Abstract
Inspired by the astonishing \(7\sigma\) discrepancy between the recent CDFII measurement and the standard model prediction on the mass of Wboson, we investigate the \(\lambda '\)corrections to the vertex of \(\mu \rightarrow \nu _\mu e\bar{\nu _e}\) decay in the context of the Rparity violating minimal supersymmetric standard model. These corrections can raise the Wboson mass independently. Combined with recent Zpole and kaon decay measurements, \(m_W \lesssim 80.37\) GeV can be reached. We find that these vertex corrections cannot explain the CDF result entirely at the \(2\sigma\) and even \(3\sigma\) levels. However, these corrections together with the oblique contributions can be accordant with the CDFII result and relevant bounds at the \(3\sigma\) level.
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 matterantimatter 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 uptodate direct evidence shows that the NP exists, there are still indirect ways, e.g., studying the loopeffects of NP on lowenergy processes or electroweak observables, like the precision measurement of the Wboson mass.
Recently, the Collider Detector at Fermilab (CDF) collaboration at Tevatron reported a high precision measurement on the mass of Wboson with the CDFII 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 Rparity violation (RPV), especially including the \(\lambda '\hat{L} \hat{Q} \hat{D}\) superpotential term, which can explain the Bphysics anomalies in the neutral current^{Footnote 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 selfenergy 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 Rparity 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 Wboson 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 Rparity violating MSSM
As we know, the Wboson mass can be determined from the muon decay with the relation (see, e.g., Refs. [68,69,70])
which comprises the three precise inputs, the Zboson mass \(m_Z\), the Fermi constant \(G_\mu\), and the fine structure constant \(\alpha\). Here the oneloop corrections to \(\Delta r\) can be expressed as
where the SM part \(\Delta r^{\text {SM}}\) is derived first in Refs. [71, 72]. Within the NP part, the selfenergy of the renormalized Wboson 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 selfenergy sector at the oneloop level. The corrections to the vertex and box involve charginos and neutralinos. Among these oneloop contributions in the MSSM, the dominant contribution to \(m_W\) is the oneloop diagrams involving pure squarks. This dominant part in \(h^s\) can be expressed by [68]
where \(\theta _W\) is the Weinberg angle and the definition of mixing angle \(\theta _{\tilde{q}}\) is referred to Ref. [68] and the function \(F_0(x,y)=x+y\frac{2xy}{xy} \log \frac{x}{y}\) 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 Rparity violating MSSM (RPVMSSM) and can omit \(h^s\) and \(h^b\) in the particular scenario.
In RPVMSSM, 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
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 CabibboKobayashiMaskawa (CKM) matrix. In this work, we restrict the index k of the superfield \(\hat{D}_k\) to the single value 3.
Including the oneloop contribution from the RPVMSSM, the \(W \ell _l \nu _{i}\)vertex is described by the following Lagrangian
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])
where \(x_t \equiv m^2_t/m^2_{\tilde{b}_R}\) and the loop function \(f_W(x) \equiv \frac{1}{x1} + \frac{(x2) \log x}{(x1)^2}\) and other nondominant parts are eliminated. This dominant contribution is from the vertex engaged by the righthanded sbottom \(\tilde{b}_R\) (Fig. 1a) while the vertex involving lefthanded squarks (Fig. 1b) provides nondominant 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 oneloop \(\lambda '\)contribution to \(\Delta r\) only comes from \(h'_{aa}\) (the index a here is restricted to 1 or 2 at a time).
Given the purpose of this work is to investigate that to what degree, the pure \(\lambda '\) contribution, \(h'_{aa}\), can accommodate the new Wboson mass data. We can further write down the prediction of the Wboson from the pure\(\lambda '\) contributions^{Footnote 3} as
with Eqs. (1) and (6). It is clear from Eq. (7) that the righthanded 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 lefthanded 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 Wboson mass at the \(2\sigma\) level, we need \(h'_{aa}\) to fulfill \(6.34
Also, \(6.34
stay in the region
because \(R^W_{\text {NP/SM}}\) can be calculated as \(1+2h'_{aa}\), with Eq. (5). Then, we compare Eq. (10) with the Wboson 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 Zvertex. The Zboson 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
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]
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 Zdecay, this model can also make looplevel 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
where the coupling \(\delta g_{\nu }^{\text {SM}}=\frac{1}{2}\) and the formulas of \(\delta g_{\nu }^{(\prime )ij}\) is given by
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
Due to that \(h'_{aa}\leqslant 0\), Eq. (15) induces the \(2(3)\sigma\) bounds \(4.6(6.0)
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.
Different from the pureRPV case that only parameters \((\tilde{\lambda }'_{133},m_{\tilde{b}_R})\) are focused on, in the following we further consider nondecoupled masses of stops and gauginos, and the parameters are collected in Table 1. Then, we utilize FeynHiggs2.18.1 [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 Higgslike boson in \(122
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 Wmass 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]
Parameters 
Sets 
Parameters 
Sets 

