Lepton Flavor Violation in Semileptonic Observables

I. Plakias${}^{\,a}$, O. Sumensari${}^{\,a}$

a Université Paris-Saclay, CNRS/IN2P3, IJCLab, 91405 Orsay, France

###### Abstract

In this paper, we perform a comprehensive study of Lepton Flavor Violation
(LFV) in semileptonic transitions in the framework of an Effective Field
Theory with general flavor structure. We account for the Renormalization Group
Equations (RGEs), which induce non-trivial correlations between the different
types of LFV processes. In particular, we show that these loop effects are
needed to improve the bounds on several coefficients that are not efficiently
constrained at tree level. For illustration, we consider a few concrete
scenarios, with predominant couplings to third-generation quarks, and we
explore the correlations between the various tree- and loop-level constraints
on these models. As a by-product, we also provide expressions for several
semileptonic LFV meson decays by using the latest determinations of the
relevant hadronic form factors on the lattice.

###### Contents

  1. 1 Introduction
  2. 2 EFT approach for LFV
    1. 2.1 Low-Energy EFT
    2. 2.2 SM EFT
    3. 2.3 Summary
  3. 3 Low-energy probes
    1. 3.1 $q_{l}\to q_{k}\ell_{i}\ell_{j}$
      1. 3.1.1 Leptonic decays: $P\to\ell_{i}\ell_{j}$
      2. 3.1.2 Leptonic decays: $V\to\ell_{i}\ell_{j}$
      3. 3.1.3 Semileptonic decays: $P\to P^{\prime}\ell_{i}^{-}\ell_{j}^{+}$
      4. 3.1.4 Semileptonic decays: $P\to V\ell_{i}^{-}\ell_{j}^{+}$
    2. 3.2 $\tau\to\ell M$
      1. 3.2.1 Pseudoscalar mesons: $\tau\to\ell P$
      2. 3.2.2 Vector mesons: $\tau\to\ell V$
    3. 3.3 $\ell_{j}\to\ell_{i}\gamma$ and $\ell_{j}\to\ell_{i}\ell_{k}\ell_{k}$
    4. 3.4 $\mu N\to eN$
  4. 4 High-energy probes
    1. 4.1 $Z\to\ell_{i}\ell_{j}$
    2. 4.2 $h\to\ell_{i}\ell_{j}$
    3. 4.3 $t\to c\ell_{i}\ell_{j}$
    4. 4.4 $pp\to\ell_{i}\ell_{j}$
  5. 5 Numerical results
    1. 5.1 LFV from top-quark loops
    2. 5.2 Correlations in concrete models
  6. 6 Conclusion
  7. 7 Acknowledgments
  8. A Conventions
  9. B LEFT basis for $b\to s\ell_{i}\ell_{j}$
  10. C Matching to the SM EFT
  11. D Semileptonic differential formulas
    1. D.1 $P\to P^{\prime}\ell_{i}^{-}\ell_{j}^{+}$
    2. D.2 $P\to V\ell_{i}^{-}\ell_{j}^{+}$
  12. E SM Yukawa running
  13. F Tree-level mediators

##  1 Introduction

Lepton Flavor Violation (LFV) is strictly forbidden by the accidental
symmetries in the Standard Model (SM). However, these symmetries are not
necessarily respected by higher-dimensional operators appearing beyond the SM,
which are suppressed by inverse powers of a heavy scale $\Lambda$. Indeed, the
observation of neutrino masses and oscillation is the first indication that
Lepton Number might be broken, which can be described by dimension-five
operators (suppressed by $1/\Lambda$) [1]. The smallness of neutrino masses,
however, guarantees that LFV in charged processes remains heavily suppressed
unless new dynamics beyond neutrino masses are present [2]. Therefore, these
processes are clean probes of New Physics, as their observation would be
unambiguous evidence of new phenomena, typically arising through dimension-six
operators (suppressed by $1/\Lambda^{2}$).

Considerable progress is expected from current and future experiments
searching for LFV in both leptonic and semileptonic processes, improving the
sensitivity by at least one order of magnitude in several channels [3]. These
include the various experiments targeting the $\mu\to e$ transition, namely
MEG-II [4], Mu2E [5], Mu3E [6] and COMET [7], as well as Belle-II that will
improve our sensitivity to both leptonic and hadronic LFV $\tau$-lepton decays
[8]. There has also been an increasing effort by LHCb and Belle to improve the
limits of several LFV decays of $B$-mesons [9, 10]. Furthermore, experiments
with kaons at NA62 [11] and KOTO [12], and charmed hadrons at BES-III [13],
are also expected to improve their sensitivity to LFV decays in the near
future.

New Physics contributions to LFV processes can be fully characterized by an
Effective Field Theory (EFT) defined above the electroweak scale, which must
be invariant under the SM gauge symmetry $SU(3)_{c}\times SU(2)_{L}\times
U(1)_{Y}$, namely the SM EFT [14]. The leading effective operators that induce
LFV at low energies are the leptonic dipoles ($\psi^{2}HX$), as well as four-
fermion leptonic and semileptonic operators ($\psi^{4}$) [15]. The remarkable
experimental sensitivity of LFV searches in specific channels implies that
some of these operators can be better probed through radiative corrections,
instead of their direct tree-level contributions to low-energy processes. An
example is the operator $(\bar{\mu}\gamma^{\mu}e)(\bar{q}\gamma^{\mu}q)$, with
$q\in\\{s,c,b,t\\}$, which can be more efficiently probed through its
contributions at one-loop to $\mu\to 3e$ rather than by studying semileptonic
processes at tree level [16, 17]. Therefore, it is clear that radiative
corrections must be systematically included to assess the full potential of
LFV processes to discover New Physics. This program has been carried out to a
large extent in the leptonic sector [16, 18, 17]. However, analogous
systematic study for semileptonic transitions is still missing, although the
one-loop anomalous dimensions are known both in the low-energy EFT [16, 19]
and in the SM EFT [20] (see Ref. [21, 22] for the first steps in this
direction).

