1 Introduction

Observations of galaxy rotation curves indicate that the circular velocity remains constant as the radius r increases [1]. This suggests the presence of a halo with mass distribution \(M(r)\propto r\) in galaxies. The halo is constructed by invisible matter known as dark matter (DM), which contributes approximately five times more than visible matter as confirmed by the analysis of the cosmic microwave background (CMB) [2, 3]. The properties of DM indicate that it must be electrically neutral, color neutral, long-lived and only involved in weak interactions [4, 5]. In the standard model (SM), the only potential DM candidates are the neutrinos, but neutrinos in the SM are massless, hence new physics (NP) beyond the SM is required to provide reasonable DM candidates. Furthermore, the relic density of tiny but nonzero massive neutrinos which account for neutrino oscillation experiments, is significantly lower than the relic density of non-baryonic matter \(\Omega h^2=0.1186\pm 0.002\) [6, 7]. This underscores the need for new DM candidates beyond the SM. Based on current observations, the DM candidate can take the form of either scalar or fermion. Among the various DM candidates, weakly interacting massive particles (WIMPs) are particularly popular [8], as they have the potential to be observed in direct detection experiments [9,10,11,12,13], indirect detection experiments [14, 15], and collider experiments [16,17,18].

The lightest neutralino in supersymmetry (SUSY) models is one of the most attractive candidates for WIMP dark matter (DM). It is electrically neutral, color neutral, long-lived, and serves as an ideal cold DM candidate. In the minimal SUSY extension of the SM (MSSM), the relic density and detection potential of the lightest neutralino as a DM candidate are analyzed in references [19,20,21,22,23,24,25,26,27]. These analyses demonstrate that the parameter space of the MSSM is strongly constrained when considering collider, astrophysics and muon anomalous magnetic dipole moment (MDM) constraints. Therefore, the DM sector coupled with the existence of nonzero neutrino masses, call for the consideration of an extended SUSY scenario. We focus on analyzing the properties of new DM candidates in the \(B-L\) symmetric SUSY model (B-LSSM), where B represents the baryon number and L represents the lepton number [28,29,30,31,32,33]. In comparison to the MSSM, there are more neutralinos in the B-LSSM, which indicates a larger pool of potential DM candidates.

The local gauge group of the B-LSSM is extended as \(SU(3)_C\otimes SU(2)_L\otimes U(1)_Y\otimes U(1)_{B-L}\), the invariance under \(U(1)_{B-L}\) gauge group imposes the R-parity conservation which is assumed in the MSSM to avoid proton decay. And R-parity conservation can be maintained if \(U(1)_{B-L}\) symmetry is broken spontaneously [34]. Three right-handed neutrinos and two singlet scalars with nonzero \(B-L\) charge are introduced in the B-LSSM. The tiny neutrino masses can be acquired naturally by the so-called type-I seesaw mechanism when the introduced singlet scalars receive vacuum expect values (VEVs) after the symmetry breaking. As can be noted, there are three additional neutralinos in the B-LSSM comparing with the MSSM, which are the supper partners of two singlet scalars and a new \(Z'\) gauge boson. These new neutralinos have the potential to serve as DM candidates. In this study, we analyze the properties of these newly proposed DM candidates in the B-LSSM, and investigate whether their relic density and cross sections with nucleon align with the experimental measurements. It is widely acknowledged that the relic density of Bino, Wino, or Higgsino DM candidates in the MSSM is hard to match experimental measurements when considering muon anomalous MDM constraints. Therefore, we incorporate the observed muon anomalous MDM constraints in our analysis.

The paper is organized as follows. Section 2 presents the framework of the B-LSSM, the neutralino mass matrix, the relic density of the DM candidate, and the cross section of DM candidates scattering with nucleons. The numerical results are calculated and analyzed in Sect. 3. Finally, a brief summary is given in Sect. 4.

2 The new DM candidates in the B-LSSM

This section presents the framework of the B-LSSM, calculates the relic density for potential DM candidate and the cross section of DM candidate scattering with nucleon.

2.1 The B-LSSM

The gauge group of B-LSSM is extended to include an additional \(U(1)_{B-L}\) local gauge group, which results in the introduction of a new \(Z'\) gauge boson. In addition, three generations of right-handed neutrinos \(\hat{\nu }_R\sim (1,1,0,1/2)\) and two chiral singlet superfields \(\hat{\eta }_1\sim (1,1,0,-1)\), \(\hat{\eta }_2\sim (1,1,0,1)\) are introduced, and the superpotential is given by

$$\begin{aligned} W= & {} Y_u^{ij}\hat{Q}_i\hat{H}_2\hat{U}_j^c+\mu H_1 H_2-Y_d^{ij}\hat{Q}_i\hat{H}_1\hat{D}_j^c-Y_e^{ij}\hat{L}_i\hat{H}_1\hat{E}_j^c\nonumber \\{} & {} +Y_\nu ^{ij}\hat{L}_i\hat{H}_2\hat{\nu }_{R,j} -\mu '\hat{\eta }_1\hat{\eta }_2+Y_x^{ij}\nu _{R,i}\hat{\eta }_1\nu _{R,j}. \end{aligned}$$
(1)

