Study of dark matter in the extended BLMSSM

There are strong evidences for existence of dark matter in some experiments at present. However, the question is that we do not have a reasonable explanation for dark matter in the framework of the Standard Model(SM) of particle physics. It is necessary to extend the SM in order to explain the dark matter. According to the current possible existence conditions of dark matter, we choose $\chi^0_L$ and $\tilde{Y}$ as candidates for dark matter in the EBLMSSM. We study the dominant annihilation processes in detail, including $\bar{\chi}^0_L\chi^0_L(\bar{\tilde{Y}}\tilde{Y})\rightarrow \bar{l}^Il^I$ and $\bar{\chi}^0_L\chi^0_L(\bar{\tilde{Y}}\tilde{Y})\rightarrow \bar{\nu}^I\nu^I$. And we calculate their annihilation cross section $\sigma$ and relic density $\Omega_D h^2$. Then we analyze the limitations of dark matter relic density on the parameters of the EBLMSSM.


I. INTRODUCTION
Astronomers are convinced that there are an amazing amount of dark matter in the universe by some astronomical observations and theoretical derivations. The earliest and perhaps the most convincing evidence of the existence of dark matter until today comes from observations of the rotation curves of galaxies [1], namely the graph of circular velocities of stars and gas as a function of their distance from the galactic center. This observation is inconsistent with our calculations using Newtonian dynamics. Therefore, it can be inferred that there are a lot of invisible matter in the universe, which we call dark matter(DM) [2][3][4].
If you want to know more evidences of the existence of dark matter, you can find them in Ref [5][6][7][8][9]. Dark matter is widespread and abundant, and it accounts for about 23% of the universe, while the common baryon matter only accounts for about 4% [10,11]. However, the standard model(SM) of the particle physics can not provide a convincing explanation.
So studying dark matter is a meaningful and interesting work to explore new physics beyond SM.
Although we don't know what dark matter is and how it exists, we can deduce some of its properties by analyzing and calculating data. First, dark matter particles have to be both electrically and color neutral. Second, it only participates in weak interactions. Finally it must remain stable or have a long life, otherwise it will decay into other particles [12,13].
Only neutrinos in SM can have the correct interaction properties, but it is now known that the masses of neutrinos are too small to constitute the main component of dark matter.
Based on the above conditions, there are no suitable dark matter candidates in SM. What is certain is that the existence of suitable dark matter particles is beyond SM. In recent years, weakly-interacting massive particles(WIMPS) have become the most popular candidates for cold dark matter. Many experiments are detecting it, so experimental limitations have been strengthening.
In fact, not only the problem of dark matter but also many other phenomena are not explained well by SM, and these problems and phenomena may become strong evidences for the exploration of new physics. The minimal supersymmetric extension of the standard model(MSSM) is a popular theory to explain these anomalies. And physicists in various countries have been studying it for many years. BLMSSM where the baryon and lepton numbers are local gauge symmetries spontaneously broken at the TeV scale is the simple extension of MSSM. Since the matter-antimatter asymmetry in the universe naturally, the baryon number(B) is broken. On the other hand, considering the neutrino oscillation experiment, the heavy majorana neutrinos contained in the seesaw mechanism can induce the tiny neutrino masses, therefore, the lepton number(L) is also expected to be broken. The proton remains stable and R-parity is not conserved. Consequently the predictions and bounds for the collider experiments should be changed in the BLMSSM.
Although BLMSSM can explain many anomalies, we find that the exotic leptons in BLMSSM is not heavy enough. The diagonal elements of their mass matrix is zero. Therefore, the masses of the exotic leptons are only related to the four parameters Y e4 , Y e5 , υ d , and υ u . Here υ d and υ u are the vacuum expectation values(VEVs) of two Higgs doublets H d and H u . They need to satisfy the equation υ 2 d + υ 2 u = υ ∼ 250GeV. At the same time, the Yukawa couplings Y e4 and Y e5 are not large parameters. It can be calculated that the masses of the exotic leptons are approximately 100GeV. With the advancement of high energy physics experiments, this boundary will soon be ruled out in the future. It is related to whether the BLMSSM can continue to exist. Therefore, in order to get heavy exotic leptons, we add two exotic Higgs superfields which are SU(2) singlets Φ N L and ϕ N L with VEVs of υ N L andῡ N L to the BLMSSM. In this way, the mass matrix of the exotic leptons become Eq.(6). These diagonal elements can be large and contribute to mass, so the masses of exotic leptons become heavy enough. Because the heavy exotic leptons should be unstable, the two superfields Y and Y ′ are introduced. On the other hand, the fourth and fifth generation leptons are mixed, which is different from the BLMSSM. It is obvious that the first four terms of W B and W L in Eq.(1) are exactly corresponding. We call this new model EBLMSSM which is an extending of BLMSSM. Fortunately, we also find several new candidates for cold dark matter in the EBLMSSM. In our previous work, we have studied Y as dark matter candidate in the EBLMSSM [21]and we study χ 0 L andỸ in this paper. Here, we study dark matter annihilating into leptons and light neutrinos in the EBLMSSM.
The biggest clue is the observation of the relic density for dark matter in the universe, which is a strong constraint on the model to explain dark matter until now. The latest experimental observations show that the dark matter relic density is 0.1186±0.0020 [35].
Furthermore, there are constraints on the mass of Higgs. In general, the EBLMSSM meet the above constraints. In our work, the DM relic density should satisfy 0.1186±0.0020 within 3σ range. It is strictly limited, which results in the parameters in the EBLMSSM to vary only in a narrow range. Our work is only to provide a possibility to explain dark matter, and also to provide a direction for indirect detection experiments.
The remaining of the paper is organized as follows: In section II we introduce the EBLMSSM model in detail. In section III we discuss the relic density in the universe at present and we calculate annihilation section of dark matter candidates. Section IV is focused on the numerical analysis. In section V, we give our conclusions.

