Indirect Detection of Self-Interacting Asymmetric Dark Matter
IIndirect Detection of Self-Interacting Asymmetric Dark Matter
Lauren Pearce
Department of Physics and Astronomy, University of California, Los Angeles, CA 90095-1547, USA
Alexander Kusenko
Department of Physics and Astronomy, University of California, Los Angeles, CA 90095-1547, USA andKavli IPMU (WPI), University of Tokyo, Kashiwa, Chiba 277-8568, Japan
Self-interacting dark matter resolves the issue of cuspy profiles that appear in non-interactingcold dark matter simluations; it may additionally resolve the so-called “too big to fail” problemin structure formation. Asymmetric dark matter provides a natural explanation of the comparabledensities of baryonic matter and dark matter. In this paper, we discuss unique indirect detectionsignals produced by a minimal model of self-interacting asymmetric scalar dark matter. Throughthe formation of dark matter bound states, a dark force mediator particle may be emitted; the decayof this particle may produce an observable signal. We estimate the produced signal and explicitlydemonstrate parameters for which the signal exceeds current observations.
I. INTRODUCTION
A number of scenarios have been put forth for ex-plaining why the amounts of dark matter and ordinarymatter are relatively close to each other, within oneorder of magnitude. One of the popular approachesis to consider dark matter with a conserved particlenumber and a particle-antiparticle asymmetry related tomatter-antimatter asymmetry [1–21]. (For a recent re-view, see, e.g., Ref. [22].) Such asymmetric dark matterdoes not annihilate at present, and, as long as it is sta-ble, no indirect detection signals are expected in gammarays or neutrinos. At the same time, some inconsisten-cies between numerical simulations of cold dark matter(CDM) and the observations hint at the possibility of self-interacting dark matter [23–36]. Indeed, interactions ofdark-matter particles can facilitate the momentum trans-fer and angular momentum transfer in halos, hence cre-ating cored rather than cuspy density profiles in bothdwarf spheroidal galaxies and in larger halos. There is avariety of particle-physics candidates for self-interactingdark matter, which include, e.g., hidden-sector particleswith gauge [30], or Yukawa interactions [37], as well asnon-topological solitons with a large enough geometricalsize [26].In this paper we will show that, if dark matter isboth asymmetric and self-interacting, then its detectionin gamma rays is possible. Although asymmetric darkmatter particles do not annihilate, their interactions canresult in emission of quanta of the field that mediatesthe self-interaction. We will investigate several methodsof producing these quanta and decay paths; on the pro-duction side, we particularly consider emissions occuringin elastic scattering (as bremsstrahlung), or in the eventsof two interacting particles forming a bound state. Thelatter is plausible because a number of models have em-ployed attractive self-interaction, such as, for exampleYukawa fields.To illustrate the possibilities of indirect detection, wewill consider a fairly generic model of scalar dark matter S interacting by means of exchange of some lighter scalarfield σ , both of which are singlets of the Standard Modelgauge group. The mediator field σ can have a nonzeromixing with the Higgs boson, and, therefore the σ bosoncan decay into photons and other Standard Model par-ticles, even if its coupling to Standard Model particlesare otherwise highly suppressed. This decay ultimatelyproduces the signal detectable by gamma-ray telescopes.The paper is organized as follows. In section II, we dis-cuss the relevant features of the particle physics modelunder consideration. In section III, we consider thebound states formation and the prospects of indirect de-tection of dark matter forming bound states. Finally,in section IV, we discuss possible signals from bremm-strahlung of self-interacting dark matter. II. THE MODEL
We begin this section by introducing the minimal par-ticle physics model that we will use. Following this, wediscuss properties of the dark matter halo within theMilky Way. We then discuss the relevant constraints onthe parameters present in the model in general, althoughdetailed discussion of the implementation of these con-straints is contained in later sections where the signalfrom bremsstrahlung and bound state formation is ex-plicitly calculated. Then finally we discuss the decays ofthe dark force mediator.
A. Dark Sector
We begin with a specific model of the relevant parti-cle physics. We supplement the Standard Model with acomplex scalar SU C (3) × SU L (2) × U Y (1) singlet S ; wealso introduce a global U S (1) symmetry under which S → e iα S S † → e − iα S † . (1) a r X i v : . [ h e p - ph ] M a y Without a loss of generality we may assume S particlescarry unit U S (1) charge. Due to charge conservation, the S particles are completely stable.We will assume that dark matter is composed of the S particles, and the correct abundance is generated insome process similar to or combined with baryogenesis. We do not assume that dark matter is necessarily a ther-mal relic. To make this dark matter self-interacting, weintroduce an additional scalar field σ which is a singletunder all the gauge symmetries, as well as U S (1). Themost general potential, after the Standard Model gaugesymmetry is spontaneously broken, is V = M h σ + m S S † S + m σ σ + m h h ) + A h ( h ) + A σ σ + A σS S † Sσ + A σh ( h ) σ + A hσ σ h + A hS S † Sh + λ S ( S † S ) + λ σ σ + λ hS S † S ( h ) + λ h h ) + λ σS σ S † S + λ σh σ ( h ) . (2)The Higgs field h and the σ field carry identical quan-tum numbers and therefore mix. The true mass eigen-values are m , = 12 (cid:18) m h + m σ ± (cid:113) ( m h − m σ ) + M (cid:19) (3)and the eigenstates are φ = cos( θ M / h + sin( θ M / σ (4) φ = − sin( θ M / h + cos( θ M / σ (5)where the mixing angle istan( θ M ) = M m h − m σ . (6)We will require that the mixing between the Higgs fieldand the σ field be small; this can be accomplished bysetting the free parameter M appropriately. Then wemay speak of the σ fields and Higgs fields as approximatemass eigenstates with masses m h and m σ respectively;this allows the mass m σ to be small even though no lightscalar boson has been observed.We assume that any interactions between the dark sec-tor particles ( S , S † , and σ ) and the particles of the Stan-dard Model are highly suppressed.Finally, we note that the relevant unitless coupling todescribe the Yukawa interaction between the S and σ particles is α = A σS / πm S . This can be established intwo ways. Regarding bound states, it is well-known thatthe Bethe-Salpeter equation reproduces the ground stateenergy of the hydrogen atom. Therefore, one may deter-mine α by setting the binding energy of the lowest boundstate, A σS / π m S , equal to α m S /
4, which includesthe correction for identical particles. Secondly, one mayconsider the non-relativistic limit of two particle scat-tering. We recall that quantum-field-theoretic wavefunc-tions include a normalization factor of 1 / √ m S for each S particle. Therefore, the relevant prefactor before theoverlap integral for one particle exchange is 4 πα = A σS / m S , which again gives α = A σS / πm S . If one insteaddefines α = A σS / πm S , as in [38] for example, then addi-tional factors of 4 must be introduced in other equations,e.g., the bound state mass. B. Dark Matter in the Milky Way
Let us now discuss the assumptions that we will makeregarding the properties of dark matter in the Milky Wayhalo. First, we assume that the correct abundance of S particles is determined by some process that is similarto baryogenesis or related to baryogenesis, as in modelsreviewed in Ref. [21]. The absence of antiparticles intoday’s universe eliminates the possibility of a signal from SS † annihilation.We use the Navarro-Frenk-White profile [39] to approx-imate the spatial mass distribution of dark matter, ρ ( r ) = ρ ( r/R s )(1 + r/R s ) . (7)We do not expect this profile to be accurate near thecenter of the galaxy; indeed, one of the motivations ofself-interacting dark matter is to remove the cusp presentat r = 0 in the NFW profile. Therefore, we will cut offour intergrals at scales of 1 kpc. We emphasize that ourresults are not dependent on the sharp cusp present inthe NFW profile. The parameters ρ and R s are relatedto the virial mass, virial radius, and concentration by R s = r vir Cρ = M vir ln(1 + C ) − C/ (1 + C ) 14 πR s . (8)For the Milky Way, we use the parameters M vir = 1 . · M (cid:12) , r vir = 258 kpc, and C = 12 [40]. This gives R s = 3 . · GeV − and ρ = 1 . · − GeV .In calculating the cross sections for bremsstrahlungemission of σ particles and bound state formation, wewill need to average over the relative velocities of theparticles. Therefore, we need the velocity distribution P ( v ( r )) as a function of the distance from the center ofthe galaxy. If the dark matter has virialized, then its av-erage circular velocity should decrease near the galacticcenter, except for a small region near the supermassiveblack hole. However, the dark matter radial velocity pro-file and dispersion are currently unknown.Because of these uncertainties, we will instead use aMaxwellian distribution with the effective temperature T eff chosen such that the average velocity is 220 km / s.We note that simulations support the assumption of alocally Gaussian velocity distribution even for cold darkmatter [41], and the isothermal approximation is betterfor self-interacting dark matter [42], [43]. The velocitydistribution for two non-relativistic S particles is P ( v , v ) dv dv = (4 π ) (cid:18) m S πT eff (cid:19) e − m ( v + v ) / T eff v v dv dv . (9)In terms of the total velocity v T = v + v and therelative velocity v rel = v − v , the distribution is P ( v rel , v T ) dv rel dv T = (4 π ) (cid:18) m S πT eff (cid:19) · e − m S ( v + v T ) / T eff v v T dv rel dv T . (10)We integrate over the total velocity to find the relativevelocity distribution (in a reference frame at rest withrespect to the Milky Way). P ( v rel ) dv rel = 4 π √ (cid:18) m S πT eff (cid:19) / e − m S v / T eff v dv rel . (11)Because the S particles are moving non-relativistically,this distribution also applies to their center of momen-tum frame. We observe that this is peaked at slightlylarger velocities than the velocity distribution of a singleparticle. C. A General Discussion of Constraints
