#  $W$ boson mass shift and muon magnetic moment in the Zee model

Talal Ahmed Chowdhury<EMAIL_ADDRESS>Department of Physics, University of
Dhaka, P.O. Box 1000, Dhaka, Bangladesh  The Abdus Salam International Centre
for Theoretical Physics, Strada Costiera 11, I-34014, Trieste, Italy  Julian
Heeck<EMAIL_ADDRESS>Department of Physics, University of Virginia,
Charlottesville, Virginia 22904-4714, USA  Shaikh Saad<EMAIL_ADDRESS>Department of Physics, University of Basel, Klingelbergstrasse 82, CH-4056
Basel, Switzerland  Anil Thapa<EMAIL_ADDRESS>Department of Physics,
University of Virginia, Charlottesville, Virginia 22904-4714, USA

###### Abstract

The CDF collaboration at Fermilab has recently reported a new precision
measurement of the $W$ boson mass showing a substantial $7\sigma$ deviation
from the Standard Model prediction. Moreover, Fermilab has recently confirmed
the longstanding tension in the $(g-2)_{\mu}$ measurement. We propose a
unified solution to these deviations within the simplest radiative neutrino
mass model: the Zee model. Our analysis establishes non-trivial links between
the origin of neutrino mass, the $(g-2)_{\mu}$ anomaly, and the $W$ boson mass
shift while being consistent with lepton flavor violation and all other
experimental constraints. We find that the mass spectrum of the physical
scalars must be hierarchical to be consistent with the $W$ boson mass shift;
remarkably, this is also the key to resolving the $(g-2)_{\mu}$ tension.
Furthermore, this mass splitting offers a unique same-sign dimuon signal
through which our model can be tested at the LHC.

##  I Introduction

The $W$ boson mass, $M_{W}$, is a precisely measured quantity, and even a
slight deviation from the predicted value would hint towards physics beyond
the Standard Model (SM). In the SM electroweak (EW) fit, $M_{W}$ is predicted
from the following relation Awramik:2003rn :

$\displaystyle M_{W}^{2}|_{\rm
fit}=\frac{M_{Z}^{2}}{2}\left[1+\left(1-\frac{\sqrt{8}\pi\alpha_{\text{EM}}(1-\Delta
r)}{G_{F}M_{Z}^{2}}\right)^{\frac{1}{2}}\right],$ (1)

involving several precisely measured quantities: the fine structure constant
$\alpha_{\text{EM}}$, the Fermi constant $G_{F}$, and the mass of the $Z$
boson $M_{Z}$. $\Delta r=\Delta r(M_{W},M_{Z},m_{t},\ldots)$ depends on the
specific particle content of the model and vanishes to leading order. The SM
prediction for $\Delta r$ includes complete one- and two-loop contributions
(and partial higher-order corrections), leading to

$\displaystyle M_{W}^{\rm SM}=(80.357\pm 0.004)\;\textrm{GeV},$ (2)

where the associated uncertainty originates from unknown higher-order
corrections Awramik:2003rn . This prediction is in agreement with the most up-
to-date PDG value ParticleDataGroup:2020ssz

$\displaystyle M_{W}^{\rm PDG}$ $\displaystyle=(80.379\pm
0.012)\;\textrm{GeV},$ (3)

at the $2\sigma$ confidence level.

Recently, the CDF collaboration at Fermilab has reported CDF:2022hxs  a new
precision measurement of $M_{W}$ using their full $8.8$ fb-1 data set,
yielding

$\displaystyle M_{W}^{\textrm{CDF-2022}}=(80.4335\pm 0.0094)\;\textrm{GeV},$
(4)

which _deviates_ from the SM prediction by $7\sigma$, clearly indicating the
presence of new physics (NP).

Another important observable in the SM that has been measured in experiments
with unprecedented accuracy, the muon’s anomalous magnetic moment (AMM), also
shows a large deviation from the SM prediction. This longstanding discrepancy
in the muon AMM measured at BNL in 2006 Bennett:2006fi  has recently been
confirmed by a new measurement performed at Fermilab Abi:2021gix . Combined,
they correspond to a large $4.2\sigma$ disagreement with the SM prediction
Aoyama:2020ynm  that requires a NP contribution

$\displaystyle\Delta a_{\mu}^{\textrm{NP}}=(2.51\pm 0.59)\times 10^{-9}.$ (5)

In addition to these two recent anomalies, one of the major shortcomings of
the SM is its prediction of massless neutrinos. Several experiments have
firmly established non-zero values of neutrino masses
ParticleDataGroup:2020ssz , although the origin of the neutrino mass is still
unknown.

