#  Constraining New Physics models from $\mu\to e$ observables in bottom-up
EFT

Marco Ardu  E-mail address<EMAIL_ADDRESS>Departament de Fısica
Teorica, Universitat de Valencia, Dr. Moliner 50, E-46100 Burjassot & IFIC,
Universitat de Valencia & CSIC, E-46071, Paterna, 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 sensitivity to $\mu\to e$ processes by
several orders of magnitude, and could observe lepton flavour-changing contact
interactions for the first time. In this paper, we investigate what could be
learned about New Physics from the measurements of these $\mu\to e$
observables, using a bottom-up effective field theory (EFT) approach and
focusing on three popular models with new particles around the TeV scale (the
type II seesaw, the inverse seesaw and a scalar leptoquark). We showed in a
previous publication that $\mu\to e$ observables have the ability to rule out
these models because none can fill the whole experimentally accessible
parameter space. In this work we give more details on our EFT formalism and
present more complete results. We discuss the impact of some observables
complementary to $\mu\to e$ transitions (such as the neutrino mass scale and
ordering, and LFV $\tau$ decays) and draw attention to the interesting
appearance of Jarlskog-like invariants in our expressions for the low-energy
Wilson coefficients.

##  I Introduction

The observed neutrino mass matrix $[m_{\nu}]$ ParticleDataGroup:2022pth
requires New Physics beyond the Standard Model (SM) which is lepton flavour-
changing. Observing lepton flavour-changing processes other than neutrino
oscillations, would therefore give complementary information on New Physics in
the lepton sector.

Flavour-changing contact interactions among charged leptons (which we refer to
as LFV), have not yet been observed, but upcoming experiments aim to improve
the sensitivity to a few processes in the $\mu\to e$ sector by orders of
magnitude, and to probe a wide palet of $\tau\to l$ processes with lesser
sensitivity. This is summarised in Table 1; it suggests that LFV could be
discovered in $\mu\to e$, while $\tau\to l$ is more promising for
distinguishing among models. For a review of $\mu\to e$ LFV, see e.g. KO .

The aim of this project is to explore what can be learned about New Physics in
the lepton sector from observations of $\mu\to e\gamma,\mu\to e\bar{e}e$
and/or $\mu A\to\\!eA$. For instance, it would be ideal if the data could
indicate properties of the New Physics model, such as whether new particles
interact with lepton doublets or singlets or both, whether LFV occurs amoung
SM particles at loop or tree level, or whether LFV is related to $[m_{\nu}]$,
baryogenesis or New Physics in the quark flavour sector.

In order to quantify what $\mu\to e$ data could tell us about models, we study
these questions in a bottom-up EFT approach, assuming
$\Lambda_{NP}\mathrel{\vbox{\hbox{$>$}\nointerlineskip\hbox{$\sim$}}}$ TeV. We
translate the data from the experimental scale to $\Lambda_{NP}$ using EFT,
then match three “representative” models to the experimentally allowed Wilson
coefficient space, and explore which differences among the models can be
identified by the data.

Model predictions for LFV have been widely studied; in particular, there is a
large literature devoted to calculating LFV rates in neutrino mass models (for
a review, see e.g Calibbi:2017uvl ; Ardu:2022sbt ; or for example
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 ; Magg:1980ut ; Lazarides:1980nt ; Schechter:1980gr ;
Mohapatra:1980yp ; Chun:2003ej ; Kakizaki:2003jk ; Akeroyd:2009nu ;
Dinh:2012bp ; Ferreira:2019qpf ; Calibbi:2022wko ; Wyler:1982dd ;
Mohapatra:1986aw ; Mohapatra:1986bd ; Ilakovac:1994kj ; Tommasini:1995ii ;
Ibarra:2011xn ; Alonso:2012ji ; Abada:2015oba ; Coy:2018bxr ;
Crivellin:2022cve ; Zhang:2021jdf ) and other Standard Model extensions such
as leptoquarks Buchmuller:1986zs ; Dorsner:2016wpm ; Bauer:2015knc ;
Sakaki:2013bfa ; Cai:2017wry ; Angelescu:2018tyl ; Lee:2021jdr ; Shi:2019gxi ;
Desai:2023jxh ; Bordone:2020lnb . We hope that our bottom-up EFT approach
could give a complementary perspective on the well-studied relations between
models and observables. Our study differs from top-down analyses in that
firstly, we suppose that upcoming $\mu\to e$ experiments will measure 12
Wilson coefficients, and not just three rates. So we are using a more
optimistic/futuristic parametrisation of the LFV observables, using as input
everything they could tell us. This allowed us to show in a previous
publication Ardu:2023yyw , that the three models we consider could be ruled
out by upcoming data. Secondly, model studies frequently scan over the model
parameter space; this allows to estimate correlations among LFV observables,
but the results depend on the choice of measure on model parameter space. We
circumvent the issue of measure and the need to scan by parametrising the
models in terms of “Jarlskog-like” invariants Jarlskog:1985ht , which
contribute to the observables. Also, the operator coefficients are allowed to
be complex, which is consistent with the hints of leptonic CP violation in
neutrino observations T2K:2019bcf .

This manuscript gives more complete results than presented in Ardu:2023yyw  —
in particular highlighting the appearance of model “invariants” in the
coefficients at the experimental scale, and explores the impact of some
complementary observables on the model predictions for $\mu\to e$. Section II
reviews $\mu\to e$ observables and the EFT formalism implemented here, then
Section III summarises the three TeV-scale models that we consider, which are
the type II seesawMagg:1980ut ; Lazarides:1980nt ; Schechter:1980gr ;
Mohapatra:1980yp , an inverse seesaw Wyler:1982dd ; Mohapatra:1986aw ;
Mohapatra:1986bd , and a scalar leptoquarkBuchmuller:1986zs ; Dorsner:2016wpm
; Bauer:2015knc ; Sakaki:2013bfa ; Cai:2017wry ; Angelescu:2018tyl ;
Lee:2021jdr ; Shi:2019gxi ; Desai:2023jxh ; Bordone:2020lnb  which can fit the
$R_{D}$ anomaly BaBar:2012obs ; Belle:2015qfa ; Belle:2019rba ; LHCb:2023zxo ;
LHCb:2023cjr . The matching of the models to the EFT is relegated to Appendix
C. Section IV gives the twelve observable Wilson coefficients at the
experimental scale (including the coefficients for $\mu A\to\\!eA$ on heavy
targets, missing in Ardu:2023yyw ), expressed in terms of “invariant”
combinations of model and SM parameters. Section V explores the interplay of
$\mu\to e$ flavour change with some complementary observables in the models we
consider. Then we discuss what we learned about bottom-up reconstruction in
Section VI, and conclude in Section VII.

##  II  Observables and Notation

In a bottom-up perspective, one starts from data and how to parametrise it,
which is reviewed in Section II.1. Section II.2 reviews our EFT formalism; it
allows to obtain expressions for the observable Wilson coefficients in terms
of operator coefficients at the weak scale, which are given in Appendix B.

###  II.1 Observables

Some models considered here generate a Majorana mass matrix for the three
light neutrinos. At energies below the weak scale (taken $\simeq m_{W}$), it
can be included in the Lagrangian as ParticleDataGroup:2022pth

$\delta{\cal
L}_{<m_{W}}=-\overline{\nu_{\alpha}}\frac{[m_{\nu}]^{\alpha\beta}}{2}\nu^{c}_{\beta}+h.c.$
(II.1)

where greek indices indicate the charged lepton mass eigenstate basis, square
brackets indicate a matrix, and $[m_{\nu}]_{\alpha\beta}=U_{\alpha
i}m_{i}U_{\beta i}$ for $m_{i}$ real and positive. The leptonic mixing matrix
$U$ is parametrised as in ParticleDataGroup:2022pth  by three mixing angles,
one “Dirac” phase and two “Majorana” phases. In our study, we approximate the
Dirac phase $\delta\approx 3\pi/2$ T2K:2023smv , and take the Majorana phases
as free parameters. The Majorana phases and the lightest neutrino mass
$m_{min}$ affect the elements of $[m_{\nu}]$, but not the flavour-changing
elements of

$[m_{\nu}m^{\dagger}_{\nu}]_{\alpha\beta}=[UD_{m}D_{m}U^{\dagger}]_{\alpha\beta}-m_{j}^{2}[UU^{\dagger}]_{\alpha\beta}$
(II.2)

for $\alpha\neq\beta\in\\{e,\mu,\tau\\}$ and $D_{m}={\rm
Diag}\,(m_{1},m_{2},m_{3})$, because the Majorana phases cancel and the
$|m_{i}^{2}-m_{j}^{2}|$ are determined in neutrino oscillations.

For LFV, we focus on $\mu\to e$ interactions that are otherwise flavour
diagonal (so we do not consider weak meson decays such as
$K\to\mu^{\pm}e^{\mp}$). A feature of this sector is that there are very
restrictive bounds on a handful of processes, and the experimental sensitivity
is planned to improve by several orders of magnitude in the next few years
(see Table 1). This differs from the $\tau\to l$ $(l\in\\{\mu,e\\})$ sector,
where a large variety of LFV $\tau$ decays currently have BRs of order
$10^{-8}$, and Belle II aims for sensitivities ${\cal O}(10^{-9}\to
10^{-10})$. So from an EFT perspective, the $\tau\to l$ sector allows to
independently measure almost every Wilson coefficient up to a New Physics
scale $\mathrel{\vbox{\hbox{$>$}\nointerlineskip\hbox{$\sim$}}}10$ TeV (at
tree level)Banerjee:2022xuw ; whereas in the $\mu\to e$ sector, fewer
coefficients are probed at greater accuracy, motivating the use of
Renormalization Group Equations (RGEs) in the $\mu\to e$ sector.

Process | Current bound on BR | Upcoming Sensitivity
---|---|---
$\mu\to e\gamma$ |  $<3.1\times 10^{-13}$ MEG:2016leq  |  ${\sim 10^{-14}}$ MEG II:2018kmf
$\mu\to\bar{e}ee$ |  $<1.0\times 10^{-12}$ SINDRUM:1987nra  |  $10^{-14}\to 10^{-16}$ Blondel:2013ia
$\mu Ti\to eTi$ |  $<6.1\times 10^{-13}$ Wintz:1998rp  |  $\sim 10^{-16}$ COMET:2009qeh ; Mu2e:2014fns
$\mu Au\to eAu$ |  $<7\times 10^{-13}$ SINDRUMII:2006dvw  | (? $\to 10^{-18})$ Kuno:2005mm ; CGroup:2022tli )
$\tau\to l\gamma$ |  $<3.3\times 10^{-8}$ tau1  |  $\mathrel{\vbox{\hbox{$<$}\nointerlineskip\hbox{$\sim$}}}10^{-8}$belle2t3l
$\tau\to 3l$ |  $<{\rm few}\times 10^{-8}$ tau2  | few$\times 10^{-10}$belle2t3l
… | … | …
Table 1: Some $\mu\to e$ processes, with current experimental bounds, and the
estimated reach of upcoming (and future) experiments. For $\mu\to e\gamma$,
MEG II MEG II:2018kmf  aims to reach $BR\sim 6\times 10^{-14}$. The Mu3e
experiment Blondel:2013ia  aims for $BR(\mu\to e\bar{e}e)\sim 10^{-14}$ with
the current beam and data-taking starting in 2025, then $\sim 10^{-16}$ with a
beam upgrade Aiba:2021bxe . The COMET COMET:2009qeh  and Mu2e Mu2e:2014fns
experiments will search for $\mu\to e$ conversion on light nuclei where the
$\mu$-$A$ bound state is long-lived; a different experimental approach, eg the
PRISM/PRIME Kuno:2005mm  or ENIGMA CGroup:2022tli  proposals would be required
to improve also the sensitivity to heavy nuclei with shorter lifetimes. Some
$\tau\to l$ processes are included for comparaison, with Belle II expectations
for a luminosity of 50 ab-1 belle2t3l .

The three processes we consider are $\mu\to e\gamma$, $\mu\to e\bar{e}e$ and
$\mu A\to\\!eA$. In the latter, a $\mu^{-}$ is captured by a nucleus, where it
can transform into an electron via various interactions; we restrict to those
which are coherent across the nucleus, or “spin-independent”111We neglect $\mu
e\gamma\gamma$ operators which can contribute to $\mu A\to\\!eA$
Davidson:2020ord . (some details are given in Appendix A). At the experimental
scale, these three processes can be parametrised by the dimensionless operator
coefficients $\\{C\\}$ of the following Lagrangian KO ; KKO ; Davidson:2020hkf
; Davidson:2022nnl :

$\displaystyle\delta{\cal L}_{<m_{W}}$ $\displaystyle=$
$\displaystyle\frac{1}{v^{2}}\sum_{X\in\\{L,R\\}}{\Big{(}}C^{e\mu}_{D,X}(m_{\mu}\overline{e}\sigma^{\alpha\beta}P_{X}\mu)F_{\alpha\beta}+C^{e\mu
ee}_{S,XX}(\overline{e}P_{X}\mu)(\overline{e}P_{X}e)+C^{e\mu
ee}_{V,LX}(\overline{e}\gamma^{\alpha}P_{L}\mu)(\overline{e}\gamma_{\alpha}P_{X}e)$
(II.3) $\displaystyle+C^{e\mu
ee}_{V,RX}(\overline{e}\gamma^{\alpha}P_{R}\mu)(\overline{e}\gamma_{\alpha}P_{X}e)+C^{e\mu}_{Al,X}{\cal
O}_{Al,X}+C^{e\mu}_{Au\perp,X}{\cal O}_{Au\perp,X}{\Big{)}}+h.c.$

where ${\cal O}_{Al}$ is the combination of operators contributing to SI
$\mu\to e$ conversion on light targets222These operators are refered to as
${\cal O}_{Alight}$ and ${\cal O}_{Aheavy}$ in Davidson:2022nnl ; Ardu:2023yyw
. such as Titanium (used by SINDRUM SINDRUMII:2006dvw ; Wintz:1998rp ) or
Aluminium(to be used by COMET COMET:2009qeh  and Mu2e Mu2e:2014fns ), and
${\cal O}_{Au\perp}$ is an independent combination probed by heavy targets
such as Gold (used by SINDRUM SINDRUMII:2006dvw ). At the experimental scale,
these operators would describe $\mu\to e$ interactions with nucleons, which
can be matched to operators involving quarks at a scale $\sim$ 2 GeV. We use
the quark basis here, because it is more convenient for comparing to models.
At 2 GeV, the operators are333Notice that the quark currents are not chiral,
which introduces factors of 2 as given in Eq. (A.2).

$\displaystyle{\cal O}_{Al,X}$ $\displaystyle\simeq$
$\displaystyle(\overline{e}P_{X}\mu){\Big{(}}0.692(\overline{u}u)+0.699(\overline{d}d)+0.0341(\overline{s}s)+0.00440(\overline{c}c)+0.00128(\overline{b}b){\Big{)}}$
(II.4)
$\displaystyle+(\overline{e}\gamma^{\alpha}P_{X}\mu){\Big{(}}0.125(\overline{u}\gamma_{\alpha}u)+0.128(\overline{d}\gamma_{\alpha}d){\Big{)}}$
$\displaystyle{\cal O}_{Au,X}$ $\displaystyle\simeq$
$\displaystyle\cos\theta_{A}{\cal O}_{Al,X}+\sin\theta_{A}{\cal
O}_{Au\perp,X}$ (II.5) $\displaystyle{\cal O}_{Au\perp,X}$
$\displaystyle\simeq$
$\displaystyle-(\overline{e}P_{X}\mu){\Big{(}}0.2(\overline{u}u)+0.1(\overline{d}d)+0.008(\overline{s}s)+0.001(\overline{c}c)+0.0003(\overline{b}b){\Big{)}}$
(II.6)
$\displaystyle+(\overline{e}\gamma^{\alpha}P_{X}\mu){\Big{(}}0.56(\overline{u}\gamma_{\alpha}u)+0.8(\overline{d}\gamma_{\alpha}d){\Big{)}}$

where $\theta_{A}$ is the misalignement angle between Gold and Aluminium, and
Appendix A briefly reviews these results.

The Branching Ratios (BRs) can be written as

$\displaystyle BR(\mu\to e\gamma)$ $\displaystyle=$ $\displaystyle
384\pi^{2}(|C^{e\mu}_{DL}|^{2}+|C^{e\mu}_{DR}|^{2})$ (II.7) $\displaystyle
BR(\mu\to e\bar{e}e)$ $\displaystyle=$ $\displaystyle\frac{|C^{e\mu
ee}_{S,LL}|^{2}+|C^{e\mu ee}_{S,RR}|^{2}}{8}+2|C^{e\mu
ee}_{V,RR}+4eC^{e\mu}_{D,L}|^{2}+2|C^{e\mu ee}_{V,LL}+4eC^{e\mu}_{D,R}|^{2}$
$\displaystyle+(64\ln\frac{m_{\mu}}{m_{e}}-136)(|eC^{e\mu}_{D,R}|^{2}+|eC^{e\mu}_{D,L}|^{2})+|C^{e\mu
ee}_{V,RL}+4eC^{e\mu}_{D,L}|^{2}+|C^{e\mu ee}_{V,LR}+4eC^{e\mu}_{D,R}|^{2}$
$\displaystyle BR_{SI}(\mu Al\to eAl)$ $\displaystyle=$ $\displaystyle
B_{Al}(|d_{Al}C^{e\mu}_{DR}+C_{Al,L}|^{2}+|d_{Al}C^{e\mu}_{DL}+C_{Al,R}|^{2})$
(II.9) $\displaystyle BR_{SI}(\mu Au\to eAu)$ $\displaystyle=$ $\displaystyle
B_{Au}(|d_{Au}C^{e\mu}_{DR}+C_{Au,L}|^{2}+|d_{Au}C^{e\mu}_{DL}+C_{Au,R}|^{2})$
(II.10)

where only the Spin Independent contribution to $\mu A\to\\!eA$ is included
based on the results of KKO  (we do not use the recent results of
Haxton:2022piv ), the $B_{A}$ are target-nucleus dependent constants discussed
in Appendix A, and $d_{A}\equiv I_{A,D}/(4|\vec{u}_{A}|)$ are given after Eqn
(A.16). In all cases, the outgoing electrons are approximated as chiral
because they are relativistic. The final states containing electrons of
different chirality therefore do not interfere, giving independent constraints
on the coefficients generating those final states. The experimental limits on
$\mu\to e\gamma$ and $\mu\to e\bar{e}e$ therefore constrain eight coefficients
— the two dipoles and six four-lepton coefficients— to be in the vicinity of
zero. The correlation matrix and resulting bounds are discussed in
Davidson:2018kud  and we repeat the bounds here for completeness:

$\displaystyle|C_{D,X}|^{2}\leq\frac{B_{\mu\to
e\gamma}}{384\pi^{2}+e^{2}(64\ln\frac{m_{\mu}}{m_{e}}-136)\frac{B_{\mu\to
e\gamma}}{B_{\mu\to e\bar{e}e}}}$
$\displaystyle|C_{V,XX}|^{2}\leq\frac{B_{\mu\to
e\bar{e}e}}{2}\left(1+\frac{32e^{2}B_{\mu\to e\gamma}}{205e^{2}B_{\mu\to
e\gamma}+384\pi^{2}B_{\mu\to e\bar{e}e}}\right)$
$\displaystyle|C_{V,XY}|^{2}\leq B_{\mu\to
e\bar{e}e}\left(1+\frac{16e^{2}B_{\mu\to e\gamma}}{205e^{2}B_{\mu\to
e\gamma}+384\pi^{2}B_{\mu\to e\bar{e}e}}\right)$ (II.11)

where $B_{process}$ is the experimental upper bound on the Branching Ratio for
the process, and $e^{2}(64\ln\frac{m_{\mu}}{m_{e}}-136)$ is written as
$205e^{2}$. Although the discussion of Davidson:2018kud  implicitly supposed
the coefficients were real, the same bounds apply for complex coefficients,
because the real and imaginary components of the coefficients are constrained
to lie inside the same ellipse around the origin.

