We give a review on the thermal average of the annihilation cross-sections
for $3\rightarrow 2$ and general higher-order processes. Thermal average of
higher order annihilations highly depend on the velocity of dark matter,
especially, for the case with resonance poles. We show such examples for scalar
dark matter in gauged $Z_3$ models.Comment: 5 pages, 2 figures, Prepared for the proceedings of the 13th
  International Conference on Gravitation, 3-7 July 201

Choi, Soo-Min

Lee, Hyun Min

Seo, Min-Seok

EPJ Web of Conferences

English

arXiv

We give a review on the thermal average of the annihilation cross-sections for 3 → 2 and general higher-order processes. Thermal average of higher order annihilations highly depend on the velocity of dark matter, especially, for the case with resonance poles. We show such examples for scalar dark matter in gauged Z3 models

Soo-Min Choi

Hyun Min Lee

Min-Seok Seo

EDP Sciences OAI-PMH repository (1.2.0)

SIMP dark matter and its cosmic abundances

We give a review on the thermal average of the annihilation cross-sections for 3 → 2 and general higher-order processes. Thermal average of higher order annihilations highly depend on the velocity of dark matter, especially, for the case with resonance poles. We show such examples for scalar dark matter in gauged Z3 models.</jats:p

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SIMP dark matter and its cosmic abundancesSoo-Min Choi1,, Hyun Min Lee1,, and Min-Seok Seo2,1Department of Physics, Chung-Ang University, Seoul 06974, Korea.2Center for Theoretical Physics of the Universe, Institute for Basic Science, 34051 Daejeon, Korea.Abstract. We give a review on the thermal average of the annihilation cross-sections for3→ 2 and general higher-order processes. Thermal average of higher order annihilationshighly depend on the velocity of dark matter, especially, for the case with resonancepoles. We show such examples for scalar dark matter in gauged Z3 models.Prepared for the proceedings of the 13th International Conference on Gravitation,Ewha Womans University, Korea, 3-7 July 2017.1 IntroductionDark matter has been hinted by missing masses in galaxies and galaxy clusters and supported byindirect evidences such as Cosmic Microwave Background(CMB) or gravitational lensing effect. Somany experiments have been carried out to search the dark matter, through direct detection, indirectdetection and collider search. Although we do not know characteristics of dark matter yet, we knowthe non-gravitational interaction of dark matter is getting more constrained.The most popular candidate for dark matter is Weakly Interacting Massive Particle(WIMP), andWIMP has 2 → 2 annihilation channels into the Standard Model particles. WIMP relic density isdetermined when the 2→ 2 annihilation channels are frozen out. There have been a lot of experimentsundertaken to find the WIMP, but we have not found any experimental evidence to date.However, to solve the small scale problems of collisionless cold dark matter, people considerthe self-interaction between dark matters as one of the solutions. Small scale problems stem fromthe differences between observation and simulation of the dark matter profiles, the density of sub-halos, and the number of sub-halos. The popular candidate for self-interacting dark matter is StronglyInteracting Massive Particle(SIMP) [1]. SIMP has a 3 → 2 self-annihilation process between darkmatter particles determining the current relic density when it is frozen-out. Unlike the WIMP, SIMPdoesn’t need to have a large coupling with the Standard Model particles. The mass of SIMP is orderof 100 MeV scale so it is much lighter than WIMP.In order to accurately determine the relic density of dark matter, a thermal averaged cross-sectionof the annihilation channel is required. For the non-relativistic dark matter, the cross-section can bee-mail: soominchoi90@gmail.come-mail: hminlee@cau.ac.kre-mail: minseokseo57@gmail.com© The Authors, published by EDP Sciences. This is an open access article distributed under the terms of the Creative Commons Attribution License 4.0 (http://creativecommons.org/licenses/by/4.0/).EPJ Web of Conferences 168, 06009 (2018) https://doi.org/10.1051/epjconf/201816806009Joint International Conference of ICGAC-XIII and IK-15 on Gravitation, Astrophysics and Cosmologyexpanded as powers of dark matter relative velocity and the