# Neutrino charge radius and electromagnetic dipole moments via scalar and
vector leptoquarks

A. Bolaños–Carrera  Tecnologico de Monterrey, School of Engineering and
Sciences, Avenida Eugenio Garza Sada 2501, CP 64849, Monterrey, Nuevo Leon,
Mexico.  M. Guiot-Lomelí  Gilberto Tavares–Velasco  Facultad de Ciencias
Físico Matemáticas, Benemérita Universidad Autónoma de Puebla, Apartado Postal
1152, Puebla, Pue., México

###### Abstract

The one-loop contribution of scalar and vector leptoquarks (LQs) to the
electromagnetic properties (NEPs) of massive Dirac neutrinos is presented via
an effective Lagrangian approach, with emphasis on the effective neutrino
charge radius (NCR), which has never been calculated and is obtained by the
background field formalism in a Yang-Mills-like scenario for gauge LQs.
Analytical results for nonzero neutrino mass are presented in terms of both
Feynman-parameter integrals and Passarino-Veltman scalar functions, which can
be useful to obtain the NEPs of heavy neutrinos, out of which approximate
expressions are obtained for light neutrinos. For the numerical analysis we
concentrate on the only renormalizable scalar and vector LQ representations
that do not need extra symmetries to forbid tree-level proton decay.
Constraints on the parameter space consistent with current experimental data
are then discussed and it is found that the LQ representations
$\widetilde{R}_{2}$ and $U_{1}$ could yield the largest contributions to the
NEPs provided that they have couplings to both left- and right-handed
neutrinos of the order of $O(1)$. For a LQ mass of $1.5$ TeV, the magnetic
dipole moment (MDM) of the tau neutrino can be of the order of $10^{-9}$
$\mu_{B}$, whereas its neutrino electric dipole moment (EDM) can reach values
as high as $10^{-20}$-$10^{-19}$ ecm. On the other hand, the NCR can reach
values up to $10^{-35}$ cm2 regardless of the neutrino flavor and even in the
absence of right-handed neutrinos. In the latter scenario, the EDM vanishes
and the contribution to neutrino MDM would be negligible, of the order of
$10^{-14}$ $\mu_{B}$ for the tau neutrino, whereas those for the muon and
electron neutrinos would be about two and seven orders of magnitude smaller,
respectively. Our estimates could be severely suppressed due to a possible
suppression of the LQ coupling constants.

##  I Introduction

The neutrino interaction with the photon is governed by the neutrino
electromagnetic properties (NEPs) [1, 2, 3, 4], which can only arise at the
one-loop level or higher orders in perturbation theory and have been long the
focus of considerable interest in the literature [5, 6, 7, 8] since they could
allow us to determine the Dirac or Majorana nature of neutrinos [4] and also
hint new physics effects in neutrino experiments[8, 9, 10, 11]. The neutrino
electromagnetic vertex function consistent with Lorentz and electromagnetic
gauge invariance can be written as [12]

$\Gamma^{\mu}\bar{=}ie\left(\gamma^{\mu}\left(F_{1}^{V}(q^{2})+F_{2}^{A}(q^{2})\gamma^{5}\right)+\frac{i}{2m_{\nu}}\sigma^{\mu\nu}q_{\nu}\left(F^{V}_{2}(q^{2})+F^{A}_{2}(q^{2})\gamma^{5}\right)+q^{\mu}\left(F^{V}_{3}(q^{2})+F^{A}_{3}(q^{2})\gamma^{5}\right)\right),$
(1)

where $q^{2}$ is the photon squared transfer momentum, $F_{1}^{V}$ is the
electric charge form factor and $F_{1}^{A}$ the anapole form factor, whereas
the chirality-flipping form factors $F_{2}^{V}$ and $F_{2}^{A}$ determine the
$CP$-conserving magnetic dipole moment (MDM) $\mu_{\nu}$ and the
$CP$-violating electric dipole moment (EDM) $d_{\nu}$ as follows

$\mu_{\nu}=e\frac{F_{2}^{V}(0)}{2m_{\nu}},$ (2)

and

$d_{\nu}=-ie\frac{F_{2}^{A}(0)}{2m_{\nu}}.$ (3)

Both dipole form factors vanish for a Majorana neutrino, which can only have
nonvanishing anapole form factor and transition dipole form factors [4, 3,
13], whereas a Dirac neutrino can have nonvanishing anapole and dipole form
factors [14, 3]. Thus, the observation of a neutrino MDM would be a clear
evidence that neutrinos are Dirac particles.

The standard model (SM) augmented with massive Dirac neutrinos predicts a non-
vanishing neutrino MDM at the one-loop level, which in units of the Bohr
magneton $\mu_{B}=e/(2m_{e})$ is given by [2, 3]

$\mu_{\nu_{\alpha}}=\frac{3eG_{F}m_{\nu_{\alpha}}}{8\sqrt{2}\pi^{2}}\simeq
3.1\times 10^{-19}\left(\frac{m_{\nu_{\alpha}}}{1\,{\rm eV}}\right)\,\mu_{B},$
(4)

whereas the EDM vanishes at the one-loop level, though transition EDMs can be
nonvanishing.

Although the neutrino electric charge form factor vanishes for an on shell
photon $F_{1}^{V}(0)=0$ except in theories with electrically millicharged
neutrinos [6], its derivative can be nonvanishing and the so-called neutrino
charge radius (NCR) is defined by

$\langle
r^{2}\rangle_{\nu}=6\frac{dF_{1}^{V}(q^{2})}{dq^{2}}\Big{|}_{q^{2}=0},$ (5)

which has been the source of some controversy [15, 16] since it is a gauge-
dependent quantity as it was shown long ago in the context of the SM [17, 18,
19, 20, 15]. Therefore, the NCR would not represent a valid observable
quantity as opposed to the static dipole moments[21]. We will dwell on this
issue below.

It is well-known that, contrary to on shell Green’s functions, off shell ones
can be plagued with several pathologies, such as dependence on the gauge-
fixing parameter (GFP) $\xi$, the gauge fixing procedure, and the presence of
ultraviolet divergences [22]. This stems from the fact that off shell Green’s
functions do not represent $S$-matrix elements and cannot be associated with
observable quantities. Such pathologies disappear once the corresponding off
shell vertex is inserted into a physical process, which must be gauge
independent and ultraviolet finite. Much effort has gone into tackling the NCR
puzzle and defining a well-behaved quantity that can serve as a probe of the
SM at neutrino experiments. In particular, in Ref. [21] the pinch (PT) [23,
24, 25, 22] was used to obtain a well-behaved $\langle r^{2}\rangle_{\nu}$ by
inserting the neutrino electromagnetic vertex into $e^{-}\nu\to e^{-}\nu$
scattering and judiciously adding up the contributions to the
$F_{1}^{V}(q^{2})$ form factor from vertex, self-energy and box diagrams,
which allows one to cancel the GFP-dependent and process-dependent terms. This
requires to perform the calculation of several of Feynman diagrams in a
general $R_{\xi}$ gauge. It has been argued that the result obtained this way
renders an effective gauge-independent NCR that is a valid observable quantity
[15]. Along these lines, it was shown that there is a correspondence at least
up to the two-loop level between GFP-independent off shell Green’s functions
extracted from a physical process through the PT and those obtained by the
background field method (BFM) via the Feynman-’t Hooft gauge [26, 27]. This
procedure alleviates the task of computing a plenty of Feynman diagrams in a
general $R_{\xi}$ gauge. For instance, in the SM the NCR can be obtained
through the BFM from the $\bar{\nu}\nu\gamma$ proper vertex, which to leading
order in the mass of the charged lepton $\ell$ receives the contributions from
the Feynman diagrams of Fig. 1. Although some authors have also considered the
GFP independent contribution to the slope of $F_{1}^{V}(q^{2})$ from the
$Z-\gamma$ self-energy, it has been argued by the authors of Ref. [21] that
such a contribution belongs to the electroweak charge form factor rather than
the electric charge one.

Figure 1: Feynman diagrams that contribute to leading order in the charged
lepton mass to the NCR at the one-loop level in the SM in the BFM.

As a result, the one-loop contribution to the effective NCR in the SM is [21,
28, 12, 29]

$\langle
r^{2}\rangle_{\nu_{\ell}}=\frac{G_{F}}{4\sqrt{2}\pi^{2}}\left(3-2\log\left(\frac{m_{\ell}^{2}}{m_{W}^{2}}\right)\right)\simeq\left\\{\begin{array}[]{lcr}-0.83\times
10^{-32}\;\text{cm}^{2}&\text{for}&\nu_{e},\\\ -0.48\times
10^{-32}\;\text{cm}^{2}&\text{for}&\nu_{\mu},\\\ -0.30\times
10^{-32}\;\text{cm}^{2}&\text{for}&\nu_{\tau},\\\ \end{array}\right.$ (6)

where $m_{\ell}$ is the mass of the corresponding charged lepton.

In this work we present a calculation of the one-loop contributions to NEPs in
leptoquark (LQ) models. For the NCR we use the BFM and the approach of Refs.
[15, 29]. LQs are hypothetical spin-0 or spin-1 particles predicted by several
new physics theories [30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43,
44], which are peculiar as they carry both lepton and baryon number, thereby
interacting simultaneously to leptons and quarks and giving rise to a rich
phenomenology. Such particles were first predicted in the $SU(4)_{R}\times
SU(4)_{L}\times SU(4^{\prime})$ model of Pati and Salam [30], which postulates
that lepton number is the fourth color quantum number, but they also appear
naturally in grand unification theories (GUT) [31, 32, 33, 34, 35, 36],
theories with composite fermions [37, 38, 39], superstring-inspired $E_{6}$
models [40, 41], technicolor models [42, 43, 44], etc. LQs can give rise to
new interesting effects, out of which the most dramatic is the appearance of
tree-level LQ-diquark couplings that can trigger rapid proton decay [45] and
constrain severely the LQ mass unless ad-hoc symmetries are invoked to
preserve proton stability.

The late 1990s saw a boom of interest in LQs prompted by the apparent anomaly
in $e^{+}p$ scattering reported by the H1 [46] and ZEUS [47] collaborations,
which could be explained by the presence of LQ particles, but such an interest
faded away once no further confirmation of a SM deviation was found in
subsequent analyses [48]. Very recently, however, the interest in LQs has
renewed as they can explain the apparent lepton flavor universality violating
(LFUV) anomalies in semi-leptonic $b$-hadron decays, namely the $R_{D,D^{*}}$
and $R_{K,K^{*}}$ anomalies, but can also provide a solution to the muon $g-2$
discrepancy: a plethora of LQ models constructed to this end have been
proposed [49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65,
66]. Even more, LQs can generate small neutrino masses radiatively [67, 56,
65, 68, 69, 70, 65]. Although the latest LHC-b data seem to exclude the
$R_{K,K^{*}}$ anomaly [71], LQs predict a rich phenomenology and it is worth
an updated study of their contributions to NEPs. Along these lines, the
neutrino MDM has already been calculated in the framework of LQ models [72,
73, 74, 75], but an up-to-date analysis is in order given the recent
developments in LQ models. As far as LQ contributions to the NCR, to our
knowledge no known calculation exists yet.

In the experimental arena, bounds on the neutrino MDM and NCR have been
obtained from several experiments. We list in Table 1 the 90% CL limits
obtained from several experiments. Other recent bounds such as those obtained
[76] by using the COHERENT experiment data [77] are not shown as they are
looser.

Table 1: 90% CL upper bounds (or allowed range) on neutrino MDM and NCR for each flavor from several neutrino experiments: XENONnT [78], Borexino [79], GEMMA [80], DONUT [81], LSND [82], TEXONO [83], and solar [84]. Flavor | $|\mu_{\nu}|\ (\times 10^{-11}\mu_{B})$ |  $\left\langle r_{\nu}^{2}\right\rangle\ \big{(}\times 10^{-32}$cm${}^{2}\big{)}$
---|---|---
$\nu_{e}$ | $\begin{array}[]{l}0.63\ \text{(XENONnT)}\\\ 3.9\ \text{(Borexino)}\\\ 2.9\ \text{(GEMMA)}\end{array}$ | $\begin{array}[]{l}[-45,\ 3.0]\ \text{(XENONnT)}\\\ [0.82,\ 1.27]\ \text{(Solar)}\\\ [-5.94,\ 8.28]\ \text{(LSND)}\\\ [-4.2,\ 6.6]\ \ \text{(TEXONO)}\end{array}$
$\nu_{\mu}$ | $\begin{array}[]{l}1.37\ \text{(XENONnT)}\\\ 5.8\ \text{(Borexino)}\\\ \end{array}$ | $\begin{array}[]{l}[-45,\ 52]\ \text{(XENONnT)}\\\ [-9,\ 31]\ \text{(Solar)}\\\ 1.2\ \text{(CHARM-II)}\\\ [-4.2,\ 0.48]\ \text{(TEXONO)}\end{array}$
$\nu_{\tau}$ | $\begin{array}[]{l}1.24\times 10^{4}\ \text{(XENONnT)}\\\ 5.8\times 10^{4}\ \text{(Borexino)}\\\ 3.9\times 10^{4}\ \text{(DONUT)}\end{array}$ | $\begin{array}[]{l}[-40,\ 45]\ \text{(XENONnT)}\\\ [-9,\ 31]\ \text{(Solar)}\end{array}$

As far as the neutrino EDM, no experimental limit exist yet, but several
indirect and theoretical bounds have been obtained. For the electron and muon
neutrino EDMs, the most stringent limits are $d_{\nu_{e},\nu_{\mu}}\lesssim
10^{-21}$ ecm [85], whereas for the tau neutrino bounds of the order of
$10^{-17}-10^{-18}$ ecm have been obtained in the context of some new physics
scenarios: $d_{\nu_{\tau}}\lesssim\times 10^{-17}$ ecm in a model-independent
approach (naturalness) [86], $d_{\nu_{\tau}}\lesssim\times 10^{-17}$ ecm in
effective Lagrangians [87], $d_{\nu_{\tau}}\lesssim 10^{-18}-10^{-20}$ ecm in
vector-like multiplet models [88], etc. Also, it was found [89] that the ILC
and CLIC would allow one to test $d_{\nu_{\tau}}$ values up to the $10^{-19}$
ecm level for center-of-mass energies of $500$ to $3000$ GeV.

The rest of this work is organized as follows. In Section II we present an
overview of some minimal renormalizable models that predict scalar and vector
LQs at the TeV scale and have no proton decay at the tree-level. Section III
is devoted to present the calculation of the electromagnetic properties of a
massive Dirac neutrino arising from both vector and scalar LQs in the most
general way, whereas Sec. IV is devoted to the discussion of the current
constraints on the LQ parameter space from experimental data along with a
numerical estimate of the NEPs of SM neutrinos. Finally, the conclusion and
outlook are presented in Sec. V. The lengthy formulas for the analytical
results of the LQ contributions to the NEPs are presented in Appendix A and
some results for the LQ contributions to several observables useful to
constrain the LQ parameter space are presented in Appendix B.

##  II Minimal models with scalar and vector LQs

We are interested in the electromagnetic properties of massive Dirac neutrinos
and assume that there are right-handed neutrinos that could interact with LQ
particles and SM quarks but are sterile to the weak force. We will not focus
on the mechanism of neutrino mass generation, though there are several LQ
models that address this problem [67, 56, 65, 68, 69, 70, 65]. Instead of
analyzing a particular LQ model in all its complexity, it is more convenient
to study the LQ effects on low-energy experiments in a model-independent
fashion via an effective Lagrangian. Apart from $SU(3)_{C}\times
SU(2)_{L}\times U(1)_{Y}$ gauge invariance, extra symmetries must be assumed
to forbid dangerous diquark couplings that can induce rapid proton decay at
tree-level, thereby pushing the LQ masses up to the ultra high-energy scale:
for instance, proton decay sets a limit of $10^{16}$ GeV on the mass of GUT
vector LQs [45, 32]. Along these lines, the authors of Ref. [90] found all the
$SU(3)_{C}\times SU(2)_{L}\times U(1)_{Y}$ LQ representations with
renormalizable couplings to SM fermion bilinear operators respecting both
baryon and lepton number conservation: it turns out that there are only five
scalar and five vector LQ representations of such a kind [90], though there
are one extra scalar and one extra vector representations when right-handed
neutrinos are included [91, 92]. Out of all these twelve LQ representations,
only ten of them can couple to neutrinos, thereby yielding one-loop
contributions to NEPs. Such representations are shown in Table 2, where apart
from the spin, fermion number, and $SU(3)_{c}\times SU(2)_{L}\times U(1)_{Y}$
gauge quantum numbers of each LQ representation, we also include the charge of
their corresponding components and indicate which of them can couple to left-
and/or right-handed neutrinos.

Table 2: $SU(3)_{c}\times SU(2)_{L}\times U(1)_{Y}$ LQ representations with renormalizable couplings to bilinear fermion operators, including right-handed neutrinos, respecting both baryon and lepton number conservation [90, 91, 92]. We show the corresponding spin $S$, fermion number $F=3B+L$, and $SU(3)_{c}\times SU(2)_{L}\times U(1)_{Y}$ gauge quantum numbers (GQNs). We also include the electric charge of the LQ components of each representation and indicate between parenthesis if they couple to left-handed $(L)$ neutrinos, right-handed $(R)$ neutrinos, or both of them (LR) [92]. In the last column we show the representations that provide renormalizable LQ models that do not need to invoke extra symmetries to forbid proton decay in perturbation theory.  Representation | $S$ | $F$ | GQNs | LQ electric charge ($|e|$) | No extra symmetries to forbid proton decay
---|---|---|---|---|---
$S_{3}$ | $0$ | $-2$ | $(\bar{3},3,1/3)$ |  $1/3$ $(L)$, $4/3$, $-2/3$ $(L)$ | ✗
$R_{2}$ | $0$ | $0$ | $(3,2,7/6)$ |  $5/3$, $2/3$ $(L)$ | ✓
$\widetilde{R}_{2}$ | $0$ | $0$ | $(3,2,1/6)$ |  $2/3$ $(R)$, $-1/3$ $(LR)$ | ✓
$S_{1}$ | $0$ | $-2$ | $(\bar{3},1,1/3)$ |  $1/3$ $(LR)$ | ✗
$\bar{S}_{1}$ | $0$ | $-2$ | $(\bar{3},1,-2/3)$ |  $-2/3$ $(R)$ | ✗
$U_{3}$ | $1$ | $0$ | $(3,3,2/3)$ |  $2/3$ $(L)$, $-1/3$ $(L)$, $5/3$ | ✓
$V_{2}$ | $1$ | $-2$ | $(\bar{3},2,5/6)$ |  $1/3$ $(L)$, $4/3$ | ✗
$\widetilde{V}_{2}$ | $1$ | $-2$ | $(\bar{3},2,-1/6)$ |  $1/3$ $(R)$, $-2/3$ $(LR)$ | ✗
$U_{1}$ | $1$ | $0$ | $(3,1,2/3)$ |  $2/3$ $(LR)$ | ✓
$\bar{U}_{1}$ | $1$ | $0$ | $(3,1,-1/3)$ |  $-1/3$ $(R)$ | ✗

Another approach was followed by the authors of Refs. [93, 94], who found the
effective LQ models based on $SU(3)_{c}\times SU(2)_{L}\times U(1)_{Y}$
representations with renormalizable couplings to fermion bilinear operators
that do not need to invoke any extra global symmetry to forbid baryon number
violation in perturbation theory, thereby forbidding tree-level proton decay
via either diquark couplings or triple and quadruple LQ self-interactions.
These models are thus still phenomenologically viable at the TeV scale as they
have no severe constraints on the LQ masses and couplings from proton decay.
It was found that the only models of this kind are those comprised by the
either one or several of the following four LQ representations: the two scalar
representations $R_{2}$ and $\widetilde{R}_{2}$ [93] and the two vector
representations $U_{1}$ and $U_{3}$ [94]. Models with one or several of these
four LQ representations have been the focus of attention recently since, apart
from providing a renormalizable framework and predicting a rich phenomenology
at the TeV scale, they can explain the LFUV anomalies in $B$-meson decays as
well as the muon $g-2$ anomaly in regions of the parameter space still allowed
by the current experimental constraints from meson decays, electroweak
precision parameters, and direct searches at the LHC [95, 96, 97, 98, 99, 99,
100, 101, 50, 102, 54].