The dipole coefficient is not constrained by $\mu A\to eA$ because it
contributes in interference, but the bound on $|C_{Al,X}|^{2}$ (and in
principle $|C_{Au\perp,X}|^{2}$) is affected in the expected way by the
independent bounds on the dipole:

$\displaystyle|C_{Al,X}|^{2}\leq\frac{B_{\mu Al\to
eAl}}{B_{Al}}+\frac{d_{Al}^{2}B_{\mu\to e\gamma}}{384\pi^{2}}~{}~{}.$

In this work, we do not consider future improvements in the experimental reach
for $\mu A\to\\!eA$ on heavy targets, so the current MEG bound on the dipole
coefficient ensures that it is negligible in $\mu Au\to eAu$. The current
bound on Gold therefore implies

$\displaystyle|C_{Au\perp,X}|^{2}\leq\frac{{B}_{\mu Au\to
eAu}}{B_{Au}\sin^{2}\theta_{A}}+\frac{\cos^{2}\theta_{A}{B}_{\mu Al\to
eAl}}{B_{Al}\sin^{2}\theta_{A}}$ (II.12)

where $\theta_{A}$ is the misalignement angle between Gold and Aluminium and
given after Eq. (A.15).

The current bounds on the twelve observable coefficients are given in Table 2,
as well as the bounds that could be set, if upcoming experiments do not
observe $\mu\to e$ processes. The $\mu A\to\\!eA$ bounds given here differ
from our previous paper Ardu:2023yyw  because ${\cal O}_{Au\perp,X}$ is here
defined in terms of operators with quarks, so differs from the nucleon
definition of Ardu:2023yyw . The matching of quarks with nucleons is discussed
in Appendix A.

| current bound | upcoming | process
---|---|---|---
$|C_{D,X}|$ | $9.0\times 10^{-9}$ | $\sim 10^{-9}$ | $\mu\to e\bar{e}e$
$|C_{V,XX}|$ | $7.0\times 10^{-7}$ | $\sim 10^{-8}$ | $\mu\to e\bar{e}e$
$|C_{V,XY}|$ | $1.0\times 10^{-6}$ | $\sim 10^{-8}$ | $\mu\to e\bar{e}e$
$|C_{S,XX}|$ | $2.8\times 10^{-6}$ | $\sim 10^{-8}$ | $\mu\to e\bar{e}e$
$C_{Al,X}$ | $4.3\times 10^{-9}$ | $7\times 10^{-11}$ | $\mu{\rm Ti}\to e{\rm Ti}$
$C_{Au\perp,X}$ | $7.4\times 10^{-8}$ | ${7.4\times 10^{-8}}$ | $\mu{\rm Au}\to e{\rm Au}$
Table 2:  Current and possible future bounds on the observable coefficients,
with $X,Y\in\\{L,R\\},X\neq Y$. The current bound on $C_{Al,X}$ is from
$\mu{\rm Ti}\to e{\rm Ti}$ Wintz:1998rp , and the future bound assumes BR(
$\mu{\rm Al}\to e{\rm Al})\leq 10^{-16}$. The current and future bounds on
$C_{Au\perp,X}$ are identical because we do not include the reach of the
proposed PRISM/PRIME and ENIGMA experiments among our upcoming results.

In this theoretical study, we optimistically consider that these twelve
coefficients can be measured with the same reach that they can be constrained,
given in Table 2. Polarising the muon could allow to distinguish between
coefficients of operators with an L vs an R projector in the $\bar{e}-\mu$
bilinear in $\mu\to e\gamma$ and $\mu\to e\bar{e}e$KO  as well as in $\mu
A\to\\!eA$ Kuno:1987dp . And in the case of $\mu\to e\bar{e}e$, the angular
distributions of the three-particle final state allow to determine the
magnitude of various coefficients and some phases, as discussed in Okadameee .
Vector operators ${\cal O}_{VXX}$ and scalar operators ${\cal O}_{SYY}$ (for
$Y\neq X$) induce the same angular distributions, but could in principle be
distinguished by measuring final state helicities Okadameee . Our models do
not generate the scalar operators within the reach of upcoming experiments
(see Subsection V.3).

###  II.2 bottom-up EFT

For an introduction to EFT, see for instance Georgi ; burashouches ;
LesHouches . Our EFT consists of an operator basis and Renormalisation Group
Equations (RGEs) to look after scale evolution. The previous section presented
an operator basis describing the observables at low energy; a more complete
set of operators is required in translating the observables from the
experimental to the New Physics scale, because the RGEs mix operators.

Our New Physics scale is at $\Lambda_{NP}\approx$ TeV, so we need a
QED$\times$QCD invariant basis of operators below the weak scale, included in
the Lagrangian as

$\displaystyle\delta{\cal L}_{<m_{W}}$ $\displaystyle=$
$\displaystyle-\overline{\nu_{\alpha}}\frac{[m_{\nu}]^{\alpha\beta}}{2}\nu^{c}_{\beta}+\sum_{n\geq
2}\sum_{A,\zeta}\frac{C_{A}^{\zeta}\mathcal{O}_{A}^{(4+n)\zeta}}{v^{n}}+h.c.$
(II.13)

where $v=174$ GeV, the superscript $\zeta$ gives the flavour indices, and the
operator subscripts indicate the Lorentz structure and particle content. So
for instance

$\mathcal{O}_{V,XY}^{e\mu
uu}=(\overline{e}\gamma^{\alpha}P_{X}\mu)(\overline{u}\gamma_{\alpha}P_{Y}u)~{}~{}~{},~{}~{}~{}X,Y\in\\{L,R\\}.$

We write the SM Yukawa matrices as $[Y_{e}],[Y_{u}]$ and $[Y_{d}]$, and the
Yukawa eigenvalue of fermion $f$ as $y_{f}$.

As discussed in Appendix C.1, we did not find a simple and reliable recipe to
express the coefficients of the Lagrangian (II.13) in terms of model
parameters, because a TeV is not so far from the weak scale. The SMEFT
operator basis BW ; polonais  and RGEs JMT  are appropriate above the weak
scale, but the EFT expansions (in operator dimension and
loop$\times$logarithm) are not converging fast. In principle, we could match
the model directly to the Lagragian (II.13), but that requires to calculate
many loop diagrams. Finally, we opt to present the matching results in the
basis of Eq. (II.13). So we will not be using the SMEFT basis, but for
comparing to the literature, we implement it as

$\displaystyle\delta{\cal L}_{SMEFT}$ $\displaystyle=$
$\displaystyle-\frac{C_{5}^{\alpha\beta}}{2v}(\overline{\ell_{\alpha}}\varepsilon
H^{*})(\ell^{c}_{\beta}\varepsilon
H^{*})+\sum_{A,\zeta}\frac{C_{A}^{\zeta}\mathcal{O}_{A}^{\zeta}}{v^{2}}+h.c.$
(II.14)

where $[m_{\nu}]^{\alpha\beta}=C_{5}^{\alpha\beta}v$.

The low energy EFT includes all LFV operators of dimension six, and some
relevant operators of dimension seven (see Davidson:2020hkf  for a list). In
this low-energy EFT, we ensure that operators appear only once by requiring
$\zeta=e\mu...$. Operators of dimension five and six ($n=1,2$) are included in
SMEFT, but without the $+h.c.$ for hermitian operators. In SMEFT, we follow
the convention that each flavour index in $\zeta$ runs over all three
generations (so some operators are repeated in the SMEFT Lagrangian, causing
some factor of 2 or 4 differences between coefficients in SMEFT vs the low-
energy EFT).

The doublet and singlet leptons are in the charged lepton mass eigenstates
$\\{e,\mu,\tau\\}$Ardu:2021koz , which can differ from the diagonal-Yukawa
basis as discussed in Appendix C.2. The singlet quarks are labelled by their
flavour, and the quark doublets are in the $u$-type mass basis, with
generation indices that run $1\to 3$.

In quantum field theories, the coefficients of renormalizable and non-
renormalizable operators evolve with scale. For non-renormalisable operators,
this evolution of the operator coefficients lined up in a row vector $\vec{C}$
can be described as

$\displaystyle\mu\frac{d}{d\mu}\vec{C}\equiv\frac{d}{dt}\vec{C}=\vec{C}[\Gamma]+\vec{C}[\vec{X}]\vec{C}^{\dagger}+...$
(II.15)

where $t=\log\mu$, the effects of renormalisable interactions on the non-
renormalisable operators are described by the matrix $[\Gamma]$, and
$[\vec{X}]$ is a three-index tensor that schematically represents the effect
of non-renormalisable interactions on the evolution of other non-
renormalisable interactions (for instance, the mixing of a pair of four-
fermion operators into another four-fermion operator via a fish diagram).

The RGEs we implement automatically for the QED$\times$ QCD-invariant EFT are
an improved leading log approximation to $[\Gamma]$ Crivellin:2017rmk , where
most of the anomalous dimension are at one-loop, augmented by the two-loop
vector to dipole mixing (because this mixing vanishes at one loop and is
${\cal O}(10^{-3})$ Ciuchini:1993fk ). We also consider the mixing of two
dimension six operators into a dimension eight operator described by
$[\vec{X}]$, and when relevant, include these contributions by hand in
matching the model to EFT at $\Lambda_{NP}$. Since the dimension eight
operators have two additional Higgs legs, they are particularily relevant when
the Higgs has ${\cal O}(1)$ couplings to loop particles, such as the top quark
in the leptoquark model. These dimension eight contributions are discussed
more fully in Section V.3 and Appendix C.

Solving the RGEs allows to translate operator coefficients from the
experimental scale to the New Physics scale — where the coefficients can be
calculated in a model. We aim to resum the QCD running, and include the
electroweak effects in perturbation theory (${\cal O}(\alpha\log)$ and
occasionally ${\cal O}(\alpha^{2}\log^{2})$). If the matrix $[\Gamma]$ were
scale-independent, then the solution would be $\sim\exp\Gamma t$; however,
$[\Gamma]$ decribes SM loop corrections, and SM parameters run in various ways
with scale. For $[\Gamma]$ scale-dependent, the RGEs can be solved by scale-
ordering, analogous to the familiar time-ordering that allows to solve a
similiar equation for the time-translation operator in quantum field theory.
We include the scale-dependence of $\alpha_{s}$ (at one loop), and the
associated running of quark Yukawas, but neglect the running due to other SM
couplings (including the top Yukawa). So the effect of QCD is to renormalise
some coefficients and rescale some electroweak anomalous dimensions, such that
the first terms in the perturbative solution of the RGE (II.15) are

$\displaystyle C_{J}(m)$ $\displaystyle\simeq$ $\displaystyle
C_{J}(\Lambda)\eta_{m}^{a_{J}}-C_{K}(\Lambda)\eta_{m}^{a_{K}}\widetilde{\Gamma}_{KJ}\ln\frac{\Lambda}{m}$
(II.16)

where $\widetilde{\Gamma}_{KJ}=f_{KJ}\Gamma_{KJ}(m)$ Bellucci:1981bs  with
$\Gamma$ the electroweak anomalous dimension matrix, no sum on $J,K$ and :

$\displaystyle
f_{KJ}=\frac{(1-\eta_{m}^{a_{J}-a_{K}-a_{I}+1})}{(1+a_{J}-a_{K}-a_{I})(1-\eta_{m})}~{}~{},~{}~{}~{}\eta_{m}=\frac{\alpha_{s}(m(m))}{\alpha_{s}(\Lambda_{NP})}~{}~{},~{}~{}~{}\eta=\frac{\alpha_{s}(2{\rm
GeV})}{\alpha_{s}(\Lambda_{NP})}$ (II.17)

where $a_{J}=-\gamma^{s}_{J}/2\beta_{0}$ for $\alpha_{s}\gamma^{s}_{J}/4\pi$
the QCD anomalous dimension of the coefficients $C_{J}$ ($a_{T}=-4/23$,
$a_{S}=a_{D}=12/23$ with 5 flavours), $\beta_{0}$ is the 1-loop QCD beta
function coefficient, and the parameters in the electroweak anomalous
dimension cause it to run as
$\Gamma_{KJ}(m)=\eta^{a_{I}}\Gamma_{KJ}(\Lambda)$. Finally, once the RGEs have
been solved, a coefficient at the experimental scale (for instance
$C_{D,X}(m_{\mu})$), can be written at $\Lambda_{NP}$ as a weighted sum of the
coefficients of operators that can contribute via loops to $\mu\to e\gamma$.

It can be shown Davidson:2020hkf  that almost every operator involving 3 or 4
legs that induces a $\mu\to e$ interaction (but no other flavour change)
contributes to the amplitude for $\mu\to e\gamma$, $\mu\to e\bar{e}e$ and/or
$\mu A\to\\!eA$ suppressed at most by a factor of order $10^{-3}$. This
suggests that the RGEs are relevant to include, because they ensure that a
small handful of processes are sensitive to almost any $\mu\to e$ operator.
Despite that almost all operators can contribute, there remain only 12
constraints. Rather than dealing with a large correlation matrix corresponding
to the usual operator basis, we use a scale-dependent basis that corresponds
to the twelve experimentally probed directions. This was proposed in
Davidson:2020hkf , and the recipe we follow is outlined in Davidson:2022nnl .

The RGEs can be solved to express an operator coefficient at the experimental
scale, eg the dipole coefficient, in terms of operator coefficients at the NP
scale:

$C_{D}(m_{\mu})\simeq\vec{C}(\Lambda_{NP})\cdot\vec{e}_{D}~{}~{}~{}.$

The directions in coefficient space corresponding to the twelve vectors
$\\{e_{O}\\}$ then form a basis for the observable subspace at $\Lambda_{NP}$.
The elements of most of these vectors at or just above the electroweak scale
are given in Davidson:2020hkf  (the coefficient combinations probed by $(\mu
Al\to eAl)$ and $(\mu Au\to eAu)$ are given for completeness in Appendix B).

##  III  The models

The three TeV-scale New Physics models that we consider are the type II and
inverse seesaw models, and a scalar leptoquark. These models are selected for
their diverse444Ref. Kanemura:2015cca  makes an interesting classification of
Majorana neutrino mass models into three sets, based on the flavour structures
that are multiplied in the definition of $[m_{\nu}]$. However, both seesaw
models enter the same set. lepton-flavour-changing predictions: LFV is
controlled by the neutrino mass matrix in the type II seesaw, is independent
of the neutrino mass matrix in the inverse seesaw, and the leptoquark can
mediate $\mu A\to\\!eA$ at tree level, as well as addressing anomalies in the
quark flavour sector.

The type II seesaw model Mohapatra:1980yp ; Magg:1980ut ; Schechter:1980gr ;
Lazarides:1980nt  is an economical neutrino mass model, where the SM particle
content is extended with a colour-singlet, SU(2) triplet scalar $\Delta$, of
hypercharge $Y=+1$ (in the normalization where the lepton doublets have
$Y=-1/2$). The SM Lagrangian at the mass scale of the triplet is augmented by

$\displaystyle\delta{\cal L}_{\Delta}$ $\displaystyle=$
$\displaystyle(D_{\rho}\Delta^{I})^{\dagger}D^{\rho}\Delta^{I}-M_{\Delta}^{2}|\Delta|^{2}+\frac{1}{2}\left(f_{\alpha\beta}\,\overline{\ell^{c}_{\alpha}}(i\tau_{2})\tau_{I}\ell_{\beta}\Delta^{I}+M_{\Delta}\lambda_{H}\,H^{T}(i\tau_{2})\tau_{I}H\Delta^{*I}+{\rm
h.c.}\right)$ (III.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 $\ell$ 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 couplings. A feature of
this model, that is shared with some other neutrino mass models, is that the
SU(2) singlet leptons do not acquire new interactions, so LFV is expected to
involve the doublet leptons. The phenomenology of the type II seesaw has been
widely studied at colliders Akeroyd:2005gt ; FileviezPerez:2008jbu ;
Melfo:2011nx ; Freitas:2014fda ; Ghosh:2017pxl ; Antusch:2018svb  and for low-
energy LFV Konetschny:1977bn ; Chun:2003ej ; Ma:2000xh ; Kakizaki:2003jk ;
Akeroyd:2009nu ; Dinh:2012bp ; Ferreira:2019qpf ; Calibbi:2022wko ;
Mandal:2022zmy .

In Effective Field Theory, the neutrino mass matrix generated in the type II
seesaw model can be obtained by matching, at $M_{\Delta}$, the tree-level
diagram in the left panel of Figure 5 onto the dimension five neutrino mass
operator appearing in Eq. (II.14). This gives a neutrino mass matrix

$\displaystyle[m_{\nu}]^{\alpha\beta}\simeq\frac{[f]^{\alpha\beta*}\lambda_{H}v^{2}}{M_{\Delta}}$
(III.2)

which can also be obtained in the model by minimising the potential for the
Higgs and the electrically neutral component of the triplet =
$(\Delta^{1}+i\Delta^{2})/\sqrt{2}\,$, which obtains a vev
$-\lambda_{H}v^{2}/(\sqrt{2}M_{\Delta})$. This model is reputed to be
predictive for LFV, because the lepton flavour-changing couplings
$f^{\alpha\beta}$ are proportional to the light neutrino mass matrix. However,
LFV is not suppressed by the small neutrino mass scale, because lepton number
change involves $[f]^{\alpha\beta*}\lambda_{H}$, so for a sufficiently small
Higgs to triplet coupling, $\lambda_{H}\sim 10^{-12}$, it is possible to have
$M_{\Delta}\sim$ TeV and $f^{\alpha\beta}$ of ${\cal O}(1)$.

The second model we consider is the inverse (type I) seesaw Wyler:1982dd ;
Mohapatra:1986aw ; Mohapatra:1986bd , which, like the type II seesaw,
naturally generates small Majorana neutrino masses from new particles that can
be at the TeV-scale with ${\cal O}(1)$ LFV couplings. In this model, $n$ gauge
singlet Dirac fermions, $\Psi_{a}^{T}=(S_{a},N_{a})$, are added to the SM
particle content, with approximately lepton number conserving interactions.
Lepton number changing interactions can be included via Yukawa couplings
and/or Majorana masses(see Fernandez-Martinez:2022gsu  for a “basis-
independent” discussion of the options); we choose to allow small Majorana
masses for $S_{a}$ and write the Lagrangian as

$\delta\mathcal{L}_{NS}=i\overline{N}\not{\partial}N+i\overline{S}\not{\partial}S-\left(Y^{\alpha
a}_{\nu}(\overline{\ell}_{\alpha}\tilde{H}N_{a})+M_{a}\overline{S}_{a}N_{a}+\frac{1}{2}\mu_{ab}\overline{S_{a}}S^{c}_{b}+{\rm
h.c}\right),$ (III.3)

where $a,b$ run from 1..$n$, $N_{a}$ and $S_{a}$ are respectively right- and
left-handed and in the eigenbases of $M$, $Y_{\nu}$ is a complex $3\times n$
dimensionless matrix, we take $M_{a}$ of ${\cal O}$(TeV), and $\mu$ is an
$n\times n$ Majorana mass matrix with $\mu_{ab}\ll M_{c}$. For vanishing
$\mu$, lepton number is conserved, and the $N_{a}$ combine with the $S_{a}$
into Dirac singlet neutrinos, which can have lepton flavour changing
interactions $Y_{\nu}$ with the SM doublets. Like the type II seesaw, this
model is expected to induce LFV among doublet leptons, but unlike the type II
seesaw, it has several non-degenerate heavy new particles, and will induce low
energy LFV processes via loop diagrams because the flavour-changing $Y_{\nu}$
couples $\ell$ to two heavy particles.

