CTPU-PTC-22-10

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

Junichiro Kawamuraa<EMAIL_ADDRESS>and Stuart Rabyb<EMAIL_ADDRESS>

a Center for Theoretical Physics of the Universe, Institute for Basic Science
(IBS), Daejeon 34051, Korea

b Department of Physics, Ohio State University, Columbus, Ohio, 43210, USA

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.

##  1 Introduction

Recently, the CDF collaboration reported a new result of the precise $W$ boson
mass measurement [1],

$\displaystyle m_{W}^{\mathrm{CDF}}=80.4335~{}(94)~{}\mathrm{GeV}.$ (1)

This value is larger than the combination of the previous measurements
$m_{W}^{\mathrm{PDG}}=80.379~{}(12)~{}\mathrm{GeV}$ and the Standard Model
(SM) prediction $m_{W}^{\mathrm{SM}}=80.361~{}(6)~{}\mathrm{GeV}$ [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 [3, 4, 5, 6,
7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26,
27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45,
46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64,
65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76].

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
$U(1)^{\prime}$ gauge symmetry which was proposed in Refs. [77, 78]. 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 333
Models with VL fermions and a $U(1)^{\prime}$ symmetry for the anomalies are
proposed in Refs. [79, 80, 81, 82, 83, 84]. Explanations for $\Delta a_{\mu}$
by VL leptons are proposed in e.g. Refs. [85, 86, 87, 88, 89, 90]. . The
recent FNAL measurement [91] confirmed the long-standing discrepancy of the
muon $g-2$ between the experimental value [92] and the SM prediction [93, 94,
95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110,
111, 112], and the current discrepancy is $\Delta a_{\mu}=2.51~{}(59)\times
10^{-9}$. Yet another discrepancy from the SM is found in the measurements of
rare semi-leptonic $B$ meson decays [113, 114, 115, 116, 117, 118, 119, 120,
121, 122, 123, 124, 125, 126, 127, 128, 129], $b\to s\ell\ell$. In this paper,
we focus on the VL leptons and ${Z^{\prime}}$ boson in this model to show that
$m_{W}$ and $\Delta a_{\mu}$ can be explained simultaneously. As we have shown
in Refs. [77, 78], this model can easily accommodate the $b\to s\ell\ell$
anomaly when the ${Z^{\prime}}$ boson is sufficiently light and strongly
couples to muons, so that $\Delta a_{\mu}$ is explained. Therefore, our model
will provide a unified explanation for the recent three anomalies, $m_{W}$,
$\Delta a_{\mu}$ and $b\to s\ell\ell$.

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 are 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.

##  2 Model

###  2.1 Mass matrices

We briefly introduce our model, with particular interests in VL leptons, see
Refs. [77, 78] 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

$\displaystyle\mathcal{L}\supset$ $\displaystyle\
-m_{L}\overline{L}_{R}L_{L}-m_{E}\overline{E}_{R}E_{L}-m_{N}\overline{N}_{R}N_{L}-\frac{1}{2}M_{R}\overline{\nu}_{R}^{c}\nu_{R}$
(2) $\displaystyle\
+y_{2}\overline{\mu}_{R}\ell_{L}H+y_{n}\overline{\nu}_{R}\ell_{L}\tilde{H}+\lambda_{L}\Phi\overline{L}_{R}\ell_{L}-\lambda_{E}\Phi^{*}\overline{\mu}_{R}E_{L}-\lambda_{N}\Phi^{*}\overline{\nu}_{R}N_{L}$
$\displaystyle\
+{\lambda_{e}}\overline{E}_{R}L_{L}H-{\lambda^{\prime}_{e}}\overline{L}_{R}\tilde{H}E_{L}+{\lambda_{n}}\overline{N}_{R}L_{L}\tilde{H}+{\lambda^{\prime}_{n}}\overline{L}_{R}{H}N_{L}+h.c.,$

where $\tilde{H}:=i\sigma_{2}H^{*}=(H_{-}^{*},-H_{0}^{*})$. The $SU(2)_{L}$
indices are contracted by $i\sigma_{2}$. We introduce the Majorana mass term
of $\nu_{R}$ for the type-I see-saw mechanism. After the scalar fields
developing their vacuum expectation values (VEVs),
$v_{H}:=\langle{H_{0}}\rangle$ and $v_{\Phi}:=\langle{\Phi}\rangle$, the Dirac
mass matrices for the leptons are given by

$\displaystyle\overline{\mathbf{e}}_{R}\mathcal{M}_{e}\mathbf{e}_{L}:=$
$\displaystyle\
\begin{pmatrix}\overline{\mu}_{R}&\overline{E}_{R}&\overline{E}_{R}^{\prime}\end{pmatrix}\begin{pmatrix}y_{2}v_{H}&0&\lambda_{E}v_{\Phi}\\\
0&{\lambda_{e}}v_{H}&m_{E}\\\
\lambda_{L}v_{\Phi}&m_{L}&{\lambda^{\prime}_{e}}v_{H}\end{pmatrix}\begin{pmatrix}\mu_{L}\\\
E^{\prime}_{L}\\\ E_{L}\end{pmatrix},$ (3)
$\displaystyle\overline{\mathbf{n}}_{R}\mathcal{M}_{n}\mathbf{n}_{L}:=$
$\displaystyle\
\begin{pmatrix}\overline{\nu}_{R}&\overline{N}_{R}&\overline{N}_{R}^{\prime}\end{pmatrix}\begin{pmatrix}y_{n}v_{H}&0&\lambda_{N}v_{\Phi}\\\
0&{\lambda_{n}}v_{H}&m_{N}\\\
\lambda_{L}v_{\Phi}&m_{L}&{\lambda^{\prime}_{n}}v_{H}\end{pmatrix}\begin{pmatrix}{\nu}_{L}\\\
{N}^{\prime}_{L}\\\ N_{L}\end{pmatrix}.$ (4)

Table 1:  Matter contents. Electric charge of fermion $f$ is $Q_{f}=T_{f}^{3}+Y_{f}/2$. The $SU(2)_{L}$ doublets have components; $\ell_{L}=(\nu_{L},\mu_{L})$, $H=(H_{0},H_{-})$, $L_{L}=(N_{L}^{\prime},E_{L}^{\prime})$ and $\overline{L}_{R}=(-\overline{E}_{R}^{\prime},\overline{N}_{R}^{\prime})$.  | $\ell_{L}$ | $\overline{\mu}_{R}$ | $\overline{\nu}_{R}$ | $H$ | $L_{L}$ | $\overline{E}_{R}$ | $\overline{L}_{R}$ | $E_{L}$ | $N_{L}$ | $\overline{N}_{R}$ | $Z^{\prime}$ | $\Phi$
---|---|---|---|---|---|---|---|---|---|---|---|---
$SU(2)_{L}$ | $\mathbf{2}$ | $\mathbf{1}$ | $\mathbf{1}$ | $\mathbf{2}$ | $\mathbf{2}$ | $\mathbf{1}$ | $\mathbf{2}$ | $\mathbf{1}$ | $\mathbf{1}$ | $\mathbf{1}$ | $\mathbf{1}$ | $\mathbf{1}$
$U(1)_{Y}$ | $-1$ | $2$ | $0$ | $-1$ | $-1$ | $2$ | $1$ | $-2$ | $0$ | $0$ | $0$ | $0$
$U(1)^{\prime}$ | $0$ | $0$ | $0$ | $0$ | $-1$ | $1$ | $1$ | $-1$ | $-1$ | $1$ | $0$ | $-1$

The mass basis is defined as

$\displaystyle\hat{\mathbf{e}}_{L}:=U_{e_{L}}^{\dagger}\mathbf{e}_{L},\quad\hat{\mathbf{e}}_{R}:=U_{e_{R}}^{\dagger}\mathbf{e}_{R},\quad\hat{\mathbf{n}}_{L}:=U_{n_{L}}^{\dagger}\mathbf{n}_{L},\quad\hat{\mathbf{n}}_{R}:=U_{n_{R}}^{\dagger}\mathbf{n}_{R},$
(5)

where unitary matrices diagonalize the mass matrices as

$\displaystyle
U_{e_{R}}^{\dagger}\mathcal{M}_{e}U_{e_{L}}=\begin{pmatrix}m_{\mu}&0&0\\\
0&m_{E_{1}}&0\\\ 0&0&m_{E_{2}}\end{pmatrix},\quad
U_{n_{R}}^{\dagger}\mathcal{M}_{n}U_{n_{L}}=\begin{pmatrix}\cdot&\cdot&\cdot\\\
0&m_{N_{1}}&0\\\ 0&0&m_{N_{2}}\end{pmatrix}.$ (6)

The masses are increasingly ordered. Here, $\cdot$’s in the first row of the
neutrino mass matrix will be irrelevant after $\nu_{R}$ is integrated out.
Neglecting $\mathcal{O}\left({v_{H}}\right)$ entries, the VL lepton masses are
given by

$\displaystyle m_{E_{1}}\sim
M_{E}:=\sqrt{m_{E}^{2}+\lambda_{E}^{2}v_{\Phi}^{2}},\quad m_{N_{1}}\sim
m_{N},\quad m_{E_{2}}\sim m_{N_{2}}\sim
M_{L}:=\sqrt{m_{L}^{2}+\lambda_{L}^{2}v_{\Phi}^{2}},$ (7)

when $M_{E},m_{N}<M_{L}$ and hence $L\sim(E_{2},N_{2})$.

We define the Dirac fermions as

$\displaystyle\hat{\mathbf{e}}:=\left(\mu,E_{1},E_{2}\right),\quad\hat{\mathbf{n}}:=\left(\nu,N_{1},N_{2}\right),$
(8)

where

$\displaystyle\left[\mathbf{e}\right]_{i}:=\left(\left[\hat{\mathbf{e}}_{L}\right]_{i},\left[\hat{\mathbf{e}}_{R}\right]_{i}\right),\quad\left[\mathbf{n}\right]_{i}:=\left(\left[\hat{\mathbf{n}}_{L}\right]_{i},\left[\hat{\mathbf{n}}_{R}\right]_{i}\right),$
(9)

with $i=1,2,3$. We expand the neutral scalar fields as

$\displaystyle
H_{0}=v_{H}+\frac{1}{\sqrt{2}}\left(h+ia_{h}\right),\quad\Phi=v_{\Phi}+\frac{1}{\sqrt{2}}\left(\chi+ia_{\chi}\right),$
(10)

where $h$ and $\chi$ are the physical real scalar fields, while the pseudo-
scalar components $a_{h}$ and $a_{\chi}$ are absorbed by the $Z$ and
${Z^{\prime}}$ bosons, respectively.

###  2.2 Interactions

The $Z$ and $W$ boson couplings are given by

$\displaystyle\mathcal{L}_{Z,W}=$ $\displaystyle\
Z_{\mu}\sum_{f=\mathbf{e},\mathbf{n}}\overline{f}\gamma^{\mu}\left(g^{Z}_{f_{L}}P_{L}+g^{Z}_{f_{R}}P_{R}\right)f+\Bigl{[}W_{\mu}\overline{\mathbf{n}}\gamma^{\mu}\left({g}^{W}_{L}P_{L}+{g}^{W}_{R}P_{R}\right)\mathbf{e}+h.c.\Bigr{]},$
(11)

where

$\displaystyle
g^{Z}_{\mathbf{e}_{A}}=\frac{g}{2c_{W}}\left(-\mathcal{E}_{A}+2s_{W}^{2}\mathbf{1}_{{3}}\right),\quad
g^{Z}_{\mathbf{n}_{A}}=\frac{g}{2c_{W}}\mathcal{N}_{A},\quad
g^{W}_{A}=\frac{g}{\sqrt{2}}h_{A},$ (12)

for $A=L,R$. Here,

$\displaystyle\mathcal{E}_{A}:=U_{e_{A}}^{\dagger}Q_{A}U_{e_{A}},\quad\mathcal{N}_{A}:=U_{n_{A}}^{\dagger}Q_{A}U_{n_{A}},\quad
h_{A}:=U_{n_{A}}^{\dagger}Q_{A}U_{e_{A}},$ (13)

where $Q_{L}=\mathrm{diag}(1,1,0)$ and $Q_{R}=\mathbf{1}_{{3}}-Q_{L}$. $P_{L}$
($P_{R}$) are the chiral projections onto the left- (right-) handed fermions.