\(\tan \beta\) 
15 
\(M_{\tilde{L}_{1,2,3}}=M_{\tilde{E}_{1,2,3}}\) 
2000 
\(\mu\) 
1000 
\(M_{\tilde{Q}_{1,2}}=M_{\tilde{U}_{1,2}}=M_{\tilde{D}_{1,2}}\) 
\(10^4\) 
\(M_1\) 
500 
\(M_{\tilde{Q}_{3}}\) 
\(1500\sim 3000\) 
\(M_2\) 
1000 
\(M_{\tilde{U}_3}\) 
\(1500\sim 10^4\) 
\(M_3\) 
5000 
\(M_{\tilde{D}_3}\) 
\(1300\sim 3000\) 
\(M_A\) 
2000 
\(A_{u,c}=A_{d,s}=A_l\) 
1500 
\(m_t\) 
173.3 
\(A_t\), \(A_b\) 
\(5000\sim 5000\) 
4 Conclusions
In this paper, inspired by the astonishing \(7\sigma\) discrepancy between the CDFII measurement and the SM prediction on the mass of Wboson, we performed a phenomenological analysis on the muon decay that is relevant to the W mass under the framework of RPVMSSM, to access whether such a deviation can be accommodated by this NP model. We focused on the oneloop 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 RPVMSSM. The numerical results shown in Fig. 2 imply that pure \(\lambda '\)contributions in the RPVMSSM 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.
Availability of data and materials
All data generated or analyzed during this study are included in this published article.
Notes

Some anomalies are observed in the \(b\rightarrow s\mu ^+\mu ^\) decays include \(P'_5\) [56], the branching fraction of \(B_s \rightarrow \phi \mu ^+ \mu ^{}\) [57], etc. The ratios \(R_{K^{(*)}}\) in the \(b\rightarrow s\ell ^+\ell ^\) (\(\ell =e,\mu\)) processes, have been reported recently by the LHCb Collaboration [58] that they are in agreement with the SM predictions, and this new result overturns the previous ones which show anomalies in \(R_{K^{(*)}}\) [59,60,61].

In this scenario, the \(\lambda '\)contributions to the \(\mu \rightarrow \nu _i e\bar{\nu _i}\) through Z penguin vanish as well.

There are always contributions from the original MSSM framework, while we can set sufficiently heavy masses of lefthanded squarks, sleptons, and gauginos in soft breaking terms to screen these effects.
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Acknowledgements
We thank Chengfeng Cai and Seishi Enomoto for valuable discussions. This work is supported in part by the National Natural Science Foundation of China under Grant Nos. 11875327 and 12275367, the Fundamental Research Funds for the Central Universities, and the Sun YatSen University Science Foundation. F.C. is also supported by the CCNUQLPL Innovation Fund (QLPL2021P01).
Funding
This research was supported in part by the National Natural Science Foundation of China under Grant Nos. 11875327 and 12275367, the Fundamental Research Funds for the Central Universities, and the Sun YatSen University Science Foundation. F.C. is also supported by the CCNUQLPL Innovation Fund (QLPL2021P01).
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Authors and Affiliations
School of Physics and Electronic Information, Shangrao Normal University, Shangrao, 334001, China
MinDi Zheng
School of Physics, Sun YatSen University, Guangzhou, 510275, China
MinDi Zheng, FengZhi Chen & HongHao Zhang
Key Laboratory of Quark and Lepton Physics (MOE), Central China Normal University, Wuhan, 430079, China
FengZhi Chen
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M.D. contributed to the study conception and design. All authors commented on previous versions of the manuscript. All authors read and approved the final manuscript.
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