# Leptoquark-vectorlike quark model for the CDF $m_{W}$, $(g-2)_{\mu}$,
$R_{K^{(\ast)}}$ anomalies
and neutrino mass

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  Shaikh
Saad<EMAIL_ADDRESS>Department of Physics, University of Basel,
Klingelbergstrasse 82, CH-4056 Basel, Switzerland

###### Abstract

Very recently, a substantial $7\sigma$ deviation of the $W$-boson mass from
the Standard Model (SM) prediction has been reported by the CDF collaboration.
Furthermore, the Muon g-2 Experiment recently confirmed the longstanding
tension in $(g-2)_{\mu}$. Besides, the updated result from the LHCb
collaboration found evidence for the breaking of lepton universality in
beauty-quark decays, which shows a $3.1\sigma$ discrepancy and is consistent
with their previous measurements. Motivated by several of these drawbacks of
the SM, in this work, we propose a model consisting of two scalar leptoquarks
and a vectorlike quark to simultaneously address the $W$-boson mass shift, the
$(g-2)_{\mu}$, and anomalies in the neutral current transitions of the
$B$-meson decays. The proposed model also sheds light on the origin of
neutrino mass and can be fully tested at the future colliders.

##  I Introduction

The $W$-boson mass, $M_{W}$, is a precisely measured quantity, and even a
slight deviation from the predicted value would hint toward physics beyond the
Standard Model (SM). The SM predicts $M_{W}^{\rm SM}=(80.357\pm
0.004)\;\textrm{GeV}$, which agrees with the most up-to-date PDG value
$M_{W}^{\rm PDG}=(80.379\pm 0.012)\;\textrm{GeV}$ at the $2\sigma$ confidence
level ParticleDataGroup:2020ssz . Very recently, the CDF collaboration has
reported a new precision measurement of $M_{W}$ using their full $8.8$ fb-1
data set that yields CDF:2022hxs

$\displaystyle M_{W}^{\textrm{CDF-2022}}=(80.4335\pm 0.0094)\;\textrm{GeV},$
(1)

which deviates from the SM prediction by $7\sigma$, clearly indicating the
presence of new physics (NP) 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 ; Chowdhury:2022moc ; Zeng:2022lkk ; Du:2022fqv ; Ghorbani:2022vtv
; Bhaskar:2022vgk ; Baek:2022agi ; Cao:2022mif ; Borah:2022zim ; Batra:2022org
; Lee:2022gyf ; Cheng:2022aau ; Addazi:2022fbj ; Heeck:2022fvl ;
Abouabid:2022lpg ; Batra:2022pej ; Benbrik:2022dja ; Cai:2022cti ;
Zhou:2022cql ; Gupta:2022lrt ; Wang:2022dte ; Barman:2022qix ; Kim:2022hvh ;
Kim:2022xuo ; Dcruz:2022dao ; Isaacson:2022rts ; Botella:2022rte ; He:2021yck
; He:2022zjz ; Popov:2022ldh .

The muon’s anomalous magnetic moment (AMM) $\Delta a_{\mu}=(g-2)_{\mu}/2$, is
another quantity that has recently been measured with unprecedented accuracy
in the Muon g-2 experiment Abi:2021gix . The result from this experiment is in
complete agreement with the previously measured value at BNL Bennett:2006fi .
When these two results are combined, it shows a large $4.2\sigma$ discrepancy
compared to the SM prediction Aoyama:2020ynm  (for original works, see Refs.
Aoyama:2012wk ; Aoyama:2019ryr ; Czarnecki:2002nt ; Gnendiger:2013pva ;
Davier:2017zfy ; Keshavarzi:2018mgv ; Colangelo:2018mtw ; Hoferichter:2019mqg
; Davier:2019can ; Keshavarzi:2019abf ; Kurz:2014wya ; Melnikov:2003xd ;
Masjuan:2017tvw ; Colangelo:2017fiz ; Hoferichter:2018kwz ; Gerardin:2019vio ;
Bijnens:2019ghy ; Colangelo:2019uex ; Blum:2019ugy ; Colangelo:2014qya ):

$\displaystyle\Delta a_{\mu}=(2.51\pm 0.59)\times 10^{-9},$ (2)

hinting towards physics beyond the SM (BSM); for a recent review, see e.g.
Athron:2021iuf .

In addition, lepton flavor universality (LFU) violating $B$-meson decays have
been persistently observed in a series of experiments LHCb:2017avl ;
LHCb:2019hip ; Belle:2019oag ; BELLE:2019xld ; LHCb:2021trn . The most
noteworthy deviation is observed in neutral-current transitions associated
with the $R_{K}-R_{K^{*}}$ ratios, which are defined as:

$\displaystyle R_{K}=\frac{Br\left(B\to K\mu^{+}\mu^{-}\right)}{Br\left(B\to
Ke^{+}e^{-}\right)},\;\;\;R_{K^{\ast}}=\frac{Br\left(B\to
K^{\ast}\mu^{+}\mu^{-}\right)}{Br\left(B\to K^{\ast}e^{+}e^{-}\right)}.$ (3)

LFU in the SM predicts these ratios to be unity with uncertainties less than
$1\%$. However, the most precise measurement by LHCb LHCb:2021trn  finds a
deficit with a significance of $3.1\sigma$ for $R_{K}$-ratio. There are
several other related observables for which LHCb also found deficits with
respect to the SM prediction, which are of order $\mathcal{O}(1.5-3.5)\sigma$;
for a comprehensive list, see e.g. CAMALICH20221 . However, if only the
theoretically clean observables: $R_{K},R_{K^{*}}$ ratios and
$BR(B_{s}\to\mu^{+}\mu^{-})$ are taken into account, the data is found to be
in $4.2\sigma$ tension with the SM Geng:2021nhg , for a recent review, see
Angelescu:2021lln . On the other hand, when both theoretically clean and dirty
observables are considered, global analyses show preferences compared to the
SM hypothesis with pulls more than $7\sigma$ (see Altmannshofer:2021qrr ;
Geng:2021nhg ; Alguero:2021anc ; Hurth:2021nsi ; Isidori:2021vtc ;
Kowalska:2019ley ; Ciuchini:2021smi ; DAmico:2017mtc  for theoretical
assumptions and data included in these fits).

On top of these downsides mentioned above, neutrinos remain massless in the
SM. On the contrary, several experiments discovered non-zero masses of the
neutrinos via observations of neutrino oscillations Super-Kamiokande:1998kpq ;
Super-Kamiokande:2001ljr ; SNO:2002tuh ; KamLAND:2002uet ; KamLAND:2004mhv ;
K2K:2002icj ; MINOS:2006foh . This work proposes a simultaneous explanation of
the $W$-boson mass shift, the tension in the $(g-2)_{\mu}$, and the anomalies
in the neutral current transitions in the $B$-meson decays, as well as
neutrino oscillation data. The proposed model employs two scalar leptoquarks
(LQs Buchmuller:1986zs ; Dorsner:2016wpm ): $\widetilde{R}_{2}\sim(3,2,1/6)$
and $S_{3}\sim(\overline{3},3,1/3)$ and a vectorlike quark (VLQ)
$\psi\sim(3,2,-5/6)$. Non-zero mixing between $\widetilde{R}_{2}$ and $S_{3}$
LQs leads to loop corrections to $W$-boson self-energy explaining the CDF
anomaly. Utilizing this same mixing, the $(g-2)_{\mu}$ receives a large NP
contribution via the mass flip of the VLQ inside the loop. The $S_{3}$ LQ,
with its interactions with the SM fermions, addresses the discrepancies in the
rare decays of $B$-mesons based on the neutral current $b\to s\ell\ell$
transitions. Furthermore, non-zero mixing between $\widetilde{R}_{2}$ and
$S_{3}$ LQs is also responsible for generating neutrino mass at the one-loop
order, and the model put forward in this work can be tested in the ongoing and
future experiments.

##  II Proposal

In this work, we propose a new leptoquark-vectorlike quark model that contains
three BSM particles: (i) an iso-doublet LQ, $\widetilde{R}_{2}(3,2,1/6)$, (ii)
an iso-triplet LQ, $S_{3}(\overline{3},3,1/3)$, and (iii) an iso-doublet
vectorlike quark, $\psi(3,2,-5/6)$. Here, the quantum numbers are shown under
the SM gauge group $SU(3)\times SU(2)\times U(1)$. Furthermore, we assign a
baryon number of $1/3$ ($-1/3$) to $\psi,\widetilde{R}_{2}$ ($S_{3}$). The
corresponding component fields of these particles are defined in the following
way:

$\displaystyle\tau.S_{3}=\begin{pmatrix}S^{1/3}&\sqrt{2}S^{4/3}\\\
\sqrt{2}S^{-2/3}&-S^{1/3}\end{pmatrix},$ (4)
$\displaystyle\widetilde{R}_{2}=\begin{pmatrix}R^{2/3}\\\
R^{-1/3}\end{pmatrix},\;\;\psi=\begin{pmatrix}\psi^{-1/3}\\\
\psi^{-4/3}\end{pmatrix}.$ (5)

Yukawa sector:– The relevant part of the Yukawa Lagrangian is given by,

$\displaystyle\mathcal{L_{Y}}\supset$
$\displaystyle\;y^{S}_{i}\;\overline{Q^{c}_{L}}_{i}\epsilon\left(\tau.S_{3}\right)L_{L}+\hat{y}^{\psi}_{L}\;\overline{\psi}_{L}\widetilde{R}_{2}\ell_{R}+y^{R}\overline{d}_{R}L_{L}\epsilon\widetilde{R}_{2}$
$\displaystyle+\hat{y}^{\psi}_{R}\;\overline{L}_{L}\left(\tau.S_{3}\right)\psi_{R}+m_{\psi}\;\overline{\psi}_{L}\psi_{R}+h.c.$
(6)

One more term is allowed by the gauge symmetries:
$y^{\prime}\overline{\psi}_{L}d_{R}\epsilon H^{\ast}$. Once the EW symmetry is
broken, it generates a mixing between $\psi^{-1/3}$ and $d^{-1/3}$ via
$\overline{\psi}_{L}d_{R}v/\sqrt{2}$, for which we assume the Yukawa coupling
to be negligibly small. Therefore, the mass generation of the SM fermions
remain unaltered. Note that, baryon number assignments as described above
forbid two terms:
$\mathcal{L_{Y}}\not\supset\overline{\psi^{c}_{R}}u_{R}\widetilde{R}_{2}$ and
$\mathcal{L_{Y}}\not\supset\overline{Q^{c}_{L}}\epsilon\left(\tau_{3}.S_{3}\right)Q_{L}$.

First we focus on a muon-philic scenario, then the Yukawa couplings in Eq. (6)
take the following form:

$\displaystyle y^{S}=\begin{pmatrix}0&0&0\\\ 0&y^{S}_{s\mu}&0\\\
0&y^{S}_{b\mu}&0\end{pmatrix},\;\;\;\hat{y}^{\psi}_{L}=\begin{pmatrix}0\\\
y^{\psi}_{L}\\\ 0\end{pmatrix},\;\;\;\hat{y}^{\psi}_{R}=\begin{pmatrix}0\\\
y^{\psi}_{R}\\\ 0\end{pmatrix}.$ (7)

These forms of $\hat{y}^{\psi}_{L,R}$ are required to avoid excessive cLFV and
provide large NP contribution to $(g-2)_{\mu}$. The texture of $y^{S}$ is
chosen to explain the anomalies in the neutral current transitions, to be
discussed later in the text. For the simplicity of our work, we take all model
parameters to be real.

Scalar sector:– The relevant terms in the scalar potential take the following
form:

$\displaystyle V\supset
m^{2}_{R}\widetilde{R}_{2}^{\dagger}\widetilde{R}_{2}+m^{2}_{S}S^{\dagger}_{3}S_{3}+\left\\{\mu\;H^{\dagger}\left(\tau.S_{3}\right)\widetilde{R}_{2}+h.c.\right\\}.$
(8)

The cubic coupling $\mu$ leads to mixing between $\widetilde{R}_{2}$ and
$S_{3}$ components, which is crucial in addressing both the $W$-boson mass and
$(g-2)_{\mu}$ anomalies within the proposed model. Remarkably, the existence
of this cubic term also allows neutrinos to have non-zero masses at the one-
loop order. In this theory, $\mu$ is one of the most important parameters, and
in the limit $\mu\to 0$, one gets $\Delta m_{W}\to 0$, $\Delta a_{\mu}\to 0$,
as well as $m_{\nu}\to 0$. Note that, to reduce the number of parameters and
for the simplicity of our study, scalar quartic couplings are assumed to be
somewhat smaller and are not included in Eq.(8). Consequently, all mass
splittings are only a function of the trilinear coupling $\mu$.

From the above potential, the mass matrices in the $\\{S^{Q},R^{Q}\\}$ basis
are given by,

$\displaystyle M^{2}_{2/3}=\begin{pmatrix}m^{2}_{S}&\mu v\\\ \mu
v&m^{2}_{R}\end{pmatrix},\;\;\;M^{2}_{1/3}=\begin{pmatrix}m^{2}_{S}&-\mu
v/\sqrt{2}\\\ -\mu v/\sqrt{2}&m^{2}_{R}\end{pmatrix},$ (9)

with $v=246$ GeV. We denote the weak and mass eigenstates with $X$ and
$\hat{X}$, respectively, which are related by,

$\displaystyle S^{\pm Q}=c_{x}\hat{S}^{\pm Q}-s_{x}\hat{R}^{\pm Q},$ (10)
$\displaystyle R^{\pm Q}=s_{x}\hat{S}^{\pm Q}+c_{x}\hat{R}^{\pm Q},$ (11)

with $x=\theta,\phi$ for $Q=2/3,1/3$. Masses and mixing for $\hat{X}^{Q}$
states then take the form,

$\displaystyle
m^{2}_{\hat{S}_{Q},\hat{R}_{Q}}=\frac{1}{2}\bigg{\\{}m^{2}_{S}+m^{2}_{R}\pm\left[(m^{2}_{S}-m^{2}_{R})^{2}+a_{Q}\mu^{2}v^{2}\right]^{1/2}\bigg{\\}},$
(12) $\displaystyle\sin 2x=\frac{b_{Q}\mu
v}{m^{2}_{\hat{S}_{Q}}-m^{2}_{\hat{R}_{Q}}},$ (13)

where, $a_{Q}(b_{Q})=4(2)$ and $2(-\sqrt{2})$ for $Q=2/3$ and $1/3$,
respectively. In the above analysis, we have adopted the convention of
$m_{S}>m_{R}$.

