev.
D95 (2017), no. 4 043527, [ arXiv:1609.02151 ].[274] L. Forestell, D. E. Morrissey, and K. Sigurdson,
Non-Abelian Dark Forces andthe Relic Densities of Dark Glueballs , Phys. Rev.
D95 (2017), no. 1 015032,[ arXiv:1605.08048 ].[275] K. K. Boddy, J. L. Feng, M. Kaplinghat, and T. M. P. Tait,
Self-InteractingDark Matter from a Non-Abelian Hidden Sector , Phys. Rev.
D89 (2014),no. 11 115017, [ arXiv:1402.3629 ].[276] A. Ibarra, D. Tran, and C. Weniger,
Indirect Searches for Decaying DarkMatter , Int. J. Mod. Phys.
A28 (2013) 1330040, [ arXiv:1307.6434 ].[277] A. A. Abdo et al.,
Fermi LAT Search for Photon Lines from 30 to 200 GeVand Dark Matter Implications , Phys. Rev. Lett. (2010) 091302,[ arXiv:1001.4836 ].[278] G. Vertongen and C. Weniger,
Hunting Dark Matter Gamma-Ray Lines withthe Fermi LAT , JCAP (2011) 027, [ arXiv:1101.2610 ].[279]
Fermi-LAT
Collaboration, M. Ackermann et al.,
Fermi LAT Search for DarkMatter in Gamma-ray Lines and the Inclusive Photon Spectrum , Phys. Rev.
D86 (2012) 022002, [ arXiv:1205.2739 ].[280] A. Esmaili, A. Ibarra, and O. L. G. Peres,
Probing the stability of superheavydark matter particles with high-energy neutrinos , JCAP (2012) 034,[ arXiv:1205.5281 ].[281] L. Dugger, T. E. Jeltema, and S. Profumo,
Constraints on Decaying DarkMatter from Fermi Observations of Nearby Galaxies and Clusters , JCAP (2010) 015, [ arXiv:1009.5988 ].[282] M. Cirelli, P. Panci, and P. D. Serpico,
Diffuse gamma ray constraints onannihilating or decaying Dark Matter after Fermi , Nucl. Phys.
B840 (2010)284–303, [ arXiv:0912.0663 ]. 355283] L. Zhang, C. Weniger, L. Maccione, J. Redondo, and G. Sigl,
ConstrainingDecaying Dark Matter with Fermi LAT Gamma-rays , JCAP (2010) 027,[ arXiv:0912.4504 ].[284]
Fermi-LAT
Collaboration, G. Zaharijas, A. Cuoco, Z. Yang, and J. Conrad,
Constraints on the Galactic Halo Dark Matter from Fermi-LAT DiffuseMeasurements , PoS
IDM2010 (2011) 111, [ arXiv:1012.0588 ].[285]
Fermi-LAT
Collaboration, G. Zaharijas, J. Conrad, A. Cuoco, and Z. Yang,
Fermi-LAT measurement of the diffuse gamma-ray emission and constraintson the Galactic Dark Matter signal , Nucl. Phys. Proc. Suppl. (2013)88–93, [ arXiv:1212.6755 ].[286]
VERITAS
Collaboration, E. Aliu et al.,
VERITAS Deep Observations of theDwarf Spheroidal Galaxy Segue 1 , Phys. Rev.
D85 (2012) 062001,[ arXiv:1202.2144 ]. [Erratum: Phys. Rev.D91,no.12,129903(2015)].[287] M. G. Baring, T. Ghosh, F. S. Queiroz, and K. Sinha,
New Limits on theDark Matter Lifetime from Dwarf Spheroidal Galaxies using Fermi-LAT , Phys. Rev.
D93 (2016), no. 10 103009, [ arXiv:1510.00389 ].[288]
CTA Consortium
Collaboration, M. Actis et al.,
Design concepts for theCherenkov Telescope Array CTA: An advanced facility for ground-basedhigh-energy gamma-ray astronomy , Exper. Astron. (2011) 193–316,[ arXiv:1008.3703 ].[289] M. Pierre, J. M. Siegal-Gaskins, and P. Scott, Sensitivity of CTA to darkmatter signals from the Galactic Center , JCAP (2014) 024,[ arXiv:1401.7330 ]. [Erratum: JCAP1410,E01(2014)].[290] A. U. Abeysekara et al.,
Sensitivity of the High Altitude Water CherenkovDetector to Sources of Multi-TeV Gamma Rays , Astropart. Phys. (2013) 26–32, [ arXiv:1306.5800 ]. 356291] T. K. Sako, K. Kawata, M. Ohnishi, A. Shiomi, M. Takita, and H. Tsuchiya,
Exploration of a 100 TeV gamma-ray northern sky using the Tibet air-showerarray combined with an underground water-Cherenkov muon-detector array , Astropart. Phys. (2009) 177–184, [ arXiv:0907.4589 ].[292] M. Ahlers and K. Murase, Probing the Galactic Origin of the IceCube Excesswith Gamma-Rays , Phys. Rev.
