Inelastic extra $U(1)$ charged scalar dark matter

The null results in dark matter direct detection experiments imply the present scalar dark matter (DM) annihilation cross section to bottom quark pairs through the Higgs boson exchange is smaller than about $10^{-31}$ cm$^3/$s for a wide DM mass range, which is much smaller than the required annihilation cross section for thermal relic DM. We propose models of a thermal relic DM with the present annihilation cross section being very suppressed. This property can be realized in an extra $U(1)$ gauge interacting complex scalar DM, where the thermal DM abundance is determined by coannihilation through the gauge interaction while the present annihilation is governed by Higgs bosons exchange processes. An interaction between DM and the extra $U(1)$ breaking Higgs field generates a small mass splitting between DM and its coannihilating partner so that coannihilation becomes possible and also the $Z'$-mediated scattering off with a nucleon in direct DM search becomes inelastic. We consider scalar dark matter in $U(1)_{B-L}, U(1)_{(B-L)_3}$ and $U(1)_{L_\mu-L_\tau}$ extended models and identify viable parameter regions. We also discuss various implications to future DM detection experiments, the DM interpretation of the gamma-ray excess in the globular cluster 47 Tucanae, the muon anomalous magnetic moment, the Hubble tension and others.


I. INTRODUCTION
Weakly interacting massive particle (WIMP) is a primary candidate for the dark matter (DM) in the Universe. An appealing property of WIMP is its complementarity: If WIMP is a thermal relic from the early Universe, the abundance of DM is determined by the annihilation cross section of the order of picobarn for WIMP annihilation into standard model (SM) particles at thermal freeze-out. This implies that WIMP annihilation occurs even today in a DM dense region with a similar magnitude of annihilation cross section, resulting in an excess of spectrum in various cosmic rays, such as gamma rays, neutrinos and charged particles. The Fermi-LAT has published limits on the DM annihilation cross section into final states generating gamma rays [1,2]. Those limits have been expressed, in particular, annihilation modes into bb pair and τ − τ + pair, because these could be dominant annihilation modes in a large class of WIMP models. In fact, for a WIMP mass smaller than about 100 GeV, the obtained upper bound on the cross section is found to be smaller than that required for the thermal DM abundance of about 1 pb [1,2].
Interactions of WIMP are constrained by the so-called direct DM detection experiments as well, through scattering processes between DM and a nucleon. Through those scatterings, WIMPs in our galactic halo are experimentally detectable. Various direct DM detection experiments such as the LUX [3], PandaX-II [4], and XENON1T [5] have not detected a significant signal, which sets the upper bound on the cross section to be smaller than O(10 −9 ) pb for the wide range of DM mass. One may recognize a large hierarchical discrepancy of magnitude between the cross section for direct DM detection and that for annihilation.
As we will explicitly show in the next section, a theoretical interpretation of the small scattering cross section of WIMP with a nucleon implies that the present DM annihilation cross section for most of the DM mass range seems smaller than about 10 −31 cm 3 /s which is far below the sensitivity of the current and near-future observations. This would lead to incompatibility between the thermal DM abundance and such too small present DM annihilation cross sections. Thus, null detection of WIMP seems to confront with the desired annihilation cross section for thermal freeze-out.
Before we proceed discussion, we note several ways out of the above argument. The first is based on our assumption that the relevant scalar-type operator, χχqq, between WIMP χ and quarks q is present and unsuppressed. We usually expect that this operator is obtained after integrating out the Higgs boson in the SM. That argument is robust as long as the SM Higgs boson is a mediator. An exception is the case that WIMP and a mediator couple with not quarks but leptons only, and thus the operator χχqq does not exist. See, for example, Refs. [6,7]. Another case is that the operator is absent or very suppressed because WIMPnucleon scattering processes occur in nonrelativistic regime. A pseudoscalar mediator dark matter [8] or pseudo-Nambu-Goldstone boson dark matter [9] is such an example. The scattering cross section with a nucleon for these DMs has been studied in detail in Refs. [10][11][12][13][14] and Refs. [15][16][17][18][19], respectively. Note that only WIMP-nucleon scattering cross section is suppressed, while WIMP annihilation cross sections both today and in the early Universe can be about 1 pb as usual. Provided that the SM Higgs boson is a mediator, the second is based on the assumption that the annihilation process for freeze-out is same as that at present and kinematically s-wave. If an annihilation process for freeze-out is p-wave dominated, the present annihilation cross section is suppressed by the velocity squared v 2 ∼ 10 −6 while that in the early Universe is not suppressed because the relative velocity is not so small as v ∼ 0.1. A similar but moderated suppression can occur in the vicinity of the resonance pole due to a difference between relative velocities at present and in the early Universe. This was utilized in Refs. [20][21][22] to account for the Galactic Center gamma-ray excess [23][24][25]. Thus, for example, if the freeze-out annihilation mode is p-wave and the present annihilation is dominated by another mode of s-wave (e.g., as in Ref. [26]), the scenario is still consistent.
There is yet another famous mechanism of different annihilation modes, that is coannihilation at freeze-out. The neutralino-stau coannihilation in supersymmetric models is a well-known example.
This paper is organized as follows. In the next section, we show that the current constraints by the null result of DM direct detection experiments imply that the present annihilation cross section into bb is typically smaller than O(10 −31 ) cm 3 /s. In Sec. III, we introduce generic models for an extra U(1) gauge interacting scalar DM, where the scalar DM with the extra U(1) breaking Higgs field is also introduced, as a preparation for discussion based on specific extra U(1) models in the following sections. In the models, thermal DM abundance is determined by coannihilation through the gauge interaction while the present annihilation is governed by Higgs bosons exchange processes. We investigate three specific U(1) models: 3 in Sec. V and U(1) Lµ−Lτ in Sec. VI, respectively. Section VII is devoted to summary.

