Resummed Photon Spectra for WIMP Annihilation
Matthew Baumgart, Timothy Cohen, Ian Moult, Nicholas L. Rodd, Tracy R. Slatyer, Mikhail P. Solon, Iain W. Stewart, Varun Vaidya
PPrepared for submission to JHEP
CALT-TH-2017-066LA-UR-17-31169MIT-CTP 4959
Resummed Photon Spectra for WIMP Annihilation
Matthew Baumgart, , Timothy Cohen, Ian Moult, , Nicholas L. Rodd, Tracy R. Slatyer, Mikhail P. Solon, Iain W. Stewart, and Varun Vaidya Department of Physics, Arizona State University, Tempe, AZ 85287 New High Energy Theory Center, Rutgers University, Piscataway, NJ 08854 Institute of Theoretical Science, University of Oregon, Eugene, OR 97403 Berkeley Center for Theoretical Physics, University of California, Berkeley, CA 94720 Theoretical Physics Group, Lawrence Berkeley National Laboratory, Berkeley, CA 94720 Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, MA 02139 Walter Burke Institute for Theoretical Physics, California Institute of Technology, Pasadena, CA 91125 Theoretical Division, MS B283, Los Alamos National Laboratory, Los Alamos, NM 87545
Abstract:
We construct an effective field theory (EFT) description of the hard photon spec-trum for heavy WIMP annihilation. This facilitates precision predictions relevant for linesearches, and allows the incorporation of non-trivial energy resolution effects. Our frameworkcombines techniques from non-relativistic EFTs and soft-collinear effective theory (SCET),as well as its multi-scale extensions that have been recently introduced for studying jet sub-structure. We find a number of interesting features, including the simultaneous presence ofSCET I and SCET II modes, as well as collinear-soft modes at the electroweak scale. We de-rive a factorization formula that enables both the resummation of the leading large Sudakovdouble logarithms that appear in the perturbative spectrum, and the inclusion of Sommerfeldenhancement effects. Consistency of this factorization is demonstrated to leading logarith-mic order through explicit calculation. Our final result contains both the exclusive and theinclusive limits, thereby providing a unifying description of these two previously-consideredapproximations. We estimate the impact on experimental sensitivity, focusing for concretenesson an SU (2) W triplet fermion dark matter – the pure wino – where the strongest constraints aredue to a search for gamma-ray lines from the Galactic Center. We find numerically significantcorrections compared to previous results, thereby highlighting the importance of accountingfor the photon spectrum when interpreting data from current and future indirect detectionexperiments. a r X i v : . [ h e p - ph ] A p r ontents – 2 – Introduction
The discovery of the dark matter (DM) particle(s) is one of the central goals of the highenergy physics program. While the Weakly Interacting Massive Particle (WIMP) paradigmwith DM masses of order the electroweak scale ∼ GeV has received the most attention,it is also a reasonable possibility that the WIMP could be much heavier. The canonicalexample is the neutral component of a new Majorana SU (2) W triplet fermion – this wino DMwill be the concrete example studied here, although many of the results presented below willhold for a wide class of heavy WIMPs. Assuming no other new states are present, the winomass is the only free parameter in this model. The wino is a prototypical heavy WIMP: acalculation of the relic density for winos annihilating to electroweak gauge bosons (includingthe impact of the charged wino states via the Sommerfeld enhancement [1–5]) yields a massof around 3 TeV. The wino as DM is motivated both from a “complete” theory perspective inthe context of split supersymmetry [6–13], but it is also interesting due to its economy, i.e. ,minimal DM [5, 14–17].Multi-TeV WIMPs are unobservable at the LHC: 14 TeV projected limits on winos are inthe few hundred GeV range, and they will even be challenging to find at a future collider [18,19]. Furthermore, the cross section at direct detection experiments suffers an accidentalcancellation between the spin-0 and spin-2 contributions, yielding a rate that is near theneutrino floor [20–22]. The one known channel that holds promise for detecting multi-TeVwinos is via astrophysical searches for their annihilation products. Annihilation to photonscould provide a very clean signal visible to ground-based air Cherenkov array telescopes [5,23, 24], and constraints from the observed flux of antiproton cosmic rays can also be relevant,but require modeling of cosmic-ray propagation and backgrounds [25]. In particular, a searchfor line photons by the HESS experiment [26] provides a powerful constraint for thermal winoswith mass near 3 TeV, although this is subject to large uncertainties from the unknown shapeof the DM density profile in the inner Galaxy [23]. Furthermore, there are many upcomingexperimental searches which could discover heavy WIMPs via indirect detection of gammarays, including new data from HESS [27, 28], HAWC [29–31], CTA [32], VERITAS [33–35],and MAGIC [36, 37]. We would therefore like to have reliable theoretical predictions forthe particle physics contribution to the cross section over a wide range of DM masses. Onekey feature of these ground-based experiments is that their resolution for line searches is notparticularly sharp, implying that finite bin effects should be accounted for when making aprecise prediction of the annihilation cross section. A main goal of the present work is toaddress this.It is by now well understood that the calculation of the annihilation rate is complicated bythe presence of multiple hierarchical scales, namely m W and M χ . For models with M χ (cid:29) m W ,this separation of scales invalidates the standard perturbative expansion, introducing a num-ber of effects that must be treated to all orders, in particular Sommerfeld enhancement,which resums terms of the form ( α W M χ /m W ) k [2, 3, 5, 38, 39], and Sudakov double log-arithms α W log ( M χ /m W ) [40–46]. These can be conveniently treated using effective field– 3 –heory (EFT) techniques, which allow for a systematic expansion in m W /M χ (cid:28) , and theidentification of universal behavior in this limit. This has attracted recent attention, result-ing in calculations from different groups, with differing assumptions. Two groups [42, 43, 46]resummed the logarithms that appear assuming the final state was specified as γ γ or γ Z (referred to here as exclusive), while [41, 44, 45] calculated a resummed cross section usingthe operator product expansion (OPE) and assuming a γ + X final state (referred to hereas inclusive). Due to these differing assumptions, distinct conclusions were reached on theimportance of the logarithmically enhanced terms.In reality, the situation is more subtle and lies somewhere in between these two extremes.Due to the finite energy resolution of the detector, the state X recoiling against the detectedphoton, which we take to have energy E γ , is not forced to be a single electroweak boson.However, X is constrained to lie near the light cone, namely it is a jet . In this region it is wellknown that the standard OPE breaks down, and a more complicated factorization, describingthe dynamics of the radiation within the jet, is required. Explicitly, this introduces anothersmall parameter (1 − z ) (cid:28) , where z = E γ M χ ∈ [0 , , (1.1)controls the distance from the endpoint, thereby further complicating the perturbative struc-ture. In particular, large logarithms of (1 − z ) appear. We will refer to these as endpointlogarithms since they become important as z → . The importance of these endpoint loga-rithms in the DM case was noticed in [45] where an attempt was made to extend the OPEbased expansion beyond its region of validity into the endpoint region. However, this frame-work did not provide a way to exponentiate these logarithms. Their resummation is one ofthe goals of this paper.In this paper we develop a comprehensive EFT framework to compute the photon spec-trum for annihilating (or decaying) DM. We use the soft-collinear effective theory (SCET)[52–54], and its recent extensions developed for treating similar multi-scale problems in jet sub-structure, to factorize the dynamics at the scales m W (electroweak breaking scale), M χ (1 − z ) (soft scale), M χ √ − z (jet scale), and M χ (hard scale). In order to perform the resummation,we will need to refactorize the cross section using techniques for multi-modal field theories [55–59]. All large logarithms present in the cross section are then captured by renormalizationgroup evolution between the relevant scales. The end result is a completely factorized de-scription that allows for systematically improvable calculations of the photon spectrum. Inthis paper we will use this framework to compute the resummed spectrum for pure wino an-nihilation. The extension to Higgsinos and more general representations will be left for futurework.An example of the result from our calculation is shown in Fig. 1. Here we have plotted Similar effects have also been seen in fixed order calculations of χ χ → W + W − γ in the WIMP DMliterature [47–51]. – 4 – . . . . . . z cut h σ v i × [ c m / s ] Thermal Wino Cross Section
InclusiveExclusive H E SS R e s o l u t i o n Figure 1 . The resummed cross section as a function of the experimental resolution parameter z cut for a TeV wino, showing the transition between the fully inclusive ( z cut = 0 ) and the fully exclusive( z cut = 1 ) cases. For z cut ∼ . - . , as relevant for the HESS experiment, the prediction is half waybetween the two limiting cases, emphasizing the importance of properly treating z cut . the cumulative spectrum, σ ( z cut ) = (cid:90) z cut d z d σ d z . (1.2)A value of z cut = 0 corresponds to the fully inclusive case, and z cut = 1 to the fully exclusivecase. As a benchmark, we have taken the wino mass to be 3 TeV – a wider range of massesare presented below in Sec. 6. Here we see the impact of resumming the endpoint logarithms:there is the known factor of . difference between the exclusive and inclusive calculations,and when we take z cut ∼ . − . (which is motivated by the HESS energy resolution), wefind that the prediction falls almost half way between the inclusive and exclusive limits. Thisdemonstrates the importance of the study presented below.An outline of this paper is as follows. In Sec. 2 we carefully review the kinematics of indi-rect detection, highlighting the different regions of the photon spectrum, the appropriate fieldtheoretic techniques that are required for their description, and the differing approximationsmade in previous presentations. In Sec. 3 we review the different effective theories that we– 5 –ill make use of in our analysis, namely non-relativistic DM effective theory (NRDM) andSCET. In Sec. 4 we present our factorization formula for the region m W (cid:28) M χ (1 − z ) (cid:28) M χ .We describe in detail the multi-step matching procedure used in its derivation, and the phys-ical role of the different functions appearing in the factorization. In Sec. 5 we perform theLL resummation, and derive a compact analytic expression for the resummed spectrum. InSec. 5.3 we show that our EFT reproduces the resummation in both the OPE region, andthe exclusive endpoint by taking appropriate limits, hence tying together different results inthe literature. In Sec. 6 we present numerical results for the case of wino DM, comparingwith previous results obtained using the exclusive and inclusive calculations, allowing us todemonstrate that properly accounting for the finite resolution has a numerically significanteffect. In Sec. 7 we estimate the impact of our newly derived predictions on indirect detectionconstraints using a simplified mock analysis of the HESS data. We conclude in Sec. 8. Twoappendices are provided: in Appendix A, we provide many technical aspects of the one-loopcalculations presented in the text, and Appendix B demonstrates the minimal impact of pho-tons from cascade decays ( e.g. χ χ → W + W − → many γ s) on our mock reanalysis of theHESS data. Guide for the Reader
We anticipate that our audience’s interests span from the technical aspects of the EFT-basedcalculation to an interest in the implications for the indirect detection experimental pre-dictions. We therefore provide two road maps for navigating this paper, depending on theexpertise of the reader. For the EFT enthusiasts, the main technical details of the factor-ization are presented in Secs. 3-5. While we have attempted to make the presentation asself contained as possible, in particular by reviewing the relevant technology, these sectionsnecessarily assume a higher level of familiarity with EFT techniques, and are as such moremathematically intensive. These sections provide the details which yield the final prediction,but can be skipped without affecting one’s big picture understanding of this work.For the reader interested primarily in the results, and the resolution of previous differingapproximations and conclusions in the literature, we recommend Sec. 2.1 and Secs. 6-7. Sec. 2.1emphasizes the physical differences between the different approximations previously madein the literature, and explains the necessity of pursuing our approach to derive a completeunderstanding for the range of parameters of interest to current and future experiments.The main results of our study are shown in graphical form in Sec. 6, where we highlight thenumerical impact of the resummation of logarithms of z cut , and compare with numerical resultsfrom previous approximations. This clearly illustrates the importance of properly includingthe finite resolution of the experiments. Finally, the impact of our updated numerical resultson DM exclusions are given in Sec. 7. – 6 – a) (b) (c) Figure 2 . (a) Fully exclusive production, which contributes only at the endpoint where z = 1 . Onlyvirtual corrections are present. (b) Operator Product Expansion for γ + X with m X ∼ M χ . Here thestate X has a large invariant mass and can be integrated out. (c) The endpoint region, m X (cid:28) M χ .Here the measurement on the final state X constrains it to have a small invariant mass. This impliesthat X cannot be integrated out and must be treated as a dynamical object in the EFT. In all cases,the dashed lines dressing the annihilating DM represent the Sommerfeld enhancement. In this section, we discuss in detail the kinematics of heavy DM decay or annihilation tophotons as relevant for indirect detection. We carefully analyze all relevant scales, identifyingregions where large ratios of scales exist, which will give rise to logarithms that need to beresummed. This analysis will also make clear the differences between the previous studiesin the literature. We will also highlight how collinear-soft modes appear in the broken the-ory, highlighting the distinction with the case of the naively similar B → X s γ that has beenthoroughly treated in the literature (see e.g. [60–64]). The discussion of this section is com-pletely independent of the details of the DM, allowing us to simultaneously consider decayand annihilation, and depends only on the kinematics of indirect detection. We consider for concreteness the annihilation of two nearly stationary DM particles of mass M χ decaying to γ + X , where the γ is assumed to be detected by the experiment. Here X denotes all final state radiation apart from the photon. The case of DM decay for a particle ofmass M χ is identical. We use a dimensionless variable z to characterize the energy fractionof the photon E γ = M χ z , (2.1)or equivalently, m X = 4 M χ (1 − z ) , (2.2)– 7 –here m X is the invariant mass of the final state X . The result of the calculation will bea differential cross section as a function of z , which will be integrated from z = z cut → .Depending on the value of z cut , a number of different field theoretic descriptions are required: • Exclusive Final State ((1 − z cut ) = 0) [42, 43, 46]: Here the final state is exactly spec-ified, either γ γ or γ Z , and we have z cut = 1 . Electroweak Sudakov double logarithms, log (2 M χ /m W ) , appear in the perturbative expansion. See Fig. 2a. • Inclusive Final State ((1 − z cut ) ∼ [41, 44, 45]: Here the final state is γ + X , andthe final state X is fully inclusive. This implies that m X is large, such that the state X can be integrated out using a local OPE [67]. See Fig. 2b. • Endpoint Region (0 < (1 − z cut ) (cid:28) ): In this region, the invariant mass of the finalstate m X → and as such it cannot be integrated out using a local OPE. The photonof interest is taken to lie along one lightcone. Then X consists of collimated high energyradiation along an orthogonal light cone, with transverse spread p T ∼ M χ √ − z , aswell as isotropic soft radiation with E ∼ M χ (1 − z ) . The standard OPE approach isnot sufficient, and a more complicated factorization theorem describing the dynamics ofthe soft and collinear radiation is required [68]. Deriving an analogous factorization forthe case of WIMP annihilation is one of the main results of this paper. In this region,Sudakov double logarithms, log (1 − z ) appear in addition to electroweak Sudakov doublelogarithms log (2 M χ /m W ) . See Fig. 2c.We can now determine which of the above regions are most relevant to model the inputphoton spectrum for a search for DM lines. In principle, if the energy resolution of thedetector is sufficiently precise, the appropriate cross section would only include the exclusivefinal state consisting of a photon and a single recoiling electroweak boson. In this case,the kinematics dictate that this condition is equivalent to requiring z (cid:38) . ( . ) for M χ ∼ GeV (10 TeV). The corresponding energy resolution is well beyond the capabilitiesof existing detectors. For example, translating the Gaussian width of the resolution quoted inthe HESS line search [28] to a hard cut, would naively imply that z cut varies from 0.83 to 0.89as M χ goes from 500 GeV to 10 TeV. This range additionally implies that we are outside theinclusive region, such that factors of log (1 − z ) are potentially large and resummation shouldbe performed. We conclude then that the theory which best describes the line observationsmade by air Cherenkov telescopes has a state X that is recoiling against the photon with m X (cid:28) M χ , i.e. , the endpoint region EFT. The theoretical descriptions of the matching to theexclusive region, as well as the OPE region, are also important for a complete description ofthe spectrum. We will see that these limits arise naturally from our endpoint EFT. At this level of discussion, namely the description of kinematics, the different regions are identical to thosefor B → X s γ and related processes. In the B -physics literature, the endpoint region, which will be the focusof this paper, is also referred to as the shape function region [60, 65, 66]. – 8 – a) I n c r e a s i n g V i r t u a li t y H (b) Figure 3 . (a) A schematic depiction of the relevant modes in the effective theory for DM annihilationnear the endpoint. Modes which are sensitive to the mass of the electroweak bosons (broken theory) arein zig-zag, while those that behave as effectively massless (unbroken theory) are curvy. (b) Rapiditiesand virtualities of the modes describing the final state. The complicated modal structure of the EFTis driven by the simultaneous presence of the scales M χ , and m W , as well as the constraint on themass of the final state. Having determined that experimental considerations drive us to focus on the endpoint region,next we describe the relevant kinematics. This will expose the corresponding modes thatwill be required to construct the EFT description. These modes are shown schematicallyin Fig. 3, along with their virtualities and rapidities. Our goal in this section is twofold.First, this discussion will motivate the EFTs introduced in Sec. 3. Second, it will allow us toprovide context and highlight the new features of the factorization needed here in a physicalmanner, motivating the technical discussion of Sec. 4. The later sections will then provide acomprehensive mathematical treatment, to complement the simple picture that follows fromkinematic arguments.We begin with the kinematics of the initial state, namely the annihilating DM. The DM inthe halo has a typical velocity v ∼ − , so a non-relativistic description is appropriate. TheDM will be modeled as heavy sources (in analogy with heavy quark EFT or non-relativisticQCD) emitting ultra-soft radiation, as shown in Fig. 3a. There is one well known compli-cation in the heavy mass limit. Winos carry electroweak charge such that the Sommerfeldenhancement due to the exchange of electroweak gauge bosons must be included. This canbe appropriately accounted for in the non-relativistic DM (NRDM) EFT by including therelevant potentials, see Sec. 3.1. A feature of the NRDM EFT is that it allows a factorizationof the Sudakov corrections from the Sommerfeld effects.– 9 –he final state is more complicated, and a full characterization will require a multi-modalEFT. Recapping the discussion above, as z cut → the final state consists of both a jet ofcollimated energetic particles and wide angle low energy radiation. As is well known, thiscan be captured by SCET. However, due to the multi-scale nature of the problem, we willshow that additional modes, illustrated in Fig. 3, will be required to fully factorize all thelogarithms. The origin of the multi-modal structure, and its complexity compared to thatseen in previous approaches to heavy WIMP annihilation, can be understood from kinematicarguments. Specifically, logarithms appear due to two types of phase space restrictions: • Kinematic Restrictions on Final States of Massless Particles: These includekinematic restrictions via event shape observables, such as thrust, or restrictions fromkinematics that force one into an endpoint region, as in B → X s γ , and have beendiscussed above. EFT descriptions in these cases typically involve three scales: the hardscale, which in our case will be M χ ; the scale of the transverse momenta of particles inthe jet (whose modes are called collinear), namely M χ √ − z ; and the energy scale ofsoft radiation, namely M χ (1 − z ) . This class of problems is well understood and can betreated using SCET I , discussed in Sec. 3.2. The radiation in the final state is factorizedinto energetic modes, referred to as collinear ( c ), which comprise the dynamics of thejet, and wide angle low energy radiation, referred to as ultrasoft ( us ). Decomposed intolight cone coordinates ( n · p, ¯ n · p, p ⊥ ) (see Eq. (3.7)), along the direction of the jet, thesemodes have momentum scaling as p c ∼ M χ (cid:0) , λ , λ (cid:1) , p us ∼ M χ (cid:0) λ , λ , λ (cid:1) ; λ = √ − z . (2.3) • Exclusive Final States of Massive Particles:
These include the classic massiveSudakov form factor [71], and more recently, exclusive electroweak production [72–77],and the exclusive approximation for DM annihilation discussed above [42, 43, 46]. Herethere are two relevant mass scales, namely the hard ( h ) scale M χ , and the scale ofthe massive boson, m W . Problems of this type can be treated using an SCET II theory,discussed in Sec. 3.2. The relevant modes in the effective theory are collinear ( c ) and soft( s ) modes. Decomposed into light cone coordinates (see Eq. (3.7)), along the directionof the jet, these modes have momentum scaling as p c ∼ M χ (cid:0) , λ , λ (cid:1) , p s ∼ M χ (cid:0) λ, λ, λ (cid:1) ; λ = m W M χ . (2.4)Note the distinction in scaling between the ultrasoft and soft modes. While in this case Here we mean massless in perturbation theory, as relevant for scales appearing in logarithms in the weakcoupling expansion. Other mass scales can appear non-perturbatively, for example, hadron mass effects inQCD event shapes have been studied in [69, 70]. Note that here and throughout the text, when we describe the scaling of modes we indicate only theparametric scaling as a function of the relevant scales in the problem, namely M χ , m W , and − z . Any O (1) numerical factors do not modify this scaling, and are therefore neglected. – 10 –he collinear and soft modes are at the same virtuality p ∼ M χ λ , they are separatedin rapidity. This explains the appearance of the rapidity axis in Fig. 3b.The annihilation of WIMP DM in the endpoint region is a more complicated problem,since it involves the physics of both types of restrictions. There is both a constraint on thefinal state radiation, as well as the presence of the mass scale of the electroweak bosons and themeasurement of just the photon state from among the SU (2) × U (1) gauge bosons. Indeed,we will find that all the scales (in both rapidity and virtuality) present in both individualcases will appear. This is illustrated in Fig. 3b, which shows the modes that live at each ofthese mass and rapidity scales. We will show how to factorize the dynamics at each of thesescales when large hierarchies are present, thereby facilitating resummation. The final forminvolves a component where the gauge boson can be treated as massless, so that the scale isset by the final state kinematic restriction, and a component where the relevant scale is m W .For example, the description of the final state jet will be split into a massless jet function,described using standard techniques in SCET I , as well as a function describing the dynamicsat the scale m W , using SCET II .In addition to these SCET I and SCET II ingredients, we will show that an extra mode isrequired to achieve the fully factorized result. This mode has a virtuality µ ∼ m W , but it hasa large momentum component along the direction of the recoiling photon of size M χ (1 − z ) (the momentum scale of the soft function): p cs ∼ M χ (1 − z ) (cid:0) λ , , λ (cid:1) , λ = m W M χ (1 − z ) . (2.5)In the case that both M χ (1 − z ) (cid:28) M χ and m W / ( M χ (1 − z )) (cid:28) , these modes are neither(ultra)soft, or collinear, i.e. , they do not appear in either SCET I or SCET II EFTs, but areinstead an example of collinear-soft modes, see Sec. 3.2. Our factorization formula allowsfor the separate treatment of these collinear-soft modes, which allows us to resum all largelogarithms, but also ensures continuity of the cross section as we move away from the endpointregion, where these modes are no longer distinguishable from the standard soft modes. It is thesimultaneous presence of the scales M χ (1 − z ) and m W that gives rise to the presence of thesecollinear-soft modes – they would not appear if only a subset of the scales were present. Thestructure of the results presented below shares similarities with the factorization formulaefor jet substructure observables, where a measurement in addition to the mass has beenperformed [55–59, 78–80].The complete description of the final state therefore combines the
SCET I collinear andultrasoft modes with the SCET II soft and collinear modes in the direction of the jet, along We will typically use a dimensionful rapidity, ν , as in Fig. 3b. This should be thought of in analogy withthe dimensional regularization scale, µ , and is introduced in Sec. 3.2 where we discuss the regularization ofrapidity singularities. Here we have argued for the existence of collinear-soft modes based only on kinematics. The fact thatthese modes are actually required is also related to the fact that there are external states with electroweakcharges, as will be discussed in Sec. 4.2. – 11 –ith the collinear-soft modes describing additional radiation along the direction of the photon.Each of these will yield distinct functions in our factorization formula Eq. (4.1), implying thateach of these functions has a clear physical origin in terms of the scales of the problem. Thisseemingly complicated description is in fact a significant simplification, since the descriptionof the dynamics at any one of these scales has been reduced to its elemental form. In the nextsection, we will introduce the EFT ingredients, and in Sec. 4 we give the technical details ofthe factorization.