The gauge interactions with the ${Z^{\prime}}$ boson in the mass basis are
defined as

$\displaystyle\mathcal{L}_{{Z^{\prime}}}=$ $\displaystyle\
Z^{\prime}_{\mu}\sum_{f=\mathbf{e},\mathbf{n}}\overline{f}\gamma^{\mu}\left({g}^{Z^{\prime}}_{f_{L}}P_{L}+{g}^{Z^{\prime}}_{f_{R}}P_{R}\right)f,$
(14)

where the coupling matrices are given by

$\displaystyle g^{Z^{\prime}}_{\mathbf{e}_{A}}=$ $\displaystyle\
{g^{\prime}}U_{e_{A}}^{\dagger}Q^{\prime}_{e}U_{e_{A}},\quad
g^{Z^{\prime}}_{\mathbf{n}_{A}}={g^{\prime}}U_{n_{A}}^{\dagger}Q^{\prime}_{n}U_{n_{A}},$
(15)

with $Q^{\prime}_{e}=Q^{\prime}_{n}=\mathrm{diag}(0,-1,-1)$. ${g^{\prime}}$ is
the gauge coupling constant for $U(1)^{\prime}$.

The Yukawa interactions are given by

$\displaystyle-\mathcal{L}_{Y}=\frac{1}{\sqrt{2}}\sum_{S=h,\chi}\sum_{f=\mathbf{e},\mathbf{n}}S\overline{f}\;Y^{S}_{f}P_{L}f+h.c.,$
(16)

where

$\displaystyle Y^{h}_{\mathbf{e}}=$ $\displaystyle\
U_{e_{R}}^{\dagger}\begin{pmatrix}y_{e}&0&0\\\ 0&{\lambda_{e}}&0\\\
0&0&{\lambda^{\prime}_{e}}\end{pmatrix}U_{e_{L}},$ $\displaystyle\quad
Y^{\chi}_{\mathbf{e}}=$ $\displaystyle\
U_{e_{R}}^{\dagger}\begin{pmatrix}0&0&\lambda_{E}\\\ 0&0&0\\\
\lambda_{L}&0&0\end{pmatrix}U_{e_{L}},$ (17) $\displaystyle
Y^{h}_{\mathbf{n}}=$ $\displaystyle\
U_{n_{R}}^{\dagger}\begin{pmatrix}y_{n}&0&0\\\ 0&{\lambda_{n}}&0\\\
0&0&{\lambda^{\prime}_{n}}\end{pmatrix}U_{n_{L}},$ $\displaystyle\quad
Y^{\chi}_{\mathbf{n}}=$ $\displaystyle\
U_{n_{R}}^{\dagger}\begin{pmatrix}0&0&\lambda_{N}\\\ 0&0&0\\\
\lambda_{L}&0&0\end{pmatrix}U_{n_{L}}.$ (18)

###  2.3 Couplings at the leading order

We show the structures of the coupling matrices at the leading order in
$\mathcal{O}\left({m_{\mu}/v_{\Phi}}\right)$. Hereafter, we take
$\lambda_{e}=\lambda_{n}=0$ for simplicity, so that the $Z$ and $W$ boson
couplings to the SM leptons do not sizably deviate from the SM values. The
case of $\lambda_{e},\lambda_{n}\neq 0$ are discussed in Appendix A. The
${Z^{\prime}}$ couplings are given by

$\displaystyle
g^{Z^{\prime}}_{\mathbf{e}_{L}}={g^{\prime}}U_{L}^{\dagger}Q_{L}^{\prime}U_{L}\sim$
$\displaystyle\
-{g^{\prime}}\begin{pmatrix}s_{L}^{2}&-c_{L}s_{L}c_{e_{L}}&c_{L}s_{L}s_{e_{L}}\\\
-c_{L}s_{L}c_{e_{L}}&s_{e_{L}}^{2}+c_{L}^{2}c_{e_{L}}^{2}&s_{L}^{2}s_{e_{L}}c_{e_{L}}\\\
c_{L}s_{L}s_{e_{L}}&s_{L}^{2}s_{e_{L}}c_{e_{L}}&c_{L}^{2}s_{e_{L}}^{2}+c_{e_{L}}^{2}\end{pmatrix},$
(19) $\displaystyle
g^{Z^{\prime}}_{\mathbf{e}_{R}}={g^{\prime}}U_{R}^{\dagger}Q_{R}^{\prime}U_{R}\sim$
$\displaystyle\
-{g^{\prime}}\begin{pmatrix}s_{E}^{2}&-c_{E}s_{E}s_{e_{R}}&-c_{E}s_{E}c_{e_{R}}\\\
-c_{E}s_{E}s_{e_{R}}&c_{e_{R}}^{2}+c_{E}^{2}s_{e_{R}}^{2}&-s_{E}^{2}s_{e_{R}}c_{e_{R}}\\\
-c_{E}s_{E}c_{e_{R}}&-s_{E}^{2}s_{e_{R}}c_{e_{R}}&c_{E}^{2}c_{e_{R}}^{2}+s_{e_{R}}^{2}\end{pmatrix},$
(20)

where

$\displaystyle c_{X}:=\frac{m_{X}}{M_{X}},\quad
s_{X}:=\frac{\lambda_{X}v_{\Phi}}{M_{X}},\quad X=L,E.$ (21)

Here, $c_{e_{A}}$ and $s_{e_{A}}$ ($A=L,R$) are the angles to diagonalize the
VL mass matrix, defined in Eq. (49). These are approximately given by

$\displaystyle c_{e_{A}}\simeq 1-\frac{\eta_{e_{A}}^{2}}{2},\quad
s_{e_{A}}\simeq\eta_{e_{A}},\quad\eta_{e_{L}}:=\frac{{\lambda^{\prime}_{e}}v_{H}M_{L}}{M_{L}^{2}-M_{E}^{2}},\quad\eta_{e_{R}}:=\frac{{\lambda^{\prime}_{e}}v_{H}M_{E}}{M_{L}^{2}-M_{E}^{2}},$
(22)

for $\eta_{e_{A}}\ll 1$. Those for the neutrino couplings
$g^{{Z^{\prime}}}_{\mathbf{n}_{A}}$ are given by replacing $s_{E}\to 0$,
$c_{E}\to 1$ and $e_{A}\to n_{A}$. The isospin parts of the $Z$ and $W$ boson
couplings are given by

$\displaystyle\mathcal{E}_{A}\sim\begin{pmatrix}\delta_{AL}&0&0\\\
0&c_{e_{A}}^{2}&-c_{e_{A}}s_{e_{A}}\\\ 0&-c_{e_{A}}s_{e_{A}}&s_{e_{A}}^{2}\\\
\end{pmatrix},\quad\mathcal{N}_{A}\sim\begin{pmatrix}\delta_{AL}&0&0\\\
0&c_{n_{A}}^{2}&-c_{n_{A}}s_{n_{A}}\\\ 0&-c_{n_{A}}s_{n_{A}}&s_{n_{A}}^{2}\\\
\end{pmatrix},$ (23)

and

$\displaystyle h_{A}\sim\begin{pmatrix}\delta_{AL}&0&0\\\
0&c_{e_{A}}c_{n_{A}}&-c_{n_{A}}s_{e_{A}}\\\
0&-c_{e_{A}}s_{n_{A}}&s_{e_{A}}s_{n_{A}}\\\ \end{pmatrix},$ (24)

for $A=L,R$. Here, $\delta_{LA}=1~{}(0)$ 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
$\mathcal{O}\left({m_{\mu}/v_{\Phi}}\right)$. 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 $\chi$ are respectively given by

$\displaystyle Y^{h}_{\mathbf{e}}\sim$ $\displaystyle\
\begin{pmatrix}y_{2}c_{L}c_{E}&\mathcal{O}\left({y_{2}}\right)&\mathcal{O}\left({y_{2}}\right)\\\
\mathcal{O}\left({y_{2}}\right)&s_{e_{L}}c_{e_{R}}{\lambda^{\prime}_{e}}&c_{e_{L}}c_{e_{R}}{\lambda^{\prime}_{e}}\\\
\mathcal{O}\left({y_{2}}\right)&-s_{e_{L}}s_{e_{R}}{\lambda^{\prime}_{e}}&-c_{e_{L}}s_{e_{R}}{\lambda^{\prime}_{e}}\end{pmatrix},$
(25) $\displaystyle Y^{\chi}_{\mathbf{e}}\sim$ $\displaystyle\
\begin{pmatrix}0&c_{E}s_{e_{L}}\lambda_{E}&c_{E}c_{e_{L}}\lambda_{E}\\\
c_{L}c_{e_{R}}\lambda_{L}&s_{E}s_{e_{L}}s_{e_{R}}\lambda_{E}+s_{L}c_{e_{L}}c_{e_{R}}\lambda_{L}&s_{E}c_{e_{L}}s_{e_{R}}\lambda_{E}-s_{L}s_{e_{L}}c_{e_{R}}\lambda_{L}\\\
-c_{L}s_{e_{R}}\lambda_{L}&s_{E}s_{e_{L}}c_{e_{R}}\lambda_{E}-s_{L}c_{e_{L}}s_{e_{R}}\lambda_{L}&s_{E}c_{e_{L}}c_{e_{R}}\lambda_{E}+s_{L}s_{e_{L}}s_{e_{R}}\lambda_{L}&\end{pmatrix}.$
(26)

Those for the neutrinos are given by replacing $y_{2}\to 0$, $e\to n$ and
$E\to N$. Note that $m_{\mu}\sim y_{2}c_{L}c_{E}v_{H}$, so the Yukawa coupling
to the Higgs boson is also not changed from the SM value. The SM Higgs
couplings to the SM and VL leptons are again suppressed by the small muon
Yukawa coupling. The $\chi$ boson coupling to the SM muon vanishes if we
neglect the muon mass.

##  3 Phenomenology

###  3.1 Oblique parameters and $W$ boson mass

In this model, the VL lepton contributions to the $T$ parameter, one of the
oblique parameters [130, 131], is given by [132],

$\displaystyle 16\pi s_{W}^{2}c_{W}^{2}T=$ $\displaystyle\
\sum_{a,\beta}\left\\{\left(\left|{h^{L}_{a\beta}}\right|^{2}+\left|{h^{R}_{a\beta}}\right|^{2}\right)\theta_{+}(y_{a},y_{\beta})+2\mathrm{Re}\left(h^{L}_{a\beta}h^{R*}_{a\beta}\right)\theta_{-}(y_{a},y_{\beta})\right\\}$
(27) $\displaystyle\
-\sum_{a<b}\left\\{\left(\left|{\mathcal{N}^{L}_{ab}}\right|^{2}+\left|{\mathcal{N}^{R}_{ab}}\right|^{2}\right)\theta_{+}(y_{a},y_{b})+2\mathrm{Re}\left(\mathcal{N}^{L}_{ab}\mathcal{N}^{R*}_{ab}\right)\theta_{-}(y_{a},y_{b})\right\\}$
$\displaystyle\
-\sum_{\alpha<\beta}\left\\{\left(\left|{\mathcal{E}^{L}_{\alpha\beta}}\right|^{2}+\left|{\mathcal{E}^{R}_{\alpha\beta}}\right|^{2}\right)\theta_{+}(y_{\alpha},y_{\beta})+2\mathrm{Re}\left(\mathcal{E}^{L}_{\alpha\beta}\mathcal{E}^{R*}_{\alpha\beta}\right)\theta_{-}(y_{\alpha},y_{\beta})\right\\},$

where $a,b=1,2,3$ ($\alpha,\beta=1,2,3$) run over the charged (neutral)
leptons. The other oblique parameters, $2\pi S$ and $-2\pi U$, are obtained by
replacing the functions, $\theta_{\pm}\to\chi_{\pm}$, except for the first
line in the $S$ parameter which should be replaced as
$\theta_{\pm}\to\psi_{\pm}$. The loop functions are defined in Ref. [41]. For
an order of magnitude estimation, taking $s_{e_{A}},s_{n_{A}}=0$ and
$m_{L^{0}}\simeq m_{L^{-}}$, we have

$\displaystyle 16\pi
s_{W}^{2}c_{W}^{2}T\sim\frac{2(m_{L^{-}}^{2}-m_{L^{0}}^{2})^{2}}{3m_{Z}^{2}M_{L}^{2}}\sim$
$\displaystyle\
\frac{2v_{H}^{4}}{3m_{Z}^{2}M_{L}^{2}}\left(\frac{{\lambda^{\prime}_{e}}^{2}}{1-M_{E}^{2}/M_{L}^{2}}-\frac{{\lambda^{\prime}_{n}}^{2}}{1-M_{N}^{2}/M_{L}^{2}}\right)^{2},$
(28)

where $m_{L^{-}}$ and $m_{L^{0}}$ are the masses of doublet-like charged and
neutral VL leptons, respectively. $\eta_{e_{A}},\eta_{n_{A}}\ll 1$ is assumed
in the second equality. Hence,

$\displaystyle T\sim
0.13\times\left(\frac{{\lambda^{\prime}_{e}}^{2}}{1-M_{E}^{2}/M_{L}^{2}}-\frac{{\lambda^{\prime}_{n}}^{2}}{1-M_{N}^{2}/M_{L}^{2}}\right)^{2}\left(\frac{250~{}\mathrm{GeV}}{M_{L}}\right)^{2},$
(29)

where we used $s_{W}^{2}=0.23121$ [2]. The $W$ boson mass is related to the
oblique parameters as [133, 134] 444 We neglect the difference in the
definitions of the oblique parameters. The length of slopes of self-energies
are taken to be finite value [132], but these effects are less than $5\%$ in
the parameter space studied in this paper, and thus not important.