In this paper, we will perform a comprehensive analysis of LFV in semileptonic
observables, accounting for the one-loop Renormalization Group Equations
(RGEs) in the low-energy EFT and the SM EFT. We will demonstrate that such
loop effects induce non-trivial correlations between the different types of
processes. Moreover, we will provide general expressions for the relevant
(semi)leptonic meson decays that can be studied experimentally, using the
latest inputs for the needed hadronic matrix elements. We will also revisit
leptonic processes, since they can provide valuable constraints on
semileptonic operators through operator mixing, including the known two-loop
contributions. To compare the sensitivity of different observables, we will
consider a specific EFT scenario with operators containing only third-
generation quarks, which contributes to various transitions through RGE
evolution above and below the electroweak scale. Finally, we will also briefly
discuss a few concrete New Physics scenarios to illustrate the relevance of
the loop effects that we compute.

The remainder of this paper is organized as follows. In Sec. 2, we define our
EFT Lagrangian and we discuss the relevant RGE effects in the low-energy EFT
and the SM EFT. In Sec. 3, we study the low-energy probes of LFV in meson and
lepton decays. In Sec. 4, we discuss the high-energy probes of LFV in Higgs,
top-quark and $Z$-boson decays, as well as indirect constraints obtained from
the high-energy tails of $pp\to\ell_{i}\ell_{j}$, with $i\neq j$. In Sec. 5,
we perform a numerical study with a specific EFT scenario and discuss the
relevance of our loop constraints to concrete New Physics models. Our findings
are briefly summarized in Sec. 6.

##  2 EFT approach for LFV

In this Section, we define our effective approach and describe the RGE effects
that are relevant for LFV observables. In Sec. 2.1, we formulate the low-
energy EFT, invariant under $SU(3)_{c}\times U(1)_{\mathrm{em}}$, which will
be used to compute the low-energy observables in Sec. 5. In Sec. 2.2, we
consider the SM EFT [14], which gives the appropriate description of New
Physics arising well above the electroweak scale, as we assume in this paper.
The RGE contributions that we consider in our phenomenological analysis are
summarized in Sec. 2.3 (see also Figs. 2 and 3).

###  2.1 Low-Energy EFT

We start by defining our low-energy effective Lagrangian

$\mathcal{L}_{\mathrm{LEFT}}=\sum_{I}\dfrac{{C}_{I}}{v^{2}}\,{O}_{I}\,,$ (1)

where $v=(\sqrt{2}G_{F})^{-1/2}$, $G_{F}$ is the Fermi constant, and $C_{I}$
denotes the low-energy effective coefficients of the operators invariant under
$SU(3)_{c}\times U(1)_{\mathrm{em}}$ . Below the electroweak scale, the
lowest-dimension operators that can violate lepton flavor are [15]

  * i)

Operators of semileptonic type ($\psi^{4}$):

$\displaystyle\begin{split}{O}_{\begin{subarray}{c}V_{XY}\\\
ijkl\end{subarray}}^{(q)}=\big{(}\bar{\ell}_{i}\gamma_{\mu}P_{X}\ell_{j}\big{)}&\big{(}\bar{q}_{k}\gamma^{\mu}P_{Y}q_{l}\big{)}\,,\qquad\quad{O}_{\begin{subarray}{c}S_{XY}\\\
ijkl\end{subarray}}^{(q)}=\big{(}\bar{\ell}_{i}P_{X}\ell_{j}\big{)}\big{(}\bar{q}_{k}P_{Y}q_{l}\big{)}\,,\\\\[3.99994pt]
{O}_{\begin{subarray}{c}T_{X}\\\
ijkl\end{subarray}}^{(q)}&=\big{(}\bar{\ell}_{i}\sigma_{\mu\nu}P_{X}\ell_{j}\big{)}\big{(}\bar{q}_{k}\sigma^{\mu\nu}P_{X}q_{l}\big{)}\,,\end{split}$
(2)

where $X,Y\in\\{L,R\\}$, $q\in\\{u,d\\}$ and $\\{i,j,k,l\\}$ denote flavor
indices.

  * ii)

Four-fermion leptonic operators ($\psi^{4}$):

$\displaystyle{O}_{\begin{subarray}{c}V_{XY}\\\
{ijkk}\end{subarray}}^{(\ell)}$
$\displaystyle=\big{(}\bar{\ell}_{i}\gamma_{\mu}P_{X}\ell_{j}\big{)}\big{(}\bar{\ell}_{k}\gamma^{\mu}P_{Y}\ell_{k}\big{)}\,,\qquad\quad{O}_{\begin{subarray}{c}S_{XX}\\\
ijkk\end{subarray}}^{(\ell)}=\big{(}\bar{\ell}_{i}P_{X}\ell_{j}\big{)}\big{(}\bar{\ell}_{k}P_{X}\ell_{k}\big{)}\,,$
(3)

where $X,Y\in\\{L,R\\}$, as above, and $k$ can take any value. For $i\neq
k\neq j$, there is an additional type of tensor operator that can be written
down,