Although the $R_{2}$, $\widetilde{R}_{2}$, $U_{1}$, and $U_{3}$
representations can give contributions to NEPs, only $\widetilde{R}_{2}$ and
$U_{1}$ can have couplings to both left- and right-handed neutrinos. In
particular, the $U_{1}$ representation has been the source of attention
recently as emerges naturally from the minimal realization of the Pati-Salam
model [30] and provides a solution to the LFUV anomalies in $B$-meson decays
[103, 104, 105, 106, 107, 101, 108, 109, 110, 111, 94, 112, 96, 113], for
which models based on the $\widetilde{R}_{2}$ have also been studied [114,
115, 108, 52]. Notice however that the scalar ${S}_{1}$ and vector
$\widetilde{V}_{2}$ representations, with fermion number 2, can also can have
couplings to both left- and right-handed neutrinos, so in order to have a
comprehensive calculation we will consider the scalar representations
$\widetilde{R}_{2}$ and $S_{1}$ as well as the vector representations $U_{1}$
and $\widetilde{V}_{2}$ since they provide the most general LQ couplings to
neutrinos and quarks that can induce contributions to NEPs at the one-loop
level.

We will present below an overview of the LQ couplings necessary for our
calculation.

###  II.1 Scalar LQ interactions

####  II.1.1 $\widetilde{R}_{2}$ and $S_{1}$ neutrino couplings

In the following, we will refrain from presenting the LQ interactions with
quarks and charged leptons as they are not relevant for our calculation and
can be found elsewhere [90, 92]. Therefore, the dimension-4 Yukawa lagrangian
for the $\widetilde{R}_{2}$ and $S_{1}$ representations that yield LQ
interactions with a quark and both left- and right-handed neutrinos can be
written as [90, 92]

$\displaystyle\mathcal{L}_{\widetilde{R}_{2}}$
$\displaystyle\supset-\widetilde{Y}^{RL}_{2\,ij}\overline{d}^{{}^{\prime}i}_{R}i\widetilde{R}_{2}^{T}\tau_{2}L_{L}^{{}^{\prime}j}+\widetilde{Y}^{LR}_{2\,ij}\overline{Q}^{{}^{\prime}i}_{L}\widetilde{R}_{2}\nu^{{}^{\prime}j}_{R}+{\rm
H.c.},$ (7)

and

$\displaystyle\mathcal{L}_{S_{1}}$ $\displaystyle\supset
Y^{LL}_{1\,ij}\overline{Q^{\prime}}^{C\,i}_{L}i\tau_{2}L^{{}^{\prime}j}_{L}S_{1}+Y^{\overline{RR}}_{1\,ij}\overline{d}^{{}^{\prime}C\,i}_{R}\nu^{{}^{\prime}j}_{R}S_{1}+{\rm
H.c.},$ (8)

where as customary $L_{L}^{{}^{\prime}i}$ and $Q_{L}^{{}^{\prime}i}$ are
$SU(2)_{L}$ left-handed lepton and quark doublets, respectively, whereas
$\nu_{R}^{{}^{\prime}i}$ and $q_{R}^{{}^{\prime}i}$ are $SU(2)$ singlets, with
the subscripts $i$ and $j$ being family indices . The left-side (right-side)
superscript of the Yukawa matrices $Y$ and $Z$ stands for the chirality of the
corresponding quark (lepton) multiplet.

We now write the LQ in terms of their components and rotate to the mass
eigenstates of the fermions via the transformations
$e^{i^{\prime}}_{L}=e^{i}_{L}$, $d^{{}^{\prime}i}_{L}=d^{i}_{L}$,
$u^{i^{\prime}}_{L}=V_{ik}u^{k}_{L}$,
$\nu^{{}^{\prime}i}_{L}=U_{ik}\nu^{k}_{L}$ (down alignment), where $V$ and $U$
are the Cabbibo-Kobayashi-Maskawa (CKM) and the Pontecorvo-Maki-Nakagawa-
Sakata (PMNS) mixing matrices, respectively. This leads to the interactions of
the $S_{1}$ and $\widetilde{R}_{2}$ representations with neutrinos and quarks

$\displaystyle\mathcal{L}_{\widetilde{R}_{2}}$
$\displaystyle\supset\widetilde{R}_{2}^{2/3}V_{ik}\widetilde{Y}^{\overline{LR}}_{2kj}\bar{u}^{i}_{L}\nu^{j}_{R}+\widetilde{R}_{2}^{-1/3}\left(U_{kj}\widetilde{Y}^{RL}_{2ik}\bar{d}^{i}_{R}\nu^{j}_{L}+\widetilde{Y}^{\overline{LR}}_{2ij}\bar{d}^{i}_{L}\nu^{j}_{R}\right)+{\rm
H.c.},$ (9)

and

$\displaystyle\mathcal{L}_{S_{1}}$ $\displaystyle\supset
S_{1}^{1/3}\left(U_{kj}Y_{1\,ik}^{LL}\bar{d}_{L}^{Ci}\nu_{L}^{j}+Y^{\overline{RR}}_{1\,ij}\bar{d}_{R}^{Ci}\nu_{R}^{j}\right)+{\rm
H.c.}$ (10)

For completeness we present in Table 3 the LQ interactions to a neutrino-quark
pair of all the scalar and vector LQs arising from all the representations of
Table 2.

Table 3: Interactions of the scalar and vector LQs arising from the representations of Table 2 with a quark and left- and right-handed neutrinos.  Representation | Neutrino interactions
---|---
$S_{3}$ | $-(Y_{3}^{LL}U)_{ij}\bar{d}_{L}^{Ci}\nu_{L}^{j}S_{3}^{1/3}+\sqrt{2}(V^{T}Y^{LL}_{3}U)_{ij}\bar{u}_{L}^{Ci}\nu_{L}^{j}S_{3}^{-2/3}+{\rm H.c.}$
$R_{2}$ | $(Y^{RL}_{2}U)_{ij}\bar{u}^{i}_{R}\nu^{j}_{L}{R}_{2}^{2/3}+{\rm H.c.}$
$\widetilde{R}_{2}$ | $(\widetilde{Y}^{RL}_{2}U)_{ij}\bar{d}^{i}_{R}\nu^{j}_{L}\widetilde{R}_{2}^{-1/3}+\widetilde{Y}^{\overline{LR}}_{2\,ij}\bar{d}^{i}_{L}\nu^{j}_{R}\widetilde{R}_{2}^{-1/3}+(V\widetilde{Y}^{\overline{LR}}_{2})_{ij}\bar{u}^{i}_{L}\nu^{j}_{R}\widetilde{R}_{2}^{2/3}+{\rm H.c.}$
$S_{1}$ | $(Y_{1}^{LL}U)_{ij}\bar{d}_{L}^{Ci}\nu_{L}^{j}S_{1}^{1/3}+Y^{\overline{RR}}_{1\,ij}\bar{d}_{R}^{Ci}\nu_{R}^{j}S_{1}^{1/3}+{\rm H.c.}$
$\bar{S}_{1}$ | $\overline{Y}^{\overline{RR}}_{1\,ij}\bar{u}_{R}^{Ci}\nu_{R}^{j}\bar{S}_{1}^{-2/3}+{\rm H.c.}$
$U_{3}$ | $(VX^{LL}_{3}U)_{ij}\bar{u}^{i}_{L}\gamma^{\mu}\nu^{j}_{L}U^{2/3}_{3\,\mu}+\sqrt{2}(X^{LL}_{3}U)_{ij}\bar{d}^{i}_{L}\gamma^{\mu}\nu^{j}_{L}U^{-1/3}_{3\,\mu}+{\rm H.c.}$
$V_{2}$ | $-(X_{2}^{RL}U)_{ij}\bar{d}_{R}^{Ci}\gamma^{\mu}\nu_{L}^{j}V_{2\,\mu}^{1/3}+{\rm H.c.}$
$\widetilde{V}_{2}$ | $-(\widetilde{X}_{2}^{RL}U)_{ij}\bar{u}_{R}^{Ci}\gamma^{\mu}\nu_{L}^{j}\widetilde{V}^{-2/3}_{2\,\mu}+(V^{T}\widetilde{X}_{2}^{\overline{LR}})_{ij}\bar{u}_{L}^{Ci}\gamma^{\mu}\nu_{R}^{j}\widetilde{V}^{-2/3}_{2\,\mu}-\widetilde{X}^{\overline{LR}}_{2\,ij}\bar{d}_{L}^{Ci}\gamma^{\mu}\nu_{R}^{j}\widetilde{V}^{1/3}_{2\,\mu}+{\rm H.c.},$
$U_{1}$ | $(VX^{LL}_{1}U)_{ij}\bar{u}^{i}_{L}\gamma^{\mu}\nu^{j}_{L}U^{2/3}_{1\,\mu}+X^{\overline{RR}}_{1\,ij}\bar{u}^{i}_{R}\gamma^{\mu}\nu^{j}_{R}U^{2/3}_{1\,\mu}$
$\widetilde{U}_{1}$ | $X^{\overline{RR}}_{1\,ij}\bar{d}_{R}^{i}\gamma^{\mu}\nu_{R}^{j}\widetilde{U}_{1\,\mu}^{-1/3}+{\rm H.c.}$

For the purposes of our calculation, we will consider a generic interaction of
a scalar LQ $\Phi^{k}$ of electric charge $Q_{k}=k$, in units of $|e|$, to
quarks and Dirac neutrinos of the form

$\displaystyle\mathcal{L}^{\Phi^{k}q^{i}\nu^{\alpha}}=$
$\displaystyle\bar{q}^{i}\left(\zeta^{0}_{L\,i\alpha}P_{L}+\zeta^{0}_{R\,i\alpha}P_{R}\right)\nu^{\alpha}\Phi^{k}+{\rm
H.c.},$ (11)

where the quark $q^{i}$ is of up type (down type) for the scalar LQ
$\Phi^{2/3}$ $(\Phi^{-1/3})$. As usual $P_{L,R}$ are the left- and right-
handed chirality projectors and the LQ coupling constants
$\zeta^{0}_{L\,i\alpha}$ and $\zeta^{0}_{R\,i\alpha}$, where the superscripts
stands for the LQ spin and the column (row) index is denoted by a Latin (Greek
letter) and it is associated with the quark (lepton) family, will be assumed
as complex and can be read off Table 3. Notice however that the scalar LQ
representation $S_{1}$ gives rise to interactions with charge conjugate quark
fields of the form

$\displaystyle\mathcal{L}^{\Phi^{k}q^{C,i}\nu^{\alpha}}$
$\displaystyle=\bar{q}^{C,i}\left(\tilde{\zeta}^{0}_{L\,i\alpha}P_{L}+\tilde{\zeta}^{0}_{R\,i\alpha}P_{R}\right)\nu^{\alpha}\Phi^{k\,*}+{\rm
H.c.},$ (12)

where again the quark $q^{i}$ is of up type (down type) for the scalar LQ
$\Phi^{2/3}$ $(\Phi^{-1/3})$. We will discuss below that our results for the
NEPs obtained from the interaction (11) can be used mutatis mutandi to obtain
the contributions of the scalar LQs that obey the interaction (12).

####  II.1.2 Electromagnetic interactions of scalar LQs

As far as the couplings of the scalar doublets to the photon are concerned,
they can be straightforwardly obtained from the kinetic Lagrangian, which for
a scalar LQ multiplet $\Phi$ can be generically written as

${\cal
L}_{\text{kin}}^{\text{LQ}}=\frac{1}{2}(D_{\mu}\Phi)^{\dagger}D^{\mu}\Phi,$
(13)

where the $SU(2)\times U(1)$ covariant derivative is given by

$D_{\mu}=\partial_{\mu}+igI^{l}W^{l}_{\mu}+ig^{\prime}Y_{\Phi}B_{\mu},$ (14)

where $l$ runs from 1 to 3 and $I^{k}$ are matrices in the $SU(2)$
representation of $\Phi$: $I^{l}=0$ for $SU(2)$ singlets, $I^{l}=\tau^{l}$
($l=1,2,3$), with $\tau^{l}$ being the Pauli matrices, for $SU(2)$ scalar LQ
doublets, and $\left(I^{l}\right)_{ij}=-i\epsilon_{ijl}$ ($i,j,l=1,2,3$) for
$SU(2)$ scalar LQ triplets. As usual $B_{\mu}$ and $W^{i}_{\mu}$ are the
abelian and nonabelian gauge fields.

After rotating to the mass eigenstates, we obtain the couplings of the LQ
multiplet components $\Phi^{k}$ to the photon, which can be written as

$\mathcal{L}^{A\Phi^{k}\Phi^{k\dagger}}=ieQ_{k}A^{\mu}\left(\Phi^{k\dagger}\partial_{\mu}\Phi^{k}-\Phi^{k}\partial_{\mu}\Phi^{k\dagger}\right).$
(15)

Apart from the usual SM Feynman rules, the remaining ones necessary for our
calculation can be obtained from the above Lagrangians and are presented in
Fig. 2.

Figure 2: Generic Feynman rules for the interactions of a scalar LQ $\Phi^{k}$
necessary for the calculation of NEPs. Here $k$ stands for the LQ electric
charge in units of $|e|$. The usual Feynman rules for the particle propagators
and the SM interactions are not shown.

###  II.2 Vector LQ interactions

####  II.2.1 $U_{1}$ and $\widetilde{V}_{2}$ neutrino couplings

As for the vector $U_{1}$ and $\widetilde{V}_{2}$ representations, their
interactions with the left- and right-handed neutrinos follow from the
current-sector Lagrangian and can be written as follows

$\displaystyle\mathcal{L}_{U_{1}}$
$\displaystyle\supset\left(X^{LL}_{1\,ij}\bar{Q}^{{}^{\prime}i}_{L}\gamma^{\mu}L^{{}^{\prime}j}_{L}+X^{\overline{RR}}_{1\,ij}\bar{u}^{{}^{\prime}i}_{R}\gamma^{\mu}\nu^{{}^{\prime}j}_{R}\right)U_{1\mu}+\text{H.c}.,$
(16)

and

$\displaystyle\mathcal{L}_{\widetilde{V}_{2}}$
$\displaystyle\supset\widetilde{X}^{RL}_{2\,ij}\bar{u}^{{}^{\prime}C\,i}_{R}\gamma^{\mu}\widetilde{V}_{2\,\mu}i\tau_{2}L^{{}^{\prime}j}_{L}+\widetilde{X}^{\overline{LR}}_{2\,ij}\bar{Q}^{{}^{\prime}C\,i}_{L}\gamma^{\mu}i\tau_{2}\widetilde{V}_{2\,\mu}\nu^{{}^{\prime}j}_{R}+\text{H.c}.,$
(17)

where again the coupling constants can be taken as complex quantities in
general, but they are fixed to gauge coupling constants in the case of gauge
LQs. Note that the $\widetilde{V}_{2}$ representation induces diquark
couplings and the couplings of the $\widetilde{V}_{2}^{2/3}$ and
$\widetilde{V}_{2}^{-1/3}$ components can be severely constrained from proton
decay unless an ad hoc symmetry is invoked to achieve proton stability, but we
consider this representation to present the most general calculation of vector
LQ contributions to NEPs.

After rotating to the mass eigenstates we obtain the following interactions of
vector LQs with left- and right-handed neutrinos

$\displaystyle\mathcal{L}_{U_{1}}$ $\displaystyle\supset
U^{2/3}_{1\,\mu}\left(V_{im}U_{kj}X^{LL}_{1\,mk}\bar{u}^{i}_{L}\gamma^{\mu}\nu^{j}_{L}+X^{\overline{RR}}_{1\,ij}\bar{u}^{i}_{R}\gamma^{\mu}\nu^{j}_{R}\right)+\text{H.c.},$
(18)

and

$\displaystyle\mathcal{L}_{\widetilde{V}_{2}}$
$\displaystyle\supset\widetilde{V}^{-2/3}_{2\,\mu}\left(-U_{ik}\widetilde{X}^{RL}_{2\,kj}\bar{u}_{R}^{C\,i}\gamma^{\mu}\nu_{L}^{j}+V^{T}_{ik}\widetilde{X}^{\overline{LR}}_{2\,kj}\bar{u}_{L}^{C\,i}\gamma^{\mu}\nu_{R}^{j}\right)-\widetilde{V}^{1/3}_{2\,\mu}\widetilde{X}^{\overline{LR}}_{2\,ij}\bar{d}_{L}^{Ci}\gamma^{\mu}\nu_{R}^{j}+{\rm
H.c.},$ (19)

Again, we will consider a generic interaction of vector LQ $V^{k}$ of electric
charge $Q_{k}=k$, in units of $|e|$, to charge $Q_{k}$ quarks and neutrinos of
the form

$\displaystyle\mathcal{L}^{V^{k}_{\mu}q^{i}\nu^{\alpha}}=$ $\displaystyle
V^{k}_{\mu}\bar{q}^{i}\gamma^{\mu}\left(\zeta^{1}_{L\,i\alpha}P_{L}+\zeta^{1}_{R\,i\alpha}P_{R}\right)\nu^{\alpha}+{\rm
H.c.},$ (20)

where the quark $q^{i}$ is of up type (down type) for the vector LQ $V^{2/3}$
$(V^{-1/3})$. The LQ coupling constants $\zeta^{1}_{L\,i\alpha}$ can be read
off from Table 3 for all the vector LQ representations of Table 2. As in the
case of scalar LQs, we will see below that the results for NEPs obtained from
(20) will also allows one to obtain the results for the contribution arising
from the $\widetilde{V}_{2}$ representation, which yields an interaction of
the form

$\displaystyle\mathcal{L}^{V^{k}_{\mu}q^{C,i}\nu^{\alpha}}=$
$\displaystyle\bar{q}^{C,i}\gamma^{\mu}\left(\tilde{\zeta}^{1}_{L\,i\alpha}P_{L}+\tilde{\zeta}^{1}_{R\,i\alpha}P_{R}\right)\nu^{\alpha}V^{k\dagger}_{\mu}+{\rm
H.c.}$ (21)

####  II.2.2 Electromagnetic interactions of vector LQs

As far as the LQ electromagnetic couplings, for simplicity we will consider
below vector LQs that are arise from a gauge theory spontaneously broken.
Following our model-independent approach we consider that once the gauge group
of the ultraviolet (UV) completion has been broken into the SM gauge group,
the gauge LQ interactions with the SM gauge bosons are given by the most
general renormalizable $SU(2)\times U(1)$ invariant Lagrangian for a gauge LQ
$V_{\mu}$ [92, 116, 117]

${\cal
L}^{V_{\mu}}=-\frac{1}{2}V^{\dagger}_{\mu\nu}V^{\mu\nu}-ig^{\prime}V^{\dagger\mu}B_{\mu\nu}V^{\nu}-igV^{\dagger\mu}\vec{W}_{\mu\nu}V^{\nu},$
(22)

where $V_{\mu\nu}=D_{\mu}V_{\nu}-D_{\nu}V_{\mu}$, with the $SU(2)_{L}\times
U(1)_{Y}$ covariant derivative given in Eq. (14), also
$\vec{W}_{\mu\nu}=I^{k}W_{\mu\nu}^{k}$, with the matrices $I^{k}$ being
defined above.

Finally, after the gauge symmetry has broken to the $U(1)_{\rm em}$ group and
all the gauge fields have been rotated to their mass eigenstates, we arrive at
the following interaction Lagrangian of a pair of gauge LQs $V^{k}_{\mu}$ to
the photon [118]

${\cal
L}^{AV^{k\dagger}V^{k}}=-\frac{1}{2}V^{k\,\dagger}_{\mu\nu}V^{k\,\mu\nu}-iQ_{k}eV^{k\,\dagger}_{\mu}F^{\mu\nu}V^{k}_{\nu},$
(23)

where $F_{\mu\nu}$ is the electromagnetic strength tensor and

$V^{k}_{\mu\nu}=D^{\rm em}_{\mu}V^{k}_{\nu}-D^{\rm
em}_{\nu}V^{k}_{\mu}=\left(\partial_{\mu}+iQ_{k}eA_{\mu}\right)V^{k}_{\nu}-\left(\partial_{\nu}+iQ_{k}eA_{\nu}\right)V^{k}_{\mu}.$
(24)

The generic Feynman rules in the unitary gauge for a gauge LQ are presented in
Fig. 3, though they are only useful to calculate the static neutrino dipole
moments, which are gauge-independent quantities.

Figure 3: Generic Feynman rules for a gauge LQ $V^{k}_{\mu}$ in the unitary
gauge necessary for the calculation of the neutrino static dipole moments.
Here $k$ stands for the LQ electric charge in units of $|e|$ and the
$AV^{k\dagger}V^{k}$ vertex function is $\Gamma^{\rm
UG}_{\mu,\alpha,\beta}(k_{1},k_{2},k_{3})=(k_{3}-k_{2})_{\mu}g_{\alpha\beta}+(k_{1}-k_{3})_{\alpha}g_{\beta\mu}+(k_{2}-k_{1})_{\beta}g_{\mu\alpha}$.
The usual Feynman rules for the particle propagators and SM interactions are
not shown.