The inverse seesaw models induce LFV Ilakovac:1994kj ; Tommasini:1995ii ;
Ibarra:2011xn ; Alonso:2012ji ; Abada:2015oba ; Coy:2018bxr ;
Crivellin:2022cve ; Zhang:2021jdf , they can lead to non-unitarity of the
lepton mixing matrix Antusch:2014woa ; Fernandez-Martinez:2016lgt ;
Chrzaszcz:2019inj ; Blennow:2023mqx , and the singlets could be discovered at
colliders for suitable mass ranges delAguila:2007qnc ; Atre:2009rg ;
Deppisch:2015qwa ; Banerjee:2015gca ; Antusch:2016vyf ; Das:2018usr ;
Mekala:2022cmm . With small $\mu_{ab}$, the leading contribution to the active
neutrino mass matrix is

$\displaystyle[m_{\nu}]^{\alpha\beta}\simeq[Y_{\nu}M^{-1}\mu
M^{-1}Y_{\nu}^{T}]^{\alpha\beta}v^{2}~{}~{}~{}.$ (III.4)

So the flavour-changing $Y_{\nu}$ can be ${\cal O}$(1) because small
$\mu_{ab}$ gives small $m_{\nu}$, and the flavour change is expected to be
independent of the active neutrino mass matrix as can be seen for $n$ = 3 by
solving Eq. (III.4) for $\mu_{ab}$.

Finally, our third model includes a leptoquark — for a review of this class of
coloured and charged bosons, see e.g. Dorsner:2016wpm — which is chosen to fit
the anomalies in $R_{D^{*}}$ and/or $R_{D}$ BaBar:2012obs ; Belle:2015qfa ;
Belle:2019rba ; LHCb:2023zxo ; LHCb:2023cjr . It is an SU(2)-singlet scalar
denoted $S_{1}$ in the notation of Buchmuller:1986zs  – not to be confused
with the singlet fermions $\\{S_{a}\\}$ of Eq. (III.3) – and has interactions

$\displaystyle{\cal L}_{S}$ $\displaystyle=$
$\displaystyle(D_{\rho}S_{1})^{\dagger}D^{\rho}S_{1}-m_{LQ}^{2}S_{1}^{\dagger}S_{1}+(-\lambda^{\alpha
j}_{L}\overline{\ell}_{\alpha}i\tau_{2}q_{j}^{c}+\lambda^{\alpha
j}_{R}\overline{e}_{\alpha}u_{j}^{c})S_{1}+(\lambda^{\alpha
j*}_{L}\overline{q^{c}}_{j}i\tau_{2}\ell_{\alpha}+\lambda^{\alpha
j*}_{R}\overline{u^{c}}_{j}e_{\alpha})S_{1}^{\dagger}$ (III.5)
$\displaystyle+\lambda_{4}H^{\dagger}HS_{1}^{\dagger}S_{1}+...$

where 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
$+\lambda^{\alpha j}_{L}\overline{e_{L}}(u_{L})^{c}S_{1}$. The leptoquark mass
is $m_{LQ}\approx$ TeV, consistent with the CMS and ATLAS searches CMSLQ ;
ATLASLQ  which exclude leptoquarks with sub-TeV masses that are pair-produced
via strong interactions, and decay to specific final states. Some leptoquark-
Higgs interactions are included in Eq. (III.5) because they appear in the
matching results of Appendix C.3.3, but their contributions to LFV observables
are negligible assuming perturbative couplings. This Lagrangian does not lead
to neutrino masses (as mentioned in Appendix C.3.3), but features $\mu
A\to\\!eA$ at tree level and LFV interactions for singlet and doublet leptons
(so unlike the seesaw models, it induces scalar and tensor operators).

The low-energy phenomenology of leptoquarks (see eg Bauer:2015knc ;
Sakaki:2013bfa ; Cai:2017wry ; Angelescu:2018tyl ; Lee:2021jdr ; Shi:2019gxi ;
Desai:2023jxh ; Bordone:2020lnb ) attracted attention in recent years due to
various B-physics anomalies. In particular, the excesses in the ratios
BaBar:2012obs ; Belle:2015qfa ; Belle:2019rba ; LHCb:2023zxo ; LHCb:2023cjr

$\displaystyle R_{X}=\frac{BR(B\to X_{c}\bar{\tau}\nu)}{BR(B\to
X_{c}\bar{\ell}\nu)}$ (III.6)

where $X_{c}=D,D^{*}...$, could indicate a new charged-current four-fermion
interaction involving $b$ and $c$ quarks, a $\tau$ and a (tau) neutrino. The
$S_{1}$ leptoquark can generate this interaction with various Lorentz
structures, while preserving lepton flavour (if $S_{1}$ only interacts with
$\tau$’s and $\nu_{\tau}$’s, one could attribute $\tau$ flavour to the
leptoquark.) So fitting $R_{D}$ and/or $R_{D^{*}}$ involves leptoquark
coupling constants that differ from the $\lambda_{X}^{eq}$, $\lambda_{X}^{\mu
q}$ that are relevant for $\mu\to e$ processes, and in this manuscript, we
neglect the $\lambda_{X}^{\tau q}$ couplings and possible correlations among
$\mu\to e$ and $\tau\to l$ processes that could arise in this model555It is
clear that if the leptoquark has couplings $\lambda_{X}^{eq}$,
$\lambda_{X}^{\mu q}$ and $\lambda_{X}^{\tau q}$, then it can mediate $\tau\to
e$, $\tau\to\mu$ and $\mu\to e$ processes. Furthermore, if $|\lambda_{X}^{\tau
q}|\sim 0.2$ to fit the B-physics anomalies, then experimental bounds
$B_{(\tau\to l)}$ on $\tau\to l$ processes could constrain
$\lambda_{X}^{lq}<B_{(\tau\to l)}/\lambda_{X}^{\tau q}$. In all the cases we
checked, we find  $\frac{B_{(\tau\to e)}}{\lambda_{X}^{\tau
q}}\frac{B_{(\tau\to\mu)}}{\lambda_{X}^{\tau q}}>B_{(\mu\to e)}$ that is, that
$\mu\to e$ processes can probe smaller $\lambda_{X}^{lq}$ couplings than
$\tau\to l$ can exclude.. The LFV predictions of the $S_{1}$ leptoquark have
been discussed, for instance, in Crivellin:2020mjs ; Bigaran:2020jil .

##  IV The observables coefficients in terms of model parameters

In this section, we give the coefficients of the observable Lagrangian (II.3)
as a function of model and SM parameters. Some of these results were already
presented in Ardu:2023yyw . The coefficients are expressed in terms of $\mu\to
e$ flavour-changing combinations of model and SM parameters, which we refer to
as “invariants”. These are a convenient stepping-stone between the models and
the observables, because they concisely identify the masses and couplings
constants that the observables depend on, and give the functional dependance.
We briefly discuss the invariants in Section IV.1 and list the coefficients in
Section IV.2.

###  IV.1 Comments on invariants

Invariants were introduced Jarlskog:1985ht ; Bernabeu:1986fc  as products of
Lagrangian coupling constants (or matrices), in order to have a Lagrangian-
basis-independent measure of symmetry breaking in a model. However, it may be
unclear how these elegant constructions relate to observable $S$-matrix
elements for symmetry-violating processes, because the mass and scale-
dependence of $S$-matrix elements can be intricate.

For instance, the original invariant constructed to measure CP violation in
the quark sector of the Standard Model can be written in terms of Lagrangian
Yukawa matrices as Jarlskog:1985ht ; Bernabeu:1986fc ; Branco:1999fs

$\displaystyle\frac{1}{3}{\rm
Tr}{\Big{\\{}}\left[Y_{u}Y_{u}^{\dagger},Y_{d}Y_{d}^{\dagger}\right]^{3}{\Big{\\}}}$
$\displaystyle=$ $\displaystyle
2iJ(y_{t}^{2}-y_{c}^{2})(y_{t}^{2}-y_{u}^{2})(y_{c}^{2}-y_{u}^{2})(y_{b}^{2}-y_{s}^{2})(y_{b}^{2}-y_{d}^{2})(y_{s}^{2}-y_{d}^{2})~{}~{},~{}~{}$
(IV.1) $\displaystyle{\rm with}~{}~{}J$ $\displaystyle=$ $\displaystyle{\rm
Im}\\{V_{us}V_{cs}^{*}V_{cb}V_{ub}^{*}\\}$

but it is unclear at what scale to evaluate the quark masses (or equivalently,
yukawa couplings). This invariant can be compared to the parameter
$\epsilon_{K}$ (see eg burashouches  for a brief review) that contributes to
CP-violation in $K-\bar{K}$ mixing, and for which the result at next-to or
next-to-next-to leading log has been expressed in a rephasing invariant form
in Brod:2019rzc . Restricting to the leading log QCD corrections for
simplicity, one can write

$\displaystyle\epsilon_{K}$ $\displaystyle\propto$ $\displaystyle
J{\Big{(}}{\rm
Re}\\{V_{td}V_{ts}^{*}V_{ud}^{*}V_{us}\\}\eta^{tt}S(x_{t})+2|V_{ud}|^{2}|V_{us}|^{2}\eta^{ut}(S(x_{c})-S(x_{c},x_{t})){\Big{)}}$
(IV.2)

where $x_{Q}=m_{Q}(m_{Q})/m_{W}$, the Inami-Lim functionsInami:1980fz  are

$\displaystyle S(x_{t})$ $\displaystyle=$
$\displaystyle\frac{4x_{t}-11x^{2}_{t}+x_{t}^{3}}{4(1-x_{t})^{2}}-\frac{3x_{t}^{2}}{2(1-x_{t})^{3}}\ln
x_{t}$ $\displaystyle S(x_{c})-S(x_{c},x_{t})$ $\displaystyle=$ $\displaystyle
x_{c}\left(1-\ln\frac{x_{t}}{x_{c}}\right)+\frac{3x_{c}x_{t}}{4(1-x_{t})}+\frac{3x_{c}x_{t}^{2}}{4(1-x_{t})^{2}}\ln
x_{t}~{}~{}~{},$

and $\eta^{pq}$ are the appropriate QCD corrections for the Inami-Lim
functions (see Brod:2019rzc ). So one sees that there is only a remote
relation between the Lagrangian invariant of Eq. (IV.1), and Eq. (IV.2), which
allows a numerically precise prediction for the observable $\epsilon_{K}$,

In the models considered here, we obtain expressions for the observable
coefficients at $m_{\mu}$ (which are $\approx$ S-matrix elements) in terms of
“invariants” , which encode the dependence of LFV observables on running SM
and model coupling constants and masses. For instance, in the type II seesaw
model, photon penguin diagrams (contributing to $\mu\to e\bar{e}e$) generate
the coefficient $C_{V,LR}^{e\mu ee}$ proportional to the invariant :

$\displaystyle\frac{v^{2}}{M_{\Delta}^{2}}\,[m_{\nu}\ln\frac{[\tilde{m}_{e}\tilde{m}_{e}^{\dagger}]}{M_{\Delta}^{2}}\,m_{\nu}^{\dagger}]^{e\mu}\,$
$\displaystyle=\,$
$\displaystyle\frac{v^{2}}{M_{\Delta}^{2}}\left([m_{\nu}m_{\nu}^{\dagger}]^{e\mu}\ln\frac{m_{\mu}^{2}}{M_{\Delta}^{2}}+[m_{\nu}]^{e\tau}[m_{\nu}^{*}]^{\mu\tau}\ln\frac{\tilde{m}_{\tau}^{2}}{m_{\mu}^{2}}+[m_{\nu}]^{ee}[m_{\nu}^{*}]^{\mu
e}\ln\frac{\tilde{m}_{e}^{2}}{m_{\mu}^{2}}\right)~{}~{}~{},$ (IV.3)

where $[\tilde{m}_{e}]$ is the charged lepton mass matrix, but with its
eigenvalues replaced by
$m_{\alpha}\to\tilde{m}_{\alpha}=\max\\{m_{\alpha},q^{2}\\}$, which is the
kinematic cutoff of the logarithm, with $4m_{e}^{2}\leq
q^{2}\mathrel{\vbox{\hbox{$<$}\nointerlineskip\hbox{$\sim$}}}m_{\mu}^{2}$ the
(four-momentum)2 of the photon. (So the cutoff of the $\tau$ loop is
$m_{\tau}$, but $\tilde{m}_{e}\approx m_{\mu}$ so the last term in the
parentheses of Eq. (IV.3) vanishes, or equivalently there is no long-distance
contribution to the matrix element, because $q^{2}\sim m_{\mu}^{2}$ in most of
the phase space Marciano:1977wx .) Eq. (IV.3) exemplifies our relatively
simple invariants, constructed by multiplying matrices, which encode the
correct scale evolution of coupling constants and masses, provided that the
scale separations are large enough for EFT to be reliable666But notice the
impact of phase space: the tilde on $m_{e}$ is from kinematics that the
Lagrangian does not know about, so the Lagrangian invariant would include a
“large” $\ln\frac{m_{\mu}}{m_{e}}$ that the coefficient does not contain
because the large log is only relevant in a small phase space..

However, our simple invariants apply only for models with a single mass scale
for LFV; for models with many heavy LFV particles around the scale
$\Lambda_{NP}$, such as the inverse seesaw, the operator coefficients obtained
in matching are not linear products of matrices in flavour space (see $eg$
Eq.s IV.10, IV.12). Constructions that involve Inami-Lim functions of mass
matrix eigenvalues are no doubt also “invariant” under Lagrangian basis
transformations, but like Eq. (IV.2), this invariance is not manifest.

Our invariants have other attractive features, beyond the correct scale-
dependence to parametrise $S$-matrix elements. As expected they measure
$\mu\to e$ flavour change in the model, and they also identify the products of
model parameters relevant to observables. This second feature allows to
circumvent the necessity of scanning over model parameters. For instance, in
the inverse seesaw model, $[Y_{\nu}]$ is a 3$\times$3 matrix of unknown
complex numbers, so naively one must scan over them all, and possibly impose
some texture. However, in the case of degenerate singlets, there are only two
invariants — $[Y_{\nu}Y_{\nu}^{\dagger}]_{e\mu}$ and
$[Y_{\nu}Y_{\nu}^{\dagger}Y_{\nu}Y_{\nu}^{\dagger}]_{e\mu}$ — which are
complex numbers of magnitude 0$\to$1\. This raises the question whether one
could reconstruct the model Lagrangian from a sufficient number of invariants?

Finally, the invariant of Eq. (IV.3) illustrates an interesting dynamical
mechanism to break properties of the model. Recall that
$[m_{\nu}m_{\nu}^{\dagger}]^{e\mu}$ is a function only of neutrino oscillation
parameters— which are measured — so the second term of the second expression
exhibits the log-induced dependence on the unknown Majorana phases and
neutrino mass scale. This logarithmic breaking of a relation between model-
matrices is reminiscent of the “log-GIM” mechanism in the quark sector
Gavela:1981sk , where $\Delta F=1$ FCNC operators can be mediated by similar
penguin diagrams at ${\cal O}(G_{F}\alpha_{e}\ln m_{W}/m_{c})$, with a charm
quark in the loop.777 In the lepton sector below the weak scale, the
(quadratic) GIM mechanism suppresses LFV amplitudes $\propto
m^{2}_{\nu}/m_{W}^{2}$, so implementing log-GIM would be very interesting
($\ln(m_{\nu}/m_{W})\gg m^{2}_{\nu}/m_{W}^{2}$). However, this appears
difficult Hernandez-Tome:2018fbq ; Blackstone:2019njl , because there are no
“penguin diagrams” in the SM with a massless gauge boson attached to a
neutrino loop, and because the logarithm in log-GIM is cutoff by the largest
of the mass in the loop, or the external momentum traversing the loop, and
this latter is of order charged lepton masses in LFV processes.

###  IV.2 Predictions at the experimental scale

This section lists the model predictions for the observable coefficients of
the Lagrangian (II.3), partially presented in Ref. Ardu:2023yyw . The model
parameters are given in the Lagrangians of Section III. The expressions given
here occasionally differ from Ardu:2023yyw , because we made numerically
insignificant modifications of the lower cutoff of some logarithms, in order
to obtain more elegant invariants with the physically correct cutoffs.

####  IV.2.1 $\mu\to e\gamma$

In the inverse seesaw model, the dipole coefficients are

$\displaystyle C^{e\mu}_{D,R}$ $\displaystyle\simeq$
$\displaystyle-\frac{e}{32\pi^{2}}[Y_{\nu}M^{-1}(M^{\dagger})^{-1}Y_{\nu}^{\dagger}]^{e\mu}v^{2}{\left(1-16\frac{\alpha_{e}}{4\pi}\ln\frac{M}{m_{\mu}}\right)}~{}~{},$
(IV.4)

where the parentheses include the flavour-universal ${\cal O}(10\%)$ QED
running, and $M$ is the singlet mass scale. The couplings $Y^{\alpha a}_{\nu}$
can be of order one — which could be especially motivated in the $\tau$
sector—so ${\cal O}(Y^{4})$ combinations can be larger than ${\cal O}(Y^{2})$,
and could appear at dimension eight when two additional Higgs legs on the
sterile neutrino line are replaced by the Higgs vev, or at two-loop when the
Higgs legs are closed. We estimate the $\mathcal{O}(Y^{4})$ terms to be
suppressed with respect to the coefficient in Eq. IV.4 by $\sim v^{2}/M^{2}$
or $\sim 1/(16\pi^{2})$, and we therefore expect them to not modify
significantly the correlations between the $\mu\to e$ observables that the
model can predict Ardu:2023yyw .

For the type II seesaw,

$\displaystyle C^{e\mu}_{D,R}$ $\displaystyle\simeq$
$\displaystyle\frac{3e}{128\pi^{2}}{\Big{[}}\frac{[m_{\nu}m_{\nu}^{\dagger}]_{e\mu}}{\lambda^{2}_{H}v^{2}}\left(1-16\frac{\alpha_{e}}{4\pi}\ln\frac{M_{\Delta}}{m_{\mu}}\right)+\frac{\alpha_{e}}{\pi}\frac{116}{27}\left(\ln\frac{M_{\Delta}}{m_{\mu}}\frac{[m_{\nu}m_{\nu}^{\dagger}]_{e\mu}}{\lambda_{H}^{2}v^{2}}-\ln\frac{m_{\tau}}{m_{\mu}}\frac{[m_{\nu}^{*}]_{\mu\tau}[m_{\nu}]_{e\tau}}{\lambda_{H}^{2}v^{2}}\right){\Big{]}}$
(IV.5) $\displaystyle\simeq$
$\displaystyle\frac{3e}{128\pi^{2}}{\Big{[}}\frac{[m_{\nu}m_{\nu}^{\dagger}]_{e\mu}}{\lambda^{2}_{H}v^{2}}{\Big{(}}1+\frac{32}{27}\frac{\alpha_{e}}{4\pi}\ln\frac{M_{\Delta}}{m_{\tau}}{\Big{)}}+\frac{116\alpha_{e}}{27\pi}\ln\frac{m_{\tau}}{m_{\mu}}\sum_{\alpha\in
e\mu}\frac{[m^{*}_{\nu}]_{\mu\alpha}[m_{\nu}]_{e\alpha}}{\lambda_{H}^{2}v^{2}}{\Big{]}}$
(IV.6)

where the second term in the bracket in Eq. (IV.5) arises from the two-loop
vector to dipole mixing Ciuchini:1993fk . In both seesaw models, the
coefficient $C^{e\mu}_{D,L}$ is suppressed by a factor $m_{e}/m_{\mu}$, and
can be obtained from Eq.s (IV.6,IV.4) by multiplying by $m_{e}/m_{\mu}$.

