APCTP Pre2022 - 010

# An alternative gauged $U(1)_{R}$ symmetric model
in light of the CDF II $W$ boson mass anomaly

Keiko I. Nagao<EMAIL_ADDRESS>Okayama University of Science, Faculty of
Science, Department of Physics, Ridaicho 1-1, Okayama, 700-0005, Japan
Takaaki Nomura<EMAIL_ADDRESS>College of Physics, Sichuan University,
Chengdu 610065, China  Hiroshi Okada<EMAIL_ADDRESS>Asia Pacific
Center for Theoretical Physics (APCTP) - Headquarters San 31, Hyoja-dong, Nam-
gu, Pohang 790-784, Korea  Department of Physics, Pohang University of Science
and Technology, Pohang 37673, Republic of Korea

###### Abstract

We consider an explanation of CDF II W bosom mass anomaly by $Z-Z^{\prime}$
mixing with $U(1)_{R}$ gauge symmetry under which right-handed fermions are
charged. It is found that $U(1)_{R}$ is preferred to be leptophobic to
accommodate the anomaly while avoiding other experimental constraints. In such
a case we require extra charged leptons to cancel quantum anomalies and the SM
charged leptons get masses via interactions with the extra ones. These
interactions also induce muon $g-2$ and lepton flavor violations. We discuss
muon $g-2$, possible flavor constraints, neutrino mass generation via inverse
seesaw mechanism, and collider physics regarding $Z^{\prime}$ production for
parameter space explaining the W boson mass anomaly.

##  I Introduction

Precision measurements of electroweak observable are good test of the standard
model(SM) and would provide a hint of beyond the SM. CDFII collaboration
recently reported updated result of the SM charged-gauge boson (W boson) mass
CDF:2022hxs

$\displaystyle m_{W}=(80.433\pm 0.0064_{\rm stat}\pm 0.0069_{\rm syst})\ {\rm
GeV},$ (1)

which deviates from the SM prediction by 7$\sigma$, where the SM prediction
indicates $m_{W}=(80.357\pm 0.006)\ {\rm GeV}$. The disagreement also appears
to the previous global combination of data from LEP, CDF, D0, and ATLAS where
they give the mass range of $m_{W}=(80.379\pm 0.012)\ {\rm GeV}$
ParticleDataGroup:2018ovx . This anomaly suggests new physics (NP) beyond the
SM Fan:2022dck ; Lu:2022bgw ; Athron:2022qpo ; Yuan:2022cpw ; Strumia:2022qkt
; Yang:2022gvz ; deBlas:2022hdk ; diCortona:2022zjy ; Du:2022pbp ;
Tang:2022pxh ; Blennow:2022yfm ; Cacciapaglia:2022xih ; Sakurai:2022hwh ;
Fan:2022yly ; Liu:2022jdq ; Lee:2022nqz ; Cheng:2022jyi ; Song:2022xts ;
Bagnaschi:2022whn ; Paul:2022dds ; Bahl:2022xzi ; Asadi:2022xiy ;
DiLuzio:2022xns ; Athron:2022isz ; Gu:2022htv ; Heckman:2022the ; Babu:2022pdn
; Heo:2022dey ; Du:2022brr ; Cheung:2022zsb ; Crivellin:2022fdf ; Endo:2022kiw
; Biekotter:2022abc ; Balkin:2022glu ; Krasnikov:2022xsi ; Ahn:2022xeq ;
Han:2022juu ; Zheng:2022irz ; Kawamura:2022uft ; Ghoshal:2022vzo ;
FileviezPerez:2022lxp ; Nagao:2022oin ; Kanemura:2022ahw ; Mondal:2022xdy ;
Zhang:2022nnh ; Borah:2022obi ; Chowdhury:2022moc ; Arcadi:2022dmt ;
Cirigliano:2022qdm ; Carpenter:2022oyg ; Popov:2022ldh ; Ghorbani:2022vtv ;
Du:2022fqv ; Bhaskar:2022vgk ; Batra:2022org ; Cao:2022mif ; Zeng:2022lkk ;
Baek:2022agi ; Borah:2022zim ; Almeida:2022lcs ; Cheng:2022aau ; Heeck:2022fvl
; Addazi:2022fbj ; Lee:2022gyf ; Cai:2022cti ; Benbrik:2022dja ; Yang:2022qgs
; Batra:2022pej ; Tan:2022bip ; Abouabid:2022lpg ; Chen:2022ocr ; Zhou:2022cql
; Gupta:2022lrt ; Basiouris:2022wei ; Wang:2022dte ; Botella:2022rte ;
Barman:2022qix ; Kim:2022hvh ; Li:2022gwc ; Isaacson:2022rts ; Evans:2022dgq ;
Chowdhury:2022dps ; Kim:2022zhj ; Lazarides:2022spe ; Senjanovic:2022zwy ;
Ghosh:2022zqs ; Li:2022eby ; Rodriguez:2022wix ; Ramirez:2022zpk ;
Kawamura:2022fhm ; Afonin:2022cbi ; Allanach:2022bik ; Xue:2022mde ;
Rizzo:2022jti ; VanLoi:2022eir ; YaserAyazi:2022tbn ; Chakrabarty:2022voz ;
CentellesChulia:2022vpz ; Kim:2022xuo , and can be interpreted as the
deviation of $\Delta T$ oblique parameter Peskin:1991sw ; Peskin:1990zt ,
where the oblique parameters are zero in the SM.

One of the straightforward explanations of the anomaly can be realized by
introducing extra $U(1)$ gauge symmetry where the SM Higgs field is charged
under it. Then $\Delta T$ is shifted by effect of mass mixing between the SM
$Z$ and a new neutral gauge boson $Z^{\prime}$. In particular one of the
minimal scenarios is the explanation by $Z^{\prime}$ with right-handed
$U(1)_{R}$ symmetry Jung:2009jz . The symmetry is originally proposed by
chiral anomaly cancellations with three right-handed neutrinos, and it is well
testable at the International Linear Collider (ILC) or the Large Hadron
Collider(LHC) due to observing the difference of chirality Nomura:2017tih ;
Nomura:2017abh ; Nomura:2018mwr ; Nomura:2017ezy ; Chao:2017rwv ; Seto:2020jal
. However the W boson anomaly cannot be explained if charged-leptons have
nonzero $U(1)_{R}$ charge because of the constraints from $W$ and $Y$ oblique
parameters Strumia:2022qkt . Then, only the possibility to explain the W boson
anomaly along this idea is that only the SM Higgs field and quarks have
nonzero charge under $U(1)_{R}$ symmetry. Note that another possibility is
$U(1)_{H}$ case where only Higgs doublet is charged under it. In this case we
need another Higgs doublet to induce the SM fermion masses and two Higgs
doublets can also contribute to $\Delta T$ parameter at loop level. Remarkably
in our setting $W$ boson anomaly is explained by purely $Z$–$Z^{\prime}$
mixing effect.

In this letter, we extend the original $U(1)_{R}$ model to make the
$Z^{\prime}$ to be leptophobic in order to explain this anomaly avoiding other
electroweak precision tests. Then extra $SU(2)$ singlet charged leptons are
required to cancel quantum anomalies. As a result the masses of the SM charged
leptons are obtained via interactions between the SM lepton and the extra
charged leptons. Such interactions also induce lepton anomalous magnetic
moments and lepton flavor violations (LFVs) at loop level Lee:2021gnw ;
Guedes:2022cfy . In addition, we include discussion of active Majorana
neutrino mass matrix via inverse seesaw scenario Nomura:2018mwr .

This letter is organized as follows. In Sec. II, we introduce our model and
show relevant formulas for phenomenology. In Sec. III, we show our
phenomenological analysis of muon $g-2$, LFVs and collider physics. Finally we
devote the summary of our results and the conclusion.

|  $Q_{L}^{a}$ |  $u_{R}^{a}$ |  $d_{R}^{a}$ |  $L_{L}^{a}$ |  $e_{R}^{a}$ |  $N_{R}^{a}$ |  $N_{L}^{a}$ |  $E_{L}^{a}$ |  $E_{R}^{a}$ |  $H$ |  $\varphi_{1}$ |  $\varphi_{2}$
---|---|---|---|---|---|---|---|---|---|---|---|---
$SU(3)_{C}$ | $\bm{3}$ | $\bm{3}$ | $\bm{3}$ | $\bm{1}$ | $\bm{1}$ | $\bm{1}$ | $\bm{1}$ | $\bm{1}$ | $\bm{1}$ | $\bm{1}$ | $\bm{1}$ | $\bm{1}$
$SU(2)_{L}$ | $\bm{2}$ | $\bm{1}$ | $\bm{1}$ | $\bm{2}$ | $\bm{1}$ | $\bm{1}$ | $\bm{1}$ | $\bm{1}$ | $\bm{1}$ | $\bm{2}$ | $\bm{1}$ | $\bm{1}$
$U(1)_{Y}$ | $\frac{1}{6}$ | $\frac{2}{3}$ | $-\frac{1}{3}$ | $-\frac{1}{2}$ | $-1$ | $0$ | $0$ | $-1$ | $-1$ | $\frac{1}{2}$ | $0$ | $0$
$U(1)_{R}$ | $0$ | $1$ | $-1$ | $0$ | $0$ | $1$ | $0$ | $0$ | $-1$ | $1$ | $-1$ | $-\frac{1}{2}$
Table 1:  Charge assignments of the our fields under $SU(3)_{C}\times
SU(2)_{L}\times U(1)_{Y}\times U(1)_{R}$, where its upper index $a$ is the
number of family that runs over $1-3$.