A. Dark matter elastic scattering with nuclei
The spin-independent (SI) DM scattering cross section with nucleus (N) made of Z protons (p) and A − Z neutrons (n) is given by [27] for a real scalar DM S with the mass m S . The effective coupling with a proton f p and a neutron f n is expressed, by use of the hadronic matrix element, as where i = p, n, and α q is an effective coupling of the DM particle with a q-flavor quark in the effective operator, which is obtained by integrating out mediator particles from the original Lagrangian. For the origin of such an operator, we consider the scalar interaction terms and Yukawa interactions with quarks 1 , where v ≃ 246 GeV is the vacuum expectation value (VEV) of the SM Higgs field and h is the SM Higgs boson with the mass of m h ≃ 125 GeV.

B. Annihilation cross section for indirect signal
The present annihilation cross section (σv) 0 of the scalar DM particle (S) is given by its s-wave component of the annihilation cross section, e.g., by the limit of v → 0. From the interaction in Eq. (4) relevant to the direct detection, we also obtain the present day annihilation cross section to bb as where m b and Γ h denote the mass of bottom quark and the total decay width of h, respectively.

C. Comparison of the Fermi bound and direct search bounds
Fermi-LAT Collaboration has shown that WIMP annihilation cross section to bb should be less than 10 −26 cm 3 /s for the WIMP mass 100 GeV [1,2]. Considering the fact that both the WIMP annihilation to bb (5) and the WIMP scattering with a nucleon (1) originate from the same interactions (4), one may expect that the bound from direct DM search experiments also sets a severe upper limit on the WIMP annihilation cross section.
We show in Fig Naive expectation based on the above consideration is that little cosmic gamma-ray excess can be generated by WIMP annihilation whose cross section should be very small due to the null results in direct DM detection experiments. One might expect another mediator Higgs boson could relax the bound as the SM Higgs boson does at the WIMP mass around 62 GeV. In Refs. [20][21][22], we have studied this possibility in the context of Galactic Center excess [23][24][25] and shown that not only can the predicted annihilation cross section not be suppressed by orders of magnitude but also the second mediator in type-II two Higgs doublet model confronts with the LHC results [22]. 2 As we have declared, we take account of only Higgs boson(s) exchange processes in this paper. However, we will not consider Majorana fermion DM because its annihilation processes through s-channel Higgs boson exchange is p-wave and suppressed by its velocity squared v 2 ∼ 10 −6 . For an explicit calculation, see Refs. [28][29][30] for example.

