Higgs portal vector dark matter for $\mathinner{\mathrm{GeV}}$ scale $\gamma$-ray excess from galactic center

We show that the $\mathinner{\mathrm{GeV}}$ scale $\gamma$-ray excess from the direction of the Galactic Center can be naturally explained by the pair annihilation of Abelian vector dark matter (VDM) into a pair of dark Higgs bosons $VV \rightarrow \phi \phi$, followed by the subsequent decay of $\phi$ into $\phi \rightarrow b\bar{b} , \tau \bar{\tau} $. All the processes are described by a renormalizable VDM model with the Higgs portal, which is naturally flavor-dependent. Some parameter space of this scenario can be tested at the near future direct dark matter search experiments such as LUX and XENON1T.

We should note that it is the shape of γ spectrum from dark matter annihilation that mainly matters rather than the precise value for σv since there is a large uncertainty in the density profile of dark matter near the Milky Way center. As long as σv (ρ DM /M DM ) 2 is at the right amount, a good fit can be achieved for bb final state. Actually, bb does not need to be the only annihilation channel, it was shown [10] that flavor-dependent annihilations can also fit the data well. Such kind of flavor-dependent annihilations may indicate a Higgs-like scalar mediator, since Higgs-like scalar will couple with the heaviest particle it can couple to.
The required cross section is very close to the canonical value for neutral thermal relic dark matter. It can be achieved either s-wave annihilation or p-wave annihilation with schannel resonance at present. However, in the latter case, the resonance band is likely to be very narrow that leads to a severe fine-tuning, which is not that attractive. With this consideration, perhaps the simplest scenario for dark matter model that can explain the γ-ray excess would be those involving scalar mediator with Higgs portal interaction(s), since in this case the scalar mediator will couple strongly to the bb, the heaviest particles kinematically producible 1 . Then, one can imagine the following simple scenarios of DM having s-wave annihilation channel: 1. Singlet scalar dark matter (SSDM): a real scalar mediator [15] 2. Singlet fermion dark matter (SFDM): a pseudo-scalar mediator [16][17][18]  3. Singlet vector dark matter (SVDM): a real scalar mediator [19][20][21] Note that the structure of above scenarios can be realized easily when DM is charged under a dark gauge symmetry which is broken to, for example, a discrete Z 2 or Z 3 symmetry. Hence those scenarios would also work equally well. For other recent proposals of DM models to address the GeV γ-ray spectrum, see Refs. [13,14,22,23].
Potentially the most important constraint on those singlet dark matter models may come from direct search experiments, for example, LUX [24]. However the existence of extra scalar boson mediating dark and visible sectors via Higgs portal interaction(s) has a significant effect on direct searches if the mass of the extra non-SM Higgs is not very different from that of SM Higgs [25], and the constraint from direct searches can be satisfied rather easily. Note that this feature is not captured at all in effective field theory approach, and it is important to work on the minimal renormalizable and unitary Lagrangian for physically sensible results 3 .
In this paper, we revisit SVDM scenario with Higgs portal in the context of the the γ-ray excess from the Galactic Center, and show that the SVDM model can naturally explain it, while satisfying all of known constraints coming from CMB, Fermi-LAT γ-ray search and LHC experiments. We also show that the parameter space relevant for the γ-ray excess can be probed by the near future direct dark matter search experiment, for example LUX and XENON1T.
This paper is organized as follows. In Section 2, we recapitulate the renormalizable SVDM model with Higgs portal. In Section 3, various relevant constraints on the model are discussed, including relic density estimation, vacuum stability, collider bounds, CMB and direct detection cross section, etc., and we show that our model can explain the γ-ray excess from the galactic center without any conflict with other cosmological and astrophysical observations. In Section 4, our conclusion is drawn.

The renormalizable SVDM with Higgs portal
Let us consider a Abelian vector boson dark matter 4 , X µ , which is assumed to be a gauge boson associated with Abelian dark gauge symmetry U (1) X . The simplest model will be defined with a complex scalar dark Higgs field Φ only, and no other extra fields. The VEV of Φ breaks U (1) X spontaneously and generate the mass for X µ through the standard Higgs mechanism (see also Ref. [28]): in addition to the SM Lagrangian which includes the Higgs potential term 3 See Refs. [20,25] for the original discussions on this point, and Ref. [26] for more discussion on the correlation between the invisible Higgs branching ratio and the direct detection cross section in the Higgs portal SFDM and SVDM models, 4 The Abelian VDM was first considered in Ref. [27] where the VDM mass assumed to be generated either by the Stückelberg or by dark Higgs mechanism, but the role of dark Higgs boson was ignored within effective field theory (EFT). However, in the presence of the dark Higgs boson, the resulting VDM phenomenology can be vastly different from the one in the VDM model of EFT. See Ref. [20] for more detailed discussion.
The covariant derivative is defined as is the U (1) X charge of Φ and we will take Q Φ = 1 throughout the paper.
Assuming that the U (1) X -charged complex scalar Φ develops a nonzero VEV, v Φ , and thus breaks U (1) X spontaneously, we would have Therefore the Abelian vector boson X µ gets mass M X = g X |Q Φ |v Φ . And the hidden sector Higgs field (or dark Higgs field) φ(x) will mix with the SM Higgs field h(x) through the Higgs portal λ ΦH term, resulting in two neutral Higgs-like scalar bosons. The mixing matrix O between the two scalar fields is defined as where s α (c α ) ≡ sin α(cos α), h, φ are the interaction eigenstates and H i (i = 1, 2) are the mass eigenstates with masses m i , respectively. The mass matrix in the basis (h, φ) can be written in terms either of Lagrangian parameters or of the physical parameters as follows: Note that one can take m 1 , m 2 and α are independent parameters. In the small mixing limit which is of our interest, the mass eigenstates are approximated to the interaction eigenstates as (H 2 , H 1 ) ≈ (h, φ), and we use (h, φ) to represent quantities associated with (H 2 , H 1 ) from now on.

