1 Introduction

In the past decades, the observation of striking agreement between the standard model (SM) predictions and the experimental results in a vast number of particle interactions has shown up the powerful predicted capacity of the SM. However, the SM is not the final answer to the particle physics, as it is unable to explain several phenomena, including the matter-antimatter asymmetry, the origin of neutrino mass, the hierarchy problem, and the candidate of dark matter. These strongly call for some new physics (NP) beyond the SM. Although no up-to-date direct evidence shows that the NP exists, there are still indirect ways, e.g., studying the loop-effects of NP on low-energy processes or electroweak observables, like the precision measurement of the W-boson mass.

Recently, the Collider Detector at Fermilab (CDF) collaboration at Tevatron reported a high precision measurement on the mass of W-boson with the CDF-II detector. The measured value is given by \(m_W^{\text {CDF}}=80.4335\pm 0.0094~\text {GeV}\) [1] with better precision than all other previous measurements and is \(7\sigma\) from the SM prediction \(m_W^\text {SM}=80.357\pm 0.006~\text {GeV}\) [2]. If the measurement is confirmed in the future, such an astonishing tension will undoubtedly be a strong challenge to the SM. After this exciting \(m_W^\text {CDF}\) reported, plentiful theoretical researches [3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38] have emerged in a short time.

Before this profound result, there are already some anomalies indicating the clues of NP, e.g., the recent average values of the observables \(R_{D^{(*)}}\), reported by the Heavy Flavor Averaging Group [39,40,41], are about \(3.2\sigma\) away from the corresponding SM predictions [42,43,44,45,46,47,48,49,50], considering the \(R_{D}\) and \(R_{D^*}\) total correlation \(-0.29\). To explain these anomalies, there are numerous phenomenological studies combined with the \(m_W^{\text {CDF}}\) measurement in different models (e.g., see Refs. [34, 51,52,53,54,55]). In this work, we utilize the minimal supersymmetric standard model (MSSM) extended by the R-parity violation (RPV), especially including the \(\lambda '\hat{L} \hat{Q} \hat{D}\) superpotential term, which can explain the B-physics anomalies in the neutral currentFootnote 1 or/and the charged one (see, e.g., Refs. [62,63,64,65]). Thus, further investigations on this framework for the \(m_W^{\text {CDF}}\) explanation are necessary. Although it is found that the MSSM can provide some parameter points which can raise \(m_W\) into the \(2\sigma\) accordance region [9], mainly through bosonic self-energy contributions relevant to the oblique corrections [66, 67], the stop mass in the solution with \(m_{\tilde{t}}\lesssim 1\) TeV is not suit for general collider search scenarios. Thus, it is also worth studying other corrections to \(m_W\) in the extended MSSM framework, considering the general bounds for colored sparticle masses at the Large Hadron Collider (LHC). Above all, we will study corrections to the vertex \(W\ell \nu\) from the R-parity violating interaction \(\lambda '\hat{L} \hat{Q} \hat{D}\) and get an enhancement to \(m_W\), which is independent of the oblique corrections.

This paper is organized as follows. In Section 2, we introduce the vertex corrections to the W-boson mass in the MSSM framework extended by RPV. Then, we show the numerical results and discussions in Section 3. Our conclusions are presented in Section 4.

2 The contribution to \(m_W\) from the R-parity violating MSSM

As we know, the W-boson mass can be determined from the muon decay with the relation (see, e.g., Refs. [68,69,70])

$$\begin{aligned} \frac{m^2_W}{m^2_Z}=\frac{1}{2}+\sqrt{\frac{1}{4}-\frac{\pi \alpha }{\sqrt{2} G_\mu m^2_Z} (1+\Delta r)}, \end{aligned}$$
(1)

which comprises the three precise inputs, the Z-boson mass \(m_Z\), the Fermi constant \(G_\mu\), and the fine structure constant \(\alpha\). Here the one-loop corrections to \(\Delta r\) can be expressed as

$$\begin{aligned} \Delta r=\Delta r^{\text {SM}}+h^{s}+h^{v}+h^{b}, \end{aligned}$$
(2)

