Self-interacting dark matter with a vector mediator: kinetic mixing with U(1)$_{(B-L)_3}$ gauge boson

A spontaneously broken hidden U(1)$_h$ gauge symmetry can explain both the dark matter stability and the observed relic abundance. In this framework, the light gauge boson can mediate the strong dark matter self-interaction, which addresses astrophysical observations that are hard to explain in collisionless cold dark matter. Motivated by flavoured grand unified theories, we introduce right-handed neutrinos and a flavoured $B - L$ gauge symmetry for the third family U(1)$_{(B-L)_3}$. The unwanted relic of the U(1)$_h$ gauge boson decays into neutrinos via the kinetic mixing with the U(1)$_{(B - L)_3}$ gauge boson. Indirect detection bounds on dark matter are systematically weakened, since dark matter annihilation results in neutrinos. However, the kinetic mixing between U(1)$_{(B - L)_3}$ and U(1)$_Y$ gauge bosons are induced by quantum corrections and leads to an observable signal in direct and indirect detection experiments of dark matter. This model can also explain the baryon asymmetry of the Universe via the thermal leptogenesis. In addition, we discuss the possibility of explaining the lepton flavour universality violation in semi-leptonic $B$ meson decays that is recently found in the LHCb experiment.


Introduction
The nature of dark matter (DM) is an longstanding mystery in cosmology and particle physics. If DM consists of some new particle, it needs to be long-lived and its relic density should explain the observed amount. A simple framework to explain these aspects is to introduce a gauge symmetry U(1) h , which is broken to some discrete group. The DM particle is stable due to the unbroken discrete group and the correct relic density is obtained through the DM annihilation into light gauge bosons Z h .
The light gauge boson mediates a DM self-interaction, which can address small-scale issues in structure formation of collisionless cold dark matter (see, e.g., Refs. [1,2]). The selfinteraction is velocity-dependent as astrophysical observations prefer [3]. The cross section per mass is σ/m 1 cm 2 /g in (dwarf) galaxies to explain, e.g., diversity in galaxy rotation curves [4][5][6] (see also Ref. [7]). Meanwhile it diminishes to σ/m 0.1 cm 2 /g in galaxy clusters to be compatible, e.g., with the inferred core of relaxed galaxy clusters [3] (see also Ref. [8]). The circular velocity in galaxy clusters is of order v ∼ 1000 km/s, while that in dwarf galaxies is of order v ∼ 30km/s. It implies that a velocity-dependent self-interaction is preferred.
Z h tends to be stable, while one can make the U(1) h -breaking scalar decay into two Z h 's. Thermally produced Z h may overclose the Universe if it is stable. One may introduce a kinetic mixing between Z h and the U(1) Y gauge boson B so that Z h can decay into an electron-positron pair or photons. However, late-time DM annihilation followed by the Z h decay is largely disfavored by indirect detection constraints, e.g. from cosmic microwave background (CMB) anisotropies (see, e.g., Ref. [9]). One way to avoid the overclosure of Z h and these constraints is to make it decay only into standard model (SM) neutrinos.
A similar line of constructing a viable self-interacting DM model was pursued in Ref. [10], where U(1) h is identified as a flavoured lepton gauge symmetry U(1) Lµ−Lτ . The MeV-scale L µ − L τ gauge boson decays predominantly into neutrinos since charged lepton channels are kinematically forbidden. On other other hand, the gauge coupling needs to be rather small to satisfy constraints from muon anomalous magnetic moment and thus the mediated selfinteraction is also small. This is why the MeV-scale U(1) Lµ−Lτ -breaking scalar was considered as a scalar mediator of the DM self-interaction. In this paper, we propose a self-interacting DM model with a vector-mediator [11][12][13][14][15][16][17][18].
We consider flavoured U(1) B−L gauge symmetries; we introduce a B−L gauge symmetry U(1) (B−L) i for each family (i = 1, 2, 3) in the SM sector. The anomaly cancellation implies that there is a right-handed neutrino N R i in each family. This model could be extended to grand unified theories such as [SO (10)] 3 [19] (see also Ref. [20]). We assume that U(1) (B−L) 3 is spontaneously broken around the electroweak scale. MeV-scale Z h decays predominantly into neutrinos through kinetic mixing between U(1) h and U(1) (B−L) 3 since channels into quarks and charged leptons are not kinematically allowed. In our model, we make the mass of the U(1) h -breaking scalar larger than 2 × m Z h so that the scalar field can decay into two Z h 's. Quantum corrections give a kinetic mixing between U(1) (B−L) 3 and U(1) Y gauge bosons because there are bicharged particles in the SM sector. Because of these kinetic mixings, our DM can be detected by direct detection experiments in the near future.
This paper is organized as follows. First, we specify our model of DM and flavoured U(1) (B−L) i . We introduce a spontaneously broken U(1) h gauge symmetry in the dark sector. The U(1) (B−L) 3 is spontaneously broken around the electroweak scale so that the kinetic mixing with Z (B−L) 3 leads to the decay of Z h into SM neutrinos. In Sec. 3, we discuss the cosmology of this model. In particular, we discuss that there are two candidates of DM in this model: the vector-like fermion in the hidden sector and N R 3 . The former one has the self-interaction through the massive gauge boson exchange and is assumed to be the dominant component of DM. Then, we discuss the compatibility with the present collider experiments and future detectability in Sec. 4. Sec. 5 is devoted to the conclusion.