D90 (2014), no. 2 023010, [ arXiv:1309.4077 ].[293] H.-C. Cheng, W.-C. Huang, X. Huang, I. Low, Y.-L. S. Tsai, and Q. Yuan,
AMS-02 Positron Excess and Indirect Detection of Three-body Decaying DarkMatter , JCAP (2017), no. 03 041, [ arXiv:1608.06382 ].[294] S. Ting,
The First Five Years of the Alpha Magnetic Spectrometer on theInternational Space Station. The First Five Years of the Alpha MagneticSpectrometer on the International Space Station , .[295] J. Petrović, P. D. Serpico, and G. Zaharijaš,
Galactic Center gamma-ray"excess" from an active past of the Galactic Centre? , JCAP (2014),no. 10 052, [ arXiv:1405.7928 ].[296] E. Carlson and S. Profumo,
Cosmic Ray Protons in the Inner Galaxy and theGalactic Center Gamma-Ray Excess , Phys. Rev.
D90 (2014), no. 2 023015,[ arXiv:1405.7685 ].[297] T. Linden,
Talk given at COSMO-14, August 25-29 , talk given at COSMO-14,August 25-29 (2014).[298] C. Gordon and O. Macias, Can Cosmic Rays Interacting With MolecularClouds Explain The Galactic Center Gamma-Ray Excess? , PoS
CRISM2014 (2015) 042, [ arXiv:1410.7840 ].[299] I. Cholis, D. Hooper, and T. Linden,
Challenges in Explaining the GalacticCenter Gamma-Ray Excess with Millisecond Pulsars , JCAP (2015),no. 06 043, [ arXiv:1407.5625 ]. 357300] A. Alves, S. Profumo, F. S. Queiroz, and W. Shepherd,
Effective field theoryapproach to the Galactic Center gamma-ray excess , Phys. Rev.
D90 (2014),no. 11 115003, [ arXiv:1403.5027 ].[301] A. Berlin, D. Hooper, and S. D. McDermott,
Simplified Dark Matter Modelsfor the Galactic Center Gamma-Ray Excess , Phys. Rev.
D89 (2014), no. 11115022, [ arXiv:1404.0022 ].[302] S. Ipek, D. McKeen, and A. E. Nelson,
A Renormalizable Model for theGalactic Center Gamma Ray Excess from Dark Matter Annihilation , Phys.Rev.
D90 (2014), no. 5 055021, [ arXiv:1404.3716 ].[303] C. Cheung, M. Papucci, D. Sanford, N. R. Shah, and K. M. Zurek,
NMSSMInterpretation of the Galactic Center Excess , Phys. Rev.
D90 (2014), no. 7075011, [ arXiv:1406.6372 ].[304] T. Gherghetta, B. von Harling, A. D. Medina, M. A. Schmidt, and T. Trott,
SUSY implications from WIMP annihilation into scalars at the GalacticCenter , Phys. Rev.
D91 (2015) 105004, [ arXiv:1502.07173 ].[305] M. Pospelov, A. Ritz, and M. B. Voloshin,
Secluded WIMP Dark Matter , Phys. Lett.
B662 (2008) 53–61, [ arXiv:0711.4866 ].[306] A. Martin, J. Shelton, and J. Unwin,
Fitting the Galactic Center Gamma-RayExcess with Cascade Annihilations , Phys. Rev.
D90 (2014), no. 10 103513,[ arXiv:1405.0272 ].[307] M. Abdullah, A. DiFranzo, A. Rajaraman, T. M. P. Tait, P. Tanedo, andA. M. Wijangco,
Hidden on-shell mediators for the Galactic Center 𝛾 -rayexcess , Phys. Rev.
D90 (2014) 035004, [ arXiv:1404.6528 ].[308] P. Ko, W.-I. Park, and Y. Tang,
Higgs portal vector dark matter for
GeV scale 𝛾 -ray excess from galactic center , JCAP (2014) 013, [ arXiv:1404.5257 ].358309] M. Freytsis, D. J. Robinson, and Y. Tsai,
Galactic Center Gamma-Ray Excessthrough a Dark Shower , Phys. Rev.
D91 (2015), no. 3 035028,[ arXiv:1410.3818 ].[310] I. Z. Rothstein, T. Schwetz, and J. Zupan,
Phenomenology of Dark Matterannihilation into a long-lived intermediate state , JCAP (2009) 018,[ arXiv:0903.3116 ].[311] M. Baumgart, C. Cheung, J. T. Ruderman, L.-T. Wang, and I. Yavin,
Non-Abelian Dark Sectors and Their Collider Signatures , JHEP (2009)014, [ arXiv:0901.0283 ].[312] Y. Nomura and J. Thaler, Dark Matter through the Axion Portal , Phys. Rev.
D79 (2009) 075008, [ arXiv:0810.5397 ].[313] J. Mardon, Y. Nomura, D. Stolarski, and J. Thaler,
Dark Matter Signals fromCascade Annihilations , JCAP (2009) 016, [ arXiv:0901.2926 ].[314] L. Bergstrom, T. Bringmann, I. Cholis, D. Hooper, and C. Weniger,
Newlimits on dark matter annihilation from AMS cosmic ray positron data , Phys.Rev. Lett. (2013) 171101, [ arXiv:1306.3983 ].[315] X.-L. Chen and M. Kamionkowski,
Particle decays during the cosmic darkages , Phys. Rev.
D70 (2004) 043502, [ astro-ph/0310473 ].[316] N. Padmanabhan and D. P. Finkbeiner,
Detecting dark matter annihilationwith CMB polarization: Signatures and experimental prospects , Phys. Rev.