where the SM part \(\Delta r^{\text {SM}}\) is derived first in Refs. [71, 72]. Within the NP part, the self-energy of the renormalized W-boson is denoted by \(h^s\), and the vertex and box corrections to the \(\mu \rightarrow \nu _\mu e\bar{\nu _e}\) decay are denoted by \(h^v\) and \(h^b\), respectively. In the MSSM, the pure squarks (sleptons) only engage the self-energy sector at the one-loop level. The corrections to the vertex and box involve charginos and neutralinos. Among these one-loop contributions in the MSSM, the dominant contribution to \(m_W\) is the one-loop diagrams involving pure squarks. This dominant part in \(h^s\) can be expressed by [68]

$$\begin{aligned} (h^s)_{\text {dom}}= & {} -\frac{3G_{\mu } \cos \theta ^2_W}{8\sqrt{2}\sin \theta ^2_W\pi ^2} \Big [ - \sin ^2\theta _{\tilde{t}}\cos ^2 \theta _{\tilde{t}} F_0 (m^2_{\tilde{t}_1}, m^2_{\tilde{t}_2}) - \sin ^2\theta _{\tilde{b}}\cos ^2 \theta _{\tilde{b}} F_0 (m^2_{\tilde{b}_1}, m^2_{\tilde{b}_2})\nonumber \\{} & {} + \cos ^2 \theta _{\tilde{t}}\cos ^2 \theta _{\tilde{b}} F_0 (m^2_{\tilde{t}_1}, m^2_{\tilde{b}_1}) + \cos ^2 \theta _{\tilde{t}}\sin ^2 \theta _{\tilde{b}} F_0 (m^2_{\tilde{t}_1}, m^2_{\tilde{b}_2})\nonumber \\{} & {} + \sin ^2 \theta _{\tilde{t}}\cos ^2 \theta _{\tilde{b}} F_0 (m^2_{\tilde{t}_2}, m^2_{\tilde{b}_1}) + \sin ^2 \theta _{\tilde{t}}\sin ^2 \theta _{\tilde{b}} F_0 (m^2_{\tilde{t}_2}, m^2_{\tilde{b}_2})\Big ], \end{aligned}$$
(3)

where \(\theta _W\) is the Weinberg angle and the definition of mixing angle \(\theta _{\tilde{q}}\) is referred to Ref. [68] and the function \(F_0(x,y)=x+y-\frac{2xy}{x-y} \log \frac{x}{y}\) with the extra properties \(F_0(m^2,m^2)=0\) and \(F_0(m^2,0)=m^2\). Thus, one can see that \(h^s\) is sensitive to the mass splitting between the isospin partners due to the factor \(\cos ^2 \theta _{\tilde{t}} \cos ^2 \theta _{\tilde{b}}\). Obviously, \(h^s\) can be negligible when the soft breaking masses \(M_{\tilde{Q}_i}\) are sufficiently heavy compared to the chiral mixing. In this work, we focus on the vertex corrections \(h^v\) affected by the \(\lambda '\)-coupling in the R-parity violating MSSM (RPV-MSSM) and can omit \(h^s\) and \(h^b\) in the particular scenario.

In RPV-MSSM, the \(\lambda '\)-superpotential term \(\mathcal{W}=\lambda '_{ijk} \hat{L}_i \hat{Q}_j \hat{D}_k\) leads to the related Lagrangian in the mass basis

$$\begin{aligned} \mathcal{L}^{\text {LQD}}= & {} \lambda '_{ijk} \left( \tilde{\nu }_{Li} \bar{d}_{Rk} d_{Lj} + \tilde{d}_{Lj} \bar{d}_{Rk} \nu _{Li} + \tilde{d}_{Rk}^*\bar{\nu }_{Li}^c d_{Lj} \right) \nonumber \\{} & {} - \tilde{\lambda }'_{ijk} \left( \tilde{l}_{Li} \bar{d}_{Rk} u_{Lj} + \tilde{u}_{Lj} \bar{d}_{Rk} l_{Li} + \tilde{d}_{Rk}^*\bar{l}_{Li}^c u_{Lj} \right) + {\text {h.c.}}, \end{aligned}$$
(4)

