Nonholomorphic Higgsino Mass Term Effects on Muon g − 2 and Dark Matter Relic Density in Flavor Symmetry-Based Minimal Supersymmetric Standard Model

Abstract

We investigate the phenomenological effects of the nonholomorphic (NH) higgsino mass term, ${\mu }^{\prime }$, within the minimal supersymmetric standard model (MSSM) extended by a non-abelian flavor symmetry, referred to as the sNHSSM. This flavor symmetry enables a substantial reduction in the number of free parameters inherent to the MSSM, streamlining them from a large set to just eight. Our study explores the interplay between cold dark matter (CDM) relic density (${\mathsf{\Omega }}_{\mathrm{CDM}}{h}^2$) and the anomalous magnetic moment of the muon, ${(g-2)}_{\mu }$. We study correlations among the theoretical parameters that emerge from this interplay and are further constrained by experimental data such as the Higgs boson mass, B-physics observables, and the charge and color breaking minima constraints. Moreover, our findings reveal that incorporating the NH higgsino mass term opens up new regions of parameter space that were previously inaccessible.

1. Introduction

The minimal supersymmetric standard model (MSSM) [1,2,3,4,5,6] has the potential to solve many of the lingering questions of the Standard Model (SM) [7,8,9,10] but involves over 105 free parameters, making phenomenological predictions difficult. The constrained MSSM (CMSSM) [11] reduces this to five parameters through ad hoc assumptions. However, recent findings from CMS and ATLAS have significantly constrained the CMSSM’s parameter space [12]. Consequently, researchers are investigating alternative models. One such approach involves introducing non-universal Higgs masses while keeping the other scalar masses universal [13,14,15,16], a framework known as NUHM2. Another approach, which also incorporates non-universal Higgs masses, assumes that the first two generations of scalar fermions share the same mass, while the third generation can have different masses. This scenario is commonly referred to as NUHM3 [17,18,19]. The flavor symmetry-based MSSM (sMSSM) [20,21,22] presents yet another perspective, incorporating the same parameter set as NUHM3—thus introducing only two additional parameters compared to the CMSSM—but also enforcing a non-abelian flavor symmetry (H).

While the soft SUSY-breaking (SSB) sector of the MSSM generally follows the holomorphicity condition, the inclusion of nonholomorphic (NH) terms is theoretically plausible. Incorporating NH terms into the SSB sector of the MSSM introduces additional interactions and couplings into the theory [23,24]. This can significantly impact the particle spectrum, phenomenology, and experimental observables. This enhanced framework is referred to as the nonholomorphic supersymmetric standard model (NHSSM). This approach not only expands the theoretical possibilities within supersymmetry but also offers potential solutions to discrepancies observed between MSSM predictions and experimental results [25,26,27,28,29,30,31,32,33,34,35,36,37]. The NHSSM can be further extended by incorporating a non-abelian flavor symmetry H, as was carried out for the MSSM. We will refer to this extended model as the sNHSSM.

One of the key advantages of the MSSM is the presence of a cold dark matter (CDM) candidate, and predicting the CDM relic density ${\mathsf{\Omega }}_{\mathrm{CDM}}{h}^2$ in agreement with experimental measurements [38,39] has long been a goal for scientists.

Additionally, there has been growing interest in the long-standing anomaly in the ${(g-2)}_{\mu }$ measurement. The latest evaluation by the Muon $g-2$ collaboration at Fermilab [40,41], combined with earlier results from the Brookhaven E821 experiment [42], shows a $5.1\sigma$ deviation from the SM prediction [43].

However, a recent re-evaluation of isospin-breaking (IB) corrections to $e^{+}{e}^{-}$ and $\tau$-decay di-pion data [44] has narrowed the discrepancy to $2.7\sigma$. Futhermore, when employing the latest Budapest–Marseille–Wuppertal (BMW) results for the hadronic vacuum polarization [45] and light-by-light scattering effects [46], the discrepancy is reduced to below $1\sigma$ [47].

Numerous studies in the literature have investigated the phenomenological implications of the flavor symmetry-based MSSM (sMSSM), including the ${(g-2)}_{\mu }$ anomaly and predictions for ${\mathsf{\Omega }}_{\mathrm{CDM}}{h}^2$ [21,22,48,49]. The interplay between ${\mathsf{\Omega }}_{\mathrm{CDM}}{h}^2$ and ${(g-2)}_{\mu }$ offers valuable insights for the MSSM [50]. Introducing nonholomorphic (NH) SSB terms expands the parameter space, potentially opening regions that were previously excluded in the MSSM [31].

This work focuses on calculating ${\mathsf{\Omega }}_{\mathrm{CDM}}{h}^2$, ${(g-2)}_{\mu }$, and $M_W$ within the sNHSSM framework. The primary goal is to determine whether the ${(g-2)}_{\mu }$ anomaly and ${\mathsf{\Omega }}_{\mathrm{CDM}}{h}^2$ can be explained within the sNHSSM, and if it provides an advantage over the sMSSM.