The right-handed neutrinos acquire Majorana masses as the two singlet scalars receive VEVs \(u_1,\;u_2\) after the symmetry breaking

$$\begin{aligned} \eta _1= & {} \frac{1}{\sqrt{2}}(u_1+\textrm{Re}\eta _1+i \textrm{Im} \eta _1),\nonumber \\ \eta _2= & {} \frac{1}{\sqrt{2}}(u_2+\textrm{Re}\eta _2+i \textrm{Im} \eta _2). \end{aligned}$$
(2)

Combining with the Dirac mass terms acquired after the two doublet scalars

$$\begin{aligned}{} & {} H_1=\left( \begin{array}{c}H_1^1\\ H_1^2\end{array}\right) \sim (1,2,-1/2,0),\;\nonumber \\ {}{} & {} H_2=\left( \begin{array}{c}H_2^1\\ H_2^2\end{array}\right) \sim (1,2,1/2,0) \end{aligned}$$
(3)

receive VEVs \(v_1,\;v_2\)

$$\begin{aligned} H_1^1= & {} \frac{1}{\sqrt{2}}(v_1+\textrm{Re}H_1^1+i \textrm{Im} H_1^1),\nonumber \\ H_2^2= & {} \frac{1}{\sqrt{2}}(v_2+\textrm{Re}H_2^2+i \textrm{Im} H_2^2), \end{aligned}$$
(4)

the observed tiny neutrino masses can be obtained by the so-called type I see-saw mechanism in the B-LSSM. For convenience, we can define \(v^2=v_1^2+v_2^2\), \(u^2=u_1^2+u_2^2\), \(\tan \beta =v_2/v_1\), \(\tan \beta '=u_2/u_1\), and the new \(Z'\) gauge boson mass can be approximated as \(M_{Z'}\approx g_B u\) where \(g_B\) is the gauge coupling constant corresponding to \(U(1)_{B-L}\). The soft breaking terms of the model can be written as

$$\begin{aligned} \mathcal {L}_\textrm{soft}= & {} \Big [-\frac{1}{2}(M_1\tilde{\lambda }_{B} \tilde{\lambda }_{B}+M_2\tilde{\lambda }_{W} \tilde{\lambda }_{W}+M_3\tilde{\lambda }_{g} \tilde{\lambda }_{g}\nonumber \\{} & {} +2M_{BB^{'}}\tilde{\lambda }_{B^{'}} \tilde{\lambda }_{B}+M_{B^{'}}\tilde{\lambda }_{B^{'}} \tilde{\lambda }_{B^{'}}) \nonumber \\{} & {} - B_\mu H_1H_2 -B_{\mu ^{'}}\tilde{\eta }_1 \tilde{\eta }_2 +T_{u,ij}\tilde{Q}_i\tilde{u}_j^cH_2\nonumber \\{} & {} +T_{d,ij}\tilde{Q}_i\tilde{d}_j^cH_1+ T_{e,ij}\tilde{L}_i\tilde{e}_j^cH_1+T_{\nu }^{ij} H_2 \tilde{\nu }_i^c \tilde{L}_j \nonumber \\{} & {} + T_x^{ij} \tilde{\eta }_1 \tilde{\nu }_i^c \tilde{\nu }_j^c+h.c.\Big ]-m_{\tilde{\nu },ij}^2(\tilde{\nu }_i^c)^* \tilde{\nu }_j^c\nonumber \\{} & {} -m_{\tilde{q},ij}^2\tilde{Q}_i^*\tilde{Q}_j-m_{\tilde{u},ij} ^2(\tilde{u}_i^c)^*\tilde{u}_j^c-m_{\tilde{\eta }_1}^2 |\tilde{\eta }_1|^2\nonumber \\{} & {} - m_{\tilde{\eta }_2}^2 |\tilde{\eta }_2|^2-m_{\tilde{d},ij}^2(\tilde{d}_i^c)^*\tilde{d}_j^c -m_{\tilde{L},ij}^2\tilde{L}_i^*\tilde{L}_j\nonumber \\{} & {} -m_{\tilde{e},ij}^2(\tilde{e}_i^c)^*\tilde{e}_j^c -m_{H_1}^2|H_1|^2-m_{H_2}^2|H_2|^2, \end{aligned}$$
(5)

with \(\lambda _{B}, \lambda _{B^{'}}\) denoting the gaugino of \(U(1)_Y\) and \(U(1)_{B-L}\) respectively.

It is worth noting that the B-LSSM introduces three additional neutralinos, which correspond to the super partners of two singlet scalars and a new \(Z^\prime \) gauge boson. These neutralinos have the potential to serve as cold DM candidates. On the basis \((\tilde{\lambda }_B,\tilde{\lambda }_W,\tilde{\lambda }_{H_1^1},\tilde{\lambda }_{H_2^2},\tilde{\lambda }_{B'},\tilde{\lambda }_{\eta _1},\tilde{\lambda }_{\eta _2})\), the mass matrix of neutralinos in the B-LSSM can be written as