##  II Model setup and Constraints

Here, we review our model. We introduce three vector-like neutral fermions
$N_{L,R}$ and singly-charged fermions $E_{L,R}$, where only $N_{R}$ and
$E_{R}$ has nonzero $U(1)_{R}$ charge with $1$ and $-1$, respectively. It
suggests that masses of charged-lepton arise not via the SM type but via
mixings among singly-charged fermions. One can straightforwardly confirm the
typical four patterns of chiral anomaly cancellations per one generation;
$[U(1)_{R}]=[U(1)_{R}]^{3}=[U(1)_{R}]^{2}[U(1)_{Y}]=[U(1)_{R}][U(1)_{Y}]^{2}=0$.
111In our paper, quark sector is the same as the SM except for the fact that
right-handed ones interact with $Z^{\prime}$ boson. As for scalar sector, we
introduce an isospin singlet fields $\varphi_{1}$ and $\varphi_{2}$ with
$U(1)_{R}$ charge $-1$ and $-\frac{1}{2}$ to spontaneous $U(1)_{R}$ symmetry
breaking where SM-like Higgs $H$ also has nonzero charge under $U(1)_{R}$ in
order to induce Yukawa Lagrangian. Each of VEVs is denoted by $\langle
H\rangle\equiv[0,v/\sqrt{2}]^{T}$ and $\langle\varphi_{1(2)}\rangle\equiv
v^{\prime}_{1(2)}/\sqrt{2}$. All the field contents and their assignments are
summarized in Table 1.

###  II.1 Lagrangian and scalar masses

The relevant lepton Yukawa Lagrangian under these symmetries is given by

$\displaystyle-{\cal
L}_{Y}=(y_{\ell})_{ab}\bar{L}^{a}_{L}HE^{b}_{R}+(y_{E})_{aa}\varphi_{1}\bar{E}^{a}_{L}E^{a}_{R}+(m_{Ee})_{ab}\bar{E}^{a}_{L}e^{b}_{R}$
$\displaystyle\qquad\quad+(y_{D})_{ab}\bar{L}^{a}_{L}\tilde{H}N^{b}_{R}+(y_{N})_{aa}\varphi_{1}\bar{N}^{a}_{L}N^{a}_{R}+(M_{N_{L}})_{ab}\bar{N}^{a}_{L}N^{cb}_{L}+{\rm
h.c.},$ (2)

where $\tilde{H}\equiv i\sigma_{2}H$, and upper(lower) indices $(a,b)=1$-$3$
for fields(Yukawa or mass matrix) are the number of families, and $y_{E},\
y_{N}$ can be diagonal matrix without loss of generality.

The scalar potential is given by

$\displaystyle{\cal V}=$
$\displaystyle\mu^{2}_{1}|\varphi_{1}|^{2}-\mu^{2}_{2}|\varphi_{2}|^{2}-\mu^{2}_{H}|H|^{2}+\lambda_{H}|H|^{4}+\lambda_{1}|\varphi_{1}|^{4}+\lambda_{2}|\varphi_{2}|^{4}-\mu_{3}(\varphi_{1}^{*}\varphi_{2}\varphi_{2}+h.c.)$
$\displaystyle+\lambda_{3}|\varphi_{1}|^{2}|H|^{2}+\lambda_{4}|\varphi_{2}|^{2}|H|^{2}+\lambda_{5}|\varphi_{1}|^{2}|\varphi_{2}|^{2}.$
(3)

The scalar fields are parameterized as

$\displaystyle H=\left[\begin{array}[]{c}w^{+}\\\
\frac{v+r+iz}{\sqrt{2}}\end{array}\right],\quad\varphi_{1,2}=\frac{v^{\prime}_{1,2}+r^{\prime}_{1,2}+iz^{\prime}_{1,2}}{\sqrt{2}},$
(6)

where $w^{+}$ and $z$ are massless Nambu-Goldstone(NG) bosons which are
absorbed by the SM gauge bosons $W^{+}$ and $Z$, and one linear combination of
$z^{\prime}_{1,2}$ corresponds to NG boson abosorbed by an extra $Z^{\prime}$
boson from $U(1)_{R}$. The VEVs are obtained from tadpole conditions
$\frac{\partial{\cal V}}{\partial v}=\frac{\partial{\cal V}}{\partial
v^{\prime}_{1}}=\frac{\partial{\cal V}}{\partial v^{\prime}_{2}}=0$. We obtain
condition for $v^{\prime}_{1}$ from $\frac{\partial{\cal V}}{\partial
v^{\prime}_{1}}=0$ such that

$\mu_{1}^{2}v^{\prime}_{1}+\lambda_{1}v^{\prime
3}_{1}-\frac{1}{2\sqrt{2}}\mu_{3}v^{\prime
2}_{2}+\lambda_{5}v^{\prime}_{1}v^{\prime 2}_{2}=0.$ (7)

Here the VEV of $\varphi_{1}$ is approximately given by

$v^{\prime}_{1}\simeq\frac{1}{2\sqrt{2}}\frac{\mu_{3}v^{\prime
2}_{2}}{\mu_{1}^{2}+\lambda_{5}v^{\prime 2}_{2}},$ (8)

where we assume $v^{\prime}_{1}$ is much smaller than $v^{\prime}_{2}$ and
$v^{\prime 3}_{1}$ term is ignored. The hierarchy of VEVs is consistently
achieved by choosing $\mu_{3}v^{\prime}_{2}\ll\mu_{1}^{2}$ and/or
$\mu_{3}v^{\prime}_{2}\ll\lambda_{5}v^{\prime 2}_{2}$. This VEV hierarchy is
necessary to explain W boson mass anomaly and to obtain sizable muon $g-2$ at
the same time, as we discuss below. In this case, $z^{\prime}_{2}$ corresponds
to the NG boson to be absorbed by $Z^{\prime}$ boson.

In our analysis, we assume $\lambda_{4}$ and $\lambda_{5}$ to be negligibly
small for simplicity, and only $r$ and $r^{\prime}_{1}$ mixes. Then, we obtain
the mass matrix for CP even scalar, $m_{R}^{2}$, in the basis of
$(r,r^{\prime}_{1})$, where the mass eigenstates $\\{h,H\\}$ is found to be
$(r,r^{\prime}_{1})^{T}=O_{R}(h,H)^{T}$, and mass eigenvalues are given by
$m_{h,H}^{2}=O_{R}^{T}m_{R}^{2}O_{R}$. $m_{R}^{2}$ and $O_{R}$ are obtained as

$\displaystyle m_{R}^{2}$
$\displaystyle\simeq\left[\begin{array}[]{cc}2v^{2}\lambda_{H}&vv^{\prime}_{1}\lambda_{3}\\\
vv^{\prime}_{1}\lambda_{3}&\mu_{1}^{2}\\\
\end{array}\right],O_{R}=\left[\begin{array}[]{cc}c_{\theta}&s_{\theta}\\\
-s_{\theta}&c_{\theta}\\\ \end{array}\right],$ (13)

where $c_{\theta}(s_{\theta})$ stands for $\cos\theta(\sin\theta)$ with
$s_{2\theta}=\frac{2vv^{\prime}_{1}\lambda_{3}}{m^{2}_{h}-m^{2}_{H}}$. The
mass eigenvalues are also calculated such that

$m_{h,H}^{2}\simeq(v^{2}\lambda_{H}+\mu_{1}^{2})\mp\sqrt{(v^{2}\lambda_{H}-\mu_{1}^{2})^{2}+v^{2}v^{\prime
2}_{1}\lambda_{3}^{2}}.$ (14)

Here $h\equiv h_{SM}$ is the SM Higgs, therefore, $m_{h}=$125 GeV. The mixing
effect for CP-even scalar is constrained by the measurements of Higgs
production cross section and its decay branching ratio at the LHC, and
$s_{a}\lesssim 0.3$ is provided by the current data pdg . The mass of
$r^{\prime}_{2}$ is approximately given by
$m_{r^{\prime}_{2}}\simeq\sqrt{2\lambda_{2}}v^{\prime}_{2}$ which is supposed
to be much heavier than $m_{H}$.

