Dark Matter Freeze-out via Catalyzed Annihilation

We present a new paradigm of dark matter freeze-out, where the annihilation of dark matter particles is catalyzed. We discuss in detail the regime that the depletion of dark matter proceeds via $2\chi \to 2A'$ and $3A' \to 2\chi$ processes, in which $\chi$ and $A'$ denote dark matter and the catalyst respectively. In this regime, the dark matter number density is depleted polynomially rather than exponentially (Boltzmann suppression) as in classic WIMPs and SIMPs. The paradigm applies for a secluded weakly interacting dark sector with a dark matter in the $\text{MeV-TeV}$ mass range. The catalyzed annihilation paradigm is compatible with CMB and BBN constraints, with enhanced indirect detection signals.


I. Introduction
The existence of dark matter (DM) is well established with ample evidence from cosmological and astrophysical observations [1]. Though, the nature of dark matter is still unknown. To solve this puzzle, tremendous efforts have been devoted to searching for dark matter candidates and studying production mechanisms of dark matter in the early universe. Among all the mechanisms that reproduce the observed abundance of dark matter, the possibility of thermal dark matter where dark matter keeps in thermal equilibrium with Standard Model (SM) particles in the early universe is especially popular and compelling.
For massive thermal dark matter, DM particles remain in thermal and chemical equilibrium while relativistic. As the universe cools down, DM particles are depleted via some certain processes and the abundance of dark matter goes down. These processes freeze-out when their interaction rate falls below the expansion rate of the universe, and consequently DM abundance is settled. There are essentially two kinds of process leading to depletion of DM particles in the literature. The first one is that DM particles annihilate into other particles, mostly SM particles. The other one is via number-changing process of dark matter. For the former case, the most studied scenario is self-annihilation process [2], e.g. 2DM → 2SM. Especially, weakly interacting massive particles (WIMPs) that naturally reproduce correct relic abundance attracted extensive attentions [3][4][5]. Other variations on the self-annihilation case include co-annihilation [6][7][8], semi-annihilation [9] and so on [6,[10][11][12][13][14][15][16][17][18][19][20][21]. Whereas, for the number-changing process, the most studied process is 3DM → 2DM annihilation, dubbed as strongly interaction massive particles (SIMPs) [22,23]. Subsequently, other number-changing process are proposed and discussed, including Z 2 -symmetric SIMPs [24,25], co-SIMPs [26], etc.
In this Letter, we propose a new pattern of dark matter burning in the early universe beside the two aforemen- * cyxing@pku.edu.cn † shzhu@pku.edu.cn tioned kinds of process, where the abundance of dark matter is determined by catalyzed processes. In catalyzed processes, there are some other particles beside DM that act as the catalyst. The catalyst can enhance the rate of dark matter burning, yet the catalyst itself is not consumed in the reaction [27][28][29]. It provides with an alternative reaction pathway to make the reaction happen without changing the reactants and products. Specifically, in this work we study a simple regime of catalyzed annihilation with two processes leading to depletion of DM particles: 2χ → 2A and 3A → 2χ, where χ and A denote dark matter and the catalyst respectively. We shown in Figure 1 a depiction of how these annihilation channels result in depopulation of DM particles, that is, three 2χ → 2A processes together with two 3A → 2χ effectively deplete two DM particles. Note that the assisted annihilation [20,21,30] are not catalyzed reactions since the assisters are consumed in the reaction. The co-SIMP process SM + χ + χ → SM + χ [26] is not catalyzed reaction either, since χ + χ → χ is kinetically forbidden and it's groundless to discuss enhancement of this unphysical process. Same thing happens to Ref. [31]. We acknowledge that catalyzed processes are also considered in the Big Bang Nucleosynthesis (BBN) [32]. The observed DM abundance can be reproduced in the catalyzed annihilation paradigm for a wide mass range of dark matter. We emphasis that the thermal evolution in the catalyzed annihilation paradigm is unique. Different with WIMPs and SIMPs, where DM number density n χ tracks Boltzmann distribution and shrinks exponentially before freeze-out, the catalyzed annihilation could lead to a polynomial suppression of n χ as the universe cools down, where s and T denote entropy density and temperature of the universe. Thus, the catalyzed annihilation lasts longer and freezes-out at late times. To reproduce correct relic abundance, the cross section of DM annihilation 2χ → 2A should be enhanced since there is less time to redshift to today [15], which corresponds to enhanced indirect detection signals.

