$W$ mass in a model with vector-like leptons and $U(1)^\prime$

We study the effects of vector-like leptons on the $W$ boson mass in a model with a vector-like $U(1)^\prime$ gauge symmetry. This model provides simultaneous explanations for the recent anomalies in the muon anomalous magnetic moment and the semi-leptonic decays of $B$ mesons. We found that the recent result of the $W$ boson mass precise measurement at CDF can be explained if the charged (neutral) vector-like lepton is lighter than 250 (80) GeV. The light vector-like leptons may not be excluded by collider experiments if these decay to a physical mode of the $U(1)^\prime$ breaking scalar field.


Introduction
Recently, the CDF collaboration reported a new result of the precise W boson mass measurement [1], m CDF W = 80.4335 (94) GeV. ( This value is larger than the combination of the previous measurements m PDG W = 80.379 (12) GeV and the Standard Model (SM) prediction m SM W = 80.361 (6) GeV [2].Since the announcement of the result, the explanations for the new W boson mass and its relations to other physics have been studied extensively .
In this work, we point out that the shift of the W boson mass can be explained in a model with vector-like (VL) leptons and an extra U (1) gauge symmetry which was proposed in Refs.[78,79].The model provides a simultaneous explanation for the anomalies in the muon anomalous magnetic moment, g−2, and the semi-leptonic decays of the B meson 3 .The recent FNAL measurement [97] confirmed the long-standing discrepancy of the muon g − 2 between the experimental value [98] and the SM prediction [99][100][101][102][103][104][105][106][107][108][109][110][111][112][113][114][115][116][117][118], and the current discrepancy is ∆a µ = 2.51 (59) × 10 −9 .Yet another discrepancy from the SM is found in the measurements of rare semi-leptonic B meson decays [119][120][121][122][123][124][125][126][127][128][129][130][131][132][133][134][135], b → s .In this paper, we focus on the VL leptons and Z boson in this model to show that m W and ∆a µ can be explained simultaneously.As we have shown in Refs.[78,79], this model can easily accommodate the b → s anomaly when the Z boson is sufficiently light and strongly couples to muons, so that ∆a µ is explained.Therefore, our model will provide a unified explanation for the recent three anomalies, m W , ∆a µ and b → s .It will turn out that the VL leptons should be lighter than those to explain ∆a µ .Hence we will study the observables which would be changed from the SM values due to the light VL leptons.
The rest of this paper is organized as follows.The model is briefly introduced in Sec. 2, and then the observables in our analysis are discussed in Sec. 3. The result of a numerical analysis is shown and the LHC signals are discussed in Sec. 4. Section 5 is devoted to summary.Diagonalization of the mass matrices is discussed in Appendix A, and the formula for the three-body decays are shown in Appendix B.

Mass matrices
We briefly introduce our model, with particular interests in VL leptons, see Refs.[78,79] for more details.The matter content relevant to the discussion in this work is shown in Table 1.We assume that the VL leptons mix with only the SM leptons in the second generation to prevent flavor violations.The mass terms and the Yukawa couplings are given by where . The SU (2) L indices are contracted by iσ 2 .We introduce the Majorana mass term of ν R for the type-I see-saw mechanism.After the scalar fields develop their vacuum expectation values (VEVs), v H := H 0 and v Φ := Φ , the Dirac mass matrices for the leptons are given by The mass basis is defined as where unitary matrices diagonalize the mass matrices as The masses are increasingly ordered.Here, •'s in the first row of the neutrino mass matrix will be irrelevant after ν R is integrated out.Neglecting O (v H ) entries, the VL lepton masses are given by when M E , m N < M L and hence L ∼ (E 2 , N 2 ).We define the Dirac fermions as where with i = 1, 2, 3. We expand the neutral scalar fields as where h and χ are the physical real scalar fields, while the pseudo-scalar components a h and a χ are absorbed by the Z and Z bosons, respectively.