$\displaystyle\frac{\delta m_{W}^{2}}{m_{W}^{2}|_{\mathrm{SM}}}=$
$\displaystyle\
\frac{\alpha_{e}}{c_{W}^{2}-s_{W}^{2}}\left(-\frac{1}{2}S+c_{W}^{2}T+\frac{c_{W}^{2}-s_{W}^{2}}{4s_{W}^{2}}U\right)\sim
0.001\times\left(\frac{T}{0.1}\right),$ (30)

where $T\gg S,U$ is assumed in the second equality. Therefore, from Eq. (29),
the shift may be explained if
${\lambda^{\prime}_{e}},{\lambda^{\prime}_{n}}\sim\mathcal{O}\left({1}\right)$
and $M_{L}\sim 250~{}\mathrm{GeV}$.

###  3.2 Muon $g-2$

The muon anomalous magnetic moment $\Delta a_{\mu}$ is shifted by the 1-loop
effects via the ${Z^{\prime}}$ and $\chi$ bosons. This is approximately given
by [135, 87]

$\displaystyle\Delta
a_{\mu}\sim-\frac{m_{\mu}{\lambda^{\prime}_{e}}v_{H}}{16\pi^{2}m_{{Z^{\prime}}}^{2}}\tilde{C}_{LR},\quad\tilde{C}_{LR}:=\frac{\lambda_{L}\lambda_{E}m_{L}m_{E}}{M_{L}M_{E}}\left(\frac{G_{Z}(x_{L})-G_{Z}(x_{E})}{x_{L}-x_{E}}+\frac{m_{Z^{\prime}}^{2}}{2m_{\chi}^{2}}\frac{{y_{L}}G_{{S}}(y_{L})-{y_{R}}G_{{S}}(y_{R})}{y_{L}-y_{R}}\right),$
(31)

where $x_{L}:=M_{L}^{2}/m_{Z^{\prime}}^{2}$,
$x_{E}:=M_{E}^{2}/m_{Z^{\prime}}^{2}$, $y_{L}:=M_{L}^{2}/m_{\chi}^{2}$ and
$y_{E}:=M_{E}^{2}/m_{\chi}^{2}$. The exact formula and the loop functions are
shown in Refs. [77, 78]. The value of $\Delta a_{\mu}$ is estimated as

$\displaystyle\Delta a_{\mu}\sim 5\times
10^{-9}\times{\lambda^{\prime}_{e}}\left(\frac{500~{}\mathrm{GeV}}{m_{Z^{\prime}}}\right)^{2}\left(\frac{\tilde{C}_{LR}}{-0.01}\right).$
(32)

Hence, the ${Z^{\prime}}$ boson and VL leptons with mass of order
$500~{}\mathrm{GeV}$ can explain $\Delta a_{\mu}$.

###  3.3 $h\to\gamma\gamma$ and muon trident process

The decay rate of $h\to\gamma\gamma$ 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 $h\to\gamma\gamma$ to that in the SM as

$\displaystyle
R_{\gamma\gamma}:=\frac{\Gamma(h\to\gamma\gamma)}{\Gamma(h\to\gamma\gamma)_{\mathrm{SM}}}=\left|{1+\sum_{i=1,2}\left(\left[Y^{h}_{\mathbf{e}}\right]_{E_{i}E_{i}}\frac{v_{H}}{m_{E_{i}}}\right)\frac{A^{H}_{1/2}(\tau_{E_{i}})}{A_{\mathrm{SM}}}}\right|^{2},$
(33)

where $\tau_{I}=m_{H}^{2}/(4m_{I}^{2})$ for the lower case $I=E_{1},E_{2}$,
and the SM contribution $A_{\mathrm{SM}}$ is given by [136]

$\displaystyle
A_{\mathrm{SM}}:=A_{1}^{H}(\tau_{W})+\sum_{f}N^{f}_{c}Q_{f}^{2}A^{H}_{1/2}(\tau_{f}).$
(34)

Here, $f$ runs over all the SM fermions. $Q_{f}$ and $N_{c}^{f}$ are the
electric charge and the number of colors of the fermion $f$, respectively. The
current experimental value is $1.11\pm 0.095$ [2].

The ${Z^{\prime}}$ boson can induce the so-called neutrino trident process,
$\nu_{\mu}N\to\nu_{\mu}\mu^{+}\mu^{-}N$ [137, 138, 139, 140, 141, 142]. The
cross section for this process at the CCFR experiment is estimated as [142,
143]

$\displaystyle
R_{\mathrm{CCFR}}:=\frac{\sigma_{\text{CCFR}}}{\sigma^{\text{SM}}_{\text{CCFR}}}\simeq\frac{(1+4s_{W}^{2}+\Delta
g^{V}_{\mu\mu\mu\mu})^{2}+1.13(1-\Delta
g^{A}_{\mu\mu\mu\mu})^{2}}{(1+4s_{W}^{2})^{2}+1.13},$ (35)

where the ${Z^{\prime}}$ boson contributions $\Delta g_{\mu\mu\mu\mu}^{V,A}$
are given by

$\displaystyle\Delta g^{V,A}_{\mu\mu\mu\mu}=$ $\displaystyle\
\frac{\sqrt{2}}{G_{F}\cdot
2m_{Z^{\prime}}^{2}}\left[{g}^{Z^{\prime}}_{\mathbf{n}_{L}}\right]_{\nu_{\mu}\nu_{\mu}}\left(\left[g^{Z^{\prime}}_{\mathbf{e}_{R}}\right]_{\mu\mu}\pm\left[g^{Z^{\prime}}_{\mathbf{e}_{L}}\right]_{\mu\mu}\right)\sim\frac{s_{L}^{2}(s_{E}^{2}\pm
s_{L}^{2})}{2\sqrt{2}G_{F}v_{\Phi}^{2}}.$ (36)

The experimentally observed rate is
$\sigma_{\text{CCFR}}/\sigma^{\text{SM}}_{\text{CCFR}}=0.82\pm 0.28$ at 95%
C.L 555 For the calculation of $R_{\mathrm{CCFR}}$, we used
$s_{W}^{2}=0.23129$ as in Ref. [142]. .

###  3.4 $b\to s\mu\mu$ anomaly

We emphasize that the recent anomaly in the $b\to s\ell\ell$ process can
easily be explained in this model. The Wilson coefficients for the semi-
leptonic operators are given by

$\displaystyle C_{9,10}\sim$ $\displaystyle\
-\frac{\pi}{2\sqrt{2}\alpha_{e}G_{F}}\frac{1}{V_{tb}V_{ts}^{*}}\left(\frac{\lambda_{E}^{2}}{M_{E}^{2}}\pm\frac{\lambda_{L}^{2}}{M_{L}^{2}}\right)\frac{g^{{Z^{\prime}}}_{sb}}{{g^{\prime}}},$
(37)

where $V_{tb}$ and $V_{ts}$ are the CKM elements, and $g^{Z^{\prime}}_{sb}$ is
the ${Z^{\prime}}$ boson coupling constant to $sb$ in the left-handed
interaction. The Wilson coefficient $C_{9}$ is estimated as

$\displaystyle
C_{9}\sim-0.8\times\left(\frac{200~{}\mathrm{GeV}}{m_{\mathrm{VLL}}}\right)^{2}\left(\frac{g^{{Z^{\prime}}}_{sb}/{g^{\prime}}}{10^{-4}}\right),\quad\frac{1}{m_{\mathrm{VLL}}^{2}}:=\frac{\lambda_{E}^{2}}{M_{E}^{2}}+\frac{\lambda_{L}^{2}}{M_{L}^{2}},$
(38)

whereas the current favored values is $C_{9}\in[-1.0,-0.5]$ depending on the
value of $C_{10}$ [144, 145, 146, 147, 148, 149, 150]. Thus, the $b\to
s\ell\ell$ anomaly can be explained even with the small ${Z^{\prime}}$
coupling to quarks, as long as those to the VL leptons are sizable to explain
the shift in $m_{W}$ and $\Delta a_{\mu}$. With such small couplings to the
${Z^{\prime}}$ boson, the flavor violations such as $B_{s}$-$\overline{B}_{s}$
mixing will not deviate from the SM prediction due to the small couplings, as
we have explicitly shown in Refs. [77, 78]. Further, the production cross
sections at the LHC will be so small that the di-muon signal is much below the
current limit [151].

##  4 Numerical results and LHC signals

###  4.1 $\chi^{2}$ fitting

Figure 1:  Scattering plots on the observables in our analysis. The cross and
circle points are $100<m_{E_{1}}<175~{}\mathrm{GeV}$ and
$m_{E_{1}}>175~{}\mathrm{GeV}$, respectively. All the points explain the
$m_{W}$ and $\Delta a_{\mu}$ within $1\sigma$, and $R_{\mathrm{CCFR}}$ and
$R_{\gamma\gamma}$ within $2\sigma$. The error bars on the left panel are the
$1\sigma$ uncertainties, and the dash horizontal line on the right panel is
95% C.L. limit on $R_{\mathrm{CCFR}}$. The central value of
$R_{\gamma\gamma}=1.10$ is outside of the figure. The benchmark points, A,B,C,
lie on top of each other in the left panel, while B and C are
indistinguishable in the right panel.

Figure 2:  Scattering plots the Yukawa couplings
$({\lambda^{\prime}_{e}},{\lambda^{\prime}_{n}})$ and the VL lepton masses
$(m_{E_{1}},m_{N_{1}})$. The cross and circle points are
$100<m_{E_{1}}<175~{}\mathrm{GeV}$ and $m_{E_{1}}>175~{}\mathrm{GeV}$,
respectively.

Figure 3:  Scattering plots on the masses of ${Z^{\prime}}$ boson and the VL
leptons. The cross and circle points are $100<m_{E_{1}}<175~{}\mathrm{GeV}$
and $m_{E_{1}}>175~{}\mathrm{GeV}$, respectively. The diagonal lines are
respectively $m_{{Z^{\prime}}}=m_{E_{1}}$ and $m_{\chi}=m_{E_{1}}$ on the left
and right panel.

Figure 4:  Scattering plots on the oblique parameters. The cross and circle
points are $100<m_{E_{1}}<175~{}\mathrm{GeV}$ and
$m_{E_{1}}>175~{}\mathrm{GeV}$, respectively. The solid (dashed) lines are
$1\sigma$ ($2\sigma$) range of the $W$ boson mass measured by the CDF for
$U=0$ and $T=0$ on the left and right panel, respectively.

