Galactic center $\gamma$-ray excess in hidden sector DM models with dark gauge symmetries: local $Z_{3}$ symmetry as an example

Hidden sector dark matter (DM) models with local dark gauge symmetry make a natural playground for the $\gamma$-ray excess from the galactic center (GC). In this paper, we first discuss in detail the GC $\gamma$-ray excess in scalar dark matter model with local $Z_{3}$ symmetry which was recently proposed by the present authors. Within this model, scalar DM with mass $30-70$GeV is allowed due to newly-opened (semi-)annihilation channels of DM pair into dark Higgs $\phi$ and/or dark photon $Z^{'}$ pair, and can fit the $\gamma$-ray spectrum from the GC. Then we argue that the GC gamma ray excess can be easily accommodated in hidden sector dark matter models where DM is stabilized by local gauge symmetries, due to the presence of dark Higgs (and also dark photon for Abelian dark gauge symmetry). As by-products, the Higgs portal interaction between the dark Higgs and the SM Higgs boson can improve the EW vacuum stability up to Planck scale, and enable the Higgs inflation scenario to have potentially large tensor-to-scalar ratio, $r\sim O(0.01 - 0.1)$.


I. INTRODUCTION
Recently, analysis of data from the Fermi Gamma-Ray Space Telescope has shown significant signal of γ-ray from the Galactic Center (GC) [1], which was also identified in the past studies [2][3][4][5][6][7][8]. Astrophysical explanation such as millisecond pulsars has been discussed [9]. However, it seems that signal from light dark matter annihilation is more favored [1]. Interpretation that dark matter annihilates directly to standard model(SM) fermions indicates the thermal cross section σv ann is at the order of 10 −26 cm 3 /s [1]. Both specific models and general frameworks have been investigated for this γ-ray signal  since then.
DM models for the GC γ-ray can be generally divided into two categories. In the first category, DM pair annihilates directly into SM fermion pairs [10][11][12][13][14][21][22][23][24][25][26][27][28], which is expected to be inconsistent with current DM constraints from collider and direct searches (see ref. [32], for example). In the second category, DM annihilates to two or more on-shell particles [15][16][17][18][19][20], which in turn decay to SM fermions. A natural scenario of the second category can be realized in DM models with dark/hidden sectors and singlet portals (Higgs portal and kinetic mixing). The dark sector interacts with SM particles through Higgs-portal and/or kinetic mixing couplings, which can easily evade the collider and direct search bounds.
In this paper, we discuss the GC γ-ray signal in scalar dark matter model with local Z 3 symmetry as an example. In this model the dark sector has local U (1) X gauge symmetry which is spontaneously broken into Z 3 subgroup, and two new particles (dark higgs and massive dark photon) are introduced in the model simlutanesouly. As discussed in [33], this model has rich phenomenologies which are qualitatively different from global Z 3 model [34] due to these two new particles. A novel feature in this model, semi-annihilation [35][36][37], can break the tight correlation between relic abundance and DM-nucleon scattering cross section, allowing light DM with mass below 125GeV in sharp contrast to global Z 3 case [34]. Also local Z 3 symmetry guarantees stability of scalar DM even in the presence of nonrenormalizable higher dimensional operators. Stabilizing DM particle via local gauge symmetries has a number of virtues, as noticed in recent works [33,[38][39][40][41].
After the detalied discussion of the GC γ-ray signal in scalar dark matter model with local Z 3 symmetry, we generalize our findings to general hidden sector DM models with local dark gauge symmetries. There are new particles, namely dark gauge bosons and dark Higgs boson, in addition to the DM, whose interactions are completely dictated by local dark gauge symmetry and renormalizability. And the renormalizable Higgs portal will be generically present in almost all the cases, thermalize DM relic density efficiently even for non Abelian dark gauge symmetries (see Refs. [39,[42][43][44][45][46][47][48][49][50] for some interesting DM models based on non Abelian dark gauge symmetries), and also play a very important role for direct and indirect detections of DM.
For Abelian dark gauge symmetry, there will be an additional portal from the gauge kinetic mixing. Very often one assumes massive dark photon with gauge kinetic mixing, and consider constraints from direct detection, thermal relic density and indirect detection signature altogether only for the gauge kinetic mixing operator. However, kinetic mixing is possible only for the U (1) dark gauge symmetry, and not for non Abelian ones. And the couplings between the DM particle and the dark photon critically depend on the U (1) X charge assignments to the DM and the dark Higgs fields (for example, compare the local Z 3 model [33] and the local Z 2 model [51]), which is often overlooked in many works. Within local dark gauge theory, it is not consistent to give dark gauge boson mass by hand, since it breaks local dark gauge symmetry explicitly. It is important to introduce dark Higgs field or some nonperturbative dynamical symmerty breaking mechanism for generating dark photon mass. Also it is important to keep all the operators allowed by local dark gauge symmetry (and renormalizability), including the Higgs portal interaction. Otherwise, the resulting phenomenology could be misleading and sometimes even wrong.