Constraints
Our VDM interacts with SM sector via Higgs portal interaction. This means that it is subject to constraints from CMB observations, direct/indirect DM searches, and collider experiments. However, for 30 GeV m V 80 GeV, constraints from CMB [29] and indirect searches [6,[30][31][32] can be easily satisfied in our scenario as far as there is no enhancement of annihilation rate relative to the one at freeze-out. So, in this section we consider only low energy phenomenology, direct detection and relic density.

Vacuum stability
The mixing between Higgs fields (H and Φ) causes a tree-level shift of λ H relative to that of SM in such a way that the relation holds. Hence, for m φ < m h one obtains λ H even smaller than that of SM, and vacuum instability of SM Higgs potential becomes worse. So, it is better to take α as small as possible.
Although tree-level mixing does not work, vacuum instability can be improved by the additional contribution of λ ΦH to the β-function of λ H , as shown in Section 3.4. Then, the tachyon-free condition, λ ΦH < 2 √ λ Φ λ H , results in λ ΦH 0.07 for α and m φ in the range of our interest. It might be large enough to improve the vacuum stability. The exact lower bound on λ ΦH that stabilizes the EW vacuum up to Planck scale depends on the precise values of top quark mass and the strong coupling constant, the detailed discussion of which is beyond this paper.

Collider bound
For m V < m h /2, the SM Higgs boson can decay into two VDM which is invisible. Recent analysis from collider experiments showed that the branching fraction of the Higgs boson into invisible particles should be constrained as [33] Br inv h < 0.51 (3.2) However the bound was extracted for a effective-field-theoretic (EFT) VDM model. In a renormalizable complete theory like the one we are considering, more parameters are involved than EFT model. Hence, instead of Eq. (3.2), we use which is an approximation obtained from the result of Ref. [34], and Br non−SM h is the branching fractions of the Higgs decay to DMs and non-SM Higgs. In our SVDM scenario, Br non−SM h is given by In the second line of the above equation, we assumed the first term dominates over the others in the small mixing limit.

Using Eqs. (3.3) and (3.4)
, we can constrain the allowed ranges of g X and α as shown in the white region of the left-panel of Fig. 1. Note that in Fig. 1, the mixing angle is constrained to be α 7 × 10 −2 for m φ = 60 GeV. The the upper-bound of α is lowered down for a lighter m φ . Note that the current LHC, LUX or the future XENON1T experiments cover only a part of the allowed parameter region in (α, g X ). There is ample region of parameter space which cannot be explored directly in any experiments.

Direct detection
For 30 GeV m V 80 GeV, LUX experiment for direct detection of WIMP imposes a strong upper bound on the spin-independent (SI) dark matter-proton scattering cross section [24] as: σ SI p (7 − 9) × 10 −46 cm 2 (3.9) The SI-elastic scattering cross section for VDM to scatter off a proton target is given by where µ V = m V m p /(m V +m p ) and f p = 0.326 [35] (see Ref. [36] for more recent analysis) was used. Note that m φ ∼ m h results in some amount of cancellation between contributions of φ and h to σ SI p . As the result, the LUX bound can be satisfied rather easily for g X s α c α 10 −2 . As shown in Fig. 1, direct detection experiments leave a wide range of parameter space uncovered. This is unfortunate since it implies that the model cannot be entirely cross checked by other physical observables.   Figure 3. Dominant s/t-channel production of H 1 s that decay dominantly to b +b

