Analyzing of singlet fermionic dark matter via the updated direct detection data

We revisit the parameter space of singlet fermionic cold dark matter model in order to determine the role of the mixing angle between the standard model Higgs and new singlet one. Furthermore, we restudy the direct detection constraints with the updated and new experimental data. As an important conclusion, this model is completely excluded by recent XENON100, PandaX II and LUX data.


II. THE MODEL
The most minimal extension of the SM, including a CDM candidate, is achieved by adding a gauge singlet fermion. We can consider the singlet fermion to play the dark matter role (SFCDM) provided that it has a very weak interaction with the SM particles because it must respect the relic abundance condition. To accommodate this in a renormalizable manner, a singlet Higgs S, in addition to the usual Higgs doublet, is needed as mediator between SFCDM and the SM particles [5,7]. The Lagrangian for the SFCDM model can be decomposed as follows: where L SM is the SM Lagrangian and L hid denotes the hidden sector Lagrangian, Here, L ψ and L S are the free Lagrangians of SFCDM, and the singlet Higgs, The last term in Eq. (2) is due to the interaction between the SFCDM and singlet Higgs with coupling constant g s . In Eq. (1), L int is related to the interaction between the new singlet Higgs and the SM doublet one We have H = 1 with v 0 and x 0 being the vacuum expectation values (VEV) of the SM Higgs and singlet Higgs, respectively. We define the fields h and s as the fluctuation around the VEVs of them. Therefore, after symmetry breaking we have and We can obtain the mass eigenstates by diagonalizing the mass matrix as follows: where θ is a mixing angle which depends on the parameters of the Lagrangian (1). One naturally expects that | cos θ| > 1 2 , so that h 1 is the SM Higgs-like scalar, while h 2 is the singlet-like one. The singlet fermion has mass m ψ = m ψ0 + g S x 0 , which is an independent parameter in the model. The VEV of our singlet Higgs, x 0 , is completely determined by minimization of the total potential (including SM and singlet Higgs potentials) as follows: where λ 0 is the self-interaction coupling constant of the SM Higgs. There are seven independent parameters, in addition to the SM ones, in this model: {m ψ , g S , m 0 , λ 1 , λ 2 , λ 3 , λ 4 }. After spontaneous symmetry breaking we encounter a new set of parameters, m ψ , g S , second Higgs mass m h2 , λ 1 , λ 2 , λ 3 , λ 4 and the mixing angle between Higgs bosons θ, which is not an independent parameter.

A. The cross section
In the SFCDM model, at tree level, pairs of singlet fermions can annihilate into SM particles, including pairs of massive fermions and gauge bosons, and also two and three Higgs bosons. We have listed the corresponding Feynman diagrams in Fig. 1. These diagrams are at leading order, so we should respect the perturbation criteria in our calculations.
We have calculated the corresponding cross section of this annihilation process. According to Fig. 1, while the annihilations into the fermions and gauge bosons proceed only through the s-channel, the annihilation into Higgs bosons occurs via the s-, t-and u-channels. The total annihilation cross section times the relative velocity v can be written as follows: where the σv SM is given by where λ f is 3 (1) for quarks (leptons), Γ hj refers to the decay widths of h j and d j = (s − m 2 hj ) 2 + m 2 hj Γ 2 hj (j = 1, 2). Here, we have used the abbreviations s 1 ≡ sin θ and s 2 ≡ cos θ. The last two terms in Eq. (9) are the annihilation cross sections into two and three Higgs bosons, respectively. To obtain these cross sections we should derive g jkl and g jklm corresponding to the vertex factors of them. For j = k we get Note that g jkl and g jklm are symmetric under permutation of their subscripts and j, k, l, m = 1, 2. Therefore, one can derive the annihilation cross section into two Higgs bosons as follows: where s .
Although the annihilation cross section into three Higgs bosons is suppressed due to its narrow phase space integral, to have a complete and more precise calculation we take it into account. For this term we have III. COMPUTATIONS