Interactions
The Z and W boson couplings are given by where for A = L, R. Here, where ) are the chiral projections onto the left-(right-) handed fermions.The gauge interactions with the Z boson in the mass basis are defined as where the coupling matrices are given by with Q e = Q n = diag(0, −1, −1).g is the gauge coupling constant for U (1) .The Yukawa interactions are given by where

Couplings at the leading order
We show the structures of the coupling matrices at the leading order in O (m µ /v Φ ).Hereafter, we assume λ e , λ n ∼ y 2 , so that the muon mass is explained without fine-tuning, and the Z and W boson couplings to the SM leptons do not sizably deviate from the SM values.The formulas with the sub-leading terms are explicitly shown in Appendix A. The Z couplings are given by where Here, c e A and s e A (A = L, R) are the angles to diagonalize the VL mass matrix, defined in Eq. (60).
These are approximately given by for η e A 1.Those for the neutrino couplings g Z n A are given by replacing s E → 0, c E → 1 and e A → n A .The isospin parts of the Z and W boson couplings are given by and for A = L, R. Here, δ AL = 1 (0) for A = L (R).Thus, the SM lepton couplings to the Z and W bosons are the same as the SM ones, and the off-diagonal couplings of the SM and VL fermions are vanishing, up to O (m µ /v Φ ).This is in contrast to the model studied in Ref. [20], and hence the W -mass could be addressed in this model solely by the VL leptons.
The Yukawa couplings to the scalars h and χ are respectively given by Those for the neutrinos are given by replacing y 2 → 0, e → n and E → N .Note that m µ ∼ (y 2 c L c E + λ e s L s E )v H , so the Yukawa coupling to the Higgs boson is also not changed from the SM value for heavy VL leptons.The SM Higgs couplings to the SM and VL leptons are again suppressed by the small muon Yukawa coupling.The χ boson coupling to the SM muon vanishes if we neglect the muon mass.

Oblique parameters and W boson mass
In this model, the VL lepton contributions to the T parameter, one of the oblique parameters [136,137], is given by [138], where a, b = 1, 2, 3 (α, β = 1, 2, 3) run over the neutral (charged) leptons.These subscripts are the elements of the couplings matrices, e.g.h L aβ := [h L ] aβ .The other oblique parameters, 2πS and −2πU , are obtained by replacing the functions, θ ± → χ ± , except for the first line in the S parameter which should be replaced as θ ± → ψ ± .The loop functions are defined in Ref. [41].For an order of magnitude estimation, taking s e A , s n A = 0 and m L 0 m L − , we have where m L − and m L 0 are the masses of doublet-like charged and neutral VL leptons, respectively.η e A , η n A 1 is assumed in the second equality.Hence, where we used s 2 W = 0.23121 [2].The W boson mass is related to the oblique parameters as [139,140] where T S, U and µ 0 are assumed in the second equality.We include the shift from the W boson coupling to the muon at tree-level, h L 11 =: h L νµ =: 1 − µ .Hereafter, we write the indices for the SM leptons by ν/µ instead of 1, so that these are not confused with the first generations.Note that µ is positive in our model as shown in Eq. ( 61).Therefore, from Eq. ( 29), the shift may be explained if λ e , λ n ∼ O (1), M L ∼ 250 GeV and

Muon g − 2
The muon anomalous magnetic moment ∆a µ is shifted by the 1-loop effects via the Z and χ bosons.This is approximately given by [93,141] where The exact formula and the loop functions are shown in Refs.[78,79].The value of ∆a µ is estimated as Hence, the Z boson with mass of order 500 GeV can explain ∆a µ .