###  II.2 Oblique parameters

Oblique parameters come from $Z_{SM}-Z^{\prime}$ mixing. Thus, we firstly
discuss this effect. Since $H$ has nonzero $U(1)_{R}$ charge, there is mixing
between $Z_{SM}$ and $Z^{\prime}$. The resulting mass matrix in basis of
$(Z_{SM},Z^{\prime})$ is given by

$\displaystyle m_{Z_{SM}Z^{\prime}}^{2}$
$\displaystyle\simeq\frac{1}{4}\left[\begin{array}[]{cc}(g_{1}^{2}+g_{2}^{2})v^{2}&-2\sqrt{g_{1}^{2}+g_{2}^{2}}g^{\prime}v^{2}\\\
-2\sqrt{g_{1}^{2}+g_{2}^{2}}g^{\prime}v^{2}&4g^{\prime 2}(v^{2}+v^{\prime
2}_{2})\\\
\end{array}\right]=m_{Z^{\prime}}^{2}\left[\begin{array}[]{cc}\epsilon_{1}^{2}&-\epsilon_{1}\epsilon_{2}\\\
-\epsilon_{1}\epsilon_{2}&1+\epsilon_{2}^{2}\\\ \end{array}\right],$ (19)

where $m_{Z_{SM}}\equiv\frac{\sqrt{g_{1}^{2}+g_{2}^{2}}v}{2}$,
$m_{Z^{\prime}}\equiv g^{\prime}v^{\prime}_{2}$,
$\epsilon_{1}\equiv\frac{m_{Z_{SM}}}{m_{Z^{\prime}}}$,
$\epsilon_{2}\equiv\frac{v}{v^{\prime}_{2}}$, $g_{1}$, $g_{2}$, and
$g^{\prime}$ are gauge coupling of $U(1)_{Y}$, $SU(2)_{L}$, and $U(1)_{R}$,
respectively. Note that we ignored $v^{\prime}_{1}$ in $m_{Z^{\prime}}$
formula due to the relation $v^{\prime}_{1}\ll v^{\prime}_{2}$. Then its mass
matrix is diagonalized by the two by two mixing matrix $V$ as
$Vm_{Z_{SM}Z^{\prime}}^{2}V^{T}\equiv{\rm Diag}(m^{2}_{Z},m^{2}_{Z_{R}})$,
where we work under $\epsilon_{2}^{2}\ll 1$ and

$\displaystyle m^{2}_{Z}$ $\displaystyle\approx
m_{Z_{SM}}^{2}(1-\epsilon_{2}^{2}),\ m^{2}_{Z_{R}}\approx
m_{Z^{\prime}}^{2}(1+\epsilon_{1}^{2}\epsilon_{2}^{2}),$ (20) $\displaystyle
V$ $\displaystyle\approx\left[\begin{array}[]{cc}c_{Z}&s_{Z}\\\
-s_{Z}&c_{Z}\\\
\end{array}\right],\quad\theta_{Z}=\frac{1}{2}\tan^{-1}\left[\frac{2\epsilon_{1}\epsilon_{2}}{1+\epsilon_{2}^{2}-\epsilon_{1}^{2}}\right].$
(23)

The $Z_{R}$ mass can be approximated as $m_{Z_{R}}\simeq
m_{Z^{\prime}}=g^{\prime}v^{\prime}_{2}$ since $\epsilon_{1,2}$ is small. Thus
gauge coupling $g^{\prime}$ is almost fixed if we choose values of $m_{Z_{R}}$
and $v^{\prime}$.

In our case, only $\Delta T$ is nonzero induced via $Z-Z^{\prime}$ mixing
thanks to zero $U(1)_{R}$ charges of $L_{L},\ e_{R}$ and defined by

$\displaystyle\Delta T=\frac{1}{\alpha_{\rm
em}}\frac{m_{Z_{SM}}^{2}-m_{Z}^{2}}{m_{Z_{SM}}^{2}}\simeq\frac{\epsilon^{2}_{2}}{\alpha_{\rm
em}},$ (24)

where we have used Eq. (20) in the last part of the above equation. Thus,
$\Delta T$ is straightforwardly given by inserting $\epsilon_{2}\equiv
v/v^{\prime}_{2}$, $v^{\prime}_{2}=m_{Z^{\prime}}/g^{\prime}$,
$v=2m_{Z}\cos\theta_{W}/g_{2}$ with $\theta_{W}$ being the Weinberg angle and
simply given by Strumia:2022qkt  222Notice here that the other valid oblique
parameters $\Delta S,\ \Delta W,\ \Delta Y$ are zero.

$\displaystyle\Delta T\simeq\frac{v^{2}}{\alpha_{\rm em}}\frac{g^{\prime
2}}{m_{Z^{\prime}}^{2}}=\frac{4m_{Z}^{2}\cos^{2}\theta_{W}}{g_{2}^{2}\alpha_{\rm
em}}\frac{g^{\prime 2}}{m_{Z^{\prime}}^{2}}.$ (25)

Note also that contribution to $\Delta T$ at loop level is negligible since
our new particles are $SU(2)$ singlet. Although mixing between the SM charged
lepton and new charged lepton discussed below can affect the $\Delta T$ at
loop level it is highly suppressed by small mixing angle. Thus our relevant
parameters for $\Delta T$ are only new gauge coupling $g^{\prime}$ and
$Z^{\prime}$ mass $m_{Z^{\prime}}$. From global fit including CDF II W boson
mass with $S\sim 0$ we obtain $1\sigma$ range of $T$ as

$0.09\leq\Delta T\leq 0.14.$ (26)

Then it is found from Eq. (25):

$20\ {\rm
TeV}\lesssim\frac{m_{Z^{\prime}}}{g^{\prime}}(=v^{\prime}_{2})\lesssim 31\
{\rm TeV},$ (27)

where we used central value of Eq. (1) for $m_{W}$. This range is allowed by
LEP constraints ALEPH:2013dgf  and dijet searches at the LHC Dobrescu:2021vak
; ATLAS:2019fgd ; CMS:2019gwf . Here, we emphasize that our model only
provides a sizable contribution to $\Delta T$ via $Z$–$Z^{\prime}$ mixing.
Thus modification of $W$ boson mass is characterized by $\Delta T$ that is
estimated by $Z^{\prime}$ mass, new gauge coupling and other electroweak
observables $\\{m_{Z},\theta_{W},\alpha_{\rm em}\\}$. It is the advantage of
our model that other oblique parameters are not modified.

Here we comment on the situation in the case of original $U(1)_{R}$ model. In
this case, we have contributions to other oblique parameters $W$ and $Y$ due
to $Z^{\prime}$ interactions with charged leptons Cacciapaglia:2006pk . The
oblique parameters, including $W$ and $Y$ are defined from effective
Lagrangian

$\mathcal{L}=-\frac{1}{2}W^{3}_{\mu}\Pi_{33}(p^{2})W^{3\mu}-\frac{1}{2}B_{\mu}\Pi_{00}(p^{2})B^{\mu}-W^{3}_{\mu}\Pi_{30}(p^{2})B^{\mu}-W^{+}_{\mu}\Pi_{WW}(p^{2})W^{-\mu},$
(28)

where $W^{3}_{\mu}$, $B^{\mu}$ and $W^{\pm}$ are gauge fields corresponding to
the third component of $SU(2)_{L}$, $U(1)_{Y}$ and $W$ boson respectively, and
$p^{2}$ is the momentum square carried by the gauge field. The oblique
parameters $W$ and $Y$ are then given by

$W=\frac{m^{2}_{W}}{2}\frac{d^{2}}{d(p^{2})^{2}}\Pi_{33}|_{p^{2}=0},\quad
Y=\frac{m^{2}_{W}}{2}\frac{d^{2}}{d(p^{2})^{2}}\Pi_{00}|_{p^{2}=0}.$ (29)

It is shown in ref. Cacciapaglia:2006pk  that $W=Y=0$ is obtained when leptons
are not charged under new $U(1)$ symmetry, which is our case. Otherwise, the
values of $W$ and $Y$ have similar order as $\hat{T}\equiv\alpha_{\rm em}T$
and the LHC constraints are $|W|\lesssim 1.8\times 10^{-4}$ and $|Y|\lesssim
2.0\times 10^{-4}$ at $1\sigma$ confidence level ATLAS:2017fih ; CMS:2022yjm ;
Strumia:2022qkt . Thus the parameter region realizing $\Delta T$ in Eq. (26)
is excluded by the constraints. Therefore we need leptophobic $U(1)_{R}$ to
explain the W boson anomaly.

###  II.3 Charged-lepton sector

The charged-lepton mass matrix is obtained via mixing among the singly-charged
fermions, and the form is give by Dey:2019cts

$\displaystyle\left(\begin{array}[]{c}\bar{e}_{L}\\\
\bar{E}_{L}\end{array}\right)^{T}{\cal M_{E}}\left(\begin{array}[]{c}e_{R}\\\
E_{R}\end{array}\right)=\left(\begin{array}[]{c}\bar{e}_{L}\\\
\bar{E}_{L}\end{array}\right)^{T}\left[\begin{array}[]{cc}0&m_{eE}\\\
m_{Ee}&M_{E}\\\ \end{array}\right]\left(\begin{array}[]{c}e_{R}\\\
E_{R}\end{array}\right),$ (40) $\displaystyle{\cal M_{E}}{\cal
M_{E}}^{\dagger}=\left[\begin{array}[]{cc}m_{eE}m_{eE}^{\dagger}&m_{eE}M_{E}\\\
M_{E}m_{eE}^{\dagger}&M_{E}^{2}+m_{Ee}M_{Ee}^{\dagger}\\\
\end{array}\right],\quad{\cal M_{E}}^{\dagger}{\cal
M_{E}}=\left[\begin{array}[]{cc}m_{Ee}^{\dagger}m_{Ee}&m_{Ee}^{\dagger}M_{E}\\\
M_{E}m_{Ee}&M_{E}^{2}+m_{eE}^{\dagger}m_{eE}\\\ \end{array}\right],$ (45)

where $m_{eE}\equiv y_{\ell}v/\sqrt{2}$, $M_{E}\equiv
y_{E}v^{\prime}/\sqrt{2}$. The mass matrix is diagonalized by the
transformation $(e_{L(R)},E_{L(R)})\to V_{L(R)}\ell_{L(R)}$. Thus we can
obtain diagonalization matrices $V_{L}$ and $V_{R}$ which respectively
diagonalize $M_{E}M_{E}^{\dagger}$ and $M_{E}^{\dagger}M_{E}$ as
$V_{L}^{\dagger}{\cal M_{E}}{\cal M_{E}}^{\dagger}V_{L}=V_{R}^{\dagger}{\cal
M_{E}}^{\dagger}{\cal M_{E}}V_{R}={\rm diag}|D_{E_{a}}|^{2}$(a=1-6), where the
first three mass eigenstates correspond to the SM charged-leptons. We then
write ${\rm diag}D_{E_{a}}={\rm
diag}(m_{e},m_{\mu},m_{\tau},M_{E_{1}},M_{E_{2}},M_{E_{3}})$.

