Dark Higgs Channel for FERMI GeV $\gamma$-ray Excess

Dark Higgs is very generic in dark matter models where DM is stabilized by some spontaneously broken dark gauge symmetries. Motivated by the FERMI observation of $\sim$GeV scale $\gamma$-ray excess from the galactic center (GC), we investigate a scenario that a pair of dark matter $X$ annihilates into a pair of dark Higgs $H_2$, which subsequently decays into standard model particles through its mixing with SM Higgs boson. Besides the two-body decay of $H_2$, we also include multibody decay channels of the dark Higgs. We find that the best-fit point is around $M_X\simeq 95.0$GeV, $M_{H_2}\simeq 86.7$GeV, $\langle \sigma v\rangle\simeq 4.0\times 10^{-26}\textrm{cm}^3\textrm{/s}$ and gives a p-value $\simeq 0.40$. Implication of this result is described in the context of dark matter models with dark gauge symmetries. Since such a dark Higgs boson is very difficult to produce at colliders, indirect DM detections of cosmic $\gamma$-rays could be an important probe of dark sectors, complementary to collider searches.


I. INTRODUCTION
Firm evidences for dark matter (DM) come exclusively from the gravitational interaction at the moment. A popular scenario in particle physics models, DM as weaklyinteracting massive particles (WIMP), generally predicts that DM should have a mass between O(GeV)−O(TeV), with a weak-scale annihilation cross section around 3×10 −26 cm 3 /s. If those annihilation final states go to standard model particles eventually, there might be notable excesses in cosmic rays and gamma ray searches.
As a first step, it is natural to investigate the GeV excess through annihilation channels that a pair of DM goes to two SM particles directly, such as qq, cc, bb, tt, l ∓ l ± , gg, hh, W W, ZZ and also their different combinations with some branching fractions. After all, no new particle has been found yet at the LHC, except the Higgs boson. This method has provided valuable information for the favored DM mass and annihilation cross section ranges. Discussions has been extended to cascade two-body decay through new mediators, such as Z and dark Higgs H 2 , which are ubiquitous in new physics beyond SM. In particular, light mediator (M Z ,H 2 < 1GeV) [61] and heavy Z [33] cases have been investigated thoroughly.
This work is intended to investigate GeV scale gamma-ray excess in models where DM annihilates into a pair of heavy dark Higgs (> 1GeV) which subsequently could decay into multi-body final states such as W W * where W * is a virtual W boson. The aim is to provide the ranges of the favored dark Higgs mass, DM mass and the annihilation cross section, which might be useful for particle physics model building, such as hidden sector DM models with local dark gauge symmetries. This work differs from most previous investigation in one essential aspect: we take into account consistently all possible decay modes for heavy dark Higgs, not restricted to its two-body decays.
This paper is organized as follows. In Section. II, we briefly discuss the theoretical motivation and establish our formalism and notations. In Section. III, we present our numerical results on the best-fit parameters. Finally, we give a summary.

II. FORMALISM
We shall consider the following annihilation channel for self-conjugate DM X, Here H 2 denotes the dark Higgs, distinguishing it from the SM-like Higgs H 1 with M H 1 125 GeV. H 2 can decay into SM particles through its small mixing with H 1 . The mixture between H 2 and H 2 can be easily achieved in particle physics model building.
For example, a real scalar DM X and a complex scalar Φ (dark Higgs that breaks local dark gauge symmetries) can have the following interactions, where H is the SM Higgs doublet. After gauge (or even possible global) symmetry breaking, where v h and v φ are the vacuum expectation values, two neutral scalars h and φ will mix with each other through the Higgs portal coupling λ φH , resulting in two mass eigenstates H 1 and H 2 with in terms of the mixing angle α.
The above Lagrangian is just one example of many DM models with Higgs portal. One can also consider the case with a real scalar φ, Again after the electroweak symmetry breaking, h and φ are mixed. We can also consider a model with fermionic X with All the above models can easily evade current experimental bounds when α is very small (see discussion in Ref. [85] for example). If one assumes all other possible new particles are heavy, DM X will dominantly annihilate into H 2 's. To be as general as possible, we shall just work with the effective operator, X 2 H 2 (Xγ 5 XH 2 2 for fermionic X), and consider the annihilation process in Fig. 1, assuming that other particles in the dark sector are all heavy enough.
The produced H 2 's could be either relativistic or non-relativistic for M H 2 M X or M H 2 M X , respectively. Different kinematics might lead to significant differences in the gamma-ray spectra. Moreover, depending on the mass of H 2 , M H 2 , H 2 can dominantly decay into 2 or 3 standard model particles. In our numerical calculation, we use PYTHIA-6.4 [86] to simulate and tabulate dN f γ /dE γ for the interesting ranges of M X and M H 2 . In particular, we focus on M X ≥ 5 GeV and M H 2 ≥ 1 GeV.
The general differential flux of the gamma-ray from the annihilation of self-conjugate DM is given by where σv f ann is the velocity-averaged annihilation cross section for the annihilation channel f , dN f γ /dE γ is prompt gamma-ray spectrum, r = r 2 + r 2 − 2r r cos θ, r is the distance to earth from the DM annihilation point, r 8.5kpc for solar system and θ is the observation angle between the line-of-sight and the center of Milky Way. An extra factor 1/2 needs to be included for non-self-conjugate DM, such as complex scalars or Dirac fermions. In our considered case, we have only one annihilation channel, X + X → H 2 + H 2 . for fermionic X or X µ X µ H 2 2 for vector X). The actual annihilation process may occur through s or t channel, and contact interaction. Details in the gray bubble depend on various ultraviolet completions. The produced H 2 s can have two-, three-or even four-body decay channels.
For DM density distribution, we use the following generalized NFW profile [87], with parameters r c 20kpc and ρ 0.4GeV/cm 3 . We shall adopt the index γ = 1.26 if not stated otherwise.