$\displaystyle\qquad{O}_{\begin{subarray}{c}T_{X}\\\
ijkk\end{subarray}}^{(\ell)}$
$\displaystyle=\big{(}\bar{\ell}_{i}\sigma_{\mu\nu}P_{X}\ell_{j}\big{)}\big{(}\bar{\ell}_{k}\sigma^{\mu\nu}P_{X}\ell_{k}\big{)}\,,\leavevmode\nobreak\
\qquad\quad(i\neq k\neq j)\,,$ (4)

and, for $X\neq Y$, there are additional scalar operators,

$\displaystyle\quad{O}_{\begin{subarray}{c}S_{XY}\\\
ijkk\end{subarray}}^{(\ell)}$ $\displaystyle\overset{X\neq
Y}{=}\big{(}\bar{\ell}_{i}P_{X}\ell_{j}\big{)}\big{(}\bar{\ell}_{k}P_{Y}\ell_{k}\big{)}\,,\leavevmode\nobreak\
\qquad\qquad\quad(i\neq k\neq j)\,,$ (5)

which would be redundant via Fierz relations if $k=i$ or $k=j$.

  * iii)

The electromagnetic dipole operators ($\psi^{2}X$),

$\displaystyle{O}_{\begin{subarray}{c}D_{X}\\\
ij\end{subarray}}^{(\ell)}=m_{\ell}\,\big{(}\bar{\ell}_{i}\sigma_{\mu\nu}P_{X}\ell_{j}\big{)}\,F^{\mu\nu}\,.$
(6)

where $X\in\\{L,R\\}$ and $m_{\ell}$ is the mass of the heaviest lepton
appearing in the operator, which we assume to be $m_{\ell_{j}}$ in the
following.

  * iv)

The gluonic operators ($\psi^{2}X^{2}$), 111Note that similar operators can be
defined with the $U(1)_{\mathrm{em}}$ field-strength tensors, which would be
relevant to processes such as $\mu\to e\gamma\gamma$ that are not considered
in this paper, see e.g. Ref. [23, 24].

$\displaystyle{O}_{\begin{subarray}{c}{G}_{X}\\\ ij\end{subarray}}^{(\ell)}$
$\displaystyle=\dfrac{\alpha_{s}}{4\pi}\dfrac{m_{\ell}}{v^{2}}\,\big{(}\bar{\ell}_{i}P_{X}\ell_{j}\big{)}G_{a\,\mu\nu}{G}^{a\,\mu\nu}\,,\qquad\quad{O}_{\begin{subarray}{c}\widetilde{G}_{X}\\\
ij\end{subarray}}^{(\ell)}=\dfrac{\alpha_{s}}{4\pi}\dfrac{m_{\ell}}{v^{2}}\,\big{(}\bar{\ell}_{i}P_{X}\ell_{j}\big{)}G_{a\,\mu\nu}\widetilde{G}^{a\,\mu\nu}\,,$
(7)

where $X\in\\{L,R\\}$, $m_{\ell}\equiv m_{\ell_{j}}$ is the mass of the
heaviest lepton in the operator, as before, and we define the dual field-
strength as
$\widetilde{G}^{a\,\mu\nu}=1/2\,\varepsilon^{\alpha\beta\mu\nu}{G}^{a}_{\alpha\beta}$.

Our notation for the SM fields is specified in Appendix A. Flavor indices of
semileptonic operators are assigned with the first two indices corresponding
to the lepton flavor and the last two to the quark ones. Furthermore, the
vector Wilson coefficients must satisfy the following Hermiticy condition,

$\displaystyle C_{\begin{subarray}{c}V_{XY}\\\ ijkl\end{subarray}}^{(q)}$
$\displaystyle=C_{\begin{subarray}{c}V_{XY}\\\
jilk\end{subarray}}^{(q)\,\ast}\,,$ (8)

whereas the scalar and tensor coefficients satisfy

$\displaystyle C_{\begin{subarray}{c}S_{LL}\\\ ijkl\end{subarray}}^{(q)}$
$\displaystyle=C_{\begin{subarray}{c}S_{RR}\\\
jilk\end{subarray}}^{(q)\,\ast}\,,\qquad\qquad\quad
C_{\begin{subarray}{c}S_{LR}\\\
ijkl\end{subarray}}^{(q)}=C_{\begin{subarray}{c}S_{RL}\\\
jilk\end{subarray}}^{(q)\,\ast}\,,$ (9) $\displaystyle
C_{\begin{subarray}{c}T_{LL}\\\ ijkl\end{subarray}}^{(q)}$
$\displaystyle=C_{\begin{subarray}{c}T_{RR}\\\
jilk\end{subarray}}^{(q)\,\ast}\,,\qquad\qquad\quad
C_{\begin{subarray}{c}T_{LR}\\\
ijkl\end{subarray}}^{(q)}=C_{\begin{subarray}{c}T_{RL}\\\
jilk\end{subarray}}^{(q)\,\ast}\,.$ (10)

To avoid redundancies in the operator basis, we will express the observables
only in terms of Wilson coefficients with the leptonic indices $i,j$ ordered
as $i<j$. From the four-lepton coefficients $C_{V_{XX}}^{(\ell)}$, we choose
only to write the ones with indices $ijkk$, where $i<j$ and $k$ can be any
flavor index, since we are only interested in operators that violate lepton
flavors by one unit, i.e. with $\Delta L_{i}=-\Delta L_{j}=-1$.

Figure 1: Finite one-loop contribution from scalar coefficients $C_{S_{XY}}$
to the gluonic operators $C_{G_{X}}^{(\ell)}$ defined in Eq. (7), which is
relevant e.g. for the phenomenology of $\mu\to e$ conversion in nuclei.