Thus far, we have introduced a model which providesa viable dark matter candidate. We introduced severalparameters in the Lagrangian describing our model (e.g.,the self-interaction coupling). These parameters cannotbe set arbitrarily; there are numerous constraints theymust satisfy, from both astrophysics and particle physics.In this subsection, we will give only a general discussionof these constraints; the specifics of how the constraintsare implemented will be discussed when particular valuesfor the coupling constants are chosen, which will be doneseperately for bremsstrahlung emission and bound stateformation. Since we aim to demonstrate that this modelproduces an observable indirect detection signal, we de-mand that it satisfy all experimental constraints exceptthose from indirect detection experiments.First, we require that this model make only insignifi-cant modifications to the branching ratio for the decaysof the Higgs boson. We forbid the decay h → SS † byrequiring m S > m h / ≈
63 GeV, using the recent Higgsmass measurements [44], [45]. The decay h → σσ can be arbitrarily suppressed by taking A σh to be sufficientlysmall; this parameter is not used elsewhere in our analy-sis. We also demand that the mixing angle θ M be smallenough that the apparant branching ratio for h → σ isless than the branching ratio for the h → γγ decay.There are many well-known bounds on the self-interaction cross section of dark matter. As explainedin [37], these constraints more appropriately restrict σ T ,the momentum transfer cross section. (For identical par-ticles, the closely-related viscosity cross section should beused instead [46]. In the limit m S ¯ v/m σ (cid:29)
1, which willbe valid for our parameters, these differ by O (1) [46]and so we will ignore this complication.) The bulletcluster bound requires σ SS /m S < ∼ . / g [47], andbounds from the evaporation of galactic halos favor σ SS /m S < ∼ . / g [35]. These bounds appear to be in con-flict with the prefered range to eliminate cuspy profiles, .
56 cm / g < ∼ σ SS /m S < ∼ . / g [24, 25].However, because these bounds affect vastly differentscales, they may be resolved by considering a velocitydependent into the cross section, as naturally arises inthe Yukawa exchange of a light boson [48], [37]. Fur-thermore, such a cross section may additionally solve the“too big to fail” problem [34, 35, 49]. For an attrac-tive Yukawa potential, as we have introduced above, thebounds are consistent for Yukawa interactions providedthat the masses m S and m σ satisfy particular relationsgiven in [50],[51]. The precise constraint is a function of v max = (cid:112) αm σ /πm S , the velocity at which vσ T peaksat a transfer cross section equal to σ maxT = 22 . /m σ .Additional bounds on the self-interaction cross sec-tion arise from observations of halo ellipticity, as intro-duced in [52], although these bounds are quite model-dependent. Yukawa couplings are discussed in [31], whichuses the observed elliptical shape of the dark matter haloof galaxy NGC 720. If the self-interaction between the S particles is too strong, the energy transfer from these col-lisions makes the halo spherical instead of elliptical. Ref-erence [31] presented analyses for masses up to 4 TeV;however, we will consider masses above this. Further-more, as has been noted by [35], these bounds may infact be somewhat weaker due corrections from the triax-ial distribution of dark matter outside of the core; how-ever, as they note, more detailed simulations are requiredto firmly establish this conclusion. We discuss these is-sues in more detail in appendix A, in which we extendthe halo ellipticity bounds to the relevant mass range.Direct detection experiments such as XENON100 [53]and CDMS [54] have set an upper bound on the crosssection for the interaction between S particles and nucle-ons; because this interaction occurs through the exchangeof a Higgs boson, this constrains A hS . The stability ofneutron stars generally imposes stronger constraints on A hS [55–58] but these constraints do not apply to scalardark matter with masses at the TeV scale or above [59].We will not use A hS in our analysis; therefore, it can beset arbitrarily small. These constraints can also constrainthe quartic interaction between self-interacting dark mat-ter [60]; we may also set this aritrarily small because itwill not be used in our analysis. We note that while wecan arbitrarily suppress the S -nucleon interaction whichoccurs through the exchange of a Higgs boson, there isan additional diagram in which the S boson emits a σ boson, which turns into a Higgs boson via mixing, whichis then absorbed by the nucleon. Although we are notfree to arbitrarily suppress this diagram, as one mightexpect, this cross section is beneath current direct de-tection limits; we discuss it in more detail in AppendixB. D. Decays of the Dark Force Carrier Particle
From the constraints discussed above, we have seenthat the mass m σ must be relatively small. However,these dark force mediator particles are not necessary sta-ble, and their decays can potentially produce detectablesignals. Later in the paper we will discuss how these σ bosons are produced (for example, though bound stateformation or bremsstrahlung); in this section, we willsimply discuss their decays irrespective of their produc-tion. Due to the nonzero σ -Higgs mixing, a σ particlehas the same decay modes as the Higgs boson, providedthat they are kinematically allowed. The amplitudes aresuppressed by the σ -Higgs mixing parameters. Since themass m σ must be small, we consider the decays σ → γγ and σ → e + e − .For m σ ∼ a few MeV, the dominant decay is σ → e + e − . The decay rate in the rest frame of the σ boson isΓ e + e − = g W m e m σ sin ( θ M / πm W (cid:18) − m e m σ (cid:19) / (12)where g W is the weak coupling constant.If m σ < m e , the decay σ → e + e − is kinematicallyforbidden, and instead the dominant decay is σ → γγ .In the σ particle’s rest frame, the decay rate isΓ γγ = sin (cid:18) θ M (cid:19) α g W π m σ m W (cid:12)(cid:12)(cid:12)(cid:12) (cid:88) i N ci e i F i (cid:12)(cid:12)(cid:12)(cid:12) . (13)In this equation, α ≈ / g W is again the weakcoupling constant, N ci is the number of color states ofthe particle in the loop, and e i is this particle’s electriccharge. The dominant contributions to the loop will befrom electrons, up quarks, and down quarks, for which F = − τ (1 + (1 − τ ) f ( τ )) (14)where τ = 4 m i /m h , and f ( τ ) = (cid:16) arcsin (cid:16)(cid:112) /τ (cid:17)(cid:17) τ ≥ − (ln( η + /η − ) − ıπ ) τ < η ± = 1 ± √ − τ . We note that in a reference frame in which the σ particle is moving with speed v , its gamma factor is γ = (1 − v ) / and its lifetime is τ = γ/ Γ . For the product of decay to be observed, the parti-cles must decay in flight before they travel the distance ≈ σ bosons because the collisions are rare(see Appendix C).To determine the signal produced by our model, wealso need to know the width of the energy distributionof the decay products. The above decays are two bodydecays; therefore, in the rest frame of the σ particle theenergy spectrum of the decay products is a sharp line at m σ /
2. This energy spectrum must be boosted into theMilky Way reference frame, in which the σ particles aremoving with speed v = (cid:112) E σ − m σ /E σ . Because the σ boson is spinless, the energy distribution is flat.For decays to an electron and positron, the energy dis-tribution is P ( E e ) = 1 (cid:112) ( E σ − m σ )(1 − m e /m σ ) (16)for E e between the values of E e , max , E e , min = E σ ± (cid:112) ( E σ − m σ )(1 − m e /m σ )2 . Similarly, for decays to two photons, the energy distri-bution is P ( E γ ) = 1 (cid:112) E σ − m σ (17)for E e between the values of E γ, max , E γ, min = E σ ± (cid:112) E σ − m σ . III. DARK MATTER BOUND STATES
It has previously been observed that many modelsof self-interacting dark matter, including supersymmet-ric models, permit the existence of dark matter boundstates [38]. The same reference notes that the decay ofemitted force carrier particles could, in theory, producea signal for indirect detection experiments. Therefore,we will begin by explicitly calculating the produced sig-nal for the above asymmetric scalar dark matter model.We will establish that it is indeed possible to produce asignal above current observational bounds, establishingthe possibility of indirect detection of asymmetric self-interacting dark matter. However, we will further showthat the limit α (cid:28)
1, as taken in [38], does not producea detectable signal.
