# Distinguishing models with $\mu\to e$ observables

Marco Ardu  E-mail address<EMAIL_ADDRESS>LUPM, CNRS, Université
Montpellier Place Eugene Bataillon, F-34095 Montpellier, Cedex 5,
France111From the 1st of October at Departament de Física Tèorica, Universitat
de València, Dr. Moliner 50, E-46100 Burjassot (València), Spain Sacha
Davidson  E-mail address<EMAIL_ADDRESS>LUPM, CNRS, Université
Montpellier Place Eugene Bataillon, F-34095 Montpellier, Cedex 5, France
Stéphane Lavignac  E-mail address<EMAIL_ADDRESS>Institut de
Physique Théorique, Université Paris Saclay, CNRS, CEA, F-91191 Gif-sur-
Yvette, France

###### Abstract

Upcoming experiments will improve the reach for the lepton flavour violating
(LFV) processes $\mu\to e\gamma$, $\mu\to e\bar{e}e$ and $\mu A\to eA$ by
orders of magnitude. We investigate whether this upcoming data could rule out
some popular TeV-scale LFV models (the type II seesaw, the inverse seesaw and
a scalar leptoquark) using a bottom-up EFT approach involving twelve Wilson
coefficients that can in principle all be determined by experimental
measurements. In this 12-dimensional coefficient space, each model can only
predict points in a specific subspace; for instance, flavour change involving
singlet electrons is suppressed in the seesaw models, and the leptoquark
induces negligible coefficients for 4-lepton scalar operators. Using the fact
that none of these models can populate the whole region accessible to upcoming
experiments, we show that $\mu\to e$ experiments have the ability to rule them
out.

##  I Introduction and review

###  I.1 Introduction

Searches for New Physics(NP) in the lepton sector are of great interest,
because such NP is required by neutrino masses, it could fit some current
anomalies (such as $(g-2)_{\mu}$ Muong-2:2021ojo  and observations in $B$
meson physics BaBar:2012obs ; Belle:2015qfa ; Belle:2019rba ; LHCb:2023zxo ;
LHCb:2023cjr ), and because leptons do not have strong interactions, so the
observables are relatively clean. In this manuscript, we assume that this
leptonic New Physics is heavy, and parametrise it in EFT Georgi ; burashouches
; LesHouches .

Lepton Flavour change (LFV) in the $\mu\\!\to\\!e$ sector is promising for the
discovery of leptonic NP, because the experimental sensitivity is already
good, and is expected to improve by several orders of magnitude in the near
future (see table 1). However, few processes are constrained, so the current
experimental bounds only set about a dozen constraints on Wilson coefficients
DKY . One can therefore wonder whether future observations of $\mu\\!\to\\!e$
flavour change could distinguish among the multitude of models that predict
LFV.

process | current sensitivity | future
---|---|---
$\mu\to e\gamma$ |  $<4.2\times 10^{-13}$(MEG TheMEG:2016wtm ) |  $\sim 10^{-14}$ (MEG II MEGII )
$\mu\to e\bar{e}e$ |  $<1.0\times 10^{-12}$(SINDRUM Bellgardt:1987du ) |  $\sim 10^{-16}$ (Mu3e Mu3e )
$\mu$Au$\to e$Au |  $<7\times 10^{-13}$(SINDRUM II Bertl:2006up ) | ? $\to 10^{-(18\to 20)}$ (PP/AMF PP ; CGroup:2022tli )
$\mu$Ti$\to e$Ti |  $<6.1\times 10^{-13}$(SINDRUM II Wintz:1998rp ) |  $\sim 10^{-16}$ (COMET COMET , Mu2e mu2e )
$\tau\to l+\dots$ |  $\mathrel{\vbox{\hbox{$<$}\nointerlineskip\hbox{$\sim$}}}10^{-8}$(Babar/Belle) tau1 ; Belle:2007cio  |  $\sim 10^{-(9\to 10)}$(BelleII) belle2t3l
Table 1: Current bounds on the branching ratios for various lepton flavour
changing processes, and the estimated reach of upcoming experiments.

Predictions for $\mu\\!\to\\!e$ LFV have been widely studied over several
decades in a multitude of models, such as the supersymmetric type I seesaw,
the supersymmetric type II seesaw, supersymmetric flavour models, left-right
symmetric models, two Higgs doublet models, the inverse seesaw and its
supersymmetric version, warped extra dimensions, the littlest Higgs model with
T parity, unparticle physics, radiative neutrino mass models, spontaneous
lepton number violation, low-scale flavour models, and many others (see e.g.
Refs. Hisano:1995cp ; Antusch:2006vw ; Rossi:2002zb ; Altmannshofer:2009ne ;
Cirigliano:2004mv ; Omura:2015nja ; Arganda:2014dta ; Deppisch:2004fa ;
Agashe:2006iy ; Blanke:2007db ; Aliev:2007qw ; Cai:2017jrq ; Escribano:2021uhf
; Lopez-Ibanez:2021yzu , and for recent reviews Refs. Calibbi:2017uvl ;
Ardu:2022sbt ). Top-down analyses – which start from the model to predict
observables – frequently show correlations among branching ratios, often
resulting from scans over model parameter space. In our bottom-up EFT
perspective, starting from the data, we address a different question: can
observations distinguish among models?

In this manuscript, we focus on three models with new heavy particles around
the TeV scale. The first two are neutrino mass models : the TeV-scale version
of the type II seesaw mechanism Magg:1980ut ; Lazarides:1980nt ;
Schechter:1980gr ; Mohapatra:1980yp  and the inverse type I seesaw
Wyler:1982dd ; Mohapatra:1986aw ; Mohapatra:1986bd , whose predictions for LFV
processes have been studied, mainly in the top-down approach, by many authors
(see e.g. Refs. Chun:2003ej ; Kakizaki:2003jk ; Akeroyd:2009nu ; Dinh:2012bp ;
Barrie:2022ake  for the type II and Refs. Tommasini:1995ii ; Ilakovac:1994kj ;
Ibarra:2011xn ; Dinh:2012bp ; Alonso:2012ji ; Abada:2015oba ; Coy:2018bxr ;
Abada:2021zcm ; Zhang:2021jdf ; Coy:2021hyr ; Granelli:2022eru ;
Crivellin:2022cve  for the inverse seesaw, where Refs. Coy:2018bxr ;
Zhang:2021jdf ; Coy:2021hyr ; Crivellin:2022cve  follow an EFT approach). Both
these models have the additional attraction of being able to generate the
baryon asymmetry of the Universe via leptogenesis Fukugita:1986hr  (for a
review, see Ref. Davidson:2008bu ). While, in the type II seesaw case, thermal
leptogenesis requires a triplet mass above $10^{10}\,{\rm GeV}$ or so
Hambye:2005tk ; AristizabalSierra:2014nzr ; Lavignac:2015gpa , a TeV-scale
scalar triplet with non-minimal coupling to gravity can lead to successful
leptogenesis Barrie:2021mwi  through the Affleck-Dine mechanism Affleck:1984fy
. The inverse seesaw model, on the other hand, features TeV-scale sterile
neutrinos which can generate the baryon asymmetry of the Universe through
resonant leptogenesis Pilaftsis:2003gt ; Blanchet:2009kk ; BhupalDev:2014pfm ;
daSilva:2022mrx  or ARS leptogenesis Akhmedov:1998qx ; Asaka:2005pn ;
Klaric:2021cpi ; Hernandez:2022ivz . The last model is an SU(2) singlet
leptoquark which can fit the $R_{D}$ anomaly BaBar:2012obs ; Belle:2015qfa ;
Belle:2019rba ; LHCb:2023zxo ; LHCb:2023cjr , as discussed by many authors
(see e.g. Refs. Sakaki:2013bfa ; Cai:2017wry ; Angelescu:2018tyl ; Lee:2021jdr
). The leptoquark differs from the neutrino mass models in that at tree level,
it generates 2lepton-2quark operators (which mediate $\mu\\!\to\\!e~{}{\rm
conversion}$), and in that it couples to SU(2) singlet fermions of the SM.

We apply bottom-up EFT to explore whether $\mu\\!\to\\!e$ LFV can distinguish
among models, starting from the observation that the data could determine 12
operator coefficients, and not just the three branching ratios. We consider
this 12-dimensional coefficient space, and ask whether the volume accessible
to upcoming experiments can be filled by each of three models. So we aim to
identify the region of the ellipse that a model cannot occupy; an observation
in this region would rule the model out. Our study is performed in an EFT
framework inspired by C+C , and differs from top-down analyses, in that we do
not scan over model parameter space for which we do not known the measure, and
because we take the data to be 12 Wilson coefficients. A more complete
analysis and technical details will appear in a subsequent publication pl2 .

Our EFT framework is briefly summarised in the next subsection. In the
following three sections, we present and discuss three models of New Physics
at the TeV scale: the type II seesaw in section II, the inverse type I seesaw
in section III, and a leptoquark in section IV. Section V compares the models
and summarises the results.

###  I.2 review

We consider three processes – $\mu\to e\gamma$, $\mu\to e\bar{e}e$ and Spin-
Independent222Spin-Dependent $\mu A\to\\!eA$ SDmuc ; Hoferichter:2022mna  is
also possible, but analogously to WIMP scattering, is relatively suppressed.
(SI)$\mu A\to\\!eA$ – because they are complementary C+C , and because the
experimental sensitivity could improve significantly in coming years. They can
be parametrised at the experimental scale by the Lagrangian KO :

$\displaystyle\delta{\cal L}$ $\displaystyle=$
$\displaystyle\frac{1}{v^{2}}\sum_{X\in
L,R}{\Big{[}}C^{e\mu}_{D,X}(m_{\mu}\overline{e}\sigma^{\rho\sigma}P_{X}\mu)F_{\rho\sigma}+C^{e\mu
ee}_{S,XX}(\overline{e}P_{X}\mu)(\overline{e}P_{X}e)+C^{e\mu
ee}_{VXR}(\overline{e}\gamma^{\rho}P_{X}\mu)(\overline{e}\gamma_{\rho}P_{R}e)$
(I.1) $\displaystyle+C^{e\mu
ee}_{VXL}(\overline{e}\gamma^{\rho}P_{X}\mu)(\overline{e}\gamma_{\rho}P_{L}e)+C_{Alight,X}{\cal
O}_{Alight,X}+C_{Aheavy,X}{\cal O}_{Aheavy,X}{\Big{]}}+{\rm h.c}$

where the twelve coefficients $C$ are dimensionless complex numbers,
$X\in\\{L,R\\}$, $\frac{1}{v^{2}}=2\sqrt{2}G_{F}$ (so $v=$174 GeV), and ${\cal
O}_{Alight,X}$ and ${\cal O}_{Aheavy,X}$ are respectively the four-fermion
operator combinations that induce $\mu A\to\\!eA$ on light nuclei like
Titanium or Aluminium , and an operator combination probed by heavy targets
like Gold. Expressions for these operators are given in the Appendix.

coefficient | current bound | future bound | process
---|---|---|---
$C_{D,X}^{e\mu}$ | $1.0\times 10^{-8}$ | $10^{-9}$ | $\mu\to e\gamma$
$C_{V,XX}^{e\mu ee}$ | $0.7\times 10^{-6}$ | $10^{-8}$ | $\mu\to e\bar{e}e$
$C_{V,XY}^{e\mu ee}$ | $1.0\times 10^{-6}$ | $10^{-8}$ | $\mu\to e\bar{e}e$
$C_{S,XX}^{e\mu ee}$ | $2.8\times 10^{-6}$ | $10^{-8}$ | $\mu\to e\bar{e}e$
$C_{Alight,X}$ | $5.0\times 10^{-8}$ | $10^{-10}$ | $\mu{\rm Ti}\to e{\rm Ti}$
$C_{Aheavy\perp,X}$ | $2\times 10^{-7}$ | $10^{-9}$ | $\mu{\rm Au}\to e{\rm Au}$
Table 2: Current bounds on the operator coefficients of the Lagrangian (I.1)
at the experimental scale $m_{\mu}$ ($X=L,R$), and estimated reach of upcoming
experiments.