We note that DM models are also constrained by indirect detections. Stringent limits come from anti-proton, positrons and radio signal [29][30][31]. However, such constraints from indirect signals vary for different DM annihilation channels and depend sensitively on various astrophysical parameters as well: for example, propagation parameters for anti-proton, local DM density for positron and DM density profile at small radii r < 5pc for radio signal [30], respectively. Currently, conservative limits still allow viable space for DM explanation of γray excess. Due to the large uncertainties for astrophysical parameters, we shall not impose such constraints in our discussion but will show how anti-proton flux depends on such kind of propagation parameters as an example.
This paper is organized as following. We first introduce the scalar SM model with local Z 3 symmetry briefly in Section. II, establishing the notations for later discussion. Then we focus on the γ-ray spectrum from Z 3 scalar DM (semi-)annihilation from the GC and compare with the data in Section. III. In Sec. IV, we generalized our finding from local Z 3 scalar DM model to general hidden sector DM models with dark gauge symmetries. Finally, we summarize the results in Sec. V.

II. SCALAR DM MODEL WITH LOCAL Z 3 SYMMETRY
In this section, we give a brief introduction of scalar dark matter model with local Z 3 symmetry [33]. The dark sector has a local U (1) X gauge symmetry which is spontaneously broken into Z 3 by the nonzero VEV of dark Higgs φ X . This can be realized with two complex scalar fields, with the U (1) X charges equal to 1 and 1/3, respectively. Then one can write down renormalizable Lagrangian for the SM fields and the dark sector fields, X µ , φ X and X: where the covariant derivative associated with the gauge fieldX µ is defined as D µ ≡ ∂ µ − ig X Q X X µ . The coupling λ 3 is chosen as real and positive, since one can always redefine the field X and absorb the phase of λ 3 . The vacuum phase we are interested in should have the following structures: where only H and φ X have non-zero vacuum expectation values (VEVs). This vacuum will break electroweak symmetry into U (1) em , and U (1) X symmetry into local Z 3 , which stabilizes the scalar field X and make it absolutely stable DM even in the presence of nonrenormalizable operators [33]. Expanding the scalar fields around Eq. (2.2), lead to the mixing of two scalars, h and φ, with the mass eigenstates H 1 as and H 2 and the mixing angle α. We shall identify H 1 as the recent discovered Higgs boson with M H 1 125GeV and treat M H 2 as a free parameter. In principle, H 2 could be either heavier or lighter than H 1 . However, we shall take M H 2 ≤ 80GeV in the following discussion, aiming at explaining the galactic center gamma ray excess from XX → H 2 H 2 . After EW and dark gauge symmetry breaking, neutral gauge fields can mix with each other, and the physical fields (A µ , Z µ , Z µ ) are defined as We have defined the new parameters: From Eq. (2.5) we can observe that the SM particles charged under SU (2) L and/or U (1) Y now also have interaction with Z µ . And particles in the dark sector also have interaction with Z µ due to the kinetic mixing beweenB µ andX µ . The physical masses for four vector bosons in our model are given by . (2.8) Unlike those models based on global Z 3 symmetry [34] in which light dark matter (m DM 125) is generally excluded by LUX direct search experiment except for resonance regime, models with local gauge symmetry can still allow such light dark matter due to the new opened annihilation channels, see Ref. [33] for details. In this paper, we only focus on the indirect signatures of local Z 3 DM in terms of γ-ray, anti-proton and positron fluxes.
Feynman diagrams for XX semi-annihilation into H 2 and Z .

III. γ-RAY FROM DM (SEMI-)ANNIHILATION
In the section, we discuss the γ-ray spectrum from dark matter (semi-)annihilation in scalar DM model with local Z 3 symmetry. We shall focus on the channels showed in Figs. 1 and 2, where H 2 and Z decay into standard model particles. DM annihilation directly into SM particle such as XX → Z * or H * 2 →f f, are suppressed by the small mixing parameters, α and . For the parameter regions we are interested in, we can take α and to be smaller than 10 −4 , which are definitely allowed by direct searches so far. For simplicity, we also assume vanishing λ φH and λ HX . Nonvanishing λ φH and λ HX will not change our discussion qualitatively, and both parameters are constrained by DM direct searches, higgs invisible decay and collider searches which are beyond the scope of this paper.
