In this review, we focus on dark matter production from thermal freeze-out
with forbidden channels and SIMP processes. We show that forbidden channels can
be dominant to produce dark matter depending on the dark photon and / or dark
Higgs mass compared to SIMP.Comment: 5 pages, Prepared for the proceedings of the 13th International
  Conference on Gravitation, 3-7 July 201

Choi, Soo-Min

Kang, Yoo-Jin

Lee, Hyun Min

EPJ Web of Conferences

English

arXiv

In this review, we focus on dark matter production from thermal freeze-out with forbidden channels and SIMP processes. We show that forbidden channels can be dominant to produce dark matter depending on the dark photon and / or dark Higgs mass compared to SIMP

Soo-Min Choi

Yoo-Jin Kang

Hyun Min Lee

EDP Sciences OAI-PMH repository (1.2.0)

Forbidden Channels and SIMP Dark Matter

In this review, we focus on dark matter production from thermal freeze-out with forbidden channels and SIMP processes. We show that forbidden channels can be dominant to produce dark matter depending on the dark photon and / or dark Higgs mass compared to SIMP.</jats:p

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Forbidden Channels and SIMP Dark MatterSoo-Min Choi1,, Yoo-Jin Kang1,2,, and Hyun Min Lee1,1Department of Physics, Chung-Ang University, Seoul 06974, Korea2Center for Theoretical Physics of the Universe, Institute for Basic Science, Daejeon 34051, KoreaAbstract. In this review, we focus on dark matter production from thermal freeze-outwith forbidden channels and SIMP processes. We show that forbidden channels can bedominant to produce dark matter depending on the dark photon and / or dark Higgs masscompared to SIMP.1 IntroductionIn the Universe, there are diverse evidences for dark matter such as galaxy rotation curves, gravita-tional lensing and Cosmic Microwave Background (CMB). Especially, CMB from Planck satellitedata infer that the averaged relic density of dark matter is ΩDMh2 = 0.1198 ± 0.0015 [1].Various dark matter candidates have been introduced including Weakly Interacting Massive Parti-cles (WIMPs). WIMPs are well-motivated but carry too small self-interaction to explain small-scaleproblems. Small-scale problems come from difference between numerical simulations with collision-less dark matter and observations from galaxies. They include core-cusp problem, missing satelliteproblem and too-big-to-fail problem. Solutions to small-scale problems require dark matter as self-interaction strong as σself/mDM = 0.1 − 10cm2/g [2, 3]. In this sense, dark matter annihilating withforbidden channels can be important. In the forbidden channels, masses of final states are larger thandark matter masses. These channels are forbidden at zero temperature but open at high temperaturein the early Universe [4]. The forbidden channel needs dark matter with sizable self-interaction, sincethey have Boltzmann suppression factor. It is discussed in section 3.1 in this review.Strongly Interacting Massive Particles (SIMP) [5] have started drawing some attention recentlybecause they can allow large self-interacting of light dark matter. The relic density of SIMP is deter-mined by 3→ 2 self annihilations in the thermal freeze-out [6]. However, the Boltzmann suppressionfactor makes SIMP be on the verge of violating unitarity or perturbativity. Also it