The non-observation of $\mu\\!\to\\!e$ processes constrains the coefficients
in eqn (I.1) to sit in an 12$d$ ellipse at the origin DKY  (see Table 2 for
the current bounds, and estimated sensitivities of upcoming experiments).
Observations could in principle determine the magnitude of each coefficient:
if the decaying muon is polarised KO  (which can also be possible for $\mu
A\to\\!eA$ Kadono:1986zz ; Kuno:1987dp ) then the chirality of the $\mu\to e$
bilinear can be determined, and asymmetries and angular distributions in
$\mu\to e\bar{e}e$ can distinguish among the four-lepton operators that
contribute Okadameee . Some relative phases can also be measured Okadameee .
Finally, changing the target material in $\mu A\to\\!eA$ allows to probe
different combinations of vector and scalar coefficients on protons or
neutrons KKO ; Haxton:2022piv ; current theoretical accuracy allows to obtain
independent information from at least two targets DKY ; Davidson:2022nnl , so
in this manuscript we focus on light targets like Titanium (used by SINDRUMII
Bertl:2006up ; Wintz:1998rp ) or Aluminium (the target for the upcoming Mu2e
and COMET experiments). The complementary constraints that can be obtained
with Gold (used by SINDRUMII Bertl:2006up ), will be discussed in pl2 . With
theoretical optimism, we assume the coefficients can be distinguished to the
reach of upcoming expts.

We take the New Physics scale $\Lambda_{NP}\sim$ TeV for the three models
considered here. The coefficients are evolved from the experimental scale
$\sim m_{\mu}$ to $\Lambda_{NP}\sim$ TeV in the broken electroweak theory,
using the “Leading Order” RGEs of QED and QCD Crivellin:2017rmk ; C+C
(starting respectively at $m_{\mu}$ and 2 GeV), for the operator basis of C+C
. This includes the leading log-enhanced loops of QED and QCD via the RGEs,
and loop diagrams with the $W$,$Z$ or Higgs can contribute in the matching. We
prefer this approach over matching to SMEFT at the weak scale, because it
allows to resum QCD333It is convenient to use 5 flavours at all scales,
because the results for 5 or 6 flavours are numerically similar. between the
experimental scale and $\Lambda_{NP}$, and avoids the issue that $v$/TeV is
not large, implying that the SMEFT expansions in $1/\Lambda_{NP}^{2}$ and
$\alpha^{n}\log$ may not converge quickly 444This approximation may also
double count some electroweak contributions that we think are higher order, as
will be discussed in pl2 .. This RG evolution gives the 12-dimensional ellipse
at $\Lambda_{NP}$. We then match onto each of the models in turn (at tree
level in the EFT), and explore whether they can fill the ellipse.

In relating models to observables, it is convenient to use as stepping stones
the coupling constant combinations that appear in Wilson coefficients, because
they parametrise the $\mu\\!\to\\!e$ LFV. For instance, tree level exchange of
a leptoquark interacting with $u$ quarks, matches onto an coefficient
$\propto\lambda^{eu}\lambda^{\mu u*}$, and the loop diagram of figure 1(b) is
$\propto[ff^{*}Y_{e}]_{e\mu}$. We refer to these combinations as “invariants”
(a la Jarlskog), because they are related to $S$-matrix elements, and
therefore should be independent of some Lagrangian redefinitions.

The operator coefficients can of course be complex, and in some cases the
relative phases are observable (for instance in asymmetries in $\mu\to
e\bar{e}e$ Okadameee ). However, for plotting purposes, it is common to
approximate the coefficients as real. In our analysis, the coefficients are
complex, but we plot either the absolute values or the real parts; the phases
will be discussed in pl2 .

##  II  Type II seesaw

The first model we consider is the type II seesaw mechanism Magg:1980ut ;
Lazarides:1980nt ; Schechter:1980gr ; Mohapatra:1980yp , which generates
neutrino masses via the tree level exchange of an SU(2) triplet scalar
$\Delta$. In this model, the Yukawa matrix is directly proportional to the
observed neutrino mass matrix, so it is predictive of flavour structure – for
instance fixing some ratios between $\tau\to l$ and $\mu\to e$ transitions –
and its LFV signatures have been widely studied Chun:2003ej ; Kakizaki:2003jk
; Akeroyd:2009nu ; Dinh:2012bp ; Barrie:2022ake .

We assume the triplet scalar is at the TeV scale, so could be produced at
current and future colliders and lead to particular signatures Akeroyd:2005gt
; FileviezPerez:2008jbu ; Melfo:2011nx ; Freitas:2014fda ; Ghosh:2017pxl ;
Antusch:2018svb . It could also affect Higgs physics Arhrib:2011uy ;
Arhrib:2011vc  and contribute to electroweak observables such as $m_{W}$
CDF:2022hxs .

The SM Lagrangian is augmented by the following interactions

$\displaystyle\delta{\cal L}_{\Delta}$ $\displaystyle=$
$\displaystyle(D_{\rho}\Delta^{I})^{\dagger}D^{\rho}\Delta^{I}-M_{\Delta}^{2}|\Delta|^{2}+\frac{1}{2}\left(f_{\alpha\beta}\,\overline{\ell^{c}_{\alpha}}(i\tau_{2})\tau_{I}\ell_{\beta}\Delta^{I}+M_{\Delta}\lambda_{H}\,H^{T}(i\tau_{2})\tau_{I}H\Delta^{*I}+{\rm
h.c.}\right)$ (II.1)
$\displaystyle+\lambda_{3}(H^{\dagger}H)(\Delta^{I*}\Delta^{I})+\lambda_{4}{\rm
Tr}(\Delta^{I*}\tau_{I}\tau_{J}\tau_{K}\Delta^{K})(H^{\dagger}\tau_{J}H)+\dots~{}~{},$

where $\Delta$ is the colour-singlet, $SU(2)$-triplet scalar of hypercharge
$Y_{\Delta}=+1$, $\ell$ are the left-handed SU(2) doublets, $M_{\Delta}$ is
the triplet mass which we take $\sim$ TeV, $f$ is a symmetric complex $3\times
3$ matrix proportional to the light neutrino mass matrix and whose indices
$\alpha,\beta$ run over $\\{e,\mu,\tau\\}$, $\\{\tau_{I}\\}$ are the Pauli
matrices, and the $\lambda$’s are real dimensionless couplings555$\lambda_{H}$
can be taken real with no loss of generality.. The dots on the right-hand side
of Eq. (II.1) stand for scalar interactions that are not relevant for LFV
processes. We also find negligible contributions to LFV operators from the
triplet-Higgs interactions assuming perturbative $\lambda_{3,4}$.
Consequently, these contributions will not be included in the subsequent
discussion.

We match the model to EFT at the scale $M_{\Delta}\sim$TeV, generating a
neutrino mass matrix $[m_{\nu}]_{\alpha\beta}=U_{\alpha i}m_{i}U_{\beta i}$
via the tree-level exchange of the triplet between pairs of leptons and
Higgses:

$[m_{\nu}]_{\alpha\beta}\simeq 0.03\ {\rm eV}\
f^{*}_{\alpha\beta}\frac{\lambda_{H}}{10^{-12}}\frac{{\rm
TeV}}{M_{\Delta}}~{}~{}.$ (II.2)

Exchanging the triplet among four leptons matches onto one of the LFV
coefficients of eqn (I.1), which induces $\mu\to e\bar{e}e$:

$\displaystyle C^{e\mu ee}_{V,LL}\simeq\frac{v^{2}}{2M^{2}_{\Delta}}f_{\mu
e}f^{*}_{ee}=\frac{[m^{*}_{\nu}]_{\mu
e}[m_{\nu}]_{ee}}{2\lambda_{H}^{2}v^{2}}~{}~{}.$ (II.3)

So the small ratio $m_{\nu}/v$ can be obtained by suppressing $\lambda_{H}$,
while leaving unconstrained $fv/M_{\Delta}$, which controls the magnitude of
LFV. However, the triplet Yukawa matrix $[f]$ is proportional to $[m_{\nu}]$,
so its flavour structure can be determined from neutrino oscillation data
ParticleDataGroup:2022pth . The only unknowns are the mass $m_{min}$ of the
lightest neutrino, two Majorana phases, and the Hierarchy (Normal =
$m_{3}>m_{2}>m_{1}$, Inverted = $m_{3}<m_{1}<m_{2}$). We use eqn (II.2) in
order to express $[f]$ in terms of $[m_{\nu}]$.

$\mu_{L}$$e_{L}$$f$$f$$l_{L}$ (a) QED penguin mixing the tree-level 4$\ell$
vector operators to vector four fermion operators involving fermions $f$
($l=e,\mu,\tau$)

$\mu$$\ell_{e}$$H$$\gamma$$\ell$$\Delta^{a}$ (b) Matching onto the $\mu\to e$
photon dipole

$\mu_{L}$$e_{L}$$\gamma$$l_{L}$ (c) One of many diagrams for vector-to-dipole
mixing in the two-loop RGEs ($l=e,\mu,\tau$)

Figure 1:  Loop contributions to $\mu\to e$ operators and to their mixing in
the type II seesaw model.