There are 10 parameters in our model,

$\displaystyle
x=({g^{\prime}},v_{\Phi},m_{\chi},m_{L},m_{E},m_{N},\lambda_{L},\lambda_{E},{\lambda^{\prime}_{e}},{\lambda^{\prime}_{n}}),$
(39)

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
${\lambda_{e}}={\lambda_{n}}=0$ to keep the $SU(2)_{L}$ gauge coupling SM-
like, as well as $\lambda_{N}=y_{n}=0$ since these are irrelevant for
phenomenology after integrating out $\nu_{R}$. To find points which can
explain the anomalies, we minimize the $\chi^{2}$ function

$\displaystyle\chi^{2}(x):=\sum_{I}\frac{(y_{I}(x)-y^{0}_{I})^{2}}{\sigma_{I}^{2}},$
(40)

where $y_{I}=(\Delta a_{\mu},m_{W},R_{\gamma\gamma},R_{\mathrm{CCFR}})$.
$y^{0}_{I}$ and $\sigma_{I}$ are the experimental central value and its
$1\sigma$ error, respectively. The experimental values are $\Delta
a_{\mu}=(2.51\pm 0.59)\times 10^{-9}$, $m_{W}=80.4335\pm 0.0094$,
$R_{\gamma\gamma}=1.11\pm 0.095$ and $R_{\mathrm{CCFR}}=0.82\pm 0.14$ 666 For
$R_{\mathrm{CCFR}}$, the error is chosen such that the $2\sigma$ deviation
corresponds to the 95% C.L limit. . Here, we assume that the $W$ boson mass is
given by the CDF result. In our analysis, we restrict the parameter space to
be

$\displaystyle v_{\Phi}\in[100,2000]~{}\mathrm{GeV},$ $\displaystyle
m_{L},m_{E}\in[100,1500]~{}\mathrm{GeV},$ (41) $\displaystyle
m_{\chi}\in[63,1500]~{}\mathrm{GeV}$ $\displaystyle
m_{N}\in[40,1500]~{}\mathrm{GeV},$ (42)

and

$\displaystyle{g^{\prime}}\in[0,0.35],\quad\lambda_{L},\lambda_{E},{\lambda^{\prime}_{e}},{\lambda^{\prime}_{n}}\in[-1,1].$
(43)

Here, the lower bounds of $m_{\chi}$ is chosen so that $h\to\chi\chi$ is
kinematically forbidden. The $U(1)^{\prime}$ gauge constant ${g^{\prime}}$ is
required to be smaller than 0.35, so that it is perturbative up to
$10^{16}~{}\mathrm{GeV}$ in the full model with VL quarks [77]. We minimize
the $\chi^{2}$ function from random initial points in the parameter range. We
further require $m_{E_{1}},m_{Z^{\prime}}>100~{}\mathrm{GeV}$ and
$m_{N_{1}}>45~{}\mathrm{GeV}$ for the fitted points, so that the collider
limits would be avoided and $Z\to N_{1}N_{1}$ is kinematically forbidden.

We calculated branching fractions of the ${Z^{\prime}}$ boson and the VL
leptons. For the ${Z^{\prime}}$ 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 3.4. For the decays of VL leptons, we
calculated the two-body decays to a SM, $\chi$, ${Z^{\prime}}$ boson and a
lepton. The three-body decays via off-shell ${Z^{\prime}}$ 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 $\chi^{2}$
fitting. On these plots, the cross and circle points are
$100<m_{E_{1}}<175~{}\mathrm{GeV}$ and $m_{E_{1}}>175~{}\mathrm{GeV}$,
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. $E_{1}\to N_{1}e\nu_{e}$ is a part of
$E_{1}\to N_{1}W$ for $m_{E_{1}}-m_{N_{1}}<m_{W}$.

The scatter plots for the observables are shown in Fig. 1. On the left panel,
the gray error bars are the $1\sigma$ errors along the axes. We see that
$\Delta a_{\mu}$ and $m_{W}$ are simultaneously explained on these points
within $1\sigma$. From the right panel, $R_{\gamma\gamma}$ can deviate from
unity at most $1\%$, and hence this is about $1\sigma$ below the experimental
central value of $1.10$. The neutrino trident process can be induced from the
${Z^{\prime}}$ boson exchange, and there are points close to the upper bound
of $1.10$, depicted by the gray dashed line. Altogether, all of the points
explain the anomalies in $m_{W}$ and $\Delta a_{\mu}$ without conflicting with
the current limits on $R_{\gamma\gamma}$ and $R_{\mathrm{CCFR}}$.

The left panel of Fig. 2 shows the scatter plots on
$({\lambda^{\prime}_{e}},{\lambda^{\prime}_{n}})$. There are points only where
$|{\lambda^{\prime}_{n}}|\sim 1.0$. 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 $m_{E_{1}}>175~{}\mathrm{GeV}$
have $E_{1}\to N_{1}W$ as the dominant decay mode.

Figure 3 shows the scatter plots on the $m_{E_{1}}$ and $m_{Z^{\prime}}$
($m_{\chi}$) plane on the left (right) panel. The gray line is
$m_{E_{1}}=m_{{Z^{\prime}}}$ or $m_{\chi}$. The upper bound on
$m_{Z^{\prime}}$ is about $600~{}\mathrm{GeV}$ to explain the muon $g-2$,
while $\chi$ can be much heavier. $E_{1}\to N_{1}W$ is the dominant decay mode
even if $m_{{Z^{\prime}}}<m_{E_{1}}$ due to the larger gauge coupling constant
$g>{g^{\prime}}$.

Figure 4 shows the scatter plots for the oblique parameters. The solid
(dashed) lines correspond to $1\sigma$ ($2\sigma$) errors of $m_{W}$ of the
CDF measurement when $U=0$ and $S=0$ on the left and right panel,
respectively. For $m_{E_{1}}>175~{}\mathrm{GeV}$, the points are found only
around $(S,T,U)\sim(0.01,0.15,0.02)$, so the $W$ boson mass shift is mainly
explained by the $T$ parameter. Different values of the oblique parameters are
allowed for $m_{E_{1}}<175~{}\mathrm{GeV}$. These patterns of the oblique
parameters will be useful to distinguish the parameter space of our model.

Table 2:  Benchmark points benchmark | A | B | C | D
---|---|---|---|---
$g^{\prime}$ | 0.3500 | 0.3495 | 0.3227 | 0.3251
$m_{Z^{\prime}}$ GeV | 302.8449 | 122.0818 | 146.2380 | 440.1635
$m_{\chi}~{}\mathrm{GeV}$ | 1500.0000 | 675.7500 | 138.2121 | 63.0000
$m_{E_{1}}~{}\mathrm{GeV}$ | 150.6716 | 163.0642 | 144.8669 | 192.0898
$m_{E_{2}}~{}\mathrm{GeV}$ | 212.2732 | 220.3620 | 353.3552 | 1729.7359
$m_{N_{1}}~{}\mathrm{GeV}$ | 70.3855 | 45.5943 | 77.2179 | 63.6683
$m_{N_{2}}~{}\mathrm{GeV}$ | 251.1165 | 273.7983 | 251.7450 | 266.9969
$R_{\gamma\gamma}$ | 1.0008 | 1.0003 | 1.0002 | 1.0001
$R_{\mathrm{CCFR}}$ | 1.0028 | 1.0016 | 1.0020 | 1.0088
$m_{W}~{}\mathrm{GeV}$ | 80.4336 | 80.4335 | 80.4335 | 80.4312
$\Delta a_{\mu}$$\times 10^{9}$ | 2.5161 | 2.5064 | 2.5036 | 2.5102
$S$ | 0.0328 | 0.0066 | 0.0356 | 0.0225
$T$ | 0.1700 | 0.1459 | 0.1725 | 0.1580
$U$ | 0.0173 | 0.0264 | 0.0160 | 0.0173
$\mathrm{Br}(Z^{\prime}\to\mu\mu)$ | 0.1904 | 0.1808 | 0.4522 | 0.0668
$\mathrm{Br}(Z^{\prime}\to\nu\nu)$ | 0.0029 | 0.0001 | 0.0332 | 0.0507
$\mathrm{Br}(N_{1}\to\chi\nu)$ | 0.0000 | 0.0000 | 0.0000 | 0.9977
$\mathrm{Br}(N_{1}\to Z^{\prime}\nu)$ | 0.9832 | 0.9996 | 0.9956 | 0.0023
$\mathrm{Br}(N_{1}\to W\mu)$ | 0.0168 | 0.0004 | 0.0044 | 0.0000
$\mathrm{Br}(E_{1}\to\chi\mu)$ | 0.0000 | 0.0000 | 0.9980 | 0.0197
$\mathrm{Br}(E_{1}\to Z^{\prime}\mu)$ | 0.3526 | 0.5049 | 0.0019 | 0.0000
$\mathrm{Br}(E_{1}\to WN_{1})$ | 0.0000 | 0.4951 | 0.0000 | 0.9803
$\mathrm{Br}(E_{1}\to\mathrm{SM})$ | 0.6474 | 0.0000 | 0.0001 | 0.0000

Table 2 shows the benchmark points 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, ${Z^{\prime}}\mu$, $\chi\mu$ and $N_{1}W$, respectively. Here, the
SM particles include $Z\mu,h\mu$ and $W\nu$. These points are plotted on Figs.
1, 2, 3 and 4. The branching fraction of the ${Z^{\prime}}$ boson to the SM
leptons are shown in this table. The ${Z^{\prime}}$ boson will also decay to
the VL leptons if it is kinematically allowed 777 Signals same as the leptonic
cascade decay ${Z^{\prime}}\to E_{1}\mu$ at the LHC is studied in Ref.[152].
This search could constrain our parameter space if the production cross
section is sufficiently large. . The lightest VL neutrino $N_{1}$ decays to a
SM neutrino and a $\chi$ boson if $m_{\chi}<m_{N_{1}}$ as in the point (D),
while it decays to a ${Z^{\prime}}$ boson in the other cases. Since
$m_{N_{1}}<m_{W}$, the decays to a $W$ boson is always suppressed compared
with that to a ${Z^{\prime}}$ boson.

###  4.2 LHC signals

We found that $m_{E_{1}}\lesssim 250~{}\mathrm{GeV}$ is required to explain
the shift of $m_{W}$. Such a light VL lepton may be constrained by LHC
searches, depending on its decay.

  1. (A)

$E_{1}\to\mathrm{SM}$
The VL lepton would decay to a SM boson and a SM lepton. If $E_{1}$ is
doublet-like, the latest limit is about $800~{}\mathrm{GeV}$ [153, 154]. Even
for the singlet-like case, the limit is about 200 GeV [155] using the run-1
data [156]. Since $E_{1}$ is mostly doublet-like, the VL lepton lighter than
$250~{}\mathrm{GeV}$ may be excluded by the current data. However, these decay
modes may be suppressed because
$[g^{Z}_{e_{L,R}}]_{E_{1}\mu}=\mathcal{O}\left({m_{\mu}/m_{E_{1}}}\right)$,
and hence the other decay mode will coexist, or even dominate the decays.
Actually, $E_{1}\to\mathrm{SM}$ is sizable only on the point (A), and there is
a sizable fraction to $E_{1}\to{Z^{\prime}}\mu$.

  2. (B)

$E_{1}\to{Z^{\prime}}\mu$
In Ref. [157], it is shown that the limit for the doublet-like VL lepton is
about 500 (1200) GeV for
$\mathrm{BR}\left({{Z^{\prime}}}\to{\mu\mu}\right)=2/3$ and
$\mathrm{BR}\left({E_{1}}\to{{Z^{\prime}}\mu}\right)=0.1~{}(1.0)$ by the
signal with four muons or more [158]. Although the ${Z^{\prime}}$ boson was
assumed to be on-shell in Ref. [157], the limits on the VL lepton would be
similar even for $m_{E_{1}}<m_{{Z^{\prime}}}$. At the point (B), this decay
mode dominates the others and
$\mathrm{Br}(E_{1}\to{Z^{\prime}}\mu\to\mu\mu\mu)\sim 0.5$, so this case may
also be excluded.

