aainstitutetext: National Research and Innovation Agency, Kompleks Puspiptek
Serpong, South Tangerang 15314, Indonesiabbinstitutetext: Department of
Physics, University of Basel, Klingelbergstrasse 82, CH-4056 Basel,
Switzerlandccinstitutetext: Department of Physics, University of Virginia,
Charlottesville, Virginia 22904-4714, USA

# A flavor-inspired radiative neutrino mass model

J. Julio, b Shaikh Saad, c and Anil Thapa<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

One of the most important discoveries in particle physics is the observation
of nonzero neutrino masses, which dictates that the Standard Model (SM) is
incomplete. Moreover, several pieces of evidence of lepton flavor universality
violation (LFUV), gathered in the last few years, hint toward physics beyond
the SM. TeV-scale scalar leptoquarks are the leading candidates for explaining
these flavor anomalies in semileptonic charged and neutral current B-decays,
the muon, and the electron magnetic dipole moments that can also participate
in neutrino mass generation. In this work, we hypothesize that neutrino masses
and LFUV have a common new physics origin and propose a new two–loop neutrino
mass model that has the potential to resolve some of these flavor anomalies
via leptoquarks and offers rich phenomenology. After deriving the neutrino
mass formula for this newly-proposed model, we perform a detailed numerical
analysis focusing on neutrino and charged lepton flavor violation
phenomenology, where the latter provides stringent constraints on the Yukawa
couplings and leptoquark masses. Finally, present and future bounds on the
model’s parameter space are scrutinized with exemplified benchmark scenarios.

##  1 Introduction

The observation of neutrino oscillations was the first direct hint that the
Standard Model of particle physics is imperfect and must be extended. Lepton
flavor universality (LFU), a solid prediction of the SM, can be easily
violated in the beyond SM (BSM) models, where the particles preferentially
couple to certain generations of leptons. In the last several years,
indications of LFU violation (LFUV) have been observed in both $b\to
s\ell\ell$ and $b\to c\ell\nu$ processes. Observables associated with these
transitions are the well-known $R_{K^{(\ast)}}$ and $R_{D^{(\ast)}}$ ratios,
respectively. LFU in the SM predicts the former ratio to be unity with
uncertainties less than $1\%$. A deficit in this neutral-current transition
has been observed consistently over the years in several experiments, and LHCb
recently updated their measurements LHCb:2021trn  that increased the
significance of the deviation. Moreover, with yet another observed deviation
in $Br\left(B^{0}_{s}\to\mu^{+}\mu^{-}\right)$ LHCb:2015wdu ; LHCb:2017rmj ;
ATLAS:2018cur ; CMS:2019bbr , the combined significance of the deviation is
uplifted to $4.7\sigma$. On the other hand, the $R_{D^{(\ast)}}$ ratio differs
from unity due to the substantial mass difference between tauon and muon. An
enhancement of this charged-current transition is reported by several
experimental measurements, which combinedly leads to approximately $3\sigma$
deviation from the SM value Na:2015kha ; Aoki:2016frl .

Besides, there has been a longstanding tension between the theoretical
prediction of the anomalous magnetic dipole moment (AMDM) of the muon
$(g-2)_{\mu}$ and the value measured at the BNL E821 experiment Bennett:2006fi
. The FNAL E989 experiment Abi:2021gix  has recently announced its result,
which has a smaller uncertainty and is fully compatible with the previous best
measurement. Together, these two experiments show a remarkably large deviation
with a significance of $4.2\sigma$ with respect to the theory prediction
Aoyama:2020ynm . Various new physics models are proposed to explain the
observed significant departure. For a most recent review see Ref.
Athron:2021iuf . The SM prediction given in Ref. Aoyama:2020ynm  is based on
the estimate of the leading-order hadronic vacuum polarization contribution,
evaluated from a data-driven approach. On the other hand, if recent lattice
computations Borsanyi:2020mff ; Ce:2022kxy ; Alexandrou:2022amy  are
considered, then the tension reduces to $1.5\sigma$ from $4.2\sigma$. 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 SM predictions Keshavarzi:2020bfy ; Crivellin:2020zul ;
DiLuzio:2021uty ; Ce:2022eix .

On top of that, the electron AMDM $(g-2)_{e}$ is also measured in the
experiments with an unprecedented level of accuracy. Recently, improved
measurement Parker:2018vye  of the fine-structure constant utilizing Caesium
atom shows a $-2.4\sigma$ deviation in comparison with the direct experimental
measurement Hanneke:2008tm . Lately, these anomalies in the lepton AMDMs have
gained a lot of attention in the theory community; for simultaneous explains
of the muon and the electron AMDMs in various BSM frameworks, see e.g. Refs.
Giudice:2012ms ; Davoudiasl:2018fbb ; Crivellin:2018qmi ; Liu:2018xkx ;
Dutta:2018fge ; Han:2018znu ; Crivellin:2019mvj ; Endo:2019bcj ;
Abdullah:2019ofw ; Bauer:2019gfk ; Badziak:2019gaf ; Hiller:2019mou ;
CarcamoHernandez:2019ydc ; Cornella:2019uxs ; Endo:2020mev ;
CarcamoHernandez:2020pxw ; Haba:2020gkr ; Bigaran:2020jil ; Jana:2020pxx ;
Calibbi:2020emz ; Chen:2020jvl ; Yang:2020bmh ; Hati:2020fzp ; Dutta:2020scq ;
Botella:2020xzf ; Chen:2020tfr ; Dorsner:2020aaz ; Arbelaez:2020rbq ;
Jana:2020joi ; Chua:2020dya ; Chun:2020uzw ; Li:2020dbg ; DelleRose:2020oaa ;
Kowalska:2020zve ; Hernandez:2021tii ; Bodas:2021fsy ; Cao:2021lmj ;
Mondal:2021vou ; CarcamoHernandez:2021iat ; Han:2021gfu ; Escribano:2021css ;
CarcamoHernandez:2021qhf ; Chang:2021axw ; Chowdhury:2021tnm ;
Bharadwaj:2021tgp ; Borah:2021khc ; Bigaran:2021kmn ; Jana:2021jjm ;
Li:2021wzv ; Biswas:2021dan ; Barman:2021xeq ; Chowdhury:2022jde . It is
noteworthy to point out that a more recent measurement of fine-structure
constant utilizing Rubidium atom Morel:2020dww  shows somewhat consistent with
the direct measurement of $a_{e}$ Hanneke:2008tm . This new result
Morel:2020dww  finds $\Delta a_{e}=+1.6\sigma$, indicating a $\sim 5\sigma$
disagreement between these two experiments (Parker:2018vye  and Morel:2020dww
). Therefore, the electron $g-2$ situation requires clarification from future
experiments.

All these flavor anomalies mentioned above are strongly pointing toward
physics beyond the SM. Interestingly, the prime candidates to solve these
flavor anomalies are leptoquarks (LQs), i.e., hypothetical particles that
combine the properties of leptons and quarks (for a recent review on LQs, see
Ref. Dorsner:2016wpm ). The existence of LQs are highly motivated since
particles of this type are naturally predicted by Grand Unified Theories.
Explanation of flavor anomalies Dorsner:2013tla ; Sakaki:2013bfa ;
Duraisamy:2014sna ; Hiller:2014yaa ; Buras:2014fpa ; Gripaios:2014tna ;
Freytsis:2015qca ; Pas:2015hca ; Bauer:2015knc ; Fajfer:2015ycq ;
Deppisch:2016qqd ; Li:2016vvp ; Becirevic:2016yqi ; Becirevic:2016oho ;
Sahoo:2016pet ; Bhattacharya:2016mcc ; Duraisamy:2016gsd ; Barbieri:2016las ;
Crivellin:2017zlb ; DAmico:2017mtc ; Hiller:2017bzc ; Becirevic:2017jtw ;
Cai:2017wry ; Alok:2017sui ; Sumensari:2017mud ; Buttazzo:2017ixm ;
Crivellin:2017dsk ; Guo:2017gxp ; Aloni:2017ixa ; Assad:2017iib ;
DiLuzio:2017vat ; Calibbi:2017qbu ; Chauhan:2017uil ; Cline:2017aed ;
Sumensari:2017ovu ; Biswas:2018jun ; Muller:2018nwq ; Blanke:2018sro ;
Schmaltz:2018nls ; Azatov:2018knx ; Sheng:2018vvm ; Becirevic:2018afm ;
Hati:2018fzc ; Azatov:2018kzb ; Huang:2018nnq ; Angelescu:2018tyl ;
DaRold:2018moy ; Balaji:2018zna ; Bansal:2018nwp ; Mandal:2018kau ;
Iguro:2018vqb ; Fornal:2018dqn ; Kim:2018oih ; deMedeirosVarzielas:2019lgb ;
Zhang:2019hth ; Aydemir:2019ynb ; deMedeirosVarzielas:2019okf ;
Cornella:2019hct ; Datta:2019tuj ; Popov:2019tyc ; Bigaran:2019bqv ;
Hati:2019ufv ; Coy:2019rfr ; Balaji:2019kwe ; Crivellin:2019dwb ; Cata:2019wbu
; Altmannshofer:2020axr ; Cheung:2020sbq ; Saad:2020ucl ; Saad:2020ihm ;
Dev:2020qet ; Crivellin:2020ukd ; Crivellin:2020tsz ; Gherardi:2020qhc ;
Babu:2020hun ; Bordone:2020lnb ; Crivellin:2020mjs ; Crivellin:2020oup ;
Hati:2020cyn ; Dorsner:2021chv ; Angelescu:2021lln ; Marzocca:2021azj ;
Crivellin:2021egp ; Perez:2021ddi ; Crivellin:2021ejk ; Zhang:2021dgl ;
Bordone:2021usz ; Carvunis:2021dss ; Marzocca:2021miv ; BhupalDev:2021ipu ;
Allwicher:2021rtd ; Wang:2021uqz ; Bandyopadhyay:2021pld ; Qian:2021ihf ;
Fischer:2021sqw ; Gherardi:2021pwm ; Crivellin:2021lix ; London:2021lfn ;
Bandyopadhyay:2021kue ; Husek:2021isa ; Afik:2021xmi ; Belanger:2021smw ;
Chowdhury:2022dps ; Heeck:2022znj ; Julio:2022bue  requires these particles to
have masses of order TeV. Remarkably, these TeV scale scalar leptoquarks
(SLQs) can also participate in neutrino mass generation.

In this work, we hypothesize that neutrino masses and LFUV have a common new
physics origin. Motivated by this unified framework, we propose a new
radiative neutrino mass generation model where scalar leptoquarks, at the
leading order, induce tiny neutrino masses as two–-loop quantum corrections.
If these LQs reside close to the TeV scale, in addition to incorporating
neutrino oscillation data, the proposed model has the potential to address the
flavor anomalies mentioned above. However, addressing flavor anomalies demands
some of the Yukawa couplings to be of order unity. Therefore, with the TeV-
scale LQs, charged lepton flavor violating (cLFV) processes are inevitable,
which are also clear signals of new physics. In what follows, we first provide
the details of our model and derive the neutrino mass matrix in a general
gauge. From the derived neutrino mass formula, we carry out a comprehensive
phenomenological study of the neutrino sector as well as cLFV, which provides
the most stringent constraints on the model parameters. Specifically, we
investigate a few minimal benchmark scenarios with a limited number of Yukawa
parameters without assuming any strong hierarchy among them. We scrutinize
these textures for their ability to satisfy neutrino observables and assess
cLFV processes with a detailed numerical study using Markov chain Monte Carlo
analysis. Finally, we illustrate how $(g-2)_{\mu}$, the most prominent flavor
anomalies, can be addressed while satisfying all LFV constraints and neutrino
oscillation data.

This paper is organized in this way: our newly-proposed model of neutrino mass
is introduced in Section 2, and the detailed derivation of the neutrino mass
formula is given in Section 3. In Sec. 4, we work out in details all charged
lepton flavor violating processes that occur in this model and present the
results in Sec. 5. Finally, we give our conclusion in Section 6.

##  2 Proposed model

Figure 1: New physics operator leading to non-zero neutrino masses in our
proposed model. Here $f_{i}$ represents SM fermions. From the left-most
(right-most) vertex, it is clear that $R_{2}$ ($S_{1}$) carries $F=0$
($F=-2$), where $F=3B+L$ is known as the fermion number.

The new neutrino mass model proposed in this work consists of the following
three BSM scalar multiplets:

$\displaystyle R_{2}(3,2,7/6)$ $\displaystyle\equiv
R=\left(\begin{array}[]{c}R^{5/3}\\\ R^{2/3}\end{array}\right),$ (2.3)
$\displaystyle S_{1}(\overline{3},1,+1/3)$ $\displaystyle\equiv S=S^{1/3},$
(2.4) $\displaystyle\xi_{3}(3,3,2/3)$
$\displaystyle\equiv\xi=\left(\begin{array}[]{cc}\frac{\xi^{2/3}}{\sqrt{2}}&\xi^{5/3}\\\
\xi^{-1/3}&-\frac{\xi^{2/3}}{\sqrt{2}}\end{array}\right).$ (2.7)

Numbers in parentheses stand for quantum numbers of each field under
$SU(3)_{C}\times SU(2)_{L}\times U(1)_{Y}$ gauge groups.