The type II seesaw will also induce other LFV coefficients given in the
Lagrangian (I.1). Tree-level triplet exchange matches onto 4$\ell$ operators
with $\mu$ and $\tau$ bilinears, and these combine with eqn (II.3) in a
“penguin”, as illustrated in figure 1(a), to generate, for instance

$\displaystyle C^{e\mu ee}_{V,LR}$ $\displaystyle=$
$\displaystyle\frac{\alpha_{e}}{4\pi}\left[\frac{[m^{\dagger}_{\nu}m_{\nu}]_{\mu
e}}{\lambda_{H}^{2}v^{2}}\log\left(\frac{M_{\Delta}}{m_{\tau}}\right)+\sum_{\alpha\in
e,\mu}\frac{[m^{*}_{\nu}]_{\mu\alpha}[m_{\nu}]_{e\alpha}}{\lambda_{H}^{2}v^{2}}\log\left(\frac{m_{\tau}}{m_{\mu}}\right)\right]$
(II.4)

This loop arises in the RGEs, or equivalently, is log-enhanced. The logarithm
is cut off at low energy by the experimental scale ($m_{\mu}$) or the mass of
the lepton in the loop, so the $\tau$ is not included in the loop between
$m_{\tau}\to m_{\mu}$. This is interesting, because
$[m^{\dagger}_{\nu}m_{\nu}]_{\mu e}$, which appears in the first term of eqn
(II.4) is determined by neutrino oscillation parameters666Here and in the rest
of this paper, we assume $\delta=3\pi/2$, a value consistent with the hints
for CP violation in the lepton sector from the T2K experiment T2K:2023smv .

$[m^{\dagger}_{\nu}m_{\nu}]_{\mu e}\sim
i\sin\theta_{13}\Delta_{atm}^{2}~{}~{},$

so any dependence of $C^{e\mu ee}_{V,LR}$ on the unknown neutrino mass scale
or Majorana phases can only arise from the second term. This same penguin
diagram also generates a loop correction to $C_{VLL}^{e\mu ee}$ of eqn (II.3),
and contributes to $\mu A\to\\!eA$ on the proton

$\displaystyle\Delta C^{e\mu ee}_{V,LL}=C^{e\mu
ee}_{V,LR}~{}~{}~{},~{}~{}~{}C^{e\mu pp}_{V,L}=-2C^{e\mu ee}_{V,LR}$ (II.5)

where $C^{e\mu pp}_{Alight,L}=\frac{1}{2}C^{e\mu pp}_{V,L}+...$. The
coefficient on neutrons, $C^{e\mu nn}_{V,L}$, vanishes at the order we
calculate.

Finally, the dipole coefficients are induced by one loop matching(see figure
1(b)), and shrink marginally in running down to the experimental scale, while
being regenerated at two loop 777The two-loop diagrams Ciuchini:1993fk ;
Degrassi:1998es ; Czarnecki:2002nt ; Crivellin:2017rmk  are included here
because they are “leading order” in the RGEs, and because they are numerically
significant – for instance, in the electroweak contribution to $(g-2)_{\mu}$,
the log-enhanced 2-loop contribution is $\sim$ 1/4 of the 1-loop matching
part. as illustrated in figure 1(c):

$\displaystyle C^{e\mu}_{D,R}$ $\displaystyle\simeq$
$\displaystyle\frac{3e}{128\pi^{2}}{\Big{[}}\frac{[m_{\nu}m_{\nu}^{\dagger}]_{e\mu}}{\lambda^{2}_{H}v^{2}}{\Big{(}}1+\frac{32}{27}\frac{\alpha_{e}}{4\pi}\ln\frac{M_{\Delta}}{m_{\tau}}{\Big{)}}+\frac{116\alpha_{e}}{27\pi}\ln\frac{m_{\tau}}{m_{\mu}}\sum_{\alpha\in
e\mu}\frac{[m_{\nu}]_{\mu\alpha}[m^{*}_{\nu}]_{e\alpha}}{|\lambda_{H}|^{2}v^{2}}{\Big{]}}$
(II.6)

where the first (leading) term is independent of the neutrino mass scale and
Majorana phases. The second term of eqn (II.6), which is of ${\cal
O}(\alpha_{e})$ with respect to the first, depends on the neutrino mass scale
and Majorana phases due to removing the $\tau$ from the loop below $m_{\tau}$,
as for the penguin diagram.

The other 8 coefficients in the Lagrangian of eqn (I.1) will be discussed
further in pl2 . The coefficient on Gold, $C_{Aheavy,L}$, should be predicted
in the type II seesaw, where $\mu A\to\\!eA$ rates are related to the $n/p$
ratio. The remaining coefficients are suppressed: for instance the dipole
$C^{e\mu}_{D,L}$ should be $\approx\frac{m_{e}}{m_{\mu}}C^{e\mu}_{D,R}$, as
expected in neutrino mass models where the new particles only interact with
lepton doublets (the chirality-flip is via SM Yukawa interactions). Similarly,
the operators with flavour-change involving singlet leptons
($C_{S,RR},C_{S,LL},C_{V,RL},C_{V,RR},C_{Alight,R},C_{Aheavy,R}$) are not
discussed here, because they are Yukawa-suppressed. So we already see that the
type II seesaw predicts that more than half the coefficients of eqn (I.1) are
negligible; however, many of these predictions are generic to models where the
New Particles interact only with lepton doublets.

The type II seesaw is expected to predict additional relations between the
Wilson coefficients of eqn (I.1), because the flavour structure of LFV is
controlled by the neutrino mass matrix. This should allow to predict ratios of
coefficients, despite that the overall magnitude of LFV is unknown. We focus
on the remaining three coefficients, given in eqns (II.3,II.4,II.6). These
formulae suggest the model prefers a hierarchy $10^{-3}:10^{-2}:1$ between the
dipole, penguin-induced and tree-level coefficients; however, we aim to
identify regions of coefficient space that the model cannot predict, not what
it prefers.

We observe that the tree-level four-lepton coefficient $C_{V,LL}^{e\mu ee}$
(see eqn II.3) can vanish, either for $[m_{\nu}]_{ee}\to 0$ in NH for
$m_{min}\sim\Delta_{sol}$ (as is familiar from neutrinoless double
$\beta$-decay), or for $[m_{\nu}]_{e\mu}\to 0$, which can occur for any
$m_{min}\mathrel{\vbox{\hbox{$>$}\nointerlineskip\hbox{$\sim$}}}\Delta_{sol}$
in NH and IH by suitable choice of both Majorana phases. If $C_{VLL}^{e\mu
ee}$ vanishes with $[m_{\nu}]_{\mu e}$, the dipole to penguin ratio is
predicted:

$\displaystyle\frac{C^{e\mu}_{D,R}}{C^{e\mu
ee}_{V,LR}}\approx\frac{3e}{32\pi\alpha_{e}\log\frac{M_{\Delta}}{m_{\tau}}}\sim\frac{2}{\pi^{2}}\,\overset{+100\%}{\underset{-30\%}{~{}}}~{}~{}~{}.$
(II.7)

because in this case the Majorana phase and neutrino mass scale dependent
terms of the penguin and dipole are proportional to $|[m_{\nu}]_{\mu e}|$ (see
the second terms of eqns II.4 and II.6). This prediction is also approximately
obtained when $C_{VLL}^{e\mu ee}$ vanishes with $[m_{\nu}]_{ee}$, because the
second term of the penguin coefficient (which depends on Majorana phases and
the mass scale, see eqn II.4) is
$\mathrel{\vbox{\hbox{$<$}\nointerlineskip\hbox{$\sim$}}}1/2$ of the first
term, whereas the dipole is numerically unaffected.

It is also the case that the “penguin-induced” coefficient of eqn (II.4), as
well as the dipole coefficient eqn (II.6), can separately vanish for specific
choices of both Majorana phases and the neutrino mass scale in the appropriate
range (However, a high neutrino mass scale
$m_{min}\mathrel{\vbox{\hbox{$>$}\nointerlineskip\hbox{$\sim$}}}.2$ eV is
required for the dipole coefficient to vanish.). So in all limits where one of
the three coefficients $C_{V,LL}^{e\mu ee}$, $C^{e\mu ee}_{V,LR}$ or
$C^{e\mu}_{D,R}$ vanishes, the ratio of the non-vanishing coefficients is
constrained888The values of the neutrino parameters that lead to cancellations
in the coefficients are sensitive to the triplet mass. Therefore, the
predictions/expectations for the non-vanishing coefficients ratio may
significantly depend on the assumption $M_{\Delta}\sim 1$ TeV.. However, the
large coefficient ratios that arise when the dipole or penguin vanishes may be
beyond the sensitivity of upcoming experiments.

In order to graphically represent the area of coefficient space that the type
II seesaw model cannot reproduce, we plot the magnitudes $|C^{e\mu}_{D,R}|$,
$|C^{e\mu ee}_{V,LR}|$ and $|C^{e\mu ee}_{V,LL}|$ in spherical coordinates,
with on the $\hat{z}$ axis $|C^{e\mu ee}_{V,LL}|\propto\cos\theta$. The
current bounds and the reach of upcoming experiments are given in table 2,
which imply that upcoming experiments could probe

$\displaystyle\tan\theta\equiv\frac{\sqrt{|C^{e\mu}_{D,R}|^{2}+|C^{e\mu
ee}_{V,LR}|^{2}}}{|C^{e\mu ee}_{V,LL}|}:10^{-3}\to
10~{}~{}~{},~{}~{}~{}\tan\phi\equiv\frac{|C^{e\mu}_{D,R}|}{|C^{e\mu
ee}_{V,LR}|}:10^{-2}\to 10$ (II.8)

Figure 2 illustrates (as empty) the regions of the tree/penguin/dipole
coefficient space that are inaccessible to the type II seesaw model. The
vertical bar represents the correlation between the dipole and penguin when
the tree contribution shrinks, given in eqn (II.7). For large tree
contribution, the penguin contribution can shrink when the second term of eqn
(II.4), $\propto|[m_{\nu}]_{\mu e}|$, cancels the first. This happens for
values of the unconstrained neutrino parameters (the lightest neutrino mass
and the Majorana phases) that enhance the tree-level coefficient $|C^{e\mu
ee}_{V,LL}|$, so that
$\tan\theta\mathrel{\vbox{\hbox{$<$}\nointerlineskip\hbox{$\sim$}}}10^{-3}$ –
this gives the upper bound to the red region. Finally, for generic values of
the Majorana phases, the tree coefficient is large with respect to the
penguin-induced coefficients and the dipole, which corresponds to the blue
region at $\tan\theta\to 0$ and
$\tan\phi\mathrel{\vbox{\hbox{$<$}\nointerlineskip\hbox{$\sim$}}}2/\pi^{2}$.
In this manuscript, we leave the neutrino mass scale free, so can obtain
$\tan\phi\to 10^{-2}$ by increasing $m_{min}$ to
$\mathrel{\vbox{\hbox{$>$}\nointerlineskip\hbox{$\sim$}}}0.2$ eV; we will
study the impact of complementary observables – such as the cosmological bound
on the neutrino mass scale – in Ref. pl2 .

Figure 2:  The white regions indicate ratios of operator coefficients that the
type II seesaw cannot predict, as discussed after eqn (II.8), where
$\tan\theta$ and $\tan\phi$ are defined. Upcoming experiments are sensitive to
the plotted ranges of the ratios. These estimates are independent of the
neutrino mass hierarchy and mass scale; the star and circle are respectively
in the regions predicted in NH and IH for $m_{min}=0$.

