Dark Matter Freeze-out via Catalyzed Annihilation
DDark Matter Freeze-out via Catalyzed Annihilation
Chuan-Yang Xing ∗ and Shou-hua Zhu
1, 2, 3, † Department of Physics and State Key Laboratory of Nuclear Physics and Technology, Peking University, Beijing 100871, China Collaborative Innovation Center of Quantum Matter, Beijing, 100871, China Center for High Energy Physics, Peking University, Beijing 100871, China
We present a new paradigm of dark matter freeze-out, where the annihilation of dark matterparticles is catalyzed. We discuss in detail the regime that the depletion of dark matter proceedsvia 2 χ → A (cid:48) and 3 A (cid:48) → χ processes, in which χ and A (cid:48) denote dark matter and the catalystrespectively. In this regime, the dark matter number density is depleted polynomially rather thanexponentially (Boltzmann suppression) as in classic WIMPs and SIMPs. The paradigm applies fora secluded weakly interacting dark sector with a dark matter in the MeV-TeV mass range. Thecatalyzed annihilation paradigm is compatible with CMB and BBN constraints, with enhancedindirect detection signals. I. Introduction
The nature of dark matter (DM) is still unknown.Though, the observed abundance of DM may give somehints on its coupling with other particles, especially Stan-dard Model (SM) particles. For thermal DM, in the earlyuniverse, if DM is very weakly coupled and decouplesfrom the thermal bath while relativistic, it should berather light to reproduce the observed relic abundance, m DM ∼ Ω h . , which is not favoured [1–3]. For heavierDM, there has to be some DM depletion processes thatare intense enough to freeze-out at late times so thatDM is not over-produced. This motivates the studiesand experimental detection of mechanisms of dark matterfreeze-out.There are essentially two kinds of process leading todepletion of DM particles in the literature. The firstone is that DM particles annihilate into other parti-cles, mostly thermal bath particles. The other one isvia number-changing process. For the former case, themost studied scenario is self-annihilation process [4]. Es-pecially, weakly interacting massive particles (WIMPs)that naturally reproduce correct relic abundance attractedextensive attentions [5–7]. Other variations on this in-clude co-annihilation [8–10], forbidden annihilation [8, 11],semi-annihilation [12], secluded annihilation [13, 14] andso on [15–23]. Whereas, for the number-changing pro-cess, the most studied process is 3DM → Z -symmetric SIMPs [26, 27], co-SIMPs [28], etc.In this work, we propose a novel kind of process ofdark matter burning in the early universe. Consider anearly secluded dark sector with a stable DM (denoted as χ in this work) and a mediator (denoted as A (cid:48) ). The twoprocesses that lead to the depletion of DM are 2 χ → A (cid:48) and 3 A (cid:48) → χ . We shown in Figure 1 a depiction of ∗ [email protected] † [email protected] + A ' FIG. 1. Schematic illustration of the catalyzed annihilationof DM χ (red line) with a catalyst A (cid:48) (blue line). Three2 χ → A (cid:48) processes plus two 3 A (cid:48) → χ processes effectivelydeplete the number of dark matter particles by two. how these annihilation channels result in depopulation ofDM particles, that is, three 2 χ → A (cid:48) processes togetherwith two 3 A (cid:48) → χ effectively deplete two dark matterparticles. On the other hand, these two processes merelychange the comoving number density of the mediatorduring the period that they dominate DM burning. Themediator acts similar to a catalyst in a chemical reactionand we dub the processes as catalyzed annihilation of darkmatter. We acknowledge that catalyzed processes are alsoconsidered in the Big Bang Nucleosynthesis (BBN) [29]and nuclear fusion [30, 31].In order for this paradigm to work, there are severalrequirements listed as follows: • The dark sector is nearly secluded. • The catalyst is long-lived ( (cid:38) − s). • The catalyst is slightly lighter than DM (1
We show in Figure 2 a typical thermal history of DMthat freezes-out via catalyzed annihilation in the earlyuniverse. For now, we are focused on the regime thatthe mass ratio of DM and the catalyst r ≡ m χ /m A (cid:48) isno larger than 1 .