Our analysis utilizes the tools SARAH  [51,52,53,54,55], SPheno  [56], micrOMEGAs [57,58,59], and the SARAH Scan and Plot (SSP) [60] framework. We first used SARAH to generate the MSSM source code for SPheno, enabling tasks such as spectrum generation and the calculation of low-energy observables, including ${(g-2)}_{\mu }$, B-physics observables, $\Delta a_{\mu }$, and the W boson mass using SPheno. The results from SPheno were then fed into micrOMEGAs to compute the relic density of dark matter.

This paper is structured as follows: In Section 2, we outline the key features of the sNHSSM. Details of the calculations related to low-energy observables, such as ${(g-2)}_{\mu }$, are provided in Section 3. The computational framework and numerical results are discussed in Section 4, while our conclusions are presented in Section 5.

2. Flavor Symmetry-Based MSSM

2.1. SSB Sector of the MSSM

The MSSM is the simplest form of supersymmetric theories that can be built using the particle content of the SM. The complete setup for the SSB parameters is described by  [1,2,3,4,5,6].

$$\begin{array}{ccc}\hfill -{\mathcal{L}}_{\mathrm{soft}}& =& {\left(m_{\tilde{Q}}^2\right)}_i^j{\tilde{q}}_L^{†i}{\tilde{q}}_{Lj}+{\left(m_{\tilde{u}}^2\right)}_j^i{\tilde{u}}_{Ri}^{*}{\tilde{u}}_R^j+{\left(m_{\tilde{d}}^2\right)}_j^i{\tilde{d}}_{Ri}^{*}{\tilde{d}}_R^j\hfill \\ & & +{\left(m_{\tilde{L}}^2\right)}_i^j{\tilde{l}}_L^{†i}{\tilde{l}}_{Lj}+{\left(m_{\tilde{e}}^2\right)}_j^i{\tilde{e}}_{Ri}^{*}{\tilde{e}}_R^j\hfill \\ & & +{\tilde{m}}_1^2{h}_1^{†}{h}_1+{\tilde{m}}_2^2{h}_2^{†}{h}_2+(B_{\mu }{h}_1{h}_2+\mathrm{h}.\mathrm{c}.)\hfill \\ & & +(A_d^{ij}{h}_1{\tilde{d}}_{Ri}^{*}{\tilde{q}}_{Lj}+A_u^{ij}{h}_2{\tilde{u}}_{Ri}^{*}{\tilde{q}}_{Lj}+A_l^{ij}{h}_1{\tilde{e}}_{Ri}^{*}{\tilde{l}}_{Lj}\hfill \\ & & +{\displaystyle \frac{1}{2}}{M}_1{\tilde{B}}_L{\tilde{B}}_L+{\displaystyle \frac{1}{2}}{M}_2{\tilde{W}}_L^a{\tilde{W}}_L^a+{\displaystyle \frac{1}{2}}{M}_3{\tilde{G}}^a{\tilde{G}}^a+\mathrm{h}.\mathrm{c}.).\hfill \end{array}$$

(1)

Here, $m_{\tilde{Q}}^2$ and $m_{\tilde{L}}^2$ are $3\times 3$ matrices in family space, with indices i and j representing generations, and they describe the SSB masses for the left-handed squark doublets ${\tilde{q}}_L$ and slepton doublets ${\tilde{l}}_L$ associated with $SU\left(2\right)$. Similarly, the matrices $m_{\tilde{u}}^2$, $m_{\tilde{d}}^2$, and $m_{\tilde{e}}^2$ represent the SSB masses for the right-handed up-type squarks ${\tilde{u}}_R$, down-type squarks ${\tilde{d}}_R$, and charged sleptons ${\tilde{e}}_R$, which are $SU\left(2\right)$ singlets. The matrices $A_u$, $A_d$, and $A_l$ also form $3\times 3$ matrices, corresponding to the trilinear couplings for up-type squarks, down-type squarks, and charged sleptons, respectively. The parameter $\mu$ denotes the Higgs mixing term, while ${\tilde{m}}_1$, ${\tilde{m}}_2$, and $B_{\mu }$ are SSB parameters related to the Higgs sector. The parameters $h_1$ and $h_2$ refer to the two Higgs doublets, and $M_1$, $M_2$, and $M_3$ define the mass parameters for the bino, wino, and gluino, respectively.

2.2. The Flavor Symmetry-Based NHSSM

In any supersymmetric framework, the superpotential must preserve holomorphicity, a condition that is consistently upheld when defining SSB terms within the MSSM. However, despite this traditional adherence, it is theoretically possible for NH terms to exist. In its simplest form, the SSB sector for the NH terms can be written as follows:

$$-{\mathcal{L}}_{\mathrm{soft}}^{\mathrm{NH}}=A_{ij}^{\prime D}{h}_2{\tilde{d}}_{Ri}^{*}{\tilde{q}}_{Lj}+A_{ij}^{\prime U}{h}_1{\tilde{u}}_{Ri}^{*}{\tilde{q}}_{Lj}+A_{ij}^{\prime E}{h}_2{\tilde{e}}_{Ri}^{*}{\tilde{l}}_{Lj}+{\mu }^{\prime }{\tilde{h}}_1{\tilde{h}}_2$$