In this section we briefly review the different EFTs that we will use, primarily to establishour notation. Our use of non-relativistic (NR) field theories will be standard in the contextof QCD [81–83] (for reviews, see [84–86]), and will focus on aspects relevant for annihilatingDM (for applications of NRDM EFT to the scattering of DM with nucleon targets, see [20,21, 87, 88]). As we review SCET, we will highlight necessary extensions that are perhaps lessfamiliar.
In the NRDM EFT, large fluctuations of the DM field χ about a particular velocity v areintegrated out. The non-relativistic DM is described by a field χ v with a label velocity v ,just as in heavy quark EFT [89, 90]. Here v is a dimensionless four vector describing thevelocity of the DM, which for concreteness we will take to be v = (1 , , , . The freedomin the choice of v is represented in the EFT as a symmetry known as reparametrizationinvariance [91, 92]. The dynamics of χ v describe the residual fluctuations of the heavy state,as in non-relativistic QCD. The EFT captures the interactions of the non-relativistic particleswhose momenta p µ = ( E, (cid:126)p ) scale as soft ( M χ v, M χ v ) , ultrasoft ( M χ v , M χ v ) , and potential ( M χ v , M χ v ) . The ultrasoft modes describe radiation, while the soft modes give rise to therunning of potentials.The leading power interactions of the heavy DM particle(s) with the ultrasoft radiationcan be eliminated using a field redefinition χ ( r ) v → S ( r ) v χ ( r ) v [41–46], where S ( r ) v ( x ) = P exp ig (cid:90) −∞ d s v · A aus ( x + sv ) T a ( r ) , (3.1)where P denotes path ordering, g is the relevant gauge coupling, and T a ( r ) is the generator forthe DM representation r . Furthermore, soft radiation is not required at the order to which wework. This implies that all dynamical radiation in NRDM is completely captured by Wilsonlines along the directions of the heavy particles, greatly simplifying the field theory treatment.– 12 –fter decoupling the soft radiation, the leading power Lagrangian is given by L (0)NRDM = χ † v (cid:32) i v · ∂ + (cid:126) ∇ M χ (cid:33) χ v + ˆ V (cid:104) χ v , χ † v (cid:105) ( m W,Z ) , (3.2)which describes the interactions of the heavy particles as the sum of a kinetic and potentialterm. The potential ˆ V describes potential exchanges of the W, Z, γ , and its explicit form can befound in Ref. [4]. Note that going to higher orders and powers is well understood in the contextof NRQCD (see e.g. Refs. [93, 94]). The dynamics of the heavy particles are governed by lowenergy matrix elements evaluated with the above Lagrangian. Since this is a non-relativisticdescription, the number of heavy particles is fixed, and there exists an associated Schrödingerequation. These low energy matrix elements give rise to the Sommerfeld enhancement, whichmust be included when computing the DM cross section. We will therefore briefly review thestructure of the low energy matrix elements and the Sommerfeld factors.
Since we have chosen to work with pure wino DM, the model includes a Majorana fermionDM candidate χ , and an electrically charged fermion χ ± . For the calculation of the Som-merfeld factors, we include a mass splitting, that is neglected when performing the Sudakovresummation. Including this splitting is important as it plays a role in determining the posi-tions of the Sommerfeld resonances. For winos, electroweak corrections yield a mass splitting δ ≡ M χ ± − M χ (cid:39) . MeV [95].In our formalism, the Sommerfeld enhancement will be captured by low energy matrixelements of the heavy annihilating particles. As discussed in Sec. 4 where we derive thefactorization formula, the following matrix elements appear F a (cid:48) b (cid:48) ab = (cid:68)(cid:0) χ χ (cid:1) S (cid:12)(cid:12)(cid:12)(cid:0) χ a (cid:48) Tv iσ χ b (cid:48) v (cid:1) † (cid:12)(cid:12)(cid:12) (cid:69)(cid:68) (cid:12)(cid:12)(cid:12)(cid:0) χ aTv iσ χ bv (cid:1)(cid:12)(cid:12)(cid:12)(cid:0) χ χ (cid:1) S (cid:69) , (3.3)where T denotes transpose, σ is the second Pauli matrix, and the external state is given bythe S -wave combination ( χ χ ) S . Here the color indices a, b, a (cid:48) , b (cid:48) = 1 , , , and we have theusual relations χ = χ and χ ± = ( χ ∓ iχ ) / √ . In terms of the charge eigenstates, we willfind that the relevant components of F a (cid:48) b (cid:48) ab are (cid:68) (cid:12)(cid:12)(cid:12) χ Tv iσ χ v (cid:12)(cid:12)(cid:12)(cid:0) χ χ (cid:1) S (cid:69) = 4 √ M χ s , (3.4) (cid:68) (cid:12)(cid:12)(cid:12) χ + Tv iσ χ − v (cid:12)(cid:12)(cid:12)(cid:0) χ χ (cid:1) S (cid:69) = 4 M χ s ± , where the Sommerfeld enhancement is captured by the factors s and s ± , which must beevaluated non-perturbatively. In practice we do this by numerically solving the associatedSchrödinger equation. We summarize some of the most important aspects here; a detaileddiscussion can be found in Appendix A of [23]. For other detailed studies of both phenomeno-logical and formal aspects of Sommerfeld enhancement, we refer the reader to Refs. [96–100].– 13 –he first step in solving for the Sommerfeld factors is to compute a wavefunction (cid:0) ψ i (cid:1) j ,where the index i labels the asymptotic state and j is the component index for the resultingsolution, and the indices i, j = 1 , refer to the (00) , (+ − ) states respectively. A discussion ofthe relevant boundary conditions can be found in Ref. [23]. Once the solutions ψ have beenobtained, the Sommerfeld enhancement matrix is s ij = (cid:0) ψ i ( ∞ ) (cid:1) j . (3.5)In practice, one must choose a velocity when computing s ij . As is well known, the Sommerfeldenhancement saturates at low velocities, and we have checked that this occurs for the rangerelevant for DM annihilations, i.e. , v (cid:46) − , for the wino mass range of interest. Therefore,we can neglect any velocity profile dependence, and treat all velocity dependence as constantfor the parameter range of interest.Once we know s ij , using Eq. (3.4) we can then determine the relevant components of F a (cid:48) b (cid:48) ab given in Eq. (3.3). From this point, the annihilation cross section can be computed as σ = (cid:88) a (cid:48) b (cid:48) ab F a (cid:48) b (cid:48) ab ˆ σ a (cid:48) b (cid:48) ab ( z cut ) , (3.6)where ˆ σ a (cid:48) b (cid:48) ab ( z cut ) denotes the resummed perturbative cross section as a function of z cut ,whose computation is the subject of this paper (see Eq. (4.1) below).As a final comment, we note that we have glossed over the fact that we will be working in atheory with a spontaneously broken gauge symmetry, as opposed to standard NRQCD. Therewill several manifestations of this fact. First, and most trivially, it impacts the Sommerfeldenhancement calculation, as well as the color algebra, due to the identification of a color indexfor the external photon. More non-trivially, a significant portion of this paper (see in particularSec. 4) will relate to the refactorization of the function describing wide angle soft radiation,including that from the incoming DM particles. This is required, since m W introduces anotherscale for the soft radiation in addition to that imposed by the final state measurement. Soft-Collinear Effective Theory (SCET) [52–54] will provide the framework for describingradiation in the final state. SCET describes the dynamics of soft and collinear radiation inthe presence of a hard scattering. While originally developed for applications to QCD withmassless gauge bosons, the formalism was extended to the electroweak sector with massivegauge bosons in [72–74]. In what follows, we will provide a brief review of the featuresof SCET that will be used for our heavy DM annihilation process (along with a few moregeneral comments).
SCET is a theory of both soft and collinear particles. Collinear particles have a large momen-tum along a particular light-like direction, while soft particles have a small momentum, and– 14 –o preferred direction. For each relevant light-like direction, we define two reference vectors n µ and ¯ n µ such that n = ¯ n = 0 and n · ¯ n = 2 . The typical choice of n µ = (1 , , , and ¯ n µ = (1 , , , − will be used below. The freedom in the choice of n , as in the case of v for non-relativistic EFTs, is represented in the EFT through a reparameterization invari-ance [101, 102]. Any four-momentum p can be decomposed with respect to n µ as p µ = ¯ n · p n µ n · p ¯ n µ p µ ⊥ . (3.7)The SCET expansion is defined by a formal power counting parameter λ (cid:28) , which isdetermined by the measurements or kinematic restrictions imposed on the radiation. Thenthe momenta for the different particles in the EFT scale asCollinear : (cid:0) n · p, ¯ n · p, p ⊥ (cid:1) ∼ Q (cid:0) λ , , λ (cid:1) , Soft : (cid:0) n · p, ¯ n · p, p ⊥ (cid:1) ∼ Q (cid:0) λ, λ, λ (cid:1) , (3.8)Ultrasoft : (cid:0) n · p, ¯ n · p, p ⊥ (cid:1) ∼ Q (cid:0) λ , λ , λ (cid:1) , where Q is a typical scale of the hard interaction. A theory with collinear and ultrasoft modesis typically referred to as SCET I , while that with collinear and soft modes is referred to as SCET II [103]. In order to expand the full theory fields around a particular direction, the momenta aredecomposed into label ˜ p µ and residual k µ components p µ = ˜ p µ + k µ = ¯ n · ˜ p n µ p µ ⊥ + k µ . (3.9)Then for a collinear particle, ¯ n · ˜ p ∼ Q and ˜ p ⊥ ∼ λQ , while k ∼ λ Q describes small fluctuationsabout the label momentum. EFT modes with momenta of definite scaling are obtained byperforming a multipole expansion of the full theory fields. SCET involves independent gaugebosons for each collinear direction A n, ˜ p ( x ) , which are labeled by their collinear direction n andtheir large label momentum ˜ p , as well as (ultra)soft gauge boson fields A ( u ) s ( x ) . Independentgauge symmetries are enforced for each set of fields. Overlap between different regions isremoved by the zero-bin procedure [106]. This ensures that there is no double counting ofmomentum regions.The leading power SCET Lagrangian takes the form L SCET = L hard + L dyn = L (0)hard + L (0) + L (0) G . (3.10)Here L (0)hard contains the hard scattering operators and is determined by an explicit matching In the presence of Glauber modes, soft modes are always required to run the Glauber potentials [104, 105].Whether or not ultrasoft modes are required depends on the physical observable in question. The standard formalism also incorporates collinear scalars and fermions as well. These are not requiredfor the calculation presented here, so we will not discuss them. – 15 –alculation. The Lagrangian L (0) describes the universal leading power dynamics of the softand collinear modes and can be found in Refs. [52–54]. Finally, L (0) G is the leading powerGlauber Lagrangian [104], which describes the leading power coupling of soft and collineardegrees of freedom through potential operators. We will not need to consider it in this paper.Hard scattering operators involving collinear fields are constructed out of products ofcollinear gauge invariant fields [52, 53]. The gauge invariant gauge boson operator is given by B µn ⊥ ( x ) = 1 g (cid:104) W † n ( x ) iD µn ⊥ W n ( x ) (cid:105) . (3.11)Here D n ⊥ is the collinear gauge covariant derivative, and W n is a collinear Wilson line W n ( x ) = (cid:34) (cid:88) perms exp (cid:16) − g ¯ n · P ¯ n · A n ( x ) (cid:17)(cid:35) , (3.12)where P µ is an operator that returns the label momentum. The collinear Wilson line, W n ( x ) ,is localized with respect to the residual position x so that B µn ⊥ ( x ) can be treated as localgauge boson fields from the perspective of the ultrasoft degrees of freedom. For the leadingpower calculation presented here, ultrasoft and soft fields will not appear explicitly in ourhard scattering operators, other than through Wilson lines via the field redefinition B aµn ⊥ → Y abn B bµn ⊥ , (3.13)which is performed in each collinear sector. For a general representation, r , the ultrasoftWilson line is defined by Y ( r ) n ( x ) = P exp ig (cid:90) −∞ d s n · A aus ( x + sn ) T a ( r ) , (3.14)where as before P denotes path ordering. This so-called BPS field redefinition has the effectof decoupling ultrasoft and collinear degrees of freedom at leading power [111]. We will alsoneed soft Wilson lines, S ( r ) n ( x ) = P exp ig (cid:90) −∞ d s n · A as ( x + sn ) T a ( r ) . (3.15) Note that when the label momentum is large compared to the virtuality of the EFT modes, it is convenientto use a mixed position/momentum space representation space Wilson line, where the label is in momentumspace and the residual fluctuations are in position space. Otherwise, Wilson lines will be written in positionspace, e.g.
Eq. (3.1). It is also possible to formulate SCET entirely in position space, see e.g.
