We point out that annihilation of dark matter in the galactic halo can be
enhanced relative to that in the early universe due to a Breit-Wigner tail, if
the dark matter annihilates through a pole just below the threshold. This
provides a new explanation to the "boost factor" which is suggested by the
recent data of the PAMELA, ATIC and PPB-BETS cosmic-ray experiments.Comment: 4 pages, 4 figure

/SLAC

/Tokyo U. /Tokyo U., IPMU

/Tokyo U., IPMU /UC, Berkeley /LBL, Berkeley

Ibe, Masahiro

Murayama, Hitoshi

Yanagida, T.T.

English

arXiv

We point out that annihilation of dark matter in the galactic halo can be enhanced relative to that in the early universe due to a Breit-Wigner tail, if the dark matter annihilates through a pole just below the threshold. This provides a new explanation to the 'boost factor' which is suggested by the recent data of the PAMELA, ATIC and PPB-BETS cosmic-ray experiments

Yanagida, T. T.

UNT (University of North Texas) Digital Library

Work supported in part by US Department of Energy contract DE-AC02-76SF00515SLAC-PUB-13479IPMU08-0097Breit–Wigner Enhancement of Dark Matter AnnihilationMasahiro Ibe,1 Hitoshi Murayama,2,3, 4 and T. T. Yanagida5, 21SLAC National Accelerator Laboratory, Menlo Park, CA 940252Institute for the Physics and Mathematics of the Universe, University of Tokyo, Kashiwa 277-8568, Japan3Department of Physics, University of California, Berkeley, CA 947204Theoretical Physics Group, Lawrence Berkeley National Laboratory, Berkeley, CA 947205Department of Physics, University of Tokyo, Tokyo 113-0033, Japan.(Dated: November 29, 2008)We point out that annihilation of dark matter in the galactic halo can be enhanced relative tothat in the early universe due to a Breit–Wigner tail, if the dark matter annihilates through a polejust below the threshold. This provides a new explanation to the “boost factor” which is suggestedby the recent data of the PAMELA, ATIC and PPB-BETS cosmic-ray experiments.Dark matter of the universe has been discussed since1933, yet its nature still remains elusive [1]. Seventy-fiveyears later we only managed to restrict its mass between10−31 and 1050 GeV, demonstrating our lack of under-standing. However, the thermal relic of a Weakly Inter-acting Massive Particle (WIMP) is arguably best theo-retically motivated because it has the same mass scaleas the anticipated new physics that would explain whyour universe is a superconductor (electroweak symme-try breaking). Hopes are high to discover WIMPs at theforthcoming LHC experiments, to detect them directly insensitive underground experiments, as well as to observesignals of WIMP annihilations from the galactic centeror the halo in high-energy cosmic rays.Recent observations of the PAMELA [2], ATIC [3], andPPB-BETS [4] experiments strongly suggest the exis-tence of a new source of positrons (and electrons) in cos-mic rays. The most interesting interpretation of theseresults is the annihilation of the dark matter with a massat the TeV scale. However, such interpretation requiresthat the annihilation cross section of the dark matter inthe galactic halo is much larger (by a factor of O(100))than the one appropriate to explain the dark matter relicdensity precisely measured by the WMAP experiment [5].The enhancement of the dark matter annihilation inthe galactic halo is called a “boost factor.” So far, therehave been several proposals to explain the origin of theboost factor both from astrophysics such as the enhancedlocal dark matter density, and from particle physics suchas the Sommerfeld enhancement due to an attractiveforce among the dark matter particles [6].In this letter, we propose a new explanation of theboost factor. We consider the dark matter which annihi-lates via a narrow Breit–Wigner resonance just below thethreshold. When the resonance mass is just below twicethe dark matter mass, the annihilation cross section be-comes sensitive to the velocity of the dark matter. In sucha case, the time evolution of the dark matter abundanceis quite different from the one in the usual non-resonantcase of annihilation, and we find that the annihilationcross section in the halo is enhanced compared to whatis inferred from the relic abundance. As we will show,the cross section required from the dark matter densitycan be large enough to explain the PAMELA, ATIC, andPPB-BETS results, and hence, we do not need in ourproposal any additional boost factor due to an overdenseregion in the halo or the Sommerfeld enhancement.Cross Section Just Above a Pole. In this study, weassume