Model
We introduce three right-handed neutrinos N R i and flavoured U(1) (B−L) i gauge symmetries (i = 1, 2, 3) that are spontaneously broken by vacuum expectation values (VEVs) of Φ i at the energy scale of v φ i . We make the third family of right-handed neutrino stable by introducing a Z 2 symmetry. We also introduce another complex scalar field Ψ and a vector-like fermion pair χ andχ that are charged under a hidden gauge symmetry U(1) h . The field Ψ is assumed to obtain a nonzero VEV to break U(1) h spontaneously at the energy scale of v ψ . The charge of Ψ is taken to be three in units of that of χ to forbid Yukawa interactions with χ orχ. The charge assignment of the newly introduced particles are summarized in Table 1, where we omit the first and second families for simplicity. We denote the gauge bosons of U(1) (B−L) i and U(1) h by Z (B−L) i and Z h , respectively. The Lagrangian is given by where L kin represents the canonical kinetic terms including the gauge interactions and H is the SM Higgs field. The scalar fields are assumed to be unstable at the origins of the potentials and obtain nonzero VEVs v φ i and v ψ at the stable minima. We denote the perturbations around the minima as φ i and ψ as After the spontaneous symmetry breaking (SSB), the gauge bosons Z (B−L) i and Z h obtain masses such as The SM neutrinos obtain small masses via the seesaw mechanism.
Because of the flavoured symmetry, the proper structure of Yukawa interactions cannot be generated in the simplest setup. To generate the proper Yukawa matrices, one may introduce (I) U(1) (B−L) 3 -charged scalars in addition to U(1) (B−L) 3 -neutral vector-like fermions that mix with the SM quarks and leptons [19]; or (II) an additional Higgs doublet that is charged under U(1) (B−L) 3 [33,34]. We do not go into further detail about these possibilities in this paper and just assume that the following flavour structure arises appropriately.
The SM Yukawa matrices can be diagonalized by a unitary rotation for each field: f = U f f . Although each unitary matrix is not observable in the SM, except for U † u L U d L = V CKM and U † e L U ν L = U PMNS , it affects the interactions with the Z (B−L) 3 boson. The interactions with the Z (B−L) 3 boson are given by In this paper, we simply assume that the rotations of the right-handed fermions are suppressed and the 2-3 family rotations of the left-handed fermions exist in addition to V CKM and U PMNS such as where R 23 (θ) is a rotation in the 23 sector by an angle θ. In particular we assume that R 13 (θ) does not arise so that Z h does not decay into electrons via the kinetic mixing with Z (B−L) 3 .
3 Cosmology of the model