II. INTRODUCTION OF THE MODEL
In this section, we briefly introduce the basic characteristics of EBLMSSM. It is the [15][16][17][18]. Compared with the BLMSSM, the EBLMSSM includes four new superfields and some new particles. In order to generate large mass for the exotic leptons, we need to introduce the two new superfields (Φ N L and ϕ N L ) with nonzero vacuum expectation values ( υ N L andῡ N L ). At the same time, the other two new superfields (Y and Y ′ ) also are added to keep the heavy exotic leptons unstable [21]. The superpotential of EBLMSSM is given by where W M SSM represents the superpotential of the MSSM. W B and W X are same as the terms in the BLMSSM. W L is different from BLMSSM for adding the first four items in Eq.(1) [19,20]. W Y has some new couplings including the lepton-exotic lepton-Y coupling and lepton-exotic slepton-Ỹ coupling. Furthermore, we can also acquire lepton-slepton- can enhance the impact of lepton flavor violating [14]. In short, these new couplings and new parameters enrich the lepton physics to a certain degree. In addition, study of dark matter has been promoted. It provides a new possibility for explaining dark matter. Besides the above mentioned problems, we can also study many other new physics problems in the EBLMSSM. Of course, these are our future work. There are one Majorana fermion(χ 0 L ), two Dirac fermions (Ỹ andX), and two scalar particles (Y and X) as good dark matter candidates in EBLMSSM. Among them, three particles(X, X and Y ) has been discussed in previous work [16,21]. The other particles(X 0 L andỸ ) will be discussed in this paper. Based on the new introduced superfields Φ N L , ϕ N L , Y and Y ′ in the EBLMSSM, the soft breaking terms are as follows is the soft breaking terms of the BLMSSM discussed in our previous work. SU (2) In the EBLMSSM, some mass matrices are different from BLMSSM because of the introduced superfields Φ N L and ϕ N L . We list some mass matrices and new couplings as following.
If you want to know more, whether it is the mass matrix or the coupling, you can find it in our previous work [22,23].
A. the mass matrices

The lepton neutralino mass matrix in the EBLMSSM
In the EBLMSSM, λ L , the superpartner of the new lepton type gauge boson Z µ L , mixes with the SUSY superpartners ( The mass matrix M L can be diagonalized by the rotation matrix Z N L . Then, we can have Here, X 0 represent the mass egeinstates of the lepton neutralino.

The exotic lepton mass matrix in the EBLMSSM
The mass matrix for the exotic leptons reads as

TheỸ mass matrix in the EBLMSSM
The mass term for superfieldỸ in the Lagrangian is given out Here mỸ (the mass ofỸ )=µ Y .
The couplings for lepton neutralino-new gauge boson Z µ L -lepton neutralino read as