However, to obtain the NCR we need to make some assumptions about the UV
completion of the LQ model since the calculation must be performed via the BFM
to obtain a gauge-independent result. As far as the LQ interactions with the
SM gauge bosons are concerned, as shown in Ref. [117], where the BFM formalism
was used to obtain the Feynman rules for the new singly and doubly charged
gauge bosons arising in an $SU(3)_{L}\times U(1)_{X}$ model, once the extended
gauge symmetry is spontaneously broken into $SU(2)_{L}\times U(1)_{Y}$, the
vertex functions for the trilinear and quartic gauge boson couplings share the
same Lorentz structure, which stems from the fact that they all obey
$SU(2)_{L}\times U(1)_{Y}$ symmetry. Even more, the couplings of any charged
gauge boson and its associated pseudo-Goldstone boson must obey
electromagnetic gauge invariance. Thus, the vertex function for the coupling
of a photon with a pair of charged gauge bosons $AVV^{\dagger}$ is identical
for any charged gauge boson except for the electric charge factor, and the
same is true for the vertex function of the $AG_{V}G_{V}^{\dagger}$ coupling,
where $G_{V}$ is the pseudo-Goldstone boson associated with the charged gauge
boson (note that in the BFM there is no mixed tree-level $AVG_{V}^{\dagger}$
coupling). Thus, the Feynman-rules for the $AVV^{\dagger}$ and
$AG_{V}G_{V}^{\dagger}$ couplings, with $V$ a gauge LQ and $G_{V}$ its
associated pseudo-Goldstone boson, must be analogue to those presented in
[117] and [26] for singly and doubly charged gauge bosons after replacing the
LQ electric charge. Nevertheless, for the interactions of the LQ and its
pseudo-Goldstone boson with the SM fermions we do need to know more details of
the UV completion of the LQ model since the coupling constants of Eq. (20) are
fixed to the gauge coupling constants. To obtain an estimate of the magnitude
of the LQ contributions to the NCR we will assume a gauge LQ with left-handed
coupling inspired in the model of [110, 119]. The corresponding Feynman rules
are presented in Fig. 3, where we also include the Feynman rules for the
coupling of a pseudo-Goldstone boson to a neutrino-quark pair, with the
coupling constants being given in Table 4.

Figure 4: Feynman rules in the Feynman-’t Hooft gauge of the BFM, which together with those of Fig. 3 are necessary for the calculation of the contribution of a gauge LQ to the NCR. Note that in this gauge the $AVV^{\dagger}$ vertex function is given by $\Gamma^{\rm BFM}_{\mu,\alpha,\beta}(k_{1},k_{2},k_{3})=(k_{3}-k_{2})_{\mu}g_{\alpha\beta}+(k_{1}-k_{3}-k2)_{\alpha}g_{\beta\mu}+(k_{2}-k_{1}+k3)_{\beta}g_{\mu\alpha}$. In addition, the left- and right handed coupling constants of the gauge LQ and its associated pseudo-Goldstone boson to the fermions must be fixed to the gauge constants. Thus, inspired in the model of Ref. [110, 119] we assume a simple renormalizable gauge LQ model where the coupling constants of a gauge LQ and its associated pseudo-Goldstone bosons to a neutrino-quark pair are given as in Table 4.  Table 4: Left- and right-handed coupling constants for the interactions of a gauge LQ and its associated pseudo-Goldstone boson to the a neutrino-quark pair, inspired in the model of Ref. [110]. Here $g_{4}$ stands for a gauge coupling constant.  Vertex | Left-handed couplings | Right-handed couplings
---|---|---
$V^{k}\bar{q}^{i}\nu^{\alpha}$ | $\zeta^{1}_{L\,i\alpha}=\dfrac{g_{4}}{\sqrt{2}}\beta_{i\alpha}$ | $\zeta^{1}_{R\,i\alpha}=0$
$G_{V}^{k}\bar{q}^{i}\nu^{\alpha}$ | $\bar{\zeta}^{1}_{L\,i\alpha}=\dfrac{g_{4}}{\sqrt{2}}\dfrac{m_{q_{i}}}{m_{\rm LQ}}\beta_{i\alpha}$ | $\bar{\zeta}^{1}_{R\,i\alpha}=\dfrac{g_{4}}{\sqrt{2}}\dfrac{m_{\nu_{\alpha}}}{m_{\rm LQ}}\beta_{i\alpha}$

##  III LQ contribution to NEPs

We turn to present our calculation, which was worked out by the Feynman-
parameter integration technique with the help of the FeynCalc package [120,
121] to perform the Dirac algebra. An independent evaluation was done by means
of the Passarino-Veltman reduction scheme via the FeynCalc and Package-X
routines [122], which allowed us to cross-check our analytical results.

We first write the most general form of the contribution to the MDM of
neutrino $\nu_{\alpha}$ from either a scalar ($\Phi$) or vector ($V$) LQ as
follows in Bohr magneton units

$\displaystyle\mu_{\nu_{\alpha}}^{s}$ $\displaystyle=\frac{N_{c}Q_{\rm
LQ}m_{e}}{16\pi^{2}m_{\rm
LQ}}\sum_{i}\left(\left(\frac{m_{\nu_{\alpha}}}{m_{\rm
LQ}}\right)\left(\left|\zeta^{s}_{L\,i\alpha}\right|{}^{2}+\left|\zeta^{s}_{R\,i\alpha}\right|{}^{2}\right)f^{s}(x_{q_{i}})+\left(\frac{m_{q_{i}}}{m_{\rm
LQ}}\right){\rm
Re}\left(\zeta^{s}_{L\,i\alpha}\zeta^{{s}*}_{R\,i\alpha}\right)g^{s}(x_{q_{i}})\right)\;\mu_{B},$
(25)

where the superscript $s$ stands for the LQ spin, whereas $Q_{\rm LQ}$ and
$m_{LQ}$ denote its electric charge and mass. We also define
$x_{q_{i}}=m_{q_{i}}^{2}/m_{\rm LQ}^{2}$, with the internal quark $q_{i}$
being of type up (down) for $Q_{\rm LQ}=2/3$ ($Q_{\rm LQ}=-1/3$). Also

$\displaystyle f^{s}(x)$ $\displaystyle=\sum_{j}f^{s}_{j}(x),$ (26)
$\displaystyle g^{s}(x)$ $\displaystyle=\sum_{j}g^{s}_{j}(x),$ (27)

where the subscript $j$ stands for the partial contribution of each Feynman
diagram. The LQ coupling constants $\zeta^{s}_{L\,i\alpha}$ and
$\zeta^{s}_{R\,i\alpha}$ can be read off Table 3 for the scalar and vector LQs
arising from the representations of Table 2.

As for the contribution of a spin-$s$ LQ to the neutrino electric dipole
moment, it requires complex LQ couplings and can be written as follows

$\displaystyle d_{\nu_{\alpha}}^{s}$ $\displaystyle=-\frac{eN_{c}Q_{\rm
LQ}}{32\pi^{2}m_{\rm LQ}}\sum_{i}\left(\frac{m_{q_{i}}}{m_{\rm LQ}}\right){\rm
Im}\left(\zeta^{s}_{L\,i\alpha}\zeta^{s*}_{R\,i\alpha}\right)g^{s}(x_{q_{i}}),$
(28)

where the $g^{s}(x)$ function and its partial contributions $g^{s}_{i}(x)$ are
the same of Eq. (25).

It is worth noting that the results for the static MDM and EDM can be safely
obtained by setting $q^{2}=0$ from the start of the calculation as long as
Feynman-parameter integration is used, but a careful limiting procedure is
required when the result is worked out by the Passarino-Veltman reduction
method.

As far as the neutrino charge radius is concerned, the calculation is more
involved as it is necessary to obtain first the neutrino charge form factor
$F_{1}^{V}(q^{2})$ for nonvanishing $q^{2}$ and then calculate its derivative.
A Taylor expansion around $q^{2}=0$ yields

$F^{V^{\prime}}(q^{2})=q^{2}\frac{dF^{V}(q^{2})}{dq^{2}}\Big{\|}_{q^{2}=0}+O\left(q^{4}\right),$
(29)

where the first term is absent because of the fact that the neutrino charge
form factor vanishes for an on shell photon $F^{V}(0)=0$, which was explicitly
verified through our calculation. From the above expansion we obtain the
following contribution to the NCR from a spin-$s$ LQ

$\displaystyle\langle r^{2}\rangle_{\nu_{\alpha}}^{s}=\frac{N_{c}Q_{\rm
LQ}}{16\pi^{2}m_{\rm
LQ}^{2}}\sum_{i}\left(\left(\left|\zeta^{s}_{L\,i\alpha}\right|{}^{2}+\left|\zeta^{s}_{R\,i\alpha}\right|{}^{2}\right)\hat{f}^{s}(x_{q_{i}})+\left(\frac{m_{q_{i}}m_{\nu_{\alpha}}}{m_{\rm
LQ}^{2}}\right){\rm
Re}\left(\zeta^{s}_{R\,i\alpha}\zeta^{s*}_{R\,i\alpha}\right)\hat{g}^{s}(x_{q_{i}})\right),$
(30)

where the $f^{s}(x)$ and $g^{s}(x)$ obey relationships similar to Eqs. (26)
and (27).

We have obtained results for the $r^{s}_{k}(x)$ functions
($r=f,\,g,\,\hat{f},\,\hat{g}$) for nonzero neutrino mass in terms of both
Feynman-parameter integrals and Passarino-Veltman scalar functions, which we
present in Appendix A and can be useful to obtain the NEPs of hypothetical
heavy Dirac neutrinos. Approximate results were also obtained to leading order
in the neutrino mass, which showed a good agreement with the numerical
evaluation of the Feynman-parameter integrals and the Passarino-Veltman
results. The latter were evaluated via the LoopTools package [123]. Such
approximate results provide a good estimate for the NEPs of light neutrinos
and are presented below.

Notice also that although the interaction of other scalar LQ to a neutrino-
quark pair involves the Majorana-type Feynman rule of Eq. (12), which require
special treatment [124, 125], after some algebra the corresponding
contributions to NEPs can be shown to be identical to those obtained from the
interaction of Eq. (11) (see for instance [126], where the anomalous MDM of
the muon was calculated in models with new scalar and vector particles) and
the same is true for the contribution of vector LQs arising from Eq. (21),
which can be obtained from the contribution arising from Eq. (20). A similar
situation was observed in the calculation of the LQ contribution to the decays
$f_{i}\to f_{j}\gamma$ and $t\to c\gamma\gamma$ [127, 128].

###  III.1 Scalar LQ contribution

In the models with scalar LQs we are interested in, the leading order
contributions to the neutrino dipole moments are obtained from the Feynman
diagrams of Fig. 5, which are clearly gauge-independent and yield ultraviolet
finite MDM and EDM.

Figure 5: Feynman diagrams that contribute to the neutrino dipole moments and
the NCR at the one-loop level in models with scalar LQs.

On the other hand, the contribution to the neutrino charge form factor
$F_{1}^{V}(q^{2})$ is ultraviolet divergent, though its derivative is finite
and so is the contribution to the NCR. Notice that $F_{1}^{V}(q^{2})$ can also
receive a contribution from the $\gamma-Z$ self-energy diagram of Fig. 6, but
as already noted such a contribution must be added to the effective neutrino
electroweak charge radius rather than the effective NCR according to Ref.
[29]. Therefore, we will not discuss such a contribution here.

Figure 6: Feynman diagram for the $\gamma-Z$ self-energy contribution to the
neutrino charge form factor $F_{1}^{V}(q^{2})$ and also to its slope in models
with scalar LQ. This contribution must not be included in the NCR according to
[29].

The contributions of a scalar LQ to the neutrino MDM are given trough the
$f^{0}_{i}(x)$ and $g^{0}_{i}(x)$ functions of Eqs. (25) and can be
approximated for light neutrinos as

$\displaystyle f^{0}_{1}(x)$
$\displaystyle=-\frac{1}{6\left(x-1\right){}^{4}}\left(\left(\left(x-6\right)x+3\right)x+6x\log\left(x\right)+2\right),$
(31) $\displaystyle f^{0}_{2}(x)$
$\displaystyle=-\frac{1}{6\left(x-1\right){}^{4}}\left(\left(x-1\right)\left(x\left(2x+5\right)-1\right)-6x^{2}\log\left(x\right)\right),$
(32)

and

$\displaystyle g^{0}_{1}(x)$
$\displaystyle=-\frac{1}{\left(x-1\right){}^{3}}\left(x^{2}-4x+2\log\left(x\right)+3\right),$
(33) $\displaystyle g^{0}_{2}(x)$
$\displaystyle=-\frac{1}{\left(x-1\right){}^{3}}\left(x^{2}-2x\log\left(x\right)-1\right).$
(34)

The latter functions are also useful to compute the contribution of a scalar
LQ to the neutrino EDM of Eq. (28).

As far as the contributions of a scalar LQ to the NCR, the $\hat{f}^{0}(x)$
and $\hat{g}^{0}(x)$ functions of Eq. (30) are given as follows for light
neutrinos

$\displaystyle\hat{f}^{0}_{1}(x)$
$\displaystyle=\frac{1}{12\left(x-1\right){}^{4}}\left(6\left(2-3x\right)\log\left(x\right)-\left(x-1\right)\left(x\left(7x-29\right)+16\right)\right),$
(35) $\displaystyle\hat{f}^{0}_{2}(x)$
$\displaystyle=-\frac{1}{12\left(x-1\right){}^{4}}\left(\left(x-1\right)\left(x\left(11x-7\right)+2\right)-6x^{3}\log\left(x\right)\right),$
(36)

and

$\displaystyle\hat{g}^{0}_{1}(x)$
$\displaystyle=-\frac{1}{3\left(x-1\right){}^{5}}\left(\left(x-1\right)\left(\left(x-8\right)x-17\right)+6\left(3x+1\right)\log\left(x\right)\right),$
(37) $\displaystyle\hat{g}^{0}_{2}(x)$
$\displaystyle=\frac{1}{3\left(x-1\right){}^{5}}\left(\left(\left(9-17x\right)x+9\right)x+6\left(x+3\right)x^{2}\log\left(x\right)-1\right).$
(38)

###  III.2 Vector LQ contribution

We first calculate the contribution of a vector LQ to the static neutrino MDM
and EDM, which are gauge independent and can thus be straightforwardly
computed in any gauge, for instance in the unitary one, where the contributing
Feynman diagrams are the ones shown in the upper row of Fig. 7.

Figure 7: Feynman diagrams for the contribution to NEPs from gauge LQ
$V^{k}_{\mu}$. In the unitary gauge there are contributions of the upper row
diagrams only, whereas the lower row diagrams are the extra contributions that
appear in the Feynman-’t Hooft gauge of the BFM. Here $G_{V}$ stands for the
pseudo-Goldstone boson associated to $V$.

The results for the $f^{1}_{k}(x)$ and $g^{1}_{k}(x)$ functions for light
neutrinos read as follows

$\displaystyle f^{1}_{1}(x)$
$\displaystyle=\frac{1}{6\left(x-1\right){}^{4}}\left(18x^{2}\log\left(x\right)-\left(x-1\right)\left(x\left(x\left(5x-9\right)+30\right)-8\right)\right),$
(39) $\displaystyle f^{1}_{2}(x)$
$\displaystyle=-\frac{1}{6\left(x_{q}-1\right){}^{4}}\left(\left(x-1\right)\left(x\left(x\left(4x-45\right)+33\right)-10\right)+18x^{3}\log\left(x\right)\right),$
(40) $\displaystyle g^{1}_{1}(x)$
$\displaystyle=\frac{1}{\left(x-1\right){}^{3}}\left(x^{3}+3x-6x\log\left(x\right)-4\right),$
(41) $\displaystyle g^{1}_{2}(x)$
$\displaystyle=\frac{1}{\left(x-1\right){}^{3}}\left(\left(x-1\right)\left(\left(x-11\right)x+4\right)+6x^{2}\log\left(x\right)\right).$
(42)

As far as the calculation of the NCR is concerned, the gauge LQ contribution
must be obtained via the BFM in the Feynman-’t Hooft gauge. Apart from the
Feynman diagrams with an internal gauge LQ, there are two additional Feynman
diagrams where the gauge LQ is replaced by its associated pseudo-Goldstone
boson, as shown in the lower row of Fig. 7.

According to the coupling constants assumed in Table 4, the gauge LQ
contribution to the NCR can be written as in Eq. (30) but with vanishing
right-handed couplings, namely, $\zeta^{1}_{R\,i\alpha}=0$. It means that
there are only contributions to the NCR via the $\hat{f}_{i}(x)$ functions,
which are given by

$\displaystyle\hat{f}^{1}_{1}(x)$
$\displaystyle=\frac{1}{6\left(x_{q}-1\right){}^{4}}\left(\left(x-1\right)\left(x\left(25x-29\right)-2\right)-6\left(6x^{2}-9x+2\right)\log\left(x\right)\right),$
(43) $\displaystyle\hat{f}^{1}_{2}(x)$
$\displaystyle=\frac{1}{6\left(x-1\right){}^{4}}\left(6x^{2}\left(5x-6\right)\log\left(x\right)-\left(x-1\right)\left(x\left(43x-65\right)+16\right)\right),$
(44) $\displaystyle\hat{f}^{1}_{3}(x)$
$\displaystyle=\frac{x}{12\left(x-1\right){}^{4}}\left(6\left(2-3x\right)\log\left(x\right)-\left(x-1\right)\left(x\left(7x-29\right)+16\right)\right),$
(45) $\displaystyle\hat{f}^{1}_{4}(x)$
$\displaystyle=\frac{x}{12\left(x-1\right){}^{4}}\left(6x^{3}\log\left(x\right)-\left(x-1\right)\left(x\left(11x-7\right)+2\right)\right),$
(46)

where again the subscript $i$ stands for the contribution of the $i$-th
diagram of Fig. 7.

Again we do not consider the contribution of the $\gamma-Z$ self-energy of
Fig. 6 with the scalar LQs replaced by the gauge LQ and its associated pseudo-
Goldstone boson for the reason already explained.

Note that for a very heavy LQ, namely, $x_{q_{i}}\ll 1$ the effective NCR for
a gauge LQ with the couplings of Table 4 can be approximated as

$\displaystyle\langle r^{2}\rangle_{\nu_{\alpha}}^{1}$
$\displaystyle\simeq\frac{N_{c}Q_{\rm LQ}}{16\pi^{2}m_{\rm
LQ}^{2}}\sum_{i}\left|\zeta^{1}_{L\,i\alpha}\right|{}^{2}\left(3-2\log\left(x_{q_{i}}\right)\right),$
(47)

which reduces to the SM result of Eq. (6) after the replacements:
$\left|\zeta^{1}_{L\,i\alpha}\right|{}^{2}\to\dfrac{g^{2}}{2}=2\sqrt{2}m_{W}^{2}G_{F}$,
$m_{\rm LQ}\to m_{W}$, $m_{q_{i}}\to m_{e}$, $Q_{\rm LQ}\to 1$, and $N_{c}\to
1$.

##  IV Numerical Analysis

We now turn to the numerical analysis of the LQ contribution to NEPs, for
which we need to discuss the current constraints on the LQ mass and couplings
consistent with experimental limits on high precision observables and direct
searches at the LHC. We will consider a LQ model with one LQ representation
that is not primarily meant to solve the LFUV anomalies in $B$-meson decays or
the $(g-2)_{\mu}$ anomaly, which may require more ingredients such as extra LQ
representations, new symmetries, and fine tuning. Note also that such
anomalies could be explained by another mechanism in the LQ UV completion or
disappear in future measurements as appears to be the case with the
$R_{K^{*},K}$ anomaly [71].

###  IV.1 Constraints on the masses of scalar and vector LQs

The most recent constraints on the masses of both scalar and vector LQs have
been obtained from direct searches at the CERN LHC by the CMS and ATLAS
collaborations using the data for proton collisions at $\sqrt{s}=13$ TeV [129,
130, 131, 132, 133, 134, 135, 136, 137, 138, 139] via single and double
production of LQs decaying into a quark and a lepton, namely,
$pp\to\Phi\bar{\ell}\to q\ell\bar{\ell}$ and $pp\to\Phi^{\dagger}\Phi\to
q\bar{q}\ell\bar{\ell},q\bar{q}\nu\bar{\nu}$. The constraints obtained this
way are rather model dependent since it is usually assumed that LQs only
couple to fermions of one generation and have a dominant decay channel, though
LQs that can couple to fermions of distinct generations have also been
considered recently [131, 133, 134]. In addition, since the oblique parameters
strongly constrain the mass splitting of a $SU(2)$ multiplet [140], LQs of the
same $SU(2)$ representation are assumed mass degenerate.

####  IV.1.1 Charge $2/3$ LQs

The ATLAS and CMS collaborations [135, 141, 139] set a lower bound on the mass
of a third-generation scalar LQ ranging between 900 and 1450 GeV for a
branching ratio of the $b\tau$ decay channel ranging from 10 to 100 percent.
On the other hand, for a vector LQ in the Yang-Mills scenario the lower mass
bound goes from 1400 to about 1950 GeV for a branching ratio of the $b\tau$
decay channel ranging from 10 to 100 percent [141, 139], whereas in the so
called minimal coupling scenario the respective lower mass bounds are about
300 GeV less stringent [141, 139].