A. Choice of parameters
We begin by discussing in more detail the constraintsthat our parameters must satisfy. To facilitate the for-mation of bound states, we desire a large coupling A σS ;we will choose α = A σS / πm S = 2. Although calcu-lations in the strongly-interacting regime are notorouslydifficult, the astrophysical bounds determined in [31] and[50] may be extrapolated to these regimes; in both refer-ences, the transfer cross section used includes correctionsfor the strongly-interacting regime, as the authors note.We will also consider α = 1 and show that this is notsufficient to produce an observable signal.First, we ensure that our parameters are consistentwith ellpitical halos; as discussed in [31], the restrictionon α becomes weaker as m σ is increased. In AppendixA, we have determined the minimum m σ for which wemay consistently choose α = 2 as a function of m S . Wechoose m S = 4 TeV, which requires that we choose m σ > ∼
30 MeV; we will take m σ = 40 MeV. Similarly, for m S = 4 TeV and α = 1, we must satisfy m σ > ∼
20 MeV;we choose m σ = 25 MeV.We must also ensure that our parameters are consis-tent with the astrophysical data. The velocity for which vσ T = σ maxT is v max = (cid:112) αm σ /πm S ; this is 1100 km / sand 2000 km / s respectively for the two sets of parame-ters above. To be consistent with astrophysical data, wethen must have σ maxT /m S ≤
100 GeV − [50]; the abovenumbers correspond to . − and 9 GeV − .In order to produce a bound state, a real σ particlemust be emitted; therefore, we must also have m σ (cid:28) B ,where the binding energy B = α m S /
4. For the first setof parameters chosen, the binding energy is 4 TeV, andfor the second set of parameters, B = 1 TeV. For both, m σ (cid:28) B . B. Production of Bound States
The rate of formation of bound states, neglectingcharge depletion, is given by dN BS dt = (cid:90) n S ( r ) σ BS v rel dV (18)where n S ( r ) = ρ ( r ) /m S is the number density of S parti-cles and σ BS is the cross secton for bound state formation.Because the S particles do not escape to infinity, this can-not be approximated using the Born approximation. InAppendix D we present a calculation of this cross sectionas a function of the relative momentum | p | = µv rel where µ is the reduced mass of the system. (This cross sectionis calculated by adapting the derivation for positroniumformation given in [61] to the case for the exchange ofa spinless boson.) The distribution of relative momen-tum of the incoming particles can be found from equa-tion (11) and is given by equation (D19). Averaging the above equation over the relative momentum gives dN BS dt = (cid:90) n S ( r ) dV · (cid:90) (cid:90) | p rel | m S σ ( | p rel | ) P ( | p | ) d | p rel | . (19)The first set of parameters discussed above correspondsto a cross section of 4 . · − GeV, which gives the rate dN BS /dt = 2 . · GeV. In one year, 9 . · boundstates are formed, which means that during the lifetimeof the Milky Way, 1 . · would have formed. This isindeed negligible in comparison to the total number of S particles between 1 kpc and 8 kpc, which is 7 . · .This justifies our neglect of charge depletion.If we decrease α to 1, then the cross section drops bytwo orders of magnitude, to 5 . · − GeV − . Therate is also two orders of magnitude smaller, dN BS /dt =2 . · GeV. Again, we may neglect charge depletion. C. σ Boson Production and Decay
The S particles do not interact electromagnetically.Therefore, when a bound state is formed, the excess en-ergy is carried off by a light σ particle. Although thebinding energy is large enough that a Higgs boson couldbe emitted instead, the σ particle dominates because itis lighter and has a stronger coupling to the S particles.Due to the nonzero σ -Higgs mixing, this σ particle hasa non-zero probability to transform into a Higgs bosonwhich then decays. Given our choices for m σ , the domi-nant decay is σ → e + e − .In the rest frame of the Milky Way, the σ particleswill have a typical energy equal to the binding energy;the additional energy the σ particle may carry from thekinetic energy of the non-relativistic S particles is negligi-ble. Using equation (12), we find that the lifetime of the σ boson, in the Milky Way’s rest frame, is 1 . · GeV,or 9 .
61 s, for the first set of parameters. The distancethat they travel before decaying is 10 m, which is sig-nificantly less than both the distance from the galacticcenter to the solar system and the mean free path calcu-lated in Appendix C. For the second set of parameters,the lifetime is 9 . · GeV = 6 .
17 s.As explained in section II D, the resulting electrons andpositrons have a flat energy distribution; their spectrumis dN e dE e dt = 1 (cid:112) ( B − m σ )(1 − m e /m σ ) dN BS dt . (20)These electrons and photons have typical energies onor just below the TeV scale; they lose energy throughsynchrotron radiation and inverse Compton scatteringrapidly, within about 1 kpc [62]. Therefore, few of theseparticles will be observed near Earth. D. Scattering of High Energy Electrons andPositrons
There are three sources of background photons:the cosmic microwave background (CMB) radiation,starlight, and the starlight reprocessed by dust (includingthe extragalactic background light, which is the starlightre-emitted by dust outside Milky Way). Outside the cen-tral molecular zone, the cosmic microwave backgroundradiation dominates the photon number density [63]. Ifthe cross section exhibited a rapid growth with energy,the higher-energy optical photons could have played arole, but this is not the case for the IC cross section.For the signal from distances between 1 kpc and 8 kpcfrom the galactic center, one may safely neglect scatter-ing from photons other than CMB photons. Due to thesmallness of the IC mean free path, other propagationeffects are not significant.In Appendix E, we calculate the final energy distri-bution for a scattering photon using the Klein-Nishinacross section; we then average over the appropriate en-ergy distribution for the electrons and positrons producedby dark force mediator decays. The rate of the produc-tion of these photons is the rate of production of thehigh energy fermions themselves. The end result of thiscalculation is described by equation (E15), which givesthe number of photons produced by this process per unitenergy per unit time; from this, we can find the flux ofgamma rays at the solar system.The production of dark force mediator particles resultsin an isotropic flux of these particles about the galacticcenter; similarly, we expect the flux of their decay prod-ucts and the scattered photons to be isotropic about thegalactic center. Therefore, the photon flux per unit areais well-approximated by an equivalent point source atthe galactic center. We can find the average flux perunit area, per unit solid angle by further dividing by 2 π ,since the signal will appear to come from the hemispherecentered on the galactic center. We note that this is anaverage; as a function of solid angle, we expect the sig-nal to be greater near the galactic center and less furtheraway from it. We also note that this is only an approx-imation to the true diffuse flux, meant to demonstratethat a detectable signal is possible. (As discussed above,we neglect the contribution from the galactic center itselfand include only the contributions from decays outsidethe inner kiloparsec.) If such a signal were to be observed,more careful analysis should be done before attemptingto fit this scenario to the data.Furthermore, we note that the production of the darkforce mediator bosons σ scales as the density squared;this increases as one approaches the galactic center. Sincethe sigma bosons only travel 10 m before decaying intothe fermions which scatter the CMB photons, the signalis dominated by the innermost region we consider. Sincewe have cutoff our calculation at an inner radius of 1kpc to avoid the known cusp in the NFW profile, thesignal comes predominantly from the region near this cut. E Φ (cid:0) G e V / c m ss t (cid:1) Energy of Scattered Photons E (GeV) FERMI data α = 1 α = 2 FIG. 1. Depending on the model parameters, the signal canrange from undetectable to already excluded. The signal isshown for two values of α , as shown in the legend, and for m S = 4 TeV, m σ = 40 MeV for α = 2, m σ = 25 MeVfor α = 1. For comparison, the data from Fermi LAT spacetelescope are also shown [64]. Therefore, the point source approximation is better thanone may naively expect. (We remind our reader that dueto this cutoff, this calculation produces an approximatelower bound on the signal strength.)We find that the average flux over the hemisphere cen-tered on the galactic center, neglecting the galactic centeritself, is Φ = dN γ, tot dE dt · π st · π (8 kpc) . (21)To compare with the the sensitivity of the Fermi-LATGamma Ray Telescope, we evaluate E Φ; this functionis plotted in Fig. 1. The signal for α = 1 is peakedat a lower energy and falls off more sharply, as we wouldexpect because the binding energy is smaller. We see that α = 2 produces a signal that is one order of magnitudelarger than the values measured by Fermi-LAT, but thesignal produced by α = 1 is two orders of magnitudetoo small. Therefore, we conclude that sufficiently largecouplings may produce a detectable signal. This suggeststhat WIMPonium models [38], which assume α (cid:28) m σ , provided that σ → e + e − remains the dominant decay. However, this param-eter is highly constrained by the astrophysical boundsdiscussed in section II C. Finally, we also note that thissignal depends relatively weakly on the cutoff we imposedto avoid the center cusp of the NFW profile. If we cut offthe integral at 1 pc instead of 1 kpc, the signal would onlybe about 20 percent greater, although the point sourceapproximation would be more accurate. E. Possibility of a Positron or Electron Excess
We will briefly discuss the possibility of producing adetectable positron or electron signal within this model.This is a particularly interesting question because of thepositron excess observed by PAMELA [65], which wasconfirmed by Fermi-LAT [66] and more recently AMS-II [67]. In order to travel from the galactic center to thesolar system relatively unimpeded, the fermions wouldneed to be lower energy that those discussed above, whichlost significant energy due to inverse Compton scattering.Energy loss due to inverse Compton scattering is some-what suppressed for energies on the GeV scale; therefore,we will briefly discuss the difficulties of producing elec-trons and positrons on this scale.To produce a significant number of dark force mediatorparticles, we desire to keep α relatively large in order tomaintain a large cross section for bound state formation.However, the energy of the fermions produced by thedecay of the dark force mediator depends only on themass of the S particles and α . Therefore, to produce10 GeV-scale positrons and electrons, we must decrease m S to be on the scale of a few hundred GeV. This isbelow the scale typically discussed in the WIMPoniumliterature.Naively, these parameters appear to run into difficul-ties with the halo ellipticity bounds such as in [31]; forexample, m S = 100 GeV with α = 2 appears to re-quire m σ = 232 MeV, for which the dominant decay isto muons instead of e + e − . (However, it should be notedthat the analytic approximation for the cross section be-gins to break down at m σ ∼
100 MeV.) This appearsto eliminate the possibility of an observable electron orpositron excess.However, a more detailed analysis of the bounds by[35] suggests that these bounds should be about an or-der of magnitude weaker. In Appendix A, we haveparametrized this uncertainty with the parameter F ,which is one if the considerations of [35] are ignored. Ifone assumes F ∼ .