The SM Higgs is denoted as $H(1,2,1/2)=\left(H^{+},H^{0}\right)^{T}$. This
scalar field will get a nonzero vacuum expectation value (vev) $v\equiv\langle
H\rangle=174$ GeV during the spontaneous breaking of the electroweak (EW)
symmetry. The fermion sector does not change. It contains the same particle
content as in the SM

$\displaystyle L_{i}(1,2,-1/2)=\begin{pmatrix}\nu_{iL}\\\
e_{iL}\end{pmatrix},\quad Q_{i}(3,2,1/6)=\begin{pmatrix}u_{iL}\\\
d_{iL}\end{pmatrix},$ $\displaystyle
e_{i}^{c}(1,1,1),~{}~{}~{}u_{i}^{c}(\bar{3},1,-2/3),~{}~{}~{}d_{i}^{c}(\bar{3},1,+1/3),$
(2.8)

where $i$ indicates generation index and $\psi^{c}\equiv C\bar{\psi_{R}}^{T}$
denotes the charge conjugate of the right-handed field.

Among three new scalars introduced, only $R$ and $S$ can be considered as LQs.
They interact with the SM fermions through the following Yukawa interactions

$\displaystyle\mathcal{L}_{Y}^{\rm new}=$ $\displaystyle
f^{L}_{ij}u^{c}_{i}R\cdot
L_{j}+f^{R}_{ij}R^{\dagger}Q_{i}e^{c}_{j}+y^{L}_{ij}SQ_{i}\cdot
L_{j}+y^{R}_{ij}u^{c}_{i}e^{c}_{j}S^{\dagger}+{\text{h}.c.}$ (2.9)

In order to avoid unnecessary cluttered notation, we have used “$\cdot$” to
denote an SU(2) contraction, e.g., $L\cdot Q\equiv L^{a}Q^{b}\epsilon_{ab}$
with $\epsilon$ being the antisymmetric tensor
($\epsilon_{12}=-\epsilon^{12}=1$) and $a,b=1,2$ being SU(2) indices.

All terms in Eq. (2.9) conserve both baryon ($B$) and lepton ($L$) numbers.
This can be seen, for instance, by setting $(B,L)$ to $(1/3,-1)$ and
$(-1/3,-1)$ for $R$ and $S$ fields, respectively. Such assignments forbid
diquark terms, $QQS^{\dagger}$ and $u^{c}d^{c}S$, despite being allowed by the
gauge symmetry. Note that it is important to have a globally conserved $B$, or
else a rapid proton decay will take place in our theory. On the contrary,
lepton number $L$ is broken, as required to generate non-zero neutrino mass,
by the following non-trivial terms in the scalar potential:

$\displaystyle V\supset$ $\displaystyle\;\lambda
S^{\dagger}H^{T}\epsilon\xi^{\dagger}H+\mu R^{\dagger}\xi H$ $\displaystyle=$
$\displaystyle~{}\lambda
S^{\dagger}\left[-\sqrt{2}\xi^{-2/3}H^{+}H^{0}-\xi^{1/3}H^{0}H^{0}+\xi^{-5/3}H^{+}H^{+}\right]$
$\displaystyle+\mu\bigg{\\{}\frac{1}{\sqrt{2}}R^{-5/3}\xi^{2/3}H^{+}+R^{-5/3}\xi^{5/3}H^{0}+R^{-2/3}\xi^{-1/3}H^{+}-\frac{1}{\sqrt{2}}R^{-2/3}\xi^{2/3}H^{0}\bigg{\\}}+{\text{h}.c.}$
(2.10)

One should have noticed that terms in Eq. (2.10) still conserve the baryon
number with $\xi$ field carrying opposite (same) baryon number as that of $S$
($R$) field. In the above equation, the two parameters, i.e., $\lambda$ and
$\mu$, can be made real by absorbing their respected phases into scalar
fields.

The $\Delta L=2$ effective operators, depicted in Fig. 1, must contain the
product of $\Delta F=0$ and $\Delta F=2$ couplings, i.e., any combination of
$y^{L,R}f^{L,R}$, so there are four kinds of $\Delta L=2$ operators that can
be generated within this model after integrating out heavy scalar states. It
is required that such operators involve gauge bosons, or else they will vanish
by $SU(2)$ symmetry. The four operators will contain $(HD^{\mu}H)$ multiplied
by the following combinations:

(i) |  $(u^{c}HD_{\mu}L)(LQ)$,
---|---
(ii) |  $(u^{c}HD_{\mu}L)\bar{e^{c}}\bar{u^{c}}$,
(iii) |  $(\bar{e^{c}}D_{\mu}\bar{Q}H)(LQ)$,
(iv) |  $(\bar{e^{c}}HD_{\mu}\bar{Q})\bar{u^{c}}\bar{e^{c}}$.

Note that the $SU(2)$ contraction occurs on fields inside parentheses. In
addition, the covariant derivative can also act on other fields. All but the
operator (iv) will lead to neutrino masses at two-loop level.

Scalar terms in Eq. (2.10) will cause mixing among $\xi^{1/3}-S^{1/3}$,
$\xi^{2/3}-R^{2/3}$, and $\xi^{5/3}-R^{5/3}$ components. The mass matrices of
leptoquarks relevant for neutrino mass generation are

$\displaystyle M^{2}_{\chi^{2/3}}=\begin{pmatrix}m_{\xi}^{2}&-\lambda v^{2}\\\
-\lambda v^{2}&m_{R}^{2}\end{pmatrix},$ $\displaystyle
M^{2}_{\chi^{1/3}}=\begin{pmatrix}m_{\xi}^{2}&-\mu v/\sqrt{2}\\\ -\mu
v/\sqrt{2}&m_{S}^{2}\end{pmatrix},$ (2.11)

where $m_{R}$ and $m_{\xi}$ are the bare masses of $R$ and $\xi$,
respectively. These two mass matrices can be diagonalized by performing the
following rotations

$\displaystyle\text{diag.}\left(M_{1}^{2},M_{2}^{2}\right)$
$\displaystyle=U_{\phi}M^{2}_{\chi^{2/3}}U^{T}_{\phi},$ (2.12)
$\displaystyle\text{diag.}\left(M_{3}^{2},M_{4}^{2}\right)$
$\displaystyle=U_{\theta}M^{2}_{\chi^{1/3}}U^{T}_{\theta},$ (2.13)

where

$\displaystyle U_{x}=\begin{pmatrix}c_{x}&s_{x}\\\ -s_{x}&c_{x}\end{pmatrix}.$
(2.14)

Here $c_{x},s_{x}$ stand for $\cos x,\sin x$. In terms of scalar mass
parameters, the two mixing angles are given by

$\displaystyle\tan 2\theta=\frac{-2\lambda
v^{2}}{m_{\xi}^{2}-m_{S}^{2}},\quad\tan 2\phi=\frac{-\sqrt{2}\mu
v}{m_{\xi}^{2}-m_{R}^{2}}.$ (2.15)

Furthermore, the mass eigenvalues of $\chi^{2/3}_{1,2}$ and $\chi^{1/3}_{1,2}$
are found to be

$\displaystyle
M_{1,2}^{2}=\frac{1}{2}\left[m_{\xi}^{2}+m_{R}^{2}\pm\sqrt{(m_{\xi}^{2}-m_{R}^{2})^{2}+2\mu^{2}v^{2}}\right],$
(2.16) $\displaystyle
M_{3,4}^{2}=\frac{1}{2}\left[m_{\xi}^{2}+m_{S}^{2}\pm\sqrt{(m_{\xi}^{2}-m_{S}^{2})^{2}+4\lambda^{2}v^{4}}\right].$
(2.17)

Note that $M^{2}_{1,2}$ and $M^{2}_{3,4}$ can be the larger or the smaller of
the two mass eigenvalues. They are defined such that

$\displaystyle
M_{1}^{2}\cos^{2}\phi+M_{2}\sin^{2}\phi=M_{3}^{2}\cos^{2}\theta+M_{4}\sin^{2}\theta.$
(2.18)

In terms of mass eigenvalues, we come out with an alternative way of writing
Eq. (2.15), namely

$\displaystyle\sin 2\theta=\frac{-2\lambda
v^{2}}{M_{3}^{2}-M_{4}^{2}}\quad{\rm and}\quad\sin 2\phi=\frac{-\sqrt{2}\mu
v}{M_{1}^{2}-M_{2}^{2}},$ (2.19)

from which both $\lambda$ and $\mu$ can be written in terms of mass
eigenvalues

$\displaystyle\sqrt{2}\lambda v=\frac{-(M_{3}^{2}-M_{4}^{2})}{v}\sin 2\theta,$
(2.20) $\displaystyle\mu=\frac{-(M_{1}^{2}-M_{2}^{2})}{v}\sin 2\phi.$ (2.21)

Having rotated the scalars into their mass eigenstates, we now do the same for
Yukawa interactions of Eq. (2.9). Without loss of generality, we can define
all couplings in Eq. (2.9) in the charged lepton mass diagonal basis. If we
assume further that the up-type quarks be diagonal as well, Eq. (2.9) becomes

$\displaystyle\mathcal{L}\supset$
$\displaystyle~{}f^{L}_{ij}u^{c}_{i}\left[(U_{\psi})_{a2}\chi^{5/3}_{a}e_{Lj}-(U_{\phi})_{a2}\chi^{2/3}_{a}\nu_{Lj}\right]$
$\displaystyle+f^{R}_{ij}\left[(U_{\psi})_{a2}\chi^{-5/3}_{a}u_{Li}+(U_{\phi})_{a2}\chi^{-2/3}_{a}V_{ik}d_{Lk}\right]e^{c}_{j}$
$\displaystyle+y^{L}_{ij}(U_{\theta})_{a2}\left(u_{Li}e_{Lj}-V_{ik}d_{Lk}\nu_{Lj}\right)\chi^{1/3}_{a}$
$\displaystyle+y^{R}_{ij}(U_{\theta})_{a2}u^{c}_{i}e^{c}_{j}\chi^{-1/3}_{a},$
(2.22)

where $V$ is the Cabibbo-Kobayashi-Maskawa mixing matrix. Note that the
results are not affected by changing the up-type to the down-type diagonal
basis. Since two choices of the Yukawa coupling textures, namely, “up-type”
and “down-type” mass-diagonal basis are widely used in the literature, in the
following text, we provide the neutrino mass formula in both these scenarios.

As one can see from Fig. 1, the neutrino mass generation requires interaction
between LQs and $W$ boson, originating from the $SU(2)$ covariant derivatives

$\displaystyle
D_{\mu}R=\left(\partial_{\mu}-i\frac{g}{2}W^{a}_{\mu}\sigma^{a}-i\frac{7}{6}g^{\prime}B_{\mu}\right)R,$
$\displaystyle
D_{\mu}\xi=\partial_{\mu}\xi-i\frac{g}{2}\left[W_{\mu}^{a}\sigma^{a},\xi\right]-i\frac{2g^{\prime}}{3}B_{\mu},$
(2.23)

where $g,g^{\prime}$ are the $SU(2)$ and $U(1)_{Y}$ gauge couplings and
$\sigma^{a}$ are Pauli matrices. Based from Eq. (2.23), the
$\xi^{2/3}$–$\xi^{1/3}$–$W$ vertex can be derived from the triplet kinetic
term, that is,

$\displaystyle(D_{\mu}\xi)^{\dagger}(D^{\mu}\xi)$ $\displaystyle\to
ig\left[\xi^{-1/3}\partial_{\mu}\xi^{-2/3}-(\partial_{\mu}\xi^{-1/3})\xi^{-2/3}\right]W^{+\mu}.$
(2.24)

After rotating the corresponding fields to their mass eigenstates, we obtain

$\displaystyle\mathcal{L}^{\rm cc}_{\rm scalar}=$
$\displaystyle~{}~{}ig(U_{\theta})_{a1}(U_{\phi})_{b1}$
$\displaystyle\times\left[(\partial_{\mu}\chi^{-1/3}_{a})\chi^{-2/3}_{b}-\chi^{-1/3}_{a}(\partial_{\mu}\chi^{-2/3}_{b})\right]W^{+\mu}.$
(2.25)

In this model, we work in the general $R_{\xi}$ gauge, so we need to know the
LQ interactions with the Goldstone boson. By using Eqs. (2.10) and (2.19), the
LQs–Goldstone interactions are found to be

$\displaystyle\mathcal{L}$ $\displaystyle\supset
g\left(\frac{M_{b}^{2}-M_{a+2}^{2}}{m_{W}}\right)(U_{\theta})_{a1}(U_{\phi})_{b1}\chi^{-1/3}_{a}\chi^{-2/3}_{b}H^{+}+{\rm
h.c.}.$ (2.26)

##  3 Neutrino mass formula

Armed with all interactions given in Eqs. (2.22), (2.25), and (2.26), we are
ready to construct diagrams leading to neutrino masses. Since there are three
different coupling products, there will be three subgroups contributing to
neutrino masses. Each contribution is presented in Fig. 2. Since neutrino
masses are Majorana in nature, in addition to the diagrams shown, there is
another set of diagrams with internal particles replaced by their charge
conjugates. The sum of the two sets of diagrams will result in the neutrino
mass matrix being symmetric.

Figure 2: All two–loop diagrams leading to neutrino mass generation. These
Feynman diagrams are presented in the mass eigenstate basis.