The constraints are more severe when it is imposed the condition that a
solution to the LFUV anomalies in $B$-meson decays is imposed. In this
scenario, the ATLAS collaboration has searched for pair produced scalar and
vector LQs decaying into a quark of the third generation accompanied by a
lepton of the second and first generations [137, 138]: for the mass of a
charge $2/3$ scalar LQ decaying into the $b\mu$ ($be$) pair a lower bound of
about 1440 (1460) GeV was found. Along the same line, the CMS collaboration
has reported [136] a search for third-generation charge $2/3$ scalar and
vector LQs decaying into a bottom quark plus a $\tau$ lepton via single and
pair LQ production: when the LQ coupling to the $b\tau$ pair is of the order
of unity, the lower bound on the mass of a scalar (vector) LQ is about 1250
GeV (1530 GeV), whereas for a LQ coupling of the order of 2.5 the
corresponding lower mass limit is about 1370 GeV (1960 GeV) for a scalar
(vector) LQ.

####  IV.1.2 Charge $-1/3$ LQs

ATLAS has searched for this type of LQs [142] via pair production and its
decay into a $\tau$ lepton plus a charm or a lighter quark. A charge $-1/3$ LQ
with mass below 1.3 TeV is excluded as long as the branching fraction for the
$\tau c$ decay channel is of one hundred percent. The CMS collaboration also
searched for scalar and vector LQs decaying into a neutrino-quark pair and
found the bounds of 1110 GeV for a scalar LQ and $1475$-$1810$ GeV for a
vector LQ [143].

In the scenario where a solution to the LFUV anomalies is required, the ATLAS
collaboration has searched for this type of LQ via the decay into $t\mu$
($te$) and set the lower bound of about 1380 GeV (1370 GeV); again, the bounds
on the mass of vector LQs are more stringent[137]: for a charge $-1/3$ vector
LQ decaying into $t\mu$ ($te$), the lower bound is 1980 GeV (1900 GeV) in the
Yang-Mills scenario, but such a bound relaxes to 1710 GeV (1620 GeV) in the
minimal coupling scenario.

We present a summary of the above constraints in Table 5. For our numerical
analysis we will consider LQs with masses from $1.2$ to $2$ TeVs.

Table 5: Current lower bounds on the mass of scalar and vector LQs from direct searches by the ATLAS and CMS collaborations [129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139]. In scenario I the condition of a solution to the LFUV anomalies in $B$-meson decays is not imposed, whereas in scenario II such a solution is required indeed. See the text and References therein for the assumptions made to obtain such limits. LQ charge | Scenario | Scalar LQ mass (GeV) | Vector LQ mass (GeV)
---|---|---|---
$2/3$ | I | 900-1450 | 1400-1950
$2/3$ | II | 1250-1450 | 1460-1960
$-1/3$ | I | 1300 |  $1475$-$1810$
$-1/3$ | II | 1370-1380 | 1620-1980

###  IV.2 Realistic scenarios for the LQ coupling constants

In order to obtain a realistic estimation of the NEPs we must consider
scenarios for the LQ couplings to fermion bilinears consistent with the most
up-to-date constraints on high-precision experimental observables. We will
focus on those scenarios that could allow for the largest values for the LQ
coupling constants as they can give the largest contributions to NEPs.

LQ particles can have dangerous effects on electroweak precision observable
quantities, which may yield strong constraints on the LQ masses and couplings.
As far as LQ couplings to fermions are concerned, we will assume the most
general scenario where LQs can couple to fermions of all generations. In
particular, LQ couplings to the first-generation fermions are strongly
constrained by low-energy processes, such as atomic parity violation [144,
145, 146], universality in leptonic pion decays [147, 146, 145], $\mu-e$
conversion [148], flavor changing Kaon decays [148, 149], as well as
$K^{0}-\overline{K}^{0}$ and $D^{0}-\overline{D}^{0}$ mixings [147, 146, 150].
We will thus assume an extra symmetry to forbid the LQ couplings to the first
generation of fermions while still allowing nonzero couplings to the fermions
of the second and third generations, as has been customary done in the
literature.

An analysis of the constraints on the coupling constants of all the LQ
representations of Table 2 is beyond the scope of this work. Thus, for our
analysis we will focus only on the scalar $\widetilde{R}_{2}$ and vector
$U_{1}$ representations as they are two of the LQ representations that can
yield the largest possible values of NEPs as will be shown below.

####  IV.2.1 The $\widetilde{R}_{2}$ representation

This LQ representation can give a solution to the anomalies in $B$-meson
decays, but additional representations are required to explain the discrepancy
in the muon $g-2$ [114]: it turns out that apart from the charge $-1/3$ LQ
that only couples to quarks and neutrinos, this representation yields a charge
$2/3$ chiral LQ that can couple to down quarks and charged leptons, thereby
inducing a contribution to the muon $g-2$ lacking of an enhancing chirality-
flipping term.

We will content ourselves with discussing the constraints on the LQ couplings
in a simple model analyzed where only one $\widetilde{R}_{2}$ LQ
representation is introduced with both left- and right handed couplings to
neutrinos. Constraints on these couplings can be obtained from the
$B_{s}-\overline{B}_{s}$ mass difference and the decays $b\to
s\mu^{-}\mu^{+}$, $B\to K\nu\nu$, $\tau\to\mu\gamma$, $\tau\to\mu\phi$ as well
as the semileptonic meson decays such $B\to\tau\nu$, $D_{s}\to\tau\nu$, etc.
The analytical expressions for the contribution of the
$\widetilde{R}_{2}^{2/3}$ LQ to these observable quantities can be found in
[108], but for easy reference they are presented in Appendix B. In our
analysis we will assume that the PMNS mixing matrix is approximately diagonal,
which means that the $\zeta^{0}_{L}$ and $\zeta^{0}_{R}$ couplings from Eq.
(11) can be identified with the Yukawa couplings $\widetilde{Y}^{RL}_{2}$ and
$\widetilde{Y}^{\overline{RR}}_{2}$, respectively, of Table 3. We will also
assume the following ansatz to avoid the strong constraints from parity
violation

${\zeta}^{0}_{L}\simeq\begin{pmatrix}0&0&0\\\
0&\zeta^{0}_{L\,2\mu}&\zeta^{0}_{L\,2\tau}\\\
0&\zeta^{0}_{L\,3\mu}&\zeta^{0}_{L\,3\tau}\end{pmatrix},$ (48)

with a similar expression for ${\zeta}^{0}_{R}$.

Since we are interested in the scenario that can yield the largest estimates
for the NEPs, we will consider small values for the coupling constants
$\zeta^{0}_{L\,2\alpha}$ and $\zeta^{0}_{R\,2\alpha}$, below the $0.1$ level,
which will allow for large values of $\zeta_{L\,3\alpha}$ and
$\zeta_{R\,3\alpha}$. We randomly scan over sets of values of the nonzero
$\zeta^{0}_{L,R}$ matrix elements and found those sets that fulfil the
experimental limits on the $B_{s}-\overline{B}_{s}$ mass difference as well as
the decays $b\to s\mu^{-}\mu^{+}$ and $B\to K\nu\nu$, along with the bounds
$\zeta^{0}_{L\,3\alpha},\,\zeta^{0}_{R\,3\alpha}\leq 1$, which are meant to
avoid the breakdown of perturbation theory and are stronger than the usual
upper bound of $4\pi$ to evade the constraints from direct searches at the
LHC. Although we include in our analysis other observables such as the lepton
flavor violating (LFV) tau decays $\tau\to\mu\gamma$ and $\tau\to\mu\phi$, as
well as semileptonic meson decays, they do not yield useful constraints for
small $\zeta^{0}_{L\,2\alpha}$ and $\zeta^{0}_{R\,2\alpha}$. We show in Fig. 8
the allowed values of the LQ couplings in the
$\zeta^{0}_{L\,2\mu}\zeta^{0}_{L\,2\tau}\zeta^{0}_{R\,2\mu}\zeta^{0}_{R\,2\tau}$
vs
$\zeta^{0}_{L\,3\mu}\zeta^{0}_{L\,3\tau}\zeta^{0}_{R\,3\mu}\zeta^{0}_{R\,3\tau}$
for LQ masses of 1.2 TeV and 1.5 TeVs. We can conclude that there are allowed
values of the product
$\zeta^{0}_{L\,3\mu}\zeta^{0}_{L\,3\tau}\zeta^{0}_{R\,3\mu}\zeta^{0}_{R\,3\tau}$
close to the unity, which means that there are regions of the parameter space
where $\zeta^{0}_{L\,3\alpha}$ and $\zeta^{0}_{R\,3\alpha}$ can be close to
one simultaneously, as can be observed in the zoomed area. Note also that such
values are also consistent with the model-independent bounds on the LQ
couplings from direct searches at the LHC compiled in [151].

Figure 8: Allowed values of the Yukawa couplings of the $\widetilde{R}_{2}$
representation in the plane
$\zeta^{0}_{L\,2\mu}\zeta^{0}_{L\,2\tau}\zeta^{0}_{R\,2\mu}\zeta^{0}_{R\,2\tau}$
vs
$\zeta^{0}_{L\,3\mu}\zeta^{0}_{L\,3\tau}\zeta^{0}_{R\,3\mu}\zeta^{0}_{R\,3\tau}$
for LQ masses of $1.2$ TeVs (green points) and $1.5$ TeVs (red points). We use
the experimental limits on the $B_{s}-\overline{B}_{s}$ mass difference as
well as the decays $b\to s\mu^{-}\mu^{+}$, $B\to K\nu\nu$, $\tau\to\mu\gamma$,
and $\tau\to\mu\phi$. The $\zeta^{0}_{L\,2\alpha}$ and
$\zeta^{0}_{R\,2\alpha}$ ($\alpha=\mu,\,\tau$) couplings were taken to be
below $0.1$ to allow for large values of the $\zeta^{0}_{L\,3\alpha}$ and
$\zeta^{0}_{R\,3\alpha}$ couplings, for which we impose the perturbativity
bound $\zeta^{0}_{L\,3\alpha},\,\zeta^{0}_{R\,3\alpha}\leq 1$, which is also
meant to evade the constraints from direct searches at the LHC.

####  IV.2.2 The $U_{1}$ representation

This representation is appealing as it gives no tree-level contribution to
observables such as the $B_{s}-\overline{B}_{s}$ mass difference and the very
constrained decay $B\to K\nu\bar{\nu}$, though loop contributions can be
significant. Nevertheless, finding constraints on the $U_{1}$ couplings can be
troublesome as its contribution to radiative corrections can be plagued with
quadratic divergences, thereby requiring a specific UV completion to tackle
this issue. As already mentioned, a few relatively simple UV completions into
which the $U_{1}$ representation can be embedded have been discussed in Refs.
[103, 106, 96, 110, 152, 153, 119], where the authors try to address some
dangerous effects on low energy observables that can push the LQ mass well
beyond the TeV scale such as in the original Pati-Salam model. Those models
contain new particles, apart from the $U_{1}$ LQ, that contribute to low-
energy observables, thus some assumptions must be made to avoid tension with
experimental measurements.

We will consider a model with a lone vector $U_{1}$ LQ that only couples to
the fermions of the second and third generations in the scenario where either
there are no right-handed neutrinos or the corresponding coupling constants
are negligible, such as in the model of Ref. [119]. For the LQ couplings to
fermions we will use $\zeta^{1}_{L\,i\alpha}=g_{4}\beta_{i\alpha}/\sqrt{2}$
and $\zeta^{1}_{L\,i\alpha}=0$ with $g_{4}\sim 3$ [119]. Again, we consider
$\zeta^{1}_{L\,2\mu},\,\zeta^{1}_{L\,2\tau}\leq 0.1$ to let the
$\zeta^{1}_{L\,3\alpha}$ couplings reach its largest allowed values, below the
perturbativity bound of $1$. We then scan over sets of random
$\zeta^{1}_{L\,i\alpha}$ values and select those sets consistent with the
experimental limits on the decays $b\to s\mu^{-}\mu^{+}$, $B\to
K\tau^{-}\tau^{+}$, $B^{+}\to K^{+}\tau^{\pm}\mu^{\mp}$, and $\tau\to\mu\phi$.
Other $b$-hadron decays such as $B\to\tau\nu$, $B_{c}\to\tau\nu$,
$B\to\tau^{-}\tau^{+}$, and $B_{s}\to\tau^{-}\mu^{+}$ yield no useful
constraints. The corresponding expressions are given in Appendix B for easy
reference. We show in Fig. 9 the allowed values for the LQ couplings in the
$\zeta^{1}_{L\,2\mu}\zeta^{1}_{L\,2\tau}$ vs
$\zeta^{1}_{L\,3\mu}\zeta^{1}_{L\,3\tau}$ plane for LQ masses of 1.5 TeV and
1.7 TeVs. We observe that the product
$\zeta^{1}_{L\,3\mu}\zeta^{1}_{L\,3\tau}$ can reach values of the order of
$O(1)$ for $\zeta^{1}_{L\,2\mu}\zeta^{1}_{L\,2\tau}$ below $10^{-4}$, which
means that the $\zeta^{1}_{L\,2\mu}$ and $\zeta^{1}_{L\,2\tau}$ couplings are
allowed to take on values of the order of $O(1)$ simultaneously, as can be
observed in the zoomed area. As in the case of the $\widetilde{R}_{2}$
representations, the allowed values for the $U_{1}$ coupling constants are
consistent with the model-independent bounds on the LQ couplings from direct
searches at the LHC compiled in Ref. [151].

Note also that in the models of [103, 106, 96, 110, 152, 153, 119] there are
extra particles such as new gauge and scalar bosons as well as additional
fermions, which can give new contributions to low energy observables. For
instance in the model of [110] there are two extra neutral gauge bosons
$g^{\prime}$ and $Z^{\prime}$, as well as new scalar bosons and fermions,
whereas the model presented in [103] also includes right-handed neutrinos. In
those works, a complete treatment of the constraints on the parameter space
from experimental data was performed with the purpose of addressing the LFUV
anomalies in $B$-meson decays. In general, one can assume that the LQ
couplings to the quarks of the second family are negligible, which allows for
large couplings to the quarks of the third generation.

Figure 9: Allowed values for the Yukawa couplings of the $U_{1}$
representation in the plane $\zeta^{1}_{L\,2\mu}\zeta^{1}_{L\,2\tau}$ vs
$\zeta^{1}_{L\,3\mu}\zeta^{1}_{L\,3\tau}$ for a LQ masses of $1.5$ TeVs (green
points) and $1.7$ TeVs (red points) from the experimental limits on the decays
$b\to s\mu^{-}\mu^{+}$, $\tau\to\mu\phi$. The $\zeta^{1}_{L\,2\alpha}$
($\alpha=\mu,\,\tau$) couplings were taken to be below $0.1$ to find the
largest allowed values of the $\zeta^{1}_{L\,3\alpha}$ couplings, for which we
impose the perturbativity bound $\zeta^{1}_{L\,3\alpha}\leq 1$. We consider
that either there are no right-handed neutrinos or the corresponding coupling
constants are too small.

Therefore, in the most promising scenario we can assume values for the
couplings of scalar and vector LQs to quarks and fermions of the order of
$O(1)$, which would lead to the best scenario for the magnitude of the LQ
contribution to the NEPs of the tau and muon neutrino, whereas those of the
electron neutrino would be strongly suppressed as the couplings to the first
generation fermions would only arise from quark mixing.

###  IV.3 Behavior of the LQ contribution to NEPs

The general behavior of the LQ contribution to NEPs can be inferred from Eqs.
(25), (28), and (30), which show the presence of chirality-flipping $(\rm CF)$
and chirality-conserving $(\rm CC)$ terms: the MDM and NCR have the following
structure

$\mu^{s}_{\nu_{\alpha}}=a^{s}_{\rm
CC}(|\zeta^{s}_{L3\alpha}|^{2}+|\zeta^{s}_{R3\alpha}|^{2})+a^{s}_{\rm CF}{\rm
Re}(\zeta^{s}_{L3\alpha}\zeta^{s*}_{R3\alpha})\equiv[a_{\rm CC},a_{\rm
CF}]_{\alpha}^{s},$ (49)

and

$\langle r^{2}\rangle^{s}_{\nu_{\alpha}}=b^{s}_{\rm
CC}(|\zeta^{s}_{L3\alpha}|^{2}+|\zeta^{s}_{R3\alpha}|^{2})+b^{s}_{\rm CF}{\rm
Re}(\zeta^{s}_{L3\alpha}\zeta^{s*}_{R3\alpha})\equiv[b_{\rm CC},b_{\rm
CF}]_{\alpha}^{s},$ (50)

whereas for the EDM we have

$d^{s}_{\nu_{\alpha}}=\widetilde{a}^{s}_{\rm CF}\,{\rm
Im}(\zeta^{s}_{L3\alpha}\zeta^{s*}_{R3\alpha})\equiv[\widetilde{a}_{\rm
CF}]_{\alpha}^{s}.$ (51)

While $a^{s}_{\rm CF}$ and $\widetilde{a}^{s}_{\rm CF}$ and $b^{s}_{\rm CC}$
are proportional to the virtual quark mass, $a^{s}_{\rm CC}$ and $b^{s}_{\rm
CF}$ are proportional to the neutrino mass. Thus in the case of both dipole
moments, the largest contributions could arise from the $U_{1}$ and
$\widetilde{V}_{2}$ representations via a charge $2/3$ LQ accompanied by the
top quark as long as there are right-handed neutrinos. Although the scalar
$\widetilde{R}_{2}$ and $S_{1}$ representations yield a charge $-1/3$ LQ that
can also have both left- and right-handed couplings to neutrinos, their
dominant contributions to neutrino dipole moments would arise from a loop with
the much lighter bottom quark. On the other hand, the NCR would be
nonvanishing even in absence of right-handed neutrinos as it receives its
dominant contribution from the $b^{s}_{\rm CC}$ term, whereas the $b^{s}_{\rm
CF}$ term yields a negligible contribution. Thus, the NCR would not be very
sensitive to both the mass of the virtual quark or the neutrino mass, which
means that it could be enhanced if there are contributions from several LQs
and quarks.

To illustrate these features of the NEPs, for simplicity we consider scalar
and vector LQs that can couple to the three SM neutrino flavors $\nu_{\alpha}$
($\alpha=e,\,\mu\,\tau$), but to the third generation quarks only. For the
effective neutrino masses we will use the bounds reported by the PDG
Collaboration [154]:

$\displaystyle m^{\text{eff}}_{\nu_{e}}$ $\displaystyle<0.8\times
10^{-9}\;\text{GeV},$ (52) $\displaystyle m^{\text{eff}}_{\nu_{\mu}}$
$\displaystyle<1.9\times 10^{-3}\;\text{GeV},$ (53) $\displaystyle
m^{\text{eff}}_{\nu_{\tau}}$ $\displaystyle<1.82\times 10^{-2}\;\text{GeV},$
(54)

where the bound on the electron neutrino mass was obtained at $90$ %CL [155],
whereas those on the tau and muon neutrino masses were obtained at $95$ %CL
[156, 157].

We now show in Table 6 the partial contributions of scalar and vector LQs of
charge $2/3$ and $-1/3$ to the NEP of the tau neutrino for three values of the
LQ mass in the bracket notation of Eqs. (49)-(51).

