Fermionic dark matter with pseudo-scalar Yukawa interaction

We consider a renormalizable extension of the standard model whose fermionic dark matter (DM) candidate interacts with a real singlet pseudo-scalar via a pseudo-scalar Yukawa term while we assume that the full Lagrangian is CP-conserved in the classical level. When the pseudo-scalar boson develops a non-zero vacuum expectation value, spontaneous CP-violation occurs and this provides a CP-violated interaction of the dark sector with the SM particles through mixing between the Higgs-like boson and the SM-like Higgs boson. This scenario suggests a minimal number of free parameters. Focusing mainly on the indirect detection observables, we calculate the dark matter annihilation cross section and then compute the DM relic density in the range up to $m_{\text{DM}} = 300$ GeV. We then find viable regions in the parameter space constrained by the observed DM relic abundance as well as invisible Higgs decay width in the light of 125 GeV Higgs discovery at the LHC. We find that within the constrained region of the parameter space, there exists a model with dark matter mass $m_{\text{DM}} \sim 38$ GeV annihilating predominantly into $b$ quarks, which can explain the Fermi-LAT galactic gamma-ray excess.


Introduction
The search for deciphering the identity of the dark matter (DM) has been under intense scrutiny since long ago, see reviews in [1,2]. There is a strong confidence that about 25 per cent of the matter content of the Universe is made of a new very long lived particle or particles, the so called dark matter [3]. The search for DM signals divides into direct detection and indirect detection methods. The former approach relies on the DM scattering with ordinary matter while the latter avenue depends on the dark matter annihilation processes.
In case there is a sensible interaction of DM with ordinary matter, in direct detection approach, the experiment is set up so as to measure the recoil energy of the nuclei induced by DM scattering off nucleons [4,5]. In this regards, the first results of LUX dark matter experiment [6] although finds no evidence for DM interaction it provides us with impressive bounds on the DM-nucleon scattering cross section in a wide range of DM mass. Along the same line, dark matter results from XENO100 experiment [7] again finds low spinindependent DM-nucleon scattering rate.
On the other hand, within the indirect detection method, lies the accurate measurement of the dark matter density. The Planck experiment recently obtained the cold dark matter (CDM) density based on high precision measurement of the acoustic peaks in the cosmic microwave background [3] Ω CDM h 2 = 0.1196 ± 0.0031 .
In compatible with the Planck result, WMAP temperature and polarization data including low multipoles [8] provided us with the cold dark matter density as Ω CDM h 2 = 0.1138 ± 0.0045 . (68% C.L.) .
In addition, as indirect detection, the dark matter pair annihilation can produce potentially measurable anomalous gamma-rays, cosmic rays and also neutrinos. Gamma-rays are particularly interesting and may be observable by the Fermi Gamma-Ray Space Telescope [9]. A promising place on the sky to look for the gamma-rays are the central region of the Milky Way which contains a high density of DM and it is relatively close to us, see for instance [10,11,12]. Besides the possibility of direct and indirect detection of DM, it is plausible to have DM production at particle colliders [13,14,15].
Motivated by the observational developments discussed above, the question now is about the nature of the dark matter. So, let us take a look at the theoretical side. As it is well-known the Standard Model (SM) of particle physics is lacking a proper candidate for DM. A copious number of theories beyond the SM exist which propose some kind of dark particle candidates generically called weakly interacting massive particles (WIMPs) in order to explain the observed relic density.
The most vastly investigated DM candidate as a WIMP is the lightest supersymmetric particle (LSP) which is a stable particle in supersymmetric (SUSY) models with conserved R-parity, see [16] and references therein. However, being well motivated theoretically, the presence of large number of free parameters have hindered our predictivity within SUSY models. On the contrary, in models with universal extra dimensions (UED), the size of the extra dimension is the only parameter of the model which dominates the physics [17]. The lightest Kaluza-Klein particle whose mass is the inverse of the compactification radius, is the DM candidate which is stable due to the conserved Kaluza-Klein parity. Relying on a new symmetry at the TeV scale dubbed T-parity, in the little higgs model introduced in [18] to cure the little hierarchy problem, emerges the lightest T-odd new particle which may serve as a WIMP. The minimal extension of the SM is the addition of a gauge singlet real scalar field [19] or a gauge singlet complex scalar field [20] to the SM with Z 2 -parity imposed on the new fields to ensure their stability as DM candidates. A minimal dark matter model is also introduced in [21] in which the new fermionic or bosonic field has only gauge interaction. Moreover, a minimal extension of the SM is constructed by the inclusion of a hidden sector incorporating a gauge singlet scalar field and a gauge singlet fermionic field [22]. The fermionic field interacts with the SM fields only through the singlet scalar field while the latter has triplet and quartic scalar interactions with the SM higgs doublet. Since the new fermion is assumed to be charged under a global unbroken U(1) symmetry while the SM fields are neutral under the same symmetry, there is no direct interaction between the singlet fermion and SM particles. In this model singlet fermion is the DM candidate.
Recently, motivated by the null result from direct detection of DM, it has been thought that the WIMP dark matter may interact in such a way as to leave a trace in one experiment but not necessarily in another one. One example in this regards, is a model put forward in [23] suggesting a new type of dark matter dubbed, coy dark matter. In the model it is shown that the proposed dark matter can explain the observed extended gamma-ray flux originating from the galactic center without expecting any signals from direct detection or elsewhere.
In this article we propose a minimal extension of the SM in which a fermionic dark matter interacts with a pseudo-scalar mediator via a pseudo-scalar Yukawa interaction. The DM candidate is charged under a new global U(1) symmetry but all the SM fields are neutral. We find that there is a viable region in the parameter space in which the observed relic density as well as the extended galactic gamma-ray excess can be explained simultaneously but given the current experimental bounds no signal from direct detection is expected.
The structure of the paper is as follows. In Section. 2 a fermionic dark matter model is constructed and limits on the couplings are discussed. Constrains on the model parameters from the invisible Higgs decay is given in Section. 3. Section. 4 is devoted to the relic density calculation and finding viable regions in the parameter space. A formula is derived for the DM-nucleus elastic scattering in Section. 5. Gamma-ray emission from DM self-annihilation is calculated within the model and comparison with data is made in Section. 6. We finish up with a conclusion in Section. 7.