Dark matter relic density
The observed GeV scale γ-ray spectrum may be explained if DM annihilates mainly into bb with a velocity-averaged annihilation cross section close to the canonical value of thermal relic dark matter. This implies that 30 GeV m V 40 GeV in case of the s-channel annihilation (Fig. 2) scenario. It is also possible to produce bb with the nearly same energy from the decay of highly non-relativistic φ which is produced from the annihilation of DM having mass of 60 GeV m V 80 GeV (Fig. 3). In both cases, it is expected to have ττ and cc productions too in the final states, because H 1 will decay into them with branching ratios about 7% and 3%.
In the process of Fig. 2, the thermally-averaged annihilation cross section of VDM is given by where m f is the mass of a SM fermion f . Note that Eq. (3.11) is suppressed by a factor s 2 α m 2 f . Hence a large enough annihilation cross section for the right amount of relic density can be achieved only around the resonance region. However in the resonance region the annihilation cross section varies a lot, as the Mandalstam s-variable varies from the value at freeze-out to the value in a dark matter halo at present. Therefore, this process can not be used for the GeV scale γ-ray spectrum from the galactic center.
On the other hand, in the process of Fig. 3 for m φ < m V 80 GeV, the thermallyaveraged annihilation cross section of VDM is given by σv rel tot = σv rel ff + σv rel φφ (3.12) where σv rel φφ ≃ 1 16πs (3.14) Note that, if we consider the off-resonance region with 2m V ≁ m h , the contribution of the s-channel H 2 mediation can be ignored and σv rel φφ does not depend neither s α nor m f . Hence a right size of annihilation cross section can be obtained by adjusting mostly g X and (m V − m φ ) /m V , with the negligible mixing angle dependence. Fig. 4 shows the relic density  at present 5 as a function of m V for m φ = 75 GeV and g X = 0.2 and the mixing angle α = 0.1. From Fig. 4, we note that the mass of our VDM is constrained to be m h /2 < m V , since SM-Higgs resonance should be also avoided. And the velocity-averaged annihilation cross section at present epoch can be close to that of freeze-out only for m φ m V . Note also that, as shown in Fig. 5, in order to match to the observed γ-ray spectrum, we need m φ ∼ m V to avoid boosted φ.
In the region of 60 GeV m φ ∼ m V 80 GeV, the SM Higgs boson decay into VDM is suppressed by the phase space factor or kinematically forbidden. Hence the collider bound 0.1 1 10 100 0.1 1 10 100 Figure 5. Illustration of γ spectra from different channels. The first two cases give almost the same spectra while in the third case γ is boosted so the spectrum is shifted to higher energy.
on the invisible decay of SM Higgs is irrelevant, but the mixing angle is still constrained by the signal strength of SM channels such that α 0.4 [34]. A remark is in order for the present annihilation cross section to obtain observed GeV scale γ-ray. Compared to the case of 30 GeV m V 40 GeV, the present number density of dark matter for 60 GeV m V 80 GeV is smaller by a factor of about a half, but each annihilation produces two pairs of bb. Hence, the expected flux which is proportional to the square of DM number density is smaller by about a half. However, there are various astrophysical uncertainties in the estimation of required annihilation cross section. In particular, a small change of the inner slope of DM density profile is enough to compensate the difference of about factor two. In addition, as discussed in Refs. [10], the GeV scale γ-ray data fits well to cross sections proportional to the square of the mass of the final state SM particles. This kind of flavor-dependence is an intrinsic nature of our SVDM scenario, thanks to the Higgs portal interaction. Therefore, with these points in mind, SVDM with mass of 60 GeV m V 80 GeV can be a natural source of the GeV scale γ-ray excess from the direction of the galactic center.

Comparison with other Higgs portal DM models
In regard of the GeV scale γ-ray excess from the galactic center, SSDM can work equally well as our SVDM scenario. One difference from SVDM is the additional Higgs portal interaction of SSDM with SM Higgs, which can improve the vacuum instability problem of SM Higgs potential better than SVDM scenario.
Contrary to SSDM or SVDM, SFDM with a real scalar mediator results in p-wave schannel annihilation. In addition, the t-channel annihilation cross section is approximately in the low momentum limit. Since (m DM − m φ ) /m DM ≪ 1 is needed in order to avoid a boosted φ, such a t-channel annihilation in SFDM scenario is suppressed by an additional factor 1 − m 2 φ /m 2 DM relative to the case of SSDM and SVDM. Hence SFDM needs a pseudo-scalar mediator and it makes model a bit complicated (see for example Ref. [18]).
Contrary to the case of SFDM where a wide range of pseudo-scalar mass is allowed, the requirement of the t-channel annihilation of DM near threshold in SSDM and SVDM constrains the mass of non-SM Higgs φ to be within a narrow range of m h /2 m φ 80 GeV

Conclusion
In this paper, we revisited the singlet vector dark matter (SVDM) model with Higgs portal in order to see if it can explain the observed GeV scale γ-ray spectrum from the galactic center by the annihilation of dark matter mainly to bb or to two non-SM light Higgses which decay subsequently and dominantly to bb. We found that the Higgs portal SVDM scenario can naturally explain the γ-ray spectrum while providing a right amount of relic density for m h /2 < m V 80 GeV and (m V − m φ )/m V ≪ 1 with m V and m φ being the masses of VDM and non-SM Higgs boson. This implies that the mass of the non-SM extra Higgs is constrained to be within a narrow range of m h /2 m φ 80 GeV (4.1) which can be focused on in dedicated searches of the second Higgs at future collider experiments although a null result due to very small mixing angle α is also possible. The dark gauge coupling is contained to be g X ∼ 0.2 for the right amount of relic density while taking α to be small enough to satisfy direct DM search bound. Unfortuantely the LUX or XENON1T cannot explore the entire parameter space of the SVDM explaining the GeV-scale γ-ray from the galactic center. The instability of SM vacuum could be improved due to the additional loop contribution of an extra scalar field.