As mentioned before, in evaluating the neutrino mass diagrams, we work in the
$R_{\xi}$ gauge. Therefore, the neutrino mass diagrams will contain gauge
parameter $\xi$ dependent terms (not to be confused with $\xi$ multiplet).
Such terms will later disappear after we sum over all diagrams. The resulted
neutrino mass matrix in the up-type quark diagonal basis can be written as

$\displaystyle(\mathcal{M}_{\nu})_{ji}=\frac{3g^{2}m_{t}}{\sqrt{2}(16\pi^{2})^{2}}$
$\displaystyle\left\\{\left[y^{L}_{mj}V_{ml}V^{\ast}_{kl}(D_{u})_{k}f^{L}_{ki}+f^{L}_{kj}(D_{u})_{k}V^{\ast}_{kl}V_{ml}y^{L}_{mi}\right]\hat{I}_{jkl}\right.$
$\displaystyle\left.+\frac{m_{\tau}}{m_{t}}(D_{\ell})_{j}\left[y^{R\ast}_{kj}f^{L}_{ki}+f^{L}_{kj}y^{R*}_{ki}\right]\tilde{I}_{jk}\right.$
$\displaystyle\left.+\frac{m_{\tau}}{m_{t}}(D_{\ell})_{j}\left[f^{R\ast}_{kj}V^{\ast}_{kl}V_{ml}y^{L}_{mi}+y^{L}_{mj}V_{ml}V^{\ast}_{kl}f^{R\ast}_{ki}\right]\bar{I}_{jl}\right\\}.$
(3.27)

Here the factor of 3 accounts for the exchange of color states inside the
loops, whereas $D_{u}$ and $D_{\ell}$ are the normalized mass matrices of up-
type quarks and charged leptons, respectively

$\displaystyle
D_{u}=\text{diag.}\left(\frac{m_{u}}{m_{t}},\frac{m_{c}}{m_{t}},1\right),~{}~{}D_{\ell}=\text{diag.}\left(\frac{m_{e}}{m_{\tau}},\frac{m_{\mu}}{m_{\tau}},1\right).$
(3.28)

Similarly, in the basis where down-type quark mass matrix is diagonal, we have

$\displaystyle(\mathcal{M}_{\nu})_{ji}=$
$\displaystyle\frac{3g^{2}m_{t}}{\sqrt{2}(16\pi^{2})^{2}}\left\\{\left[y^{L}_{lj}V^{\ast}_{kl}(D_{u})_{k}f^{L}_{ki}+f^{L}_{kj}(D_{u})_{k}V^{\ast}_{kl}y^{L}_{li}\right]\hat{I}_{jkl}\right.$
$\displaystyle+\frac{m_{\tau}}{m_{t}}(D_{\ell})_{j}\left[y^{R\ast}_{kj}f^{L}_{ki}+f^{L}_{kj}y^{R*}_{ki}\right]\tilde{I}_{jk}$
$\displaystyle\left.+\frac{m_{\tau}}{m_{t}}(D_{\ell})_{j}\left[f^{R\ast}_{lj}y^{L}_{li}+y^{L}_{lj}f^{R\ast}_{li}\right]\bar{I}_{jl}\right\\}.$
(3.29)

One should note that Eqs. (3.27) and (3.29) are equivalent, as one can recover
the latter, for instance, by redefining $y^{L}\to V^{\ast}y^{L}$ and $f^{R}\to
V^{\ast}f^{R}$. This reflects the basis independence mentioned previously.

The loop integrals shown in Eqs. (3.27) and (3.29), i.e.,
$\hat{I}_{jkl},\tilde{I}_{jk}$, and $\bar{I}_{jl}$, indicate the contribution
of each subgroup. Each of them is defined as

$\displaystyle\hat{I}_{jkl}=$
$\displaystyle(16\pi^{2})^{2}\left[I^{(1)}_{kl}+I^{(2)}_{kl}+I^{(3)}_{jk}+I^{(4)}_{jk}\right],$
$\displaystyle\tilde{I}_{jk}=$
$\displaystyle(16\pi^{2})^{2}\left[I^{(5)}_{jk}+I^{(6)}_{jk}\right],$ (3.30)
$\displaystyle\bar{I}_{jl}=$
$\displaystyle(16\pi^{2})^{2}\left[I^{(7)}_{jl}+I^{(8)}_{jl}\right],$

with $I^{(n)}_{ij}$ denoting the dimensionless loop function for the $n$-th
diagram, that is,

$\displaystyle I^{(1)}_{kl}=~{}$
$\displaystyle-(U_{\theta})_{a1}(U_{\theta})_{a2}(U_{\phi})_{b1}(U_{\phi})_{b2}\int\frac{d^{4}k}{(2\pi)^{4}}\int\frac{d^{4}q}{(2\pi)^{4}}\frac{1}{k^{2}-m^{2}_{W}}$
$\displaystyle\times\frac{1}{(q+k)^{2}-m^{2}_{u_{k}}}\frac{1}{(q+k)^{2}-M^{2}_{b}}\frac{1}{q^{2}-m^{2}_{d_{l}}}\frac{1}{q^{2}-M^{2}_{a+2}}$
$\displaystyle\times\left[\not{q}(2\not{q}+\not{k})+\frac{\not{q}\not{k}}{k^{2}}\left(-1+\xi\frac{k^{2}-m^{2}_{W}}{k^{2}-\xi
m^{2}_{W}}\right)k\cdot(2q+k)\right]\;,$ (3.31) $\displaystyle
I^{(2)}_{kl}=~{}$
$\displaystyle(U_{\theta})_{a1}(U_{\theta})_{a2}(U_{\phi})_{b1}(U_{\phi})_{b2}\int\frac{d^{4}k}{(2\pi)^{4}}\int\frac{d^{4}q}{(2\pi)^{4}}\frac{1}{k^{2}}\left(\frac{M_{b}^{2}-M_{a+2}^{2}}{m^{2}_{W}}\right)\left(1+\xi\frac{m^{2}_{W}}{k^{2}-\xi
m^{2}_{W}}\right)$
$\displaystyle\times\left(1+\frac{\not{q}\not{k}}{q^{2}-m_{d_{l}}^{2}}\right)\frac{1}{(q+k)^{2}-m^{2}_{u_{k}}}\frac{1}{q^{2}-M^{2}_{a+2}}\frac{1}{(q+k)^{2}-M^{2}_{b}}\;,$
(3.32) $\displaystyle I^{(3)}_{jk}=~{}$
$\displaystyle-(U_{\theta})_{a1}(U_{\theta})_{a2}(U_{\phi})_{b1}(U_{\phi})_{b2}\int\frac{d^{4}k}{(2\pi)^{4}}\int\frac{d^{4}q}{(2\pi)^{4}}\frac{1}{k^{2}}\left(1+\frac{m^{2}_{e_{j}}}{k^{2}-m^{2}_{e_{j}}}\right)$
$\displaystyle\times\frac{1}{k^{2}-m^{2}_{W}}\frac{1}{q^{2}-M^{2}_{a+2}}\frac{1}{(q+k)^{2}-M^{2}_{b}}\frac{1}{(q+k)^{2}-m^{2}_{u_{k}}}$
$\displaystyle\times\left[(2\not{q}+\not{k})\not{k}+\left(-1+\xi\frac{k^{2}-m^{2}_{W}}{k^{2}-\xi
m^{2}_{W}}\right)k\cdot(2q+k)\right]\;,$ (3.33) $\displaystyle
I^{(4)}_{jk}=~{}$
$\displaystyle(U_{\theta})_{a1}(U_{\theta})_{a2}(U_{\phi})_{b1}(U_{\phi})_{b2}\int\frac{d^{4}k}{(2\pi)^{4}}\int\frac{d^{4}q}{(2\pi)^{4}}\frac{1}{k^{2}}\left(\frac{m^{2}_{e_{j}}}{k^{2}-m^{2}_{e_{j}}}\right)\left(\frac{M_{b}^{2}-M_{a+2}^{2}}{m^{2}_{W}}\right)$
$\displaystyle\times\left(1+\xi\frac{m^{2}_{W}}{k^{2}-\xi
m^{2}_{W}}\right)\frac{1}{q^{2}-M^{2}_{a+2}}\frac{1}{(q+k)^{2}-M^{2}_{b}}\frac{1}{(q+k)^{2}-m^{2}_{u_{k}}}\;,$
(3.34) $\displaystyle I^{(5)}_{jk}=~{}$
$\displaystyle-(U_{\theta})_{a1}(U_{\theta})_{a2}(U_{\phi})_{b1}(U_{\phi})_{b2}\int\frac{d^{4}k}{(2\pi)^{4}}\int\frac{d^{4}q}{(2\pi)^{4}}$
$\displaystyle\times\frac{1}{k^{2}-m_{W}^{2}}\frac{1}{k^{2}-m_{e_{j}}^{2}}\frac{1}{q^{2}-M_{a+2}^{2}}\frac{1}{(q+k)^{2}-M_{b}^{2}}\frac{1}{(q+k)^{2}-m_{u_{k}}^{2}}$
$\displaystyle\times\left[(2\not{q}+\not{k})(\not{q}+\not{k})+\frac{\not{k}(\not{q}+\not{k})}{k^{2}}\left(-1+\xi\frac{k^{2}-m_{W}^{2}}{k^{2}-\xi
m_{W}^{2}}\right)k\cdot(2q+k)\right]\;,$ (3.35) $\displaystyle
I^{(6)}_{jk}=~{}$
$\displaystyle(U_{\theta})_{a1}(U_{\theta})_{a2}(U_{\phi})_{b1}(U_{\phi})_{b2}\int\frac{d^{4}k}{(2\pi)^{4}}\int\frac{d^{4}q}{(2\pi)^{4}}\frac{\not{k}(\not{q}+\not{k})}{k^{2}}\left(\frac{M_{b}^{2}-M_{a+2}^{2}}{m_{W}^{2}}\right)$
$\displaystyle\times\left(1+\xi\frac{m_{W}^{2}}{k^{2}-\xi
m_{W}^{2}}\right)\frac{1}{k^{2}-m_{e_{j}}^{2}}\frac{1}{q^{2}-M_{a+2}^{2}}\frac{1}{(q+k)^{2}-M_{b}^{2}}\frac{1}{(q+k)^{2}-m_{u_{k}}^{2}}\;,$
(3.36) $\displaystyle I^{(7)}_{jl}=~{}$
$\displaystyle(U_{\theta})_{a1}(U_{\theta})_{a2}(U_{\phi})_{b1}(U_{\phi})_{b2}\int\frac{d^{4}k}{(2\pi)^{4}}\int\frac{d^{4}q}{(2\pi)^{4}}\frac{1}{k^{2}-m_{W}^{2}}\frac{1}{k^{2}-m_{e_{j}}^{2}}$
$\displaystyle\times\frac{1}{(q+k)^{2}-m_{d_{l}}^{2}}\frac{1}{(q+k)^{2}-M_{a+2}^{2}}\frac{1}{q^{2}-M_{b}^{2}}$
$\displaystyle\times\left[(2\not{q}+\not{k})(\not{q}+\not{k})+\frac{\not{k}(\not{q}+\not{k})}{k^{2}}\left(-1+\xi\frac{k^{2}-m_{W}^{2}}{k^{2}-\xi
m_{W}^{2}}\right)k\cdot(2q+k)\right]\;,$ (3.37) $\displaystyle
I^{(8)}_{jl}=~{}$
$\displaystyle(U_{\theta})_{a1}(U_{\theta})_{a2}(U_{\phi})_{b1}(U_{\phi})_{b2}\int\frac{d^{4}k}{(2\pi)^{4}}\int\frac{d^{4}q}{(2\pi)^{4}}\frac{\not{k}(\not{q}+\not{k})}{k^{2}}\left(\frac{M_{b}^{2}-M_{a+2}^{2}}{m_{W}^{2}}\right)$
$\displaystyle\times\left(1+\xi\frac{m_{W}^{2}}{k^{2}-\xi
m_{W}^{2}}\right)\frac{1}{k^{2}-m_{e_{j}}^{2}}\frac{1}{(q+k)^{2}-m_{d_{l}}^{2}}\frac{1}{(q+k)^{2}-M_{a+2}^{2}}\frac{1}{q^{2}-M_{b}^{2}}\;.$
(3.38)

The cancellation of terms containing the gauge parameter $\xi$ can be inferred
directly from Eqs. (3.31)-(3.38). To see how this cancellation takes place, it
is desirable to express $k\cdot(2q+k)$, which appears in all $W$-mediated
diagrams, as

$\displaystyle k\cdot(2q+k)=$
$\displaystyle\left[(q+k)^{2}-M_{b}^{2}-q^{2}+M_{a+2}^{2}\right]+M_{b}^{2}-M_{a+2}^{2}.$
(3.39)

Terms inside parentheses will cancel the LQ propagators, particularly those
appearing in diagrams 1, 3, and 5. Thus, they will vanish by the orthogonality
of LQ mixing matrices. The remaining terms, which are proportional to
$M_{b}^{2}-M_{a+2}^{2}$, will make such loop terms have the same coefficients
but opposite signs with the corresponding Goldstone loop integrals, allowing
the cancellation of $\xi$-dependent terms. For diagram 7, due to momentum
switch between $\chi^{1/3}$ and $\chi^{2/3}$ (see Fig. 2), we have instead
$k\cdot(2q+k)\to M_{a+2}^{2}-M_{b}^{2}$. The cancellation of $\xi$-dependent
terms in this case too can be foreseen right away. The gauge parameter
cancellation indicates further that all two-loop diagrams presented in Fig. 2
are the complete set of diagrams generating neutrino masses at the lowest
order.