A theory model for dark matter
We consider a renormalizable extension of the Standard Model (SM) Lagrangian with one Dirac fermion field χ and one real pseudo-scalar field φ. The new fields are SM gauge singlets and the fermionic field is charged under a global U(1) DM symmetry. Since all the SM fields are singlet under the global symmetry, the SM particles interact with the dark sector only via the Higgs portal.
The model Lagrangian therefore consists of the following parts: where L Dark introduces the singlet Dirac field which does not undergo any mixing with the SM fermions due to the presumed global U(1) DM symmetry of the singlet fermion with and L φ is a renormalizable Lagrangian for the pseudo-scalar boson as The interaction Lagrangian itself, L int , consists of a pseudo-scalar Yukawa term and an interaction term incorporating SM-higgs doublet and singlet pseudo-scalar, It is seen that the Lagrangian L is invariant under the parity transformation φ(t, x) → −φ(t, − x) and χ(t, x) → γ 0 χ(t, − x) in which the scalar field φ carries odd parity. The Higgs field, H, is a SM SU(2) L scalar doublet. On the other hand, the SM Higgs potential introduces the quartic self coupling of the Higgs field as The Higgs field develops a non-zero vacuum expectation value (vev) which gives rise to the electroweak spontaneous symmetry breaking. The fluctuation about the vev is described by the scalar fieldh such that where v H = 246 GeV. In addition we assume in this model that the pseudo-scalar singlet also acquires a non-zero vev as This consequently leads to the spontaneously breaking of parity. The global U(1) DM symmetry is conserved even after the spontaneous symmetry breaking and thus, this ensures the stability of the fermionic singlet which is a necessary condition for a proper dark matter candidate.
From the minimization condition of the potential, i.e., we can express two parameters of the model in terms of the vevs and quartic coupling by the relations We now turn back to the Lagrangian and pick out entries of the mass matrix associated with the SM-higgs field,h, and the scalar field S, and in which to obtain the relations above we have used Eq. (9). We then indicate the mass eigenstates h and ρ as following by defining the mass mixing angle θ, where, The two neutral Higgs-like scalars h and ρ given as admixtures of SM higgsh and scalar S, have reduced couplings to the SM particles by a factor sin θ or cos θ. The corresponding mass eigenvalues are given by where the upper sign (lower sign) corresponds to m h (m ρ ). In the following we assume that h is the eigenstate of the SM higgs with m h = 125 GeV and ρ corresponds to the eigenstate of the singlet scalar. It is possible to obtain the quartic couplings in terms of higgs masses, vevs and mixing angle The stability of the potential puts constrains on the quartic couplings as λ > 0, λ H > 0 and λλ H > 6λ 2 1 . One more restriction on the couplings comes from the perturbativity requirement of the model which demands |λ i | < 4π. The set of independent free parameters in the model are considered to be m χ , m ρ , g, θ and v φ . We use the relations given in Eq. (16) to display in Fig. 1 the dependency of the couplings on the mixing angle, m ρ and v φ . Two different values are chosen for the scalar boson mass, m ρ = 400 and 500 GeV while for both cases we take v H = 246 GeV. Comparison between our results in the left and right panel are made for two different values of v φ , namely 600 GeV and 1000 GeV. We find out that both conditions, λλ H > 6λ 2 1 and |λ i | < 4π are well fulfilled for the above parameter set.