###  II.4 New contribution to $Z\to\ell\bar{\ell}$

Due to the mixing between the exotic singly-charged fermions and the SM
leptons, we have new contribution to $Z\to\ell_{i}\bar{\ell}_{j}$. Their
kinetic Lagrangian in terms of mass eigenvectors is given by

$\displaystyle\frac{g_{2}}{c_{W}}\left[\left(-\frac{1}{2}\sum_{a=1}^{3}V^{\dagger}_{L_{ia}}V_{L_{aj}}+s_{W}^{2}\delta_{ij}\right)\bar{\ell}_{L_{i}}\gamma^{\mu}\ell_{L_{j}}+s_{W}^{2}\delta_{ij}\bar{\ell}_{R_{i}}\gamma^{\mu}\ell_{R_{j}}\right]Z_{\mu},$
(46)

where $s_{W},\ c_{W}$ are short-hand notations of Weinberg angles;
$\sin\theta_{W},\ \cos\theta_{W}$, which are rewritten in terms of $g_{1},\
g_{2}$. Then, the decay rate of $Z\to\ell_{i}\bar{\ell}_{j}$ is given by

$\displaystyle\Gamma(Z\to\ell_{i}\bar{\ell}_{j})\simeq\frac{g_{2}^{2}}{24\pi
c_{W}^{2}}m_{Z}\left[\left|-\frac{1}{2}\sum_{a=1}^{3}V^{\dagger}_{L_{ia}}V_{L_{aj}}+s_{W}^{2}\delta_{ij}\right|^{2}+s_{W}^{4}\delta_{ij}\right],$
(47)

where we assume $m_{Z}>>m_{\ell}$. Here, one confirms that the SM contribution
is derived when $\sum_{a=1}^{3}V^{\dagger}_{L_{ia}}V_{L_{aj}}=1$;

$\displaystyle\Gamma(Z\to\ell_{i}\bar{\ell}_{j})_{SM}\simeq\frac{g_{2}^{2}}{24\pi
c_{W}^{2}}m_{Z}\left[\frac{1}{4}-s_{W}^{2}+2s_{W}^{4}\right]\delta_{ij}.$ (48)

Thus, the new contribution of branching ratios are given by

$\displaystyle\Delta{\rm
BR}(Z\to\ell_{i}\bar{\ell}_{j})=\frac{\Gamma(Z\to\ell_{i}\bar{\ell}_{j})-\Gamma(Z\to\ell_{i}\bar{\ell}_{j})_{SM}}{\Gamma_{Z}^{\rm
tot}}\quad{\rm for\ i=j},$ (49) $\displaystyle{\rm
BR}(Z\to\ell_{i}\bar{\ell}_{j})=\frac{\Gamma(Z\to\ell_{i}\bar{\ell}_{j})}{\Gamma_{Z}^{\rm
tot}}\quad{\rm for\ i\neq j},$ (50)

where the total $Z$ decay width $\Gamma_{Z}^{\rm tot}=2.4952\pm 0.0023$ GeV
pdg .

The current bounds on the lepton-flavor-(conserving)changing $Z$ boson decay
branching ratios(BRs) at 95 % CL are given by pdg :

$\displaystyle\Delta{\rm BR}(Z\to e^{\pm}e^{\mp})<\pm 4.2\times 10^{-5}~{},\
\Delta{\rm BR}(Z\to\mu^{\pm}\mu^{\mp})<\pm 6.6\times 10^{-5}~{},$
$\displaystyle\Delta{\rm BR}(Z\to\tau^{\pm}\tau^{\mp})<\pm 8.3\times
10^{-5}~{},\ {\rm BR}(Z\to e^{\pm}\mu^{\mp})<7.5\times 10^{-7}~{},$
$\displaystyle{\rm BR}(Z\to e^{\pm}\tau^{\mp})<9.8\times 10^{-6}~{},\ {\rm
BR}(Z\to\mu^{\pm}\tau^{\mp})<1.2\times 10^{-5}~{}.$ (51)

Figure 1: The diagram that gives dominant contributions to
$\ell\to\ell^{\prime}\gamma$ and muon $g-2$ in flavor basis. The $\times$ mark
indicates mass (mixing) insertion.

###  II.5 Lepton flavor violations and muon $g-2$

Due to mixing between the SM charged-lepton and heavier leptons, we have
nonzero LFVs and muon $g-2$ via $y_{\ell}$. The current experimental upper
bounds on LFVs are given by MEG:2016leq ; MEG:2013oxv

$\displaystyle{\rm BR}(\mu\rightarrow e\gamma)$ $\displaystyle\leq 4.2\times
10^{-13},\quad{\rm BR}(\tau\rightarrow\mu\gamma)\leq 4.4\times
10^{-8},\quad{\rm BR}(\tau\rightarrow e\gamma)\leq 3.3\times 10^{-8}~{}.$ (52)

On the other hand, new results on the muon $(g-2)$ were recently published by
the E989 collaboration at Fermilab Muong-2:2021ojo :

$\displaystyle a^{\rm FNAL}_{\mu}=116592040(54)\times 10^{-11}.$ (53)

Combined with the previous BNL result, this means that the muon $(g-2)$
deviates from the SM prediction by 4.2$\sigma$ level Muong-2:2021ojo ;
Aoyama:2012wk ; Aoyama:2019ryr ; Czarnecki:2002nt ; Gnendiger:2013pva ;
Davier:2017zfy ; Keshavarzi:2018mgv ; Colangelo:2018mtw ; Hoferichter:2019mqg
; Davier:2019can ; Keshavarzi:2019abf ; Kurz:2014wya ; Melnikov:2003xd ;
Masjuan:2017tvw ; Colangelo:2017fiz ; Hoferichter:2018kwz ; Gerardin:2019vio ;
Bijnens:2019ghy ; Colangelo:2019uex ; Blum:2019ugy ; Colangelo:2014qya ;
Hagiwara:2011af :

$\displaystyle\Delta a^{\rm new}_{\mu}=(25.1\pm 5.9)\times 10^{-10},$ (54)

and it could be a verfiable signature of the physics beyond the SM.

In our scenario, the relevant Lagrangian to induce LFVs and muon $g-2$ in
terms of mass eigenstate is given by 333Even though we have a contribution
from a new gauge boson $Z^{\prime}$ to these phenomenologies, these are
subdominant for the value of $g^{\prime}/m_{Z^{\prime}}$ in the range of Eq.
(27). We have checked it numerically.

$\displaystyle\mathcal{L}_{LFV}=(Y_{\alpha\beta}c_{\theta}-\tilde{Y}_{\alpha\beta}s_{\theta})h\bar{\ell}_{\alpha}P_{R}\ell_{\beta}+(Y_{\alpha\beta}s_{\theta}+\tilde{Y}_{\alpha\beta}c_{\theta})H\bar{\ell}_{\alpha}P_{R}\ell_{\beta}+h.c.,$
(55) $\displaystyle
Y_{\alpha\beta}\equiv\frac{1}{\sqrt{2}}\sum_{a=1,2,3}\sum_{b=1,2,3}(V^{\dagger}_{L})_{\alpha,a}(y_{\ell})_{ab}(V_{R})_{b+3,\beta},$
(56)
$\displaystyle\tilde{Y}_{\alpha\beta}\equiv\frac{1}{\sqrt{2}}\sum_{a=1,2,3}\sum_{b=1,2,3}(V^{\dagger}_{L})_{\alpha,a+3}(y_{E})_{aa}(V_{R})_{a+3,\beta}.$
(57)

Then, the dominant contributions to LFVs and muon $g-2$ at one-loop level are
given by