##  III Inverse Type I seesaw

In this section, we consider the inverse type I seesaw model Wyler:1982dd ;
Mohapatra:1986aw ; Mohapatra:1986bd , which generates neutrino masses via the
exchange of heavy gauge-singlet fermions. Like the type II seesaw, the model
can generate LFV without Lepton Number Violation, so LFV rates are not
suppressed by small neutrino masses. However, unlike the type II case, the
flavour-changing couplings are disconnected from the neutrino mass matrix, and
several heavy new particles are added, with potentially different masses.

We add to the SM $n$ pairs of gauge singlet fermions $N,S$ of opposite
chirality, with the interactions

$\delta\mathcal{L}_{NS}=i\overline{N}\not{\partial}N+i\overline{S}\not{\partial}S-\left(Y^{\alpha
a}_{\nu}(\overline{\ell}_{\alpha}\tilde{H}N_{a})+M_{ab}\overline{S}_{a}N_{b}+\frac{1}{2}\mu_{ab}\overline{S_{a}}S^{c}_{b}+{\rm
h.c}\right),$ (III.1)

where $Y_{\nu}$ is a complex $3\times n$ dimensionless matrix and $M,\mu$ are
$n\times n$ mass matrices. If Lepton Number is attributed to $\ell,N$ and $S$,
then only $\mu$ is Lepton Number Violating. Upon the spontaneous breaking of
the electroweak symmetry, the neutrino mass Lagrangian reads (suppressing
flavour indices)

$\mathcal{M}_{\nu
N}=\overline{\begin{pmatrix}\nu_{L}&N^{c}&S\end{pmatrix}}\begin{pmatrix}0&m_{D}&0\\\
m_{D}^{T}&0&M^{T}\\\ 0&M&\mu\end{pmatrix}\begin{pmatrix}\nu^{c}_{L}\\\ N\\\
S^{c}\end{pmatrix}+{\rm h.c.}$ (III.2)

which, in the seesaw limit ($Y_{\nu}v=m_{D}\ll M$), give the following active
neutrino masses

$m_{\nu}=m_{D}(M^{-1})\mu(M^{T})^{-1}m^{T}_{D}.$ (III.3)

while for $M\gg\mu$ the $N,S$ pairs have pseudo-Dirac masses determined by the
eigenvalues of $M$. Neutrino masses and oscillation parameters can be obtained
by adjusting the lepton number breaking matrix $\mu$ for arbitrary choices of
the Yukawas $Y_{\nu}$ and sterile masses $M$. This contrasts with the
“vanilla” type I seesaw expectation of GUT scale sterile neutrinos or
suppressed Yukawas Minkowski:1977sc ; Gell-Mann:1979vob ; Yanagida:1979as ;
Glashow:1979nm ; Mohapatra:1979ia , which give negligible contributions to LFV
observables.

In the following, we consider $M\sim$ TeV and allow $Y_{\nu}$ to vary in the
parameter space allowed by current LFV searches and other experimental
constraints. Low-scale type I seesaw models can be directly probed via the
production of the heavy neutral leptons at colliders delAguila:2007qnc ;
Atre:2009rg ; Deppisch:2015qwa ; Banerjee:2015gca ; Antusch:2016vyf ;
Das:2018usr ; Mekala:2022cmm , or indirectly through the active-sterile
neutrino mixing (or the associated non-unitarity of the effective $3\times 3$
lepton mixing matrix), which affect electroweak precision observables,
universality ratios and lepton flavor violating processes Antusch:2014woa ;
Fernandez-Martinez:2016lgt ; Chrzaszcz:2019inj ; Blennow:2023mqx .

###  III.1 $\mu\to e$ LFV

$\mu_{L}$$H$$H$$f$$f$$e_{L}$$N$$Z$$\mu_{L}$$f$$f$$e_{L}$$N$$H^{-}$$\gamma$

(a) $\mathcal{O}(Y_{\nu}Y^{\dagger}_{\nu})$ $Z,\gamma$ penguin diagram
contributing to $\mu\to e$ four-vector operators.

$\mu_{L}$$H$$H$$f$$f$$e_{L}$$N$$N$$Z$ (b) Penguin diagram proportional to four
neutrino Yukawas contributing to the $\mu\to e$ four-fermion operators

$\mu_{L}$$H$$e_{L}$$e_{L}$$H$$e_{L}$$N$$N$ (c) Box diagrams matching onto the
$\mu\to e_{L}\overline{e_{L}}e_{L}$ vector

$\mu_{R}$$e_{L}$$H$$\gamma$$N$$H$ (d) Matching onto the $\mu\to e$ dipole
operator

Figure 3: Matching contributions to $\mu\to e$ operators in the inverse
seesaw. The diagrams illustrate the relevant interactions that are generated,
but other diagrams may also contribute to the same operators

Large lepton flavour violating transitions are among the distinctive features
of the inverse seesaw Tommasini:1995ii ; Ilakovac:1994kj ; Ibarra:2011xn ;
Dinh:2012bp ; Alonso:2012ji ; Abada:2015oba ; Coy:2018bxr ; Abada:2021zcm ;
Zhang:2021jdf ; Coy:2021hyr ; Granelli:2022eru ; Crivellin:2022cve . In this
manuscript, we focus on the contact interactions that are relevant for $\mu\to
e$ observables and aim at determining the region of the EFT coefficient space
that the model cannot reach.

The LFV transitions we are interested in occurs in this model via loops, as we
illustrate in Figure 3. The four-fermion operator coefficients are obtained in
matching out the heavy singlets in penguin and box diagrams. The vector four-
fermion coefficients $C^{e\mu ff}_{V,LX}$, receive contributions from penguin
diagrams shown in Figures 3(a) and 3(b), which are respectively ${\cal
O}(Y_{\nu}Y_{\nu}^{\dagger})$ and ${\cal
O}(Y_{\nu}Y_{\nu}^{\dagger}Y_{\nu}Y_{\nu}^{\dagger})$. We include the second
diagram following Crivellin:2022cve , who observed that the contributions
$\propto Y^{4}_{\nu}$ could be relevant for $Y_{\nu}$ $\mathcal{O}(1)$. The
box diagrams of Fig. 3(c) also match onto vector four-lepton operators, while
the diagrams of Fig. (3(d)) match onto the $\mu\to e$ dipole. Similarly to the
type II seesaw model of section II, the new states couple to the left-handed
doublets, so the operators featuring LFV currents with electron singlets are
suppressed by the electron Yukawa. As a result, the model matches onto
four999We are not considering here the operator combination that is probed by
$\mu A\to eA$ on heavy targets because the upcoming experiments will use
Aluminum. The combination on heavy targets can be predicted by measuring the
four operator coefficients.  of the operators in Eq. (I.1):

$\displaystyle C^{e\mu ee}_{V,LR}$ $\displaystyle\simeq$ $\displaystyle
v^{2}\frac{\alpha_{e}}{4\pi}\left(1.5[Y_{\nu}M_{a}^{-2}\left(\frac{11}{6}+\log\left(\frac{m^{2}_{W}}{M_{a}^{2}}\right)\right)Y_{\nu}^{\dagger}]_{e\mu}-2.7[Y_{\nu}(Y^{\dagger}_{\nu}Y_{\nu})_{ab}\frac{1}{M^{2}_{a}-M^{2}_{b}}\log\left(\frac{M^{2}_{a}}{M^{2}_{b}}\right)Y_{\nu}^{\dagger}]_{e\mu}+\mathcal{O}\left(\frac{\alpha_{e}}{4\pi}\right)\right)$
$\displaystyle C^{e\mu}_{Alight,L}$ $\displaystyle\simeq$ $\displaystyle
v^{2}\frac{\alpha_{e}}{4\pi}\left(-0.6[Y_{\nu}M_{a}^{-2}\left(\frac{11}{6}+\log\left(\frac{m^{2}_{W}}{M_{a}^{2}}\right)\right)Y_{\nu}^{\dagger}]_{e\mu}+1.1[Y_{\nu}(Y^{\dagger}_{\nu}Y_{\nu})_{ab}\frac{1}{M^{2}_{a}-M^{2}_{b}}\log\left(\frac{M^{2}_{a}}{M^{2}_{b}}\right)Y_{\nu}^{\dagger}]_{e\mu}+\mathcal{O}\left(\frac{\alpha_{e}}{4\pi}\right)\right)$
$\displaystyle C^{e\mu ee}_{V,LL}$ $\displaystyle\simeq$ $\displaystyle
v^{2}\frac{\alpha_{e}}{4\pi}\bigg{(}-1.8[Y_{\nu}M_{a}^{-2}\left(\frac{11}{6}+\log\left(\frac{m^{2}_{W}}{M_{a}^{2}}\right)\right)Y_{\nu}^{\dagger}]_{e\mu}+2.7[Y_{\nu}(Y^{\dagger}_{\nu}Y_{\nu})_{ab}\frac{1}{M^{2}_{a}-M^{2}_{b}}\log\left(\frac{M^{2}_{a}}{M^{2}_{b}}\right)Y_{\nu}^{\dagger}]_{e\mu}+$
$\displaystyle+2.5Y^{ea}_{\nu}Y^{*\mu
a}_{\nu}Y^{eb}_{\nu}Y^{*eb}_{\nu}\frac{1}{M^{2}_{a}-M^{2}_{b}}\log\left(\frac{M^{2}_{a}}{M^{2}_{b}}\right)+\mathcal{O}\left(\frac{\alpha_{e}}{4\pi}\right)\bigg{)}$
$\displaystyle C^{e\mu}_{D,R}$ $\displaystyle\simeq$
$\displaystyle-\frac{v^{2}}{2}\left(\frac{\alpha_{e}}{4\pi
e}\right)[Y_{\nu}M^{-2}Y_{\nu}^{\dagger}]_{e\mu},$ (III.4)

where $a,b$ are summed over the number $n$ of sterile neutrinos. We include
the finite part of the penguin diagrams shown in Fig.3(a) because the ratios
of sterile masses and the electroweak scale involved in the logarithms are not
large (as we discuss in the introduction). Higher-order terms in the
$\alpha_{e}/(4\pi)$ expansion are neglected because they are small and would
require including some two-loop diagrams for a consistent treatment.
Consequently, the results presented in Eq.(III.4) are reliable at the
$\lesssim\text{few}\times\mathcal{O}(1\%)$ level.