5. As is shown in the figure, there areessentially four stages in the thermal history of DM:1.
Equilibrium stage.
Both DM and the catalyst A (cid:48) stay in chemical equilibrium with the thermal bath. n χ (cid:39) ¯ n χ , n A (cid:48) (cid:39) ¯ n A (cid:48) . (3) n χ,A (cid:48) denote the number densities and ¯ n χ,A (cid:48) are theequilibrium densities of DM and A (cid:48) with zero chem-ical potential. In the non-relativistic limit, we have¯ n χ,A (cid:48) = g χ,A (cid:48) (cid:16) m χ,A (cid:48) T π (cid:17) / e − m χ,A (cid:48) /T [32], where g denotes number of internal degrees of freedom.2. Chemical stage.
During this period, though DMand the catalyst are chemically decoupled from thethermal bath, they can still maintain chemical equi-librium with each other via the 2 χ ↔ A (cid:48) process. n χ / ¯ n χ (cid:39) n A (cid:48) / ¯ n A (cid:48) . (4)
10 100 1000 100010 - - - - - equilibrium chemical catalyzed freeze - out FIG. 2. Thermal evolution of DM (solid red) and the catalyst(solid blue). The dashed curves denote equilibrium yieldsfor DM and the catalyst respectively, while the dashed blackcurve shows the approximate DM number density during thecatalyzed annihilation stage n appχ = (cid:112) n A (cid:48) (cid:104) σ v (cid:105) / (cid:104) σ v (cid:105) fromEq. 5. The parameters are taken for the model presented inSection IV. Catalyzed annihilation.
After the chemical decou-pling of DM and the catalyst, the catalyzed annihila-tion takes over. In this stage, the 3 A (cid:48) → χ processdominates over 2 A (cid:48) → χ annihilation since the lat-ter is exponentially suppressed at low temperature.The rate of 3 A (cid:48) → χ process turns comparablewith the 2 χ → A (cid:48) annihilation rate. (cid:104) σ v (cid:105) n χ (cid:39) (cid:10) σ v (cid:11) n A (cid:48) . (5)We used (cid:104) σ v (cid:105) and (cid:10) σ v (cid:11) to denote the thermallyaveraged cross section of 2 χ → A (cid:48) and 3 A (cid:48) → χ processes respectively.4. Freeze-out and the catalyst decays.
As the universeexpands, the rate of the catalyzed annihilation de-scends and freezes-out. The catalyst is expected todecay after the freeze-out.The equilibrium stage ends when the rates of thenumber-changing processes inside the dark sector fall be-hind the expansion of the universe. The dominate number-changing process is 3 A (cid:48) → χ for moderately large massratio r (cid:38) .