We are, then, left with gauge-independent terms. It is straightforward to
evaluate the integrals, from which we get

$\displaystyle\hat{I}_{jkl}=~{}$ $\displaystyle-\frac{1}{4}\sin 2\theta\sin
2\phi\sum_{a,b=1}^{2}(-1)^{a+b}\frac{1}{(M_{b}^{2}-m_{u_{k}}^{2})(M_{a+2}^{2}-m_{d_{l}}^{2})}\int_{0}^{1}dx\int_{0}^{\infty}dt\frac{t}{t+m_{W}^{2}}$
$\displaystyle\times\left\\{xt\left[6x-5+\left(\frac{M_{b}^{2}-M_{a+2}^{2}}{m_{W}^{2}}\right)\right]\left[\ln\frac{\Delta(x,t;M_{b},M_{a+2})}{\Delta(x,t;M_{b},m_{d_{l}})}-\ln\frac{\Delta(x,t;m_{u_{k}},M_{a+2})}{\Delta(x,t;m_{u_{k}},m_{d_{l}})}\right]\right.$
$\displaystyle\quad-4\left[A(x;M_{b},M_{a+2})\ln\frac{\Delta(x,t;M_{b},M_{a+2})}{m_{W}^{2}}+A(x;m_{u_{k}},m_{d_{l}})\ln\frac{\Delta(x,t;m_{u_{k}},m_{d_{l}})}{m_{W}^{2}}\right.$
$\displaystyle\quad\left.-A(x;M_{b},m_{d_{l}})\ln\frac{\Delta(x,t;M_{b},m_{d_{l}})}{m_{W}^{2}}-A(x;m_{u_{k}},M_{a+2})\ln\frac{\Delta(x,t;m_{u_{k}},M_{a+2})}{m_{W}^{2}}\right]$
$\displaystyle\quad+\left(\frac{M_{a+2}^{2}-m_{d_{l}}^{2}}{t+m_{e_{j}}^{2}}\right)\left[(2x-1)+\left(\frac{M_{b}^{2}-M_{a+2}^{2}}{m_{W}^{2}}\right)\right]t\ln\frac{\Delta(x,t;M_{b},M_{a+2})}{\Delta(x,t;m_{u_{k}},M_{a+2})}\bigg{\\}},$
(3.40) $\displaystyle\tilde{I}_{jk}=~{}$ $\displaystyle\frac{1}{4}\sin
2\theta\sin
2\phi\sum_{a,b=1}^{2}(-1)^{a+b}\frac{1}{M_{b}^{2}-m_{u_{k}}^{2}}\int_{0}^{1}dx\int_{0}^{\infty}dt\frac{t}{(t+m_{W}^{2})(t+m_{e_{j}}^{2})}$
$\displaystyle\times\left\\{\left[1-6x-\left(\frac{M_{b}^{2}-M_{a+2}^{2}}{m_{W}^{2}}\right)\right](1-x)t\ln\frac{\Delta(x,t;M_{b},M_{a+2})}{\Delta(x,t;m_{u_{k}},M_{a+2})}\right.$
$\displaystyle\left.-4A(x;M_{b},M_{a+2})\ln\frac{\Delta(x,t;M_{b},M_{a+2})}{m_{W}^{2}}+4A(x;m_{u_{k}},M_{a+2})\ln\frac{\Delta(x,t;m_{u_{k}},M_{a+2})}{m_{W}^{2}}\right\\},$
(3.41) $\displaystyle\bar{I}_{jl}=~{}$ $\displaystyle-\frac{1}{4}\sin
2\theta\sin
2\phi\sum_{a,b=1}^{2}(-1)^{a+b}\frac{1}{M_{a+2}^{2}-m_{d_{l}}^{2}}\int_{0}^{1}dx\int_{0}^{\infty}dt\frac{t}{(t+m_{W}^{2})(t+m_{e_{j}}^{2})}$
$\displaystyle\times\left\\{\left[1-6x-\left(\frac{M_{a+2}^{2}-M_{b}^{2}}{m_{W}^{2}}\right)\right](1-x)t\ln\frac{\Delta(x,t;M_{a+2},M_{b})}{\Delta(x,t;m_{d_{l}},M_{b})}\right.$
$\displaystyle\left.-4A(x;M_{a+2},M_{b})\ln\frac{\Delta(x,t;M_{a+2},M_{b})}{m_{W}^{2}}+4A(x;m_{d_{l}},M_{b})\ln\frac{\Delta(x,t;m_{d_{l}},M_{b})}{m_{W}^{2}}\right\\},$
(3.42)

where only terms relevant to neutrino masses are kept. In those loop integral
expressions, we have introduced a parameter

$\displaystyle\Delta(x,t;m,M)\equiv x(1-x)t+A(x;m,M),$ (3.43)

with $A(x;m,M)\equiv xm^{2}+(1-x)M^{2}$.

Figure 3: The plot of loop integral of each diagram. The solid line represents
the case with the top quark inside the loop, while the dashed line represents
the case with the charm quark. It is obvious that the loop integrals depend
mildly on fermion masses. Here we use $M_{S}=1200$ GeV, $M_{R}=1501$ GeV,
$\mu=10$ GeV, and $\lambda=1$.

All loop integrals are finite, and thus can be calculated numerically. In
addition, the contributions of light fermion masses are negligible. Therefore,
they can be simply omitted from the integrals, which is demonstrated in Fig.
3.

##  4 Lepton flavor violation

All couplings presented in Eq. (2.9) naturally generate lepton-flavor
violating (LFV) processes, which are strongly constrained. In this section, we
will use those constraints to scrutinize our model. For the sake of
compactness, we write the Eq. (2.9) as

$\displaystyle{\cal
L}=\bar{\ell}_{i}\left(\lambda_{L}^{ij}P_{R}+\lambda_{R}^{ij}P_{L}\right)q_{j}\phi^{\ast}+{\rm
h.c.}.$ (4.44)

In this notation, we define $\ell$ as the charged leptons with $Q_{\ell}=-1$
and subscripts $L,R$ on $\lambda$’s indicate the chirality of such fields. It
is then straightforward to see that there are three types of $\lambda$’s
within this model. The mapping of $\lambda_{L,R}$ into couplings given in Eq.
(2.9) is shown in Table I.

$q,\phi$ | $\lambda_{L},\lambda_{R}$
---|---
Up-mass diagonal | Down-mass diagonal
$u,R^{5/3}$ | $(f^{L})^{\dagger},(f^{R})^{T}$ | $(f)^{L})^{\dagger},(f^{R})^{T}V^{\dagger}$
$u^{c},S^{1/3}$ | $(y^{L})^{\dagger},(y^{R})^{T}$ | $(y^{L})^{\dagger}V^{T},(y^{R})^{T}$
$d,R^{2/3}$ | $0,(f^{R})^{T}$ | $0,(f^{R})^{T}V$
Table I: Mapping of Yukawa couplings of Eq. (2.9) into $\lambda_{L,R}$. Note
that, in this table, $u^{c}$ is defined as charge conjugate of up-type quarks,
namely $u^{c}=C\bar{u}^{T}$.

###  4.1 $\ell_{i}\to\ell_{k}+\gamma$ decay

Due to flavor-violating nature of Eq. (2.9), lepton flavor violating processes
in general are expected to occur within this model. The first process we
consider is $\ell_{i}\to\ell_{k}+\gamma^{\ast}$ transition, whose effective
Lagrangian

$\displaystyle\mathcal{L}_{\ell_{i}\to\ell_{k}+\gamma^{*}}=$
$\displaystyle~{}-m_{\ell_{i}}\bar{\ell}_{k}\sigma^{\alpha\beta}\left(\tfrac{e}{2}A_{2L}^{\ast}P_{L}+\tfrac{e}{2}A_{2R}^{\ast}P_{R}\right)\ell_{i}F_{\alpha\beta}$
$\displaystyle~{}-\bar{\ell}_{k}\gamma^{\alpha}\left(eA_{1L}^{\ast}P_{L}+eA_{1R}^{\ast}P_{R}\right)\ell_{i}A^{\beta}\left(q^{2}g_{\alpha\beta}-q_{\alpha}q_{\beta}\right)+{\rm
h.c.},$ (4.45)

where $F_{\alpha\beta}=\partial_{\alpha}A_{\beta}-\partial_{\beta}A_{\alpha}$
is the electromagnetic field tensor and $q=p_{\ell_{i}}-p_{\ell_{k}}$ is the
photon momentum transfer. At the lowest order, this kind of processes arises
through penguin-type diagrams exchanging LQs. It is worth noting that, owing
to the Ward-Takahashi identity, only dipole terms survive in a lepton decay
into an on-shell photon. Its decay width is given by Lavoura:2003xp

$\displaystyle\Gamma(\ell_{i}\to\ell_{k}\gamma)=$
$\displaystyle~{}\frac{\alpha_{em}}{4}m_{\ell_{i}}^{5}\left(|A_{2L}|^{2}+|A_{2R}|^{2}\right),$
(4.46)

where $\alpha_{em}=e^{2}/4\pi$ is the electromagnetic fine-structure constant,
$m_{\ell_{i}}$ is the decaying lepton mass, and

$\displaystyle A_{2L}=$
$\displaystyle~{}\frac{3}{16\pi^{2}}\sum_{\lambda,\phi}\frac{1}{M_{\phi}^{2}}\bigg{\\{}\left[\lambda^{ij}_{R}\lambda^{kj\ast}_{R}+\frac{m_{\ell_{k}}}{m_{\ell_{i}}}\lambda^{ij}_{L}\lambda^{kj\ast}_{L}\right]\left[Q_{q}F_{1}(x_{j})+Q_{\phi}F_{2}(x_{j})\right]$
$\displaystyle\qquad+\frac{m_{q_{j}}}{m_{\ell_{i}}}\lambda^{ij}_{L}\lambda^{kj\ast}_{R}\left[Q_{q}F_{3}(x_{j})+Q_{\phi}F_{4}(x_{j})\right]\bigg{\\}},$
$\displaystyle A_{2R}=$ $\displaystyle~{}A_{2L}(L\leftrightarrow R).$ (4.47)

In the above equation, 3 is the color factor and
$x_{j}=m_{q_{j}}^{2}/M_{\phi}^{2}$. The summation is performed over all
possible couplings and leptoquark fields, as given in Table I. Quantities
$Q_{q}$ and $Q_{\phi}$ are the corresponding quark and LQ electric charges,
which obey $Q_{\ell}+Q_{\phi}=Q_{q}$ with $Q_{\ell}=-1$. Functions $F_{1}(x)$
and $F_{3}(x)$ are evaluated from diagrams emitting a photon from the quark
line, whereas $F_{2}(x)$ and $F_{4}(x)$ from diagrams emitting a photon from
the LQ line. They all are given by

$\displaystyle F_{1}(x)=$ $\displaystyle~{}\frac{2+3x-6x^{2}+x^{3}+6x\ln
x}{6(1-x)^{4}},$ $\displaystyle F_{2}(x)=$
$\displaystyle~{}\frac{1-6x+3x^{2}+2x^{3}-6x^{2}\ln x}{6(1-x)^{4}},$
$\displaystyle F_{3}(x)=$ $\displaystyle~{}\frac{-3+4x-x^{2}-2\ln
x}{(1-x)^{3}},$ $\displaystyle F_{4}(x)=$ $\displaystyle~{}\frac{1-x^{2}+2x\ln
x}{(1-x)^{3}}.$ (4.48)

Due to the possibility of simultaneous existence of both $\lambda_{L}$ and
$\lambda_{R}$ in this model, see Table I, we can have a chirality-enhanced
process, especially when the top quark is inside the loop. This will lead to
severe constraints on the Yukawa couplings. Current and future rates of this
kind of processes are presented in Table II.

$\ell_{i}\to\ell_{k}\gamma$ | Present bound | Future sensitivity
---|---|---
$\mu\to e\gamma$ |  $4.2\times 10^{-13}$ MEG:2016leq  |  $6\times 10^{-14}$ Baldini:2013ke
$\tau\to e\gamma$ |  $3.3\times 10^{-8}$ BaBar:2009hkt  |  $\sim 10^{-9}$ Aushev:2010bq
$\tau\to\mu\gamma$ |  $4.4\times 10^{-8}$ BaBar:2009hkt  |  $\sim 10^{-9}$ Aushev:2010bq
Table II: Current experimental bounds on the $BR(\ell_{i}\to\ell_{k}\gamma)$.
Future sensitivities are presented on the last column.

