\section{Introduction}
\label{sec:intro}

The measurement of the anomalous magnetic moment of the muon, $a_\mu=(g-2)_\mu/2$, from the Muon $g-2$ Collaboration  at Fermilab, when combined with the value reported by the E821 Collaboration at the Brookhaven National Laboratory, deviates from the standard model (SM) prediction by $4.2\sigma$,
\begin{eqnarray}
\Delta a_\mu = a_\mu^{\rm exp} - a_\mu^{\rm SM} = (25.1 \pm 5.9) \times 10^{-10},
\end{eqnarray}
although more precise nonperturbative QCD calculation is required in the theory side. The deviation may be explained by new physics (NP) which couples to the muon.


The $U(1)_{L_\mu-L_\tau}$ extension of the SM gauge group has received much attention as the contribution of the new gauge boson, $Z'$, can solve the discrepancy~\cite{Baek:2001kca}. However the experimental data for the cross section of the neutrino trident production, the scattering of the muon neutrino off heavy 
nuclei producing a $\mu^+\mu^-$ pair, or the process  $\nu_\mu N \to \nu_\mu N \mu^+ \mu^-$ in the SM, turn out to be very stringent in constraining the parameter space of $U(1)_{L_\mu-L_\tau}$ models.  The CHARM-II Collaboration and the CCFR Collaboration detected the events with the cross sections
\begin{eqnarray}
\frac{\sigma^{\rm CHARM-II}}{\sigma^{\rm SM}}=1.58 \pm 0.57 \quad\text{and}\quad
\frac{\sigma^{\rm CCFR}}{\sigma^{\rm SM}}=0.82 \pm 0.28,
\end{eqnarray}
which are in good agreement with the SM expectations. 
Applying the 2$\sigma$ upper bound from the CCFR result, the authors of~\cite{Altmannshofer:2014pba} found that only a small region with $m_{Z'} \sim {\cal O}(10^{1\pm 1})$ MeV and the gauge coupling $\sim {\cal O}(10^{-4})$ can explain the $\Delta a_\mu$ in 
the  minimal model of $U(1)_{L_\mu-L_\tau}$.
The model, however, needs to be extended, for example, to incorporate the neutrino masses to explain the neutrino oscillation data~\cite{Baek:2015mna,Baek:2015fea}.

Recently the CDF Collaboration measured the W-boson mass~\cite{CDF:2022hxs}
\begin{eqnarray}
m_W = 80,433.5 \pm 9.4 \, {\rm MeV},
\end{eqnarray}
which is in tension with the SM prediction $m_W^{\rm SM} = 80,357 \pm 6$ MeV from the precision electroweak data by 7$\sigma$.
This may also call for NP if it is confirmed by the future experiments~\cite{Fan:2022dck,Lu:2022bgw,Athron:2022qpo,Yuan:2022cpw,Strumia:2022qkt,Yang:2022gvz,deBlas:2022hdk,Du:2022pbp,Tang:2022pxh,Cacciapaglia:2022xih,Blennow:2022yfm,Sakurai:2022hwh,Fan:2022yly,Liu:2022jdq,Lee:2022nqz,Cheng:2022jyi,Bagnaschi:2022whn,Paul:2022dds,Bahl:2022xzi,Asadi:2022xiy,DiLuzio:2022xns,Athron:2022isz,Gu:2022htv,Babu:2022pdn,Heo:2022dey,Du:2022brr,Cheung:2022zsb,Crivellin:2022fdf,Endo:2022kiw,Biekotter:2022abc,Balkin:2022glu,Ahn:2022xeq,Han:2022juu,Zheng:2022irz,Kawamura:2022uft,Ghoshal:2022vzo,Perez:2022uil,Mondal:2022xdy,Zhang:2022nnh,Borah:2022obi,Chowdhury:2022moc,Arcadi:2022dmt,Cirigliano:2022qdm,Carpenter:2022oyg,Popov:2022ldh,Ghorbani:2022vtv,Du:2022fqv,Bhaskar:2022vgk}.