that the dark matter with mass m annihilatesvia a narrow resonance R. For a simplicity, we considera scalar resonance, although generalization to arbitraryspins is straight forward. The general formula for thescattering cross section via a resonance R is given byσ =16πE2cmβ̄iβiM2Γ2(E2cm − M2)2 + M2Γ2BiBf , (1)where M and Γ are the mass and the decay rate of theresonance R, respectively. Two body initial and finalstates are assumed and β̄i =√1 − 4m2/M2 is the initialstate phase space factor evaluated on the resonance whileβi =√1 − 4m2/E2cm at the center of mass energy ofthe collision. The branching fractions of the resonanceinto the initial and final states are denoted by Bi andBf , respectively. Note that since we are assuming anunphysical pole, i.e., 2m > M , Bi and β̄i are not physicaland should be understood as analytic continuations ofthose quantities from the physical region, 2m < M . Evenso, a combination Bi/β̄i is well-defined in both regions,and hence, the above cross section is also well-definedeven in the unphysical pole case.The dark matter annihilation in the early universemust be thermally averaged. On the other hand, thedark matter annihilation in the galactic halo is averagedover the velocity distribution, which can be approximatedby the Maxwellian distribution. In both cases, the darkmatter is non-relativistic. Therefore, in either case, wecan use the Gaussian average,〈σvrel〉 =1(2πv20)3∫d~v1d~v2e−(~v21+~v22)/2v202σβi, (2)where ~v1,2 are the velocities of the initial states and wehave used vrel = 2βi. The non-relativistic approximationgivesE2cm = 4m2 + m2(~vrel)2, (~vrel = ~v1 − ~v2). (3)June 20092Now, let us consider the annihilation process near anarrow resonance, i.e.,M2 = 4m2(1 − δ), |δ| ≪ 1. (4)Note that positive δ implies that the pole is just belowthe threshold of the dark matter annihilation. With thisnotation, we can rewrite the above cross section as,σ =16πM2β̄iβiγ2(δ + ~v2rel/4)2 + γ2BiBf , (5)where we have defined,γ = Γ/M. (6)Furthermore, we have verified that we can approximatethe Gaussian integral reasonably well by〈σvrel〉 ≃32πM2β̄iγ2(δ + ξ v20)2 + γ2BiBf , (7)where a parameter ξ ≈ 1/√2 gives the best fit to thenumerical results for v0 ≪ 1 and δ > 0. This expressionshows that the denominator is dominated by the ~v2rel termfor |δ|, γ ≪ ξ v20 , while the other terms dominate whenthe velocity is much smaller. Therefore, the cross sec-tion is sensitive to the parameters δ and γ, and have anenhanced behavior at the lower temperature for small δand γ. This main point of this letter can be seen easilyin a schematic plot in Fig. 1.Note that the approximation given in Eq. (7) is not agood one for the dark matter with a rather large veloc-ity, i.e., v0 & 0.1 [7]. The approximation also becomesworse around the pole in the physical region, possible ifδ < 0. In the following analysis, we mainly consider theunphysical pole δ > 0. As we will see below, for this case,the resultant dark matter density is mostly determinedby the dynamics of the dark matter at v0 ≪ 1, and hencethe approximated cross section works quite well. We willlater also briefly discuss the case of a physical pole, δ < 0.Time Evolution of Dark Matter Density. Now let usconsider the time evolution of the relic density of the darkmatter. The most notable feature of our proposal is thatthe annihilation process does not freeze out even afterthe “freeze-out” time for the usual non-resonant annihi-lation models. That is, the interaction rate can be largerthan the Hubble expansion rate even after a “freeze-out”time, since the cross section is enhanced at the lowertemperature. In the following, to avoid confusion, wedefine the “freeze-out” time x̃f by the usual meaning atwhich the yield of the dark matter Y = n/s starts de-viating from the value in the thermal equilibrium, i.e.,Y − YEQ = O(YEQ). As we will show, the actual freeze-out time is much later than x̃f . In this section, we mainlyconsider the unphysical pole, δ > 0.Following Ref. [8], section 5.2, we write down theBoltzmann equation of the yield of the dark matter [10],dYdx= − λx2(δ2 + γ2)(δ + ξ x−1)2 + γ2(Y 2 − Y 2EQ). (8)−4 −2 0 2 4 6 8 10vrel2/(10−3c2)Breit-Wignersmall v0large v0FIG. 1: A schematic plot that shows dispersion in relativevelocity v2rel and an unphysical pole in the cross section atv2rel < 0 (below threshold). It is clear that a smaller dispersionv0 gives a larger overlap with the Breit–Wigner tail in thecross section and hence an enhanced averaged cross section.Here, we have used the following definitionsλ =[s(T )H(T )]T=m〈σvrel〉T=0 =√8π245g∗ MPL m σ0 ,σ0 = 〈σvrel〉T=0 =32πBiBfM2β̄iγ2δ2 + γ2,YEQ =454√2 π7/2(gig∗)x3/2e−x,x =mT= v−20 . (9)The parameter g∗ (gi) is the number of the degreesof freedom for massless particles (dark matter), respec-tively. Note that we have used the reduced Planck scaleMPL ≃ 2.4 × 1018 GeV.As we have defined, the “freeze-out” time x̃f is deter-mined by Y −YEQ = O(YEQ), and hence, the value of x̃fis not so sensitive to the parameters δ and γ and mainlydetermined by the exponential suppression factor in YEQ.Thus, we can expect that the value of x̃f is comparable tothe freeze-out temperature xf in the usual non-resonantannihilation models, i.e., x̃f ≈ xf ≈ O(10).Unlike the non-resonant case where the annihilationcross section stays constant 〈σvrel〉 once non-relativistic,however, the annihilation cross section here increases astemperature drops. As a result, the annihilation processdoes not freeze out even for x > x̃f and the dark matterkeeps annihilating until the temperature comes down toxb ≃ ξ−1 × max [δ, γ]−1 ≫ x̃f . (10)Below this temperature, Eq. (8) reduces todYdx= − λx2Y 2, (11)3and we obtain an asymptotic solution,Y∞ ≃1λxb ≃1λ× max [δ, γ]−1 . (12)In the Fig. 2, we show the time evolution of the yieldY of the dark matter for a given parameter set. Here,we have used the numerical result of the Gaussian av-erage of the cross section [11]. We also show the timeevolution of the yield with the approximate cross sec-tion given in Eq. (7), Yapp. As we have expected, theyield deviates from YEQ at x̃f = O(10), while the actualfreeze-out occurs at xb ≫ x̃f . From the figure, we seethat Y < Yapp during x̃f < x < xb. This means thatthe averaged cross section at x̃f is somewhat larger thanthe approximate one, while the final result is determinedby the late time dynamics where the approximate crosssection works well. The figure shows that the above ap-proximate asymptotic solution gives a good estimate ofthe resultant yield of the dark matter.It is worth comparing the asymptotic solution inEq. (12) with the asymptotic solution in the usual non-resonant (S-wave) annihilation models [8],Y∞ ≃1λxf , (13)with xf = O(10). These two solutions show that the darkmatter abundance in our proposal is larger by a factorxbxf≃ max [δ, γ]−1O(10)(14)when we assume the same cross section at the zero tem-perature for both models. In the Fig. 2, we have showedthe time evolution of the yield in the usual non-resonantannihilation models assuming the same cross section atthe zero temperature σ0 used in Y (i.e., λ = 109). Asexpected, the yield in the non-resonant annihilation ismore suppressed compared to the resonant annihilation.The physical reasons of this enhancement are as follows.First, in our proposal, the cross section is suppressed by(x̃f/xb)2 at x̃f , which results in a relatively larger abun-dance in a period of x > x̃f . Second, the annihilationprocess is relatively less effective during x̃f < x < xbcompared to the usual annihilation case, although theannihilation does not freeze out in that period.Before closing this section, we mention the model witha physical pole, i.e., δ < 0. In this case, the cross sec-tion given in Eq. (7) poorly approximates the thermal av-eraged cross section [7]. Especially, the thermal averagecan pick up the cross section at the pole, v2rel = 4|δ| whenthe temperature is rather high, i.e., x−1 ≫ |δ|. Thus, theaveraged cross section can be much higher than that ex-pected in the unphysical pole where the cross section issuppressed by x2(δ2+γ2) at high temperature. Thus, theannihilation cross section at x̃f , which typically satisfies1 10 100 1000 10000x=m/TY=n/sγ<<10−3x f x bYEQYYnon-resδ=+10−3x f~Y∞YappFIG. 2: The time evolution of the yield Y of the dark mat-ter in terms of the parameter x = m/T for given values of δand γ (the solid line). In this figure, we assume δ > 0 andthe pole is not in the physical region. The long-dashed linelabeled Yapp is the evolution with the approximated cross sec-tion in Eq. (7). The dashed line labeled Y∞ is the asymptoticsolution Y∞ given in Eq. (12). The short-dashed line repre-sents the equilibrium yield YEQ. The dash-dotted line labeledYnon−res shows the time evolution of the yield in the usual(non-resonant) freeze-out assuming the same cross section atthe low temperature σ0 used in Y (see Eq. (13)). The boostfactor is the asymptotic value of Y/Ynon−res.1 10 100 1000 10000x=m/TY=n/sγ<<10−3x f x bYEQYYnon-resδ=−10−3x f~Y∞FIG. 3: The time evolution of the yield of the dark matter Yfor δ < 0 (the solid line). Everything else is the same as inFig. 