based on work with A. Vincent and J. Cline Phys.Rev. D81 (2010) 083512 Phys.Rev. D82 (2010) 123519
COSMIC RAY ANOMALIES, GAMMA RAY CONSTRAINTS AND SUBHALOS IN MODELS OF DARK MATER ANNIHILATION
Wei XUE McGill University
1 OUTLINE
Cosmic Anomaly
Dark Matter Subhalos outside
Dark Matter subhalos inside
Particle Physics realization
2 Cosmic Ray Anomaly
Charged Cosmic Particles
Positron fraction(>10Gev) PAMELA
Electron+Positron Peaking around 500Gev Fermi
3 Dark Matter The Phenomena have no known astrophysics origin (Could be pulsar?)
Attractive explanation: 1 Tev WIMP DM + DM → messengers fields φ (on shell) → SM particles. mφ<2mproton Sommerfeld enhancement to cross section <σv>=BF × 3×10-26cm3s-1
Two issues: Gamma Ray? substructure?
4 FromPhysics N-body System Simulation ★Via Lactea II 1 DM main halo in the centre 100 inside the visible galaxy Correlated 20,047 System resolved DM subhalos Liouville’s conservation law(too much information) ★Extend as far out as 4,000 kpc Our Visible galaxy is 40 kpc aross
5 Subhalos contributions
Positive contribution Large Number 20,000 subhalos High density small dispersion velocity sub-substructure
Negative contribution long distance to the milky way
6 GALPROP+Via Lactea II GALPROP Numerically solve diffusion equation of electrons and positrons d ∂ ψe (x, p,t)=Qe (x,E)+ (D(E) ψe (x, p,t)) + [b(x,E)ψ (x, p,t)] dt ± ± ∇ · ∇ ± ∂E ± Source 2 r α 3 Main halo: ρEin(r)=ρsexp 1 −α rs − the annihilation rate of diffuse DM particles of the host subhalo i halo. " ! Subhalos:For a given subhalo i at a distance !i from the edge of i the diffusion zone of the galaxy, the flux of e+ or e− on thissimilar boundary takes density the form profile, i ∞ 2 2 8 kpc dΦi dN r ρi diffusion zone = BSH!σv" (!i) 2 2 dr (3) dE dE !0 !i MDM 20 kpc where BSH is an average boost factor for the subhalos due to Sommerfeld enhancement for example, and ρ (r) is the Change the boundary conditioni FIG. of 1: GALPROP Geometry of a subhalo shining by leptons on the bound- mass density profile of the subhalo. The unboosted cross ary of diffusion zone of the galaxy. section is assumed to be !σv" =3× 10−26 cm3 s−1 in usingaccordance the with subhalos the standard assumption data that from the DM Via Lactea II abundance was determined by freeze-out starting from a ity dispersion is at a maximum, through the relation thermal density. In a more exact treatment, the boost ∼ rs = rvmax /2.212, while the prefactor scales with the factor would be velocity dependent [25, 26] and appear ∼ 2 2 7 within the average over DM velocities indicated by the maximum velocity vmax as ρs = vmax/(0.897 · 4πrs G). brackets in !σv". Moreover each subhalo in general has a To incorporate the contribution (3) to the lepton flux different boost factor since the velocity dispersions that from the subhalos in GALPROP, we add delta func- tion source terms to q(r, z, p) for the cylindrical surface determine BSH depend on the size of the subhalo [13]. For this preliminary study, we simply parametrize the effect bounding the diffusion zone, as illustrated in fig. 1: by an average boost factor, where the averaging includes dΦi the sum over all subhalos as well as the integration over qdisk =2δ(z ± h/2) cos θi dE velocities. &i The energy spectrum dN/dE of electrons from the DM dΦi qband =2δ(r − R) sin θi (5) annihilations is taken for simplicity to be a step function dE −1 &i at the interaction point, dN/dE = MDMΘ(MDM − E0), where E0 is the energy immediately following the an- where h and R are respectively the height and radius of nihilation. We are interested in models where the DM the cylinder. The factor of 2 corrects for the fact that particles initially annihilate into two hidden sector gauge sources in GALPROP have no directionality, whereas the or Higgs bosons, each of which subsequently decays into flux impinging on the surface is inward. The sum is over + − e e [3]. The four-body phase space would thus be a the 20,048 resolved subhalos in the Via Lactea II simu- more exact expression for dN/dE, but the step function lation. In addition, the sources were averaged over the has the correct qualitative shape and is simpler to imple- azimuthal angle φ because GALPROP assumes cylindri- ment in GALPROP. cal symmetry in its 2D mode (and the 3D mode runs too The energy of the electron at the edge of the galaxy is slowly for our purposes). Finally, the distance !i must reduced from its initial value E0 by scattering with CMB be corrected for subhalos that are close to the diffusion photons before reaching the galaxy (starlight, infrared zone; rather than the distance to the center of the galaxy, radiation and synchrotron radiation are only important it should be the distance to the cylindrical boundary. On in the inner galaxy [27]), according to the loss equation average, this is a reduction by 17 kpc compared to the 2 2 dE/d! = −κE [9] where κ =(4σT /3me) uCMB =6.31 × distance to the galactic center. −7 −1 8π ! 2 10 kpc GeV , σT = 3 (αEM /mec) is the Thomson Although most subhalos were located outside of the 3 cross-section and uCMB =0.062 eV/cm is the present diffusion zone, there are 143 lying inside, whose contri- energy density of the CMB. It is convenient to write the bution required special treatment. Assuming approxi- −1 + − solution in the inverted form: E0 =(−κ! +1/E(!)) mate isotropy, we took their entire flux of e + e to be for substitution into dN/dE. Numerically, we find that pumped into the diffusion zone from the boundary rather the losses outside the diffusion zone make a small correc- than from their individual positions. Treating them in tion, and that the distinction between E0 and E(!) is not this manner allowed us to group their contribution with important here. that of the other subhalos and thus consider a single av- Each subhalo is characterized by a density profile that erage boost factor for all subhalos. This approximation has been fit to the Einasto form would break down if one subhalo happened to be very close to our position in the galaxy, but treating such a α 2 r case would anyway require going beyond the standard ρi = ρs,i exp − − 1 (4) " α ## rs,i $ $% cylindrical symmetry (2D) mode of GALPROP and us- ing the much slower 3D mode. We believe this treatment with α =0.17 [15]. The scale radius is found to be is conservative in the sense that it should only underes-
proportional to the radius rvmax at which the veloc- timate the contributions of the nearby subhalos. Mainhalo+Subhalo better fit to PAMELA and Fermi with 2.2 Tev WIMP when subhalos are included(BFSH=3744,BFMH=92)
Larger mass is because of larger propagation distance. The energy loss is due to Inverse Compton Scattering with CMB, IR and Starlight 5
3 10 Subhalo and main halo annihilation
) Main halo only 1 − 2 sr
1 10 − s 2 − m 2 2 2
! 10
1 Total Flux 10 Subhalo Contribution
dN/dE (GeV Main Halo Contribution 3
E Astrophysical Background Fermi Data 1 2 3 10 10 10 0.5 1 1.5 2 2.5 3 3.5 4 Energy (GeV) DM mass (TeV)
FIG. 3: Same as fig. 2, but now including subhalo contribu- FIG. 6: Combined χ2 for the Fermi and PAMELA data as tions to the lepton fluxes. afunctionofthedarkmattermass,fortheunconstrained background. Dashed (blue) line: main halo DM annihilation 8 only. Solid (red) line: subhalo and main halo contributions combined.
3 10
−1 Subhalo and main halo annihilation 10 Main halo only
2 10 positron fraction
−2 positron fraction 2 ! 10 background PAMELA data 1 10 1 2 3 10 10 10 Energy (GeV)
FIG. 4: PAMELA data and predicted positron fraction of the 0 best main-halo-only fit to Fermi and PAMELA data, with an 10 unconstrained background. 0.5 1 1.5 2 2.5 3 3.5 4 DM mass (TeV)
2 + − FIG. 7: χ versus MDM for the Fermi e + e data using the GALPROP constrained background. Dashed (blue) line: main halo only. Solid (red) line: subhalos plus main halo.