Table 6: Contributions of vector and scalar LQs of charge $2/3$ and $-1/3$ to the NEPs of the tau neutrino. We assume that the dominant contribution arises from the third generation quarks and present the results in the bracket notation of Eqs. (49)-(51). For the contribution of gauge LQs to the NCR we assume the couplings of Table 4. LQ spin | LQ charge | $m_{\rm LQ}$ | $\mu_{\nu_{\tau}}$ | $d_{\nu_{\tau}}$ | $\langle r^{2}\rangle_{\nu_{\tau}}$
---|---|---|---|---|---
| $[|e|]$ | [TeV] | $[\mu_{B}]$ | [ecm] |  $[$cm${}^{2}]$
0 | $2/3$ | $1.2$ | $[3.64\times 10^{-14},4.71\times 10^{-9}]$ | $[-9.13\times 10^{-20}]$ | $[-8.67\times 10^{-36},4.73\times 10^{-42}]$
| $-1/3$ | $1.2$ | $[-2.04\times 10^{-14},-2.05\times 10^{-10}]$ | $[3.98\times 10^{-21}]$ | $[1.66\times 10^{-35},-6.51\times 10^{-44}]$
| $2/3$ | $1.5$ | $[2.4\times 10^{-14},3.42\times 10^{-9}]$ | $[-6.63\times 10^{-20}]$ | $[-6.42\times 10^{-36},1.98\times 10^{-42}]$
| $-1/3$ | $1.5$ | $[-1.31\times 10^{-14},-1.37\times 10^{-10}]$ | $[2.66\times 10^{-21}]$ | $[1.11\times 10^{-35},-2.67\times 10^{-44}]$
| $2/3$ | $2.0$ | $[1.39\times 10^{-14},2.23\times 10^{-9}]$ | $[-4.32\times 10^{-20}]$ | $[-4.27\times 10^{-36},6.4\times 10^{-43}]$
| $-1/3$ | $2.0$ | $[-7.35\times 10^{-15},-8.13\times 10^{-11}]$ | $[1.58\times 10^{-21}]$ | $[6.61\times 10^{-36},-8.44\times 10^{-45}]$
1 | $2/3$ | $1.2$ | $[2.43\times 10^{-13},-6.21\times 10^{-9}]$ | $[1.2\times 10^{-19}]$ | $[3.68\times 10^{-35},0]$
| $-1/3$ | $1.2$ | $[-1.22\times 10^{-13},8.07\times 10^{-11}]$ | $[-1.56\times 10^{-21}]$ | $[-4.35\times 10^{-35},0]$
| $2/3$ | $1.5$ | $[1.56\times 10^{-13},-3.99\times 10^{-9}]$ | $[7.74\times 10^{-20}]$ | $[2.55\times 10^{-35},0]$
| $-1/3$ | $1.5$ | $[-7.84\times 10^{-14},5.17\times 10^{-11}]$ | $[-1.\times 10^{-21}]$ | $[-2.88\times 10^{-35},0]$
| $2/3$ | $2.0$ | $[8.78\times 10^{-14},-2.25\times 10^{-9}]$ | $[4.37\times 10^{-20}]$ | $[1.58\times 10^{-35},0]$
| $-1/3$ | $2.0$ | $[-4.41\times 10^{-14},2.91\times 10^{-11}]$ | $[-5.63\times 10^{-22}]$ | $[-1.69\times 10^{-35},0]$

We first analyze the contributions of a charge $2/3$ scalar LQ. We observe
that the respective contribution to the tau neutrino MDM can reach values as
large as $10^{-9}$ $\mu_{B}$ at most as long as there are couplings to right-
handed neutrinos, in which case there would be a nonzero EDM of the order of
$10^{-20}$ ecm at most. The latter vanishes in the scenario with no right-
handed neutrinos, whereas the respective contribution to the MDM plunges
dramatically up to the $10^{-14}$ $\mu_{B}$ level. On the other hand, the
contribution of a charge $2/3$ scalar LQ to the NCR could be of the order of
$10^{-35}$ cm2 regardless of the presence or absence of right-handed
neutrinos. As for the contributions of a charge $-1/3$ scalar LQ to the
neutrino dipole moments, they would be about one order of magnitude smaller
than those of the charge $2/3$ scalar LQ, whereas the size of the NCR would
remain the same as it is almost insensitive to the LQ charge and mass, with
the main distinction between the contributions of distinct LQs being a change
of sign.

We also note that, in general, the vector LQ contributions to the NEPs of the
tau neutrino have a similar order of magnitude than the contributions of
scalar LQs for similar values of the LQ masses, though they can be of opposite
sign. Bearing in mind that the constraints on the masses of vector LQs are
stronger than the ones of scalar LQs, it would seem that the contributions to
NEPs from vector LQs would be more suppressed than those of scalar LQs, but
there are only a slight change in the NEPs as the LQ increases from 1.2 to 2
TeVs.

Let us now consider more specifically the scenarios posed by the scalar and
vector LQ representations of Table 2 assuming the presence of right-handed
neutrinos, which would yield the largest contributions to the MDM and a
nonvanishing EDM. We show in Table 7 the estimates for the contributions of
such representations to the NEPs of SM neutrinos for a LQ mass of 1.5 TeVs.
Since the scalar LQ representations $\widetilde{R}_{2}$ and $S_{1}$, along
with the vector representations $U_{1}$ and $\widetilde{V}_{2}$, are the only
ones that yield a LQ that can have simultaneous couplings to both left- and
right-handed neutrinos, they could give the largest contributions to the
neutrino MDM, which in the best scenario could be of the order of
$10^{-10}$-$10^{-9}$. Such representations could also allow for an EDM of the
order of $10^{-19}$-$10^{-20}$ ecm provided that there are complex LQ
couplings. All other LQ representations cannot couple to right-handed
neutrinos, so they could give a contribution of the order of $10^{-14}$ to the
tau neutrino MDM, whereas the contributions to the muon and electron neutrino
MDMs would be two and seven orders of magnitude smaller, respectively. As far
as the NCR is concerned, it is insensitive to the neutrino mass and to whether
or not the LQs couple to right-handed neutrinos, so all the LQ representations
could yield a contribution of the order of $10^{-35}$ cm2 for the three
neutrino flavors, regardless of the LQ charge and the accompanying quark. Note
however that the above estimates must take into account the fact that the
coupling constants are not expected to be flavor blind, as shown in our
analysis of the bounds on the LQ coupling constants of the $\widetilde{R}_{2}$
and $U_{1}$ representations, thus an extra suppression is expected: according
to the assumptions made to obtain constraints on the LQ coupling constants,
the NEPs of the electron neutrino would be negligible.

Table 7: Estimate for the contributions to NEPs of the LQ representations of Table 2 according to their couplings to left- and right-handed neutrinos for a LQ with a mass of about 1.5 TeVs. It is assumed that a third-generation quark runs into the loop. The coupling constants are those associated with the specific LQ representation and neutrino flavor, and the same estimate is valid for the three neutrino flavors when only one value for the corresponding NEP is shown.  Representation |  $(\mu_{\nu_{\tau}}$, $\mu_{\nu_{\mu}}$, $\mu_{\nu_{e}}$) | $(d_{\nu_{\tau}},\,d_{\nu_{\mu}},\,d_{\nu_{e}})$ | $(\langle r^{2}\rangle_{\nu_{\tau}},\,\langle r^{2}\rangle_{\nu_{\mu}},\,\langle r^{2}\rangle_{\nu_{e}})$
---|---|---|---
| [$\mu_{B}$] | [ecm] | [cm${}^{2}]$
$S_{3}$, $R_{2}$ | $(10^{-14},\,10^{-16},\,10^{-21})\,\times|\zeta^{0}_{L\,3\alpha}|^{2}$ | $-$ | $10^{-35}\times|\zeta^{0}_{L\,3\alpha}|^{2}$
$\widetilde{R}_{2}$, $S_{1}$ | $10^{-10}\times{\rm Re}(\zeta^{0}_{L\,3\alpha}\zeta^{0\,*}_{R\,3\alpha})$ | $10^{-21}\times{\rm Im}(\zeta^{0}_{L\,3\alpha}\zeta^{0\,*}_{R\,3\alpha})$ | $10^{-35}\times\left(|\zeta^{0}_{L\,3\alpha}|^{2}+|\zeta^{0}_{R\,3\alpha}|^{2}\right)$
$\overline{S}_{1}$ | $(10^{-14},\,10^{-16},\,10^{-21})\,\times|\zeta^{0}_{R\,3\alpha}|^{2}$ | $-$ | $10^{-35}\times|\zeta^{0}_{R\,3\alpha}|^{2}$
$U_{3}$, $V_{2}$ | $(10^{-13},\,10^{-15},\,10^{-20})\,\times|\zeta^{1}_{L\,3\alpha}|^{2}$ | $-$ | $10^{-35}\times|\zeta^{1}_{L\,3\alpha}|^{2}$
$U_{1}$, $\widetilde{V}_{2}$ | $10^{-9}\times{\rm Re}(\zeta^{1}_{L\,3\alpha}\zeta^{1\,*}_{R\,3\alpha})$ | $10^{-19}\times{\rm Im}(\zeta^{1}_{L\,3\alpha}\zeta^{1\,*}_{R\,3\alpha})$ | $10^{-35}\times\left(|\zeta^{1}_{L\,3\alpha}|^{2}+|\zeta^{1}_{R\,3\alpha}|^{2}\right)$
$\overline{U}_{1}$ | $(10^{-13},\,10^{-15},\,10^{-20})\,\times|\zeta^{1}_{R\,3\alpha}|^{2}$ | $-$ | $10^{-35}\times|\zeta^{1}_{R\,3\alpha}|^{2}$

We now consider a charge $-1/3$ scalar LQ and a charge $2/3$ vector LQ, as the
ones arising from the $\widetilde{R}_{2}$ and $U_{1}$ representations, in the
scenario with right-handed neutrinos. All the features of the behavior of the
NEPs described above are best illustrated in Fig. 10, where we show the
contours of the LQ contributions to the NEPs of the tau neutrino as functions
of the LQ mass and the LQ coupling constants.

Figure 10: Contour plots for the contributions to the NEPs of the tau neutrino
of a scalar LQ of charge $-1/3$ (top row) and vector LQ of charge $2/3$
(bottom row) as functions of the LQ mass and the indicated LQ coupling
constant factors. We assume that LQs can couple to both left- and right handed
neutrinos and have neglected the $\rm CC$ ($\rm CF$) term for
$\mu_{\nu_{\alpha}}$ ($\langle r^{2}\rangle_{\nu_{\alpha}}$). It is also
assumed that the only contribution arises from virtual quarks of the third
generation.

Finally we consider the scenario with no right-handed neutrinos and again
assume that there are only contribution to the NEPs from the loops with the
third-generation quarks. We show the contours of the contributions to the MDM
of SM neutrinos from charge $2/3$ scalar and vector LQs, as the ones arising
from the $\widetilde{R}_{2}$ and $U_{1}$ representations, in Fig. 11 as
functions of the LQ mass and the squared left-handed LQ coupling constants. In
this scenario the EDM vanishes, whereas the NCR exhibits almost the same
behavior to that shown in Fig. 10 for the three neutrino flavors. Again, note
that according to the assumptions made to obtain constraints on the LQ
coupling constants, the MDM of the electron neutrino would be negligible.

Figure 11: Contour plots for the contributions to the MDM of the tau neutrino
(left), muon neutrino (center), and electron neutrino (right) from charge
$2/3$ scalar (top row) and vector LQs (bottom row) as functions of the LQ mass
and the indicated LQ coupling constant factors in the scenario with no right-
handed neutrinos. We assume that there is only contribution from virtual
quarks of the third generation.

###  IV.4 Comparison with experimental constraints

Let us finally compare our estimates for the LQ contributions to the NEPs with
the experimental constraints of Table 1. While the LQ contributions to the
neutrino MED are slightly below the experimental constraints, those to the NCR
are two orders of magnitude smaller. On the other hand, our prediction for the
EDM is similar to the existing indirect limits. It is also worth assessing a
possible enhancement to the NEPs in hypothetical LQ models with several LQs:
even in the case that the distinct LQ contributions to the NEPs add
constructively, am enhancement of several orders of magnitude cannot be
expected since all the partial contributions would be about the same order of
magnitude.

##  V Summary

In this work we have presented a calculation of the one-loop contribution of
scalar and vector LQ models to the static electromagnetic properties of
massive Dirac neutrinos, namely, the effective NCR and the magnetic and
electric dipole moments. We do not make any assumption on the mechanism of
neutrino mass generation and consider the effective Lagrangian approach of
Buchmuller, Ruckl, and Wyler for the $SU(3)_{c}\times SU(2)_{L}\times
U(1)_{Y}$ scalar and vector LQ representations that have renormalizable
couplings to fermion bilinears, including right-handed neutrinos. Analytical
results are presented in terms of both Feynman-parameter integrals and
Passarino-Veltman scalar functions in the case of nonzero neutrino mass, which
to our knowledge have never been reported in the literature and could be
useful to obtain the LQ contributions to the NEPs of hypothetical heavy
neutrinos. From such general results, simple expressions are obtained in the
limit of light neutrinos. Since the slope of the neutrino charge form factor
is in general a gauge-dependent quantity, special care was taken to obtain a
well-behaved NCR: while the contribution of scalar LQs does not depend on the
gauge-fixing parameter, that of a vector LQ was calculated via the BFM, which
in the Feynman-’t Hooft gauge yields an identical result to that obtained
through the more laborious PT, which is known to yield well-behaved off shell
Green’s functions. For the numerical evaluation we focus on the scalar and
vector LQ models that are renormalizable and do not need extra symmetries to
forbid proton decay at the tree-level, thereby still being phenomenologically
viable at the TeV scale, though we also discuss the potential contributions of
other LQ representations. It is found that the largest contributions to the
NEPs of the SM neutrinos may arise from the scalar LQ representation
$\widetilde{R}_{2}$ and the vector representation $U_{1}$, which can have
simultaneous couplings to both left- and right-handed neutrinos. We then
analyze the current constraints of the parameter space of such LQ models from
experimental data and found that LQ couplings to quarks and neutrinos of the
order of $O(1)$ are still allowed. In this scenario, the $\widetilde{R}_{2}$
and $U_{1}$ representations could yield contributions to the MDM of the tau
neutrino of the order of $10^{-9}$ $\mu_{B}$ at most for a LQ mass of 1.5 TeV,
whereas a neutrino EDM of the order of $10^{-20}-10^{-19}$ ecm would be
induced provided that the LQ couplings have a complex phase. On the other
hand, when there are no right-handed neutrinos, the EDM vanishes and the
corresponding LQ contributions to the MDM would be of the order of $10^{-14}$
for the tau neutrino, whereas the corresponding contributions to the MDM of
the muon and electron neutrinos would be about two and seven orders of
magnitude smaller. As for the LQ contribution to the NCR, it can reach values
up to $10^{-35}$ cm2 regardless the neutrino flavor and even in the absence of
right-handed neutrinos. Our estimates may have a severe suppression if the LQ
couplings to quarks and fermions are below the $O(1)$ level.

###### Acknowledgements.

We acknowledge support from Conacyt (Mexico). G. Tavares-Velasco also
acknowledges partial support from Sistema Nacional de Investigadores (Mexico)
and Vicerrectoría de Investigación y Estudios de Posgrado de la Benémerita
Universidad Autónoma de Puebla.

##  Appendix A Analytical results for the LQ contributions to NEPs

In this appendix we present the $f^{s}_{k}(x)$, $g^{s}_{k}(x)$,
$\hat{f}^{s}_{k}(x)$, and $\hat{g}^{s}_{k}(x)$ functions ($k=1,2$ and $s=0,1$)
of Eqs. (25), (28), and (30) in terms of both Feynman-parameter integrals and
Passarino-Veltman scalar functions for both scalar and vector LQs and nonzero
neutrino mass.

###  A.1 Feynman parameter results

In terms of Feynman-parameter integrals, the $f^{s}_{k}(x)$, $g^{s}_{k}(x)$,
$\hat{f}^{s}_{k}(x)$, and $\hat{g}^{s}_{k}(x)$ functions can be cast in the
form

$r^{s}_{k}(x)=\int_{0}^{1}\frac{R^{s}_{k}(x,y)}{((1-y)(x-yx_{\nu_{\alpha}})+y)^{a}}dy,$
(55)

where $a=1$ for $r^{s}_{k}(x)=f^{s}_{k}(x)$ and $g^{s}_{k}(x)$, whereas $a=2$
for $r^{s}_{k}(x)=\hat{f}^{s}_{k}(x)$ and $\hat{g}^{s}_{k}(x)$. The respective
$R^{s}_{k}(x,y)$ functions are given below for both scalar ($s=0$) and vector
($s=1$) LQs. Note that the dependence of the $r^{s}_{k}(x)$ and $R^{s}_{k}(x)$
functions on $x_{\nu_{\alpha}}$ is not written out explicitly. Here
$x_{\nu_{\alpha}}=m_{\nu_{\alpha}}^{2}/m_{\rm LQ}^{2}$, where $m_{\rm LQ}$
stands for the LQ mass.

####  A.1.1 Scalar LQ contribution

For the contribution of a scalar leptoquark to the neutrino MDM and EDM we
obtain the following expressions for the $F^{s}_{k}(x,y)$ and $G^{s}_{i}(x,y)$
functions:

$\displaystyle F^{0}_{k}(x,y)$ $\displaystyle=\frac{1}{2}(1-y)y\,h_{k}(y),$
(56) $\displaystyle G^{0}_{k}(x,y)$
$\displaystyle=\frac{1}{3}(1-y)\,h_{k}(y).$ (57)

where

$h_{k}(y)=\left\\{\begin{array}[]{lcr}1-y&&k=1,\\\ y&&k=2.\end{array}\right.$
(58)

As far as the contribution of a scalar LQ to the NCR, the
$\hat{F}^{0}_{k}(x,y)$ and $\hat{G}^{0}_{k}(x,y)$ functions can be written as

$\displaystyle\hat{F}^{0}_{1}(x,y)$
$\displaystyle=\frac{1}{2}(y-1)^{3}\left(3(x+y^{2}{x_{\nu_{\alpha}}})-2y\left(x+{x_{\nu_{\alpha}}}-1\right)\right),$
(59) $\displaystyle\hat{F}^{0}_{2}(x,y)$
$\displaystyle=\frac{1}{2}y^{3}\left((1-y)(x+y{x_{\nu_{\alpha}}})+y\right),$
(60)

and

$\displaystyle\hat{G}^{0}_{1}(x,y)$ $\displaystyle=2y(y-1)^{3},$ (61)
$\displaystyle\hat{G}^{0}_{2}(x,y)$ $\displaystyle=-2(y-1)y^{3}.$ (62)

The integration of the $r^{0}_{k}(x)$ functions is straightforward when
$x_{\nu_{\alpha}}$ is neglected and yield the results of Eqs. (31), (33),
(35), and (37).

####  A.1.2 Vector LQ contribution

As far as the contribution of a vector LQ to the neutrino MDM and EDM, the
$F^{1}_{k}(x,y)$ and $G^{1}_{k}(x,y)$ functions are given as follows

$\displaystyle F^{1}_{k}(x,y)$
$\displaystyle=\left(y\left(y\left(x+{x_{\nu_{\alpha}}}+2\right)-3x-{x_{\nu_{\alpha}}}+2\right)+2x\right)h_{k}(y),$
(63) $\displaystyle G^{1}_{k}(x,y)$
$\displaystyle=-2\left(y\left((2y-3){x_{\nu_{\alpha}}}+4-x\right)+x+{x_{\nu_{\alpha}}}\right)h_{k}(y),$
(64)

where the $h_{k}(y)$ function is given in Eq. (58).

On the other hand, the $\hat{F}^{1}_{k}(x,y)$ functions associated with the
contributions of the Feynman diagrams of Fig. 7 can be written as

$\displaystyle\hat{F}^{1}_{1}(x,y)$
$\displaystyle=(y-1)\left(y\left(y\left(-2y\left(x+{x_{\nu_{\alpha}}}-1\right)+x+3y^{2}{x_{\nu_{\alpha}}}+{x_{\nu_{\alpha}}}+2\right)-2\left(x+{x_{\nu_{\alpha}}}-1\right)\right)+3x\right),$
(65) $\displaystyle\hat{F}_{2}^{1}(x,y)$
$\displaystyle=y^{2}\left(y\left(7x+6({x_{\nu_{\alpha}}}-1)-y\left(x+(y+5){x_{\nu_{\alpha}}}-1\right)+{x_{\nu_{\alpha}}}\right)-6x\right),$
(66) $\displaystyle\hat{F}_{3}^{1}(x,y)$
$\displaystyle=\frac{1}{2}(y-1)^{3}\left(3y^{2}x_{\nu_{\alpha}}\left(x+x_{\nu_{\alpha}}\right)-2y\left(x\left(4x_{\nu_{\alpha}}-1\right)+x^{2}+\left(x_{\nu_{\alpha}}-1\right)x_{\nu_{\alpha}}\right)+3x\left(x+x_{\nu_{\alpha}}\right)\right),$
(67)

and

$\displaystyle\hat{F}_{4}^{1}(x,y)$
$\displaystyle=\frac{y^{3}}{2}\left(-y^{2}x_{\nu_{\alpha}}\left(x+x_{\nu_{\alpha}}\right)+y\left(4xx_{\nu_{\alpha}}-x^{2}+x+x_{\nu_{\alpha}}^{2}+x_{\nu_{\alpha}}\right)+x\left(x-3x_{\nu_{\alpha}}\right)\right).$
(68)

We note that for the gauge LQ couplings of Table 4 there are no contributions
to Eq. (30) of the $\hat{G}^{1}_{k}(x,y)$ functions.