The couplings withỸ
For another dark matter candidateỸ , in addition to the interactions with exotic sleptons(L ′ ) and leptons(l), there are also interactions with exotic neutrinos(Ñ ′ ) and neutrinos(ν), whose couplings are in the following form Y interacts with Z µ L andỸ , whose coupling is in the following form The new gauge boson Z µ L couples with leptons and neutrinos, whose couplings can be find in our previous work [21].
Here n is the number density of the dark matter, σ is the annihilation cross section of the particle, v is the relative velocity of the annihilating particles, H is the expansion rate of the Universe and n 0 is the dark matter number density in thermally equilibrium. Finally, we can obtain an iterative equation about x F for solving Eq. (14).
Here, M pl = 1.22 × 10 19 GeV is the Planck mass and g * is the number of the relativistic degrees of freedom with mass less than T F . We can calculate cross section and the term σv in the Eqs. (14)(15) can be written as FIG. 2: Feynman diagrams for theỸỸ → Z µ L →l I l I (ν I ν I ) andỸỸ →L ′ (Ñ ′ ) →l I l I (ν I ν I ) at the tree-level.
Notice a and b are the first two coefficients in the Taylor expansion of the annihilation cross section [21,28,31]. We begin with a brief calculation formula of the present relic density(Ω D h 2 ) of DM candidates, assuming that the mass m D as well as the annihilation cross section σ are known. Furthermore, neglecting terms which are O(v 4 ), we give the expression of x F and we can calculate Ω D h 2 [32][33][34]by A. a and b of χ 0 L We give the most important lepton neutralino (χ 0 L ) annihilation diagrams whose final states are leptons and light neutrinos. For a complete list of all tree level processes(in FIG.1), we can calculate a and b (in the low velocity limit) by using the couplings in Eqs. (8)(9)(10).
We define above terms to simplify the following formulas. They are all expressions related to couplings. Next we write the concrete expression of a and b. 1, 2, 3, 4) . (20) Because the mass of the light neutrino is too small, we regard it as zero. When we compute the term b, we find the specific form of b is tedious and complicated. To simplify the results, we perform Taylor expansion on m l m D and retain it to the second order.

B. a and b ofỸ
Similarly, we can also calculate the results of a and b forỸ . We give the dominant contribution to the annihilation cross section come froml I l I andν I ν I . The tree diagrams are shown in FIG.2. To simplify the results, we use the following assumptions: Using the couplings in Eqs. (11)(12)(13) The relic density and The mass rotation matrices corresponding to χ 0 L ,L,ν,L ′ andÑ ′ are Z N L , ZL, Zν, ZL ′ and ZÑ ′ .

IV. NUMERICAL RESULTS
We are now in a position to present some numerical results. Current data imply that dark mater is five times more prevalent than normal matter and accounts for about a quarter of the universe. In section I, we give precisely the constrain of the relic density of cold non-baryonic dark matter and it is Ω D h 2 = 0.1186 ± 0.0020 [35]. Next we will discuss Ωh 2 of the χ 0 L andỸ . To obtain a more transparent numerical results, we adopt the following assumptions on parameter space: Here i=1,2,3, µ L =µ N L =0.8TeV. The lightest m χ 0 L mass is denoted by m D , m χ 0 L =m D . We see from FIG.3 that in order to constrain the region of parameter space in EBLMSSM we need satisfy experimental results of Ω D h 2 . We study relic density Ω D h 2 and x F versus M L . The grey area is the experimental results of Ω D h 2 in 3σ and the solid line represents µ N L is related to the non-diagonal parts of the χ 0 L mass matrix in the EBLMSSM. In addition to the parameters of Eq.(24), we also let M L =2TeV and µ L =1.1TeV. As can be seen from FIG.4, when µ N L increases, Ω D h 2 shows a downward trend. Different from the diagonal element M L is that σv increases as µ N L increases. As sensitive parameters µ N L is limited to 1248-1256GeV when Ω D h 2 satisfies the experiment bounds. It's range is about half of the parameter M L . Besides, the curve of x F is very slowly rising. x F is related to σv and increases as σv increases. This is the reason for the decrease in Ω D h 2 .

B. Numerical result ofỸ
To obtain the numerical results, we adopt the following parameters as The parameters that are repeated with the subsection A are not listed, they can all be Y e4 is the Yukawa coupling constant that can influence the mass matrix of exotic slepton.
When Y e4 gradually becomes larger, Ω D h 2 also increases accordingly. And x F is limited to 24.6-24.8, almost no change within the scope of meeting the relic density boundaries of dark matter. In general, the changes in both the dashed line and the solid line are flat and stable.
The reason is that m D is taken as a fixed value in this figure. We choose the lightest χ 0 L andỸ as dark matter candidates due to that they are consistent with the characteristics of cold dark matter. Then we research the relic density of χ 0 L and Y . In rational parameter space, Ω D h 2 can match the experiment bounds. And based on experimental data we can give confine on sensitive parameters. The EBLMSSM inevitably will be a feature of many particle physics models beyond the standard model. We believe the results presented here may generally be useful in the study of such models and of their cosmological consequences.