III. MODEL
A consequence of the above discussion is how those constraints can be compatible with the annihilation cross section of the order of 10 −26 cm 3 /s for thermal freeze-out in the early Universe. The present annihilation cross section (σv) 0 of a WIMP, which is relevant for the indirect DM detection, is given by its s-wave component of the self-pair annihilation cross section, i.e., by the limit of v → 0. Note that in general the thermal averaged cross section σv at the early Universe required to reproduce Ωh 2 ≃ 0.1 [46] is not necessarily the same as the present annihilation cross section (σv) 0 .
In the following, we will show that an extra U(1) gauge interaction and its breaking provide large enough annihilation cross section for thermal freeze-out only and do not induce present annihilation processes much. Then, not only the observed Ωh 2 can be reproduced but also the expected WIMP-nucleon scattering cross section and the present WIMP pair annihilation cross section is small enough to be consistent with the null results in those DM search experiments. In this section, we summarize the general aspect of scalar DM interacting through an extra U(1) interaction before we discuss the details for specific U(1) models in the following sections.  We consider an extend SM based on the gauge group In addition to the SM model particles, we introduce right-handed neutrinos (N R ) and two SM singlet Higgs fields (φ 1 and φ 2 ) charged under the new extra U(1) gauge symmetry. We identify φ 1 with a DM candidate while φ 2 is a Higgs field responsible to break the extra U(1) gauge symmetry. Here, the subscripts 1 and 2 stand for those gauge charges.
The particle content is listed on Table I. Charges of fermions under the specific gauge group must be assigned to make the model free from anomalies. The scalar potential is expressed as [31,32] with Φ being the SM Higgs field. All parameters in the potential (6) are taken to be real and positive.
Here, we emphasize that, in previous works on an extra gauged U(1) charged scalar DM [33][34][35][36][37][38], the scalar DM does not interact with the U(1) breaking Higgs field through a trilinear coupling due to its gauge charge, and the φ 1 φ 1 φ † 2 corresponding term in our scalar potential (6) is absent. On the other hand, the presence and the effect of this term is essential in this paper, as we will show below.

B. Dark matter mass and interactions
At the U(1) and the electroweak (EW) symmetry breaking vacuum, the SM Higgs field and the U(1) Higgs field are expanded around those VEVs, v and v 2 , as where those VEVs are determined by the stationary conditions: Mass terms of particles are expressed as where g ′ is the U(1) gauge coupling. The physical states (ϕ and ϕ 2 ) are diagonalized to the mass eigenstates (h and H) with masses m h and m H as   ϕ For a small mixing angle α, h is identified with the SM-like Higgs boson. In fact, we will take α ≃ 0.001 in the following analysis. With the U(1) and the EW symmetry breaking, the Z ′ boson, S and P acquire their masses, respectively, as Note that the parameter A controls the mass splitting between S and P . Since we take A positive, S is lighter than P and becomes the DM candidate. Here, we note that the mass degeneracy of coannihilating particles are accidental in many cases of coannihilation including that in supersymmetric models. On the other hand, the mass degeneracy between S and P in our model would be reasonable because both are originally in the same multiplet with the common mass.
Three point interaction terms among S(P ) and h(H) are expressed as The Yukawa interactions can then be written as Gauge interaction of the DM particle is expressed as and similarly all generation quarks and leptons also interact to Z ′ with corresponding charges. The absence of Z ′ -DM-DM coupling means that the Z ′ -mediating DM scattering off with a nucleon is inelastic and ineffective for the mass splitting larger than the energy transfer in the scatterings [39,40]. Hence, the DM-nucleon scattering in our model is Higgs We here count the number of additional free parameters in our model except for the SM gauge and Yukawa couplings. One is the extra gauge coupling g ′ and the scalar potential (6) contains ten parameters. λ 1 in the potential (6)