  3. (C)

$E_{1}\to\chi\mu$
This is the dominant decay mode on the point (C). This case may not be
excluded if $\chi$ decays to quarks. The $\chi$ coupling to two SM fermions is
induced only by the fermion mass effects, so $\chi\to bb$ can be the dominant
mode 888 For $m_{\chi}>2m_{t}$, $\chi\to tt$ may be the dominant decay mode,
so the signal would be constrained by the 4 top quarks search [159]. Such
heavy $\chi$ boson is, however, always heavier than $E_{1}$, and
$E_{1}\to\chi\mu$ is kinematically forbidden. . Another possibility is
$\chi\to hh^{*}$ via the quartic interaction $|H|^{2}|\Phi|^{2}$ in the scalar
potential. Since the decays to fermions are suppressed, this mode can dominate
over the others. In this case, we expect $4h+\mu\mu$ from the $E_{1}$ pair
production. Yet another possibility is loop-induced decays such as
$\chi\to\gamma\gamma$ and $gg$, which may be particularly important for
$m_{\chi}<m_{h}$. 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.

  4. (D)

$E_{1}\to WN_{1}$
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
$N_{1}\to{Z^{\prime}}\nu$ or $N_{1}\to\chi\nu$. At the point (D),
$N_{1}\to\chi\nu$ dominates, so the signal is $E_{1}E_{1}\to
WW+\chi\chi+\nu\nu$. 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 $\geq
4\mu$ search [157], 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
$\mathrm{Br}(N_{1}\to{Z^{\prime}}\nu\to\mu\mu\nu)$ 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 $\chi$ 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 $\chi$ 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.

##  5 Summary

In this work, we studied the $W$ boson mass in the SM extension with VL
leptons and a $U(1)^{\prime}$ 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 $b\to s\ell\ell$
decay simultaneously [77, 78]. We explicitly studied the $W$ boson mass and
$\Delta a_{\mu}$ induced by the VL leptons and the ${Z^{\prime}}$ boson, and
we found points which can explain both anomalies. Since $b\to s\ell\ell$ can
easily be explained by tiny couplings with the SM quarks, as discussed in Sec.
3.4, 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 ${Z^{\prime}}$ or $\chi$
boson. If the VL lepton decays through the ${Z^{\prime}}$ boson, which may
decay to di-muons, there will be strong constraints from the searches for
signals with four muons or more [157]. Hence, the decay to the $\chi$ boson
should dominate the decay of the VL lepton, which may be achieved by the mass
hierarchy $m_{\chi}<m_{E_{1},N_{1}}<m_{{Z^{\prime}}}$. The $\chi$ 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 $\chi$ boson in the full model with VL quarks is
our future work.

## Acknowledgment

The work of J.K. is supported in part by the Institute for Basic Science
(IBS-R018-D1), and the Grant-in-Aid for Scientific Research from the Ministry
of Education, Science, Sports and Culture (MEXT), Japan No. 18K13534. The work
of S.R. is supported in part by the Department of Energy (DOE) under Award No.
DE-SC0011726.

##  Appendix A Case of $\lambda_{e},\lambda_{n}\neq 0$

We first discuss the diagonalization of the charged lepton mass matrix Eq. (3)
for ${\lambda^{\prime}_{e}}\neq 0$. The diagonalization matrix is decomposed
as

$\displaystyle U_{L}=U_{L}^{0}U_{L}^{1}U_{L}^{2},\quad
U_{R}=U_{R}^{0}U_{R}^{1}U_{R}^{2}.$ (44)

The unitary matrices are given by

$\displaystyle U_{L}^{0}=\begin{pmatrix}c_{L}&s_{L}&0\\\ -s_{L}&c_{L}&0\\\
0&0&1\end{pmatrix},\quad U_{R}^{0}=\begin{pmatrix}c_{E}&s_{E}&0\\\
-s_{E}&c_{E}&0\\\ 0&0&1\end{pmatrix},$ (45) $\displaystyle U_{L}^{1}=$
$\displaystyle\
\begin{pmatrix}1-y_{42}^{2}\dfrac{v_{H}^{2}}{2M_{E}^{2}}&y_{22}y_{24}\dfrac{v_{H}^{2}}{M_{L}^{2}}-y_{42}{\lambda^{\prime}_{e}}\dfrac{v_{H}^{2}}{M_{E}M_{L}}&y_{42}\dfrac{v_{H}}{M_{E}}\\\
-y_{22}y_{24}\dfrac{v_{H}^{2}}{M_{L}^{2}}+y_{42}{\lambda^{\prime}_{e}}\dfrac{v_{H}^{2}}{M_{E}M_{L}}&1&0\\\
-y_{42}\dfrac{v_{H}}{M_{E}}&0&1-y_{42}^{2}\dfrac{v_{H}^{2}}{2M_{E}^{2}}\end{pmatrix}+\mathcal{O}\left({m_{\mu}^{3}}\right),$
(46) $\displaystyle U_{R}^{1}=$ $\displaystyle\
\begin{pmatrix}1-y_{24}^{2}\dfrac{v_{H}^{2}}{2M_{L}^{2}}&y_{22}y_{42}\dfrac{v_{H}^{2}}{M_{E}^{2}}-y_{24}{\lambda^{\prime}_{e}}\dfrac{v_{H}^{2}}{M_{E}M_{L}}&y_{24}\dfrac{v_{H}}{M_{L}}\\\
-y_{22}y_{42}\dfrac{v_{H}^{2}}{M_{E}^{2}}+y_{24}{\lambda^{\prime}_{e}}\dfrac{v_{H}^{2}}{M_{E}M_{L}}&1&0\\\
-y_{24}\dfrac{v_{H}}{M_{L}}&0&1-y_{24}^{2}\dfrac{v_{H}^{2}}{2M_{L}^{2}}\end{pmatrix}+\mathcal{O}\left({m_{\mu}^{3}}\right),$
(47)

and

$\displaystyle U_{L}^{2}=\begin{pmatrix}1&0&0\\\ 0&c_{e_{L}}&-s_{e_{L}}\\\
0&s_{e_{L}}&c_{e_{L}}\end{pmatrix},\quad U_{R}^{2}=\begin{pmatrix}1&0&0\\\
0&s_{e_{R}}&c_{e_{R}}\\\ 0&c_{e_{R}}&-s_{e_{R}}\end{pmatrix},$ (48)

where

$\displaystyle\begin{pmatrix}s_{e_{R}}&c_{e_{R}}\\\
c_{e_{R}}&-s_{e_{R}}\end{pmatrix}\begin{pmatrix}y_{44}v_{H}&M_{E}\\\
M_{L}&{\lambda^{\prime}_{e}}v_{H}\end{pmatrix}\begin{pmatrix}c_{e_{L}}&-s_{e_{L}}\\\
s_{e_{L}}&c_{e_{L}}\end{pmatrix}=\mathrm{diag}\left(m_{E_{1}},m_{E_{2}}\right).$
(49)

Here, $y_{22}=c_{L}c_{E}y_{2}+s_{L}s_{E}{\lambda_{e}}$,
$y_{24}=c_{E}s_{L}y_{2}-s_{E}c_{L}{\lambda_{e}}$,
$y_{42}=c_{L}s_{E}y_{2}-c_{E}s_{L}{\lambda_{e}}$ and
$y_{44}=s_{L}s_{E}y_{2}+c_{L}c_{E}{\lambda_{e}}$. The unitary matrices
$U_{L,R}^{1}$ block diagonalize the SM and VL leptons up to
$\mathcal{O}\left({y_{2}^{2}v_{H}^{2}}\right)$ when we assume
$\lambda_{e}\sim\mathcal{O}\left({y_{2}}\right)$. These are simply identity
matrices if we neglect $\mathcal{O}\left({m_{\mu}/v_{\Phi}}\right)$. The
unitary matrices $U_{{L,R}}^{0}$ keeps the $SU(2)_{L}$ gauge couplings
unchanged, while $U_{{L,R}}^{1}$ do change the couplings as

$\displaystyle\left(U_{L}^{1}\right)^{\dagger}Q_{L}U_{L}^{1}=$ $\displaystyle\
\begin{pmatrix}1-y_{42}^{2}\dfrac{v_{H}^{2}}{M_{E}^{2}}&0&y_{42}\dfrac{v_{H}}{M_{E}}\\\
0&1&0\\\
y_{42}\dfrac{v_{H}}{M_{E}}&0&y_{42}^{2}\dfrac{v_{H}^{2}}{M_{E}^{2}}\end{pmatrix},\quad\left(U_{R}^{1}\right)^{\dagger}Q_{R}U_{R}^{1}=\begin{pmatrix}y_{24}^{2}\dfrac{v_{H}^{2}}{M_{L}^{2}}&0&-y_{24}\dfrac{v_{H}}{M_{L}}\\\
0&0&0\\\
-y_{24}\dfrac{v_{H}}{M_{L}}&0&1-y_{24}^{2}\dfrac{v_{H}^{2}}{M_{L}^{2}}\end{pmatrix}.$
(50)

For ${\lambda_{e}}v_{H}\lesssim m_{\mu}$, the correction to the muon coupling
deviates from the SM value within $\mathcal{O}\left({10^{-6}}\right)$ for
$\mathcal{O}\left({100}\right)~{}\mathrm{GeV}$ VL lepton masses, while it can
be $\mathcal{O}\left({1}\right)$ for
$\lambda_{e}\sim\mathcal{O}\left({1}\right)$ with the fine-tuning for
$y_{22}v_{H}\sim m_{\mu}$. Note that $U_{e_{L,R}}^{2}$ 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 $\mathcal{M}_{n}$ as

$\displaystyle\widetilde{\mathcal{M}}_{n}=\mathcal{M}_{n}V_{L}^{0}=\begin{pmatrix}m_{D}&\cdot&\cdot\\\
0&0&\widetilde{m}_{N}\\\ 0&\widetilde{M}_{L}&m_{n}\\\ \end{pmatrix},\quad
V_{L}^{0}=\begin{pmatrix}1&0&0\\\ 0&c_{N}&s_{N}\\\
0&-s_{N}&c_{N}\end{pmatrix}\begin{pmatrix}\tilde{c}_{L}&\tilde{s}_{L}&0\\\
-\tilde{s}_{L}&\tilde{c}_{L}&0\\\ 0&0&1\end{pmatrix},$ (51)

where $m_{D}=\tilde{c}_{L}y_{n}v_{H}+\tilde{s}_{L}s_{N}\lambda_{N}v_{\Phi}$,
$\widetilde{m}_{N}=\sqrt{m_{N}^{2}+\lambda_{n}^{2}v_{H}^{2}}$,
$\widetilde{m}_{L}=c_{N}m_{L}-s_{N}{\lambda^{\prime}_{n}}v_{H}$,
$m_{n}=s_{N}m_{L}+c_{N}{\lambda^{\prime}_{n}}v_{H}$ and
$\widetilde{M}_{L}=\sqrt{\widetilde{m}_{L}^{2}+\lambda_{L}^{2}v_{\Phi}^{2}}$.
The mixing angles are defined as

$\displaystyle c_{N}:=\frac{m_{N}}{\widetilde{m}_{N}},\quad
s_{N}:=\frac{\lambda_{n}v_{H}}{\widetilde{m}_{N}},\quad\tilde{c}_{L}:=\frac{\widetilde{m}_{L}}{\widetilde{M}_{L}}\quad\tilde{s}_{L}:=\frac{\lambda_{L}v_{\Phi}}{\widetilde{M}_{L}}.$
(52)

The dots in the first low are linear combinations of $y_{n}v_{H}$ and
$\lambda_{N}v_{\Phi}$, which are irrelevant for our discussion after
integrating out the right-handed neutrino $\nu_{R}$. The active neutrino mass
is given by $m_{D}^{2}/M_{R}$. The unitary matrices to diagonalize the
neutrino mass is given by

$\displaystyle U_{n_{L}}=V_{L}^{0}\begin{pmatrix}1&0&0\\\
0&c_{n_{L}}&-s_{n_{L}}\\\ 0&s_{n_{L}}&c_{n_{L}}\end{pmatrix},\quad
U_{n_{R}}=\begin{pmatrix}1&0&0\\\ 0&s_{n_{R}}&c_{n_{R}}\\\
0&c_{n_{R}}&-s_{n_{R}}\end{pmatrix},$ (53)

up to $v_{\Phi}/M_{R}\ll 1$, where

$\displaystyle\begin{pmatrix}s_{n_{R}}&c_{n_{R}}\\\
c_{n_{R}}&-s_{n_{R}}\end{pmatrix}\begin{pmatrix}0&\widetilde{m}_{N}\\\
\widetilde{M}_{L}&m_{n}\end{pmatrix}\begin{pmatrix}c_{n_{L}}&-s_{n_{L}}\\\
s_{n_{L}}&c_{n_{L}}\end{pmatrix}=\mathrm{diag}\left(m_{N_{1}},m_{N_{2}}\right)$
(54)

With the unitary matrix $U_{n_{L}}$, the SM muon neutrino coupling to the $Z$
boson is rescaled as $c_{L}^{2}+c_{N}^{2}s_{L}^{2}$. Therefore,
$s_{N}\sim\lambda_{n}v_{H}/m_{N}\ll 1$ is required for the SM-like $Z$ boson
coupling to muon neutrinos.

