Class of Higgs-portal Dark Matter models in the light of gamma-ray excess from Galactic center

Recently the study of anomalous gamma-ray emission in the regions surrounding the galactic center has drawn a lot of attention as it points out that the excess of $\sim 1-3$ GeV gamma-ray in the low latitude is consistent with the emission expected from annihilating dark matter. The best-fit to the gamma-ray spectrum corresponds to dark matter (DM) candidate having mass in the range $\sim 31-40$ GeV annihilating into $b\bar{b}$-pair with cross-section $\langle \sigma v \rangle = (1.4-2.0)\times 10^{-26}\;\textrm{cm}^3 \textrm{sec}^{-1}$. We have shown that the Higgs-portal dark matter models in presence of scalar resonance (in the annihilation channel) are well-suited for explaining these phenomena. In addition, the parameter space of these models also satisfy constraints from the LHC Higgs searches, relic abundance and direct detection experiments. We also comment on real singlet scalar Higgs-portal DM model which is found to be incompatible with the recent analysis.


Introduction
Gamma-ray emission from the galactic center (GC) and the inner galaxy regions as found in the Fermi-LAT data has gained a lot of attention from the perspective of dark matter (DM) searches. Past studies [1,2,3,4,5,6,7,8] have pointed out a spatially extended excess of ∼ 1−3 GeV gamma rays from the regions surrounding the galactic center, the morphology and spectrum of which is best fitted with that predicted from the annihilations of a 31 − 40 GeV WIMP (weakly interacting massive particle) dark matter (DM) candidate annihilating mostly to b-quarks (or a ∼ 7 − 10 GeV WIMP annihilating significantly to τleptons). Gamma rays from the galactic center is specially interesting because the region is predicted to contain very high densities of dark matter. Alternative explanations such as gamma-ray excess originating from thousands of unresolved millisecond pulsars have been disfavored since the signal extends well beyond the boundaries of the central stellar cluster. A more recent scrutiny of the morphology and spectrum of the anomalous gamma-ray emission in order to identify the origin has confirmed that the signal is very well fitted by a 31-40 GeV dark matter particle annihilating to bb with an annihilation cross section of σv = (1.4 − 2.0) × 10 −26 cm 3 sec −1 (normalized to a local dark matter density of 0.3 GeV cm −3 ) [9], which is accidentally close to the weak cross-section for producing correct relic abundance.
Email addresses: tanu@prl.res.in (Tanushree Basak), tanmoym@prl.res.in (Tanmoy Mondal) The excess seen in the gamma ray spectrum at the low latitude region can be well explained in a simple dark matter model, where the DM dominantly annihilates into quark pairs with cross-section in the desired range for obtaining correct relic abundance. Already a handful of particle physics model of dark matter [10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29] have been proposed to explain the reported gamma-ray excess. Among these some are focused on various Higgs-portal dark matter models [11,12,20]. These kind of models are simply interesting because they enjoy a special feature of scalar resonances, provided dark matter mass is half of the scalar mass(es). This resonant feature is crucial as it enhances the annihilation cross-section.
In this letter, we have studied a class of Higgs-portal dark matter models to explain the reported excess. We showed that the simplest Higgs-portal model, i.e, the real singlet scalar extension of the Standard model (SM), is inconsistent with a 30 − 40 GeV dark matter, because of the absence of resonance. Another Higgs-portal model considered in this letter is the so-called Singlet fermionic dark matter (SFDM) model, which consist of SM alongwith a hidden sector with a gauge singlet scalar and a Diracfermion singlet, acting as a potential DM candidate. We analyse the parameter space of this model owing to constraints from LHC bound on SM-Higgs, relic density and direct detection of DM. We found this model to be consistent as well with the requirements to explain the galactic center γ-ray excess. The last model we consider is the minimal U (1) B−L extension of the SM with a SM singlet scalar S and three right-handed (RH) neutrinos. The third generation RH-neutrino, which is a Majorana fermion, serves as a viable DM candidate as an artifact of Z 2 -symmetry. The parameters like DM coupling with the SM-Higgs boson and scalar mixing are subject to the constraints from the LHC Higgs searches apart from other observational constraints on dark matter. However, annihilation of Majorana fermionic dark matter through a scalar resonance is velocity suppressed. But, the presence of a very narrow scalar resonance in the DM annihilation channel lifts the cross sections considerably via Breit-Wigner enhancement at later times and makes the model compatible with the recent analysis.