###  4.2 Lepton 3-body decay

This kind of processes also occurs at loop level, consisting of photon- and
$Z$-penguin diagrams as well as the box diagrams. For the photon-mediated
processes, one just needs to attach the photon leg in Eq. (4.45) with a
$\ell^{+}_{l}\ell^{-}_{l}$ pair. Now, the photon is off shell, both $A_{2L,R}$
and $A_{1L,R}$ contribute to the photon-induced effective Lagrangian, written
as Kuno:1999jp

$\displaystyle{\cal
L}_{\ell_{i}^{-}\to\ell_{k}^{-}\ell_{l}^{+}\ell_{l}^{-}}^{\gamma-\rm
penguin}=$
$\displaystyle~{}-m_{\ell_{i}}\bar{\ell}_{k}\sigma^{\alpha\beta}\left(\tfrac{e}{2}A_{2L}^{\ast}P_{L}+\tfrac{e}{2}A_{2R}^{\ast}P_{R}\right)\ell_{i}F_{\alpha\beta}$
$\displaystyle~{}\left[\bar{\ell}_{k}\gamma^{\alpha}\left(e^{2}A_{1L}^{\ast}P_{L}+e^{2}A_{1R}^{\ast}P_{R}\right)\ell_{i}\right]\left[\bar{\ell}_{l}\gamma_{\alpha}P_{L}\ell_{l}+\bar{\ell}_{l}\gamma_{\alpha}P_{R}\ell_{l}\right].$
(4.49)

It is also straightforward to evaluate $A_{1L},A_{1R}$, which are given by

$\displaystyle A_{1L}=$
$\displaystyle~{}\frac{3}{16\pi^{2}}\sum_{\lambda,\phi}\frac{1}{M_{\phi}^{2}}\lambda_{L}^{ij}\lambda^{kj\ast}_{L}\left[Q_{q}G_{1}(x_{j})+Q_{\phi}G_{2}(x_{j})\right],$
$\displaystyle A_{1R}=$ $\displaystyle~{}A_{1L}(L\leftrightarrow R),$ (4.50)

with

$\displaystyle G_{1}(x)=$
$\displaystyle~{}\frac{16-45x+36x^{2}-7x^{3}+6(2-3x)\ln x}{36(1-x)^{4}},$
$\displaystyle G_{2}(x)=$
$\displaystyle~{}\frac{2-9x+18x^{2}-11x^{3}+6x^{3}\ln x}{36(1-x)^{4}}.$ (4.51)

In addition to the aforementioned photonic diagrams, one can also have
$Z$-penguin interactions. They are given by

$\displaystyle{\cal
L}_{\ell_{i}^{-}\to\ell_{k}^{-}\ell_{l}^{+}\ell_{l}^{-}}^{Z-\rm penguin}=$
$\displaystyle\left[\bar{\ell}_{k}\gamma^{\alpha}\left(e^{2}Z_{L}^{\ast}P_{L}+e^{2}Z_{R}^{\ast}P_{R}\right)\ell_{i}\right]\left[g_{L}^{(\ell)}\bar{\ell}_{l}\gamma_{\alpha}P_{L}\ell_{l}+g_{R}^{(\ell)}\bar{\ell}_{l}\gamma_{\alpha}P_{R}\ell_{l}\right]+{\rm
h.c.},$ (4.52)

with

$\displaystyle Z_{L}=$
$\displaystyle~{}\frac{3}{16\pi^{2}}\sum_{\lambda,\phi}\frac{\lambda^{ij}_{L}\lambda^{kj\ast}_{L}}{m_{Z}^{2}\cos^{2}\theta_{W}\sin^{2}\theta_{W}}$
$\displaystyle\qquad\qquad\times\left\\{g_{R}^{(q_{j})}C_{1}(x_{j})+g_{L}^{(q_{j})}C_{2}(x_{j})-\left[g^{(\phi)}+g^{(\ell_{k})}_{L}\right]\left[C_{1}(x_{j})+C_{2}(x_{j})\right]\right\\},$
$\displaystyle Z_{R}=$ $\displaystyle~{}Z_{L}(L\leftrightarrow R).$ (4.53)

In deriving $Z_{L,R}$, we have neglected terms proportional to $m_{\ell_{i}}$.
Here $g^{(f)}_{L,R}=T_{3f_{L,R}}-Q_{f}\sin^{2}\theta_{W}$, with $\theta_{W}$
being the weak mixing angle, and

$\displaystyle C_{1}(x)$ $\displaystyle=$
$\displaystyle\frac{-1+x^{2}+2(-2+x)x\ln x}{4(1-x)^{2}},$ $\displaystyle
C_{2}(x)$ $\displaystyle=$ $\displaystyle\frac{x(1-x+\ln x)}{(1-x)^{2}}.$
(4.54)

Similarly, for the box diagrams, we have

$\displaystyle{\cal
L}_{\ell_{i}^{-}\to\ell_{k}^{-}\ell_{l}^{+}\ell_{l}^{-}}^{\rm box}=$
$\displaystyle~{}~{}e^{2}B_{1L}^{\ast}[\bar{\ell}_{k}\gamma^{\alpha}P_{L}\ell_{i}][\bar{\ell}_{l}\gamma_{\alpha}P_{L}\ell_{l}]+e^{2}B_{2L}^{\ast}[\bar{\ell}_{k}\gamma^{\alpha}P_{L}\ell_{i}][\bar{\ell}_{l}\gamma_{\alpha}P_{R}\ell_{l}]$
$\displaystyle+e^{2}B_{3L}^{\ast}[\bar{\ell}_{k}P_{L}\ell_{i}][\bar{\ell}_{l}P_{L}\ell_{l}]+(L\leftrightarrow
R)+{\rm h.c.}$ (4.55)

Note that we do not list box operators in the form of $(S\mp P)\times(S\pm
P)$, with $S,P$ indicating scalar and pseudoscalar bilinears. This is because
they can always be Fierz reordered into the form of $(V\mp A)\times(V\pm A)$
operator, which is already included. The corresponding Wilson’s coefficients
are found to be

$\displaystyle e^{2}B_{1L}=$
$\displaystyle~{}\frac{3}{16\pi^{2}}\sum_{\lambda,\phi}\frac{1}{M_{\phi}^{2}}\lambda^{ij}_{L}\lambda^{kj\ast}_{L}|\lambda_{L}^{kn}|^{2}b_{1}(x_{j},x_{n}),$
$\displaystyle e^{2}B_{2L}=$
$\displaystyle\frac{3}{16\pi^{2}}\sum_{\lambda,\phi}\frac{1}{M_{\phi}^{2}}\left[\lambda_{L}^{ij}\lambda_{L}^{kj\ast}|\lambda_{R}^{kn}|^{2}b_{1}(x_{j},x_{n})-\tfrac{1}{2}\lambda_{L}^{ij}\lambda_{R}^{kj\ast}\lambda_{R}^{kn}\lambda_{L}^{kn\ast}b_{2}(x_{j},x_{n})\right],$
$\displaystyle e^{2}B_{3L}=$
$\displaystyle\frac{3}{16\pi^{2}}\sum_{\lambda,\phi}\frac{1}{M_{\phi}^{2}}\lambda_{L}^{ij}\lambda_{R}^{kj\ast}\lambda_{L}^{kn}\lambda_{R}^{kn\ast}b_{2}(x_{j},x_{n}),$
$\displaystyle B_{iR}=$ $\displaystyle~{}B_{iL}(L\leftrightarrow
R),\quad{i=1,2,3}.$ (4.56)

One can see that $B_{2L}$ has two terms, but they do not come from the same
set of couplings. The first term, coming from momenta of internal quarks,
similar to $B_{1L}$, has $(V-A)\times(V+A)$ type. The second one comes through
the internal quark chirality flip, which is then Fierz reordered. That
explains why it picks the factor of $-1/2$. The loop functions $b_{1}$ and
$b_{2}$ are determined to be

$\displaystyle b_{1}(x_{j},x_{n})=$ $\displaystyle-\tfrac{1}{2}\int
dt\frac{t^{2}}{(t+1)^{2}(t+x_{j})(t+x_{n})},$ $\displaystyle
b_{2}(x_{j},x_{n})=$ $\displaystyle\sqrt{x_{j}x_{n}}\int
dt\frac{t}{(t+1)^{2}(t+x_{j})(t+x_{n})}.$ (4.57)

The factor of $-1/2$ comes from $\int
k^{\mu}k^{\nu}=\tfrac{1}{2}g^{\mu\nu}\int k^{2}$, which is later Wick rotated.
It is straightforward to evaluate these integrals, yielding

$\displaystyle b_{1}(x,y)=\left\\{\begin{array}[]{c}\frac{-1+x^{2}-2x\ln
x}{2(1-x)^{3}}~{}{\rm for}~{}y=x\\\ \\\ \frac{-1+x-x\ln x}{2(1-x)^{2}}~{}{\rm
for}~{}y=0\\\ \\\ -\tfrac{1}{2}~{}{\rm for}~{}x=y=0\end{array}\right.$ (4.63)
$\displaystyle b_{2}(x,y)=\frac{-2x+2x^{2}-(1+x)x\ln x}{(1-x)^{3}}~{}{\rm
for}~{}y=x.$ (4.64)

Combining all interactions mentioned before, we can write the most general
$\ell_{i}^{-}\to\ell_{k}^{-}\ell_{l}^{+}\ell_{l}^{-}$ effective Lagrangian,
namely

$\displaystyle{\cal L}_{\ell_{i}^{-}\to\ell_{k}^{-}\ell_{l}^{+}\ell_{l}^{+}}=$
$\displaystyle~{}-m_{\ell_{i}}\bar{\ell}_{i}\sigma^{\alpha\beta}\left(\tfrac{e}{2}A_{2R}P_{L}+\tfrac{e}{2}A_{2L}P_{R}\right)\ell_{k}F_{\alpha\beta}$
$\displaystyle~{}-\left[g_{1}(\bar{\ell}_{i}P_{R}\ell_{k})(\bar{\ell}_{l}P_{R}\ell_{l})+g_{2}(\bar{\ell}_{i}P_{L}\ell_{k})(\bar{\ell}_{l}P_{L}\ell_{l})\right.$
$\displaystyle\quad+g_{3}(\bar{\ell}_{i}\gamma^{\alpha}P_{R}\ell_{k})(\bar{\ell}_{l}\gamma_{\alpha}P_{R}\ell_{l})+g_{5}(\bar{\ell}_{i}\gamma^{\alpha}P_{L}\ell_{k})(\bar{\ell}_{l}\gamma_{\alpha}P_{L}\ell_{l})$
$\displaystyle\quad\left.+g_{5}(\bar{\ell}_{i}\gamma^{\alpha}P_{R}\ell_{k})(\bar{\ell}_{l}\gamma_{\alpha}P_{L}\ell_{l})+g_{6}(\bar{\ell}_{i}\gamma^{\alpha}P_{L}\ell_{k})(\bar{\ell}_{l}\gamma_{\alpha}P_{R}\ell_{l})\right]+\rm
h.c.,$ (4.65)

where we have followed the notation of Ref. Kuno:1999jp . The coefficients
$g_{1},...g_{6}$ consist of all contributions from photon, $Z$, and box
diagrams, which are given by

$\displaystyle g_{1}=-e^{2}B_{3L},\quad g_{2}=-e^{2}B_{3R},$ (4.66)
$\displaystyle
g_{3}=-e^{2}\left(A_{1R}+Z_{R}g_{R}^{(\ell)}+B_{1R}\right),\quad
g_{4}=-e^{2}\left(A_{1L}+Z_{L}g_{L}^{(\ell)}+B_{1L}\right),$ $\displaystyle
g_{5}=-e^{2}\left(A_{1R}+Z_{R}g_{L}^{(\ell)}+B_{2R}\right),\quad
g_{6}=-e^{2}\left(A_{1L}+Z_{L}g_{R}^{(\ell)}+B_{2L}\right).$ (4.67)

From here we can calculate the decay width Kuno:1999jp ; Abada:2014kba

$\displaystyle\Gamma(\ell^{-}_{i}\to\ell^{-}_{k}\ell^{+}_{l}\ell^{-}_{l})=$
$\displaystyle~{}\frac{m_{\ell_{i}}^{5}}{512\pi^{3}}$
$\displaystyle\times\bigg{[}\frac{1}{12(1+\delta_{kl})}\left(|g_{1}|^{2}+|g_{2}|^{2}\right)+\frac{1+\delta_{kl}}{3}\left(|g_{3}|^{2}+|g_{4}|^{2}\right)+\frac{1}{3}\left(|g_{5}|^{2}+|g_{6}|^{2}\right)$
$\displaystyle\qquad+\left[\frac{16}{3}\ln\frac{m_{\ell_{i}}}{m_{\ell_{n}}}-\frac{2}{3}(12-\delta_{kl})\right]\left(|e^{2}A_{2L}|^{2}+|e^{2}A_{2R}|^{2}\right)$
$\displaystyle\qquad+\frac{4e^{2}}{3}{\rm
Re}\bigg{\\{}A_{2R}\left[(1+2\delta_{kl})g_{4}^{\ast}+g_{6}^{\ast}\right]+A_{2L}\left[(1+2\delta_{kl})g_{3}^{\ast}+g_{5}^{\ast}\right]\bigg{\\}}\bigg{]}.$
(4.68)

In the case of $\ell^{-}_{i}\to\ell^{+}_{k}\ell^{-}_{n}\ell^{-}_{n}$ decay
(i.e., $k\neq n$), only box diagrams contribute. The decay width is found to
be Abada:2014kba

$\displaystyle\Gamma(\ell^{-}_{i}\to\ell^{+}_{k}\ell^{-}_{n}\ell^{-}_{n})=$
$\displaystyle\frac{m_{\ell_{i}}^{5}}{512\pi^{3}}\left[\frac{1}{24}\left(|g_{1}|^{2}+|g_{2}|^{2}\right)+\frac{2}{3}\left(|\tilde{g}_{3}|^{2}+|\tilde{g}_{4}|^{2}\right)+\frac{1}{3}\left(|\tilde{g}_{5}|^{2}+|\tilde{g}_{6}|^{2}\right)\right],$
(4.69)

where we have defined $\left.\tilde{g}_{i}=g_{i}\right|_{A_{1L,R}=Z_{L,R}=0}$
for $i=3,...,6$. We present current bounds and projected sensitivities of
these processes in Table III.