##  Appendix B Decay widths

We calculate the three-body decay, $F\to f_{3}V^{*}\to
f_{3}\overline{f}_{2}f_{1}$, of fermions via a off-shell vector boson $V$
whose couplings are given by,

$\displaystyle\mathcal{L}=V_{\mu}\overline{f}_{1}\gamma^{\mu}\left(g_{L}^{12}P_{L}+g_{R}^{12}P_{R}\right)f_{2}+V_{\mu}\overline{f}_{3}\gamma^{\mu}\left(g_{L}^{3F}P_{L}+g_{R}^{3F}P_{R}\right)F.$
(55)

The partial width is given by

$\displaystyle\ \Gamma\left(F\to
f_{3}\overline{f}_{2}f_{1}\right)=\frac{m_{F}}{768\pi^{3}}\left(\left|{g_{L}^{12}}\right|^{2}+\left|{g^{12}_{R}}\right|^{2}\right)$
(56) $\displaystyle\
\times\int^{(1-\sqrt{y})^{2}}_{0}dt\frac{\beta(t)}{(t-z)^{2}}\left[\left(\left|{g_{L}^{3F}}\right|^{2}+\left|{g_{R}^{3F}}\right|^{2}\right)\left\\{(1-y)^{2}+(1+y)t-2t^{2}\right\\}-12\mathrm{Re}\left(g_{L}^{3F}g_{R}^{3F*}\right)\sqrt{y}\;t\right],$

where $\beta(t)=\sqrt{t^{2}-2(1+y)t+(1-y)^{2}}$, $y=m_{3}^{2}/m_{F}^{2}$ and
$z=m_{V}^{2}/m_{F}^{2}$. 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 $f_{2}$.