$$\begin{aligned}{} & {} M_{\chi ^0}=\left( \begin{array}{ccccccc}M_1 &{} 0 &{} -\frac{1}{2}g_1 v_1&{} \frac{1}{2}g_1 v_2 &{} M_{BB'} &{} 0 &{} 0\\ 0 &{} M_2 &{} \frac{1}{2}g_2 v_1 &{} -\frac{1}{2}g_2 v_2 &{} 0 &{} 0 &{} 0\\ -\frac{1}{2}g_1 v_1 &{} \frac{1}{2}g_2 v_1 &{} 0 &{} -\mu &{} -\frac{1}{2}g_{YB} v_1 &{} 0 &{} 0\\ \frac{1}{2}g_1 v_2 &{} -\frac{1}{2}g_2 v_2 &{} -\mu &{} 0 &{} \frac{1}{2}g_{YB} v_2 &{} 0 &{} 0\\ M_{BB'} &{} 0 &{} -\frac{1}{2}g_{YB} v_1 &{} \frac{1}{2}g_{YB} v_2 &{} M_{BL} &{} -g_B u_1 &{} g_B u_2\\ 0 &{} 0 &{} 0 &{} 0 &{} -g_Bu_1 &{} 0 &{} -\mu '\\ 0 &{} 0 &{} 0 &{} 0 &{} g_B u_2 &{} -\mu ' &{} 0\end{array}\right) , \end{aligned}$$
(6)

where \(g_{YB}\) is the gauge coupling constant arising from the gauge kinetic mixing effects in the models with two Abelian gauge groups. For gaugino mass terms \(M_1,\;M_2,\;\mu \) appeared in Eq. (6), we take \(M_0\equiv \mu =2M_1=2M_2\) in the following analysis for simplicity. In order to obtain the analytical masses of the additional neutralinos approximately, we neglect the mixing effects between the MSSM sector with the B-LSSM sector in Eq. (6), i.e. we take \(M_{BB'}=0,\;g_{YB}=0\) and \(M_{BL}\ll u,\;\mu _B\ll u\). Under these assumptions, the masses corresponding to \(\tilde{\lambda }_{B'},\tilde{\lambda }_{\eta _1},\tilde{\lambda }_{\eta _2}\) can be expressed as:

$$\begin{aligned}{} & {} M_{\chi ^0_{\tilde{B}'}}\approx M_{Z'}+\frac{1}{2}M_{BL}+\frac{\tan \beta '}{\tan ^2\beta '+1}\mu _B,\;\nonumber \\{} & {} M_{\chi ^0_{\tilde{\eta }_1}}\approx \frac{2\tan \beta '}{\tan ^2\beta '+1}\mu _B,\nonumber \\{} & {} M_{\chi ^0_{\tilde{\eta }_2}}\approx M_{Z'}-\frac{1}{2}M_{BL}-\frac{\tan \beta '}{\tan ^2\beta '+1}\mu _B. \end{aligned}$$
(7)

The updated experimental data on searching \(Z'\) gauge boson shows \(M_{Z'}>4.05\;\textrm{TeV}\) at \(95\%\) confidence level [35]. And Eq. (7) indicates that the super partners of \(Z'\) gauge boson and \(\eta _2\) is significantly heavy and unlikely to serve as the LSP for potential dark matter candidates.Footnote 1 However, \(M_{\chi ^0_{\tilde{\eta }_1}}\) is equal to \(\mu _B\) approximately and can be considered as a potential DM candidate in the B-LSSM. Therefore, we focus on \(\chi ^0_{\tilde{\eta }_1}\), the super partner of \(\eta _1\), as the DM candidate in the B-LSSM and investigate its relic density and scattering cross section with nucleon to see weather it can satisfy the experimental observations. For simplicity, the neutralinos in the B-LSSM are denoted by \(\chi _i^0,\;(i=1,...,7)\) in the following analysis, and the corresponding masses have \(M_{\chi _i^0}<M_{\chi _j^0}\) when \(i<j\).

The co-annihilation processes involving charginos should be considered in the calculations of DM relic density. The chargino sector in the B-LSSM is same as the one in the MSSM, and the corresponding mass matrix can be written as

$$\begin{aligned}{} & {} M_{\chi ^\pm }= \left( \begin{array}{cc}M_2,&{}\frac{1}{\sqrt{2}}g_2 v_2\\ \frac{1}{\sqrt{2}}g_2 v_1,&{}\mu \end{array}\right) . \end{aligned}$$
(8)

2.2 The relic density and direct detection of DM

We adopt the methods described in Ref. [36] to determine the relic density of DM. The abundances of new particles \(\chi _i\;(i=1,...,N)\) beyond the SM with \(m_i<m_j\) for \(i<j\) are determined by the Boltzmann equations

$$\begin{aligned} \frac{d n_i}{d t}= & {} -3Hn_i-\sum _{j,X}[\langle \sigma _{ij} v\rangle (n_in_j-n_i^\textrm{eq}n_j^\textrm{eq})\nonumber \\ {}{} & {} -(\langle \sigma '_{ij} v\rangle n_in_X-\langle \sigma '_{ji} v\rangle n_jn_X)\nonumber \\{} & {} -\Gamma _{ij}(n_i-n_i^\textrm{eq})], \end{aligned}$$
(9)

where H is the Hubble parameter, \(n_i^\textrm{eq}\) is the \(\chi _i\) equilibrium number density, v is the “relative velocity”, \(\langle \sigma _{ij} v\rangle \) is the thermal average of the corresponding annihilation cross section which can be Taylor expanded as \(\langle \sigma _{ij} v\rangle =a_{ij}+b_{ij}\langle v^2\rangle \), and

$$\begin{aligned}{} & {} \sigma _{ij}=\sigma (\chi _i\chi _j\rightarrow XX'),\nonumber \\{} & {} \sigma '_{ij}=\sigma (\chi _iX\rightarrow \chi _j X'),\nonumber \\{} & {} \Gamma _{ij}=\sigma (\chi _i\rightarrow \chi _jXX'). \end{aligned}$$
(10)