Refs. [107, 108],although we will not use the position space formalism here. Here we give the explicit result for an incoming Wilson line. Depending on whether particles are incomingour outgoing, different Wilson lines must be used. When done correctly, the BPS field redefinition accountsfor the full path of the particles [109, 110]. – 16 –inally, the refactorization of the soft sector (see Sec. 4.2.3 below) will require the inclusionof collinear-soft modes from SCET + [55–59]. Collinear-soft modes have both a collinear andsoft scaling p cs ∼ Q ˜ λ (cid:0) λ , , λ (cid:1) , (3.16)where λ and ˜ λ are distinct power counting parameters. Such modes first appeared in cal-culations of jet substructure when multiple simultaneous measurements are made on a jet[55–59]. This introduces additional scales, implying the need for both λ and ˜ λ . For contrast,the measurement of a single observable, such as the mass of a jet, only introduces a singlescale; the mass can either fix the angular spread of the mode, resulting in a collinear mode,or it can fix the energy of the mode, resulting in soft or ultrasoft modes, but it cannot fixboth, as required for collinear-soft modes. In our case, the collinear-soft modes will arise dueto the presence of both the mass scale of the final state m X , and the mass scale of electroweaksymmetry breaking m W . Our study provides a new application of collinear-soft modes.Since the collinear-soft modes arise from a refactorization of the soft sector, they coupleeikonally and their interactions can be absorbed using additional Wilson lines defined as X ( r ) n ( x ) = P exp ig (cid:90) −∞ d s n · A acs ( x + sn ) T a ( r ) , (3.17)and V ( r ) n ( x ) = P exp ig (cid:90) −∞ d s ¯ n · A acs ( x + s ¯ n ) T a ( r ) . (3.18)This notation is chosen to reflect that the X Wilson lines will arise from a BPS field redefi-nition, similar to the Y Wilson lines in
SCET I (and X precedes Y in the alphabet), and the V Wilson lines are generated by integrating out interactions with particles in the ¯ n direction,similar to the W Wilson lines that accompany the collinear fields (and V precedes W in thealphabet). As with (ultra) soft fields, at the order to which we work, collinear-soft fields willappear only in Wilson lines. For example, they will arise from the BPS field redefinition, whichallows the all orders decoupling of interactions between collinear-soft and collinear particles.This is identical to the transformation in Eq. (3.13) but with a collinear-soft Wilson line. Fora more detailed discussion of the BPS field redefinition for collinear-soft fields, see [55]. SCET allows for the resummation of large logarithms through the renormalization group (RG)evolution of matrix elements of collinear, (ultra)soft, collinear-soft fields. Since we will useboth
SCET I and SCET II , this RG evolution can be either in virtuality, µ , or rapidity, ν – 17 –112–114]. We use the regulator of [113, 114], modifying the Wilson lines as S n ( x ) = (cid:34) (cid:88) perms exp (cid:32) − gn · P ω | P z | − η/ ν − η/ n · A s ( x ) (cid:33)(cid:35) , (3.19) W n ( x ) = (cid:34) (cid:88) perms exp (cid:32) − g ¯ n · P ω | ¯ n · P| − η/ ν − η/ ¯ n · A n ( x ) (cid:33)(cid:35) , (3.20)Here ν is a rapidity scale, analogous to µ in dimensional regularization, η is the regulatingparameter, and P z returns the z -component of the label momentum. This allows us to definea dimensional regularization-like RG in terms of ν . Here ω is a formal bookkeeping parameterwhich satisfies ν ∂∂ν ω ( ν ) = − η ω ( ν ) , lim η → ω ( ν ) = 1 . (3.21)For convenience, we set ω = 1 throughout our calculations since it can be trivially restored.Rapidity divergences for the collinear-soft modes will also be regulated with the appropriatelymodified versions of Eqs. (3.19) and (3.20).In our factorization, we will encounter functions that satisfy both multiplicative andconvolutional renormalization group equations. For a function F ( µ, ν ) which is renormalizedby a multiplicative factor Z F ( µ, ν ) , we have F bare = Z F ( µ, ν ) F ( µ, ν ) , (3.22)from which we derive the RG equationsdd log µ F ( µ, ν ) = γ µF ( µ, ν ) F ( µ, ν ) , dd log ν F ( µ, ν ) = γ νF ( µ, ν ) F ( µ, ν ) , (3.23)with γ µF ( µ, ν ) = − Z F ( µ, ν ) dd log µ Z F ( µ, ν ) , γ νF ( µ, ν ) = − Z F ( µ, ν ) dd log ν Z F ( µ, ν ) . (3.24)Convolutional renormalization in a variable τ takes the form F bare ( τ ) = (cid:90) d τ (cid:48) Z F ( τ − τ (cid:48) ; µ, ν ) F ( τ (cid:48) ; µ, ν ) , (3.25)giving rise to the RG equationsdd log µ F ( τ ; µ, ν ) = (cid:90) d τ (cid:48) γ µF ( τ − τ (cid:48) ; µ, ν ) F ( τ (cid:48) ; µ, ν ) , (3.26)– 18 –d log ν F ( τ ; µ, ν ) = (cid:90) d τ (cid:48) γ νF ( τ − τ (cid:48) ; µ, ν ) F ( τ (cid:48) ; µ, ν ) , (3.27)where the anomalous dimensions are given by γ µF ( τ ; µ, ν ) = − (cid:90) d τ (cid:48) Z − F ( τ − τ (cid:48) ; µ, ν ) dd log µ Z F ( τ (cid:48) ; µ, ν ) , (3.28) γ νF ( τ ; µ, ν ) = − (cid:90) d τ (cid:48) Z − F ( τ − τ (cid:48) ; µ, ν ) dd log ν Z F ( τ (cid:48) ; µ, ν ) . (3.29)Convolutional RG equations are most easily treated in a conjugate space (we will use Laplacespace below), in which they are multiplicative.The RG evolution can be used to run functions from their natural scale, where all largelogarithms are minimized, to an arbitrary scale. The independence of the RG path is guaran-teed by the fact that the anomalous dimensions sum to zero, schematically (cid:88) F γ Fµ = 0 , (cid:88) F γ Fν = 0 , (3.30)where the sum is over the functions F that appear in the factorization formula, along withthe fact that evolution in µ and ν commutes: (cid:20) dd log µ , dd log ν (cid:21) = 0 . (3.31)The consistency of the anomalous dimensions will provide a strong check on our calculation.We will use the path independence to choose a particularly simple path to resum all largelogarithms in the EFT, see Fig. 6 below. In this section, we present the factorization formula for the endpoint region of heavy WIMPannihilation – this is one of the main results of this paper. We focus here on the short-distance component of the cross section, denoted ˆ σ ( z cut ) in Eq. (3.6). As discussed below, thelong-distance contributions, i.e. , the Sommerfeld enhancement, also arise naturally from thefactorization of the matrix elements presented in this section; we refer the reader to Sec. 3.1.1for the details of how these factors are (numerically) computed.In Sec. 4.1, we present the factorization formula, and discuss each of its components inturn. This section is aimed at readers without a technical EFT background, and as suchemphasizes the physical content of each ingredient. In Sec. 4.2, we provide the technicaldiscussion of the multi-stage matching used to derive the factorization formula, emphasizingthe operator definitions for the functions and key aspects of the refactorization. Tree level andone-loop results for all functions in both the intermediate and final EFT, as well as details ofthe calculations can be found in Appendix A.– 19 – .1 Factorization Overview The main result of this section is a factorization formula for the photon spectrum in theendpoint region. We find that the differential cross section for the heavy WIMP annihilation χ χ → γ + X factorizes in the limit that z → asd ˆ σ LL d z = H ( M χ , µ ) J γ ( m W , µ, ν ) J ¯ n ( m W , µ, ν ) S ( m W , µ, ν ) × H J ¯ n ( M χ , − z, µ ) ⊗ H S ( M χ , − z, µ ) ⊗ C S ( M χ , − z, m W , µ, ν ) , (4.1)where z is defined in Eq. (2.1), and we use ⊗ to denote a convolution between the functionsin the second line, as explained in detail below. Here ˆ σ denotes the short-distance componentof the cross section in Eq. (3.6) with suppressed initial/final state indices. The indices are tobe contracted with the matrix element F a (cid:48) b (cid:48) ab in Eq. (3.3). This function also arises naturallywhen considering the factorization of the cross section, but to keep our discussion focused onthe Sudakov factors, we will not consider F a (cid:48) b (cid:48) ab in this section. When we present the finalcross section results in Sec. 5.2, F a (cid:48) b (cid:48) ab will be included. The LL superscript indicates thatthis factorization as written is only true for the leading logarithmic contributions. Beyondthis order additional functions are required, as will be described in this section.The iterative matching procedure used to derive this result is shown schematically inFig. 4. In the first stage, we match onto a standard SCET theory, leading to the standardfactorization into functions that describe the underlying hard scattering ( H ), the collinearradiation along the jet ( J (cid:48) ¯ n ) and photon ( J γ ) directions, and soft radiation ( S (cid:48) ). In the secondstage, we match onto a (electroweak symmetry breaking) theory with massive soft and collinearmodes. In particular, this manifests as a refactorization of the soft function S (cid:48) into thefunctions H S , S and C S , and of the jet function J (cid:48) ¯ n into the functions H J ¯ n and J ¯ n – theseadditional functions are described below.The final EFT description consists of a collection of independent sectors, each corre-sponding to the functions appearing in the factorization formula Eq. (4.1). The procedure forfactorizing the full cross section into these functions is illustrated in Fig. 4. The interpretationof each of the functions is discussed in the following, which is organized by the characteristicscale µ for these sectors. In particular, we separate it into two classes of functions, namelythose that depend on m W , and those that do not.The first class of functions depend on scales far above the electroweak scale, µ (cid:29) m W , andare thus independent of electroweak symmetry breaking effects. • H ( M χ , µ ) describes the underlying hard scattering process of χ χ → γ γ, γ Z , and in-cludes contributions from modes with virtuality µ ∼ M χ . • H J ¯ n ( M χ , − z, µ ) describes collinear radiation along the jet direction with virtuality µ ∼ M χ √ − z such that it contributes to the final state mass.– 20 – n c r e a s i n g V i r t u a li t y H H Figure 4 . A schematic of the multistage matching procedure used to derive the factorization formulafor heavy WIMP annihilation in the endpoint region. The jet and soft functions appearing in the firststage of matching are refactorized into components that depend either on m W , or on the phase spacerestriction implemented by z . • H S ( M χ , − z, µ ) describes soft wide-angle radiation with virtuality µ ∼ M χ (1 − z ) suchthat it contributes to the final state mass.The second class of functions encode electroweak symmetry breaking effects, and have µ ∼ m W , so that the gauge fields are treated as massive. Additionally, these functions all dependon a rapidity renormalization scale ν . • J γ ( m W , µ, ν ) describes the final state photon, and results purely from modes with energy E γ and virtuality µ ∼ m W . This function receives only virtual corrections, since thefinal state is exactly specified. • S ( m W , µ, ν ) describes homogenous soft radiation with virtuality µ ∼ m W such that itdoes not contribute to the final state mass. • C S ( M χ , − z, m W , µ, ν ) describes radiation that is simultaneously soft and collinear tothe photon direction. The momentum for this radiation has collinear scaling, virtuality µ ∼ m W , and contributes to the final state mass. • J ¯ n ( m W , µ, ν ) describes collinear radiation along the jet direction with virtuality µ ∼ m W such that it does not contribute to the final state mass.– 21 –his full factorization simultaneously involves functions from NRDM, SCET I , SCET II , andSCET + , and resummation requires RG evolution in both virtuality and rapidity.For the analysis here, we will be interested in resumming only the leading logs (LL). Ourapproach to the factorization persists at higher logarithmic order. However, as written, therefactorization of the soft function S (cid:48) is only valid at LL order. The origin of this effect, as wellas the mechanism for disentangling these scales, is akin to the case of non-global logarithms(NGLs), and is discussed in Sec. 4.2.3.While we will present the factorization formula using the concrete example of an SU (2) W triplet of Majorana fermions, this choice merely affects the particular spin and charge structureof the operators involved, and as such the main features of the factorization and the relevantmodes are universal. The same factorization will also apply, e.g. to the annihilation of heavySU (2) W doublets or the decay of a heavy dark bound state [115]. Furthermore, some ofthe structure is generic to situations where event shape observables are measured on jets ofmassive radiation, and thus variants of Eq. (4.1) may find applications for future high energycolliders [116, 117]. In this section, we discuss the derivation of the factorization formula given in Eq. (4.1). InSec. 4.2.1 we present the first stage of matching, including the structure of the hard scat-tering operators, the factorization of the Hilbert space and measurement function for softand collinear modes, and the matrix element definitions of the functions. In Sec. 4.2.2 andSec. 4.2.3 we present the details for the second stage of matching, namely the refactorization ofthe collinear and soft sectors. For the soft sector, we give a detailed discussion of the relevantsoft and colinear-soft modes.
We begin by determining the hard scattering Lagrangian in SCET, denoted by L hard inEq. (3.10). This is done through matching the full theory consisting of the Standard Modeland an SU (2) W triplet of Majorana fermions onto SCET, and is identical to the fully exclusivecase [42, 43, 46]. The Lagrangian describing the hard scattering is L (0) hard = (cid:88) r =1 C r ( M χ , µ ) O r = (cid:88) r =1 C r ( M χ , µ ) (cid:16) χ aTv iσ χ bv (cid:17) (cid:16) Y abcdr B icn ⊥ B jd ¯ n ⊥ (cid:17) i (cid:15) ijk ( n − ¯ n ) k , (4.2)with the Wilson line structures Y abcd = δ ab (cid:16) Y cen Y de ¯ n (cid:17) , Y abcd = (cid:16) Y aev Y cen (cid:17)(cid:16) Y bfv Y df ¯ n (cid:17) , (4.3)– 22 –btained through the BPS field redefinition. The Wilson coefficients C r are IR finite, andindependent of the scale m W . Performing a tree-level matching at the scale µ ∼ M χ , we find C ( µ ) = − C ( µ ) = − π α W ( µ ) M χ . (4.4)The C r ( µ ) encode the underlying hard scattering process and determine the hard function H ( M χ , µ ) appearing in our factorization formula, as will be defined in Eq. (4.9). Togetherwith L dyn in Eq. (3.10), the hard scattering operators in Eq. (4.2) describe the annihilationat scales µ (cid:46) M χ .The factorization formula for the cross section for χ χ → γ + X depends on the squaredmatrix elements of these hard scattering operators. For contrast, in the exclusive case thereare only virtual contributions, and thus the factorization can be done at the level of theamplitude [42, 43, 46]. In the present analysis, there are both real and virtual contributionsthat are sensitive to m W as well as the scales imposed by the endpoint restrictions though z .These low-energy dynamics are not yet factorized at this stage.First, we consider the factorization of the Hilbert space for the final state | X (cid:105) . Since thesoft and collinear modes are decoupled, the final state can be written as (cid:12)(cid:12) X (cid:11) = (cid:12)(cid:12) X s (cid:11) (cid:12)(cid:12) X c (cid:11) . (4.5)Next, we expand out the contributions to the final state mass m X , (1 − z ) = 14 M χ m X = 14 M χ (cid:32) (cid:88) i ∈ X s p µi + (cid:88) i ∈ X c p µi (cid:33) = 24 M χ (cid:32) (cid:88) i ∈ X s p µi (cid:33) · (cid:32) (cid:88) i ∈ X c p µi (cid:33) + 14 M χ (cid:32) (cid:88) i ∈ X c p µi (cid:33) + O ( λ )= 24 M χ (cid:88) i ∈ X s ¯ n · p i + 14 M χ (cid:32) (cid:88) i ∈ X c p µi (cid:33) + O ( λ ) ≡ (1 − z s ) + (1 − z c ) + O ( λ ) , (4.6)which shows that contributions to the final state radiation from soft and collinear modes canbe separated to leading power. The last line in Eq. (4.6) defines the contributions from the softand collinear modes as (1 − z s ) and (1 − z c ) , respectively, and demonstrates the factorization ofthe final state restriction. This allows us to define soft and collinear measurement operators, (cid:99) M s and (cid:99) M c , as (cid:99) M s (cid:12)(cid:12) X s (cid:11) = 12 M χ (cid:88) i ∈ X s ¯ n · p i (cid:12)(cid:12) X s (cid:11) , (cid:99) M c | X c (cid:105) = 14 M χ (cid:32) (cid:88) i ∈ X c p µi (cid:33) (cid:12)(cid:12) X c (cid:11) . (4.7)– 23 –hese measurement operators can be written in terms of the energy momentum tensor ofeither the full or effective theories [118–120]. Here their role will simply be to return the valueof the observable for a particular perturbative state in momentum space.With the above ingredients, we can algebraically manipulate the cross section into afactorized form involving matrix elements of either soft or collinear fields. These matrixelements will be coupled together both through color indices and the convolutions that arepresent as a result of enforcing the measurements. This procedure is standard (see, e.g. thereview [121]) and we simply give the final result. At the first stage of matching, the differentialcross section with factorized dynamics in SCET is given in terms of the hard function H , thejet functions J (cid:48) ¯ n and J γ for X and the photon respectively, and the soft function S (cid:48) asd ˆ σ d z = (cid:90) d z s d z c δ (1 + z − z c − z s ) H ij ( M χ ) J (cid:48) ¯ n ( M χ , − z c , m W ) J γ ( m W ) S (cid:48) ij (1 − z s , m W ) ≡ H ij ( M χ ) J γ ( m W ) J (cid:48) ¯ n ( M χ , − z, m W ) ⊗ S (cid:48) ij (1 − z, m W ) , (4.8)where we have suppressed the color indices and the dependence on the RG scales µ and ν forsimplicity. As in Eq. (4.1), we have used ⊗ to denote the convolution in z . The convolutionarises due to the fact that the total invariant mass of the final state is a sum over the soft andcollinear sectors, see Eq. (4.6).The functions labeled with a superscript prime are those that require further factorization.Note that the J (cid:48) ¯ n and S (cid:48) functions still depend on both the m W and (1 − z ) scales. Thiscomplication did not occur for the fully exclusive case, where the above factorization wassufficient since there is no intermediate scale (1 − z ) . The refactorization of the jet and softfunctions will be discussed in Sec. 4.2.2 and Sec. 4.2.3.Next, we provide field-theoretic definitions for the functions appearing in Eq. (4.8). Thehard function is defined in terms of the Wilson coefficients of the hard scattering operators inEq. (4.2) as H ij = C ∗ i C j . (4.9)The soft function is a vacuum matrix element of the soft Wilson lines Y r in Eq. (4.2), S (cid:48) ij (1 − z s , m W , µ, ν ) = (cid:68) (cid:12)(cid:12)(cid:12) ¯T Y † i (0) δ (cid:16) (1 − z s ) − (cid:99) M s (cid:17) T Y j (0) (cid:12)(cid:12)(cid:12) (cid:69) , (4.10)where the color indices are suppressed, T and ¯T denote time ordering and anti-time orderingrespectively, and the Y r factors are the products of Wilson lines defined in Eq. (4.3). Thecomponents of the soft function with explicit color indices are S (cid:48) a (cid:48) b (cid:48) ab = (cid:28) (cid:12)(cid:12)(cid:12)(cid:12) (cid:16) Y kn Y dk ¯ n (cid:17) † δ (cid:16) (1 − z s ) − (cid:99) M s (cid:17) (cid:16) Y jn Y dj ¯ n (cid:17)(cid:12)(cid:12)(cid:12)(cid:12) (cid:29) δ a (cid:48) b (cid:48) δ ab , – 24 – (cid:48) a (cid:48) b (cid:48) ab = (cid:28) (cid:12)(cid:12)(cid:12)(cid:12) (cid:16) Y f (cid:48) n Y dg (cid:48) ¯ n Y a (cid:48) f (cid:48) v Y b (cid:48) g (cid:48) v (cid:17) † δ (cid:16) (1 − z s ) − (cid:99) M s (cid:17) (cid:16) Y fn Y dg ¯ n Y afv Y bgv (cid:17)(cid:12)(cid:12)(cid:12)(cid:12) (cid:29) ,S (cid:48) a (cid:48) b (cid:48) ab = (cid:28) (cid:12)(cid:12)(cid:12)(cid:12) (cid:16) Y kn Y dk ¯ n (cid:17) † δ (cid:16) (1 − z s ) − (cid:99) M s (cid:17) (cid:16) Y gn Y df ¯ n Y agv Y bfv (cid:17)(cid:12)(cid:12)(cid:12)(cid:12) (cid:29) δ a (cid:48) b (cid:48) ,S (cid:48) a (cid:48) b (cid:48) ab = (cid:28) (cid:12)(cid:12)(cid:12)(cid:12) (cid:16) Y f (cid:48) n Y dg (cid:48) ¯ n Y a (cid:48) f (cid:48) v Y b (cid:48) g (cid:48) v (cid:17) † δ (cid:16) (1 − z s ) − (cid:99) M s (cid:17) (cid:16) Y kn Y dk ¯ n (cid:17)(cid:12)(cid:12)(cid:12)(cid:12) (cid:29) δ ab , (4.11)where the color indices are explicit, but we have dropped the arguments and scale dependenceof the functions for simplicity. Here, as well as in the expressions below, we keep the timeordering convention and the dependence on x = 0 implicit. Note that the color index corresponds to the photon final state.The indices i, j in the hard and soft functions span the space of the operators given inEq. (4.2) and are contracted with each other as H ij S (cid:48) ij . To reduce the number of indicesappearing in later formulas, we introduce the following notation: H ≡ H , H ≡ H , H ≡ H = H ,S (cid:48) ≡ S (cid:48) , S (cid:48) ≡ S (cid:48) , S (cid:48) ≡ S (cid:48) + S (cid:48) , (4.12)such that H ij S (cid:48) ij = H i S (cid:48) i .The jet functions for the recoiling jet X and the photon are color-singlet matrix elementsof collinear fields. Explicitly, we have J (cid:48) dd (cid:48) ¯ n (cid:0) M χ , − z c , m W , µ (cid:1) = (cid:68) (cid:12)(cid:12)(cid:12) B d (cid:48) ¯ n ⊥ δ (cid:16) (1 − z c ) − (cid:99) M c (cid:17) δ (cid:0) M χ − ¯ n · P (cid:1) δ (cid:0) (cid:126) P ⊥ (cid:1) B d ¯ n ⊥ (cid:12)(cid:12)(cid:12) (cid:69) ,J γ (cid:0) m W , µ, ν (cid:1) = (cid:68) (cid:12)(cid:12)(cid:12) B cn ⊥ (cid:12)(cid:12)(cid:12) γ (cid:69)(cid:68) γ (cid:12)(cid:12)(cid:12) B cn ⊥ (cid:12)(cid:12)(cid:12) (cid:69) , (4.13)where (cid:126) P ⊥ returns the perpendicular component of the label momentum. As discussed above,this is the final form for J γ , but the jet function for X will require further factorization – weturn to this in the next section. As currently formulated, the jet function J (cid:48) ¯ n in Eq. (4.8) results from dynamics at both thescale M χ √ − z and the scale m W . To be able to resum logarithms of m W / ( M χ √ − z ) , whichcan become large as we move towards the endpoint, we must factorize these two scales. Thisfactorization is similar to that performed in the fully exclusive case, where one is separating M χ from m W using a hard matching coefficient that is independent of the IR scale m W , alongwith jet and soft functions which describe the dynamics at the scale m W . Here we will writethe jet function J (cid:48) ¯ n ( M χ , − z, m W , µ, ν ) as a hard matching coefficient H J ¯ n ( M χ , − z, µ ) , anda jet function J ¯ n ( m W , µ, ν ) . – 25 –he collinear state, X c , factorizes into two types of collinear modes as (cid:12)(cid:12) X c (cid:11) = (cid:12)(cid:12) X c z (cid:11) (cid:12)(cid:12) X c W (cid:11) , (4.14)where c z is in the Hilbert space containing the collinear modes that are sensitive to themeasurement enforced as a function of z , while c W is in the Hilbert space that containsthe modes with mass m W . This follows from the same logic as the standard hard-collinearfactorization. Here the c z modes which contribute to the jet mass measurement have thestandard scaling for an SCET I collinear mode associated with the mass measurement, p c z ∼ M χ (cid:0) λ , , λ (cid:1) , λ = √ − z . (4.15)The modes sensitive to the m W scale are standard SCET II collinear modes at the scale m W ,with scaling p c W ∼ M χ (cid:0) λ , , λ (cid:1) , λ = m W M χ , (4.16)and do not contribute to the mass of the final state at leading power.The factorization of the measurement function is trivial since, at leading power, the low-energy collinear modes have an invariant mass p c W ∼ m W (cid:28) M χ (1 − z ) , and do not contributeto the mass of the final state. We therefore only have (cid:99) M c z (cid:12)(cid:12) X c z (cid:11) = 14 M χ (cid:88) i ∈ X cz p µi (cid:12)(cid:12) X c z (cid:11) . (4.17)The separation of collinear modes through Eqs. (4.14) and (4.17) allows us to fully factorizethe jet function as J (cid:48) ¯ n (cid:0) M χ , − z, m W , µ, ν (cid:1) = H J ¯ n (cid:0) M χ , − z, µ (cid:1) J ¯ n (cid:0) m W , µ, ν (cid:1) + O (cid:18) m W M χ √ − z (cid:19) . (4.18)This factorization is a power expansion in m W / ( M χ √ − z ) . The matching coefficient H J ¯ n can be evaluated in the unbroken theory with massless electroweak bosons, and is IR finite dueto the mass measurement. The dependence on the electroweak scale is completely capturedby the function J ¯ n ( m W , µ, ν ) . Next, we turn to the refactorization of the soft function S (cid:48) . The goal is to have separate EFTsfor the dynamics at scales µ ∼ M χ (1 − z ) and µ ∼ m W . Comparing to the discussion of thejet refactorization in the previous section, the physics of the soft sector is more interesting, aslogarithms due to collinear-soft modes appear.