Thermal leptogenesis
We can generate the lepton asymmetry by the thermal leptogenesis via the decay of the first and second family right-handed neutrinos. We assume that the reheating temperature after inflation is as high as the mass of the lighter one among these neutrinos so that they can be produced from the thermal plasma. The lepton asymmetry can be generated by their decay. Since the B + L symmetry is broken by the non-perturbative effect, we can generate the baryon asymmetry from the lepton asymmetry. The observed baryon asymmetry can be explained when the lighter one is heavier than about 10 9 GeV [25].
If the third family right-handed neutrino has a Yukawa interaction with the SM particles, the B−L symmetry violating interaction may be in equilibrium after the thermal leptogenesis and the lepton asymmetry may be washed out. To avoid this washout effect, we impose a Z 2 symmetry on N R 3 . As a result, it is stable and can be a DM candidate.
If we do not introduce the Z 2 symmetry on N R 3 , the Yukawa coupling with the SM fields should be small enough to suppress the washout effect. The decay rate of N R 3 is given by is a Yukawa matrix for the interaction between N R i and the SM lepton doublet L j . The washout effect should not be efficient, Γ N R 3 H, until the temperature of the Universe decreases to the mass of N R 3 . Thus we require to avoid the washout effect.

Dark matter
There are two DM candidates in our model. We identify χ andχ as the dominant component of DM, while N R 3 is the subdominant component. Their thermal relic densities are determined as with the s-wave annihilation cross section times relative velocity (σ i v).

Weakly-interacting DM: N R 3
The annihilation of N R 3 proceeds through the U(1) (B−L) 3 gauge interaction and the Yukawa interaction with Φ 3 . We found in Ref. [30] that the dominant process is a s-wave annihilation channel The cross section is given by . The resulting amount of N R 3 can be then estimated as We assume that N R 3 is the subdominant component of DM: (Ω DM h 2 ) obs 0.12. Then we obtain The Yukawa coupling y R 3 cannot be arbitrary large because of the Unitarity bound. One may also require that the Landau pole does not appear below the Planck scale, which leads to y R 3 1.2 [30]. Then we obtain v φ 3 2.4 TeV from Eq. (3.6). Although the dominant annihilation channel is s-wave and its cross section is not suppressed at the galactic center, N R 3 is not the dominant component of the DM and hence its indirect detection signals can be neglected.

Self-interacting DM: χ andχ
For χ andχ, the annihilation cross section is given by [35] The total abundance of χ andχ is twice larger than Eq. (3.3) because there are χ andχ, each abundance of which is determined by the thermal freeze out. The resulting amount of χ andχ can be then estimated as 1 (3.9) The massive gauge boson Z h mediates the self-interaction of χ andχ. It is convenient to use the transfer cross section defined by [37] When one computes σ T , one encounters three regimes [37]: In the Born regime, one can rely on the perturbative calculation and find an analytic expression in Refs. [37,39,40]. In the classical and resonance regimes, one needs to solve the Schrödinger equation to take into account non-perturbative effects related to multiple exchange of Z h . Meanwhile, fitting formulas can be found in the classical regimes [37,40,41,41,42]. In the resonance regime, an approximate formula can be obtained in the Hulthén potential [37].
Kinematics of dwarf and low-surface brightness galaxies indicate that σ T /m χ ≈ 1 − 10 cm 2 /g for the DM velocity of order 30 km/s [3]. On the other hand, observations of galaxy clusters prefer σ/m χ 0.1 cm 2 /g for the velocity of order 1000 km/s [3]. If the cross section saturates this upper bound, we can also explain the inferred density cores in the galaxy clusters [8]. The desirable parameter region is mostly in the resonance regime (see, e.g., Ref. [10]), where the parameter dependence of the self-scattering cross section is non-trivial. In this paper, we do not pin down the precise values of m χ and m Z h because they are not sensitive to other observables. Instead, we simply use an approximate formulas found in Ref. [37] with the replacement of cos θ → |cos θ| in Eq. (3.11) (see footnote 2) to check if the self-interaction cross section is within a desirable range. We find that the above constraints can be satisfied when m Z h ≈ 10-100 MeV and m χ ≈ 10-100 GeV.