In this paper we show that a minimal extension of the $U(1)_{L_\mu-L_\tau}$ allows electroweak scale $Z'$ to explain the $\Delta a_\mu$ and solves the CDF W-mass anomaly at the same time. In Section~\ref{sec:model} we introduce the model. In Section~\ref{sec:gm2} we show that there is large region of parameter space which can solve $\Delta a_\mu$
while evading the neutrino trident production experiments. In Section~\ref{sec:ST} we calculate the oblique parameters $S$ and $T$ and show that the excess in the $W$-mass with respect to the SM prediction can be accommodated.

\section{The Model}
\label{sec:model}

We introduce vector-like (VL) leptons which transform like $L \sim (1, 2, -1/2, 2)$ and $E \sim (1, 1, -1, 2)$ in the gauge 
group $SU(3)_C \times SU(2)_L \times U(1)_Y \times U(1)_{L_\mu-L_\tau}$. The $U(1)_{L_\mu-L_\tau}$ symmetry is
broken spontaneously when the new scalar field $\Phi \sim (1,1,0,1)$ develops the vacuum expectation value (VEV).
By definition the second (the third) generation 
left-handed doublet $\ell_\mu$ ($\ell_\tau$) and right-handed singlet $\mu_R (\tau_R)$ leptons of the SM also have $U(1)_{L_\mu-L_\tau}$ charge $+1 (-1)$. All the rest SM fields are neutral under the new $U(1)$ group. Then the bare masses of the VL leptons and the new Yukawa interactions are allowed
\begin{eqnarray}
{\cal L} \supset && -m_L \overline{L} L - m_E \overline{E} E   \nonumber\\
&& -y_L  \overline{L} P_R E H -y_E \overline{L} P_L E H -\lambda_L  \overline{L} P_L \ell_\mu \Phi -\lambda_E \overline{\mu} P_L E \Phi^* + H.c., 
\end{eqnarray}
where $H \sim (1,2,1/2,0)$ is the SM Higgs doublet scalar. We assume the new Yukawa couplings are real parameters.
We can decompose the VL doublets as $L_{L(R)}=(\widetilde{N}, \widetilde{E})^T_{L(R)} $. 