– 26 –onsider the possible classes of soft modes with virtuality µ ∼ m W . The virtuality of thesoft modes with scaling p S (cid:48) ∼ M χ (1 − z )(1 , , can be lowered uniformly to yield modes with p S ∼ ( m W , m W , m W ) . When acting on these states, the measurement function in Eq. (4.10)can be expanded as δ (cid:16) (1 − z s ) − (cid:99) M s (cid:17) = δ (1 − z s ) + O (cid:18) m W M χ (1 − z ) (cid:19) . (4.19)We conclude that these soft modes do not contribute to the measurement, which allows asimplification of the operator structure. As an explicit example, the soft functions S (cid:48) and S (cid:48) become S (cid:48) a (cid:48) b (cid:48) ab → S a (cid:48) b (cid:48) ab = δ a (cid:48) b (cid:48) δ ab δ (1 − z s ) , (4.20) S (cid:48) a (cid:48) b (cid:48) ab → S a (cid:48) b (cid:48) ab = δ (1 − z s ) (cid:18) δ a (cid:48) b (cid:48) (cid:68) (cid:12)(cid:12)(cid:12) (cid:0) Y † ¯ n (cid:1) e Y aev Y bfv Y f ¯ n (cid:12)(cid:12)(cid:12) (cid:69) + δ ab (cid:68) (cid:12)(cid:12)(cid:12) (cid:0) Y † ¯ n (cid:1) e Y a (cid:48) ev Y b (cid:48) fv Y f ¯ n (cid:12)(cid:12)(cid:12) (cid:69)(cid:19) , (4.21)where we have used the unitarity of the Wilson lines. These new functions S i are now inde-pendent of m W . Physically, the simplification (collapse) of the Wilson lines occurs because themeasurement operator has been expanded away, implying that the refactorized soft functionsare now inclusive. However, we are still specifying the photon as the final state, and thereforeviolate the assumptions of the Bloch-Nordsieck [122] or KLN [123, 124] theorems, as originallypointed out in [125–127]. This explains why the Wilson lines in S do not completely simplify,as compared to S where the Wilson line dependence has collapsed to the unit operator leavingbehind only color and kinematic factors.It is clear from the collapse of the Wilson lines that the modes p S are not sufficient tocomplete the picture. In particular, the divergences associated with m W , for example in S (cid:48) ,should be reproduced after factorization, but the function S in Eq. (4.20) does not have sucha divergence. Interestingly, however, there is a second possibility for lowering the virtualityof the soft modes down to the scale m W : keep their momentum component along the photondirection fixed, but decrease their angle (increase their collinearity) with respect to the photon.These modes are shown schematically in Fig. 5. Such modes then have the scaling p c S ∼ M χ (1 − z ) (cid:0) , λ , λ (cid:1) , λ = m W M χ (1 − z ) . (4.22)These modes have a virtuality µ ∼ m W , but, like the original soft modes with momentum p S ,have a large momentum component M χ (1 − z ) . This is an example of the collinear-soft modesdiscussed in Sec. 3.2, which arise from the simultaneous presence of the two scales M χ (1 − z ) and m W .These arguments imply that the Hilbert space of the soft sector factorizes into soft modes– 27 – n c r e a s i n g V i r t u a li t y (a) (b) Figure 5 . (a) The refactorization of the soft function into collinear-soft and soft functions at differentrapidity scales. (b) The relevant modes required for the refactorization of the soft function are collinear-soft modes, which are collimated along the direction of the photon, and wide angle soft modes, whichare isotropic. with uniform scaling and collinear-soft modes as (cid:12)(cid:12) X S (cid:48) (cid:11) = (cid:12)(cid:12) X S (cid:11) (cid:12)(cid:12) X c S (cid:11) . (4.23)The soft modes do not contribute to the measurement, while the collinear-soft modes aresensitive to a measurement function (cid:99) M c S (cid:12)(cid:12) X c S (cid:11) = 12 M χ (cid:88) i ∈ X cS ¯ n · p i (cid:12)(cid:12) X c S (cid:11) . (4.24)The most interesting aspect of these collinear-soft modes is that they contribute to themeasurement of the final state mass through their large component, which is independentof their virtuality. To our knowledge, this type of collinear-soft mode has not previouslyappeared in the literature. For example, in the case of thrust [128] or other SCET I eventshapes, the definition of the measurement guarantees that it is always the small componentof the momentum of a particle that is measured.Using the measurement function in Eq. (4.24), the Wilson lines that make up the collinear-soft function do not collapse, but are instead expanded assuming the momentum scaling for thecollinear-soft modes. Since the collinear-soft modes are boosted along the photon’s direction n , the v and ¯ n Wilson lines appear to collapse down to the ¯ n direction. The collinear-softfunction is therefore given as a product of Wilson lines C S (cid:0) M χ , − z c , m W , µ, ν (cid:1) = (cid:68) (cid:12)(cid:12)(cid:12)(cid:0) X n V n (cid:1) † δ (cid:16) (1 − z c ) − (cid:99) M c S (cid:17) X n V n (cid:12)(cid:12)(cid:12) (cid:69) , (4.25)where the X and V Wilson lines were defined in Sec. 3.2, and implicitly include rapidity– 28 –egulators. We have suppressed color indices for simplicity. Explicit expressions with colorindices will be given below. To regulate rapidity singularities for the collinear-soft Wilsonlines, we do not expand the regulator, using the full | k z | − η dependence. Performing thenaive power expansion of the regulator yields unregulated rapidity divergences in the collinear-soft sector. This choice of regulator defines the zero-bin structure [106] of the collinear-softsector, and we find that non-trivial zero-bins are present, which must be correctly incorporatedto remove overlap. This is described in more detail in Appendix A. Strict power countingcan be preserved by introducing a boost parameter β , and using the regulator | k z | − η →| k + + β k − | − η [104].Having discussed the modes that are required to describe the physics at the scale m W , wenext explain how to refactorize the soft function into a matching coefficient that describes thedynamics at the scale M χ (1 − z ) , and a soft and jet function that describe the dynamics at thescale m W . This is more complicated than for the jet function. The complication emerges dueto the existence of a hierarchy in energy but not in angle for the homogeneous soft modes thatlive at the scales m W and M χ (1 − z ) . Hence, any emission at the scale M χ (1 − z ) , which canbe at an arbitrary angle, eikonalizes from the perspective of the emissions at the scale m W ,and is described as a new Wilson line source. In this way, an infinite number of operators isgenerated in the matching (although only a finite number appear at any order in α W ). Thissituation is familiar from the case of NGLs [129], where there exist multiple hierarchical softscales. Due to the generation of these new sources, the resummation of NGLs is governed bythe non-linear BMS equation [130]. In the present case, however, the measurement functionfor the modes at the scale m W is expanded, and what is generated are Bloch-Nordsieck orKLN violating NGLs. We are not aware of these appearing previously in the literature. Whileit is possible that these take a simple form, or completely cancel, they first contribute at NLLorder. Here we restrict ourselves to LL accuracy, and so we will not discuss this higher orderstructure any further. We leave the study of them using existing formulations of NGLs infactorization [58, 131–134] for future work.At LL order, we do not need to consider the generation of additional Wilson lines inthe matching. Nevertheless, the general structure of the refactorized function can becomecomplicated since four Wilson lines appear in each of the soft and collinear-soft functions, andmixing between these color structures can be generated beyond tree-level. In the most generalcase, the refactorization takes the form S (cid:48) aba (cid:48) b (cid:48) i (cid:0) M χ , − z, m W , µ, ν (cid:1) = H S,ij (cid:0) M χ , − z, µ (cid:1) (cid:104) C S (cid:0) M χ , − z, m W , µ, ν (cid:1) S (cid:0) m W , µ (cid:1)(cid:105) aba (cid:48) b (cid:48) j × (cid:20) O (cid:18) m W M χ (1 − z ) (cid:19)(cid:21) . (4.26)This refactorization, along with the scales of each of the functions, is shown in Fig. 5. Thefunctions C S and S each carry eight color (triplet) indices. Two of these sixteen color indicesare identified as carrying the quantum number of the photon, and the rest are contracted as to– 29 –eave the overall indices aba (cid:48) b (cid:48) , which are contracted with the initial state wavefunction factors.In Eq. (4.26), we are using the notation introduced in Eq. (4.12); the index i enumerates thecolor structures in the soft function before refactorization, i.e. , i = 1 , , . The index j sumsover the color structures in the soft function after refactorization.Instead of writing down a complete basis, we construct the color structures explicitly fromthe top down by explicitly refactorizing the soft function S (cid:48) . This requires us to supplementthe operators written in Eq. (4.11) above with those that appear at one-loop, to ensure thatthe RG closes. Fortunately, only a limited basis of color structures is required at this order.The color structures are derived in Appendix A.2. Here we simply state the results for therefactorization of the soft functions. We denote the combined collinear-soft and soft functionsas ˜ S aba (cid:48) b (cid:48) j = (cid:0) C S S (cid:1) aba (cid:48) b (cid:48) j , (4.27)and they are ˜ S aba (cid:48) b (cid:48) = (cid:68) (cid:12)(cid:12)(cid:12) (cid:16) X f (cid:48) n V dg (cid:48) n (cid:17) † δ (cid:16) (1 − z s ) − (cid:99) M c S (cid:17) (cid:16) X fn V dgn (cid:17) (cid:12)(cid:12)(cid:12) (cid:69) δ f (cid:48) g (cid:48) δ a (cid:48) b (cid:48) δ fg δ ab , ˜ S aba (cid:48) b (cid:48) = (cid:68) (cid:12)(cid:12)(cid:12) (cid:0) X cen V Aen (cid:1) † δ (cid:16) (1 − z s ) − (cid:99) M c S (cid:17) X c (cid:48) g (cid:48) n V A (cid:48) g (cid:48) n (cid:12)(cid:12)(cid:12) (cid:69)(cid:68) (cid:12)(cid:12)(cid:12) (cid:104) S cn S c (cid:48) n S a (cid:48) A (cid:48) v S aAv (cid:105) (cid:12)(cid:12)(cid:12) (cid:69) δ bb (cid:48) , ˜ S aba (cid:48) b (cid:48) = (cid:68) (cid:12)(cid:12)(cid:12) (cid:16) X cen V B (cid:48) en (cid:17) † δ (cid:16) (1 − z s ) − (cid:99) M c S (cid:17) X c (cid:48) g (cid:48) n V A (cid:48) g (cid:48) n (cid:12)(cid:12)(cid:12) (cid:69) × (cid:16)(cid:68) (cid:12)(cid:12)(cid:12) (cid:104) S cn S c (cid:48) n S a (cid:48) A (cid:48) v S b (cid:48) B (cid:48) v (cid:105) (cid:12)(cid:12)(cid:12) (cid:69) δ ab + (cid:68) (cid:12)(cid:12)(cid:12) (cid:104) S cn S c (cid:48) n S aA (cid:48) v S bB (cid:48) v (cid:105) (cid:12)(cid:12)(cid:12) (cid:69) δ a (cid:48) b (cid:48) (cid:17) . (4.28)Here we have made the color structure explicit, but we have dropped the arguments and scaledependence of the functions for simplicity. The collinear-soft function reproduces the m W dependent IR divergences of the soft function. Additionally, for the RG to close we will needthe following operator ˜ S aba (cid:48) b (cid:48) = (cid:68) (cid:12)(cid:12)(cid:12) (cid:16) X f (cid:48) n V df (cid:48) n (cid:17) † δ (cid:16) (1 − z s ) − (cid:99) M c S (cid:17) (cid:16) X fn V dfn (cid:17)(cid:12)(cid:12)(cid:12) (cid:69) δ a (cid:48) a δ b (cid:48) b , (4.29)which has a vanishing tree-level matching coefficient, but will appear in the mixing that resultsas we RG evolve the functions. The refactorized functions ˜ S aba (cid:48) b (cid:48) and ˜ S aba (cid:48) b (cid:48) have a trivialsoft sector, while the functions ˜ S aba (cid:48) b (cid:48) and ˜ S aba (cid:48) b (cid:48) have non-trivial collinear-soft and softcomponents. The final result is the factorization formula in Eq. (4.26) with index j summedover j = 1 , , , . In Sec. 5 the hard coefficients H S from tree-level matching will be givenexplicitly. – 30 – Leading Log Resummation for the Endpoint Region
Having stated the factorization formula, and discussed the physical intuition that underlies it,this section tackles the resummation of large logarithms of m W / ( M χ (1 − z )) , m W / ( M χ √ − z ) ,and m W /M χ . In Sec. 5.1, we present the one-loop anomalous dimensions obtained by com-puting the real and virtual corrections to the factorized functions presented in the previoussection. We also check consistency conditions for these anomalous dimensions (namely thatthey sum to zero), thus verifying our factorization formula at the one-loop level. In Sec. 5.2,we describe a simplified resummation path sufficient for LL order and then solve the RGEs andcollect all the resummation factors necessary for obtaining the final resummed cross section.The culmination of this work is Eq. (5.30). Explicit calculations are given in Appendix A. InSec. 5.3, we demonstrate that our result recovers both the exclusive and inclusive limits. In the results for the anomalous dimensions presented below, we only keep the double logpieces that are required for resummation at LL accuracy. The hard function H i ( M χ , µ ) onlyhas a µ anomalous dimension, γ Hµ,ij = − C A ˜ α W log (cid:18) µ (2 M χ ) z (cid:19) δ ij , (5.1)where C A is the SU (2) W quadratic Casimir invariant for the adjoint representation (explicitly C A = 2 ), i, j = 1 , , , and the structure of the RGE is diagonal. Here, and throughout thissection we will use ˜ α W = α W / (4 π ) to simplify the results. Furthermore, we can set z → to leading power, so that the hard function is independent of the infrared measurement.The same anomalous dimension, but derived at the level of the amplitude, was obtained forexclusive heavy WIMP annihilation [42, 43, 46].The photon jet function J γ ( m W , µ, ν ) consists of only virtual diagrams, and is computedin the broken theory. An example diagram is . (5.2)Here the dashed line indicates the final state cut, which puts the single identified photon onshell. We find that the µ and ν anomalous dimensions are given by γ J γ µ = 8 C A ˜ α W log (cid:18) ν M χ (cid:19) , γ J γ ν = 8 C A ˜ α W log (cid:18) µm W (cid:19) . (5.3)For the recoiling jet function J ¯ n ( m W , µ, ν ) , the low scale matrix element is fully inclusive.– 31 –xamples of real and virtual diagrams are + . (5.4)Due to its fully inclusive nature, we find that it has no anomalous dimension in µ or ν . Instead,these dependences are entirely captured by the matching coefficient H J ¯ n ( M χ , − z, µ ) , whichis described by the same diagrams but at the high scale. The dashed line again represents thefinal state cut, which at NLO can contain one or two particles. Since the one-loop correctionto the jet function is a plus distribution, the RG evolution takes a simpler form in Laplacespace. We will use s to denote the Laplace variable conjugate to M χ (1 − z ) . We find itsanomalous dimension to be γ H J ¯ n µ = 8 C A ˜ α W log (cid:18) µ s M χ (cid:19) . (5.5)For the soft function, the relevant one-loop diagrams are represented by , (5.6)where the electroweak boson can attach to any of the crosses, and the double lines denoteWilson lines. We have drawn the two v Wilson lines, which correspond to the annihilatingheavy WIMPs, as distinct directions for visual clarity. The collinear-soft function has a similarstructure, except the incoming Wilson lines are contracted to lie in the same direction . (5.7)As discussed in Sec. 4.2.3, the general case is complicated by a proliferation of color structuresthat mix beyond tree-level. For simplicity, we will consider, by top-down construction, onlythe functions that appear in our analysis at LL order. The µ RGE for the ˜ S functions is amatrix equation dd log µ ˜ S = ˆ γ ˜ Sµ ˜ S , (5.8)where ˜ S denotes the vector ˜ S i . The explicit form of the anomalous dimension matrix at– 32 –ne-loop is given by ˆ γ ˜ Sµ = 4 C A ˜ α W − ν s µ s − ν s − log µ s − µ s µ s − ν s
00 0 0 − ν s , (5.9)which exhibits a non-trivial mixing structure. The ν RGE is given bydd log ν ˜ S = ˆ γ ˜ Sν ˜ S , (5.10)where the matrix is diagonal ˆ γ ˜ Sν = − C A ˜ α W log (cid:18) µm W (cid:19) . (5.11)The interpretation of the scales appearing in the function ˜ S = C S S requires some caresince this is a combined object. While both the C S and S functions have a natural scale µ = m W (see the ν anomalous dimension given in Eq. (5.11)), the scale µ = 1 /s appears inthe logarithms of the µ anomalous dimension in Eq. (5.9). This can be understood from theconsistency of the RG, since the µ running of C S and S must combine to yield the natural scaleof H S , namely µ = 1 /s . Despite its confusing appearance, this appearance of /s provides anon-trivial check on our refactorization.One further important feature of the anomalous dimensions in Eq. (5.10) is that at LLorder, all rapidity anomalous dimensions vanish for µ = m W . We will exploit this featurein Sec. 5.2 by choosing a resummation path where all rapidity evolution is done at the scale µ = m W , eliminating the need for a non-trivial rapidity evolution.For the matching coefficients H S,ij of the soft sector we havedd log µ H
S,ij = γ H S µ,jk H S,ik , (5.12)where the explicit results at one-loop order aredd log µ H S, = 0 , (5.13)dd log µ H S, = − C A ˜ α W log( µ s ) H S, , dd log µ H S, = 4 C A ˜ α W log( µ s ) H S, , dd log µ H S, = 8 C A ˜ α W log( µ s ) H S, , dd log µ H S, = − C A ˜ α W log( µ s ) H S, . Now we are in the position to verify our factorization formula by checking consistency– 33 –elations among the anomalous dimensions. For the anomalous dimensions of the functionsbefore the refactorization of the jet and soft functions, we have the relations γ J γ ν + 13 γ S (cid:48) ν,ii = 0 , γ Hµ,ii + γ J γ µ + γ J (cid:48) ¯ n µ + 13 γ S (cid:48) µ,ii = 0 , (5.14)which involves the anomalous dimensions for the soft and jet functions before refactorization,given by γ S (cid:48) µ,ij = − C A ˜ α W log( ν s ) δ ij ,γ S (cid:48) ν,ij = − C A ˜ α W log (cid:18) µm W (cid:19) δ ij ,γ J (cid:48) ¯ n µ = 8 C A ˜ α W log (cid:18) µ s M χ (cid:19) . (5.15)As in the case of the hard function, the RG structure for the soft functions S (cid:48) i is diagonal. Usingthe anomalous dimensions in Eqs. (5.1), (5.3), and (5.15), one can check that the relations inEq. (5.14) are indeed satisfied.For the anomalous dimensions after refactorization, we have the consistency relations γ J (cid:48) ¯ n µ = γ H J ¯ n µ , γ S (cid:48) µ,ii δ kl = γ ˜ Sµ,kl + γ H S µ,lk , γ S (cid:48) ν,ii δ kl = γ ˜ Sν,kl , (5.16)where k, l = 1 , , , . One can check that these relations are satisfied using Eqs. (5.5), (5.9),(5.10), (5.13), and (5.15). We now have all the necessary ingredients to provide an analytic expression for the resummedspectrum at LL accuracy. As discussed in Sec. 3.2, the resummation can be simplified bymaking a judicious choice of path in the ( µ, ν ) plane. Our choice is illustrated in Fig. 6.Due to the refactorization of the soft function S (cid:48) into the soft and collinear-soft functions,each of which have a complicated color structure, and whose renormalization will involve colormixing, the renormalization group structure is quite complicated for a generic path. However,this can be avoided by noting that at µ = m W , the rapidity anomalous dimensions of thesoft and collinear-soft functions given in Eq. (5.10) vanish at LL order. Hence, we take thefunctions at their natural scale – H with µ = M χ , H J ¯ n with µ = (cid:112) M χ /s , and H S with– 34 – igure 6 . A schematic of the resummation path in the ( µ, ν ) plane used to perform the resummation.We choose to run all functions to ( µ, ν ) = ( m W , /s ) . This particular choice of path eliminates theneed to separately run the soft and collinear-soft functions in rapidity at LL order. This independencein rapidity at the scale m W is depicted by the light blue box. µ = 1 /s – and run them all down to µ = m W . Finally, at µ = m W , we can then trivially runthe soft, collinear-soft, and jet functions to the same rapidity. This choice of path providesa significant simplification since we can simply compute the µ anomalous dimensions for thefunctions H , H J ¯ n and H S . Beyond LL accuracy, this is no longer possible, and the fullfactorization that we have developed in this paper must be utilized.There is one additional subtlety regarding the evolution structure that has been glossedover in Fig. 6, but that requires care to reproduce the correct behavior in the limit z → .Recall that in deriving our factorization, which is summarized in Fig. 4, we have assumed thehierarchy M χ (1 − z ) (cid:29) M χ √ − z (cid:29) m W , (5.17)which allows us to factorize the dynamics at the scale m W from that at the scales M χ √ − z and M χ (1 − z ) . However, at z = 1 − m W / (2 M χ ) the soft scale hits the scale m W and at z = 1 − m W / (2 M χ ) the jet scale hits the scale m W . In this small region near the endpoint,our EFT is technically speaking invalidated. Physically, the constraint on the final statebecomes so restrictive that the jet is composed of a single boson. Due to the intrinsic IRcutoff set by electroweak symmetry breaking, it is unphysical for these scales to go below thescale m W . Instead, we must introduce a Θ -function in the RG evolution, which ensures thatthe running only contributes in the region where the scales are above m W . As we will see, withthis modification, our EFT will correctly transition to the exclusive endpoint calculation. Thischoice of scales is implemented in (1 − z ) space. Therefore, in Laplace space we take arbitrary– 35 –cales µ H J ¯ n and µ H S ( µ H can be set to its canonical value since it is z independent) transformto cumulative space where we can implement our scale setting as a function of (1 − z cut ) ,and then differentiate to obtain the resummed spectrum. Note that in the following, we willalways use z cut when discussing the cumulative space, as per the definition of Eq. (1.2).The RG equations can now be solved in the usual manner. For the hard functions H and H J ¯ n , we derive the evolution kernels U H (cid:0) M χ , m W (cid:1) = exp (cid:18) − C A ˜ α W log (cid:18) m W M χ (cid:19)(cid:19) , (5.18) U H J ¯ n (cid:0) µ H J ¯ n , m W (cid:1) = exp (cid:18) C A ˜ α W (cid:18) log (cid:18) m W (cid:114) s M χ (cid:19) − log (cid:18) µ H J ¯ n (cid:114) s M χ (cid:19)(cid:19)(cid:19) , where the first and second arguments of the kernels denote the scales we are running between,starting from the natural scale of the relevant function, and ending at µ ∼ m W . For the hardfunction H S , we need to solve the system of RG equations in Eq. (5.13) in order to run from µ = µ H S down to µ = m W . We find that H S, ( m W ) = H S, ( µ H S ) , (cid:32) H S, ( m W ) H S, ( m W ) (cid:33) = (cid:32) U H S ( µ H S , m W ) 02 (1 − U H S ( µ H S , m W )) / (cid:33) (cid:32) H S, ( µ H S ) H S, ( µ H S ) (cid:33) , (cid:32) H S, ( m W ) H S, ( m W ) (cid:33) = (cid:32) U H S ( µ H S , m W ) 0(1 − U H S ( µ H S , m W )) / (cid:33) (cid:32) H S, ( µ H S ) H S, ( µ H S ) (cid:33) , (5.19)where U H S (cid:0) µ H S , m W (cid:1) = exp (cid:0) − C A ˜ α W (cid:0) log ( m W s ) − log ( µ H S s ) (cid:1)(cid:1) . (5.20)These kernels resum all leading double logarithms.To put together the resummed cross section, we need the tree-level values of the hardfunction H , see Eq. (4.12), H tree1 = π α W M χ , H tree2 = π α W M χ , H tree3 = − π α W M χ , (5.21)the hard-soft functions H S , see Eq. (4.26), H tree S, = 1 , H tree S, = 2 , H tree S, = 1 , (5.22)– 36 –nd collinear-soft functions ˜ S , see Eq. (4.27), (cid:16) ˜ S aba (cid:48) b (cid:48) (cid:17) tree = δ a (cid:48) b (cid:48) δ ab , (cid:16) ˜ S aba (cid:48) b (cid:48) (cid:17) tree = δ a δ a (cid:48) δ bb (cid:48) , (cid:16) ˜ S aba (cid:48) b (cid:48) (cid:17) tree = δ a δ b δ a (cid:48) b (cid:48) + δ a (cid:48) δ b (cid:48) δ ab , (cid:16) ˜ S aba (cid:48) b (cid:48) (cid:17) tree = δ a (cid:48) a δ bb (cid:48) . (5.23)In order to express the final result, we need to include one final piece, the Sommerfeldenhancement which is encoded in the wavefunction factor F a (cid:48) b (cid:48) ab introduced in Eq. (3.3). Therequired contractions are (cid:16) ˜ S aba (cid:48) b (cid:48) (cid:17) tree F a (cid:48) b (cid:48) ab = 16 M χ (cid:12)(cid:12) √ s + 2 s ± (cid:12)(cid:12) , (cid:16) ˜ S aba (cid:48) b (cid:48) (cid:17) tree F a (cid:48) b (cid:48) ab = 32 M χ (cid:12)(cid:12) s (cid:12)(cid:12) , (cid:16) ˜ S aba (cid:48) b (cid:48) (cid:17) tree F a (cid:48) b (cid:48) ab = 16 M χ (cid:0) √ s + 2 s ± (cid:1) ∗ × √ s + c.c. , (cid:16) ˜ S aba (cid:48) b (cid:48) (cid:17) tree F a (cid:48) b (cid:48) ab = 32 M χ (cid:12)(cid:12) s | + 32 M χ (cid:12)(cid:12) s ± | , (5.24)where we have used the tree-level values of the functions ˜ S i and the expressions for thewavefunction factor F a (cid:48) b (cid:48) ab in terms of the Sommerfeld factors s and s ± (see Eq. (3.5) inSec. 3.1.1). Upon expanding the product H i ( m W ) H S,ij ( m W ) ˜ S j ( m W ) in terms of the evolutionkernels in Eq. (5.18) and Eq. (5.19) and using the tree-level results in Eqs. (5.21), (5.22), (5.23),we find z d σ LL d z = π α W sin θ W M χ v LP − (cid:40) U H (2 M χ , m W ) U H J ¯ n ( µ H J ¯ n , m W ) (cid:18) | s | (cid:0) − U H S ( µ H S , m W ) (cid:1) + 2 | s ± | (cid:0) U H S ( µ H S , m W ) (cid:1) + 2 √
