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

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## Abstract

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 a 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.

## 1 Introduction

There exist several pieces of evidence that indicate the highest fraction of matter in the universe is composed of unknown particles called dark matter (DM) (see [1, 2]). The baryonic matter contributes only less than 5% of the universe content. While the standard model (SM) is very successful in the experimental tests, it does not predict any appropriate candidate for DM. Hence, many authors have been convinced that we need a model beyond the SM. The evidence hints that the DM candidates should be mostly stable, non-baryonic, massive, non-relativistic and have insignificant or very weak interactions with other particles (see [3] for a discussion of the conditions of DM candidates and their properties). The DM particles with these properties are often called cold DM (CDM) or weakly interacting massive particles (WIMPs). Since no signal, predicted by any theory beyond SM, has been confirmed experimentally, it is reasonable to consider the most minimal extension of the SM to explain DM. Singlet fermionic CDM (SFCDM) is a minimal extension of the SM which proposes a singlet fermion as an appropriate candidate for CDM [4, 5, 6, 7].

One can achieve a renormalizable theory for SFCDM if the SM is extended by a singlet fermion as CDM and a singlet scalar Higgs boson as a mediator. For the SFCDM masses below 100 GeV, the relic abundance constraint and the direct detection bounds have been studied in [5, 6]. An almost comprehensive study of the parameter space of SFCDM has been performed in [7]. The SFCDM annihilation into two photons under the relic abundance constraint has been obtained and compared with Fermi-Lat bounds for masses below 200 GeV in [8]. From the Higgs coupling measurements, the mixing angle is constrained at 95% CL to be \(\sin \theta \lesssim 0.4\) [9], independent of the second Higgs mass. The analysis of Ref. [10], by the electroweak precision tests, implies slightly stronger constraints in the relevant mass range; for example one finds \(\sin \theta \le 0.32\) for the second Higgs mass about 750 GeV at 95% CL. In addition, for this mass of the Higgs, it has been shown that \(\sin \theta \) is constrained to be less than 0.1, and this constraint is also put on any scenario where the new scalar is somehow involved in electroweak symmetry breaking [11]. In this paper, we restudy the parameter space of the SFCDM, focusing on the role of the Higgs mixing angle and compare our results with latest experimental data. We take the SM and singlet Higgs mass to be 125 and 750 GeV, respectively. The former is fixed by earlier ATLAS [19] and CMS [20] results. For the latter, due to the above statements for the Higgs mixing angle, we choose 750 TeV as an interesting mass.^{1} Of course, as we shall state in Sect. 3.1, for the other masses between the range about 500–1000 GeV our general results and discussions do not get altered.

Furthermore, there are several experiments which report the measured cross section for direct detection of dark matter, recently, such as the XENON100, LUX, COUPP, PICO, EDELWIESS II, PandaX II and Darkside Collaborations. In this paper, using the most updated direct detection data reported by some of these experiments and considering the issues on the mixing angle mentioned above, we reanalyze the parameter space by imposing the relic abundance condition. We shall see that the entire parameter space is excluded by XENON100 [21], PandaX II [22], and LUX [23].

We have organized the paper as follows: in Sect. 2 the renormalizable model for a SFCDM is briefly reviewed. In Sect. 3, we obtain the coupling constant by imposing the relic abundance condition, then we calculate the scattering cross section of SFCDM from the nucleon and explore the parameter space using the most recent direct detection data. Finally, we summarize our discussion and conclusions in the last section.

## 2 The model

*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:

*h*and

*s*as the fluctuation around the VEVs of them. Therefore, after symmetry breaking we have

### 2.1 The cross section

*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:

## 3 Computations

### 3.1 The relic density

To study the allowed parameter space consistent with the relic abundance constraint obtained by WMAP observations [18], the SM Higgs boson mass is fixed to 125 GeV according to the 2012 CMS and ATLAS results [19, 20] and the other Higgs mass to 750 GeV.^{2} Although the variations of the \(\lambda \) 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 25,000 sample models randomly in the whole parameter space. Namely, in addition to the \(\lambda \), we take \(\theta \) and \(m_\psi \) to be free. In the other two investigations, each of which concerned with 10,000 sample models, we set \(\theta =0.1\) and \(\theta =0.01\). 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 \(\theta \) by the contour plot of Fig. 3. This figure shows that for \(\theta <0.1\) 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 \(g_s>1\).

### 3.2 Direct detection

*q*, in the following effective Lagrangian:

*t*-channel by intermediating a Higgs boson, \(\alpha _q\) can be derived:

Using \(g_s\) as obtained in the previous subsection for \(\theta =0.1\) 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 [21], PandaX II [22], LUX [23], PICO-60 [25] and Darkside-50 [26] Collaborations.

## 4 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 [19] and CMS [20] 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 25,000 sample models randomly in the whole parameter space. In fact, in addition to the \(\lambda \), we take \(\theta \) and \(m_\psi \) 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 \(m_\psi \) larger than about 750 GeV. Two other investigations with fixed \(\theta =0.1\) and \(\theta =0.01\), each of which with 10,000 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 \(m_\psi \) and \(\theta \) through Fig. 3. We see that there exist limited regions (300 GeV \(<m_\psi <400\) GeV and 700 GeV \(<m_\psi <1000\) GeV) in which \(\theta <0.1\) 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 \(m_\psi \) for two various choices of \(\theta \). We have compared our results with different experimental data. According to this figure, the entire parameter space is excluded by XENON100 [21], LUX [23] and PandaX II [22]. For more comparison, we have also shown the recent experiments PICO-60 [25] and DarkSide-50 [26] in this figure.

## Footnotes

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