After the breaking of $SU(2)_L \times U(1)_Y$ and $U(1)_{L_\mu-L_\tau}$ symmetries by $\langle H^0 \rangle = v/\sqrt{2}$ and
$\langle \Phi \rangle = v_\Phi/\sqrt{2}$, respectively,
the fermion mass matrix in the basis $(\mu, \widetilde{E}, E)$ is in the form,
\begin{eqnarray}
{\cal L}_{\rm mass}&=& -m_L \overline{\widetilde{N}} \widetilde{N}  \nonumber\\
&& -(\overline{\mu'} \; \; \overline{\widetilde{E}} \; \; \overline{E}) 
\begin{pmatrix}
m'_\mu & 0 & {\lambda_E v_\Phi \over \sqrt{2}}\\
{\lambda_L v_\Phi \over \sqrt{2}} & m_L & {y_E v \over \sqrt{2}} \\
0 & {y_L v \over \sqrt{2}} & m_E
\end{pmatrix}
P_L
\begin{pmatrix}
\mu' \\
\widetilde{E} \\
E
\end{pmatrix} + H.c.
\end{eqnarray}
Denoting the $3 \times 3$ charged lepton mass as ${\cal M}$,
we can diagonalize it by biunitary transformation,
\begin{eqnarray}
V_R^\dagger {\cal M} V_L = \hat{\cal M},
\end{eqnarray}
where  $V_{L,R}$ are real orthogonal matrices, and  $\hat{\cal M}$ is diagonal with positive eigenvalues.
We denote the mass eigenstates as $( E_1, E_2, E_3)$ with masses $(M_1, M_2, M_3)$ with $\mu \equiv E_1$ and $m_\mu\equiv M_1$.
We consider the situation where the mass terms generated from the Higgs mechanism are much smaller than
VL bare masses: $m_\mu, \lambda_{L(E)} v_\Phi/\sqrt{2}, y_{L(E)} v/\sqrt{2} \ll m_L, m_E$.
We generically denote their ratio as $\theta$ so we have $\theta \ll {\cal O}(1)$.
In the leading approximation we get 
\begin{eqnarray}
V_L &=& 
\begin{pmatrix}
1  &  {\lambda_L v_\Phi \over \sqrt{2} m_L} & 
   {m_\mu \lambda_E v_\Phi \over \sqrt{2} m_E^2} -{\lambda_L v_\Phi (m_L y_L + m_E y_E)v \over 2 m_E (m_L^2-m_E^2)} \\
- {\lambda_L v_\Phi \over \sqrt{2} m_L}   & 1 & -{(m_L y_E + m_E y_L) v \over \sqrt{2} (m_L^2 -m_E^2)} \\
- {m_\mu \lambda_E v_\Phi \over \sqrt{2} m_E^2} +{\lambda_L v_\Phi y_L v \over 2 m_L m_E}  
    & {(m_L y_E + m_E y_L) v \over \sqrt{2} (m_L^2 -m_E^2)} & 1
\end{pmatrix},  \nonumber\\
V_R &=& 
\begin{pmatrix}
1  &   {m_\mu \lambda_L v_\Phi \over \sqrt{2} m_L^2} +{\lambda_E v_\Phi (m_L y_E + m_E y_L)v \over 2 m_L (m_L^2-m_E^2)} &
{\lambda_E v_\Phi \over \sqrt{2} m_E}    \\
- {m_\mu \lambda_L v_\Phi  \over \sqrt{2} m_L^2}+ {y_L v \lambda_E v_\Phi \over 2 m_L m_E}   & 1 & -{(m_L y_L + m_E y_E) v \over \sqrt{2} (m_L^2 -m_E^2)} \\
-{ \lambda_E v_\Phi \over \sqrt{2} m_E} &   {(m_L y_L + m_E y_E) v \over \sqrt{2} (m_L^2 -m_E^2)} & 1
\end{pmatrix}.
\end{eqnarray}
We see that the off-diagonal terms 
$V_{L_{12(21)}}$, $V_{L_{23(32)}}$, $V_{R_{13(31)}}$, and $V_{R_{23(32)}}$ are suppressed by $\theta$,  
while $V_{L_{13(31)}}$ and $V_{R_{12(21)}}$ by $\theta^2$.
These mixing terms can make significant contributions to $(g-2)_\mu$ and $W$-mass.

\section{The Muon $(g-2)$}
\label{sec:gm2}

Without introducing the VL leptons, the minimal model predicts the $Z'$ contribution to the muon $(g-2)$ to be~\cite{Baek:2001kca}
\begin{eqnarray}
a_\mu^{Z'} = \frac{g_X^2 m_\mu^2}{4 \pi^2} \int_0^1 \frac{x^2(1-x)}{m_{Z'}^2 (1-x) + m_\mu^2 x}.
\end{eqnarray}
In the presence of the VL leptons which mix with $\mu$ the new contribution can be enhanced due to the chirality flip in the
VL lepton line with enhancement factor $M_{1(2)}/m_\mu$, which can be huge.  
In Fig.~\ref{feynman_gm2} we show the Feynman diagrams for the $Z'$ contributions to $(g-2)_\mu$.
There are also the corresponding diagrams with $Z'$ replaced by $Z$. But the $Z$-mediated $\mu$-mixing with $E_{2,3}$ are naturally suppressed by higher powers of $\theta$
than that of $Z'$-mediated $\mu$-mixing with $E_{2,3}$.
This allows the model to evade constraints from the precisely measured $Z-\mu-\mu$ couplings. 
In a good approximation we obtain~\cite{Leveille:1977rc}. 
\begin{eqnarray}
a_\mu &\simeq& \frac{g_X^2 m_\mu}{16 \pi^2 m_{Z'}^2} \sum_{i=2,3}  M_i V_{L_{1i}} V_{R_{1i}}.
\end{eqnarray}


Fig.~\ref{fig:trident} shows the prediction of $(g-2)_\mu$ in the $(m_{Z'}, g_X)$-plane along with constraints.
The thick red line represents $a_\mu^{Z'} =25.1 \times 10^{-10}$, explaining the central value of $\Delta a_\mu$, while
the region enclosed by the thin red lines are favored by the $(g-2)_\mu$ measurements at 1$\sigma$. 
For this plot we fixed $m_L=1$ TeV, 	$m_R=1.5$ TeV, $\lambda_L = y_L =0.5$, and $\lambda_R = y_R =-0.5$.
The light gray region is excluded  at 95\% C.L. by the CCFR measurement of the neutrino trident production cross section. The dark gray region is disfavored by the observation of the SM $Z$ boson decay into four leptons at the CMS Collaboration at the LHC~\cite{CMS:2012bw}.