Class of Higgs-portal dark matter models
The basic feature of Higgs-portal model is that all the interactions of DM are mediated through Higgs(es) and the presence of scalar resonance plays a crucial role in determining the correct relic abundance. Here, we will discuss a class of Higgs-portal DM model in the light of the recent analysis [9] of the excess gamma-ray emission in the Fermi-bubble.

Scalar Singlet extension of SM
The scalar singlet extension of SM [30,31,32,33,34,35,36] is the most simplified Higgs-portal model to account for a WIMP candidate. The real singlet S ′ , stabilized by odd Z 2 -parity, acts as a viable DM candidate. It interacts only with the SM Higgs boson through the renormalizable interaction term present in the lagrangian, The mass of the DM after EWSB becomes, The coupling between DM and SM-Higgs, i.e, λ S ′ is constrained from the invisible decay width of Higgs boson when m S ′ m h /2, such that BR(h → SS) 0.20 [37]. Figure. 1 shows the contours of invisible branching ratio of the SM Higgs boson in λ S ′ − m DM plane. Region above red-dashed line is excluded as in the region the invisible branching ratio of the SM Higgs is more than 20%. Blue-solid, green-dotted and purple-dot-dashed contours show the allowed region if the invisible branching ratio is 25%, 30% and 35% respectively. As expected, the more invisible decay, the higher values of λ S ′ are allowed. For example, λ S ′ must be 8 × 10 −3 if 20% of the SM Higgs decays invisibly .

Relic Abundance
The relic abundance of DM can be formulated as [38], where thermal averaged value of DM annihilation cross-section times relative velocity. σv ann can be obtained using the well known formula [39], where prime denotes differentiation with respect to s ( √ s is the center of mass energy) and evaluated at s = 2m DM 2 . The function w(s) is same as defined in [40].
In order to fit the spectrum of the gamma-ray emission near the galactic center, one requires a WIMP of mass ∼ 31 − 40 GeV, which dominantly annihilates into final state bb through the s-channel exchange of the SM-Higgs boson . Also we choose, λ S ′ ≃ 0.007 as a benchmark value. We obtain that σv bb = (0.92−2.17)×10 −30 cm 3 /s , which cannot fit the observed gamma-ray signal. We also found that such a WIMP candidate cannot produce the required relic-abundance unless a scalar resonance is present i.e, when, m S ′ ≃ m h /2 ∼ 62 GeV. Also Ref. [35] has mentioned that for m S ′ < m h /2, the parameter space is severely restricted from both LHC and direct detection constraints. We conclude that the singlet scalar DM with mass around 31-40 GeV is incompatible with the dark matter interpretation for the gamma ray excess from GC.

Singlet fermionic dark matter model
The singlet fermionic dark matter (SFDM) model is a renormalizable extension of SM with a hidden sector containing a scalar singlet Φ s and a singlet Dirac fermion ψ [41,42]. Here, the singlet fermionic dark matter ψ, interacts with the SM sector via the singlet Φ s which mixes with the SM-Higgs doublet Φ. Therefore, this is also an example of Higgs-portal model. The lagrangian of the SFDM model is given as, where, After EWSB, the singlet field Φ s can be written as, The two scalar eigenstates are denoted as, where, H 2 is identified as the SM-Higgs boson and we consider the case when, m H2 > m H1 . Now, the mass of the DM is given by, m DM = m ψ + λ ψS x, with m ψ as a free parameter. In order to explain the observed gamma-ray excess in the low latitude, we consider the following set of parameters, m DM ∼ 31 GeV, m H1 ≃ 2m DM . The DM interaction strength depends on the parameter λ DM = λ ψS . Thus, the two parameters λ DM and scalar mixing cos α play crucial role in DM phenomenology. Here, the scalar mixing angle and DM-coupling are subject to various constraints like LHC bound on SM-Higgs boson, relic abundance of DM and upper bound on the DM-nucleon scattering cross section.