C. Thermal relic abundance
We estimate the thermal relic abundance of the real scalar DM, S, by solving the Boltz- where H and n EQ are the Hubble parameter and the DM number density at thermal equilibrium, respectively [41]. In our model, the main annihilation mode is coannihilation SP → ff through s-channel Z ′ exchange for m Z ′ > m S and the annihilation mode SS → Z ′ Z ′ by u(t)-channel P exchange for m Z ′ < m S (see Appendixes for the formula we employ in our analysis). We use the effective thermal averaged annihilation cross section and suppressed by the destructive interference between two. In addition, for m Z ′ < m S , the annihilation mode SS → Z ′ Z ′ by u(t)-channel P exchange opens and has significant s-wave component, which is expressed as in the m P → m S limit. Hence, the m Z ′ < m S case is well constrained by Fermi-LAT or other DM indirect detection experiments. First, we consider a model based on the gauge group SU (3) [44,45]. In addition to the SM particle content with three generations (i = 1, 2, 3), three RH neutrinos (N i R (i = 1, 2, 3)) and two U(1) B−L Higgs fields (φ 1 and φ 2 ) are introduced. In the presence of the three RH neutrinos, the model is free from all the gauge and mixed gauge-gravitational anomalies. The particles and those charges are listed on Table II.

B. Thermal relic abundance
We show in Fig. 2 the relation between m S and m Z ′ drawn with blue lines in order to reproduce the observed DM relic abundance Ωh 2 ≃ 0.1 [46]. Since the so-called unitarity bound sets an upper limit on a WIMP mass to be less than a hundred TeV [47], the curve is grown up to the WIMP mass of one hundred TeV. The experimental bound on the mass of Z ′ in the U(1) B−L model has been derived based on the LEP and Tevarton data [48] as well as the latest LHC Run 2 results [49,50] that are expressed as a brown shaded region. There are two cases reproducing the observed DM abundance: one is m S ∼ several TeV by the resonant annihilation through s-channel Z ′ exchange and the other is due to the SS → Z ′ Z ′ annihilation for m S > 10 TeV. For a smaller g B−L , DM is overabundant and no solution to Ωh 2 ≃ 0.1 is found.

C. Prospect for direct and indirect dark matter detection experiments
We show, in the left panel of Fig. 3, the spin-independent cross section, σ SI , as a function of the DM mass for the Z ′ funnel region in Fig. 2, along with the black solid curve and the dashed orange curve being the XENON1T(2018) limit and the neutrino background level [51], respectively. It is clear that the predicted cross section lies below the neutrino background. We show in the right panel of Fig. 3 [52][53][54][55]. The particles and those charges are listed on Table III.
Yukawa couplings are given by The experimental constraints on this model comes from lepton universality derived by the  1.005 ± 0.013(stat.) ± 0.022(syst.) [57]. Since the decay width ratio can be expressed as where m Υ is the mass of Υ, we have constraints on g (B−L) 3 and m Z ′ .

B. Thermal relic abundance
We show, in Fig. 4, conditions to reproduce the observed DM relic abundance Ωh 2 ≃ 0.1 in a (m S − m Z ′ ) plane. The purple, blue, light-blue and cyan correspond to g (B−L) 3 = 1, 0.7, 0.3 and 0.1, respectively. One can easily find the case that SS → Z ′ Z ′ annihilation is available 5 For a Z ′ boson search at the LHC, we refer the results with ditau final state [58]. Using a Z ′ boson production cross section from bb annihilation presented in, for example, Ref. [59] and BR(Z ′ → τ + τ − ) ∼ 0.5, we conclude that the parameter set used in our analysis is consistent with the current LHC results. For future search reach with a large luminosity, see Ref. [60] and a part of parameter space of the (B − L) 3 model will be explored.
for m S > m Z ′ as well as Z ′ resonant enhanced regions along the line m S ≃ m Z ′ /2.