III. NUMERICAL ANALYSIS
We first show three cases for the gamma-ray spectrum in Fig. 2. The vertical axis marks the conventional where ∆Ω indicates the region of interest. The 24 data points we used to compare with are from Ref. [10], denoted as CCW hereafter. As we can see, different parameter sets can give different spectrum shape, especially in the high energy regime. When the branching ratios of H 2 → γγ, Zγ are increasing, we can see the gamma lines more easily around E M H 2 /2. Since the annihilation cross section is at order of 10 −26 cm 3 /s and the branching ratios of H 2 → γγ, Zγ are around 0.2% at most, the considered parameters are still consistent with constraint from gamma-line searches.
We now use the χ 2 function and find its minimum to find out the best fit: where µ i and f i are the predicted and measured fluxes in the i-th energy bin respectively, and Σ is the 24×24 covariance matrix. We take the numerical values for f i and Σ from CCW [10].
Minimizing the χ 2 against f i with respect to M X , M H 2 and σv gives the best-fit points, and then two-dimensional 1σ, 2σ and 3σ contours are defined at ∆χ 2 ≡ χ 2 − χ 2 min = 2.3, 6.2 and 11.8, respectively. Three illustrative cases for gamma-ray spectra in contrast with CCW data points [10]. All masses are in GeV unit and σv with cm 3 /s. Line shape around E M H 2 /2 is due to decay modes, H 2 → γγ, Zγ.
gives χ 2 min 22.0, with the corresponding p-value equal to 0.40. We also notice that there are two separate regimes, one in the low mass region and the other in high mass region. The higher mass region is basically aligned with M H 2 M X since otherwise a highly-boosted H 2 would give a harder gamma-ray spectrum. In this region, H 2 mostly decays into bb. As one increases the mass of H 2 , H 2 → W ± l ∓ ν, H 2 → Zl ± l ∓ , H 2 → γγ and H 2 → γZ become more and more important, and all of them give harder gamma-ray spectra either due to the leptonic final states or the gamma lines. Eventually, χ 2 increases significantly when M H 2 ≥ 150GeV.
In the low mass region, the contours are scattered but centered around M H 2 10GeV and such a light H 2 most likely decays into bb, cc and τ + τ − . Since cc and τ + τ − would give harder spectra than bb does, we would need a lower M X to fit the data, which is exactly what we see in Fig. 2 (dotted curve). Increasing the branching ratios of cc and τ + τ − would require a even lower M X .
In the right panel, we show a special case in which M H 2 M X , so that the produced H 2 s are non-relativistic. In such a case, the d.o.f. is now 22. The best-fit parameters are 125GeV also give a good-fit. This point is equivalent to the channel that DM X annihilates into SM Higgs, which has been already found in previous study [54,60].
In the left panel of Fig. 4, we fix the mass of dark Higgs to the best-point value, M H 2 = 86.7, and vary M X and σv . We show 1σ, 2σ and 3σ contours in terms of solid(black),   dashed(blue) and long-dashed(red) curves, respectively. To compare with bb channel, we also present 3σ region in the right panel of Fig. 4

IV. SUMMARY
In the letter, we have explored a possibility that the GeV scale γ-ray excess from the galactic center is due to DM pair annihilation into a pair of dark Higgs, followed by the dark   Table I). This information could be important inputs in dark matter models with dark Higgs boson.
At this stage we cannot make any strong statement about the existence of dark Higgs with mass close to the DM mass ∼ 95GeV. However dark Higgs is very generic in DM models where DM is stabilized by some spontaneously broken local (or even global) dark gauge symmetries [26,27,41,69,[88][89][90][91][92][93][94][95][96]. Since the dark Higgs boson is a SM singlet scalar, it is very difficult to find it at colliders. It is simply more difficult to produce them at colliders when the mixing angle is small. It is very amusing to notice that indirect DM detection experiments can be more sensitive to such a dark Higgs than the collider experiments. Compared with more popular dark photon scenario, it is more natural to have flavor dependent couplings of dark Higgs boson to the SM fermions, since its couplings are basically the same as those of the SM Higgs boson modulo the mixing angle effect. This fact makes much easier DM model building [26,27]. It remains to be seen whether this fit survives in the future data sets, and if there would be any indication of such a dark Higgs from the future collider experiments.