Composite Asymmetric Dark Matter with a Dark Photon Portal

Asymmetric dark matter (ADM) is an attractive framework relating the observed baryon asymmetry of the Universe to the dark matter density. A composite particle in a new strong dynamics is a promising candidate for ADM as the strong dynamics naturally explains the ADM mass in the GeV range. Its large annihilation cross section due to the strong dynamics leaves the asymmetric component to be dominant over the symmetric component. In such composite ADM scenarios, the dark sector has a relatively large entropy density in the early Universe. The large dark sector entropy results in the overclosure of the Universe or at best contradicts with the observations of the cosmic microwave background and the successful Big-Bang Nucleosynthesis. Thus, composite ADM models generically require some portal to transfer the entropy of the dark sector into the Standard Model sector. In this paper, we consider a dark photon portal with a mass in the sub-GeV range and kinetic mixing with the Standard Model photon. We investigate the viable parameter space of the dark photon in detail, which can find broad applications to dark photon portal models. We also provide a simple working example of composite ADM with a dark photon portal. Our model is compatible with thermal leptogenesis and $B - L$ symmetry. By taking into account the derived constraints, we show that the parameter space is largely tested by direct detection experiments.


I. INTRODUCTION
The existence of dark matter (DM) has been overwhelmingly established by a wide range of cosmological and astrophysical observations. Nevertheless, its nature remains elusive other than several: its electromagnetic interaction is feeble; it is cold enough to cluster along the primordial gravitational potential; and it is stable at least over the age of the Universe. Identification of the nature of DM is one of the most important challenges of modern particle physics (see, e.g., Refs. [1][2][3]).
In the present Universe, the mass density of DM is about five times larger than that of Standard Model (SM) baryon [4]. This coincidence can be naturally explained when the DM number has the same origin as baryon asymmetry of the Universe and the DM particle mass is in the GeV range. Such a framework is called asymmetric dark matter (ADM) (see, e.g., Refs. [5][6][7]). In this paper, we consider an ADM model where composite DM is charged under the B − L gauge symmetry.
The B − L gauge symmetry is one of the most plausible extensions of the SM. The quantum anomalies of the B − L gauge symmetry are canceled when three SM singlet righthanded neutrinos are introduced. The B − L extended SM naturally explains the observed tiny neutrino masses via the seesaw mechanism [8][9][10][11][12]. Furthermore, decay of right-handed neutrino explains baryon asymmetry via thermal leptogenesis [13] (see also Refs. [14][15][16] for reviews). As DM carries a B − L charge, there can be a portal interaction of B − L asymmetry into the confining dark sector.
Composite ADM has several advantages [17,18]. The DM mass in the GeV range naturally arises by the strong dynamics without fine-tuning. A large annihilation cross section of composite DM makes the symmetric part of relic DM negligible and thus the DM density is dominated by the asymmetric component proportional to B − L asymmetry. DM decay through the portal is suppressed by a hierarchy between the strong dynamics scale and the portal scale.
As we will see, we reach a simple model with a QCD-like SU(3) strong interaction and a QED-like U(1) gauge interaction. We assume that the entropy of the dark sector is released to the SM sector through kinetic mixing of dark photon to SM photon. We show that such a model can be tested via direct detection experiments of DM.
The organization of the paper is as follows. In Sec. II, we construct a composite B − L ADM model in a bottom-up approach. In Sec. III, we discuss cosmological constraints on the model. The final section is devoted to our conclusions.
We consider a SU(N c ) D gauge dynamics referred to as the QCD . There are N f -flavors of vector-like dark quarks (Q i ,Q i ) (i = 1 · · · N f ) with B − L charges of (q B−L , −q B−L ). We assume that the masses of dark quarks, are smaller than the QCD scale Λ QCD . Below the dynamical scale, dark quarks are confined into dark mesons and dark baryons.
By assuming spontaneous chiral symmetry breaking, we expect that the lightest mesons are pseudo-Nambu-Goldstone modes, i.e., dark pions. The dark pions obtain masses of m π = O( m 1, 2 Λ QCD ). Dark baryons carry a B − L charge and the lightest ones are good ADM candidates. In this paper, we assume that dark baryons with the lowest spin are lighter than those with higher spins, while the detailed mass spectrum does not change the following discussion qualitatively. The annihilation cross section of dark baryons into dark mesons is quite large due to the strong dynamics, with which the symmetric part of relic DM is negligibly small [19][20][21][22][23]. As a result, the DM abundance is naturally dominated by the asymmetric component. In our scenario, we assume that B − L asymmetry is generated by thermal leptogenesis when the cosmic temperature is around the right-handed neutrino mass M R 10 10 GeV [14][15][16]. The right-handed neutrinos couple to the SM particles via where H and L denote the SM Higgs and lepton doublets, respectively. We remark that M R encapsulates the effects of spontaneous breaking of B − L with a B − L charge of −2.