II. Catalyzed Freeze-out
In order for the catalyzed annihilation paradigm to work, there are several requirements listed as follows: • The dark sector is nearly secluded.
• Annihilation channels as in Figure 1.
The dark sector should be secluded so that the annihilation channels to SM particles freeze-out before the catalyzed annihilation. The catalyst is lighter than DM and long-lived so that its number density is large and the 3A → 2χ process is not suppressed, which ensures that the catalyzed annihilation happens. If A decays fast, the paradigm recovers to the secluded DM regime [11,12]. Besides, the catalyzed annihilation will heat up the dark sector. For simplicity, we assume the dark sector could scatter with SM particles intensely enough to maintain thermal equilibrium with SM particles. We show in Figure 2 a typical thermal history of dark matter that freezes-out via catalyzed annihilation. For now, we are focused on the regime that the mass ratio of dark matter and the catalyst r ≡ m χ /m A is no larger than 1.5. As is shown in the figure, there are four stages in the thermal evolution: 1. Equilibrium stage. Both χ and A stay in chemical equilibrium due to the number-changing processes in the dark sector. The dominate number-changing process is 3A → 2χ for r 1.1. Other processes with DM in the initial state, e.g. χA A → χA , are suppressed and negligible, since n χ n A .
n χ,A denote the number densities andn χ,A are the equilibrium densities. In the non-relativistic 2. Chemical stage. χ and A are chemically decoupled from equilibrium, but they can still maintain chemical equilibrium with each other via the 2χ ↔ 2A process.
3. Catalyzed annihilation. As the rate of 2A → 2χ (inverse process of 2χ → 2A ) descends exponentially at low temperature, the 3A → 2χ process dominates over it. The evolution of DM number density is now controlled by the catalyzed annihilation, i.e. 2χ → 2A and 3A → 2χ. Before freeze-out, the rates of the 2χ → 2A and 3A → 2χ reactions are much larger than the rate of change of n χ , as well as the Hubble rate and rates of other reactions (see in Eq. 10). Thus, neglecting the subdominant terms, we get an approximate relation, We used σ 2 v and σ 3 v 2 to denote the thermally averaged cross sections of 2χ → 2A and 3A → 2χ respectively. In this stage, since y A ≡ n A /s is practically constant and σ 2 v and σ 3 v 2 are polynomial functions of T , Eq. 4 indicates that n χ is polynomially suppressed. It is similar to the scaling of the number density of the assisting particle after DM freeze-out in Ref. [21].
4. Freeze-out. As the universe expands, the rate of the catalyzed annihilation descends and dark matter freezes-out.
The equilibrium stage ends when the rate of 3A → 2χ falls below Hubble constant H. The temperature of departure from equilibrium T c can be determined approximately with, We note that the annihilation channels to SM particles or the 3A → 2A process can also deplete dark sector particles and T c could be altered if these channels freeze-out later. The ending of the chemical stage is insignificant since the freeze-out temperature T f and relic abundance can be estimated without it. Lastly, the catalyzed annihilation freezes-out when the rate drops below H. T f is determined by, The relic abundance of DM can be estimated approximately in the same spirit of WIMPs [15,33], where we solve for x c with Eq. 5 first. Simplifying Eq. 5, we find, where M Pl is Planck mass. With x c determined in Eq. 8, we can solve for n A . Substituting the result into Eq. 6, we get, The subscripts m, c, f in the equations above mark the temperatures, T = m χ , T c , T f , respectively, for the quantities, including entropy density s, Hubble constant H, effective degrees of freedom g 1 and the thermally averaged cross sections. Note that if x c is delayed due to annihilation to SM particles or 3A → 2A , Eq. 9 and Eq. 8 should be modified to include these processes and DM will freeze-out earlier in this case with a smaller relic abundance. Based on the partial wave unitarity limit [34], χ v , we can estimate the upper bound of DM mass from Eq. 7 for the catalyzed annihilation paradigm. With x f 100, we deduce m χ 100TeV. Compared to SIMP dark matter that lives in the MeV scale [22], it is compelling 1 We neglect the differences between effective entropy degrees of freedom g ,s and effective energy degrees of freedom g as in Ref.
[15] to notice that 3 → 2 process can apply to such a heavy dark matter. In order to study the thermal evolution and DM freezeout in a quantitative way, we turn to the Boltzmann equations. As is discussed above, we neglect the subdominant 3 → 2 annihilation channels, including χA A → χA , χχA → A A , χχA → χχ, χχχ → χA and assume 3A → 2A is subdominant. If A decays to SM particles, the Boltzmann equations reads, The yield y χ,A ≡ n χ,A /s can be solved numerically and are shown in Figure 2.