Dark radiation
We assume that the mass of the dark Higgs boson ψ is larger than twice that of Z h , so that it can decay into two Z h 's. We make Z h unstable by introducing a kinetic mixing between U(1) (B−L) 3 and U(1) h : where F (B−L) 3 and F h denote the field strengths of U(1) (B−L) 3 and U(1) h , respectively. Then Z h can decay into tau-neutrinos ν τ via the mixing with Z (B−L) 3 . We remark that the decay of Z h into muons µ or taus τ is kinematically forbidden for m Z h 200 MeV. The other decay of Z h into electrons e is suppressed since Z (B−L) 3 does not directly couple to e under our assumption of the Yukawa structure [see Eq. (2.9)]. Thus the late time DM annihilation into Z h results in ν τ and thus is harmless.
The decay rate can be estimated as We require that Z h decays into ν τ long before the neutrino decoupling; otherwise only the temperature of ν τ is enhanced by the decay of Z h and the energy density of ν τ may exceed the upper bound on that of dark radiation. This can be satisfied when Γ Z h H| T =1 MeV , where H is the Hubble expansion rate at temperature T . It gives the lower bound on the mixing parameter as Even if Z h decays into ν τ long before the neutrino decoupling, the thermalized Z h can still enhance only the temperature of ν τ after the neutrino decoupling. This constraint is evaded for m Z h 10 MeV [10,43]. Furthermore, one needs to take account of Z h possibly dominating the energy density of the Universe. It takes place if the decay rate of the dark Higgs boson is much smaller than the Hubble expansion rate when the temperature is comparable to the mass of dark Higgs boson, If this condition is not satisfied, the amount of the entropy production due to the Z h decay can be estimated as where a i (a f ) is the scale factor, s i (s f ) is the entropy density before (after) the Z h domination, and T d is the decay temperature of Z h . The constraint (3.15) can be avoided if the generated baryon asymmetry is larger than the observed value by this factor. This can be realized when the first and second right-handed neutrinos are heavier than 10 9 GeV at least by the same factor. Here we comment on another possible mechanism of the entropy production, which could be relevant in models with spontaneous symmetry breaking. As for a dynamics of U(1) h breaking in the hidden sector, thermal inflation may occur at the time of the phase transition if the mass of the gauge boson m Z h is many orders of magnitude larger than that of the symmetry-breaking field m ψ . This effect washes out the baryon asymmetry, so that we should avoid such a thermal inflation. We discuss the condition to avoid thermal inflation in Appendix A and check that it does not occur in our model. However, we note that it is non-trivial in other models with hierarchical mass scales.