$\displaystyle{\rm
BR}(\ell_{\beta}\to\ell_{\alpha}\gamma)\simeq\frac{12\pi^{2}C_{\beta\alpha}}{(4\pi)^{4}m_{\ell_{\beta}}^{2}G_{F}^{2}}(|a_{L_{\alpha\beta}}|^{2}+|a_{R_{\alpha\beta}}|^{2}),$
(58) $\displaystyle\Delta
a_{\mu}\simeq-\frac{m_{\mu}}{(4\pi)^{2}}(a_{L_{22}}+a_{R_{22}}),$ (59)
$\displaystyle
a_{L_{\alpha\beta}}\approx-\frac{1}{4}{(Y^{\dagger}_{\alpha\rho}c_{\theta}-\tilde{Y}^{\dagger}_{\alpha\rho}s_{\theta})D_{E_{\rho}}(Y^{\dagger}_{\rho\beta}c_{\theta}-\tilde{Y}^{\dagger}_{\rho\beta}s_{\theta})}F(h_{1},D_{E_{\rho}})$
$\displaystyle-\frac{1}{4}{(Y^{\dagger}_{\alpha\rho}s_{\theta}+\tilde{Y}^{\dagger}_{\alpha\rho}c_{\theta})D_{E_{\rho}}(Y^{\dagger}_{\rho\beta}s_{\theta}+\tilde{Y}^{\dagger}_{\rho\beta}c_{\theta})}F(h_{2},D_{E_{\rho}}),$
(60) $\displaystyle
a_{R_{\alpha\beta}}\approx-\frac{1}{4}{(Y_{\alpha\rho}c_{\theta}-\tilde{Y}_{\alpha\rho}s_{\theta})D_{E_{\rho}}(Y_{\rho\beta}c_{\theta}-\tilde{Y}_{\rho\beta}s_{\theta})}F(h_{1},D_{E_{\rho}})$
$\displaystyle-\frac{1}{4}{(Y_{\alpha\rho}s_{\theta}+\tilde{Y}_{\alpha\rho}c_{\theta})D_{E_{\rho}}(Y_{\rho\beta}s_{\theta}+\tilde{Y}_{\rho\beta}c_{\theta})}F(h_{2},D_{E_{\rho}}),$
(61) $\displaystyle
F(m_{h_{i}},D_{E_{\rho}})\approx\frac{m_{h_{i}}^{4}-4m_{h_{i}}^{2}D_{E_{\rho}}^{2}+3D_{E_{\rho}}^{4}-2m_{h_{i}}^{2}(m_{h_{i}}^{2}-2D_{E_{\rho}}^{2})\ln\left[\frac{m_{h_{i}}^{2}}{D_{E_{\rho}}^{2}}\right]}{(m_{h_{i}}^{2}-D_{E_{\rho}}^{2})^{3}},$
(62)

where $C_{21}\approx 1$, $C_{31}\approx 0.1784$, $C_{32}\approx 0.1736$,
$h_{1}\equiv h$, $h_{2}\equiv H$. The dominant contribution is obtained from
the diagram in Fig. 1 that does not have chiral suppression.

###  II.6 Neutrino mass via inverse seesaw mechanism

After spontaneous gauge symmetry breaking, we obtain neutral fermion mass
matrix in the basis of $(\nu^{c}_{L},N_{R},N^{c}_{L})$ as follows

$M_{N}=\left[\begin{array}[]{ccc}0&m_{D}&0\\\ m^{T}_{D}&0&M\\\
0&M&M_{N_{L}}\end{array}\right],$ (63)

where $m_{D}\equiv vy_{D}/\sqrt{2}$ and $M\equiv
y_{N}v^{\prime}_{1}/\sqrt{2}$. When mass parameters satisfy $M_{N_{L}}\ll
m_{D}\lesssim M$ active neutrino mass can be approximately written by

$m_{\nu}\simeq m_{D}M^{-1}M_{N_{L}}(M^{T})^{-1}m^{T}_{D}.$ (64)

The neutrino mass matrix is diagonalized by a unitary matrix $U_{\nu}$;
$D_{\nu}=U^{T}_{\nu}m_{\nu}U_{\nu}$ with $D_{\nu}\equiv{\rm
diag}(m_{1},m_{2},m_{3})$. Including charged lepton mixing matrix, the PMNS
matrix is defined by $U_{PMNS}\equiv V^{\dagger}_{L}U_{\nu}$.

We discuss constraint from non-unitarity which is described by a matrix
$U^{\prime}_{PMNS}$. This matrix is typically parametrized by the form of

$\displaystyle
U^{\prime}_{PMNS}\equiv\left(1-\frac{1}{2}FF^{\dagger}\right)U_{PMNS},$ (65)

where $F\equiv(M^{T})^{-1}m_{D}$ is a hermitian matrix. The global constraints
on elements of $|FF^{\dagger}|$ are found combining several experimental
results such as the SM $W$ boson mass $M_{W}$, the effective Weinberg angle
$\theta_{W}$, several ratios of $Z$ boson fermionic decays, invisible decay of
$Z$, electroweak universality, measured Cabbibo-Kobayashi-Maskawa, and lepton
flavor violations Fernandez-Martinez:2016lgt . The result is then given by
Agostinho:2017wfs

$\displaystyle|FF^{\dagger}|\leq\left[\begin{array}[]{ccc}2.5\times
10^{-3}&2.4\times 10^{-5}&2.7\times 10^{-3}\\\ 2.4\times 10^{-5}&4.0\times
10^{-4}&1.2\times 10^{-3}\\\ 2.7\times 10^{-3}&1.2\times 10^{-3}&5.6\times
10^{-3}\\\ \end{array}\right].$ (69)

If we require $|F|\sim 10^{-5}$ conservatively, $M_{N_{L}}\sim 1-10$ GeV can
reproduce active neutrino mass scale. Also we require $y_{D}\sim 10^{-4}$ for
$M=\mathcal{O}(1)$ TeV. Observed neutrino mixing can be easily obtained since
we have sufficient number of free parameters $\\{y_{D},y_{N},M_{N_{L}}\\}$.

##  III Numerical analysis and phenomenological consequences

In this section we carry out numerical study to estimate muon $g-2$ and LFVs.
We also calculate $Z^{\prime}$ production cross section for parameter space
explaining W boson mass anomaly and show phenomenological implications of our
scenario at hadron collider experiments.

###  III.1 Parameter scan for muon $g-2$ and LFVs

Here we discuss muon $g-2$ taking into account charged lepton masses and LFV
constraints in our model performing numerical analysis. The dominant
contribution to muon $g-2$ comes from the diagram in Fig. 1 since it has
chirality change inside loop giving heavy charged lepton mass factor. The
relevant free parameters are

$\\{(y_{\ell})_{ab},(y_{E})_{aa},(m_{Ee})_{ab},\sin\theta,m_{H}\\},$ (70)

where $a,b=1-3$. We then scan these free parameters globally to search for
best fit value of muon $g-2$. In Table 2, we show our benchmark point(BP)
found by numerical analysis. We find that large value of Yukawa coupling
$y_{E}$ is preferred to obtain sizable muon $g-2$ while exotic heavy charged
lepton masses are around 700 GeV. Then values of $v^{\prime}_{1}$ and $m_{H}$
are preferred to be around electroweak scale. This is the reason why we need
two singlet scalars $\varphi_{1}$ and $\varphi_{2}$ to explain both muon $g-2$
and W boson mass anomaly as the required scale of these VEVs are different;
$v^{\prime}_{1}\ll v^{\prime}_{2}$. We also show ${\rm
BR}(\ell\to\ell^{\prime}\gamma)$ and find that those of $\mu\to e\gamma$ and
$\tau\to\mu\gamma$ are close to the current upper limit. Thus it could be
tested in future measurements. Furthermore we show $(\Delta){\rm
BR}(Z\to\ell\ell^{\prime})$ values that might be also tested in future
precision measurements for $Z$ boson decay.

Input
---
$v^{\prime}_{1}/$GeV | $284$
$m_{H}/$GeV | $245$
$\sin\theta$ | 0.250
$[(y_{\ell})_{11},(y_{\ell})_{12},(y_{\ell})_{13}]$ | $[-0.000512,\ 0.00520,\ 0.00236]$
$[(y_{\ell})_{21},(y_{\ell})_{22},(y_{\ell})_{23}]$ | $[-0.000105,\ 0.000624,\ 0.0643]$
$[(y_{\ell})_{31},(y_{\ell})_{32},(y_{\ell})_{33}]$ | $[0.000148,\ 0.0145,\ 0.0647]$
$[(m_{eE})_{11},(m_{eE})_{12},(m_{eE})_{13}]/$GeV | $[0.674,\ 16.8,\ 5.92]$
$[(m_{eE})_{21},(m_{eE})_{22},(m_{eE})_{23}]/$GeV | $[-16.2,\ -46.3,\ 22.3]$
$[(m_{eE})_{31},(m_{eE})_{32},(m_{eE})_{33}]/$GeV | $[-12.2,\ 69.5,\ 19.8]$
$[(y_{E})_{11},(y_{E})_{22},(y_{E})_{33}]$ | $[-3.26,\ 3.43,\ -3.41]$
Output
$[m_{E_{1}},m_{E_{2}},m_{E_{3}}]/$GeV | $[655,\ 690,\ 694]$
$\Delta a_{\mu}$ | $2.11\times 10^{-9}$
${\rm BR}(\mu\to e\gamma)$ | $3.59\times 10^{-13}$
${\rm BR}(\tau\to e\gamma)$ | $9.91\times 10^{-11}$
${\rm BR}(\tau\to\mu\gamma)$ | $2.22\times 10^{-8}$
$\Delta{\rm BR}(Z\to e^{\pm}e^{\mp})$ | $-1.55\times 10^{-9}$
$\Delta{\rm BR}(Z\to\mu^{\pm}\mu^{\mp})$ | $-6.11\times 10^{-7}$
$\Delta{\rm BR}(Z\to\tau^{\pm}\tau^{\mp})$ | $-3.82\times 10^{-5}$
$\Delta{\rm BR}(Z\to e^{\pm}\mu^{\mp})$ | $6.27\times 10^{-17}$
$\Delta{\rm BR}(Z\to e^{\pm}\tau^{\mp})$ | $3.83\times 10^{-14}$
$\Delta{\rm BR}(Z\to\mu^{\pm}\tau^{\mp})$ | $3.64\times 10^{-11}$
Table 2: Benchmark point that explains muon $g-2$. Figure 2: The $1\sigma$
region to explain W boson mass anomaly on $\\{m_{Z^{\prime}},g^{\prime}\\}$
plane where the red colored line indicates the value providing central value
of $\Delta T$ in Eq. (26).  Figure 3: The products of $Z^{\prime}$ production
cross section and the BRs for $jj$ and $E\bar{E}(N\bar{N})$ modes at the LHC
14 TeV where we chose $m_{E(N)}=700$ GeV for three generations.