\(X,\;X'\) denote the SM particles. The Boltzmann equation determining the abundance of DM can be obtained from Eq. (9) as

$$\begin{aligned}{} & {} \frac{d n}{d t}=-3Hn-\sum _{i=1,j=1}^N[\langle \sigma _{ij} v\rangle (n_in_j-n_i^\textrm{eq}n_j^\textrm{eq})]. \end{aligned}$$
(11)

where \(n=\sum _{i=1}^N n_i\) because all \(\chi _i\) which survive annihilation eventually decay into the LSP \(\chi _1\). Defining

$$\begin{aligned}{} & {} r_i=n_i^\textrm{eq}/n^\textrm{eq}=\frac{g_i(1+\Delta _i)^{3/2}e^{ -x\Delta _i}}{g_\textrm{eff}},\;\;\nonumber \\{} & {} \Delta _i=(m_i-m_1)/m_1,\nonumber \\{} & {} g_\textrm{eff}=\sum _{i=1}^N g_i(1+\Delta _i)^{3/2}e^{-x\Delta _i},\;\;x\equiv \frac{m}{T}, \end{aligned}$$
(12)

where T denotes temperature, \(g_i\) is the number of degrees of freedom of \(\chi _i\). Equation (11) can be rewritten as

$$\begin{aligned}{} & {} \frac{d n}{d t}=-3Hn-\langle \sigma _\textrm{eff} v\rangle (n^2-n^2_\textrm{eq}), \end{aligned}$$
(13)

where

$$\begin{aligned} \sigma _\textrm{eff}= & {} \sum _{ij}^N r_ir_j\nonumber \\ {}= & {} \sum _{ij}^N \sigma _{ij}\frac{g_ig_j}{g_\textrm{eff}}(1+\Delta _i)^{3/2}(1+\Delta _j)^{3/2}e^{-x(\Delta _i+\Delta _j)}.\nonumber \\ \end{aligned}$$
(14)

The DM freeze-out temperature \(X_F\) can be obtained by

$$\begin{aligned}{} & {} X_F=\ln \frac{0.038 g_\textrm{eff} M_{PL} M_{DM}\langle \sigma _\textrm{eff} v\rangle }{g_*^{1/2} X_F^{1/2}}, \end{aligned}$$
(15)

with \(M_\textrm{DM}\) denoting the DM mass, \(M_{PL}=1.22\times 10^{19}\;\textrm{GeV}\), and \(g_*\) being the total number of effectively relativistic degrees of freedom at the time of freeze-out. Finally, the relic density of DM \(\chi _1\) can be written as

$$\begin{aligned}{} & {} \Omega h^2=\frac{1.07\times 10^9 x_F}{g_\star ^{1/2}M_\textrm{PL}(a_{11}I_a+3b_{11}I_b/x_F)}, \end{aligned}$$
(16)

where

$$\begin{aligned}{} & {} I_a=\frac{x_F}{a_{11}}\int _{x_F}^\infty x^{-2} a_\textrm{eff} dx,\nonumber \\{} & {} I_b=\frac{2x_F^2}{a_{11}}\int _{x_F}^\infty x^{-3} b_\textrm{eff} dx, \end{aligned}$$
(17)

and \(a_\textrm{eff}\;(b_\textrm{eff})\) can be obtained by replacing \(\sigma _\textrm{eff}\), \(\sigma _{ij}\) in Eq. (14) by \(a_\textrm{eff}\;(b_\textrm{eff})\), \(a_{ij}\;(b_{ij})\) respectively.

As analyzed above, the newly possible DM candidate in the B-LSSM is the super partner of the scalar field \(\eta _1\). The dominant self-annihilation and co-annihilation processes considered in our calculations are

$$\begin{aligned}{} & {} (1):\;\chi _j^0\chi _k^0\rightarrow \bar{u}_i u_i,\;\bar{d}_i d_i,\;\bar{l}_i l_i,\;\bar{\nu }_i \nu _i,\;WW,\;ZZ,\;hh,\;\nonumber \\{} & {} \quad \quad hZ\;(i=1,2,3,\;j,k=1,2,3),\end{aligned}$$
(18)
$$\begin{aligned}{} & {} (2):\;\chi _j^0\chi _k^\pm \rightarrow \bar{u}_i d_i,\;\bar{l}_i \nu _i,\;ZW,\;hW,\;\nonumber \\{} & {} \quad \quad rW\;(i=1,2,3,\;j=1,2,3,\;k=1),\end{aligned}$$
(19)
$$\begin{aligned}{} & {} (3):\;\chi _j^\pm \chi _k^\pm \rightarrow \bar{u}_i u_i,\;\bar{d}_i d_i,\;\bar{l}_i l_i,\;\bar{\nu }_i \nu _i,\;WW,\;ZZ,\;hh,\;\nonumber \\{} & {} \qquad \gamma \gamma ,\;hZ,\;\gamma Z,\gamma h\;(i=1,2,3,\;\; j,k=1). \end{aligned}$$
(20)

The results shown in Ref. [37] indicates that the co-annihilation processes involving particle \(\chi _{_X}\) are highly suppressed when \(M_{\chi _{_X}}>1.5 M_\textrm{DM}\), hence we can neglect the contributions from co-annihilation processes to the relic density safely when the involving particles are heavier than \(1.5 M_{\chi _1^0}\).