We can see much larger region of the parameter space explaining the $\Delta a_\mu$ is allowed while evading the experimental constraints from the trident neutrino production than in Fig.~2 of \cite{Altmannshofer:2014pba}. Especially the electroweak scale $m_{Z'}$ is allowed except for the region disfavored by the CMS experiment. 
Although the process of the neutrino trident production has contributions from the VL leptons, their contributions are always subleading due to the fact that there is no enhancement proportional to $M_2,M_3$ contrary to the $(g-2)_\mu$. And the contribution from the virtual muon in the diagram is dominant, and we neglected the VL contribution in the calculation of the cross section.
	
\begin{figure}
	\centerline{\includegraphics[width=0.5\linewidth]{feynman_gm2.pdf}}
	\hspace{0.5cm}
	\caption{Feynman diagram for the muon $(g-2)$.}
\label{feynman_gm2}
\end{figure}

\begin{figure}
	\centerline{\includegraphics[width=0.5\linewidth]{gm2_trident.pdf}}
	\hspace{0.5cm}
	\caption{The prediction of $(g-2)_\mu$ in the $(m_{Z'},g_X)$-plane.
	The thick red line represents $a_\mu^{Z'} =25.1 \times 10^{-10}$, explaining the central value of $\Delta a_\mu$, while
	the region enclosed by the thin red lines are favored by the $(g-2)_\mu$ measurements at 1$\sigma$. 
	For this plot we fixed $m_L=1$ TeV, 	$m_R=1.5$ TeV, $\lambda_L = y_L =0.5$, and $\lambda_R = y_R =-0.5$.
	The light gray region is excluded  at 95\% C.L. by the CCFR measurement of the neutrino trident production cross section. The dark gray region is disfavored by the observation of the SM $Z$ boson decay into four leptons at the CMS Collaboration at the LHC.}
\label{fig:trident}
\end{figure}