(2)

The matrices $A_{ij}^{\prime U}$, $A_{ij}^{\prime D}$, and $A_{ij}^{\prime E}$ represent the NH trilinear couplings for up-type squarks, down-type squarks, and charged sleptons, respectively. The NH higgsino mass parameter ${\mu }^{\prime }$, as introduced in Equation (2), modifies the neutralino and chargino mass matrices, which are given by

$$\mathbf{Y}=\left(\begin{array}{cccc}{M}_1& 0& -M_Z{s}_{\mathrm{w}}cos\beta & M_Z{s}_{\mathrm{w}}sin\beta \\ 0& M_2& M_Z{c}_{\mathrm{w}}cos\beta & -M_Z{c}_{\mathrm{w}}sin\beta \\ -M_Z{s}_{\mathrm{w}}cos\beta & M_Z{c}_{\mathrm{w}}cos\beta & 0& -(\mu +{\mu }^{\prime })\\ M_Z{s}_{\mathrm{w}}sin\beta & -M_Z{c}_{\mathrm{w}}sin\beta & -(\mu +{\mu }^{\prime })& 0\end{array}\right).$$

(3)

$$\mathbf{X}=\left(\begin{array}{cc}{M}_2& \sqrt{2}{M}_{W}sin\beta \\ \sqrt{2}{M}_{W}cos\beta & (\mu +{\mu }^{\prime })\end{array}\right).$$

(4)

Although the NH trilinear couplings may influence the couplings of the SUSY particles, the NH parameter ${\mu }^{\prime }$ does not affect the couplings of the SUSY particle, but rather only modifies the neutralino and chargino mass matrices. This characteristic renders it particularly significant for observables involving neutralinos and charginos, such as ${\mathsf{\Omega }}_{\mathrm{CDM}}{h}^2$. In our current investigation, the NH trilinear couplings hold less relevance, while the interplay between $\mu$ and ${\mu }^{\prime }$ emerges as potentially crucial.

2.3. Flavor Symmetry-Based NHSSM

Equation (1) contains over 105 free parameters, making it difficult to produce viable phenomenological predictions in the MSSM. However, in the CMSSM, this large number of parameters is drastically reduced to just five by assuming that scalar masses and gaugino masses are unified at the GUT scale. Despite the drastic reduction in the number of parameters, experimental results from the LHC have severely constrained the CMSSM parameter space. To overcome the limitations of the CMSSM, a class of supersymmetric models has been developed where the SSB Lagrangian is defined purely by symmetry considerations. This framework, known as the flavor symmetry-based MSSM (sMSSM), imposes two key symmetry conditions on the SSB Lagrangian. First, the parameters respect a grand unified symmetry, such as $\mathtt{SO}\left(10\right)$, and second, a non-abelian flavor symmetry H acts on the three particle generations, suppressing Flavor-Changing Neutral Currents (FCNCs) mediated by SUSY particles. Specific examples of these models [61,62,63,64,65] are constructed with flavor symmetries like $\mathtt{SU}{\left(2\right)}_H$ and $\mathtt{SO}{\left(3\right)}_H,$ where the generations transform as $\mathbf{2}+\mathbf{1}$ and $\mathbf{3}$ representations, respectively. In $\mathtt{SU}{\left(2\right)}_H$, flavor-violating D-terms are suppressed using an interchange symmetry, while the $\mathtt{SO}{\left(3\right)}_H$ framework naturally avoids D-term problems. In this context, it is sufficient to group the first two generations in a multiplet to resolve the SUSY flavor problem, with the third family treated as a singlet. Hence, both $\mathbf{2}+\mathbf{1}$ and $\mathbf{3}$ assignments are valid.

Compatibility with $\mathtt{SO}\left(10\right)$ offers significant advantages for SSB, reducing the soft masses for sfermions from fifteen (under the SM gauge symmetry) to just three [20], lowering the number of free parameters. It also provides a symmetry-based reason for gaugino mass unification, cutting the free parameters from three to one. The sMSSM adds two more free parameters compared to the CMSSM, bringing the total to seven. The resulting parameter space is equivalent to that of NUHM3; hence, the forthcoming analysis is equally relevant to the NUHM3 model. The concept of flavor symmetry can also be incorporated into the NHSSM framework to reduce the number of input parameters. By setting the NH trilinear couplings to zero or equal to the holomorphic trilinear couplings, the only additional parameter in the model would be ${\mu }^{\prime }$. Thus, the input parameters in the sNHSSM will comprise the following:

$$\{m_{0_{1,2}},m_{0_3},M_{1/2},A_0,tan\beta ,\mu ,m_A,{\mu }^{\prime }\}.$$

where $m_{0_{1,2}}$ represents the SSB mass parameter for the first two families of sfermions, $m_{0_3}$ for the third generation, $M_{1/2}$ for the SSB gaugino mass, and $A_0$ for the SSB trilinear coupling. The parameters $tan\beta$ and $M_A$ represent the ratio of the vacuum expectation values of the two Higgs doublets and the mass of the $\mathcal{CP}$-odd Higgs boson A, respectively.