2.3 The direct detection of DM

DM detection experiments set strict constraints on the DM-nucleon scattering cross section, where the most stringent constraints are applied to the spin-independent (SI) scattering. The lower bound on the SI scattering cross section from the LUX-ZEPLIN (LZ) experiments [38] reads \(6.5\times 10^{-48}\;\textrm{cm}^2\) for \(M_\textrm{DM}=30\;\textrm{GeV}\), while the one reads \(2.58\times 10^{-47}\;\textrm{cm}^2\) for \(M_\textrm{DM}=28\;\textrm{GeV}\) from the recent XENONnT experimental results [39]. The LZ experiments provide the most stringent constraints at present, so we incorporate the constraints derived from the latest LZ experimental results into our analysis.

In the B-LSSM, the effective Lagrangian for SI scattering can be written as

$$\begin{aligned} \mathcal {L}_{SI}= & {} \left[ \frac{(C_{\bar{f} h_i f}^L+C_{\bar{f} h_i f}^R)(C_{\overline{\tilde{\chi }^1_0} h_i \tilde{\chi }_0^1}^L+C_{\overline{\tilde{\chi }_0^1} h_i \tilde{\chi }_0^1}^R)}{-4m_{h_i}^2}\right. \nonumber \\ {}{} & {} \left. +\frac{C_{\bar{f}\tilde{f}\chi ^0}^L C_{\bar{f}\tilde{f}\chi ^0}^R}{-4m_{\tilde{f}}^2}\right] \overline{\tilde{\chi }^1_0}\tilde{\chi }_0^1 \bar{f} f\nonumber \\\equiv & {} C_{SI}^f\overline{\tilde{\chi }^1_0}\tilde{\chi }_0^1 \bar{f} f, \end{aligned}$$
(21)

with f denoting quark, \(\tilde{f}\) is the super partner of quark f, and \(C_{abc}^{L,R}\) (with a, b, c denoting the interactional particles) are the left-handed, right-handed part of coupling abc respectively. The cross section of DM-nucleon SI scattering is [40]

$$\begin{aligned}{} & {} \sigma _{SI}^{\tilde{\chi }_0^1-n}=\frac{4M_{\chi _0^1}^2 m_{n}^2}{\pi (M_{\chi _0^1}+m_{n})^2}|f_{n}|^2, \end{aligned}$$
(22)

where \(m_n\) is the neuron mass, and [41,42,43]

$$\begin{aligned} f_{n}= & {} \sum _{f=u,d,s}f_{Tf}^{n}\frac{C_{SI}^fm_{n}}{m_f}\nonumber \\ {}{} & {} + \frac{2}{27}\left( 1-\sum _{f=u,d,s}f_{Tf}^{n}\right) \sum _{f=c,b,t}\frac{C_{SI}^fm_{n}}{m_f},\nonumber \\ f_{Tu}^n= & {} 0.0110,\;f_{Tu}^n=0.0273,\;f_{Tu}^n=0.0447. \end{aligned}$$
(23)

3 Numerical results

In this section, we present and analyze the numerical results of the relic density, the cross section of SI scattering with nucleon of new DM candidate in the B-LSSM. As input parameters [6], we take \(m_W=80.385\textrm{GeV},\;m_Z=90.1876\textrm{GeV},\;\alpha _{em}(m_Z)=1/128.9,\;\alpha _s(m_Z)=0.118\), the fermion coupling constant \(G_F=1.1664\times 10^{-5}\textrm{GeV}^{-2}\), the lepton masses \(m_e=0.511\;\textrm{MeV}\), \(m_\mu =0.105\;\textrm{GeV}\), \(m_\tau =1.78\;\textrm{GeV}\), the quark masses \(m_u=2.3\;\textrm{MeV}\), \(m_c=1.28\;\textrm{GeV}\), \(m_t=173.5\;\textrm{GeV}\), \(m_d=4.80\;\textrm{MeV}\), \(m_s=0.095\;\textrm{GeV}\), \(m_b=4.65\;\textrm{GeV}\). The measured Higgs boson mass at \(3\sigma \) level errors reads [6]

$$\begin{aligned}{} & {} m_h=125.09\pm 0.72\;\textrm{GeV}. \end{aligned}$$
(24)
Fig. 1
figure 1

Taking \(M_B=M_{BB'}=0.5\;\textrm{TeV}\), \(\tan \beta =20\) and \(\tan \beta '=1.15\), \(\Omega h^2\) (a) and \(\sigma _\textrm{SI}^{\chi _1^0-n}\) (b) versus \(\Delta \) are plotted, where the black solid, black dashed, black dotted lines denote the results for \(\mu _B=0.1,\;0.4,\;0.7\;\textrm{TeV}\) respectively, the gray area in a denotes the experimental \(3\sigma \) interval of \(\Omega h^2\), the red solid, red dashed, red dotted lines in b denote the LZ experimental upper bounds on \(\sigma _\textrm{SI}^{\chi _1^0-n}\) for \(\mu _B=0.1,\;0.4,\;0.7\;\textrm{TeV}\) respectively