The four operator coefficients generated by the model are linear combinations
of four contractions of the neutrino Yukawas and sterile masses. As the number
of relevant $\mu\to e$ observables equals the number of invariant matrix
elements upon which the observables depend, it seems that the model can
predict any observation – i.e points in the 4-dimensional space of the
operator coefficients– with suitable choices of the $Y_{\nu}$ and $M$
matrices. The number of invariants could be reduced if the sterile neutrinos
are nearly degenerate 101010A motivation for considering this limit comes from
the baryon asymmetry of the Universe, which can be generated from resonant
leptogenesis with highly degenerate TeV-scale sterile neutrinos (see e.g.
Refs. Pilaftsis:2003gt ; Blanchet:2009kk ; BhupalDev:2014pfm ; daSilva:2022mrx
), or from the CP-violating oscillations of nearly degenerate sterile
neutrinos (a mechanism known as ARS leptogenesis Akhmedov:1998qx ) with masses
in the GeV Asaka:2005pn  to multi-TeV Klaric:2021cpi  range.. In this limit,
the combination entering the $\mathcal{O}(Y_{\nu}Y^{\dagger}_{\nu})$ penguin
contributions aligns with the matrix elements parameterizing the dipole
coefficient. Indeed, by expanding $M^{2}_{a}/M^{2}=1+x_{a}$ for small $x_{a}$,
we have that

$\frac{1}{M_{a}^{2}}\left(\frac{11}{6}+\log\left(\frac{m_{W}^{2}}{M^{2}_{a}}\right)\right)=\frac{1}{M^{2}(1+x_{a})}\left(\frac{11}{6}+\log\left(\frac{m_{W}^{2}}{M^{2}}\right)-\log(1+x_{a})\right)=\frac{1}{M^{2}}\left(\frac{11}{6}+\log\left(\frac{m_{W}^{2}}{M^{2}}\right)+\mathcal{O}(x_{a})\right).$
(III.5)

If the mass-splitting between the heavy singlets is $\lesssim v^{2}$, the
error introduced by the degenerate approximation is a dimension eight
$v^{2}/M^{2}$ suppressed contribution, that, for TeV scale sterile masses,
would be approximately of the same order of the neglected
$\mathcal{O}(\alpha_{e}/4\pi)$ corrections. Similarly, the leading order term
in the $x_{a}$ expansion of the mass function that enters in the
$\mathcal{O}(Y_{\nu}Y_{\nu}^{\dagger}Y_{\nu}Y_{\nu}^{\dagger})$ penguin and in
the boxes is

$\frac{1}{M^{2}_{a}-M^{2}_{b}}\log\left(\frac{M^{2}_{a}}{M^{2}_{b}}\right)=\frac{1}{M^{2}}\left(1+\mathcal{O}(x_{a},x_{b})\right),$
(III.6)

so that in the nearly degenerate limit, we find111111We remind that the
operator coefficients depend logarithmically on the scale of the new states,
which we take to be $\sim$ TeV.

$\displaystyle C^{e\mu}_{D,R}(m_{\mu})$
$\displaystyle\simeq-10^{-3}\frac{v^{2}}{M^{2}}(Y_{\nu}Y^{\dagger}_{\nu})_{e\mu}$
$\displaystyle C^{e\mu}_{Alight,L}(m_{\mu})$
$\displaystyle\simeq\frac{v^{2}}{M^{2}}\left(1\times
10^{-3}(Y_{\nu}Y^{\dagger}_{\nu})_{e\mu}+6.6\times
10^{-4}(Y_{\nu}Y^{\dagger}_{\nu}Y_{\nu}Y^{\dagger}_{\nu})_{e\mu}\right)$
$\displaystyle C^{e\mu ee}_{V,LR}(m_{\mu})$
$\displaystyle\simeq\frac{v^{2}}{M^{2}}\left(-2.8\times
10^{-3}(Y_{\nu}Y^{\dagger}_{\nu})_{e\mu}-1.6\times
10^{-3}(Y_{\nu}Y^{\dagger}_{\nu}Y_{\nu}Y^{\dagger}_{\nu})_{e\mu}\right)$
$\displaystyle C^{e\mu ee}_{V,LL}(m_{\mu})$
$\displaystyle\simeq\frac{v^{2}}{M^{2}}\left(3.3\times
10^{-3}(Y_{\nu}Y^{\dagger}_{\nu})_{e\mu}(1+0.56(Y_{\nu}Y^{\dagger}_{\nu})_{ee})+1.55\times
10^{-3}(Y_{\nu}Y^{\dagger}_{\nu}Y_{\nu}Y^{\dagger}_{\nu})_{e\mu}\right)$
(III.7)

Despite the large number of free parameters in the inverse seesaw model, even
in the degenerate limit, the coefficients of the $\mu\to e$ operators can now
be determined by just two invariant contractions of the neutrino Yukawa
matrix. Being linear combinations of two invariants, the correlations of the
operator coefficients that the model can predict are restricted: by measuring
two (complex) coefficients, it would be possible to predict the others.
Focusing on the first three operators of Eq. (III.7), we find that

(a) The light-gray sphere illustrate the experimentally allowed ellipse in the
$C^{e\mu}_{D,R}(m_{\mu}),C^{e\mu ee}_{Alight,L}(m_{\mu}),C^{e\mu
ee}_{V,LR}(m_{\mu})$ space. We consider the real parts of the coefficients and
normalise to the current upper bound. If the sterile neutrinos are nearly
degenerate, the model can cover the region defined in Eq. (III.8), which
correspond to the blue plane.

(b) The light-gray sphere illustrate the experimentally allowed ellipse in the
$C^{e\mu}_{D,R}(m_{\mu}),C^{e\mu ee}_{Alight,L}(m_{\mu}),C^{e\mu
ee}_{V,LL}(m_{\mu})$ space. We consider the real parts of the coefficients and
normalise to the current upper bound. If the sterile neutrinos are nearly
degenerate, the model can cover the region defined in Eq. (III.9) and
correspond to the volume delimited by the two blue planes (see Fig. 4(c))

(c) Zoom of Figure 4(b). The model can cover only the region enclosed by the
two planes.

Figure 4: Parameter space covered by the inverse seesaw (with degenerate
sterile neutrinos) in the low-energy operator coefficient space

$C^{e\mu
ee}_{V,LR}(m_{\mu})=-2.4C^{e\mu}_{Alight,L}(m_{\mu})+0.02C^{e\mu}_{D,R}(m_{\mu})$
(III.8)

In the purely left-handed $\mu\to 3e$ vector the magnitude of the coefficient
multiplying the matrix element $(Y_{\nu}Y^{\dagger}_{\nu})_{e\mu}$ is
dependent on the real and positive parameter $(Y_{\nu}Y^{\dagger}_{\nu})_{ee}$
arising from the box diagram contribution. However, since the Yukawas are
assumed to be perturbative $(Y_{\nu}Y^{\dagger}_{\nu})_{ee}\lesssim 1$, we can
similarly find that

$C^{e\mu
ee}_{V,LL}(m_{\mu})=2.4C^{e\mu}_{Alight,L}(m_{\mu})+c_{d}C^{e\mu}_{D,R}(m_{\mu})$
(III.9)

where $-1.99\lesssim c_{d}\lesssim-0.57$. The correlations described by Eq.s
(III.8) and (III.9) hold, within the accuracy of our calculations, for general
complex coefficients. To visually represent the parameter space accessible to
the inverse seesaw model, we consider the real parts of the coefficients and
plot the corresponding planes in the 3D space of low-energy coefficients. By
normalizing each coefficient to the upper limit imposed by current
experimental searches, the allowed region of parameter space correspond to the
interior of a sphere. The inverse seesaw model (with nearly degenerate sterile
neutrinos) can sit in the intersection of this region with the planes defined
by Eq. (III.8) and Eq. (III.9), as illustrated in Figure 4. Since the dipole
coefficient in Eq. (III.9) is unknown but bounded, the model can cover the
volume enclosed by the two extreme planes

##  IV Leptoquark

This section studies the $\mu\\!\to\\!e$ predictions of an SU(2) singlet
leptoquark of hypercharge $Y=1/3$ that could fit the the RD anomaly
BaBar:2012obs ; Belle:2015qfa ; Belle:2019rba ; LHCb:2023zxo ; LHCb:2023cjr ;
Sakaki:2013bfa ; Cai:2017wry ; Angelescu:2018tyl ; Lee:2021jdr , which is an
excess of $b\to c\bar{\tau}\nu$ events. Requiring the leptoquark to fit
$R_{D}$ fixes the mass to be O(TeV) and restricts the quantum numbers, but our
$\mu\\!\to\\!e$ interactions are independent of the couplings that contribute
to $R_{D}$. Unlike the models of the previous sections, the leptoquark couples
to both lepton doublets and singlets, and can mediate $\mu A\to\\!eA$ at tree
level – but does not generate neutrino masses.

The SU(2)-singlet leptoquark is denoted $S_{1}$ Buchmuller:1986zs  (not to be
confused with the singlet fermions $\\{S_{a}\\}$ of the previous section),
with interactions:

$\displaystyle{\cal L}_{S}$ $\displaystyle=$
$\displaystyle(D_{\rho}S_{1})^{\dagger}D^{\rho}S_{1}-m_{LQ}^{2}S_{1}^{\dagger}S_{1}+(-\lambda^{\alpha
j}_{L}\overline{\ell}_{\alpha}i\tau_{2}q_{j}^{c}+\lambda^{\alpha
j}_{R}\overline{e}_{\alpha}u_{j}^{c})S_{1}+(\lambda^{\alpha
j*}_{L}\overline{q^{c}}_{j}i\tau_{2}\ell_{\alpha}+\lambda^{\alpha
j*}_{R}\overline{u^{c}}_{j}e_{\alpha})S_{1}^{\dagger}$
$\displaystyle+{\rm~{}Higgs~{}interactions}$

where the leptoquark mass is $m_{LQ}\simeq$ TeV, the generation indices are
$\alpha\in\\{e,\mu,\tau\\}$ and $j\in\\{u,c,t\\}$, and the sign of the doublet
contraction is taken to give $+\lambda^{\alpha
j}_{L}\overline{e_{L}}(u_{L})^{c}S_{1}$. Like in the type II model of section
II, the leptoquark-Higgs interactions are neglected because their
contributions to LFV observables are negligible assuming perturbative
couplings.

Leptoquarks are strongly interacting, so can be readily produced at hadron
colliders; the current LHC searches impose
$m_{LQ}\mathrel{\vbox{\hbox{$>$}\nointerlineskip\hbox{$\sim$}}}$ 1-2 TeV
ParticleDataGroup:2022pth . Also, their peculiar Yukawa interactions
connecting quarks to leptons, can predict diverse quark and/or lepton flavour-
changing processes. For instance, non-zero $\lambda^{\mu u}_{X}$,
$\lambda^{\mu c}_{X},\lambda^{eu}_{X}$ and $\lambda^{ec}_{X}$ induce $\mu\to
e$ processes on a $u$ and $c$ quark currents – which we study here – and also
induce LFV $D$ decays with $e^{\pm}\mu^{\mp}$ in the final state. In addition,
$S_{1}$ will mediate $\Delta F=2$ four-quark operators via box diagrams which
can contribute to meson-anti-meson mixing UTfit:2007eik . We did not find
relevant constraints on the LFV interactions of $S_{1}$ from quark flavour
physics, but will discuss in more detail the complementarity of quark and
lepton observables in pl2 .