$\ell_{i}\to\ell_{k}\ell_{m}\ell_{n}$ | Present bound | Future sensitivity
---|---|---
$\mu\to eee$ |  $1.0\times 10^{-12}$ BELLGARDT19881  |  $\sim 10^{-16}$ Blondel:2013ia
$\tau\to eee$ |  $2.7\times 10^{-8}$ Hayasaka:2010np  |  $\sim 10^{-9}$ Aushev:2010bq
$\tau\to\mu\mu\mu$ |  $2.1\times 10^{-8}$ Hayasaka:2010np  |  $\sim 10^{-9}$ Aushev:2010bq
$\tau^{-}\to e^{-}\mu\mu$ |  $2.7\times 10^{-8}$ Hayasaka:2010np  |  $\sim 10^{-9}$ Aushev:2010bq
$\tau^{-}\to\mu^{-}ee$ |  $1.8\times 10^{-8}$ Hayasaka:2010np  |  $\sim 10^{-9}$ Aushev:2010bq
$\tau^{-}\to e^{+}\mu^{-}\mu^{-}$ |  $1.7\times 10^{-8}$ Hayasaka:2010np  |  $\sim 10^{-9}$ Aushev:2010bq
$\tau^{+}\to\mu^{+}e^{-}e^{-}$ |  $1.5\times 10^{-8}$ Hayasaka:2010np  |  $\sim 10^{-9}$ Aushev:2010bq
Table III: Current experimental bounds on the
$BR(\ell_{i}\to\ell_{k}\ell_{m}\ell_{n})$. Future sensitivities are presented
on the last column.

###  4.3 Lepton anomalous magnetic-dipole moment

This quantity arises from the following interaction

$\displaystyle
T=\frac{e}{2m_{\mu}}F(q^{2})\bar{u}(p_{2})\sigma^{\alpha\beta}iq_{\beta}u(p_{1})\epsilon_{\alpha}(q),$
(4.70)

with $q=p_{2}-p_{1}$ and $\Delta a_{\ell}=F(q^{2}=0)$, evaluated at one-loop
penguin diagrams. This gives

$\displaystyle\Delta
a_{\ell}=-\frac{3}{16\pi^{2}}\sum_{\lambda,\phi}\frac{m_{\ell}^{2}}{M_{\phi}^{2}}$
$\displaystyle\bigg{\\{}\left[|\lambda^{\ell j}_{L}|^{2}+|\lambda^{\ell
j}_{R}|^{2}\right]\left[Q_{q}F_{1}(x_{j})+Q_{\phi}F_{2}(x_{j})\right]$
$\displaystyle+\frac{m_{q_{j}}}{m_{\ell}}{\rm Re}(\lambda^{\ell
j}_{L}\lambda^{\ell
j\ast}_{R})\left[Q_{q}F_{3}(x_{j})+Q_{\phi}F_{4}(x_{j})\right]\bigg{\\}}.$
(4.71)

###  4.4 $\mu$-$e$ conversion in nuclei

For this process, we are interested in the so-called coherent processes, that
is, no change in nucleon state during the transition happens. The relevant
interactions, therefore, can be written as Kitano:2002mt

$\displaystyle{\cal L}_{\rm eff}=$
$\displaystyle~{}-m_{\mu}\bar{\mu}\sigma^{\alpha\beta}\left(\tfrac{e}{2}A_{2R}P_{L}+\tfrac{e}{2}A_{2L}P_{R}\right)eF_{\alpha\beta}$
$\displaystyle~{}-\frac{1}{4}\left[\left(g_{RS}^{(q)}\bar{\mu}P_{R}e+g_{LS}^{(q)}\bar{\mu}P_{L}e\right)\bar{q}q+\left(g_{RV}^{(q)}\bar{\mu}\gamma^{\alpha}P_{R}e+g_{LV}^{(q)}\bar{\mu}\gamma^{\alpha}P_{L}e\right)\bar{q}\gamma_{\alpha}q\right]+{\rm
h.c.}$ (4.72)

The corresponding Wilson’s coefficients are given by

$\displaystyle g_{RS}^{(q)}=T_{RL}^{(q)};\quad g_{LS}^{(q)}=T_{LR},$
$\displaystyle
g_{RV}^{(q)}=T_{RR}^{(q)}-2e^{2}\left[-2Q_{q}A_{1R}+Z_{R}(g^{(q)}_{R}+g^{(q)}_{L})+B_{q}\right],$
$\displaystyle
g_{LV}^{(q)}=T_{LL}^{(q)}-2e^{2}\left[-2Q_{q}A_{1L}+Z_{L}(g^{(q)}_{R}+g^{(q)}_{L})+B_{q}\right],$
(4.73)

with

$\displaystyle
T_{XY}^{(q)}=\sum_{\lambda,\phi}\frac{\lambda_{X}^{2q}\lambda_{Y}^{1q\ast}}{M_{\phi}^{2}},$
(4.74) $\displaystyle
B_{q}=\frac{1}{16\pi^{2}}\sum_{\lambda,\phi}\frac{\lambda_{L}^{2j}\lambda_{L}^{1j\ast}}{M_{\phi}^{2}}\left[(\lambda_{L}^{\dagger}\lambda_{L})^{qq}+(\lambda_{R}^{\dagger}\lambda_{R})^{qq}\right]b_{1}(x_{j},0).$
(4.75)

Using these expressions, we can calculate the $\mu$-$e$ transition rate

$\displaystyle\Gamma_{\mu-e~{}\rm conv.}$ $\displaystyle=$
$\displaystyle\tfrac{1}{4}\left|\tfrac{e}{2}A_{2R}D+\tilde{g}_{LS}^{(p)}S^{(p)}+\tilde{g}_{LS}^{(n)}S^{(n)}+\tilde{g}_{LV}^{(p)}V^{(p)}+\tilde{g}_{LV}^{(n)}V^{(n)}\right|^{2}$
(4.76) $\displaystyle+(L\leftrightarrow R),$

where $D,S^{(p,n)},V^{(p,n)}$ are the overlap integrals for each operator.
Their values in the unit of $m_{\mu}^{5/2}$ are given in Kitano:2002mt . The
effective couplings $\tilde{g}^{(N)}_{LK,RK}$ with $K=S,V$ are given by

$\displaystyle\tilde{g}_{LK,RK}^{(N)}=\sum_{q}G_{K}^{(q,N)}g_{LK,RK}^{(q)}.$
(4.77)

The sum runs over all quark flavors for $K=S$ and runs over valence quarks for
$K=V$. Numerically, $G_{V}^{(u,p)}=G_{V}^{(d,n)}=2$,
$G_{V}^{(d,p)}=G_{V}^{(u,n)}=1$, and $G_{S}^{(u,p)}=G_{S}^{(d,n)}=5.1$,
$G_{S}^{(d,p)}=G_{S}^{(u,n)}=4.3$, $G_{S}^{(s,p)}=G_{S}^{(s,n)}=2.5$, while
those of heavy quarks are negligible. As in the previous LFV cases, we present
current bounds and future sensitivities of this process in Table IV.

Nucleus | Present bound | Future sensitivity
---|---|---
Gold |  $7\times 10^{-13}$ SINDRUMII:2006dvw  | $-$
Titanium |  $4.3\times 10^{-12}$ DOHMEN1993631  |  $\sim 10^{-18}$ unPUB
Aluminum | $-$ |  $10^{-15}-10^{-18}$ Pezzullo:2017iqq
Table IV: Current experimental bounds on the $BR(\mu-e)$ conv. in the nuclei.
Future sensitivities are presented on the last column.

##  5 Results

In the previous section, we derived all charged lepton flavor violating
processes that can be used to constrain the model’s parameter space. Since
addressing flavor anomalies demands TeV-scale leptoquarks with some of the
couplings being of order one, bounds from lepton flavor violating processes
provide the most stringent constraints on the Yukawa couplings, which we
explore in this section in great details.

###  5.1 Case studies

The neutrino mass formula given in Eq. (3.27) (or Eq. (3.29)) consists of four
different Yukawa couplings $y^{L,R}$ and $f^{L,R}$, which are a priori
arbitrary $3\times 3$ matrices. The parameter space is quite broad; therefore,
we choose a few specific benchmark scenarios and perform a detailed numerical
analysis. Since the terms in the second and the third lines in the neutrino
mass formula Eq. (3.27) (or Eq. (3.29)) are proportional to $m_{\tau}/m_{t}$,
for Yukawa couplings of a similar order, these terms can be completely
neglected. This is why, for our numerical study, we stick to the simplified
scenario where $f^{R}$ and $y^{R}$ also provide sub-leading contributions to
LFV unless otherwise explicitly mentioned. To further reduce the parameters,
we assume vanishing Yukawa couplings with the first generation quarks, i.e.,
$y^{L}_{1i},f^{L}_{1i}=0$.

Among the few predictive cases that we consider, in the following, we first
discuss the most minimal scenario consisting of six non-zero Yukawa parameters
$y^{L}_{3j},f^{L}_{3j}\neq 0$ that provides an excellent fit to the neutrino
oscillation data. As will be discussed later in the text, further parameter
space reduction fails to fit neutrino observables with their respective
$2\sigma$ values. Considering the loop integral behavior discussed above and
the suppression of $m_{\tau}/m_{t}$, in the case of no large hierarchy among
Yukawa couplings, it is an excellent approximation to keep only the third
generation of quarks. Then the neutrino mass matrix formula, in this case,
becomes

$\displaystyle(\mathcal{M}_{\nu})_{ji}\simeq\frac{3g^{2}m_{t}}{\sqrt{2}(16\pi^{2})^{2}}\left[y^{L}_{3j}f^{L}_{3i}+f^{L}_{3j}y^{L}_{3i}\right]\hat{I}_{j33}.$
(5.78)

The above formula applies to both the up-quark and down-quark mass diagonal
bases. This is because, in the limit we are working, in the up-quark mass
diagonal basis Eq. (3.27), the loop integrals are flavor independent. On the
other hand, in the down-quark mass diagonal basis Eq. (3.29), the remaining
factor is $V_{tb}\approx 1$. One can see from this formula that the neutrino
mass matrix is reduced into a rank two matrix, whose determinant vanishes.
Thus, this specific texture predicts that one of the neutrinos is massless,
although both neutrino mass orderings, i.e., normal hierarchy (NH) and
inverted hierarchy (IH), can be admitted. The neutrino mass matrix in this
form can nicely fit oscillation data.

Before diving into numerics, we first demonstrate that the undetermined Yukawa
couplings appearing in the above neutrino mass formula can be fully expressed
in terms of neutrino observables and as a function of LQ masses and mixing
parameters. By following the parametrization described in Cordero-
Carrion:2018xre ; Cordero-Carrion:2019qtu  (for alternative parameterizations,
see also, Ref. Cai:2014kra ; Hagedorn:2018spx ), we determine these Yukawa
couplings appearing in Eq. (5.78). To do so, the neutrino mass matrix is
diagonalized as follows:

$\displaystyle\mathcal{M}_{\nu}=U^{*}\begin{pmatrix}m_{1}&0&\\\ 0&m_{2}&0\\\
0&0&m_{3}\end{pmatrix}U^{\dagger},$ (5.79)

where $U$ is the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) mixing matrix and
$m_{i}$ are neutrino mass eigenvalues given by

$\displaystyle m_{1}=0,\;m_{2}=\sqrt{\Delta m^{2}_{21}},\;m_{3}=\sqrt{\Delta
m^{2}_{31}},$ (5.80)

for NH and

$\displaystyle m_{1}=\sqrt{-\Delta m^{2}_{32}-\Delta
m^{2}_{21}},\;m_{2}=\sqrt{-\Delta m^{2}_{32}},\;m_{3}=0,$ (5.81)

for IH. Now the neutrino mass matrix given in Eq. (5.78) can be re-written as

$\displaystyle\mathcal{M}_{\nu}=a_{0}\left(Y^{T}_{a}\hat{m}Y_{b}+Y^{T}_{b}\hat{m}Y_{a}\right),$
(5.82) $\displaystyle
a_{0}=\frac{3g^{2}}{\sqrt{2}(16\pi^{2})^{2}},\;\;\hat{m}=m_{t}\hat{I}_{j33}.$
(5.83)

Here, $Y_{a}=y^{L}_{3i}$ and $Y_{b}=f^{L}_{3i}$ are row matrices. Utilizing
this form, the two unknown Yukawa coupling matrices can be entirely determined
by the known values Esteban:2020cvm  of neutrino observables, SM fermion
masses, and as a function of scalar masses and mixings that run through the
loops. For NH, we have

$\displaystyle
Y^{T}_{a}=\frac{16\pi^{2}}{3^{1/2}2^{1/4}g}\begin{pmatrix}i\;r_{2}\;U^{*}_{12}+r_{3}\;U^{*}_{13}\\\
i\;r_{2}\;U^{*}_{22}+r_{3}\;U^{*}_{23}\\\
i\;r_{2}\;U^{*}_{32}+r_{3}\;U^{*}_{33}\end{pmatrix},$ (5.84) $\displaystyle
Y^{T}_{b}=\frac{16\pi^{2}}{3^{1/2}2^{1/4}g}\begin{pmatrix}-i\;r_{2}\;U^{*}_{12}+r_{3}\;U^{*}_{13}\\\
-i\;r_{2}\;U^{*}_{22}+r_{3}\;U^{*}_{23}\\\
-i\;r_{2}\;U^{*}_{32}+r_{3}\;U^{*}_{33}\end{pmatrix},$ (5.85)

Similarly, for IH, the solution for $Y^{a,b}$ takes the following forms

$\displaystyle
Y^{T}_{a}=\frac{16\pi^{2}}{3^{1/2}2^{1/4}g}\begin{pmatrix}r_{1}\;U^{*}_{11}+i\;r_{2}\;U^{*}_{12}\\\
r_{1}\;U^{*}_{21}+i\;r_{2}\;U^{*}_{22}\\\
r_{1}\;U^{*}_{31}+i\;r_{2}\;U^{*}_{32}\end{pmatrix},$ (5.86) $\displaystyle
Y^{T}_{b}=\frac{16\pi^{2}}{3^{1/2}2^{1/4}g}\begin{pmatrix}r_{1}\;U^{*}_{11}-i\;r_{2}\;U^{*}_{12}\\\
r_{1}\;U^{*}_{21}-i\;r_{2}\;U^{*}_{22}\\\
r_{1}\;U^{*}_{31}-i\;r_{2}\;U^{*}_{32}\end{pmatrix}.$ (5.87)

In both hierarchies, we define $r_{i}=\left(m_{i}/\hat{m}\right)^{1/2}$.