$W$-boson mass shift:– The effects of NP phenomena on the electroweak (EW)
gauge sector are parameterized in terms of oblique parameters Peskin:1990zt ;
Peskin:1991sw  $S$, $T$, and $U$. Then, the shift in the $W$-boson mass from
the NP can be calculated as a function of these oblique parameters
Grimus:2008nb ,

$\displaystyle m_{W}^{2}=m^{2}_{W,\rm
SM}\bigg{\\{}1+\frac{\alpha_{em}\left(c_{w}^{2}T-\frac{1}{2}S+\frac{c_{w}^{2}-s_{w}^{2}}{4s_{w}^{2}}U\right)}{c_{w}^{2}-s_{w}^{2}}\bigg{\\}}.$
(14)

When the new CDF data is taken into account in a global electroweak precision
fit, the oblique parameters would deviate from their previous (PDG) SM
predictions, which several studies have already analyzed Lu:2022bgw ;
Asadi:2022xiy ; Bagnaschi:2022whn , and updated the $2\sigma$ allowed ranges
for $S$, $T$, and $U$ parameters in light of the CDF result. By incorporating
these new sets of values of oblique parameters in our numerical analysis, we
find that for our model with TeV scale LQs, the mass splitting of the mixed LQ
states must be of order $\Delta m_{LQ}\sim\mathcal{O}(100)$ GeV to be
compatible with the result reported by CDF collaboration. It is noteworthy to
mention that while the CDF II data alone shows a $7\sigma$ deviation; taking
the World Average that includes previous measurements (that are compatible
with the SM) from the Tevatron and LEP experiments as well as LHC would reduce
the tension somewhat (for quantitative analysis, see, for example Lu:2022bgw ;
Asadi:2022xiy ; Bagnaschi:2022whn ).

In our model, NP contributions to these parameters originate from the mass
splittings among the component fields as a result of mixing between the same
charged states from $\widetilde{R}_{2}$ and $S_{3}$. We obtain the following
one-loop correction to the $T$-parameter for our model,

$\displaystyle\Delta T$ $\displaystyle=\frac{N_{c}}{16\pi
s^{2}_{W}m^{2}_{W}}\bigg{\\{}\left(s_{\phi}s_{\theta}-\sqrt{2}c_{\phi}c_{\theta}\right)^{2}\hat{F}\left[m_{\hat{S}^{-2/3}},m_{\hat{S}^{-1/3}}\right]$
$\displaystyle+\left(s_{\phi}c_{\theta}+\sqrt{2}c_{\phi}s_{\theta}\right)^{2}\hat{F}\left[m_{\hat{R}^{-2/3}},m_{\hat{S}^{-1/3}}\right]$
$\displaystyle+\left(c_{\phi}s_{\theta}+\sqrt{2}s_{\phi}c_{\theta}\right)^{2}\hat{F}\left[m_{\hat{S}^{-2/3}},m_{\hat{R}^{-1/3}}\right]$
$\displaystyle+\left(c_{\phi}c_{\theta}-\sqrt{2}s_{\phi}s_{\theta}\right)^{2}\hat{F}\left[m_{\hat{R}^{-2/3}},m_{\hat{R}^{-1/3}}\right]$
$\displaystyle+\left(\sqrt{2}c_{\phi}\right)^{2}\hat{F}\left[m_{\hat{S}^{1/3}},m_{S^{-4/3}}\right]$
$\displaystyle+\left(-\sqrt{2}s_{\phi}\right)^{2}\hat{F}\left[m_{\hat{R}^{1/3}},m_{S^{-4/3}}\right]$
$\displaystyle-\frac{c_{\phi}^{2}s_{\phi}^{2}}{2}\hat{F}\left[m_{\hat{S}^{1/3}},m_{\hat{R}^{-1/3}}\right]-\frac{c_{\theta}^{2}s_{\theta}^{2}}{2}\hat{F}\left[m_{\hat{S}^{2/3}},m_{\hat{R}^{-2/3}}\right]\bigg{\\}},$
(15)

with,

$\displaystyle\hat{F}(m_{1},m_{2})=m_{1}^{2}+m^{2}_{2}-\frac{2m_{1}^{2}m^{2}_{2}}{m_{1}^{2}-m^{2}_{2}}\log\left(\frac{m^{2}_{1}}{m^{2}_{2}}\right).$
(16)

In contrast, we find that the NP contribution to $\Delta S$ is small compared
to $\Delta T$, which in our model, takes the following form:

$\displaystyle\Delta S$ $\displaystyle=\frac{N_{c}}{\pi
m_{Z}^{2}}\bigg{\\{}-\frac{1}{3}{\cal
B}_{22}\left[m_{Z}^{2},m^{2}_{S^{4/3}},m^{2}_{S^{4/3}}\right]$
$\displaystyle+\frac{1}{48}(3+c_{2\theta})(1+3c_{2\theta}){\cal
B}_{22}\left[m_{Z}^{2},m^{2}_{\hat{S}^{2/3}},m^{2}_{\hat{S}^{2/3}}\right]$
$\displaystyle+\frac{1}{96}(9-20c_{2\theta}+3c_{4\theta}){\cal
B}_{22}\left[m_{Z}^{2},m^{2}_{\hat{R}^{2/3}},m^{2}_{\hat{R}^{2/3}}\right]$
$\displaystyle-\frac{1}{24}(1+3c_{2\phi})s^{2}_{\phi}{\cal
B}_{22}\left[m_{Z}^{2},m^{2}_{\hat{S}^{1/3}},m^{2}_{\hat{S}^{1/3}}\right]$
$\displaystyle+\frac{1}{24}(-1+3c_{2\phi})c^{2}_{\phi}{\cal
B}_{22}\left[m_{Z}^{2},m^{2}_{\hat{R}^{1/3}},m^{2}_{\hat{R}^{1/3}}\right]$
$\displaystyle+\frac{1}{8}s^{2}_{2\theta}{\cal
B}_{22}\left[m_{Z}^{2},m^{2}_{\hat{S}^{2/3}},m^{2}_{\hat{R}^{2/3}}\right]$
$\displaystyle+\frac{1}{8}s^{2}_{2\phi}{\cal
B}_{22}\left[m_{Z}^{2},m^{2}_{\hat{S}^{1/3}},m^{2}_{\hat{R}^{1/3}}\right]\bigg{\\}},$
(17)

here, the expression for the loop function ${\cal
B}_{22}(q^{2},m_{1}^{2},m_{2}^{2})$ is defined as He:2001tp  (with
$x_{k}=m^{2}_{k}/q^{2}$):

$\displaystyle{\cal
B}_{22}(q^{2},m_{1}^{2},m_{2}^{2})=\frac{q^{2}}{24}\bigg{\\{}2\ln
q^{2}+\ln(x_{1}x_{2})+$
$\displaystyle\left[\left(x_{1}-x_{2}\right)^{3}-3\left(x_{1}^{2}-x_{2}^{2}\right)+3\left(x_{1}-x_{2}\right)\right]\ln\frac{x_{1}}{x_{2}}$
$\displaystyle-\left[2(x_{1}-x_{2})^{2}-8(x_{1}+x_{2})+\frac{10}{3}\right]-$
$\displaystyle\left[(x_{1}-x_{2})^{2}-2(x_{1}+x_{2})+1\right]h(x_{1},x_{2})-6H(x_{1},x_{2})\bigg{\\}},$
(18)

with

$\displaystyle
H(x_{1},x_{2})=\frac{x_{1}+x_{2}}{2}-\frac{x_{1}x_{2}}{x_{1}-x_{2}}\ln\frac{x_{1}}{x_{2}},$
(19)

and

$\displaystyle h(x_{1},x_{2})$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{ll}-2\sqrt{\Delta}\left[\tan^{-1}\frac{x_{1}-x_{2}+1}{\sqrt{\Delta}}-\tan^{-1}\frac{x_{1}-x_{2}-1}{\sqrt{\Delta}}\right],\\\
0,\\\
\sqrt{-\Delta}\ln\frac{x_{1}+x_{2}-1+\sqrt{-\Delta}}{x_{1}+x_{2}-1-\sqrt{-\Delta}},\end{array}\right.$
(23) $\displaystyle\Delta$ $\displaystyle=$ $\displaystyle
2(x_{1}+x_{2})-(x_{1}-x_{2})^{2}-1\,,$ (24)

for $\Delta>0$, $\Delta=0$, and $\Delta<0$, respectively. For previous works
on leptoquarks effects in EW oblique parameters, see, for example, Refs.
Froggatt:1991qw ; Keith:1997fv ; Dorsner:2019itg ; Dorsner:2020aaz ;
Crivellin:2020ukd .

Figure 1: Leading order NP contribution to muon AMM (in the weak basis).
Photon can be attached to either fermion line or the scalar line.

Muon AMM:– In this theory, the AMM of the muon receives NP contributions as
shown in Fig. 1, which can be expressed as Leveille:1977rc ; Crivellin:2018qmi
,

$\displaystyle\Delta
a_{\mu}=-\frac{m_{\mu}N_{c}}{4\pi^{2}}\sum_{k}\bigg{\\{}\Gamma^{L\ast}_{k}\Gamma^{R}_{k}\frac{m_{\psi}}{m^{2}_{\phi_{k}}}F_{k}\left(\frac{m^{2}_{\psi}}{m^{2}_{\phi_{k}}}\right)+$
$\displaystyle\left[|\Gamma^{L}_{k}|^{2}+|\Gamma^{R}_{k}|^{2}\right]\frac{m_{\mu}}{m^{2}_{\phi_{k}}}G_{k}\left(\frac{m^{2}_{\psi}}{m^{2}_{\phi_{k}}}\right)\bigg{\\}},$
(25)

here sum is taken over
$k=\\{\hat{S}^{2/3},\hat{R}^{2/3},\hat{S}^{1/3},\hat{R}^{1/3}\\}$ for which we
define
$\\{\Gamma^{R}_{k},\Gamma^{L}_{k}\\}=\\{y^{\psi}_{L}s_{\theta},\sqrt{2}y^{\psi}_{R}c_{\theta}\\}$,
$\\{y^{\psi}_{L}c_{\theta},-\sqrt{2}y^{\psi}_{R}s_{\theta}\\}$,
$\\{y^{\psi}_{L}s_{\phi},-y^{\psi}_{R}c_{\phi}\\}$, and
$\\{y^{\psi}_{L}c_{\phi},y^{\psi}_{R}s_{\phi}\\}$, respectively. While
$\hat{X}^{2/3}$ is propagating in the loop $Q_{\psi}=-1/3$, and for
$\hat{X}^{-1/3}$ it is $Q_{\psi}=-4/3$. And the functions $F(x),G(x)$ are
given by,

$\displaystyle
F(x)=f(x)+Q_{\psi}g(x),\;\;G(x)=\widetilde{f}(x)+Q_{\psi}\widetilde{g}(x),$
(26) $\displaystyle f(x)=\frac{x^{2}-1-2x\log
x}{4(x-1)^{3}},\;\;g(x)=\frac{x-1-\log x}{2(x-1)^{2}},$ (27)
$\displaystyle\widetilde{f}(x)=\frac{2x^{3}+3x^{2}-6x+1-6x^{2}\log
x}{24(x-1)^{4}},\;\;\widetilde{g}(x)=\frac{1}{2}f(x).$ (28)

By considering only the dominant chirally-enhanced terms, $(g-2)_{\mu}$ takes
the following simpler form:

$\displaystyle\Delta a_{\mu}$
$\displaystyle=\frac{3m_{\mu}m_{\psi}y^{\psi}_{L}y^{\psi}_{R}}{8\pi^{2}}\bigg{\\{}s_{2\phi}\left[\frac{F\left(\frac{m^{2}_{\psi}}{m^{2}_{\hat{S}^{1/3}}}\right)}{m^{2}_{\hat{S}^{1/3}}}-\frac{F\left(\frac{m^{2}_{\psi}}{m^{2}_{\hat{R}^{1/3}}}\right)}{m^{2}_{\hat{R}^{1/3}}}\right]$
$\displaystyle-\sqrt{2}s_{2\theta}\left[\frac{F\left(\frac{m^{2}_{\psi}}{m^{2}_{\hat{S}^{2/3}}}\right)}{m^{2}_{\hat{S}^{2/3}}}-\frac{F\left(\frac{m^{2}_{\psi}}{m^{2}_{\hat{R}^{2/3}}}\right)}{m^{2}_{\hat{R}^{2/3}}}\right]\bigg{\\}}.$
(29)

If the external photon leg is removed from the Feynman diagram Fig. 1, then
the corresponding diagram contributes to the muon mass. Such a chirally-
enhanced contribution can in principle generate a large mass correction
$\delta m_{\ell}$ to the lepton Czarnecki:2001pv ; Fuyuto:2018scm ;
Athron:2021iuf . Theoretically, this can be absorbed by adjusting the pole
mass of the lepton, $m^{0}_{\ell}$, which, however, introduces large degree of
fine-tuning. To avoid fine-tuned solution, in this work, we adopt the
criterion $\delta m_{\ell}\lesssim m^{0}_{\ell}$ Bigaran:2021kmn .