23 ( s s ∗ ± + s ∗ s ± ) (cid:0) − U H S ( µ H S , m W ) (cid:1)(cid:19)(cid:41) . (5.25)Here LP − denotes the inverse Laplace transform. The prefactors are determined by tree-levelmatching to full theory, and we have suppressed the arguments of the evolution kernels.At LL accuracy, the cumulative distribution, σ LL ( z cut ) = (cid:90) z cut d z d σ LL d z , (5.26)can be obtained setting s = 1 / (2 M χ (1 − z cut )) in the Laplace space expression for the crosssection, and inserting a / (2 M χ ) for the measure. At the level of the cumulative, we can now– 37 –xplicitly set our canonical scales as µ H J ¯ n = 2 M χ √ − z cut Θ (cid:16) M χ √ − z cut − m W (cid:17) + m W Θ (cid:16) m W − M χ √ − z cut (cid:17) ,µ H S = 2 M χ (1 − z cut ) Θ (cid:16) M χ (1 − z cut ) − m W (cid:17) + m W Θ (cid:16) m W − M χ (1 − z cut ) (cid:17) . (5.27)This implements the physical constraint that the jet and soft scales never go below the scale m W . For a more sophisticated analysis, smooth transition functions could be used insteadof Θ -functions. This is often done to transition from resummation to fixed order, where thesmooth transition functions are referred to as profiles [135]. Here we content ourselves withthis simple choice of scales. This simple choice of profiles also allows us to give a closedform analytic result for the differential spectrum involving the Θ -functions. With this choiceof scale, the evolution kernels appearing in the cross section, now also explicitly involve the Θ -functions that cut off their evolution as appropriate. For example, for the jet functionevolution kernel, we have U H J ¯ n (cid:0) µ H J ¯ n , m W (cid:1) = exp (cid:18) C A ˜ α W log (cid:18) m W M χ √ − z cut (cid:19)(cid:19) Θ (cid:0) M χ √ − z cut − m W (cid:1) + Θ (cid:0) m W − M χ √ − z cut (cid:1) , (5.28)which becomes unity for m W ≥ M χ √ − z cut . The soft function evolution kernel is com-pletely analogous.Combining all the ingredients, we arrive at the final expression for the cumulative crosssection at LL accuracy σ LL ( z cut ) = 4 | s ± | σ tree e − ˜ α W L χ Θ(1 − z cut )+ σ tree e − ˜ α W L χ (cid:40) (cid:16) − F + F e ˜ α W L J ( z cut ) (cid:17) Θ (cid:18) − m W M χ − z cut (cid:19) + (cid:16) − F + F e ˜ α W L J ( z cut ) (cid:17) Θ (cid:18) z cut − m W M χ (cid:19) Θ (cid:18) − m W M χ − z cut (cid:19) + (cid:16) − F + F e ˜ α W ( L J ( z cut ) − L S ( z cut ) ) (cid:17) Θ (cid:18) − m W M χ − z cut (cid:19) (cid:41) . (5.29)Here the Θ -functions explicitly enforce that none of the functions are RG evolved below thescale m W , as emphasized above, and are a crucial part of the final result. Each of the functionsappearing in this expression, as well as their physical significance will be defined shortly.We can now obtain the differential spectrum by taking the derivative of Eq. (5.29) withrespect to (1 − z cut ) . The differentiation of the cumulative result must be performed carefullydue to the presence of the Θ -functions, which when differentiated give rise to δ -functions.However, all the δ -functions explicitly cancel, except for the δ -function for the fully exclu-– 38 –ive contribution. Carefully performing the differentiation, we obtain the final result for thedifferential spectrum:d σ LL d z = 4 | s ± | σ tree e − ˜ α W L χ δ (1 − z )+ 4 σ tree e − ˜ α W L χ (cid:26) C A ˜ α W F (cid:16) L S ( z ) − L J ( z ) (cid:17) e ˜ α W (cid:0) Θ J L J ( z ) − Θ S L S ( z ) (cid:1) − C A ˜ α W F L J ( z ) e ˜ α W L J ( z ) (cid:27) . (5.30)This simple formula provides the resummation of all logarithmically enhanced terms to thespectrum at LL accuracy.As before, to simplify the notation, we have written this expression with ˜ α W = α W / (4 π ) .This result is composed of several pieces with clear physical significance, each of which wenow explain. The tree-level cross section σ tree = π α W sin θ W M χ v , (5.31)appears as an overall multiplicative factor, as does the standard massive Sudakov form factorwith logarithm L χ = log (cid:18) m W M χ (cid:19) . (5.32)The double logarithmic asymptotics is governed by the cusp anomalous dimension [136], inthis case at one-loop, Γ = 4 C A , (5.33)where we recall that C A is the Casimir of the adjoint representation of SU (2) . Explicitly, inour normalization, C A = 2 . In Eq. (5.30) we have written Γ in the exponent to emphasizethat it is the cusp that controls the anomalous dimensions, but used the explicit form ofEq. (5.33) in the prefactors.The first term in the Eq. (5.30) is localized at z = 1 . Only the Sommerfeld factor | s ± | appears since the tree-level process is the annihilation of the charged states χ ± . The secondterm describes the non-trivial z dependence. Here the combination of Sommerfeld factors F = 43 (cid:12)(cid:12) s (cid:12)(cid:12) + 2 (cid:12)(cid:12) s ± (cid:12)(cid:12) + 2 √ (cid:0) s s ∗ ± + s ∗ s ± (cid:1) ,F = − (cid:12)(cid:12) s (cid:12)(cid:12) + 2 (cid:12)(cid:12) s ± (cid:12)(cid:12) − √ (cid:0) s s ∗ ± + s ∗ s ± (cid:1) , (5.34)– 39 –ppear. The perturbative dynamics are controlled by the two logarithms L J ( z ) = log (cid:18) m W M χ √ − z (cid:19) , L S ( z ) = log (cid:18) m W M χ (1 − z ) (cid:19) , (5.35)associated with the jet and soft scales, respectively. For convenience, we have also defined Θ functions associated with the range of the soft and collinear scales Θ J = Θ (cid:18) − m W M χ − z (cid:19) , Θ S = Θ (cid:18) − m W M χ − z (cid:19) . (5.36)In addition to the Sudakov logarithms, the z dependence is controlled by the functions L J ( z ) = L J − z Θ J , L S ( z ) = L S − z Θ S , (5.37)which capture the power divergence in − z , and the subscript is standard notation denotingthat these contain a single power of the logarithm. The presence of the / (1 − z ) factor givesthe expected leading power scaling for the cross section. The power divergence for the softlogarithm is cutoff at z = 1 − m W / (2 M χ ) and for the jet logarithm at z = 1 − m W / (2 M χ ) .These physical cutoffs arise from the value of z at which the soft and jet scales hit the scale m W , where the running must be turned off, as has been discussed above. We note that in themassless theory, the power law divergences would be regulated as plus distributions. Instead,here m W explicitly cuts off the divergence at a finite distance from the endpoint.There is a physical interpretation for each of the different terms in Eq. (5.30). The firstterm, which is localized at the endpoint, corresponds to the fully exclusive cross section, whilethe other terms describe deviations from the endpoint associated with either soft or collinearradiation. With this understanding of the correct treatment of the scales as we transition tothe fully exclusive endpoint, and how they are implemented in our final factorization formula,in the next section we show that our LL expression in the endpoint region correctly reproducesthe LL in both the exclusive and OPE regions. Firstly, however, note that expanding Eq. (5.30)to fixed order, setting the Sommerfeld factor to its tree-level result | s | = 1 , and dropping Θ -functions, we find d σ d z = 4 α W sin θ W M χ v log (cid:16) M χ (1 − z ) m W (cid:17) − z + O ( α W ) . (5.38)This result agrees with the O ( α W ) logarithm derived in the fixed order calculation of [50]. In this section, we demonstrate that our EFT acts as a mother theory which includes boththe exclusive ( z cut → ) and inclusive ( z cut → ) results as limiting cases of our resummedexpression Eq. (5.30). It is important to note that the expansions performed here differ from– 40 –revious calculations such that power corrections are not expected to be identical. However,this complication is avoided here due to the simple structures that are present at LL order.The focus of this section will be showing how to take these two limits analytically. Sec. 6 willprovide a numerical study of the theoretical error that results from scale variation. To obtain the inclusive limit of the total cross section, we simply integrate the differentialcross section given in Eq. (5.30) from z = 0 to the endpoint z = 1 . Explicitly, σ incl = (cid:90) d z d σ d z = (cid:90) d z | s ± | σ tree e − ˜ α W L χ δ (1 − z )+ (cid:90) d z σ tree e − ˜ α W L χ (cid:26) C A ˜ α W F (cid:16) L S ( z ) − L J ( z ) (cid:17) e ˜ α W (cid:0) Θ J L J ( z ) − Θ S L S ( z ) (cid:1) − C A ˜ α W F L J ( z ) e ˜ α W L J ( z ) (cid:27) . (5.39)Performing the integral, we have σ incl = σ tree (cid:16) F + F e − Γ ˜ α W L χ (cid:17) = σ tree (cid:32) | s | f − + 2 | s ± | f + + 2 √ (cid:0) s s ∗ ± + c.c (cid:1) f − (cid:33) , (5.40)where in the last line we have introduced the notation of [41] f ± = 1 ± e − Γ ˜ α W L χ . (5.41)This is precisely the result obtained in [41, 44, 45], demonstrating that we reproduce theinclusive limit to LL order. Note that the signature of interest for experiments like HESS, where the experimental reso-lution has a width σ (cid:29) m W / (4 M χ ) , includes a contribution from the exclusive line and theendpoint spectrum. It is therefore important that we are also able to reproduce the resummedfully exclusive cross section from our factorization. This can be accomplished by integratingEq. (5.30) from z = 1 − m W / (4 M χ ) to z = 1 , which corresponds to a kinematic requirementsuch that only the exclusive final state is possible since both the jet and soft scales are set bythe electroweak boson mass. This demonstrates that for the case where the experimental Note that for z > − m W / (4 M χ ) , Eq. (5.30) is proportional to a delta function for exclusive production,namely δ (1 − z ) . It is important to note that we have power expanded away any mass dependence that would – 41 –esolution has a width δ (cid:29) m W / (4 M χ ) , we have provided the complete description as relevantexperimentally (with the additional caveats discussed in Appendix B).When integrating from z = 1 − m W / (4 M χ ) to the endpoint both L J and L S are zero.Therefore, we can we trivially integrate the δ (1 − z ) dependent term to find σ excl = (cid:90) − m W M χ d z d σ d z = 4 | s ± | σ tree e − ˜ α W L χ . (5.42)This agrees with the exclusive calculation at leading log accuracy performed in [42, 43]. Thefact that we reproduce this result makes it straightforward to convolve the resummed photonspectrum with the experimental resolution – no merging between different results is required.In this sense, our EFT acts as a mother theory that completely describes the photon spectrumfor heavy WIMP annihilation at LL order. In this section, we provide a numerical study of our final prediction for the spectrum byevaluating Eq. (5.30) for wino DM. This allows us to explore the relative contributions fromthe line annihilation and the endpoint spectrum for different choices of the DM mass. Wewill also show the cumulative spectra, as given to LL accuracy in Eq. (5.29), which providesintuition for the finite bin effects that are relevant to realistic experiments. Then in Sec. 7we will provide a mock reanalysis of the HESS line search, and will convolve our predictedspectrum with the Gaussian line shape assumed by HESS.As shown explicitly in Sec. 5.3, our resummed spectrum analytically reproduces the fullyexclusive and the fully inclusive limits, so that we can additionally study the transition betweenthese approximations. This clarifies the disparate conclusions that have been drawn usingthese different approaches. In particular, the exclusive calculations of [42, 43, 46] claimed areduction factor of ∼ . when compared with the tree-level cross section for a TeV wino.For contrast, the inclusive calculation of [41, 44, 45] found a reduction of only a few percent.Physically, this results from the fact that an increasingly exclusive constraint on the final stateimplies there will be less cancellation between the virtual and real corrections (for discussionsin the context of electroweak logarithms, see e.g. [137, 138]). The proper interpretation of theexperimental limits depends on how rapidly the transition between the exclusive and inclusivecross sections occurs. Our EFT analysis provides a complete and decisive resolution of thisissue. Interestingly, we find that the experimental values of current interest to the HESS linesearch, z cut ∼ . - . , lies right in a transition region between the two limiting cases. Thisemphasizes the need to properly treat the impact of finite resolution, as we will do in the next lead to kinematic differences between the γ γ and γ Z final states. We therefore are implicitly assuming that thefinite resolution function sufficiently smears these differences such that they are not experimentally relevant. – 42 – . . . . . . z = E γ /M χ − − − − z d h σ v i / d z [ c m / s ] Differential Cross Section M χ [TeV] Figure 7 . The z weighted differential endpoint cross section as a function of z for three choices ofthe wino mass. Note that the delta function contribution due to the exclusive annihilation process isnot included for clarity of presentation. The error bands are due to scale variation as discussed in thetext. section. However, before moving to our mock reanalysis of the HESS search, we will providesome numerical results along with an estimate of the impact of scale uncertainty.In Fig. 7 we show the differential spectrum z d (cid:104) σv (cid:105) / d z for several values of the DMmass. The delta function contribution from the exclusive process is not included. We seethat the endpoint tracks the mass of the DM as expected. Furthermore, the contributionfrom the resummed continuum grows as the DM mass is increased. However, this effect ismitigated by the strong mass dependence of the overall cross section, both due to Sommerfeldenhancement and the overall /M χ scaling, which explains why the 3 TeV result lies aboveboth the 1 TeV and 10 TeV results. The kink in the TeV distribution is a result of the Θ -functions appearing in the choice of scales, as discussed in Sec. 5.2 (in reality, there arekinks in all the distributions, but they are only visible by eye for the TeV distribution). Thiskink is ultimately unphysical and could be removed by a smooth choice of scales, but is wellwithin our uncertainty bands.The uncertainty bands in Fig. 7 are the result of varying the renormalization scales cor-responding to the natural scales of the functions appearing in our factorization. Due to ourchoice of renormalization path, we simply vary the µ scale of the different functions by a factorof two about their natural scales.An alternative numerical representation of our results is provided in Fig. 8, where we plotthe cumulative cross section as a function of the z cut , for several values of the DM mass. Here– 43 – . . . . . . z cut h σ v i × [ c m / s ] Cumulative Cross Section M χ [TeV] Figure 8 . The cumulative cross section as a function of z cut for three choices of the wino mass. Theexclusive contribution is included here. The error bands are due to scale variation as discussed in thetext. we do include the delta function contribution that yields the exclusive annihilation process,which accounts for the finite value when z cut = 1 . The uncertainty bands are computed usingthe same prescription for the scale variation performed for Fig. 7.The two endpoints, namely z cut = 1 and z cut = 0 , correspond to the fully exclusiveand fully inclusive limits, respectively. Interestingly, for the experimentally relevant range z cut ∼ . - . , the cumulative cross section takes an intermediate value approximately mid-way between the two extremes. This implies that for these values of z cut , logarithms of theresolution are playing an important role, in keeping with the conclusions of the fixed or-der calculation in the inclusive limit [45]. Theoretically robust results require the all-ordersresummation of logarithms from finite bin effects, as has been done here for the first time. The resummed photon spectra derived above have clear implications for heavy DM linesearches. In particular, thermal wino annihilations would produce TeV scale photons. Whenthese photons strike the Earth’s atmosphere, they initiate a detectable shower of particles thatpersists to the surface. Exactly reconstructing the energy of the incident photon from the re-sultant shower is impossible, and as such any real instrument will need to account for finite Another case where a careful treatment of endpoint contributions will be relevant is Higgsino DM, asdemonstrated in [45]. We leave this study to future work. – 44 –nergy resolution effects associated with the spread of possible reconstructed energies given asingle true energy.As discussed at the outset, the strongest constraints on the wino are due to HESS ob-servations of the Galactic Center [26, 28]; updated limits are expected shortly involving thefull HESS I dataset [139, 140]. Line searches are typically designed to be model-independent,and thus assume that only the line emission is relevant (although some specific non-line hardspectra have also been tested [28, 141]). As we demonstrated in Fig. 8 above, photons awayfrom the endpoint contribute to a finite bin at a non-trivial level. This is especially true forHESS, where the effective z cut ∼ . − . depending on the incident energy. Furthermore, theline analysis of HESS is not a bin-based counting experiment but requires subtraction of anunknown background, which is modeled by a smooth function. The presence of signal photonsat even lower energies may bias the data-driven background model if this signal spectrum isnot correctly modeled, further modifying the limit.The goal of this section is to estimate how much including the correct shape and normal-ization of the resummed spectrum would be expected to change the HESS limit, relative tothe case of a pure line.It is important to emphasize that the results presented in this section are approximate,and should not be taken as updated limits on the wino. At issue is that the full datasetHESS used to construct their limits in Ref. [28] is not public. What we will show are resultsfrom a simplified mock version of that analysis, using a Gaussian likelihood rather than the fulllikelihood, which has been validated to yield comparable limits when assuming exclusive lineemission. We can then explore how the various conclusions are modified when we include theendpoint emission spectrum. The conclusion is that the additional emission should strengthenlimits on the wino by a O (1) factor. This provides motivation for future experimental analysesto include these contributions when determining limits.This section contains three parts. First, we review how to map from DM model param-eters, including the relevant astrophysical inputs, to a prediction for the number of photonsthat HESS would observe. Then we apply this formalism to demonstrate the range of param-eters that HESS can constrain. Finally, we outline our mock analysis procedure and presentapproximate results showing the impact of our resummed spectra on current constraints. In order to determine the sensitivity to wino DM, we need a prediction for the number ofphotons that should arrive at an experiment as a function of the DM parameters. This can bederived using the canonical indirect detection formula, which specifies the differential energyflux arriving at the detector, ROI d Φ γ d E = J (cid:104) σv (cid:105) π M χ d N γ d E , (7.1)where Ω ROI ≡ (cid:82) ROI d Ω . – 45 –he particle physics contribution (cid:104) σv (cid:105) / (8 π M χ ) d N γ / d E depends on the velocity averagedtotal annihilation cross section (cid:104) σv (cid:105) , which is summed over all final states involving a photon,and the average photon spectrum per annihilation, d N γ / d E , which can be written as d N γ d E = (cid:88) f Br f d N fγ d E , (7.2)where the f index refers to the different final states with associated branching fractions Br f and photon spectra d N fγ / d E . Since the spectrum here is the result of resumming multipleelectroweak final states (not including the photons that result from decay of unstable W ± and Z bosons, see Appendix B for a discussion), we will only refer to the total averaged quantityd N γ / d E for the remainder of this section.The remaining ingredient is the so-called J -factor, which is an astrophysical input. Itis determined by the distribution of the DM along the line of sight in the region of interest(ROI) under consideration. It additionally accounts for the fact that two particles must findeach other for for annihilation to occur; the J -factor depends on the number density squaredas J = (cid:82) ROI d s d Ω ρ ( s, Ω)Ω
ROI , (7.3)where ρ DM is the Milky Way DM mass distribution, s is the distance from Earth along theline of sight, and Ω gives the coordinates on the sky within the ROI. Note that as written,the J -factor has units of TeV · cm − , and in particular there are no units of sr due to thedenominator in Eq. (7.3). We caution, however, that a number of other conventions are inuse. For a fixed ROI, J is then in principle determined by the Milky Way DM profile. Unfor-tunately, the shape of ρ DM is very uncertain near the Galactic Center, see e.g. [143], and inparticular within the ROI of the HESS search of Ref. [28]. For the case of the wino, once themass is fixed the cross section is fully specified. Therefore, one can translate limits on winoannihilations into a constraint on J , as done in Fig. 10 below.It is also of interest to fix a prototypical value for J and then set a limit on the annihilationcross section, since this is how these constraints are typically presented. For this purpose weadopt the Einasto profile, the default profile assumed in the HESS analyses, which is given by ρ Einasto ( r ) ∝ exp (cid:20) − α (cid:18)(cid:18) rr s (cid:19) α − (cid:19)(cid:21) , (7.4)where r is the distance from the center of the halo, and following [144], by default we take α = 0 . , r s = 20 kpc, and then normalise the profile so that we reproduce the local DMdensity of .