Full size image

It is widely recognized that the measured muon anomalous MDM should be taken into account when considering the lightest neutralino as the DM candidate. The experimental average for the deviation between the experimental measurement and SM theoretical prediction of the muon anomalous MDM is given by [44,45,46,47]

$$\begin{aligned} \Delta a_{\mu }=a_{\mu }^\textrm{exp}-a_{\mu }^\textrm{SM}=(25.1\pm 5.9)\times 10^{-10}. \end{aligned}$$
(25)

Based on our previous analysis [48, 49], the measured muon anomalous MDM prefers the slepton masses are not very heavy than \(1.5\;\textrm{TeV}\) approximately, hence we take the three generations of slepton masses equal to \(1\;\textrm{TeV}\) for simplicity. For the trilinear Higgs slepton couplings, we define \(T_e\equiv \textrm{diag}(Y_{e_1},\;Y_{e_2},\;Y_{e_3}) A_e\) with \(A_e=0.1\;\textrm{TeV}\), where \(Y_{e_i}\;(i=1,2,3)\) is the corresponding Yukawa coupling constant of \(e_i\). In order to coincide with the measured B meson rare decays branching rations [50,51,52]

$$\begin{aligned}{} & {} Br(\bar{B}\rightarrow X_s \gamma )=(3.49\pm 0.19)\times 10^{-4},\nonumber \\{} & {} Br(B_s^0\rightarrow \mu ^+\mu ^-)=(2.9_{-0.6}^{+0.7})\times 10^{-9}, \end{aligned}$$
(26)

we take the charged Higgs mass \(M_{H^\pm }=1.5\;\textrm{TeV}\) [53], the squark masses are \(2\;\textrm{TeV}\), the trilinear Higgs squark couplings have \(T_u\equiv \textrm{diag}(Y_{u_1},\;Y_{u_2},\;Y_{u_3}) A_q\), \(T_d\equiv \textrm{diag}(Y_{d_1},\;Y_{d_2},\;Y_{d_3}) A_q\) with \(A_q=0.1\;\textrm{TeV}\), where \(Y_{q_i}\;(q=u,d,\;i=1,2,3)\) is the corresponding Yukawa coupling constant of \(q_i\).

Fig. 2
figure 2

Taking \(\tan \beta '=1.15\), \(\tan \beta =20\), \(\mu _B=0.4\;\textrm{TeV}\), \(\Delta =84\;\textrm{MeV}\), \(\Omega h^2\) (a), \(\sigma _\textrm{SI}^{\chi _1^0-n}\) (b) and \(M_{\chi _2^0}-M_{\chi _1^0}\) (c) versus \(M_{BB'}\) are plotted, where the black solid, black dashed, black dotted lines denote the results for \(M_{BL}=0,\;1,\;2\;\textrm{TeV}\) respectively, the gray area in a denotes the experimental \(3\sigma \) interval of \(\Omega h^2\), the red solid line in b denotes the LZ experimental upper bounds on \(\sigma _\textrm{SI}^{\chi _1^0-n}\) for \(\mu _B=0.4\;\textrm{TeV}\)

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As mentioned above, \(M_{Z'}>4.05 \;\textrm{TeV}\) with \(95\%\) confidence level and we take \(M_{Z'}=5.0 \;\textrm{TeV}\) without losing generality because the contributions from \(Z'\) are suppressed by its heavy mass. The new gauge coupling constants \(g_B\), \(g_{YB}\) are taken as \(g_B=0.4\), \(g_{YB}= - 0.4\) in the calculations. All parameters fixed above affect the relic density and cross section scattering with nucleon of the new DM candidate in the B-LSSM negligibly. For the gaugino mass term \(M_0\equiv \mu =2M_1=2M_2\), we take \(M_0=2(\mu _B+\Delta )\) where \(\Delta \) reflects the mass differences between \(M_{\chi _1^0}\) with \(M_{\chi _2^0}\), \(M_{\chi _3^0}\) or \(M_{\chi _1^\pm }\) directly because \(M_{\chi _1^0}\approx \mu _B\), \(M_{\chi _2^0}\approx M_1\), \(M_{\chi _3^0}\approx M_2\), \(M_{\chi _1^\pm }\approx M_2\).

In order to see the effects of \(\mu _B\) and \(\Delta \), we take \(M_B=M_{BB'}=0.5\;\textrm{TeV}\), \(\tan \beta =20\), \(\tan \beta '=1.15\) and plot \(\Omega h^2\) versus \(\Delta \) in Fig. 1a, where the solid, dashed, dotted lines denote the results for \(\mu _B=0.1,\;0.4,\;0.7\;\textrm{TeV}\) respectively, the gray area denotes the experimental \(3\sigma \) interval of \(\Omega h^2\). Similarly, the results of SI DM-nucleon scattering cross section \(\sigma _\textrm{SI}^{\chi _1^0-n}\) versus \(\Delta \) are plotted in Fig. 1b, where red solid, red dashed, red dotted lines denote the LZ experimental upper bounds on \(\sigma _\textrm{SI}^{\chi _1^0-n}\) for \(\mu _B=0.1,\;0.4,\;0.7\;\textrm{TeV}\) respectively.