1, then a small region of parameterspace remains which is consistent with the halo elliptic-ity bounds and m σ is small enough (50 to 80 MeV) thatthe decay to e + e − dominates.The analysis proceeds as above, up to the point whereone calculates the inverse Compton scattering. For theselower energy electrons and positron, we do not expect in-verse Compton scattering to be a significant effect. How-ever, other effects can influence the shape of the spec-trum observed at Earth; for example, we must be partic-ularly concerned with positron annihilation, which willgenerically decrease the detected positron fraction. Wedo expect these particles to lose energy due to bremm-strahlung. A detailed analysis could be run using cosmicray propagation software such as GALPROP.However, we believe that it is unlikely that the re-sulting spectrum could be tuned to reproduce the ob-served positron excess observed in PAMELA [65], Fermi-LAT [66], and AMS-II [67]. The energy spectrum of fermions produced through the decay of dark force me-diator particles is flat, and while this spectrum will nodoubt be modified by a detailed analysis of propagationfrom the galactic center to the solar system, we think theresulting E Φ is unlikely to be as flat as that observed byPAMELA. Furthermore, the positron excess extends tohigher energies beyond that which can be accomodatedby our model. Hence, we conclude that such models areunable to account for this observed excess.Finally, we note that it may be possible to adjust theparameters so that an excess of positrons or electronsabove PAMELA’s observations is produced, althoughagain careful analysis of the propagation of said fermionswould be necessary. If such an excess can be produced,we would expect it decrease at or before the TeV scale, atwhich point the spectrum would be limited due to inverseCompton scattering energy losses. Since no such behav-ior is observed in PAMELA’s spectrum, one would trans-late this into bounds on the parameters. However, sincethe parameter space in which such a signal is potentialpossible is already quite small, any resulting constraints(if any) would be quite weak.
IV. BREMSSTRAHLUNG EMISSION OF DARKFORCE MEDIATORS
In this section, we will discuss the signal produced bybremmstrahlung emission of a σ boson which decays tophotons. First, we choose our free parameters consistentthe constraints discussed in II C. Then we calculate therate at which σ bosons are produced via bremsstrahlungnear the galactic center. Finally, we determine the fluxat the Earth and compare with observations by the IN-TEGRAL experiment. We will see that while the fluxexceeds the flux produced via bound state production,the resulting signal is significantly beneath observationallimits, due to the substantial background at lower ener-gies. A. Choice of parameters
In bremsstrahlung emission, the emitted σ boson willcarry an energy comparable to the kinetic energy of the S particles. Since these have a velocity of order 10 − , thismeans that the typical energy scale of bremsstrahlungemission will be six orders of magnitude below m S . Wewill require m σ (cid:28) ¯ v m S / σ bosons. This will influence the implementation of theconstraints discussed in section II C. (We note, however,that due to the contribution of the tail of the relativevelocity distribution, we do not necessarily expect a sharpcutoff at the average kinetic energy.)The first constraint comes from Ref. [50], which de-termines the condition for astrophysical observations tobe consistent with dark matter whose self-interaction isdescribed by a Yukawa potential. The precise constraintis a function of v max = (cid:112) αm σ /πm S , the velocity atwhich vσ T peaks at a transfer cross section equal to σ maxT = 22 . /m σ . If v max ∼
10 km / s, then the astrophys-ical constraints are consistent if 22 . /m σ m S < ∼
35 cm / g = 16000 GeV − [51]; we will verify that we satisfy thiscondition below. Combining this with m σ (cid:28) ¯ v m S / m S (cid:29) (cid:18) . · v · − (cid:19) / = 1 . . (22)Let us choose m S = 10 TeV and m σ = . A σS . We extended the anal-ysis of Ref. [31] to m S = 10 TeV in Appendix A; thisshowed that for m σ = . m S = 10 TeV, weprefer to take α = A σS / πm S < ∼ .