DM direct and indirect detection constraints
There may be couplings between scalar fields like λ HΦ i |H| 2 |Φ i | 2 and λ HΨ |H| 2 |Ψ| 2 . Since they are irrelevant in the above discussion, we take the loop induced values as natural choices. For example, the former interaction arises at the two loop level as λ HΦ 3 ∼ y 2 t α 2 (B−L) 3 /(4π) 2 . It results in the mixing between the SM Higgs and φ, which leads to spin-independent N R 3nucleon scatterings. However, N R 3 is the subdominant component of DM and hence easily evades the constraint from the direct detection experiments for DM. For the same reason, the indirect detection constraint on N R 3 is also weakened.
The kinetic mixing between the U(1) Y and U(1) (B−L) 3 gauge bosons arises at the oneloop level: where g Y is the U(1) Y gauge coupling. Here µ is the energy scale considered and Λ is a cutoff scale at which the kinetic mixing vanishes. When U(1) Y is unified into a non-abelian gauge symmetry, we should take Λ to be the grand unification scale of order 10 16 GeV. 4 Through the kinetic mixings parametrized by 1 and 2 , we obtain the following effective interaction at a low energy scale: where p represents the proton. The coefficient b p is given by where e is the electromagnetic charge and θ W is the Weinberg angle. 4 We implicitly consider SU(5)×U(1) (B−L) 3 ×U(1) h gauge theory as an effective theory, where SU(5) breaks down to the SM gauge groups at the GUT scale. In this case, the kinetic mixing between the U(1)Y and U(1) h gauge bosons are forbidden above the GUT scale while the one between the U(1) (B−L) 3 and U(1) h gauge bosons is allowed by the symmetry, SU(5)×U(1) (B−L) 3 ×U(1) h . Although the former one can be induced after the GUT symmetry breaking, it depends on the detail of the model. In this paper, we assume that the mixing parameter is suppressed enough so that we can avoid the constraints coming from DM direct detection experiments.
For a given nucleus A Z N , the coefficient of the coupling is given by b N = Zb p , where we neglect the contribution comes from the neutron. Then the spin-independent χ-nucleon scattering cross section is given by (see, e.g., Ref. [45]), where µ N is the χ-nucleon reduced mass and F 2 FI,T (E R ) is the form factor defined such that F 2 FI,T (0) = 1. This is just below the present upper bound reported by XENON1T for m χ = 20 − 100 GeV [46]. XENONnT [47], DarkSide-20k [48], and LUX-ZEPLIN [49] can search DM with a cross section smaller by a factor of order 10. DARWIN can detect DM if the cross section is above the neutrino coherent scattering cross section [50], which is around 10 −49 cm 2 for m χ 20 − 100 GeV.
As we stressed, the late-time annihilation of the dominant component of DM, χ and χ, predominantly results in ν τ . Since the detection of neutrino signals is quite challenging and the constraint is very weak [51], this does not lead to observable effects on astrophysical experiments.

Collider constraints
As discussed in Sec. 2, we assume that the appropriate flavour structure of the SM Yukawa matrices comes from some UV physics (see, e.g., Ref [19]). The CKM and PMNS matrices are attributed to the up-type left-handed quarks and leptons, respectively. We allow for additional 2-3 family rotations of left-handed quarks and leptons.

No additional physical phase
First we discuss the constraints from collider experiments, when there are no additional physical phases, θ = θ q = 0, following Ref. [30].
A relevant constraint on the Z (B−L) 3 mass comes from the lepton flavor universality violation in Υ decays. The lepton flavor universality ratio is modified in the presence of Z (B−L) 3 as where m Υ (≈ 9.46 GeV) is the Upsilon mass and α = e 2 /(4π). The BaBar experiment places the constraint on this ratio as R τ µ = 1.005 ± 0.013(stat.) ± 0.022(syst.) [52]. In the limit of . This constraint, α (B−L) 3 10 −3 for m Z (B−L) 3 ∼ 200 GeV, would not be quite stringent when compared to others. 5 The kinetic mixing between the Z (B−L) 3 and U(1) Y gauge bosons changes the mass eigenstates and interactions of the vector mesons. In particular it leads to a shift in the SM Z boson mass. In the mass basis, the physical mass of the SM Z boson is given by where m Z 0 = m W ± / cos θ W (see, e.g., Ref. [56]). The mass of the SM Z boson is tightly constrained by electroweak precision measurements and is consistent with the SM prediction.
In this paper we require that the mass difference is smaller than the current experimental uncertainty of 0.0021 GeV [57]. The kinetic mixing parameter 1 can be written in terms of α (B−L) 3 by using Eq. (3.19). Then we can plot a constraint in the α (B−L) 3 -m Z (B−L) 3 plane as shown by the blue region in Fig. 1. The orange region in the upper left corner is excluded by the Υ decay measurement. We also plot a green region which is excluded by the flavour physics which we will discuss in Sec. 4.2. The dashed lines are a couple of reference values of the parameters we will use in Fig. 2. The Higgs-portal interaction λ HΦ 3 |H| 2 |Φ 3 | 2 may provide indirect signals of Higgs invisible decays if the SM Higgs can decay into N R 3 N R 3 , Z (B−L) 3 Z (B−L) 3 , or φ 3 φ 3 . The constraint, however, can be easily evaded unless the Higgs-portal coupling is as large as O(1).