−1 10 ) − 4.2. Constrained background + e + /(e
+ We performed a second analysis by taking the elec- e −2 tron and positron backgrounds to be those predicted by 10 GALPROP. In this case, although there is no good simul- positron fraction taneous fit to the combined PAMELA and Fermi data, background PAMELA data we nevertheless find that SH contributions improve the fit. In rough agreement with ref. [6], we find that the 1 2 3 10 10 10 PAMELA data require a boost factor several times higher Energy (GeV) than that needed to fit the Fermi data. 2 The plots of χ versus MDM, for both the MH-only and FIG. 5: Same as fig. 4, but now including subhalo contribu- MH+SH models, are shown respectively for the Fermi tions to the lepton fluxes. and PAMELA data in figures 7 and 8. It is striking that Gamma Ray
Best DM annihilation models predict much larger gamma ray fluxes near the galactic center(GC) from Final state radiation (Bresstrahlung) and Inverse Compton Scattering(ICS)
Most Constrints come from ICS here
We used the first year of Fermi LAT difuse gamma ray(Aug 8 2008 to Aug 25 2009) data available from NASA to constrain the allowable DM annhihilaiton
9 AddingNumerical subhalos and still explaining Result PAMELA and Fermi only slightly reduce the gamma ray constraints
Density of DM main halo in the GC is still high
DM profile and final state(4e,4μ,π,e) has some impact on it, but not enough
10 What about local subhalo?
100(s) subhalos inside the galaxy from Via Lactrea
small velocity, high density and close to us
PAMELA and Fermi excess can come from the close subhalo
We are within 3 kpc of subhalo centre, but further away than 20pc.
11 Where is the subhalo SH1 - SH4 are from Via Lactea , and SH5 is engineered by choosing parameters close to SH4. SH5 is with a higher density (so lower BF)
4 gamma rays 2 5 10
electrons 0 10
3 Predicted/observed flux
ï2 10 1 2
ï2 ï1 0 1 10 10 10 10 distance from Earth to SH center (kpc)
12 The five subhalos
They are atypical in the sense of needing a higher - than average central density. a large rs is unlikely at small distances from the GC due to tidal disruption.
Each subhalo was situated along an optimal axis, namely that connecting the earth to the GC.
The biggest contribution of Gamma Ray is from final -state bremsstrahlung rather than ICS.
13 11
−9 10 main halo scenario close subhalo (SH5) 0.2 kpc away −3 0.5 kpc away 10 −10 1 kpc away 60 GeV Fermi upper bound (500 GeV 10 bound too high to be seen) ) 1
− −4
sr 10 1 − s −11 2
− 10 cm # 1
− −5 10
−12 10 dE (MeV
" −6 /d 10 ! d
−13 10 single subhalo 500 GeV dipole −7 10 single subhalo 50 GeV dipole
−2 −1 0 1 2 −14 10 10 10 10 10 10 0 10 20 30 40 50 60 70 80 90 distance from Earth to SH center (kpc) galactic latitude (degrees) Figure 11: Dipole anisotropy δ of the cosmic ray electron Figure 10: Dependence of predicted gamma ray fluxes on and positron flux predicted by SH5 if it saturates the Fermi galactic latitude b,intheregion 9◦ < ! < 9◦ at E = 23 excess. BackgroundClose cosmic Subhalo ray electrons and positrons anisotropy are − included, and taken to be isotropic. δ increases monotonically GeV, the most constraining energy bin for the main halo sce- 11 nario. Black: main halo scenario (Einasto profile, BF = 110) with energy from the red line (60 GeV) to the black line (500 Dashed: subhalo 5, as specified in Table II. Background ICS GeV). −9 is included in both predictions, but signal is dominated by 10 main halo scenario close subhalo (SH5) 0.2 kpc away final state radiation. Dots are the Fermi data for that region −3 0.5 kpc away 10 and energy. −10 where C is the standard1 kpc away dipole power of the measured 60 GeV Fermi upper bound (500 GeV 10 1 bound too high to be seen) )
1 electron and positron flux in the sky. The Fermi LAT − −4
sr 10 1 − s −11 collaboration [47] have presented upper bounds on this 2 − 10 < −3 cm # 1 quantity. These range from δ 3 10 at E 60 GeV
− e would contribute less than 0.1% of the local DM density. Fermi Dipole anisotropy −5 up to δ < 9 10−2 at E 500∼ × GeV. Given10 a$ diffusive −12 e From previous works, we infer that extragalactic 10 dE (MeV ∼ of× electron and$ positron " model, this can be computed [47]: −6 bounds on this scenario are not as strong as the ones /d 10 ! d we have computed above. Bounds from dwarf spheroidal −13 10 # 3D(E) ne single subhalo 500 GeV dipole −7 galaxies could plausibly be important since the veloc- δ = |∇ |, 10 (22) single subhalo 50 GeV dipole c ne ity dispersions are of the same order as what is required −2 −1 0 1 2 −14 10 10 10 10 10 − 10 for our subhalo enhancement, i.e. 10 50 km s 1 0 10 20 30 40 50 60 70 80 90 distance from Earth to SH center (kpc) where D(E)galactic is thelatitude (degrees) diffusion coefficient (3) and ne is the [44]. However, the most stringent Fermi∼ − LAT bounds density of cosmic ray electrons and positrons,Figure including 11: Dipole anisotropy δ of the cosmic ray electron [45] from such galaxies put the upper limit on DM anni-Figureastrophysical 10: Dependence backgrounds. of predicted gamma Taking ray fluxes the on backgroundand positron to flux be predicted by SH5 if it saturates the Fermi hilation into a 2µ final state at around BF = 3000 if onlygalactic latitude b,intheregion 9◦ < ! < 9◦ at E = 23 excess. Background cosmic ray electrons and positrons are isotropic, we computed− the dipole anisotropy in the case final-state radiation is considered, and around 300 if ICSGeV, the most constraining energy bin for the main halo sce- included, and taken to be isotropic. δ increases monotonically nario.of Black: a single main halo close scenario subhalo (Einasto producing profile, BF = enough 110) with electrons energy from to the red line (60 GeV) to the black line (500 bounds are included as well. [46] computed the cosmo-Dashed:explain subhalo the 5, as Fermi specified excess. in Table II. In Background every case ICSδ fallsGeV). well below 14 logical dark matter annihilation bounds for the same 2µis includedbounds. in both Results predictions, for but SH5 signal are is presented dominated by in Figure 11. The final state scenario, and find that BF larger than 300 isfinal state radiation. Dots are the Fermi data for that region and energy.anisotropy rises monotonically with energy,where fromC 601 is GeV the standard dipole power of the measured excluded at the 90% confidence level. This is using the (red line) to 500 GeV (black line). electron and positron flux in the sky. The Fermi LAT results of the Millennium II structure formation simula- collaboration [47] have presented upper bounds on this < −3 tion, and is indeed model-dependent. Extrapolation towould contribute less than 0.1% of the local DM density. quantity. These range from δ 3 10 at Ee 60 GeV < −2 ∼ × $ From previous works, we infer that extragalactic up to δ 9 10 at Ee 500 GeV. Given a diffusive the 4µ scenario is independent of astrophysics. We can 5. PARTICLE PHYSICS REALIZATIONSmodel, this∼ can× be computed$ [47]: therefore take the results of [5, 33] who have construcedbounds on this scenario are not as strong as the ones we have computed above. Bounds from dwarf spheroidal 3D(E) # n bounds on both channels. They show that FSR boundsgalaxiesIn could the plausibly previous be important sections since we the have veloc- identified scenar- δ = |∇ e|, (22) c n are consistently an order of magnitude weaker in the 4µity dispersionsios where are of subhalos the same order could as what provide is required the observed excess e for our subhalo enhancement, i.e. 10 50 km s−1 case, given the softer photon spectrum in this scenario. PAMELA and Fermi leptons,∼ − from a purelywhere phenomeno-D(E) is the diffusion coefficient (3) and ne is the We can therefore take these extragalactic results to be[44]. However, the most stringent Fermi LAT bounds density of cosmic ray electrons and positrons, including [45] fromlogical such galaxies perspective. put the upper In particular, limit on DM certain anni- valuesastrophysical for the backgrounds. Taking the background to be far less constraining than the stringent bounds from thehilationannihilation into a 2µ final cross state at section around boostBF = 3000 factors if only are neededisotropic, for we the computed the dipole anisotropy in the case center of our own galaxy. final-statesubhalos, radiation and is considered, upper bounds and around for 300 that if ICS of theof a main single halo close subhalo producing enough electrons to Finally, we verify that this model does not saturatebounds are included as well. [46] computed the cosmo- explain the Fermi excess. In every case δ falls well below logical(depending dark matter annihilation upon assumptions bounds for the about same its 2µ density profile) bounds on dipole anisotropy of the cosmic ray e+ + e− bounds. Results for SH5 are presented in Figure 11. The final statewere scenario, derived. and It find is that interestingBF larger to than ask 300 whether is anisotropy simple rises par- monotonically with energy, from 60 GeV spectrum. The dipole anisotropy can be defined as excludedticle at physics the 90% confidence models with level. boost This is factors using the from(red Sommerfeld line) to 500 GeV (black line). resultsenhancement of the Millennium can II bestructure consistent formation with simula- these requirements. tion, and is indeed model-dependent. Extrapolation to C The simplest possibility for model building is dark mat- δ =3 1 , (21)the 4µ scenario is independent of astrophysics. We can 5. PARTICLE PHYSICS REALIZATIONS thereforeter take that the annihilates results of [5, 33] into who light have construced scalar or vector bosons, ! 4π bounds on both channels. They show that FSR bounds In the previous sections we have identified scenar- are consistently an order of magnitude weaker in the 4µ ios where subhalos could provide the observed excess case, given the softer photon spectrum in this scenario. PAMELA and Fermi leptons, from a purely phenomeno- We can therefore take these extragalactic results to be logical perspective. In particular, certain values for the far less constraining than the stringent bounds from the annihilation cross section boost factors are needed for the center of our own galaxy. subhalos, and upper bounds for that of the main halo Finally, we verify that this model does not saturate (depending upon assumptions about its density profile) bounds on dipole anisotropy of the cosmic ray e+ + e− were derived. It is interesting to ask whether simple par- spectrum. The dipole anisotropy can be defined as ticle physics models with boost factors from Sommerfeld enhancement can be consistent with these requirements. C1 The simplest possibility for model building is dark mat- δ =3 , (21) ter that annihilates into light scalar or vector bosons, ! 4π Particle Physics Realization certain values for the cross section BF are needed for the subhalos upper bound of BF for main halo
Is there simple Particle Physics model can be consistent with the requirements.