This work proposes a simultaneous explanation of all the puzzles mentioned
above within the Zee model Zee:1980ai ; Zee:1985id  – the simplest radiative
neutrino mass model. We show that a consistent explanation of the CDF $W$
boson mass measurement rules out a degenerate mass spectrum of the NP scalars.
It rather dictates a mass hierarchy among the scalars, which is also crucial
to resolving the longstanding tension in the $(g-2)_{\mu}$. Another
consequence of this mass splitting is that it gives rise to a same-sign dimuon
signal – a novel feature of this model that can be uniquely probed at the LHC.
By exploring the parameter space of the proposed model, we illustrate how to
address these puzzles while being consistent with all current experimental
data: collider bounds, charged lepton flavor violation (cLFV), EW precision
observables, and low-energy measurements.

##  II Model

The simplest radiative neutrino mass model, the Zee model Zee:1980ai ;
Zee:1985id , utilizes two Higgs doublets $H_{1,2}$. Nonzero neutrino masses
arise at the one-loop level when a singly charged scalar $\eta^{+}$ is
introduced to this two-Higgs-doublet model framework. In the Higgs basis
(scalar potential given in Appendix-I), only one neutral Higgs acquires a
nonzero vacuum expectation value, and the doublets are parameterized as,

$\displaystyle H_{1}=\begin{pmatrix}G^{+}\\\
\frac{v+\phi_{1}^{0}+iG^{0}}{\sqrt{2}}\end{pmatrix},$ $\displaystyle
H_{2}=\begin{pmatrix}H^{+}_{2}\\\
\frac{\phi_{2}^{0}+iA}{\sqrt{2}}\end{pmatrix},$ (6)

where $G^{+}$ and $G^{0}$ are the Goldstones, $H^{+}_{2}$ and
$\\{\phi_{1}^{0},\phi_{2}^{0},A\\}$ are the physical scalars. The vacuum
expectation value of $H_{1}$, $v\simeq 246$ GeV, governs the EW symmetry
breaking.

We work in the CP-conserving alignment limit Branco:2011iw , where the SM
Higgs $\phi_{1}^{0}\approx h$ decouples from the new CP-even Higgs
($\phi_{2}^{0}\approx H$), traditionally denoted by $\cos(\beta-\alpha)=0$,
$\beta-\alpha$ being the relevant mixing angle in two-Higgs-doublet models
defined as: $\sin
2(\alpha-\beta)=\frac{2v^{2}\lambda_{6}}{m^{2}_{H}-m^{2}_{h}}$.
Correspondingly, the masses of these states are given by,

$\displaystyle m^{2}_{h}$ $\displaystyle=\lambda_{1}v^{2},\ \ \ \
m^{2}_{H}=\mu^{2}_{22}+\frac{v^{2}}{2}(\lambda_{3}+\lambda_{4}+\lambda_{5}),$
(7) $\displaystyle m^{2}_{A}$ $\displaystyle=m^{2}_{H}-v^{2}\lambda_{5},\ \ \
\ m^{2}_{H^{+}_{2}}=m^{2}_{H}-\frac{v^{2}}{2}(\lambda_{4}+\lambda_{5}).$ (8)

The above relations show that these mass eigenstates can be made hierarchical,
which is the key to a simultaneous resolution of $W$ boson mass and
$(g-2)_{\mu}$.

For simplicity, we consider the second doublet $H_{2}$ to be leptophilic in
nature. The leptonic Yukawa part of the Lagrangian then reads

$\displaystyle-$ $\displaystyle\mathcal{L}_{Y}\supset
Y_{E}\bar{L}H_{1}\ell_{R}+Y\bar{L}H_{2}\ell_{R}+{\rm h.c.},$ (9)

suppressing flavor indices. In the chosen alignment limit, the Yukawa coupling
$Y_{E}$ associated to $H_{1}$ is responsible for generating masses of the
charged leptons, i.e.,
$Y_{E}=\sqrt{2}\;\mathrm{diag}(m_{e},m_{\mu},m_{\tau})/v\equiv\sqrt{2}m_{E}/v$,
while $Y$ determines neutrino observables as well as the $(g-2)_{\mu}$.