Note, in particular, that the CP-odd and CP-even gluonic operators
$\smash{{O}_{G_{X}}^{(\ell)}}$ and $\smash{{O}_{\widetilde{G}_{X}}^{(\ell)}}$
defined in Eq. (7) are of higher dimension. We list them among the other
operators since they can be produced via an anomaly triangle when heavy quarks
are integrated out from the low-energy EFT at one-loop, cf. Fig. 1. The shifts
of the gluonic coefficients induced by the heavy quarks read [25]

$\displaystyle C_{G_{X}}^{(\ell)}$ $\displaystyle\to
C_{G_{X}}^{(\ell)}-\frac{v}{3m_{\ell}}\sum_{q=c,b,t}\frac{v}{m_{q}}\big{(}C_{S_{XL}}^{(q)}+C_{S_{XR}}^{(q)}\big{)}\,,$
(11) $\displaystyle C_{\widetilde{G}_{X}}^{(\ell)}$ $\displaystyle\to
C_{\widetilde{G}_{X}}^{(\ell)}-\frac{iv}{2m_{\ell}}\sum_{q=c,b,t}\frac{v}{m_{q}}\big{(}C_{S_{XL}}^{(q)}-C_{S_{XR}}^{(q)}\big{)}\,,$
(12)

where $X\in\\{L,R\\}$ and we have kept the first term in the $1/m_{q}$
expansion. These finite matching contributions will be considered in our
phenomenological analysis since they are the leading contributions to gluonic
operators in our setup, having implications to processes such as $\mu N\to eN$
(see also Ref. [26]).

#### Operator mixing

The one-loop running of the LFV low-energy EFT has been computed in full
generality Ref. [16] (see also Ref. [19]). Besides the known renormalization
of the scalar and tensor semileptonic operators by QCD [27], it has been shown
that the QED running induces a nontrivial mixing of LFV operators, cf. Fig. 2.
For instance, the operator
$\smash{(\bar{\ell}_{i}\gamma^{\mu}P_{X}\ell_{j})(\bar{q}\gamma_{\mu}q)}$
mixes into
$\smash{(\bar{\ell}_{i}\gamma^{\mu}P_{X}\ell_{j})(\bar{\ell}_{k}\gamma_{\mu}\ell_{k})}$
via a penguin diagram depicted in Fig. 2. Such effects allow us to constrain
semileptonic operators with same-flavor quarks by using the experimental
limits on $\ell_{j}^{-}\to\ell_{i}^{-}\ell_{k}^{+}\ell_{k}^{-}$, with $i\leq
k<j$. Another important example is the one-loop mixing of
$\smash{(\bar{\ell}_{i}\sigma^{\mu\nu}P_{X}\ell_{j})(\bar{q}\sigma_{\mu\nu}P_{X}q)}$
into the dipoles
$\smash{(\bar{\ell}_{i}\sigma^{\mu\nu}P_{X}\ell_{j})F_{\mu\nu}}$, which induce
chirality-enhanced contributions to $\ell_{j}\to\ell_{i}\gamma$, as shown in
Fig. 2 [16, 28, 29].

In the following, we consider LFV operators with fixed lepton flavor indices
$i<j$. The _leading logarithm_ solution of the RGEs from the electroweak scale
($\mu_{\mathrm{ew}}$) to the relevant low-energy scale ($\mu_{\mathrm{low}}$)
is given by

$C_{I}(\mu_{\mathrm{low}})\simeq
C_{I}(\mu_{\mathrm{ew}})+\dfrac{\gamma_{JI}}{16\pi^{2}}\log(\mu_{\mathrm{low}}/\mu_{\mathrm{ew}})\,C_{J}(\mu_{\mathrm{ew}})\,,$
(13)

where $\gamma_{JI}$ are the elements of the one-loop anomalous-dimension
matrix, which is defined via
$16\pi^{2}\,\mu\mathrm{d}C_{I}/\mathrm{d}\mu=\gamma_{JI}\,C_{J}$. These
_leading logarithm_ effects can be schematically written in our case as
follows [16],

$\displaystyle\begin{pmatrix}\vec{\bm{C}}_{V}\\\\[3.00003pt]
\vec{\bm{C}}_{S}\\\\[3.00003pt] \vec{\bm{C}}_{T}\\\\[3.00003pt]
\vec{\bm{C}}_{D}\\\\[3.00003pt]
\vec{\bm{C}}_{G}\end{pmatrix}_{\mu_{\mathrm{low}}}\overset{\tiny
1-\mathrm{loop}}{\simeq}\begin{pmatrix}U_{VV}&0&0&0&0\\\\[1.50002pt]
0&U_{SS}&U_{ST}&0&U_{SG}\\\\[1.50002pt] 0&U_{TS}&U_{TT}&0&0\\\\[1.50002pt]
0&0&U_{DT}&U_{DD}&0\\\\[1.50002pt]
0&0&0&0&U_{GG}\end{pmatrix}\cdot\begin{pmatrix}\vec{\bm{C}}_{V}\\\\[1.50002pt]
\vec{\bm{C}}_{S}\\\\[1.50002pt] \vec{\bm{C}}_{T}\\\\[1.50002pt]
\vec{\bm{C}}_{D}\\\\[1.50002pt]
\vec{\bm{C}}_{G}\end{pmatrix}_{\mu_{\mathrm{ew}}}\,,$ (14)

where we collect the $\Delta L_{i}=-\Delta L_{j}$ Wilson coefficients in the
vectors