1, since the catalyst is exponentially moreabundant than DM at low temperature n A (cid:48) (cid:29) n χ . Otherprocesses with DM in the initial state, e.g. χA (cid:48) A (cid:48) → χA (cid:48) ,are suppressed and negligible. Subsequently, we can de-termine the temperature of departure from equilibrium T c approximately, (cid:10) σ v (cid:11) n A (cid:48) (cid:39) H ( n A (cid:48) + n χ ) , (6)where H is Hubble constant. We note that the annihila-tion channels to SM particles or the 3 A (cid:48) → A (cid:48) processcan also result in depletion of dark sector particles. Thedeparture from equilibrium could be determined by thesechannels if they freeze-out later. Similarly, the catalyzedannihilation freezes-out when the rate drops below Hubbleconstant. The freeze-out temperature T f , therefore, isdetermine by, (cid:104) σ v (cid:105) n χ (cid:39) (cid:10) σ v (cid:11) n A (cid:48) (cid:39) Hn χ . (7)Note the difference between Eq. 6 and Eq. 7.The relic abundance of DM can be estimated approxi-mately in the same spirit of WIMPs [33],Ω χ = m χ s H m ρ c s m √ g (cid:63),m √ g (cid:63),f x f (cid:104) σ v (cid:105) , (8)where x f ≡ m χ /T f . The subscripts m and f mark thetemperatures, T = m χ and T = T f respectively, for thequantities, including entropy density s , Hubble constant H and effective degrees of freedom g (cid:63) . ρ c denotes criticalenergy density and s is the entropy density today. Weemphasize that although T c deduced from Eq. 6 does notshow explicitly in Eq. 8, the freeze-out temperature T f isdependent on T c and DM relic abundance changes if thedeparture from equilibrium is delayed due to annihilationchannels to SM particles or 3 A (cid:48) → A (cid:48) process.Based on the partial wave unitarity limit [34], σ v ≤ πm χ v , we can estimate the upper bound of DM mass fromEq. 8 for the catalyzed annihilation paradigm. With x f (cid:38) m χ (cid:46) . (9)Compared to SIMP dark matter that lives in MeVscale [24], it is intriguing to notice that 3 → → χA (cid:48) A (cid:48) → χA (cid:48) , χχA (cid:48) → A (cid:48) A (cid:48) , χχA (cid:48) → χχ , χχχ → χA (cid:48) and assume3 A (cid:48) → A (cid:48) is subdominant. If A (cid:48) decays to SM particles,the Boltzmann equations reads,˙ n χ + 3 Hn χ = − (cid:104) σ v (cid:105) (cid:18) n χ − ¯ n χ n A (cid:48) ¯ n A (cid:48) (cid:19) (10)+ 2 (cid:10) σ v (cid:11) (cid:32) n A (cid:48) − ¯ n A (cid:48) n χ ¯ n χ (cid:33) , ˙ n A (cid:48) + 3 Hn A (cid:48) = +2 (cid:104) σ v (cid:105) (cid:18) n χ − ¯ n χ n A (cid:48) ¯ n A (cid:48) (cid:19) − (cid:10) σ v (cid:11) (cid:32) n A (cid:48) − ¯ n A (cid:48) n χ ¯ n χ (cid:33) − (cid:104) Γ A (cid:48) (cid:105) ( n A (cid:48) − ¯ n A (cid:48) ) . Note the conventions adopted in Eq. 10. We only showthe number of difference in the initial and final state for We neglect the differences between effective entropy degrees offreedom g (cid:63),s and effective energy degrees of freedom g (cid:63) as inRef. [17] FIG. 3. Curves of relic abundance Ω χ h for m χ = 350GeV(solid red) and m χ = 1000GeV (solid blue) with respect todifferent mass ratio. For dashed colored curves, the 4 A (cid:48) → χ process is neglected. The dashed black curve denotes theobserved DM relic abundance [36]. The parameters are takenfor the model presented in Section IV. each particle, i.e. the number of host particle for eachequation in the final state minus that in the initial state.Other factors, including the initial state symmetry factorand a factor 2 if the process is not self-conjugate, areabsorbed in the definition of thermal average [23, 35].The yield of DM and the catalyst y χ,A (cid:48) ≡ n χ,A (cid:48) /s can besolved numerically and are shown in Figure 2. III. Mass Ratio
In previous section, we concentrated on the mass ratio r ≤ .
5. In fact, the catalyzed annihilation paradigmcan go beyond this limit. Firstly, when the mass ratiois slightly larger than 1 .