Then, part of B − L asymmetry is propagated into the dark sector through the portal interaction,  [24,25]).
The portal interaction eventually decouples around where M PL 2.4 × 10 18 GeV denotes the reduced Planck scale. Then, B − L number is conserved independently in the SM sector and in the dark sector, making DM particle quasistable up to the portal interaction. Decay through the portal interaction is suppressed by powers of Λ QCD /M * . In ADM with strong annihilation, the DM mass is determined by the ratio of B − L asymmetry between the DM and SM sectors A DM /A SM as where we used the ratio between between A SM and the baryon asymmetry observed today, A SM /A B = 97/30 [26]. In the composite model, the effective number of massless degrees of freedom in the dark sector is sizable in the early Universe. Thus, if some dark pions are stable, they overclose the Universe or contribute to the effective number of neutrino degrees of freedom N eff too much, depending on their masses [27]. In this paper, we introduce a U (1) D gauge dynamics QED , under which d. As dark quarks are charged under U (1) D , dark mesons annihilate into dark photons. We assume that dark photon obtains a mass m γ by the Higgs mechanism in the dark sector and has kinetic mixing with SM photon: where F and F are the field strengths of SM photon A and dark photon A , respectively. Dark photon decays into SM particles through kinetic mixing with SM photon with the rate of Here, α denotes the QED fine-structure constant. If dark photon decays only into electron and positron, N ch = 1. To make the above thermal history available, we arrange the masses as where m e denotes the electron mass. If the mixing parameter is too small, the entropy of the dark sector is not released to the SM sector efficiently, which results in too much dark radiation. As dark baryon charged under U (1) D interacts with SM proton via dark photon exchange, direct detection experiments provide upper bounds on . In the next section, we will identify a viable parameter region of (m γ , ) by taking the following model as an example.
A. N c = 3 case As we find the minimal model with N c = 2 and N f = 2 rather subtle as shown in appendix A, here, let us consider the case with N c = 3 and N f = 2. In Table I, we show the charge assignment of dark quarks. As the QCD and QED charge assignment is parallel to the SM one, it is apparently free from quantum anomalies and we can use the analogy to QCD. 1 In this case, dark pions are and dark baryons are We summarize hadron mass formulas in appendix B. We emphasize that the QED charge assignment in Table I is the unique choice (up to trivial normalization) that makes one of the dark baryon neutral and allows the following portal interaction. The lowest dimensional portal interaction is given by which requires q B−L = 1/3. Below the mass scale of M R , the above portal interaction results in an effective interaction, and hence, M * in Eq. (3) should be identified as (M 2 * M R /y N ) 1/3 . We assume that T D is below the right-handed neutrino mass scale and is above the decoupling temperature of the Sphaleron process. The ratio of the B − L asymmetries between the dark and SM sectors is given by [25], It leads to m DM = 8.5 GeV [see Eq. (5)], for which we take Λ QCD ∼ 10 × Λ QCD with Λ QCD ∼ 200 MeV denoting the QCD scale. By arranging m 1 and m 2 , one can take dark neutron lighter or heavier than dark proton. We consider a dark pion mass of O(10-100) MeV or larger since the dark photon mass is in this range as we will see in the next section. We assume that the n -p mass difference, m n − m p = O(m 1,2 ) (see appendix B)), is smaller than the dark pion mass, m π = O( m 1, 2 Λ QCD ). The portal interaction in Eq. (12) leads to decay of dark neutron into dark pion and SM neutrino. Neutrino flux measurements by the Super-Kamiokande (SK) collaboration bound the portal scale from below as M * 10 8.5 GeV [25] (see also Ref. [32]). Dark proton property depends on the charge of the U (1) D Higgs boson H D . For charge −2, the Z 2 subgroup of U (1) D remains unbroken, with which π ± becomes stable and p becomes quasi-stable up to the portal interaction. Since m π > m n − m p , n is also quasi-stable up to the portal interaction. In the following, we consider this case for the sake of simplicity of the analysis, although the case with charge −1 can also be viable as discussed in appendix C.
Before closing this section, let us notice that there can be a B −L neutral portal operator, for q B−L = 1/3. When both the portals in Eqs. (12) and (14) are effective in the thermal bath with the temperature below M R , baryon asymmetry generated by thermal leptogenesis is washed out. To avoid this problem, we assume that M * in Eq. (14) is much larger than that in Eq. (12). This can be realized, for instance, as follows. One introduces a scalar quark φ that transforms as a fundamental representation of SU (3) D with the QED charge of 2/3 and B − L charge of −2/3. With this charge assignment, the portal interaction in Eq. (11) is generated via,  14) is generated via which leads to a larger suppression scale,