As aforementioned, for the AMM of the muon, we adopt the theoretical estimate
Eq. (2) quoted in the 2020 White Paper Aoyama:2020ynm . The SM prediction
given in Aoyama:2020ynm  is based on the estimate of the leading-order
hadronic vacuum polarization (HVP) contribution ($a^{\textrm{hvp}}_{\mu}$),
evaluated from a data-driven approach (using dispersion integral involving
hadronic cross-section data). This method has an error of $0.6\%$, subject to
experimental uncertainties associated with measured cross-section data. On the
other hand, lattice QCD calculations for $a^{\textrm{hvp}}_{\mu}$, generally
face numerous technical challenges. Despite that, a recent lattice
computation, namely by the BMW collaboration Borsanyi:2020mff  quotes an error
of only $0.8\%$. If this computation of $a^{\textrm{hvp}}_{\mu}$ is
considered, then the combined measurement from E989 and E821, when compared to
the SM prediction, reduces to $1.5\sigma$ from $4.2\sigma$. Moreover, two more
groups, CLS/Mainz group Ce:2022kxy  and ETMC Alexandrou:2022amy  just recently
released their lattice computations which show consistency with the BMW
result. However, the results of Ce:2022kxy  is somewhat in tension with the
previous lattice computations for light-quarks by the RBC/UKQCD collaboration
RBC:2018dos  and ETMC Giusti:2021dvd . Instead of considering the HVP from the
data-driven method, if the result from Ce:2022kxy  is adopted, the tension
between the SM prediction for $(g-2)_{\mu}$ and experiment would be reduced to
$2.9\sigma$. Concerning the other new lattice result of Alexandrou:2022amy ,
it agrees with the BMW (CLS/Mainz) group at the level of $1.0\sigma$
($1.3\sigma$). However, if these new lattice results hold, they point towards
a large $\sim 4.2\sigma$ discrepancy with the low-energy $e^{+}e^{-}\to$
hadrons cross-section data with respect to Standard Model (SM) predictions
(see also Keshavarzi:2020bfy ; Crivellin:2020zul ; DiLuzio:2021uty ;
Ce:2022eix ).

Figure 2: Feynman diagram leading to $b\to s\mu^{+}\mu^{-}$ transition.

$R_{K}-R_{K^{\ast}}$ anomalies:– The effective Hamiltonian responsible for
processes of the form $B\to K^{(\ast)}\ell^{+}\ell^{\prime-}$ can be described
by,

$\displaystyle\mathcal{H}^{dd\ell\ell}_{eff}=-\frac{4G_{F}}{\sqrt{2}}V_{tj}V^{\ast}_{ti}\left(\sum_{X=9,10}C_{X}^{ij,\ell\ell^{\prime}}\mathcal{O}_{X}^{ij,\ell\ell^{\prime}}\right)+h.c.,$
(30)

where the effective operators are defined as,

$\displaystyle\mathcal{O}_{9}^{ij,\ell\ell^{\prime}}=\frac{\alpha}{4\pi}\left(\overline{d}_{i}\gamma^{\mu}P_{L}d_{j}\right)\left(\overline{\ell}\gamma_{\mu}\ell^{\prime}\right),\;$
(31)
$\displaystyle\mathcal{O}_{10}^{ij,\ell\ell^{\prime}}=\frac{\alpha}{4\pi}\left(\overline{d}_{i}\gamma^{\mu}P_{L}d_{j}\right)\left(\overline{\ell}\gamma_{\mu}\gamma_{5}\ell^{\prime}\right).$
(32)

Now, the part of the Lagrangian relevant for $R_{K^{(\ast)}}$-observable is
the first term given in Eq (6). We choose to work with the ‘down-type
diagonal’ flavor ansatz for which the CKM matrix enters in the interactions
associated with the up-type quarks. Then $S_{3}$ couplings to SM fermions
contain a term of the form:

$\displaystyle\mathcal{L}_{S_{3}}\supset$
$\displaystyle\left(-\sqrt{2}y^{S}\right)_{ij}\overline{d^{c}_{L}}_{i}\mu_{Lj}S^{4/3}+\;h.c.,$
(33)

which leads to $b\to s\mu^{+}\mu^{-}$ transition as shown in Fig 2. After
integrating out the heavy leptoquark and combining the Yukawa part of the
Lagrangian associated to $S_{3}$ as given above, the relevant Wilson
coefficients containing in Eq. (30) at the LQ mass scale generating such
neutral current processes take the form,

$\displaystyle\Delta C_{9}^{\mu\mu}=-\Delta
C_{10}^{\mu\mu}=\frac{v^{2}}{V_{tb}V_{ts}^{\ast}}\frac{\pi}{\alpha_{em}}\frac{y^{S}_{b\mu}\left(y^{S}_{s\mu}\right)^{*}}{M_{4/3}^{2}}.$
(34)

A global fit to the data that includes all $b\to s\mu\mu$ observables, the
$R_{K^{(*)}}$ ratios, and $B_{s}\to\mu^{-}\mu^{+}$ branching ratio prefers
$\Delta C_{9}^{\mu\mu}=-\Delta C_{10}^{\mu\mu}=-0.39\pm 0.07$
Altmannshofer:2021qrr  (see also Carvunis:2021jga ; Geng:2021nhg ).

Figure 3: Feynman diagram leading to non-zero neutrino masses (in the weak
basis).

Neutrino mass:– Non-zero mixing between the $\widetilde{R}_{2}$ and $S_{3}$
LQs and BSM Yukawa interactions of the SM fermions with these LQs give rise to
neutrino oscillations Dorsner:2017wwn ; Julio:2022ton  in this theory (for LQ
effects in leptonic processes, see e.g. Crivellin:2020tsz ; Crivellin:2020mjs
). Feynman diagram that leads to non-zero neutrino mass is shown in Fig. 3,
and the neutrino mass formula takes the following form Dorsner:2017wwn :

$\displaystyle\mathcal{M}^{\nu}_{ij}$ $\displaystyle=\frac{\sin
2\phi}{32\pi^{2}}\sum_{k=d,s,b}m_{k}\left[(y^{R})_{ki}(y^{S})_{kj}+(y^{R})_{kj}(y^{S})_{ki}\right]$
$\displaystyle\times\left[\frac{m^{2}_{\hat{S}_{1/3}}\ln{\frac{m^{2}_{k}}{m^{2}_{\hat{S}_{1/3}}}}}{m^{2}_{\hat{S}_{1/3}}-m^{2}_{k}}-\frac{m^{2}_{\hat{R}_{1/3}}\ln{\frac{m^{2}_{k}}{m^{2}_{\hat{R}_{1/3}}}}}{m^{2}_{\hat{R}_{1/3}}-m^{2}_{k}}\right].$
(35)

Since down-type quark masses are much smaller than the LQ masses, the above
formula can be further simplified,

$\displaystyle\mathcal{M}^{\nu}\approx\frac{\sin
2\phi}{16\pi^{2}}\log\left(\frac{m_{\hat{R}_{1/3}}}{m_{\hat{S}_{1/3}}}\right)\bigg{\\{}(y^{R})^{T}m_{D}y^{S}+(y^{S})^{T}m_{D}y^{R}\bigg{\\}}.$
(36)

Collider constraints:– At LHC, $\widetilde{R}_{2}$ and $S_{3}$ LQs can be pair
produced Diaz:2017lit ; Dorsner:2018ynv  via gluon-fusion
$pp\to\textrm{LQ}\;\textrm{LQ}^{\dagger}$. Once produced, each of these LQs
would decay to SM fermions. Several searches for LQ pairs have been made at
ATLAS and CMS for different final states with or without neutrinos. The
strongest constraints for our scenario come from decay of these LQs to a third
generation quark and a second generation charged lepton, namely $b\mu$ and
$t\mu$. For $100\%$ branching ratio to $pp\to b\overline{b}\mu^{+}\mu^{-}$
$(pp\to t\overline{t}\mu^{+}\mu^{-})$ channel, LHC provides a lower bound of
$m_{\textrm{LQ}}\gtrsim$ 1.7 (1.5) TeV ATLAS:2020dsk ; ATLAS:2020xov . For
similar processes with third generation charged lepton, the corresponding
bounds are $m_{\textrm{LQ}}\gtrsim$ 1 TeV $(pp\to
b\overline{b}\tau^{+}\tau^{-})$ and $m_{\textrm{LQ}}\gtrsim$ 1.4 TeV $(pp\to
t\overline{t}\tau^{+}\tau^{-})$, respectively ATLAS:2019qpq ; ATLAS:2021oiz .

The single production of LQ becomes only relevant for larger Yukawa couplings
to the first and second-generation quarks, which is not the case in our
scenario. For a similar reason, non-resonant dipleton searches at the LHC do
not provide strong constraints for the parameter space we are interested in.

VLQs can also be pair produced at the LHC through gluon-fusion. If LQs are
lighter that VLQs, then each VLQ would mostly decay to a muon and a LQ leading
to
$pp\to\overline{t}t\mu^{+}\mu^{-}\ell^{+}\ell^{-},\;\overline{b}b\mu^{+}\mu^{-}\ell^{+}\ell^{-}$.
Processes of these types have been previously considered in Ref.
Dobrescu:2016pda . We take the LHC bounds on VLQ from CMS:2018zkf ;
ATLAS:2018mpo  that typically correspond to $m_{\psi}\gtrsim 1.3-1.4$ TeV.

On the other hand, if LQs are heavier, then pair produced VLQs would still
decay to SM fermion final states as mentioned above, via effective 4-fermion
operator of the form
$y^{\psi}y^{\textrm{LQ}}/M^{2}_{\textrm{LQ}}\;\left(\overline{\psi}\ell\right)\left(\ell
q\right)$; for details see Ref. Dobrescu:2016pda ; where $y^{\textrm{LQ}}$ and
$y^{\psi}$ represent generic Yukawa couplings of the LQs and VLQ. For TeV
scale VLQ with order one couplings, the decay is prompt even for LQs masses as
heavy as 100 TeV. However, if the corresponding LQ couplings with the SM
fermions are very small, then the VLQ can become long-lived and form R-hadrons
Farrar:1978xj . For such a scenario, by comparing with the limits on
production cross-sections for gluinos and squarks, lower bound of
$m_{\psi}\gtrsim 1.5$ TeV on the VLQ mass is obtained in Ref. Criado:2019mvu .

Future collider prospects:– Here we point out that future experiments such as
multi-TeV muon collider (MuC) InternationalMuonCollider:2022qki ;
Jindariani:2022gxj ; Aime:2022flm ; MuonCollider:2022xlm  and 100 TeV future
circular hadron collider (FCC-hh) FCC:2018byv ; FCC:2018vvp ; Bernardi:2022hny
will probe the entire parameter space of the theory relevant for $B$-meson
anomaly. The most efficient way to probe this scenario is via the predominant
LQ interactions with the muons since the new physics cannot appear at an
arbitrarily high energy scale ($2\to 2$ fermion scattering amplitudes
associated to $b\to s\ell\ell$ transitions saturate the unitarity bound below
80 TeV DiLuzio:2017chi ).

From pair-production, FCC-hh will rule out LQ masses almost up to 10 TeV
Azatov:2022itm . On the other hand, Drell–Yan (DY) $pp\to\mu\overline{\mu}$
from non-resonant $t$-channel contribution, a large portion of the parameter
space in the Yukawa - mass plane will be ruled out Azatov:2022itm  leaving
part of the parameter space unconstrained (corresponding to $y_{b\mu}\approx
y_{s\mu}$ that minimizes the contribution $pp\to\mu\overline{\mu}$).

Figure 4: Expected values of the $\mu$ parameter to incorporate CDF mass shift
of the $W$-boson; see text for details. The $1\sigma$ and $2\sigma$ regions
are obtained from Ref. Lu:2022bgw  that performs a global electroweak fit,
including the CDF II data. For clarity, part of the parameter space is zoomed
in for three different ranges of $\mu$ parameter. The red star represents a
particular benchmark point discussed later in the text.

For muon colliders, the Inverted Drell-Yan (IDY) channel $\mu\overline{\mu}\to
jj$ will constrain the Yukawa - mass plane, which is similar to FCC-hh
scenario (with DY processes). Remarkably, MuC could directly observe an
$s$-channel resonance in the $\mu q\to\mu j$ (due to the quark content inside
the muon) for masses up to $\sim s^{1/2}_{0}$, which would be the most
promising on-shell process at muon colliders. When the LQ pair-production,
IDY, and $\mu\mu\to\mu j$ processes are combined, MuC10 will probe the entire
parameter space Azatov:2022itm .

Results and Discussion:– One of the most important parameters in this model is
the scalar cubic coupling $\mu$, which mixes the two LQs. As described above,
for $\mu\to 0$, $(g-2)_{\mu},\Delta m_{W},m_{\nu}\to 0$. First, we explicitly
demonstrate the required range of $\mu$ to correctly reproduce the electroweak
oblique parameters $(\Delta T,\Delta S)$ consistent with the recent CDF II
measurement. This is portrayed in Fig. 4 by randomly varying the LQ mass
parameters in the ranges $m_{R}\in(1,5)$ TeV, $m_{S}-m_{R}\in(1,500)$ GeV. In
making this plot, $\mu$ is restricted to vary in the range $\mu\lesssim 0.7$
TeV; this upper bound is chosen here for the sake of clarity and illustration.
The $1\sigma$ and $2\sigma$ ranges that favor CDF II data in the $(\Delta
T,\Delta S)$ plane are taken from Ref. Lu:2022bgw  that performed a global fit
to the electroweak data. This figure shows that $\mu\gtrsim 350$ GeV is
essential to resolve the anomaly. Note that in our scenario with only the
$\mu$ parameter responsible for splitting the masses of the LQ states, CDF
anomaly can only be addressed at the $2\sigma$ confidence limit (C.L.). It is
because the $\Delta S$-parameter given in Eq. (17) turns out to be always
tiny, however, addressing CDF anomaly within $1\sigma$ C.L. demands $\Delta
S\in[0.03,0.27]$, whereas it is $\Delta S\in[-0.048,0.35]$ within $2\sigma$
C.L. In this figure, the red star represents a particular benchmark scenario
discussed later in the text.