D72 (2005) 023508, [ astro-ph/0503486 ].[317] T. R. Slatyer, N. Padmanabhan, and D. P. Finkbeiner,
CMB Constraints onWIMP Annihilation: Energy Absorption During the Recombination Epoch , Phys. Rev.
D80 (2009) 043526, [ arXiv:0906.1197 ].[318] M. S. Madhavacheril, N. Sehgal, and T. R. Slatyer,
Current Dark MatterAnnihilation Constraints from CMB and Low-Redshift Data , Phys. Rev.
D89 (2014) 103508, [ arXiv:1310.3815 ].359319] J. Fan and M. Reece,
Simple dark matter recipe for the 111 and 128 GeVFermi-LAT lines , Phys. Rev.
D88 (2013), no. 3 035014, [ arXiv:1209.1097 ].[320] Y. Gao, A. V. Gritsan, Z. Guo, K. Melnikov, M. Schulze, and N. V. Tran,
Spin determination of single-produced resonances at hadron colliders , Phys.Rev.
D81 (2010) 075022, [ arXiv:1001.3396 ].[321] N. Arkani-Hamed and N. Weiner,
LHC Signals for a SuperUnified Theory ofDark Matter , JHEP (2008) 104, [ arXiv:0810.0714 ].[322] C. Cheung, J. T. Ruderman, L.-T. Wang, and I. Yavin, Lepton Jets in(Supersymmetric) Electroweak Processes , JHEP (2010) 116,[ arXiv:0909.0290 ].[323] B. Patt and F. Wilczek, Higgs-field portal into hidden sectors , hep-ph/0605188 .[324] J. March-Russell, S. M. West, D. Cumberbatch, and D. Hooper, Heavy DarkMatter Through the Higgs Portal , JHEP (2008) 058, [ arXiv:0801.3440 ].[325] K. R. Dienes, C. F. Kolda, and J. March-Russell, Kinetic mixing and thesupersymmetric gauge hierarchy , Nucl. Phys.
B492 (1997) 104–118,[ hep-ph/9610479 ].[326] A. Falkowski, J. Juknevich, and J. Shelton,
Dark Matter Through theNeutrino Portal , arXiv:0908.1790 .[327] R. Essig et al., Working Group Report: New Light Weakly Coupled Particles , arXiv:1311.0029 .[328] B. Holdom, Two U(1)’s and Epsilon Charge Shifts , Phys. Lett.
B166 (1986)196.[329] M. J. Strassler and K. M. Zurek,
Echoes of a hidden valley at hadron colliders , Phys. Lett.
B651 (2007) 374–379, [ hep-ph/0604261 ].360330] T. Han, Z. Si, K. M. Zurek, and M. J. Strassler,
Phenomenology of hiddenvalleys at hadron colliders , JHEP (2008) 008, [ arXiv:0712.2041 ].[331] D. Hooper, N. Weiner, and W. Xue, Dark Forces and Light Dark Matter , Phys. Rev.
D86 (2012) 056009, [ arXiv:1206.2929 ].[332] A. Berlin, P. Gratia, D. Hooper, and S. D. McDermott,
Hidden Sector DarkMatter Models for the Galactic Center Gamma-Ray Excess , Phys. Rev.
D90 (2014), no. 1 015032, [ arXiv:1405.5204 ].[333] J. M. Cline, G. Dupuis, Z. Liu, and W. Xue,
The windows for kineticallymixed Z’-mediated dark matter and the galactic center gamma ray excess , JHEP (2014) 131, [ arXiv:1405.7691 ].[334] J. Liu, N. Weiner, and W. Xue, Signals of a Light Dark Force in the GalacticCenter , JHEP (2015) 050, [ arXiv:1412.1485 ].[335] J. M. Cline, G. Dupuis, Z. Liu, and W. Xue, Multimediator models for thegalactic center gamma ray excess , Phys. Rev.
D91 (2015), no. 11 115010,[ arXiv:1503.08213 ].[336]
AMS
Collaboration, M. Aguilar et al.,
Electron and Positron Fluxes inPrimary Cosmic Rays Measured with the Alpha Magnetic Spectrometer on theInternational Space Station , Phys. Rev. Lett. (2014) 121102.[337]
AMS
Collaboration, L. Accardo et al.,
High Statistics Measurement of thePositron Fraction in Primary Cosmic Rays of 0.5-500 GeV with the AlphaMagnetic Spectrometer on the International Space Station , Phys. Rev. Lett. (2014) 121101.[338] M. Cirelli, M. Kadastik, M. Raidal, and A. Strumia,
Model-IndependentImplications of the E+-, Anti-Proton Cosmic Ray Spectra on Properties ofDark Matter , Nucl. Phys.
B813 (2009) 1–21, [ arXiv:0809.2409 ]. [Addendum:Nucl. Phys.B873,530(2013)]. 361339] J. Mardon, Y. Nomura, and J. Thaler,
Cosmic Signals from the HiddenSector , Phys. Rev.
D80 (2009) 035013, [ arXiv:0905.3749 ].[340]
HESS
Collaboration, A. Abramowski et al.,
Search for dark matterannihilation signatures in H.E.S.S. observations of Dwarf Spheroidal Galaxies , Phys. Rev.