$\displaystyle\vec{\bm{C}}_{V}$
$\displaystyle=\Big{(}\vec{{C}}_{\begin{subarray}{c}V_{LL}\\\
ij\end{subarray}}^{(d)}\,,\leavevmode\nobreak\
\vec{{C}}_{\begin{subarray}{c}V_{LR}\\\
ij\end{subarray}}^{(d)}\,,\leavevmode\nobreak\
\vec{{C}}_{\begin{subarray}{c}V_{LL}\\\
ij\end{subarray}}^{(u)}\,,\leavevmode\nobreak\
\vec{{C}}_{\begin{subarray}{c}V_{LR}\\\
ij\end{subarray}}^{(u)}\,,\leavevmode\nobreak\
\vec{{C}}_{\begin{subarray}{c}V_{LL}\\\
ij\end{subarray}}^{(\ell)}\,,\leavevmode\nobreak\
\vec{{C}}_{\begin{subarray}{c}V_{LR}\\\ ij\end{subarray}}^{(\ell)}\Big{)}\,,$
(15) $\displaystyle\vec{\bm{C}}_{S}$
$\displaystyle=\Big{(}\vec{{C}}_{\begin{subarray}{c}S_{LL}\\\
ij\end{subarray}}^{(d)}\,,\leavevmode\nobreak\
\vec{{C}}_{\begin{subarray}{c}S_{LR}\\\
ij\end{subarray}}^{(d)}\,,\leavevmode\nobreak\
\vec{C}_{\begin{subarray}{c}S_{LL}\\\
ij\end{subarray}}^{(u)}\,,\leavevmode\nobreak\
\vec{{C}}_{\begin{subarray}{c}S_{LR}\\\
ij\end{subarray}}^{(u)}\,,\leavevmode\nobreak\
\vec{C}_{\begin{subarray}{c}S_{LL}\\\
ij\end{subarray}}^{(\ell)}\,,\leavevmode\nobreak\
\vec{{C}}_{\begin{subarray}{c}S_{LR}\\\ ij\end{subarray}}^{(\ell)}\Big{)}\,,$
(16) $\displaystyle\vec{\bm{C}}_{T}$
$\displaystyle=\Big{(}\vec{{C}}_{\begin{subarray}{c}T_{L}\\\
ij\end{subarray}}^{(d)}\,,\leavevmode\nobreak\
\vec{{C}}_{\begin{subarray}{c}T_{L}\\\
ij\end{subarray}}^{(u)}\,,\leavevmode\nobreak\
\vec{{C}}_{\begin{subarray}{c}T_{L}\\\ ij\end{subarray}}^{(\ell)}\Big{)}\,,$
(17) $\displaystyle\vec{\bm{C}}_{D}$
$\displaystyle=\Big{(}C_{\begin{subarray}{c}D_{L}\\\
ij\end{subarray}}\Big{)}\,,$ (18) $\displaystyle\vec{\bm{C}}_{G}$
$\displaystyle=\Big{(}C_{\begin{subarray}{c}G_{L}\\\
ij\end{subarray}}\,,\leavevmode\nobreak\
\widetilde{C}_{\begin{subarray}{c}G_{L}\\\ ij\end{subarray}}\Big{)}\,,$ (19)

where lepton flavor indices $i,j$ are fixed, and the sub-vectors with all
possible quark flavor indices are defined e.g. at the $\mu=\mu_{\mathrm{EW}}$
scale as 222Note, in particular, that the up-type quark coefficients should
not contain the top-quark as it has been integrated out. Furthermore, one
should use the basis defined in Sec. 2.1 to avoid redundancies related to
Fierz identities.

$\vec{{C}}_{\begin{subarray}{c}V_{LL}\\\
ij\end{subarray}}^{(d)}=\Big{(}C_{\begin{subarray}{c}V_{LL}\\\
ij11\end{subarray}}^{(d)}\,,\leavevmode\nobreak\
C_{\begin{subarray}{c}V_{LL}\\\
ij12\end{subarray}}^{(d)}\,,\leavevmode\nobreak\
C_{\begin{subarray}{c}V_{LL}\\\
ij13\end{subarray}}^{(d)}\,,\leavevmode\nobreak\
C_{\begin{subarray}{c}V_{LL}\\\
ij21\end{subarray}}^{(d)}\,,\leavevmode\nobreak\
C_{\begin{subarray}{c}V_{LL}\\\
ij22\end{subarray}}^{(d)}\,,\leavevmode\nobreak\
C_{\begin{subarray}{c}V_{LL}\\\
ij23\end{subarray}}^{(d)}\,,\leavevmode\nobreak\
C_{\begin{subarray}{c}V_{LL}\\\
ij31\end{subarray}}^{(d)}\,,\leavevmode\nobreak\
C_{\begin{subarray}{c}V_{LL}\\\
ij32\end{subarray}}^{(d)}\,,\leavevmode\nobreak\
C_{\begin{subarray}{c}V_{LL}\\\ ij33\end{subarray}}^{(d)}\Big{)}\,.$ (20)

Similar definitions hold for the four-lepton operators, where we only consider
operators that violate lepton flavor by one unit, e.g.,

$\vec{{C}}_{\begin{subarray}{c}V_{LL}\\\
ij\end{subarray}}^{(\ell)}=\Big{(}C_{\begin{subarray}{c}V_{LL}\\\
ij11\end{subarray}}^{(\ell)}\,,\leavevmode\nobreak\
C_{\begin{subarray}{c}V_{LL}\\\
ij22\end{subarray}}^{(\ell)}\,,\leavevmode\nobreak\
C_{\begin{subarray}{c}V_{LL}\\\ ij33\end{subarray}}^{(\ell)}\Big{)}\,.$ (21)

Expressions for the operators with a flipped chirality can be obtained from
Eq. (14) via the trivial replacement $L\leftrightarrow R$.