93 although thismay be loosened somewhat. If we choose to saturatethis bound, we find A σS = 68 TeV. These values give v max ≈
50 km / s, self-consistent with our initial assump-tion that v max ∼
10 km / s.The bremsstrahlunged σ boson can be emitted by ei-ther of the S particles; however, it can also be emitted bythe σ boson exchanged between the S particles. Thesediagrams involve the coupling A σ , which is thus far un-constrained. In order to enhance the signal, we will sat-urate the perturbativity bound, taking A σ = 3 . A σ = 0, whichis equivalent to neglecting the two diagrams on the rightof Fig. 2. While this certainly won’t help to increase oursignal, the properties of the signal will be qualitativelydifferent in the two cases in interesting ways. B. Production of σ Bosons throughBremsstrahlung
Next, we must know the cross section forbremsstrahlung emission of a soft σ boson, whichinvolves evaluating the 10 diagrams shown in Fig. 2.The derivation of this cross section, including averagingover the relative velocity of the incoming particles, iscontained in Appendix F. For the parameters givenabove, the cross section is σ = . − . Thisis the same order of magnitude as the α = 2 crosssection for bound state formation; we note, however,that we did not have to increase the coupling α into thenon-perturbative regime in order to reach this value. Ingeneral, as we would expect, the bremsstrahlung crosssections are indeed large in comparison to the boundstate formation cross section.We shoud note that this cross section does not includeany enhancement due to Sommerfeld factors; this contri-bution will be discussed later. Also, although naive esti-mates would suggest a large enhancement, in this regimethe Sommerfeld factor may be unreliable and a properresummation suggests that any enhancement is at most O (1) to O (10)[68]. This is discussed in somewhat moredetail after the calculation of the cross section in Ap-pendix F.The rate of production of bremsstrahlung σ bosons is dN σ dt = (cid:90) (cid:90) n S ( r ) v rel σ brem ( v rel ) P ( v rel ) dV dv rel (23)where n S ( r ) = ρ ( r ) /m S is the number density of darkmatter S particles, and ρ ( r ) is given by equation (7).We have also averaged over the relative velocity of the S bosons, and the integration extends from 1 kpc to 8 kpc,the distance from the solar system to the galactic center.For the given parameters, dN σ /dt = 3 . · GeV,or 4 . · s − . As might be expected, for A σ = 0,we find the lower rate dN σ /dt = 1 . · GeV − =2 . · s − . The fact that the cross section drops bytwo orders of magnitude shows that at A σ = 2 . σ boson isemitted by the exchanged σ boson dominate. Since thesediagrams are absent for A σ = 0, we expect the signalsproduced to have qualitative differences.We will show that bremsstrahlung will not producea detectable signal, while we found that for sufficientlylarge couplings bound state formation can. Since this isperhaps a surprising result, one may find it beneficial tocompare with the calculation of the bound state signalat each step to determine why this is so. We emphasize,however, that such comparisons must be made carefully,since the bound state calculations were performed in adifferent region of parameter space. We wish to empha-size that for any fixed perturbative value of α , the rate ofbremsstrahlung production will always be much greaterthan the rate of bound state formation, as one wouldexpect. However, if one compares the value of dN σ /dt found above with dN BS /dt given in the previous section,which are evaluated at different parameters, one findsthat dN BS /dt is larger by about an order of magnitude,even though we have chosen parameters such that thecross sections are comparable. This is a result of taking m S = 10 TeV here as opposed to 4 TeV above; increasing m S decreases the number density n S ( r ).Next we observe that the spectrum of the emitted σ bosons per SS → SSσ event is given by dN σ dE σ = 1 σ brem dσ brem dE σ . (24)As we might expect, this spectrum is sharply peaked at600 keV, which is on the same scale as the kinetic energy.The spectrum of the produced σ bosons per unit time is d N σ dt dE σ = (cid:18)(cid:90) n S ( r ) dV (cid:19) (cid:90) v rel dσ brem dE σ P ( v rel ) dv rel . (25) C. Decay of the σ Bosons and Resulting Signal
For m σ = . σ boson is σ → γγ , which is decribed by equation (13). FIG. 2. These diagrams contribute to the emission of a bremsstrahlung σ boson. The solid lines represent S bosons, whilethe dashed lines represent σ bosons. The top line represents t -channel scattering, while the bottom line represents u -channelscattering. If we assume the mixing angle between the σ boson andthe Higgs boson is 10 − , then the typical lifetime of theproduced σ bosons is 10 s, during which they travelabout 10 m, which is significantly less than the 10 mbetween the galactic center and the solar system.The spectrum of the photons produced by the decayof the bremsstrahlung σ bosons is given by d N γ dE γ dt = 2 (cid:90) d N σ dt dE σ P ( E γ , E σ ) dE σ (26)where the distribution of photon energies, as a functionof the initial σ boson energies, is given by equation (17).(The E γ dependence appears in evaluating the Heavisidestep functions.) As we would expect, this spectrum ispeaked around 300 keV. We note that the tail decreasesless rapidly as A σ is decreased. As a result, the signal for A σ = 0 will be skewed torwards higher energies.The production of dark force mediator particles resultsin an isotropic flux of these particles about the galacticcenter; similarly, we expect the flux of their decay prod-ucts to be isotropic about the galactic center. Therefore,the photon flux per unit area is well-approximated by anequivalent point source at the galactic center. We canfind the average flux per unit area, per unit solid angleby further dividing by 2 π , since the signal will appear tocome from the hemisphere centered on the galactic cen-ter. We note that this is an average; as a function of solidangle, we expect the signal to be greater near the galacticcenter and less further away from it. We also note thatthis is only an approximation to the true diffuse flux,meant to demonstrate that a detectable signal is possible(and we remind the reader that we are already neglictingthe contribution from the galactic center itself).We find that the average flux at the solar system isΦ = 14 πd · π st d N γ dE γ dt , (27)where d = 8 kpc is the distance from the galactic centerto the solar system. Since we have calculated the number of produced σ bosons out to a radius of 8 kpc, the signalwe calculate here comes from the hemisphere centeredon the galactic center, which explains the 2 π st. Wenote that this is the average over the hemisphere; theflux will be somewhat greater towards the galactic centerand somewhat less towards the edges; however, this is arelatively small effect, contributing perhaps an order ofmagnitude increase as we approach the center.Again, with the same caveats as above, let us comparewith the bound state case. The photon energies here arespread out over the scale of 100 keV, whereas the photonsignal for the bound state production is spread over thescall of 100 GeV. However, a single high energy fermionproduces about 10 ∼ GeV-scale photons throughscattering off of the CMB, while each σ boson producedthrough bremsstrahlung produces a mere 2 photons. Asa result, the estimated ratio of fluxes is Φ brem / Φ BS ∼ or 10 . We note that since the two scenarios are in dif-ferent regions in parameter space, this cannot be inter-pretted as the ratio of actual bremsstrahlung-producedphotons to bound state produced photons in the galaxy.The relevant energy scale for bremsstrahlung emissionis on the scale of hundreds of keV, while the relevantenergy scale for bound state emission is on the scale ofa hundred GeV. Astrophysical backgrounds are signifi-cantly larger at this smaller scale; the SPI on the INTE-GRAL experiment records E γ Φ on the order of 1 to 10keV / cm s st for energies 20 keV and 1000 keV [69]. Theflux of produced photons cannot be distinguished fromthis large backaground.The resulting signal is shown in fig. 3; as we ex-pect, it is about 8 orders of magnitude smaller than thebound state formation signal. More importantly, thelarger A σ = 3 . A σ = 2 . A σ = 0. We observe that without these dia-0 E γ Φ (cid:0) k e V / c m ss t (cid:1) E γ (keV) α = . , A σ = 3 . α = . , A σ = 0 MeV FIG. 3. The flux of gamma rays produced by bremsstrahlungemission of σ particles and their subsequent decay for m S =10 TeV, m σ = . grams, the signal is significantly smaller, but it is peakedat higher energies.We have noted above that the calculated cross sectiondoes not include a Sommerfeld enhancement, becausesome analysis suggest that such large factors are unre-liable [68]. Even if we assume that the naive Sommerfeldfactor given by S = απ/v − exp( − απ/v ) (28)is accurate to arbitrarily large scales, this enhancementis not sufficient to produce a detectable signal. For theparameters in the range discussed, the enhancement is oforder 10 or 10 , which is still too small to produce theseven orders of magnitude amplification required for thesignal to be detectable.The signal can be increased by increasing the cou-plings; and indeed, as discussed in Appendix A, thereis some uncertainty in the halo ellipticity bounds. Toproduce a detectable signal requires increasing the cou-pling α to ∼ , well outside the perturbative regimeand far beyond what can be made consistent with thehalo ellipticity bounds. It is true that A σ is unrestrictedby astrophysical bounds, but in order to amplify the twodiagrams it appears in to the scale of INTEGRAL’s ob-servations, we would need to take A σ /m σ ∼ , whichis unreasonably large.Therefore, we conclude that bremsstrahlung emissionof dark force mediator particles cannot produce de-tectable signals, although the photon flux is generallysignificantly larger than bound state production. Onemight consider the idea that even if the signal producednear the galactic center is not detectable, perhaps such processes enhance the gamma ray or x-ray emission ofnearby dwarf galaxies sufficiently to be observable; how-ever, a simple estimate reveals that this is not the case.Even if the signal calculated above, for the Milky Waygalaxy, was somehow shrunk into a dwarf galaxy 40 kpcfrom us which covered a 3 ◦ by 3 ◦ patch of the sky, thenumber of counts expected in an ideal 1 m detector isof order 10 − keV − s − , which is again well below thebackground emission. V. CONCLUSIONS
We have considered indirect detection signals producedby a minimal asymmetric self-interacting dark matter.Due to the U S (1) asymmetry, the typical indirect detec-tion signal from dark matter annihilation is absent in thismodel. However, we demonstrated that asymmetric self-interacting dark matter can, in fact, produce a strongsignal from the processes accompanying the formation ofbound states, as has been discussed in the WIMPoniumliterature. We have found that signals are possible forsufficiently large couplings. This effect makes possible in-direct detection of asymmetric self-interacting dark mat-ter. The spectrum of gamma rays can help distinguishcollisionless dark matter from self-interacting dark mat-ter. We have performed explicit calculations for severalsets of parameters; showing that for α = 2, m S = 4 TeV,and m σ = 40 MeV the signal would be detectable. How-ever, we have shown that this signal is detectable onlyin the strongly interacting regime, by showing that if α is decreased to 1 (keeping m S constant), the resultingsignal is not detectable.Then we have discussed, albeit briefly, the possibil-ity that this model could, in a narrow region of pa-rameter space, produce a detectable excess in electronsand/or positrons. Additionally, we have also consideredthe signal produced by the bremsstrahlung emission ofthe σ boson. This was calculated for two points in pa-rameter space ( m S = 10 , m σ = . , α = .