Figure 5: Dependence of $(g-2)_{\mu}$ on the scalar cubic coupling $\mu$. See
text for details.

The non-trivial functional dependence of $(g-2)_{\mu}$ on $\mu$ is given in
Eq. (29), which we graphically illustrate in Fig. 5. Here (for Figs. 5, 6, and
8), we randomly scan over the relevant parameters in the ranges:
$m_{S,R}\in[1,15]$ TeV, $\mu\in[0.1,5]$ TeV, $m_{\psi}\in[1,10]$ TeV, and
$-y^{\psi}_{L}\cdot y^{\psi}_{L}\in[10^{-5},1]$. The points in green (yellow)
correspond to solutions that simultaneously satisfy the $(g-2)_{\mu}$ anomaly
at the $1\sigma$ ($2\sigma$) and the CDF anomaly at the $2\sigma$ C.L. The red
points that are consistent with CDF anomaly at the $2\sigma$ C.L, however,
fail to reproduce the muon AMM within its $2\sigma$ values. As can be seen
from Fig. 5, reproducing correct value of $\Delta a_{\mu}$ requires $\mu$ in
between $\mathcal{O}(0.5)$ to $\mathcal{O}(3)$ TeV. As shown in Baker:2021yli
, even though the current LHC measurements CMS:2020xwi ; ATLAS:2020fzp  of
$h\to\mu\mu$ allow a large trilinear coupling, future colliders such as the
FCC may be able to measure this coupling and constraint the theory parameter
space.

Figure 6: Correlations between the product of the Yukawa couplings
$|y^{\psi}_{L}\cdot y^{\psi}_{R}|$ and the mass $m_{\psi}$ of the VLQ. See
text for details.  Figure 7: Unique signature of the proposed model.

Moreover, in Fig. 6, we depict the correlations between the product of the
Yukawa couplings $|y^{\psi}_{L}\cdot y^{\psi}_{R}|$ and the mass of the VLQ,
$m_{\psi}$. This figure exhibits the required range of the Yukawa couplings as
a function of $m_{\psi}$. Correctly replicating the measured value of the muon
AMM given in Eq. (2) calls for a somewhat small mass $m_{\psi}\lesssim 3$ TeV
for the VLQ. Due to these low-mass vectorlike quarks, our model can be
distinctively searched for at the LHC. Since the LQs as well as the VLQ
predominantly couple to the muons, see Eq. (7), pair-produced VLQs will lead
to a unique signature $pp\to 4\mu+t\overline{t}$, either via on-shell or off-
shell leptoquarks (see discussion above), as shown in Fig. 7. However, a
dedicated collider study is beyond the scope of this work. Branching ratios to
other up-type quark final states are expected to be suppressed by CKM
elements, and diagrams with neutrinos in the final states are not shown.

Even though both $(g-2)_{\mu}$ and $W$-boson mass shift heavily depend on the
$\mu$-parameter, they are largely insensitive to the scale of the LQ masses.
From a naive estimation, $\Delta a_{\mu}\sim
3m_{\mu}m_{\psi}\theta^{2}_{\textrm{LQ}}y^{2}_{\textrm{LQ}}/(16\pi^{2}m^{2}_{\textrm{LQ}})\ln[m^{2}_{\psi}/m^{2}_{\textrm{LQ}}]$,
where a sum over all LQ contributions must be taken with appropriate signs
(the relative signs play important role). Then, for TeV scale VLQ, with order
unity Yukawa couplings and maximal mixing, LQ masses as heavy as
$m_{\textrm{LQ}}\sim\mathcal{O}(10)$ TeV still provide correct order $\Delta
a_{\mu}$, as depicted in Fig. 8.

Figure 8: Correlations between the $\mu$ parameter and the masses of the LQs;
see text for details. For an example, here we have chosen $Q=2/3$; the other
case with $Q=-1/3$ shows indistinguishable behavior.

Finally, we exemplify how this model can address the anomaly in the neutral
current transitions of the $B$-meson decays as well as incorporate neutrino
oscillation data. This, however, requires a careful fit to neutrino
observables via minimization of a $\chi^{2}$-function; a random scan over the
parameters is not sufficient. This is why, we demonstrate the viability of our
model with a specific benchmark for which we choose $m_{S}=2$ TeV, $m_{R}=1.8$
TeV and mass of the VLQ is $m_{\psi}=1.5$ TeV and $\mu=0.62$ TeV. This
specific benchmark point along with $y^{\psi}_{L}y^{\psi}_{R}=-0.3$
corresponds to $\Delta a_{\mu}=2.5\times 10^{-9}$ (consistent with Eq. (2) at
the $1\sigma$ C.L.) and $[\Delta T,\Delta S]=[0.14038,-8\cdot 10^{-6}]$ (this
point represents the red star shown in Fig. 4). Note, however, that this
fitting procedure is highly non-trivial since the same Yukawa couplings
addressing the $R_{K}-R_{K^{\ast}}$ anomalies also enter in neutrino
observables. In fact, with the texture for $y^{S}$ with only two non-zero
elements as given in Eq. (7), neutrino masses and mixings utilizing the
formula Eq. (36) cannot be accommodated. To generate viable neutrino masses
and mixings, a few more entries must be introduced in $y^{S}$, which would
lead to both charged lepton flavor violation as well as flavor violations in
the quark sector Saad:2020ucl ; Saad:2020ihm ; Julio:2022bue  (see also
Pas:2015hca ; Cheung:2016fjo ; Deppisch:2016qqd ; Cai:2017wry ; Guo:2017gxp ;
Hati:2018fzc ; Datta:2019tuj ; Popov:2019tyc ; Dev:2020qet ; Babu:2020hun ;
Julio:2022ton ).

Figure 9: The results of random scans showing the correlations between
$BR[\tau\to\mu\gamma]$ and $R^{\nu\overline{\nu}}_{K^{(*)}}$. Current (future)
bound on $BR[\tau\to\mu\gamma]$ from BaBar Collaboration BaBar:2009hkt  (Super
B Factory Aushev:2010bq ) is shown by shaded light-gray area (vertical dashed
white line). For the observable $R^{\nu\overline{\nu}}_{K^{(*)}}$, current
experimental limit from Belle collaboration Belle:2017oht  is presented by
shaded light-gray area. The regions bounded by dotted, dot-dashed, and dashed
white (horizontal) lines depict the projected reach ($1\pm 0.25$) for
$R^{\nu\overline{\nu}}_{K^{(*)}}$ at Belle II Belle-II:2018jsg  for 5
$ab^{-1}$ of data with $1\sigma$, $2\sigma$, and $3\sigma$ C.L., respectively,
assuming the best-fit value is SM-like.

Our detailed numerical analysis shows that a non-zero $23$-block in $y^{S}$ is
insufficient to satisfy neutrino oscillation data. Therefore, we also
introduce a non-zero $31$-entry, which is constrained from $\mu-e$ transition
(through CKM rotations). The most crucial cLFV for the given texture is
$\tau\to\mu\gamma$, via which this model may be probed at the upcoming
experiment Aushev:2010bq . In addition, various other flavor violating
processes are considered in our numerical analysis that includes
$Z\to\ell\ell,\ell\to\ell^{{}^{\prime}}\ell^{{}^{\prime\prime}}\ell^{{}^{\prime\prime\prime}}$,
$\tau$ decays into mesons, and several meson decay observables as well as
meson-antimeson oscillations; for details, see e.g. Saad:2020ucl ;
Saad:2020ihm . The observables that provide the most stringent constraints on
the model parameters are summarized in Appendix-A. As aforementioned, the
$23$-block in $y^{S}$ plays an important role in fitting neutrino observables;
some of its entries address the $R_{K^{(*)}}$ anomalies and are required to be
sizable. By randomly varying these couplings, interrelations between
$BR[\tau\to\mu\gamma]$ and $B\to K^{(*)}\nu\overline{\nu}$ observables are
depicted in Fig. 9 along with their respective experimental bounds. Clearly,
the prospective measurement of $B\to K^{(*)}\nu\overline{\nu}$ signal at Belle
II experiment shows a promising avenue to test our model.

In addition to the above-mentioned parameters, from a combined fit, we obtain
the following Yukawa couplings addressing $R_{K}-R_{K^{\ast}}$ anomalies as
well as neutrino oscillation data:

$\displaystyle y^{S}$ $\displaystyle=\left(\begin{array}[]{ccc}0&0&0\\\
0&0.01216&-0.01495\\\ 0.07729&0.19480&0.07439\\\ \end{array}\right),$ (40)
$\displaystyle y^{R}$
$\displaystyle=10^{-7}\left(\begin{array}[]{ccc}1.2583&0.29906&-0.72677\\\
0.21836&-0.60107&-0.03121\\\ 0.12504&-0.88446&-1.49160\\\ \end{array}\right).$
(44)

This fit corresponds to the following neutrino observables:

$\displaystyle(m_{1},m_{2},m_{3})=(1.59\times
10^{-2},8.60,50.27)\;\textrm{meV},$ (45)
$\displaystyle(\sin^{2}\theta_{12},\sin^{2}\theta_{23},\sin^{2}\theta_{13})=(0.309,0.574,0.02224),$
(46)

which are in excellent agreement with experimental data and satisfy all flavor
constraints.

UV completion:– Before concluding, we briefly discuss the possible ultraviolet
(UV) complete model of the proposed scenario. For demonstration, we choose
$SU(5)$ GUT, which is the minimal simple group containing the entire SM gauge
group, i.e., $SU(5)\supset SU(3)_{c}\times SU(2)_{L}\times U(1)_{Y}$. As well
known, the minimal $SU(5)$ GUT, namely, the Georgi–Glashow model Georgi:1974sy
is incompatible with the observed charged fermion masses. To overcome this
drawback, Georgi and Jarlskog proposed Georgi:1979df  to include a $45_{H}$.
Interestingly, $45_{H}$ Higgs contains the $S_{3}$ LQ; $45\supset(3,3,-1/3)$.
On the other hand, $\widetilde{R}_{2}$ LQ can be embedded in a new Higgs in
the $10_{H}$ dimensional representation; $10\supset(3,2,1/6)$. Finally, the
BSM fermion $\psi$ can emerge from $24_{F}$ dimensional fermionic
representation since $24\supset(3,2,-5/6)+(\overline{3},2,5/6)$.

Within this framework, the Yukawa couplings giving rise to $(g-2)_{\mu}$
originate from two independent interactions, $10_{F}24_{F}10^{\ast}_{H}$ and
$\overline{5}_{F}24_{F}45_{H}$. Recall that in $SU(5)$ GUT, the SM fermions
are embedded in $\overline{5}_{F}+10_{F}$ representations, and in the
Georgi–Glashow model, the scalar $5_{H}$ contains the SM Higgs. Finally, the
cubic coupling of Fig. 1 emanates from the interaction term of the form
$10^{\ast}_{H}5_{H}45_{H}$. The Yukawa coupling responsible for addressing the
$B$-meson anomaly in the neutral current transition arises from
$10_{F}\overline{5}_{F}45^{\ast}_{H}$ interaction. In addition to the last two
interaction terms as aforementioned, neutrino mass generation utilizes another
Yukawa coupling with the $\widetilde{R}_{2}$ LQ that appears from
$\overline{5}_{F}\overline{5}_{F}10_{H}$ term in the Lagrangian.

One requires additional fine-tuning on top of the usual doublet-triplet
splitting to keep the required multiplets ($S_{3},\widetilde{R}_{2},\psi$)
light compared to their partners which we assume to reside at the GUT scale.
Further tuning of parameters would be required to keep more states light to
achieve successful gauge coupling unification, which we do not attempt to
address in this work. For the scalars there are enough number of free
parameters in the scalar potential to achieve this, for details, see, for
example, Ref. Saad:2019vjo . For the BSM fermion sector, there exist only two
free parameters ($\mathcal{L}\supset m_{F}24^{2}_{F}+\lambda
24_{F}^{2}24_{H}$) that determine the mass of all the multiplets. To be
precise, $m_{(1,1,0)}=m_{F}-X$, $m_{(1,3,0)}=m_{F}-3X$,
$m_{(8,1,0)}=m_{F}+2X$, and $m_{(3,2,-5/6)}=m_{F}-X/2$, where
$X=\lambda\;v_{\textrm{GUT}}/\sqrt{30}$ and the vev of the Higgs in the
adjoint representation is $\langle
24_{H}\rangle=\textrm{diag}(2,2,2,-3,-3)v_{\textrm{GUT}}/\sqrt{30}$ that
breaks the GUT symmetry to the SM gauge group. It is straightforward to see
that in the limit of $X\to 2m_{F}$ (this leads to an additional fine-tuning on
top of the well known doublet-triplet splitting in generic GUTs), only
$\psi+c.c.$ remain light, whereas rest of the components of $24_{F}$ lives
close to the GUT scale if the natural choice $m_{F}\sim v_{\textrm{GUT}}$ is
assumed.

Finally, we comment on proton decay constraints. The $\widetilde{R}_{2}$ LQ
does not contain di-quark couplings, hence does not lead to proton decay.
However, $S_{3}$ LQ, in general, has both lepton-quark as well as quark-quark
couplings leading to proton decay. The latter coupling originates from terms
of the form $10_{F}10_{F}45_{H}$; as long as this coupling is forbidden (or
highly suppressed) and $S_{3}$ LQ does not mix with any other fields that
contain di-quark couplings, $S_{3}$ LQ can be kept light without any conflict
with the stringent experimental limits on proton decay. Both of these
requirements can be satisfied within this framework, leading to additional
fine-tuning. Furthermore, the texture zeros in the Yukawa couplings that we
consider may be arranged by imposing flavor symmetries, which, however, is
beyond the scope of this work.