For the leptoquark, which interacts with singlet and doublet leptons, the
dipole coefficients are

$\displaystyle\frac{m_{LQ}^{2}}{v^{2}}C^{e\mu}_{D,X}(m_{\mu})$
$\displaystyle\simeq$
$\displaystyle\frac{e[\lambda_{Y}\lambda^{\dagger}_{Y}]^{e\mu}}{128\pi^{2}}\left(1-16\frac{\alpha_{e}}{4\pi}\ln\frac{m_{LQ}}{m_{\mu}}\right)+\frac{2\alpha^{2}_{e}}{9\pi^{2}e}\left[\lambda_{Y}\ln\frac{m_{LQ}}{\tilde{m}_{Q}}\lambda^{\dagger}_{Y}\right]^{e\mu}$
(IV.7) $\displaystyle-\frac{\alpha_{e}}{2\pi
ey_{\mu}}f_{TD}\left[\lambda_{Y}Y_{u}\left(\eta^{a_{T}}_{\tilde{m}_{Q}}\ln\frac{m_{LQ}}{\tilde{m}_{Q}}-\frac{5}{4}\right)\lambda^{\dagger}_{X}\right]^{e\mu}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$

where $X\neq Y\in\\{L,R\\}$, $f_{TD}$, $a_{T}$ and $\eta$ are related to the
QCD running and defined at Eq. II.17, and the $\tilde{m}_{Q}$ serving as lower
cutoff for the logarithms (here and further in the manuscript) is

$\tilde{m}_{Q}={\rm max}\\{m_{Q}(m_{Q}),2~{}{\rm GeV}\\}$

because the quarks are matched to nucleons at 2 GeV. The first term in Eq.
(IV.7) is the matching contribution (times its QED running),the second term is
the 2-loop mixing of tree vector operators into the dipole, and the third term
is the one loop mixing of tensor operators to dipoles.

The last term of Eq. (IV.7) requires some discussion, because the first log-
enhanced term in the parentheses arises in the RGEs between $m_{LQ}\to
m_{\mu}$, but the second “finite”, or not-log-enhanced term is formally of
higher order in EFT. It is included because it is of comparable magnitude to
the log-enhanced term in the case of internal top quarks — that is, as
discussed in Appendix C.1, the scale ratio $m_{LQ}/m_{t}$ is not large, so our
matching conditions at $\Lambda_{NP}$ are constructed to reproduce the results
of matching to a QCD$\times$QED invariant EFT at $m_{W}$. And finally,
although the finite part is only required for the top quark, a quark-flavour-
summed expression is given in order to retain the “invariant” formulation.
(The light quark contributions are numerically negligible in this expression,
which is fortunate because their QCD running is also not correct.)

We do not consider dimension 8 contributions to the dipole, because they are
suppressed $\propto v^{2}/M^{2}$, and do not allow to circumvent the
parametric suppression of the dimension six term. For instance, in the inverse
seesaw, such terms also have the loop and $y_{\mu}$ suppression applying to
(IV.4).

####  IV.2.2 $\mu\to e\bar{e}e$

The decay $\mu\to e\bar{e}e$ can be mediated at the experimental scale by the
dipole operators, and vector and scalar four-lepton operators (see Eq. (II.1).
We do not give results for the scalar coefficients, because they are
effectively vanishing: in matching, all three models induce coefficients that
are smaller than the upcoming experimental sensitivity, and SM interactions
that could transform some other LFV operator into a scalar are suppressed by
lepton Yukawas, so negligible as well. Unfortunately, scalar coefficients
$C_{S,XX}$ are indistinguishable from vectors $C_{V,YY}$ in the angular
distribution of $\mu\to e\bar{e}e$, so the absence of scalar coefficients in
these models would be challenging to test.

For the type II seesaw, the vector four-lepton coefficients arise at tree
level, as illustrated in the middle diagram of Figure 5, and in the RGEs via
QED penguin diagrams:

$\displaystyle C^{e\mu ee}_{V,LL}$ $\displaystyle\simeq$
$\displaystyle\frac{[m^{*}_{\nu}]_{\mu
e}[m_{\nu}]_{ee}}{2\lambda_{H}^{2}v^{2}}+{\frac{\alpha_{e}}{{3}\pi\lambda_{H}^{2}v^{2}}{\Big{[}}m^{\dagger}_{\nu}\ln\left(\frac{M_{\Delta}}{\tilde{m}_{\alpha}}\right)m_{\nu}{\Big{]}}_{\mu
e}}$ (IV.8) $\displaystyle C^{e\mu ee}_{V,LR}$ $\displaystyle\simeq$
$\displaystyle{\frac{\alpha_{e}}{3\pi\lambda_{H}^{2}v^{2}}{\Big{[}}m^{\dagger}_{\nu}\ln\left(\frac{M_{\Delta}}{\tilde{m}_{\alpha}}\right)m_{\nu}{\Big{]}}_{\mu
e}}~{}~{}~{}.$ (IV.9)

where $\alpha$ is the index of the intermediate charged lepton. The operators
${\cal O}^{e\mu ff}_{V,RR}$ and ${\cal O}^{e\mu ff}_{V,RL}$, where the
flavour-change is among singlet leptons, have coefficients below upcoming
experimental sensitivities because they are suppressed $\propto y_{e}y_{\mu}$.
This is also the case for the inverse seesaw.

In the case of the inverse seesaw, both vector operators arise via loop
diagrams, with propagating singlets and Higgses:

$\displaystyle C^{e\mu ee}_{V,LL}$ $\displaystyle\simeq$ $\displaystyle
v^{2}\frac{\alpha_{e}}{4\pi}\bigg{(}-1.8[Y_{\nu}M_{a}^{-2}\left(\frac{11}{6}+\ln\left(\frac{m^{2}_{W}}{M_{a}^{2}}\right)\right)Y_{\nu}^{\dagger}]_{e\mu}+2.7[Y_{\nu}(Y^{\dagger}_{\nu}Y_{\nu})_{ab}\frac{1}{M^{2}_{a}-M^{2}_{b}}\ln\left(\frac{M^{2}_{a}}{M^{2}_{b}}\right)Y_{\nu}^{\dagger}]_{e\mu}$
(IV.10) $\displaystyle+2.5Y^{ea}_{\nu}Y^{*\mu
a}_{\nu}Y^{eb}_{\nu}Y^{*eb}_{\nu}\frac{1}{M^{2}_{a}-M^{2}_{b}}\ln\left(\frac{M^{2}_{a}}{M^{2}_{b}}\right)+\mathcal{O}\left(\frac{\alpha_{e}}{4\pi}\right)\bigg{)}$
$\displaystyle\to$ $\displaystyle\frac{v^{2}}{M^{2}}\left(3.3\times
10^{-3}(Y_{\nu}Y^{\dagger}_{\nu})_{e\mu}(1+0.56(Y_{\nu}Y^{\dagger}_{\nu})_{ee})+1.55\times
10^{-3}(Y_{\nu}Y^{\dagger}_{\nu}Y_{\nu}Y^{\dagger}_{\nu})_{e\mu}\right)$
(IV.11) $\displaystyle C^{e\mu ee}_{V,LR}$ $\displaystyle\simeq$
$\displaystyle
v^{2}\frac{\alpha_{e}}{4\pi}\bigg{(}1.5[Y_{\nu}M_{a}^{-2}\left(\frac{11}{6}+\ln\left(\frac{m^{2}_{W}}{M_{a}^{2}}\right)\right)Y_{\nu}^{\dagger}]_{e\mu}-2.7[Y_{\nu}(Y^{\dagger}_{\nu}Y_{\nu})_{ab}\frac{1}{M^{2}_{a}-M^{2}_{b}}\ln\left(\frac{M^{2}_{a}}{M^{2}_{b}}\right)Y_{\nu}^{\dagger}]_{e\mu}$
(IV.12)
$\displaystyle+\mathcal{O}\left(\frac{\alpha_{e}}{4\pi}\right)\bigg{)}$
$\displaystyle\to$ $\displaystyle\frac{v^{2}}{M^{2}}\left(-2.8\times
10^{-3}(Y_{\nu}Y^{\dagger}_{\nu})_{e\mu}-1.6\times
10^{-3}(Y_{\nu}Y^{\dagger}_{\nu}Y_{\nu}Y^{\dagger}_{\nu})_{e\mu}\right)$
(IV.13)

where the first expression for a coefficient is for arbitrary singlet masses
$\mathrel{\vbox{\hbox{$>$}\nointerlineskip\hbox{$\sim$}}}$ TeV , and after the
arrow is the simplified formula when the singlets mass2 differences are less
than $v^{2}$ Ardu:2023yyw . In the first expression, the first two terms arise
from $Z$ and $\gamma$ penguins above the electroweak scale (the Higgs
propagates in the loop), and the last one is from boxes.

Finally the leptoquark can generate flavour-changing lepton currrents
involving either singlet or doublet leptons. We give here the coefficient for
left-handed leptons ; the result for singlets is obtained by interchanging
$L\leftrightarrow R$:

$\displaystyle\frac{m_{LQ}^{2}}{v^{2}}C_{V,LX}^{e\mu ee}(m_{\mu})$
$\displaystyle\simeq$
$\displaystyle-\frac{N_{c}}{64\pi^{2}}[\lambda_{L}\lambda_{L}^{\dagger}]^{e\mu}[\lambda_{X}\lambda_{X}^{\dagger}]^{ee}\left(1\mp
12\frac{\alpha_{e}}{4\pi}\ln\frac{m_{LQ}}{m_{\mu}}\right){+}\frac{\alpha_{e}}{3\pi}\left[\lambda_{L}\ln\frac{m_{LQ}}{m_{Q}}\lambda_{L}^{\dagger}\right]^{e\mu}$
$\displaystyle-g^{e}_{X}\frac{N_{c}}{16\pi^{2}}\left[\lambda_{L}Y_{u}\left(\ln\frac{m_{LQ}}{m_{Q}}-\frac{5}{6}\right)Y_{u}^{\dagger}\lambda^{\dagger}_{L}\right]^{e\mu}$

where $g^{e}_{L}=-1+2\sin^{2}\theta_{W}$, $g^{e}_{R}=2\sin^{2}\theta_{W}$, the
first term represents the box diagram at $m_{LQ}$ (and its QED running to
$m_{\mu}$, with $-$/$+$ for $X=$/$\neq L$), 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, and the last term is the
contribution of the $Z$-penguins without the negligible QED running. Since we
only retain the part of the $Z$-penguin that is proportional to
$y_{Q}^{2}$(see Appendix C.4), the dominant contribution arises from the top
quark, where the “finite” (not log-enhanced) part of the diagrams is included
because the logarithm is not large (see the discussion after Eq. (IV.7) or in
Appendix C.1).

####  IV.2.3 $\mu{\rm Al}\to e{\rm Al}$

The SINDRUMII experiment searched for $\mu A\to\\!eA$ on Titanium ($Z=22$) and
Gold ($Z=79$), setting the bounds listed in Table 1. Upcoming experiments will
start with an Aluminium target, which probes a similar combination of
coefficients as Titanium in the analysis of KKO . So this section gives
expressions for the coefficient on Aluminium $C_{Al,X}$, expressed in the
quark operator basis, where the conversion ratio on Aluminium is given in Eq.
(A.16).

The photon penguin diagrams in the type II seesaw model generate a vector
$\mu\to e$ operator on $u$ and $d$ quarks, giving

$\displaystyle{C}^{e\mu}_{Al,L}$ $\displaystyle\simeq$
$\displaystyle-\frac{\alpha_{e}}{72\pi\lambda_{H}^{2}v^{2}}{\Big{[}}m^{\dagger}_{\nu}\ln\left(\frac{M_{\Delta}}{\tilde{m}_{\alpha}}\right)m_{\nu}{\Big{]}}_{\mu
e}~{}~{}~{},$ (IV.15)

where $\tilde{m}_{\tau}=m_{\tau}$ and $\tilde{m}_{\alpha}=m_{\mu}$ for
$\alpha\in\\{e,\mu\\}$ because the logarithm is cut off by the momentum
transfer from the leptons to the nucleusRaidal:1997hq , which is of order
$m_{\mu}$. Notice that the penguin diagram of Figure 5 generates a
2-lepton-2-quark operator at scales above 2 GeV, where quarks are matched to
nucleons, then it continues to mix into a 2-lepton-2-proton operator, which is
why the logarithm cuts off at $m_{\mu}$.

The inverse seesaw generates vector operators, via $Z$ and $\gamma$ penguins
above the weak scale:

$\displaystyle{C}^{e\mu}_{Al,L}$ $\displaystyle\simeq$ $\displaystyle
v^{2}\frac{\alpha_{e}}{4\pi}\bigg{(}-0.05[Y_{\nu}M_{a}^{-2}\left(\frac{11}{6}+\ln\left(\frac{m^{2}_{W}}{M_{a}^{2}}\right)\right)Y_{\nu}^{\dagger}]_{e\mu}+0.09[Y_{\nu}(Y^{\dagger}_{\nu}Y_{\nu})_{ab}\frac{1}{M^{2}_{a}-M^{2}_{b}}\ln\left(\frac{M^{2}_{a}}{M^{2}_{b}}\right)Y_{\nu}^{\dagger}]_{e\mu}~{}~{}~{}~{}$
$\displaystyle+$
$\displaystyle\mathcal{O}\left(\frac{\alpha_{e}}{4\pi}\right)\bigg{)}$
$\displaystyle\to$ $\displaystyle\frac{v^{2}}{M^{2}}\left(8.6\times
10^{-5}(Y_{\nu}Y^{\dagger}_{\nu})_{e\mu}+5.6\times
10^{-5}(Y_{\nu}Y^{\dagger}_{\nu}Y_{\nu}Y^{\dagger}_{\nu})_{e\mu}\right)$
(IV.17)

For the leptoquark, which induces scalar and vector $2l2q$ operators, we
obtain

$\displaystyle{C}^{e\mu}_{Al,L}$ $\displaystyle\simeq$
$\displaystyle{\Big{(}}0.032\lambda^{eu}_{L}\lambda^{\mu
u*}_{L}.\left(1{+}\frac{{2}\alpha}{\pi}\ln\frac{m_{LQ}}{2{\rm
GeV}}\right)+0.033\left(\frac{g^{2}}{64\pi^{2}}\lambda^{eu}_{L}\lambda^{\mu
u*}_{L}\ln\frac{m_{LQ}}{m_{W}}\right)$
$\displaystyle-0.086\frac{\alpha}{6\pi}\left[\lambda_{L}\ln\frac{m_{LQ}}{\tilde{m}_{Q}}\lambda^{\dagger}_{L}\right]^{e\mu}-4.1\times
10^{-4}\left[\lambda_{L}Y_{u}\left(\ln\frac{m_{LQ}}{\tilde{m}_{Q}}-\frac{5}{6}\right)Y_{u}^{\dagger}\lambda^{\dagger}_{L}\right]^{e\mu}$
$\displaystyle-\eta^{a_{S}}\left(0.16\lambda^{eu}_{L}\lambda^{\mu
u*}_{R}+\frac{0.035m_{N}}{27m_{c}}\lambda^{ec}_{L}\lambda^{\mu
c*}_{R}\right)(1+\frac{13\alpha}{6\pi}\ln\frac{m_{LQ}}{2{\rm
GeV}})+\frac{0.094m_{N}}{27m_{t}}\lambda^{et}_{L}\lambda^{\mu
t*}_{R}{\Big{)}}\times\frac{v^{2}}{m_{LQ}^{2}}\ ,$

where are included the tree vector coefficient on $u$ quarks with its QED
running, the electroweak box contribution to the $d$ vector, the QED then $Z$
penguin contributions to the $u$ and $d$ vectors (where we took $V_{ud}\simeq
1,\sin^{2}\theta_{W}\simeq 1/4$), and the scalar $u$, $c$ and $t$
contributions (where the QCD running of the top contribution is negligible).
As in Eq. IV.2.2, the $Z$-penguin contribution only includes the diagrams
$\propto y_{Q}^{2}$, with their “finite parts”.

####  IV.2.4 $\mu A\to\\!eA$ on heavy targets

Changing the target in $\mu A\to\\!eA$ allows to probe a different combination
of operator coefficients KKO . This is discussed quantitatively in Appendix A,
where Eq.s (A.12, A.13, A.14) give the operators probed by light and heavy
targets in the quark basis. The SINDRUM experiment searched for $\mu
A\to\\!eA$ on Gold (see Table 1), and there are plans based on the proposal of
Ref. Kuno:2005mm  to build experiments that could probe $\mu A\to\\!eA$ on
heavy targets (see Table 1). However, we consider these experiments to be to
far in the future for the purposes of our study, so we suppose that the data
for Gold remains the bound of SINDRUM given in Table 1. The operator probed by
Gold can be decomposed into the operator probed by light targets, plus the
remaining part, approximately given in Eq (II.6). In this section, we discuss
the coefficient of this orthogonal part, which can be written as

$\displaystyle C_{Au\perp,L}$ $\displaystyle\simeq$
$\displaystyle-{\Big{(}}0.2C^{e\mu uu}_{SR}+0.1C^{e\mu dd}_{SR}+0.0075C^{e\mu
ss}_{SR}+0.001C^{e\mu cc}_{SR}+0.0003C^{e\mu bb}_{SR}{\Big{)}}$ (IV.19)
$\displaystyle+0.56C^{e\mu uu}_{VL}+0.7956C^{e\mu dd}_{VL}~{}~{}~{}.$

The type II seesaw only generated a vector operator on protons, (no scalar
operators, and no vector operator on neutrons), so once the coefficient is
measured on a first target, it can be predicted on any other. That is, Gold
probes approximately the same four-fermion operator as light targets, and
since the dipole and proton vector coefficients are weighted by approximately
1/4 and 1/2 in both the amplitudes on Gold and Aluminium, the ratio of the
rates is

$\frac{BR(\mu Au\to eAu)}{BR(\mu Al\to
eAl)}\approx\frac{\widetilde{B}_{Au}}{\widetilde{B}_{Al}}\approx 2\pm{\cal
O}(10\%)~{}~{},$

where $\widetilde{B}_{Al}$ and $\widetilde{B}_{Au}$ are given after Eq. (A.5).