Going beyond the leading-logarithm approximation can be important since
coefficients such as the dipoles $C_{D_{L(R)}}$ are subject to very stringent
experimental constraints from $\mu\to e\gamma$ and $\mu N\to eN$ [30], which
are far more constraining than the ones on LFV mesons decays, and which will
be considerably improved in the coming years. These constraints can supersede
the direct ones on operators that are not efficiently constrained at the
leading order, provided that they contribute at higher loop orders to dipoles
[31]. A noticeable example is the vector four-fermion operators, which mix
into the dipoles at the two-loop level, yielding [16] 333This is a well-known
effect in the quark-sector, where the effective coefficient $C_{7}$ mixes into
$C_{9}$ at two-loop order in QCD, cf. e.g. Ref. [32]. See Ref. [16] for a
brief discussion on the renormalization-scheme dependence of these two-loop
contributions.

$\displaystyle C_{\begin{subarray}{c}D_{L}\\\
ij\end{subarray}}(\mu_{\mathrm{low}})\propto\dfrac{e^{3}Q_{f}^{2}Q_{l}N_{c}^{(f)}}{(16\pi^{2})^{2}}\log\bigg{(}\dfrac{\mu_{\mathrm{low}}}{\mu_{\mathrm{ew}}}\bigg{)}C_{\begin{subarray}{c}V_{XY}\\\
ijkk\end{subarray}}^{(f)}(\mu_{\mathrm{ew}})\,,.pdf$ (22)

where $f\in\\{u,d,l\\}$ denotes a fermion, with electric charge $Q_{f}$ and
color number $N_{c}^{(f)}$. For instance, for the $\mu\to e$ transition and
for a heavy fermion $f\in\\{c,\tau,b,t\\}$, the two-loop constraint from
$\mu\to e\gamma$ obtained in this way can be competitive with the one-loop
constraint from $\mu\to eee$ [16]. Another relevant example is the scalar
$C_{S_{LR}}^{(q)}$, which can be Fierzed into a vector operator that also
mixes into $C_{D_{L}}$ at two-loop order, thus being constrained by $\mu\to
e\gamma$ [16].

Figure 2: Schematic representation of one- and two-loop diagrams that can
induce non-diagonal operator-mixing via QED and QCD below the electroweak
scale [16], cf. Eq. (14) and Eqs. (22)–(23). The blue dots represent
insertions of specific $d=6$ operators appearing in Eq. (14).

Besides the above-mentioned (single logarithm) two-loop effects, there are
also double-logarithm contributions that are relevant to phenomenology. The
most relevant one is the product of the one-loop matrices $U_{DT}\times
U_{TS}$, which induces the one-loop mixing of a scalar into a tensor operator,
which then mixes into the dipole at one-loop again [23],

$\displaystyle C_{\begin{subarray}{c}D_{L}\\\
ij\end{subarray}}(\mu_{\mathrm{low}})\propto\frac{e^{3}Q_{q}^{2}Q_{\ell}N_{c}^{(q)}}{(16\pi^{2})^{2}}\dfrac{m_{q}}{m_{\ell}}\Big{[}\log\Big{(}\frac{\mu_{\text{low}}}{\mu_{\text{ew}}}\Big{)}\Big{]}^{2}C^{(q)}_{\begin{subarray}{c}S_{LL}\\\
ijkk\end{subarray}}(\mu_{\mathrm{ew}})\,.$ (23)

The loop-level constraints on $C_{S_{LL}}^{(q)}$ from
$\ell_{j}\to\ell_{i}\gamma$ can also supersede the ones from tree-level
processes despite the additional suppression by $1/(16\pi^{2})$. In this case,
the chirality enhancement ($\propto m_{q}/m_{\ell}$) plays an important role
in making this contribution sizable if $q$ is a heavy quark.

Even though the two-loop anomalous-dimension matrix in the low-energy EFT is
not yet fully known for a consistent analysis of these processes at order
$1/(16\pi^{2})^{2}$, we opt to account for these contributions, by using the
expressions from Ref. [16], since they can be the dominant effects in some
cases. However, the constraints depending on these effects should be more
carefully reassessed when the full two-loop anomalous-dimensions calculations
will become available. Finally, we remind the reader that the RGE
contributions that we consider below the electroweak scale are summarized in
Fig. 2.

###  2.2 SM EFT

The dimension $d=6$ SMEFT Lagrangian can be written as

$\mathcal{L}_{\mathrm{SMEFT}}^{(6)}=\sum_{i}\dfrac{\mathcal{C}_{I}}{\Lambda^{2}}\,\mathcal{O}_{I}\,,$
(24)

where $\mathcal{C}_{I}$ denotes the relevant Wilson coefficients and $\Lambda$
stands for the EFT cutoff. There are only a few types of $d=6$ operators
defined above the electroweak scale that can violate lepton flavors by one
unit:

  * i)

The semileptonic operators ($\psi^{4}$) collected in Table 1, which contribute
at tree level to the various semileptonic transitions;

  * ii)

The four-fermion leptonic operators ($\psi^{4}$):