In addition to the Yukawa couplings given in Eq. (9), the Zee model contains
one more Yukawa interaction associated with the singly charged scalar
$\eta^{+}\sim(1,1,1)$,

$\displaystyle-\mathcal{L}_{Y}\supset f_{ij}L_{i}\varepsilon
L_{j}\eta^{+}+{\rm h.c.},$ (10)

where $f_{ij}$ is antisymmetric in flavor indices. Eqs. (9) and (10), together
with the following cubic term in the scalar potential (rest of the terms are
not relevant to our study),

$\displaystyle-V\supset\mu H_{1}\varepsilon H_{2}\eta^{-}+{\rm h.c.},$ (11)

break the lepton number by two units and lead to a Majorana neutrino mass
(left diagram in Fig. 1) matrix Zee:1980ai

$\displaystyle M_{\nu}$ $\displaystyle=\underbrace{\frac{\sin
2\omega}{16\pi^{2}}\ln\left(\frac{m^{2}_{h^{+}}}{m^{2}_{H^{+}}}\right)}_{\equiv
a_{0}}\left(fm_{E}Y-Y^{T}m_{E}f\right),$ (12)

where $\sin 2\omega=\sqrt{2}v\mu/(m^{2}_{h^{+}}-m^{2}_{H^{+}})$ gives the
mixing angle $\omega$ between the singly charged scalars, and $h^{+}$ and
$H^{+}$ represent the mass eigenstates.

(a) (b)

Figure 1: (a) Radiative neutrino mass diagram in the Zee Model. (b)
Corrections to $\Delta a_{\ell}$ in the Zee Model from the neutral scalars
$\phi=H,A$. The charged scalar contribution to $\Delta a_{\ell}$ is obtained
by replacing neutral (charged lepton) with charged Higgs (neutrino).

##  III Correction to $W$ boson mass

EW precision constraints can be expressed in terms of the oblique parameters
$S$, $T$, and $U$ Peskin:1990zt ; Peskin:1991sw . In the scenario we consider,
an extended Higgs sector, the NP effects enter mainly through gauge boson
self-energies and $U$ is suppressed compared to $S$ and $T$. We have
implemented the one-loop corrections to these oblique parameters for our model
(see Appendix-II). In order to get the correct neutrino mass scale, the mixing
angle $\omega$ between the two charged states is typically expected to be
small. As a result, the flavor eigenstate of the singly charged scalar can be
approximated by its mass eigenstate (however, while computing the neutrino
mass, we properly account for the mixing effects). In this limit, its effect
on the oblique parameters is typically small.

The $W$ boson mass can be calculated as a function of these oblique
parameters, given by Grimus:2008nb

$M_{W}^{2}=M^{2}_{W,\rm
SM}\left[1+\frac{\alpha_{em}\left(c_{W}^{2}T-\frac{1}{2}S+\frac{c_{W}^{2}-s_{W}^{2}}{4s_{W}^{2}}U\right)}{c_{W}^{2}-s_{W}^{2}}\right].$
(13)

Here $\theta_{W}$ represents the Weinberg angle. Since the EW parameters in
the SM are closely related, one expects some observables in the global EW fit
may suffer from new tensions once the new CDF data is taken into account. Ref.
Asadi:2022xiy  has performed a thorough global fit to relevant EW observables
including the new CDF data and quoted $2\sigma$ allowed ranges for the oblique
parameters that consistently explain the CDF result, which we incorporate in
our analysis. We find that the mass of the charged scalar $H^{+}$ must be
split from those of the neutral scalars $H$ and $A$ to explain the preferred
$S$, $T$, $U$ ranges and therefore the $W$ boson mass measurement
consistently, as illustrated in Fig. 2. For a light $H^{+}$, we require
$m_{A,H}>m_{H^{+}}$, while a heavy $H^{+}$ can also support the opposite
hierarchy, $m_{H^{+}}>m_{A,H}$. In order to explain the muon’s AMM, we are
interested in the region $m_{H}<m_{A}\lesssim m_{H^{+}}$. With this hierarchy,
the CDF measurement together with other electroweak data imposes a lower limit
on $m_{H}$ of around $170\,\mathrm{GeV}$; $m_{H}$ can be pushed to slightly
lower values away from the alignment limit $\cos(\beta-\alpha)=0$, but not
significantly. For demonstration purpose we assumed $\sin\omega=0$ in Fig. 2
since a tiny mixing angle $\omega$ is required to reproduce the correct
neutrino mass scale. The lower limit on $m_{H}$ has important consequences for
the resolution of $(g-2)_{\mu}$ within this model.