Constraints from LHC
Observation of SM-like Higgs boson at LHC by CMS [43] and ATLAS [44] collaboration will constrain this mixing angle severely. The signal strength or reduction factor of a particular channel can be defined as: where, σ Hi and BR Hi→xx are the production cross section of H i , and the branching ratio of H i → xx respectively.
Similarly, σ SM Hi and BR SM Hi→xx are the corresponding quantities of the SM-Higgs. Using eq. 10 one obtains, where, Γ SM Hi denotes the total decay width of the SM-Higgs boson and Γ Hid Hi is the invisible decay width (H i → 2 DM). The invisible decay width of the SM Higgs reads as Since, m DM < m H2 /2, we can constrain the DM coupling λ DM from the invisible decay width of SM-Higgs boson. Figure.2 shows the allowed range of λ DM with mass of DM for different invisible branching ratio of the SM-Higgs boson, assuming the width of the Higgs to SM fermions as 4.21 MeV. We observe that for m DM ∼ 30 GeV, if BR inv ≥ 20% (35%) then DM-coupling, λ DM should be less than 0.06 (0.075). Again, the signal strength (as defined in eqs. 10-11) depends on the scalar mixing angle. Constraining r 2 to be ≤ 0.9 (or 0.8), we obtain the allowed range of scalar mixing cos α as a function of m DM for a particular value of DM-coupling.

Constraints from relic density and direct detection
We obtain the relic abundance (using Eqn. 2) of the dark matter in agreement with WMAP-9 year result [45] and PLANCK [46], only near resonance where, m DM = m h1 /2 ∼ 31 GeV. Dominant contribution to relic density comes from final-state bb annihilation with cross-section σv ≃ 1.7 × 10 −26 cm 3 sec −1 , which is also in the desired range for explaining galactic center γ-ray excess. We observe that as we decrease λ DM , the annihilation crosssection is also decreased. But, if we approach very near the resonance region, i.e, m H1 − 2m DM ∼ O(10 −4 ), the annihilation cross-section can be enhanced significantly, which counter-balance the previous effect. However, if we are slightly away from resonance we need to have λ DM ∼ 10 −2 , to get correct relic.
The scattering cross-section (spin-independent) for the dark matter off a proton or neutron as, where, m r is the reduced mass defined as, 1/m r = 1/m DM + 1/m p,n and f p,n is the hadronic matrix element, given by The f-values are given in [47]. Here, a q is the effective coupling constant between the DM and the quark. An approximate form of a q /m q can be recast as : In order to be consistent with the latest exclusion limit on σ SI p as specified by LUX [48], Xenon 100 [49,50], we require σ SI p 10 −45 cm 2 . In Figure. 3, we show the contour of σ SI p = 10 −45 cm 2 (red-solid). It indicates that λ DM should be small enough (in the range of ∼ 10 −4 − 10 −5 ) to satisfy the required value of σ SI p . As argued before, very near resonance region, for λ DM ∼ 10 −4 , also gives correct relic density. The contour of relic abundance has been shown in Figure. 3 by the blue-dot-dashed line.

Minimal U (1) B−L gauge extension of SM
The minimal U (1) B−L extension of the SM [51,52,53,40] contains in addition to SM : a SM singlet S with B − L charge +2, three right-handed neutrinos N i R (i = 1, 2, 3) having B − L charge -1. The assignment of Z 2 -odd charge ensures the stability of N 3 R [54,55] which qualified as a viable DM candidate. Scalar Lagrangian of this model can be written as, where the potential term is, with Φ and S as the SM-scalar doublet and singlet fields, respectively. After spontaneous symmetry breaking (SSB) the singlet scalar field can be written as, with v B−L real and positive. The mass eigenstates (H 1 , H 2 ) are linear combinations of φ and φ ′ with mixing angle α. We identify H 2 as the SM-like Higgs boson with mass 125.5 GeV. We choose v B−L ≃ 4 TeV, in accordance with the constraint on the mass of Z ′ -boson [56]. The scalar mixing angle, α can be expressed as: The RH neutrinos interact with the singlet scalar field S through interaction term of the lagrangian: Here we define, λ DM as the coupling between DM candidate and the SM Higgs boson, which is effectively the Yukawa coupling of the N 3 R . Thus, the mass of dark matter is given by,

Constraints from LHC
As λ DM is suppressed by B − L symmetry breaking VEV, the invisible decay width remains very small (∼ 0.5%) for DM mass ∼ 30 − 40 GeV.
On the other hand, the decay width of the SM Higgs decays to light scalar boson is where g H 2 H 1 H 1 is defined in [40]. In order to have H 2 as a SM Higgs boson we require r 2 ≥ 0.9 (0.8) and correspondingly r 1 ≤ 0.1 (0.2). We have obtained that r 2 being ≥ 0.9 (0.8) restricts the choice of scalar mixing such that, cos α ≥ 0.96 (0.94) for m DM ∼ 31 GeV.

