W boson mass in the NP models with extra U (1) gauge group

The precise measurement of the W boson mass is closely related to the contributions of new physics (NP), which can signiﬁcantly constrain the parameter space of NP models, particularly those with an additional U (1) local gauge group. The inclusion of a new Z ′ gauge boson and gauge couplings in these models can contribute to the oblique parameters S , T , U and W boson mass at tree level. Taking into account the eﬀects of kinetic mixing, we calculate and analyze the oblique parameters S , T , U and W boson mass in such NP models in this study. It is found that the kinetic mixing eﬀects can make signiﬁcant contributions to the W boson mass, which can satisfy the recently measured W boson mass at CDF II or ATLAS by choosing appropriate values of gauge coupling constants and extra U (1) group charges of leptons or scalar doublets. In addition, if the leptonic Yukawa couplings are invariant under the extra U (1) local gauge group, these contributions can be eliminated by redeﬁning the gauge boson ﬁelds through eliminating the neutral currents involving charged leptons, even with nonzero kinetic mixing eﬀects.


I. INTRODUCTION
The imposing of gauge invariant in the Standard Model (SM) requires the introducing of gauge bosons.Due to the fact that the Lagrangian involving Higgs doublet in the SM is invariant under the SU(2) L ⊗ U(1) Y gauge transformation, the electroweak gauge bosons acquire nonzero masses through the spontaneous symmetry breaking (SSB).This SSB mechanism also provides mass to all massive particles in the SM.Therefore, highprecision measurements of gauge boson masses are crucial for testing SM and searching for NP phenomena.Recently, the CDF II measurement [1] of the W boson mass in 2022 is approximately 7σ away from the SM prediction [2] M SM W = 80354 ± 7 MeV, And M CDF II W is not compatible with the world average which is based on the measurements at LEP-2 [3], Tevatron [4,5] and the LHC [6,7].The measured W boson mass anomaly at CDF II has prompted numerous studies attempting to explain it within various NP models .In a recent update, the ATLAS collaboration reported a measurement of the W boson mass based on data from proton-proton (pp) collisions at a center-of-mass energy √ s = 7 TeV, the result reads [33] Interestingly, the updated result from the ATLAS collaboration shows agreement with the SM prediction, with no observed deviation from the SM expectation.This underscores the importance of obtaining a proper W boson mass that can satisfy experimental measurements in NP models, which is crucial for NP phenomenology studies.
Among the various NP models, those with an extra U( Due to the presence of two Abelian groups, the gauge kinetic mixing can occur and can be induced through RGEs even if it is set to zero at M GUT [47][48][49][50][51][52].
This leads to the covariant derivatives in this type of NP model being expressed as where Y, X are the hypercharge and U(1) X charge respectively, g Y is the measured hypercharge coupling constant, g X is the coupling constant corresponding to U(1) X gauge group, g Y X is the coupling constant arises from the gauge kinetic mixing effect, while g XY can always be rotated to 0 as long as the two Abelian gauge groups are unbroken.
The kinetic terms of vector bosons Π 33 (p 2 ), Π 00 (p 2 ), Π 30 (p 2 ), Π W W (p 2 ) can be defined by the effective Lagrangian [53] Π ij (p 2 ) (ij = 33, 00, 30, W W ) in Eq. ( 6) can be expanded at p 2 = 0 because the NP is generally considered to be very heavy, and Π ij (p 2 ) can be written as where the higher order terms can be neglected safely.The oblique parameters S, T, U are defined as where s W ≡ sin θ W , c W ≡ cos θ W , and θ W is the Weinberg angle.
Generally, the Π matrix of neutral vector bosons can be written in the basis where effect even X H = 0 because of the existence of kinetic mixing effect.In general, the kinetic mixing effect can also be described by the kinetic mixing parameter ǫ The kinetic terms of the Lagrangian above can be normalized by where The normalization arises the nonzero mixing terms of the mass matrix for (B µ , X µ ) Combining Eq. ( 9) with Eq. ( 12), we can obtain Hence, it is equivalent to describe the kinetic mixing effect by Eq. ( 9) and Eq. ( 12), we will take the forms defined in Eq. ( 9) to carry out the following analysis.