This parametrization is sometimes useful to fix the undetermined Yukawa
parameters of the theory. In our detailed numerical analysis, we consider not
only the benchmark (BM) scenario as mentioned above but also a few variations
of it that include reducing as well as extending the number of parameters.
Particularly, all case studies we examine are summarized in the following:

  1. $\bullet$

Texture given in Eq. (5.78) with $f^{L}_{3j},y^{L}_{3j}\neq 0$. We study both
NH and IH, which we label as NH-I and IH-I, respectively. Both cases provide a
good fit to neutrino data.

  2. $\bullet$

A more minimal variation of the scenario mentioned above is to choose at least
one of $f^{L}_{3j}=0$ or $y^{L}_{3j}=0$, leading to vanishing $({\cal
M}_{\nu})_{jj}$. Coupled with the fact that the lightest neutrino is massless,
this restriction clearly does not work for NH. Interestingly, for IH, the case
with $f^{L}_{32}=0$ or $y^{L}_{32}=0$ can still be fitted within $3\sigma$
experimental values of the neutrino observables. We demonstrate this by
choosing $f^{L}_{32}=0$ and label it as IH-II.

  3. $\bullet$

In the cases mentioned above, with all zero entries in the first and the
second rows, the ($31$)-entries of both coupling matrices are required to be
comparable with the other entries to provide a good fit, which subsequently
leads to large $\mu\to e\gamma$. Consequently, these cases demand large LQ
masses to be consistent with the non-observation of LVF. In search for a
minimal texture that is also compatible with $\sim\mathcal{O}(1)$ TeV LQs, we
explore a scenario with $f^{L}_{31}=0$ but introduce nonzero couplings
$f^{L}_{2j},y^{L}_{2j}$ for at least one $j$. Now, although $({\cal
M}_{\nu})_{11}=0$, the determinant of the neutrino mass matrix is no longer
zero, so a viable neutrino fit, which is compatible with NH, can be obtained.
(A vanishing (11)-element of neutrino mass matrix cannot be realized in IH
case.) For demonstration purpose, we choose to have nonzero $f^{L}_{23}$ and
$y^{L}_{23}$, while other $y^{L}_{2j},f^{L}_{2j}$ are simply set to zero. The
two working benchmarks are labeled as NH-II (up-diagonal basis) and NH-III
(down-diagonal basis).

###  5.2 Numerical analysis

Our numerical study is based on $\chi^{2}$ analysis, and the
$\chi^{2}$-function is defined as

$\displaystyle\chi^{2}=\sum_{i}\left(\frac{T_{i}-E_{i}}{\sigma_{i}}\right)^{2},$
(5.88)

where $\sigma_{i}$ represents experimental $1\sigma$ uncertainty; $T_{i}$ and
$E_{i}$ represent the theoretical prediction and the experimental central
value for the $i$-th observable, respectively. In the above equation, $i$ is
summed over five observables: two neutrino mass squared differences and three
mixing angles. For the simplicity of our work, we consider all parameters to
be real; hence, we do not attempt to fit the CP-violating Dirac phase in the
neutrino sector, which can be trivially done by turning on phases of these
couplings. Neutrino oscillation data used in our fit are summarized in Table
V. Once a good fit to data is obtained from $\chi^{2}$ analysis, we perform a
Markov chain Monte Carlo (MCMC) analysis to explore the parameter space
(consistent with neutrino observables) and inspect lepton flavor violation for
which we varied the non-zero Yukawa couplings and LQ masses in the ranges
$[-1,1]$ and $[1,100]$ TeV, respectively.

|  | Normal Ordering | Inverted Ordering
---|---|---|---
|  bfv $\pm 1\sigma$ |  $3\sigma$ range |  bfv $\pm 1\sigma$ |  $3\sigma$ range
| $\sin^{2}\theta_{12}$ | $0.304_{-0.012}^{+0.013}$ | $0.269\to 0.343$ | $0.304_{-0.012}^{+0.012}$ | $0.269\to 0.343$
| $\sin^{2}\theta_{23}$ | $0.573_{-0.023}^{+0.018}$ | $0.405\to 0.620$ | $0.578_{-0.021}^{+0.017}$ | $0.410\to 0.623$
| $\sin^{2}\theta_{13}$ | $0.02220_{-0.00062}^{+0.00068}$ | $0.02034\to 0.02430$ | $0.02238_{-0.00062}^{+0.00064}$ | $0.02053\to 0.02434$
|  $\frac{\Delta m^{2}_{21}}{10^{-5}}$ eV2 | $7.42_{-0.20}^{+0.21}$ | $6.82\to 8.04$ | $7.42_{-0.20}^{+0.21}$ | $6.82\to 8.04$
|  $\frac{\Delta m^{2}_{3\ell}}{10^{-3}}$ eV2 | $2.515_{-0.028}^{+0.028}$ | $2.431\to 2.599$ | $-2.498_{-0.029}^{+0.028}$ | $-2.584\to-2.413$
Table V: Neutrino oscillation parameters taken from Ref. Esteban:2020cvm .
Here, $\Delta m^{2}_{31}>0$ for NH and $\Delta m^{2}_{32}<0$ for IH. Here
’bfv’ represents best fit values obtained from global fit Esteban:2020cvm .

Sample fits obtained from our numerical procedure that is consistent with
neutrino observables are presented below for each of the cases listed above
(here, we have defined $m_{0}=a_{0}\hat{m}$):

$\displaystyle\textrm{\bf IH-I:}\;\;m_{0}=0.0576\;\textrm{eV},$ (5.89)
$\displaystyle f^{L}=\left(\begin{array}[]{ccc}0&0&0\\\ 0&0&0\\\
-0.1962&-0.6212&-0.7730\\\
\end{array}\right),\;\;\;y^{L}=\left(\begin{array}[]{ccc}0&0&0\\\ 0&0&0\\\
0.8239&-0.2074&-0.0508\\\ \end{array}\right).$ (5.96)
$\displaystyle\textrm{\bf IH-II:}\;\;m_{0}=0.0592\;\textrm{eV},$ (5.97)
$\displaystyle f^{L}=\left(\begin{array}[]{ccc}0&0&0\\\ 0&0&0\\\
-0.7134&0&-0.1701\\\
\end{array}\right),\;\;\;y^{L}=\left(\begin{array}[]{ccc}0&0&0\\\ 0&0&0\\\
0.1931&0.7401&-0.8472\\\ \end{array}\right).$ (5.104)
$\displaystyle\textrm{\bf NH-I:}\;\;m_{0}=0.0418\;\textrm{eV},$ (5.105)
$\displaystyle f^{L}=\left(\begin{array}[]{ccc}0&0&0\\\ 0&0&0\\\
-0.0680&-0.4836&-0.8209\\\
\end{array}\right),\;\;\;y^{L}=\left(\begin{array}[]{ccc}0&0&0\\\ 0&0&0\\\
0.2549&-0.6442&-0.2462\\\ \end{array}\right).$ (5.112)
$\displaystyle\textrm{\bf NH-II:}\;\;m_{0}=24.0268\;\textrm{eV},$ (5.113)
$\displaystyle f^{L}=\left(\begin{array}[]{ccc}0&0&0\\\ 0&0&-0.9747\\\
0&0.00974&-0.03489\\\
\end{array}\right),\;\;\;y^{L}=\left(\begin{array}[]{ccc}0&0&0\\\
0&0&0.9871\\\ 0.01048&-0.05672&-0.08842\\\ \end{array}\right).$ (5.120)
$\displaystyle\textrm{\bf NH-III:}\;\;m_{0}=8.4371\;\textrm{eV},$ (5.121)
$\displaystyle f^{L}=\left(\begin{array}[]{ccc}0&0&0\\\ 0&0&-0.4931\\\
0&0.0199&-0.01547\\\
\end{array}\right),\;\;\;y^{L}=\left(\begin{array}[]{ccc}0&0&0\\\
0&0&0.4419\\\ 0.05253&-0.05396&-0.1469\\\ \end{array}\right).$ (5.128)

The corresponding fit values of neutrino observables are collected in Table
VI, and the resulting neutrino mass matrices are shown in the following:

$\displaystyle\mathcal{M}^{\textrm{IH-I}}_{\nu}=\left(\begin{array}[]{ccc}-0.01864&-0.02716&-0.03614\\\
-0.02716&0.01485&0.01106\\\ -0.03614&0.01106&0.004534\\\
\end{array}\right)\;\textrm{eV},$ (5.132)
$\displaystyle\mathcal{M}^{\textrm{IH-
II}}_{\nu}=\left(\begin{array}[]{ccc}-0.01633&-0.03129&0.03387\\\
-0.03129&0&-0.007460\\\ 0.03387&-0.007460&0.01708\\\
\end{array}\right)\;\textrm{eV},$ (5.136)
$\displaystyle\mathcal{M}^{\textrm{NH-I}}_{\nu}=\left(\begin{array}[]{ccc}-0.001451&-0.003323&-0.008053\\\
-0.003323&0.02606&0.02710\\\ -0.008053&0.02710&0.01691\\\
\end{array}\right)\;\textrm{eV},$ (5.140)
$\displaystyle\mathcal{M}^{\textrm{NH-
II}}_{\nu}=\left(\begin{array}[]{ccc}0&0.002452&-0.008786\\\
0.002452&-0.02655&0.02686\\\ -0.008786&0.02686&-0.01697\\\
\end{array}\right)\;\textrm{eV},$ (5.144)
$\displaystyle\mathcal{M}^{\textrm{NH-
III}}_{\nu}=\left(\begin{array}[]{ccc}0&0.00885&0.002236\\\
0.00885&-0.01818&-0.02705\\\ 0.002236&-0.02705&-0.02534\\\
\end{array}\right)\;\textrm{eV}.$ (5.148)

Quantity | IH-I | IH-II | NH-I | NH-II | NH-III
---|---|---|---|---|---
$\sin^{2}\theta_{12}$ | 0.304 | 0.327 | 0.305 | 0.304 | 0.304
$\sin^{2}\theta_{23}$ | 0.578 | 0.593 | 0.572 | 0.574 | 0.448
$\sin^{2}\theta_{13}$ | 0.02239 | 0.022611 | 0.02223 | 0.02237 | 0.02234
$\Delta m^{2}_{21}\times 10^{5}$ eV2 | 7.425 | 7.408 | 7.425 | 7.4111 | 7.428
$\Delta m^{2}_{3l}\times 10^{3}$ eV2 | -2.498 | -2.498 | 2.515 | 2.514 | 2.513

Table VI: Neutrino mass-squared differences and mixing angles obtained from
fits for all the cases studied in this work. Here we have defined $\Delta
m^{2}_{3l}=\Delta m^{2}_{31}>0$ for NH and $\Delta m^{2}_{3l}=\Delta
m^{2}_{32}<0$ for IH.

From our detailed numerical scan over the parameter space using MCMC analysis,
we obtain interrelationships among various observables: correlations between
neutrino mixing parameters, bounds on LQ masses from LFV processes, and
correlations among different LVF processes are presented in Figs. 4-9.

Figure 4: Correlations between $\sin^{2}\theta_{12}$ and $\sin^{2}\theta_{23}$
for the five different textures we study. See text for details.

For both the NH and IH cases, a global fit to neutrino oscillation data has
two local minima for the mixing angle $\theta_{23}$, the one with
$\theta_{23}>45^{\circ}$ being the lower one Esteban:2020cvm  (for those
without SuperKamiokande data). Whereas for NH, these two minima are almost
identical (in the sense of $\Delta\chi^{2}$ measure), however, they
significantly differ for IH, and $\theta_{23}>45^{\circ}$ case is highly
preferred to $\theta_{23}<45^{\circ}$. This feature is clearly visible in the
upper left panel in Fig. 4 for the texture IH-I.