where the generation indices \(i,j,k=1,2,3\), while the color ones are omitted, and “c” indicates the charge conjugated fermions. In this paper, all the repeated indices are defaulted to be summed over unless otherwise stated. The relation between \(\lambda '\) and \(\tilde{\lambda }'\) is \(\tilde{\lambda }'_{ijk}=\lambda '_{ij'k} K^{*}_{jj'}\) with K being the Cabibbo-Kobayashi-Maskawa (CKM) matrix. In this work, we restrict the index k of the superfield \(\hat{D}_k\) to the single value 3.

Including the one-loop contribution from the RPV-MSSM, the \(W \ell _l \nu _{i}\)-vertex is described by the following Lagrangian

$$\begin{aligned} \mathcal{L}^W_{\text {eff}}=\frac{g}{\sqrt{2}} \bar{\ell _l} \gamma ^\mu P_L (\delta _{li}+h_{li}) \nu _i W^{-}_\mu + {\text {h.c.}}, \end{aligned}$$
(5)

where g is the \({\text {SU}}(2)_L\) gauge coupling, and the correction part \(h_{li}\) from the \(\lambda '\)-contributions is given by (as the analogy to the formula in Ref. [73])

$$\begin{aligned} h'_{li} = -\frac{3}{64\pi ^2} x_t f_W(x_t) \tilde{\lambda }'^{*}_{l33} \tilde{\lambda }'_{i33}, \end{aligned}$$
(6)

where \(x_t \equiv m^2_t/m^2_{\tilde{b}_R}\) and the loop function \(f_W(x) \equiv \frac{1}{x-1} + \frac{(x-2) \log x}{(x-1)^2}\) and other non-dominant parts are eliminated. This dominant contribution is from the vertex engaged by the right-handed sbottom \(\tilde{b}_R\) (Fig. 1a) while the vertex involving left-handed squarks (Fig. 1b) provides non-dominant effects and can be eliminated. Then, we consider the \(\lambda '\)-correction only to the \(W\mu \nu\)-vertex or to the \(We\nu\)-vertex at a time. This can be easily achieved by setting one of the couplings \((\tilde{\lambda }'_{133}, \tilde{\lambda }'_{233})\) dominant while neglecting the rest. Given this “single coefficient dominance” scenario, the \(\lambda '\)-corrections to the \(\mu \rightarrow \nu _\mu e\bar{\nu _e}\) box also vanish,Footnote 2 then the one-loop \(\lambda '\)-contribution to \(\Delta r\) only comes from \(h'_{aa}\) (the index a here is restricted to 1 or 2 at a time).

Fig. 1
figure 1

The diagrams of the \(W\ell \nu\)-vertex and \(Z\ell \ell ^{(')}\)-vertex (shown as examples) in the RPV-MSSM

Full size image

Given the purpose of this work is to investigate that to what degree, the pure \(\lambda '\) contribution, \(h'_{aa}\), can accommodate the new W-boson mass data. We can further write down the prediction of the W-boson from the pure-\(\lambda '\) contributionsFootnote 3 as

$$\begin{aligned} (m^{\lambda '}_W/{\text {GeV}})=80.357-15.6387 h'_{aa}=80.357+0.0743 x_t f_W(x_t) \left| \tilde{\lambda }'_{a33} \right| ^2 \end{aligned}$$
(7)

with Eqs. (1) and (6). It is clear from Eq. (7) that the right-handed sbottom mass \(m_{\tilde{b}_R}\) and the coupling \(\tilde{\lambda }'_{a33}\) are related to \(\lambda '\)-correction of \(m_W\).