93 with A σ = 3 . A σ = 0) to demonstrate two limits ofspectrum shape. However, we have shown that althoughthe flux of gamma rays can be rather large, the result-ing signal is actually quite small and significantly belowbackgrounds.This work was supported by DOE Grant DE-FG03-91ER40662 and by the World Premier International Re-search Center Initiative (WPI Initiative), MEXT, Japan. Appendix A: Extension of Bounds From HaloEllipticity
As was noted in section II C, one constraint on theself-interaction of dark matter arises from the observedellipticity of dark matter halos. In this appendix, we willextend the result of [31] to higher dark matter masses. Ashas been noted by [35], these bounds may in fact be some-1 α m S (TeV) m σ = 3 MeV , F = 1 m σ = . , F = 1 m σ = . , F = . FIG. 4. A plot of the critical coupling α = A σS / πm S asa function of m S . Couplings below the critical coupling areconsistent with the elliptical shape of dark matter halos. what weaker due corrections from the triaxial distribu-tion of dark matter outside of the core; however, as theynote, more detailed simulations are required to firmly es-tablish this conclusion. Therefore, we will parametrizeour uncertainty by the coefficient F ; the numerical simu-lations presented in [35] could be interpretted as favoring F ∼ . k = (cid:90) d v d v f ( v ) f ( v )( n S v rel F σ T )( v /v ) (A1)where σ T is the momentum-transfer cross section, givenby σ T = (cid:82) d Ω ( dσ/d
Ω)(1 − cos( θ )), and f ( v ) is the darkmatter velocity distribution. The analytic fit for σ T , thedistribution functions, and the relevant parameters forNGC 720 are all available in [31]. In this reference, theyproduce plots of numerical results for m S up to 4 TeV.However, we will need to consider masses above this, andtherefore, we extend their results to higher masses. Wenote that quantum corrections to the cross section be-come important if the limit m S ¯ v/m σ (cid:29) α = A σS / πm S to m S = 12 TeV. We showthe results for m σ = . m σ = 3 MeV. We also consider F = 1 and F = . m σ = . m S = 10 TeV, this bound with F = 1 requires α ≤ . m σ ( M e V ) m S (TeV) α = 1 , F = 1 α = 2 , F = 1 α = 1 , F = . α = 2 , F = . FIG. 5. The minimum value of m σ for which α = 2 or α = 1is consistent with elliptical halos. However, it is substantially loosened to α ≤ . F = . m σ increases. Therefore, we calculate the minimum m σ forwhich α = 2 or α = 1 is consistent with the observed haloellipticity, as a function of m S . The results are shown inFig. 5. Again we see that taking F = . Appendix B: S -Nucleon Interaction Cross Section As was noted in section II C, the dominant interac-tion between the S boson and nucleons, which is relevantfor direct detection experiments, typically occurs throughthe exchange of a single Higgs boson. This diagram isproportional to A σh , which is otherwise unconstrained inour model, and therefore we can arbitrarily decrease thiscoupling, thus killing the signal.However, as we note in the text, there is another dia-gram which becomes dominant at sufficiently small valuesof the coupling: the S boson may emit a σ boson, whichtransforms into a Higgs boson via mixing and couples toa nucleon. This diagram involves the coupling A σS andthe mixing angle θ M . We cannot take either of these pa-rameters to zero without eliminating the signal, althoughthe mixing angle may be quite small. Thus one cannotarbitrarily decrease the S -nucleon cross section; there isa minimum value set by this diagram. In this Appendix,we will show that the contribution of this diagram is in-deed quite small, as expected, and causes no tension withdirect detection constraints.We note that the oscillation time scale, which is givenby τ osc = 2 πE/ ∆ m , is generally many orders of magni-tude smaller than the interaction time scale, which can2be estimated by considering the overlap of the wavefunc-tions. Consequently, averaging over the “detector scale”(nucleon size), along with the source location, will sim-ply give a factor of 1 /
2. (This is in contrast with certainneutrino oscillation experiments, for which τ osc may belarge in comparison to other experimental scales, due tothe small ∆ m . In our scenario, ∆ m ∼ m h .)The S particles under consideration are generally muchheavier than the protons; we will masses between 4 and10 TeV. Therefore, in the center of momentum referenceframe the S particles will be approximately stationary,while the protons approach at speeds of approximately220 km / s. The momentum transfer is approximately2 m p v = 1 . − i M ≈ u im S ¯ v A σS cos (cid:18) θ M (cid:19) m q v sin (cid:18) θ M (cid:19) u, where v is the vacuum expectation value of the Higgsboson and u , ¯ u are spinnors for the proton. This yields |M| ≈ m S ¯ v A σS m q m n v cos (cid:18) θ M (cid:19) sin (cid:18) θ M (cid:19) . Because the velocities are non-relativistic, the initial en-ergy squared is approximately ( m S + m n ) ≈ m S , whichgives an approximate cross section σ ≈ πm S · m S ¯ v A σS m q m n v cos (cid:18) θ M (cid:19) sin (cid:18) θ M (cid:19) . Let us consider one of the sets of parameters usedin the bound state cross section; m S = 4 TeV and A σS = 20 TeV, corresponding to α = 2. For the av-erage effective mass of a quark, we use 3 MeV, and wechoose θ M = 10 − . This gives σ ∼ − GeV − . Allof our other choices for parameters give a cross sectionbelow this value. This is well beneath the limits fromdirect detection experiments, which are 10 − cm or10 − GeV − [53], [54]. There is a significant uncertaintyin the contribution of the s -quark to the effective quarkmass. Since the Higgs coupling to s is much greater thanthe couplings to u and d , even a relatively small contribu-tion of the sea quarks with higher masses can dominatethe cross section. The measured s quark contribution,manifest as the nuclear pion-nucleon sigma term, is un-certain, and the resulting uncertainty in the cross sectioncan be as large as an order of magnitude [70]. However,even at the upper edge of the range, the cross sectiondoes not reach the present lowest cross sections acces-sible in experiment. Hence, at present, direct detectionexperiments do not constrain the scenario we have con-sidered. σS S SσS SSσ σ FIG. 6. The scattering of σ particles on dark matter. Appendix C: Mean Free Path of Dark ForceMediator Particles
As was noted in section II D, the mean free path for the σ particles in the galaxy must be greater than the dis-tance they would travel before decaying; otherwise, con-stant scattering can act like a quantum Zeno experimentthat prevents the decay. In this appendix, we present thecalculation for the mean free path, and show that it isgreater than the distance from the galactic center to thesolar system for the relevant regions of parameter space.Since the quartic coupling λ σS can be made aribtrarilysmall, we will assume that the scattering is dominated bythe Sσ interaction mediated by an S boson; there are twodiagrams that contribute, which are shown in Fig. 6.We assume that in the lab frame, the σ particle is mov-ing relativistically with energy E σ , while the S particle ismoving non-relativistically with velocity v of order 10 − .We do not assume any relation between E σ and the ki-netic energy of the S particle. Since the cross section isa relativistic invariant, we may evaluate it in the centerof momentum frame, which under the above assumptionsis attained by boosting by β = E σ / ( E σ + m S ). Keepingonly the largest terms, we find that the initial and finalfour-momenta in the CM frame are p µσ,i = ( γβm S , , , γβm S ) p µS,i = ( γm S , , , − γβm S ) p µσ,f = γβm S , γβm S sin( θ ) , , γβm S cos( θ )) p µS,f = ( γm S , − γβm S sin( θ ) , , − γβm S cos( θ ) (C1)where we have used the fact that the collision is elastic.The matrix element is − ı M = − A σS m S − ( p S,i − p σ,f ) − A σS m S − ( p S,i + p σ,i ) = − A σS γ m S (cid:18) θ )(1 + β )(1 + β cos( θ )) (cid:19) . (C2)The cross section is given by σ = 164 π (cid:90) |M| γ (1 + β ) m S d Ω= A σS πγ (1 + β ) m S (cid:90) π (cid:18) θ )1 + β cos( θ ) (cid:19) sin( θ ) dθ = A σS πm S β + (1 − β ) ln((1 − β ) / (1 + β )) γ β (1 + β ) (C3)3The mean free path is (cid:96) = ( σn S ) − , where n S can befound using equation (7). Since n S depends on r , themean free path will also depend on r ; it is the smallestas we approach the galactic center. Let us consider sometypical parameters. For m S = 5 TeV, A σS = 3 TeV,and E σ = 1 TeV, the mean free path at 1 pc is of order10 m. If we decrease E σ to 1 MeV, the mean free pathincreases to order 10 m (at 1 pc again). These valuesare all much greater than the 10 m between the galac-tic center and the solar system; therefore, requiring thatthe σ particles decay before reaching the solar systemprovides a stronger bound as claimed. Appendix D: Bound State Formation Cross Section