The gauge couplings involving charged leptons can be written as Y L , Y E are the hypercharges of the left-handed and right-handed components of leptons respectively, they can be normalized as And X L , X E , X H are the U(1) X charges of left-handed components of leptons, right-handed components of leptons and scalar doublets respectively.Since the most strict constraints come from the precision measurements performed at e + e − colliders (such as LEP1, LEP2 and etc), we choose to eliminate the neutral currents involving charged leptons in Eq. ( 14), which can be done by redefining the vector fields Then the oblique parameters S, T, U can be obtained from Eq. ( 8) as As can be seen in Eqs.(16)(17)(18) that S = T = U = 0 when X E − X L + X H = 0, it indicates the leptonic Yukawa couplings are invariant under the U(1) X symmetry in this case which avoids the need of a model to generate the lepton masses.And for nonzero X E − X L + X H , other mechanisms are needed to generate the lepton masses, such as the U(1) H model [54] and etc.In addition, we neglect the Z couplings to neutrinos or quarks in our calculations approximately, because they are measured much less accurately than Z couplings to charged leptons, and all effects involving charged leptons are included in the approximation.The W boson mass with local U(1) X gauge group can be written as [55,56] Eqs. (16)(17)(18) indicate that the U(1) X charges of leptons, denoted as X E and X L , can affect the theoretical predictions for the W boson mass.Consequently, we will consider X E and X L as free parameters to explore their effects in the following analysis.Typically, X E and X L are subject to constraints enforced by chiral anomaly cancellations.For NP models with an extra U(1) X local gauge group, there are four distinctive chiral anomaly cancellation patterns for per generation [20] f Here, f represents various fermion species, encompassing the left-handed lepton doublet L, the left-handed quark doublet Q, the right-handed lepton singlets Ê and N (heavy neutrinos), and the right-handed quark singlets û and d.It is evident from Eq. ( 20) that the phenomena of chiral anomaly cancellations can be realized by assigning approximate values to X N , X Q , X u , and X d while considering X E and X L as unconstrained.We focus on illustrating the effects of extra U(1) X local gauge group on the W boson mass, hence we do not explore the chiral anomaly cancellations detailedly for different X E , X L in specific NP models.

III. NUMERICAL ANALYSES
Based on the calculations in Sec.II, the numerical results are computed and presented in this section.As input parameters [57], M SM W is taken as 80.354GeV, the Z boson mass is M Z = 91.1876GeV, the fine-structure constant α em (m Z ) = 1/128.9,and the fermion For Z ′ gauge boson, the large amount of data collected at the LHC provides great potential to search Z ′ directly.Currently, ATLAS [58] and CMS [59] present direct searches on Z ′ gauge boson through the channel pp → Z ′ → e + e − , µ + µ − based on the data collected in proton−proton collisions at a centre-of-mass energy 13 TeV.In addition, there is a measurement of t t pair production cross section with 35.9 fb −1 data from CMS [60] and 139 fb −1 data from ATLAS [61] through the channel pp → Z ′ → t t, and ATLAS [62] also present a search for new resonances decaying into a pair of jets through the channel pp → Z ′ → q q, including b b.Although the Z ′ gauge boson is not observed so far, severe constraints are set on Z ′ boson mass, which depends on the U(1) symmetry and the corresponding gauge coupling strength generally.For example, for a sequential Standard Model (SSM), the latest experimental constraint on Z ′ SSM boson (which has the same fermion couplings as the SM Z boson) mass is M Z ′ > ∼ 5.1 TeV [58,59].For an E 6 −motivated Grand Unification model, the additional gauge bosons are required to be heavier than about 4.1 TeV for Z ′ Ψ and 4.6 TeV for Z ′ χ experimentally [58,59].The new Z ′ boson can also make contributions to B s − Bs mixing and the process B s → µ + µ − , which depends on the additional U(1) symmetry.For example, the contributions from Z ′ boson to B s → µ + µ − for the U(1) B−L symmetry are highly suppressed by its heavy mass [63], while Z ′ boson can make important contributions to B s − Bs mixing and B s → µ + µ − at the tree level for the U(1) Lµ−Lτ symmetry [64] and the flavor dependent U(1) F symmetry [65,66].In addition, how to probe the new introduced Z ′ boson in future also depends on the U(1) symmetry and the relevant couplings.For example, Z ′ will be produced at the LHC if Z ′ have nonzero couplings with quarks, which indicates the searches for high-mass dielectron, dimuon, dijet resonances at the future HL-LHC are effective to probe Z ′ boson directly; Z ′ has the potential to be probed by the di-muon production at the future high-energy electron-positron Linear Colliders if Z ′ have nonzero couplings with charged leptons [67], and Z ′ also may be produced at future muon collider [68][69][70] in this case.In the following analysis, we consider the constraints on Z ′ gauge boson mass M Z ′ and TeV. FIG.1: Taking , where the gray areas denote the ATLAS 2σ interval, the green areas denote the CDF II 2σ interval, the black, red, blue lines denote the results for M Z ′ = 4.2, 7, 10 TeV respectively, and the solid, dashed, dotted lines denote the results for g X = 0.1.0.4, 0.7 respectively.