3 Numerical results and discussions

In this section, we investigate the explanation of \(m_W^{\text {CDF}}\) combined with the relevant constraints. At first, we concentrate on the pure \(\lambda '\)-effects assuming the soft breaking masses of gauginos and left-handed squarks (sleptons) are sufficiently heavy, and then, only the model parameters \((\tilde{\lambda }'_{a33},m_{\tilde{b}_R})\) are involved. If the pure \(\lambda '\)-contribution (see Eq. (7)) can explain the new W-boson mass at the \(2\sigma\) level, we need \(h'_{aa}\) to fulfill \(-6.34<h'_{aa}\times 10^3<-3.47\). With Eq. (6), this bound provides

$$\begin{aligned} 0.7313< x_t f_W(x_t) \left| \tilde{\lambda }'_{a33} \right| ^2<1.3357. \end{aligned}$$
(8)

Also, \(-6.34<h'_{aa}\times 10^3<-3.47\) let the ratio

$$\begin{aligned} R^W_{\text {NP/SM}}\equiv \Gamma (W\rightarrow \ell _a\nu _a)_{\text {NP}}/\Gamma (W\rightarrow \ell _a\nu _a)_{\text {SM}} \end{aligned}$$
(9)

stay in the region

$$\begin{aligned} -12.7<(R^{W}_{\text {NP/SM}}-1)\times 10^{3}<-6.9, \end{aligned}$$
(10)

because \(R^W_{\text {NP/SM}}\) can be calculated as \(1+2h'_{aa}\), with Eq. (5). Then, we compare Eq. (10) with the W-boson partial width ratios \(R^W_{l/l'}\equiv \Gamma (W\rightarrow l\nu )/\Gamma (W\rightarrow l'\nu )\), and their experimental results are given as \(R^W_{\mu /e}=0.996\pm 0.008\), \(R^W_{\tau /\mu }=1.008\pm 0.031\), and \(R^W_{\tau /e}=1.043\pm 0.024\) [74]. It is found that the \(m_W\) explanation demands much stronger bounds, whenever the NP exists in the \(\mu\) or e channel (the \(\tau\) flavor is assumed decoupled with the NP for simplicity).

As shown in Fig. 1c and d, the NP effects on the \(W\ell \nu\)-vertex will also inevitably affect the Z-vertex. The Z-boson partial width ratios \(R^Z_{l/l'}\equiv \Gamma (Z\rightarrow ll)/\Gamma (Z\rightarrow l'l')\) are measured as \(R^Z_{\mu /e}=1.0001\pm 0.0024\), \(R^Z_{\tau /\mu }=1.0010\pm 0.0026\), and \(R^Z_{\tau /e}=1.0020\pm 0.0032\) [74], which all constrain the coupling \(g_{\ell _L}\) in the effective Lagrangian

$$\begin{aligned} \mathcal{L}^Z_{\text {eff}}=\frac{g}{\cos \theta _W} \bar{\ell }_i \gamma ^\mu \left[ g_{\ell _L}^{ij} P_L + g_{\ell _R}^{ij} P_R \right] \ell _j Z_\mu , \end{aligned}$$
(11)

where \(g_{\ell _L}^{ij}=\delta ^{ij}g_{\ell _L}^{\text {SM}}+\delta g_{\ell _L}^{ij}+\delta g_{\ell _L}^{\prime ij}\) and \(g_{\ell _R}^{ij}=\delta ^{ij}g_{\ell _R}^{\text {SM}}\), with \(g_{\ell _L}^{\text {SM}}=-\frac{1}{2}+\sin ^2\theta _W\) and \(g_{\ell _R}^{\text {SM}}=\sin ^2\theta _W\). The formulas of \(\delta g_{\ell _L}^{ij}\) and \(\delta g_{\ell _L}^{\prime ij}\) are [75]

$$\begin{aligned} (32\pi ^2) \delta g_{\ell _L}^{ij}= & {} 3 \tilde{\lambda }'_{j33} \tilde{\lambda }'^{*}_{i33} \biggl \{-x_t (1 + \log x_t) + \frac{m_Z^2}{18 m^2_{\tilde{b}_R}}\biggl [(11 - 10 \sin ^2\theta _W)\nonumber \\{} & {} + (6 - 8 \sin ^2\theta _W)\log x_t + \frac{1}{10}(-9 + 16 \sin ^2\theta _W)\frac{m_Z^2}{m_t^2} \biggr ] \biggl \},\nonumber \\ (32\pi ^2) \delta g_{\ell _L}^{\prime ij}= & {} \tilde{\lambda }'_{j33} \tilde{\lambda }'^{*}_{i33} \frac{m^2_Z}{m^2_{\tilde{t}_L}} \biggl [ \left( -\frac{2}{3} \sin ^2\theta _W \right) \left( \log \left( \frac{m^2_Z}{m^2_{\tilde{t}_L}} \right) - i \pi - \frac{1}{2}\right) + \left( -\frac{1}{6} + \frac{1}{9} \sin ^2\theta _W \right) \biggr ]. \end{aligned}$$
(12)