In this appendix, we calculate the cross section for two S particles to form a bound state through the exchangeof σ bosons. We emphasize that because the S particlesform a bound state, they do not escape to infinity, andtherefore the Born approximation is not applicable. Wenote that, although the coupling is strong, we are in theclassical regime, because m S ¯ v/m σ (cid:29)
1; therefore we donot need to include additional quantum corrections suchas those calculated numerically in [46].We will approximate the σ boson as massless. Thecross section for non-relativistic electrons and positronsto form a bound state through photon exchange was cal-culated in [61]; we adapt this derivation for scalar fields.The matrix element is M = − i (cid:90) Ψ ∗ f ( r , r ) (cid:32) (cid:88) n =1 , A n e − i k · r n (cid:33) · Ψ i ( r , r ) d r d r (2 π ) δ ( E i − E f − E σ ) . (D1)In this equation, r and r are the locations of the two S particles respectively, Ψ f is the wavefunction of thebound state, and Ψ i is the wavefunction for the two in-coming S particles. The factor e − i k · r n represents thewavefunction of the σ particle, and the sum is over thetwo S particles it can couple to. In this equation, thewavefunctions have the standard normalization in quan-tum field theory; however, since we are interested in thenon-relativistic limit, let us use wavefunctions that arenormalized to one. Then the matrix element is M = − i A σS m S (cid:90) Ψ ∗ f ( r , r ) (cid:0) e − i k · r + e − i k · r (cid:1) · Ψ i ( r , r ) d r d r (2 π ) δ ( E i − E f − E σ ) . (D2)Next we define R = r + r r = r − r (D3)and write the wavefunctions asΨ i ( r , r ) = e ı Q · R Ψ i ( r )Ψ f ( r , r ) = e ı P · R Ψ f ( r ) (D4) where Q = p + p is the total momentum of the initialparticles. Similarly, P is the momentum of the boundstate. After performing the d R integral, we have M = − i A σS m S (cid:90) Ψ ∗ f ( r ) (cid:16) e i k · r / + e − i k · r / (cid:17) Ψ i ( r ) d r · (2 π ) δ ( E i − E f − E σ ) δ ( Q − k − P ) . (D5)The reduced matrix element is¯ M = (cid:90) Ψ ∗ f ( r ) (cid:16) e ı k · r / + e − ı k · r / (cid:17) Ψ i ( r ) d r (D6)and the differential probability is dW = T V (2 π ) E σ A σS m S δ ( E i − E f − E σ ) δ ( Q − k − P ) · | ¯ M| | k | d | k | d Ω d P, (D7)where V is the normalized volume, T is the interactiontime, and d Ω is the solid angle for the σ particle. The re-maining integrals enforce momentum and energy conser-vation; we may perform them by directly imposing theseconstraints in our calculation. The transition probabilityper unit volume and unit time is dw = A σS m S | k | d Ω2 E σ (2 π ) | ¯ M| . (D8)If m σ (cid:28) B then E σ ≈ | k | , and this simplifies to dw = A σS m S | k | d Ω2(2 π ) | ¯ M| (D9)The differential cross section is dσ = dw/v rel where v rel is the relative velocity of the particles in the initial state.We define the relative momentum by p = µ v rel where µ = m S / | p | is also the momentumof one of the incoming particles in the center of momen-tum frame; we will now specialize to this frame. (We notethat the cross section is Lorentz invariant, and thereforestill applicable to other reference frames.) Then dσ = A σS m S | k | d Ω | p | (2 π ) | ¯ M| . (D10)The free S particles do not escape to infinity; theyexist only in the initial state. Therefore, at large r , Ψ i ( r )must be a superposition of a plane wave and an outgoingspherical Coulomb wave. (Although our interaction isnot electromagnetic, the appropriate asymptote is still aspherical Coulomb wave in the approximation that m σ (cid:28) m S .) The appropriate wavefunction to use is [71] (alsodiscussed in [61])Ψ i ( r ) = e πζ/ Γ(1 − ıζ ) F ( ıζ, , ı ( pr − p · r )) e ı p · r (D11)where ζ = A σS m S / | p | m S = A σS / | p | , and F is theconfluent hypergeometrical function. This has the samenormalization as a plane wave. We note that the cross4section will be very sensitive to the ratio A σS / | p | ∼ A σS /m S as a consequence of the exponential. We adapt thehydrogen ground state wavefunction for Ψ f ( r ); again,this is accurate in the approximation that m σ is negligi-ble. Ψ f = (cid:114) η π e − rη , (D12)where η = ζ | p | = A σS /
4; this is the radius of the boundstate. The reduced matrix element is¯ M = (cid:114) η π e πζ/ Γ(1 − ıζ ) (cid:90) e ı p · r − rη (cid:16) e ı k · r / + e − ı k · r / (cid:17) · F ( ıζ, , ı ( pr − p · r )) d r. (D13)To evaluate the integral, we differentiate the identity[72] (cid:90) e ı ( p − κ ) · r − ηr F ( ıζ, , ı ( pr − p · r )) d rr = 4 π [ | κ | + ( η − ı | p | ) ] − ıζ [( p − κ ) + η ] − ıζ (D14)with respect to η . The result is (cid:90) e ı ( p − κ ) · r − ηr F ( ıζ, , ı ( pr − p · r )) d r = 8 π [ | κ | + ( η − ı | p | ) ] − ıζ [( p − κ ) + η ] − ıζ · (cid:20) ζ ( η − ı | p | )[( p − κ ) + η ][ | κ | + ( η − ı | p | ) ] − ıη (1 − ıζ ) (cid:21) ≡ g ( κ, χ ) (D15)where χ is the angle between p and κ . We observe that g ( κ, π − χ ) = g ( − κ, χ ). If the angle between k and p isΥ, the reduced matrix element is¯ M = (cid:114) η π e πζ/ Γ(1 − ıζ ) (cid:18) g (cid:18) | k | , Υ (cid:19) + g (cid:18) − | k | , Υ (cid:19)(cid:19) (D16)This can be evaluated numerically. The last remainingunknown quantity in (D10) is | k | , which can be foundfrom the energy conservation equation2 m + | p | m = (2 m − B ) + | k | m − B ) + | k | (D17)where we have noted that in the center of momentumreference frame, the bound state also has momentum | k | .We find the total cross section by numerically integrating(D10). We will also average over a relative momentumdistribution P ( | p | ); the total cross section is given by σ BS = (cid:90) (cid:90) A σS m S | k || ¯ M| | p | (2 π ) P ( | p | ) d | p | π sin(Υ) d Υ(D18) We note that in the non-relativistic limit the momentumdifference of the two particles is independent of referenceframe; therefore we can calculate P ( | p | ) in any conve-nient frame even though we specialized to the center ofmomentum reference frame above. The total cross sec-tion is, of course, Lorentz invariant. Using equation (11),we find the relative momentum distribution P ( | p | ) d | p | = 4 π √ m S (cid:18) m S πT eff (cid:19) / e −| p | /m S T eff | p | d | p | . (D19) Appendix E: Scattering From CMB Photons
When we calculated the signal produced by boundstate formation, we found that the dark force mediatorbosons σ decayed into TeV-scale fermions. These loseenergy due to scattering with CMB photons, as notedin section III D. In this appendix, we produce a calcu-lation of the spectrum of gamma rays produced by thisscattering.First, we will show that we can neglect the energy lossdue to synchotron radiation, which is described by dE e dt = − b sync E e (E1)where the unitless coefficient b sync is given by b sync = 4 σ T m e B π (E2) σ T is the Thomson cross section. Since we considera spherical region extending from 1 kpc to 8 kpc,very few of the fermions will be created in the galac-tic plane. Therefore, the appropriate magnetic field is1 µ G [73], [74], which gives b sync = 6 · − .The energy loss of a single fermion due to inverseCompton scattering is described by the equation dE e dt = − b ICS E e (E3)where now the unitless coefficient is b ICS = 4 σ KN w ph m e . (E4) σ KN is the Klein-Nishina cross section, which reduces tothe Thomson cross section when relativistic correctionsare negligible. Since this is applicable for scattering withCMB photons, b ICS = 5 . · − and is approximatelyindependent of energy. (For the parameters with α = 1,we have 5 . · − instead.) Since this is two orders ofmagnitude larger than the corresponding value for syn-chrotron radiation, we may neglect energy loss due tosynchrotron radiation.Therefore, we calculate the photon energy spectrumfrom inverse Compton scattering with CMB photons.5The cosmic microwave background radiation is a black-body at T CMB = 2 .