15 χ X enhancement on particle mass and velocity. First of all, we note that for mV 0, the potential becomes ... Coulomb-like. In this case→ the Schr¨odinger equation can be solved analytically; the resulting Sommerfeld enhancement is: χ X¯ πα S = (1 e−πα/β)−1. (3) FIG. 1: Ladder diagram giving rise to the Sommerfeld β − enhancement for χχ → XX annihilation, via the exchange of gauge bosons. For very small velocities (β 0), the boost S πα/β:thisiswhytheSommerfeldenhancementis→ $ often referred as a 1/v enhancement. On the other hand, S 1whenα/β 0, as one would expect. II. THE SOMMERFELD ENHANCEMENT → → It should however be noted that the 1/v behaviour breaks down at very small velocities. The reason is Dark matter annihilation cross sections in the low- that the condition for neglecting the Yukawa part of velocity regime can be enhanced through the so-called the potential is that the kinetic energy of the collision “Sommerfeld effect” [9, 10, 11, 12, 13, 14, 15]. This should be much larger than the boson mass mV times 2 non-relativistic quantum effect arises because, when the coupling constant α,i.e.,mβ αmV,andthis % the particles interact through some kind of force, their condition will not be fulfilled for very small values of wave function is distorted by the presence of a po- β.Thisisalsoevidentifweexpandthepotentialin 2 tential if their kinetic energy is low enough. In the powers of x = mVr;then,neglectingtermsoforderx language of quantum field theory, this correspond to or smaller, the Schr¨odinger equation can be written as the contribution of “ladder” Feynman diagrams like (the prime denotes the derivative with respect to x): the one shown in Fig. 1 in which the force carrier is α ψ β2 α exchanged many times before the annihilation finally ψ"" + = + ψ, (4) occurs. This gives rise to (non-perturbative) correc- ε x !− ε2 ε " tions to the cross section for the process under con- having defined ε = mV/m. The Coulomb case is sideration. The actual annihilation cross section times 2 velocity will then be: recovered for β αε,orexactlytheconditionon the kinetic energy% stated above. It is useful to de- ∗ ∗ σv = S (σv) (1) fine β αmV/m such that β β is the velocity 0 ≡ % regime where# the Coulomb approximation for the po- where (σv)0 is the tree level cross section times veloc- tential is valid. ity, and in the following we will refer to the factor S as the “Sommerfeld boost” or “Sommerfeld enhance- Another simple, classical interpretation of this result Sommerfeld ment”enhancement1. is the following. The range of the Yukawa interaction is given by R m−1.Thenthecrossingtimescaleis $ V The non-relativistic particlesIn this sectionare moving we will study in this some process in a semi- given by tcross R/v 1/βmV.Ontheotherhand, quantitative way using a simple case, namely that of the dynamical$ time scale$ associated to the potential potential. The wave function is distorted by the 3 3 aparticleinteractingthroughaYukawapotential.We is tdyn R m/α m/αmV.Thenthecondition ∗$ $ attractive potential. Or considerSumming a dark matter over particleall the of ladder mass m.Let ψ(r) β β is# equivalent to#tcross tdyn,i.e.,thecrossing be the reduced two-body wave function for the s-wave time% should be much smaller than' the dynamical time- Feynman diagram (QFT)annihilation; in the non-relativistic limit, it will obey scale. Finally, we note that since in the Coulomb case the radial Schr¨odinger equation: S 1/β for α β,theregionwheretheSommerfeld χ ∼ % ∗ X enhancement on particle mass and velocity. First of enhancement actually has a 1/v behaviour is β 2 all, we note1 d thatψ(r for) mV 0, the potential becomes ' → 2 β α.Itisinterestingtonoticethatthisregion ... Coulomb-like. In2 this caseV ( ther)ψ Schr¨odinger(r)= m equationβ ψ(r), (2) ' < can be solvedm dr analytically;− the resulting− Sommerfeld does not exist at all when m mV/α. enhancement is: ∼ χ ¯ X where β is the velocity of the particle and V (r)= ∗ πα −πα/β −1 α −mVr S = (1 e ) . (3) The other interesting regime to examine is β β . FIG. 1: Ladder diagram giving rise to the Sommerfeld r e is an attractiveβ − Yukawa potential mediated ' → − Following the discussion above, this corresponds to enhancement for χχ XX annihilation, via the exchangeby a boson of mass mV. of gaugeConsider bosons. a DM particle withFor verya U(1) small velocities coupling (β 0), the to boost a Sdark the potential energy dominating over the kinetic term. → $ πα/β:thisiswhytheSommerfeldenhancementis Referring again to the form (4) for x 1ofthe gauge boson of mass μ(O(1)Gev)Theoften Sommerfeld referred as a enhancement 1/v enhancement.S Oncan the be other calculated by ' hand, S 1whenα/β 0, as one would expect. Schr¨odinger equation, this becomes: II. THE SOMMERFELD ENHANCEMENT solving the→ Schr¨odinger→ equation with the boundary conditionIt shouldd howeverψ/dr = beim notedβψ thatas ther 1/v behaviour.Eq.(2)canbe α "" α ψ α Using realistic velocity easilydistributionbreaks solved down numerically.at very smalland velocities. It correct is however→∞ The reason useful, is to con- ψ + = ψ. (5) Dark matter annihilation cross sections in the low- that the condition for neglecting the Yukawa part of ε x ε velocitythis regime typically can be enhanced throughpredicts the so-called toosider themuch some potential particular isenhancement that the kineticlimits energy in order of the. to collision gain some qual- “Sommerfeld effect” [9, 10, 11, 12, 13, 14, 15]. This should be much larger than the boson mass mV times The positiveness of the right-hand side of the equation itative insight into the dependence2 of the Sommerfeld non-relativistic quantum effect arises because, when the coupling constant α,i.e.,mβ αmV,andthis Gamma ray constraints are immediately saturated.% points to the existence of bound states. In fact, this the particles interact through some kind of force, their condition will not be fulfilled for very small values of wave function is distorted by the presence of a po- β.Thisisalsoevidentifweexpandthepotentialin equation has the same form as the one describing the 2 16 tential if their kinetic energy is low enough. In the powers of x = mVr;then,neglectingtermsoforderx hydrogen atom. Then bound states exist when α/ε language of quantum field theory, this correspond to1 or smaller, the Schr¨odinger equation can be written as is an even integer, i.e. when: # the contribution of “ladder” Feynman diagrams like In(the the prime case denotes of repulsive the derivative forces, with the respect Sommerfeld to x): “enhance- ment” can actually be S<1, although we will not consider the one shown in Fig. 1 in which the force carrier is 2 2 this possibility"" here.α ψ β α exchanged many times before the annihilation finally ψ + = + ψ, (4) m =4mVn /α,n=1, 2,... (6) occurs. This gives rise to (non-perturbative) correc- ε x !