lowest order terms of relative velocity arethe most important. This applies equally to high order annihilations such as SIMP, especially whenthere is a higher partial wave or resonance pole.In this article, we discuss the thermal averaged cross-section of high order annihilations for SIMPand perform a valid thermal average of the velocity dependent cross-section such as in the case withresonance pole.2 Resonance and thermal averageIn order to obtain the relic density of dark matter, we need to calculate a thermal average of cross-section with the integration of a momenta for initial particles. For the non-relativistic dark matter,if the cross-section has a resonance of Breit-Wigner form and the sum of masses for initial particlesare almost the same as the resonance mass, the thermal averaged cross-section is very sensitive to thevelocities of initial dark matter particles. So we need to consider exactly the momentum dependenceof the cross-section. In the paper of Gondolo and Gelmini [2], they showed the exact results of thermalaverage only for 2 → 2 cross-section.Recently, SIMP and higher order annihilating models have shed light on solving the small scaleproblems [4–10]. To get the right relic density, the exact calculation of thermal average for n → 2(n ≥ 3) cross-section is also very important. So in this article, we show the results of more generalthermal average of both non-resonant and resonant cross-sections.3 Thermal average of SIMPAn interesting way for producing the self-interacting dark matter is through higher order annihilations,namely, the initial particles are three or more in annihilation channels. A good example of self-interacting dark matter with higher order annihilation is SIMP. The relic density of SIMP is governedby the Boltzmann equation,n˙χ + 3Hnχ = −〈σv2〉(n3χ − nχn2eq) (1)whereσv2 =12E12E22E3∫d4p4(2pi)32E4d4p5(2pi)32E5|M|2(2pi)4δ4(p1 + p2 + p3 − p4 − p5). (2)The cross-section with 5-point interactions should be thermal averaged to get the exact relic den-sity. Thermal averaged 3 → 2 cross-section is very similar with 2 → 2 cross-section. We just needto consider one more initial dark matter. For the initial particles with the same masses, the thermalaveraged 3 → 2 cross-section is given by [9],〈σv2〉 =∫d3v1d3v2d3v3 δ3(v1 +v2 +v3)(σv2) e−12 x(v21+v22+v23)∫d3v1d3v2d3v3 δ3(v1 +v2 +v3) e−12 x(v21+v22+v23). (3)where delta function comes from center of mass frame of initial particles. If the 5-point interaction isnon-resonant, then the cross-section can be expanded by the velocities of initial particles like,(σv2) = a0 + a1(v21 + v22 + v23) + a(1)2 (v21 + v22 + v23)2 + a(2)2 (v41 + v42 + v43) + · · · . (4)2EPJ Web of Conferences 168, 06009 (2018) https://doi.org/10.1051/epjconf/201816806009Joint International Conference of ICGAC-XIII and IK-15 on Gravitation, Astrophysics and Cosmologyexpanded as powers of dark matter relative velocity and the lowest order terms of relative velocity arethe most important. This applies equally to high order annihilations such as SIMP, especially whenthere is a higher partial wave or resonance pole.In this article, we discuss the thermal averaged cross-section of high order annihilations for SIMPand perform a valid thermal average of the velocity dependent cross-section such as in the case withresonance pole.2 Resonance and thermal averageIn order to obtain the relic density of dark matter, we need to calculate a thermal average of cross-section with the integration of a momenta for initial particles. For the non-relativistic dark matter,if the cross-section has a resonance of Breit-Wigner form and the sum of masses for initial particlesare almost the same as the resonance mass, the thermal averaged cross-section is very sensitive to thevelocities of initial dark matter particles. So we need to consider exactly the momentum dependenceof the cross-section. In the paper of Gondolo and Gelmini [2], they showed the exact results of thermalaverage only for 2 → 2 cross-section.Recently, SIMP and higher order annihilating models have shed light on solving the small scaleproblems [4–10]. To get the right relic density, the exact calculation of thermal average for n → 2(n ≥ 3) cross-section is also very important. So in this article, we show the results of more generalthermal average of both non-resonant and resonant cross-sections.3 Thermal average of SIMPAn interesting way for producing the