Constrain from invisible Higgs decay
Within the Standard Model the total decay width of the Higgs boson is Γ SM Higgs ≈ 4 MeV for a Higgs mass of 125 GeV. For light dark matter mass such that m χ < m h 2 and additionally having the condition m ρ > m h /2, there is only one new channel open for the Higgs decay which is kinematically allowed, We therefore expect the modification of the total decay width of the Higgs boson as The invisible branching ratio of the Higgs decay for various channels are investigated recently in the light of 125 GeV Higgs discovery at the LHC in [24,25,26]. In [26] a conservative experimental upper limit for the invisible branching fraction of the Higgs boson is achieved, B inv 0.35. Thus we can obtain from Eq. (18) an upper limit for the invisible Higgs decay width, Given the decay width of the Higgs boson into two dark matter particles in Eq. (17) we derive an upper limit for the product |g tan θ| as In case we consider light scalar boson with m ρ < m h /2, there is one more possible decay channel for SM-higgs decay with in which Our numerical examination shows that the effect of the decay h → ρρ on the upper bound of |g tan θ| is essentially negligible.

Dark Matter Relic Density
The problem of dark matter is an interesting instance of freeze-out in the early Universe. It is in fact the question of what happens when dark particles (χ) go out of equilibrium. The pair annihilation of dark particles into pairs of SM particles (χχ →XX) and the inverse processes play a central role in our treatment based on the Boltzmann transport equation in an expanding Universe. The reason relies on the fact that only the annihilation and production reactions can change the number of dark particles in the comoving volume.
In thermal equilibrium, annihilation of dark particles take place with the same rate as their creation processes occur. However, an expanding Universe cools down and reaches a point (T≪ m DM ) in which the dark particle interactions freeze out. In fact at the freezeout temperature the annihilation rate of dark particles drops below the Hubble expansion rate. On the other hand, at temperature T≪ m DM , dark particle production reactions are Boltzmann suppressed since only a small portion ofXX have enough kinetic energy to produce aχχ pair. After freeze-out, the number density, n χ does not change with time asymptotically. We can thus determine the present value of the relic density by solving numerically the evolution equation.
Taking into account the considerations sketched above, the time evolution of the number density of the singlet dark matter in departure from equilibrium is governed by the Boltzmann equation as The second term in the left-hand side is the dilution due to the expanding Universe, where H is the Hubble parameter. In the expression σ ann v rel thermal averaging is understood because particles annihilate with random thermal velocities and directions. The thermal average of the annihilation cross section times the relative velocity at temperature T is obtained by integration over the center of mass energy √ s as in which K 1,2 are modified Bessel functions of first and second rank. The number of possible annihilation channels at the limit of zero velocity depends on the mass of the dark particle. In the proposed model, at tree level in perturbation theory, the annihilation processes occur with exchanging a SM-higgs field, h or a ρ boson field. We consider a range of mass for DM where a pair of dark particles may annihilate through s-channel into a pair of SM fermions (quarks and leptons) and a pair of gauge bosons (W + W − , ZZ) and also through s-, t-and u-channel into hh or ρρ. We provide the necessary cross section formulas in the Appendix. We implement our model into the program CalcHEP [27] and calculate the annihilation cross sections and as a cross check on our implementation we find agreement with the results given in the Appendix. In the present article we analyze the relic density of DM by employing the program MicrOMEGAs [28] which solves the Boltzmann equation numerically. MicrOMEGAs in turn uses the program CalcHEP to calculate all the relevant cross sections.