3. Calculation of Low-Energy Observables

SUSY Contributions to ${(g-2)}_{\mu }$

As mentioned earlier, the discrepancy between the current world average (WA) SM predictions [43,66] and CDF measurements stands at $5.1\sigma$, given by

$$\begin{array}{c}\hfill \Delta a_{\mu }^{\mathrm{WA}}=a_{\mu }^{\exp }-a_{\mu }^{\mathrm{SM},\mathrm{WA}}=(24.9\pm 4.8)\times {10}^{-10}.\end{array}$$

(5)

Similarly, for IB corrections, we have

$$\Delta a_{\mu }^{\mathrm{IB}}=a_{\mu }^{\exp }-a_{\mu }^{\mathrm{SM},\mathrm{IB}}=(14.8_{-5.4}^{+5.1})\times {10}^{-10},$$

(6)

while the BMW results yield

$$\Delta a_{\mu }^{\mathrm{BMW}}=a_{\mu }^{\exp }-a_{\mu }^{\mathrm{SM},\mathrm{BMW}}=(0.4\pm 4.2)\times {10}^{-10}.$$

(7)

The value of $\Delta a_{\mu }^{\mathrm{BMW}}$ aligns with zero, suggesting no significant deviation between the SM prediction and experimental measurements, thereby leaving minimal scope for new physics effects. On the other hand, $\Delta a_{\mu }^{\mathrm{BMW}}$ demands moderate contributions from beyond the SM physics, while $\Delta a_{\mu }^{\mathrm{WA}}$ points to even larger contributions. This underscores the unresolved nature of the ${(g-2)}_{\mu }$ anomaly. Nevertheless, a comprehensive exploration of potential new physics contributions to ${(g-2)}_{\mu }$ could shed light on this anomaly and delineate the relevant parameter space for such theories. This discrepancy can be addressed by incorporating the MSSM contributions to ${(g-2)}_{\mu }$. The primary MSSM contributions stem from loops involving sleptons, charginos, and neutralinos, as illustrated in Figure 1. The corrections from the Feynman diagrams shown in Figure 1 can be summarized as in [67,68,69,70].

$$\Delta a_{\mu }^{\mathrm{MSSM}}=\Delta a_{\mu }^{\tilde{B}{\tilde{\mu }}_L{\tilde{\mu }}_R}+\Delta a_{\mu }^{(\tilde{B}-\tilde{H}){\tilde{\mu }}_L}+\Delta a_{\mu }^{(\tilde{H}-\tilde{B}){\tilde{\mu }}_R}+\Delta a_{\mu }^{(\tilde{H}-\tilde{W}){\tilde{\mu }}_L}+\Delta a_{\mu }^{(\tilde{W}-\tilde{H}){\tilde{\nu }}_{\mu }}$$

(8)

where

$$\begin{array}{cc}\hfill \Delta a_{\mu }^{\tilde{B}{\tilde{\mu }}_L{\tilde{\mu }}_R}& ={\displaystyle \frac{g_Y^2}{8{\pi }^2}}{\displaystyle \frac{m_{\mu }^2\mu tan\beta }{M_1^3}}{F}_b\left({\displaystyle \frac{m_{{\tilde{\mu }}_L}^2}{M_1^2}},{\displaystyle \frac{m_{{\tilde{\mu }}_R}^2}{M_1^2}}\right)\hfill \end{array}$$

(9)

$$\begin{array}{cc}\hfill \Delta a_{\mu }^{(\tilde{H}-\tilde{B}){\tilde{\mu }}_R}& ={\displaystyle \frac{-g_Y^2}{8{\pi }^2}}{\displaystyle \frac{m_{\mu }^2{M}_1\mu tan\beta }{m_{{\tilde{\mu }}_R}^4}}{F}_b\left({\displaystyle \frac{M_1^2}{m_{{\tilde{\mu }}_R}^2}},{\displaystyle \frac{{\mu }^2}{m_{{\tilde{\mu }}_R}^2}}\right)\hfill \end{array}$$

(10)

$$\begin{array}{cc}\hfill \Delta a_{\mu }^{(\tilde{B}-\tilde{H}){\tilde{\mu }}_L}& ={\displaystyle \frac{g_Y^2}{16{\pi }^2}}{\displaystyle \frac{m_{\mu }^2{M}_1\mu tan\beta }{m_{{\tilde{\mu }}_L}^4}}{F}_b\left({\displaystyle \frac{M_1^2}{m_{{\tilde{\mu }}_L}^2}},{\displaystyle \frac{{\mu }^2}{m_{{\tilde{\mu }}_L}^2}}\right)\hfill \end{array}$$

(11)

$$\begin{array}{cc}\hfill \Delta a_{\mu }^{(\tilde{H}-\tilde{W}){\tilde{\mu }}_L}& ={\displaystyle \frac{-g^2}{16{\pi }^2}}{\displaystyle \frac{m_{\mu }^2{M}_2\mu tan\beta }{m_{{\tilde{\mu }}_L}^4}}{F}_b\left({\displaystyle \frac{M_2^2}{m_{{\tilde{\mu }}_L}^2}},{\displaystyle \frac{{\mu }^2}{m_{{\tilde{\mu }}_L}^2}}\right)\hfill \end{array}$$