Figure 1a shows that \(\Omega h^2\) increases quickly with the increasing of \(\Delta \), because the self-annihilation cross section of \(\chi ^0_{\tilde{\eta }_1}\) is small due to its tiny interactions with the SM particles, and the relic density of \(\chi ^0_{\tilde{\eta }_1}\) depends on the co-annihilation processes dominantly while the cross sections of co-annihilation processes are proportional to \(e^{-x (\Delta _i+\Delta _j)}\) as shown in Eq. (14). In addition, \(\Delta \) is limited strictly by considering \(\Omega h^2\) in the experimental \(3\sigma \) interval for given DM mass (i.e. for given \(\mu _B\)). It is obvious in Fig. 1b that the obtained \(\sigma _\textrm{SI}^{\chi _1^0-n}\) is well below the LZ experimental upper bounds in the considered parameter space. \(\sigma _\textrm{SI}^{\chi _1^0-n}\) mainly depends on the DM mass and increases slowly with the increasing of \(\Delta \).

Fig. 3
figure 3

Taking \(M_B=M_{BB'}=0.5\;\textrm{TeV}\), \(\mu _B=0.4\;\textrm{TeV}\), \(\Delta =84\;\textrm{MeV}\), \(\Omega h^2\) (a), \(\sigma _\textrm{SI}^{\chi _1^0-n}\) (b) and \(M_{\chi _2^0}-M_{\chi _1^0}\) (c) versus \(\tan \beta '\) are plotted, where the black solid, black dashed, black dotted lines denote the results for \(\tan \beta =10,\;20,\;30\) respectively, the gray area in a denotes the experimental \(3\sigma \) interval of \(\Omega h^2\), the red solid line in b denotes the LZ experimental upper bounds on \(\sigma _\textrm{SI}^{\chi _1^0-n}\) for \(\mu _B=0.4\;\textrm{TeV}\)

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In deriving Eq. (7), we set \(M_{BB'}=0\) and make the approximate assumptions \(M_{BL}\ll u,\;\mu _B\ll u\). In fact, the new gaugino mass terms \(M_{BB'}\), \(M_{BL}\) can also influence the mass differences \(M_{\chi _2^0}-M_{\chi _1^0}\), thereby affecting the relic density of the new DM candidate \(\chi ^0_{\tilde{\eta }_1}\). Taking \(\tan \beta '=1.15\), \(\tan \beta =20\), \(\mu _B=0.4\;\textrm{TeV}\), \(\Delta =84\;\textrm{MeV}\), the results of \(\Omega h^2\) versus \(M_{BB'}\) are plotted in Fig. 2a, where the gray area denotes the experimental \(3\sigma \) interval of \(\Omega h^2\), the solid, dashed, dotted lines denote the results for \(M_{BL}=0,\;1,\;2\;\textrm{TeV}\) respectively. Similarly, \(\sigma _\textrm{SI}^{\chi _1^0-n}\) and \(M_{\chi _2^0}-M_{\chi _1^0}\) versus \(M_{BB'}\) are plotted in Fig. 2b and c respectively, where the red solid line in Fig. 2b denotes the LZ experimental upper bounds on \(\sigma _\textrm{SI}^{\chi _1^0-n}\) for \(\mu _B=0.4\;\textrm{TeV}\).

Figure 2a shows that the relic density \(\Omega h^2\) initially increases and then decreases with the decreasing of \(|M_{BB'}|\). From Fig. 2c, it can be seen that the mass difference \(M_{\chi _2^0}-M_{\chi _1^0}\) increases with the decreasing of \(|M_{BB'}|\) which leads to the increases of \(\Omega h^2\) in Fig. 2a. On the other hand, \(|M_{BB'}|\) can also affect the \(\chi _i^0\;(i=2,\;3)\) coupling strength with SM particles which leads to the decreasing of \(\Omega h^2\) in Fig. 2a when \(|M_{BB'}|\) approaches to 0. Figure 2b demonstrates that the SI DM-nucleon scattering cross section \(\sigma _\textrm{SI}^{\chi _1^0-n}\) is well below the upper bounds set by the LZ experiments.

As shown in Eq. (6), \(\tan \beta \) and \(\tan \beta '\) can potentially influence the mass matrix of neutralinos. In order to see the effects of \(\tan \beta \), \(\tan \beta '\), we take \(\mu _B=0.4\;\textrm{TeV}\), \(\Delta =84\;\textrm{MeV}\), \(M_B=M_{BB'}=0.5\;\textrm{TeV}\) and plot \(\Omega h^2\), \(\sigma _\textrm{SI}^{\chi _1^0-n}\), \(M_{\chi _2^0}-M_{\chi _1^0}\) versus \(\tan \beta '\) in Fig. 3a–c respectively, where the black solid, black dashed, black dotted lines denote the results for \(\tan \beta =10,\;20,\;30\) respectively, the gray area in Fig. 3a denotes the experimental \(3\sigma \) interval of \(\Omega h^2\), the red solid line in Fig. 3b denotes the LZ experimental upper bounds on \(\sigma _\textrm{SI}^{\chi _1^0-n}\) for \(\mu _B=0.4\;\textrm{TeV}\).