Here we can define \(B^{ij} \equiv (32\pi ^2) (\delta g_{\ell _L}^{ij}+\delta g_{\ell _L}^{\prime ij})\) and further get the bound \(|B^{aa}|<0.35(0.53)\) at the \(2(3)\sigma\) level. Given that the mass of \(\tilde{t}_L\) is set sufficiently heavy, the \(\delta g_{\ell _L}^{\prime ij}\) part can be eliminated.

As to the invisible Z-decay, this model can also make loop-level contributions to the \(Z\rightarrow \nu \bar{\nu }\), i.e., \(\ell\) exchanged with \(\nu\) and u(\(\tilde{u}_L\)) exchanged by d(\(\tilde{d}_L\)) in Fig. 1c, d. Then, the effective number of light neutrinos \(N_\nu\), which is defined by \(\Gamma _{\text {inv}}=N_\nu \Gamma _{\nu \bar{\nu }}^{\text {SM}}\) [76], will constrain the couplings via

$$\begin{aligned} N_\nu =\left| 1+\frac{\delta g_{\nu }^{aa}+\delta g_{\nu }^{\prime aa}}{\delta g_{\nu }^{\text {SM}}} \right| ^2+2, \end{aligned}$$
(13)

where the coupling \(\delta g_{\nu }^{\text {SM}}=\frac{1}{2}\) and the formulas of \(\delta g_{\nu }^{(\prime )ij}\) is given by

$$\begin{aligned} (32\pi ^2) \delta g_{\nu }^{ij}= & {} \lambda '_{j33} \lambda '^{*}_{i33}\frac{m^2_Z}{m^2_{\tilde{b}_R}}\biggl [ \left( -1+\frac{2}{3} \sin ^2\theta _W \right) \left( \log \left( \frac{m^2_Z}{m^2_{\tilde{b}_R}} \right) - i \pi - \frac{1}{3} \right) + \left( -\frac{1}{12} + \frac{4}{9} \sin ^2\theta _W \right) \biggr ],\nonumber \\ (32\pi ^2) \delta g_{\nu }^{\prime ij}= & {} \lambda '_{j33} \lambda '^{*}_{i33} \frac{m^2_Z}{m^2_{\tilde{b}_L}} \biggl [\left( \frac{1}{3} \sin ^2\theta _W \right) \left( \log \left( \frac{m^2_Z}{m^2_{\tilde{b}_L}} \right) - i \pi - \frac{1}{2}\right) + \left( -\frac{1}{6} + \frac{4}{9} \sin ^2\theta _W \right) \biggr ]. \end{aligned}$$
(14)

Then, the measurement \(N_\nu ^{\text {exp}}=2.9840(82)\) [76] will make constraints.

Except the purely leptonic decays of W/Z boson, the \(\mu \rightarrow e\bar{\nu }_e\nu _\mu\) and \(\tau \rightarrow \ell \bar{\nu }_{\ell }\nu _\tau\) decays, which contain the \(W\ell \nu\)-vertex, should also be considered. The fraction ratios \(\mathcal{B}(\tau \rightarrow \mu \bar{\nu }_\mu \nu _\tau )/\mathcal{B}(\tau \rightarrow e\bar{\nu }_e\nu _\tau )\), \(\mathcal{B}(\tau \rightarrow e\bar{\nu }_e\nu _\tau )/\mathcal{B}(\mu \rightarrow e\bar{\nu }_e\nu _\mu )\), and \(\mathcal{B}(\tau \rightarrow \mu \bar{\nu }_\mu \nu _\tau )/\mathcal{B}(\mu \rightarrow e\bar{\nu }_e\nu _\mu )\) make the bounds [77] as