In the inverse seesaw, $\mu A\to\\!eA$ on heavy targets could give
complementary information, because the $Z$ penguin contribution generates
vector $\mu\to e$ operators on both protons and neutrons:

$\displaystyle{C}^{e\mu}_{Au\perp,L}$ $\displaystyle\simeq$ $\displaystyle
v^{2}\frac{\alpha_{e}}{4\pi}\bigg{(}-0.5[Y_{\nu}M_{a}^{-2}\left(\frac{11}{6}+\ln\left(\frac{m^{2}_{W}}{M_{a}^{2}}\right)\right)Y_{\nu}^{\dagger}]_{e\mu}+0.8[Y_{\nu}(Y^{\dagger}_{\nu}Y_{\nu})_{ab}\frac{1}{M^{2}_{a}-M^{2}_{b}}\ln\left(\frac{M^{2}_{a}}{M^{2}_{b}}\right)Y_{\nu}^{\dagger}]_{e\mu}~{}~{}~{}~{}$
$\displaystyle+$
$\displaystyle\mathcal{O}\left(\frac{\alpha_{e}}{4\pi}\right)\bigg{)}$
$\displaystyle\to$ $\displaystyle\frac{v^{2}}{M^{2}}\left(1.1\times
10^{-4}(Y_{\nu}Y^{\dagger}_{\nu})_{e\mu}+4.9\times
10^{-4}(Y_{\nu}Y^{\dagger}_{\nu}Y_{\nu}Y^{\dagger}_{\nu})_{e\mu}\right)$
(IV.21)

The coefficient is nonetheless a combination of the same invariants that
feature in the other $\mu\to e$ operators, and so can be predicted with a
combination of $\mu\to e$ observations. For instance, in the nearly degenerate
limit, it can be written as a linear combination of the light targets
coefficient and the dipole.

In the leptoquark model, we neglect the scalar coefficients on $d,s$ and $b$
quarks, because the model does not generate scalar operators with down-type
quarks at tree level, and the estimates of Appendix C.3.3 suggest that the
loop-induced coefficients are below experimental sensitivity. So we obtain

$\displaystyle C_{Au\perp,L}$ $\displaystyle\simeq$
$\displaystyle\left(\eta^{a_{S}}\left(0.1\frac{\lambda^{eu}_{L}\lambda^{\mu
u*}_{R}}{2}+0.001\frac{\lambda^{ec}_{L}\lambda^{\mu
c*}_{R}}{2}\right)\left(1+\frac{13\alpha}{6\pi}\ln\frac{m_{LQ}}{2{\rm
GeV}}\right)\right.$ $\displaystyle+0.14\lambda^{eu}_{L}\lambda^{\mu
u*}_{L}(1+\frac{2\alpha}{\pi}\ln\frac{m_{LQ}}{2{\rm GeV}})$
$\displaystyle+0.4\frac{g^{2}}{32\pi^{2}}[\lambda_{L}V]^{ed}[V^{\dagger}\lambda^{\dagger}_{L}]^{d\mu}-0.32\frac{\alpha}{9\pi}[\lambda_{L}\log\frac{m_{LQ}}{m_{Q}}\lambda^{\dagger}_{L}]^{e\mu}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
$\displaystyle\left.-\frac{N_{c}}{16\pi^{2}}{\Big{(}}0.28(g_{L}^{u}+g_{R}^{u})+0.4(g_{L}^{d}+g_{R}^{d}){\Big{)}}[\lambda_{L}Y_{u}\left(\log\frac{m_{LQ}}{m_{Q}}-\frac{5}{6}\right)Y_{u}^{\dagger}\lambda^{\dagger}_{L}]^{e\mu}\right)\times\frac{v^{2}}{m_{LQ}^{2}}$

where the terms, in order, are the scalar up and charm quark contribution with
their QED and QCD running, the tree vector up quark contribution with its QED
running, the leptoquark-$W$ box matching onto the vector operator for down
quarks (neglecting the QED running), then the QED penguin contribution to the
vector coefficients for $u$ and $d$ quarks, and finally the $Z$ penguin
contributions to both $u$ and $d$ vector coefficients.

We claim that in this leptoquark model, ${C}_{Au\perp,X}$ is independent from
$C_{Al,X}$, implying that $C_{Au\perp,X}$ could be just below the current
SINDRUMII bound (see Table 2) even if $\mu\to e$ flavour change is not
observed in upcoming experiments. Furthermore, we anticipate that this will
remain true, even if the definition of ${\cal O}_{Au\perp}$ changes as
theoretical calculations are updated (the definition of the ${\cal O}_{A}$s in
References Haxton:2022piv ; KKO  appears different.). The point is that
cancellations can occur, for instance among vector and scalar coefficients in
the coefficient on Aluminium, allowing the coefficient on Gold to be
relatively large.

##  V Phenomenology

The type II seesaw model is reputed to be predictive, because the lepton
flavour-changing couplings of the scalar boson $\Delta$ are proportional to
the observed neutrino mass matrix. However, we observed in Ardu:2023yyw  that
knowing the neutrino oscillation parameters does not predict the observable
$\mu\to e$ coefficients. So Sections V.1 and V.2 explore the connections
between $\mu\to e$ processes and other lepton flavour- and number-changing
observables in the type II seesaw. The remaining two subsections respectively
discuss how $\mu\to e\gamma$ constraints suppress dimension eight operators in
the leptoquark model (Section V.3), and the impact of allowing operator
coefficients to be complex(Section V.4).

###  V.1 The neutrino mass scale in the type II seesaw

In this section, we explore how the predictions of the type II seesaw model
for $\mu\to e$ observables are influenced by the lightest neutrino mass scale,
denoted as $m_{\rm min}.$

The lightest neutrino mass plays a crucial role in determining the relevant
coefficients in the model, namely $C^{e\mu}_{D,R}$, $C^{e\mu ee}_{V,LL}$, and
$C^{e\mu ee}_{V,LR}\propto C_{Al,L},\ C_{Au\perp,L}$. The tau mass cut-off in
the two-loop vector mixing and one-loop photon penguin introduces a term
$\propto[m^{*}_{\nu}]_{\mu\tau}[m_{\nu}]_{e\tau}$ (see Eq. IV.5 and IV.9) that
gives rise to the dipole and $C^{e\mu ee}_{V,LR}$ dependence on the unknown
neutrino parameters. Additionally, the $\mu\to 3e_{L}$ vector depends on
$m_{\rm min}$ from both the tree-level
$\propto[m_{\nu}]_{ee}[m^{*}_{\nu}]_{\mu e}$ and the photon penguin
contributions.

The magnitude of these unknown terms increases with the lightest neutrino
mass, making them more relevant when $m_{\rm min}$ is large. For example, if
we allow $m_{\rm min}\sim 0.2$ eV, the two-loop vector-to-dipole mixing can
reach the size of the one-loop matching contribution to the dipole, and
certain choices of Majorana phases could allow for a cancellation that
suppresses the dipole coefficient. Although this has not been pointed out in
the literature before (the dipole cannot vanish in the type II seesaw at the
leading order), the cancellation requires a high neutrino mass scale $m_{\rm
min}$. Values $m_{\rm min}\sim 0.2$ eV are compatible with the laboratory
constraint extracted from the tritium decay end-point, which yields
$\sqrt{\sum_{i}m^{2}_{i}|U_{ei}|^{2}}<0.8$ eV ($90\%$ CL) KATRIN:2021uub , but
are disfavored by the cosmological bounds on the neutrino masses sum $\sum
m_{i}$. Assuming $\Lambda$CDM, CMB data constrain the sum to be $\sum
m_{i}<0.26$ eV ($95\%$ CL), while combined with the BAO measurements the
constraint is stricter $\sum m_{i}<0.12$ eV ($95\%$ CL) Planck:2018vyg .

(a)

(b)

Figure 1: $\frac{|C^{e\mu ee}_{V,LR}|}{|C^{e\mu}_{D,R}|}$ and $\frac{|C^{e\mu
ee}_{V,LL}|}{|C^{e\mu}_{D,R}|}$ as functions of $m_{\rm min}$ for normal and
inverted ordering in blue and red respectively, for all possible values of the
Majorana phases. We consider only the one-loop matching contribution to the
dipole since for $m_{\rm min}\lesssim 0.2$ eV the two-loop vector to dipole
mixing is negligible. The shaded blue region corresponds to the values of
$m_{\rm min}$ above the cosmology preferred upper bound $m_{\rm min}\lesssim
0.04$ eV.

By imposing the cosmological upper bound $m_{\rm min}\lesssim 0.04$ eV, the
$\mu\to e$ coefficients are also constrained. In this case, the dipole is
approximately unaffected by the two-loop contribution and is completely
determined from the neutrino oscillation parameters, apart from the overall
LFV scale. When the dipole coefficient is non-vanishing, the ratios

$\frac{|C^{e\mu ee}_{V,LL}|}{|C^{e\mu}_{D,R}|},\qquad\frac{|C^{e\mu
ee}_{V,LR}|}{|C^{e\mu}_{D,R}|},$ (V.1)

are finite for any value of the Majorana phases. Figures 1(a) and 1(b)
illustrate that by imposing the upper-bound $m_{\rm min}\lesssim 0.04$ eV and
allowing the Majorana phases to vary freely, these ratios are bounded from
above

$m_{\rm min}\leq 0.04\ {\rm eV}\quad\to\quad\frac{|C^{e\mu
ee}_{V,LL}|}{|C^{e\mu}_{D,R}|}\leq\begin{cases}4.3\times 10^{3}\ &{\rm(IO)}\\\
2\times 10^{3}\ &{\rm(NO)}\\\ \end{cases},\qquad\frac{|C^{e\mu
ee}_{V,LR}|}{|C^{e\mu}_{D,R}|}\leq\begin{cases}28\ &{\rm(IO)}\\\ 21\
&{\rm(NO)}\\\ \end{cases}$ (V.2)

The cosmological bound is generally insufficient to constrain the ratios from
below, as there exist $m_{\rm min}$ values for both $\mu\to 3e$ vectors such
that the operator coefficient can vanish. If $m_{\rm min}$ is measured, for
instance by observing the neutrinoless double beta decay, additional
information can be obtained for the $\mu\to e$ coefficients. The penguin
cannot vanish for $m_{\rm min}\lesssim 0.02$ eV regardless of the ordering,
while the $\mu\to 3e_{L}$ vector is also non-vanishing for $m_{\rm
min}\lesssim 10^{-3}$ eV:

$m_{\rm min}\lesssim 0.02\ {\rm eV}\qquad\to\quad\begin{cases}4.7\
&{\rm(IO)}\\\ 8.5\ &{\rm(NO)}\\\ \end{cases}\leq\frac{|C^{e\mu
ee}_{V,LR}|}{|C^{e\mu}_{D,R}|}\leq\begin{cases}15\ &{\rm(IO)}\\\ 10\
&{\rm(NO)}\\\ \end{cases}$ $m_{\rm min}\lesssim 10^{-3}\ {\rm
eV}\qquad\to\quad\begin{cases}17\ &{\rm(IO)}\\\ 20\ &{\rm(NO)}\\\
\end{cases}\leq\frac{|C^{e\mu
ee}_{V,LL}|}{|C^{e\mu}_{D,R}|}\leq\begin{cases}2.3\times 10^{3}\ &{\rm(IO)}\\\
62\ &{\rm(NO)}\\\ \end{cases}$ (V.3)

Therefore, if the ratio of coefficients $\frac{|C^{e\mu
ee}_{V,LR}|}{|C^{e\mu}_{D,R}|}$ or $\frac{|C^{e\mu
ee}_{V,LL}|}{|C^{e\mu}_{D,R}|}$ were observed outside the ranges of Eq. (V.3),
it would be possible to infer a lower bound on the neutrino mass scale (if it
originates from the type II seesaw mechanism). Since these ranges are
significantly narrower in the NO case, measuring the ordering (which is
expected to be determined in the upcoming years) would be particularly useful
to pinpoint the interplay between the lightest neutrino mass and the $\mu\to
e$ predictions in the type II seesaw. Furthermore, if the mass ordering is
normal, future beta decay experiments might be able to constrain $m_{\rm
min}\lesssim 0.02\,\mbox{eV}$ (the Project 8 experiment Project8:2022wqh  aims
at a 90% C. L. sensitivity of $40\,\mbox{meV}$ on the effective neutrino mass
in beta decay, which corresponds to $m_{\rm min}\simeq 0.04\,\mbox{eV}$). In
this case, measuring the ratio of coefficients $\frac{|C^{e\mu
ee}_{V,LR}|}{|C^{e\mu}_{D,R}|}$ outside the range $[8.5,10]$ would exclude the
type II seesaw model.

###  V.2  $\tau$ LFV in the type II seesaw

In this section, we explore whether the type II seesaw model can be
predictive, when one considers flavour-changing decays of muons and taus. In
particular, we focus on observations of $\tau\to 3l$ decays, if one does not
see $\mu\to e$ decays. So in practise in this section, the Majorana phases are
fixed to ensure that the tree contribution to $\mu\to e\bar{e}e$ vanishes, and
we study the $\tau\to 3l$ rates as a function of the neutrino mass scale.

There are four $\Delta$LF$=$1 $\tau\to 3l$ decays ($\tau\to\mu\bar{\mu}\mu$,
$\to e\bar{e}e$, $\to e\bar{\mu}\mu$ and $\to e\bar{e}\mu$) and two that are
$\Delta$LF$=$2 ($\to e\bar{\mu}e$, and $\to\mu\bar{e}\mu$), as given in
Kitano:2000fg . They can all be mediated at tree level by $\Delta$ exchange as
illustrated in the middle diagram of Figure 5. The branching ratios for the
four $\Delta$LF$=$1 decays are analogous to Eq (II.1)

$\displaystyle BR(\tau\to l_{L}\overline{l}l)$ $\displaystyle=$ $\displaystyle
0.18{\Big{\\{}}2|C^{l\tau ll}_{V,LL}+4eC^{l\tau}_{D,R}|^{2}$
$\displaystyle+(64\ln\frac{m_{\tau}}{m_{l}}-136)|eC^{l\tau}_{D,R}|^{2}+|C^{l\tau
ll}_{V,LR}+4eC^{l\tau}_{D,R}|^{2}{\Big{\\}}}$ $\displaystyle
BR(\tau\to\tilde{l}_{L}\overline{l}l)$ $\displaystyle=$ $\displaystyle
0.18{\Big{\\{}}|C^{\tilde{l}\tau ll}_{V,LL}+4eC^{\tilde{l}\tau}_{D,R}|^{2}$
$\displaystyle+(64\ln\frac{m_{\tau}}{m_{l}}-136)|eC^{\tilde{l}\tau}_{D,R}|^{2}+|C^{\tilde{l}\tau
ll}_{V,LR}+4eC^{\tilde{l}\tau}_{D,R}|^{2}{\Big{\\}}}$ $\displaystyle
BR(\tau\to l_{L}\overline{\tilde{l}}l)$ $\displaystyle=$ $\displaystyle
0.18|C^{l\tau l\tilde{l}}_{V,LL}|^{2}$ (V.6)

where we restrict to a fixed chirality of the flavour-changing lepton
bilinear, $l,\tilde{l}\in\\{e,\mu\\},l\neq\tilde{l}$ and the factor 0.18
accounts for the hadronic $\tau$ decays: $\Gamma(\tau\to
e\bar{\nu}\nu)/\Gamma(\tau\to{\rm all})\simeq 0.18$. The final pair of
branching ratios (Eq. V.6) are $\Delta$LF$=$2; we assume such coefficients run
with QED like other four-lepton vector operators. The current experimental
constraints (see Table 1) are of order

$\displaystyle BR(\tau\to 3l)$
$\displaystyle\mathrel{\vbox{\hbox{$<$}\nointerlineskip\hbox{$\sim$}}}$
$\displaystyle{\rm few}\times 10^{-8}~{}{\rm(now)}\to
10^{-10}~{}~{}{\rm(BelleII)}~{}~{}.$

Figure 2: Ratios BR($\tau\to 3l$)/BR($\mu\to e\bar{e}e$) in the type II seesaw
model as a function of $m_{1}$ in inverted ordering (so
$m_{1}\geq\sqrt{\Delta^{2}_{atm}}$) for vanishing $[m_{\nu}]_{e\mu}$. This
illustrates that Belle II could observe specific $\tau\to 3l$ decays, even if
$\mu\to e\bar{e}e$ and $\mu\to e\gamma$ are not observed at upcoming
experiments because the tree contribution to $\mu\to e\bar{e}e$ vanishes with
$[m_{\nu}]_{e\mu}$. Some $\tau\to 3l$ processes (in black) also vanish at tree
level, and we include $\tau\to e\gamma$ and $\tau\to\mu\gamma$ to illustrate
our claim that they are undetectable at Belle II in the Type II seesaw
model(the BR$(l_{i}\to l_{j}\gamma)$ are also divided by BR$(\mu\to
e\bar{e}e)$).  Figure 3:  Same as Fig. 2 for normal ordering in the case where
$\mu\to e\bar{e}e$ is suppressed because $[m_{\nu}]_{e\mu}$ vanishes (so
$m_{\rm
min}=m_{1}\mathrel{\vbox{\hbox{$>$}\nointerlineskip\hbox{$\sim$}}}\sqrt{\Delta^{2}_{sol}}$).

The type II seesaw can predict $\tau\to 3l$ within the reach of Belle II, in
spite of the current bound BR$(\mu\to e\bar{e}e)\leq 10^{-12}$. For instance,
in normal ordering with vanishing lightest neutrino mass $m_{1}$

$\displaystyle\frac{BR(\mu\to e\bar{e}e)}{BR(\tau\to\mu\bar{\mu}\mu)}$
$\displaystyle\sim$ $\displaystyle
5\left|\frac{s_{13}\Delta_{sol}}{\sqrt{2}\Delta_{atm}}\right|^{2}\sim 10^{-3}$

However, in the coming years, the Mu3e experiment will improve the sensitivity
to $\mu\to e\bar{e}e$, so the allowed range for the $\tau\to l$ rates will
depend on the results of Mu3e. To be concrete, we suppose that $\mu\to e$
flavour change is not observed at MEG II or Mu3e. So in the following, we
suppose that the Majorana phases (and the lightest neutrino mass $m_{min}$)
are fixed such that $C_{VLL}^{e\mu ee}$ vanishes at tree level, implying that
the $\mu\to e\bar{e}e$ rate is suppressed by ${\cal O}(\alpha^{2})$, because
it is mediated by the dipole and the penguin-induced $C_{VLR}^{e\mu ee}$.
Recall that the tree-level contribution to the coefficient $C_{VLL}^{e\mu
ee}$, which is proportional to $[m]^{*}_{\mu e}[m]_{ee}$, can vanish with
$[m]_{ee}$ in normal ordering (NO) for
$m_{min}\mathrel{\vbox{\hbox{$<$}\nointerlineskip\hbox{$\sim$}}}\Delta_{sol}$
(as is well known from neutrinoless double beta decay Vergados:2012xy ), and
it can also vanish with $[m]_{e\mu}$ both in normal and inverted ordering for
$m_{min}\mathrel{\vbox{\hbox{$>$}\nointerlineskip\hbox{$\sim$}}}\Delta_{sol}$.

There are six $\tau\to 3l$ decays, whose tree-level amplitudes are
proportional to products of only five neutrino mass matrix elements. This
suggests at least one relation among these decays for generic $m_{min}$ and
Majorana phases —but this relation could be difficult to test in general,
because $m_{min}$ and the Majorana phases can accidentally suppress almost any
element of the neutrino mass matrix (as we saw for $[m_{\nu}]_{e\mu}$). This
section considers a scenario where either $[m_{\nu}]_{e\mu}$ or
$[m_{\nu}]_{ee}$ vanishes in order to suppress $\mu\to e$ rates, so testable
predictions can be expected.