###  III.2 Signature of $Z^{\prime}$ production at collider

We consider signature of $Z^{\prime}$ production at hadron collider
experiments focusing on parameter space that can explain W boson mass anomaly.
In Fig. 2, we show the $1\sigma$ region to explain W boson mass anomaly on
$\\{m_{Z^{\prime}},g^{\prime}\\}$ plane where the red colored line indicates
the value providing central value of $\Delta T$ in Eq. (26). For the parameter
region we estimate cross section of $pp\to Z^{\prime}$ process using CalcHEP
Belyaev:2012qa  implementing relevant gauge interactions.

Our $Z^{\prime}$ dominantly decays into
$\\{q\bar{q},N_{R}\bar{N}_{R},E_{R}\bar{E}_{R}\\}$ modes. The decay width is
estimated by

$\Gamma(Z^{\prime}\to f\bar{f})=\frac{g^{\prime
2}N_{c}}{12\pi}m_{Z^{\prime}}\sqrt{1-\frac{4m_{f}^{2}}{m^{2}_{Z^{\prime}}}}\left(1-\frac{m_{f}^{2}}{m^{2}_{Z^{\prime}}}\right)$
(71)

where $N_{c}$ is color degrees of freedom. BRs of $Z^{\prime}$ decay can be
estimated by the width. In Fig. 3, we show products of cross section and the
BRs for $jj$ and $E\bar{E}(N\bar{N})$ modes where we chose $m_{E(N)}=700$ GeV
for three generations. We thus obtain sizable cross section that can be tested
at future LHC experiments. Extra fermions $E_{a}$ and $N_{a}$ dominantly decay
into the SM lepton with boson as

$\displaystyle E_{a}\to\ell_{i}h,$ (72) $\displaystyle
N_{a}\to\nu_{i}h,\nu_{i}Z,\ell_{i}W,$ (73)

where lepton flavor dependence is determined by structure of Yukawa couplings.
Therefore signatures of our $Z^{\prime}$ are dijet, top quark pair and SM
leptons with higgs or gauge bosons. More detailed numerical analysis is beyond
the scope of this paper and left as future work.

##  IV Summary and Conclusions

We have proposed a model with leptophobic $U(1)_{R}$ gauge symmetry to explain
CDF W boson mass anomaly. The shift of W boson mass is realized via neutral
gauge boson mass which induces non-zero oblique $\Delta T$ parameter. Also we
have shown that $U(1)_{R}$ should be leptophobic since parameter region to
explain W bosom mass anomaly is excluded by the LHC constraints on $W$ and $Y$
oblique parameters if $Z^{\prime}$ is coupled to the SM charged leptons. It is
then found that $Z^{\prime}$ mass and new gauge coupling should satisfy 20 TeV
$\lesssim m_{Z^{\prime}}/g^{\prime}\lesssim$ 31 TeV to accommodate CDF W bosom
mass.

In our model charged lepton masses are induced via interactions between the SM
leptons and extra charged leptons. These interactions also induce muon $g-2$
and LFVs at loop level. We have investigated muon $g-2$ and LFVs taking into
current experimental constraints. It is found that some LFV BRs, $\mu\to
e\gamma$ and $\tau\to\mu\gamma$, tends to be close to current upper limits
when we explain muon $g-2$ and the BRs can be tested in future experiments. In
addition we have discussed realization of active neutrino mass and mixings via
inverse seesaw mechanism. Moreover we have discussed collider physics focusing
on $Z^{\prime}$ production and its decay at the LHC. We have found that
sizable cross section can be expected considering parameter region explaining
W boson mass anomaly.

## Acknowledgments

KIN was supported by JSPS Grant-in-Aid for Scientific Research (A) 18H03699,
(C) 21K03562, (C) 21K03583, Okayama Foundation for Science and Technology, and
Wesco Scientific Promotion Foundation. This research was supported by an
appointment to the JRG Program at the APCTP through the Science and Technology
Promotion Fund and Lottery Fund of the Korean Government. This was also
supported by the Korean Local Governments - Gyeongsangbuk-do Province and
Pohang City (H.O.). H. O. is sincerely grateful for KIAS and all the members.
The work was also supported by the Fundamental Research Funds for the Central
Universities (T. N.).