C. Prospect for direct and indirect dark matter detection experiments
The expected spin-independent cross section and the expected present DM pair annihilation cross section as a function of the DM mass satisfying the DM abundance are shown in Fig. 5 for g (B−L) 3 = 0.3 and in Fig. 6 for g (B−L) 3 = 0.03. In both figures, blue curves are for m S < m Z ′ while green curves are for m S > m Z ′ . We can find that an expected present annihilation cross section is O(10 −30 ) cm 3 /s for its wide mass range.
We may apply our scenario to explain an excess in γ-ray emission from the globular cluster 47 Tucanae, which could be interpreted as DM annihilation with the mass about 34   6 Millisecond pulsars interpretation has also been pointed out [63].   [64,65] . This is interesting because the extra gauge boson may solve [66,67] discrepancy between the experimental result and the SM prediction on the muon anomalous magnetic moment [68][69][70][71]. Two RH neutrinos (N 2 R and N 3 R ) are also charged under the U(1) Lµ−Lτ and two U(1) Lµ−Lτ Higgs fields (φ 1 and φ 2 ) are introduced. The particles and those charges are listed on Table IV.

B. Thermal relic abundance
We show, in Fig. 7, experimental constraints, a parameter region favored to solve the muon anomalous magnetic moment and the contours to reproduce the observed DM relic abundance Ωh 2 ≃ 0.1 for various m S values with m P − m S = 0.01m S in (m Z ′ , g µ−τ ) plane, where g µ−τ is the U(1) gauge coupling. The light gray region is excluded by the BABAR experiment [72]. The gray region is constrained by nonobservation of the neutrino trident processes [73,74]. The LHC bound has been studied in Refs. [56,75,76] and will be exhibited as a brown-shaded region in the plot. The vermilion region is favored to account for the discrepancy in the anomalous magnetic moment of muon [66,67,73]. The big bang nucleosynthesis bound has been obtained as m Z ′ 5 MeV by demanding ∆N eff < 0.7 [77,78]. On the other hand, it is also recently pointed out that the "Hubble tension", which is the discrepancy between the values of the Hubble constant as determined from local measurements [79,80] and estimated from the temperature anisotropies of the cosmic microwave background [46], can be relaxed in the U(1) Lµ−Lτ model by increasing the number of relativistic degrees of freedom as ∆N eff ≃ 0.2 [81]. The turquoise, cyan, blue and navy contours correspond to m S = 0.1, 1, 10 and 100 GeV, respectively. A sharp drop is due to the rapid annihilation by s-channel Z ′ resonance pole. For a lighter mass region of Z ′ as m Z ′ < m S , the annihilation mode SS → Z ′ Z ′ is dominant. As we will see in Fig. 8, for a wide mass range of m S 100 GeV, the m Z ′ < m S case is constrained by the Fermi-LAT bound on the present DM annihilation cross section SS → Z ′ Z ′ followed by the decay of Z ′ → τ + τ − , µ + µ − and νν. One exception is that, for a very small mass case, e.g., m S ≃ 0.1 GeV drawn by the turquoise curve, it is free from the constraint because Z ′ cannot decay into charged leptons but do into only neutrinos. Remarkably, such a DM mass range with m Z ′ ≃ O(0.01) GeV can solve the muon g − 2 anomaly, explain the feature of high energy neutrino spectrum [82,83] measured by the IceCube Collaboration [84] as well as relax the Hubble tension. In addition, a heavier Z ′ mass > 10 GeV in the L µ − L τ model is also favored to solve the so-called b → sµ + µ − anomaly [85,86] reported by the LHCb [87][88][89][90] and the Belle [91] experiments. In comparison to our scalar DM, L µ − L τ charged fermion DM [92,93] with such a Z ′ mass suffers from the compatibility of the thermal abundance and the LHC bound, unless the charge is larger than 3 [94].

C. Prospect for direct and indirect dark matter detection experiments
We show in Fig. 8  The constraints [95] and future prospects [96,97] on sub-GeV WIMP have been studied for annihilation into e − e + . However, those cannot directly be applied to our case, since S does not annihilate into e − e + and Z ′ decays into only neutrinos as long as the mass of Z ′ is lighter than the muon mass. GeV is particularly interesting, because the discrepancy of the muon anomalous magnetic moment as well as the Hubble tension can be relaxed.

Annihilation into neutrinos
for the m P → m S limit.