Numerical Analysis: A first look
We investigate here the viable region in the parameter space of the proposed model concerning the indirect detection of the fermionic dark matter along with implications from invisible Higgs decay at LHC. Our analysis is performed with values for the quartic couplings which meet the constrains from vacuum stability and perturbativity condition.
As a first numerical look we calculate the relic abundance as a function of DM mass between 30 GeV and 300 GeV for two different values of ρ boson mass, 400 GeV and 500 GeV. The constrain from invisible higgs decay is not imposed here. The results depicted in Fig. 2 for three choices of g sin θ = 0.02, 0.08 and 0.48 show some correct characteristic features. One can see that the relic density drops fast for all set of parameters with m ρ = 400 GeV, at DM mass close to 62 GeV and 200 GeV corresponding to the exchange of a SMhiggs and a singlet scalar, respectively. This sounds reasonable because the annihilation cross section get enhanced at resonance regions and since Ωh 2 ∝ (σv) −1 , a dip in the relic density should appear. Moreover, we expect some important effects on the relic density when m χ ≈ m W and m χ ≈ m Z since at these masses two new channels now open up for DM to annihilate into. These effects show up in all the plots in Fig. 2 at threshold values where the annihilation cross section increases and therefore make the relic density to decrease. One more additional study is done on the impact of the quartic couplings on the relic abundance. This can be done by adopting two distinct values for v φ , namely 600 GeV and 1000 GeV. We know already that only at large enough DM mass where two new channelsχχ → hh andχχ → ρρ open, the size of v φ becomes important as the relations in Eq. (16) imply. In Fig. 2 the results when compared between left panel and right panel indicate that for smaller value of the quartic couplings (corresponding to v φ = 1000 GeV) the relic density grows more significantly at DM mass close to 300 GeV when m ρ = 400 GeV as anticipated.

Viable parameter space
Taking into account the correct relic abundance of DM, we scan the parameter space over two ranges of the DM mass, namely m χ < m h /2 and m h /2 < m χ < 200 GeV. We have chosen this way because the invisible higgs decay put constrain on g sin θ for m χ < m h /2 but not on larger DM masses. We first report on our results concerning the lower range mass, m χ < m h /2 in Fig. (3) where we take for the mixing angle such that sin θ = 0.0026 and generate random values for g with 0 < g < 10. The dominant DM annihilation channels are into final statesbb and τ + τ − and in case we consider the region m ρ m χ , DM annihilation into ρρ will take over at smaller values for g sin θ. As one important outcome, it is apparent from the figure that there is no allowed mass value for DM in the parameter space when m ρ is far larger than m χ . This can be explained in terms of annihilation cross sections for two reactions χχ →f f andχχ → ρρ. When the latter reaction becomes kinematically closed, in order for the total cross section to compensate the lack, it should pick up large values of g sin θ which may exceed allowed values. Now, we look at the higher range for the DM mass, 60 GeV < m χ < 200 GeV. Our results are provided by Fig. (4) for two distinct interval for m ρ , namely, 100 GeV < m ρ < 160 GeV in the left panel and 250 GeV < m ρ < 550 GeV in the right panel. For both intervals we set sin θ = 0.1 and randomly generate 0 < g < 10 and then single out the allowed region in the plane (g sin θ, m χ ). We can notice from the figures that at ρ boson masses less than 160 GeV, the coupling g is allowed to pick out values larger than unity up  to DM mass of about 150 GeV. For larger ρ boson masses (250 GeV < m ρ <550 GeV) the coupling g is allowed to exceed unity only at m χ 80 GeV and m χ 170 GeV. Finally, in Fig. 5 we scan the viable region in (m χ , m ρ ) space for reasonable values for the Yukawa coupling g, 0 < g < 1 and a choice for the mixing angle such that sin θ = 0.1.