EW observables
The mixing of the second generation leptons and the VL states can change the SM predictions of the EW boson couplings.The Fermi constant G µ determined by the muon decay is given by at the tree-level, where G µ = 1.1663787 × 10 −5 GeV 2 [2].The partial width of the W boson is given by where the muon and neutrino masses are neglected.In the SM, the gauge coupling constant and Γ W are given by g(m Z ) = 0.65184 (18) [142] and Γ W = 2.0895 (8) GeV [143], respectively.We use these SM values for numerical analysis, since the branching fraction, ∝ g 2 m W /Γ W , is approximately independent of g and m W .The Z boson partial decay widths are given by where the Z boson couplings to the electron and tau neutrinos are set to the SM values.The leading QED effect is included for Z → µµ.The asymmetry parameter A µ and A µ F B are given by Since the Z boson couplings to the electrons are the SM-like, A e = A SM e = 0.1468 (03) [2].We shall use the SM value for the weak angle, s 2 W = 0.23155 [2] to calculate A µ .Similarly to the SM gauge bosons, h → µµ can deviate from the SM value.We define the ratio of the width, The decay rate of h → γγ will deviate from the SM value due to the loop effects mediated by the charged VL leptons.We define the ratio of the width of h → γγ to that in the SM as where τ I = m 2 H /(4m 2 I ) for the lower case I = E 1 , E 2 , and the SM contribution A SM is given by [144] A Here, f runs over all the SM fermions.Q f and N f c are the electric charge and the number of colors of the fermion f , respectively.Assuming that the production cross sections are the same as in the SM, the current experimental values are R µµ = 1.19 ± 0.34 and R γγ = 1.10 ± 0.07 [2].
For h L νµ = 1, the SM gauge boson couplings are not lepton flavor universal.We shall consider the decays of Z, W bosons and τ , where We also study the ratio of the Z/W boson decays to tau and muon, Γ (Z → τ τ ) /Γ (Z → µµ) and Γ (W → τ ν) /Γ (W → µν) which are given by the inverse of Eq. ( 42) in our model.

b → sµµ anomaly
We emphasize that the recent anomaly in the b → s process can easily be explained in this model.The Wilson coefficients for the semi-leptonic operators are given by where V tb and V ts are the CKM elements, and g Z sb is the Z boson coupling constant to sb in the left-handed interaction.The Wilson coefficient C 9 is estimated as whereas the current favored values is C 9 ∈ [−1.0, −0.5] depending on the value of C 10 [152][153][154][155][156][157][158].Thus, the b → s anomaly can be explained even with the small Z coupling to quarks, as long as those to the VL leptons are sizable to explain the shift in m W and ∆a µ .With such small couplings to the Z boson, the flavor violations such as B s -B s mixing will not deviate from the SM prediction due to the small couplings, as we have explicitly shown in Refs.[78,79].Further, the production cross sections at the LHC will be so small that the di-muon signal is much below the current limit [159].

Cabibbo angle anomaly
As pointed out in Refs.[160,161], the shift of the W boson coupling to the muon could explain the recent Cabibbo angle anomaly, which may be caused by the disagreement between values V us determined from beta and Kaon decays.Let us consider the observable [160] where V Kµ us (V β ud ) is the value of the CKM element determined from the Kaon (β) decay.The CKM elements without superscript are those in our model which are assumed to be unitary 6 .Here, V Kµ us = 0.2252 (5), V ub = 4 × 10 −3 [2] and V ud = 0.97373 (09) [160].The current measured value is R(V us ) = 0.9891 (27) [160].Since µ > 0 in this model, the tension can not be resolved by the mixing with the VL leptons.We shall not include R(V us ) in our χ 2 analysis because it can not be explained by the mixing with VL leptons, and could be explained by mixing with VL quarks.