makes SIMP be ina tension with the bounds from Bullet cluster and halo shapes in the most of the parameter space [7].In this review, we consider a complex scalar dark matter, dark photon and dark Higgs. We discussdark matter can be produced from thermal freeze-out with forbidden channels and/or SIMP processes.In forbidden channels, annihilation cross section can be suppressed by a Boltzmann factor whichdepends on mass difference, ∆i = (mi − mDM)/mDM, between dark matter and dark photon or darkHiggs. We suggest a model with gauged Z3 symmetry for forbidden and SIMP dark matter. Specifice-mail: soominchoi90@gmail.come-mail: yoojinkang91@gmail.come-mail: hmlee71@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, 06010 (2018) https://doi.org/10.1051/epjconf/201816806010Joint International Conference of ICGAC-XIII and IK-15 on Gravitation, Astrophysics and Cosmologyrange of mass differences allow dark matter to have a smaller self-interaction being compatible withthe relic density, as compared to the SIMP case. In our model, the self-coupling contributes to bothSIMP and forbidden channels.2 Model for self-interacting dark matterWe consider a complex scalar dark matter χ and dark Higgs φ in dark local U(1)d model with chargeqχ = +1 and qφ = +3 each. Local U(1)d symmetry is spontaneously broken to Z3 which makes darkmatter stable. Then, the dark Higgs φ has non-zero vacuum expectation value 〈φ〉 = vd/√2 [8, 9].The Lagrangian for SM singlet scalars, χ and φ, and the SM Higgs doublet H, is given byL = −14VµνVµν − 12 sin ξVµνBµν + |Dµφ|2 + |Dµχ|2 + |DµH|2 − V(φ, χ,H) (1)where Vµν = ∂µVν − ∂νVµ, Dµφ = (∂µ − iqφgdVµ)φ, Dµχ = (∂µ − iqχgdVµ)χ, with dark gauge couplinggd, and DµH = (∂µ − ig′YHBµ − 12 igTaWaµ)H. The second term means mixing between dark photon Z′and SM gauge boson Z. This V(φ, χ,H) = VDM + VSM is the scalar potential and VSM is the StandardModel Higgs potential. VDM isVDM = −m2φ|φ|2 + m2χ|χ|2 + λφ|φ|4 + λχ|χ|4 + λφχ|φ|2|χ|2 +( √23!κφ†χ3 + h.c.)+λφH |φ|2|H|2 + λχH |χ|2|H|2. (2)After the dark Higgs is expanded as φ = (vd + hd)/√2, a triple coupling for χ is obtained. Thiscoupling makes dark matter annihilate by 3 → 2 process.3 Thermal freeze-outWe discuss the thermal production of light dark matter from process in combination with dark Higgsquartic coupling and Z′ gauge coupling with the 2→2 forbidden channels and the 3→2 annihilationSIMP both by dark matter self-interactions. From that, we can find which process would be dominantprocess to determine the relic density depending on the mass difference between dark matter and darkphoton and/or dark Higgs. Moreover, depending on dark photon and dark Higgs masses, annihilationinto a pair of the SM particles also can be dominant. The SM-annihilation is discussed in the originalpaper [10]. In this review, we focus on comparison between forbidden channels and SIMP channels.3.1 Forbidden channelsAlthough the dark photon and/or dark Higgs are heavier than dark matter, they can still contribute tothe relic density for dark matter through the forbidden channels. We consider the case where darkphoton and / or dark Higgs are light. We assume that dark photon and/or dark Higgs are in kineticequilibrium during freeze-out process.If we consider the annihilation cross section for forbidden channels for example χχ∗ → Z′Z′ andχχ∗ → Z′χ∗, the detailed