− ε2 ε " tions to the cross section for the process under con- having defined ε = mV/m. The Coulomb case is sideration. The actual annihilation cross section times 2 velocity will then be: recovered for β αε,orexactlytheconditionon the kinetic energy% stated above. It is useful to de- ∗ ∗ σv = S (σv) (1) fine β αmV/m such that β β is the velocity 0 ≡ % regime where# the Coulomb approximation for the po- where (σv)0 is the tree level cross section times veloc- tential is valid. ity, and in the following we will refer to the factor S as the “Sommerfeld boost” or “Sommerfeld enhance- Another simple, classical interpretation of this result ment” 1. is the following. The range of the Yukawa interaction is given by R m−1.Thenthecrossingtimescaleis $ V In this section we will study this process in a semi- given by tcross R/v 1/βmV.Ontheotherhand, quantitative way using a simple case, namely that of the dynamical$ time scale$ associated to the potential 3 3 aparticleinteractingthroughaYukawapotential.We is tdyn R m/α m/αmV.Thenthecondition ∗$ $ consider a dark matter particle of mass m.Letψ(r) β β is# equivalent to#tcross tdyn,i.e.,thecrossing be the reduced two-body wave function for the s-wave time% should be much smaller than' the dynamical time- annihilation; in the non-relativistic limit, it will obey scale. Finally, we note that since in the Coulomb case the radial Schr¨odinger equation: S 1/β for α β,theregionwheretheSommerfeld enhancement∼ actually% has a 1/v behaviour is β∗ 2 ' 1 d ψ(r) 2 β α.Itisinterestingtonoticethatthisregion 2 V (r)ψ(r)= mβ ψ(r), (2) ' m dr − − does not exist at all when m < mV/α. ∼ where β is the velocity of the particle and V (r)= The other interesting regime to examine is β β∗. α e−mVr is an attractive Yukawa potential mediated ' − r Following the discussion above, this corresponds to by a boson of mass mV. the potential energy dominating over the kinetic term. Referring again to the form (4) for x 1ofthe The Sommerfeld enhancement S can be calculated by Schr¨odinger equation, this becomes: ' solving the Schr¨odinger equation with the boundary condition dψ/dr = imβψ as r .Eq.(2)canbe α ψ α →∞ ψ"" + = ψ. (5) easily solved numerically. It is however useful to con- ε x ε sider some particular limits in order to gain some qual- itative insight into the dependence of the Sommerfeld The positiveness of the right-hand side of the equation points to the existence of bound states. In fact, this equation has the same form as the one describing the hydrogen atom. Then bound states exist when α/ε 1 In the case of repulsive forces, the Sommerfeld “enhance- is an even integer, i.e. when: # ment” can actually be S<1, although we will not consider 2 this possibility here. m =4mVn /α,n=1, 2,... (6) Effective Boost Factor It is not obvious that one can find models with the desired BF for subhalos and main halo. leptophilic DM is a subdominant component of the total DM, comprising some fraction 1/f.
Cross section <σv>∼α2/M2
Parametrize α=√f αth, By solving Boltzmann eq., the relic density is proportional to <σv>-1 Therefore, the rate of annihilations goes like ρ2σ ∝ 1/f. Accordingly, we define an effective BF S S¯ = f 17 Theoretical fits
Given α ⇒ boost factors for main and subhalos
The working example(larger f is needed for a cuspy main halo) 15
Isothermal, f = 50, V = 201 km/s Einasto profile, f = 500, V = 277 km/s max max 5 3 SH1 SH2 subhalos 2.5 4 subhalos " " SH4 SH3 _ _ S S
! ! 10
10 2 SH5 3 log log SH4 1.5 MH 2 MH 1 SH5 main halo main halo 1 0.5 1 1.5 2 0 0.5 1 1.5 2 µ (GeV) µ (GeV)
Figure 15: Predicted effective boost factors S¯ as a function of gauge boson mass (solid curves) and target values (or upper limit in case of main halo, dashed curves) to explain! " PAMELA/Fermi lepton observations and Fermi gamma ray constraints. Pair 18 of dashed curves for main halo (MH) correspond respectively to 1 and 2σ upper limits. Left panel is for f =100,Vmax =277 km/s, Einasto main halo profile; right is for f =25,V,max =201km/s,isothermalmainhaloprofile.Subhalosarethoseof table II. Points which satisfy all constraints are those where subhalo curves intersect their corresponding dashed linewhilethe main halo curve lies below its dashed lines.
One advantage of requiring large f is that the corre- -21 sponding dilution of the DM density by 1/f insures that the model satisfies stringent CMB constraints from anni- hilations in the early universe changing the optical depth -22 µ
[53–55], as pointed out in [14]. The CMB constraint is / s ) +Fermi - CMB4 reionizationconstraint 3 shown in fig. 16, along with the PAMELA/Fermi allowed µ 4µ IC constraint regions from ref. [33] for 4e and 4µ final states. The 4e -23 v (cm PAMELA - 4
case is allowed by the CMB constraint, but 4µ is ruled # +Fermi - 4e 4e IC constraint out. Because our model has at most a fraction of 0.45 of muons in the final state, it is probably already safe, but log -24 the additional weakening of the bound by the factor 1/f ensures that this will be the case. Similarly, our scenario PAMELA - 4e -25 overcomes the no-go result of ref. [10], which pointed out 2 2.5 3 3.5 4 that Sommerfeld enhanced annihilation in the early uni- log M (GeV) verse leads to constraints on the MH boost factor which are lower than those needed to explain the lepton anoma- Figure 16: Allowed regions for PAMELA and Fermi excess lies. Our MH boost factor can satisfy these constraints leptons, and upper bounds from inverse Compton gamma since the MH is no longer considered to be the source of rays, from ref. [33], for Einasto profile with α =0.17 and the excess leptons. rs =20kpc.CMBconstraintisfromref.[53].
6. DISCUSSION AND CONCLUSIONS
The Sommerfeld enhancement is nearly saturated for We have shown that gamma ray constraints on the low velocities of the subhalos at these large values of leptophilic annihilating dark matter are significantly αg 0.1 0.35, so their S¯ curves are nearly overlapping stronger than in previous studies, when we take into except∼ at− the smallest gauge# $ boson masses. The main account the contributions to inverse Compton scatter- halo boost factor is not saturated on the other hand, and ing from primary and secondary electrons and positrons, lies below the FSR bound for most values of µ.Wehave before including excess leptons from the DM annihila- chosen the gauge couplings, parametrized by f, to nearly tion. We attribute part of this difference to the method saturate the FSR bound. By taking larger αg (larger f), of solving the diffusion equation (1) — fully numeri- the bounds could be satisfied by a larger margin. But cal rather than semi-analytic — meaning that the (r, z) this would also reduce the S¯ values of the subhalos space-dependence of the diffusion coefficient is taken into by a similar amount, making# it$ more difficult to get a account. The difference between the predicted and ob- large enough lepton signal from SH1 SH3. SH4 and SH5 served spectra of gamma rays is greatly reduced, leaving would remain robust possible explanations.− little room for new contributions. Because of this, even Summary We can get better fit to the lepton data by including all the DM subhalos, but the gamma ray constraints are still too strong close subhalo can explain PAMELA/Fermi, also consistent with Fermi LAT diffuse gamma ray survey
Caveat: Need large, dense subhalos
A realistic U(1) model typically produce too much enhancement. This can be solved if only part of the DM can annihilate to SM particles in this channel.