In matching the leptoquark onto the QCD$\times$QED-invariant EFT at $m_{LQ}$,
vector ($\propto\lambda^{*}_{R}\lambda_{R}$, $\lambda^{*}_{L}\lambda_{L}$),
and scalar/tensor ($\propto\lambda^{*}_{R}\lambda_{L}$,
$\lambda^{*}_{L}\lambda_{R}$) operators are generated at tree-level. We only
consider the subset which are quark flavour-diagonal and $\mu\\!\to\\!e$
flavour-changing. The model matches onto vector four-fermion operators of the
form $(\bar{e}\gamma^{\rho}P_{X}\mu)(\bar{f}\gamma_{\rho}P_{Y}f)$ (where
$X,Y\in\\{L,R\\}$ and $f$ any lepton or quark) via “penguin” diagrams (see
figure 5(a)), and also can generate vector four lepton operators via box
diagrams as in figure 5(b). Finally, the dipole operators can be generated via
the last diagram of figure 5. This collection of operators at the leptoquark
mass scale is schematically represented in figure 6 as the top row of boxes
and ovals.

$\mu_{L}$$H$$H$$f$$f$$e_{L}$$t$$S_{1}$$Z$ (a) Penguin diagram contributing to
the $\mu\to e$ four-fermion operators.

$\mu_{L}$$S_{1}$$e_{L}$$e_{L}$$S_{1}$$e_{L}$$u_{j}$$u_{j}$

(b) Example box diagram matching onto the $\mu\to e_{L}\overline{e_{L}}e_{L}$
vector

.

$\mu_{R}$$e_{L}$$H$$\gamma$$u_{j}$$S_{1}$ (c) Illustrative contribution
matching to the dipole operator.

Figure 5: Representative diagrams for the matching of the leptoquark onto
four-fermion operators, and the dipole.

Several of the operators generated in matching out the leptoquark are present
in the Lagrangian of eqn (I.1). For instance, $S_{1}$ matches onto vector
and/or scalar $\bar{e}$-$\mu$-$\bar{u}$-$u$ operators, which give large
contributions to $\mu A\to\\!eA$. In addition, the log-enhanced loops change
the predictions significantly: the coefficients of scalar and tensor quark
operators respectively grow and shrink due to QCD, and QED loops can cause
some ${\cal O}(1)$ mixing, such as the top and charm tensors into the dipole,
or the $u$-tensor into the $u$-scalar. The effect of the RGEs is represented
by lines in figure 6.

Figure 6: A schematic representation of how the leptoquark generates $\mu\to
e\gamma$, $\mu\to e\bar{e}e$ and $\mu A\to\\!eA$. The top row represents
classes of coefficients, generated in matching out the leptoquark, of given
Lorentz structure and particle type, for any flavours and chiralities. The
boxes correspond to operators with coefficients of ${\cal
O}(\lambda^{2}/m_{LQ}^{2})$, whereas the ovals have suppressed coefficients
$\sim{\cal O}(\lambda^{2}/[16\pi^{2}m_{LQ}^{2}])$, or $\sim{\cal
O}(\lambda^{4}/[16\pi^{2}m_{LQ}^{2}])$. The bottom row of boxes are the six
observable coefficients (for fixed $e$ chirality in the $\mu\\!\to\\!e$
bilinear) of eqn (I.1). Lines represent the transformation between the
$m_{LQ}$ and experimental scale; a straight line means the observable
coefficient can be directly obtained in matching. Operator mixing is
represented as wavy lines: a thick line indicates an ${\cal O}(1)$
contribution of at least one operator from the class to the observable; a thin
line indicates a more suppressed ${\cal O}(\alpha)$ contribution.

At the experimental scale, the contribution of the $S_{1}$ leptoquark to the
dipole coefficients is

$\displaystyle\frac{m_{LQ}^{2}}{v^{2}}C^{e\mu}_{D,R}(m_{\mu})$
$\displaystyle\simeq$
$\displaystyle\frac{e[\lambda_{L}\lambda^{\dagger}_{L}]_{e\mu}}{128\pi^{2}}\left(1-16\frac{\alpha_{e}}{4\pi}\ln\frac{m_{LQ}}{m_{\mu}}\right)+\frac{2\alpha^{2}_{e}}{9\pi^{2}e}[\lambda_{L}\ln\frac{m_{LQ}}{m_{Q}}\lambda^{\dagger}_{L}]_{e\mu}-\frac{\alpha_{e}}{2\pi
ey_{\mu}}\left([\lambda_{L}[Y_{u}\tilde{f}^{Q}\ln\frac{m_{LQ}}{m_{Q}}]\lambda^{\dagger}_{R}]_{e\mu}\right)~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
(IV.1)

where the first term is the matching contribution (times its QED running), the
second term is the 2-loop mixing of tree vector operators into the dipole, the
last term is the RG-mixing of tensor operators to dipoles, and the $m_{Q}$
serving as lower cutoff for the logarithms (here and further in the
manuscript) is max$\\{m_{Q},2~{}{\rm GeV}\\}$ 121212We neglect the estimates
of Dekens:2018pbu .. For $C^{e\mu}_{D,L}$, one interchanges $R\leftrightarrow
L$. The QCD running of the quark tensor operator is intricated with the QED
mixing to the dipole Bellucci:1981bs ; Buchalla:1989we , which induces a
quark-flavour-dependent rescaling $\tilde{f}^{Q}\simeq\\{1,1.4\\}$ for
$\\{t,c\\}$ quarks.

The leptoquark also generates vector four-lepton operators (for
$X\in\\{L,R\\}$)

$\displaystyle\frac{m_{LQ}^{2}}{v^{2}}C_{V,LX}^{e\mu}(m_{\mu})$
$\displaystyle\simeq$
$\displaystyle-\frac{N_{c}}{64\pi^{2}}[\lambda_{L}\lambda_{L}^{\dagger}]_{e\mu}[\lambda_{X}\lambda_{X}^{\dagger}]_{ee}\left(1\mp
12\frac{\alpha_{e}}{4\pi}\ln\frac{m_{LQ}}{m_{\mu}}\right){+}\frac{\alpha_{e}}{3\pi}[\lambda_{L}\ln\frac{m_{LQ}}{m_{Q}}\lambda_{L}^{\dagger}]_{e\mu}$
$\displaystyle{-}g^{e}_{X}\frac{N_{c}}{16\pi^{2}}[\lambda_{L}Y_{u}Y_{u}^{\dagger}\lambda^{\dagger}_{L}]_{e\mu}\log\frac{m_{LQ}}{m_{t}}$

where $g^{e}_{L}=-1+2s_{W}^{2}$, $g^{e}_{R}=2s_{W}^{2}$, the first term
represents the box diagram at $m_{LQ}$ (and its QED running to $m_{\mu}$, with
$-$/$+$ for X=/$\neq$Y) which is represented as the $V,4l$ oval at the top of
figure 6 connecting to the $V_{XY}$ box at the bottom, the second term is the
log-enhanced photon penguin that mixes the tree operators ${\cal
O}_{VLL}^{QQ}$ (for $Q\in\\{u,c,t\\}$) into 4-lepton operators (represented in
figure 6 as a thin wavy line between the $V,2l2u$ and $V_{XY}$ boxes), and the
last term is the contribution of the $Z$-penguins shown in figure 5(a) (the
$V2l2f$ oval of figure 6), not including the negligible effect of the RGEs.

The scalar 4-lepton coefficient $C_{SXX}^{e\mu ee}$ can be generated via a box
diagram, with Higgs insertions on the internal quark lines (so the coefficient
can be significant for internal top quarks); however, the coupling constant
combination that appears on the flavour-changing line is already strictly
constrained by $\mu\to e\gamma$. So this coefficient has a very small
contribution to $\mu\to e$ processes, and we neglect it.

A classic signature of leptoquarks is $\mu A\to\\!eA$, which can be mediated
at tree level, via scalar or vector operators involving first generation
quarks. The branching ratio on light targets like Titanium or Aluminium, can
be written

$\displaystyle\sqrt{\frac{BR^{exp}_{Ti}}{250}}$
$\displaystyle\mathrel{\vbox{\hbox{$>$}\nointerlineskip\hbox{$\sim$}}}$
$\displaystyle{\Big{|}}0.250C_{D,R}(m_{\mu})+0.37\lambda^{eu}_{L}\lambda^{\mu
u*}_{L}.\left(1{+}\frac{{2}\alpha}{\pi}\log\right)+0.39\left(\frac{g^{2}}{64\pi^{2}}\lambda^{eu}_{L}\lambda^{\mu
u*}_{L}\log\frac{m_{LQ}}{m_{W}}\right)$
$\displaystyle-\frac{\alpha}{6\pi}\left[\lambda_{L}\log\frac{m_{LQ}}{m_{Q}}\lambda^{\dagger}_{L}\right]^{e\mu}-\frac{3}{{64}\pi^{2}}[\lambda_{L}Y_{u}Y_{u}^{\dagger}\lambda^{\dagger}_{L}]^{e\mu}\left(\log\frac{m_{LQ}}{m_{W}}+\frac{5}{6}\right)$
$\displaystyle-\eta\left(1.95\lambda^{eu}_{L}\lambda^{\mu
u*}_{R}+\frac{0.41m_{N}}{27m_{c}}\lambda^{ec}_{L}\lambda^{\mu
c*}_{R}\right)+\eta(m_{t})\frac{1.07m_{N}}{27m_{t}}\lambda^{et}_{L}\lambda^{\mu
t*}_{R}{\Big{|}}\times\frac{v^{2}}{m_{LQ}^{2}}\ ,$

where in order, the terms are: the dipole coefficient given in eqn (IV.1), the
tree vector coefficient on $u$ quarks with its QED running (represented as the
upper left box of figure 6), the electroweak box contribution to the $d$
vector, the QED then $Z$ penguin (see Figure 5(a)) contributions to the $u$
and $d$ vectors (where we took $V_{ud}=1,\sin^{2}\theta_{W}=1/4$), and the
scalar $u$, $c$ and $t$ contributions. Most of the scalar top contribution
comes from a loop-induced flavour changing Higgs coupling (in SMEFT, ${\cal
O}^{e\mu tt}_{LEQU}$ mixing into ${\cal O}^{e\mu}_{EH}$), which generates
scalar quark operators of all quark flavours.) The tensor to scalar mixing is
neglected, because the model generates tensor coefficients that are
proportional to the scalars. The QCD and QED running of the scalars is
contained in $\eta$:

$\eta=\left[\frac{\alpha_{s}(m_{LQ})}{\alpha_{s}(2{\rm
GeV})}\right]^{-12/23}\\!\\!\\!\\!\\!\\!\times(1+\frac{13\alpha}{6\pi}\ln\frac{m_{LQ}}{2{\rm
GeV}})\approx
1.79~{}~{}~{},~{}~{}~{}\eta(m_{t})=\left[\frac{\alpha_{s}(m_{LQ})}{\alpha_{s}(m_{t})}\right]^{-12/23}\\!\\!\\!\\!\\!\\!\times(1+\frac{13\alpha}{6\pi}\ln\frac{m_{LQ}}{m_{t}})\approx
1.11~{}~{}~{}.$

It is easy to see that the operator coefficients of eqn (I.1) depend on more
than 12 different combinations of the leptoquark couplings, so the only
prediction of the leptoquark model is that the four-lepton scalar coefficients
$C_{S,XX}^{e\mu ee}$ are negligible (as discussed after eqn IV); the model
could fit any observed values for the remaining 10 coefficients.