39 GeV cm − at our location which is 8.5 kpc from the Galactic Center. Anotherfrequently invoked DM distribution is the Navarro-Frenk-White (NFW) profile [145], which This is sometimes defined as the spectrum per DM particle, which differs by a factor of 2. For a recent review of the conventions used for indirect detection, see Appendix A of [142]. – 46 –akes the form ρ NFW ( r ) ∝ r/r s )(1 + r/r s ) , (7.5)where again we take r s = 20 kpc. We will also make use of the NFW profile (including thepossibility of a non-trivial core) when interpreting our results below.Finally, putting this all together results in the differential energy flux arriving at thedetector, ROI d Φ γ d E , which has units of photons · cm − · s − · TeV − · sr − . This quantitycan be converted into a predicted number of photons (per unit area per unit time) arriving atthe experiment from DM annihilation by first multiplying by the solid angle of the consideredROI, Ω ROI , and then integrating over the energy range determined by the experimental search.This photon flux Φ γ has units of photons · cm − · s − . Converting this to the actual number ofphotons depends on the experimental effective area and time over which the ROI is observed;a larger detector and longer observations will result in more observed photons. For HESS, theeffective area is ∼ cm at 1 TeV and current searches make use of 112 hours of observationsof the Galactic Center, yielding sensitivity to fluxes ∼ − cm − s − . We can then constrainthe DM model using this prediction for the number of photons as an input to a likelihoodanalysis. Before we give the details of and results from our mock analysis, it is useful to discuss howwe are mapping from the theory prediction to the experimental constraints. The subtletyarises because the original search was performed under the assumption that the annihilationsignature is a line; in this case, by definition all photons have the same energy. The spectrumof a typical WIMP can be decomposed into two contributionsd N γ d E ∼ line + continuum . (7.6)The line is due to exclusive annihilations to γ γ and γ Z . Since our interest here is in heavyWIMPs, we will neglect the fact that the finite Z mass causes E γ = M χ − m W / (4 M χ ) < M χ for the photons that result from the γ Z process, and will combine these line contributionsusing (cid:104) σv (cid:105) line ≡ (cid:104) σv (cid:105) γγ + 12 (cid:104) σv (cid:105) γZ , (7.7)with E γ = M χ for all line photons.The continuum receives many contributions. In the DM literature, this is usually sepa-rated into photons from “internal bremsstrahlung” [47–51], as well as final and initial stateradiation, on one hand, and those photons that result from the cascade decay chain of un-stable particles on the other hand. The decay processes can yield many final state photonswith a broad energy spectrum. Our endpoint calculation for winos resums the non-decayperturbative processes, and as such it does not include the additional continuum photons thatresult from the decay of the W ± and Z . However, this contribution is demonstrated to have– 47 –ittle impact on the limits for heavy winos in Appendix B. This conclusion is intuitive sincethe photons from the W ± /Z cascade decays are much lower energy than the exclusive andendpoint contributions. Therefore, we model the continuum as only being due to the end-point contributions, which we denote with E ( E ) , and using Eq. (5.30) the LL result is givenexplicitly by E LL ( E ) = 1 (cid:104) σv (cid:105) line d (cid:104) σv (cid:105) d E − δ (cid:0) E − M χ (cid:1) (7.8) = 2 | s ± | M χ (cid:26) C A ˜ α W F (cid:16) L S ( z ) − L J ( z ) (cid:17) e ˜ α W (cid:0) L J ( z ) − ( L S ( z )) (1 − z ) (cid:1) − C A ˜ α W F L J ( z ) e ˜ α W L J ( z ) (cid:27) , where as usual, z = E/M χ . The resulting spectrum per annihilation isd N γ d E = (cid:104) σv (cid:105) line (cid:104) σv (cid:105) (cid:16) δ ( E − M χ ) + E ( E ) (cid:17) , (7.9)such that (cid:104) σv (cid:105) line / (cid:104) σv (cid:105) is the branching fraction to line photons. Note that our calculationpredicts not only the shape of the endpoint contribution, but also the relative normalizationof this with respect to the line spectrum. Putting these details together, we arrive at thetheory prediction (cid:18) d Φ γ d E (cid:19) ideal = J Ω ROI (cid:104) σv (cid:105) line π M χ (cid:104) δ ( E − M χ ) + E ( E ) (cid:105) , (7.10)which is idealized in the sense that it neglects experimental effects.As such we are still missing one ingredient, which is the fact that we need to convolvethis with the experimental energy resolution. We can describe the energy resolution via aconvolution function Σ( E − E (cid:48) ) , where E (cid:48) is the true photon energy and E is the reconstructedvalue, and the spectrum an experiment would measure is (cid:18) d Φ γ d E (cid:19) smeared = J Ω ROI (cid:104) σv (cid:105) line π M χ (cid:90) M χ d E (cid:48) Σ (cid:0) E (cid:48) − E (cid:1)(cid:104) δ (cid:0) E (cid:48) − M χ (cid:1) + E (cid:0) E (cid:48) (cid:1)(cid:105) . (7.11)The HESS collaboration has published a model for Σ( E − E (cid:48) ) which we use here, a Gaussianthat is peaked near the true energy with a width that varies from 17% at 0.5 TeV and 11% at10 TeV. We interpolate in between these values using the log of the energy, and find a width ∼ % at 3 TeV.HESS can constrain the overall normalization of Eq. (7.11); in terms of the theory pre-– 48 –iction, this can be interpreted as a constraint on the quantity C HESS = J Ω ROI (cid:104) σv (cid:105) line π M χ . (7.12)However, it is critical to specify the assumed energy spectrum E ( E ) (in addition to a linecontribution) when deriving a HESS constraint on the cross section. For the following com-parisons, we will use the LL endpoint spectrum computed in this work, so that (cid:18) d Φ γ d E (cid:19) HESS = C HESS (cid:90) M χ d E (cid:48) Σ (cid:0) E (cid:48) − E (cid:1)(cid:104) δ (cid:0) E (cid:48) − M χ (cid:1) + E LL (cid:0) E (cid:48) (cid:1)(cid:105) , (7.13)where C HESS is the coefficient that is constrained using the HESS data, we take E LL (cid:0) E (cid:48) (cid:1) fromEq. (7.8), and Σ( E (cid:48) − E ) is as discussed above. In the next section, we will interpret theHESS data as a constraint on C HESS using a mock analysis, and will then convert this intoan approximate constraint on winos using Eq. (7.12). We will either use Eq. (5.42) to predict (cid:104) σv (cid:105) line for a given mass in order to set a constraint on J , or we will assume the Einastoprofile which gives us J and then constrain the cross section (cid:104) σv (cid:105) line . We will also provide aconstraint on the core size, using the NFW profile modified to include a core.Note that we can test the effects of ignoring the non-line endpoint contributions by simplysetting E ( E ) = 0 ; up to the approximations in our analysis required by not having the fulllikelihood available, this should reproduce the limits stated in Ref. [28]. This allows us todirectly compare constraints on the line only and the line plus endpoint spectrum, therebyhighlighting the impact of our main result Eq. (5.30). The next section outlines the detailsof our mock analysis and provides approximate constraints on either the cross section or the J -factor. Using the procedure described in the previous section, one can in principle interpret the HESSdata as a constraint on wino DM annihilations. As the data collected by the instrument isnot public, we are not able to provide a full and precise update of the constraints on winos.Instead, we will perform a simplified mock version of the HESS analysis in order to estimatethe impact of the corrections calculated here on the resulting limits. Our mock analysis canroughly recover the published line limits in the case where we take E ( E ) = 0 above. Wewill then extend the analysis to include the endpoint contributions, demonstrating that theystrengthen the limits by an O (1) factor.Our mock analysis is based on a simplified version of the analysis performed in Ref. [28].Figure 1 of that work provides the measured flux and the associated uncertainty as a functionof energy in their ROI near the Galactic Center. We digitized this dataset and used it asthe input to a Gaussian likelihood analysis. We note that since HESS is an instrument thatcounts the number of incident photons, the Poisson likelihood should in principle be used.However, the number of counts cannot be exactly reconstructed from the publicly released– 49 –ux data. The non-Gaussian nature of this dataset is made manifest by the asymmetric errorbars that are particularly clear at higher energies. We approximately included the asymmetryin the likelihood by using the upper error bars to determine the likelihood contribution frombins where our model prediction exceeded the data, and the lower error bars for bins whereour model prediction fell below the data. We found that this approach gave better agreementwith the published HESS results than symmetrizing the error bars.The dataset d i is defined using energy bins with associated index i , where the digitizedHESS flux gives a central value µ i and error σ i , chosen (between the upper and lower error bars)in the manner described above. The prediction m i ( θ ) is a function of the model parameters θ . The DM-signal contribution to the model is computed using Eq. (7.11). We will treat thistheory flux as being a function of the DM mass, M χ , the line photon cross section, (cid:104) σv (cid:105) line ,and the J -factor. As emphasized above, given M χ we can either calculate (cid:104) σv (cid:105) line and thenconstrain J , or assume a value of J and turn this into a constraint on (cid:104) σv (cid:105) line .Even in the most optimistic DM scenario, the events collected by HESS will not be solelydue to DM annihilation. Firstly, there is a substantial flux of cosmic rays colliding with theatmosphere, which can mimic gamma-ray signals. Secondly, there will be genuine gamma-rays due to high-energy astrophysical processes, such as protons in the inner galaxy collidingwith gas and producing energetic neutral pions which decay to gamma-rays. The expectedflux from cosmic-rays and astrophysical sources of gamma-rays is not well understood in theHESS energy range, and as such Ref. [28] parametrized the background contribution usingthe following seven parameter model: (cid:18) d Φ γ d E (cid:19) bkg = a (cid:18) E (cid:19) − . (cid:20) P (cid:18) log (cid:20) E (cid:21)(cid:19) + β G (cid:18) log (cid:20) E (cid:21)(cid:19)(cid:21) ,P ( x ) ≡ exp( a x + a x + a x ) ,G ( x ) ≡ (cid:112) π σ x exp (cid:20) − ( x − µ x ) σ x (cid:21) . (7.14)The background is then specified by the seven parameters θ bkg = { a , a , a , a , β, µ x , σ x } .Combining the signal and background, we arrive at our full model prediction of m i ( θ ) = (cid:34)(cid:18) d Φ γ d E (cid:0) M χ , (cid:104) σv (cid:105) line , J (cid:1)(cid:19) Smeared + (cid:18) d Φ γ d E ( θ bkg ) (cid:19) bkg (cid:35)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) E = E i , (7.15)so that the model is specified by three signal and seven background parameters. From here,given the HESS dataset described above, d = { d i } = { µ i , σ i } , we can write down our assumedGaussian likelihood function as L (cid:0) d | θ (cid:1) = (cid:89) i (cid:113) π σ i exp (cid:20) − ( m i ( θ ) − µ i ) σ i (cid:21) . (7.16)– 50 – . . . . . M χ [TeV] − − − − h σ v i li n e [ c m / s ] Estimated Limits
Line (HESS Published)Line (this ref)+ endpoint (this ref)LL h σv i line Figure 9 . The approximate constraints on the line annihilation cross section as a function of theDM mass for the Einasto profile using our mock reanalysis of the HESS line search. The dotted lineassumes the line-only spectrum and the dashed line assumes the full endpoint + line spectrum. Weadditionally provide the LL resummed prediction (including the Sommerfeld enhancement) for theline annihilation. Under these assumptions, the wino would be excluded when the LL prediction isabove the HESS full constraint. We also show the published HESS line limit in dots to demonstratethe extent to which our line-only analysis reproduces their result.
In order to restrict our likelihood to be a function of only the signal parameters, we eliminatethe nuisance parameters using the profile likelihood method, L (cid:0) d | θ sig (cid:1) = L (cid:0) d | θ sig , ˆ θ bkg (cid:1) , (7.17)where the hat indicates evaluating the function at the values of θ bkg that maximize thelikelihood (see [146] for a review).With this reduced likelihood, we can then define a test statistic for upper limits as afunction of M χ on either (cid:104) σv (cid:105) line or J . To begin with, we can fix J and set a limit on (cid:104) σv (cid:105) line .To determine the fixed value of J , we use Eq. (7.3) assuming an Einasto profile as given inEq. (7.4). The ROI for this dataset was a 1 ◦ circle around the Galactic Center, with theGalactic plane masked for latitudes less than 0.3 ◦ , which yields J (cid:39) . × TeV cm − . (7.18)– 51 – . . . . . M χ [TeV] J - f a c t o r [ T e V / c m ] Estimated J-factor Constraint
Line (this ref)+ endpoint (this ref)
Figure 10 . The approximate constraints on the J -factor as a function of the DM mass, assuming theline only spectrum and the full endpoint + line spectrum, as derived from our mock reanalysis of theHESS line search. Fixing this value, we define the test statistic as q (cid:104) σv (cid:105) line ( M χ ) ≡ (cid:104) log L ( d | M χ , (cid:104) σv (cid:105) line ) − log L ( d | M χ , (cid:100) (cid:104) σv (cid:105) line ) (cid:105) (cid:104) σv (cid:105) line ≥ (cid:100) (cid:104) σv (cid:105) line (cid:104) σv (cid:105) line < (cid:100) (cid:104) σv (cid:105) line , (7.19)where again a hat denotes the value that maximizes the likelihood. Using this test statistic,the 95% limit on (cid:104) σv (cid:105) line is then determined by solving for q (cid:104) σv (cid:105) line ( M χ ) = − . , and is shownin Fig. 9. In this figure we have also shown the prediction for the wino cross section – if thesewere exact limits and if the DM distribution followed an Einasto profile in the inner galaxy,then the wino would be excluded when this prediction is above the mock limit curve.This figure also contains the published HESS limits, taken from Fig. 4 of [28]. The extentto which our line-only limits disagree with the published values highlights that our mockanalysis is not exact and thus should not be taken as the true limit on wino DM. Neverthelessthe figure does make it clear that the addition of the endpoint contributions can lead to anon-trivial enhancement on the sensitivity. For this reason, the effects calculated in this workrepresent an important contribution that should be included in future searches for heavy DMannihilation.Alternatively, for limits on J , we fix (cid:104) σv (cid:105) line to the exclusive wino prediction appropriate– 52 – . . . . . M χ [TeV] C o r e S i ze [ k p c ] Estimated Core Size Constraint
Line (this ref)+ endpoint (this ref) . . . . . . Figure 11 . The NFW core size required to save the wino as derived using our mock analysis. Thisfigure follows from the J -factor constraints given in Fig. 10. At a given mass, the constraint on J canbe converted into a core size limit by calculating the corresponding cored NFW J -factor in the HESSanalysis ROI. For a thermal wino at 3 TeV the estimated constraints improve from 0.70 to 0.99 kpcwhen including the endpoint contributions. This core size is beginning to be probed in both numericaland astrophysical settings. We again emphasize that these constraints should be only taken as anestimate. for that mass using Eq. (5.30), and in a similar notation to [147], define our test statistic as q J ( M χ ) ≡ (cid:104) log L (cid:0) d | M χ , J (cid:1) − log L (cid:0) d | M χ , ˆ J (cid:1)(cid:105) J ≥ ˆ J J < ˆ J . (7.20)As for the cross section, this test statistic allows us to establish the 95% limit at a given massthrough the relation q J ( M χ ) = − . , and the result is shown in Fig. 10. In this case we haverepeated the analysis with and without the endpoint contributions calculated in this work,with the impact of our calculation being as much as a factor of 3 improvement in the limit,and a factor of ∼ O (1) improvements in the limits on (cid:104) σv (cid:105) line or the GalacticCenter J -factor. Finally, we emphasize that the search for the wino is reaching a level ofsensitivity such that O (1) factors are important. One way to see this, is by converting thelimits into a statement on how large a core in the Milky Way DM density profile is requiredto save the wino from the HESS constraints.– 53 –or concreteness, we use a cored version of the NFW profile, following [24]. For r > r core ,we take the NFW profile as defined in Eq. (7.5); when r ≤ r core , we set the profile to a constantvalue ρ NFW ( r core ) , such that the density profile is flat within the core radius. Restrictingourselves to cores smaller than 8.5 kpc, the presence of a core reduces the associated J -factorof the halo. In this way we can directly convert J -factor limits into a corresponding constrainton r core , which we show in Fig. 11. From this, we can see that for a thermal wino at exactly3 TeV, the estimated core constraint increases from 0.70 kpc to 0.99 kpc when including theadditional photons from the endpoint spectrum.The values constrained in Fig. 11 turn out to be at the edge of the core sizes that arebeginning to be probed by a combination of numerical simulations and data. On the numericalside, it was shown that recent simulations of Milky Way-like halos in simulations includingthe effects of baryons, can potentially contain cores up to O (1) kpc [148]. The total DM massin the Galactic Bulge region can be estimated from observations of stars in the Bulge [149],and disfavors a canonical NFW profile with a core size larger than ∼ In this paper we have developed a comprehensive effective field theory framework to computethe photon spectrum for annihilating (or decaying) DM. We provided a new factorizationformula, which allows for a resummation of all large logarithmic contributions, properly treat-ing the effects due to electroweak symmetry breaking, the experimental resolution on the γ + X final state, and the Sommerfeld enhancement. We have computed the relevant one-loopanomalous dimensions, showing the consistency of the factorization formula at this order. Wehave shown that the contribution from the spectrum has a numerically important effect forexperimental searches of interest, e.g. gamma-ray line searches from the HESS telescope. Ourfinal result is a compact analytic expression for the differential spectrum at LL accuracy, whichcan easily be convolved with experimental resolution functions to provide realistic predictions.Our EFT can be interpreted as a mother theory that includes as particular limits thefully exclusive and fully inclusive cases. The framework presented here correctly describesthe transition between these two limits, allowing us to understand how Sudakov double log-arithms impact the spectrum as a function of the experimental resolution. It also allows usto assess the extent to which a fully exclusive or fully inclusive approximation, as had beenpreviously considered in the literature, is appropriate. Interestingly, we find that for the rangeof resolution parameters applicable for current and near future experiments, the result is inter-mediate between the fully exclusive and fully inclusive predictions. This resolves the differingconclusions obtained in the literature, and provides a unifying picture of the importance ofSudakov resummation for indirect detection searches. We have estimated the impact on theinterpretation of current searches by providing a mock reanalysis of the HESS data, and we– 54 –nd that we are probing core sizes in a region where precise calculations of the particle physicscomponents are relevant.Now that this paper has established an EFT framework for describing the photon spec-trum resulting from DM annihilation, one can extend this work in a number of future di-rections. It would be of formal interest to understand the structure of the factorization andresummation at higher logarithmic order. Although the electroweak couplings are small, sig-nificantly improved uncertainties have been observed at NLL [42, 43, 46], implying that NLLis likely the highest order that is relevant. Additionally, the explicit NLO calculations pro-vided in [42, 46] demonstrate that higher order terms that are not logarithmically enhancedare numerically unimportant, justifying our choice to neglect them.There are also additional phenomenological applications. One could extend these resultsto other heavy DM models, e.g. a thermal Higgsino, a mixed wino-higgsino, or minimal DMcandidates. In many of these cases, the constraints can be different [24, 151, 152], implyingthat a dedicated analysis is warranted. From the point of view of extending the work presentedhere, the underlying EFT is unchanged, but one must modify the Sommerfeld calculation andthe explicit values for the hard matching coefficients and anomalous dimensions.A simple heavy DM candidate provides a viable and phenomenologically relevant ex-planation for the observed relic abundance that could show up in current or future indirectdetection searches. This work casts the prediction for the photon spectrum that can resultfrom this class of models in a theoretically robust setting, where perturbation theory can bemaintained by performing resummation of all large double logarithms. If a signal of heavyDM annihilation appears, this work will be critical to interpreting it. Acknowledgments
We thank Martin Bauer, Marco Cirelli, Richard Hill, Emmanuel Moulin, Duff Neill, andLucia Rinchiuso for useful discussions. MB is supported by the U.S. Department of Energy,under grant number DE-SC0003883. TC is supported by the U.S. Department of Energy,under grant number DE-SC0018191. IM is supported by the U.S. Department of Energy,under grant number DE-AC02-05CH11231 and the LBNL LDRD program. NLR and TRSare supported by the U.S. Department of Energy, under grant numbers DE-SC00012567 andDE-SC0013999. MPS is supported by the U.S. Department of Energy, under grant numberDE-SC0011632. IWS is supported by the Office of Nuclear Physics of the U.S. Departmentof Energy under the Grant No. DE-SCD011090 and by the Simons Foundation through theInvestigator grant 327942. VV was supported by the Office of Nuclear Physics of the U.S.Department of Energy under the Grant No. Contract DE-AC52-06NA25396 and through theLANL LDRD Program. – 55 – ppendicesA One-Loop Calculations
In this Appendix we provide details of the calculation of the one-loop anomalous dimensionsfor the different functions appearing in the factorization formula, or provide references wherethey can be obtained from known results. Details of the refactorization are provided, andrelevant integrals used in the calculation are also collected.