##  III Conclusion

In this work, we proposed a simple new physics scenario to simultaneously
address several puzzles that cannot be accounted for by the Standard Model
alone. The model is comprised of two scalar leptoquarks and a vectorlike
quark. One of the most crucial parameters in this theory is the mixing
parameter between the two leptoquarks. This mixing generates neutrino masses
via quantum corrections at one loop, provides additional contributions to the
$W$-boson mass consistent with recent CDF measurement, and plays a non-trivial
role in incorporating the longstanding tension in the muon anomalous magnetic
moment. The vectorlike quark, assisted with both the leptoquarks, gives rise
to the required sizeable new physics contributions to the $(g-2)_{\mu}$ via
chirally enhanced terms proportional to its mass. Furthermore, the iso-triplet
leptoquark is responsible for accounting for the deviations observed
persistently in the $R_{K^{(*)}}$ ratios. By performing a numerical analysis,
we have illustrated how to consistently resolve all these mysteries mentioned
above by keeping flavor violations under control. Moreover, the model is
within reach of the current and future upgrades of the LHC and has the
potential to be fully probed by the future colliders such as future circular
hadron collider and multi-TeV muon collider.

##  Appendix A Constraints on model parameters

In this appendix, we provide relevant expressions of all the NP contributions
to various flavor violating processes. Since the Yukawa couplings of the
$\widetilde{R}_{2}$ LQ are small as required for neutrino mass generation,
here we focus on the constraints associated with the $S_{3}$ LQ. Since we have
chosen flavor conserving Yukawa couplings of the vectorlike-quark, $\psi$ does
not lead to flavor violation.

### LFV: $\ell\to\ell^{\prime}\gamma$

In our model, the dominant LFV process arise from
$\ell\to\ell^{\prime}\gamma$. The effective Lagrangian leading to such
radiative decays of the charged leptons is given by Lavoura:2003xp ,

$\displaystyle\mathcal{L}_{\ell\to\ell^{\prime}\gamma}=\frac{e}{2}\overline{\ell^{\prime}}i\sigma^{\mu\nu}F_{\mu\nu}\left(\sigma_{L}^{\ell\ell^{\prime}}P_{L}+\sigma_{R}^{\ell\ell^{\prime}}P_{R}\right)\ell.$
(47)

Then the branching ratios associated to these process are calculated by the
following formula Lavoura:2003xp :

$\displaystyle
Br(\ell\to\ell^{\prime}\gamma)=\frac{\tau_{\ell}\;\alpha\;m^{3}_{\ell}}{4}\left(|\sigma^{\ell\ell^{\prime}}_{L}|^{2}+|\sigma^{\ell\ell^{\prime}}_{R}|^{2}\right),$
(48)

where $\tau_{\ell}$ is the lifetime of the initial state lepton. The
expressions of $\sigma_{L,R}$ are as follows Lavoura:2003xp ; Dorsner:2016wpm
:

$\displaystyle\sigma_{L,S_{3}}^{if}=\frac{iN_{c}}{16\pi^{2}M^{2}_{3}}m_{i}\left\\{(Vy^{S})^{*}_{qf}(Vy^{S})_{qi}\frac{-1}{12}+(y^{S})^{*}_{qf}y^{S}_{qi}\frac{1}{3}\right\\},$
(49)
$\displaystyle\sigma_{R,S_{3}}^{if}=\frac{iN_{c}}{16\pi^{2}M^{2}_{3}}m_{f}\left\\{(Vy^{S})^{*}_{qf}(Vy^{S})_{qi}\frac{-1}{12}+(y^{S})^{*}_{qf}y^{S}_{qi}\frac{1}{3}\right\\}.$
(50)

Here $V$ is the Cabibbo-Kobayashi-Maskawa (CKM) mixing matrix. The current
experimental limits on these processes are TheMEG:2016wtm ; Aubert:2009ag :

$\displaystyle Br\left(\mu\to e\gamma\right)<4.2\times 10^{-13},$ (51)
$\displaystyle Br\left(\tau\to e\gamma\right)<3.3\times 10^{-8},$ (52)
$\displaystyle Br\left(\tau\to\mu\gamma\right)<4.4\times 10^{-8}.$ (53)

Among these, $\tau\to\mu\gamma$ provides the most stringent constraint in our
scenario, and the respective future sensitivity is of order
$BR[\tau\to\mu\gamma]\sim 10^{-9}$ Aushev:2010bq .

### Z decays: $Z\to\ell\ell^{\prime}$

Leptonic decays of the $Z$-boson receive contributions from the LQs that
constraint the Yukawa couplings. These processes are explained with the
following effective Lagrangian:

$\displaystyle\delta\mathcal{L}^{Z\to\ell\ell^{\prime}}_{eff}=\frac{g}{\cos\theta_{W}}\sum_{f,i,j}\overline{f}\gamma^{\mu}\left(g^{ij}_{f_{L}}P_{L}+g^{ij}_{f_{R}}P_{R}\right)f_{j}Z_{\mu}.$
(54)

Here $g$ is the $SU(2)_{L}$ gauge coupling; $g^{ij}$ are dimensionless
couplings measured with great accuracy at the LEP Tanabashi:2018oca  that
provide stringent constraints on the associated Yukawa couplings for a fixed
LQ mass. NP contributions to these dimensionless couplings can be expressed as
follows Arnan:2019olv :

$\displaystyle Re\left[\delta g^{\ell}_{L,R}\right]^{ij}=$
$\displaystyle\dfrac{3w^{u}_{tj}(w^{u}_{ti})^{\ast}}{16\pi^{2}}\Bigg{[}(g_{u_{L,R}}-g_{u_{R,L}})\dfrac{x_{t}(x_{t}-1-\log
x_{t})}{(x_{t}-1)^{2}}\Bigg{]}$
$\displaystyle+\dfrac{x_{Z}}{16\pi^{2}}\sum_{q=u,c}w^{u}_{qj}(w^{u}_{qi})^{\ast}\Bigg{[}g_{u_{L,R}}\left(\log
x_{Z}-\frac{1}{6}\right)+\frac{g_{\ell_{L,R}}}{6}\Bigg{]}$
$\displaystyle+\dfrac{x_{Z}}{16\pi^{2}}\sum_{q=d,s,b}w^{d}_{qj}(w^{d}_{qi})^{\ast}\Bigg{[}g_{d_{L,R}}\left(\log
x_{Z}-\frac{1}{6}\right)+\frac{g_{\ell_{L,R}}}{6}\Bigg{]}.$ (55)

For $\delta g_{L}$ ($\delta g_{R}$), $w^{u}_{ij}=-(V^{*}y^{S})_{ij}$,
$w^{d}_{ij}=-\sqrt{2}y^{S}_{ij}$ ($w^{u}_{ij}=0$, $w^{d}_{ij}=0$) for $S_{3}$
LQ. The LEP collaboration provides the following limits on these NP
contributions ALEPH:2005ab :

$\displaystyle Re[\delta g^{ee}_{L}]\leq 3.0\times 10^{-4},$ (56)
$\displaystyle Re[\delta g^{\mu\mu}_{L}]\leq 1.1\times 10^{-3},$ (57)
$\displaystyle Re[\delta g^{\tau\tau}_{L}]\leq 5.8\times 10^{-4}.$ (58)

###  $\mu\to e$ conversion

The $S_{3}$ LQ mediates $\mu\to e$ transition in nuclei at the tree-level, and
the rate of which can be calculated from the following formula Kitano:2002mt :

$\displaystyle CR(\mu-e)=\frac{\Gamma^{\mu-e}}{\Gamma_{\text{capture}}(Z)},$
(59)
$\displaystyle\Gamma^{\mu-e}\,=\,2\,G_{F}^{2}\,\left|(2V^{(p)}+g_{LV}^{(u)}V^{(n)})g_{LV}^{(u)}\right|^{2},$
(60) $\displaystyle
g_{LV}^{(u)}=\frac{-2v^{2}}{m_{S_{3}}^{2}}\,(V^{\ast}y^{S})_{u\ell^{\prime}}\,(V^{\ast}y^{S})^{*}_{u\ell}.$
(61)

Here, $\Gamma_{\text{capture}}(Z)$ is the total capture rate for a nucleus
with atomic number $Z$, which is $13.07\times 10^{6}$ $s^{-1}$ for gold, and
the corresponding nuclear form factors in units of $m_{\mu}^{5/2}$ are given
by $V^{(p)}=0.0974$, $V^{(n)}=0.146$ Kitano:2002mt . The current sensitivity
implies Bertl:2006up ,

$\displaystyle\text{CR}(\mu-e)<7\times 10^{-13}$ (62)

whereas the future projected sensitivity is expected to make almost four
orders of magnitude improvement over the current limit, i.e., $CR(\mu\to
e)<10^{-16}$ Kurup:2011zza ; Cui:2009zz ; Chang:2000ac ; Adamov:2018vin ;
Bartoszek:2014mya ; Pezzullo:2018fzp ; Bonventre:2019grv .

###  $B\to K^{(*)}\nu\overline{\nu}$ decays

Contributions to the left-handed currents in the $b\to s\ell\ell$ process
unavoidably imply contributions to $B\to K^{(*)}\nu\overline{\nu}$ decays
which are well constrained by experiments. The $S_{3}$ LQs can induce $B\to
K^{(*)}\nu\overline{\nu}$ decay at the tree-level via $d_{k}\to
d_{j}\nu\overline{\nu}$ processes. The Wilson coefficient responsible for such
decays associated with $b\to s$ transition takes the form,

$\displaystyle C^{fi}_{L}=\frac{\pi
v^{2}}{2V_{tb}V^{*}_{ts}\alpha}\frac{y^{S}_{bi}(y^{S})^{*}_{sf}}{M^{2}_{3}}.$
(63)

Then the branching ratio for $B\to K^{(*)}\nu\overline{\nu}$ can be expressed
as Buras:2014fpa :

$\displaystyle
R^{\nu\overline{\nu}}_{K^{(*)}}=\frac{1}{3|C^{\text{SM}}_{L}|^{2}}\sum_{i,f=1}^{3}\left|\delta^{fi}C^{\text{SM}}_{L}+C^{fi}_{L}\right|^{2},$
(64)

where $C^{\text{SM}}_{L}=-1.47/\sin^{2}\theta_{W}$ is the SM contribution. The
Belle collaboration limits these ratios to be $R^{\nu\overline{\nu}}_{K}<3.9$
and $R^{\nu\overline{\nu}}_{K^{*}}<2.7$ Grygier:2017tzo .

###  $B^{0}_{s}-\overline{B^{0}_{s}}$ oscillation

$S_{3}$ contributes to meson-antimeson mixing, and this NP contribution to
$B^{0}_{s}-\overline{B^{0}_{s}}$ mixing can be described by the following
effective LagrangianMarzocca:2018wcf :

$\displaystyle\mathcal{L}^{\Delta
B=2}_{eff}=-(C^{\text{SM}}_{1}+C^{NP}_{1})\left(\overline{b}_{L}\gamma_{\mu}s_{L}\right)^{2}.$
(65)

Here the SM part is
$C^{\text{SM}}_{1}=2.35/(4\pi^{2})\;\left(V_{tb}V^{*}_{ts}G_{F}m_{W}\right)^{2}$
Lenz:2010gu  and the NP contribution at the heavy scale ($\Lambda$) is given
by Dorsner:2016wpm ; Bobeth:2017ecx ; Marzocca:2018wcf ; Crivellin:2019dwb ,

$\displaystyle
C^{NP}_{1}=\frac{1}{128\pi^{2}}\frac{5}{M^{2}_{3}}\left(\sum_{\ell}{y^{S}_{b\ell}}^{\ast}y^{S}_{s\ell}\right)^{2}.$
(66)

Here we neglect the evolution of $C^{NP}_{1}$ from high scale to the $m_{w}$
scale, which is only relevant for precision calculation. Then the mass
difference is given by:

$\displaystyle\Delta m_{B_{s}}^{SM+NP}=\Delta
m_{B_{s}}^{\text{SM}}\left|1+\frac{C_{1}^{NP}}{C_{1}^{\text{SM}}}\right|,$
(67)

where the SM prediction is $\Delta m_{B_{s}}^{\text{SM}}=(18.3\pm 2.7)\times
10^{12}s^{-1}$ Bona:2006sa ; Jubb:2016mvq . This mass difference has been
measured in the experiments Bona:2008jn ; Tanabashi:2018oca  with great
accuracy, leading to strong constraints on the NP contribution UTfit:2007eik :

$\displaystyle|C^{NP}_{1}|\;\;<2.01\times 10^{-5}\;\textrm{TeV}^{-2}.$ (68)