The γ-ray flux from self-conjugate dark matter (semi-)annihilation is determined by particle physics factors, σv ann and dN γ /dE γ , and dark matter density profile ρ from astrophysics: where r is the distance to the earth from the DM annihilation point, r = r 2 + r 2 − 2r r cos θ, r 8.5kpc and θ is the observation angle between the line of sight and the center of Milky Way. An extra factor 1/2 has to be included when dark matter X annihilates with its anti-particleX.
We use the DM density profile parametrized as ρ (r) = ρ r r with r c 20kpc, ρ 0.3GeV/cm 3 and γ = 1 for NFW profile. Define a dimensionless function I, we show how I depends on the power index γ in Fig. 3. Because of the large uncertainty of ρ, it is not that meaningful to quantify the exact value for σv ann in Eq. (3.1), as long as it is at the order 10 −26 cm 3 /s. We can roughly estimate σv ann in our model by fixing index γ = 1.26. For self-conjugate DM annihilation, it was shown in [1] that σv ann 1.7(1.1) × 10 −26 cm 3 /s can fit the gamma spectrum well for bb(democratic) channel. It is then straightforward to get σv ann 6.8(4.4) × 10 −26 cm 3 /s for complex scalar or Dirac fermion DM that annihilates firstly to two on-shell particles H 2 (Z ), which in turn decay into the SM particles. In the following discussion, we shall fix index γ = 1.26 and treat σv ann as free 1 . This also means that at this stage we do not consider the relic abundance precisely. Due to the mixing among neutral gauge fields, Z can decay into SM fermion pairs with branching ratios depending on flavours. For instance, the couplings for Z µf R γ µ f R and Z µf L γ µ f L are respectively where Y is the U (1) Y hypercharge and T 3 is related to SU (2) L . On the other hand, the dark Higgs H 2 couples to the standerd model fermions in the same way as the SM Higgs does, except that the couplings are rescaled by the mixing angle factor sin α. Therefore the branching ratios of H 2 into the SM particles will be similar to those of the SM Higgs boson. The shape of γ-ray spectrum, or dN γ /dE γ , depends on the mass parameters of the involved particles, m X , m H 2 and m Z , and the relative contributions of each annihilation channel in Figs. 1 and 2 which also depends on the parameters, λ 3 , λ φX , g X . Here for simplicity, we discuss two illustrative cases at the boundary, either 100% to H 2 or Z at the final state. A large parameter space exists between these two extreme cases. Dedicated analysis would require multi-dimension χ 2 fitting, which is beyond the scope of this paper. We use micrOMEGAs-3 [52] for our numerical calculation and to generate the γ-ray, anti-proton and positron spectra.
We show the γ-ray spectra in Fig. 4 from H 2 (left panel) and Z (right panel). Since we are discussing the boundary cases, we choose the mass of H 2 /Z to be close to m X . It is seen that m X is around 70GeV for H 2 case while m X ∼ 30 for Z case. This is due to the fact that light H 2 mainly decays to bb which give a softer γ-ray spectrum, compared to Z 's decay to charged lepton pairs. For 30GeV m X 70GeV, we can adjust the relative contributions from H 2 /Z -channel, and it is anticipated that this can be easily achieved by shifting m H 2 , m Z , λ 3 , λ φX and g X .
We also studiedp and e + spectra, as shown in Fig. 5, and checked if our choice of parameters is compatible with these cosmic ray spectra or not. Charged particles generated  [53] for anti-proton and [54] for positron fluxes, using the database [55].
from DM (semi-)annihilation propogate to earth, subjected to diffusion, synchroton radiation and inverse Compton scattering. We use the micrOMEGAs-3 [52] for calculate their spectra, with MIN model being used for anti-proton propagation. As we can see thatp fluxes are at the same order while e + can have huge differences in the considered cases. This is due to the decay of Z that can produce much harder e + , which can be also used to distinguish different models. In Fig. 5 we only show the signal from DM (semi-)annihilation. When added with the background, these fluxes can be compared with the data from PAMELA [53,54]. The constraints fromp and e + can provide important complementary information for models explaining γ-ray excess. It should be pointed out that potentially stringent constraints from indirect detections of cosmic rays depend sensitively on astrophysical parameters involved in the calculations of cosmic ray production and propagation.