self-interacting dark matter is through higher order annihilations,namely, the initial particles are three or more in annihilation channels. A good example of self-interacting dark matter with higher order annihilation is SIMP. The relic density of SIMP is governedby the Boltzmann equation,n˙χ + 3Hnχ = −〈σv2〉(n3χ − nχn2eq) (1)whereσv2 =12E12E22E3∫d4p4(2pi)32E4d4p5(2pi)32E5|M|2(2pi)4δ4(p1 + p2 + p3 − p4 − p5). (2)The cross-section with 5-point interactions should be thermal averaged to get the exact relic den-sity. Thermal averaged 3 → 2 cross-section is very similar with 2 → 2 cross-section. We just needto consider one more initial dark matter. For the initial particles with the same masses, the thermalaveraged 3 → 2 cross-section is given by [9],〈σv2〉 =∫d3v1d3v2d3v3 δ3(v1 +v2 +v3)(σv2) e−12 x(v21+v22+v23)∫d3v1d3v2d3v3 δ3(v1 +v2 +v3) e−12 x(v21+v22+v23). (3)where delta function comes from center of mass frame of initial particles. If the 5-point interaction isnon-resonant, then the cross-section can be expanded by the velocities of initial particles like,(σv2) = a0 + a1(v21 + v22 + v23) + a(1)2 (v21 + v22 + v23)2 + a(2)2 (v41 + v42 + v43) + · · · . (4)So there are SO(9) invariant terms and general terms of the form, (v21)n(v22)m(v23)l. If the 3 → 2cross-section has only SO(9) invariant terms, the corresponding thermal average is〈σv2〉 = 12x3∞∑l=0all!∫ ∞0dη ηl+2e−xη= a0 + 3a1x−1 + 6a2x−2 + · · · . (5)with η ≡ 12 (v21 + v22 + v23). Otherwise, if the cross-section depends on the general terms, then,〈(v21)n(v22)m(v23)l〉 =3√3x3pi∫ ∞0dv1v21∫ ∞0dv2v22(v21)n(v22)m××∫ +1−1d cos θ12(v21 + v22 + 2v1v2 cos θ12)l e−x(v21+v22+v1v2 cos θ12),especially,〈(v21)n〉 =(43)nΓ(n + 32 )Γ( 32 )x−n = 〈(v22)n〉 = 〈(v23)n〉.When the 3 → 2 process is due to resonance, then the resulting cross-section has also a Breit-Wigner form like,〈σv2〉 =〈 ∞∑l=0b(l)Rl!ηlγR(R − 23η)2 + γ2R〉=34pix3∞∑l=0b(l)Rl!Gl(zR; x), (6)whereGl(zR; x) = Re[ ipi∫ ∞0dηηl+2e−xη32 zR − η]= (−1)l ∂l∂xlG0(zR; x), (7)R ≡ m2R−9m2DM9m2DM, γR ≡ mRΓR9m2DM and zR ≡ R+ iγR. When γR  1,G0(zR; x) ≈94 2Re− 32 xRθ(R). So the thermalaveraged cross-section is given by〈σv2〉R ≈ 2716pi2Rx3e−32 xRθ(R)∞∑l=0b(l)Rl!(32)l lR. (8)Because the phase space integral in the thermal average for SIMP depends on the higher order of darkmatter velocity than in the case of WIMP, SIMP case highly depends on the resonance mass (2R) thanWIMP case (1/2R ).We can generalize the above results of the thermal average of SIMP processes for non-degenerateinitial particle masses and even higher order annihilation like n→ 2, where n > 3. Even if the massesof the initial particles are different, we can calculate the thermal averaged cross-section. Especially,when the cross-section is s-wave, the result can be obtained by m→ (m1+m2+m3)/3 in the definitionof x in eq.(12), (13) and (14). For even higher order annihilations, we can get the general p-waveresults by using SO(3n) symmetry.3EPJ Web of Conferences 168, 06009 (2018) https://doi.org/10.1051/epjconf/201816806009Joint International Conference of ICGAC-XIII and IK-15 on Gravitation, Astrophysics and Cosmologyχχχχχ∗h′Figure 1. One example of the dark Higgs resonance channel of the discrete Z3 gauged symmetry dark mattermodel.4 Model for resonant SIMPAs a concrete model, we show the discrete Z3 gauged model for scalar dark matter. In this model,there are dark Higgs and dark matter which are charged under the dark U(1) gauge symmetry. A darkmatter triple coupling κ is generated after the dark U(1) is broken down to Z3. The details of the modelare in the references [7, 10]. Because there are dark Higgs resonance channels as shown in figure 1,we should use the results near resonance. In the non-relativistic limit, the effective 3 → 2 annihilationcross-section with a dark Higgs resonance has the form,(σv2)Z3 = C3γR(R − 23η)2 + γ2R, (9)and C3 is given byC3 =√5κ212βχm5χ(1 +λφχv′2m2χ)2. (10)Definition of βχ is√1 − 4m2χ/m2h′ . Because the three-body decay (h′ → χχχ) is suppressed by extraphase space with 2R/(4pi2), the decay width of dark Higgs is determined mainly by two-body decay(h′ → χχ∗). From eq.(6), the thermal averaged cross-section of eq.