Non-zero new physical phases
Next we allow the additional physical phases to be non-zero. The following discussion is based on Ref. [19].

Semi-leptonic B decays
To study semi-leptonic B decays, it is convenient to use the following effective Hamiltonian at the low energy: After integrating out Z (B−L) 3 , we obtain the deviation from the SM contributions for µ such as where V tb denotes the tb component of V CKM and so on. Hereafter in this subsection, we also use s θq = sin θ q and so on. The LHCb experiment reported a lepton flavour universality violation in semi-leptonic B decays [31,32], which is represented by the ratio of (4.13) The tension with the SM prediction is around the 4σ level [58][59][60][61][62][63]. This can be explained by the Z (B−L) 3 contribution when δC µ 9 ∈ [−0.81, −0.48] (1σ interval) [64]. Using |V tb | ≈ 1.0 and |V ts | ≈ 3.9 × 10 −2 [57], we then require (4.14) The B meson can decay also into neutrinos: B → K ( * ) νν. The deviation from the SM contribution is given by The ratio to the SM prediction is given by where C (SM) ν ≈ −6.35 [65]. The experimental upper bound is R νν < 4.3 at the 90% CL [66,67], which gives Combining this with Eq. (4.14), we obtain |s θ l | 0.18 .

Lepton flavour violation
Lepton flavour violating processes are also induced by Z (B−L) 3 interactions. The most important effective interaction is which gives the τ decay into 3µ. The resulting branching ratio is given by We simply estimate the constraint by replacing g of Fig. 1 of Ref. [82] with our g (B−L) 3 s 4 θ l /3 2 , where a factor of 3 2 comes from a difference of the charge between the models. The resulting constraint is g (B−L) 3 s 4 θ l 6 × 10 −2 for m Z (B−L) 3 = 20-30 GeV, while it is two orders of magnitude weaker for m Z (B−L) 3 70 GeV. We find that this constraint is negligible in most of the parameter region we are interested in.
Because of the kinetic mixing, Z (B−L) 3 can be produced by hadron colliders via the Drell-Yan process, which leads to a clear dilepton signal with an invariant mass about the Z (B−L) 3 boson mass. In Ref. [83], they considered the case where a hidden gauge boson is coupled with the SM sector only via the kinetic mixing and estimated that the 8 TeV LHC with 20 f b −1 luminosity put a constraint on the kinetic mixing parameter as 1 0.005-0.01 for the gauge boson mass range of 10-70 GeV. In our model, the Z (B−L) 3 boson can be produced via the kinetic mixing 7 and decay into µμ via the flavour mixing. The constraint can be interpreted as a bound on 1 s 2 θ l , where s 2 θ l comes from the branching ratio into µμ. However, this does not give a strong constraint on s θ l when α (B−L) 3 10 −4 . It was also discussed that the constraint will be improved by a factor about 5 by using 3000 fb −1 of 14 TeV data. In this case, the high-luminosity LHC would observe a dilepton signal for The Z (B−L) 3 gauge boson can also be produced by lepton colliders through the kinetic mixing 1 and its decay signal can be searched by the future e + e − colliders, such as Circular Electron Positron Collider (CEPC) [84], International Linear Collider (ILC) [85,86], and Future Circular Collider (FCC-ee) [87]. The relevant process is e + e − → γZ (B−L) 3 followed by Z (B−L) 3 → µ + µ − . Projected constraints were discussed in Ref. [88] in the case where the dark photon couples to the SM particles only via the kinetic mixing with the U(1) Y gauge boson. The upper bound on the kinetic mixing parameter was found to be about 0.003. Again, we could interpret their result in the same way as discussed above. The future lepton colliders would observe a signal of Z (B−L) 3 gauge boson for α (B−L) 3 ∼ 10 −4 .