For $m_{ee}\to 0$, one of the $\Delta LF$ = 2 processes vanishes (at the order
we calculate):

$\displaystyle BR(\tau\to e\bar{\mu}e)\to 0~{}~{}~{},$ $\displaystyle{\rm
for}~{}[m_{\nu}]_{ee}\to 0~{}~{}~{}~{}.$ (V.7)

Similarly, the tree contribution to $\tau\to e\bar{e}e$ vanishes, so that only
the dipole and penguin contributions to this decay remain. So a signature of
the type II seesaw with vanishing $[m_{\nu}]_{ee}$, is that these decays are
suppressed simultaneously with the matrix element for neutrinoless double beta
decay.

In the case where $\mu\to e\bar{e}e$ is suppressed because $[m_{\nu}]^{e\mu}$
vanishes (and
$m_{min}\mathrel{\vbox{\hbox{$>$}\nointerlineskip\hbox{$\sim$}}}\sqrt{\Delta^{2}_{sol}}$),
there are identities among the tree-level coefficients
${C}_{V,LL}^{l\tau\sigma\rho}\propto m_{l\sigma}m_{\tau\rho}^{*}$ ( which by
default dominate the rates for the decays $\tau\to l\overline{\rho}\sigma):$

$\displaystyle\frac{m_{ee}}{m_{\mu\mu}}=\frac{{C}_{V,LL}^{e\tau
ee}}{C_{V,LL}^{\mu\tau\mu e}}=\frac{{C}_{V,LL}^{e\tau
e\mu}}{C_{V,LL}^{\mu\tau\mu\mu}}~{}~{}~{},~{}~{}~{}\frac{m^{*}_{\tau
e}}{m^{*}_{\tau\mu}}=\frac{{C}_{V,LL}^{\mu\tau\mu
e}}{C_{V,LL}^{\mu\tau\mu\mu}}=\frac{{C}_{V,LL}^{e\tau ee}}{C_{V,LL}^{e\tau
e\mu}}=\frac{{C}_{V,LL}^{e\tau\mu e}}{C_{V,LL}^{e\tau\mu\mu}}~{}~{}~{}.$ (V.8)

As a result, for values of the Majorana phases fixed to suppress $\mu\to
e\bar{e}e$ (and for compatible $m_{min}$), ratios of $\tau\to 3l$ decays are
predicted. We plot in Figs. 3 (for normal ordering) and 2 (for inverted
ordering) the rates of various $\tau\to 3l$ decays, normalised to the rate for
$\mu\to e\bar{e}e$888In the type II seesaw, the overall magnitude of LFV
$\sim|f^{2}v^{2}/M_{\Delta}^{2}|$ is unknown, so we plot ratios of rates. This
unknown corresponds to $\lambda_{H}$ in our parametrisation., for the case
where $\mu\to e\bar{e}e$ is suppressed by $[m_{\nu}]_{e\mu}\to 0$. This shows
Chun:2003ej ; Akeroyd:2009nu  that Belle II could see $\tau\to 3l$ decays for
BR($\mu\to
e\bar{e}e)\mathrel{\vbox{\hbox{$<$}\nointerlineskip\hbox{$\sim$}}}10^{-16}$
and BR$(\mu\to
e\gamma)\mathrel{\vbox{\hbox{$<$}\nointerlineskip\hbox{$\sim$}}}10^{-14}$. If
several $\tau\to 3l$ decays are observed (including $\tau\to e\bar{e}e$), we
may compare their relative branching ratios with the predictions of the type
II seesaw shown in Figs. 2 and 3, and be able to either exclude the model or
deduce some constraints on $m_{1}$ and the mass ordering. Regarding $\mu
A\to\\!eA$, we estimate that BR$(\mu Al\to eAl)\sim 10^{2}$ BR$(\mu\to
e\bar{e}e)$ in the case of vanishing $[m_{\nu}]_{e\mu}$ in the type II seesaw,
because $C_{V,LR}^{e\mu ee}$ and $\tilde{C}_{Al,L}$ are both induced by the
photon penguin, see Eqs. (IV.9,IV.15).

Finally, the decays $\tau\to e\gamma$ and $\tau\to\mu\gamma$ will not be
observed at Belle II if neutrino masses arise via the type II seesaw model,
because this requires $\mu\to e\gamma$ or $\mu\to e\bar{e}e$ larger than the
current constraints (this conclusion is fully general and does not assume that
the tree-level contribution to $\mu\to e\bar{e}e$ vanishes). The dipole
coefficients $C_{D,R}^{l\tau}$, at the experimental scale $m_{\tau}$, can be
written analogously to Eq. (IV.6), with the index replacement $e\to l$ and
$\mu\to\tau$, and without the second term, because the RG running ends at
$m_{\tau}$ for all flavours in the loop. At the order we calculate, the $\tau$
dipole coefficients are therefore given by
$[m_{\nu}m_{\nu}^{\dagger}]_{l\tau}$, so are independent of the neutrino mass
scale and Majorana phases (see Eq. II.2), and the ratio

$\frac{BR(\tau\to\mu\gamma)}{BR(\tau\to
e\gamma)}\simeq\left|\frac{[m_{\nu}m_{\nu}^{\dagger}]_{\mu\tau}}{[m_{\nu}m_{\nu}^{\dagger}]_{e\tau}}\right|^{2}\sim\frac{1}{2s^{2}_{13}}$

is predicted. However, in order to be within the reach of Belle II
(BR$\mathrel{\vbox{\hbox{$>$}\nointerlineskip\hbox{$\sim$}}}10^{-9}$, see
Table 1), these branching ratios need to be much larger than BR($\mu\to
e\gamma)$, which can be engineered via a cancellation in $C_{D,R}^{e\mu}$ for
specific Majorana phases at a large neutrino mass scale. However, when this
cancellation arises, the model predicts a larger branching ratio for $\mu\to
e\bar{e}e$ than $\tau\to l\gamma$, because the coefficient $C_{VLL}^{e\mu
ee}\propto m_{1}^{2}$ arises at tree level.

###  V.3 The dipole constraints on boxes for the leptoquark

The leptoquark Lagrangian of Eq. (III.5) allows the leptoquark to interact
with doublet and singlet leptons, so it can induce lepton flavour-changing
dipole, tensor and scalar operators, without any suppression by the lepton
Yukawas. Nonetheless, we neglect four-lepton scalar operators in this model,
because the coefficients are suppressed below the upcoming experimental reach
by the dipole constraint. This section aims to show that suppression.

(0,60)(25,60) (60,60)(25,60) (60,60)(90,60) (90,30)(60,30) (25,30)(60,30)
(25,30)(0,30)
$\mu_{X}$$l_{Y}$$e_{Y}$$l_{X}$$S_{1}$$S_{1}$$H$$H$(30,30)(30,60)5
(60,60)(60,30)5 (45,60)(45,80)3 (45,30)(45,10)3

(0,60)(30,45) (60,45)(90,60) (45,45)(15,0,360) (90,30)(60,45) (25,45)(0,30)
$\mu_{X}$$l_{Y}$$e_{Y}$$l_{X}$(30,45)5.7 (60,45)5.7 $H$$H$(45,60)(45,80)3
(45,30)(45,10)3

(0,60)(30,45) (60,45)(90,60) (45,45)(15,0,360) (90,30)(60,45) (25,45)(0,30)
$\mu_{X}$$l_{Y}$$e_{Y}$$l_{X}$(30,45)5.7 (60,45)5.7 xx

Figure 4:  A diagram that generates the four-lepton scalar operator ${\cal
O}_{SXX}^{e\mu ll}$ in the model (on the left), in the RGEs of SMEFT (centre)
and in the RGEs of the QCD$\times$QED invariant EFT (on the right). (There is
another diagram with $l_{Y}\leftrightarrow e_{Y}$.)

We are interested in scalar four-lepton operators ${\cal O}_{S,XX}$, such as
those in the observable Lagrangian of Eq. (II.3). These operators occur at
dimension eight in SMEFT with a pair of Higgs legs, for instance in the form
$(\overline{\ell}_{e}H\mu_{R})(\overline{\ell}_{e}He_{R})$ and can be
generated in the leptoquark model via box diagrams with Higgs legs, as
illustrated on the left in Figure 4. Equivalently, these operators are
generated in the dimension six${}^{2}\to$ dimension eight RGEs of SMEFT or the
low energy EFT, respectively by the fish diagrams at the centre or right of
Figure 4. It is straightforward to see from the model diagram, that

$C_{SRR}^{e\mu ee}\propto
N_{c}[\lambda_{L}Y^{*}_{u}\lambda_{R}^{\dagger}]^{e\mu}[\lambda_{L}Y^{*}_{u}\lambda_{R}^{\dagger}]^{ee}\frac{v^{4}}{32\pi^{2}m_{LQ}^{4}}\log$
(V.9)

so that this coefficient could be $\sim 3\times 10^{-5}$ for leptoquark
couplings of ${\cal O}(1)$ to the top quark. This is marginally above the
current experimental bound, which is $|C_{SRR}^{e\mu ee}|\leq 2.8\times
10^{-6}$ from SINDRUM SINDRUM:1987nra . So now we want to show that the
product $\lambda_{L}^{\alpha t}\lambda_{R}^{\beta t*}$ is constrained to be
$\mathrel{\vbox{\hbox{$<$}\nointerlineskip\hbox{$\sim$}}}0.01$ for almost any
combination of lepton flavours $\alpha,\beta$.

In the presence of both the $\lambda_{L}$ and $\lambda_{R}$ interactions, the
leptoquark matches onto $2l2q$ tensor operators involving the up-type quarks
(see Eq C.11). These tensor operators then mix to the dipole (see the last
term of Eq. IV.7), generating an “invariant” that is not identical to the one
appearing in Eq (V.9):

$\Delta C_{DR}^{\alpha\beta}\propto\frac{\alpha}{\pi
y_{\beta}}[\lambda_{L}Y_{u}^{*}\eta_{m_{Q}}\ln\frac{m_{LQ}}{m_{Q}}\lambda_{R}^{\dagger}]^{\alpha\beta}\sim\frac{\alpha}{\pi
y_{\beta}}{\Big{(}}{\cal O}{\Big{(}}\lambda^{\alpha
t}_{L}y_{t}\lambda_{R}^{\beta t*}{\Big{)}}+{\cal O}{\Big{(}}\lambda^{\alpha
c}_{L}y_{c}\lambda_{R}^{\beta c*}{\Big{)}}+...{\Big{)}}~{}~{}.$ (V.10)

However, the term $\propto y_{t}$ in
$[\lambda_{L}Y_{u}^{*}\eta_{m_{Q}}\ln\frac{m_{LQ}}{m_{Q}}\lambda_{R}^{\dagger}]^{e\mu}$
is of comparable magnitude to the term $\propto y_{t}$ of
$[\lambda_{L}Y_{u}^{*}\lambda_{R}^{\dagger}]^{e\mu}$ from Eq (V.9).

Now we want to argue that this term
$\mathrel{\vbox{\hbox{$<$}\nointerlineskip\hbox{$\sim$}}}y_{c},y_{\tau}$,
because if its larger it exceeds the experimental bound on $\Delta
C_{DR}^{\alpha\beta}$, so must be small enough to cancel against next biggest
term. In this argument, we assume that $|\lambda_{X}^{\sigma
Q}|\mathrel{\vbox{\hbox{$<$}\nointerlineskip\hbox{$\sim$}}}1$, and use a
different normalisation of the dipole operator:

$\displaystyle\widetilde{\cal O}_{D}^{\alpha\beta}$ $\displaystyle=$
$\displaystyle\overline{\ell}_{\alpha}H\sigma^{\rho\sigma}P_{R}e_{\beta}F_{\rho\sigma}$
(V.11)

where the lepton Yukawa is removed. This is more convenient for comparing the
experimental bounds MEG:2016leq ; Belletmg ; Babartmgteg ; Morel:2020dww ;
Muong-2:2021ojo ; Roussy:2022cmp ; Muong-2:2008ebm  on NP contributions to the
coefficient of the redefined dipole operator, expressed as a matrix in flavour
space:

$\displaystyle|[\widetilde{C}_{D}]^{\alpha\beta}|\leq\left[\begin{array}[]{ccc}2.3\times
10^{-6}&6\times 10^{-12}&7\times 10^{-8}\\\ 6\times 10^{-12}&4\times
10^{-7}&7\times 10^{-8}\\\ 7\times 10^{-8}&7\times
10^{-8}&-\end{array}\right]$ (V.15)

where on the diagonal are the $(g-2)_{\beta}$ constraintsMorel:2020dww ;
Muong-2:2021ojo  using $|\Delta a_{\beta}|\simeq
2m_{\beta}|C_{D}^{\beta\beta}+C_{D}^{\beta\beta*}|/(ev)$, and the EDM
constraints Roussy:2022cmp ; Muong-2:2008ebm  on ${\cal
I}m\\{C_{D}^{\beta\beta}\\}\simeq-vd_{\beta}/2$ Pospelov:2005pr  are
$|C_{D}^{ee}|\mathrel{\vbox{\hbox{$<$}\nointerlineskip\hbox{$\sim$}}}2.5\times
10^{-14}$, and
$|C_{D}^{\mu\mu}|\mathrel{\vbox{\hbox{$<$}\nointerlineskip\hbox{$\sim$}}}3\times
10^{-4}$. The contribution of Eq. (V.10), with $\lambda_{L}^{\alpha
t}\lambda_{R}^{\alpha t*}$ of order 1, is larger than the bounds of Eq. (V.15)
on all the dipole coefficients, except possibly
$[\widetilde{C}_{D}]^{\tau\tau}$. So the top contribution must cancel against
the next largest contribution to the dipole, which is relatively suppressed at
least by the charm or $\tau$ Yukawa, because the dipole operator requires a
single Higgs leg. So this implies that

$\lambda_{X}^{\alpha t}\lambda_{Y}^{\alpha
t*}\mathrel{\vbox{\hbox{$<$}\nointerlineskip\hbox{$\sim$}}}10^{-2}$

for all flavour combinations $\alpha\beta$ except $\tau\tau$, so the scalar
four-lepton coefficients are below upcoming experimental sensiticity and can
be neglected.

We also explored the possibility that constraints on vector four-quark
operatorsUTfit:2007eik  — for instance from meson-antimeson oscillations—
allow to set bounds on the vector four-lepton operators. Our hope was that the
quark sector could set bounds on the eigenvalues $\\{\lambda_{i}\\}$ of the
leptoquark coupling matrices

$[\lambda_{L}]=V_{l}^{\dagger}D_{\lambda}V_{Q}$

where $D_{\lambda}=$ diag$\\{\lambda_{1},\lambda_{2},\lambda_{3}\\}$, and
$V_{l}$ and $V_{Q}$ are unitary matrices. $K^{0}\\!-\\!\overline{K^{0}}$ and
$D^{0}\\!-\\!\overline{D^{0}}$ mixing constrain two independent combinations
of $V_{Q}^{\dagger}D^{2}_{\lambda}V_{Q}$, but in order to set an upper bound
on the eigenvalues, at least one more constraint would be required, and we did
not find useful constraints involving tops.

###  V.4 Complex Coefficients—what changes?

The coefficients of the observables operator are generally complex numbers,
and it is not immediately clear whether experiments can fully determine these
coefficients when non-zero phases are present. When considering upper limits
on branching ratios, the only difference is that we have two identical
12-dimensional ellipses, each respectively in the space of the real and
imaginary parts of the coefficients. This happens because the branching
fractions are functions of the absolute values of operator coefficient
combinations, which result in a quadrature sum of the real and imaginary
components. Therefore, the two do not mix, and the rate is a combination of
the same positive defined quadratic forms, to which the upper limit can
separately apply.

The complication arises when we consider the possibility of measuring the
complex coefficients from data. Expanding the branching ratios of Eq.
(II.7)-(II.10) in terms of the (complex) coefficients of Eq. (II.3), we
find999We assume that we can distinguish the processes with electron/positrons
of different chiralities via the angular distributions. We do not discuss the
scalar operator $C^{e\mu ee}_{S,XX}$ because it is negligible in the models we
consider.

$\displaystyle BR(\mu\to e_{X}\gamma)$ $\displaystyle=$ $\displaystyle
384\pi^{2}|C^{e\mu}_{D,Y}|^{2}$ (V.16) $\displaystyle BR(\mu\to
e_{X}\overline{e}_{X}e_{X})$ $\displaystyle=$ $\displaystyle 2|C^{e\mu
ee}_{V,XX}|^{2}+32e^{2}\left(\log\left(\frac{m_{\mu}}{m_{e}}\right)-1\right)|C^{e\mu}_{D,Y}|^{2}+16e|C^{e\mu
ee}_{V,XX}C^{e\mu}_{D,Y}|\cos(\phi_{V,XX}-\phi_{D,Y})$ (V.17) $\displaystyle
BR(\mu\to e_{X}\overline{e}_{Y}e_{Y})$ $\displaystyle=$ $\displaystyle|C^{e\mu
ee}_{V,XY}|^{2}+32e^{2}\left(\log\left(\frac{m_{\mu}}{m_{e}}\right)-\frac{3}{2}\right)|C^{e\mu}_{D,Y}|^{2}+8e|C^{e\mu
ee}_{V,XY}C^{e\mu}_{D,Y}|\cos(\phi_{V,XY}-\phi_{D,Y})$ (V.18) $\displaystyle
BR(\mu A\to e_{X}A)$ $\displaystyle=$ $\displaystyle
B_{A}\left(d^{2}_{A}|C^{e\mu}_{D,Y}|^{2}+|C^{e\mu}_{A}|^{2}+2d_{A}|C^{e\mu}_{D,Y}C^{e\mu}_{A}|\cos(\phi_{A}-\phi_{D,Y})\right)$
(V.19)

where we have defined $C_{\square}=|C_{\square}|e^{i\phi_{\square}}$, $X\neq
Y$ and $A$ can be Al or Au$\perp$. The observables only depend on relative
phases, so for brevity we relabel
$\phi_{\square}-\phi_{D,Y}\to\phi_{\square}$. We thus have 10 branching
fractions that can generally depend on 18 parameters: 10 absolute values and 8
relative phases. For example, observing $\mu\to e_{X}\gamma$ and $\mu\to
e_{X}\overline{e}_{X}e_{X}$ ($\mu\to e_{X}\overline{e}_{Y}e_{Y}$) would not be
sufficient to measure the real and imaginary parts of the coefficient $C^{e\mu
ee}_{V,XX}$ ($C^{e\mu ee}_{V,XY}$). However, some observables may be directly
related to the coefficient relative phases. For instance, it has been shown in
Okadameee  that the T-odd asymmetry term

$A^{T}_{\mu\to 3e}\propto-{\rm
Im}(C_{D,R}(3C^{*}_{V,LL}-2C^{*}_{V,LR})+(L\leftrightarrow R)=3|C^{e\mu
ee}_{V,LL}C^{e\mu ee}_{D,R}|\sin\phi_{V,LL}-2|C^{e\mu ee}_{V,LR}C^{e\mu
ee}_{D,R}|\sin\phi_{V,LR}+(L\leftrightarrow R)$ (V.20)

is accessible via the angular distribution of the outgoing
electrons/positrons, and could assist in determining the relative phase of the
$\mu\to 3e$ vectors and dipole. In addition, interpreting data assuming
specific models can help in reducing the number of relevant parameters, and
measurements may still be used to find regions of parameter space that are
incompatible with the model predictions. For the three models we considered
here:

  1. 1.