Figure 2: $2\sigma$ ranges for $S$ and $T$ from Ref. Asadi:2022xiy  translated
to masses in the aligned two-Higgs-doublet model. $\cos(\beta-\alpha)=0$
denotes the alignment limit in which $h$ and $H$ do not mix Branco:2011iw .

##  IV Muon’s Magnetic Moment

If the NP contribution of $(g-2)_{\mu}$ results from a single sizable coupling
(for example, $Y_{e\mu}$), then the neutral component must be lighter than the
charged scalar to provide the desired sign for $\Delta a_{\mu}$
Lindner:2016bgg . Similarly, if a chirally enhanced contribution dominates
(for example, if both $Y_{\mu\tau}$ and $Y_{\tau\mu}$ are sizable), the CP-
even and the CP-odd states need to have hierarchical spectrum to avoid exact
cancellations in the degenerate limit Hou:2021sfl . To satisfy all these non-
trivial requirements including the new CDF measurement as well as strong
experimental constraints on $H^{+}$, we choose $m_{H}<m_{A}<m_{H^{+}}$. As
shown in Fig. 2, a small value of $m_{H}$ and larger but non-degenerate values
for $m_{A}$ and $m_{H^{+}}$ are fully compatible with the new CDF result.
Then, choosing $m_{A}$ closer to $m_{H^{+}}$ allows for a maximum splitting
between $m_{H}$ and $m_{H^{+}}$.

In general, both the neutral and charged scalars in the model contribute to
$(g-2)_{\mu}$; see the right diagram in Fig. 1 (which is computed in Appendix-
III). However, in the aforementioned hierarchical limit, the dominant
contribution arises from $H$. Moreover, the contribution from the singly
charged scalar $\eta^{+}$ can be ignored for heavier mass ($\sim{\rm TeV}$)
and small mixing angle $\omega$. The relevant $Y$ entries to generate the
needed correction to $\Delta a_{\mu}$ are as follows:

$\Delta a_{\mu}\quad\Rightarrow\quad
Y=\begin{pmatrix}.&{\color[rgb]{1,0,0}Y_{e\mu}}&.\\\ {\color[rgb]{1,0,0}Y_{\mu
e}}&{\color[rgb]{0,0,1}Y_{\mu\mu}}&{\color[rgb]{0,.5,.5}Y_{\mu\tau}}\\\
.&{\color[rgb]{0,.5,.5}Y_{\tau\mu}}&.\end{pmatrix}.$ (14)

In this work, we investigate three different possibilities:
$\\{{\color[rgb]{1,0,0}Y_{\mu e},Y_{e\mu}}\\}$-,
$\\{{\color[rgb]{0,0,1}Y_{\mu\mu}}\\}$-, and
$\\{{\color[rgb]{0,.5,.5}Y_{\mu\tau},Y_{\tau\mu}}\\}$-dominated scenarios.
Note that only the third case is chirally enhanced due to the tau-mass flip
inside the loop. Textures differing from these three are typically disfavored
from cLFV and would require significant finetuning.

##  V Low-energy Constraints/ Collider

There are various low-energy constraints on the scalar mass and relevant
Yukawa couplings. The important constraints on the Yukawa couplings $Y$ and
the masses of the doublet Higgs are cLFV from radiative decay
$\ell_{i}\to\ell_{j}\gamma$ Lavoura:2003xp ; ParticleDataGroup:2020ssz ,
trilepton decays Cai:2017jrq , muonium-antimuonium oscillation Cvetic:2005gx ,
as well as various experimental constraints obtained from
$e^{+}e^{-}\to\mu^{+}\mu^{-}H$ searches at BABAR BaBar:2016sci  and LHC
CMS:2018yxg . Moreover, LEP Electroweak:2003ram  puts a strong constraint with
an upper limit of about 30 GeV Barman:2021xeq  on the scalar mass associated
with $Y_{e\mu}$ to explain $\Delta a_{\mu}$, obtained via searches
$e^{+}e^{-}\to e^{\pm}\mu^{\mp}(H\to e^{\pm}\mu^{\mp})$. There are also
constraints from cLFV and universality in $\ell_{i}\to\ell_{j}\bar{\nu}\nu$ on
$f_{ij}$ couplings, which are relatively relaxed compared to $Y_{ij}$ in our
scenario.