\section{CDF II $W$-mass}
\label{sec:ST}

By including the CDF II $W$-mass measurement in the fitting, the authors of \cite{Strumia:2022qkt,Bagnaschi:2022whn} found that the excess can be explained by enhancing especially the oblique $T$-parameter of the electroweak precision tests.
We calculated the $S,T$-parameters in our model. The NP contribution can be obtained from the Lagrangian terms
\begin{eqnarray}
{\cal L} \supset && -{g \over \sqrt{2}} W_\mu^+ \overline{\widetilde{N}} \gamma^\mu \widetilde{E} + H.c. \nonumber\\
&&-\left( -{g' \over 2} B_\mu + {g \over 2} W^3_\mu\right) \overline{\widetilde{N}} \gamma^\mu \widetilde{N} 
-\left( -{g' \over 2} B_\mu - {g \over 2} W^3_\mu\right) \overline{\widetilde{E}} \gamma^\mu \widetilde{E} 
-\left( -g' B_\mu \right) \overline{E} \gamma^\mu E. 
\end{eqnarray}
Using $P_{L(R)} \widetilde{E} = V_{L_{2i}} P_{L(R)} E_i$, $P_{L(R)} E = V_{L_{3i}} P_{L(R)} E_i$, we get~\cite{Cynolter:2008ea}
\begin{align}
T =& \frac{1}{4\pi s_W^2 c_W^2 m_Z^2}  \Bigg[ 
2 \sum_{i=2,3} \Big\{(V_{L_{2i}}-V_{R_{2i}})^2 \widetilde{\Pi}_{LL}(M_0,M_i,0) +  V_{L_{2i}}V_{R_{2i}} \widetilde{\Pi}_{VV}(M_0,M_i,0)\Big\} \nonumber\\
&-\sum_{i,j=2,3} \Big\{ (g_L^{ij}(W^3)-g_R^{ij}(W^3))^2 \widetilde{\Pi}_{LL}(M_i,M_j,0)
+g_L^{ij}(W^3)g_R^{ij}(W^3) \widetilde{\Pi}_{VV}(M_i,M_j,0)\Big\} 
\Bigg], 
\end{align}
and
\begin{align}
S =& {1 \over \pi} \Bigg[ \widetilde{\Pi}'_{VV}(M_0,M_0,0) \nonumber\\
&- \sum_{i,j=2,3} \Big\{ (g_L^{ij}(B)-g_L^{ij}(B))(g_L^{ij}(W^3)-g_L^{ij}(W^3)\widetilde{\Pi}'_{LL}(M_i, M_j,0) \nonumber\\
&+ {1 \over 2} (g_L^{ij}(B)g_R^{ij}(W^3)+g_L^{ij}(W^3)g_R^{ij}(B))\widetilde{\Pi}'_{VV}(M_i,M_j,0)\Big\}
\Bigg],
\end{align}
where $g_{L(R)}^{ij}(W^3) = V_{{L(R)}_{2i}} V_{{L(R)}_{2j}}$ and 
$g_{L(R)}^{ij}(B) = V_{{L(R)}_{2i}} V_{{L(R)}_{2j}} +2 V_{{L(R)}_{3i}} V_{{L(R)}_{3j}}$.
The explicit expressions for the vacuum polarization amplitudes and their derivatives, $\widetilde{\Pi}$ and $\widetilde{\Pi}'$, are listed in the Appendix.
In the above calculation we neglected the contributions from the mixing between $\mu'$ and $\widetilde{E}, E$ because they are suppressed either by the small muon
mass compared to $M_2, M_3$ or by the small mixing angles. We have checked the divergences are cancelled and the final results are finite.


Fig.~\ref{fig:ST} shows the predictions of $S,T$-parameters in our model.
The black dashed (solid) ellipse is 68\% (95\%) contour of fits to the oblique parameters $S$ and $T$ including the CDF 
result~\cite{Strumia:2022qkt,Bagnaschi:2022whn}. We fixed $m_L=1.0$ TeV and $m_E=1.5$ TeV. The red (dashed blue, dotted green) line is for 
$y_L=0 (0.5, 2.0)$. We scanned $y_E \in (-3,3)$. The $T$-parameter can be largely enhanced but the $S$-parameter does not change much, which is a prediction of the model.
We can see that the excess of $m_W$ can be easily explained while the Yukawa couplings are still in perturbative
regime.


\begin{figure}
	\centerline{\includegraphics[width=0.6\linewidth]{ST.pdf}}
	\hspace{0.5cm}
	\caption{The black dashed (solid) ellipse is 68\% (95\%) contour of fits to the oblique parameters $S$ and $T$ including the 
	CDF result. We fixed $m_L=1.0$ TeV and $m_E=1.5$ TeV. The red (dashed blue, dotted green) line is for $y_L=0 (0.5, 2.0)$. We scanned $y_E \in (-3,3)$. }
\label{fig:ST}
\end{figure}

\section{Conclusions}
The anomalies in the measurements of the anomalous magnetic moment of muon and the $W$-boson mass, if confirmed by future experiments, suggest the existence of new physics beyond the standard model of particle physics. We considered a possible solution to both of these problems.
We extended the gauge group of  the standard  model to include $U(1)_{L_\mu-L_\tau}$.
We also added a vector-like lepton doublet $L$ and a vector-like lepton singlet $E$ with hypercharge, $-1/2$ and $-1$, respectively.
The extra gauge group is broken by a scalar singlet field $\Phi$. All the new particles are charged under $U(1)_{L_\mu-L_\tau}$. 

We showed that the stringent constraint on the solution to the muon $(g-2)$ in the minimal $U(1)_{L_\mu-L_\tau}$ is significantly lifted.
We also found that the excess of the $W$-boson mass can be explained in the perturbative regime.

\acknowledgments
This work was supported in part by the National Research Foundation of Korea(NRF) grant funded by
     the Korea government(MSIT), Grant No. NRF-2018R1A2A3075605.