2. There is practically no boost for this parameter set.x̃−1f ≫ |δ|, γ, is much larger than that in the case of theunphysical pole. Therefore, the yield enhancement seenin the unphysical pole is much smaller in this case.Effective Boost Factor. Finally, let us work out theboost factor for annihilation in the galactic halo. Aswe have seen, the dark matter abundance is enhancedby a factor of (xb/xf ) for δ > 0 compared to the usualnon-resonant annihilation models for a given annihilationcross section at the zero temperature, σ0. This meansthat the cross section σ0 must be larger than the one4expected in the usual models to reproduce the observeddark matter density.More explicitly, the yield given in Eq. (12) is translatedto the mass density parameter,ΩDMh2 ≃ 0.1 ×(10−9 GeV−2σ0)(xbxf), (15)where we have used xf ≃ 20 and g∗ ≃ 200. Therefore,the observed dark matter density Ωh2 ≃ 0.1 requiresσ0 ≃ 10−9 GeV−2 ×(xbxf), (16)which is much larger than what is expected in the usualannihilation case, i.e.,σ0 ≃ 10−9 GeV−2. (17)In the galactic halo, the average velocity is givenby v0 ≃ 10−3, and the cross section is well approxi-mated by the one at the zero temperature as long asv20 ≪ δ, γ. Thus, we can achieve the large annihilationcross section suggested by the PAMELA, PPB-BETS,and ATIC anomalies, 〈σvrel〉T=0 = O(10−(6−7))GeV−2for δ, γ . 10−3. Therefore, in our proposal, we can ex-plain the large annihilation cross section in the galactichalo without other boost factors.For convenience, we could define an effective boost fac-tor as the ratio between the cross sections in the usualand our models. From Eqs. (16) and (17) we obtainBF ≃ xbxf≃ max [δ, γ]−1xf≃ max [δ, γ]−1O(10). (18)In this way, we can explain the boost factor of O(100),in a model with δ, γ = O(10−3), as seen in Fig. 4.Discussion. In this letter, we showed that the boostfactor required in recent observations of cosmic ray elec-trons and positrons can be obtained if the dark matterannihilates via a narrow resonance just below the thresh-old. Because the annihilation in the early universe issuppressed by the Breit–Wigner tail, the observed darkmatter density requires a larger-than-normal cross sec-tion which can be consistent with the PAMELA, ATICand PPB-BETS results. The required cross section isachieved for 4m2 − M2, MΓ . 10−3M2.Note that the small decay width Γ ≪ 10−3M can beachieved rather easily, when the resonance R is a weaklycoupled particle. In some models, a coincidence in massescan also be naturally realized. For example, if the darkmatter is the lightest Kaluza-Klein (KK) particle and theresonance is at the second KK level, we have the relationM = 2m at the tree-level [9] [12] (see also a recent dis-cussion in Ref. [3]). Thus, in such models, we may wellhave a small mass splitting δ as a result of small radiativecorrections to the tree-level mass relation.103102101boost factorFIG. 4: The boost factor on the (δ, γ) plane. Thermal aver-age is done numerically without relying on the approximationEq. (7).The work of M.I. was supported by the U.S. De-partment of Energy under contract number DE-AC02-76SF00515. The work of H.M. and T.T.Y. was supportedin part by World Premier International Research CenterInitiative (WPI Initiative), MEXT, Japan. The work ofH.M. was also supported in part by the U.S. DOE underContract DE-AC03-76SF00098, and in part by the NSFunder grant PHY-04-57315.[1] See, e.g., H. Murayama, arXiv:0704.2276 [hep-ph].[2] O. Adriani et al., arXiv:0810.4995 [astro-ph].[3] J. Chang et al., Nature 456, 362 (2008).[4] S. Torii et al., arXiv:0809.0760 [astro-ph].[5] E. Komatsu et al. [WMAP Collaboration],arXiv:0803.0547 [astro-ph].[6] N. Arkani-Hamed, D. P. Finkbeiner, T. Slatyer andN. Weiner, arXiv:0810.0713 [hep-ph].[7] K. Griest and D. Seckel, Phys. Rev. D 43, 3191 (1991).[8] E. W. Kolb and M. S. Turner, “The Early Universe,”Frontiers in Physics, Westview Press, 1994.[9] M. Kakizaki, S. Matsumoto, Y. Sato and M. Senami,Phys. Rev. D 71, 123522 (2005) [arXiv:hep-ph/0502059].[10] The analysis of the Boltzmann equation near a resonancehas been developed in Ref. [7].[11] In the computation, we have used λ = 109. The qualita-tive behavior of the time evolution does not depend onλ, although the values of xf and x̃f slightly depend onλ.[12] In the explicit model given in the reference, the resonancemainly decays into a top pair which is not favored by thenull results of the excess of the anti-proton flux [2, 3, 4].

Breit-Wigner Enhancement of Dark Matter Annihilation

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