Figure 3: The allowed parameter space in the Yukawa coupling and mass plane
that satisfies $(g-2)_{\mu}$ at $1\sigma$ (green) and $2\sigma$ (orange) as
well as $W$ boson mass shift (cyan). cLFV constraints are shown in grey. The
pink (purple) shaded region are excluded from $e^{+}e^{-}\to\mu^{+}\mu^{-}H$
searches at BABAR BaBar:2016sci  and LHC CMS:2018yxg  assuming
BR$(H\to\mu\mu)=1\,(0.5)$ with the purple dotted line BR$(H\to\mu\mu)=1$. The
region above the black dash-dotted line shows the projected sensitivity reach
on $Y_{\mu\mu}$ from direct searches in the
$\mu^{+}\mu^{-}\to\mu^{+}\mu^{-}\left(H\to\mu^{+}\mu^{-}\right)$ channel at a
3 TeV Muon Collider assuming $\mathcal{L}=1~{}{\rm ab^{-1}}$ Barman:2021xeq .
The blue star denotes the benchmark fit for TX-II (the quoted values of
$m_{A}$ and $m_{H^{+}}$ correspond to the benchmark fit). The $5\sigma$
($3\sigma$) discovery prospect of $pp\to\mu^{\pm}\mu^{\pm}jj\not{E}_{T}$ at
the 14 TeV LHC with integrated luminosity $\mathcal{L}=1\,\mathrm{ab^{-1}}$ is
shown by dashed (dash-dotted) vertical line.

At a future muon collider (MuC), the coupling $Y_{\mu\mu}$ could be directly
accessed via searches in the
$\mu^{+}\mu^{-}\to\mu^{+}\mu^{-}(H\to\mu^{+}\mu^{-})$ channel. The sensitivity
of this channel at the projected MuC configuration of
$\sqrt{s}=1\,\mathrm{TeV}$ with the integrated luminosity
$L=1\,\mathrm{ab^{-1}}$ Delahaye:2019omf  is depicted in Fig. 3 (for a
detailed analysis see Ref. Barman:2021xeq ). There are no dedicated searches
for the neutral scalar fields decaying into the $\tau^{+}\mu^{-}$ sector by
ATLAS and CMS. Moreover, $H^{+}$ can be pair-produced via the Drell–Yan
process that decay into leptons $H^{\pm}\to\ell^{\pm}\nu$ ($\ell=e,\mu$). From
slepton searches, LHC puts a lower limit on the charged Higgs mass
$m_{H^{+}}\gtrsim 420$ GeV ATLAS:2019lff  for a $100\%$ branching ratio for
both $e^{+}e^{-}$ and $\mu^{+}\mu^{+}$ channels.

##  VI Numerical Analysis

The neutrino mass formula of Eq. (12) and the muon’s AMM share the same Yukawa
matrix $Y$; consequently, there are strong constraints on its elements from
cLFV. Furthermore, there is a non-trivial correlation between the $W$ boson
mass shift and the $(g-2)_{\mu}$ that favors a specific hierarchical structure
of the scalar bosons, as discussed above.