(12)

$$\begin{array}{cc}\hfill \Delta a_{\mu }^{(\tilde{W}-\tilde{H}){\tilde{\nu }}_{\mu }}& ={\displaystyle \frac{g^2}{8{\pi }^2}}{\displaystyle \frac{m_{\mu }^2{M}_2\mu tan\beta }{m_{{\tilde{\nu }}_{\mu }}^4}}{F}_a\left({\displaystyle \frac{M_2^2}{m_{{\tilde{\nu }}_{\mu }}^2}},{\displaystyle \frac{{\mu }^2}{m_{{\tilde{\nu }}_{\mu }}^2}}\right)\hfill \end{array}$$

(13)

with $F_a\left(x,y\right)$ and $F_b\left(x,y\right)$ defined as

$$\begin{array}{cc}\hfill F_a\left(x,y\right)& ={\displaystyle \frac{-1}{2(x-y)}}\left({\displaystyle \frac{(x-1)(x-3)+2lnx}{{(x-1)}^3}}-{\displaystyle \frac{(y-1)(y-3)+2lny}{{(y-1)}^3}}\right)\hfill \\ \hfill F_b\left(x,y\right)& ={\displaystyle \frac{-1}{2(x-y)}}\left({\displaystyle \frac{(x-1)(x+1)-2xlnx}{{(x-1)}^3}}-{\displaystyle \frac{(y-1)(y+1)-2ylny}{{(y-1)}^3}}\right)\hfill \end{array}$$

As can be seen from Equations (9)–(13), all the contributions are proportional to $\mu tan\beta$; however, the value of $\mu tan\beta$ is constrained by the vacuum stability bounds and electroweak symmetry breaking (EWSB).

4. Numerical Results

4.1. Computational Strategy

The workflow for our computation is outlined as follows. Using the Mathematica package SARAH [51,52,53,54,55], we first generated the NHSSM source code for SPheno [56]. SPheno is a tool that performs numerical calculations of SUSY mass spectra, particle decay rates, and various low-energy observables such as $\Delta a_{\mu }$. Similarly, we used SARAH to generate the micrOMEGAs [57,58,59] source code, enabling us to include dark matter constraints in our analysis. These codes were interfaced with SSP [60], a Mathematica package that streamlines parameter scans and data visualization. The following parameter set was employed for random scans within the sNHSSM framework:

$$\begin{array}{ccc}0\le & m_{0_{1,2}}& \le 5\mathrm{TeV}\\ 0\le & m_{0_3}& \le 15\mathrm{TeV}\\ 0\le & M_{1/2}& \le 2\mathrm{TeV}\\ -3\le & A_0/m_{0_3}& \le 3\mathrm{TeV}\\ 0\le & M_A& \le 10\mathrm{TeV}\\ 1\le & tan\beta & \le 60.\end{array}$$

We set the NH trilinear terms to zero. Additionally, in conjunction with the previously specified parameter ranges, we applied the following ranges for $\mu$ and ${\mu }^{\prime }$:

$$\begin{array}{ccc}0\le & \mu & \le 600\mathrm{GeV}\\ 0\le & {\mu }^{\prime }& \le 600\mathrm{GeV}\end{array}$$

These specified boundary conditions were applied at the GUT scale, except for the parameters $\mu$, ${\mu }^{\prime }$, and $M_A$, which were defined at the SUSY scale instead. Additionally, a custom code was implemented within the SSP package to calculate the constraints related to charge- and color-breaking minima (CCB), based on Refs. [33,71]. Data were then generated in accordance with the following constraints on low-energy observables:

$$\begin{array}{cc}\hfill M_h& =123-127\mathrm{GeV},\hfill \\ \hfill m_{\tilde{g}}& \ge 2.1\mathrm{TeV},\hfill \\ \hfill 1.99\times {10}^{-9}& \le \mathrm{BR}(B_s\to {\mu }^{+}{\mu }^{-})\le 3.43\times {10}^{-9}\left(2\sigma \right),\hfill \\ \hfill 3.02\times {10}^{-4}& \le \mathrm{BR}(B\to X_s\gamma )\le 3.62\times {10}^{-4}\left(2\sigma \right),\hfill \\ \hfill 0.115& \le {\mathsf{\Omega }}_{\mathrm{CDM}}{h}^2\le 0.125\left(5\sigma \right).\hfill \end{array}$$

The current theoretical uncertainty in MSSM predictions for $M_h$ is approximately $1.5\mathrm{GeV}$ [72]. However, we chose a range of $123\mathrm{GeV}\le M_h\le 127\mathrm{GeV}$. For $M_W$, we selected the $2\sigma$ range of $M_W^{\mathrm{avg}}$. The values for $\mathrm{BR}(B\to X_s\gamma )$ and $\mathrm{BR}(B_s\to {\mu }^{+}{\mu }^{-})$ were also set at the $2\sigma$ level of their experimentally measured values [73,74]. The lower limit on ${\mathsf{\Omega }}_{\mathrm{CDM}}$ can be overlooked, as other particle species may contribute to the dark matter relic abundance. Consequently, we focus solely on points that are consistent with the LSP neutralino relic density, ensuring it aligns with or is lower than the Planck measurements [75]. Nonetheless, we explore the parameter space where both upper and lower limits can be satisfied.