D90 (2014) 112012, [ arXiv:1410.2589 ].[341]
VERITAS
Collaboration, B. Zitzer,
Search for Dark Matter from DwarfGalaxies using VERITAS , in
Proceedings, 34th International Cosmic RayConference (ICRC 2015) , 2015. arXiv:1509.01105 .[342] D. Hooper, C. Kelso, and F. S. Queiroz,
Stringent and Robust Constraints onthe Dark Matter Annihilation Cross Section From the Region of the GalacticCenter , Astropart. Phys. (2013) 55–70, [ arXiv:1209.3015 ].[343] E. Storm, T. E. Jeltema, S. Profumo, and L. Rudnick, Constraints on DarkMatter Annihilation in Clusters of Galaxies from Diffuse Radio Emission , Astrophys. J. (2013) 106, [ arXiv:1210.0872 ].[344] B. von Harling and K. L. McDonald,
Secluded Dark Matter Coupled to aHidden CFT , JHEP (2012) 048, [ arXiv:1203.6646 ].[345] Y. Kuno and Y. Okada, Muon decay and physics beyond the standard model , Rev. Mod. Phys. (2001) 151–202, [ hep-ph/9909265 ].[346] L. Michel, Interaction Between Four Half Spin Particles and the Decay of the 𝜇 Meson , Proc. Phys. Soc.
A63 (1950) 514–531.[347] D. P. Finkbeiner, S. Galli, T. Lin, and T. R. Slatyer,
Searching for DarkMatter in the CMB: A Compact Parameterization of Energy Injection fromNew Physics , Phys. Rev.
D85 (2012) 043522, [ arXiv:1109.6322 ].[348] S. Galli, F. Iocco, G. Bertone, and A. Melchiorri,
Updated CMB constraints onDark Matter annihilation cross-sections , Phys. Rev.
D84 (2011) 027302,[ arXiv:1106.1528 ]. 362349] G. Hutsi, J. Chluba, A. Hektor, and M. Raidal,
WMAP7 and future CMBconstraints on annihilating dark matter: implications on GeV-scale WIMPs , Astron. Astrophys. (2011) A26, [ arXiv:1103.2766 ].[350] G. Giesen, J. Lesgourgues, B. Audren, and Y. Ali-Haimoud,
CMB photonsshedding light on dark matter , JCAP (2012) 008, [ arXiv:1209.0247 ].[351] C. Weniger, P. D. Serpico, F. Iocco, and G. Bertone,
CMB bounds on darkmatter annihilation: Nucleon energy-losses after recombination , Phys.Rev.
D87 (2013), no. 12 123008, [ arXiv:1303.0942 ].[352]
DES
Collaboration, K. Bechtol et al.,
Eight New Milky Way CompanionsDiscovered in First-Year Dark Energy Survey Data , Astrophys. J. (2015),no. 1 50, [ arXiv:1503.02584 ].[353] S. E. Koposov, V. Belokurov, G. Torrealba, and N. W. Evans,
Beasts of theSouthern Wild: Discovery of Nine Ultra Faint Satellites in the Vicinity of theMagellanic Clouds , Astrophys. J. (2015), no. 2 130, [ arXiv:1503.02079 ].[354]
DES, Fermi-LAT
Collaboration, A. Drlica-Wagner et al.,
Search forGamma-Ray Emission from Des Dwarf Spheroidal Galaxy Candidates withFermi-Lat Data , Submitted to: Astrophys. J. (2015) [ arXiv:1503.02632 ].[355] A. Geringer-Sameth, M. G. Walker, S. M. Koushiappas, S. E. Koposov,V. Belokurov, G. Torrealba, and N. W. Evans,
Indication of Gamma-RayEmission from the Newly Discovered Dwarf Galaxy Reticulum II , arXiv:1503.02320 .[356] D. Hooper and T. Linden, On the Gamma-Ray Emission from Reticulum IIand Other Dwarf Galaxies , arXiv:1503.06209 .[357] P. Blasi, The Origin of the Positron Excess in Cosmic Rays , Phys. Rev. Lett. (2009) 051104, [ arXiv:0903.2794 ].363358] L. J. Gleeson and W. I. Axford,
Solar Modulation of Galactic Cosmic Rays , Astrophys. J. (1968) 1011.[359] I. V. Moskalenko, A. W. Strong, J. F. Ormes, and M. S. Potgieter,
SecondaryAnti-Protons and Propagation of Cosmic Rays in the Galaxy and Heliosphere , Astrophys. J. (2002) 280–296, [ astro-ph/0106567 ].[360] B. Beischer, P. von Doetinchem, H. Gast, T. Kirn, and S. Schael,
Perspectivesfor indirect dark matter search with AMS-2 using cosmic-ray electrons andpositrons , New J. Phys. (2009) 105021.[361] D. Hooper and W. Xue, Possibility of Testing the Light Dark MatterHypothesis with the Alpha Magnetic Spectrometer , Phys. Rev. Lett. (2013), no. 4 041302, [ arXiv:1210.1220 ].[362] C. Evoli, D. Gaggero, D. Grasso, and L. Maccione,
Cosmic-Ray Nuclei,Antiprotons and Gamma-Rays in the Galaxy: a New Diffusion Model , JCAP (2008) 018, [ arXiv:0807.4730 ].[363] L. Maccione, C. Evoli, D. Gaggero, and D. Grasso, “DRAGON: GalacticCosmic Ray Diffusion Code.” Astrophysics Source Code Library, June, 2011.[364] C. Evoli, I. Cholis, D. Grasso, L. Maccione, and P. Ullio,
Antiprotons fromDark Matter Annihilation in the Galaxy: Astrophysical Uncertainties , Phys.Rev.