Direct detection
The WIMP-nucleus elastic scattering cross section depends on the fundamental interaction of the WIMP-quark. The quark level interaction in our model occurs via t-channel by the Higgs exchange or the singlet scalar exchange, where at low momentum transfer the interaction is given by an effective four-fermi contact Lagrangian as with We can now define the tree-level matrix element describing the scattering between the fermionic dark matter, χ and the individual nucleons N (either proton p or neutron n) We cannot evaluate the nucleonic matrix element analytically because it is not known yet how to connect the quark degrees of freedoms into the nucleonic ones through the nonpetrubative mechanism of confinement. However, it is conventionally assumed that in the limit of vanishing momentum transfer, the nucleonic matrix element with the quark current is proportional to that with nucleon current [29,30,31,32] q where The proportionality constants f N T q and f N T g incorporate the non-perturbative physics of strong interaction at low energy and m N represents the nucleon mass. To proceed we shall follow closely the discussions in [33,34,35]. We can now construct the matrix element for the dark matter-nucleus scattering in the non-relativistic limit as where χ f | q. S χ |χ i indicates the DM-spin operator in which q is the momentum transferred to the nucleus and ξ s N is the two-component spinor corresponding to the fermion N with spin s. The extra factor mT mN is inserted due to the different normalization between the target nucleus with mass m T and the nucleon with mass m N . We therefore obtain the corresponding squared matrix element averaged over initial spin states and summed over the final states as The differential cross section for DM-nucleus scattering in the non-relativistic limit reads We can then calculate the total cross section as where µ χT is the reduced mass of the DM-nucleus system. We arrive finally at our expression for the spin-independent (SI) total cross section for DM-nucleus scattering where the DM-nucleus relative velocity v ∼ O(10 −3 ). We note that the DM bilinear matrix element results in a velocity suppression in the cross section of WIMP-nucleus in the norelativistic limit. More precisely, the cross section obtained above is suppressed by a factor of v 2 /m 2 χ in comparison with the one where the DM matrix element is scalar in structure. Numerically speaking, this corresponds to a suppression of order ∼ 10 −6 and even less in the cross section. Our numerical probe over the full parameter space shows that DM scattering rate in our model is far below the minimum bands imposed on the scattering rate by the current results from LUX and XENON100 so that the dark matter particle can evade direct detection. Thus, we expect no constraints on the parameter space from direct detection of DM.
6 Gamma-ray emission from DM self-annihilation The evidence for the gamma-ray emission from a small region centered on the Galactic Center originating from annihilating dark matter was pointed out firstly in [10] based on data from Fermi Gamma-Ray Space Telescope. Further studies with confirmation on this finding can be found in [11,12,36,37,38,39]. Other sources, in particular, unresolved millisecond pulsars are also considered to explain the observed anomalous gamma emission from the Inner Galaxy [36,37,40]. However, recent studies relying on an estimated population of millisecond pulsars in the Inner Galaxy suggest that millisecond pulsars make up only a small portion (< 5% ) of the total observed gamma excess, see discussions in [41,42]. In the following we assume that the observed gamma-rays are produced as a result of DM annihilation in the Inner Galaxy. We shall then discuss in this section the gamma-ray emission from dark matter self-annihilation in the proposed fermionic model. Here, we restrict our attention to DM mass below W ± and Z threshold, thus dark matter annihilating take place only with SM fermions in the final states (χχ →f f ) via SM-higgs exchange or  Figure 6: The flux of gamma-ray excess as data points are shown at 5 • from the Galactic Center [43]. Gamma-ray spectra from dark matter annihilation into fermion pair at m DM = 38 GeV are compared for two different values of the inner slope, γ.
singlet scalar exchange. The flux of gamma-rays at Earth produced by annihilating dark matter located in the central region of the Milky Way is where the distance from the annihilation point to the earth denoted by r and r ′ is given in terms of the angle between the line of sight and the center of the galaxy as r ′ = r 2 ⊙ + r 2 − 2r ⊙ r cos θ with r ⊙ = 8.5 kpc. The photon flux depends upon two dynamical quantities, the annihilation cross section times the relative velocity, σv ann and the photon energy spectrum generated per self-annihilation into a fermion pair, dN γ /dE γ . It is assumed that the dark matter distribution is approximately spherical and thus we can give the dark matter density as a function of the distance from the Galactic Center, r. Throughout our study we use the DM density characterized as where γ = 1 is the standard NFW value for the inner slope. The scale radius chosen as r c = 20 kpc and ρ ⊙ = 0.3 GeV/cm 3 is the local dark matter density at 8.5 kpc from the Galactic Center. We employ the package MicrOMEGAs to calculate the gamma-ray spectrum. Since the astrophysical parameters involved in our computation for the gammaray flux are given with uncertainties we do not limit ourself to the region in the parameter space which precisely meet the constrains from relic abundance and invisible higgs decay width. Our results for the gamma-ray excess is presented in Fig. (6) for m DM = 38 GeV as an example, with two values for the inner slope, γ = 1.18 and 1.20. The singlet scalar mass is chosen as m ρ = 76 GeV (this is the resonance mass and enhance the cross section significantly). It turns out that the dominant annihilation channels are intobb quark pair (∼ 94%) and τ + τ − (∼ 6%) with the total annihilation cross section σv ann ∼ 1.7 × 10 −26 cm 3 s −1 consistent with the value demanded by the thermal relic. We compare our results with the data for the extended gamma-ray excess extracted from [43]. As it is evident from the plots in Fig. (6), the gamma-ray flux with γ = 1.18 gives a better fit to the data.