The kinetic mixing effect appears in any NP models with two Abelian groups, we focus on the effects of g Y X on M W with eliminating the neutral currents involving charged leptons firstly.Taking , where the gray areas denote the ATLAS 2σ interval, the green areas denote the CDF II 2σ interval, the black, red, blue lines denote the results for M Z ′ = 4.2, 7, 10 TeV respectively, and the solid, dashed, dotted lines denote the results for g X = 0.1.0.4, 0.7 respectively.The picture shows that large g X , g Y X and X E − X L + X H combined with small M Z ′ can well explain the measured M W at CDF II, while the one measured at ATLAS prefers small g X , g Y X , X E − X L + X H and large M Z ′ .In addition, a positive g Y X can increase the theoretical prediction of M W , while a negative g Y X suppresses the contributions from U(1) X gauge group to M W .The effects of g Y X are affected by the values of M Z ′ , g X , X E − X L + X H obviously, where g Y X affects M W drastically when M Z ′ is small and g X , X E − X L + X H are large.The fact is well to be understood because the effects of introducing U(1) X local gauge group are highly suppressed by large M Z ′ or very small g X , and the leptonic Yukawa couplings approach to invariant under the U(1) X symmetry as X E − X L + X H approaches to 0. Since the kinetic mixing constant g Y X may affects the theoretical predictions on M W significantly, the constraints on M Z ′ /g X from high-precision W boson mass in the NP models with extra local U(1) gauge group would be relaxed partly by considering the contributions from g Y X .
In order to see the effects of X E , X L , X H on the W boson masses, we take g B = 0.6, M Z ′ = 5 TeV and scan the following parameter space g Y X = (−0.8,0.8), ).
The allowed results of X E , X L for X E − X L + X H = 1/2, 1, 3/2 are plotted in Fig. 2  local gauge group, the ranges of X E , X L would be limited by the W boson mass measured at CDF II more strictly for smaller X E − X L + X H , while the ranges of X E , X L would be limited by the one measured at ATLAS more strictly for larger X E − X L + X H .And the W boson mass measured at CDF II or ATLAS can be satisfied for appropriate values of X E , X L , X H , g Y X , g X , M Z ′ .
Nonzero X L , X E , g Y X may affect the theoretical predictions on the branching ratio of Z → l l significantly, which would be useful for testing the models with extra U(1) gauge group in future experiments.Generally, Br(Z → l l) can be simplified by neglecting the final lepton masses as where Γ Z = 2.4952 GeV [57], v = 246.22GeV [57] and
(a1), (b1), (c1) respectively, where the blue points are obtained by considering the W boson mass measured at CDF II in 2σ interval, the red points are obtained by considering the W boson mass measured at ATLAS in 2σ interval.To illustrate the effects of kinetic mixing on the allowed results of X E , X L , the results for scanning g Y X in the range g Y X = (−0.4,0.4) are plotted in Fig.2(a2), (b2), (c2) similarly.As can be seen by comparing Fig.2(a1), (b1), (c1) with Fig.2(a2), (b2), (c2), the allowed ranges of X E , X L for g Y X = (−0.4,0.4) are narrower than the ones for g Y X = (−0.8,0.8), because in the chosen parameter space g Y X affects the numerical results obviously as concluded above.For NP models with extra U(1)
In these NP models, the introduction of a new Z ′ gauge boson corresponding the extra U(1) gauge group and new introduced gauge couplings can make contributions to the oblique parameters S, T , U at the tree level, and consequently impact the theoretical predictions on the W boson mass.Therefore, determining the appropriate gauge coupling strength and the new U(1) gauge group charges of the particle fields that can satisfy the high-precision measurements of the W boson mass is crucial in these NP models.In this work, we present a general analysis of the W boson mass in the NP models with a new extra U(1) gauge group, and the obtained results can be applied to all such NP models.
[42][43][44][45][46][39][40][41]B−L extended SM, the minimal supersymmetric model (MSSM) with local B − L gauge symmetry[34][35][36][37][38][39][40][41], the MSSM with U(1) X local gauge group[42][43][44][45][46]and etc, have attracted the attention of physicists.The paper is organized as follows.The S, T , U parameters and W boson mass in the NP models with extra U(1) local gauge group are calculated in Sec.II.The numerical results are presented and analyzed in Sec.III.Finally, a summary is made in Sec.IV.II.W BOSON MASS IN THE NP MODELS WITH EXTRA U (1) LOCAL GAUGEGROUPFor convenience, the newly introduced U(1) local gauge group can be defined as U(1) X (X = B, L, B − L and etc, where B refers to the baryon number, L refers to the lepton number), i.e. the local gauge group of this kind of NP models is extended to SU