## References

  * [1] CDF Collaboration, T. Aaltonen et al., High-precision measurement of the W boson mass with the CDF II detector, Science 376 (2022), no. 6589 170–176.
  * [2] Particle Data Group Collaboration, P. A. Zyla et al., Review of Particle Physics, PTEP 2020 (2020), no. 8 083C01.
  * [3] A. Strumia, Interpreting electroweak precision data including the $W$-mass CDF anomaly, arXiv:2204.04191.
  * [4] J. de Blas, M. Pierini, L. Reina, and L. Silvestrini, Impact of the recent measurements of the top-quark and W-boson masses on electroweak precision fits, arXiv:2204.04204.
  * [5] J. M. Yang and Y. Zhang, Low energy SUSY confronted with new measurements of W-boson mass and muon g-2, arXiv:2204.04202.
  * [6] G.-W. Yuan, L. Zu, L. Feng, and Y.-F. Cai, $W$-boson mass anomaly: probing the models of axion-like particle, dark photon and Chameleon dark energy, arXiv:2204.04183.
  * [7] P. Athron, A. Fowlie, C.-T. Lu, L. Wu, Y. Wu, and B. Zhu, The $W$ boson Mass and Muon $g-2$: Hadronic Uncertainties or New Physics?, arXiv:2204.03996.
  * [8] C.-T. Lu, L. Wu, Y. Wu, and B. Zhu, Electroweak Precision Fit and New Physics in light of $W$ Boson Mass, arXiv:2204.03796.
  * [9] Y.-Z. Fan, T.-P. Tang, Y.-L. S. Tsai, and L. Wu, Inert Higgs Dark Matter for New CDF W-boson Mass and Detection Prospects, arXiv:2204.03693.
  * [10] K. S. Babu, S. Jana, and V. P. K., Correlating $W$-Boson Mass Shift with Muon \boldmath${g-2}$ in the 2HDM, arXiv:2204.05303.
  * [11] J. J. Heckman, Extra $W$-Boson Mass from a D3-Brane, arXiv:2204.05302.
  * [12] J. Gu, Z. Liu, T. Ma, and J. Shu, Speculations on the W-Mass Measurement at CDF, arXiv:2204.05296.
  * [13] P. Athron, M. Bach, D. H. J. Jacob, W. Kotlarski, D. Stöckinger, and A. Voigt, Precise calculation of the W boson pole mass beyond the Standard Model with FlexibleSUSY, arXiv:2204.05285.
  * [14] L. Di Luzio, R. Gröber, and P. Paradisi, Higgs physics confronts the $M_{W}$ anomaly, arXiv:2204.05284.
  * [15] P. Asadi, C. Cesarotti, K. Fraser, S. Homiller, and A. Parikh, Oblique Lessons from the $W$ Mass Measurement at CDF II, arXiv:2204.05283.
  * [16] H. Bahl, J. Braathen, and G. Weiglein, New physics effects on the $W$-boson mass from a doublet extension of the SM Higgs sector, arXiv:2204.05269.
  * [17] A. Paul and M. Valli, Violation of custodial symmetry from W-boson mass measurements, arXiv:2204.05267.
  * [18] E. Bagnaschi, J. Ellis, M. Madigan, K. Mimasu, V. Sanz, and T. You, SMEFT Analysis of $m_{W}$, arXiv:2204.05260.
  * [19] Y. Cheng, X.-G. He, Z.-L. Huang, and M.-W. Li, Type-II Seesaw Triplet Scalar and Its VEV Effects on Neutrino Trident Scattering and W mass, arXiv:2204.05031.
  * [20] H. M. Lee and K. Yamashita, A Model of Vector-like Leptons for the Muon $g-2$ and the $W$ Boson Mass, arXiv:2204.05024.
  * [21] X. Liu, S.-Y. Guo, B. Zhu, and Y. Li, Unifying gravitational waves with $W$ boson, FIMP dark matter, and Majorana Seesaw mechanism, arXiv:2204.04834.
  * [22] J. Fan, L. Li, T. Liu, and K.-F. Lyu, $W$-Boson Mass, Electroweak Precision Tests and SMEFT, arXiv:2204.04805.
  * [23] K. Sakurai, F. Takahashi, and W. Yin, Singlet extensions and W boson mass in the light of the CDF II result, arXiv:2204.04770.
  * [24] R. Balkin, E. Madge, T. Menzo, G. Perez, Y. Soreq, and J. Zupan, On the implications of positive W mass shift, arXiv:2204.05992.
  * [25] T. Biekötter, S. Heinemeyer, and G. Weiglein, Excesses in the low-mass Higgs-boson search and the W-boson mass measurement, arXiv:2204.05975.
  * [26] M. Endo and S. Mishima, New physics interpretation of $W$-boson mass anomaly, arXiv:2204.05965.
  * [27] A. Crivellin, M. Kirk, T. Kitahara, and F. Mescia, Correlating $t\to cZ$ to the $W$ Mass and $B$ Physics with Vector-Like Quarks, arXiv:2204.05962.
  * [28] Y. Heo, D.-W. Jung, and J. S. Lee, Impact of the CDF $W$-mass anomaly on two Higgs doublet model, arXiv:2204.05728.
  * [29] X.-F. Han, F. Wang, L. Wang, J. M. Yang, and Y. Zhang, A joint explanation of W-mass and muon g-2 in 2HDM, arXiv:2204.06505.
  * [30] Y. H. Ahn, S. K. Kang, and R. Ramos, Implications of New CDF-II $W$ Boson Mass on Two Higgs Doublet Model, arXiv:2204.06485.
  * [31] H. Song, W. Su, and M. Zhang, Electroweak Phase Transition in 2HDM under Higgs, Z-pole, and W precision measurements, arXiv:2204.05085.
  * [32] M. Blennow, P. Coloma, E. Fernández-Martínez, and M. González-López, Right-handed neutrinos and the CDF II anomaly, arXiv:2204.04559.
  * [33] G. Cacciapaglia and F. Sannino, The W boson mass weighs in on the non-standard Higgs, arXiv:2204.04514.
  * [34] T.-P. Tang, M. Abdughani, L. Feng, Y.-L. S. Tsai, and Y.-Z. Fan, NMSSM neutralino dark matter for $W$-boson mass and muon $g-2$ and the promising prospect of direct detection, arXiv:2204.04356.
  * [35] C.-R. Zhu, M.-Y. Cui, Z.-Q. Xia, Z.-H. Yu, X. Huang, Q. Yuan, and Y. Z. Fan, GeV antiproton/gamma-ray excesses and the $W$-boson mass anomaly: three faces of $\sim 60-70$ GeV dark matter particle?, arXiv:2204.03767.
  * [36] M.-D. Zheng, F.-Z. Chen, and H.-H. Zhang, The $W\ell\nu$-vertex corrections to W-boson mass in the R-parity violating MSSM, arXiv:2204.06541.
  * [37] N. V. Krasnikov, Nonlocal generalization of the SM as an explanation of recent CDF result, arXiv:2204.06327.
  * [38] F. Arias-Aragón, E. Fernández-Martínez, M. González-López, and L. Merlo, Dynamical Minimal Flavour Violating Inverse Seesaw, arXiv:2204.04672.
  * [39] X. K. Du, Z. Li, F. Wang, and Y. K. Zhang, Explaining The New CDFII W-Boson Mass In The Georgi-Machacek Extension Models, arXiv:2204.05760.
  * [40] X. K. Du, Z. Li, F. Wang, and Y. K. Zhang, Explaining The Muon $g-2$ Anomaly and New CDF II W-Boson Mass in the Framework of (Extra)Ordinary Gauge Mediation, arXiv:2204.04286.
  * [41] J. Kawamura, S. Okawa, and Y. Omura, $W$ boson mass and muon $g-2$ in a lepton portal dark matter model, arXiv:2204.07022.
  * [42] S. Kanemura and K. Yagyu, Implication of the $W$ boson mass anomaly at CDF II in the Higgs triplet model with a mass difference, arXiv:2204.07511.
  * [43] K. I. Nagao, T. Nomura, and H. Okada, A model explaining the new CDF II W boson mass linking to muon $g-2$ and dark matter, arXiv:2204.07411.
  * [44] K.-Y. Zhang and W.-Z. Feng, Explaining $W$ boson mass anomaly and dark matter with a $U(1)$ dark sector, arXiv:2204.08067.
  * [45] L. M. Carpenter, T. Murphy, and M. J. Smylie, Changing patterns in electroweak precision with new color-charged states: Oblique corrections and the $W$ boson mass, arXiv:2204.08546.
  * [46] O. Popov and R. Srivastava, The Triplet Dirac Seesaw in the View of the Recent CDF-II W Mass Anomaly, arXiv:2204.08568.
  * [47] T. A. Chowdhury, J. Heeck, S. Saad, and A. Thapa, $W$ boson mass shift and muon magnetic moment in the Zee model, arXiv:2204.08390.
  * [48] D. Borah, S. Mahapatra, D. Nanda, and N. Sahu, Type II Dirac Seesaw with Observable $\Delta N_{eff}$ in the light of W-mass Anomaly, arXiv:2204.08266.
  * [49] Y.-P. Zeng, C. Cai, Y.-H. Su, and H.-H. Zhang, Extra boson mix with Z boson explaining the mass of W boson, arXiv:2204.09487.
  * [50] M. Du, Z. Liu, and P. Nath, CDF W mass anomaly from a dark sector with a Stueckelberg-Higgs portal, arXiv:2204.09024.
  * [51] K. Ghorbani and P. Ghorbani, $W$-Boson Mass Anomaly from Scale Invariant 2HDM, arXiv:2204.09001.
  * [52] A. Bhaskar, A. A. Madathil, T. Mandal, and S. Mitra, Combined explanation of $W$-mass, muon $g-2$, $R_{K^{(*)}}$ and $R_{D^{(*)}}$ anomalies in a singlet-triplet scalar leptoquark model, arXiv:2204.09031.
  * [53] S. Baek, Implications of CDF $W$-mass and $(g-2)_{\mu}$ on $U(1)_{L_{\mu}-L_{\tau}}$ model, arXiv:2204.09585.
  * [54] J. Cao, L. Meng, L. Shang, S. Wang, and B. Yang, Interpreting the $W$ mass anomaly in the vectorlike quark models, arXiv:2204.09477.
  * [55] D. Borah, S. Mahapatra, and N. Sahu, Singlet-Doublet Fermion Origin of Dark Matter, Neutrino Mass and W-Mass Anomaly, arXiv:2204.09671.
  * [56] A. Batra, S. K. A., S. Mandal, and R. Srivastava, W boson mass in Singlet-Triplet Scotogenic dark matter model, arXiv:2204.09376.
  * [57] S. Lee, K. Cheung, J. Kim, C.-T. Lu, and J. Song, Status of the two-Higgs-doublet model in light of the CDF $m_{W}$ measurement, arXiv:2204.10338.
  * [58] E. d. S. Almeida, A. Alves, O. J. P. Eboli, and M. C. Gonzalez-Garcia, Impact of CDF-II measurement of $M_{W}$ on the electroweak legacy of the LHC Run II, arXiv:2204.10130.
  * [59] Y. Cheng, X.-G. He, F. Huang, J. Sun, and Z.-P. Xing, Dark photon kinetic mixing effects for CDF W mass excess, arXiv:2204.10156.
  * [60] J. Heeck, W-boson mass in the triplet seesaw model, arXiv:2204.10274.
  * [61] H. Abouabid, A. Arhrib, R. Benbrik, M. Krab, and M. Ouchemhou, Is the new CDF $M_{W}$ measurement consistent with the two higgs doublet model?, arXiv:2204.12018.
  * [62] A. Batra, S. K. A, S. Mandal, H. Prajapati, and R. Srivastava, CDF-II $W$ Boson Mass Anomaly in the Canonical Scotogenic Neutrino-Dark Matter Model, arXiv:2204.11945.
  * [63] R. Benbrik, M. Boukidi, and B. Manaut, $W$-mass and 96 GeV excess in type-III 2HDM, arXiv:2204.11755.
  * [64] C. Cai, D. Qiu, Y.-L. Tang, Z.-H. Yu, and H.-H. Zhang, Corrections to electroweak precision observables from mixings of an exotic vector boson in light of the CDF $W$-mass anomaly, arXiv:2204.11570.
  * [65] Q. Zhou and X.-F. Han, The CDF W-mass, muon g-2, and dark matter in a $U(1)_{L_{\mu}-L_{\tau}}$ model with vector-like leptons, arXiv:2204.13027.
  * [66] R. S. Gupta, Running away from the T-parameter solution to the W mass anomaly, arXiv:2204.13690.
  * [67] J.-W. Wang, X.-J. Bi, P.-F. Yin, and Z.-H. Yu, Electroweak dark matter model accounting for the CDF $W$-mass anomaly, arXiv:2205.00783.
  * [68] B. Barman, A. Das, and S. Sengupta, New $W$-Boson mass in the light of doubly warped braneworld model, arXiv:2205.01699.
  * [69] J. Kim, S. Lee, P. Sanyal, and J. Song, CDF $m_{W}$ and the muon $g-2$ through the Higgs-phobic light pseudoscalar in type-X two-Higgs-doublet model, arXiv:2205.01701.
  * [70] J. Kim, Compatibility of muon $g-2$, $W$ mass anomaly in type-X 2HDM, arXiv:2205.01437.
  * [71] R. Dcruz and A. Thapa, $W$ boson mass, dark matter and $(g-2)_{\ell}$ in ScotoZee neutrino mass model, arXiv:2205.02217.
  * [72] J. Isaacson, Y. Fu, and C. P. Yuan, ResBos2 and the CDF W Mass Measurement, arXiv:2205.02788.
  * [73] T. A. Chowdhury and S. Saad, Leptoquark-vectorlike quark model for $m_{W}$ (CDF), $(g-2)_{\mu}$, $R_{K^{(\ast)}}$ anomalies and neutrino mass, arXiv:2205.03917.
  * [74] S.-S. Kim, H. M. Lee, A. G. Menkara, and K. Yamashita, The $SU(2)_{D}$ lepton portals for muon $g-2$, $W$ boson mass and dark matter, arXiv:2205.04016.
  * [75] J. Gao, D. Liu, and K. Xie, Understanding PDF uncertainty on the $W$ boson mass measurements in CT18 global analysis, arXiv:2205.03942.
  * [76] G. Lazarides, R. Maji, R. Roshan, and Q. Shafi, Heavier $W$-boson, dark matter and gravitational waves from strings in an SO(10) axion model, arXiv:2205.04824.
  * [77] J. Kawamura, S. Raby, and A. Trautner, Complete vectorlike fourth family and new U(1)’ for muon anomalies, Phys. Rev. D 100 (2019), no. 5 055030, [arXiv:1906.11297].
  * [78] J. Kawamura, S. Raby, and A. Trautner, Complete vectorlike fourth family with U(1)’ : A global analysis, Phys. Rev. D 101 (2020), no. 3 035026, [arXiv:1911.11075].
  * [79] B. Allanach, F. S. Queiroz, A. Strumia, and S. Sun, $Z^{\prime}$ models for the LHCb and $g-2$ muon anomalies, Phys. Rev. D93 (2016), no. 5 055045, [arXiv:1511.07447]. [Erratum: Phys. Rev.D95,no.11,119902(2017)].
  * [80] W. Altmannshofer, M. Carena, and A. Crivellin, $L_{\mu}-L_{\tau}$ theory of Higgs flavor violation and $(g-2)_{\mu}$, Phys. Rev. D94 (2016), no. 9 095026, [arXiv:1604.08221].
  * [81] E. Megias, M. Quiros, and L. Salas, $g_{\mu}-2$ from Vector-Like Leptons in Warped Space, JHEP 05 (2017) 016, [arXiv:1701.05072].
  * [82] S. Raby and A. Trautner, Vectorlike chiral fourth family to explain muon anomalies, Phys. Rev. D97 (2018), no. 9 095006, [arXiv:1712.09360].
  * [83] S. F. King, Flavourful $Z^{\prime}$ models for ${R}_{K^{\left(\ast\right)}}$, JHEP 08 (2017) 019, [arXiv:1706.06100].
  * [84] L. Darmé, K. Kowalska, L. Roszkowski, and E. M. Sessolo, Flavor anomalies and dark matter in SUSY with an extra U(1), JHEP 10 (2018) 052, [arXiv:1806.06036].
  * [85] A. Czarnecki and W. J. Marciano, The Muon anomalous magnetic moment: A Harbinger for ’new physics’, Phys. Rev. D64 (2001) 013014, [hep-ph/0102122].
  * [86] K. Kannike, M. Raidal, D. M. Straub, and A. Strumia, Anthropic solution to the magnetic muon anomaly: the charged see-saw, JHEP 02 (2012) 106, [arXiv:1111.2551]. [Erratum: JHEP10,136(2012)].
  * [87] R. Dermisek and A. Raval, Explanation of the Muon g-2 Anomaly with Vectorlike Leptons and its Implications for Higgs Decays, Phys. Rev. D88 (2013) 013017, [arXiv:1305.3522].
  * [88] Z. Poh and S. Raby, Vectorlike leptons: Muon g-2 anomaly, lepton flavor violation, Higgs boson decays, and lepton nonuniversality, Phys. Rev. D96 (2017), no. 1 015032, [arXiv:1705.07007].