III. Mass Ratio
In previous section, we concentrated on the mass ratio r ≤ 1.5. In fact, the catalyzed annihilation paradigm can go beyond this limit. Firstly, when the mass ratio is slightly larger than 1.5, i.e. 3m A < 2m χ , σ 3 v 2 is exponentially suppressed as the temperature goes down, where x ≡ m χ /T . During the catalyzed annihilation period, with less DM particles produced via 3A → 2χ process since the cross section is smaller, the DM number density shrinks more sharply. Consequently, the catalyzed annihilation freezes-out much earlier.
As the mass ratio grows, when r 2, it is intriguing to notice that the 4A → 2χ process may play a part in the catalyzed annihilation. To be specific, after a period of catalyzed annihilation governed by 2χ → 2A and 3A → 2χ as usual, there would be an extra stage of catalyzed annihilation predominated by 2χ → 2A and 4A → 2χ, in which the non-suppressed 4A → 2χ process takes over the role of converting A to DM particles since the cross section of 3A → 2χ is exponentially suppressed (Eq. 11), Similar to Eq. 4, we can deduce an approximate relation that holds in this stage, where σ 4 v 3 denotes the thermally averaged cross section for 4A → 2χ. The presence of 4A → 2χ is essential. If it is neglected, as is discussed previously, n χ shrinks sharply and dark matter freezes-out early. Once 4A → 2χ takes charge, the sharply falling of n χ is bent and the polynomial suppression recovers (compared to Eq. 1).
Thus, the catalyzed annihilation freezes-out at later times, leading to enhanced DM relic abundance. For even larger mass ratio, we expect the processes with more catalysts annihilating to two DM particles, e.g. 5A → 2χ, to possibly play a role in the catalyzed annihilation, especially when the dark sector is strongly coupled.
We show in Figure 3 the variation of DM relic abundance Ω χ h 2 with different mass ratio. When the mass ratio passes the critical value of 1.5, Ω χ h 2 decreases rapidly. On the other hand, for r 2, relic abundance is uplifted if 4A → 2χ process is included.

IV. A Model
The requirements for realization of the catalyzed annihilation presented in Section II can be easily met in many models. In this section, we simply present a dark photon model [36][37][38][39][40] with a Dirac fermion χ charged under a novel U (1) gauge group and A being the gauge field. The Lagrangian for the dark sector is, where / D = / ∂ − ig D / A and g D is the gauge coupling constant. The mass of the dark photon can be generated via the Higgs mechanism (or Stueckelberg mechanism [41,42]). We assume the dark Higgs boson is heavy and can be neglected. SM particles are neutral under the U (1) gauge group. The dark photon can be kinetically mixed with SM hypercharge field. is the mixing constant and θ W denotes the Weinberg angle. B µ is SM hypercharge field. Therefore, the dark sector can communicate with SM particles via the mixing and the dark photon A can decay to SM particles. should be small so that the dark photon is long-lived and acts as the catalyst. Additionally, the kinetic mixing could not keep the dark sector in thermal equilibrium with SM since is small. In order to thermalize the dark sector, we need another portal for the dark sector to interact with SM particles, which might be the dark Higgs. Anyhow, we won't model this part and simply assume that the dark sector stays in thermal equilibrium before freeze-out.
We show in Figure 4 different phases for the model in the calculation of DM relic abundance. For short-lived dark photon, before dark matter freezes-out, it simply stays in equilibrium with SM particles via the decay and inverse-decay process. When DM particles annihilates into the dark photon, it immediately decays. This is the secluded phase of the model. On the other hand, when the dark photon width Γ A is small, the catalyzed annihilation emerges. It is a continuous shift, since the decay of the dark photon can occur during the catalyzed annihilation. When the dark photon decays after DM freeze-out, Ω χ h 2 is independent with Γ A . If A is sufficiently long-lived (Γ A 10 −23 GeV), it would come to dominate the energy density of the universe. When it decays, considerable entropy is produced and DM abundance is diluted. This effect can help to circumvent the upper bound of DM mass [43][44][45] (see in Figure 4). We show in dashed gray curves for five different values of g D that reproduce the observed relic abundance in Figure 4.
The catalyzed annihilation paradigm is constrained by numerous terrestrial and celestial experiments and observations. Firstly, the residual annihilation of 2χ → 2A after freeze-out will distort the anisotropy of the Cosmic Microwave Background (CMB) since the decay products of A are electrically charged particles [46][47][48][49][50][51][52].
For light dark photon, beam dump and fixed target experiments provide great sensitivity on the mixing coupling constant [72][73][74][75]. There are also lots of new experiments [76][77][78][79] proposed in recent years that are focused on long-lived particles. Besides, the long-lived dark photon can enhance the cooling of supernova and the constraints from SN 1987A [80][81][82] is widely discussed. These bounds on the dark photon model are considered and presented in Figure 4.

V. Conclusion and discussion
We proposed a novel paradigm for thermal relic dark matter, yielding the observed relic abundance. The distinctive wisdom of the paradigm is that the dark matter freeze-out proceeds via catalyzed annihilation. We discussed in detail the scenario that the catalyzed annihilation includes 2χ → 2A and 3A → 2χ, where χ and A are dark matter and the catalyst respectively. The paradigm applies for a wide mass range of dark matter, from 1MeV to 100TeV, with a unique thermal history compared with WIMPs and SIMPs. Besides, the paradigm offers rich phenomenology including indirect DM search and long-lived particles.
We note that thermal decoupling effects can significantly modify dark matter relic abundance [16,30,[83][84][85][86]. We leave this to future works [87]. Additionally, catalyzed annihilation dominating DM abundance can go far beyond the reactions considered here and should be investigated further.