4.2. $\Delta a_{\mu }$ and ${\mathsf{\Omega }}_{\mathrm{CDM}}{h}^2$ in the sNHSSM

In a previous study [49], we investigated the ${(g-2)}_{\mu }$ discrepancy and ${\mathsf{\Omega }}_{\mathrm{CDM}}{h}^2$ within the framework of the sMSSM. We carefully examined the most suitable parameter space, revealing correlations between the input parameters and the various constraints, which we reproduce here for completeness:

$$\begin{array}{ccc}0\le & m_{0_{1,2}}& \le 3\mathrm{TeV}\\ 7\le & m_{0_3}& \le 15\mathrm{TeV}\\ 0.8\le & M_{1/2}& \le 2\mathrm{TeV}\\ 0\le & \mu & \le 570\mathrm{GeV}\end{array}$$

The constraints on $m_{0_{1,2}}$ and $\mu$ arise from considerations related to $\Delta a_{\mu }^{\mathrm{WA}}$, while those on $m_{0_3}$ and $M_{1/2}$ are derived from $M_h$ and $m_{\tilde{g}}$, respectively.

The addition of nonholomorphic SSB terms in the MSSM offers rich phenomenology. The ${\mu }^{\prime }$ term, in particular, can have very important consequences. As already mentioned, the ${\mu }^{\prime }$ term does not affect the couplings of the sparticles but only the mass matrices of the neutralinos and charginos. This, in turn, has the potential to significantly change ${\mathsf{\Omega }}_{\mathrm{CDM}}{h}^2$ while leaving other phenomenological aspects unchanged. We have specifically checked that the constraints and correlations between the parameters described above are not affected by the inclusion of ${\mu }^{\prime }$.

As we said above, the inclusion of ${\mu }^{\prime }$ affects mainly the chargino and neutralino masses; therefore, we find similar values for the other SUSY masses and parameters to the ones presented in Ref. [49]. According to those results, the relic density bounds are satisfied by neutralinos with a sizeable higgsino component; then, LSP composition and coanihilations with charginos will change with the new parameter ${\mu }^{\prime }$, leading to interesting results when both terms are below 600 GeV.

The impact of the new parameter can be seen in Figure 2, where we present the NHSSM predictions in the $M_A$-$\mu$ plane. The purple + and cyan × markers represent parameter points within the $2\sigma$ range of $\Delta a_{\mu }^{\mathrm{IB}}$ and $\Delta a_{\mu }^{\mathrm{WA}}$, respectively. Points with predictions below the IB bounds are not shown, despite being compatible with the BMW constraints. All displayed points respect the upper $5\sigma$ constraint of the Planck limit, but only the green dots satisfy the lower bound as well. Hence, only the green points can explain the Planck results, while for the remaining points, additional DM components are required. To indicate the models that can simultaneously accommodate both DM constraints and provide a solution to the ${(g-2)}_{\mu }$ problem, we mark with black (red) stars the points that satisfy the $2\sigma$ range of $\Delta a_{\mu }^{\mathrm{IB}}$ ($\Delta a_{\mu }^{\mathrm{WA}}$) while also ensuring that ${\mathsf{\Omega }}_{\mathrm{CDM}}{h}^2$ remains within $5\sigma$ of the Planck limit. In Ref. [49], it was shown that the $5\sigma$ range of ${\mathsf{\Omega }}_{\mathrm{CDM}}{h}^2$ prefers $\mu >600\mathrm{GeV}$, while $\Delta a_{\mu }^{\mathrm{WA}}$ favors values of $\mu$ below $450\mathrm{GeV}$. Consequently, there are only very narrow regions where both constraints can be simultaneously satisfied. Notably, most of the cyan × lie in the region where $\mu \le 450\mathrm{GeV}$, as before. However, after the inclusion of ${\mu }^{\prime }$, the green points can now be found in the region where $\mu <450\mathrm{GeV}$. The additional contribution to ${\mathsf{\Omega }}_{\mathrm{CDM}}{h}^2$ comes from the presence of ${\mu }^{\prime }$ in the neutralino and chargino mass matrices, making it easier to explain the $\Delta a_{\mu }^{\mathrm{WA}}$ discrepancy while still adhering to the upper and lower limits for ${\mathsf{\Omega }}_{\mathrm{CDM}}{h}^2$. This can be seen by the presence of black and red stars that satisfy the $2\sigma$ range of $\Delta a_{\mu }^{\mathrm{IB}}$ and $\Delta a_{\mu }^{\mathrm{WA}}$ while also ensuring that ${\mathsf{\Omega }}_{\mathrm{CDM}}{h}^2$ remains within $5\sigma$ of the Planck limit.