Conclusions
The main aim behind the present work has been to study a model whose DM candidate is coy in the sense that DM signals are expected for instance in indirect detection experiments but not in the direct detection experiments. We have proposed an extension of the SM in which the fermionic DM candidate interacts with a hidden sector via a pseudo-scalar Yukawa term. We have found the viable region in the parameter space given the constraints from the observed relic density and invisible Higgs decay width. Moreover, the DM-nucleus scattering cross section is calculated within the model and it turned out the cross section is velocity suppressed so that no direct detection of the DM can take place given the current sensitivity of the present underground experiments. However, we demonstrated that it is possible to find a region in the parameter space that both the observed relic abundance and the extended gamma-ray excess from the Galactic Center can be explained.

Acknowledgments
I would like to thank Hossein Ghorbani for useful discussions.

Appendix: Annihilation cross sections
We obtain the annihilation cross section of a DM pair into a pair of SM fermions as σ ann v rel (χχ →f f ) = g 2 sin 2 2θ 64π where N c is the number of color charge. The dominant contributions belong to the heavier final states bb and tt. The total cross section into a pair of gauge bosons in unitary gauge is given by σ ann v rel (χχ → W + W − , ZZ) = g 2 sin 2 2θ 64π And finally we get the following result for the annihilation scattering into two higgs bosons as with a = sin 3 θλv φ + 6 cos 3 θλ H v H + 6 sin 2 θ cos θλ 1 v H + 6 cos 2 θ sin θλ 1 v φ b = cos θ sin 2 θλv φ − 6 cos 2 θ sin θλ H v H − 6 sin 3 θλ 1 v H + 4 sin θλ 1 v H −6 cos θ sin 2 θλ 1 v φ + 2 cos θλ 1 v φ , and annihilation cross section into two ρ boson is with c = λv φ cos 3 θ − 6λ H v H sin 3 θ − 6λ 1 v H cos 2 θ sin θ + 6λ 1 v φ sin 2 θ cos θ d = λv φ sin θ cos 2 θ + 6λ H v H sin 2 θ cos θ − 6λ 1 v H sin 2 θ cos θ + 2λ 1 v H cos θ