$$\begin{aligned} \frac{1+\delta ^{\text {a}2} h'_{\mu \mu }}{1+\delta ^{\text {a}1} h'_{ee}}= & {} 1.0018(14),\nonumber \\ \frac{1}{1+\delta ^{\text {a}2} h'_{\mu \mu }}= & {} 1.0010(14), \nonumber \\ \frac{1}{1+\delta ^{\text {a}1} h'_{ee}}= & {} 1.0029(14). \end{aligned}$$
(15)

Due to that \(h'_{aa}\leqslant 0\), Eq. (15) induces the \(2(3)\sigma\) bounds \(-4.6(-6.0)<h'_{ee}\times 10^3 \leqslant 0\) or \(-1(-2.4)<h'_{\mu \mu }\times 10^3 \leqslant 0\) for a restricted to 1 or 2, respectively. Similarly, the decays of kaons and pions also make the bounds, and the most stringent ones [77] induce \(-1.4(-3.2)<h'_{ee}\times 10^3 \leqslant 0\) or \(-0.8(-1.7)<h'_{\mu \mu }\times 10^3 \leqslant 0\) at the \(2(3)\sigma\) level, which are provided by the constraints in the \(K^+\rightarrow \ell ^+\nu (\gamma )\) and \(\pi \rightarrow \ell \nu (\gamma )\) decays, respectively. However, the \(2\sigma\)-level \(m^{\text {CDF}}_W\) explanation demands \(-6.34<h'_{aa}\times 10^3<-3.47\) as mentioned before, and thus, one can see exactly the \(2\sigma\)-level explanation is already excluded by these decays of kaons and pions, while we can still investigate the degree of the NP raising \(m_W\). Given that \(h'_{\mu \mu }\) is constrained more strongly than \(h'_{ee}\), in the following we focus the NP in the e flavor. Thus, we set \(\lambda '_{133}\) dominant and other \(\lambda '\) couplings negligible. Then with the CKM rotation, \(\tilde{\lambda }'_{ijk}=\lambda '_{ij'k} K^{*}_{jj'}\), the nonzero \(\tilde{\lambda }'\) couplings \(|\tilde{\lambda }'_{133}| \approx |\lambda '_{133}|\), \(|\tilde{\lambda }'_{123}| \approx 0.04 |\lambda '_{133}|\), and \(|\tilde{\lambda }'_{113}| \approx 10^{-3} |\lambda '_{133}|\). As we consider the real number \(\lambda '_{133}\) varying in \(0\leqslant \lambda '_{133} \leqslant 3\), it is checked that the effects on the \(m^{\text {CDF}}_W\) explanation and constraints from the couplings \(\tilde{\lambda }'_{123}\) and \(\tilde{\lambda }'_{113}\) are negligible. It is worth to mention that all the model parameters are set at the scale of around TeV. Given that only (axial-)vector currents are involved in relevant processes discussed before, the couplings of these currents are kept nearly the same when the scale runs down to the electroweak scale.

Combining the bounds introduced above with the W mass explanation, the allowed regions are shown in Fig. 2. The two areas allowed by \(Z\rightarrow \ell \ell\) and kaon decays overlap almost entirely at the \(2\sigma\) level, while the \(Z\rightarrow \ell \ell\) bound is stronger at the \(3\sigma\) level. The bounds of \(N_\nu ^{\text {exp}}\) is more stringent than the former two at the \(2\sigma\) level, but the loosest at the \(3\sigma\) level. In the common region of these three observables at the \(2\sigma\) level, \(m^{\lambda '}_W\) can be raised to around 80.37 GeV at most, while it cannot reach the value to explain \(m^{\text {CDF}}_W\) as predicted. Even at the \(3\sigma\) level, there are still none common areas for \(m^{\text {CDF}}_W\) and bounds besides the one when \(m_{\tilde{b}_R} \lesssim 600\) GeV, but this mass scale is already excluded by LHC searches [78,79,80]. Therefore, we find that the pure \(\lambda '\) contributions cannot fully solve the \(m_W\) problem unless with other effects, e.g., the oblique corrections [9, 81]. Thus, we will further study the combination explanation with the \(\lambda '\)-contributions and the oblique ones of the MSSM framework.