###  A.2 Passarino-Veltman results

For completeness we also present results for the $r^{s}_{k}(x)$ functions in
terms of two-point Passarino-Veltman scalar functions. In this case we write

$r^{s}_{k}(x)=\frac{R^{s}_{k}(x)}{(1-x)^{a}x^{b}_{\nu_{\alpha}}\lambda^{c}(x,x_{\nu_{\alpha}})},$
(69)

where $\lambda(u,v)=1+u^{2}+v^{2}-2(u+v+uv)$, whereas the triads of integers
$(a,\,b,\,c)$ are $(4,\,2,\,0)$ for $f^{s}_{k}(x)$, $(3,\,1,\,0)$ for
$g^{s}_{k}(x)$, and $(1,\,2,\,2)$ for $\hat{f}^{s}_{k}(x)$ and
$\hat{g}^{s}_{k}(x)$. Note that the $R^{s}_{k}(x)$ functions are not the same
as those of Eq. (55), but we use the same notation for simplicity. We also
introduce the following dimensionless ultraviolet finite functions
$\Delta_{i}(x)$

$\displaystyle\Delta_{1}(x)$ $\displaystyle=B_{0}(0,x\,m_{\rm
LQ}^{2},x\,m_{\rm LQ}^{2})-B_{0}(0,\,m_{\rm LQ}^{2},m_{\rm LQ}^{2})=-\log(x),$
(70) $\displaystyle\Delta_{2}(x)$ $\displaystyle=B_{0}(x_{\nu_{\alpha}}m_{\rm
LQ}^{2},x\,m_{\rm LQ}^{2},m_{\rm LQ}^{2})-B_{0}(0,x\,m_{\rm LQ}^{2},m_{\rm
LQ}^{2}),$ (71)

where the arguments of the two-point Passarino-Veltman scalar functions
$B_{0}$ are given in the usual notation [120]. Note that the following two
three-point scalar functions appear throughout the calculation

$\displaystyle C_{1}(x)$ $\displaystyle=m_{\rm
LQ}^{2}C_{0}(m_{\nu_{\alpha}}^{2},m_{\nu_{\alpha}}^{2},0,m_{\rm
LQ}^{2},xm_{\rm LQ}^{2},m_{\rm LQ}^{2}),$ (72) $\displaystyle C_{2}(x)$
$\displaystyle=m_{\rm
LQ}^{2}C_{0}(m_{\nu_{\alpha}}^{2},m_{\nu_{\alpha}}^{2},0,xm_{\rm
LQ}^{2},m_{\rm LQ}^{2},xm_{\rm LQ}^{2}),$ (73)

but they can be written in terms of two-point scalar functions [158] as
follows

$\displaystyle C_{1}(x)$
$\displaystyle=\frac{1}{(x-1)\lambda(x,x_{\nu_{\alpha}})}\Big{(}(x-1)\left(x_{\nu_{\alpha}}+x-1\right)-x\left(x_{\nu_{\alpha}}-x+1\right)\Delta_{1}(x)-(x-1)\left(x_{\nu_{\alpha}}+x-1\right)\Delta_{2}(x)\Big{)},$
(74) $\displaystyle C_{2}(x)$
$\displaystyle=\frac{1}{(x-1)\lambda(x,x_{\nu_{\alpha}})}\Big{(}(x-1)\left(x_{\nu_{\alpha}}-x+1\right)-\left(x_{\nu_{\alpha}}+x-1\right)\Delta_{1}(x)-(x-1)\left(x_{\nu_{\alpha}}-x+1\right)\Delta_{2}(x)\Big{)}.$
(75)

It is also worth noticing that this calculation requires derivatives of the
two- and three-point scalar functions, which we denote as

$\displaystyle C^{\prime}_{1}(x)$ $\displaystyle=m_{\rm
LQ}^{4}C^{\prime}_{0}(m_{\nu_{\alpha}}^{2},m_{\nu_{\alpha}}^{2},q^{2},m_{\rm
LQ}^{2},xm_{\rm LQ}^{2},m_{\rm LQ}^{2})|_{q^{2}=0},$ (76) $\displaystyle
C^{\prime}_{2}(x)$ $\displaystyle=m_{\rm
LQ}^{4}C^{\prime}_{0}(m_{\nu_{\alpha}}^{2},m_{\nu_{\alpha}}^{2},q^{2},xm_{\rm
LQ}^{2},m_{\rm LQ}^{2},xm_{\rm LQ}^{2})|_{q^{2}=0},$ (77)

where the prime represents derivative with respect to $q^{2}$. Again the
$C^{\prime}_{i}(x)$ functions can be decomposed in terms of two-point scalar
functions as follows [158]

$\displaystyle C^{\prime}_{1}(x)$
$\displaystyle=\frac{1}{6(x-1)x_{\nu_{\alpha}}\lambda^{2}(x,x_{\nu_{\alpha}})}\Big{(}(x-1)x_{\nu_{\alpha}}\left(x^{2}\left(3x_{\nu_{\alpha}}-1\right)-3x\left(x_{\nu_{\alpha}}^{2}-1\right)+\left(x_{\nu_{\alpha}}-1\right){}^{3}-x^{3}\right)$
$\displaystyle+xx_{\nu_{\alpha}}\left(4x\left(x_{\nu_{\alpha}}+1\right)-\left(x_{\nu_{\alpha}}-1\right){}^{2}-3x^{2}\right)\Delta_{1}(x)-(x-1)\left(x_{\nu_{\alpha}}+x-1\right)\left(\lambda(x,x_{\nu_{\alpha}})-2xx_{\nu_{\alpha}}\right)\Delta_{2}(x)\Big{)},$
(78) $\displaystyle C^{\prime}_{2}(x)$
$\displaystyle=\frac{1}{6(x-1)x_{\nu_{\alpha}}\lambda^{2}(x,x_{\nu_{\alpha}})}\Big{(}(x-1)x_{\nu_{\alpha}}\left(3x^{2}\left(x_{\nu_{\alpha}}+1\right)-3xx_{\nu_{\alpha}}^{2}-\left(x_{\nu_{\alpha}}-1\right){}^{3}-x^{3}-x\right)$
$\displaystyle+xx_{\nu_{\alpha}}\left(2x\left(x_{\nu_{\alpha}}+2\right)-x_{\nu_{\alpha}}^{2}+4x_{\nu_{\alpha}}+x^{2}-3\right)\Delta_{1}(x)-(x-1)x\left(x_{\nu_{\alpha}}-x+1\right)\left(\lambda(x,x_{\nu_{\alpha}})-2x_{\nu_{\alpha}}\right)\Delta_{2}(x)\Big{)}.$
(79)

We also need the following two-point scalar function derivative

$B^{\prime}_{0}(q^{2},m_{a}^{2},m_{a}^{2})=\frac{1}{6m_{a}^{2}}.$ (80)

Below we present the $R^{s}_{k}(x)$ functions below for the contributions of
scalar and vector LQs.

###  A.3 Scalar LQ contribution

For the contribution of scalar LQs to the static neutrino MDM and EDM we have
the following $F^{0}_{k}(x)$ and $G^{0}_{k}(x)$ functions

$\displaystyle F^{0}_{1}(x)$
$\displaystyle=-\frac{1}{2}\Big{(}-\left(x-1\right)x_{\nu_{\alpha}}\left(2\left(3x+1\right)x_{\nu_{\alpha}}+\left(x+1\right)\left(x-1\right){}^{2}\right)-2xx_{\nu_{\alpha}}\left(\left(x+3\right)x_{\nu_{\alpha}}+\left(x-1\right){}^{2}\right)\Delta_{1}(x)$
$\displaystyle+2\left(x-1\right)\left(\left(x-1\right){}^{2}x_{\nu_{\alpha}}+\left(3x+1\right)x_{\nu_{\alpha}}^{2}+\left(x-1\right){}^{4}\right)\Delta_{2}(x)\Big{)},$
(81) $\displaystyle F^{0}_{2}(x)$
$\displaystyle=-\frac{1}{2}\Big{(}\left(x-1\right)x_{\nu_{\alpha}}\left(2x\left(x+3\right)x_{\nu_{\alpha}}+\left(x+1\right)\left(x-1\right){}^{2}\right)+2xx_{\nu_{\alpha}}\left(x\left(x+3x_{\nu_{\alpha}}-2\right)+x_{\nu_{\alpha}}+1\right)\Delta_{1}(x)$
$\displaystyle-\left(2x\left(x-1\right){}^{3}x_{\nu_{\alpha}}+2x\left(x+3\right)\left(x-1\right)x_{\nu_{\alpha}}^{2}+2\left(x-1\right){}^{5}\right)\Delta_{2}(x)\Big{)},$
(82)

and

$\displaystyle G^{0}_{1}(x)$
$\displaystyle=-2\left(-2\left(x-1\right)x_{\nu_{\alpha}}-\left(x+1\right)x_{\nu_{\alpha}}\Delta_{1}(x)+\left(x-1\right)\left(\left(x-1\right){}^{2}+2x_{\nu_{\alpha}}\right)\Delta_{2}(x)\right),$
(83) $\displaystyle G^{0}_{2}(x)$
$\displaystyle=-2\left(\left(x^{2}-1\right)x_{\nu_{\alpha}}+2xx_{\nu_{\alpha}}\Delta_{1}(x)-\left(x-1\right)\left(x\left(x+x_{\nu_{\alpha}}-2\right)+x_{\nu_{\alpha}}+1\right)\Delta_{2}(x)\right).$
(84)

For small $x_{\nu_{\alpha}}$ we obtain the following expansion to order
$O\left(x_{\nu_{\alpha}}^{2}\right)$

$\displaystyle\Delta_{2}(x)$
$\displaystyle=\frac{1}{6\left(x-1\right){}^{5}}\Big{(}3\left(x-1\right){}^{2}\left(x^{2}-2x\log\left(x\right)-1\right)x_{\nu_{\alpha}}+\left(x\left(x\left(x+9\right)-6(1+x)\log(x)-9\right)-1\right)x_{\nu_{\alpha}}^{2}\Big{)}$
$\displaystyle+O\left(x_{\nu_{\alpha}}^{4}\right),$ (85)

which after their substitution in the $r^{0}_{k}(x)$ functions yield the
results presented in Eqs. (31) and (33) at leading order in the neutrino mass.

As far as the scalar LQ contributions to the NCR are concerned, we obtain the
following $\hat{F}^{0}_{k}(x)$ and $\hat{G}^{0}_{k}(x)$ functions

$\displaystyle\hat{F}^{0}_{1}(x)$
$\displaystyle=\frac{1}{4}\Big{(}(x-1)\left(x-1-x_{\nu_{\alpha}}\right)x_{\nu_{\alpha}}\left(x_{\nu_{\alpha}}^{3}-7(x+1)x_{\nu_{\alpha}}^{2}+(x(11x+26)+11)x_{\nu_{\alpha}}-5(x-1)^{2}(x+1)\right)$
$\displaystyle-2x_{\nu_{\alpha}}\Big{(}5x^{4}-15x^{3}\left(x_{\nu_{\alpha}}+1\right)+x^{2}\left(x_{\nu_{\alpha}}\left(17x_{\nu_{\alpha}}+4\right)+15\right)+x\left(x_{\nu_{\alpha}}\left(13-9x_{\nu_{\alpha}}^{2}+x_{\nu_{\alpha}}\right)-5\right)$
$\displaystyle+2\left(x_{\nu_{\alpha}}-1\right){}^{3}x_{\nu_{\alpha}}\Big{)}\Delta_{1}(x)+2(x-1)\Big{(}\left(1-26x^{2}+x\right)x_{\nu_{\alpha}}^{3}-(20x+13)(x-1)^{3}x_{\nu_{\alpha}}$
$\displaystyle+(x(32x+5)+11)(x-1)x_{\nu_{\alpha}}^{2}-2x_{\nu_{\alpha}}^{5}+(11x+4)x_{\nu_{\alpha}}^{4}+5(x-1)^{5}\Big{)}\Delta_{2}(x)\Big{)},$
(86) $\displaystyle\hat{F}^{0}_{2}(x)$
$\displaystyle=\frac{1}{4}\Big{(}(x-1)\left(x-1-x_{\nu_{\alpha}}\right)x_{\nu_{\alpha}}\left(x_{\nu_{\alpha}}^{3}-7(x+1)x_{\nu_{\alpha}}^{2}+(x(11x+26)+11)x_{\nu_{\alpha}}-5(x-1)^{2}(x+1)\right)-2x_{\nu_{\alpha}}$
$\displaystyle\times\Big{(}5x^{4}-15x^{3}\left(x_{\nu_{\alpha}}+1\right)+x^{2}\left(x_{\nu_{\alpha}}\left(17x_{\nu_{\alpha}}+4\right)+15\right)+x\left(x_{\nu_{\alpha}}\left(13-9x_{\nu_{\alpha}}^{2}+x_{\nu_{\alpha}}\right)-5\right)+2\left(x_{\nu_{\alpha}}-1\right){}^{3}x_{\nu_{\alpha}}\Big{)}\Delta_{1}(x)$
$\displaystyle+2(x-1)\Big{(}\left(1-26x^{2}+x\right)x_{\nu_{\alpha}}^{3}-(20x+13)(x-1)^{3}x_{\nu_{\alpha}}+(x(32x+5)+11)(x-1)x_{\nu_{\alpha}}^{2}-2x_{\nu_{\alpha}}^{5}$
$\displaystyle+(11x+4)x_{\nu_{\alpha}}^{4}+5(x-1)^{5}\Big{)}\Delta_{2}(x)\Big{)},$
(87)

and

$\displaystyle\hat{G}^{0}_{1}(x)$
$\displaystyle=2(x-1)\left(x_{\nu_{\alpha}}-x-1\right)x_{\nu_{\alpha}}\left(x_{\nu_{\alpha}}^{2}-2(x+2)x_{\nu_{\alpha}}+(x-1)^{2}\right)$
$\displaystyle+2x_{\nu_{\alpha}}\left(x_{\nu_{\alpha}}^{3}-2(2x+1)x_{\nu_{\alpha}}^{2}+x(5x+6)x_{\nu_{\alpha}}+x_{\nu_{\alpha}}-2(x-1)^{2}x\right)\Delta_{1}(x)$
$\displaystyle+2(x-1)\left(3\left(3x^{2}+1\right)x_{\nu_{\alpha}}^{2}-(7x+5)(x-1)^{2}x_{\nu_{\alpha}}+x_{\nu_{\alpha}}^{4}-(5x+1)x_{\nu_{\alpha}}^{3}+2(x-1)^{4}\right)\Delta_{2}(x),$
(88) $\displaystyle\hat{G}^{0}_{2}(x)$
$\displaystyle=-2(x-1)\left(-x_{\nu_{\alpha}}+x+1\right)x_{\nu_{\alpha}}\left(x_{\nu_{\alpha}}^{2}-2(x+2)x_{\nu_{\alpha}}+(x-1)^{2}\right)$
$\displaystyle+2x_{\nu_{\alpha}}\left(x_{\nu_{\alpha}}^{3}-2(2x+1)x_{\nu_{\alpha}}^{2}+x(5x+6)x_{\nu_{\alpha}}+x_{\nu_{\alpha}}-2(x-1)^{2}x\right)\Delta_{1}(x)$
$\displaystyle+2(x-1)\left(3\left(3x^{2}+1\right)x_{\nu_{\alpha}}^{2}-(7x+5)(x-1)^{2}x_{\nu_{\alpha}}+x_{\nu_{\alpha}}^{4}-(5x+1)x_{\nu_{\alpha}}^{3}+2(x-1)^{4}\right)\Delta_{2}(x).$
(89)

###  A.4 Vector LQ contribution

As for the vector contribution to the neutrino MDM and EMD, the $F^{1}_{k}(x)$
and $G^{1}_{k}(x)$ functions are given by

$\displaystyle F^{1}_{1}(x)$
$\displaystyle=\frac{1}{4}\Big{(}(x-1)x_{\nu_{\alpha}}\left((x(x(x+5)+21)-3)x_{\nu_{\alpha}}+(x+1)(x+2)(x-1)^{2}\right)$
$\displaystyle+2xx_{\nu_{\alpha}}\left(x\left(11x_{\nu_{\alpha}}-3\right)+x_{\nu_{\alpha}}+x^{3}+2\right)\Delta_{1}(x)$
$\displaystyle+\left(2((x-5)x+1)(x-1)^{3}x_{\nu_{\alpha}}-2(x(4x+9)-1)(x-1)x_{\nu_{\alpha}}^{2}-2(x+2)(x-1)^{5}\right)\Delta_{2}(x)\Big{)},$
(90) $\displaystyle F^{1}_{2}(x)$
$\displaystyle=\frac{1}{4}\Big{(}-(x-1)x_{\nu_{\alpha}}\left(x\left(x\left((25-x)x_{\nu_{\alpha}}+x(1+x)-3\right)-x_{\nu_{\alpha}}-1\right)+x_{\nu_{\alpha}}+2\right)$
$\displaystyle-2xx_{\nu_{\alpha}}\left(x\left((4x+9)x_{\nu_{\alpha}}+x^{2}-3\right)-x_{\nu_{\alpha}}+2\right)\Delta_{1}(x)$
$\displaystyle+\left(6(2x-1)(x-1)^{3}x_{\nu_{\alpha}}+2x(11x+1)(x-1)x_{\nu_{\alpha}}^{2}+2(x+2)(x-1)^{5}\right)\Delta_{2}(x)\Big{)},$
(91)

and

$\displaystyle G^{1}_{1}(x)$
$\displaystyle=\left(3-x(1+x+x^{2})\right)x_{\nu_{\alpha}}-(x+5)xx_{\nu_{\alpha}}\Delta_{1}(x)$
$\displaystyle+(x-1)\left(x\left((6-x)x_{\nu_{\alpha}}+x^{2}-3\right)+x_{\nu_{\alpha}}+2\right)\Delta_{2}(x),$
(92) $\displaystyle G^{1}_{2}(x)$
$\displaystyle=(x-1)(7x-1)x_{\nu_{\alpha}}+2(2x+1)xx_{\nu_{\alpha}}\Delta_{1}(x)-(x-1)\left(x\left(7x_{\nu_{\alpha}}-3\right)-x_{\nu_{\alpha}}+x^{3}+2\right)\Delta_{2}(x),$
(93)

where the $\Delta_{i}(x)$ functions were defined above, but with $m_{\rm
LQ}=m_{V}$ instead.