Numerical results and LHC signals 4.1 χ 2 fitting
There are 12 parameters in our model, to be scanned in our parameter search.The SM Yukawa coupling constant y 2 is fixed to explain the muon mass for a given parameter set.We take λ N = y n = 0 since these are irrelevant R CCFR 0.82 0.14 Ref. [150,151] for phenomenology after integrating out ν R .To find points which can explain the anomalies consistently with the other observables, we minimize the χ 2 function where y I 's are observables listed in Table 2. y 0 I and σ I are the experimental central value and its 1σ error, respectively.For R CCFR , the error is chosen such that the 2σ deviation corresponds to the 95% C.L limit.Here, we assume that the W boson mass is given by the CDF result.In the SM, χ 2 SM 94 mainly originated from ∆a µ and m W as well as the 2.7σ deviation in Γ (W → τ ν) /Γ (W → µν).
In our analysis, we restrict the parameter space to be and Here, the lower bounds of m χ is chosen so that h → χχ is kinematically forbidden.The U (1) gauge constant g is required to be smaller than 0.35, so that it is perturbative up to 10 16 GeV in the full model with VL quarks [78].We minimize the χ 2 function from random initial points in the parameter range.We further require m E 1 , m Z > 100 GeV and m N 1 > 45 GeV for the fitted points, so that the collider limits would be avoided and Z → N 1 N 1 is kinematically forbidden.We calculated branching fractions of the Z boson and the VL leptons.For the Z boson decay, we calculated the two-body decays to the leptons.We neglect the decays to quarks, since the couplings to quarks will be tiny as discussed in Section 3.5.For the decays of VL leptons, we calculated the two-body decays to a SM, χ, Z boson and a lepton.The three-body decays via off-shell Z or W boson are calculated based on the formula shown in Appendix B if the corresponding on-shell two-body decay is kinematically forbidden.
Figures 1, 2, 3 and 4 show the scatter plots of the result of the χ 2 fitting.On these plots, the cross and circle points are 100 < m E 1 < 175 GeV and m E 1 > 175 GeV, respectively.The colors of points indicate the dominant decay mode of the lightest charged VL lepton E 1 .Note that the three-body decays are also classified by its off-shell boson, e.g.
The scatter plots for the observables are shown in Fig. 1.On the left panel, the gray error bars are the 1σ errors along the axes.We see that ∆a µ and m W are simultaneously explained on these points within 1σ.From the right panel, R γγ can deviate from unity at most 4%, and hence The left panel of Fig. 2 shows the scatter plots on (λ e , λ n ).There are points only where |λ n | ∼ 1.0.This indicates that the mass shift of m W is predominantly induced by the VL neutrino which can be lighter than the charged ones.The right panel shows the same plot on (m E 1 , m N 1 ).We see that the VL neutrino should be lighter than 80 GeV to explain the shift of m W , and the upper bound becomes stronger for heavier E 1 .Thus, most of the points with m E 1 > 175 GeV have E 1 → N 1 W as the dominant decay mode.
Figure 3 shows the scatter plots on the m E 1 and m Z (m χ ) plane on the left (right) panel.The gray line is m E 1 = m Z or m χ .The upper bound on m Z is about 600 GeV to explain the muon g − 2, while χ can be much heavier.E 1 → N 1 W is the dominant decay mode even if m Z < m E 1 due to the larger gauge coupling constant g > g .
Figure 4 shows the scatter plots for the oblique parameters.The solid (dashed) lines correspond to 1σ (2σ) errors of m W of the CDF measurement when U = 0 and S = 0 on the left and right panel, respectively.For m E 1 > 175 GeV, the points are found only around (S, T, U ) ∼ (0.01, 0.15, 0.02), so the W boson mass shift is mainly explained by the T parameter.Different values of the oblique parameters are allowed for m E 1 < 175 GeV.These patterns of the oblique parameters will be useful to distinguish the parameter space of our model.
The points (A), (B), (C) and (D) are chosen from the points whose E 1 decay is dominated by that to the SM particles, Z µ, χµ and N 1 W , respectively.Here, the SM particles include Zµ, hµ and W ν. These points are plotted on Figs. 1, 2, 3 and 4. The branching fraction of the Z boson to the SM leptons are shown in this table.The Z boson will also decay to the VL leptons if it is kinematically allowed 7 .The lightest VL neutrino N 1 decays to a SM neutrino and a χ boson if m χ < m N 1 as in the point (D), while it decays to a W or Z boson in the other cases.