For our numerical analysis, we consider three benchmark scenarios, each
associated with a distinct texture for the $Y$ matrix, which has the following
forms:

$\begin{pmatrix}0&{\color[rgb]{1,0,0}y_{e\mu}}&0\\\ \ast&0&\ast\\\
0&\ast&\ast\end{pmatrix},\ \begin{pmatrix}\ast&0&\ast\\\
0&{\color[rgb]{0,0,1}y_{\mu\mu}}&0\\\ 0&\ast&0\end{pmatrix},\
\begin{pmatrix}0&\ast&0\\\ \ast&0&{\color[rgb]{0,.5,.5}y_{\mu\tau}}\\\
0&{\color[rgb]{0,.5,.5}y_{\tau\mu}}&\ast\end{pmatrix},$ (15)

denoted as TX-I, TX-II, and TX-III, respectively. The elements shown in color
are sizable and dominantly contribute to $(g-2)_{\mu}$; moreover, additional
non-zero entries represented with “$\ast$” are needed to incorporate neutrino
oscillation data. These textures are chosen so that we introduce a minimum
number of non-zero entries to simultaneously satisfy $(g-2)_{\mu}$ and
observables in the neutrino sector ($\Delta m_{21}^{2},\Delta
m_{31}^{2},\sin^{2}\theta_{13},\sin^{2}\theta_{23},\sin^{2}\theta_{12}$).

TX-I: For TX-I, we find fits to neutrino observables that satisfy all cLFV.
Addressing $(g-2)_{\mu}$ via $Y_{e\mu}$ requires $m_{H}<30\,\mathrm{GeV}$ to
be consistent with bounds from LEP. This upper bound on $m_{H}$ along with
experimental constraints on $m_{H^{+}}$, however, conflicts with oblique
parameters needed to satisfy the new $W$ boson mass from CDF. This rules out
the TX-I scenario.

TX-III: This texture is consistent with oblique parameters to satisfy $W$
boson mass but inconsistent with neutrino oscillation data due to excessive
cLFV. Any other combination of non-zero entries “$\ast$” leads to even
stronger constraints from $\mu\to e\gamma$.

TX-II: The only consistent solution of $(g-2)_{\mu}$ and $M_{W}$ we obtain is
for TX-II. A benchmark fit with $m_{H}=180\,\mathrm{GeV}$ for the normal
hierarchy gives:

$\displaystyle\theta_{12}=32.3^{\circ},\ \theta_{13}=8.70^{\circ},\
\theta_{23}=46.6^{\circ},\ $ $\displaystyle m_{1}=0.014\ {\rm eV},\
\delta_{CP}=241^{\circ}.$ (16)

The observables $\Delta m_{31}^{2}$ and $\Delta m_{21}^{2}$ are fitted to
their respective central values Esteban:2020cvm . Yukawa couplings associated
with these fits are presented in Tab. 1. Relevant experimental constraints and
allowed parameter space to satisfy $(g-2)_{\mu}$ for these textures are
illustrated in Fig. 3. $\tau\to 3\mu$ emerges as the strongest cLFV signal and
should be detectable in Belle II Banerjee:2022xuw  (see Fig. 3).

$y_{\mu\mu}$ | $y_{\tau\mu}\,(10^{-3})$ | $y_{e\tau}\,(10^{-5})$ | $y_{ee}\,(10^{-6})$ | $f_{e\mu}$ | $f_{e\tau}\,(10^{-1})$ | $f_{\mu\tau}\,(10^{-7})$ | $a_{0}\,(10^{-3})$
---|---|---|---|---|---|---|---
$0.26+0.011\,i$ | $-4.86+0.23\,i$ | $2.9-0.48\,i$ | $-5.06-2.9i$ | $0.081$ | $2.5+0.23\,i$ | $4.89$ | 2.27
Table 1: Benchmark fit for texture TX-II.

This model can be uniquely probed at the LHC through a novel same-sign dimuon
signature Aiko:2019mww ; Jana:2020pxx  $pp\to\mu^{\pm}\mu^{\pm}jj\not{E}_{T}$
via vector boson fusion. This process vanishes in the degenerate limit
$m_{H}=m_{A}$. The amplitude of this process is directly proportional to the
mass splitting between $H$ and $A$, which is crucial in realizing
$(g-2)_{\mu}$. By recasting the analysis of Aiko:2019mww  for our benchmark
scenario, the discovery prospect of $pp\to\mu^{\pm}\mu^{\pm}jj\not{E}_{T}$
signal at the 14 TeV LHC is shown in Fig. 3. Corresponding to this benchmark,
$m_{H^{+}}$ up to $\sim 490\;(400)$ GeV can be discovered at
$3\sigma\;(5\sigma$) CL with an integrated luminosity
$\mathcal{L}=1\,\mathrm{ab^{-1}}$. The Zee-model region preferred by neutrino
masses, CDF, and $(g-2)_{\mu}$ can therefore be conclusively probed in
complementary ways in the near future.