III. COSMOLOGICAL CONSTRAINTS
In this section, we discuss constraints on the dark photon parameters (m γ , ) assuming the model in Sec. II A, while most of the following arguments are applicable to the model in appendix A with trivial modifications. We first discuss the requirement to satisfy the cosmic microwave background (CMB) constraints on N eff = 3.15 ± 0.23 [4].
Below T D , the temperatures of the SM and dark sectors evolve independently. Through dark photon decay and inverse decay, the dark sector recouples to the SM one. We define the recoupling scale factor a th by 3H(a th ) = where H denotes the Hubble expansion rate, K n denotes the nth order modified Bessel function of the second kind, and the dark photon decay rate at rest Γ γ is given by Eq. (7). We approximate the evolution of the dark photon temperature T γ as follows: Here a F = (41/3) 1/3 a QCD arises from the entropy conservation in the dark sector. We count all the degrees of freedom including dark Higgs before the QCD phase transition, T QCD ∼ 10 × T QCD with the SM QCD transition temperature T QCD ∼ 170 MeV, and count only dark photon after the QCD phase transition. As the details of the QCD confinement are not tractable, we simply assume that the dark photon temperature does not change inbetween.
We calculate the SM temperature as a function of a ( a th ) through the entropy conservation by using the result in Ref. [33]. The impact of resultant dark photon on N eff depends on whether the reheating temperature of the SM sector at recoupling is above the neutrino decoupling temperature, T ν-dec 3 MeV. This is because dark photon energy primarily injects only into electromagnetic particles. To judge it, we define T cr as where the left-hand (right-hand) side denotes the energy densities before (after) recopuling. We evaluate ρ γ (a th ) by using the following distribution function: 2 For T cr > T ν-dec , we consider that dark photon decay reheats the whole SM sector. Otherwise, we consider that decay reheats only electron and photon. In the latter case, we use the following photon reheating temperature T com instead of T cr : The neutrino temperature, on the other hand, is not affected by decay of the dark photon in this case.

A. Recoupling above neutrino decoupling: T cr > T ν-dec
If dark photon reheats the SM sector above the neutrino decoupling temperature, energy injection at recoupling does not cause cosmological issues. On the other hand, light dark photon changes the neutrino-to-photon temperature ratio T ν /T γ and thus N eff by heating only electron and photon after neutrino decoupling. By considering the entropy conservation in the electron, photon, and dark photon plasma and that in the neutrino plasma independently, one finds 2 This is valid when double Compton scattering and bremsstrahlung of dark proton become inefficient before dark photon becomes non-relativistic. If dark photon is in thermal bath when it becomes non-relativistic, the entropy conservation requires ρ γ ∝ a 3 / ln(a). A similar situation can be found in the freeze-out of self-interacting DM through a 3 → 2 process [34]. The resultant ρ γ (a th ) is different from our evaluation only by a small logarithmic factor.
Here, s γ (T ν-dec ) is the entropy density of dark photon at T ν-dec . As a result, the effective number of neutrino types is changed to where N ν = 3.046 [35] (3.045 in the recent analysis [36]) is the SM value. From the CMB observation, we find that the dark photon mass is bounded from below, We note that the constraint does not depend on in this case as shown in the left blue shaded region of Fig. 1.

B. Recoupling below neutrino decoupling: T cr < T ν-dec
The entropy conservation of the neutrino plasma for a > a ν-dec and that of the photon, electron, and dark photon plasma for a > a th lead to and thus The lower blue shaded region of Fig. 1 shows the resultant constraint. The upper bound on leads to the dark photon lifetime of O(1) s. In addition to the electron channel, we take account of the muon, charged pion, and charged Kaon channels in Γ γ when kinematically allowed.