D85 (2012) 123511, [ arXiv:1108.0664 ].[365] M. S. Pshirkov, P. G. Tinyakov, P. P. Kronberg, and K. J. Newton-McGee,
Deriving Global Structure of the Galactic Magnetic Field from FaradayRotation Measures of Extragalactic Sources , Astrophys. J. (2011) 192,[ arXiv:1103.0814 ].[366] G. Di Bernardo, C. Evoli, D. Gaggero, D. Grasso, and L. Maccione,
CosmicRay Electrons, Positrons and the Synchrotron Emission of the Galaxy:Consistent Analysis and Implications , JCAP (2013) 036,[ arXiv:1210.4546 ]. 364367] T. R. Jaffe, J. P. Leahy, A. J. Banday, S. M. Leach, S. R. Lowe, andA. Wilkinson,
Modelling the Galactic Magnetic Field on the Plane in 2D , Mon. Not. Roy. Astron. Soc. (2010) 1013, [ arXiv:0907.3994 ].[368]
HESS
Collaboration, J. A. Hinton,
The Status of the H.E.S.S. project , NewAstron. Rev. (2004) 331–337, [ astro-ph/0403052 ].[369] HESS
Collaboration, A. Abramowski et al.,
Search for Photon-LinelikeSignatures from Dark Matter Annihilations with H.E.S.S. , Phys. Rev. Lett. (2013) 041301, [ arXiv:1301.1173 ].[370] G. Sinnis, A. Smith, and J. E. McEnery,
HAWC: A Next generation all - skyVHE gamma-ray telescope , in
On recent developments in theoretical andexperimental general relativity, gravitation, and relativistic field theories.Proceedings, 10th Marcel Grossmann Meeting, MG10, Rio de Janeiro, Brazil,July 20-26, 2003. Pt. A-C , pp. 1068–1088, 2004. astro-ph/0403096 .[371]
HAWC
Collaboration, J. P. Harding and B. Dingus,
Dark MatterAnnihilation and Decay Searches with the High Altitude Water Cherenkov(HAWC) Observatory , in
Proceedings, 34th International Cosmic RayConference (ICRC 2015) , 2015. arXiv:1508.04352 .[372]
HAWC
Collaboration, J. Pretz,
Highlights from the High Altitude WaterCherenkov Observatory , in
Proceedings, 34th International Cosmic RayConference (ICRC 2015) , 2015. arXiv:1509.07851 .[373] T. C. Weekes et al.,
VERITAS: The Very energetic radiation imaging telescopearray system , Astropart. Phys. (2002) 221–243, [ astro-ph/0108478 ].[374] VERITAS
Collaboration, J. Holder et al.,
The first VERITAS telescope , Astropart. Phys. (2006) 391–401, [ astro-ph/0604119 ].[375] VERITAS
Collaboration, A. Geringer-Sameth,
The VERITAS Dark MatterProgram , in , 2013. arXiv:1303.1406 .365376]
MAGIC
Collaboration, J. Flix Molina,
Planned dark matter searches with theMAGIC Telescope , in
Proceedings, 40th Rencontres de Moriond on Very HighEnergy Phenomena in the Universe , pp. 421–424, 2005. astro-ph/0505313 .[377]
Fermi-LAT, MAGIC
Collaboration, M. L. Ahnen et al.,
Limits to darkmatter annihilation cross-section from a combined analysis of MAGIC andFermi-LAT observations of dwarf satellite galaxies , JCAP (2016), no. 02039, [ arXiv:1601.06590 ].[378] M. Ciafaloni, P. Ciafaloni, and D. Comelli,
Bloch-Nordsieck violatingelectroweak corrections to inclusive TeV scale hard processes , Phys. Rev. Lett. (2000) 4810–4813, [ hep-ph/0001142 ].[379] P. Ciafaloni and D. Comelli, Sudakov enhancement of electroweak corrections , Phys. Lett.
B446 (1999) 278–284, [ hep-ph/9809321 ].[380] P. Ciafaloni and D. Comelli,
Electroweak Sudakov form-factors andnonfactorizable soft QED effects at NLC energies , Phys. Lett.
B476 (2000)49–57, [ hep-ph/9910278 ].[381] J.-y. Chiu, A. Fuhrer, R. Kelley, and A. V. Manohar,
Factorization Structureof Gauge Theory Amplitudes and Application to Hard Scattering Processes atthe LHC , Phys. Rev.