19 !ank y"
20 5
Freely-varying background (Einasto) 2 2 2 MDM (TeV) χFermi χPAMELA χtotal BMH BSH MH (4e) 0.85 15.5 18.7 34.3 90.3 Release Gamma− Ray MH+SH 1.2 2.3 14.2 16.5 92.8 3774 Fixed GALPROP background (Einasto) MH (4e) 1.0 8.2 144 152 110 − MH+SH 2.2 2.1 175 177 146 1946 MH (e, µ, π) 1.2 3.8 109 112 118 Constraints− Isothermal profile (fixed background) MH (4e) 1.0 9.1 186 195 113 − MH (e, µ, π) 1.2 3.0 151 154 119 Electron-positron distribution−5 Table I: First four rows: best fit results from [11], assuming Freely-varying background (Einasto) Einasto profile. By varying the boost factors of the main halo 2 2 2 and faraway subhalos separately, we found that the fit to the MDM (TeV) χ χPAMELA χ BMH BSH Fermi total PAMELA and Fermi data from MH annihilations alone could MH (4e) 0.85 15.5 18.7 34.3 90.3 − be improved by inclusion of SH annihilations as shown. Last MH+SH 1.2 2.3 14.2 16.5 92.8 3774 two rows: new fit for isothermal profile (rs =3.2kpc,ρs = Fixed GALPROP background (Einasto) 3.0GeV/cm3), main-halo-only scenario from this work, using MH (4e) 1.0 8.2 144 152 110 the fixed GALPROP background, and same parameters as in − MH+SH 2.2 2.1 175 177 146 1946 [11]. We assume the annilation to the 4e final state, except in the cases “MH (e, µ, π)” which indicates the the process MH (e, µ, π) 1.2 3.8 109 112 118 ± ± ± − χχ BB 4$,where$ stands for e , µ or π ,with → → Isothermal profile (fixed background) branching ratios re = rµ =0.45 and rπ =0.1asexplainedin MH (4e) 1.0 9.1 186 195 113 Section 2.2. − MH (e, µ, π) 1.2 3.0 151 154 119 − Table I: First four rows: best fit results from [11], assuming Einasto profile. By varying the boost factors of the main halo In each fi,thesquarerootfactorcomesfromthephase Figure 2: Simulated steady-state distribution of electronsand and faraway subhalos separately, we found that the fit tosubhalos the space, while the gamma rest is from the squared rays matrix ele-arepositrons from from DM all annihilation the within directions the Milky Way diffu- sion zone. The galactic center is located at z =0,r =0; PAMELA and Fermi data from MH annihilations alone could ment for the decay. Below threshold, fi is defined to be be improved by inclusion of SH annihilations as shown. Last zero. For a gauge boson with a mass µ > 1GeV,wefind red corresponds to high densities, blue to low densities. Top: two rows: new fit for isothermal profile (rs =3.2kpc,ρs = leptons from the main halo only. Bottom: leptons from the 3 r = r =0.45 and r =0.1. In this case∼ the electrons 3.0GeV/cm ), main-halo-only scenario from this work, usingoutsidee µ π subhalos only, sourced from the diffusion zone boundary. Note the fixed GALPROP background, and same parameters as in produced from the final decay of the µ’s and π’s peak at that the scales are different: the peak main halo density (at [11]. We assume the annilation to the 4e final state, except a lower energy, thus requiring a slightly higher mass of the GC) is about 200 times larger than the peak subhalo den- in the cases “MH (e, µ, π)” which indicates the the process MDM =1.2 TeV in order to fit the Fermi and PAMELA sity (near the edge of the diffusion zone) χχ BB 4$,where$ stands for e±, µ± or π±,with data. This is much smaller than the well-known M → → DM branching ratios re = rµ =0.45 and rπ =0.1asexplainedinMain2.2 TeV besthalo fit in the gamma pure-muon final state [4,rays 5, 33] be-" are largely from the centre of Section 2.2. cause of the large fraction of gauge bosons still decaying The astrophysical and particle physics dependences of directly to high-energy electrons. These results are also each flux can be factorized as theshown galaxy, in Table I. because of the peak of the density profile 2 In each fi,thesquarerootfactorcomesfromthephase Figure 2: Simulated steady-state distribution of electronsand dΦmain 1 σv ρ! dN positrons from DM annihilation within the Milky Way diffu- = # $r! J¯main (10) space, while the rest is from the squared matrix ele- dE dΩ 2 4π m2 dE of DMsion zone. The galactic center is located at z =0,r =0; γ χ γ ment for the decay. Below threshold, fi is defined to be zero. For a gauge boson with a mass µ > 1GeV,wefind red corresponds to high densities, blue to low densities. Top: leptons from the main halo only. Bottom: leptons from the and r = r =0.45 and r =0.1. In this case∼ the electrons 3. GAMMA RAY COMPUTATION FROM e µ π subhalos only, sourced from the diffusion zone boundary. Note produced from the final decay of the µ’s and π’s peak at INVERSE COMPTON SCATTERING AND that the scales are different: the peak main halo density (at dΦsub 1 dN ¯ a lower energy, thus requiring a slightly higher mass of BREMSSTRAHLUNG = σv Jsub. (11) the GC) is about 200 times larger than the peak subhalo den- dEγ dΩ 2# $dEγ MDM =1.2 TeV in order to fit the Fermi and PAMELA sity (near the edge of the diffusion zone) 21 data. This is much smaller than the well-known MDM 3.1. “Prompt” gamma ray emission In each case, the J¯i factor depends only upon astrophys- 2.2 TeV best fit in the pure-muon final state [4, 5, 33] be-" (bremsstrahlung) ical inputs. The main halo J factor is defined as a line cause of the large fraction of gauge bosons still decaying The astrophysical and particle physics dependences of of sight (l.o.s.) integral of flux at each pixel: directly to high-energy electrons. These results are also each flux can be factorized as shown in Table I. Prompt gamma ray emission appears in the final stage 2 2 1 ds ρ [r(s, ψ)] dΦmain 1 σv ρ! dN ¯ main of DM annihilation,= softening# $r the leptonJ¯ spectrum.(10) The Jmain = dΩ . (12) ! 2 main ∆Ω r ρ flux can be divideddEγ dΩ into2 main4π halomχ d andEγ subhalo parts: !∆Ω !l.o.s. ! " ! # and In the case of flux originating from many distant subha- 3. GAMMA RAY COMPUTATION FROM dΦ dΦmain dΦsub los, we may treat each one as a point source of radiation. INVERSE COMPTON SCATTERING AND = + . (9) dΦsub 1 dN ¯ In this case, the diffuse flux per solid angle requires a BREMSSTRAHLUNG dEγ dΩ =dEγ dσΩv dEJsubγ dΩ. (11) dEγ dΩ 2# $dEγ
3.1. “Prompt” gamma ray emission In each case, the J¯i factor depends only upon astrophys- (bremsstrahlung) ical inputs. The main halo J factor is defined as a line of sight (l.o.s.) integral of flux at each pixel:
Prompt gamma ray emission appears in the final stage 2 1 ds ρ [r(s, ψ)] of DM annihilation, softening the lepton spectrum. The J¯ = dΩ main . (12) main ∆Ω r ρ flux can be divided into main halo and subhalo parts: !∆Ω !l.o.s. ! " ! # In the case of flux originating from many distant subha- dΦ dΦmain dΦsub los, we may treat each one as a point source of radiation. = + . (9) dEγ dΩ dEγ dΩ dEγ dΩ In this case, the diffuse flux per solid angle requires a Gamma Ray Constraints
We obtained the constraints for the MH boost factor in the case of Einasto DM profile, annihilation to 4e
BF<25(35) at 1σ(2σ) for MDM=1Tev BF<42(52) at 1σ(2σ) for MDM=2.2Tev
Increasing intermediate gauge boson to allow decay to μ and π
BF<23(38) at 1σ(2σ) for MDM=1.2Tev
And choosing a flatter isothermal DM density profile BF<62(72) at 1σ(2σ) for MDM=1.2Tev
22 Diffusion Equation Semi-analytic approach to solve the diffusion eq. d ∂ ψe (x, p,t)=Qe (x,E)+ (D(E) ψe (x, p,t)) + [b(x,E)ψ (x, p,t)] dt ± ± ∇ · ∇ ± ∂E ± Q is the source term from subhalo 1 ρ(x) 2 dN n2 dN Q = σv = DM BF σv . 2 M dE 2 0 dE Subhalo has the similar DM density distribution as the main halo. 2 r α ρ (r)=ρ exp 1 Ein s −α r − s ρs ρiso(r)= 2 1+(r/rs)