$\displaystyle\begin{split}\mathcal{O}_{\begin{subarray}{c}ll\\\
ijkk\end{subarray}}&=\big{(}\bar{l}_{i}\gamma^{\mu}l_{j}\big{)}\big{(}\bar{l}_{k}\gamma_{\mu}l_{k}\big{)}\,,\quad\,\,\mathcal{O}_{\begin{subarray}{c}le\\\
ijkl\end{subarray}}=\big{(}\bar{l}_{i}\gamma^{\mu}l_{j}\big{)}\big{(}\bar{e}_{k}\gamma_{\mu}e_{k}\big{)}\,,\quad\,\,\mathcal{O}_{\begin{subarray}{c}ee\\\
ijkl\end{subarray}}=\big{(}\bar{e}_{i}\gamma^{\mu}e_{j}\big{)}\big{(}\bar{e}_{k}\gamma_{\mu}e_{k}\big{)}\,,\end{split}$
(25)

and the leptonic dipole ($\psi^{2}HX$) operators:

$\displaystyle\qquad\mathcal{O}_{\begin{subarray}{c}eB\\\ ij\end{subarray}}$
$\displaystyle=\big{(}\bar{l}_{i}\sigma^{\mu\nu}e_{j}\big{)}HB_{\mu\nu}\,,\qquad\qquad\mathcal{O}_{\begin{subarray}{c}eW\\\
ij\end{subarray}}=\big{(}\bar{l}_{i}\sigma^{\mu\nu}\tau^{I}e_{j}\big{)}HW^{I}_{\mu\nu}\,,$
(26)

where $\tau^{I}$ ($I=1,2,3$) are the Pauli matrices. These operators become
relevant when including RGE effects which mix semileptonic operators into
purely leptonic ones.

  * iii)

The Yukawa operators ($\psi^{2}H^{3}$):

$\displaystyle\mathcal{O}_{\begin{subarray}{c}eH\\\
ij\end{subarray}}=\big{(}\bar{l}_{i}e_{j}H\big{)}\big{(}H^{\dagger}H\big{)}\,,$
(27)

which induces LFV Higgs decays after electroweak-symmetry breaking.

  * iv)

The Higgs-current operators ($\psi^{2}H^{2}D$): 444We define
$H^{\dagger}\overleftrightarrow{D}_{\mu}H\equiv
H^{\dagger}\big{(}{D}_{\mu}H\big{)}-\big{(}{D}_{\mu}H^{\dagger}\big{)}H$ and
$H^{\dagger}\overleftrightarrow{D}_{\mu}^{I}H\equiv
H^{\dagger}\tau^{I}\big{(}{D}_{\mu}H\big{)}-\big{(}{D}_{\mu}H^{\dagger}\big{)}\tau^{I}H$.

$\displaystyle\mathcal{O}_{\begin{subarray}{c}Hl\\\
ij\end{subarray}}^{(1)}=\big{(}H^{\dagger}\overleftrightarrow{D}_{\mu}H\big{)}$
$\displaystyle\big{(}\bar{l}_{i}\gamma^{\mu}l_{j}\big{)}\,,\qquad\quad\mathcal{O}_{\begin{subarray}{c}Hl\\\
ij\end{subarray}}^{(3)}=\big{(}H^{\dagger}\overleftrightarrow{D}_{\mu}^{I}H\big{)}\big{(}\bar{l}_{i}\tau^{I}\gamma^{\mu}l_{j}\big{)}\,,$
(28)
$\displaystyle\mathcal{O}_{\begin{subarray}{c}He\end{subarray}}=\big{(}H^{\dagger}\overleftrightarrow{D}_{\mu}H\big{)}\big{(}\bar{e}_{i}\gamma^{\mu}e_{j}\big{)}\,,$
(29)

which induce LFV $Z$-boson decays, in addition to LFV in semileptonic
amplitudes mediated by the $Z$-boson, which are quark-flavor conserving.

| $\psi^{4}$ | Operator
---|---
$\mathcal{O}_{lq}^{(1)}$ | $\big{(}\bar{l}_{i}\gamma^{\mu}l_{j}\big{)}\big{(}\bar{q}_{k}\gamma_{\mu}q_{l}\big{)}$
$\mathcal{O}_{lq}^{(3)}$ | $\big{(}\bar{l}_{i}\gamma^{\mu}\tau^{I}l_{j}\big{)}\big{(}\bar{q}_{k}\gamma_{\mu}\tau^{I}q_{l}\big{)}$
$\mathcal{O}_{lu}$ | $\big{(}\bar{l}_{i}\gamma^{\mu}l_{j}\big{)}\big{(}\bar{u}_{k}\gamma_{\mu}u_{l}\big{)}$
$\mathcal{O}_{ld}$ | $\big{(}\bar{l}_{i}\gamma^{\mu}l_{j}\big{)}\big{(}\bar{d}_{k}\gamma_{\mu}d_{l}\big{)}$
$\mathcal{O}_{eq}$ | $\big{(}\bar{e}_{i}\gamma^{\mu}e_{j}\big{)}\big{(}\bar{q}_{k}\gamma_{\mu}q_{l}\big{)}$
$\mathcal{O}_{eu}$ | $\big{(}\bar{e}_{i}\gamma^{\mu}e_{j}\big{)}\big{(}\bar{u}_{k}\gamma_{\mu}u_{l}\big{)}$
$\mathcal{O}_{ed}$ | $\big{(}\bar{e}_{i}\gamma^{\mu}e_{j}\big{)}\big{(}\bar{d}_{k}\gamma_{\mu}d_{l}\big{)}$
| $\psi^{4}$ | Operator $+\mathrm{h.c.}$
---|---
$\mathcal{O}_{ledq}$ | $\big{(}\bar{l}^{a}_{i}e_{j}\big{)}\big{(}\bar{d}_{k}q^{a}_{l}\big{)}$
$\mathcal{O}_{lequ}^{(1)}$ | $\big{(}\bar{l}^{a}_{i}e_{j}\big{)}\varepsilon_{ab}\big{(}\bar{q}^{b}_{k}u_{l}\big{)}$
$\mathcal{O}_{lequ}^{(3)}$ | $\big{(}\bar{l}^{a}_{i}\sigma^{\mu\nu}e_{j}\big{)}\varepsilon_{ab}\big{(}\bar{q}^{b}_{k}\sigma_{\mu\nu}u_{l}\big{)}$
Table 1: Hermitian (left) and non-Hermitian (right) $d=6$ semileptonic
operators in the SMEFT. We consider the Warsaw basis [14], and we renamed the
operator $\mathcal{O}_{qe}$ as $\mathcal{O}_{eq}$ to have lepton flavor
indices before the quark ones. The $SU(2)_{L}$ indices are denoted by $a,b$,
with $\varepsilon_{12}=-\varepsilon_{21}=+1$, $\tau^{I}$ ($I=1,2,3$) are the
Pauli matrices, and $SU(3)_{c}$ indices are omitted. Flavor indices are
denoted by $i,j,k,l$ and should be assigned as
$\mathcal{O}\equiv\mathcal{O}_{ijkl}$ in the left columns of the table. See
Appendix A for our notation for the SM fields.