Velocity dependent cross-section and Breit-Wigner enhancement
In general the annihilation of Majorana fermionic DM into SM-fermion pairs through a scalar mediator is velocity suppressed. In that case the thermally averaged annihilation cross-section can be written as, σv = a + bv 2 , where a, b are model dependent variables.
The term a comes from s-channel s-wave process, where as, b has contributions from both s-wave and p-wave. The averaged velocity v can be expressed as, v ∼ 3/x. Because of p-wave suppression, σv at the time of freezeout (x f ∼ 20) is different than that at the galactic halo (x ∼ 10 6 ). However, σv at the galactic halo can be substantially enhanced using the Breit-Wigner mechanism [58,59], where the DM annihilates through a narrow schannel resonance.
The leading annihilation channels of DM are, The s-channel resonant annihilation cross-section into final state bb (dominant) is given as, where, Γ H1 is the total decay width of H 1 .
Here, we introduce two parameters δ and γ as, Clearly, δ < 0 and δ > 0 represents the physical and unphysical pole respectively. Adopting the single-integral formula for thermally averaged cross-section, we obtain, where, K 1 (x) and K 2 (x) are the modified Bessel's function of second kind and g i is the internal degrees of freedom of dark matter particle. We again redefine s as, s = 4m 2 DM (1 + y) where, y ∝ v 2 . Eq.20 can be recast in terms of δ, γ and y as, where, y ef f ∼ max[4/x, 2|δ|] for δ < 0 and y ef f ∼ 4/x for δ > 0 case. If δ and γ are much smaller than unity, σv scales as v −4 in the limit v 2 ≫ max[γ, δ]. At smaller velocity, the thermally averaged annihilation cross-section becomes proportional to v −2 and approach towards a constant value when v 2 ≪ max[γ, δ]. We obtain the relic abundance using eqs. (2,20). Fig.4 shows the relic abundance (red curve) as a function of DM mass. The resultant relic abundance is found to be consistent with the reported value of WMAP-9 [45] (shown by the black solid line) and PLANCK experiment [46], only near resonance when, m DM ∼ (1/2) m H1 .
We have also achieved the required σv bb ∼ 1.881 × 10 − 26 cm 3 /s at the galactic halo through the Breit-Wigner enhancement given the value of parameters 1 δ ≃ −10 −3 and γ ≃ 10 −5 . Note that, the same set of parameter values have been used to compute the relic abundance.

Constraints from direct detection searches
The spin-independent scattering cross-section of DM off nucleon is obtained using eq.13. In Fig.4 the yellow region above is excluded by LUX(2013) [48]. We observe that the resultant spin-independent scattering cross-section (blue curve) lies well below the LUX exclusion limit. However, the projected sensitivity of Xenon1T experiment [57] (green-dashed line) might constrain the scenario of m DM = 31 − 40 GeV in this model.

Summary and Conclusion
The excess of γ-ray emission in the low latitude region near the galactic center can be explained by annihilation of DM (in the mass range ∼ 31 − 40 GeV) into bb, with cross-section of the order of the weak cross-section (i.e ∼ 10 −26 cm 3 sec −1 ). In this context, we have analysed a class of Higgs-portal DM models and constrain the parameter space of these models. We found that the real singlet scalar DM model is incompatible with the recent analysis. However, the singlet fermionic dark matter model can account for this phenomena apart from satisfying relic abundance criterion. Besides this, the SI-scattering crosssection can be well below the exclusion limit from LUX, Xenon 100, provided λ DM lies below ∼ 10 −4 . Also, RHneutrino DM in the minimal U (1) B−L model is well-suited for explaining the galactic-center gamma-ray excess along with satisfying other DM and collider constraints. The relic abundance is found to be consistent with the recent WMAP9 and PLANCK data only near scalar resonances, i.e, m DM ≃ m H1 /2. Here, we obtain the required σv for explaining this reported excess at the galactic center through Breit-Wigner enhancement mechanism. Although, future experiment like Xenon 1T can further restrict the parameter space of minimal U (1) B−L model.
In passing by, we would like to mention that the anti-proton data from indirect detection experiments like PAMELA [60,61], AMS-02 [62] have constrained the annihilation cross-section into hadronic (mostly bb) final states in a model independently way. But, the present exclusion limit on σv bb lies much above the reported value in Ref. [9] for DM mass in the range 31-40 GeV. However, the bound on σv bb from the projected anti-proton data of AMS-02 (see Fig.1 of Ref. [22]) can be an important discriminator of dark matter models.