It is worth noting that these constraints become less stringent when considering $\Delta a_{\mu }^{\mathrm{IB}}$, which permits smaller values of $\Delta a_{\mu }^{\mathrm{NHSSM}}$. Furthermore, since $\Delta a_{\mu }^{\mathrm{BMW}}$ is compatible with zero discrepancy, it allows the parameter space to extend to even higher values of $\mu$.

In Figure 3, we show the NHSSM predictions in the ${\mu }^{\prime }-\mu$ plane. The color coding remains the same as in the previous figure. We can see that lower values of $\mu$ and ${\mu }^{\prime }$ favor $\Delta a_{\mu }^{\mathrm{WA}}$ prediction at $2\sigma$, while the opposite applies to the $5\sigma$ range of ${\mathsf{\Omega }}_{\mathrm{CDM}}{h}^2$. Points satisfying $\Delta a_{\mu }^{\mathrm{IB}}$ ($\Delta a_{\mu }^{\mathrm{WA}}$) and ${\mathsf{\Omega }}_{\mathrm{CDM}}{h}^2$ constraints, shown as black (red) stars, are found along one of the diagonals of the plot. As we said before, the main effect of the inclusion of ${\mu }^{\prime }$ arises from the shifting of the masses of neutralinos and charginos. This is significant in our model because the $5\sigma$ range of ${\mathsf{\Omega }}_{\mathrm{CDM}}{h}^2$ is reached mainly in cases where the LSP neutralino has a significant higgsino component together with a moderate degeneracy with the NLSP chargino, as was shown in Ref. [49]. Indeed, in Figure 3 we can see that inclusion of ${\mu }^{\prime }$ increases the number of models with relic density inside Planck range, while lower values explain better the $\Delta a_{\mu }^{\mathrm{WA}}$ anomaly. However, the overlapping regions satisfying both limits are significantly increased with respect to the area found in Ref. [49]. In addition, in Figure 4 we can see that the range of masses for the LSP is similar to that found in the former reference. However, models with lower values of $\mu$ predict LSPs with large scattering cross-sections with nucleons. Therefore, as we will see in the next section, models that can explain DM predict spin-independent cross-sections that have already been excluded by direct detection experiments.

4.3. Spin-Independent WIMP–Proton Cross-Section Constraints

As we mentioned in the previous section, the LSP predicted in the framework under consideration is a well-tempered mixture of bino and higgsino. Therefore, the WIMP–nucleon cross-section can reach significant values. In Figure 5, we present our results for the spin-independent WIMP–proton cross-section, ${\sigma }_{\mathrm{SI}}^p\left(\mathrm{pb}\right)$, as a function of the lightest neutralino mass, $M_{{\tilde{\chi }}_1^0}$. We also present the current direct detection constraints from the XENONnt (dashed blue), LUX (orange), and future projections from the DARWIN (dashed purple) experiments. The left panel includes a color bar representing the values of ${\mathsf{\Omega }}_{\mathrm{CDM}}{h}^2$, while the right panel highlights parameter points satisfying the $2\sigma$ constraints of $\Delta a_{\mu }^{\mathrm{IB}}$ (purple +) and $\Delta a_{\mu }^{\mathrm{WA}}$ (cyan ×). We scale ${\sigma }_{\mathrm{SI}}^p$ by a factor of ${\mathsf{\Omega }}_{\mathrm{CDM}}{h}^2/0.114$ to account for the possibility that additional particle species may contribute to the total DM relic density.

From the left panel, it is evident that almost all parameter points satisfying Planck’s $5\sigma$ limits on ${\mathsf{\Omega }}_{\mathrm{CDM}}{h}^2$ are already excluded by the constraints from LUX and XENON experiments. This suggests that the model remains viable only if the lower bound on ${\mathsf{\Omega }}_{\mathrm{CDM}}{h}^2$ is relaxed. Meanwhile, the right panel shows that viable regions still exist where the $2\sigma$ constraints of $\Delta a_{\mu }^{\mathrm{BMW}}$, $\Delta a_{\mu }^{\mathrm{IB}}$, and $\Delta a_{\mu }^{\mathrm{WA}}$ are simultaneously satisfied. However, it should be kept in mind that $\Delta a_{\mu }^{\mathrm{BMW}}$ is compatible with zero, opening up a region of parameter space. These points correspond to those with $\Delta a_{\mu }$ below the IB region, which are not shown in Figure 5.

Finally, in 

Table 1. Three benchmark points in the sNHSSM that satisfy the $M_h$, $M_W$, and ${\sigma }_{\mathrm{SI}}^p$ constraints, along with their corresponding predictions for key observables.