As discussed above, the most minimal Yukawa texture in this theory, which is
still consistent with oscillation data, corresponds to IH-II. Due to its
minimality, this scenario fails to reproduce neutrino observables within the
experimental $2\sigma$ range; however, it can be fitted within $3\sigma$
values, as can be seen from the third column in Table VI. For this specific
texture, a tension exists to simultaneously fit $\theta_{12}$ and
$\theta_{23}$ close to their central values. This attribute is demonstrated in
the upper right panel in Fig. 4.

Moreover, for NH-I and NH-III, both $\theta_{23}>45^{\circ}$ and
$\theta_{23}<45^{\circ}$ are equally preferred (see middle left and lower
panels in Fig. 4, respectively), whereas, for the texture NH-II, MCMC analysis
returns solutions only for $\theta_{23}>45^{\circ}$ as depicted in the middle
right panel in Fig. 4.

Figure 5: $BR(\mu\to e\gamma)$ and $BR(\mu\to eee)$ as a function of the
common LQ mass. Shaded colored regions are ruled out by current data and
dotted lines represent future sensitivities.

Figure 6: $BR(\mu\to eee)$ and $BR(\mu-e)$ conversion as a function of the
common LQ mass. Shaded colored regions are ruled out by current data and
dotted lines represent future sensitivities.

To obtain a good fit to data, for textures with IH-I, IH-II, and NH-I, the
(31)-entries in $f^{L},y^{L}$ are required to be sizable and are of similar
order compared to other non-zero entries, as can be seen from fits Eqs.
(5.96)–(5.112). Due to this requirement, the LQ masses must be much above the
TeV scale to satisfy the stringent LFV processes; the most relevant process is
the $\mu\to e\gamma$. The plots of this process, as a function of LQ mass, are
presented in Fig. 5; they show that $M_{\textrm{LQ}}\gtrsim\mathcal{O}(10)$
TeV must be satisfied. On the contrary, for textures NH-II and NH-III,
$f^{L}_{31}$ is set to zero, and non-zero (23)-entries are introduced. A
successful fit to data requires (23)-entries being dominant, whereas
$y^{L}_{31}$ is somewhat small, as can be seen from fits Eqs. (5.120)-(5.128).
Consequently, the branching ratio of $\mu\to e\gamma$ is highly suppressed in
the latter two scenarios allowing for TeV-scale LQs. However, LQ mass below a
TeV is ruled out, and the lower bound on its mass comes from the most
dominating LFV processes $\mu\to eee$ and $\mu-e$ conversion, as depicted in
Fig. 6.

Further correlations among most prominent LFV processes, namely $\mu\to
e\gamma$, $\mu\to eee$, and $\mu-e$ conversion are depicted in Figs. 7-9.
These plots are made by marginalizing over all relevant parameters (LQ mass
and Yukawa couplings) in our MCMC likelihood analysis, as discussed above.
Remarkably, from these plots, it can be seen that most of the minimal textures
we have exploited in this work will be entirely ruled out by upcoming low-
energy experiments searching for LFV for
$M_{\textrm{LQ}}\lesssim\mathcal{O}(100)$ TeV.

Figure 7: Correlations between $\mu\to e\gamma$ and $\mu\to eee$. Shaded
colored regions are ruled out by current data and dotted lines represent
future sensitivities.

Figure 8: Correlations between $\mu\to e\gamma$ and $\mu-e$ conversion. Shaded
colored regions are ruled out by current data and dotted lines represent
future sensitivities.

Figure 9: Correlations between $\mu-e$ and $\mu\to eee$. Shaded colored
regions are ruled out by current data and dotted lines represent future
sensitivities.

Finally, we demonstrate how to simultaneously satisfy neutrino observables and
muon $g-2$, where, currently, $(g-2)_{\mu}$ is the most prominent flavor
anomaly that shows $4.2\sigma$ deviation from the SM prediction. As explained
above, textures IH-I, IH-II, NH-I do not allow TeV scale LQs, therefore, to
obtain a viable scenario, we consider an example with NH-II texture.

NH-II consists of zero $f^{L}_{31}$ entry, so adding a new coupling
$f^{R}_{32}$, for instance, will not induce a new contribution to $\mu\to
e\gamma$ arising from chirality-enhanced term; it only induces a weaker
$\tau\to\mu\gamma$ process. However, the benchmark provided in Eq. (5.120) is
still unsuitable for incorporating $(g-2)_{\mu}$ with a TeV-scale LQ. This
particular fit has a somewhat small $f^{L}_{32}$ element; thus, to reproduce
$(g-2)_{\mu}$, an order unity $f^{R}_{32}$ coupling is needed. Once this
required size of $f^{R}_{32}$ is included, along with the $f^{L}_{33}$
coupling present in Eq. (5.120), top-quark chirality-enhanced contribution to
$\tau\to\mu\gamma$ rate becomes too large and rules out this particular fit.
Because of that, we perform a new fit by including $(g-2)_{\mu}$ observable
along with LFV rates in the $\chi^{2}$-function to allow for a TeV scale LQ
mass. From Eq. (4.71), one can approximate the $(g-2)_{\mu}$ up to the leading
order as

$\displaystyle\Delta
a_{\mu}\simeq-\frac{3}{8\pi^{2}}\frac{m_{t}m_{\mu}}{M^{2}_{LQ}}\left[y_{32}^{L}y^{R}_{32}\left(\frac{7}{6}+\frac{2}{3}\ln\frac{m^{2}_{t}}{M^{2}_{LQ}}\right)-f^{L}_{32}f^{R}_{32}\left(\frac{1}{6}+\frac{2}{3}\ln\frac{m_{t}^{2}}{M^{2}_{LQ}}\right)\right].$
(5.149)

We obtain the following parameters from the numerical fit, i.e.,

$\displaystyle
m_{0}=11.388\;\textrm{eV},\;M_{\textrm{LQ}}=2.1\;\textrm{TeV},\;f^{R}_{32}=0.18,$
(5.150) $\displaystyle f^{L}=\left(\begin{array}[]{ccc}0&0&0\\\
0&0&-0.870445\\\ 0&-0.0319843&-0.00376365\\\
\end{array}\right),\;\;\;y^{L}=\left(\begin{array}[]{ccc}0&0&0\\\
0&0&-0.175317\\\ 0.0248746&-0.0344455&-0.0702396\\\ \end{array}\right).$
(5.157)

We verified that introduction of non-zero $f^{R}_{32}$ still provides sub-
leading contribution to the neutrino mass, and its effect can be safely
neglected. However, it has significant effect on cLFV, which we have also
incorporated. The above parameters provide a good fit to the neutrino
observables

$\displaystyle\Delta m^{2}_{21}=7.414\times 10^{-5}\textrm{eV}^{2},\;\Delta
m^{2}_{31}=2.511\times 10^{-3}\textrm{eV}^{2},$ (5.158)
$\displaystyle\sin^{2}\theta_{12}=0.305,\;\sin^{2}\theta_{23}=0.574,\;\sin^{2}\theta_{13}=0.02223,\;$
(5.159)

as well as to the muon $g-2$

$\displaystyle\Delta a_{\mu}=2.62\times 10^{-9}.$ (5.160)

Since top-quark chirality enhancement is required to fit $(g-2)_{\mu}$
consistently with sizable $f^{L,R}_{32}$ entries, as can be seen from Eq.
(5.157) that (33)-entry must be pretty small compared to the rest of the
elements to keep $\tau$ decays under control. In addition, $\mu-e$ conversion
in the gold nucleus also lies just below the current bound. Branching ratios
of these two leading processes for this fit are found to be

$\displaystyle BR(\tau\to\mu\gamma)=1.1\times
10^{-8},\;\;\;BR(\mu-e)\;\textrm{conv.}=6.1\times 10^{-13}.$ (5.161)

On the other hand, the consistency with the recent lattice results that
weakens the long-standing discrepancy in $(g-2)_{\mu}$ between experiment and
theory can be obtained by reducing the value of $f^{R}_{32}$ without affecting
the neutrino observables. For example, setting $f^{R}_{32}=0.134$ instead of
$0.18$ leads to $\Delta a_{\mu}=1.95\times 10^{-9}$ in agreement with lattice
result Borsanyi:2020mff . Such a reduced value of this coupling subsequently
decreases $\tau\to\mu\gamma$ rate; this new value of $f^{R}_{32}$ corresponds
to $BR(\tau\to\mu\gamma)=5.8\times 10^{-9}$, whereas $BR(\mu-e)$ conversion as
quoted in Eq. (5.161) remains unaltered since this coupling plays no role for
this observable.

###  5.3 Non-standard Neutrino Interactions

The LQs $R_{2}$ and $S_{1}$ couple to neutrinos and quarks (cf. Eq. (2.9)),
consequently, charged-current non-standard interactions (NSI) at tree-level
can be induced Wolfenstein:1977ue ; Proceedings:2019qno ; Babu:2019mfe . Using
the effective dimension-6 operators for NSI introduced in Ref.
Wolfenstein:1977ue , the effective NSI parameters in our model can be written
(in the up-quark mass diagonal basis) as,

$\displaystyle\varepsilon_{\alpha\beta}\ =\
\frac{3}{4\sqrt{2}G_{F}}\left(\frac{f^{L\star}_{u\alpha}f^{L}_{u\beta}}{M_{R^{2/3}}^{2}}+\frac{\hat{y}^{L}_{d\alpha}\hat{y}^{L}_{d\beta}}{M_{S^{1/3}}^{2}}\right)\,,$
(5.162)

where $\hat{y}^{L}\equiv-V^{T}y^{L}$. Note that any nonzero $y^{L}_{d\alpha}$
is in conjunction to Cabibbo rotation and induces $\hat{y}^{L}_{se}$ leading
to strong constraints, for instance, $K^{+}\to\pi^{+}\nu\nu$ with
Re[$\hat{y}^{L}_{de}\hat{y}^{L}_{se}$] $=[-3.7,8.3]\times
10^{-4}\left(M_{S_{1}}/\text{TeV}\right)^{2}$. Thus, NSI induced from $S_{1}$
LQ via $y^{L}$ Yukawa coupling is subdominant. Moreover, any Yukawa couplings
to electron and muon sector $f^{L}_{u\alpha}$ and $y^{L}_{d\alpha}$
($\alpha=e,\mu$) are subjected to stringent constraints from the non-resonant
dilepton searches Babu:2020hun ; Angelescu:2021lln  at the LHC. However, the
LHC limits on the LQ Yukawa coupling in the tau sector are weaker and in
principle be $\mathcal{O}(1)$ leading to $\epsilon_{\tau\tau}$ as large as
34.4 % Babu:2019mfe , which is within reach of long-baseline neutrino
experiments, such as DUNE Chatterjee:2021wac .

##  6 Conclusions

Neutrino oscillations were discovered almost 25 years ago, showing that
neutrinos have a mass; however, its origin remains unknown. Recently, several
pieces of evidence of lepton flavor universality violation strongly indicate
physics beyond the SM. Scalar leptoquarks are the prime candidates for
resolving all these flavor anomalies. Motivated by this, in this work, we
hypothesized that neutrino masses and flavor anomalies have a common new
physics origin and proposed a new two-loop neutrino mass model consisting of
scalar leptoquarks $(\overline{3},1,1/3)$ and $(3,2,7/6)$ along with a third
scalar $(3,3,2/3)$. Each of these scalar leptoquarks has the potential to
incorporate $R_{D^{(\ast)}}$, $(g-2)_{e}$, and $(g-2)_{\mu}$ anomalies. The
scalar leptoquark $(3,2,7/6)$ may also address anomalies in the
$R_{K^{(\ast)}}$ ratios via new physics interactions with the electron. Since
resolution to flavor anomalies requires TeV scale scalar leptoquarks with some
of the Yukawa couplings of order unity, the proposed model can be tested in
ongoing and future colliders. However, probes of lepton flavor violation in
neutrino mass models provide the most efficient way of searching for physics
beyond the Standard Model that expands far beyond the reach of colliders such
as the LHC. In this work, we have primarily focused on the neutrino
phenomenology and examined various minimal textures of the Yukawa coupling
matrices that can satisfy the neutrino oscillation data. In particular, we
have exploited five benchmark scenarios with a limited number of Yukawa
parameters of a similar order, two of which provide an inverted hierarchy for
the neutrino masses, and the rest provide a normal hierarchy. Moreover, by
performing a detailed numerical procedure, namely, the Markov chain Monte
Carlo analysis, we have studied in depth various lepton flavor violating
processes and constrained the parameter space of this theory. Our analysis
shows that for the minimal Yukawa textures considered in this work, the
current low-energy experiments provide stringent constraints on model
parameters, and near-future experiments hunting for lepton flavor violating
rare processes will rule out these scenarios for leptoquark masses below 100
TeV. Finally, we have presented a case study where neutrino observables and
the tension in the muon anomalous magnetic moment, the most prominent flavor
anomaly, are incorporated simultaneously for TeV scale leptoquark masses by
keeping lepton flavor violations under control, which requires a bit of tuning
of the Yukawa parameters.

## Acknowledgments

The work of J.J. was supported in part by the National Research and Innovation
Agency of the Republic of Indonesia via Research Support Facility Program.

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