SIMP dark matter and its cosmic abundances
SSIMP dark matter and its cosmic abundances
Soo-Min
Choi ,(cid:63) , Hyun Min
Lee ,(cid:63)(cid:63) , and Min-Seok
Seo ,(cid:63)(cid:63)(cid:63) Department of Physics, Chung-Ang University, Seoul 06974, Korea. Center 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 → Z models. Prepared for the proceedings of the 13th International Conference on Gravitation,Ewha Womans University, Korea, 3-7 July 2017.
Dark 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 e ff ect. 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 → → ff erences 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 Particles(SIMP) [1]. SIMP has a 3 → (cid:63) e-mail: [email protected] (cid:63)(cid:63) e-mail: [email protected] (cid:63)(cid:63)(cid:63) e-mail: [email protected] a r X i v : . [ h e p - ph ] A ug xpanded 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. In 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 → n → 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.
An 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 + Hn χ = −(cid:104) σv (cid:105) ( n χ − n χ n eq ) (1)where σv = E E E (cid:90) d p (2 π ) E d p (2 π ) E |M| (2 π ) δ ( p + p + p − p − p ) . (2)The cross-section with 5-point interactions should be thermal averaged to get the exact relic den-sity. Thermal averaged 3 → → → (cid:104) σv (cid:105) = (cid:82) d v d v d v δ ( (cid:126)v + (cid:126)v + (cid:126)v )( σv ) e − x ( v + v + v ) (cid:82) d v d v d v δ ( (cid:126)v + (cid:126)v + (cid:126)v ) e − x ( v + v + v ) . (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,( σv ) = a + a ( v + v + v ) + a (1)2 ( v + v + v ) + a (2)2 ( v + v + v ) + · · · . (4)o there are SO(9) invariant terms and general terms of the form, ( v ) n ( v ) m ( v ) l . If the 3 → (cid:104) σv (cid:105) = x ∞ (cid:88) l = a l l ! (cid:90) ∞ d η η l + e − x η = a + a x − + a x − + · · · . (5)with η ≡ ( v + v + v ). Otherwise, if the cross-section depends on the general terms, then, (cid:104) ( v ) n ( v ) m ( v ) l (cid:105) = √ x π (cid:90) ∞ d v v (cid:90) ∞ d v v ( v ) n ( v ) m ×× (cid:90) + − d cos θ ( v + v + v v cos θ ) l e − x ( v + v + v v cos θ ) , especially, (cid:104) ( v ) n (cid:105) = (cid:32) (cid:33) n Γ ( n + ) Γ ( ) x − n = (cid:104) ( v ) n (cid:105) = (cid:104) ( v ) n (cid:105) . When the 3 → (cid:104) σv (cid:105) = (cid:42) ∞ (cid:88) l = b ( l ) R l ! η l γ R ( (cid:15) R − η ) + γ R (cid:43) = π x ∞ (cid:88) l = b ( l ) R l ! G l ( z R ; x ) , (6)where G l ( z R ; x ) = Re (cid:20) i π (cid:90) ∞ d η η l + e − x η z R − η (cid:21) = ( − l ∂ l ∂ x l G ( z R ; x ) , (7) (cid:15) R ≡ m R − m m , γ R ≡ m R Γ R m and z R ≡ (cid:15) R + i γ R . When γ R (cid:28) G ( z R ; x ) ≈ (cid:15) R e − x (cid:15) R θ ( (cid:15) R ). So the thermalaveraged cross-section is given by (cid:104) σv (cid:105) R ≈ π(cid:15) R x e − x (cid:15) R θ ( (cid:15) R ) ∞ (cid:88) l = b ( l ) R l ! (cid:16) (cid:17) l (cid:15) 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, the SIMP case highly depends on the resonance mass( (cid:15) R )than WIMP case( (cid:15) / R ).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 → , where n >
3. Even if the massesof the initial particles are di ff erent, we can calculate the thermal averaged cross-section. Especially,when the cross-section is s-wave, the result can be obtained by m → ( m + m + m ) / χχ χχ ∗ h (cid:48) Figure 1.
One example of the dark Higgs resonance channel of the discrete Z gauged symmetry dark mattermodel As a concrete model, we show the discrete Z 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 Z . 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 e ff ective 3 → σv ) Z = C γ R ( (cid:15) R − η ) + γ R , (9)and C is given by C = √ κ β χ m χ (cid:32) + λ φχ v (cid:48) m χ (cid:33) . (10)Definition of β χ is (cid:113) − m χ / m h (cid:48) . Because the three-body decay ( h (cid:48) → χχχ ) is suppressed by extraphase space with (cid:15) R / (4 π ), the decay width of dark Higgs is determined mainly by two-body decay( h (cid:48) → χχ ∗ ). From eq.(6), the thermal averaged cross-section of eq.(9) is given by (cid:104) σv (cid:105) Z = C π x G ( z R ; x ) . (11)By using the narrow width approximation, (cid:104) σv (cid:105) Z ≈ C π(cid:15) R x e − x (cid:15) R θ ( (cid:15) R ) . (12)In figure 2, we show the parameter space for the relic density of dark matter as a function of C and (cid:15) 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 (cid:15) R = . , . , .
06. For both plots in figure 2, we assume narrow widthapproximation(NWA).
We 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 exact R = ϵ R = C Ω h NWA, m χ = ϵ R = ϵ R = ϵ R =
100 200 300 400024681012 m χ ( MeV ) κ NWA
Figure 2.
Relic density of Z dark matter as a function of C and (cid:15) R in the left. Dark matter cubic couplingsatisfying a correct relic density as a function of dark matter mass and (cid:15) 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. The 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.
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