Fig. 4
figure 4

Scanning the parameter space in Eq. (27) and keeping the muon anomalous MDM, the B meson rare decay branching ratios, the relic density and the Higgs boson mass in their respective experimental \(3\sigma \) interval, the allowed ranges of \(\Delta \), \(M_\textrm{DM}\) (a) and \(\sigma _\textrm{SI}^{\chi _1^0-n}\) versus \(M_\textrm{DM}\) (b) are plotted, where the red solid line in b denotes the LZ experimental upper bounds on \(\sigma _\textrm{SI}^{\chi _1^0-n}\) the green area denotes the neutrino floor

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To maintain consistency with the experimental \(3\sigma \) interval of the Higgs boson mass in the B-LSSM, we consider \(\tan \beta '\ge 1.1\) in the plotting. Figure 3a clearly demonstrates that the relic density \(\Omega h^2\) increases quickly with the increasing of \(\tan \beta '\), owing to the simultaneous increase in the mass difference \(M_{\chi _2^0}-M_{\chi _1^0}\) as shown in Fig. 3c. In addition, from Fig. 3a, c we can see that \(\tan \beta \) affects relic density \(\Omega h^2\) and the mass difference \(M_{\chi _2^0}-M_{\chi _1^0}\) mildly. Figure 3b shows that with the increasing of \(\tan \beta '\), the DM-nucleon SI scattering cross section \(\sigma _\textrm{SI}^{\chi _1^0-n}\) decreases to a minimum value and then increases. The specific value of \(\tan \beta '\) at which \(\sigma _\textrm{SI}^{\chi _1^0-n}\) starts to increase depends on the chosen value of \(\tan \beta \). And \(\sigma _\textrm{SI}^{\chi _1^0-n}\) is beyond the LZ experimental upper bound in our chosen parameter space when \(\tan \beta '\lesssim 1.107\).

To investigate the combined effects of \(\tan \beta \), \(\tan \beta '\), \(\mu _B\), \(M_{BL}\), \(M_{BB'}\) and \(\Delta \), we scan the following parameter space

$$\begin{aligned}{} & {} \tan \beta =(5,40),\;\tan \beta '=(1.05,1.5),\;\nonumber \\{} & {} \mu _B =(0.01,1.0)\;\textrm{TeV},\;M_{BL}=(-2,2)\;\textrm{TeV},\nonumber \\{} & {} M_{BB'}=(-2,2)\;\textrm{TeV},\;\Delta =(-0.1,0.5)\;\textrm{TeV}, \end{aligned}$$
(27)

and keep the muon anomalous MDM, the B meson rare decay branching ratios, the relic density and the Higgs boson mass in their respective experimental \(3\sigma \) interval in the scanning. Then we plot the allowed ranges of \(\Delta \), \(M_\textrm{DM}\) in Fig. 4a. \(\sigma _\textrm{SI}^{\chi _1^0-n}\) versus \(M_\textrm{DM}\) is plotted in Fig. 4b, where the red solid line in Fig. 4b denotes the LZ experimental upper bounds on \(\sigma _\textrm{SI}^{\chi _1^0-n}\) the green area denotes the neutrino floor.

It is evident from the figure that for \(\chi ^0_{\tilde{\eta }_1}\) to be the dark matter candidate in the B-LSSM, the minimum dark matter mass of approximately \(80\;\textrm{GeV}\) is required to satisfy the observed relic density. Furthermore, Fig. 4a illustrates that \(\Delta \) is limited more strictly for smaller \(M_\textrm{DM}\), and more points are excluded by the LZ experimental upper bound on \(\sigma _\textrm{SI}^{\chi _1^0-n}\) as \(M_\textrm{DM}\) increases which is shown explicitly in Fig. 4b.

4 Summary

In this work, we focus on the potential new DM candidates in the B-LSSM. Comparing with the MSSM, there are three additional neutralinos in the B-LSSM which provide more DM candidates. Considering the experimental lower bound on the \(Z'\) gauge boson mass, we find that the super partner of singlet scalar \(\eta _1\) can be considered as the DM candidate in the B-LSSM. The relic density of \(\chi ^0_{\tilde{\eta }_1}\) (i.e. \(\chi _1^0\)) and the SI scattering cross section with nucleon \(\sigma _\textrm{SI}^{\chi _1^0-n}\) are calculated. Taking into account the experimental constraints from Higgs boson mass, B meson rare decays and muon anomalous MDM, the numerical results are obtained and analyzed. It is found that the properties of new DM candidate \(\chi ^0_{\tilde{\eta }_1}\) in the B-LSSM satisfy the observed relic density \(\Omega h^2\) and comply with the latest upper bounds on \(\sigma _\textrm{SI}^{\chi _1^0-n}\). The co-annihilation processes play dominant roles on the relic density \(\Omega h^2\) which indicates the mass differences between \(\chi ^0_{\tilde{\eta }_1}\) with the other SUSY particles affect \(\Omega h^2\) obviously. The theoretical predictions on \(\Omega h^2\) and \(\sigma _\textrm{SI}^{\chi _1^0-n}\) are also influenced by the new gaugino mass terms \(M_{BL}\), \(M_{BB'}\) and new parameters \(\tan \beta '\) in the B-LSSM. In addition, for \(\chi ^0_{\tilde{\eta }_1}\) to be the DM candidate in the B-LSSM, its mass is expected to greater than \(80\;\textrm{GeV}\) approximately as shown in Fig. 4.