D80 (2009) 094013, [ arXiv:0909.0012 ].[382] A. Hryczuk and R. Iengo,
The one-loop and Sommerfeld electroweakcorrections to the Wino dark matter annihilation , JHEP (2012) 163,[ arXiv:1111.2916 ]. [Erratum: JHEP06,137(2012)].[383] M. Baumgart, I. Z. Rothstein, and V. Vaidya, Calculating the AnnihilationRate of Weakly Interacting Massive Particles , Phys. Rev. Lett. (2015)211301, [ arXiv:1409.4415 ].[384] M. Bauer, T. Cohen, R. J. Hill, and M. P. Solon,
Soft Collinear EffectiveTheory for Heavy WIMP Annihilation , JHEP (2015) 099,[ arXiv:1409.7392 ]. 366385] G. Ovanesyan, T. R. Slatyer, and I. W. Stewart, Heavy Dark MatterAnnihilation from Effective Field Theory , Phys. Rev. Lett. (2015), no. 21211302, [ arXiv:1409.8294 ].[386] M. Baumgart, I. Z. Rothstein, and V. Vaidya,
Constraints on Galactic WinoDensities from Gamma Ray Lines , JHEP (2015) 106, [ arXiv:1412.8698 ].[387] M. Baumgart and V. Vaidya, Semi-inclusive wino and higgsino annihilation toLL’ , JHEP (2016) 213, [ arXiv:1510.02470 ].[388] J. D. Wells, PeV-scale supersymmetry , Phys. Rev.
D71 (2005) 015013,[ hep-ph/0411041 ].[389] N. Arkani-Hamed and S. Dimopoulos,
Supersymmetric unification without lowenergy supersymmetry and signatures for fine-tuning at the LHC , JHEP (2005) 073, [ hep-th/0405159 ].[390] G. F. Giudice and A. Romanino, Split supersymmetry , Nucl. Phys.
B699 (2004) 65–89, [ hep-ph/0406088 ]. [Erratum: Nucl. Phys.B706,487(2005)].[391] A. Arvanitaki, N. Craig, S. Dimopoulos, and G. Villadoro,
Mini-Split , JHEP (2013) 126, [ arXiv:1210.0555 ].[392] N. Arkani-Hamed, A. Gupta, D. E. Kaplan, N. Weiner, and T. Zorawski, Simply Unnatural Supersymmetry , arXiv:1212.6971 .[393] L. J. Hall, Y. Nomura, and S. Shirai, Spread Supersymmetry with Wino LSP:Gluino and Dark Matter Signals , JHEP (2013) 036, [ arXiv:1210.2395 ].[394] T. Cohen, M. Lisanti, A. Pierce, and T. R. Slatyer, Wino Dark Matter UnderSiege , JCAP (2013) 061, [ arXiv:1307.4082 ].[395] P. Ciafaloni, D. Comelli, A. De Simone, A. Riotto, and A. Urbano,
Electroweak Bremsstrahlung for Wino-Like Dark Matter Annihilations , JCAP (2012) 016, [ arXiv:1202.0692 ].367396] C. W. Bauer, S. Fleming, and M. E. Luke,
Summing Sudakov logarithms in B—> X(s gamma) in effective field theory , Phys. Rev.
D63 (2000) 014006,[ hep-ph/0005275 ].[397] C. W. Bauer, S. Fleming, D. Pirjol, and I. W. Stewart,
An Effective fieldtheory for collinear and soft gluons: Heavy to light decays , Phys. Rev.
D63 (2001) 114020, [ hep-ph/0011336 ].[398] C. W. Bauer and I. W. Stewart,
Invariant operators in collinear effectivetheory , Phys. Lett.
B516 (2001) 134–142, [ hep-ph/0107001 ].[399] C. W. Bauer, D. Pirjol, and I. W. Stewart,
Soft collinear factorization ineffective field theory , Phys. Rev.
D65 (2002) 054022, [ hep-ph/0109045 ].[400] J.-y. Chiu, F. Golf, R. Kelley, and A. V. Manohar,
Electroweak Sudakovcorrections using effective field theory , Phys. Rev. Lett. (2008) 021802,[ arXiv:0709.2377 ].[401] J.-y. Chiu, F. Golf, R. Kelley, and A. V. Manohar,
Electroweak Corrections inHigh Energy Processes using Effective Field Theory , Phys. Rev.
D77 (2008)053004, [ arXiv:0712.0396 ].[402] J.-y. Chiu, R. Kelley, and A. V. Manohar,
Electroweak Corrections usingEffective Field Theory: Applications to the LHC , Phys. Rev.
D78 (2008)073006, [ arXiv:0806.1240 ].[403] J.-y. Chiu, A. Fuhrer, R. Kelley, and A. V. Manohar,
Soft and CollinearFunctions for the Standard Model , Phys. Rev.
D81 (2010) 014023,[ arXiv:0909.0947 ].[404] C. W. Bauer, S. Fleming, D. Pirjol, I. Z. Rothstein, and I. W. Stewart,
Hardscattering factorization from effective field theory , Phys. Rev.
D66 (2002)014017, [ hep-ph/0202088 ]. 368405] J.-Y. Chiu, A. Jain, D. Neill, and I. Z. Rothstein,
A Formalism for theSystematic Treatment of Rapidity Logarithms in Quantum Field Theory , JHEP (2012) 084, [ arXiv:1202.0814 ].[406] T. Becher and M. Neubert, Drell-Yan Production at Small 𝑞 𝑇 , TransverseParton Distributions and the Collinear Anomaly , Eur. Phys. J.