Figure 7: The $S_{1}$ leptoquark interacting only with one quark generation
predicts relations among coefficients given in Appendix B: for large positive
values of the four-fermion coefficient on light targets
$C_{Alight,L}^{e\mu}/C_{V,LL}^{e\mu ee}$ (horizontal,in log), the ratio
$C_{DR}^{e\mu}/C_{V,LL}^{e\mu ee}$ (magnitude on the vertical) is large and
positive. The four-lepton coefficients are comparable for all quark
generations.

However, the leptoquark has the interesting feature of predicting specific
patterns of $\mu\\!\to\\!e$ LFV when this occurs with only one quark
generation: in this case, all the four-lepton coefficients are of comparable
magnitude, and knowing in addition the dipole allows to predict $C_{AL,X}$ (or
vice versa). To illustrate this, we neglect the box contributions to 4-lepton
operators (discussed in more detail in pl2 ), which are subdominant for
internal $t$ quarks, but can give an ${\cal O}(1)$, same-sign contribution for
internal $u$ and $c$ quarks when $\lambda_{X}^{eQ}\sim 1$. Without the boxes,
all the coefficients are determined by four combinations of coupling
constants: $\lambda^{eQ}_{X}\lambda^{\mu Q*}_{X}$ and
$\lambda^{eQ}_{X}\lambda^{\mu Q*}_{Y}$ for $X,Y\in\\{L,R\\}$ and $Y\neq X$
(the expressions for the coefficients are given in Appendix B). This leads to
correlations among operator coefficients, as occurred in the inverse seesaw.
In figure 7, we illustrate this correlation by plotting ratios of
coefficients, rather than the 3-$d$ plots of section III (notice that the
horizontal axis is in log10 scale, but runs from negative to positive values,
so small values of $C_{Al,X}$ have been deleted at the origin).

One sees that “generically”, a leptoquark coupled to the $t$ gives a large
dipole, whereas a large $\mu A\to\\!eA$ rate is expected for leptoquarks
interacting with the up quark. However, neither of these expectations is an
unambiguous footprint of the quark flavour dominantly coupled to the
leptoquark, because $C_{Al,X}$ (resp. $C_{D,X}$) can vanish for leptoquarks
interacting with $u$ (resp. $t$) quarks. Therefore, the observation of $\mu\to
3e$ without $\mu\to e\gamma$ would not exclude an $S_{1}$ leptoquark coupling
mostly to top quarks; it would just exclude generic values of the parameters
of that model (i.e., values of the parameters that do not lead to cancellation
in some Wilson coefficients). Similarly, the observation of $\mu\to e\gamma$
but not $\mu\to e$ conversion on light nuclei would not exclude an $S_{1}$
leptoquark coupling only to up quarks.

##  V Discussion and Summary

In this manuscript, we explored whether a bottom-up EFT analysis (outlined in
section I) can give a perspective on LFV models that is complementary to top-
down studies. We emphasize that in EFT, the data for $\mu\to e\gamma$, $\mu\to
e\bar{e}e$ and $\mu A\to\\!eA$ consists of twelve Wilson coefficients (given
in Eq. I.1), and not just three branching ratios. The current experimental
null-results confine the coefficients to the interior of a 12-D ellipse
centered at the origin, and the aim of this manuscript was to determine
whether a future observation could exclude models. To address this question,
we searched for the regions of coefficient space accessible-to-future-
experiments that each model cannot reach, as an observation in that part of
the ellipse would rule the model out.

We studied three TeV-scale131313Although the EFT calculations are only
logarithmically sensitive to the choice $\Lambda_{\rm NP}\sim{\rm TeV}$, our
results may depend on this assumption, especially in the cases where
cancellations between different contributions are envisaged. models: the type
II seesaw, the inverse type I seesaw and a singlet scalar leptoquark added to
the SM. We chose the first two because they can explain neutrino masses (which
are the best motivation for LFV), while we considered the scalar leptoquark in
light of the charged current anomaly observed in $b\to cl\nu$ transitions. The
model predictions depend on combinations of NP and SM parameters which we
refer to as “invariants”, see e.g. Eqs. (III.4) and (III.7).

The type II and inverse seesaw models generate Majorana neutrino masses via
the tree-level exchange of heavy new particles, respectively a scalar triplet
and fermion singlets. Large lepton-flavour-changing rates are possible because
the models can contain LFV without lepton number violation, avoiding any
suppression by small neutrino masses. In both models the new particles
interact with lepton doublets, so the coefficients of operators with flavour-
changing currents involving singlet charged leptons are suppressed by the
lepton Yukawas141414This Yukawa also appears in $C_{DR}^{e\mu}$, but since the
operator is defined with the muon mass (see Eq. I.1), the Yukawa does not
“count” as a suppression in this case. and neglected here:

$\displaystyle C_{D,L},C^{e\mu ee}_{V,RR},C^{e\mu ee}_{V,RL},C^{e\mu
ee}_{S,RR},C^{e\mu}_{Alight,R},C^{e\mu}_{Aheavy,R},C^{e\mu ee}_{S,LL}\sim
0~{}~{}~{},~{}~{}~{}{\rm(type~{}II,~{}inverse~{}seesaw)}$

So in the twelve-dimensional space that can be probed by experiment, these
models can only occupy 5 dimensions: should one of the above coefficients be
observed (in the absence of the unsuppressed ones), then these models would be
excluded. In addition, these vanishing coefficients imply that $\mu A\to\\!eA$
only occurs via the dipole and vector interactions.

Section II shows that in the type II model, three of the remaining
coefficients, $C^{e\mu}_{Alight,L}$, $C^{e\mu}_{Aheavy,L}$ and $C^{e\mu
ee}_{V,LR}$(given in Eq II.4), arise from the same- loop diagrams and are all
proportional to the same combination of invariants. This implies that the
model occupies a line in the three-dimensional space of these three
coefficients, and that one of the three rates for $\mu\to e_{L}\gamma$,
$\mu\to e_{L}\overline{e_{R}}e_{R}$ and $\mu A\to e_{L}A$ being predicted by
the other two. The coefficients $C^{e\mu}_{D,R}$ and $C^{e\mu ee}_{V,LL}$ can
be expressed in terms of two other invariants, all of which are constructed
with the Yukawa matrix of the triplet scalar. This is proportional to the
neutrino mass matrix, so known, up to the overall magnitude, the neutrino mass
hierarchy, the lightest mass $m_{\rm min}$ and the Majorana Phases
$\alpha_{1,2}$. So although the model generates $C^{e\mu ee}_{V,LL}$ at tree
level, suggesting $\mu\to e\bar{e}e$ to discover the type II seesaw, this
coefficient can vanish (for specific values of the Majorana phases and a range
of $m_{min}$), as could $C^{e\mu}_{D,R}$ or $C^{e\mu ee}_{V,LR}$. When this
occurs, the ratio of the remaining two coefficients is restricted, so there
are combinations of observations that the type II seesaw cannot predict. This
is illustrated in Figure 2, where coefficient ratios (that correspond to
angular coordinates in the three remaining dimensions of the original ellipse)
are varied over the ranges accessible to upcoming experiments. We find that at
least one of the four-fermion coefficients is always larger than the dipole,
so that observing $\mu\to e\gamma$ with a branching ratio $Br(\mu\to
e\gamma)\gtrsim 10^{-14}$ without detecting $\mu\to 3e$ in upcoming searches
with $Br(\mu\to 3e)\gtrsim 10^{-16}$ can rule out the type II seesaw model
studied here.

Section III studies the inverse seesaw model, and shows that
$C^{e\mu}_{D,R},C^{e\mu ee}_{V,LL},C^{e\mu
ee}_{V,LR},C^{e\mu}_{Alight,L},C^{e\mu}_{Aheavy,L}$ are functions of four
invariants constructed from the neutrino Yukawa and sterile neutrino mass
matrices, as given in Eq( III.4). This implies that $Br(\mu Au\to e_{L}Au)$
could be predicted, given the rates for $\mu Al\to e_{L}Al$, $\mu\to
e_{L}\bar{e}_{L}e_{L}$, $\mu\to e_{L}\bar{e}_{R}e_{R}$ and $\mu\to
e_{L}\gamma$. The relevant contributions to these $\mu\to e$ coefficients
arise via loop diagrams in matching (no four-SM-fermion operators are
generated at tree-level), and are non-linear functions of the nondegenerate
singlet masses. The RGEs of QED just renormalise these coefficients by a few
percent, but do not generate additional invariants. So we observe that the
number of invariants is reduced, if the sterile neutrino masses can be
approximated as degenerate – as occurs for mass differences of
$\mathcal{O}(v)$, see Eq. (III.5). In this limit, the five non-negligeable
coefficients are controlled by two invariants constructed from the neutrino
Yukawas : $\mathcal{O}(Y_{\nu}Y_{\nu}^{\dagger})$ and
$\mathcal{O}(Y_{\nu}Y_{\nu}^{\dagger}Y_{\nu}Y_{\nu}^{\dagger})$. This implies
that when two coefficients are known, the remaining three are predicted,
implying, eg, that the model predicts $Br(\mu A\to\\!eA)$, from the rates for
$\mu\to e\gamma$ and $\mu\to e\bar{e}e$. In the twelve-dimensional ellipse,
our inverse seesaw model with degenerate sterile neutrinos therefore occupies
a two-dimensional subspace (see Eq.s III.8 and III.9), which we illustrate in
Figure 4 by plotting the model prediction for the real part of three
coefficients.

Finally, in section IV, we investigate the $\mu\to e$ predictions of a singlet
scalar leptoquark, selected to fit the excess of $b\to c\bar{\tau}\nu$ events
observed in the $R_{D}$ ratio. The model contributes to all but two of the
$\mu\to e$ observable coefficients with different coupling combinations,
implying that it could entirely fill 10 dimensions of the 12-D ellipse. Only
the observation of a non-zero $\mu\to 3e$ scalar coefficient $C^{e\mu
ee}_{S,XX}$ could not be explained by the leptoquark. On the other hand, the
model is more predictive when the leptoquark only interacts with one quark
generation. In this case, all the invariants become
$\propto\lambda_{X}^{eQ}\lambda_{L}^{\mu Q*}$ or
$\lambda_{X}^{eQ}\lambda_{R}^{\mu Q*}$, so once four coefficients are
measured, the remaining eight can be predicted. For a specific chirality of
the outgoing electron in the LFV current, this resembles the degenerate
inverse seesaw case, and the equations relating the coefficients are given in
Appendix B. The relations between coefficient ratios that are expected when
the leptoquark only interacts with one quark generation are illustrated in
Fig. 7.