LHC signals
We found that m E 1 250 GeV is required to explain the shift of m W .Such a light VL lepton may be constrained by LHC searches, depending on its decay.
The VL lepton would decay to a SM boson and a SM lepton.If E 1 is doublet-like, the latest limit is about 1000 GeV [163] 8 .Even for the singlet-like case, the limit is about 200 GeV [164] using the run-1 data [165].Since E 1 is mostly doublet-like, the VL lepton lighter than 250 GeV may be excluded by the current data.However, these decay modes may be suppressed because [g Z e L,R ] E 1 µ = O (m µ /m E 1 ), and hence the other decay mode will coexist, or even dominate the decays.Actually, E 1 → SM is sizable only on the point (A).
In Ref. [166], it is shown that the limit for the doublet-like VL lepton is about 500 (1200) GeV for BR (Z → µµ) = 2/3 and BR (E 1 → Z µ) = 0.1 (1.0) by the signal with four muons or more [167].Although the Z boson was assumed to be on-shell in Ref. [166], the limits on the VL lepton would be similar even for m E 1 < m Z .At the point (B), this decay mode dominates the others and Br(E 1 → Z µ → µµµ) ∼ 0.65, so this case may also be excluded.
(C) E 1 → χµ This is the dominant decay mode on the point (C).This case may not be excluded if χ decays to quarks.The χ coupling to two SM fermions is induced only by the fermion mass effects, so χ → bb can be the dominant mode 9 .Another possibility is χ → hh * via the quartic interaction |H| 2 |Φ| 2 in the scalar potential.Since the decays to fermions are suppressed, this mode can dominate over the others.In this case, we expect 4h + µµ from the E 1 pair production.Yet another possibility is loop-induced decays such as χ → γγ and gg, which may be particularly important for m χ < m h .Thus there are so many decay modes that could be sizable and many parameters independent of our discussions which are involved.Thus this case may not be excluded.
This decay mode may dominate the others when m N 1 < m E 1 which is favored to explain the m W shift.The VL neutrino N 1 then decays as N 1 → Z ν or N 1 → χν.At the point (D), N 1 → χν dominates, so the signal is E 1 E 1 → W W + χχ + νν.This signal contains many particles in the final state, but these will be relatively soft because of the smaller phase space.Thus, there may be a case in which the point (D) is not excluded by the current bounds.
We note that the pair production of N 1 is also constrained by the ≥ 4µ search [166] or the VL lepton search [163], as for E 1 pair production.Although N 1 is mostly singlet-like, there is a mixing with the doublet-like state, and thus N 1 can be pair produced from a Z boson.The light N 1 would be excluded if BR (N 1 → Z ν) or BR (N 1 → W µ) is sizable.Hence, the point (C) would also be excluded by the direct N 1 search.Therefore, the benchmark points other than the point (D) may be excluded by the current data.At the point (D), the decay of χ is important for the collider signals of both N 1 and E 1 .As we have already discussed, there are various decay modes of the χ boson because of the vanishing couplings to two SM fermions, see Eq. ( 26), and thus there may be cases which any search can not constrain.This is an interesting subject, but is beyond the scope of this paper.

Summary
In this work, we studied the W boson mass in the SM extension with VL leptons and a U (1) gauge symmetry, in which only the VL leptons carry non-zero charges.The full model with VL quarks was originally considered to explain the anomalies in the muon g − 2 and the b → s decay simultaneously [78,79].We explicitly studied the W boson mass and ∆a µ induced by the VL leptons and the Z boson, and we found points which can explain both anomalies.Since b → s can easily be explained by tiny couplings with the SM quarks, as discussed in Sec.3.5, our model provides a simultaneous explanation for the three anomalies.
The VL leptons should be light to explain the shift of m W .We found that the lightest charged (neutral) VL lepton E 1 (N 1 ) is lighter than 250 (80) GeV if absolute values of the Yukawa couplings are less than unity.In our model, the VL leptons typically decay through the Z or χ boson.If the VL lepton decays through the Z boson, which may decay to di-muons, there will be strong constraints from the searches for signals with four muons or more [166].Hence, the decay to the χ boson should dominate the decay of the VL lepton, which may be achieved by the mass hierarchy m χ < m E 1 ,N 1 < m Z .The χ boson decays to the SM particles in various ways, and thus there may be some cases for which none of the LHC searches exclude our model with the light VL leptons.Studying the decays of the χ boson in the full model with VL quarks is our future work.