C71 (2011)1665, [ arXiv:1007.4005 ].[407] A. Denner, L. Jenniches, J.-N. Lang, and C. Sturm,
Gauge-independent
𝑀 𝑆 renormalization in the 2HDM , JHEP (2016) 115, [ arXiv:1607.07352 ].[408] L. Bergstrom, G. Bertone, J. Conrad, C. Farnier, and C. Weniger, Investigating Gamma-Ray Lines from Dark Matter with Future Observatories , JCAP (2012) 025, [ arXiv:1207.6773 ].[409] D. J. Schlegel, D. P. Finkbeiner, and M. Davis,
Maps of dust IR emission foruse in estimation of reddening and CMBR foregrounds , Astrophys. J. (1998) 525, [ astro-ph/9710327 ].[410] T. D. P. Edwards and C. Weniger,
A Fresh Approach to Forecasting inAstroparticle Physics and Dark Matter Searches , arXiv:1704.05458 .[411] A. Burkert, The Structure of dark matter halos in dwarf galaxies , IAU Symp. (1996) 175, [ astro-ph/9504041 ]. [Astrophys. J.447,L25(1995)].[412] B. Diemer and A. V. Kravtsov,
A universal model for halo concentrations , Astrophys. J. (2015), no. 1 108, [ arXiv:1407.4730 ].[413]
Fermi-LAT
Collaboration, F. Acero et al.,
Development of the Model ofGalactic Interstellar Emission for Standard Point-Source Analysis of FermiLarge Area Telescope Data , Astrophys. J. Suppl. (2016), no. 2 26,[ arXiv:1602.07246 ].[414] M. Schaller, C. S. Frenk, R. G. Bower, T. Theuns, A. Jenkins, J. Schaye,R. A. Crain, M. Furlong, C. D. Vecchia, and I. G. McCarthy,
Baryon effects n the internal structure of Λ CDM haloes in the EAGLE simulations , Mon.Not. Roy. Astron. Soc. (2015), no. 2 1247–1267, [ arXiv:1409.8617 ].[415] J. S. Bullock, T. S. Kolatt, Y. Sigad, R. S. Somerville, A. V. Kravtsov, A. A.Klypin, J. R. Primack, and A. Dekel,
Profiles of dark haloes. Evolution,scatter, and environment , Mon. Not. Roy. Astron. Soc. (2001) 559–575,[ astro-ph/9908159 ].[416] E. Nezri, R. White, C. Combet, D. Maurin, E. Pointecouteau, and J. A.Hinton, gamma-rays from annihilating dark matter in galaxy clusters: stackingvs single source analysis , Mon. Not. Roy. Astron. Soc. (2012) 477,[ arXiv:1203.1165 ].[417] M. A. Sánchez-Conde and F. Prada,
The flattening of the concentration–massrelation towards low halo masses and its implications for the annihilationsignal boost , Mon. Not. Roy. Astron. Soc. (2014), no. 3 2271–2277,[ arXiv:1312.1729 ].[418] A. Molinè, M. A. Sànchez-Conde, S. Palomares-Ruiz, and F. Prada,
Characterization of subhalo structural properties and implications for darkmatter annihilation signals , Mon. Not. Roy. Astron. Soc. (2017), no. 44974–4990, [ arXiv:1603.04057 ].[419] Y. Lu, X. Yang, F. Shi, H. J. Mo, D. Tweed, H. Wang, Y. Zhang, S. Li, andS. H. Lim,
Galaxy groups in the 2MASS Redshift Survey , Astrophys. J. (2016), no. 1 39, [ arXiv:1607.03982 ].[420] I. D. Karachentsev and D. A. Makarov,
The Galaxy Motion Relative to NearbyGalaxies and the Local Velocity Field , AJ (Feb., 1996) 794.[421] I. D. Karachentsev, R. B. Tully, P.-F. Wu, E. J. Shaya, and A. E. Dolphin, Infall of nearby galaxies into the Virgo cluster as traced with HST , Astrophys.J. (2014) 4, [ arXiv:1312.6769 ].370422] A. Diaferio, M. Ramella, M. J. Geller, and A. Ferrari,
Are groups of galaxiesvirialized systems? , Astrophys. J. (June, 1993) 2035–2046.[423] P. Bhattacharjee and G. Sigl,
Origin and propagation of extremely high-energycosmic rays , Phys. Rept. (2000) 109–247, [ astro-ph/9811011 ].[424] K. Ishiwata, S. Matsumoto, and T. Moroi,
High Energy Cosmic Rays from theDecay of Gravitino Dark Matter , Phys. Rev.
D78 (2008) 063505,[ arXiv:0805.1133 ].[425] M. Grefe,
Neutrino signals from gravitino dark matter with broken R-parity .PhD thesis, Hamburg U., 2008. arXiv:1111.6041 .[426] A. Alloul, N. D. Christensen, C. Degrande, C. Duhr, and B. Fuks,
FeynRules2.0 - A complete toolbox for tree-level phenomenology , Comput. Phys.Commun. (2014) 2250–2300, [ arXiv:1310.1921 ].[427] J. Alwall, M. Herquet, F. Maltoni, O. Mattelaer, and T. Stelzer,
MadGraph 5: Going Beyond , JHEP (2011) 128, [ arXiv:1106.0522 ].[428] J. Alwall, R. Frederix, S. Frixione, V. Hirschi, F. Maltoni, O. Mattelaer, H. S.Shao, T. Stelzer, P. Torrielli, and M. Zaro, The automated computation oftree-level and next-to-leading order differential cross sections, and theirmatching to parton shower simulations , JHEP (2014) 079,[ arXiv:1405.0301 ].[429] J.-F. Fortin, J. Shelton, S. Thomas, and Y. Zhao, Gamma Ray Spectra fromDark Matter Annihilation and Decay , arXiv:0908.2258 .[430] D. P. Finkbeiner and N. Weiner, X-ray line from exciting dark matter , Phys.Rev.