73 K; therefore the photon densityper unit energy is n ph ( (cid:15) ) ≡ d N ph , CMB dV d(cid:15) = 1 π (cid:15) exp( (cid:15)/T CMB ) − (cid:15) is the energy of the unscattered photon. For in-verse Compton scattering, the number of scattered pho-tons per unit energy per unit time produced by an elec-tron or positron with Lorentz factor γ is given by [75],[76] d N γ dE dt s ( E, γ ) = (cid:90) ∞ d(cid:15) n ph ( (cid:15) ) σ KN ( E, (cid:15), γ ) (E6)where σ KN ( E, (cid:15), γ ) is σ KN ( E, (cid:15), γ ) = 3 σ T (cid:15)γ G ( q, Γ) (E7)and G ( q, Γ) = 2 q ln( q ) + (1 + 2 q )(1 − q ) + 2 ηq (1 − q )Γ = 4 (cid:15)m e η = (cid:15)Em e q = E Γ( m e − E ) . We have put a subscript on t S to remind us that this vari-able measures the time during which the fermion scattersagainst CMB photons. We use the symbol E for the fi-nal energy of the scattered photon. The Thomson limitcorresponds to Γ (cid:28) E between the followingvalues are allowed E min ( γ, (cid:15) ) = γm e Γ4 γ + Γ E max ( γ, (cid:15) ) = γm e Γ1 + Γ (E8)which we enforce by writing d N γ dE dt S ( E, γ ) = (cid:90) ∞ d(cid:15) n ph ( (cid:15) ) σ KN ( E, (cid:15), γ ) · Θ( E max ( γ, (cid:15) ) − E )Θ( E − E min ( γ, (cid:15) ))(E9)This equation gives the number of photons per unitenergy per unit time scattered by an electron or positronwith energy γm e . From the fermion energy distributiongiven in equation (16), the corresponding γ distributionis P ( γ ) = m e (cid:112) ( B − m σ )(1 − m e /m σ ) (E10)for γ between the values γ max , γ min = ± (cid:112) ( B − m σ )(1 − m e /m σ )2 m e (E11)Averaging d N γ /dE dt over the γ distribution gives d N γ dE dt S ( E ) = (cid:90) γ max γ min P ( γ ) (cid:90) ∞ d(cid:15) n ph ( (cid:15) ) σ KN ( E, (cid:15), γ )Θ( E max ( γ, (cid:15) ) − E )Θ( E − E min ( γ, (cid:15) )) . (E12) d N γ / d E (cid:0) G e V − (cid:1) Energy of Scattered Photons E (GeV) FIG. 7. This plot shows dN γ, tot /dE , described by equation(E14), evaluated for the first set of parameters ( α = 2, m S =4 TeV, m σ = 40 MeV). This equation gives us the number of photons scat-tered per electron (or positron) per unit time; however,we require the total number of photons scattered by oneelectron before it loses all of its energy. Properly, weshould integrate over t S ; this is complicated because γ isa function of t S . Therefore, we will approximate dN γ dE ≈ d NdE dt S · T, (E13)where T = 1 /b ICS E e = 1 /b ICS γm e is the relevant time-scale for energy loss. This gives dN γ dE ( E ) = 1 b ICS m e (cid:90) γ max γ min P ( γ ) γ (cid:90) ∞ d(cid:15) n ph ( (cid:15) ) · σ KN ( E, (cid:15), γ )Θ( E max ( γ, (cid:15) ) − E )Θ( E − E min ( γ, (cid:15) )) . (E14)The equation describes the total number of scatteredphotons of a particular energy, per single electron orpositron. We have evaluated this equation for the α = 2parameters and the result shown in Fig. 7 for energiesbetween 1 GeV and 10 GeV. We see that it drops offrapidly as a function of energy.To find the total number of photons per unit energyper unit time, we must multiply by the rate of productionof the high energy fermions, which gives dN γ, tot dE dt ( E ) = dN BS dt b ICS m e (cid:90) γ max γ min P ( γ ) γ (cid:90) ∞ d(cid:15) n ph ( (cid:15) ) · σ KN ( E, (cid:15), γ )Θ( E max ( γ, (cid:15) ) − E )Θ( E − E min ( γ, (cid:15) )) . (E15) Appendix F: Bremsstrahlung Cross Section
In this appendix, we derive the cross section forbremsstrahlung emission of a σ boson in SS → SS scat-6tering. Then 10 relevant tree-level diagrams are shownin Fig. 2. Note that the t and u -channel diagrams cancelto lowest order in the m σ → p and p , the outgoing momenta of the two S particles as p and p , and the outgoing momentum ofthe bremsstrahlung σ particle as p . The matrix elementis then − ı M = − A σS ( m σ − ( p − p ) )( m S − ( p + p ) ) − A σS ( m σ − ( p − p ) )( m S − ( p + p ) ) − A σS ( m σ − ( p − p ) )( m S − ( p − p ) ) − A σS ( m σ − ( p − p ) )( m S − ( p − p ) ) − A σS A σ ( m σ − ( p − p ) )( m σ − ( p − p ) )+ ( p ↔ p ) (F1)where the last term, in which the momenta p and p are switched, represents the contribution of the bottomrow of diagrams. We will specialize to the center of massframe; we note that the total cross section is a relativisticinvariant and therefore it is irrelevant what frame it iscalculated in. Without a loss of generality we write themomenta as p µ = (cid:18) m S + | p I | m S , , , | p I | (cid:19) , (F2) p µ = (cid:18) m S + | p I | m S , , , −| p I | (cid:19) , (F3) p µ = (cid:18) m S + | p | m S , | p | sin( θ ) cos( φ ) , | p | sin( θ ) sin( φ ) , | p | cos( θ )) , (F4) p µ = (cid:18) m S + p m S , | p | sin( θ ) cos( φ ) , | p | sin( θ ) sin( φ ) , | p | cos( θ )) , (F5)and p µ = (cid:16)(cid:112) m σ + | p | , | p | sin( θ ) , , | p | cos( θ ) (cid:17) . (F6)Now we turn our attention to the cross section, whichis given by σ brem = (cid:90) |M| E + E ) (2 π ) δ ( p + p − p − p − p ) dLips (F7)where the extra 1 / dLips is the Lorentz-invariantphase space for the final state particles. In particular,this is dLips = (cid:89) i =3 d p i (2 π ) E i . (F8) In the phase space denominators, we may make theapproximation E = E = E = E = m S , and weintegrate over the 3-momentum delta function, setting p = − p − p . When the S particles are non-relativistic,the energy delta function becomes δ (cid:18) | p I | m S − | p | m S − | p + p | m S − (cid:112) m σ − | p | (cid:19) (F9)Let us call the angle between p and p θ . The deltafunction enforces | p I | m S − | p | m S − | p | m S − | p || p | cos( θ ) m S − (cid:112) m σ + | p | = 0 , (F10)which can be solved for | p | in terms of | p | and θ . | p | = − | p | cos( θ )2 + 12 (cid:0) | p | cos ( θ ) − | p | + 4 | p I | − m S (cid:112) m σ + | p | (cid:17) / (F11)We must of course ensure that the result is positive. Byour choice of coordinates, the dφ integral is trivial; thisleaves the integrals over θ , φ , θ , and d | p | to be donenumerically. This integral is not infrared divergent dueto the nonzero mass of the σ boson. Since the initialmomentum in the center of momentum frame is p I = v rel / m S , the above calculation gives σ ( v rel ). We canthen average over the relative momentum σ brem = (cid:90) P ( v rel ) σ ( v rel ) dv rel (F12)using equation (11).Finally, we address Sommerfeld factors, which multi-ply the cross section and naively can have a large impactat low velocities. (Note that this is a multiplicative fac-tor in addition to the typical 1 /v behavior of the crosssection.) These describe the formation of a quasi-boundstate during the interaction; the modified cross section is σ Somm = πα/v − exp( − πα/v ) σ (F13)7For the parameters under consideration, these factors canbe extremely large, of order 10 or 10 . However, it hasbeen argued that in this regime the Sommerfeld factorgiven above is unreliable; additional diagrams beyond theladder diagrams implicitly summed in the above equationmust be taken into account and a proper resummation suggests the factors are of order O (1) to O (10) [68]. Thisis supported by some experimental evidence [77], [78],including more recent observations at BaBar [79]. 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