balance condition at high temperature are〈σv〉χχ∗→Z′Z′ =4(neqZ′ )2(neqDM)2〈σv〉Z′Z′→χχ∗= 9(1 + ∆Z′ )3e−2∆Z′ x〈σv〉Z′Z′→χχ∗ (3)2EPJ Web of Conferences 168, 06010 (2018) https://doi.org/10.1051/epjconf/201816806010Joint International Conference of ICGAC-XIII and IK-15 on Gravitation, Astrophysics and Cosmologyrange of mass differences allow dark matter to have a smaller self-interaction being compatible withthe relic density, as compared to the SIMP case. In our model, the self-coupling contributes to bothSIMP and forbidden channels.2 Model for self-interacting dark matterWe consider a complex scalar dark matter χ and dark Higgs φ in dark local U(1)d model with chargeqχ = +1 and qφ = +3 each. Local U(1)d symmetry is spontaneously broken to Z3 which makes darkmatter stable. Then, the dark Higgs φ has non-zero vacuum expectation value 〈φ〉 = vd/√2 [8, 9].The Lagrangian for SM singlet scalars, χ and φ, and the SM Higgs doublet H, is given byL = −14VµνVµν − 12 sin ξVµνBµν + |Dµφ|2 + |Dµχ|2 + |DµH|2 − V(φ, χ,H) (1)where Vµν = ∂µVν − ∂νVµ, Dµφ = (∂µ − iqφgdVµ)φ, Dµχ = (∂µ − iqχgdVµ)χ, with dark gauge couplinggd, and DµH = (∂µ − ig′YHBµ − 12 igTaWaµ)H. The second term means mixing between dark photon Z′and SM gauge boson Z. This V(φ, χ,H) = VDM + VSM is the scalar potential and VSM is the StandardModel Higgs potential. VDM isVDM = −m2φ|φ|2 + m2χ|χ|2 + λφ|φ|4 + λχ|χ|4 + λφχ|φ|2|χ|2 +( √23!κφ†χ3 + h.c.)+λφH |φ|2|H|2 + λχH |χ|2|H|2. (2)After the dark Higgs is expanded as φ = (vd + hd)/√2, a triple coupling for χ is obtained. Thiscoupling makes dark matter annihilate by 3 → 2 process.3 Thermal freeze-outWe discuss the thermal production of light dark matter from process in combination with dark Higgsquartic coupling and Z′ gauge coupling with the 2→2 forbidden channels and the 3→2 annihilationSIMP both by dark matter self-interactions. From that, we can find which process would be dominantprocess to determine the relic density depending on the mass difference between dark matter and darkphoton and/or dark Higgs. Moreover, depending on dark photon and dark Higgs masses, annihilationinto a pair of the SM particles also can be dominant. The SM-annihilation is discussed in the originalpaper [10]. In this review, we focus on comparison between forbidden channels and SIMP channels.3.1 Forbidden channelsAlthough the dark photon and/or dark Higgs are heavier than dark matter, they can still contribute tothe relic density for dark matter through the forbidden channels. We consider the case where darkphoton and / or dark Higgs are light. We assume that dark photon and/or dark Higgs are in kineticequilibrium during freeze-out process.If we consider the annihilation cross section for forbidden channels for example χχ∗ → Z′Z′ andχχ∗ → Z′χ∗, the detailed balance condition at high temperature are〈σv〉χχ∗→Z′Z′ =4(neqZ′ )2(neqDM)2〈σv〉Z′Z′→χχ∗= 9(1 + ∆Z′ )3e−2∆Z′ x〈σv〉Z′Z′→χχ∗ (3)and similarly〈σv〉χχ→Z′χ∗ = 3(1 + ∆Z′ )3e−2∆Z′ x〈σv〉Z′χ∗→χχ (4)with ∆Z′ ≡ (mZ′ − mχ)/mχ and x = mχ/T .From (4) and (5), we can see that the annihilation cross sections for forbidden channels depend one−∆Z′ x. Therefore, in the case with mχ < mZ′  mh1 , dark Higgs can’t contribute to the relic densityfor dark matter. Similarly, in the case with mχ < mh1  mZ′ , dark photon can’t contribute to the relicdensity for dark matter. However, the both