In the type II seesaw the observable coefficients are related to the neutrino
mass matrix, which is complex (given the hints of the CP violating Dirac phase
$\delta\simeq 3/2\pi$ and due the potential presence of non-zero Majorana
phases). In our analysis of the type II seesaw in Ardu:2023yyw , we identified
the region where the model can sit in the space of the angular variables
$\tan\theta=\sqrt{|C_{D}|^{2}+|C_{V,LR}|^{2}}/|C_{V,LL}|$ and
$\tan\phi=|C_{D}/C_{V,LR}|$, which depend on the absolute values of the
observable coefficients. One may wonder whether experiments can identify a
point in this space despite having complex phases, which make three branching
fractions depend on five parameters (three absolute values and two phases).
Since the flavour changing interactions with electron singlets are negligible,
the $A^{T}_{\mu\to 3e}$ asymmetry is given by the relative phases of the
$C_{V,LL},C_{V,LR}$ vectors with the $C^{e\mu}_{D,R}$ dipole. Combined with
the measurements of the branching fractions for $\mu\to e_{L}\gamma$, $\mu\to
e_{L}\overline{e}_{L}e_{L}$ and $\mu\to e_{L}\overline{e}_{R}e_{R}$, one of
five physical parameters would still remain undetermined. Taking advantage of
the fact that in the type II seesaw $C_{A}\propto C_{V,LR}$, a detection of
$\mu A\to eA$ could be used to extract the value of the last unknown,
resulting in the complete knowledge of the relevant complex coefficients
(modulo an overall phase). This also opens the interesting possibility of
taking advantage of $\mu\to e$ observables to determine the unmeasured
neutrino parameters, i.e the lightest neutrino mass and the Majorana phases.
Since the complex EFT coefficients depends on these three unknown, measuring
the coefficients could allow to infer their values. Additionally, since the
system is over-constrained, with three complex coefficients being a function
of four parameters (including the overall magnitude of LFV), we can use
experimental results to check for consistency with the type II seesaw.

  2. 2.

The operator coefficients in the inverse seesaw are given by the off-diagonal
elements of Hermitian matrices, which can be in general complex. In the case
of nearly degenerate sterile neutrinos, we have found that the coefficients
satisfy linear relations of the following form Ardu:2023yyw

$C^{e\mu ee}_{V,XY}=a_{XY}C^{e\mu}_{A,L}+b_{XY}C^{e\mu ee}_{D,R}$ (V.21)

where $a$ and $b$ are real numbers, and $XY=LL,LR$. These relations hold in
general and do not assume real coefficients. However, with non-vanishing
phases, two observables are not sufficient to fully determine the $\mu\to e$
predictions of the inverse seesaw. Observing $\mu\to e_{L}\gamma$ would give
the dipole absolute value, and measuring $\mu A\to e_{L}A$ would only yield a
combination of $|C^{e\mu}_{2q,L}|$ and dipole-four fermion relative phase.
Taking advantage of Equation V.21, a $\mu\to e_{L}\overline{e}_{R}e_{R}$
signal could allow to solve for $C^{e\mu}_{2q,L}$ as a complex number. Then,
again by means of Equation V.21 for $XY=LR$, we could predict $BR(\mu\to
e_{L}\overline{e}_{R}e_{R})$ and compare it against experimental results. We
conclude that, despite the non-zero phases, the inverse seesaw with degenerate
sterile neutrinos is predictive enough that it could be ruled out by a
combination of $\mu\to e$ observations

  3. 3.

In the leptoquark model the $\mu\to e$ coefficients depend on a number of
invariants that is larger than the number of observables. This means that the
model could already fill the experimental ellipse (with the exception of the
scalar four lepton directions) even in the case of real couplings. Allowing
complex couplings would make the model even less constrained, and thus we do
not discuss this case further.

##  VI Discussion

The purpose of bottom-up EFT is to take low-energy experimental information to
high-scale models. In this section, we discuss various aspects of this
process, in the light of the three models we considered.

####  VI.0.1  What differences among models can be identified by the data?

We showed in Ardu:2023yyw  that the data could rule out the models we
consider, because the models predict relations between the Wilson
coefficients, so are unable to fill the whole ellipse in coefficient space
that is accessible to experiments. But it would be more interesting to ask
whether $\mu\to e$ observations could identify properties of models.

A simple question is whether $\mu\to e$ data could distinguish models with LFV
couplings to either lepton doublets or singlets, vs models that interact with
both. It seems that the answer is yes. If LFV interactions involve only
doublets or singlets, the only lepton-chirality-changing interaction in the
theory is the Higgs Yukawa. So the coefficients of chirality-changing LFV
operators must be proportional to the Yukawa couplnging to an odd power. For
instance, the dipole coefficients would satisfy

$\frac{C^{e\mu}_{DR}}{C^{e\mu}_{DL}}\propto\left(\frac{m_{e}}{m_{\mu}}\right)^{\pm
1}~{}~{}~{}.$

If this relation is not satisfied, it suggests that LFV involves doublets and
singlets. If it is satisfied with exponent $-1$, then it is probable that LFV
involves doublets but not singlets. If in addition, the coefficients of other
singlet-LFV operators are negligible, it becomes very probable that LFV
involves only doublets —although it could result from accidental cancellations
in a model with LFV for doublets and singlets.

It would be interesting and useful if the data could also identify other model
properties, such as the loop order at which LFV occurs. But our models suggest
this is not possible, because loops that occur in matching the model to the
EFT are indistinguishable from small couplings, and because coefficients that
arise at tree level can be accidentally small, as occured in the type II
seesaw where $C_{VLL}^{e\mu ee}$ can vanish. For similar reasons, $\mu\to e$
observables can not distinguish NP that interacts only with leptons in its
renormalisable Lagrangian, from NP that interacts with quarks and leptons.

####  VI.0.2 What is the role of the RGEs?

There are many reasons to use Renormalisation Group Equations in EFT, as is
well-known in quark flavour physics. However, the RGEs may not be motivated in
the lepton sector, because LFV has yet to be discovered, so precise, scheme-
independent predictions are not crucial. Indeed, the RGEs are often not
included in the $\tau\to l$ sector, where the data separately constrain most
coefficients. In the $\mu\to e$ sector where there are fewer restrictive
bounds, an EFT analysis suggests that the RGEs are relevant because they mix
difficult-to-probe operators into well-constrained processes. In this section,
we explore whether this occurs in our models.

The inverse seesaw at the TeV-scale is an example of a model where the RGEs
are unneccessary, because low-energy LFV operators are generated via loop
diagrams in matching, and the RGEs just contribute an ${\cal O}(10\%)$
renormalisation. A few properties of the model contribute to this behaviour:
first, the new interactions couple one light SM particle with a heavy new
particle and a weak boson, so LFV occurs via loops that contribute in
matching. Then, there is no significant operator mixing via the RGEs, because
the photon penguin vanishes below the weak scale at one loop; that is, LFV
operators are vectors(or $\propto$ a lepton yukawa coupling) because only the
lepton doublet interacts with new particles, and vector operators only mix
into each other via the penguins. But in the one-loop penguin diagrams, the
gauge boson attaches to the Higgs, so the penguins are only present above the
weak scale, and generate $2l2q$ operators in matching.

The type II seesaw is an example of a model where the leading contribution to
some observables arises in matching, and via the RGEs for others. This
behaviour can be reproduced in EFT, and can also be obtained in model
calculations that judiciously include log-enhanced loops Dinh:2012bp .
However, the correct lower limits of the logarithms are crucial to obtain the
dependence of LFV observables on the unknown parameters of the neutrino mass
matrix in the type II seesaw model. These lower limits should be implemented
automatically in EFT, but to our knowledge were previously missing from the
literature. So it seems that the RGEs are required in this model, in order to
identify the parameter space the model cannot reach.

In the type II seesaw, a careful one-loop model calculation could include all
the terms of our leading-log EFT, because we do not resum $(\alpha\lg)^{n}$
for all $n$, but rather work to ${\cal O}(\alpha\log)$. However, our EFT also
includes the “leading” vector to dipole mixing at ${\cal O}(\alpha^{2}log)$,
which we did not find in the literature about this model. This mixing causes
the dipole to depend on the unknown neutrino parameters (Majorana phases and
$m_{min}$), thereby allowing it to vanish. It is “well known” that the 2-loop
electroweak contribution to $(g-2)_{\mu}$ is comparable to the one-loop part,
but it seems that the implications of this may not have been implemented in
all model calculations. However, it is relatively simple to implement in EFT
Crivellin:2017rmk , illustrating the first reason to do EFT: it is the simpler
way to get a more precise result.

The leptoquark is our model where the RGEs are most useful, because they allow
to simultaneously include the multitude of relevant electroweak loops and
large QCD effects in an organised fashion. The RGEs are required to obtain
model predictions, because they mix difficult-to-constrain coefficients— such
as tensor operators involving top quarks—into observable coefficients like the
dipole, while simultaneously including the QCD running of the operators. so
are required to obtain model predictions.

####  VI.0.3 Do cancellations among coefficients occur in models?

It is common to make tables listing the “sensitivity” of observables to
operator coefficients (e.g. Davidson:2020hkf ; Crivellin:2017rmk ); these
“one-at-a-time-bounds” are simple to obtain by allowing a single operator to
have a non-zero coefficient, and computing the experimental constraint upon
it. It can also be common to take these sensitivities as bounds, because it is
generally considered unlikely that models generate cancellations among
operator coefficients, especially since these coefficients run with scale.
However, such cancellations can occur, for instance via the equations of
motion, so these “sensitivities” are not true upper bounds (instead, they are
the value of the coefficient above which it could be detected).

The operator population in EFTs is often reduced via the equations of motion
(as pedagogically discussed in EoMEFT ; polonais ). This can sometimes impose
“accidental” but precise cancellations among operator coefficients. An example
is discussed in Appendix C.4: in a model, the $Z$ penguin diagrams can be
$\propto q^{2}$ (= the momentum-transfer2 of the $Z$), or $\propto v^{2}$.
Both contributions could contribute to the decay $Z\to e^{\pm}\mu^{\mp}$, but
the part $\propto q^{2}$ gives a negligible contribution to $\mu\to
e\bar{e}e$, due to the $q^{2}\sim m_{\mu}^{2}$ suppression. However if the
model is matched to SMEFT, the equations of motion are used to to transform
the $q^{2}$ part of the diagrams into four-fermion and penguin operators
(${\cal O}_{HL3}^{e\mu}$ and ${\cal O}_{HL1}^{e\mu}$), with coefficients whose
sum cancels in low-energy matrix elements where $q^{2}\to 0$. The “one at a
time bounds” on penguin and four-fermion operators miss this cancellation, so
would instead suggest that both are strictly constrained by $\mu\to
e\bar{e}e$.

In the type II seesaw model, the $Z$-penguin diagrams give negligible
contribution to $\mu\to e\bar{e}e$ because the $\propto v^{2}$ part is
suppressed by lepton Yukawas, and the $\propto q^{2}$ part is kinematically
suppressed as discussed above. So experimental observations do not exclude a
$\bar{e}\,Z\\!\\!\\!\\!/~{}\mu$ interaction just below the sensitivity of the
LHC, despite that the “one-at-a-time-bounds” suggest that they do.
Nonetheless, the model can not generate such interactions, because they are
controlled by the same model parameters as the photon penguin diagrams, which
are constrained by $\mu\to e\bar{e}e$.

So in summary, apparently accidental cancellations can occur among
coefficients in EFT; whether this affects the constraints on models is model-
dependent.

####  VI.0.4 Does it matter that coefficients are complex?

$\mu\to e$ observables define an experimentally accessible 12-dimensional
ellipse in the space of the Wilson coefficients, but these are generally
complex numbers. Although this does not complicate the analysis when imposing
bounds on the coefficients, because the ellipses for the real and imaginary
components are identical, allowing for complex phases could generally hinder
the determination of the coefficients from data. If the muon polarization and
the electron/positron angular observables could in principle identify 12 real
coefficients, unobservable directions will be present when considering the
full parameter space spanned by coefficients with non-zero phases. Measuring
for instance the branching fraction for $\mu\to e_{X}\gamma$ would identify a
circle with radius $\sqrt{BR(\mu\to e_{X}\gamma)/384\pi^{2}}$ centered at the
origin of the complex plane for the dipole coefficient $C_{DX}$, but without
further assumptions the real and imaginary parts would remain undetermined.
However, if our goal is to use data to exclude models, the model predictions
can help in reducing the number of unmeasurable direction. Interpreting data
in light of a particular model can lead to the determination of absolute
values as well as relative phases for the coefficient. The neutrino mass
models we considered in this paper are an example where this determination is
possible.

In the type II seesaw, the LFV operators are complex because their
coefficients are directly related to the neutrino mass matrix, which contains
up to three phases in the case of Majorana neutrinos. We discussed in section
V.4 how, taking advantage of the model predictions and making use of
observables directly related to the coefficient phases (see Eq. (V.20)), one
could interpret $\mu\to e$ data to determine the (complex) coefficients
predicted by the type II seesaw.

Similarly, in the inverse seesaw model, the predictions for flavour-changing
observables are determined by the magnitude of off-diagonal matrix elements,
and the operators can be complex. In section V.4, we showed that despite the
presence of operator phases, we can use the model predictions to identify
points in the experimentally accessible ellipse.

##  VII Summary

The $\mu\to e$ sector is promising for the discovery of LFV, due to the
exceptional upcoming experimental sensitivity — to three processes. So this
project explored what could be learned about the New Physics in the lepton
sector from $\mu\to e$ observations, by studying some “representative” TeV-
scale models described in Section III. We take the data to be 12 Wilson
coefficients, which can be individually constrained and distinguished in
measurements (with the exception of vector and scalar four-lepton operators
which have indistinguishable angular distributions in $\mu\to e\bar{e}e$, see
Section II).

Bottom-up EFT is an appropriate formalism to compare data with models, because
data improves slowly while models can evolve more rapidly. It also gave some
relevant effects (such as the two-loop vector to dipole mixing in the type II
seesaw, see Eq. IV.6) which we did not find in the literature. Our analysis is
at leading order in EFT, meaning that we attempt to include the largest
contribution of the model to all the operators to which the data can be
sensitive. This includes some 2-loop anomalous dimensions and some operators
which are dimension eight in SMEFT. Our notation and assumptions are
summarised in section II. Our models are located at the TeV scale in order to
profit from many complementary observables, but the ratio $m_{W}/$TeV is not
large, which implies that EFT is poorly motivated between $m_{W}\to$ TeV (see
the discussion in Appendix C.1), so in practise we match our models to the
QCD$\times$QED-invariant EFT that is relevant below $m_{W}$.

The observable operator coefficients are given in Section IV in terms of model
parameters at the TeV, which usually appear with SM parameters in elegant
combinations that recall Jarlskog invariants. This unforeseen curiosity
(discussed in Section IV.1) may be an accident of our simple leading order
analysis, which allows analytic expressions. Or possibly it indicates an
interesting new role for invariants as stepping stones in the reconstruction
of models from EFT coefficients. For instance, our models did not fulfill our
expectations: we anticipated that the type II seesaw was predictive because
the flavour-changing couplings $f_{\alpha\beta}$ (Eq.III.1) are determined by
the neutrino mass matrix, and that the inverse seesaw was unpredictive because
the $Y^{\alpha a}$ couplings (Eq. III.3) are unknown. However, Section IV.2
shows that $\mu\to e$ flavour change is controlled by two invariants in the
inverse seesaw with degenerate singlets, whereas three invariants are needed
in the type II model. In any case, it is interesting that $\mu\to e$ flavour
change in these models is controlled by a few complex numbers of magnitude
$\mathrel{\vbox{\hbox{$<$}\nointerlineskip\hbox{$\sim$}}}1$, which could be
obtained for a wide variety of flavour structures in Lagrangians.

We showed in Ardu:2023yyw  that $\mu\to e$ data has the ability to exclude the
models we consider, because they cannot fill the whole ellipse in coefficient
space accessible to upcoming experiments. In Section V, we explored the more
interesting question of whether observations could indicate a model—
specifically, the type II seesaw model — by including some complementary
observables. In Section V.1 we showed that, in the type II seesaw model, some
ratios of $\mu\to e$ Wilson coefficients can be confined to relatively narrow
intervals (depending on the mass ordering, see Eq. V.3) if the lightest
neutrino mass is small enough. Therefore, if these ratios where observed
outside the ranges quoted in Section V.1, a lower bound on the neutrino mass
scale could be inferred (assuming that neutrino masses arise from the type II
seesaw mechanism). In Section V.2, we considered $\tau$-LFV at Belle II —
still in the type II seesaw model—in the case where neither $\mu\to e\gamma$
nor $\mu\to e\bar{e}e$ are observed in upcoming experiments (see table 1). In
this case, the model makes specific predictions for $\tau$-LFV ratios that
could be observed at Belle II, as a function of the neutrino mass scale and
ordering. As a result, Belle II could contribute to constraining the neutrino
mass scale and ordering or rule out the type II seesaw model.

Finally, in the discussion section VI, we addressed some questions that arise
in a bottom-up EFT attempt to reconstruct New Physics from (low-energy) data.

#### Acknowledgements

We thank Ann-Kathrin Perrevoort for helpful comments abour observables in
$\mu\to e\bar{e}e$. 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. The work of MA is
supported by the Spanish AEI-MICINN
PID2020-113334GB-I00/AEI/10.13039/501100011033

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

References KKO ; Cirigliano:2009bz  observed that the target-dependence of the
$\mu A\to\\!eA$ rate could give information about the contributing operators.
So the aim of this Appendix is to identify the independent four-fermion
operators ${\cal O}_{Al,X}$ and ${\cal O}_{Au\perp,X}$, probed by Spin-
Independent (SI) $\mu A\to\\!eA$ on light and heavy targets. We use the
formalism of Davidson:2020ord  and use the nuclear calculation of KKO . The
operators ${\cal O}_{Al,X}$ and ${\cal O}_{Au,X}$ can be constructed in the
nucleon basis relevant at the experimental scale, or in a quark basis more
relevant for comparing to models. In the following, we first construct these
operators in the nucleon basis, where operators, coefficients and parameters
wear “tildes”, then express them in the quark basis without tildes. Notice
that normalisations change between the bases, so there are numerical
differences between, eg, $\widetilde{B}_{A}$ and $B_{A}$.

The results in this Appendix have two peculiarities, arising in the matching
of nucleons with quarks. The first is that there is a “loss of information” in
going from nucleons to quarks, because the scalar $u$ and $d$ content of a
nucleon are comparable: $\langle N|\bar{u}u|N\rangle\approx\langle
N|\bar{d}d|N\rangle$ (Or in the notation of Eq (A.8), $G_{S}^{N,u}\approx
G_{S}^{N,d}$, which is obtained both with lattice and $\chi$PT
methodsFlavourLatticeAveragingGroupFLAG:2021npn ). As a result, the
coefficients of scalar $p$ and $n$ operators need to be measured accurately,
in order to distinguish scalar coefficients involving $u$s vs $d$s. The second
curiosity is that ${\cal O}_{Au\perp,X}$ is different, when calculated in the
quark or nucleon bases. That is, Ref. Davidson:2018kud  obtained an
orthonormal basis of nucleon operators for the two-dimensional space probed by
Gold and Titanium. When these nucleon operators are matching to quarks, they
are no longer orthogonal. In Ref. Davidson:2022nnl , the operators probed by
Gold and Titanium were first matching to quarks, then decomposed into
orthonormal components. We follow the second approach here, because the aim is
to identify the independent information available about models, and that is
expressed in the quark basis in our bottom-up perspective.