## References

  * (1) Particle Data Group Collaboration, P. A. Zyla et al., “Review of Particle Physics,” PTEP 2020 no. 8, (2020) 083C01.
  * (2) CDF Collaboration, T. Aaltonen et al., “High-precision measurement of the W boson mass with the CDF II detector,” Science 376 no. 6589, (2022) 170–176.
  * (3) 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 [hep-ph].
  * (4) 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 [astro-ph.HE].
  * (5) 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 [hep-ph].
  * (6) 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 [hep-ph].
  * (7) J. M. Yang and Y. Zhang, “Low energy SUSY confronted with new measurements of W-boson mass and muon g-2,” arXiv:2204.04202 [hep-ph].
  * (8) 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 [hep-ph].
  * (9) 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 [hep-ph].
  * (10) M. Blennow, P. Coloma, E. Fernández-Martínez, and M. González-López, “Right-handed neutrinos and the CDF II anomaly,” arXiv:2204.04559 [hep-ph].
  * (11) B.-Y. Zhu, S. Li, J.-G. Cheng, R.-L. Li, and Y.-F. Liang, “Using gamma-ray observation of dwarf spheroidal galaxy to test a dark matter model that can interpret the W-boson mass anomaly,” arXiv:2204.04688 [astro-ph.HE].
  * (12) K. Sakurai, F. Takahashi, and W. Yin, “Singlet extensions and W boson mass in the light of the CDF II result,” arXiv:2204.04770 [hep-ph].
  * (13) Y. Heo, D.-W. Jung, and J. S. Lee, “Impact of the CDF $W$-mass anomaly on two Higgs doublet model,” arXiv:2204.05728 [hep-ph].
  * (14) K. Cheung, W.-Y. Keung, and P.-Y. Tseng, “Iso-doublet Vector Leptoquark solution to the Muon $g-2$, $R_{K,K^{*}}$, $R_{D,D^{*}}$, and $W$-mass Anomalies,” arXiv:2204.05942 [hep-ph].
  * (15) 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 [hep-ph].
  * (16) A. Strumia, “Interpreting electroweak precision data including the $W$-mass CDF anomaly,” arXiv:2204.04191 [hep-ph].
  * (17) J. Fan, L. Li, T. Liu, and K.-F. Lyu, “$W$-Boson Mass, Electroweak Precision Tests and SMEFT,” arXiv:2204.04805 [hep-ph].
  * (18) G. Cacciapaglia and F. Sannino, “The W boson mass weighs in on the non-standard Higgs,” arXiv:2204.04514 [hep-ph].
  * (19) 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 [hep-ph].
  * (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 [hep-ph].
  * (21) 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 [hep-ph].
  * (22) H. Song, W. Su, and M. Zhang, “Electroweak Phase Transition in 2HDM under Higgs, Z-pole, and W precision measurements,” arXiv:2204.05085 [hep-ph].
  * (23) E. Bagnaschi, J. Ellis, M. Madigan, K. Mimasu, V. Sanz, and T. You, “SMEFT Analysis of $m_{W}$,” arXiv:2204.05260 [hep-ph].
  * (24) A. Paul and M. Valli, “Violation of custodial symmetry from W-boson mass measurements,” arXiv:2204.05267 [hep-ph].
  * (25) 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 [hep-ph].
  * (26) P. Asadi, C. Cesarotti, K. Fraser, S. Homiller, and A. Parikh, “Oblique Lessons from the $W$ Mass Measurement at CDF II,” arXiv:2204.05283 [hep-ph].
  * (27) L. Di Luzio, R. Gröber, and P. Paradisi, “Higgs physics confronts the $M_{W}$ anomaly,” arXiv:2204.05284 [hep-ph].
  * (28) 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 [hep-ph].
  * (29) J. Gu, Z. Liu, T. Ma, and J. Shu, “Speculations on the W-Mass Measurement at CDF,” arXiv:2204.05296 [hep-ph].
  * (30) 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 [hep-ph].
  * (31) 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 [hep-ph].
  * (32) M. Endo and S. Mishima, “New physics interpretation of $W$-boson mass anomaly,” arXiv:2204.05965 [hep-ph].
  * (33) 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 [hep-ph].
  * (34) T. Biekötter, S. Heinemeyer, and G. Weiglein, “Excesses in the low-mass Higgs-boson search and the W-boson mass measurement,” arXiv:2204.05975 [hep-ph].
  * (35) R. Balkin, E. Madge, T. Menzo, G. Perez, Y. Soreq, and J. Zupan, “On the implications of positive W mass shift,” arXiv:2204.05992 [hep-ph].
  * (36) J. Kawamura, S. Okawa, and Y. Omura, “$W$ boson mass and muon $g-2$ in a lepton portal dark matter model,” arXiv:2204.07022 [hep-ph].
  * (37) A. Ghoshal, N. Okada, S. Okada, D. Raut, Q. Shafi, and A. Thapa, “Type III seesaw with R-parity violation in light of $m_{W}$ (CDF),” arXiv:2204.07138 [hep-ph].
  * (38) P. F. Perez, H. H. Patel, and A. D. Plascencia, “On the $W$-mass and New Higgs Bosons,” arXiv:2204.07144 [hep-ph].
  * (39) 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 [hep-ph].
  * (40) 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 [hep-ph].
  * (41) J. J. Heckman, “Extra $W$-Boson Mass from a D3-Brane,” arXiv:2204.05302 [hep-ph].
  * (42) 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 [hep-ph].
  * (43) 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 [hep-ph].
  * (44) 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 [hep-ph].
  * (45) M. Du, Z. Liu, and P. Nath, “CDF W mass anomaly from a dark sector with a Stueckelberg-Higgs portal,” arXiv:2204.09024 [hep-ph].
  * (46) K. Ghorbani and P. Ghorbani, “$W$-Boson Mass Anomaly from Scale Invariant 2HDM,” arXiv:2204.09001 [hep-ph].
  * (47) 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 [hep-ph].
  * (48) S. Baek, “Implications of CDF $W$-mass and $(g-2)_{\mu}$ on $U(1)_{L_{\mu}-L_{\tau}}$ model,” arXiv:2204.09585 [hep-ph].
  * (49) J. Cao, L. Meng, L. Shang, S. Wang, and B. Yang, “Interpreting the $W$ mass anomaly in the vectorlike quark models,” arXiv:2204.09477 [hep-ph].
  * (50) D. Borah, S. Mahapatra, and N. Sahu, “Singlet-Doublet Fermion Origin of Dark Matter, Neutrino Mass and W-Mass Anomaly,” arXiv:2204.09671 [hep-ph].
  * (51) A. Batra, S. K. A., S. Mandal, and R. Srivastava, “W boson mass in Singlet-Triplet Scotogenic dark matter model,” arXiv:2204.09376 [hep-ph].
  * (52) 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 [hep-ph].
  * (53) 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 [hep-ph].
  * (54) A. Addazi, A. Marciano, A. P. Morais, R. Pasechnik, and H. Yang, “CDF II $W$-mass anomaly faces first-order electroweak phase transition,” arXiv:2204.10315 [hep-ph].
  * (55) J. Heeck, “W-boson mass in the triplet seesaw model,” arXiv:2204.10274 [hep-ph].
  * (56) 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 [hep-ph].
  * (57) 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 [hep-ph].
  * (58) R. Benbrik, M. Boukidi, and B. Manaut, “$W$-mass and 96 GeV excess in type-III 2HDM,” arXiv:2204.11755 [hep-ph].
  * (59) 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 [hep-ph].
  * (60) 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 [hep-ph].
  * (61) R. S. Gupta, “Running away from the T-parameter solution to the W mass anomaly,” arXiv:2204.13690 [hep-ph].
  * (62) 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 [hep-ph].
  * (63) B. Barman, A. Das, and S. Sengupta, “New $W$-Boson mass in the light of doubly warped braneworld model,” arXiv:2205.01699 [hep-ph].
  * (64) 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 [hep-ph].
  * (65) J. Kim, “Compatibility of muon $g-2$, $W$ mass anomaly in type-X 2HDM,” arXiv:2205.01437 [hep-ph].
  * (66) R. Dcruz and A. Thapa, “$W$ boson mass, dark matter and $(g-2)_{\ell}$ in ScotoZee neutrino mass model,” arXiv:2205.02217 [hep-ph].
  * (67) J. Isaacson, Y. Fu, and C. P. Yuan, “ResBos2 and the CDF W Mass Measurement,” arXiv:2205.02788 [hep-ph].
  * (68) F. J. Botella, F. Cornet-Gomez, C. Miró, and M. Nebot, “Muon and electron $g-2$ anomalies in a flavor conserving 2HDM with an oblique view on the CDF $M_{W}$ value,” arXiv:2205.01115 [hep-ph].
  * (69) S.-P. He, “Leptoquark and vectorlike quark extended models as the explanation of the muon g-2 anomaly,” Phys. Rev. D 105 no. 3, (2022) 035017, arXiv:2112.13490 [hep-ph].
  * (70) S.-P. He, “A leptoquark and vector-like quark extended model for the simultaneous explanation of the $W$ boson mass and muon $g-2$ anomalies,” arXiv:2205.02088 [hep-ph].
  * (71) O. Popov and R. Srivastava, “The Triplet Dirac Seesaw in the View of the Recent CDF-II W Mass Anomaly,” arXiv:2204.08568 [hep-ph].
  * (72) Muon g-2 Collaboration, B. Abi et al., “Measurement of the Positive Muon Anomalous Magnetic Moment to 0.46 ppm,” Phys. Rev. Lett. 126 no. 14, (2021) 141801, arXiv:2104.03281 [hep-ex].
  * (73) Muon g-2 Collaboration, G. W. Bennett et al., “Final Report of the Muon E821 Anomalous Magnetic Moment Measurement at BNL,” Phys. Rev. D73 (2006) 072003, arXiv:hep-ex/0602035 [hep-ex].
  * (74) T. Aoyama et al., “The anomalous magnetic moment of the muon in the Standard Model,” Phys. Rept. 887 (2020) 1–166, arXiv:2006.04822 [hep-ph].
  * (75) 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 [hep-ph].
  * (76) T. Aoyama, T. Kinoshita, and M. Nio, “Theory of the Anomalous Magnetic Moment of the Electron,” Atoms 7 no. 1, (2019) 28.
  * (77) A. Czarnecki, W. J. Marciano, and A. Vainshtein, “Refinements in electroweak contributions to the muon anomalous magnetic moment,” Phys. Rev. D67 (2003) 073006, arXiv:hep-ph/0212229 [hep-ph]. [Erratum: Phys. Rev.D73,119901(2006)].
  * (78) C. Gnendiger, D. Stöckinger, and H. Stöckinger-Kim, “The electroweak contributions to $(g-2)_{\mu}$ after the Higgs boson mass measurement,” Phys. Rev. D 88 (2013) 053005, arXiv:1306.5546 [hep-ph].
  * (79) 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. C 77 no. 12, (2017) 827, arXiv:1706.09436 [hep-ph].
  * (80) A. Keshavarzi, D. Nomura, and T. Teubner, “Muon $g-2$ and $\alpha(M_{Z}^{2})$: a new data-based analysis,” Phys. Rev. D 97 no. 11, (2018) 114025, arXiv:1802.02995 [hep-ph].
  * (81) G. Colangelo, M. Hoferichter, and P. Stoffer, “Two-pion contribution to hadronic vacuum polarization,” JHEP 02 (2019) 006, arXiv:1810.00007 [hep-ph].
  * (82) M. Hoferichter, B.-L. Hoid, and B. Kubis, “Three-pion contribution to hadronic vacuum polarization,” JHEP 08 (2019) 137, arXiv:1907.01556 [hep-ph].
  * (83) 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 $\mathbf{\boldsymbol{\alpha}(m_{Z}^{2})}$,” Eur. Phys. J. C 80 no. 3, (2020) 241, arXiv:1908.00921 [hep-ph]. [Erratum: Eur.Phys.J.C 80, 410 (2020)].
  * (84) 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 no. 1, (2020) 014029, arXiv:1911.00367 [hep-ph].
  * (85) 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. B 734 (2014) 144–147, arXiv:1403.6400 [hep-ph].
  * (86) K. Melnikov and A. Vainshtein, “Hadronic light-by-light scattering contribution to the muon anomalous magnetic moment revisited,” Phys. Rev. D70 (2004) 113006, arXiv:hep-ph/0312226 [hep-ph].
  * (87) P. Masjuan and P. Sanchez-Puertas, “Pseudoscalar-pole contribution to the $(g_{\mu}-2)$: a rational approach,” Phys. Rev. D 95 no. 5, (2017) 054026, arXiv:1701.05829 [hep-ph].
  * (88) 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 [hep-ph].
  * (89) 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 [hep-ph].
  * (90) 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. D 100 no. 3, (2019) 034520, arXiv:1903.09471 [hep-lat].
  * (91) J. Bijnens, N. Hermansson-Truedsson, and A. Rodríguez-Sánchez, “Short-distance constraints for the HLbL contribution to the muon anomalous magnetic moment,” Phys. Lett. B 798 (2019) 134994, arXiv:1908.03331 [hep-ph].
  * (92) 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,” arXiv:1910.13432 [hep-ph].