dark photon and dark Higgs can contribute to the forbiddenchannels at the same time in the case with mχ < mZ′ ∼ mh1 . The first case are discussed below. Theother casse are discussed with the first case in the original paper [10].In the case with mχ < mZ′  mh1 , the Boltzmann equation is approximated todnDMdt+ 3HnDM ≈ = −12 〈σv〉χχ∗→Z′Z′n2DM + 2〈σv〉Z′Z′→χχ∗ (neqZ′ )2−12〈σv〉χχ→Z′χ∗n2DM + 〈σv〉Z′χ∗→χχneqZ′nDM (5)It can be rewritten with (4) and (5). Then, the relic density is determined to beΩDMh2 = 5.20 × 10−10GeV−2(g∗10.75)−1/2( x f20)e∆Z′ x f g(∆Z′ , x f ) (6)and the g(∆Z′ , x f ) is written in the original paper [10].We put 〈σv〉Z′Z′→χχ∗ = a and 〈σv〉Z′χ∗→χχ = bv2 = 6b/x. Due to e∆Z′ x f , the 2 → 2 annihilationcross section can be large and the self-scattering cross section of dark matter also can be large.3.2 SIMP processesZ′ and dark Higgs are heavier than dark matter and they have small couplings to dark matter and SMparticles, the 3 → 2 annihilation process for dark matter becomes dominant.Then, the Boltzmann equation is approximated todnDMdt+ 3HnDM ≈ −〈σv2〉3→2(n3DM − n2DMneqDM). (7)Here, the effective 3 → 2 annihilation cross section is obtained as〈σv2〉3→2 = 14(〈σv2〉χχχ∗→χ∗χ∗ + 〈σv2〉χχχ→χχ∗) ≡ α3effm5χ. (8)As a result, the dark matter relic density is given byΩDMh2 = 1.41 × 10−8GeV−2(g∗10.75)−3/4( x f20)2( αeffM1/3P mχ)−3/2. (9)From the result, the relic density depends on the ratio, mχ/αeff . The 3 → 2 self-annihilation crosssections are directly related with the self-scattering cross section, σself ∼ α2eff/m2χ. Therefore, theself-scattering cross section can be large enough.Especially, in SIMP process, we define the effective 2 → 2 annihilation cross section by〈σv〉eff,2→2 ≡ nDM〈σv2〉3→2. Then, we can rewrite as below〈σv〉eff,2→2 ≈ g(x/2pi)3T 3e−x〈σv2〉3→2. (10)3EPJ Web of Conferences 168, 06010 (2018) https://doi.org/10.1051/epjconf/201816806010Joint International Conference of ICGAC-XIII and IK-15 on Gravitation, Astrophysics and CosmologyRelic, ΔZ'=0.5Unitarity 0.1cm2/g0.01cm2/g0.001cm2/gRelic, ΔZ'=0.4200 400 600 800 10000.00.51.01.52.02.5mχ(MeV)Rgd=0.1, Δh1=10, λϕχ=0.4UnitarityStabilityPerturbativity1cm2/g0.1cm2/g10cm2/g0 50 100 150 200 2500.00.51.01.52.02.5mχ(MeV)Rgd=0.1, ΔZ'=4, Δh1=0.5, λϕχ=0.4Figure 1. Parameter space of R vs mχ, satisfying the relic density for forbidden channel (on left) and for SIMPprocess (on right) with R =√2κvd/6mχ. The black lines correspond to self-scattering cross sections, σself/mχ =0.001, 0.01 and 0.1 cm2/g for left, 0.1, 1 and 10 cm2/g for right.Ωχ h2=0.1198R=0.1, mχ=1GeVR=0.1, mχ=10MeVR=1, mχ=1GeVR=1, mχ=10MeV0.05 0.10 0.50 110-610-40.011100104106ΔZ'Ωχh2Δh1=10, gd=0.1, λϕχ=0.4Ωχ h2=0.1198R=0.1, mχ=1GeVR=0.1, mχ=10MeVR=1, mχ=1GeVR=1, mχ=10MeV0.05 0.10 0.50 110-610-40.011100104106ΔZ'Ωχh2Δh1=10, gd=0.1, λϕχ=0.4Figure 2. Dark matter relic density as a function of ∆Z′ for only forbidden channels (on left) for both SIMP andforbidden channels (on right). The central value of relic density, Ωχh2 = 0.1198, is given by black dashed lines.We have taken R = 0.1 and R = 1.0 with R =√2κvd/6mχ.In Fig.1, we note that SIMP process (on right) requires larger dark matter self-scattering cross sectionthan the case with forbidden channels (on left). Besides, the Boltzmann supression factor in (10) iscomparable with those in (4) and (5) for the forbidden channels. From