23 Five subhalos Sample subhalos from the Via Lactea Ⅱ simulation. Using them fitting the PAMELA/Fermi lepton fluxes and Fermi gamma ray fluxes. Subhalo rs (kpc) ρs log BF dmin (pc) Vmax (km/s) 1 0.01 69 4.74 33.9 2.9 2 0.1 3.46 4.34 95.5 6.7 3 3.2 0.04 3.76 178 22 4 0.9 1.27 2.35 165 36 5 1.1 2.0 1.70 170 55 Main halo, 4e channel Einasto 25 0.048 < 1.40 201 277 1.48 − − Isothermal 3.2 2.32 < 1.81 201 277 1.88 − − Main halo, 4e + 4µ +4π channel Einasto 25 0.048 < 1.36 201 277 1.45 − − Isothermal 3.2 2.32 < 1.80 201 277 1.86 − −
24 Close Subhalo vs. main hao
ï9 10 main halo scenario close subhalo (SH5) 0.2 kpc away 0.5 kpc away
ï10 1 kpc away 10
Gamma ray fluxes on ) 1 ï sr 1 ï
s ï11 2 galactic latitude b, in the ï 10 cm 1 o o ï region -9 ï14 10 0 10 20 30 40 50 60 70 80 90 galactic latitude (degrees) 25 Relic Density Constraint 13 Sommerfeld enhancement sensitively relies on α 0.035 There are two kinds of final states for annihilation of DM in this class of models: into a pair of gauge bosons -1 M = 1 TeV Model: DMBµ ,has by virtual a U(1) DM exchangegauge in symmetry. the t and u channels, or 0.03 M = 2 TeV into dark Higgs bosons h,byexchangeofagaugebosonν ε ν μ M = 3 TeV Kineticin the s channel. mixing Assuming B μ theF DM. (χ) is much heavier than the final states, the respective squared amplitudes, 0.025 ( M / TeV) There are averagedtwo kinds over initial of final and summed states over for final annihilation spins, are of # DM 4 2 g, th 1 2 4g (1 + 2v ), χχ BB " 0.02 = → 4 |M| 1 g4q2(1 v2 cos2 θ), χχ hh¯ # 2 − → " (29) 0 5 10 q is15 the U(1)where chargeq is the U(1) of h charge relative of h relative to χ. to χ (replace 2 q2 q2 for multiple Higgs bosons), θ is the scat- !i q i i i tering→ angle, and we have included the leading depen- $ Figure 13: Value of gauge coupling leading to correct ther- dence on the initial velocity v in the center of mass 2 mal relic DM density, αg,th/M ,versussquaredchargeofdark frame. The factor cos θ averages to 2/3 in the inte- Higgs bosons in U(1) model, for several values of DM mass gral over θ. In computing the associated cross section, M. it must be remembered that the 2B final state consists of identical particles, while the Higgs channel does not. 26 The total amplitude can therefore be written in the form With these ingredients, we can compute an average 1 2 =4g4(a + bv2), with Sommerfeld enhancement factor S for each subhalo: 4 |M| ! " $ 1 2 1 2 r2 2 2 3 3 1 a =1+4 i qi ,b=2(1 12 i qi )(30) dr r ρ d v1 d v2 f(v1) f(v2)S( "v1 "v2 ) − S = r1 2 | − | ! " r2 dr r2 ρ2 if we use the phase$ space for identical particles.$ ! ! r1 (28) To find the cross section relevant during freeze-out in ! The factor of 1 in the argument of S occurs because the the early universe, we thermally average the v-dependent 2 σv following ref. [52]. We include approximately the v appearing in eq. (23) through #v is half of the relative rel velocity. ρ2 is the appropriate weighting factor because effect of Sommerfeld enhancement as described there, to the rate of annihilations is proportional to σv ρ2.For obtain ! " the subhalos, the range of integration for r is from 0 2 παg to , but for the main halo we take lower and upper σv = a 1+α π M ∞ ! rel" ∼ 2M 2 g T limits r1,2 that correspond to the angular region of the % % & ' sky that is used to set the gamma ray constraints: r1 = T 4 3 M + (b 3 a) 2 + αg π T (31) 0.67 kpc and r2 =1.34 kpc. The reason is that the M − bound S<¯ 30 for the main halo comes from the gamma ( & )' ray constraint rather than from lepton production. We The terms that are subleading in αg,butenhancedby are thus interested in the boost factor relevant to the M/T, are due to the Sommerfeld correction. We ap- ◦ ◦ region 4.5 < b < 9 of galactic latitude. The distances proximate the freezeout temperature as T ∼= M/20, the of closest approach| | to the galactic center, hence largest *usual result of solving the Boltzmann equation for DM rate of γ ray production associated with these lines of in the TeV mass range, and equate σvrel to the value −26 3 ! " sight, are given by r = r" sin b. σv 0 =3 10 cm /s usually assumed to give the !correct" relic× density. This gives an implicit equation for 2 2 αg,th of the form αg = c1M σv 0/(1+c2αg), which how- 5.2. Relic Density Constraint ever quickly converges by numerically! " iterating. Fig. 13 2 displays the resulting dependence of αg,th/M on i qi The enhancement factor (23) depends rather strongly for several values of M. $ on the gauge coupling αg; therefore it is interesting to The bound that the density of the leptophilic DM com- know what constraint the relic density places upon αg. ponent not exceed the total DM density is αg > αg,th. The effect of a Sommerfeld-enhanced DM model on the We parametrize the coupling by αg = √f αg,th with relic densitie has been discussed by [51]. Notice that DM f>1 in what follows. transforming under a U(1) gauge symmetry as we have assumed must be Dirac and therefore could have a relic density through its asymmetry, similar to baryons. How- 5.3. Interpolation between 4e and mixed final states ever, unless the DM was never in thermal equilibrium, then αg should not be less than the usual value αg,th In our numerical computations with GALPROP, we leading to the correct relic density, since otherwise the considered two cases for the final state annihilation chan- thermal component will be too large. nels: either χχ 4e, applicable for gauge bosons with → 13 0.035 There are two kinds of final states for annihilation of DM in this class of models: into a pair of gauge bosons -1 M = 1 TeV Bµ, by virtual DM exchange in the t and u channels, or 0.03 M = 2 TeV into dark Higgs bosons h,byexchangeofagaugeboson M = 3 TeV in the s channel. Assuming the DM (χ) is much heavier than the final states, the respective squared amplitudes, 0.025 ( M / TeV) averaged over initial and summed over final spins, are # 13 4 2 g, th 1 2 4g (1 + 2v ), χχ BB " 0.02 = → 0.035 There are4 two|M| kinds of1 g4 finalq2(1 statesv2 cos2 forθ), annihilationχχ hh¯ of # 2 − → DM in this" class of models: into a pair of gauge bosons(29) -1 0 5 10 15 where q is the U(1) charge of h relative to χ (replace M = 1 TeV2 Bµ, by virtual DM exchange in the t and u channels, or 0.03 ! q q2 q2 for multiple Higgs bosons), θ is the scat- M = 2 TeVi i into dark Higgs→ i bosonsi h,byexchangeofagaugeboson M = 3 TeV tering angle, and we have included the leading depen- in the s channel.$ Assuming the DM (χ) is much heavier Figure 13: Value of gauge coupling leading to correct ther- dence on the initial velocity v in the center of mass than theframe. final states, The factor the cos respective2 θ averages squared to 2/3 amplitudes, in the inte- 0.025 mal relic DM density, αg,th/M ,versussquaredchargeofdark ( M / TeV) Higgs bosons in U(1) model, for several values of DM mass gral over θ. In computing the associated cross section, averaged over initial and summed over final spins, are # M. it must be remembered that the 2B final state consists of identical particles,4 while2 the Higgs channel does not. g, th 1 RelicThe total2 amplitude4g (1 +Density can 2v therefore), be writtenχχ inBB the form Constraint " 0.02 = → With these ingredients, we can compute an average4 1 2 1 44 2 2 2 2 ¯ 4|M| #=42gg(aq+(1bv ),v withcos θ), χχ hh Sommerfeld enhancement factor S for each subhalo: " |M| − → ! " 1 2 1 2 (29) r2 $ a =1+ q ,b=2(1 q )(30) 0 5 2 2 310 3 15 1 4 i i 12 i i r1 dr r ρ d v1 d v2 f(v1) f(v2)S( 2 "v1 where"v2 ) q is the U(1) charge of h relative− to χ (replace S = 2 r2 | − 2| 2 ! " ! q dr r2 ρ2 q if weq usefor the multiple phase$ space Higgs for identical bosons), particles.$θ is the scat- ! i !i r1 → i i tering(28) angle,IncludeTo find and the we cross approximately have section included relevant the during leading freeze-out the depen- effect in of Sommerfeld ! The factor of 1 in the argument of S occurs becausedence the $ onthe the early initial universe, velocity we thermallyv in average the center the v-dependent of mass Figure 13: Value of gauge coupling2 leading to correct ther- σvenhancement,following ref. [52]. We the include cross approximately section the is given by v appearing in eq. (23) through #v is half of the relativeframe. Therel factor cos2 θ averages to 2/3 in the inte- mal relic DM density,velocity.