  * [89] J. Kawamura, S. Okawa, and Y. Omura, Current status and muon $g-2$ explanation of lepton portal dark matter, JHEP 08 (2020) 042, [arXiv:2002.12534].
  * [90] Y. Bai and J. Berger, Muon g-2 in Lepton Portal Dark Matter, arXiv:2104.03301.
  * [91] Muon g-2 Collaboration, B. Abi et al., Measurement of the Positive Muon Anomalous Magnetic Moment to 0.46 ppm, Phys. Rev. Lett. 126 (2021) 141801, [arXiv:2104.03281].
  * [92] Muon g-2 Collaboration, G. W. Bennett et al., Measurement of the negative muon anomalous magnetic moment to 0.7 ppm, Phys. Rev. Lett. 92 (2004) 161802, [hep-ex/0401008].
  * [93] T. Aoyama et al., The anomalous magnetic moment of the muon in the Standard Model, Phys. Rept. 887 (2020) 1–166, [arXiv:2006.04822].
  * [94] T. Aoyama, M. Hayakawa, T. Kinoshita, and M. Nio, Complete Tenth-Order QED Contribution to the Muon $g-2$, Phys. Rev. Lett. 109 (2012) 111808, [arXiv:1205.5370].
  * [95] T. Aoyama, T. Kinoshita, and M. Nio, Theory of the Anomalous Magnetic Moment of the Electron, Atoms 7 (2019), no. 1 28.
  * [96] A. Czarnecki, W. J. Marciano, and A. Vainshtein, Refinements in electroweak contributions to the muon anomalous magnetic moment, Phys. Rev. D67 (2003) 073006, [hep-ph/0212229]. [Erratum: Phys. Rev. D73, 119901 (2006)].
  * [97] C. Gnendiger, D. Stöckinger, and H. Stöckinger-Kim, The electroweak contributions to $(g-2)_{\mu}$ after the Higgs boson mass measurement, Phys. Rev. D88 (2013) 053005, [arXiv:1306.5546].
  * [98] M. Davier, A. Hoecker, B. Malaescu, and Z. Zhang, Reevaluation of the hadronic vacuum polarisation contributions to the Standard Model predictions of the muon $g-2$ and ${\alpha(m_{Z}^{2})}$ using newest hadronic cross-section data, Eur. Phys. J. C77 (2017), no. 12 827, [arXiv:1706.09436].
  * [99] A. Keshavarzi, D. Nomura, and T. Teubner, Muon $g-2$ and $\alpha(M_{Z}^{2})$: a new data-based analysis, Phys. Rev. D97 (2018), no. 11 114025, [arXiv:1802.02995].
  * [100] G. Colangelo, M. Hoferichter, and P. Stoffer, Two-pion contribution to hadronic vacuum polarization, JHEP 02 (2019) 006, [arXiv:1810.00007].
  * [101] M. Hoferichter, B.-L. Hoid, and B. Kubis, Three-pion contribution to hadronic vacuum polarization, JHEP 08 (2019) 137, [arXiv:1907.01556].
  * [102] M. Davier, A. Hoecker, B. Malaescu, and Z. Zhang, A new evaluation of the hadronic vacuum polarisation contributions to the muon anomalous magnetic moment and to $\mathbf{\bm{\alpha}(m_{Z}^{2})}$, Eur. Phys. J. C 80 (2020), no. 3 241, [arXiv:1908.00921].
  * [103] A. Keshavarzi, D. Nomura, and T. Teubner, $g-2$ of charged leptons, $\alpha(M^{2}_{Z})$ , and the hyperfine splitting of muonium, Phys. Rev. D 101 (2020), no. 1 014029, [arXiv:1911.00367].
  * [104] A. Kurz, T. Liu, P. Marquard, and M. Steinhauser, Hadronic contribution to the muon anomalous magnetic moment to next-to-next-to-leading order, Phys. Lett. B734 (2014) 144–147, [arXiv:1403.6400].
  * [105] K. Melnikov and A. Vainshtein, Hadronic light-by-light scattering contribution to the muon anomalous magnetic moment revisited, Phys. Rev. D70 (2004) 113006, [hep-ph/0312226].
  * [106] P. Masjuan and P. Sánchez-Puertas, Pseudoscalar-pole contribution to the $(g_{\mu}-2)$: a rational approach, Phys. Rev. D95 (2017), no. 5 054026, [arXiv:1701.05829].
  * [107] G. Colangelo, M. Hoferichter, M. Procura, and P. Stoffer, Dispersion relation for hadronic light-by-light scattering: two-pion contributions, JHEP 04 (2017) 161, [arXiv:1702.07347].
  * [108] M. Hoferichter, B.-L. Hoid, B. Kubis, S. Leupold, and S. P. Schneider, Dispersion relation for hadronic light-by-light scattering: pion pole, JHEP 10 (2018) 141, [arXiv:1808.04823].
  * [109] A. Gérardin, H. B. Meyer, and A. Nyffeler, Lattice calculation of the pion transition form factor with $N_{f}=2+1$ Wilson quarks, Phys. Rev. D100 (2019), no. 3 034520, [arXiv:1903.09471].
  * [110] J. Bijnens, N. Hermansson-Truedsson, and A. Rodríguez-Sánchez, Short-distance constraints for the HLbL contribution to the muon anomalous magnetic moment, Phys. Lett. B798 (2019) 134994, [arXiv:1908.03331].
  * [111] G. Colangelo, F. Hagelstein, M. Hoferichter, L. Laub, and P. Stoffer, Longitudinal short-distance constraints for the hadronic light-by-light contribution to $(g-2)_{\mu}$ with large-$N_{c}$ Regge models, JHEP 03 (2020) 101, [arXiv:1910.13432].
  * [112] T. Blum, N. Christ, M. Hayakawa, T. Izubuchi, L. Jin, C. Jung, and C. Lehner, The hadronic light-by-light scattering contribution to the muon anomalous magnetic moment from lattice QCD, Phys. Rev. Lett. 124 (2020), no. 13 132002, [arXiv:1911.08123].
  * [113] LHCb Collaboration, R. Aaij et al., Test of lepton universality using $B^{+}\rightarrow K^{+}\ell^{+}\ell^{-}$ decays, Phys. Rev. Lett. 113 (2014) 151601, [arXiv:1406.6482].
  * [114] LHCb Collaboration, R. Aaij et al., Test of lepton universality with $B^{0}\rightarrow K^{*0}\ell^{+}\ell^{-}$ decays, JHEP 08 (2017) 055, [arXiv:1705.05802].
  * [115] LHCb Collaboration, R. Aaij et al., Search for lepton-universality violation in $B^{+}\to K^{+}\ell^{+}\ell^{-}$ decays, Phys. Rev. Lett. 122 (2019), no. 19 191801, [arXiv:1903.09252].
  * [116] Belle Collaboration, A. Abdesselam et al., Test of lepton flavor universality in ${B\to K^{\ast}\ell^{+}\ell^{-}}$ decays at Belle, arXiv:1904.02440.
  * [117] LHCb Collaboration, R. Aaij et al., Differential branching fraction and angular analysis of the decay $B_{s}^{0}\to\phi\mu^{+}\mu^{-}$, JHEP 07 (2013) 084, [arXiv:1305.2168].
  * [118] BaBar Collaboration, J. P. Lees et al., Measurement of the $B\to X_{s}l^{+}l^{-}$ branching fraction and search for direct CP violation from a sum of exclusive final states, Phys. Rev. Lett. 112 (2014) 211802, [arXiv:1312.5364].
  * [119] LHCb Collaboration, R. Aaij et al., Differential branching fractions and isospin asymmetries of $B\to K^{(*)}\mu^{+}\mu^{-}$ decays, JHEP 06 (2014) 133, [arXiv:1403.8044].
  * [120] LHCb Collaboration, R. Aaij et al., Angular analysis and differential branching fraction of the decay $B^{0}_{s}\to\phi\mu^{+}\mu^{-}$, JHEP 09 (2015) 179, [arXiv:1506.08777].
  * [121] LHCb Collaboration, R. Aaij et al., Measurement of Form-Factor-Independent Observables in the Decay $B^{0}\to K^{*0}\mu^{+}\mu^{-}$, Phys. Rev. Lett. 111 (2013) 191801, [arXiv:1308.1707].
  * [122] CMS Collaboration, V. Khachatryan et al., Angular analysis of the decay $B^{0}\to K^{*0}\mu^{+}\mu^{-}$ from pp collisions at $\sqrt{s}=8$ TeV, Phys. Lett. B753 (2016) 424–448, [arXiv:1507.08126].
  * [123] LHCb Collaboration, R. Aaij et al., Angular analysis of the $B^{0}\to K^{*0}\mu^{+}\mu^{-}$ decay using 3 fb-1 of integrated luminosity, JHEP 02 (2016) 104, [arXiv:1512.04442].
  * [124] Belle Collaboration, A. Abdesselam et al., Angular analysis of $B^{0}\to K^{\ast}(892)^{0}\ell^{+}\ell^{-}$, in Proceedings, LHCSki 2016 - A First Discussion of 13 TeV Results: Obergurgl, Austria, April 10-15, 2016, 2016\.  arXiv:1604.04042.
  * [125] Belle Collaboration, S. Wehle et al., Lepton-Flavor-Dependent Angular Analysis of $B\to K^{\ast}\ell^{+}\ell^{-}$, Phys. Rev. Lett. 118 (2017), no. 11 111801, [arXiv:1612.05014].
  * [126] ATLAS Collaboration, T. A. collaboration, Angular analysis of $B^{0}_{d}\to K^{*}\mu^{+}\mu^{-}$ decays in $pp$ collisions at $\sqrt{s}=8$ TeV with the ATLAS detector, .
  * [127] CMS Collaboration, C. Collaboration, Measurement of the $P_{1}$ and $P_{5}^{\prime}$ angular parameters of the decay $\mathrm{B}^{0}\to\mathrm{K}^{*0}\mu^{+}\mu^{-}$ in proton-proton collisions at $\sqrt{s}=8~{}\mathrm{TeV}$, .
  * [128] LHCb Collaboration, R. Aaij et al., Angular analysis of the $B^{+}\rightarrow K^{\ast+}\mu^{+}\mu^{-}$ decay, arXiv:2012.13241.
  * [129] LHCb Collaboration, R. Aaij et al., Test of lepton universality in beauty-quark decays, arXiv:2103.11769.
  * [130] M. E. Peskin and T. Takeuchi, A New constraint on a strongly interacting Higgs sector, Phys. Rev. Lett. 65 (1990) 964–967.
  * [131] M. E. Peskin and T. Takeuchi, Estimation of oblique electroweak corrections, Phys. Rev. D 46 (1992) 381–409.
  * [132] L. Lavoura and J. P. Silva, The Oblique corrections from vector - like singlet and doublet quarks, Phys. Rev. D 47 (1993) 2046–2057.
  * [133] I. Maksymyk, C. P. Burgess, and D. London, Beyond S, T and U, Phys. Rev. D 50 (1994) 529–535, [hep-ph/9306267].
  * [134] W. Grimus, L. Lavoura, O. M. Ogreid, and P. Osland, The Oblique parameters in multi-Higgs-doublet models, Nucl. Phys. B 801 (2008) 81–96, [arXiv:0802.4353].
  * [135] F. Jegerlehner and A. Nyffeler, The Muon g-2, Phys. Rept. 477 (2009) 1–110, [arXiv:0902.3360].
  * [136] A. Djouadi, The Anatomy of electro-weak symmetry breaking. I: The Higgs boson in the standard model, Phys. Rept. 457 (2008) 1–216, [hep-ph/0503172].
  * [137] W. Altmannshofer, S. Gori, M. Pospelov, and I. Yavin, Quark flavor transitions in $L_{\mu}-L_{\tau}$ models, Phys. Rev. D89 (2014) 095033, [arXiv:1403.1269].
  * [138] W. Altmannshofer, S. Gori, M. Pospelov, and I. Yavin, Neutrino Trident Production: A Powerful Probe of New Physics with Neutrino Beams, Phys. Rev. Lett. 113 (2014) 091801, [arXiv:1406.2332].
  * [139] G. Magill and R. Plestid, Neutrino Trident Production at the Intensity Frontier, Phys. Rev. D95 (2017), no. 7 073004, [arXiv:1612.05642].
  * [140] S.-F. Ge, M. Lindner, and W. Rodejohann, Atmospheric Trident Production for Probing New Physics, Phys. Lett. B772 (2017) 164–168, [arXiv:1702.02617].
  * [141] P. Ballett, M. Hostert, S. Pascoli, Y. F. Perez-Gonzalez, Z. Tabrizi, and R. Zukanovich Funchal, Neutrino Trident Scattering at Near Detectors, JHEP 01 (2019) 119, [arXiv:1807.10973].
  * [142] W. Altmannshofer, S. Gori, J. Martín-Albo, A. Sousa, and M. Wallbank, Neutrino Tridents at DUNE, arXiv:1902.06765.
  * [143] B. Zhou and J. F. Beacom, Neutrino-nucleus cross sections for W-boson and trident production, arXiv:1910.08090.
  * [144] J. Aebischer, W. Altmannshofer, D. Guadagnoli, M. Reboud, P. Stangl, and D. M. Straub, $B$-decay discrepancies after Moriond 2019, arXiv:1903.10434.
  * [145] M. Algueró, B. Capdevila, A. Crivellin, S. Descotes-Genon, P. Masjuan, J. Matias, M. Novoa Brunet, and J. Virto, Emerging patterns of New Physics with and without Lepton Flavour Universal contributions, Eur. Phys. J. C 79 (2019), no. 8 714, [arXiv:1903.09578]. [Addendum: Eur.Phys.J.C 80, 511 (2020)].
  * [146] A. K. Alok, A. Dighe, S. Gangal, and D. Kumar, Continuing search for new physics in $b\to s\mu\mu$ decays: two operators at a time, arXiv:1903.09617.
  * [147] M. Ciuchini, A. M. Coutinho, M. Fedele, E. Franco, A. Paul, L. Silvestrini, and M. Valli, New Physics in $b\to s\ell^{+}\ell^{-}$ confronts new data on Lepton Universality, arXiv:1903.09632.
  * [148] A. Datta, J. Kumar, and D. London, The $B$ Anomalies and New Physics in $b\to se^{+}e^{-}$, arXiv:1903.10086.
  * [149] K. Kowalska, D. Kumar, and E. M. Sessolo, Implications for New Physics in $b\to s\mu\mu$ transitions after recent measurements by Belle and LHCb, arXiv:1903.10932.
  * [150] A. Arbey, T. Hurth, F. Mahmoudi, D. Martinez Santos, and S. Neshatpour, Update on the b$\to$s anomalies, arXiv:1904.08399.
  * [151] ATLAS Collaboration, G. Aad et al., Search for high-mass dilepton resonances using 139 fb-1 of $pp$ collision data collected at $\sqrt{s}=$13 TeV with the ATLAS detector, Phys. Lett. B 796 (2019) 68–87, [arXiv:1903.06248].
  * [152] R. Dermisek, J. Kawamura, E. Lunghi, N. McGinnis, and S. Shin, Leptonic cascade decays of a heavy Higgs boson through vectorlike leptons at the LHC, arXiv:2204.13272.
  * [153] CMS Collaboration, A. Tumasyan et al., Inclusive nonresonant multilepton probes of new phenomena at $\sqrt{s}$ = 13 TeV, arXiv:2202.08676.
  * [154] CMS Collaboration, A. M. Sirunyan et al., Search for physics beyond the standard model in multilepton final states in proton-proton collisions at $\sqrt{s}=$ 13 TeV, JHEP 03 (2020) 051, [arXiv:1911.04968].
  * [155] R. Dermisek, J. P. Hall, E. Lunghi, and S. Shin, Limits on Vectorlike Leptons from Searches for Anomalous Production of Multi-Lepton Events, JHEP 12 (2014) 013, [arXiv:1408.3123].
  * [156] Search for New Physics in Events with Three Charged Leptons with the ATLAS detector, tech. rep., CERN, Geneva, Jul, 2013.  All figures including auxiliary figures are available at https://atlas.web.cern.ch/Atlas/GROUPS/PHYSICS/CONFNOTES/ATLAS-CONF-2013-070.
  * [157] J. Kawamura and S. Raby, Signal of four muons or more from a vector-like lepton decaying to a muon-philic Z’ boson at the LHC, Phys. Rev. D 104 (2021), no. 3 035007, [arXiv:2104.04461].
  * [158] ATLAS Collaboration, G. Aad et al., Search for supersymmetry in events with four or more charged leptons in 139 fb-1 of $\sqrt{s}$ = 13 TeV pp collisions with the ATLAS detector, JHEP 07 (2021) 167, [arXiv:2103.11684].
  * [159] ATLAS Collaboration, M. Aaboud et al., Search for four-top-quark production in the single-lepton and opposite-sign dilepton final states in pp collisions at $\sqrt{s}$ = 13 TeV with the ATLAS detector, Phys. Rev. D 99 (2019), no. 5 052009, [arXiv:1811.02305].