A. The relic density
The relic density Ω ψ h 2 , defined as the ratio of the present density of particles to the critical density, is written as follows: where σv ann is the thermally averaged annihilation cross section times the relative velocity [17]: with K 1,2 (x) being the modified Bessel functions. Here x F = m ψ /T F is the inverse freeze-out temperature, which can be determined by the following iterative equation: where g * is the effective degrees of freedom for the relativistic quantities in equilibrium [18] and M Pl = 1. is the Planck mass.
To study the allowed parameter space consistent with the relic abundance constraint obtained by WMAP observations [19], the SM Higgs boson mass is fixed to 125 GeV according to the 2012 CMS and ATLAS results [20,21]  and the other Higgs mass to 750 GeV. 2 Although the variations of the λ's have no significant impact [7], we let them vary as far as perturbation theory is correct. To find the couplings g s which satisfy the relic density condition, we first investigate about 25000 sample models randomly in the whole parameter space. Namely, in addition to λ's, we take θ and m ψ to be free. In the other two investigations, each of which concerned whit 10000 sample models, we set θ = 0.1 and θ = 0.01. We collect all of these three data sets in Fig. 2. Using our first data set, we also illustrate the role of the mixing angle θ by the contour plot of Fig. 3. This figure shows that for θ < 0.1 there is only a mass region between about 700-1000 GeV as well as a narrow one about 350 GeV, where we get g s < 1 and therefore our perturbative analysis works self-consistently. For the other regions, although obtaining g s from the relic density is not consistent with perturbation theory, we necessarily conclude that g s > 1.

B. Direct detection
In this subsection, we investigate the consistency of SFCDM with the direct detection bounds. We use the following effective Lagrangian at the hadronic level to describe the scattering of SFCDM from a nucleon: where f p and f n are the effective couplings of DM to protons and neutrons, respectively, and they are given by: with the matrix elements m p,n f (p,n) T q ≡ p,n|m qq q|p,n for q = u, d, s and f The numerical values of the hadronic matrix elements are given in [25]. Here, α q is an effective coupling constant between SFCDM and quark q, in the following effective Lagrangian: Since the scattering SFCDM and quarks proceeds through t-channel by intermediating a Higgs boson, α q can be derived: Consequently, the elastic spin-independent scattering cross section off a single nucleon becomes where m r = 1 m ψ + 1 Using g s as obtained in the previous subsection for θ = 0.1 and 0.01 we plot the direct detection cross section in Fig. 4. We also compare our result with the new updated experimental data in this figure. The data which we have used here are from the XENON100 [22], PandaX II [23], LUX [24], PICO-60 [26] and Darkside-50 [27] Collaborations.

IV. DISCUSSION AND CONCLUSIONS
The most minimal and renormalizable extension of the SM, which introduces a singlet fermion as CDM candidate, is the SFCDM model. Namely, one adds a singlet fermion as CDM and a scalar as mediator to the SM content. A comprehensive analysis of this model has been given in [7]. However, the mixing angle between the SM Higgs and singlet scalar is constrained to be less than 0.1 [11]. Therefore, we have restudied the relevant parameter space to determine the role of the mixing angle. The SM Higgs boson mass is fixed to 125 GeV according to the 2012 ATLAS [20] and CMS [21] reports and the other Higgs mass to 750 GeV as we have explained in the main body of paper. In order to find the coupling g s which satisfies the relic density condition, we first investigate about 25000 sample models randomly in the whole parameter space. In fact, in addition to λ, we take θ and m ψ to be free. The data of this study is denoted by blue points in Fig. 2. We see that g s tends to a unique value for m ψ larger than about 750 GeV. Two other investigations with fixed θ = 0.1 and θ = 0.01, each of which with 10000 sample models, have been denoted in Fig. 2 by orange and green points, respectively. For more clarification, we illustrate the behavior of g s in terms of m ψ and θ through Fig. 3. We see that there exist limited regions (300 GeV< m ψ < 400 GeV and 700 GeV< m ψ < 1000 GeV) in which θ < 0.1 and g s < 1. Furthermore, after deriving the spin-independent cross section of the elastic scattering of SFCDM from nucleon, we use the g s obtained from relic abundance condition to calculate and plot this cross section. It is illustrated through Fig. 4 in terms of m ψ for two various choices of θ. We have compared our results with different experimental data. According to this figure, the entire parameter space is excluded by XENON100 [22], LUX [24] and PandaX II [23]. For more comparison, we have also shown the recent experiments PICO-60 [26] and DarkSide-50 [27] in this figure.