αg,th/Mρ2 is,versussquaredchargeofdark the appropriate weighting factor because effect of Sommerfeld enhancement as described there, to 2 gral over θ. In computing the associated cross section, Higgs bosons in U(1)the model,rate of annihilations for several values is proportional of DM mass to σv ρ .For obtain(Cline, Frey, Fang 1008.1784) 13 ! " M. the subhalos, the range of integration for r is fromit must 0 be remembered that2 the 2B final state consists παg M to , but for the main halo we take lower and upperof identical particles,σvrel = while thea Higgs1+αg channelπ does not.0.035 There are two kinds of final states for annihilation of ∞ ! " ∼ 2M 2 T limits r1,2 that correspond to the angular region ofThe the total amplitude can therefore% % be written' in the form DM in this class of models: into a pair of gauge bosons & -1 With these ingredients,sky that is used we can to set compute the gamma an ray average constraints: 1r = 2 4 2 T Bµ, by virtual DM exchange in the t and u channels, or 1 =4g (a + bv+ ), with(b 4 a) 3 + α π M (31) M = 1 TeV 0.67 kpc and r =1.34 kpc. The reason is that4 the 3 2 g T 0.03 M = 2 TeV into dark Higgs bosons h,byexchangeofagaugeboson Sommerfeld enhancement factor2 S for each subhalo: |M| M − M = 3 TeV ¯ ! " ( & )' in the s channel. Assuming the DM (χ) is much heavier bound S<30 for the main halo comes from the gamma$ 1 2 1 2 r2 1 Thea =1+ terms that areq ,b subleading=2(1 in α ,butenhancedbyq )(30) than the final states, the respective squared amplitudes, 2 ray2 constraint3 3 rather than from lepton production. We 4 i i 12g i i 0.025 dr r ρ d v1 d v2 f(v1) f(v2)S( "v1 "v2 ) − ( M / TeV) r1 are thus interested in the boost2 factor relevant to the M/T, are due to the Sommerfeld correction. We ap- averaged over initial and summed over final spins, are S = r2 | − | # ! " region 4.5◦ < drb < r29◦ρ2of galactic latitude. The distancesif we useproximate the phase the$ space freezeout for temperature identical particles. as$T = M/20, the ! ! r1 4 2 | | * ∼ g, th 1 2 4g (1 + 2v ), χχ BB " 0.02 = of closest approach to the galactic center, hence largestTo findusual the result cross of section solving the relevant Boltzmann during equation freeze-out for DM in 1 4 2 2 2 → ¯ (28) 4 |M| # g q (1 v cos θ), χχ hh 1 rate of γ !ray production associated with these linesthe of earlyin universe, the TeV mass we thermally range, and average equate σ thevrel vto-dependent the value 2 − → The factor of in the argument of S occurs because the α=√f αth−26, f>13 ! " " (29) 2 sight, are given by r = r" sin b. σv 0 =3 10 cm /s usually assumed to give the σvrel following! " ref.× [52]. We include approximately the 0 5 10 15 where q is the U(1) charge of h relative to χ (replace v appearing in eq. (23) through #v is half of the relative correct relic density. This gives an implicit equation for 2 q2 q2 for multiple Higgs bosons), θ is the scat- 2 effect of Sommerfeld enhancement2 2 as described there, to !i q i i i velocity. ρ is the appropriate weighting factor because αg,th of the form αg = c1M σv 0/(1+c2αg), which how- tering→ angle, and we have included the leading depen- 2 ! " the rate of annihilations is5.2. proportional Relic Density to σ Constraintv ρ .For obtain ever quickly converges by numerically iterating. Fig. 13 dence$ on the initial velocity v in the center of mass Figure2 13: Value of gauge coupling leading to correct ther- ! " displays the resulting dependence of αg,th/M on q 2 the subhalos, the range of integration for r is from 0 2 mali i relic DM density, αg,th/M ,versussquaredchargeofdark frame. The factor cos θ averages to 2/3 in the inte- The enhancement factor (23) depends rather strongly for several valuesπαg of M. M Higgs bosons in U(1) model, for several values of DM mass gral over θ. In computing the associated cross section, to , but for the main halo we take lower and upper σvrel = a 1+αg π T $M. it must be remembered that the 2B final state consists ∞ on the gauge coupling αg; therefore it is interesting to ! The" bound∼ 2 thatM 2 the density of the leptophilic DM com- limits r1,2 that correspond to the angular region of the of identical particles, while the Higgs channel does not. know what constraint the relic density places upon αg. ponent not exceed the% total% DM& density' is αg > αg,th. The27 total amplitude can therefore be written in the form sky that is used toThe set eff theect of gamma a Sommerfeld-enhanced ray constraints: DMr1 model= on the We parametrizeT the coupling4 3 by αg = M√f αg,th with + (b a) + α π (31)With these ingredients, we can compute an average 1 2 4 2 3 2 g T 4 =4g (a + bv ), with 0.67 kpc and r2 relic=1 densitie.34 kpc. has been The discussed reason by is [51]. that Notice the that DM f>1 in whatM follows.− Sommerfeld enhancement factor S for each subhalo: |M| ¯ transforming under a U(1) gauge symmetry as we have ' ! " 1 2 1 2 bound S<30 for the main halo comes from the gamma ( & ) r2 $ a =1+ q ,b=2(1 q )(30) dr r2 ρ2 d3v d3v f(v ) f(v )S( 1 "v "v ) 4 i i 12 i i assumed must be Dirac and therefore could have a relic r1 1 2 1 2 2 1 2 − ray constraint rather than from lepton production. We The terms that are subleading in αg,butenhancedbyS = | − | density through its asymmetry, similar to baryons. How- 5.3. Interpolation between 4e and mixed final states! " r2 dr r2 ρ2 if we use the phase$ space for identical particles.$ ! ! r1 are thus interested in the boost factor relevant to the M/T, are due to the Sommerfeld correction. We ap- To find the cross section relevant during freeze-out in ◦ ever,◦ unless the DM was never in thermal equilibrium, (28) 1 ! the early universe, we thermally average the v-dependent region 4.5 < b The failure to satisfy the bounds drives us to consider f 14 2 #i qi = 16 mass µ, and similarly for the main halo, except here we have an upper bound on S¯ rather than a best-fit value. This bound in fact presents" # the biggest challenge to find- 2.5 ing a working particle physics model. For αg close to predicted the thermal relic density value αg,th, the predicted boost " < factor of the main halo far exceeds the bound S¯ 30, _ S " # ! even if we try to decrease S¯ by reducing αg via a∼ large 2 isothermal, V = 201 km/s " # max hidden Higgs content or by increasing the dispersion of the main halo. Fig. 14 illustrates the discrepancy for ICS upper limit 2 q =16andVmax = 277 km/s. Lower values of Vmax Einasto, V = 277 km/s i i max or q2 only make this tension worse. 1.5 ! i i As we mentioned above, even though it is not theoreti- ! 