The contribution of gauge LQs to the NCR is obtained via the
$\hat{F}^{1}_{k}(x)$ functions only given our assumptions for the coupling
constants and can be written as

$\displaystyle\hat{F}^{1}_{1}(x)$
$\displaystyle=\frac{1}{2}\Big{(}(x-1)x_{\nu_{\alpha}}\Big{(}2x^{3}\left(2x_{\nu_{\alpha}}+5\right)+2x^{2}x_{\nu_{\alpha}}\left(9x_{\nu_{\alpha}}+11\right)-2x\left(x_{\nu_{\alpha}}\left(14x_{\nu_{\alpha}}^{2}+x_{\nu_{\alpha}}+4\right)+5\right)$
$\displaystyle+\left(x_{\nu_{\alpha}}-1\right){}^{2}\left(x_{\nu_{\alpha}}\left(11x_{\nu_{\alpha}}-8\right)+5\right)-5x^{4}\Big{)}-2x_{\nu_{\alpha}}\left(x_{\nu_{\alpha}}+x-1\right)\Big{(}2x_{\nu_{\alpha}}^{3}+(x-4)x_{\nu_{\alpha}}^{2}-2x(4x+9)x_{\nu_{\alpha}}$
$\displaystyle+2x_{\nu_{\alpha}}+5(x-1)^{2}x\Big{)}\Delta_{1}(x)+2(x-1)\Big{(}(14x+19)(1-x)^{3}x_{\nu_{\alpha}}+(x(14x+5)+29)(x-1)x_{\nu_{\alpha}}^{2}$
$\displaystyle-2x_{\nu_{\alpha}}^{5}+(5x-2)x_{\nu_{\alpha}}^{4}+((13-8x)x+19)x_{\nu_{\alpha}}^{3}+5(x-1)^{5}\Big{)}\Delta_{2}(x)\Big{)},$
(94) $\displaystyle\hat{F}^{1}_{2}(x)$
$\displaystyle=\frac{1}{2}\Big{(}(x-1)x_{\nu_{\alpha}}\left(10x^{3}\left(x_{\nu_{\alpha}}-1\right)-48x^{2}x_{\nu_{\alpha}}\left(x_{\nu_{\alpha}}+1\right)+2x\left(x_{\nu_{\alpha}}-1\right){}^{2}\left(19x_{\nu_{\alpha}}+5\right)-5\left(x_{\nu_{\alpha}}-1\right){}^{4}+5x^{4}\right)$
$\displaystyle+2xx_{\nu_{\alpha}}\left(7x_{\nu_{\alpha}}^{2}-12(x+1)x_{\nu_{\alpha}}+5(x-1)^{2}\right)\left(x\left(x_{\nu_{\alpha}}+1\right)-\left(x_{\nu_{\alpha}}-1\right){}^{2}\right)\Delta_{1}(x)$
$\displaystyle-2(x-1)\Big{(}\left(-5x^{3}+63x-58\right)x_{\nu_{\alpha}}^{2}-3(2x+9)(x-1)^{3}x_{\nu_{\alpha}}+7x_{\nu_{\alpha}}^{5}-3(4x+11)x_{\nu_{\alpha}}^{4}+(x(11x-1)+62)x_{\nu_{\alpha}}^{3}$
$\displaystyle+5(x-1)^{5}\Big{)}\Delta_{2}(x)\Big{)},$ (95)
$\displaystyle\hat{F}^{1}_{3}(x)$
$\displaystyle=\frac{1}{4}\Big{(}2x_{\nu_{\alpha}}\left(2x_{\nu_{\alpha}}^{3}+(x-4)x_{\nu_{\alpha}}^{2}-2x(4x+9)x_{\nu_{\alpha}}+2x_{\nu_{\alpha}}+5(x-1)^{2}x\right)\left(x\left(2x_{\nu_{\alpha}}+1\right)-\left(x_{\nu_{\alpha}}-1\right)x_{\nu_{\alpha}}-x^{2}\right)\Delta_{1}(x)$
$\displaystyle+2(x-1)\big{(}-\left(x\left(30x^{2}+26x+5\right)+11\right)x_{\nu_{\alpha}}^{3}-2x_{\nu_{\alpha}}^{6}+(5x+4)x_{\nu_{\alpha}}^{5}+(x(5x+9)+1)x_{\nu_{\alpha}}^{4}$
$\displaystyle+(x-1)(x(8x(5x+3)-3)-13)x_{\nu_{\alpha}}^{2}-(x-1)^{3}(x(23x+15)-5)x_{\nu_{\alpha}}+5(x-1)^{5}x\big{)}\Delta_{2}(x)$
$\displaystyle-(x-1)x_{\nu_{\alpha}}\big{(}x_{\nu_{\alpha}}^{5}+(x-6)x_{\nu_{\alpha}}^{4}-2(x(7x+10)-2)x_{\nu_{\alpha}}^{3}+2(x(x(13x+24)+32)+3)x_{\nu_{\alpha}}^{2}$
$\displaystyle-(x-1)(x(x(19x+31)+3)-5)x_{\nu_{\alpha}}+5(x-1)^{3}x(x+1)\big{)}\Big{)},$
(96)

and

$\displaystyle\hat{F}^{1}_{4}(x)$
$\displaystyle=\frac{1}{4}\Big{(}(x-1)x_{\nu_{\alpha}}\big{(}-x^{4}\left(17x_{\nu_{\alpha}}+10\right)+2x^{3}x_{\nu_{\alpha}}\left(13x_{\nu_{\alpha}}-1\right)+2x^{2}\left(-13x_{\nu_{\alpha}}^{3}+7x_{\nu_{\alpha}}^{2}+x_{\nu_{\alpha}}+5\right)$
$\displaystyle+x\left(x_{\nu_{\alpha}}-1\right){}^{2}\left(x_{\nu_{\alpha}}+1\right)\left(17x_{\nu_{\alpha}}-5\right)-5\left(x_{\nu_{\alpha}}-1\right){}^{4}x_{\nu_{\alpha}}+5x^{5}\big{)}$
$\displaystyle-2xx_{\nu_{\alpha}}\big{(}x_{\nu_{\alpha}}^{5}-4(x+2)x_{\nu_{\alpha}}^{4}+3(x(2x+5)+6)x_{\nu_{\alpha}}^{3}-(x(x(4x+11)-7)+16)x_{\nu_{\alpha}}^{2}$
$\displaystyle+(x(x(x+1)(x+8)-23)+5)x_{\nu_{\alpha}}-5(x-1)^{3}x\big{)}\Delta_{1}(x)$
$\displaystyle
2(x-1)\big{(}\left(x\left(6x^{2}+x-17\right)+34\right)x_{\nu_{\alpha}}^{3}-3\left(x\left(x\left(5x^{2}+x+6\right)-19\right)+7\right)x_{\nu_{\alpha}}^{2}$
$\displaystyle-
x_{\nu_{\alpha}}^{6}+3(x+3)x_{\nu_{\alpha}}^{5}-(x(3x+10)+26)x_{\nu_{\alpha}}^{4}+(x-1)^{3}(x(15x+23)-5)x_{\nu_{\alpha}}-5(x-1)^{5}x\big{)}\Delta_{2}(x)\Big{)}.$
(97)

##  Appendix B LQ contributions to low-energy observables

In this Appendix we present the most relevant analytical expressions for the
LQ contributions to the low-energy observables useful to constrain the LQ
Yukawa couplings $\zeta^{s}_{L\,i\alpha}$ and $\zeta^{s}_{R\,i\alpha}$ of Eqs.
(11) and (20) for the LQs arisen from the scalar $\widetilde{R}_{2}$ and
vector $U_{1}$ representations. We remind the reader that the index $i$
($i=1,2$) corresponds to the quark family, whereas the index $\alpha$
($\alpha=\mu,\,\tau$) corresponds to the lepton family. We have also assumed
that the couplings to the fermions of the first generation vanish. Such
expressions can be found in several sources but the results shown below are
those compiled in [52, 108] for the scalar LQ contribution and [101] for the
vector LQ contribution.

###  B.1 $\widetilde{R}_{2}$ representation

The $\widetilde{R}_{2}$ contribution to the Wilson coefficients relevant for
the $b\to s\ell_{1}\ell_{2}$ decay in terms of the coupling constants of Eq.
(11) is

$\left(C_{9}^{\ell_{1}\ell_{2}}\right)^{\prime}=-\left(C_{10}^{\ell_{1}\ell_{2}}\right)^{\prime}=-\frac{\pi\upsilon^{2}}{2V_{tb}V^{*}_{ts}\alpha_{\rm
em}}\frac{\zeta^{0}_{L\,3\ell_{1}}\zeta^{0}_{L\,3\ell_{2}}}{m_{\rm LQ}^{2}}.$
(98)

For the mass difference of the $B_{s}-\overline{B}_{s}$ system we have

$\displaystyle\frac{\Delta m_{B_{s}}^{\rm Theor.}}{\Delta m_{B_{s}}^{\rm SM}}$
$\displaystyle=1+\frac{\eta_{1}}{16G_{F}^{2}m_{W}^{2}|V_{tb}V^{*}_{ts}|^{2}\eta_{B}S_{0}(x_{t})m_{\rm
LQ}^{2}}\Bigg{(}\left(\zeta^{0}_{L\,3\alpha}\zeta^{0*}_{L\,2\alpha}\right)^{2}+\frac{1}{2}\left(\zeta^{0}_{R\,3\alpha}\zeta^{0*}_{R\,2\alpha}\right)^{2}$
$\displaystyle-\eta_{41}\frac{3}{2}\left(\zeta^{0}_{L\,3\alpha}\zeta^{0*}_{L\,2\alpha}\right)\left(\zeta^{0}_{R\,3\alpha^{\prime}}\zeta^{0*}_{R\,2\alpha^{\prime}}\right)\left(\frac{m_{B_{s}}}{m_{b}+m_{s}}\right)^{2}\frac{B_{4}(m_{b})}{B_{1}(m_{b})}\Bigg{)},$
(99)

where again repeated indices $\alpha$ and $\alpha^{\prime}$ sum over $\mu$ and
$\tau$, $x_{t}=m_{t}^{2}/m_{W}^{2}$, and the $S_{0}(x)$ is Inami-Lim function

$S_{0}(x)=\frac{1}{4}\left(1+\frac{9}{1-x}-\frac{6}{(1-x)^{2}}-\frac{6x^{2}}{(1-x)^{3}}\log(x)\right).$
(100)

Also, we use $\eta_{1}=0.81(1)$, $\eta_{41}=4.4(1)$ [52], whereas
$B_{1}(m_{b})=3.4$ and $B_{4}(m_{b})=4.5$ for the bag parameters [159].

The corresponding contribution to the $B\to K\nu\nu$ decay can be written as

$R_{\nu\nu}=\frac{{\rm BR}(B\to K\nu\nu)^{\rm Theor.}}{{\rm BR}(B\to
K\nu\nu)^{\rm SM}}=1-\frac{1}{6C_{L}^{\rm SM}}{\rm
Re}\left(\frac{\zeta^{0}_{L\,3\alpha}\zeta^{0*}_{L\,3\alpha}}{Nm_{\rm
LQ}^{2}}\right)+\frac{1}{48{C_{L}^{\rm
SM}}^{2}}\frac{\zeta^{0}_{L\,3\alpha}\zeta^{0*}_{L\,3\alpha}\zeta^{0}_{L\,3\alpha^{\prime}}\zeta^{0*}_{L\,3\alpha^{\prime}}}{|N|^{2}m_{\rm
LQ}^{4}},$ (101)

where again repeated indices $\alpha$ and $\alpha^{\prime}$ sum over $\mu$ and
$\tau$, whereas $N=\dfrac{G_{F}V_{tb}V_{ts}^{*}\alpha_{\rm em}}{\sqrt{2}\pi}$
and $C_{L}^{\rm SM}=-6.38(10)$ [160].

The scalar LQ contributions to the LFV decays $\tau\to\mu\gamma$ and
$\tau\to\phi\gamma$ can be written as [108]

${\rm BR}(\tau\to\mu\gamma)=\frac{\alpha_{\rm
em}(m_{\tau}^{2}-m_{\mu}^{2})^{3}}{4m_{\tau}^{3}\Gamma_{\tau}}\left|\zeta^{0}_{L\,3\mu}\zeta^{0*}_{L\,3\tau}\frac{N_{c}m_{b}^{2}m_{\tau}}{96\pi^{2}m_{\rm
LQ}^{4}}\left(\frac{5}{2}+\log\left(\frac{m_{b}^{2}}{m_{\rm
LQ}^{2}}\right)\right)\right|^{2},$ (102)

and

${\rm BR}(\tau\to\mu\phi)=\frac{f_{\phi}^{2}m_{\phi}^{4}}{256\pi
m_{\tau}^{3}\Gamma_{\tau}}\left|\frac{\zeta^{0}_{L\,2\tau}\zeta^{0*}_{L\,2\mu}}{m_{\rm
LQ}^{2}}\right|^{2}\left(-1+\frac{m_{\mu}^{2}+m_{\tau}^{2}}{2m_{\phi}^{2}}+\frac{\left(m_{\mu}^{2}-m_{\tau}^{2}\right)^{2}}{2m_{\phi}^{4}}\right)\lambda^{1/2}(m_{\phi}^{2},m_{\tau}^{2},m_{\mu}^{2}),$
(103)

where the triangle function is $\lambda(x,y,z)=x^{2}+y^{2}+z^{2}-2(xy+xz+yz)$
and the $\phi$ decay constant $f_{\phi}=241(8)$ MeV.

###  B.2 $U_{1}$ representation

We consider the model of Ref. [110] where the LQ coupling constants of Eq.
(20) are given by $\zeta^{1}_{L\,\i\alpha}=g_{4}/\sqrt{2}\beta_{L\,i\alpha}$
and $\zeta^{1}_{R\,\i\alpha}=0$, with $g_{4}=2m_{\rm
LQ}\sqrt{C_{U}}/\upsilon$. Below we list the $U_{1}$ contribution to some
observables that can be useful to constrain the LQ couplings and fit the model
parameters.

The $U_{1}$ contribution to the $B\to D\tau\nu$ and $B\to D^{*}\tau\nu$ decays
reads as

$R_{D}=\frac{{\rm BR}\left(B\to D\tau\nu\right)}{{\rm BR}\left(B\to
D\tau\nu\right)_{\rm SM}}\simeq R_{D^{*}}\simeq R_{D}^{\rm
SM}\left(1+2C_{U}{\rm
Re}\left(1+\frac{V_{cs}}{V_{cb}}\beta_{L\,2\tau}\right)\right).$ (104)

For the decay $b\to s\ell^{-}\ell^{+}$ we have

$C_{9}^{\ell\ell}=-C_{10}^{\ell_{\ell}}=-\frac{2\pi}{V_{tb}V^{*}_{ts}\alpha_{\rm
em}}C_{U}\beta_{L\,2\ell}\beta^{*}_{L\,3\ell}.$ (105)

The contribution to the following $B$ meson decays can also be useful to
constrain the $U_{1}$ LQ coupling constants

${\rm BR}\left(B\to K\tau^{-}\tau^{+}\right)\simeq 1.5\times 10^{-7}+1.4\times
10^{-3}C_{U}{\rm
Re}\left(\beta_{L\,2\tau}\right)+3.5C_{U}^{2}\left|\beta_{L\,2\tau}\right|^{2},$
(106) ${\rm BR}\left(B^{+}\to
K^{+}\tau^{+}\mu^{-}\right)=8.3C_{U}^{2}\left|\beta_{L\,2\mu}\right|^{2},$
(107) ${\rm BR}\left(B^{+}\to
K^{+}\tau^{-}\mu^{+}\right)=8.3C_{U}^{2}\left|\beta_{L\,3\mu}\beta_{L\,2\tau}^{*}\right|^{2},$
(108)

and The $U_{1}$ contribution to the LFV tau decay $\tau\to\mu\phi$ is given by

${\rm
BR}(\tau\to\mu\phi)=\frac{f_{\phi}^{2}G_{F}^{2}}{16\pi\Gamma_{\tau}}m_{\tau}^{3}\left(1-\frac{m_{\phi}^{2}}{m_{\tau}^{2}}\right)^{2}\left(1+2\frac{m_{\phi}^{2}}{m_{\tau}^{2}}\right)C_{U}^{2}\left|\beta_{L\,2\tau}\beta_{L\,2\mu}^{*}\right|^{2}.$
(109)

The masses and life times of the $B$ mesons as well as the latest experimental
constraints and measurements on the $B$-meson decay modes were taken from
[154] and [161].