A Diagonalization of mass matrices
We first discuss the diagonalization of the charged lepton mass matrix Eq. ( 3).The diagonalization matrix is decomposed as The unitary matrices are given by and where Here, The unitary matrices U 1 L,R block diagonalize the SM and VL leptons up to O (y 2 2 v 2 H ) when we assume λ e ∼ O (y 2 ).These are simply identity matrices if we neglect O (m µ /v Φ ).The unitary matrices U 0 L,R keeps the SU (2) L gauge couplings unchanged, while U 1 L,R do change the couplings as For λ e v H m µ , the correction to the muon coupling deviates from the SM value within O (10 −6 ) for O (100) GeV VL lepton masses, while it can be O (1) for λ e ∼ O (1) with the fine-tuning for y 22 v H ∼ m µ .Note that U 2 e L,R does not change the SM coupling since these rotate only the VL block.
The neutrino mass matrix, Eq. ( 4), can be diagonalized as follows.First, we rotate M n as where The mixing angles are defined as The dots in the first low are linear combinations of y n v H and λ N v Φ , which are irrelevant for our discussion after integrating out the right-handed neutrino ν R .The active neutrino mass is given by m 2 D /M R .The unitary matrices to diagonalize the neutrino mass is given by up to v Φ /M R 1, where With the unitary matrix U n L , the SM muon neutrino coupling to the Z boson is rescaled as c 2 L + c 2 N s 2 L .Therefore, s N ∼ λ n v H /m N 1 is required for the SM-like Z boson coupling to muon neutrinos.

B Decay widths
We calculate the three-body decay, F → f 3 V * → f 3 f 2 f 1 , of fermions via a off-shell vector boson V whose couplings are given by, The partial width is given by where β(t) = t 2 − 2(1 + y)t + (1 − y) 2 , y = m 2 3 /m 2 F and z = m 2 V /m 2 F .Here, m 3 , m F and m V are respectively the masses of f 3 , F and V , and we neglect the masses of the fermions f 1 and f 2 .

Figure 1 :
Figure 1: Scattering plots on the observables in our analysis.The cross and circle points are 100 < m E 1 < 175 GeV and m E 1 > 175 GeV, respectively.All the points explain the m W and ∆a µ within 1σ.The error bars on the left panel are the 1σ uncertainties, and the dash horizontal line on the right panel is 95% C.L. limit on R CCFR .The central value of R γγ = 1.10 is outside of the figure.The benchmark points, (A), (B), (C), and (D), lie on top of each other in the left panel.

Figure 3 :
Figure 3: Scattering plots on the masses of Z boson and the VL leptons.The cross and circle points are 100 < m E 1 < 175 GeV and m E 1 > 175 GeV, respectively.The diagonal lines are respectively m Z = m E 1 and m χ = m E 1 on the left and right panel.

Figure 4 :
Figure 4: Scattering plots on the oblique parameters.The cross and circle points are 100 < m E 1 < 175 GeV and m E 1 > 175 GeV, respectively.The solid (dashed) lines are 1σ (2σ) range of the W boson mass measured by the CDF for U = 0 and S = 0 on the left and right panel, respectively.

Table 1 :
Matter contents.Electric charge of fermion f is Q f

Table 2 :
The list of observables and their values studied in our χ 2 analysis.

Table 3 :
Selected input parameters and branching fractions of new particles.The observables not included in our χ 2 analysis are shown in the last four rows.

Table 4 :
Values of observables at the benchmark points.The second column is the SM prediction.The degree of freedom is 15 − 12 = 3.