C. Direct detection and other constraints
Next we consider p scattering with SM p by exchanging dark photon. The p ratio to the whole DM is important for placing constraints from DM direct detection experiments. We consider that dark nucleon inelastic scattering with dark pion is in charge of keeping chemical equilibrium between dark nucleons. The p number ratio is given by It freezes out when dark pions annihilate into dark photons, T γ ∼ m π /20-30. Since m π > m n − m p (see appendix B), we consider that p accounts for half of the whole DM in Fig. 1 �� 1: Constraints on the dark photon parameters. The blue shaded regions are excluded by the cosmological constraints discussed in Secs. III A (left) and III B (bottom). We take T QCD = 1 GeV and a F /a QCD = 3, although the result barely depends on their values. The gray shaded regions are excluded by SN 1987A [37,38], beam dump experiments, and collider experiments [39]. The red lines show the upper limit on at 90% CL for m DM = 8.5 GeV from the DM direct detection experiment. We set α D = α (lower) and α D = α/100 (upper). In our analysis, we assume the Maxwell velocity distribution with the velocity dispersion of v 0 = 220 km/s, which is truncated at the Galactic escape velocity v esc = 544 km/s. The local circular velocity is also fixed to be v circ = 220 km/s with the peculiar motions of the Earth being neglected. The local DM density is fixed to a conventional value, ρ DM = 0.3 GeV/cm 3 , where half of the total DM consists of p .
(red lines). 3 Following the analysis in Ref. [47], we place the upper bound on on the dark photon parameter from the 54 ton×day exposure of PandaX-II [48]. With this exposure, no signal candidates were observed while the expected background in the signal region was 1.8 ± 0.5. This leads to an upper limit of 0.63 signal events in the signal region at 90% CL. Similar constraints are expected from the results of LUX [49] and XENON1T [50].
The figure summarizes other constraints as well (see the caption). The direct detection experiment constraint is severer than that from SN 1987A for the QED fine-structure constant α D = α. For α D = α and m γ 100 MeV, large portion of the parameter region can 3 If the dark photon mass is smaller than the dark deuterium binding energy, the dark nucleosynthesis could proceed [40][41][42][43][44][45][46]  be tested by future experiments such as XENONnT [51], LZ [52], and Darwin [53]. With a light mediator, m γ 100 MeV, the nuclear recoil energy spectrum of DM scattering is distinguishable from that of the neutrino background [54].

IV. CONCLUSIONS
The B − L gauge symmetry is one of the most authentic extensions of the SM. It provides a natural framework for the seesaw mechanism and thermal leptogenesis. In this paper, we have constructed a model of composite ADM in a bottom-up approach, where DM is charged under B − L. By assuming that the entropy of the dark sector is released to the SM sector through kinetic mixing of dark photon to SM photon, we have reached a simple model with a QCD-like SU(3) gauge theory and a QED-like U(1) gauge interaction. As part of dark baryon is charged under U(1) D , our ADM can be tested by direct detection experiments. We have found that the current direct detection constraint is severer than that from SN 1987A. A large portion of the parameter space can be tested by future experiments such as XENONnT, LZ, and Darwin.
Our ADM has interesting astrophysical implications. As the QCD is similar to the SM QCD, our ADM would have a cross section similar to that of SM nucleon, which is O(1) b and constant at low velocity, while diminishes with increasing velocity above v/c = O(10 −2 ) [55]. Such a velocity-dependent cross section could solve issues of cold dark matter structure formation on galactic scales, while satisfying the constraints from galaxy clusters (see, e.g., Ref. [56] for a review). The nature of our ADM self-scattering and its implication for structure formation are worth investigating. The lowest dimensional portal interaction is given by which require q B−L = 1. By integrating outN R , we obtain A drawback of this charge assignment is that it allows a coupling, which leads to dark quarks masses of O(M R ). Thus, the minimal charge assignment contradicts with the assumption of the model. We can take q B−L = ±1/2 to avoid the unwanted mass term in Eq. (A5). In this case, the lowest dimensional portal interaction is given by Unlike the portal interaction in Eq. (A4), this operator does not lead to decay of DM, which can be understood by the residual Z 2 symmetry of the B − L subgroup. As a result, this choice leads to a viable model of ADM. On the other hand, as a rule of thumb, a higher dimensional portal interaction requires more complex new physics to generate it.
where m u (d) is the SM up-type (down-type) quark mass. The squared mass difference of dark pions is given by The average dark (SM) nucleon mass m N (N ) are given as with the mass difference, Here, δm QED n-p = −0.178 +0.004 −0.064 GeV and κ N = 0.95 +0.08 −0.06 parameterize the electromagnetic and isospin-violating contributions, respectively [57].
induce dark proton mixing with dark neutron once the U(1) D symmetry is spontaneously broken. In this case, the heavier nucleon can decay into the lighter one and dark photon or charged leptons, depending on the mass difference m n − m p [see Eq. (B4)]. If the dark photon channel is kinematically forbidden, the lifetime of the heavier nucleon is of O(10 10 ) s for = O(10 −10 ) and its ratio to the whole DM is severely constrained at the level of O(10 −4 ) by the light element abundance [58]. The ratio is determined by dark nucleon inelastic scattering with dark photon. This interaction decouples when T γ ∼ |m n − m p |/20-30 and the resultant ratio is of O(10 −9 ) [see Eq. (27)], evading the constraint. 4 We remark that π ± obtains a vacuum expectation value through Eq. (C1). Thus, π ± mixes with the SM Higgs boson through the Higgs portal coupling, |H| 2 |H D | 2 , and hence decays into the SM fermions. Such decay modes provide an alternative route (to kinetic mixing) to transfer the entropy in the hidden sector to the SM sector (see, e.g., Ref. [59]).
A detailed discussion will be given elsewhere.