0 0.5 1 1.5 2 cally possible to make the gauge coupling weak enough to µ (GeV) solve this problem, ironically one can rescue the scenario by increasing αg beyond the thermal value, since this Figure 14: Solid lines: predicted main halo boost factor for 2 suppresses the relic density of the DM component we are thermal value of αg,withdarkHiggsbosoncharges! q = i i interested in, and thus reduces the scattering rate. Al- 16 and maximum circular velocity Vmax =277km/s.Up- √ 28 per curve is for Einasto profile, lower for isothermal. Dashed lowing αg = fαg,th decreases both the density of the line is 2σ upper limit from gamma rays produced by inverse leptophilic component and the effective boost factor by 1/f. With f 50 500, depending upon the shape of Compton scattering. The failure to satisfy this bound even ∼ − with large dark Higgs content and large Vmax drives us to the main halo DM density profile, we can satisfy the con- consider larger than thermal gauge couplings, f>1. straint on the MH and still have a large enough boost in certain hypothetical nearby subhalos for them to supply the observed lepton excess. The minimum value of f that mass µ<2mµ, or to a mixture of electrons, muons and is needed is larger for a cuspy main halo. charged pions, appropriate for decays of gauge bosons We give two working examples in figure 15, one with with mass greater than 2mπ. The relative abundance f =500andVmax = 277 km/s (the larger value advo- of e, µ and π in the mixed final state can be computed cated in ref. [50]) and assuming an Einasto profile for the from the branching fractions of the decays, discussed in main halo, and the other having f =50andVmax =201 connection with eq. (8). km/s (the more standard assumption for the velocity dis- For intermediate values of the gauge boson mass, persion), with an isothermal halo. In these figures the < 2mµ <µ 2mπ, we can use the branching ratios to in- averaged boost factor S¯ of the relevant subhalos are terpolate∼ between our maximum-allowed MH or best-fit plotted as solid lines," while# the required values of S¯ SH boost factors for the 4e case and those of the fidu- are the dashed curves. Wherever these intersect repre-" # cial e + µ + π case. The maximum allowed boost factors sents a possible value of the gauge boson mass to consis- of the main halo complying with the ICS constraints are tently explain the observed lepton excess. At the same taken from table II. To estimate the best fit boost factors time, the main halo boost factor (lowest solid curve in for the subhalos in the fiducial e + µ + π final state, we the small-µ region) must lie below the black dashed lines rescale the 4e results shown in table II by the ratio of to satifsy gamma ray constraints. The rationale for tak- best-fit boost factors for the main halo, in the MH-only ing the larger value of Vmax for the Einasto profile is that scenario. These ratios are 118/110 for the Einasto pro- larger velocities help to suppress the boost factors and file and 119/113 for the isothermal, quite close to unity, thus make it easier to satisfy the ICS constraint, so that and so the best-fit values of the SH boost factors hardly we are not forced to choose an even larger value of f. depend upon this scaling. More significant is the change The isothermal profile is less constrained. in the best-fit mass, from M =1.0 to 1.2 TeV, which In the first panel of fig. 15 with the Einasto profile, enters into the computation of the Sommerfeld enhance- only subhalos SH4 and SH5 have large enough boost fac- ment and the value of the gauge coupling (α M). We g ∼ tors to ever reach the required values. There are many use the branching ratios to interpolate M as well. For points of intersection, but mainly those for SH5 and in the MH upper bounds in the small- and large-µ regions, the mass range µ<750 MeV are consistent with the we use the values from Table II, and interpolate similarly gamma ray bounds on the main halo. For the isothermal for intermediate µ. halo, these constraints are less stringent, and it is possible to find points of intersection using f = 50 for all five of the sample subhalos, although they are much more rare 5.4. Theoretical fits for SH1 SH3 than for SH4 and SH5. In this example, the intersection− points that respect the ICS bound are < For a given value of the gauge coupling αg,wecande- restricted to µ 1 GeV. For larger values of f, all the termine the predicted boost factors as well as the desired boost factors will∼ be further suppressed, and µ>1GeV values for each subhalo, as a function of the gauge boson will become allowed for SH4 and SH5. 15 Isothermal, f = 50, V = 201 km/s Einasto profile, f = 500, V = 277 km/s max max 5 3 SH1 SH2 subhalos 2.5 4 subhalos " " SH4 SH3 _ _ S S ! ! 10 10 2 SH5 3 log log SH4 1.5 MH 2 MH 1 SH5 CMBmain halo and Relic Densitymain halo 1 0.5 1 1.5 2 0 0.5 1 1.5 2 µ (GeV) µ (GeV) Figure 15: Predicted effective boost factors S¯ as a function of gauge boson mass (solid curves) and target values (or upper limit constraints! " in case of main halo, dashed curves) to explain PAMELA/Fermi lepton observations and Fermi gamma ray constraints. Pair of dashed curves for main halo (MH) correspond respectively to 1 and 2σ upper limits. Left panel is for f =100,Vmax =277 km/s, Einasto main halo profile; right is for f =25,V,max =201km/s,isothermalmainhaloprofile.Subhalosarethoseof table II. Points which satisfy all constraints are those where subhalo curves intersect their corresponding dashed linewhilethe main halo curve lies below its dashed lines. Onethe advantage dilution of requiring of large DMf is that the corre- -21 sponding dilution of the DM density by 1/f insures that thedensity model satisfies stringentby 1/f CMB insures constraints from anni- hilations in the early universe changing the optical depth -22 µ [53–55], as pointed out in [14]. The CMB constraint is / s ) +Fermi - CMB4 reionizationconstraint that the model satisfies 3 shown in fig. 16, along with the PAMELA/Fermi allowed µ 4µ IC constraint regions from ref. [33] for 4e and 4µ final states. The 4e -23 stringent CMB v (cm PAMELA - 4 case is allowed by the CMB constraint, but 4µ is ruled # +Fermi - 4e 4e IC constraint out. Because our model has at most a fraction of 0.45 of muonsconstraints in the final state, it from is probably already safe, but log -24 the additional weakening of the bound by the factor 1/f ensureschanging that this will be the the case. optical Similarly, our depth scenario PAMELA - 4e -25 overcomes the no-go result of ref. [10], which pointed out 2 2.5 3 3.5 4 that(Slatyer, Sommerfeld enhanced etc, annihilation 0906.1197, in the early uni- log M (GeV) verse leads to constraints on the MH boost factor which are lower than those needed to explain the lepton anoma- Figure 16: Allowed regions for PAMELA and Fermi excess lies.Cirelli Our MH boost and factor Cline, can satisfy these constraints leptons, and upper bounds from inverse Compton gamma since the MH is no longer considered to be the source of rays, from ref. [33], for Einasto profile with α =0.17 and the1005.1779) excess leptons. rs =20kpc.CMBconstraintisfromref.[53]. 6. DISCUSSION AND CONCLUSIONS The Sommerfeld enhancement is nearly saturated for We have shown that gamma ray constraints on 29 the low velocities of the subhalos at these large values of leptophilic annihilating dark matter are significantly αg 0.1 0.35, so their S¯ curves are nearly overlapping stronger than in previous studies, when we take into except∼ at− the smallest gauge# $ boson masses. The main account the contributions to inverse Compton scatter- halo boost factor is not saturated on the other hand, and ing from primary and secondary electrons and positrons, lies below the FSR bound for most values of µ.Wehave before including excess leptons from the DM annihila- chosen the gauge couplings, parametrized by f, to nearly tion. We attribute part of this difference to the method saturate the FSR bound. By taking larger αg (larger f), of solving the diffusion equation (1) — fully numeri- the bounds could be satisfied by a larger margin. But cal rather than semi-analytic — meaning that the (r, z) this would also reduce the S¯ values of the subhalos space-dependence of the diffusion coefficient is taken into by a similar amount, making# it$ more difficult to get a account. The difference between the predicted and ob- large enough lepton signal from SH1 SH3. SH4 and SH5 served spectra of gamma rays is greatly reduced, leaving would remain robust possible explanations.− little room for new contributions. Because of this, even