A.1 One-Loop Calculation and Anomalous Dimensions: Intermediate EFT
We begin by giving details related to the calculation of the anomalous dimensions for theintermediate EFT, before refactorization. This will help to make clear how these anomalousdimensions, and the associated divergences, are split in the refactorized description.
Hard Function
The hard function is independent of the infrared measurement made on the final state.It can therefore be extracted directly from the literature. Although we will only considerthe LL resummation in this paper, we give the NLL anomalous dimension for completeness.The anomalous dimension matrix for ( C C ) T can be written in terms of a diagonal and anon-diagonal component as ˆ γ = 2 γ W T + ˆ γ S . (A.1)Explicit results for γ W T and ˆ γ S were given in [43], namely γ NLL W T = α W π Γ log (cid:18) M χ µ (cid:19) − α W π b + (cid:16) α W π (cid:17) Γ log (cid:18) M χ µ (cid:19) , (A.2)and ˆ γ NLL S = α W π (1 − iπ ) (cid:18) − (cid:19) − α W π (cid:18) (cid:19) . (A.3)The constants appearing in these expressions are the SU (2) Casimir C A = 2 , the one-loop β -function b = 19 / , and the relevant cusp anomalous dimensions are Γ = 4 C A and Γ =8 (cid:0) − π (cid:1) . Photon Jet Function
The photon jet function, which has a single photon as its final state, is defined in Eq. (4.13)as J γ = (cid:68) (cid:12)(cid:12)(cid:12) B cn ⊥ (0) (cid:12)(cid:12)(cid:12) γ (cid:69)(cid:68) γ (cid:12)(cid:12)(cid:12) B cn ⊥ (0) (cid:12)(cid:12)(cid:12) (cid:69) . (A.4)– 56 –valuating this function at one-loop yields J γ = − − C A α W π (cid:18) µm W (cid:19) (cid:15) (cid:18) ν M χ (cid:19) η Γ( (cid:15) ) η + α W π Γ( (cid:15) ) (cid:18) µm W (cid:19) (cid:15) (cid:20) C A − n f C ( R ) (cid:21) + O (cid:0) α W (cid:1) , (A.5)where µ and ν are the virtuality and rapidity renormalization scales respectively. Here n f denotes the number of fermion flavors. We take n f = 5 in our numerical results. The µ and ν anomalous dimensions can immediately be extracted from this result, and we find, γ nµ = 2 C A α W π log (cid:18) ν M χ (cid:19) , (A.6) γ nν = 2 C A α W π log (cid:18) µm W (cid:19) . (A.7) Recoiling Jet Function
When computing the recoiling jet function, all IR divergences are explicitly regulated bythe measurement of the final state mass. This is unlike the photon jet function, where thescale m W acts as a regulator. To compute the anomalous dimensions, it is therefore sufficientto expand away the scale m W from the beginning, simplifying the calculation. The inclusiverecoiling jet function is then defined as J (cid:48) ¯ n ( k + ) = (cid:88) X C (cid:68) (cid:12)(cid:12)(cid:12) B d ¯ n ⊥ (0) δ (cid:0) k + − P + (cid:1) δ (cid:0) M χ − P − / (cid:1) δ (cid:0) (cid:126) P ⊥ (cid:1)(cid:12)(cid:12)(cid:12) X C (cid:69)(cid:68) X C (cid:12)(cid:12)(cid:12) B d ¯ n ⊥ (0) (cid:12)(cid:12)(cid:12) (cid:69) . (A.8)We can rewrite this jet function as J (cid:48) ¯ n ( p ) = (cid:88) X C (cid:90) d x (2 π ) e i p · x (cid:68) (cid:12)(cid:12)(cid:12) B d ¯ n ⊥ (0) (cid:12)(cid:12)(cid:12) X C (cid:69)(cid:68) X C (cid:12)(cid:12)(cid:12) B d ¯ n ⊥ ( x ) (cid:12)(cid:12)(cid:12) (cid:69) . (A.9)with p = (2 M χ , k + , in order to enforce the δ -function measurement constraints. Writtenin this form, the function is completely inclusive. Therefore, we can use the optical theoremto write this as the imaginary part of the forward scattering amplitude J (cid:48) ¯ n ( p ) = 12 Im (cid:90) d x (2 π ) e i p · x (cid:68) (cid:12)(cid:12)(cid:12) T B d ¯ n ⊥ (0) B d ¯ n ⊥ ( x ) (cid:12)(cid:12)(cid:12) (cid:69) . (A.10)This jet function has been evaluated in the literature [153–155]; the one-loop result is J (cid:48) ¯ n ( p ) = δ ( p ) + α W π (cid:18) C A log( p /µ ) − b p (cid:19) + + O (cid:0) α W (cid:1) , (A.11)– 57 –here the subscript plus denotes a plus distribution, see e.g. [156] for an extensive review ofits properties. The kinematics for heavy DM annihilation imply that p = 2 M χ k + , so that J (cid:48) ¯ n ( k + ) = δ (cid:0) M χ k + (cid:1) + α W π µ (cid:32) C A log (cid:0) M χ k + /µ (cid:1) − b M χ k + /µ (cid:33) + + O (cid:0) α W (cid:1) . (A.12)To expose the simple renormalization group structure, we transform to Laplace space, wherethe Laplace conjugate variable of k + is taken to be s . Keeping only the leading log term, wefind J (cid:48) ¯ n ( s ) = 12 M χ + 2 C A α W π M χ log (cid:18) µ s e γ E M χ (cid:19) + O (cid:0) α W (cid:1) , (A.13)where γ E is the Euler-Mascheroni constant. Finally, we extract the µ anomalous dimension γ ¯ nµ = 2 C A α W π log (cid:18) µ s e γ E M χ (cid:19) . (A.14)This function has no rapidity anomalous dimension as it is a SCET I type function. Ultrasoft Function
There are four operators that contribute to the ultrasoft function in the EFT: S (cid:48) , S (cid:48) , S (cid:48) , S (cid:48) , see Eq. (4.11) above. Using the expressions below, we can then extract the LL µ and ν anomalous dimensions. We will find that each operator yields the same result, γ S (cid:48) µ = − C A α W π log (cid:0) ν s (cid:1) ,γ S (cid:48) ν = − C A α W π log (cid:18) µm W (cid:19) . (A.15)This calculation will also expose additional IR divergent contributions, which is the sign thatrefactorization is necessary.The one-loop results will be expressed in terms of several integrals, denoted in bold andlabeled with V and R for virtual and real respectively, which are defined and evaluated below.These integrals are evaluated using dimensional regularization as an IR regulator, and with therapidity regulator as defined in Sec. 3.2. The integrals that we will require in our calculationare defined as follows. The n¯n integrals are I Rn¯n = − g W (cid:90) d d (cid:96) (2 π ) d − δ + (cid:0) (cid:96) − m W (cid:1)(cid:12)(cid:12) (cid:96) + − (cid:96) − (cid:12)(cid:12) − η/ δ ( q + − (cid:96) + ) (cid:96) + (cid:96) − = − α W π (cid:18) µm W (cid:19) (cid:15) (cid:18) ν q + m W (cid:19) η/ Γ[ (cid:15) + η/ q + , (A.16)– 58 – Vn¯n = δ (cid:0) q + (cid:1) g W µ (cid:15) ν η/ (cid:90) d d (cid:96) (2 π ) d − i ( (cid:96) − m W + i (cid:12)(cid:12) (cid:96) + − (cid:96) − (cid:12)(cid:12) − η/ ( (cid:96) + + i
0) ( (cid:96) − − i − δ (cid:0) q + (cid:1) α W π (cid:18) µm W (cid:19) (cid:15) (cid:18) νm W (cid:19) η/ − η/ η Γ[ (cid:15) + η/ / − η/ / iπ δ (cid:0) q + (cid:1) α W π (cid:18) µm W (cid:19) (cid:15) Γ[ (cid:15) ] . (A.17)The iπ term in this expression arises from a Glauber contribution, and as it does not contributeat LL we will not consider it further, although we include it for completeness as it is relevantat NLL [157]. Note the expression here agrees with [158]. Continuing, the nv integrals are I Vnv = g W µ (cid:15) ν η/ (cid:90) d d (cid:96) (2 π ) d − i ( (cid:96) − m W + i (cid:12)(cid:12) (cid:96) + (cid:12)(cid:12) − η/ ( (cid:96) + + (cid:96) − − i
0) ( (cid:96) − − i − α W π (cid:18) µm W (cid:19) (cid:15) (cid:18) νm W (cid:19) η/ Γ[ (cid:15) + η/ − η/ η , (A.18) I Rnv = − g W (cid:90) d d (cid:96) (2 π ) d − δ + (cid:0) (cid:96) − m W (cid:1) δ (cid:0) q + − (cid:96) + (cid:1) ( (cid:96) + + (cid:96) − ) (cid:96) − = − α W π q + log (cid:32) (cid:112) ( q + ) + m W m W (cid:33) , (A.19)and the ¯nv integrals are I V¯nv = δ (cid:0) q + (cid:1) g W µ (cid:15) ν η/ (cid:90) d d (cid:96) (2 π ) d − i ( (cid:96) − m W + i (cid:12)(cid:12) (cid:96) + − (cid:96) − (cid:12)(cid:12) − η/ ( (cid:96) + − i
0) ( (cid:96) + + (cid:96) − − i − δ (cid:0) q + (cid:1) α W π (cid:18) µm W (cid:19) (cid:15) (cid:18) νm W (cid:19) η/ Γ[ (cid:15) + η/ − η/ η , (A.20) I R¯nv = − g W (cid:90) d d (cid:96) (2 π ) d − δ + (cid:0) (cid:96) − m W (cid:1) δ (cid:0) q + − (cid:96) + (cid:1)(cid:12)(cid:12) (cid:96) + − (cid:96) − (cid:12)(cid:12) − η/ ( (cid:96) + + (cid:96) − ) ( (cid:96) + )= I Rn¯n − I Rnv . (A.21)Next, we consider each of the four ultrasoft functions in turn. First we provide the operatordefinition, followed by the tree-level and one-loop evaluation in order to compute the anoma-lous dimensions for the different color structures of the ultrasoft function. Since we are doingthis in the EFT before refactorization, we will refer to these as ultrasoft functions and thecorresponding states as | X US (cid:105) . These ultrasoft functions will ultimately be refactorized.– 59 – S (cid:48) is defined as S (cid:48) aba (cid:48) b (cid:48) = (cid:88) X US (cid:68) (cid:12)(cid:12)(cid:12) (cid:16) Y f (cid:48) n Y dg (cid:48) ¯ n (cid:17) † (0) (cid:12)(cid:12)(cid:12) X US (cid:69) × (cid:68) X US (cid:12)(cid:12)(cid:12) δ ( q + − P + ) (cid:16) Y fn Y dg ¯ n (cid:17) (0) (cid:12)(cid:12)(cid:12) (cid:69) δ f (cid:48) g (cid:48) δ a (cid:48) b (cid:48) δ fg δ ab . (A.22)Evaluating at tree-level in Laplace space yields (cid:16) S (cid:48) aba (cid:48) b (cid:48) (cid:17) tree = δ ab δ a (cid:48) b (cid:48) , (A.23)and at one-loop yields (cid:16) S (cid:48) aba (cid:48) b (cid:48) (cid:17) = − δ ab δ a (cid:48) b (cid:48) C A (cid:16) I Rn¯n − I Vn¯n (cid:17) −−−−→
Laplace − δ ab δ a (cid:48) b (cid:48) C A α W π (cid:18) µm W (cid:19) (cid:15) Γ[ (cid:15) ] (cid:18) η + log (cid:0) ν s (cid:1)(cid:19) , (A.24)where the second line is expressed in Laplace space. Extracting the LL anomalous dimen-sions from these results yields Eq. (A.15). • S (cid:48) and S (cid:48) are defined as S (cid:48) aba (cid:48) b (cid:48) = (cid:88) X US (cid:68) (cid:12)(cid:12)(cid:12) (cid:16) Y f (cid:48) n Y dg (cid:48) ¯ n (cid:17) † (0) δ (cid:0) q + − P + (cid:1)(cid:12)(cid:12)(cid:12) X US (cid:69) × (cid:68) X US (cid:12)(cid:12)(cid:12) (cid:16) Y gn Y df ¯ n Y agv Y bfv (cid:17) (0) (cid:12)(cid:12)(cid:12) (cid:69) δ f (cid:48) g (cid:48) δ a (cid:48) b (cid:48) ,S (cid:48) aba (cid:48) b (cid:48) = S (cid:48) a (cid:48) b (cid:48) ab . (A.25)Evaluating at tree-level in Laplace space yields (cid:16) S (cid:48) aba (cid:48) b (cid:48) (cid:17) tree = δ b δ a δ a (cid:48) b (cid:48) , (A.26)and at one-loop yields (cid:16) S (cid:48) aba (cid:48) b (cid:48) (cid:17) = − δ a (cid:48) b (cid:48) (cid:104)(cid:0) − δ a δ b − δ ab (cid:1)(cid:16) I Vn¯n − I Rn¯n (cid:17) + (cid:0) δ ab − δ a δ b (cid:1)(cid:16) I Vnv + I Rnv + I V¯nv − I R¯nv (cid:17)(cid:105) – 60 – −−−→
Laplace − δ a (cid:48) b (cid:48) δ a δ b C A α W π µ (cid:15) Γ[ (cid:15) ] (cid:18) η + log (cid:0) ν s (cid:1)(cid:19) + δ a (cid:48) b (cid:48) (cid:0) δ ab − δ a δ b (cid:1) α W π log (cid:0) m W s (cid:1) , (A.27)where the second line is expressed in Laplace space. Extracting the LL anomalous dimen-sions from these results yields Eq. (A.15).This result manifests the same UV virtuality and rapidity divergences as in the case ofthe S (cid:48) operator which is why it yields the same anomalous dimension as S (cid:48) . However,we see an additional IR divergence appears in the form of log (cid:0) m W s (cid:1) . This results fromthe non-singlet nature of this operator. In order to factorize this new double log, we needto match this ultrasoft operator onto an EFT below the scale s . This allows us to separatethe scales s and m W , yielding our final fully factorized result. • S (cid:48) is defined as S (cid:48) aba (cid:48) b (cid:48) = (cid:88) X US (cid:68) (cid:12)(cid:12)(cid:12) (cid:16) Y f (cid:48) n Y dg (cid:48) ¯ n Y a (cid:48) f (cid:48) v Y b (cid:48) g (cid:48) v (cid:17) † (0) δ (cid:0) q + − P + (cid:1)(cid:12)(cid:12)(cid:12) X US (cid:69) × (cid:68) X US (cid:12)(cid:12)(cid:12) (cid:16) Y fn Y dg ¯ n Y afv Y bgv (cid:17) (0) (cid:12)(cid:12)(cid:12) (cid:69) . (A.28)Evaluating at tree-level in Laplace space yields (cid:16) S (cid:48) aba (cid:48) b (cid:48) (cid:17) tree = δ a δ a (cid:48) δ bb (cid:48) , (A.29)and at one-loop yields (cid:16) S (cid:48) aba (cid:48) b (cid:48) (cid:17) = (cid:16) − δ a δ b δ a (cid:48) b (cid:48) + δ a δ b (cid:48) δ a (cid:48) b − δ a (cid:48) δ b (cid:48) δ ab + δ b δ a (cid:48) δ ab (cid:48) (cid:17) (cid:16) I Vn¯n − I Rn¯n (cid:17) + (cid:16) δ a (cid:110) − δ a (cid:48) δ bb (cid:48) − δ b (cid:48) δ a (cid:48) b + δ b δ a (cid:48) b (cid:48) (cid:111) + δ a (cid:48) (cid:110) − δ a δ bb (cid:48) − δ b δ ab (cid:48) + δ b (cid:48) δ ab (cid:111)(cid:17)(cid:16) I V¯nv − I R¯nv + I Vnv (cid:17) −−−−→
Laplace C A δ a δ a (cid:48) δ bb (cid:48) α W π µ (cid:15) Γ[ (cid:15) ] (cid:18) η + log (cid:0) ν s (cid:1)(cid:19) + δ bb (cid:48) (cid:0) δ aa (cid:48) − δ a δ a (cid:48) (cid:1) α W π log (cid:0) m W s (cid:1) . (A.30)where the second line is expressed in Laplace space. Extracting the LL anomalous dimen-sions from these results yields Eq. (A.15). Note that although the result appears not to besymmetric in the color structure, the wavefunction F a (cid:48) b (cid:48) ab defined in Eq. (5.24) is symmetricunder the interchange a, a (cid:48) ↔ b, b (cid:48) . – 61 – .2 Calculations in the Refactorized Theory Having presented the calculations for the anomalous dimensions in the intermediate EFT, inthis section we discuss some details related to the refactorization that were skipped in thetext, and present the anomalous dimensions in the refactorized theory.