##  VII Conclusions

An exciting recent development in particle physics has been the measurement of
the $W$ boson mass by the CDF Collaboration, which shows a $7\sigma$
discrepancy with the SM prediction. On top of that, the longstanding
$(g-2)_{\mu}$ anomaly has recently been confirmed at Fermilab. These are clear
signs for physics beyond the SM. Here, we showed that a simultaneous unified
solution to these puzzles together with neutrino oscillation data can be
achieved within the Zee model, satisfying all current experimental constraints
such as collider bounds and lepton flavor violation. In this setup, lepton
flavor violation, $(g-2)_{\mu}$, and neutrino oscillations are inherently
linked, and our detailed numerical analysis finds consistency with only a
mostly muon-philic Yukawa texture that leads to testable rates for $\tau\to
3\mu$ at Belle II. Furthermore, our analysis shows that the mass spectrum of
the physical scalars must be hierarchical to be consistent with the $W$ boson
mass measured at the CDF experiment, which is also the key to solving the
$(g-2)_{\mu}$ tension. This mass splitting offers a unique same-sign dimuon
signal through which our proposed model can be tested at the LHC.

Note added: As we were completing this paper, several papers Fan:2022dck ;
Zhu:2022tpr ; Athron:2022qpo ; Du:2022pbp ; Yang:2022gvz ; deBlas:2022hdk ;
Tang:2022pxh ; Blennow:2022yfm ; Zhu:2022scj ; Sakurai:2022hwh ; Heo:2022dey ;
Cheung:2022zsb ; Lu:2022bgw ; Strumia:2022qkt ; Fan:2022yly ;
Cacciapaglia:2022xih ; Liu:2022jdq ; Lee:2022nqz ; Cheng:2022jyi ;
Song:2022xts ; Bagnaschi:2022whn ; Paul:2022dds ; Bahl:2022xzi ; Asadi:2022xiy
; DiLuzio:2022xns ; Athron:2022isz ; Gu:2022htv ; Babu:2022pdn ;
Crivellin:2022fdf ; Endo:2022kiw ; Han:2022juu ; Biekotter:2022abc ;
Balkin:2022glu ; Kawamura:2022uft ; Ghoshal:2022vzo ; Perez:2022uil ;
Nagao:2022oin ; Kanemura:2022ahw ; Heckman:2022the ; Ahn:2022xeq  appeared
that also discussed the impact of the new $M_{W}$ measurement on new physics
scenarios.

## Appendix-I

Two-Higgs-doublet model potential: Here we present the scalar potential of the
two-Higgs-doublet model in the Higgs basis Branco:2011iw ,

$\displaystyle
V(H_{1},H_{2})=\mu_{1}^{2}H_{1}^{\dagger}H_{1}+\mu_{2}^{2}H_{2}^{\dagger}H_{2}-\\{\mu_{12}^{2}H_{1}^{\dagger}H_{2}+{\rm
h.c.}\\}$
$\displaystyle\quad+\frac{\lambda_{1}}{2}(H_{1}^{\dagger}H_{1})^{2}+\frac{\lambda_{2}}{2}(H_{2}^{\dagger}H_{2})^{2}+\lambda_{3}(H_{1}^{\dagger}H_{1})(H_{2}^{\dagger}H_{2})$
$\displaystyle\quad+\lambda_{4}(H_{1}^{\dagger}H_{2})(H_{2}^{\dagger}H_{1})+\left\\{\frac{\lambda_{5}}{2}(H_{1}^{\dagger}H_{2})^{2}+{\rm
h.c.}\right\\}$
$\displaystyle\quad+\left\\{\big{[}\lambda_{6}(H_{1}^{\dagger}H_{1})+\lambda_{7}(H_{2}^{\dagger}H_{2})\big{]}H_{1}^{\dagger}H_{2}+{\rm
h.c.}\right\\}.$ (17)

## Appendix-II

Oblique parameters:

The $S$, $T$ and $U$ parameters for the Zee model in the alignment limit
Haber:2010bw  are given by Herrero-Garcia:2017xdu