Fig. 2
figure 2

The regions of \((m_{\tilde{b}_R},|\tilde{\lambda }'_{133}|)\) at the \(2\sigma\) (left) and \(3\sigma\) (right) levels. The pure \(\lambda '\)-contributions to the \(m^{\text {CDF}}_W\) explanation are denoted by the blue. The areas allowed by \(R^Z_{\ell /\ell '}\) and \(N_\nu ^{\text {exp}}\) are shown by the green and gray, respectively. The areas filled by the red are allowed by the data of \(K^+\rightarrow \ell ^+\nu (\gamma )\) decay. The dashed lines express the \(m^{\lambda '}_W\) value (GeV) enhanced by vertex corrections

Full size image

Different from the pure-RPV case that only parameters \((\tilde{\lambda }'_{133},m_{\tilde{b}_R})\) are focused on, in the following we further consider non-decoupled masses of stops and gauginos, and the parameters are collected in Table 1. Then, we utilize FeynHiggs-2.18.1 [82,83,84,85,86,87,88,89] to calculate the loop correction of MSSM part, i.e., \(h^s\), which is given as the nearly fixed value \(h^s \approx -8\times 10^{-4}\) for the parameters \(M_{\tilde{Q}_{3}}\), \(M_{\tilde{U}_3}\), \(M_{\tilde{D}_3}\), \(A_t\), and \(A_b\) varying in the ranges shown in Table 1, also keeping the mass of Higgs-like boson in \(122<m_H<128\) GeV. Then, we can further set \(M_{\tilde{Q}_3}=2.1\) TeV, \(M_{\tilde{U}_3}=10\) TeV, and \(A_t=A_b=1.5\) TeV as the benchmark point, and write down the prediction of \(m_W\) from combined contributions as

$$\begin{aligned} (m^{\text {NP}}_W/{\text {GeV}})=80.370-15.622 h'_{aa}=80.370 + 0.0742 x_t f_W(x_t)\left| \tilde{\lambda }'_{a33} \right| ^2. \end{aligned}$$
(16)

Then, the allowed regions are shown in Fig. 3. One can see that \(m_W\) can be raised to around 80.38 GeV in the \(2\sigma\)-level allowed region of the Z and kaon decays, and explaining the W-mass anomaly at \(2\sigma\) is still unachievable. However, the \(3\sigma\)-level explanation is allowed by all the bounds, within the narrow overlap near the edge of \(m^{\text {CDF}}_W\) region.

Table 1 The sets of parameters in the MSSM part. Parameters with mass dimension are given in GeV. The lower limits of squark masses refer to Ref. [78]
Full size table
Fig. 3
figure 3

Same as Fig. 2 but the dashed lines express the \(m^{\text {NP}}_W\) value (GeV) enhanced by both vertex corrections and oblique ones of the MSSM

Full size image

4 Conclusions

In this paper, inspired by the astonishing \(7\sigma\) discrepancy between the CDF-II measurement and the SM prediction on the mass of W-boson, we performed a phenomenological analysis on the muon decay that is relevant to the W mass under the framework of RPV-MSSM, to access whether such a deviation can be accommodated by this NP model. We focused on the one-loop corrections to the vertex of \(\mu \rightarrow \nu _\mu e\bar{\nu _e}\) decay, assuming that the vertex correction is only affected by a single \(\lambda '\) coupling in the RPV-MSSM. The numerical results shown in Fig. 2 imply that pure \(\lambda '\)-contributions in the RPV-MSSM are hard to accommodate the CDF measurement entirely. However, the \(\lambda '\)-corrections can help raise the prediction of W mass to be accordant with \(m^{\text {CDF}}_W\) at the \(3\sigma\) level when combined with the oblique corrections, which is shown in Fig. 3.