In the nucleon basis at the experimental scale, the lepton-nucleon operators
relevant for Spin-Independent $\mu A\to\\!eA$ can be added to the Lagrangian
as KKO

$\delta{\cal
L}=2\sqrt{2}G_{F}\sum_{N\in\\{n,p\\}}\sum_{X\in\\{L,R\\}}\left(\widetilde{C}_{S,X}^{(NN)}\widetilde{{\cal
O}}_{S,X}^{(NN)}+\widetilde{C}_{V,X}^{(NN)}\widetilde{{\cal
O}}_{V,X}^{(NN)}+h.c.\right)$ (A.1)

where $\widetilde{{\cal
O}}_{S,X}^{(NN)}=(\overline{e}P_{X}\mu)(\overline{N}N)$ and $\widetilde{{\cal
O}}_{V,X}^{(NN)}=(\overline{e}\gamma^{\alpha}P_{X}\mu)(\overline{N}\gamma_{\alpha}N)$.
Notice that the nucleon currents are not chiral: $\widetilde{{\cal
O}}_{V,X}^{(NN)}=\widetilde{{\cal O}}_{V,XL}^{(NN)}+\widetilde{{\cal
O}}_{V,XR}^{(NN)}$, so

$\widetilde{C}_{V,X}^{(NN)}=\frac{1}{2}{\Big{(}}\widetilde{C}_{V,XL}^{(NN)}+\widetilde{C}_{V,XR}^{(NN)}{\Big{)}}~{}~{}~{}.$
(A.2)

A similar relation holds for the coefficients on quarks used in $\mu
A\to\\!eA$.

The Spin-Independent conversion rate, normalised to the $\mu$ capture
rate101010 Some capture rates are given in KKO ; Suzuki:1987jf ; the capture
rates in Aluminium, Titanium and Gold are respectively $0.7054$, 2.59, and
13.07 $\times 10^{6}$ sec-1 $(\mu+A\to\nu+A^{\prime})$, can be written KKO

$\displaystyle{\rm BR}_{SI}(\mu A\to eA)$ $\displaystyle=$
$\displaystyle\frac{32G_{F}^{2}m_{\mu}^{5}}{\Gamma_{cap}}{\Big{(}}\big{|}\widetilde{C}^{pp}_{V,R}I_{A,V}^{(p)}+\widetilde{C}^{pp}_{S,L}I_{A,S}^{(p)}+\widetilde{C}^{nn}_{V,R}I_{A,V}^{(n)}+\widetilde{C}^{nn}_{S,L}I_{A,S}^{(n)}+C_{D,L}{\frac{I_{A,D}}{4}}\big{|}^{2}+\\{L\leftrightarrow
R\\}~{}{\Big{)}}~{}~{}~{}.$ (A.3)

The nucleus($A$) and nucleon($N$) -dependent “overlap integrals”
$I_{A,V}^{(N)}$, $I_{A,S}^{(N)},I_{A,D}$ correspond to the integral over the
nucleus of the lepton wavefunctions and the appropriate nucleon density; we
use here the results of KKO . We neglect smaller contributions to the $\mu\to
e$ conversion amplitude, such as the Spin-Dependent part Cirigliano:2017azj ;
Davidson:2017nrp ; Hoferichter:2022mna  which is not coherently enhanced by
the numerous nucleons, and subdominant Spin-Independent contributions arising
from, eg $\mu e\gamma\gamma$ operators Davidson:2020ord , or from tensor
contributions to the scalar coefficient Cirigliano:2017azj .

Focusing on an outgoing electron of left helicity, the Branching Ratio can be
re-expressed as Davidson:2018kud ; Davidson:2022nnl  ($\widetilde{B}_{A}$ here
is written $B_{A}$ in Davidson:2018kud )

$\displaystyle{\rm BR}_{SI}(\mu A\to e_{L}A)$ $\displaystyle=$
$\displaystyle\widetilde{B}_{A}|\vec{\tilde{C}}_{L}\cdot\hat{v}_{A,5}|^{2}$
(A.4)

where the coefficients (of operators with outgoing $e_{L}$) are lined up in a
vector $\vec{\tilde{C}}_{L}$, the overlap integrals are lined up in a five-
component vector
$\vec{v}_{A5}=(I_{A,S}^{(p)},I_{A,V}^{(p)},I_{A,S}^{(n)},I_{A,V}^{(n)},\frac{1}{4}I_{A,D})$,
and the target-dependent constants $\widetilde{B}_{A}$ are

$\widetilde{B}_{A}=\frac{6144\pi^{3}|\vec{v}_{A5}|^{2}}{2.197\Gamma_{capt,A}10^{-6}{\rm
sec}}$ (A.5)

with $\widetilde{B}_{Al}=142$,$\widetilde{B}_{Ti}=250$, and
$\widetilde{B}_{Au}=300$.

In order to identify the four-fermion operator probed by a target $A$ (as
opposed to the combination of coefficients), it is convenient to temporarily
neglect the dipole. This implies that $\mu A\to\\!eA$ on the target nucleus
$A$ probes the combination of coefficients
$\tilde{C}_{A,L}\equiv\vec{\tilde{C}}_{L}\cdot\hat{v}_{A4}$ where
$\vec{v}_{A4}$ contains only the four-fermion overlap integrals (so neglects
the dipole). So $\tilde{C}_{A,L}$ is the coefficient of the operator

$\widetilde{{\cal O}}_{A,L}\equiv\hat{v}_{A4}\cdot(\widetilde{{\cal
O}}_{S,R}^{(p)},\widetilde{{\cal O}}_{V,L}^{(p)},\widetilde{{\cal
O}}_{S,R}^{(n)},\widetilde{{\cal O}}_{V,L}^{(n)})$ (A.6)

because the Branching Ratio resulting from $\delta{\cal
L}=\tilde{C}_{A,L}\widetilde{{\cal O}}_{A,L}$ will be

${\rm BR}_{SI}(\mu A\to
e_{L}A)=\widetilde{B}_{A}|\tilde{C}_{A,L}|^{2}\frac{|\vec{v}_{A4}|^{2}}{|\vec{v}_{A5}|^{2}}$

in agreement with eqn (A.4). In general,

${\rm BR}_{SI}(\mu A\to
e_{L}A)=\widetilde{B}_{Al}\left|\frac{I_{A}^{D}}{4|v_{A5}|}C^{e\mu}_{DR}+\frac{|v_{A4}|}{|v_{A5}|}\tilde{C}_{Al,L}\right|^{2}+L\leftrightarrow
R$ (A.7)

where on light targets like Aluminium, $I_{A}^{D}/(4|v_{A5}|)=0.27$ and
$|v_{A4}|/|v_{A5}|\simeq$0.96.

In order to express the Al and Au operators in the quark basis, there is one
more step. The nucleon operators of eqn (A.1) can be matched at 2 GeV onto
light quark operators ${\cal
O}_{S,X}^{qq}=(\overline{e}P_{X}\mu)(\overline{q}q)$, ${\cal
O}_{V,X}^{qq}=(\overline{e}\gamma^{\alpha}P_{X}\mu)(\overline{q}\gamma_{\alpha}q)$
using

$\displaystyle\langle N(P_{f})|\bar{q}(x)\Gamma_{O}q(x)|N(P_{i})\rangle\simeq
G^{N,q}_{O}\langle
N(P_{f})|\bar{N}(x)\Gamma_{O}N(x)|N(P_{i})\rangle=G^{N,q}_{O}\overline{u_{N}}(P_{f})\Gamma_{O}u_{N}(P_{i})e^{-i(P_{f}-P_{i})x}~{}~{}~{}.$
(A.8)

where the parameters $G^{N,q}_{O}$ can be determined from sum rules, lattice
calculations or experiment, and we use the values summarised in
Davidson:2022nnl . Below the heavy quark ($b$ and $c$) mass scales, there is
also a two-step contribution to scalar nucleon operators from scalar quark
operators SVZ ; CKOT , via matching first onto the gluon operator
$(\bar{e}P_{R}\mu)GG$, then onto nucleons111111 The scalar operator with top
quarks also matches at $m_{t}$ onto $(\bar{e}P_{R}\mu)GG$, whose QCD running
we suppose is accounted for by the wavefunction contributions, and which at 2
GeV matches onto nucleons like for the $b$ and $c$:  $G^{NQ}_{S}\simeq
0.9\frac{2m_{N}}{27m_{Q}(m_{Q})}\frac{\alpha_{s}(m_{Q})}{\alpha_{s}(2{\rm
GeV})}~{}~{}.$ However, we do not write the top contribution because it is
absent from the EFT below the weak scale, so we include it at $m_{t}$. As a
result, the nucleon and quark coefficients are related as

$\displaystyle\widetilde{{C}}^{N}_{O,X}=\sum_{q}G_{O}^{Nq}C_{O,X}^{qq}~{}~{}~{},$
(A.9)

for $q\in\\{u,d,s,c,b,t\\}$, and $O\in\\{S,V\\}$. This allows to define quark
“overlap integrals” for target $A$ as

$\displaystyle I^{q}_{A,S}$ $\displaystyle=$ $\displaystyle
G_{S}^{pq}I_{A,S}^{p}+G_{S}^{nq}I_{A,S}^{n}$ $\displaystyle I^{q}_{A,V}$
$\displaystyle=$ $\displaystyle G_{V}^{pq}I_{A,V}^{p}+G_{V}^{nq}I_{A,V}^{n}$
(A.10)

which can be assembled into a “target vector” $\vec{u}_{A}$ in the space of
operators describing $\mu\to e$ conversion on quarks:

$\displaystyle\vec{u}_{A}$ $\displaystyle=$
$\displaystyle(I^{u}_{A,S},I^{d}_{A,S},I^{s}_{A,S},I^{c}_{A,S},I^{b}_{A,S},I^{t}_{A,S},I^{u}_{A,V},I^{d}_{A,V})~{}~{},$
(A.11) $\displaystyle\hat{u}_{Al}$ $\displaystyle=$
$\displaystyle(0.692,0.699,0.0341,0.00400,0.00121,0.0000118,0.125,0.128)~{}~{},~{}~{}|\vec{u}_{Al}|=0.397$
(A.12) $\displaystyle\hat{u}_{Ti}$ $\displaystyle=$
$\displaystyle(0.690,0.699,0.0340,0.00398,0.00121,0.0000118,0.127,0.134)~{}~{},~{}~{}|\vec{u}_{Ti}|=0.991$
(A.13) $\displaystyle\hat{u}_{Au}$ $\displaystyle=$
$\displaystyle(0.672,0.689,0.0334,0.00380,0.00118,0.0000115,0.177,0.202)~{}~{},~{}~{}|\vec{u}_{Au}|=1.923$
(A.14)

Analogously to eqn (A.6), we use $\hat{u}_{Al}\approx\hat{u}_{Ti}$ to define
${\cal O}_{Al}$ in eqn (II.4). The orthogonal direction probed by Gold,
$\hat{u}_{Au\perp}$ is defined as

$\displaystyle\vec{u}_{Au}$ $\displaystyle=$
$\displaystyle|\vec{u}_{Au}|(\cos\theta_{\rm A}\hat{u}_{Al}+\sin\theta_{\rm
A}\hat{u}_{\perp})$ $\displaystyle\Rightarrow\hat{u}_{\perp}$ $\displaystyle=$
$\displaystyle\frac{\hat{u}_{Au}-\cos\theta_{A}\hat{u}_{Al}}{\sin\theta_{A}}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\hat{u}_{\perp}$
$\displaystyle=$
$\displaystyle(-0.21,-0.099,-0.0075,-0.001,-0.0003,-0.000003,0.558,0.7956)$
(A.15)

where the operators are in the order given in Eq. (A.11),
sin$\theta_{A}\approx 0.093$ and $\theta_{A}$ is $\approx$ 5.3
degreesDavidson:2022nnl . This gives the definition of ${\cal O}_{Au\perp}$ in
eqn (II.6).

Finally, it can be convenient, in the quark operator basis, to write

$\displaystyle BR(\mu A\to\\!e_{L}A)$ $\displaystyle=$ $\displaystyle
B_{A}{\Big{|}}\vec{C}_{L}\cdot\hat{u}_{A}+C_{D,R}\frac{I_{D}}{4|\vec{u}_{A}|}{\Big{|}}^{2}$
(A.16)

where $B_{A}$ is defined as is eqn (A.5) but with
$\vec{v}_{A5}\to\vec{u}_{A}$, $B_{Al}=19363$, $B_{Ti}=32860$, $B_{Au}=24519$,
and $d_{A}=I_{D}/(4|\vec{u}_{A}|)=0.0228,0.0219,0.0246$ respectively for Al,
Ti and Au.

##  Appendix B observable coefficients at $\Lambda_{NP}$

The coefficients of the Lagrangian of Eq (II.3), which are at the experimental
scale $m_{\mu}$, are written in Davidson:2020hkf  as linear combinations of
coefficients at some higher scale taken to be $m_{W}$. These formulae allows
to identify what magnitude of which coefficients needs to be retained in
matching to the models. This Appendix gives the linear combination of
coefficients probed by $\mu A\to\\!eA$ on Aluminium, not given in
Davidson:2020hkf  and also on Gold for completeness.

Using the results of KKO , the combination of coefficients at the scale
$\Lambda$ which is probed by $\mu A\to\\!eA$ on Aluminium, for outgoing
$e_{L}$, is:

$\displaystyle\sqrt{\frac{BR^{exp}_{Al}}{\widetilde{B}_{Al}}}$
$\displaystyle\mathrel{\vbox{\hbox{$>$}\nointerlineskip\hbox{$\sim$}}}$
$\displaystyle{\Big{|}}0.266C_{D,R}(m_{\mu})+1.454C^{uu}_{V,L}+1.490C^{dd}_{V,L}-0.86\frac{8\pi
m_{N}}{9\alpha_{s}m_{t}}C_{GG,R}$ (B.1)
$\displaystyle-{\frac{\alpha}{3\pi}}{\Big{(}}2(C^{uu}_{V,L}+C^{cc}_{V,L})-(C^{dd}_{V,L}+C^{ss}_{V,L}+C^{bb}_{V,L})-(C^{ee}_{V,L}+C^{\mu\mu}_{V,L}+C^{\tau\tau}_{V,L}){\Big{)}}\widetilde{\ln}$
$\displaystyle+\frac{\alpha}{\pi}{\Big{(}}(1.5C^{dd}_{A,L}-3C^{uu}_{A,L})+\frac{1}{6}(C^{ee}_{V,L}+C^{\mu\mu}_{V,L}-C^{ee}_{A,L}-C^{\mu\mu}_{A,L}){\Big{)}}\widetilde{\ln}$
$\displaystyle+\eta^{a_{S}}\left(1+\frac{13\alpha}{6\pi}\widetilde{\ln}\right)(8.06C^{uu}_{S,R}+\frac{1.7m_{N}}{27m_{c}}C^{cc}_{S,R})+\eta^{a_{S}}\left(1+\frac{5\alpha}{3\pi}\widetilde{\ln}\right)(8.14C^{dd}_{S,R}+0.405C^{ss}_{S,R}+\frac{1.7m_{N}}{27m_{b}}C^{bb}_{S,R})$
$\displaystyle-\eta^{a_{T}}f_{TS}\frac{4\alpha}{\pi}{\Big{(}}16.12C^{uu}_{T,RR}+\frac{3.4m_{N}}{27m_{c}}C^{cc}_{T,RR}-8.14C^{dd}_{T,RR}-0.405C^{ss}_{T,RR}-\frac{1.7m_{N}}{27m_{b}}C^{bb}_{T,RR}){\Big{)}}\widetilde{\ln}{\Big{|}}$

where the result for Titanium can be approximately obtained by replacing
$\widetilde{B}_{Al}=142\longrightarrow\widetilde{B}_{Ti}=250$. The constraint
from $\mu A\to\\!eA$ on Gold is:

$\displaystyle 4.9\times 10^{-8}$
$\displaystyle\mathrel{\vbox{\hbox{$>$}\nointerlineskip\hbox{$\sim$}}}$
$\displaystyle{\Big{|}}\left(0.222C_{D,R}(m_{\mu})+1.602C^{uu}_{V,L}+1.830C^{dd}_{V,L}-0.721f_{N}^{GG}\frac{8\pi
m_{N}}{9\alpha_{s}m_{t}}C_{GG,R}\right)$
$\displaystyle-\frac{\alpha}{2\pi}\widetilde{\ln}{\Big{(}}6.41C^{uu}_{A,L}{-}3.66C^{dd}_{A,L}-0.305({C^{ee}_{V,L}}+C^{\mu\mu}_{V,L}{-C^{ee}_{A,L}}-C^{\mu\mu}_{A,L})+0.228C_{ping,1}{\Big{)}}$
$\displaystyle{+\eta^{a_{S}}{\Big{(}}1+\frac{\alpha}{4\pi}\frac{26}{3}\widetilde{\ln}{\Big{)}}}\left(6.10C^{uu}_{S,R}+\frac{1.30m_{N}}{27m_{c}}C^{cc}_{S,R}\right){+\eta^{a_{S}}{\Big{(}}1+\frac{\alpha}{4\pi}\frac{20}{3}\widetilde{\ln}{\Big{)}}}\left(6.26C^{dd}_{S,R}+0.303C^{ss}_{S,R}+\frac{1.30m_{N}}{27m_{b}}C^{bb}_{S,R}\right)$
$\displaystyle-
f_{TS}\eta^{a_{T}}\frac{4\alpha}{\pi}\widetilde{\ln}\left(2(6.100C^{uu}_{T,RR}+\frac{1.30m_{N}}{27m_{c}}C^{cc}_{T,RR})-6.258C^{dd}_{T,RR}-0.303C^{ss}_{T,RR}-\frac{1.30m_{N}}{27m_{b}}C^{bb}_{T,RR}\right){\Big{|}}$

where we used the scalar quark densities in the nucleon of Ref.
Hoferichter:2015dsa , the coefficients are at $\Lambda$, $m_{f}=m_{f}(m_{f})$,

$C^{ff}_{V,L}\equiv\frac{1}{2}(C^{e\mu ff}_{V,LR}+C^{e\mu
ff}_{V,LL})~{}~{}~{},~{}~{}~{}C^{ff}_{A,L}\equiv\frac{1}{2}(C^{e\mu
ff}_{V,LR}-C^{e\mu
ff}_{V,LL})~{}~{}~{},~{}~{}~{}C^{ff}_{S,L}\equiv\frac{1}{2}(C^{e\mu
ff}_{S,LR}+C^{e\mu ff}_{S,LL})$

$\eta$ and $f_{TS}$ are defined in Eq.(II.17),
$\widetilde{\ln}\equiv\ln(\Lambda/m_{\psi})$ where
$m_{\psi}\in\\{m_{b},2~{}{\rm GeV},m_{\mu}\\}$ is a suitably chosen lower
cutoff for the logarithm, and the dipole contribution is given at the
experimental scale (it can be written in terms of coefficients at $\Lambda$ as
given in Davidson:2020hkf ). A similar bound applies with $L$ and $R$
interchanged.

.

##  Appendix C Matching the models with the EFT

This section summarises the matching of the models with EFT. We first discuss
which EFT to match onto in Section C.1, and how we define the lepton flavour
basis in Section C.2, then give the matching of our three models onto a