this, we can infer that theforbidden channels can be dominant until ∆Z′ ∼ 0.5 and we check it exactly in the right side of Fig. 2which is in the case with mχ < mZ′  mh1 .4 ConclusionWe have considered the thermal production from freeze-out for self-interacting dark matter in thegauged Z3 model with dark photon and dark Higgs. We discussed that the forbidden channels (χχ∗ →4EPJ Web of Conferences 168, 06010 (2018) https://doi.org/10.1051/epjconf/201816806010Joint International Conference of ICGAC-XIII and IK-15 on Gravitation, Astrophysics and CosmologyRelic, ΔZ'=0.5Unitarity 0.1cm2/g0.01cm2/g0.001cm2/gRelic, ΔZ'=0.4200 400 600 800 10000.00.51.01.52.02.5mχ(MeV)Rgd=0.1, Δh1=10, λϕχ=0.4UnitarityStabilityPerturbativity1cm2/g0.1cm2/g10cm2/g0 50 100 150 200 2500.00.51.01.52.02.5mχ(MeV)Rgd=0.1, ΔZ'=4, Δh1=0.5, λϕχ=0.4Figure 1. Parameter space of R vs mχ, satisfying the relic density for forbidden channel (on left) and for SIMPprocess (on right) with R =√2κvd/6mχ. The black lines correspond to self-scattering cross sections, σself/mχ =0.001, 0.01 and 0.1 cm2/g for left, 0.1, 1 and 10 cm2/g for right.Ωχ h2=0.1198R=0.1, mχ=1GeVR=0.1, mχ=10MeVR=1, mχ=1GeVR=1, mχ=10MeV0.05 0.10 0.50 110-610-40.011100104106ΔZ'Ωχh2Δh1=10, gd=0.1, λϕχ=0.4Ωχ h2=0.1198R=0.1, mχ=1GeVR=0.1, mχ=10MeVR=1, mχ=1GeVR=1, mχ=10MeV0.05 0.10 0.50 110-610-40.011100104106ΔZ'Ωχh2Δh1=10, gd=0.1, λϕχ=0.4Figure 2. Dark matter relic density as a function of ∆Z′ for only forbidden channels (on left) for both SIMP andforbidden channels (on right). The central value of relic density, Ωχh2 = 0.1198, is given by black dashed lines.We have taken R = 0.1 and R = 1.0 with R =√2κvd/6mχ.In Fig.1, we note that SIMP process (on right) requires larger dark matter self-scattering cross sectionthan the case with forbidden channels (on left). Besides, the Boltzmann supression factor in (10) iscomparable with those in (4) and (5) for the forbidden channels. From this, we can infer that theforbidden channels can be dominant until ∆Z′ ∼ 0.5 and we check it exactly in the right side of Fig. 2which is in the case with mχ < mZ′  mh1 .4 ConclusionWe have considered the thermal production from freeze-out for self-interacting dark matter in thegauged Z3 model with dark photon and dark Higgs. We discussed that the forbidden channels (χχ∗ →Z′Z′ and/or h1h1, χχ → Z′χ∗ and/or h1χ∗), 3 → 2 self-annihilation and the Standard Model 2 → 2annihilation can contribute to determining the relic density. We found that forbidden channels canproduce light dark matter with sizable self-interation of dark matter to explain the small-scale problem.But the SIMP process favors lighter dark matter and larger self-interaction than forbidden channels.We identified the relevant thermal production mechanisms for self-interacting dark matter.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. The work of YJK was supported by IBS under the project code, IBS-R018-D1.References[1] P. A. R. Ade et al. [Planck Collaboration], Astron.Astrophys. 594 A13 (2016)[2] D. N. Spergel and P. J. Steinhardt, Phys. Rev. Lett. 84 3760 (2000); W. J. G. de Blok, Adv. Astron2010 789293 (2010); M. Boylan-Kolchin, J. S. Bullock andM. Kaplinghat, Mon. Not. Roy. 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Forbidden Channels and SIMP Dark Matter

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