The $W\ell\nu$-vertex corrections to W-boson mass in the R-parity violating MSSM

Inspired by the astonishing $7\sigma$ discrepancy between the recent CDF-II measurement and the standard model prediction on the mass of $W$-boson, we investigate the $\lambda'$-corrections to the vertex of $\mu\to\nu_\mu e\bar{\nu_e}$ decay in the context of the $R$-parity violating minimal supersymmetric standard model. These corrections can raise the $W$-boson mass independently. Combined with recent $Z$-pole 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 CDF-II result and relevant bounds at the $3\sigma$ level.

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 ℓν from the R-parity violating interaction λ ′LQD 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]) (1 + ∆r), (2.1) which comprises the three precise inputs, the Z-boson mass m Z , the Fermi constant G µ , and the fine structure constant α. Here the one-loop corrections to ∆r can be expressed as where the SM part ∆r 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 µ → ν µ eν 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] (h s ) dom = − 3G µ cos θ 2 where θ W is the Weinberg angle and the definition of mixing angle θq is referred to Ref. [68] and the function F 0 (x, y) = x + y − 2xy x−y log 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 θt cos 2 θb. Obviously, h s can be negligible when the soft breaking masses MQ i are sufficiently heavy compared to the chiral mixing. In this work, we focus on the vertex corrections h v affected by the λ ′ -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 λ ′ -superpotential term W = λ ′ ijkL iQjDk 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 λ ′ andλ ′ isλ ′ ijk = λ ′ ij ′ k K * jj ′ with K being the Cabibbo-Kobayashi-Maskawa (CKM) matrix. In this work, we restrict the index k of the superfieldD k to the single value 3.
Including the one-loop contribution from the RPV-MSSM, the W ℓ l ν i -vertex is described by the following Lagrangian where g is the SU(2) L gauge coupling, and the correction part h li from the λ ′ -contributions is given by (as the analogy to the formula in Ref. [73])  vides non-dominant effects and can be eliminated. Then, we consider the λ ′ -correction only to the W µν-vertex or to the W eν-vertex at a time. This can be easily achieved by setting one of the couplings (λ ′ 133 ,λ ′ 233 ) dominant while neglecting the rest. Given this "single coefficient dominance" scenario, the λ ′ -corrections to the µ → ν µ eν e box also vanish, 2 then the one-loop λ ′ -contribution to ∆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 λ ′ 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-λ ′ contributions 3 as because R W NP/SM can be calculated as 1 + 2h ′ aa , with Eq. (2.5). Then, we compare Eq. (3.3) with the W -boson partial width ratios R W l/l ′ ≡ Γ(W → lν)/Γ(W → l ′ ν), and their experimental results are given as R W µ/e = 0.996 ± 0.008, R W τ /µ = 1.008 ± 0.031, and R W τ /e = 1.043 ± 0.024 [74]. It is found that the m W explanation demands much stronger bounds, whenever the NP exists in the µ or e channel (the τ flavor is assumed decoupled with the NP for simplicity).
As to the invisible Z-decay, this model can also make loop-level contributions to the Z → νν, i.e., ℓ exchanged with ν and u(ũ L ) exchanged by d(d L ) in figure 1c, d. Then, the effective number of light neutrinos N ν , which is defined by Γ inv = N ν Γ SM νν [76], will constrain the couplings via where the coupling δg SM ν = 1 2 and the formulas of δg (′)ij ν is given by Then the measurement N exp ν = 2.9840(82) [76] will make constraints.
Combining the bounds introduced above with the W mass explanation, the allowed regions are shown in figure 2. The two areas allowed by Z → ℓℓ and kaon decays overlap almost entirely at the 2σ level, while the Z → ℓℓ bound is stronger at the 3σ level. The bounds of N exp ν is more stringent than the former two at the 2σ level, but the loosest at the 3σ level. In the common region of these three observables at the 2σ level, m λ ′ W can be raised to around 80.37 GeV at most, while it cannot reach the value to explain m CDF W as predicted. Even at the 3σ level, there are still none common areas for m CDF W and bounds besides the one when mb R ≲ 600 GeV, but this mass scale is already excluded by LHC searches [78][79][80]. Therefore, we find that the pure λ ′ 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 λ ′ -contributions and the oblique ones of the MSSM framework.
Different from the pure-RPV case that only parameters (λ ′ 133 , mb 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]    . (3.9) Then, the allowed regions are shown in figure 3. One can see that m W can be raised to around

Conclusions
In this paper, inspired by the astonishing 7σ 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 µ → ν µ eν e decay, assuming that the vertex correction is only affected by a single λ ′ coupling in the RPV-MSSM. The numerical results shown in figure 2 imply that pure λ ′ -contributions in the RPV-MSSM are hard to accommodate the CDF measurement entirely. However, the λ ′ -corrections can help raise the prediction of W mass to be accordant with m CDF W at the 3σ level when combined with the oblique corrections, which is shown in figure 3.