Summary of the collider constraints
Now we shall put together all the constraints discussed in this section. The result is shown in Fig. 2 TeV. This is shown in Fig. 1 as the green-shaded region. From Fig. 1, we can see that there is a certain parameter region where we can explain the lepton flavour universality violation in semi-leptonic B decays consistently with the constraints coming from the kinetic mixing.

Conclusion
We have proposed a model of DM whose stability is guaranteed by a discrete symmetry that is a subgroup of a spontaneously broken hidden U(1) h gauge symmetry. The massive gauge boson Z h is assumed to be much lighter than the DM and mediates the velocitydependent DM self-interaction that are suggested by small-scale issues in structure formation of collisionless cold dark matter. The observed abundance of DM is explained by the thermal relic via the freeze-out mechanism. Motivated by flavoured grand unified theories, we have also introduced right-handed neutrinos and flavoured B−L gauge symmetries. The unwanted relic of Z h can then decay into neutrinos via the kinetic mixing with the electroweak scale U(1) (B−L) 3 gauge boson Z (B−L) 3 . This model can also explain the baryon asymmetry of the Universe via the thermal leptogenesis.
Although the hidden sector couples to the SM sector only via a kinetic mixing with the U(1) (B−L) 3 gauge boson, it predicts detectable DM signals in direct detection experiments. Our model also predicts a relatively light U(1) (B−L) 3 gauge boson, which leads to interesting signals in collider phenomenology. In particular, we have found that we can explain the lepton flavour universality violation in semi-leptonic B meson decays recently found in LHCb experiment. The U(1) (B−L) 3 gauge boson can also be searched by the future high-luminosity LHC experiment and e + e − colliders such as CEPC, ILC, and FCC-ee. These experiments would observe signals when the fine-structure constant for U(1) (B−L) 3 is of order 10 −4 .

A Phase transition and thermal inflation
In this Appendix, we comment on the effects of the phase transition from the SSB of U(1) h , in which the dark Higgs boson Ψ develops the VEV v Ψ . 8 Since it breaks the local Abelian gauge symmetry, cosmic strings form through the phase transition. However, their effects are negligible in our model because the energy density of cosmic strings is suppressed by a factor of the VEV squared in the Planck units when compared to the total energy density of the Universe.
If the VEV of a scalar field is much larger than its mass, the potential energy before the phase transition may be much larger than the energy of the thermal plasma. In this case, the energy density of the Universe becomes dominated by the former vacuum energy and a mini inflation called a thermal inflation occurs through the phase transition [89,90]. The duration of the thermal inflation depends on the ratio between the VEV and (zero-temperature+thermal) mass of the SSB field. After the thermal inflation, the vacuum energy will be released into the radiation and the entropy production proceeds. As a result, the baryon asymmetry is diluted due to the entropy production at the time of this reheating.
Here, we give a quantitative estimate of the dilution of the thermal relic through the entropy production. The (zero temperature+thermal) potential of Ψ is given by where λ is a quartic coupling constant. The mass of ψ at the vacuum is given by √ 2λ v ψ . The thermal potential V T from ψ and the gauge boson Z h is approximately given by where q ψ (= 3) is the charge of Ψ. At a high temperature, the thermal potential dominates and Ψ is stabilized at Ψ = 0. When the temperature becomes lower than the critical temperature T c , which is given by the potential becomes unstable at Ψ = 0 and the scalar field starts to oscillate around the true vacuum at |Ψ| = v ψ / √ 2. We define a dilution factor as the ratio of the initial to the final comoving entropy density as It follows that the mass of gauge boson cannot be arbitrary larger than the mass of the SSB field. This condition is easily satisfied in our model though it is non-trivial in other models with hierarchical mass scales.