Photon Jet Function
The photon jet function J γ is only sensitive to a single scale m W , and therefore is unmod-ified under the refactorization procedure. Recoiling Jet Function
As discussed in Sec. A.1, although in the intermediate theory the recoiling jet function issensitive both to the scale M χ √ − z set by the final state measurement, as well as to the scale m W , the final state measurement regulates all singularities, and could therefore be expandedto begin with. Combining this result with the structure of the factorization J (cid:48) ¯ n (cid:0) M χ , √ − z, m W , µ (cid:1) = H J ¯ n (cid:0) M χ , √ − z, µ (cid:1) J ¯ n (cid:0) m W , µ, ν (cid:1) + O (cid:18) m W M χ √ − z (cid:19) , (A.31)we find that the one-loop result for the matching coefficient in Laplace space ( M χ √ − z → s )is given by H J ¯ n ( M χ , s, µ ) = 12 M χ + 2 C A α W π M χ log (cid:18) µ s e γ E M χ (cid:19) + O (cid:0) α W (cid:1) . (A.32)This then implies thatdd log µ J (cid:48) ¯ n (cid:0) M χ , √ − z, m W , µ (cid:1) = dd log µ H J ¯ n (cid:0) M χ , √ − z, µ (cid:1) , (A.33)which is given in Eq. (A.14), and dd log µ J ¯ n (cid:0) m W , µ, ν (cid:1) = 0 . (A.34)To the order that we work, we need just the tree level value for J ¯ n , which is J tree ¯ n (cid:0) m W , µ, ν ) = 1 . (A.35)– 62 – nomalous Dimensions for the Refactorized Ultrasoft Function Unlike for the jet functions, the refactorization of the ultrasoft function is significantlymore involved. As given in the text, the general form of the refactorization is S (cid:48) aba (cid:48) b (cid:48) i (cid:0) M χ , − z, m W , µ, ν (cid:1) = H S,ij (cid:0) M χ , − z, µ (cid:1) (cid:104) C S (cid:0) M χ , − z, m W , µ, ν (cid:1) S (cid:0) m W , µ (cid:1)(cid:105) aba (cid:48) b (cid:48) j × (cid:20) O (cid:18) m W M χ (1 − z ) (cid:19)(cid:21) . (A.36)The goal of this section will be to describe this refactorization in more details, and derive therequired anomalous dimensions.Before considering the structure of the anomalous dimensions, we must first derive thecolor structures of the collinear-soft and soft functions, which were only stated without deriva-tion in the main body of the text. The structure of the Wilson lines in the soft and collinear-softfunctions can be derived by performing the BPS field redefinition iteratively. We thereforereturn to the two amplitude level operators (see Eq. (4.2) above) O = (cid:16) χ aTv iσ χ bv (cid:17) B cn ⊥ B d ¯ n ⊥ δ ab δ cd , O = (cid:16) χ aTv iσ χ bv (cid:17) B cn ⊥ B d ¯ n ⊥ δ ac δ bd . (A.37)Next we iteratively perform the BPS field redefinition for both the collinear-soft modes andrefactorized soft modes, O = (cid:104) δ AB V Dcn X Ccn (cid:105) (cid:104) S ¯ AAv S ¯ BBv S ¯ DD ¯ n S ¯ CCn (cid:105) (cid:16) χ ¯ ATv iσ χ ¯ Bv (cid:17) B ¯ Cn ⊥ B ¯ D ¯ n ⊥ , O = (cid:104) δ BD V Acn X Ccn (cid:105) (cid:104) S ¯ AAv S ¯ BBv S ¯ DD ¯ n S ¯ CCn (cid:105) (cid:16) χ ¯ ATv iσ χ ¯ Bv (cid:17) B ¯ Cn ⊥ B ¯ D ¯ n ⊥ . (A.38)We can now derive the soft and collinear-soft functions in the standard way, by squaring theamplitude level operators and setting ¯ D = ¯ D (cid:48) , ¯ C = ¯ C (cid:48) = 3 . We find ˜ S = (cid:68) (cid:12)(cid:12)(cid:12) (cid:104) V dcn X Ccn (cid:105) (cid:104) V dc (cid:48) n X C (cid:48) c (cid:48) n (cid:105) × (cid:104) S C (cid:48) n S Cn (cid:105) δ ¯ A ¯ B δ ¯ A (cid:48) ¯ B (cid:48) (cid:12)(cid:12)(cid:12) (cid:69) , ˜ S + ˜ S = (cid:68) (cid:12)(cid:12)(cid:12) (cid:104) V B (cid:48) cn X Ccn (cid:105) (cid:104) V A (cid:48) c (cid:48) n X C (cid:48) c (cid:48) n (cid:105) × (cid:104) S Cn S ¯ A (cid:48) A (cid:48) v S ¯ B (cid:48) B (cid:48) v S C (cid:48) n (cid:105) δ ¯ A ¯ B + (cid:8) ¯ A, ¯ B ↔ ¯ A (cid:48) , ¯ B (cid:48) (cid:9)(cid:12)(cid:12)(cid:12) (cid:69) , ˜ S = (cid:68) (cid:12)(cid:12)(cid:12) (cid:104) V B (cid:48) cn X Ccn (cid:105) (cid:104) V A (cid:48) c (cid:48) n X C (cid:48) c (cid:48) n (cid:105) × (cid:104) S Cn S ¯ A (cid:48) A (cid:48) v S ¯ AB (cid:48) v S C (cid:48) n (cid:105) δ ¯ B ¯ B (cid:48) (cid:12)(cid:12)(cid:12) (cid:69) . (A.39)To simplify the notation and focus on the color structures, we have suppressed the measure-ment function.One additional complication that arises in the refactorization of the ultrasoft function, isthat there are non-trivial zero-bins [106] that must be correctly incorporated. We therefore– 63 –riefly discuss the structure of these zero-bins, and their dependence on our choice of regulator,showing through two examples how the factorization correctly reproduces the structure ofintegrands once the zero bin is included. We consider one example of a virtual integral andone example of a real integral, arising from the S (cid:48) integrand. • Consider the virtual integral (see Eq. (A.17) for the evaluation of the unexpanded integral) I Vn¯n = δ (cid:0) q + (cid:1) g W µ (cid:15) ν η/ (cid:90) d d (cid:96) (2 π ) d − i ( (cid:96) − m W + i (cid:12)(cid:12) (cid:96) + − (cid:96) − | − η/ ( (cid:96) + + i
0) ( (cid:96) − − i . (A.40)Let us now consider the collinear-soft limit ( (cid:96) + (cid:29) (cid:96) − ) of this integral. It would appear thataccording to the power counting the only effect is to drop (cid:96) − from the rapidity regulatorterm | (cid:96) + − (cid:96) − | η/ . Since the rest of the integrand is unchanged, this would lead to an un-regulated divergence as (cid:96) − → ∞ . We would then be forced to introduce a new regulator tocounter this divergence. While there are several ways to do this (a ∆ -regulator [159], forinstance), the simplest way is just to keep the original form of the rapidity regulator. Thechoice of the regulator we use will affect the zero-bin subtraction that will be needed.If we do not expand out the regulator, then the collinear-soft and soft limits of Eq. (A.40) areidentical to the full US integral. The soft-bin subtraction is implemented in the collinear-soft(CS) sector by subtracting out the soft limit of the CS integral. With this subtraction I V , CSn¯n = 0 , I V , Sn¯n = I Vn¯n , (A.41)so that we recover the full US virtual contribution. • Now, consider the real emission integral (see Eq. (A.16) for the evaluation of the unexpandedintegral) I Rn¯n = − g W (cid:90) d d (cid:96) (2 π ) d − δ + (cid:0) (cid:96) − m W (cid:1)(cid:12)(cid:12) (cid:96) + − (cid:96) − (cid:12)(cid:12) − η/ δ (cid:0) q + − (cid:96) + (cid:1) (cid:96) + (cid:96) − . (A.42)The soft limit is I R , Sn¯n = − δ (cid:0) q + (cid:1) g W (cid:90) d d (cid:96) (2 π ) d − δ + (cid:0) (cid:96) − m W (cid:1)(cid:12)(cid:12) (cid:96) + − (cid:96) − (cid:12)(cid:12) − η/ (cid:96) + (cid:96) − . (A.43)The CS limit is identical to the full integral. Applying the zero-bin subtraction to this(which turns out to be the same as the soft integral), we are left with I R , CSn¯n = I Rn¯n − I R , Sn¯n . (A.44)– 64 –hus, once again we recover the full US real contribution adding together the CS and softlimits. The zero-bin subtraction scheme then is to simply subtract the soft limit of the CSintegrals from the CS sector.The analysis of these integrals provides a non-trivial check that our factorization is indeedcorrect, and that the infrared is completely reproduced by our factorized description.Having understood the operator basis and the structure of the zero-bin subtractions, wecan now compute the anomalous dimensions of the functions arising after the factorization ofthe ultrasoft function. Here we can considerably simplify the calculation by using the choiceof resummation path described in Sec. 5.2 and shown in Fig. 6. In particular, for this pathit is not necessary to separately run the collinear-soft and soft functions. We can thereforesimplify our refactorization to S aba (cid:48) b (cid:48) ij = H S,ijkl (cid:16) C AS,k S Bl (cid:17) aba (cid:48) b (cid:48) = H S,ijkl (cid:16) ˜ S kl (cid:17) aba (cid:48) b (cid:48) , (A.45)and only compute the anomalous dimensions for the functions H S,ijkl and (cid:0) ˜ S kl (cid:1) aba (cid:48) b (cid:48) . Thisdrastically simplifies the calculation, since the structure of the color mixing for the collinear-soft and soft operators is quite involved. In the remainder of this appendix we give the explicitresults for the anomalous dimensions for H S,ijkl and (cid:0) ˜ S kl (cid:1) aba (cid:48) b (cid:48) for all relevant color channelsappearing in our factorization.For ease of notation, as in the body of the text, we will define our ultrasoft operators as,see Eq. (4.12), S (cid:48) ≡ S (cid:48) S (cid:48) ≡ S (cid:48) , S (cid:48) ≡ S (cid:48) + S (cid:48) . (A.46)In this notation, the refactorization of the ultrasoft function is given by, see Eq. (4.26), S (cid:48) aba (cid:48) b (cid:48) i = H S,ikl (cid:0) C AS,k S Bl (cid:1) aba (cid:48) b (cid:48) = H S,ij (cid:16) ˜ S j (cid:17) aba (cid:48) b (cid:48) . (A.47)The tree-level, and one-loop results, along with the µ and ν anomalous dimensions for the H and ˜ S functions appearing in the factorization are as follows: • ˜ S is defined as ˜ S aba (cid:48) b (cid:48) = (cid:88) X cS (cid:68) (cid:12)(cid:12)(cid:12) (cid:16) X f (cid:48) n V dg (cid:48) n (cid:17) † (0) (cid:12)(cid:12)(cid:12) X c S (cid:69) × (cid:68) X c S (cid:12)(cid:12)(cid:12) δ (cid:0) q + − P + (cid:1) (cid:16) X fn V dgn (cid:17) (0) (cid:12)(cid:12)(cid:12) (cid:69) δ f (cid:48) g (cid:48) δ a (cid:48) b (cid:48) δ fg δ ab , (A.48)where the soft sector Wilson lines have contracted to the identity. By inspection, theanomalous dimension for this operator is identical to S aba (cid:48) b (cid:48) , implying that H S, = 1 is theonly non-zero matching coefficient. – 65 – ˜ S is defined as ˜ S aba (cid:48) b (cid:48) = (cid:88) X cS (cid:68) (cid:12)(cid:12)(cid:12) (cid:104) X cen V B (cid:48) en δ ( q + − P + ) (cid:12)(cid:12)(cid:12) X c S (cid:69)(cid:68) X c S (cid:12)(cid:12)(cid:12) X c (cid:48) g (cid:48) n V A (cid:48) g (cid:48) n (cid:105) (cid:104) S cn S c (cid:48) n S a (cid:48) A (cid:48) v S b (cid:48) B (cid:48) v (cid:105) δ ab (cid:12)(cid:12)(cid:12) (cid:69) + (cid:0) a, b ↔ a (cid:48) , b (cid:48) (cid:1) . (A.49)At tree-level in Laplace space we have (cid:16) ˜ S aba (cid:48) b (cid:48) (cid:17) tree = δ a (cid:48) δ b (cid:48) δ ab + (cid:0) a, b ↔ a (cid:48) , b (cid:48) (cid:1) , (A.50)and at one-loop in Laplace space, we have (cid:16) ˜ S aba (cid:48) b (cid:48) (cid:17) = − δ a (cid:48) δ b (cid:48) δ ab α W π (cid:18) µm W (cid:19) (cid:15) (cid:15) log (cid:0) ν s (cid:1) + δ ab (cid:0) δ a (cid:48) b (cid:48) − δ a (cid:48) δ b (cid:48) (cid:1) α W π (cid:18) − (cid:18) µm W (cid:19) log (cid:0) µ s (cid:1) + log (cid:18) µm W (cid:19)(cid:19) + (cid:0) a, b ↔ a (cid:48) , b (cid:48) (cid:1) . (A.51)The second line here is essentially the IR piece of the term log (cid:0) m W s (cid:1) . Extracting the LLanomalous dimensions yieldsdd log µ ˜ S aba (cid:48) b (cid:48) = (cid:16) − C A α W π log (cid:0) ν s (cid:1) + 3 C A α W π log (cid:0) µ s (cid:1)(cid:17) ˜ S aba (cid:48) b (cid:48) − C A α W π log (cid:0) µ s (cid:1) ˜ S aba (cid:48) b (cid:48) , (A.52)which shows a mixing between ˜ S and ˜ S , along withdd log ν ˜ S aba (cid:48) b (cid:48) = − C A α W π log (cid:18) µm W (cid:19) ˜ S aba (cid:48) b (cid:48) . (A.53)We can now read off the matching coefficients H S, = 1 − α W π log ( µ s ) ,H S, = 2 α W π log ( µ s ) , (A.54)which immediately tells us thatdd log µ H S, = − C A α W π log (cid:0) µ s (cid:1) H S, , dd log µ H S, = 2 C A α W π log (cid:0) µ s (cid:1) H S, . (A.55)– 66 – ˜ S is defined as ˜ S aba (cid:48) b (cid:48) = (cid:88) X cS (cid:68) (cid:12)(cid:12)(cid:12) (cid:104) X cen V Aen δ ( q + − P + ) (cid:12)(cid:12)(cid:12) X c S (cid:69)(cid:68) X c S (cid:12)(cid:12)(cid:12) X c (cid:48) g (cid:48) n V A (cid:48) g (cid:48) n (cid:105) (cid:104) S cn S c (cid:48) n S a (cid:48) A (cid:48) v S aAv (cid:105) δ bb (cid:48) (cid:12)(cid:12)(cid:12) (cid:69) . (A.56)At tree-level in Laplace space we have (cid:16) ˜ S aba (cid:48) b (cid:48) (cid:17) tree = δ bb (cid:48) δ a (cid:48) δ a , (A.57)and at one-loop in Laplace space, we have (cid:16) ˜ S aba (cid:48) b (cid:48) (cid:17) = − δ bb (cid:48) δ a (cid:48) δ a α W π (cid:18) µm W (cid:19) (cid:15) (cid:15) log (cid:0) ν s (cid:1) (A.58) + δ bb (cid:48) (cid:0) δ aa (cid:48) − δ a (cid:48) δ a (cid:1) α W π (cid:18) − (cid:18) µm W (cid:19) log (cid:0) µ s (cid:1) + log (cid:18) µm W (cid:19)(cid:19) . From the color structure of this result, it is clear that another operator has been inducedat loop level, namely ˜ S aba (cid:48) b (cid:48) = (cid:88) X cS (cid:68) (cid:12)(cid:12)(cid:12) (cid:16) X f (cid:48) n V df (cid:48) n (cid:17) † (0) (cid:12)(cid:12)(cid:12) X c S (cid:69)(cid:68) X c S (cid:12)(cid:12)(cid:12) δ (cid:0) q + − P + (cid:1) (cid:16) X fn V dfn (cid:17) (0) (cid:12)(cid:12)(cid:12) (cid:69) δ aa (cid:48) δ bb (cid:48) , (A.59)which is similar to ˜ S a (cid:48) b (cid:48) ab but with a different color structure. Evaluating this operator attree-level in Laplace space yields (cid:16) ˜ S aba (cid:48) b (cid:48) (cid:17) tree = δ aa (cid:48) δ bb (cid:48) , (A.60)and at one-loop in Laplace space yields (cid:16) ˜ S a (cid:48) b (cid:48) ab (cid:17) = − δ aa (cid:48) δ bb (cid:48) C A α W π (cid:18) µm W (cid:19) (cid:15) (cid:15) (cid:18) η + log (cid:0) ν s (cid:1)(cid:19) . (A.61)Recall that the matching coefficient for this operator is at tree-level, since it did not appearin our original basis. Extracting the LL anomalous dimensions for this operator yieldsdd log µ ˜ S aba (cid:48) b (cid:48) = − C A α W π log (cid:0) ν s (cid:1) ˜ S aba (cid:48) b (cid:48) , dd log ν ˜ S aba (cid:48) b (cid:48) = − C A α W π log (cid:18) µm W (cid:19) ˜ S aba (cid:48) b (cid:48) . (A.62)– 67 –e can use these results to extract the anomalous dimension for ˜ S ,dd log µ ˜ S aba (cid:48) b (cid:48) = (cid:16) − C A α W π log (cid:0) ν s (cid:1) + 3 C A α W π log (cid:0) µ s (cid:1)(cid:17) ˜ S aba (cid:48) b (cid:48) − C A α W π log( µ s ) ˜ S aba (cid:48) b (cid:48) , dd log ν ˜ S aba (cid:48) b (cid:48) = − C A α W π log (cid:18) µm W (cid:19) ˜ S aba (cid:48) b (cid:48) . (A.63)We can then extract the Wilson coefficients, H S, = 1 − α W π log (cid:0) µ s (cid:1) ,H S, = α W π log (cid:0) µ s (cid:1) , (A.64)and their anomalous dimensionsdd log µ H S, = − α W π log (cid:0) µ s (cid:1) H S, , dd log µ H S, = 2 α W π log (cid:0) µ s (cid:1) H S, . (A.65)This provides the complete set of ingredients required for the LL resummation in theendpoint region. B Impact of Continuum Photons from Cascade Decays
In the main body of this work we presented a calculation of the internal bremsstrahlung (+initial/final state radiation), or endpoint, contribution to the wino annihilation spectrum. Aswe mentioned there, another source of photons arises from the final state decay products ofthe unstable particles that are produced by DM annihilations, such as the W ± and Z bosons.In this appendix we estimate the contribution from these additional final states, and showthat they have a small impact on the HESS constraints for the thermal wino. However, theycould be interesting for instruments searching for lower energy photons such as Fermi .In order to estimate these contributions, we have added to the line and endpoint spectrathe spectrum coming from the decay of W ± and Z bosons. The spectrum of photons thatarises from their decay is determined using PPPC4DMID [160] with electroweak correctionsturned off, whereas the branching fraction is evaluated differently for the two cases. For The electroweak corrections in PPPC4DMID include a partial accounting of the endpoint corrections thatwe determined in the main body, which they include following [161], and so we remove them to avoid doublecounting. We thank Marco Cirelli for confirming this point. This choice means we are missing the electroweakcorrections from the remainder of the continuum spectrum, however we have confirmed these effects are small – 68 – . . . . E γ [TeV] − − − E . d N / d E [ T e V . / m / s / s r ] Spectra M χ = 3 TeV
Line+ endpoint+ continuum (a) . . . E γ [TeV] − − − E . d N / d E [ T e V . / m / s / s r ] Spectra M χ = 10 TeV
Line+ endpoint+ continuum (b)
Figure 12 . The differential photon flux observed at HESS for the wino at (a) 3 TeV; and (b) 10 TeV.In each case we show, progressively, the contribution from the line only case, the endpoint contributioncalculated in the main body, and finally the continuum arising from the decay of the produced W and Z bosons. In all cases the contributions have been smeared by the HESS energy resolution. by directly comparing the spectra to the predictions of Pythia – 69 – . . . . . M χ [TeV] − − − − h σ v i li n e [ c m / s ] Estimated Limits
Line (HESS Published)Line (this ref)+ endpoint (this ref)+ continuum (this ref)LL h σv i line (a) . . . . . M χ [TeV] J - f a c t o r [ T e V / c m ] Estimated J-factor Constraint
Line (this ref)+ endpoint (this ref)+ continuum (this ref) (b)
Figure 13 . As in Figs. 9 and 10, but showing the impact of adding the continuum contribution from W and Z decays in addition to the endpoint on the constraints. In general these contributions have amuch smaller impact than that already resulting from adding in the endpoint spectrum. We cautiononce more that these are only estimated limits. – 70 –nnihilation to W + W − , the branching fraction is given by the Sommerfeld-enhanced tree-level cross section for this final state [3, 4]. Radiative corrections to this cross section,which have been shown to be small [40], are not included. To determine the Z productioncross section, we use the leading log cross section, which is given by Eq. (5.42) reweighted by c W /s W .In Fig. 12, we show the impact on the photon spectrum from DM, after convolving it withthe HESS energy resolution, when this continuum contribution is added, for two DM masses.Generically, as we approach E γ ∼ M χ , this continuum emission is a sub-dominant effect.However, at lower energies it can have substantial impact (note this spectrum is multipliedby E . which downweights the flux at lower energies). Nevertheless, such a contributionover many energy bins is hard to distinguish from the 7 parameter background model usedby HESS. These background parameters are profiled over, so that we would not expect thisadditional emission to make a sizable impact. Indeed, in Fig. 13 we demonstrate this point, byrepeating the analysis from Sec. 7.3 with the inclusion of the additional continuum photons.We note the effect of including the continuum becomes more important at higher masses, butis almost always subdominant to the impact of adding in the endpoint emission. Further, thebroad nature of the continuum emission can lead to a non-trivial interplay with the backgroundmodel in fits to the data, and in fact lead to weaker limits for some masses. For example, near M χ = 20 TeV in Fig. 13, the additional continuum emission at lower energies drives down thebest fit background model, resulting in a reduced background prediction near the dark mattermass where the line and endpoint contributions dominate, and accordingly a weaker limit.Finally we note in passing that the large contribution from the continuum may be relevantto lower energy instruments such as
Fermi -LAT. The advantage of such an approach is that wecan look at a number of different potential astrophysical sources of DM flux, each associatedwith partially uncorrelated systematics on their J -factors. In this way we can extend thesearch beyond the Galactic Center and its large uncertainties to look at potentially cleanerenvironments such as the Milky Way Dwarfs [166, 167] or even galaxy clusters [142, 168].However, note that the effective area of Fermi -LAT drops sharply at TeV energies. Thisimplies that if the DM mass is multi-TeV, the HESS constraints are generally stronger thanthose from
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