D94 (2016), no. 8 083002, [ arXiv:1402.6671 ].[431] A. Sommerfeld,
Über die Beugung und Bremsung der Elektronen , Annalen derPhysik (1931) 257–330. 371432] M. Pospelov and A. Ritz,
Astrophysical Signatures of Secluded Dark Matter , Phys. Lett.
B671 (2009) 391–397, [ arXiv:0810.1502 ].[433] E. Byckling and K. Kajantie,
N-particle phase space in terms of invariantmomentum transfers , Nucl. Phys. B9 (1969) 568–576.[434] B. P. Kersevan and E. Richter-Was, Improved phase space treatment ofmassive multi-particle final states , Eur. Phys. J.
C39 (2005) 439–450,[ hep-ph/0405248 ].[435] F. Boudjema, A. Semenov, and D. Temes,
Self-annihilation of the neutralinodark matter into two photons or a Z and a photon in the MSSM , Phys. Rev.
D72 (2005) 055024, [ hep-ph/0507127 ].[436] N. Baro, F. Boudjema, and A. Semenov,
Full one-loop corrections to the relicdensity in the MSSM: A Few examples , Phys. Lett.
B660 (2008) 550–560,[ arXiv:0710.1821 ].[437] N. Baro, F. Boudjema, G. Chalons, and S. Hao,
Relic density at one-loop withgauge boson pair production , Phys. Rev.
D81 (2010) 015005,[ arXiv:0910.3293 ].[438] A. V. Manohar and I. W. Stewart,
The Zero-Bin and Mode Factorization inQuantum Field Theory , Phys. Rev.
D76 (2007) 074002, [ hep-ph/0605001 ].[439] A. Fuhrer, A. V. Manohar, J.-y. Chiu, and R. Kelley,
Radiative Corrections toLongitudinal and Transverse Gauge Boson and Higgs Production , Phys. Rev.
D81 (2010) 093005, [ arXiv:1003.0025 ].[440] G. Passarino and M. J. G. Veltman,
One Loop Corrections for e+ e-Annihilation Into mu+ mu- in the Weinberg Model , Nucl. Phys.
B160 (1979)151–207.[441] A. Denner, H. Eck, O. Hahn, and J. Kublbeck,
Feynman rules for fermionnumber violating interactions , Nucl. Phys.
B387 (1992) 467–484.372442] R. K. Ellis and G. Zanderighi,
Scalar one-loop integrals for QCD , JHEP (2008) 002, [ arXiv:0712.1851 ].[443] G. ’t Hooft and M. J. G. Veltman, Scalar One Loop Integrals , Nucl. Phys.
B153 (1979) 365–401.[444] R. K. Ellis, Z. Kunszt, K. Melnikov, and G. Zanderighi,
One-loop calculationsin quantum field theory: from Feynman diagrams to unitarity cuts , Phys.Rept. (2012) 141–250, [ arXiv:1105.4319 ].[445] R. Mertig, M. Bohm, and A. Denner,
FEYN CALC: Computer algebraiccalculation of Feynman amplitudes , Comput. Phys. Commun. (1991)345–359.[446] V. Shtabovenko, R. Mertig, and F. Orellana, New Developments in FeynCalc9.0 , arXiv:1601.01167 .[447] H. H. Patel, Package-X: A Mathematica package for the analytic calculation ofone-loop integrals , Comput. Phys. Commun. (2015) 276–290,[ arXiv:1503.01469 ].[448] D. Yu. Bardin and G. Passarino,
The Standard Model in the Making:Precision Study of the Electroweak Interactions . 1999.[449] I. W. Stewart,
Extraction of the D* D pi coupling from D* decays , Nucl. Phys.
B529 (1998) 62–80, [ hep-ph/9803227 ].[450] A. V. Manohar and M. B. Wise,
Heavy quark physics , Camb. Monogr. Part.Phys. Nucl. Phys. Cosmol. (2000) 1–191.[451] S. Catani and M. H. Seymour, The Dipole formalism for the calculation ofQCD jet cross-sections at next-to-leading order , Phys. Lett.
B378 (1996)287–301, [ hep-ph/9602277 ]. 373452] S. Catani and M. H. Seymour,
A General algorithm for calculating jetcross-sections in NLO QCD , Nucl. Phys.
B485 (1997) 291–419,[ hep-ph/9605323 ]. [Erratum: Nucl. Phys.B510,503(1998)].[453] I. Moult, I. W. Stewart, F. J. Tackmann, and W. J. Waalewijn,
EmployingHelicity Amplitudes for Resummation , Phys. Rev.
D93 (2016), no. 9 094003,[ arXiv:1508.02397 ].[454] J.-y. Chiu, A. Fuhrer, A. H. Hoang, R. Kelley, and A. V. Manohar,
Soft-Collinear Factorization and Zero-Bin Subtractions , Phys. Rev.
D79 (2009) 053007, [ arXiv:0901.1332 ].[455] A. Denner,
Techniques for calculation of electroweak radiative corrections atthe one loop level and results for W physics at LEP-200 , Fortsch. Phys. (1993) 307–420, [ arXiv:0709.1075arXiv:0709.1075