## References

  * Bernstein and Lee [1963] J. Bernstein and T. D. Lee, Electromagnetic Form-factor of the Neutrinos, Phys. Rev. Lett. 11, 512 (1963).
  * Fujikawa and Shrock [1980] K. Fujikawa and R. Shrock, The Magnetic Moment of a Massive Neutrino and Neutrino Spin Rotation, Phys. Rev. Lett. 45, 963 (1980).
  * Shrock [1982] R. E. Shrock, Electromagnetic Properties and Decays of Dirac and Majorana Neutrinos in a General Class of Gauge Theories, Nucl. Phys. B 206, 359 (1982).
  * Kayser [1982] B. Kayser, Majorana Neutrinos and their Electromagnetic Properties, Phys. Rev. D 26, 1662 (1982).
  * Giunti and Studenikin [2009] C. Giunti and A. Studenikin, Neutrino electromagnetic properties, Phys. Atom. Nucl. 72, 2089 (2009), arXiv:0812.3646 [hep-ph] .
  * Giunti and Studenikin [2015] C. Giunti and A. Studenikin, Neutrino electromagnetic interactions: a window to new physics, Rev. Mod. Phys. 87, 531 (2015), arXiv:1403.6344 [hep-ph] .
  * Wong and Li [2005] H. T. Wong and H.-B. Li, Neutrino magnetic moments, Mod. Phys. Lett. A 20, 1103 (2005).
  * Broggini  _et al._ [2012] C. Broggini, C. Giunti, and A. Studenikin, Electromagnetic Properties of Neutrinos, Adv. High Energy Phys. 2012, 459526 (2012), arXiv:1207.3980 [hep-ph] .
  * Studenikin [2020] A. Studenikin, Electromagnetic neutrinos: New constraints and new effects in oscillations, J. Phys. Conf. Ser. 1468, 012196 (2020), arXiv:1912.12501 [hep-ph] .
  * Studenikin [2021] A. Studenikin, Electromagnetic neutrino properties: new constraints and new effects, PoS ICHEP2020, 180 (2021), arXiv:2102.05468 [hep-ph] .
  * Giunti  _et al._ [2016] C. Giunti, K. A. Kouzakov, Y.-F. Li, A. V. Lokhov, A. I. Studenikin, and S. Zhou, Electromagnetic neutrinos in laboratory experiments and astrophysics, Annalen Phys. 528, 198 (2016), arXiv:1506.05387 [hep-ph] .
  * Fujikawa and Shrock [2004] K. Fujikawa and R. Shrock, On a neutrino electroweak radius, Phys. Rev. D 69, 013007 (2004), arXiv:hep-ph/0309329 .
  * Schechter and Valle [1981] J. Schechter and J. W. F. Valle, Majorana Neutrinos and Magnetic Fields, Phys. Rev. D 24, 1883 (1981), [Erratum: Phys.Rev.D 25, 283 (1982)].
  * Mohapatra and Pati [1975] R. Mohapatra and J. C. Pati, A Natural Left-Right Symmetry, Phys.Rev. D11, 2558 (1975).
  * Bernabeu  _et al._ [2002] J. Bernabeu, J. Papavassiliou, and J. Vidal, On the observability of the neutrino charge radius, Phys. Rev. Lett. 89, 101802 (2002), [Erratum: Phys.Rev.Lett. 89, 229902 (2002)], arXiv:hep-ph/0206015 .
  * Fujikawa and Shrock [2003] K. Fujikawa and R. Shrock, Comment on ‘Observability of the neutrino charge radius’, (2003), arXiv:hep-ph/0303188 .
  * Bardeen  _et al._ [1972] W. A. Bardeen, R. Gastmans, and B. E. Lautrup, Static quantities in Weinberg’s model of weak and electromagnetic interactions, Nucl. Phys. B 46, 319 (1972).
  * Lee and Shrock [1977] B. W. Lee and R. E. Shrock, Natural Suppression of Symmetry Violation in Gauge Theories: Muon - Lepton and Electron Lepton Number Nonconservation, Phys. Rev. D 16, 1444 (1977).
  * Monyonko and Reid [1985] N. M. Monyonko and J. H. Reid, What Is the Charge Radius of the Neutrino?, Prog. Theor. Phys. 73, 734 (1985).
  * Lucio  _et al._ [1985] J. L. Lucio, A. Rosado, and A. Zepeda, A Characteristic Size for the Neutrino, Phys. Rev. D 31, 1091 (1985).
  * Bernabeu  _et al._ [2000] J. Bernabeu, L. G. Cabral-Rosetti, J. Papavassiliou, and J. Vidal, On the charge radius of the neutrino, Phys. Rev. D 62, 113012 (2000), arXiv:hep-ph/0008114 .
  * Binosi and Papavassiliou [2009] D. Binosi and J. Papavassiliou, Pinch Technique: Theory and Applications, Phys. Rept. 479, 1 (2009), arXiv:0909.2536 [hep-ph] .
  * Cornwall [1982] J. M. Cornwall, Dynamical Mass Generation in Continuum QCD, Phys. Rev. D 26, 1453 (1982).
  * Cornwall and Papavassiliou [1989] J. M. Cornwall and J. Papavassiliou, Gauge Invariant Three Gluon Vertex in QCD, Phys. Rev. D 40, 3474 (1989).
  * Papavassiliou [1990] J. Papavassiliou, Gauge Invariant Proper Selfenergies and Vertices in Gauge Theories with Broken Symmetry, Phys. Rev. D 41, 3179 (1990).
  * Denner  _et al._ [1995] A. Denner, G. Weiglein, and S. Dittmaier, Application of the background field method to the electroweak standard model, Nucl. Phys. B 440, 95 (1995), arXiv:hep-ph/9410338 .
  * Pilaftsis [1997] A. Pilaftsis, Generalized pinch technique and the background field method in general gauges, Nucl. Phys. B 487, 467 (1997), arXiv:hep-ph/9607451 .
  * Bernabeu  _et al._ [2004] J. Bernabeu, J. Papavassiliou, and J. Vidal, The Neutrino charge radius is a physical observable, Nucl. Phys. B 680, 450 (2004), arXiv:hep-ph/0210055 .
  * Papavassiliou  _et al._ [2004] J. Papavassiliou, J. Bernabeu, D. Binosi, and J. Vidal, The Effective neutrino charge radius, Eur. Phys. J. C 33, S865 (2004), arXiv:hep-ph/0310028 .
  * Pati and Salam [1974] J. C. Pati and A. Salam, Lepton Number as the Fourth Color, Phys. Rev. D 10, 275 (1974), [Erratum: Phys.Rev.D 11, 703–703 (1975)].
  * Georgi and Glashow [1974] H. Georgi and S. L. Glashow, Unity of All Elementary Particle Forces, Phys. Rev. Lett. 32, 438 (1974).
  * Georgi  _et al._ [1974] H. Georgi, H. R. Quinn, and S. Weinberg, Hierarchy of Interactions in Unified Gauge Theories, Phys. Rev. Lett. 33, 451 (1974).
  * Fritzsch and Minkowski [1975] H. Fritzsch and P. Minkowski, Unified Interactions of Leptons and Hadrons, Annals Phys. 93, 193 (1975).
  * Dimopoulos  _et al._ [1980] S. Dimopoulos, S. Raby, and L. Susskind, Light Composite Fermions, Nucl. Phys. B 173, 208 (1980).
  * Senjanovic and Sokorac [1983] G. Senjanovic and A. Sokorac, Light Leptoquarks in SO(10), Z. Phys. C 20, 255 (1983).
  * Frampton and Lee [1990] P. H. Frampton and B.-H. Lee, SU(15) GRAND UNIFICATION, Phys.Rev.Lett. 64, 619 (1990).
  * Schrempp and Schrempp [1985] B. Schrempp and F. Schrempp, LIGHT LEPTOQUARKS, Phys. Lett. B 153, 101 (1985).
  * Buchmuller [1985] W. Buchmuller, Composite Quarks and Leptons, Acta Phys. Austriaca Suppl. 27, 517 (1985).
  * Gripaios [2010] B. Gripaios, Composite Leptoquarks at the LHC, JHEP 02, 045, arXiv:0910.1789 [hep-ph] .
  * Witten [1985] E. Witten, Symmetry Breaking Patterns in Superstring Models, Nucl. Phys. B 258, 75 (1985).
  * Hewett and Rizzo [1989] J. L. Hewett and T. G. Rizzo, Low-Energy Phenomenology of Superstring Inspired E(6) Models, Phys. Rept. 183, 193 (1989).
  * Ellis  _et al._ [1981] J. R. Ellis, M. K. Gaillard, D. V. Nanopoulos, and P. Sikivie, Can One Tell Technicolor from a Hole in the Ground?, Nucl. Phys. B 182, 529 (1981).
  * Farhi and Susskind [1981] E. Farhi and L. Susskind, Technicolor, Phys. Rept. 74, 277 (1981).
  * Hill and Simmons [2003] C. T. Hill and E. H. Simmons, Strong Dynamics and Electroweak Symmetry Breaking, Phys. Rept. 381, 235 (2003), [Erratum: Phys.Rept. 390, 553–554 (2004)], arXiv:hep-ph/0203079 .
  * Langacker [1981] P. Langacker, Grand Unified Theories and Proton Decay, Phys. Rept. 72, 185 (1981).
  * Adloff  _et al._ [1997] C. Adloff _et al._ (H1), Observation of events at very high $Q^{2}$ in $ep$ collisions at HERA, Z. Phys. C 74, 191 (1997), arXiv:hep-ex/9702012 .
  * Breitweg  _et al._ [1997] J. Breitweg _et al._ (ZEUS), Comparison of ZEUS data with standard model predictions for $e^{+}p\to e^{+}X$ scattering at high x and $Q^{2}$, Z. Phys. C 74, 207 (1997), arXiv:hep-ex/9702015 .
  * Breitweg  _et al._ [1999] J. Breitweg _et al._ (ZEUS), Measurement of high $Q^{2}$ neutral current $e^{+}p$ deep inelastic scattering cross-sections at HERA, Eur. Phys. J. C 11, 427 (1999), arXiv:hep-ex/9905032 .
  * Marzocca and Trifinopoulos [2021] D. Marzocca and S. Trifinopoulos, Minimal Explanation of Flavor Anomalies: B-Meson Decays, Muon Magnetic Moment, and the Cabibbo Angle, Phys. Rev. Lett. 127, 061803 (2021), arXiv:2104.05730 [hep-ph] .
  * Angelescu  _et al._ [2021] A. Angelescu, D. Bečirević, D. A. Faroughy, F. Jaffredo, and O. Sumensari, Single leptoquark solutions to the B-physics anomalies, Phys. Rev. D 104, 055017 (2021), arXiv:2103.12504 [hep-ph] .
  * Bauer and Neubert [2016] M. Bauer and M. Neubert, Minimal Leptoquark Explanation for the $R_{D^{(*)}}$ , $R_{K}$ , and $(g-2)_{\mu}$ Anomalies, Phys. Rev. Lett. 116, 141802 (2016), arXiv:1511.01900 [hep-ph] .
  * Bečirević  _et al._ [2016a] D. Bečirević, S. Fajfer, N. Košnik, and O. Sumensari, Leptoquark model to explain the $B$-physics anomalies, $R_{K}$ and $R_{D}$, Phys. Rev. D 94, 115021 (2016a), arXiv:1608.08501 [hep-ph] .
  * Kumar  _et al._ [2019] J. Kumar, D. London, and R. Watanabe, Combined Explanations of the $b\to s\mu^{+}\mu^{-}$ and $b\to c\tau^{-}{\bar{\nu}}$ Anomalies: a General Model Analysis, Phys. Rev. D 99, 015007 (2019), arXiv:1806.07403 [hep-ph] .
  * Cheung  _et al._ [2022] K. Cheung, W.-Y. Keung, and P.-Y. Tseng, Isodoublet vector leptoquark solution to the muon g-2, RK,K*, RD,D*, and W-mass anomalies, Phys. Rev. D 106, 015029 (2022), arXiv:2204.05942 [hep-ph] .
  * Bhaskar  _et al._ [2022] A. Bhaskar, A. A. Madathil, T. Mandal, and S. Mitra, Combined explanation of W-mass, muon g-2, RK(*) and RD(*) anomalies in a singlet-triplet scalar leptoquark model, Phys. Rev. D 106, 115009 (2022), arXiv:2204.09031 [hep-ph] .
  * Saad [2020] 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, 015019 (2020), arXiv:2005.04352 [hep-ph] .
  * Saad and Thapa [2020] S. Saad and A. Thapa, Common origin of neutrino masses and $R_{D^{(\ast)}}$, $R_{K^{(\ast)}}$ anomalies, Phys. Rev. D 102, 015014 (2020), arXiv:2004.07880 [hep-ph] .
  * Da Rold and Lamagna [2021] L. Da Rold and F. Lamagna, Model for the singlet-triplet leptoquarks, Phys. Rev. D 103, 115007 (2021), arXiv:2011.10061 [hep-ph] .
  * Bhupal Dev  _et al._ [2020] P. S. Bhupal Dev, R. Mohanta, S. Patra, and S. Sahoo, Unified explanation of flavor anomalies, radiative neutrino masses, and ANITA anomalous events in a vector leptoquark model, Phys. Rev. D 102, 095012 (2020), arXiv:2004.09464 [hep-ph] .
  * Altmannshofer  _et al._ [2020] W. Altmannshofer, P. S. B. Dev, A. Soni, and Y. Sui, Addressing R${}_{D^{(*)}}$, R${}_{K^{(*)}}$, muon $g-2$ and ANITA anomalies in a minimal $R$-parity violating supersymmetric framework, Phys. Rev. D 102, 015031 (2020), arXiv:2002.12910 [hep-ph] .
  * Fuentes-Martín and Stangl [2020] J. Fuentes-Martín and P. Stangl, Third-family quark-lepton unification with a fundamental composite Higgs, Phys. Lett. B 811, 135953 (2020), arXiv:2004.11376 [hep-ph] .
  * Gherardi  _et al._ [2021] V. Gherardi, D. Marzocca, and E. Venturini, Low-energy phenomenology of scalar leptoquarks at one-loop accuracy, JHEP 01, 138, arXiv:2008.09548 [hep-ph] .
  * Bigaran and Volkas [2020] I. Bigaran and R. R. Volkas, Getting chirality right: Single scalar leptoquark solutions to the $(g-2)_{e,\mu}$ puzzle, Phys. Rev. D 102, 075037 (2020), arXiv:2002.12544 [hep-ph] .
  * Doršner  _et al._ [2020] 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, 075007 (2020), arXiv:2006.11624 [hep-ph] .
  * Babu  _et al._ [2021] K. S. Babu, P. S. B. Dev, S. Jana, and A. Thapa, Unified framework for $B$-anomalies, muon $g\textendash 2$ and neutrino masses, JHEP 03, 179, arXiv:2009.01771 [hep-ph] .
  * Crivellin  _et al._ [2021] 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, 021801 (2021), arXiv:2008.02643 [hep-ph] .
  * Aristizabal Sierra  _et al._ [2008] D. Aristizabal Sierra, M. Hirsch, and S. G. Kovalenko, Leptoquarks: Neutrino masses and accelerator phenomenology, Phys. Rev. D 77, 055011 (2008), arXiv:0710.5699 [hep-ph] .
  * Zhang [2021] D. Zhang, Radiative neutrino masses, lepton flavor mixing and muon g $-$ 2 in a leptoquark model, JHEP 07, 069, arXiv:2105.08670 [hep-ph] .
  * Cai  _et al._ [2017] Y. Cai, J. Gargalionis, M. A. Schmidt, and R. R. Volkas, Reconsidering the One Leptoquark solution: flavor anomalies and neutrino mass, JHEP 10, 047, arXiv:1704.05849 [hep-ph] .
  * Popov and White [2017] O. Popov and G. A. White, One Leptoquark to unify them? Neutrino masses and unification in the light of $(g-2)_{\mu}$, $R_{D^{(\star)}}$ and $R_{K}$ anomalies, Nucl. Phys. B 923, 324 (2017), arXiv:1611.04566 [hep-ph] .
  * LHC [2022] Test of lepton universality in $b\rightarrow s\ell^{+}\ell^{-}$ decays, (2022), arXiv:2212.09152 [hep-ex] .
  * Hewett and Rizzo [1998] J. L. Hewett and T. G. Rizzo, Don ’ $t$ stop thinking about leptoquarks: Constructing new models, Phys. Rev. D 58, 055005 (1998), arXiv:hep-ph/9708419 .
  * Chua and Hwang [1999] C.-K. Chua and W. Y. P. Hwang, The Neutrino magnetic moment induced by leptoquarks, Phys. Rev. D 60, 073002 (1999), arXiv:hep-ph/9811232 .
  * Povarov [2007] A. V. Povarov, Scalar-leptoquark contributions to the neutrino magnetic moment, Phys. Atom. Nucl. 70, 871 (2007).
  * Sánchez-Vélez [2022] R. Sánchez-Vélez, Neutrino electromagnetic properties in a Leptoquark model, (2022), arXiv:2204.01167 [hep-ph] .
  * Khan [2023a] A. N. Khan, Neutrino millicharge and other electromagnetic interactions with COHERENT-2021 data, Nucl. Phys. B 986, 116064 (2023a), arXiv:2201.10578 [hep-ph] .
  * Akimov  _et al._ [2022] D. Akimov _et al._ (COHERENT), Measurement of the Coherent Elastic Neutrino-Nucleus Scattering Cross Section on CsI by COHERENT, Phys. Rev. Lett. 129, 081801 (2022), arXiv:2110.07730 [hep-ex] .
  * Khan [2023b] A. N. Khan, Light new physics and neutrino electromagnetic interactions in XENONnT, Phys. Lett. B 837, 137650 (2023b), arXiv:2208.02144 [hep-ph] .
  * Agostini  _et al._ [2017] M. Agostini _et al._ (Borexino), Limiting neutrino magnetic moments with Borexino Phase-II solar neutrino data, Phys. Rev. D 96, 091103 (2017), arXiv:1707.09355 [hep-ex] .
  * Beda  _et al._ [2012] A. G. Beda, V. B. Brudanin, V. G. Egorov, D. V. Medvedev, V. S. Pogosov, M. V. Shirchenko, and A. S. Starostin, The results of search for the neutrino magnetic moment in GEMMA experiment, Adv. High Energy Phys. 2012, 350150 (2012).
  * Schwienhorst  _et al._ [2001] R. Schwienhorst _et al._ (DONUT), A New upper limit for the tau - neutrino magnetic moment, Phys. Lett. B 513, 23 (2001), arXiv:hep-ex/0102026 .
  * Auerbach  _et al._ [2001] L. B. Auerbach _et al._ (LSND), Measurement of electron - neutrino - electron elastic scattering, Phys. Rev. D 63, 112001 (2001), arXiv:hep-ex/0101039 .
  * Deniz  _et al._ [2010] M. Deniz _et al._ (TEXONO), Measurement of Nu(e)-bar -Electron Scattering Cross-Section with a CsI(Tl) Scintillating Crystal Array at the Kuo-Sheng Nuclear Power Reactor, Phys. Rev. D 81, 072001 (2010), arXiv:0911.1597 [hep-ex] .
  * Khan [2019] A. N. Khan, $\sin^{2}\theta_{W}$ Estimate and Neutrino Electromagnetic Properties from Low-Energy Solar Data, J. Phys. G 46, 035005 (2019), arXiv:1709.02930 [hep-ph] .
  * del Aguila and Sher [1990] F. del Aguila and M. Sher, The Electric dipole moment of the tau, Phys. Lett. B 252, 116 (1990).
  * Akama  _et al._ [2002] K. Akama, T. Hattori, and K. Katsuura, Naturalness bounds on dipole moments from new physics, Phys. Rev. Lett. 88, 201601 (2002), arXiv:hep-ph/0111238 .
  * Escribano and Masso [1997] R. Escribano and E. Masso, Improved bounds on the electromagnetic dipole moments of the $\tau$ lepton, Phys. Lett. B 395, 369 (1997), arXiv:hep-ph/9609423 .
  * Ibrahim and Nath [2010] T. Ibrahim and P. Nath, Large Tau and Tau Neutrino Electric Dipole Moments in Models with Vector Like Multiplets, Phys. Rev. D 81, 033007 (2010), [Erratum: Phys.Rev.D 89, 119902 (2014)], arXiv:1001.0231 [hep-ph] .
  * Llamas-Bugarin  _et al._ [2017] A. Llamas-Bugarin, A. Gutiérrez-Rodríguez, and M. A. Hernández-Ruíz, Probing the electromagnetic dipole moments of the tau-neutrino in the $U(1)_{B-L}$ model at the ILC and CLIC energies, Phys. Rev. D 95, 116008 (2017), arXiv:1706.03635 [hep-ph] .
  * Buchmuller  _et al._ [1987] W. Buchmuller, R. Ruckl, and D. Wyler, Leptoquarks in Lepton - Quark Collisions, Phys. Lett. B 191, 442 (1987), [Erratum: Phys.Lett.B 448, 320–320 (1999)].
  * Davies and He [1991] A. J. Davies and X.-G. He, Tree Level Scalar Fermion Interactions Consistent With the Symmetries of the Standard Model, Phys. Rev. D 43, 225 (1991).
  * Doršner  _et al._ [2016] 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, 1 (2016), arXiv:1603.04993 [hep-ph] .
  * Arnold  _et al._ [2013] J. M. Arnold, B. Fornal, and M. B. Wise, Phenomenology of scalar leptoquarks, Phys. Rev. D 88, 035009 (2013), arXiv:1304.6119 [hep-ph] .
  * Assad  _et al._ [2018] N. Assad, B. Fornal, and B. Grinstein, Baryon Number and Lepton Universality Violation in Leptoquark and Diquark Models, Phys. Lett. B 777, 324 (2018), arXiv:1708.06350 [hep-ph] .
  * Fajfer and Košnik [2016] S. Fajfer and N. Košnik, Vector leptoquark resolution of $R_{K}$ and $R_{D^{(*)}}$ puzzles, Phys. Lett. B 755, 270 (2016), arXiv:1511.06024 [hep-ph] .
  * Calibbi  _et al._ [2018] L. Calibbi, A. Crivellin, and T. Li, Model of vector leptoquarks in view of the $B$-physics anomalies, Phys. Rev. D 98, 115002 (2018), arXiv:1709.00692 [hep-ph] .
  * Crivellin  _et al._ [2019] A. Crivellin, C. Greub, D. Müller, and F. Saturnino, Importance of Loop Effects in Explaining the Accumulated Evidence for New Physics in B Decays with a Vector Leptoquark, Phys. Rev. Lett. 122, 011805 (2019), arXiv:1807.02068 [hep-ph] .
  * Datta  _et al._ [2020] A. Datta, J. L. Feng, S. Kamali, and J. Kumar, Resolving the $(g-2)_{\mu}$ and $B$ Anomalies with Leptoquarks and a Dark Higgs Boson, Phys. Rev. D 101, 035010 (2020), arXiv:1908.08625 [hep-ph] .
  * Hati  _et al._ [2019] C. Hati, J. Kriewald, J. Orloff, and A. M. Teixeira, A nonunitary interpretation for a single vector leptoquark combined explanation to the $B$-decay anomalies, JHEP 12, 006, arXiv:1907.05511 [hep-ph] .
  * Da Rold and Lamagna [2019] L. Da Rold and F. Lamagna, A vector leptoquark for the B-physics anomalies from a composite GUT, JHEP 12, 112, arXiv:1906.11666 [hep-ph] .
  * Cornella  _et al._ [2019] C. Cornella, J. Fuentes-Martin, and G. Isidori, Revisiting the vector leptoquark explanation of the B-physics anomalies, JHEP 07, 168, arXiv:1903.11517 [hep-ph] .
  * Bhaskar  _et al._ [2021] A. Bhaskar, D. Das, T. Mandal, S. Mitra, and C. Neeraj, Precise limits on the charge-2/3 U1 vector leptoquark, Phys. Rev. D 104, 035016 (2021), arXiv:2101.12069 [hep-ph] .
  * Bordone  _et al._ [2018a] M. Bordone, C. Cornella, J. Fuentes-Martin, and G. Isidori, A three-site gauge model for flavor hierarchies and flavor anomalies, Phys. Lett. B 779, 317 (2018a), arXiv:1712.01368 [hep-ph] .
  * Bordone  _et al._ [2018b] M. Bordone, C. Cornella, J. Fuentes-Martín, and G. Isidori, Low-energy signatures of the $\mathrm{PS}^{3}$ model: from $B$-physics anomalies to LFV, JHEP 10, 148, arXiv:1805.09328 [hep-ph] .
  * Barbieri  _et al._ [2016] R. Barbieri, G. Isidori, A. Pattori, and F. Senia, Anomalies in $B$-decays and $U(2)$ flavour symmetry, Eur. Phys. J. C 76, 67 (2016), arXiv:1512.01560 [hep-ph] .
  * Buttazzo  _et al._ [2017] D. Buttazzo, A. Greljo, G. Isidori, and D. Marzocca, B-physics anomalies: a guide to combined explanations, JHEP 11, 044, arXiv:1706.07808 [hep-ph] .
  * Angelescu  _et al._ [2018] A. Angelescu, D. Bečirević, D. A. Faroughy, and O. Sumensari, Closing the window on single leptoquark solutions to the $B$-physics anomalies, JHEP 10, 183, arXiv:1808.08179 [hep-ph] .
  * Bečirević  _et al._ [2016b] D. Bečirević, N. Košnik, O. Sumensari, and R. Zukanovich Funchal, Palatable Leptoquark Scenarios for Lepton Flavor Violation in Exclusive $b\to s\ell_{1}\ell_{2}$ modes, JHEP 11, 035, arXiv:1608.07583 [hep-ph] .
  * Bhattacharya  _et al._ [2017] B. Bhattacharya, A. Datta, J.-P. Guévin, D. London, and R. Watanabe, Simultaneous Explanation of the $R_{K}$ and $R_{D^{(*)}}$ Puzzles: a Model Analysis, JHEP 01, 015, arXiv:1609.09078 [hep-ph] .
  * Di Luzio  _et al._ [2017] L. Di Luzio, A. Greljo, and M. Nardecchia, Gauge leptoquark as the origin of B-physics anomalies, Phys. Rev. D 96, 115011 (2017), arXiv:1708.08450 [hep-ph] .
  * Aebischer  _et al._ [2023] J. Aebischer, G. Isidori, M. Pesut, B. A. Stefanek, and F. Wilsch, Confronting the vector leptoquark hypothesis with new low- and high-energy data, Eur. Phys. J. C 83, 153 (2023), arXiv:2210.13422 [hep-ph] .
  * Heeck and Teresi [2018] J. Heeck and D. Teresi, Pati-Salam explanations of the B-meson anomalies, JHEP 12, 103, arXiv:1808.07492 [hep-ph] .
  * Aydemir  _et al._ [2018] U. Aydemir, D. Minic, C. Sun, and T. Takeuchi, $B$-decay anomalies and scalar leptoquarks in unified Pati-Salam models from noncommutative geometry, JHEP 09, 117, arXiv:1804.05844 [hep-ph] .
  * Chen  _et al._ [2022] S.-L. Chen, W.-w. Jiang, and Z.-K. Liu, Combined explanations of B-physics anomalies, $(g-2)_{e,\mu}$ and neutrino masses by scalar leptoquarks, Eur. Phys. J. C 82, 959 (2022), arXiv:2205.15794 [hep-ph] .
  * Dey  _et al._ [2018] U. K. Dey, D. Kar, M. Mitra, M. Spannowsky, and A. C. Vincent, Searching for Leptoquarks at IceCube and the LHC, Phys. Rev. D 98, 035014 (2018), arXiv:1709.02009 [hep-ph] .
  * Gabrielli  _et al._ [2015] E. Gabrielli, L. Marzola, M. Raidal, and H. Veermäe, Dark matter and spin-1 milli-charged particles, JHEP 08, 150, arXiv:1507.00571 [hep-ph] .