5, i.e. 3 m A (cid:48) ≤ m χ , (cid:10) σ v (cid:11) isexponentially suppressed as the temperature goes down, (cid:10) σ v (cid:11) ∝ e − (2 r − x/r . (11)During the catalyzed annihilation period, with less DMparticles produced via 3 A (cid:48) → χ process since the crosssection is smaller, the number density of DM shrinksmore sharply (see in Eq. 5). Consequently, the catalyzedannihilation freezes-out earlier, much earlier than theregime of r ≤ . r (cid:46)
2, it is intriguingto notice that the 4 A (cid:48) → χ process may play a part inthe catalyzed annihilation. To be specific, after a periodof catalyzed annihilation governed by the 2 χ → A (cid:48) and3 A (cid:48) → χ processes as usual, since the 3 A (cid:48) → χ anni-hilation is exponentially suppressed, the non-suppressed4 A (cid:48) → χ process takes over the role of converting the cat-alyst to DM particles and there would be an extra stageof catalyzed annihilation predominated by the 2 χ → A (cid:48) and 4 A (cid:48) → χ processes. Similar to Eq. 5, we can deducean approximate relation that holds in this stage, (cid:104) σ v (cid:105) n χ (cid:39) (cid:10) σ v (cid:11) n A (cid:48) , (12)where (cid:10) σ v (cid:11) denotes the thermally averaged cross sectionfor 4 A (cid:48) → χ process. The thermal evolution and relicabundance of DM can be significantly modified due tothe 4 A (cid:48) → χ process. Specifically, if the 4 A (cid:48) → χ pro-cess is neglected, as is discussed previously, n χ decreasesexponentially (Eq. 11) during the catalyzed annihilationand freezes-out early. Once the 4 A (cid:48) → χ process takescharge, the exponentially falling of n χ is bent and thepolynomial suppression recovers (compared to Eq. 1). n χ ∝ s ∝ x − . (13)Therefore, the catalyzed annihilation freezes-out at latertimes, leading to an enhanced relic abundance of DM.For even larger mass ratio, we expect the processeswith more catalyst annihilating to two DM particles, e.g.5 A (cid:48) → χ , to possibly play a role in the catalyzed an-nihilation, especially when the dark sector is stronglycoupled.We show in Figure 3 the variation of relic abundance ofDM Ω χ h with different mass ratio. When the mass ratiopasses the critical value of 1 .
5, Ω χ h decreases rapidly.On the other hand, for r (cid:46)
2, relic abundance is upliftedif 4 A (cid:48) → χ process is included. IV. A Model
The requirements for realization of the catalyzed anni-hilation presented in Section I can be easily met in variouskinds of models. In this section, we simply present a darkphoton model [37–41] with a Dirac fermion χ chargedunder a novel U (1) (cid:48) gauge group and A (cid:48) being the gaugefield. The Lagrangian for the dark sector is, L DS = − F (cid:48) µν F (cid:48) µν + 12 m A (cid:48) A (cid:48) µ A (cid:48) µ + ¯ χ ( i /D − m χ ) χ, (14)where /D = /∂ − ig D /A (cid:48) and g D is the gauge couplingconstant. The mass of the dark photon can be generatedvia the Higgs mechanism (or Stueckelberg mechanism [42,43]). We assume the dark Higgs boson is heavy and canbe integrated out. SM particles are netural under the U (1) (cid:48) gauge group. Though, the dark photon can bekinetically mixed with SM hypercharge field. L mix = − (cid:15) θ W F (cid:48) µν B µν . (15) (cid:15) is the mixing constant and θ W denotes the Weinbergangle. B µ is SM hypercharge field. Therefore, the darksector can communicate with SM particles via the mixingand the dark photon A (cid:48) can decay to SM particles. Weexpect that the mixing constant (cid:15) is small so that thedark photon is long-lived and acts as the catalyst. Wenote that since the catalyzed annihilation happens withinthe dark sector, the thermal evolution of DM is generallyindependent of (cid:15) as long as it is small enough so thatthe dark photon decays after the freeze-out. In contrast,if the dark photon decays fast, the models recovers the - - - - - Secluded Catalyzed N o n - p er t u r b a t i v e BeamDumpSupernova CMB Fermi - LAT BBN
FIG. 4. Phase diagram and constraints for the dark photonmodel in the ( m χ , Γ A (cid:48) ) plane with r = 1 .