  * (93) 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,” arXiv:1911.08123 [hep-lat].
  * (94) G. Colangelo, M. Hoferichter, A. Nyffeler, M. Passera, and P. Stoffer, “Remarks on higher-order hadronic corrections to the muon g$-$2,” Phys. Lett. B 735 (2014) 90–91, arXiv:1403.7512 [hep-ph].
  * (95) P. Athron, C. Balázs, D. H. J. Jacob, W. Kotlarski, D. Stöckinger, and H. Stöckinger-Kim, “New physics explanations of $a_{\mu}$ in light of the FNAL muon g-2 measurement,” JHEP 09 (2021) 080, arXiv:2104.03691 [hep-ph].
  * (96) 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 [hep-ex].
  * (97) LHCb Collaboration, R. Aaij et al., “Search for lepton-universality violation in $B^{+}\to K^{+}\ell^{+}\ell^{-}$ decays,” Phys. Rev. Lett. 122 no. 19, (2019) 191801, arXiv:1903.09252 [hep-ex].
  * (98) Belle Collaboration, A. Abdesselam et al., “Test of Lepton-Flavor Universality in ${B\to K^{\ast}\ell^{+}\ell^{-}}$ Decays at Belle,” Phys. Rev. Lett. 126 no. 16, (2021) 161801, arXiv:1904.02440 [hep-ex].
  * (99) BELLE Collaboration, S. Choudhury et al., “Test of lepton flavor universality and search for lepton flavor violation in $B\rightarrow K\ell\ell$ decays,” JHEP 03 (2021) 105, arXiv:1908.01848 [hep-ex].
  * (100) LHCb Collaboration, R. Aaij et al., “Test of lepton universality in beauty-quark decays,” Nature Phys. 18 no. 3, (2022) 277–282, arXiv:2103.11769 [hep-ex].
  * (101) J. M. Camalich and M. Patel, “New lepton non-universal forces in flavor physics?,” Science Bulletin 67 no. 1, (2022) 1–4. https://www.sciencedirect.com/science/article/pii/S2095927321006307.
  * (102) L.-S. Geng, B. Grinstein, S. Jäger, S.-Y. Li, J. Martin Camalich, and R.-X. Shi, “Implications of new evidence for lepton-universality violation in $b\to s\ell^{-}\ell^{+}$\- decays,” Phys. Rev. D 104 no. 3, (2021) 035029, arXiv:2103.12738 [hep-ph].
  * (103) A. Angelescu, D. Bečirević, D. A. Faroughy, F. Jaffredo, and O. Sumensari, “Single leptoquark solutions to the B-physics anomalies,” Phys. Rev. D 104 no. 5, (2021) 055017, arXiv:2103.12504 [hep-ph].
  * (104) W. Altmannshofer and P. Stangl, “New Physics in Rare B Decays after Moriond 2021,” arXiv:2103.13370 [hep-ph].
  * (105) M. Algueró, B. Capdevila, S. Descotes-Genon, J. Matias, and M. Novoa-Brunet, “$b\rightarrow s\ell^{+}\ell^{-}$ global fits after $R_{K_{S}}$ and $R_{K^{*+}}$,” Eur. Phys. J. C 82 no. 4, (2022) 326, arXiv:2104.08921 [hep-ph].
  * (106) T. Hurth, F. Mahmoudi, D. M. Santos, and S. Neshatpour, “More indications for lepton nonuniversality in $b\to s\ell^{-}\ell^{+}$,” Phys. Lett. B 824 (2022) 136838, arXiv:2104.10058 [hep-ph].
  * (107) G. Isidori, D. Lancierini, P. Owen, and N. Serra, “On the significance of new physics in $b\to s\ell^{-}\ell^{+}$ decays,” Phys. Lett. B 822 (2021) 136644, arXiv:2104.05631 [hep-ph].
  * (108) K. Kowalska, D. Kumar, and E. M. Sessolo, “Implications for new physics in $b\rightarrow s\mu\mu$ transitions after recent measurements by Belle and LHCb,” Eur. Phys. J. C 79 no. 10, (2019) 840, arXiv:1903.10932 [hep-ph].
  * (109) M. Ciuchini, M. Fedele, E. Franco, A. Paul, L. Silvestrini, and M. Valli, “New Physics without bias: Charming Penguins and Lepton Universality Violation in $b\to s\ell^{+}\ell^{-}$ decays,” arXiv:2110.10126 [hep-ph].
  * (110) G. D’Amico, M. Nardecchia, P. Panci, F. Sannino, A. Strumia, R. Torre, and A. Urbano, “Flavour anomalies after the $R_{K^{*}}$ measurement,” JHEP 09 (2017) 010, arXiv:1704.05438 [hep-ph].
  * (111) Super-Kamiokande Collaboration, Y. Fukuda et al., “Evidence for oscillation of atmospheric neutrinos,” Phys. Rev. Lett. 81 (1998) 1562–1567, arXiv:hep-ex/9807003.
  * (112) Super-Kamiokande Collaboration, S. Fukuda et al., “Solar B-8 and hep neutrino measurements from 1258 days of Super-Kamiokande data,” Phys. Rev. Lett. 86 (2001) 5651–5655, arXiv:hep-ex/0103032.
  * (113) SNO Collaboration, Q. R. Ahmad et al., “Direct evidence for neutrino flavor transformation from neutral current interactions in the Sudbury Neutrino Observatory,” Phys. Rev. Lett. 89 (2002) 011301, arXiv:nucl-ex/0204008.
  * (114) KamLAND Collaboration, K. Eguchi et al., “First results from KamLAND: Evidence for reactor anti-neutrino disappearance,” Phys. Rev. Lett. 90 (2003) 021802, arXiv:hep-ex/0212021.
  * (115) KamLAND Collaboration, T. Araki et al., “Measurement of neutrino oscillation with KamLAND: Evidence of spectral distortion,” Phys. Rev. Lett. 94 (2005) 081801, arXiv:hep-ex/0406035.
  * (116) K2K Collaboration, M. H. Ahn et al., “Indications of neutrino oscillation in a 250 km long baseline experiment,” Phys. Rev. Lett. 90 (2003) 041801, arXiv:hep-ex/0212007.
  * (117) MINOS Collaboration, D. G. Michael et al., “Observation of muon neutrino disappearance with the MINOS detectors and the NuMI neutrino beam,” Phys. Rev. Lett. 97 (2006) 191801, arXiv:hep-ex/0607088.
  * (118) W. Buchmuller, R. Ruckl, and D. Wyler, “Leptoquarks in Lepton - Quark Collisions,” Phys. Lett. B191 (1987) 442–448. [Erratum: Phys. Lett.B448,320(1999)].
  * (119) I. Doršner, S. Fajfer, A. Greljo, J. F. Kamenik, and N. Košnik, “Physics of leptoquarks in precision experiments and at particle colliders,” Phys. Rept. 641 (2016) 1–68, arXiv:1603.04993 [hep-ph].
  * (120) M. E. Peskin and T. Takeuchi, “A New constraint on a strongly interacting Higgs sector,” Phys. Rev. Lett. 65 (1990) 964–967.
  * (121) M. E. Peskin and T. Takeuchi, “Estimation of oblique electroweak corrections,” Phys. Rev. D46 (1992) 381–409.
  * (122) W. Grimus, L. Lavoura, O. M. Ogreid, and P. Osland, “The Oblique parameters in multi-Higgs-doublet models,” Nucl. Phys. B801 (2008) 81–96, arXiv:0802.4353 [hep-ph].
  * (123) H.-J. He, N. Polonsky, and S.-f. Su, “Extra families, Higgs spectrum and oblique corrections,” Phys. Rev. D 64 (2001) 053004, arXiv:hep-ph/0102144.
  * (124) C. D. Froggatt, R. G. Moorhouse, and I. G. Knowles, “Leading radiative corrections in two scalar doublet models,” Phys. Rev. D 45 (1992) 2471–2481.
  * (125) E. Keith and E. Ma, “S, T, and leptoquarks at HERA,” Phys. Rev. Lett. 79 (1997) 4318–4320, arXiv:hep-ph/9707214.
  * (126) I. Doršner, S. Fajfer, and O. Sumensari, “Muon $g-2$ and scalar leptoquark mixing,” JHEP 06 (2020) 089, arXiv:1910.03877 [hep-ph].
  * (127) I. Doršner, S. Fajfer, and S. Saad, “$\mu\to e\gamma$ selecting scalar leptoquark solutions for the $(g-2)_{e,\mu}$ puzzles,” Phys. Rev. D 102 no. 7, (2020) 075007, arXiv:2006.11624 [hep-ph].
  * (128) A. Crivellin, D. Müller, and F. Saturnino, “Leptoquarks in oblique corrections and Higgs signal strength: status and prospects,” JHEP 11 (2020) 094, arXiv:2006.10758 [hep-ph].
  * (129) J. P. Leveille, “The Second Order Weak Correction to (G-2) of the Muon in Arbitrary Gauge Models,” Nucl. Phys. B137 (1978) 63–76.
  * (130) A. Crivellin, M. Hoferichter, and P. Schmidt-Wellenburg, “Combined explanations of $(g-2)_{\mu,e}$ and implications for a large muon EDM,” Phys. Rev. D98 no. 11, (2018) 113002, arXiv:1807.11484 [hep-ph].
  * (131) A. Czarnecki and W. J. Marciano, “The Muon anomalous magnetic moment: A Harbinger for ’new physics’,” Phys. Rev. D 64 (2001) 013014, arXiv:hep-ph/0102122.
  * (132) K. Fuyuto, M. Ramsey-Musolf, and T. Shen, “Electric Dipole Moments from CP-Violating Scalar Leptoquark Interactions,” Phys. Lett. B 788 (2019) 52–57, arXiv:1804.01137 [hep-ph].
  * (133) I. Bigaran and R. R. Volkas, “Reflecting on Chirality: CP-violating extensions of the single scalar-leptoquark solutions for the $(g-2)_{e,\mu}$ puzzles and their implications for lepton EDMs,” arXiv:2110.03707 [hep-ph].
  * (134) S. Borsanyi et al., “Leading hadronic contribution to the muon magnetic moment from lattice QCD,” Nature 593 no. 7857, (2021) 51–55, arXiv:2002.12347 [hep-lat].
  * (135) M. Cè et al., “Window observable for the hadronic vacuum polarization contribution to the muon $g-2$ from lattice QCD,” arXiv:2206.06582 [hep-lat].
  * (136) C. Alexandrou et al., “Lattice calculation of the short and intermediate time-distance hadronic vacuum polarization contributions to the muon magnetic moment using twisted-mass fermions,” arXiv:2206.15084 [hep-lat].
  * (137) RBC, UKQCD Collaboration, T. Blum, P. A. Boyle, V. Gülpers, T. Izubuchi, L. Jin, C. Jung, A. Jüttner, C. Lehner, A. Portelli, and J. T. Tsang, “Calculation of the hadronic vacuum polarization contribution to the muon anomalous magnetic moment,” Phys. Rev. Lett. 121 no. 2, (2018) 022003, arXiv:1801.07224 [hep-lat].
  * (138) D. Giusti and S. Simula, “Window contributions to the muon hadronic vacuum polarization with twisted-mass fermions,” PoS LATTICE2021 (2022) 189, arXiv:2111.15329 [hep-lat].
  * (139) A. Keshavarzi, W. J. Marciano, M. Passera, and A. Sirlin, “Muon $g-2$ and $\Delta\alpha$ connection,” Phys. Rev. D 102 no. 3, (2020) 033002, arXiv:2006.12666 [hep-ph].
  * (140) A. Crivellin, M. Hoferichter, C. A. Manzari, and M. Montull, “Hadronic Vacuum Polarization: $(g-2)_{\mu}$ versus Global Electroweak Fits,” Phys. Rev. Lett. 125 no. 9, (2020) 091801, arXiv:2003.04886 [hep-ph].
  * (141) L. Di Luzio, A. Masiero, P. Paradisi, and M. Passera, “New physics behind the new muon g-2 puzzle?,” Phys. Lett. B 829 (2022) 137037, arXiv:2112.08312 [hep-ph].
  * (142) M. Cè, A. Gérardin, G. von Hippel, H. B. Meyer, K. Miura, K. Ottnad, A. Risch, T. S. José, J. Wilhelm, and H. Wittig, “The hadronic running of the electromagnetic coupling and the electroweak mixing angle from lattice QCD,” arXiv:2203.08676 [hep-lat].
  * (143) A. Carvunis, F. Dettori, S. Gangal, D. Guadagnoli, and C. Normand, “On the effective lifetime of Bs→ $\mu$$\mu$$\gamma$,” JHEP 12 (2021) 078, arXiv:2102.13390 [hep-ph].
  * (144) I. Doršner, S. Fajfer, and N. Košnik, “Leptoquark mechanism of neutrino masses within the grand unification framework,” Eur. Phys. J. C 77 no. 6, (2017) 417, arXiv:1701.08322 [hep-ph].
  * (145) J. Julio, S. Saad, and A. Thapa, “A Tale of Flavor Anomalies and the Origin of Neutrino Mass,” arXiv:2202.10479 [hep-ph].
  * (146) A. Crivellin, D. Mueller, and F. Saturnino, “Correlating h→$\mu$+$\mu$\- to the Anomalous Magnetic Moment of the Muon via Leptoquarks,” Phys. Rev. Lett. 127 no. 2, (2021) 021801, arXiv:2008.02643 [hep-ph].
  * (147) A. Crivellin, C. Greub, D. Müller, and F. Saturnino, “Scalar Leptoquarks in Leptonic Processes,” JHEP 02 (2021) 182, arXiv:2010.06593 [hep-ph].
  * (148) B. Diaz, M. Schmaltz, and Y.-M. Zhong, “The leptoquark Hunter’s guide: Pair production,” JHEP 10 (2017) 097, arXiv:1706.05033 [hep-ph].
  * (149) I. Doršner and A. Greljo, “Leptoquark toolbox for precision collider studies,” JHEP 05 (2018) 126, arXiv:1801.07641 [hep-ph].
  * (150) ATLAS Collaboration, G. Aad et al., “Search for pairs of scalar leptoquarks decaying into quarks and electrons or muons in $\sqrt{s}$ = 13 TeV $pp$ collisions with the ATLAS detector,” JHEP 10 (2020) 112, arXiv:2006.05872 [hep-ex].
  * (151) ATLAS Collaboration, G. Aad et al., “Search for pair production of scalar leptoquarks decaying into first- or second-generation leptons and top quarks in proton–proton collisions at $\sqrt{s}$ = 13 TeV with the ATLAS detector,” Eur. Phys. J. C 81 no. 4, (2021) 313, arXiv:2010.02098 [hep-ex].
  * (152) ATLAS Collaboration, M. Aaboud et al., “Searches for third-generation scalar leptoquarks in $\sqrt{s}$ = 13 TeV pp collisions with the ATLAS detector,” JHEP 06 (2019) 144, arXiv:1902.08103 [hep-ex].