The propogation equation that describes the evolution of the energy distribution for charged particle a is [56] ∂ ∂z where ψ a = dn/dE is the number density of particle a per unit volume and energy. Q a is the source term from dark matter annihilation. The function K is the space diffusion coefficient which depends on the energy E: Here β is the particle velocity, R = p/q is its rigidity, and b(E) is the energy loss rate. As a concrete illustration, in Fig. 6, we show how anti-proton flux can change for different astrophysical models in table I, solving Eq. (3.6) with micrOMEGAs-3 [52]. As shown, because of these uncertainties, there is still viable parameter space that is consistent with   [57,58]. L is half of the thickness of diffusion zone for cosmic rays. such constraints (see Ref. [30] for further detailed discussion including the constraints from radiowave). Neutrinos are also produced promptly from the above DM (semi-)annihilation with the absolute flux depending on the final states. Since the σv ann ∼ 10 −26 cm 3 /s, the flux is about 3 − 4 order smaller than the current sensitivity or the limits from neutrino telescope like IceCube. Unless there is a huge boost factor from astrophysics or other mechanisms, we expect that the produced neutrinos can not be detected and therefore no meaningful constraints from neutrino flux measurements.

IV. GENERALIZATION TO HIDDEN SECTOR DM MODELS WITH LOCAL DARK GAUGE SYMMETRIES
From our discussion, it is clear that the gamma ray excess from the GC can be accommodated if there is a new particle analogous to the dark photon or dark Higgs boson with suitable mass spectra. This is in fact realized readily in a class of dark matter models with local dark gauge symmetry and thermalized by singlet portals (including Higgs portal). In such scenarios, there is almost always a SM singlet scalar from the dark sector [39], as well as dark gauge boson that couples to the dark matter particles, independent of details of dark gauge symmetry or matter contents in the dark sector.
The dark Higgs boson is a SM singlet, and will mix in general with the SM Higgs boson. It will improve the stability of the EW vacuum upto Planck scale [59], and also modify the predictions of Higgs inflation so that somewhat large tensor-to-scalar ratio r ∼ O(0.01 − 0.1) becomes possible even for m t = 173 GeV [60]. Dark Higgs will modify the signal strengths of the SM Higgs boson in a simple testable way: the signal strength will be reduced from "1" in a universal fashion, independent of production and decay channels of the SM Higgs boson, which are generic aspects of renormalziable and unitary Higgs portal DM models with local dark gauge symmetries [38? -41] or without local dark gauge symmetries [61,62]. Also nonstandard decays of the SM Higgs boson are possible, such as H → φφ, Z Z , XX, etc., including the invisible decay into a pair of DM particles (see Ref. [63] for global analysis of Higgs signal strengths in the presence of light extra singlet scalar that mixes with the SM Higgs boson).

V. CONCLUSION
In conclusion, we discussed the galactic center γ-ray excess in scalar dark matter model with local Z 3 symmetry which is the remnant of a spontaneously broken U (1) gauge symmetry. Due to the newly opened (semi-)annihilation channels involving dark Higgs and/or dark gauge boson, scalar dark matter as light as several ten GeV can still exist, the mass range which was not allowed in the global Z 3 model because of thermal relic density and direct detection constraints. In the local Z 3 scalar DM model, dark matter particle can (semi)-annihilate into the dark Higgs H 2 and/or dark photon Z which then decay to light SM fermions such as bb or τ τ , etc. The γ-ray from these light fermions can fit data reasonably well. Depending on the relative contributions of individual (semi-)annihilation channel, DM mass can vary from 30GeV to 70GeV, in a wide range of parameter space [33].
Finally we generalized this mechanism for the GeV scale γ ray excess from the GC to dark matter models with local dark gauge symmetry, and argued that one can easily accommodate the gamma excess from the GC using the dark photon and/or dark Higgs which are generically present in dark matter models with local dark gauge symmetry. In particular dark Higgs plays more important role, since the dark gauge boson coupled to the SM fields can be realized only in the Abelian dark gauge symmetry. Since there are many interesting models with non Abelian dark gauge symmetry where dark gauge bosons cannot have renormalizable couplings to the SM fields, one can invoke dark Higgs for the GC γ ray in case of nonAbelian dark gauge symmetries. Dark Higgs couples to the heavier SM fermions such as bb or τ τ , and thus is naturally flavor dependent as noticed in Ref. [15]. It is amusing to notice that DM models where DM is stabilized by local dark gauge symmetry have a number of new fields (dark gauge bosons and dark Higgs) which can be utilized for DM self-interaction or the γ ray excess from the GC, depending on the mass spectra of the new particles. Dark Higgs boson is a singlet scalar that mixes with the SM Higgs boson, and the Higgs signal strengths will be reduced from one in a universal manner independent of production and decay channels. And dark Higgs also improves the EW vacuum stability upto the Planck scale, and can make it possible for Higgs inflation to have larger tensor-toscalar ratio with r ∼ O(0.01 − 0.1). Therefore active search for the dark Higgs at colliders is clearly warranteed.