5. The solid blackcurve marks the boundary of the secluded phase and thecatalyzed phase of the model. Correct relic abundance canbe reproduced for each point in the figure by varying thevalue of g D . Especially, we show in gray dashed curves forfive different values of g D that reproduce correct relic density.The non-perturbative region is painted gray, while the color-shaded regions denote the bounds from various experimentsand observations correspondingly. secluded dark matter paradigm. Additionally, we notethat the kinetic mixing in Eq. 15 could not keep the darksector in thermal equilibrium with the thermal bath since (cid:15) is too small. In order to thermalize the dark sector, weneed another portal for the dark sector to interact withSM particles, which might be the dark Higgs. Anyhow,we won’t model this part in this work and simply assumethat the dark sector stays in thermal equilibrium beforefreeze-out.We show in Figure 4 different phases for the model inthe calculation of DM relic abundance. For short-liveddark photon, before dark matter freezes-out, it simplystays in equilibrium with the thermal bath via the decayand inverse-decay process. Thus, when DM particlesannihilates into the dark photon, it immediately decaysinto SM particles. This is the secluded phase of themodel. On the other hand, when the dark photon widthΓ A (cid:48) is small, the catalyzed annihilation emerges. It isa continuous shift, since the decay of the dark photoncan occur during the catalyzed annihilation stage. Whenthe dark photon decays after DM freeze-out, Ω χ h isindependent with Γ A (cid:48) . Figure 4 shows such a featureexplicitly. We show in dashed gray curves for five differentvalues of g D that reproduce the observed relic abundanceand the curves only bend near the phase boundary.The catalyzed annihilation paradigm is constrainedby numerous terrestrial and celestial experiments andobservations. Generally speaking, for any model thatundergoes a period of catalyzed annihilation, the residualannihilation of DM into the catalyst after freeze-out willdistort the anisotropy of the Cosmic Microwave Back-ground (CMB) if the decay products of the catalyst areelectrically charged particles [44–49]. Similarly, the signalof the annihilation of DM at present is detectable in indi-rect detection experiments [50–55]. We used bounds fromFermi-LAT experiment to constrain our model in Figure 4.The late time decay of the catalyst, on the other hand,is also stringently constrained by CMB [56–58] as wellas BBN [59–61]. For the dark photon model presentedpreviously, the lower bound of the decay width of the darkphoton is taken conservatively according to Ref. [61],Γ A (cid:48) ≥ . · − GeV . (16)We note that this bound can be evaded if the catalystdecays into neutrinos or dark radiations [62–64].For light dark photon, beam dump and fixed targetexperiments provide great sensitivity on the mixing cou-pling constant (cid:15) [65–68]. There are also lots of new experi-ments [69–72] proposed in recent years that are focused onlong-lived particles. Besides, the long-lived dark photoncan enhance the cooling of supernova and the constraintsfrom SN 1987A [73–75] is widely discussed. These boundson the dark photon model are considered and presentedin Figure 4. V. Conclusion and discussion
We proposed a novel paradigm for thermal relic darkmatter, yielding the observed relic abundance. The dis- tinctive feature of the paradigm is that the dark matterfreeze-out proceeds via catalyzed annihilation. We dis-cussed in detail the scenario that the catalyzed annihila-tion includes 2 χ → A (cid:48) and 3 A (cid:48) → χ processes, where χ and A (cid:48) are dark matter and the catalyst respectively.The paradigm applies for a wide range of dark matter,from MeV-scale to several tens of TeV.In this work, the dark sector is assumed to be in thermalequilibrium before freeze-out. Whereas, we note thatthermal decoupling effects can significantly modify darkmatter relic abundance [18, 76–80]. In fact, it can besubtle to tune the annihilation of dark sector to SMparticles to be small and the thermal scattering betweenthem to be large simultaneously, especially when thefreeze-out occurs at late time. We leave this to futureworks [81]. Acknowledgments
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