The results of this manuscript will be extended in a subsequent publication,
where also some technical details of our EFT calculations will be discussed.
We will explore the impact of complementary observables and the uses of
invariants, and we will discuss the consequences of relative complex phases
for the operator coefficients.

In summary, we find that there are observations of $\mu\to e$ processes that
could rule out the three models we considered. The type II seesaw model
predicts coefficients in part of a 3-dimensional subspace of the 12-d
coefficient space accessible to experiments. The inverse seesaw maps onto a
4-d subspace of the 12-d space, in the case of non-dengenerate sterile
neutrinos, but is more predictive for (nearly) degenerate steriles, where it
is restricted to a 2-dimensional subspace. The singlet scalar leptoquark model
does not generate sizable scalar four-lepton operators but can give arbitrary
contributions to all other Wilson coefficients, thus completely filling 10
dimensions of the 12-d ellipse. However, if the leptoquark couplings to the
electron and muon involve a single quark generation, the model predictions are
restricted to a 4-dimensional subspace.

#### Acknowledgements

We thank Luca Silvestrini for a helpful suggestion. MA was supported by a
doctoral fellowship from the IN2P3. The work of SL is supported in part by the
European Union’s Horizon 2020 research and innovation programme under the
Marie Sklodowska-Curie grant agreement No. 860881-HIDDeN.

##  Appendix A Appendix: the $\mu A\to\\!eA$ operators

The Spin-Independent $\mu\to e$ conversion rate, normalised to the $\mu$
capture rate KKO ; Suzuki:1987jf  can be written KKO

$\displaystyle{\rm BR}_{SI}(\mu A\to eA)$ $\displaystyle=$
$\displaystyle\frac{32G_{F}^{2}m_{\mu}^{5}}{\Gamma_{cap}}{\Big{[}}\big{|}\widetilde{C}^{pp}_{V,R}I_{A,V}^{(p)}+\widetilde{C}^{pp}_{S,L}I_{A,S}^{(p)}+\widetilde{C}^{nn}_{V,R}I_{A,V}^{(n)}+\widetilde{C}^{{}^{\prime}nn}_{S,L}I_{A,S}^{(n)}+C_{D,L}{\frac{I_{A,D}}{4}}\big{|}^{2}+\\{L\leftrightarrow
R\\}~{}{\Big{]}}~{}~{}~{}.$ (A.1)

where $I_{A,V}^{(N)}$, $I_{A,S}^{(N)}$ and $I_{A,D}$ are target($A$)-dependent
“overlap integrals” inside the nucleus of the lepton wavefunctions and the
appropriate nucleon density. This shows that a target probes a linear
combination of coefficients identified by the overlap integrals. With current
theoretical uncertainties on the overlap integrals, at least two independent
combinations of the coefficients-on-nucleons $\\{\tilde{C}\\}$ could be
constrained DKY . We will take these two combinations to correspond to light
and heavy nuclei.

For light targets like Aluminium or Titanium, all the four-fermion overlap
integrals are comparable, so the four-fermion operator that is probed is
approximately

$\displaystyle\mathcal{O}_{Alight,X}\approx\frac{1}{2}{\Big{(}}(\overline{e}P_{X}\mu)(\overline{p}p)+(\overline{e}\gamma^{\alpha}P_{X}\mu)(\overline{p}\gamma_{\alpha}p)+(\overline{e}P_{X}\mu)(\overline{n}n)+(\overline{e}\gamma^{\alpha}P_{X}\mu)(\overline{n}\gamma_{\alpha}n){\Big{)}}$
(A.2)

or more precisely, the KKO calculation says that the combination of
coefficients probed by Aluminium is DKY

$\displaystyle\tilde{C}_{Alight,X}$ $\displaystyle=$ $\displaystyle
0.455\widetilde{C}^{pp}_{S,X}+0.473\widetilde{C}^{pp}_{V,Y}+0.490\widetilde{C}^{nn}_{S,X}+0.508\widetilde{C}^{nn}_{V,Y})~{}~{}~{}.$

For Gold, the coefficient combination is slightly misaligned:

$\displaystyle\tilde{C}_{Aheavy,X}$ $\displaystyle=$ $\displaystyle
0.289\widetilde{C}^{pp}_{S,X}+0.458\widetilde{C}^{pp}_{V,Y}+0.432\widetilde{C}^{nn}_{S,X}+0.686\widetilde{C}^{nn}_{V,Y}~{}~{}~{},$

indeed, the operator probed by heavy targets can be written as
$\mathcal{O}_{Aheavy,X}=\cos\phi\mathcal{O}_{Alight,X}+\sin\phi\mathcal{O}_{Aheavy\perp,X}$.
Measuring the coefficient of $\mathcal{O}_{Aheavy\perp,X}$ is the new
information that can be obtained from heavy targets, but not light ones.

The definition of $\mathcal{O}_{Aheavy\perp,X}$ depends on whether it is
constructed in the nucleon EFT, or the quark EFT relevant above a scale of 2
GeV. This is because there is information loss in matching nucleons to quarks,
because the scalar densities of both $u$ and $d$ quarks in the neutron and
proton are all comparable, so the scalar $u$ and $d$ coefficients
$C_{SX}^{qq}$ are indistinguishable unless the scalar nucleon coefficients
$\widetilde{C}^{NN}_{S,X}$ are accurately measured. In addition, there is a
several-$\sigma$ discrepancy between the determinations of the scalar quark
densities in the nucleon from the lattice and pion data.

So in this manuscript we focus on $\mu A\to\\!eA$ on light targets, because
only the leptoquark induces scalar quark coefficients, and we prefer to avoid
the quark-scalar uncertainties associated with defining
$\mathcal{O}_{Aheavy\perp,X}$ . We will consider the complementary information
from heavy targets in pl2 .

##  Appendix B Appendix: If the leptoquark interacts only with one generation
of quarks

In this appendix, we give formulae for the operator coefficients in the
leptoquark model, for the case where the leptoquak interacts only with one
generation of quarks.

If the leptoquark only interacts with top quarks, one obtains:

$\displaystyle\frac{m_{LQ}^{2}}{v^{2}}C_{DR}(m_{\mu})$ $\displaystyle\simeq$
$\displaystyle 2.3\times
10^{-4}[\lambda_{L}\lambda^{\dagger}_{L}]_{e\mu}-12[\lambda_{L}\lambda^{\dagger}_{R}]_{e\mu}$
(B.1) $\displaystyle\frac{m_{LQ}^{2}}{v^{2}}C_{V,XY}^{e\mu ee}(m_{\mu})$
$\displaystyle\simeq$ $\displaystyle-4.45\times
10^{-3}[\lambda_{X}\lambda_{X}^{\dagger}]_{e\mu}[\lambda_{Y}\lambda_{Y}^{\dagger}]_{ee}{+}(1.36\times
10^{-3}-g^{e}_{Y}0.033)[\lambda_{X}\lambda^{\dagger}_{X}]_{e\mu}$ (B.2)
$\displaystyle\frac{m_{LQ}^{2}}{v^{2}}C_{Alight,L}$ $\displaystyle\simeq$
$\displaystyle-6.8\times
10^{-4}\left[\lambda_{L}\lambda^{\dagger}_{L}\right]^{e\mu}-0.0084[\lambda_{L}\lambda^{\dagger}_{L}]^{e\mu}{+}2.4\times
10^{-4}\lambda^{et}_{L}\lambda^{\mu t*}_{R}$

where $X,Y\in\\{L,R\\}$ for the vector four-lepton coefficients, for which the
contributions due to boxes are included. The three terms of the four-fermion
contribution to $\mu A\to\\!eA$ are induced by the photon and $Z$ penguins,
and the top-loop contribution to the $\mu\\!\to\\!e$ Yukawa coupling.

If the leptoquark only interacts with charm quarks, then:

$\displaystyle\frac{m_{LQ}^{2}}{v^{2}}C_{DR}(m_{\mu})$ $\displaystyle\simeq$
$\displaystyle 2.3\times
10^{-4}[\lambda_{L}\lambda^{\dagger}_{L}]_{e\mu}-0.42[\lambda_{L}\lambda^{\dagger}_{R}]_{e\mu}$
(B.3) $\displaystyle\frac{m_{LQ}^{2}}{v^{2}}C_{V,XY}^{e\mu ee}(m_{\mu})$
$\displaystyle\simeq$ $\displaystyle-4.5\times
10^{-3}[\lambda_{X}\lambda^{\dagger}_{X}]_{e\mu}[\lambda_{Y}\lambda_{Y}^{\dagger}]_{ee}{+}(4.8\times
10^{-3}-g^{e}_{Y}1.9\times 10^{-6})[\lambda_{X}\lambda^{\dagger}_{X}]_{e\mu}$
$\displaystyle\frac{m_{LQ}^{2}}{v^{2}}C_{Alight,L}$ $\displaystyle\simeq$
$\displaystyle-2.4\times
10^{-3}\left[\lambda_{L}\lambda^{\dagger}_{L}\right]^{e\mu}{-}0.02\lambda^{ec}_{L}\lambda^{\mu
c*}_{R}$

where $X,Y\in\\{L,R\\}$ in the vector four-lepton coefficients, for which the
$Z$-penguin ( the last term) is suppressed $\propto m_{c}^{2}/v^{2}$, and any
box can contribute because $(g-2)_{e}$ only constrains
$\lambda^{ec}_{L}\lambda^{ec*}_{R}<0.7$.

And finally for a leptoquark that only has $\mu\\!\to\\!e$ interactions on $u$
quarks, one obtains:

$\displaystyle\frac{m_{LQ}^{2}}{v^{2}}C_{DR}(m_{\mu})$ $\displaystyle\simeq$
$\displaystyle 2.3\times
10^{-4}[\lambda_{L}\lambda^{\dagger}_{L}]_{e\mu}-7.3\times
10^{-4}[\lambda_{L}\lambda^{\dagger}_{R}]_{e\mu}$ (B.4)
$\displaystyle\frac{m_{LQ}^{2}}{v^{2}}C_{V,XY}^{e\mu ee}(m_{\mu})$
$\displaystyle\simeq$ $\displaystyle-4.5\times
10^{-3}[\lambda_{X}\lambda^{\dagger}_{X}]_{e\mu}[\lambda_{Y}\lambda_{Y}^{\dagger}]_{ee}{+}4.8\times
10^{-3}[\lambda_{X}\lambda^{\dagger}_{X}]_{e\mu}$
$\displaystyle\frac{m_{LQ}^{2}}{v^{2}}C_{Alight,L}$ $\displaystyle\simeq$
$\displaystyle 0.38\lambda^{eu}_{L}\lambda^{\mu
u*}_{L}-2.0\eta\lambda^{eu}_{L}\lambda^{\mu u*}_{R}$ (B.5)

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