Parameter	${\mathit{P}}_1$	${\mathit{P}}_2$	${\mathit{P}}_3$
$m_{0_{1,2}}$	$2.245\mathrm{TeV}$	$0.852\mathrm{TeV}$	$1.833\mathrm{TeV}$
$m_{0_3}$	$10.9422\mathrm{TeV}$	$11.309\mathrm{TeV}$	$9.701\mathrm{TeV}$
$M_{1/2}$	$968.671\mathrm{GeV}$	$1845\mathrm{GeV}$	$1117.45\mathrm{GeV}$
$tan\beta$	$59.66$	$42.39$	$34.45$
$A_0$	$-0.194\mathrm{GeV}$	$0.254\mathrm{GeV}$	$-0.273\mathrm{GeV}$
$M_A$	$7.964\mathrm{TeV}$	$8.214\mathrm{TeV}$	$1.425\mathrm{TeV}$
$\mu$	$15.66\mathrm{GeV}$	$300.0\mathrm{GeV}$	$331.45\mathrm{GeV}$
${\mu }^{\prime }$	$71.12\mathrm{GeV}$	$564.2\mathrm{GeV}$	$528.48\mathrm{GeV}$
$m_{{\tilde{u}}_1}–m_{{\tilde{u}}_6}$	1.285–7.562 $\mathrm{TeV}$	3.064–9.379$\mathrm{TeV}$	2.541–8.160 $\mathrm{TeV}$
$M_h$	$124.153\mathrm{GeV}$	$123.67\mathrm{GeV}$	$123.66\mathrm{GeV}$
$M_W$	$80.402\mathrm{GeV}$	$80.405\mathrm{GeV}$	$80.400\mathrm{GeV}$
$\Delta a_{\mu }^{\mathrm{NHSSM}}$	$27.0\times {10}^{-10}$	$5.20\times {10}^{-10}$	$1.56\times {10}^{-10}$
${\mathsf{\Omega }}_{\mathrm{CDM}}{h}^2$	$0.004$	$0.008$	$0.117$
${\sigma }_{\mathrm{SI}}^p\left(\mathrm{pb}\right)\times {\mathsf{\Omega }}_{\mathrm{CDM}}{h}^2/0.114$	$1.99\times {10}^{-11}$	$9.78\times {10}^{-11}$	$8.04\times {10}^{-11}$

, we present three benchmark points, all of which satisfy the constraints from $M_h$, $M_W$, and the direct detection limits set by LUX. We also show the range for scalar quark mass eigenstates, denoted by $m_{{\tilde{u}}_1}–m_{{\tilde{u}}_6}$. The first point, $P_1$, falls within the $2\sigma$ range of $\Delta a_{\mu }^{\mathrm{WA}}$, but the predicted value of ${\mathsf{\Omega }}_{\mathrm{CDM}}{h}^2$ remains very low. In this case, both $\mu$ and ${\mu }^{\prime }$ are small, whereas $tan\beta$ is considerably high. Similarly, the second point, $P_2$, lies within the $2\sigma$ range of $\Delta a_{\mu }^{\mathrm{IB}}$, yet ${\mathsf{\Omega }}_{\mathrm{CDM}}{h}^2$ remains significantly lower than the Planck value. Here, $\mu$ and ${\mu }^{\prime }$ are relatively large, while $tan\beta$ remains moderate. The third point, $P_3$, predicts ${\mathsf{\Omega }}_{\mathrm{CDM}}{h}^2$ within the $5\sigma$ range; however, the $(g-2)\mu$ value is very small and is only compatible with $\Delta a_{\mu }^{\mathrm{BMW}}$.

5. Conclusions

The flavor symmetry-based minimal supersymmetric standard model (sMSSM) is proposed as an alternative to the constrained MSSM (CMSSM), which encounters difficulties aligning with LHC data. This model uses non-abelian flavor symmetry at the GUT scale to reduce free parameters to seven. Extending the sMSSM by adding a nonholomorphic (NH) soft SUSY-breaking term ${\mu }^{\prime }$ increases the input parameters to eight, forming the sNHSSM.

In this paper, we examine predictions for the muon’s anomalous magnetic moment ${(g-2)}_{\mu }$ and dark matter relic density ${\mathsf{\Omega }}_{\mathrm{CDM}}{h}^2$ in the sNHSSM. We generated the NHSSM source code for SPheno and micrOMEGAs using the Mathematica package SARAH, and then used the SSP setup for particle spectra and low-energy observable calculations, including $M_h$, $M_W$, $\Delta a_{\mu }$, and ${\mathsf{\Omega }}_{\mathrm{CDM}}{h}^2$. Our parameter scans respect constraints from the Higgs boson mass $M_h$, B-physics observables, experimental gluino mass limits, and dark matter relic density bounds.

We found values for $\mu$ lower than those in the case of the sMSSM, which allows for a larger number of models predicting ${\mathsf{\Omega }}_{\mathrm{CDM}}{h}^2$ within the $5\sigma$ range, while also predicting a sizeable contribution to $\Delta a_{\mu }^{\mathrm{NHSSM}}$. However, these models tend to predict WIMPs excluded by direct detection, so the only points predicting ${\sigma }_{\mathrm{SI}}^p$ below the LUX bounds are those compatible with the SM BMW evaluation of the ${(g-2)}_{\mu }$, which does not require a significant contribution from physics beyond the SM. For instance, we provide three benchmarks with the relevant physical observables in scenarios where the $\Delta a_{\mu }^{\mathrm{NHSSM}}$ range is computed according to three different evaluations of the deviation from the SM.