## References

  * (1) T. Aaltonen et al. [CDF], Science 376 (2022) no.6589, 170-176 doi:10.1126/science.abk1781
  * (2) M. Tanabashi et al. [Particle Data Group], Phys. Rev. D 98 (2018) no.3, 030001 doi:10.1103/PhysRevD.98.030001
  * (3) Y. Z. Fan, T. P. Tang, Y. L. S. Tsai and L. Wu, [arXiv:2204.03693 [hep-ph]].
  * (4) C. T. Lu, L. Wu, Y. Wu and B. Zhu, [arXiv:2204.03796 [hep-ph]].
  * (5) P. Athron, A. Fowlie, C. T. Lu, L. Wu, Y. Wu and B. Zhu, [arXiv:2204.03996 [hep-ph]].
  * (6) G. W. Yuan, L. Zu, L. Feng and Y. F. Cai, [arXiv:2204.04183 [hep-ph]].
  * (7) A. Strumia, [arXiv:2204.04191 [hep-ph]].
  * (8) J. M. Yang and Y. Zhang, [arXiv:2204.04202 [hep-ph]].
  * (9) J. de Blas, M. Pierini, L. Reina and L. Silvestrini, [arXiv:2204.04204 [hep-ph]].
  * (10) G. G. di Cortona and E. Nardi, [arXiv:2204.04227 [hep-ph]].
  * (11) X. K. Du, Z. Li, F. Wang and Y. K. Zhang, [arXiv:2204.04286 [hep-ph]].
  * (12) T. P. Tang, M. Abdughani, L. Feng, Y. L. S. Tsai and Y. Z. Fan, [arXiv:2204.04356 [hep-ph]].
  * (13) G. Cacciapaglia and F. Sannino, [arXiv:2204.04514 [hep-ph]].
  * (14) M. Blennow, P. Coloma, E. Fernández-Martínez and M. González-López, [arXiv:2204.04559 [hep-ph]].
  * (15) K. Sakurai, F. Takahashi and W. Yin, [arXiv:2204.04770 [hep-ph]].
  * (16) J. Fan, L. Li, T. Liu and K. F. Lyu, [arXiv:2204.04805 [hep-ph]].
  * (17) X. Liu, S. Y. Guo, B. Zhu and Y. Li, [arXiv:2204.04834 [hep-ph]].
  * (18) H. M. Lee and K. Yamashita, [arXiv:2204.05024 [hep-ph]].
  * (19) Y. Cheng, X. G. He, Z. L. Huang and M. W. Li, [arXiv:2204.05031 [hep-ph]].
  * (20) H. Song, W. Su and M. Zhang, [arXiv:2204.05085 [hep-ph]].
  * (21) E. Bagnaschi, J. Ellis, M. Madigan, K. Mimasu, V. Sanz and T. You, [arXiv:2204.05260 [hep-ph]].
  * (22) A. Paul and M. Valli, [arXiv:2204.05267 [hep-ph]].
  * (23) H. Bahl, J. Braathen and G. Weiglein, [arXiv:2204.05269 [hep-ph]].
  * (24) P. Asadi, C. Cesarotti, K. Fraser, S. Homiller and A. Parikh, [arXiv:2204.05283 [hep-ph]].
  * (25) L. Di Luzio, R. Gröber and P. Paradisi, [arXiv:2204.05284 [hep-ph]].
  * (26) P. Athron, M. Bach, D. H. J. Jacob, W. Kotlarski, D. Stöckinger and A. Voigt, [arXiv:2204.05285 [hep-ph]].
  * (27) J. Gu, Z. Liu, T. Ma and J. Shu, [arXiv:2204.05296 [hep-ph]].
  * (28) J. J. Heckman, [arXiv:2204.05302 [hep-ph]].
  * (29) K. S. Babu, S. Jana and V. P. K., [arXiv:2204.05303 [hep-ph]].
  * (30) Y. Heo, D. W. Jung and J. S. Lee, [arXiv:2204.05728 [hep-ph]].
  * (31) X. K. Du, Z. Li, F. Wang and Y. K. Zhang, [arXiv:2204.05760 [hep-ph]].
  * (32) K. Cheung, W. Y. Keung and P. Y. Tseng, [arXiv:2204.05942 [hep-ph]].
  * (33) A. Crivellin, M. Kirk, T. Kitahara and F. Mescia, [arXiv:2204.05962 [hep-ph]].
  * (34) M. Endo and S. Mishima, [arXiv:2204.05965 [hep-ph]].
  * (35) T. Biekötter, S. Heinemeyer and G. Weiglein, [arXiv:2204.05975 [hep-ph]].
  * (36) R. Balkin, E. Madge, T. Menzo, G. Perez, Y. Soreq and J. Zupan, [arXiv:2204.05992 [hep-ph]].
  * (37) N. V. Krasnikov, [arXiv:2204.06327 [hep-ph]].
  * (38) Y. H. Ahn, S. K. Kang and R. Ramos, [arXiv:2204.06485 [hep-ph]].
  * (39) X. F. Han, F. Wang, L. Wang, J. M. Yang and Y. Zhang, [arXiv:2204.06505 [hep-ph]].
  * (40) M. D. Zheng, F. Z. Chen and H. H. Zhang, [arXiv:2204.06541 [hep-ph]].
  * (41) J. Kawamura, S. Okawa and Y. Omura, [arXiv:2204.07022 [hep-ph]].
  * (42) A. Ghoshal, N. Okada, S. Okada, D. Raut, Q. Shafi and A. Thapa, [arXiv:2204.07138 [hep-ph]].
  * (43) P. Fileviez Perez, H. H. Patel and A. D. Plascencia, [arXiv:2204.07144 [hep-ph]].
  * (44) K. I. Nagao, T. Nomura and H. Okada, [arXiv:2204.07411 [hep-ph]].
  * (45) S. Kanemura and K. Yagyu, [arXiv:2204.07511 [hep-ph]].
  * (46) P. Mondal, [arXiv:2204.07844 [hep-ph]].
  * (47) K. Y. Zhang and W. Z. Feng, [arXiv:2204.08067 [hep-ph]].
  * (48) D. Borah, S. Mahapatra, D. Nanda and N. Sahu, [arXiv:2204.08266 [hep-ph]].
  * (49) T. A. Chowdhury, J. Heeck, S. Saad and A. Thapa, [arXiv:2204.08390 [hep-ph]].
  * (50) G. Arcadi and A. Djouadi, [arXiv:2204.08406 [hep-ph]].
  * (51) V. Cirigliano, W. Dekens, J. de Vries, E. Mereghetti and T. Tong, [arXiv:2204.08440 [hep-ph]].
  * (52) L. M. Carpenter, T. Murphy and M. J. Smylie, [arXiv:2204.08546 [hep-ph]].
  * (53) O. Popov and R. Srivastava, [arXiv:2204.08568 [hep-ph]].
  * (54) K. Ghorbani and P. Ghorbani, [arXiv:2204.09001 [hep-ph]].
  * (55) M. Du, Z. Liu and P. Nath, [arXiv:2204.09024 [hep-ph]].
  * (56) A. Bhaskar, A. A. Madathil, T. Mandal and S. Mitra, [arXiv:2204.09031 [hep-ph]].
  * (57) A. Batra, S. K.A., S. Mandal and R. Srivastava, [arXiv:2204.09376 [hep-ph]].
  * (58) J. Cao, L. Meng, L. Shang, S. Wang and B. Yang, [arXiv:2204.09477 [hep-ph]].
  * (59) Y. P. Zeng, C. Cai, Y. H. Su and H. H. Zhang, [arXiv:2204.09487 [hep-ph]].
  * (60) S. Baek, [arXiv:2204.09585 [hep-ph]].
  * (61) D. Borah, S. Mahapatra and N. Sahu, [arXiv:2204.09671 [hep-ph]].
  * (62) E. d. Almeida, A. Alves, O. J. P. Eboli and M. C. Gonzalez-Garcia, [arXiv:2204.10130 [hep-ph]].
  * (63) Y. Cheng, X. G. He, F. Huang, J. Sun and Z. P. Xing, [arXiv:2204.10156 [hep-ph]].
  * (64) J. Heeck, [arXiv:2204.10274 [hep-ph]].
  * (65) A. Addazi, A. Marciano, A. P. Morais, R. Pasechnik and H. Yang, [arXiv:2204.10315 [hep-ph]].
  * (66) S. Lee, K. Cheung, J. Kim, C. T. Lu and J. Song, [arXiv:2204.10338 [hep-ph]].
  * (67) C. Cai, D. Qiu, Y. L. Tang, Z. H. Yu and H. H. Zhang, [arXiv:2204.11570 [hep-ph]].
  * (68) R. Benbrik, M. Boukidi and B. Manaut, [arXiv:2204.11755 [hep-ph]].
  * (69) T. Yang, S. Qian, S. Deng, J. Xiao, L. Gao, A. M. Levin, Q. Li, M. Lu and Z. You, [arXiv:2204.11871 [hep-ph]].
  * (70) A. Batra, S. K. A, S. Mandal, H. Prajapati and R. Srivastava, [arXiv:2204.11945 [hep-ph]].
  * (71) H. B. T. Tan and A. Derevianko, [arXiv:2204.11991 [hep-ph]].
  * (72) H. Abouabid, A. Arhrib, R. Benbrik, M. Krab and M. Ouchemhou, [arXiv:2204.12018 [hep-ph]].
  * (73) T. K. Chen, C. W. Chiang and K. Yagyu, [arXiv:2204.12898 [hep-ph]].
  * (74) Q. Zhou and X. F. Han, [arXiv:2204.13027 [hep-ph]].
  * (75) R. S. Gupta, [arXiv:2204.13690 [hep-ph]].
  * (76) V. Basiouris and G. K. Leontaris, [arXiv:2205.00758 [hep-ph]].
  * (77) J. W. Wang, X. J. Bi, P. F. Yin and Z. H. Yu, [arXiv:2205.00783 [hep-ph]].
  * (78) F. J. Botella, F. Cornet-Gomez, C. Miró and M. Nebot, [arXiv:2205.01115 [hep-ph]].
  * (79) J. Kim, Phys. Lett. B 832 (2022), 137220 [arXiv:2205.01437 [hep-ph]]. Copy to ClipboardDownload
  * (80) B. Barman, A. Das and S. Sengupta, [arXiv:2205.01699 [hep-ph]].
  * (81) J. Kim, S. Lee, P. Sanyal and J. Song, [arXiv:2205.01701 [hep-ph]].
  * (82) X. Q. Li, Z. J. Xie, Y. D. Yang and X. B. Yuan, [arXiv:2205.02205 [hep-ph]].
  * (83) J. Isaacson, Y. Fu and C. P. Yuan, [arXiv:2205.02788 [hep-ph]].
  * (84) J. L. Evans, T. T. Yanagida and N. Yokozaki, [arXiv:2205.03877 [hep-ph]].
  * (85) T. A. Chowdhury and S. Saad, [arXiv:2205.03917 [hep-ph]].
  * (86) S. S. Kim, H. M. Lee, A. G. Menkara and K. Yamashita, [arXiv:2205.04016 [hep-ph]].
  * (87) G. Lazarides, R. Maji, R. Roshan and Q. Shafi, [arXiv:2205.04824 [hep-ph]].
  * (88) G. Senjanović and M. Zantedeschi, [arXiv:2205.05022 [hep-ph]].
  * (89) R. Ghosh, B. Mukhopadhyaya and U. Sarkar, [arXiv:2205.05041 [hep-ph]].
  * (90) T. Li, J. Pei, X. Yin and B. Zhu, [arXiv:2205.08215 [hep-ph]].