$\displaystyle T$ $\displaystyle=\frac{1}{16\pi
s_{w}^{2}M_{W}^{2}}\left[c_{\omega}^{2}\left(F(m_{H^{+}}^{2},m_{H}^{2})+F(m_{H^{+}}^{2},m_{A}^{2})\right)+s_{\omega}^{2}\left(F(m_{h^{+}}^{2},m_{H}^{2})+F(m_{h^{+}}^{2},m_{A}^{2})\right)-\frac{1}{2}s_{\omega}^{2}c_{\omega}^{2}F(m_{h^{+}}^{2},m_{H^{+}}^{2})\right.$
$\displaystyle\quad\left.-F(m_{H}^{2},m_{A}^{2})\right],$ (18) $\displaystyle
S$ $\displaystyle=\frac{1}{\pi M_{Z}^{2}}\left[{\cal
B}_{22}(M_{Z}^{2},m_{H}^{2},m^{2}_{A})+\frac{c_{\omega}^{2}}{2}(c_{2\omega}-3){\cal
B}_{22}(M_{Z}^{2},m_{H^{+}}^{2},m^{2}_{H^{+}})-\frac{s_{\omega}^{2}}{2}(c_{2\omega}+3){\cal
B}_{22}(M_{Z}^{2},m_{h^{+}}^{2},m^{2}_{h^{+}})\right.$
$\displaystyle\quad\left.+2s_{\omega}^{2}c_{\omega}^{2}{\cal
B}_{22}(M_{Z}^{2},m_{H^{+}}^{2},m^{2}_{h^{+}})\right],$ (19) $\displaystyle U$
$\displaystyle=-S+\frac{1}{\pi M_{W}^{2}}\left[c_{\omega}^{2}\left({\cal
B}_{22}(M_{W}^{2},m_{H^{+}}^{2},m^{2}_{A})+{\cal
B}_{22}(M_{W}^{2},m_{H^{+}}^{2},m^{2}_{H})\right)+s_{\omega}^{2}\left({\cal
B}_{22}(M_{W}^{2},m_{h^{+}}^{2},m^{2}_{A})+{\cal
B}_{22}(M_{W}^{2},m_{h^{+}}^{2},m^{2}_{H})\right)\right.$
$\displaystyle\quad-\left.2c_{\omega}^{2}{\cal
B}_{22}(M_{W}^{2},m_{H^{+}}^{2},m^{2}_{H^{+}})-2s_{\omega}^{2}{\cal
B}_{22}(M_{W}^{2},m_{h^{+}}^{2},m^{2}_{h^{+}})\right],$ (20)

where the functions $F(x_{1},x_{2})$ and ${\cal B}_{22}(z,x_{1},x_{2})$ are
given in Herrero-Garcia:2017xdu ; He:2001tp , $s_{\omega}\equiv\sin\omega$,
and $c_{\omega}\equiv\cos\omega$.

## Appendix-III

New physics contributions to $(g-2)_{\mu}$: Neutral scalar contribution to
anomalous magnetic moment at one-loop Leveille:1977rc  as shown in Fig. 1 (b)
is

$\displaystyle\Delta
a_{\ell}^{H(A)}=\frac{m_{\ell}^{2}}{32\pi^{2}}\Big{(}\left\\{|Y_{\ell
i}|^{2}+|Y_{i\ell}|^{2}\right\\}G[m_{H(A)},1]$ $\displaystyle\pm
2\frac{m_{i}}{m_{\ell}}\Re(Y_{\ell i}Y_{i\ell})G[m_{H(A)},0]\Big{)},$ (21)

where,

$\displaystyle G(M,\varepsilon)=\int_{0}^{1}\frac{x^{2}-\varepsilon
x^{3}}{m_{\ell}^{2}x^{2}+(m_{i}^{2}-m_{\ell}^{2})x+M^{2}(1-x)}\,dx.$ (22)

The contribution from the charged scalar Higgs $H^{+}$ is obtained by
replacing the neutral field with charged in Fig. 1 (b) and attaching the
photon to the charged scalar, which reads

$\Delta
a_{\ell}^{H^{+}}=\frac{m_{\ell}^{2}}{16\pi^{2}}|Y_{i\ell}|^{2}\int_{0}^{1}\frac{x^{3}-x^{2}}{m_{\ell}^{2}x^{2}+(m_{H^{+}}^{2}-m_{\ell}^{2})x}\,dx.$
(23)

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