  * (153) ATLAS Collaboration, G. Aad et al., “Search for pair production of third-generation scalar leptoquarks decaying into a top quark and a $\tau$-lepton in $pp$ collisions at $\sqrt{s}$ = 13 TeV with the ATLAS detector,” JHEP 06 (2021) 179, arXiv:2101.11582 [hep-ex].
  * (154) B. A. Dobrescu and F. Yu, “Exotic Signals of Vectorlike Quarks,” J. Phys. G45 no. 8, (2018) 08LT01, arXiv:1612.01909 [hep-ph].
  * (155) CMS Collaboration, A. M. Sirunyan et al., “Search for vector-like T and B quark pairs in final states with leptons at $\sqrt{s}=$ 13 TeV,” JHEP 08 (2018) 177, arXiv:1805.04758 [hep-ex].
  * (156) ATLAS Collaboration, M. Aaboud et al., “Search for pair production of heavy vector-like quarks decaying into high-$p_{T}$ $W$ bosons and top quarks in the lepton-plus-jets final state in $pp$ collisions at $\sqrt{s}=13$ TeV with the ATLAS detector,” JHEP 08 (2018) 048, arXiv:1806.01762 [hep-ex].
  * (157) G. R. Farrar and P. Fayet, “Phenomenology of the Production, Decay, and Detection of New Hadronic States Associated with Supersymmetry,” Phys. Lett. B 76 (1978) 575–579.
  * (158) J. C. Criado and M. Perez-Victoria, “Vector-like quarks with non-renormalizable interactions,” JHEP 01 (2020) 057, arXiv:1908.08964 [hep-ph].
  * (159) International Muon Collider Collaboration, D. Stratakis et al., “A Muon Collider Facility for Physics Discovery,” arXiv:2203.08033 [physics.acc-ph].
  * (160) S. Jindariani et al., “Promising Technologies and R&D Directions for the Future Muon Collider Detectors,” in 2022 Snowmass Summer Study.  3, 2022.  arXiv:2203.07224 [physics.ins-det].
  * (161) C. Aime et al., “Muon Collider Physics Summary,” arXiv:2203.07256 [hep-ph].
  * (162) Muon Collider Collaboration, J. de Blas et al., “The physics case of a 3 TeV muon collider stage,” arXiv:2203.07261 [hep-ph].
  * (163) FCC Collaboration, A. Abada et al., “FCC Physics Opportunities: Future Circular Collider Conceptual Design Report Volume 1,” Eur. Phys. J. C 79 no. 6, (2019) 474.
  * (164) FCC Collaboration, A. Abada et al., “FCC-hh: The Hadron Collider: Future Circular Collider Conceptual Design Report Volume 3,” Eur. Phys. J. ST 228 no. 4, (2019) 755–1107.
  * (165) G. Bernardi et al., “The Future Circular Collider: a Summary for the US 2021 Snowmass Process,” arXiv:2203.06520 [hep-ex].
  * (166) L. Di Luzio and M. Nardecchia, “What is the scale of new physics behind the $B$-flavour anomalies?,” Eur. Phys. J. C77 no. 8, (2017) 536, arXiv:1706.01868 [hep-ph].
  * (167) A. Azatov, F. Garosi, A. Greljo, D. Marzocca, J. Salko, and S. Trifinopoulos, “New Physics in $b\to s\mu\mu$: FCC-hh or a Muon Collider?,” arXiv:2205.13552 [hep-ph].
  * (168) M. J. Baker, P. Cox, and R. R. Volkas, “Radiative muon mass models and $(g-2)_{\mu}$,” JHEP 05 (2021) 174, arXiv:2103.13401 [hep-ph].
  * (169) CMS Collaboration, A. M. Sirunyan et al., “Evidence for Higgs boson decay to a pair of muons,” JHEP 01 (2021) 148, arXiv:2009.04363 [hep-ex].
  * (170) ATLAS Collaboration, G. Aad et al., “A search for the dimuon decay of the Standard Model Higgs boson with the ATLAS detector,” Phys. Lett. B 812 (2021) 135980, arXiv:2007.07830 [hep-ex].
  * (171) S. Saad and A. Thapa, “Common origin of neutrino masses and $R_{D^{(\ast)}}$, $R_{K^{(\ast)}}$ anomalies,” Phys. Rev. D 102 no. 1, (2020) 015014, arXiv:2004.07880 [hep-ph].
  * (172) S. Saad, “Combined explanations of $(g-2)_{\mu}$, $R_{D^{(*)}}$, $R_{K^{(*)}}$ anomalies in a two-loop radiative neutrino mass model,” Phys. Rev. D 102 no. 1, (2020) 015019, arXiv:2005.04352 [hep-ph].
  * (173) J. Julio, S. Saad, and A. Thapa, “Marriage between neutrino mass and flavor anomalies,” arXiv:2203.15499 [hep-ph].
  * (174) H. Päs and E. Schumacher, “Common origin of $R_{K}$ and neutrino masses,” Phys. Rev. D92 no. 11, (2015) 114025, arXiv:1510.08757 [hep-ph].
  * (175) K. Cheung, T. Nomura, and H. Okada, “Testable radiative neutrino mass model without additional symmetries and explanation for the $b\to s\ell^{+}\ell^{-}$ anomaly,” Phys. Rev. D 94 no. 11, (2016) 115024, arXiv:1610.02322 [hep-ph].
  * (176) F. F. Deppisch, S. Kulkarni, H. Päs, and E. Schumacher, “Leptoquark patterns unifying neutrino masses, flavor anomalies, and the diphoton excess,” Phys. Rev. D94 no. 1, (2016) 013003, arXiv:1603.07672 [hep-ph].
  * (177) Y. Cai, J. Gargalionis, M. A. Schmidt, and R. R. Volkas, “Reconsidering the One Leptoquark solution: flavor anomalies and neutrino mass,” JHEP 10 (2017) 047, arXiv:1704.05849 [hep-ph].
  * (178) S.-Y. Guo, Z.-L. Han, B. Li, Y. Liao, and X.-D. Ma, “Interpreting the $R_{K^{(*)}}$ anomaly in the colored Zee–Babu model,” Nucl. Phys. B928 (2018) 435–447, arXiv:1707.00522 [hep-ph].
  * (179) C. Hati, G. Kumar, J. Orloff, and A. M. Teixeira, “Reconciling $B$-meson decay anomalies with neutrino masses, dark matter and constraints from flavour violation,” JHEP 11 (2018) 011, arXiv:1806.10146 [hep-ph].
  * (180) A. Datta, D. Sachdeva, and J. Waite, “Unified explanation of $b\to s\mu^{+}\mu^{-}$ anomalies, neutrino masses, and $B\rightarrow\pi K$ puzzle,” Phys. Rev. D100 no. 5, (2019) 055015, arXiv:1905.04046 [hep-ph].
  * (181) O. Popov, M. A. Schmidt, and G. White, “$R_{2}$ as a single leptoquark solution to $R_{D^{(*)}}$ and $R_{K^{(*)}}$,” Phys. Rev. D100 no. 3, (2019) 035028, arXiv:1905.06339 [hep-ph].
  * (182) P. B. Dev, R. Mohanta, S. Patra, and S. Sahoo, “Unified explanation of flavor anomalies, radiative neutrino mass and ANITA anomalous events in a vector leptoquark model,” arXiv:2004.09464 [hep-ph].
  * (183) K. S. Babu, P. S. B. Dev, S. Jana, and A. Thapa, “Unified framework for $B$-anomalies, muon $g-2$ and neutrino masses,” JHEP 03 (2021) 179, arXiv:2009.01771 [hep-ph].
  * (184) BaBar Collaboration, B. Aubert et al., “Searches for Lepton Flavor Violation in the Decays tau+- —$>$ e+- gamma and tau+- —$>$ mu+- gamma,” Phys. Rev. Lett. 104 (2010) 021802, arXiv:0908.2381 [hep-ex].
  * (185) T. Aushev et al., “Physics at Super B Factory,” arXiv:1002.5012 [hep-ex].
  * (186) Belle Collaboration, J. Grygier et al., “Search for $\boldsymbol{B\to h\nu\bar{\nu}}$ decays with semileptonic tagging at Belle,” Phys. Rev. D 96 no. 9, (2017) 091101, arXiv:1702.03224 [hep-ex]. [Addendum: Phys.Rev.D 97, 099902 (2018)].
  * (187) Belle-II Collaboration, W. Altmannshofer et al., “The Belle II Physics Book,” PTEP 2019 no. 12, (2019) 123C01, arXiv:1808.10567 [hep-ex]. [Erratum: PTEP 2020, 029201 (2020)].
  * (188) H. Georgi and S. L. Glashow, “Unity of All Elementary Particle Forces,” Phys. Rev. Lett. 32 (1974) 438–441.
  * (189) H. Georgi and C. Jarlskog, “A New Lepton - Quark Mass Relation in a Unified Theory,” Phys. Lett. B 86 (1979) 297–300.
  * (190) S. Saad, “Origin of a two-loop neutrino mass from SU(5) grand unification,” Phys. Rev. D99 no. 11, (2019) 115016, arXiv:1902.11254 [hep-ph].
  * (191) L. Lavoura, “General formulae for $f(1)\to f(2)\gamma$,” Eur. Phys. J. C29 (2003) 191–195, arXiv:hep-ph/0302221 [hep-ph].
  * (192) MEG Collaboration, A. M. Baldini et al., “Search for the lepton flavour violating decay $\mu^{+}\rightarrow\mathrm{e}^{+}\gamma$ with the full dataset of the MEG experiment,” Eur. Phys. J. C76 no. 8, (2016) 434, arXiv:1605.05081 [hep-ex].
  * (193) BaBar Collaboration, B. Aubert et al., “Searches for Lepton Flavor Violation in the Decays $\tau^{\pm}\to e^{\pm}\gamma$ and $\tau^{\pm}\to\mu^{\pm}\gamma$,” Phys. Rev. Lett. 104 (2010) 021802, arXiv:0908.2381 [hep-ex].
  * (194) Particle Data Group Collaboration, M. Tanabashi et al., “Review of Particle Physics,” Phys. Rev. D98 no. 3, (2018) 030001.
  * (195) P. Arnan, D. Becirevic, F. Mescia, and O. Sumensari, “Probing low energy scalar leptoquarks by the leptonic $W$ and $Z$ couplings,” JHEP 02 (2019) 109, arXiv:1901.06315 [hep-ph].
  * (196) ALEPH, DELPHI, L3, OPAL, SLD, LEP Electroweak Working Group, SLD Electroweak Group, SLD Heavy Flavour Group Collaboration, S. Schael et al., “Precision electroweak measurements on the $Z$ resonance,” Phys. Rept. 427 (2006) 257–454, arXiv:hep-ex/0509008.
  * (197) R. Kitano, M. Koike, and Y. Okada, “Detailed calculation of lepton flavor violating muon electron conversion rate for various nuclei,” Phys. Rev. D66 (2002) 096002, arXiv:hep-ph/0203110 [hep-ph]. [Erratum: Phys. Rev.D76,059902(2007)].
  * (198) SINDRUM II Collaboration, W. H. Bertl et al., “A Search for muon to electron conversion in muonic gold,” Eur. Phys. J. C47 (2006) 337–346.
  * (199) COMET Collaboration, A. Kurup, “The COherent Muon to Electron Transition (COMET) experiment,” Nucl. Phys. Proc. Suppl. 218 (2011) 38–43.
  * (200) COMET Collaboration, Y. G. Cui et al., “Conceptual design report for experimental search for lepton flavor violating mu- e- conversion at sensitivity of 10**(-16) with a slow-extracted bunched proton beam (COMET),”.
  * (201) C.-H. Chang, S.-L. Chen, T.-F. Feng, and X.-Q. Li, “The Lifetime of $B_{c}$ meson and some relevant problems,” Phys. Rev. D64 (2001) 014003, arXiv:hep-ph/0007162 [hep-ph].
  * (202) COMET Collaboration, R. Abramishvili et al., “COMET Phase-I Technical Design Report,” arXiv:1812.09018 [physics.ins-det].
  * (203) Mu2e Collaboration, L. Bartoszek et al., “Mu2e Technical Design Report,” arXiv:1501.05241 [physics.ins-det].
  * (204) Mu2e Collaboration, G. Pezzullo, “Mu2e: A Search for Charged Lepton Flavor Violation in $\rm\mu N\to eN$ Conversion with a Sensitivity $<10^{-16}$,” PoS ICHEP2018 (2019) 583.
  * (205) Mu2e Collaboration, R. Bonventre, “Searching for muon to electron conversion: The Mu2e experiment at Fermilab,” SciPost Phys. Proc. 1 (2019) 038.
  * (206) A. J. Buras, J. Girrbach-Noe, C. Niehoff, and D. M. Straub, “$B\to{K}^{\left(\ast\right)}\nu\overline{\nu}$ decays in the Standard Model and beyond,” JHEP 02 (2015) 184, arXiv:1409.4557 [hep-ph].
  * (207) Belle Collaboration, J. Grygier et al., “Search for $\boldsymbol{B\to h\nu\bar{\nu}}$ decays with semileptonic tagging at Belle,” Phys. Rev. D 96 no. 9, (2017) 091101, arXiv:1702.03224 [hep-ex]. [Addendum: Phys.Rev.D 97, 099902 (2018)].
  * (208) D. Marzocca, “Addressing the B-physics anomalies in a fundamental Composite Higgs Model,” JHEP 07 (2018) 121, arXiv:1803.10972 [hep-ph].
  * (209) A. Lenz, U. Nierste, J. Charles, S. Descotes-Genon, A. Jantsch, C. Kaufhold, H. Lacker, S. Monteil, V. Niess, and S. T’Jampens, “Anatomy of New Physics in $B-\bar{B}$ mixing,” Phys. Rev. D 83 (2011) 036004, arXiv:1008.1593 [hep-ph].
  * (210) C. Bobeth and A. J. Buras, “Leptoquarks meet $\varepsilon^{\prime}/\varepsilon$ and rare Kaon processes,” JHEP 02 (2018) 101, arXiv:1712.01295 [hep-ph].
  * (211) A. Crivellin, D. Müller, and F. Saturnino, “Flavor Phenomenology of the Leptoquark Singlet-Triplet Model,” arXiv:1912.04224 [hep-ph].
  * (212) UTfit Collaboration, M. Bona et al., “Constraints on new physics from the quark mixing unitarity triangle,” Phys. Rev. Lett. 97 (2006) 151803, arXiv:hep-ph/0605213.
  * (213) T. Jubb, M. Kirk, A. Lenz, and G. Tetlalmatzi-Xolocotzi, “On the ultimate precision of meson mixing observables,” Nucl. Phys. B 915 (2017) 431–453, arXiv:1603.07770 [hep-ph].
  * (214) UTfit Collaboration, M. Bona et al., “First Evidence of New Physics in $b\longleftrightarrow s$ Transitions,” PMC Phys. A 3 (2009) 6, arXiv:0803.0659 [hep-ph].
  * (215) UTfit Collaboration, M. Bona et al., “Model-independent constraints on $\Delta F=2$ operators and the scale of new physics,” JHEP 03 (2008) 049, arXiv:0707.0636 [hep-ph].