  * (91) M. C. Rodriguez, [arXiv:2205.09109 [hep-ph]].
  * (92) E. Ramirez and P. Roig, [arXiv:2205.10420 [hep-ph]].
  * (93) J. Kawamura and S. Raby, [arXiv:2205.10480 [hep-ph]].
  * (94) S. S. Afonin, [arXiv:2205.12237 [hep-ph]].
  * (95) B. C. Allanach and J. Davighi, [arXiv:2205.12252 [hep-ph]].
  * (96) S. S. Xue, [arXiv:2205.14957 [hep-ph]].
  * (97) T. G. Rizzo, [arXiv:2206.09814 [hep-ph]].
  * (98) D. Van Loi and P. Van Dong, [arXiv:2206.10100 [hep-ph]].
  * (99) S. Yaser Ayazi and M. Hosseini, [arXiv:2206.11041 [hep-ph]].
  * (100) N. Chakrabarty, [arXiv:2206.11771 [hep-ph]].
  * (101) S. Centelles Chuliá, R. Srivastava and S. Yadav, [arXiv:2206.11903 [hep-ph]].
  * (102) M. E. Peskin and T. Takeuchi, Phys. Rev. D 46 (1992), 381-409 doi:10.1103/PhysRevD.46.381
  * (103) M. E. Peskin and T. Takeuchi, Phys. Rev. Lett. 65 (1990), 964-967 doi:10.1103/PhysRevLett.65.964
  * (104) S. Jung, H. Murayama, A. Pierce and J. D. Wells, Phys. Rev. D 81, 015004 (2010) [arXiv:0907.4112 [hep-ph]].
  * (105) T. Nomura and H. Okada, Phys. Rev. D 97 (2018) no.1, 015015 doi:10.1103/PhysRevD.97.015015 [arXiv:1707.00929 [hep-ph]].
  * (106) T. Nomura, H. Okada and H. Yokoya, arXiv:1702.03396 [hep-ph].
  * (107) T. Nomura and H. Okada, LHEP 1 (2018) no.2, 10-13 doi:10.31526/LHEP.2.2018.01 [arXiv:1806.01714 [hep-ph]].
  * (108) T. Nomura and H. Okada, arXiv:1704.03382 [hep-ph].
  * (109) W. Chao, arXiv:1707.07858 [hep-ph].
  * (110) O. Seto and T. Shimomura, JHEP 04 (2021), 025 [arXiv:2006.05497 [hep-ph]].
  * (111) H. M. Lee, J. Song and K. Yamashita, J. Korean Phys. Soc. 79 (2021) no.12, 1121-1134 [arXiv:2110.09942 [hep-ph]].
  * (112) G. Guedes and P. Olgoso, [arXiv:2205.04480 [hep-ph]].
  * (113) U. K. Dey, T. Nomura and H. Okada, Phys. Rev. D 100 (2019) no.7, 075013 doi:10.1103/PhysRevD.100.075013 [arXiv:1902.06205 [hep-ph]].
  * (114) I. Esteban, M. C. Gonzalez-Garcia, A. Hernandez-Cabezudo, M. Maltoni and T. Schwetz, JHEP 01 (2019), 106 [arXiv:1811.05487 [hep-ph]].
  * (115) E. Fernandez-Martinez, J. Hernandez-Garcia and J. Lopez-Pavon, JHEP 08 (2016), 033 doi:10.1007/JHEP08(2016)033 [arXiv:1605.08774 [hep-ph]].
  * (116) N. R. Agostinho, G. C. Branco, P. M. F. Pereira, M. N. Rebelo and J. I. Silva-Marcos, Eur. Phys. J. C 78 (2018) no.11, 895 doi:10.1140/epjc/s10052-018-6347-2 [arXiv:1711.06229 [hep-ph]].
  * (117) A. Belyaev, N. D. Christensen and A. Pukhov, Comput. Phys. Commun. 184, 1729 (2013) [arXiv:1207.6082 [hep-ph]].
  * (118) P. M. Nadolsky, H. L. Lai, Q. H. Cao, J. Huston, J. Pumplin, D. Stump, W. K. Tung and C.-P. Yuan, Phys. Rev. D 78, 013004 (2008) [arXiv:0802.0007 [hep-ph]].
  * (119) L. Cerrito, D. Millar, S. Moretti and F. Spano, arXiv:1609.05540 [hep-ph].
  * (120) P. A. Zyla et al. (Particle Data Group), Prog. Theor. Exp. Phys. 2020, 083C01 (2020).
  * (121) S. Schael et al. [ALEPH, DELPHI, L3, OPAL and LEP Electroweak], Phys. Rept. 532 (2013), 119-244 [arXiv:1302.3415 [hep-ex]].
  * (122) G. Aad et al. [ATLAS], JHEP 03 (2020), 145 [arXiv:1910.08447 [hep-ex]].
  * (123) A. M. Sirunyan et al. [CMS], JHEP 05 (2020), 033 [arXiv:1911.03947 [hep-ex]].
  * (124) B. A. Dobrescu and F. Yu, [arXiv:2112.05392 [hep-ph]].
  * (125) G. Cacciapaglia, C. Csaki, G. Marandella and A. Strumia, Phys. Rev. D 74 (2006), 033011 doi:10.1103/PhysRevD.74.033011 [arXiv:hep-ph/0604111 [hep-ph]].
  * (126) A. Tumasyan et al. [CMS], JHEP 07 (2022), 067 doi:10.1007/JHEP07(2022)067 [arXiv:2202.06075 [hep-ex]].
  * (127) M. Aaboud et al. [ATLAS], JHEP 10 (2017), 182 doi:10.1007/JHEP10(2017)182 [arXiv:1707.02424 [hep-ex]].
  * (128) A. M. Baldini et al. [MEG], Eur. Phys. J. C 76 (2016) no.8, 434 doi:10.1140/epjc/s10052-016-4271-x [arXiv:1605.05081 [hep-ex]].
  * (129) J. Adam et al. [MEG], Phys. Rev. Lett. 110 (2013), 201801 doi:10.1103/PhysRevLett.110.201801 [arXiv:1303.0754 [hep-ex]].
  * (130) B. Abi et al. [Muon g-2], Phys. Rev. Lett. 126 (2021) no.14, 141801 doi:10.1103/PhysRevLett.126.141801 [arXiv:2104.03281 [hep-ex]].
  * (131) K. Hagiwara, R. Liao, A. D. Martin, D. Nomura and T. Teubner, J. Phys. G 38, 085003 (2011) [arXiv:1105.3149 [hep-ph]].
  * (132) T. Aoyama, M. Hayakawa, T. Kinoshita and M. Nio, Phys. Rev. Lett. 109, 111808 (2012) doi:10.1103/PhysRevLett.109.111808 [arXiv:1205.5370 [hep-ph]].
  * (133) T. Aoyama, T. Kinoshita and M. Nio, Atoms 7, no.1, 28 (2019) doi:10.3390/atoms7010028
  * (134) A. Czarnecki, W. J. Marciano and A. Vainshtein, Phys. Rev. D 67, 073006 (2003) [erratum: Phys. Rev. D 73, 119901 (2006)] doi:10.1103/PhysRevD.67.073006 [arXiv:hep-ph/0212229 [hep-ph]].
  * (135) C. Gnendiger, D. Stöckinger and H. Stöckinger-Kim, Phys. Rev. D 88, 053005 (2013) doi:10.1103/PhysRevD.88.053005 [arXiv:1306.5546 [hep-ph]].
  * (136) A. Keshavarzi, D. Nomura and T. Teubner, Phys. Rev. D 97, no.11, 114025 (2018) doi:10.1103/PhysRevD.97.114025 [arXiv:1802.02995 [hep-ph]].
  * (137) G. Colangelo, M. Hoferichter and P. Stoffer, JHEP 02, 006 (2019) doi:10.1007/JHEP02(2019)006 [arXiv:1810.00007 [hep-ph]].
  * (138) M. Hoferichter, B. L. Hoid and B. Kubis, JHEP 08, 137 (2019) doi:10.1007/JHEP08(2019)137 [arXiv:1907.01556 [hep-ph]].
  * (139) A. Keshavarzi, D. Nomura and T. Teubner, Phys. Rev. D 101, no.1, 014029 (2020) doi:10.1103/PhysRevD.101.014029 [arXiv:1911.00367 [hep-ph]].
  * (140) A. Kurz, T. Liu, P. Marquard and M. Steinhauser, Phys. Lett. B 734, 144-147 (2014) doi:10.1016/j.physletb.2014.05.043 [arXiv:1403.6400 [hep-ph]].
  * (141) K. Melnikov and A. Vainshtein, Phys. Rev. D 70, 113006 (2004) doi:10.1103/PhysRevD.70.113006 [arXiv:hep-ph/0312226 [hep-ph]].
  * (142) P. Masjuan and P. Sanchez-Puertas, Phys. Rev. D 95, no.5, 054026 (2017) doi:10.1103/PhysRevD.95.054026 [arXiv:1701.05829 [hep-ph]].
  * (143) G. Colangelo, M. Hoferichter, M. Procura and P. Stoffer, JHEP 04, 161 (2017) doi:10.1007/JHEP04(2017)161 [arXiv:1702.07347 [hep-ph]].
  * (144) M. Hoferichter, B. L. Hoid, B. Kubis, S. Leupold and S. P. Schneider, JHEP 10, 141 (2018) doi:10.1007/JHEP10(2018)141 [arXiv:1808.04823 [hep-ph]].
  * (145) A. Gérardin, H. B. Meyer and A. Nyffeler, Phys. Rev. D 100, no.3, 034520 (2019) doi:10.1103/PhysRevD.100.034520 [arXiv:1903.09471 [hep-lat]].
  * (146) J. Bijnens, N. Hermansson-Truedsson and A. Rodríguez-Sánchez, Phys. Lett. B 798, 134994 (2019) doi:10.1016/j.physletb.2019.134994 [arXiv:1908.03331 [hep-ph]].
  * (147) G. Colangelo, F. Hagelstein, M. Hoferichter, L. Laub and P. Stoffer, JHEP 03, 101 (2020) doi:10.1007/JHEP03(2020)101 [arXiv:1910.13432 [hep-ph]].
  * (148) T. Blum, N. Christ, M. Hayakawa, T. Izubuchi, L. Jin, C. Jung and C. Lehner, Phys. Rev. Lett. 124, no.13, 132002 (2020) doi:10.1103/PhysRevLett.124.132002 [arXiv:1911.08123 [hep-lat]].
  * (149) G. Colangelo, M. Hoferichter, A. Nyffeler, M. Passera and P. Stoffer, Phys. Lett. B 735, 90-91 (2014) doi:10.1016/j.physletb.2014.06.012 [arXiv:1403.7512 [hep-ph]].
  * (150) M. Davier, A. Hoecker, B. Malaescu and Z. Zhang, Eur. Phys. J. C 77, no.12, 827 (2017) doi:10.1140/epjc/s10052-017-5161-6 [arXiv:1706.09436 [hep-ph]].
  * (151) M. Davier, A. Hoecker, B. Malaescu and Z. Zhang, Eur. Phys. J. C 80, no.3, 241 (2020) [erratum: Eur. Phys. J. C 80, no.5, 410 (2020)] doi:10.1140/epjc/s10052-020-7792-2 [arXiv:1908.00921 [hep-ph]].