(9) is given by〈σv2〉Z3 =34C3pix3G0(zR; x). (11)By using the narrow width approximation,〈σv2〉Z3 ≈2716C3pi2Rx3e−32 xRθ(R). (12)In figure 2, we show the parameter space for the relic density of dark matter as a function of C3 andR in the left panel. We choose dark matter mass as 100 MeV. Right panel is the parameter space fordark matter cubic coupling κ and dark matter mass satisfying the relic density of dark matter. In thisplot, we show the value of R = 0.01, 0.02, 0.06. For both plots in figure 2, we assume narrow widthapproximation (NWA).5 ConclusionsWe have shown the general results of the thermal averaged cross-sections for SIMP and higher orderannihilations. When the cross-section has high dependence on velocity and resonance pole, the exact4EPJ Web of Conferences 168, 06009 (2018) https://doi.org/10.1051/epjconf/201816806009Joint International Conference of ICGAC-XIII and IK-15 on Gravitation, Astrophysics and Cosmologyχχχχχ∗h′Figure 1. One example of the dark Higgs resonance channel of the discrete Z3 gauged symmetry dark mattermodel.4 Model for resonant SIMPAs a concrete model, we show the discrete Z3 gauged model for scalar dark matter. In this model,there are dark Higgs and dark matter which are charged under the dark U(1) gauge symmetry. A darkmatter triple coupling κ is generated after the dark U(1) is broken down to Z3. The details of the modelare in the references [7, 10]. Because there are dark Higgs resonance channels as shown in figure 1,we should use the results near resonance. In the non-relativistic limit, the effective 3 → 2 annihilationcross-section with a dark Higgs resonance has the form,(σv2)Z3 = C3γR(R − 23η)2 + γ2R, (9)and C3 is given byC3 =√5κ212βχm5χ(1 +λφχv′2m2χ)2. (10)Definition of βχ is√1 − 4m2χ/m2h′ . Because the three-body decay (h′ → χχχ) is suppressed by extraphase space with 2R/(4pi2), the decay width of dark Higgs is determined mainly by two-body decay(h′ → χχ∗). From eq.(6), the thermal averaged cross-section of eq.(9) is given by〈σv2〉Z3 =34C3pix3G0(zR; x). (11)By using the narrow width approximation,〈σv2〉Z3 ≈2716C3pi2Rx3e−32 xRθ(R). (12)In figure 2, we show the parameter space for the relic density of dark matter as a function of C3 andR in the left panel. We choose dark matter mass as 100 MeV. Right panel is the parameter space fordark matter cubic coupling κ and dark matter mass satisfying the relic density of dark matter. In thisplot, we show the value of R = 0.01, 0.02, 0.06. For both plots in figure 2, we assume narrow widthapproximation (NWA).5 ConclusionsWe have shown the general results of the thermal averaged cross-sections for SIMP and higher orderannihilations. When the cross-section has high dependence on velocity and resonance pole, the exactϵR=0.02ϵR=0.060.5 1.0 1.5 2.00.050.100.150.20C3Ωh2NWA, mχ=100MeVϵR=0.06ϵR=0.02ϵR=0.01100 200 300 400024681012mχ(MeV)κNWAFigure 2. Relic density of Z3 dark matter as a function of C3 and R in the left. Dark matter cubic couplingsatisfying a correct relic density as a function of dark matter mass and R in the right. The blue line is the relicdensity of dark matter obtained by Planck.calculation of thermal averaged cross-section is very important for the relic density of dark matter.Moreover, the results can be generalized for the non-degenerate initial particle masses and even higherannihilation channels.6 AcknowledgmentsThe work is supported in part by Basic Science Research Program through the National ResearchFoundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (NRF-2016R1A2B4008759). The work of SMC is supported in part by TJ Park Science Fellowship ofPOSCO TJ Park Foundation. MS is supported by IBS under the project code, IBS-R018-D1.References[1] Y. Hochberg, E. Kuflik, T. Volansky and J. G. Wacker, Phys. Rev. Lett. 113 171301 (2014)[2] P. Gondolo and G. Gelmini, Nucl. Phys. B 360 145. (1991)[3] K. Griest and D. Seckel, Phys. Rev. D 43 3191. (1991)[4] Y. Hochberg, E. Kuflik, H. Murayama, T. Volansky and J. G. Wacker, Phys. Rev. Lett. 115 2,021301 (2015)[5] N. Bernal, C. Garcia-Cely and R. Rosenfeld, JCAP 1504 04, 012 (2015)[6] H. M. Lee and M. S. Seo, Phys. Lett. B 748 316 (2015)[7] S. M. Choi and H. M. Lee, JHEP 1509 063 (2015)[8] S. M. Choi and H. M. Lee, Phys. Lett. B 758 47 (2016)[9] S. M. Choi, H. M. Lee and M. S. Seo, JHEP 1704 154 (2017)[10] S. M. Choi, Y. J. Kang and H. M. Lee, JHEP 1612 099 (2016)5EPJ Web of Conferences 168, 06009 (2018) https://doi.org/10.1051/epjconf/201816806009Joint International Conference of ICGAC-XIII and IK-15 on Gravitation, Astrophysics and Cosmology

SIMP dark matter and its cosmic abundances

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