Similarly to Sec. 2.1, flavor indices are denoted by Latin symbols. For
semileptonic operators, our convention is that the first two indices always
correspond to lepton flavors. For Hermitian operators such as
$\mathcal{O}_{ld}$, we further impose that lepton indices satisfy $i<j$ to
avoid redundancy in the operator basis. Moreover, we assume that down-quark
Yukawas are diagonal, i.e. the Cabibbo–Kobayashi–Maskawa (CKM) matrix $V$
appears in the upper component of
$q_{i}=[(V^{\dagger}u_{L})_{i}\,\leavevmode\nobreak\ d_{Li}]^{T}$, cf.
Appendix A.

The SMEFT Lagrangian in Eq. (24) can be matched to the one Low-Energy EFT
defined in Eq. (1), as provided at tree level in Appendix C (see also Ref.
[33]). We note, in particular, that the leptonic scalar and tensor
coefficients $C_{S_{XY}}^{(\ell)}$ and $C_{T_{X}}^{(\ell)}$ vanish at $d=6$,
only appearing at higher dimensions via insertions of the Higgs doublets,
which are needed to make the operator invariant under $SU(2)_{L}\times
U(1)_{Y}$. The same holds for the down-type tensor coefficient
$C_{T_{X}}^{(d)}$ and other four-fermion semileptonic operators, cf. Ref.
[34].

#### Operator Mixing

The different types of LFV operators listed above can mix through
renormalization group evolution [20]. For the operators that we consider, the
relevant RGE effects are induced by the electroweak and the Yukawa
interactions, which mix different operators, in addition to the QCD running
that can only change the magnitude of scalar and tensor semileptonic Wilson
coefficients. The RGE effects generated by the electroweak interactions are
necessarily flavor-conserving, whereas the Yukawa ones can induce quark-flavor
violation through the CKM matrix, as depicted in Fig. 3.

In what follows, we consider the $d=6$ LFV operators with fixed lepton flavor
indices $i<j$. 555To this order in the EFT expansion, these operators do not
mix with operators with different combinations of lepton flavors. See Ref.
[35] for the double-insertion of $d=6$ operators and their mixing into $d=8$
operators at one-loop. The leading-logarithm solution to the RGEs can then be
schematically written as follows [20],

$\displaystyle\begin{pmatrix}\vec{\bm{\mathcal{C}}}_{V}\\\\[3.00003pt]
\vec{\bm{\mathcal{C}}}_{S}\\\\[3.00003pt]
\vec{\bm{\mathcal{C}}}_{T}\\\\[3.00003pt]
\vec{\bm{\mathcal{C}}}_{D}\\\\[3.00003pt]
\vec{\bm{\mathcal{C}}}_{\mathrm{Yuk}}\\\\[3.00003pt]
\vec{\bm{\mathcal{C}}}_{H_{\ell}}\end{pmatrix}_{\mu_{\mathrm{ew}}}\overset{\tiny
1-\mathrm{loop}}{\simeq}\begin{pmatrix}\mathcal{U}_{VV}&0&0&0&0&\mathcal{U}_{VH_{\ell}}\\\\[1.99997pt]
0&\mathcal{U}_{SS}&\mathcal{U}_{ST}&0&0&0\\\\[1.99997pt]
0&\mathcal{U}_{TS}&\mathcal{U}_{TT}&\mathcal{U}_{TD}&0&0\\\\[1.99997pt]
0&0&\mathcal{U}_{DT}&\mathcal{U}_{DD}&0&0\\\\[1.99997pt]
0&\mathcal{U}_{\mathrm{Y}S}&0&0&\mathcal{U}_{\mathrm{Y}\mathrm{Y}}&0\\\\[1.99997pt]
\mathcal{U}_{H_{\ell}V}&0&0&0&0&\mathcal{U}_{H_{\ell}H_{\ell}}\end{pmatrix}\cdot\begin{pmatrix}\vec{\bm{\mathcal{C}}}_{V}\\\\[1.50002pt]
\vec{\bm{\mathcal{C}}}_{S}\\\\[1.50002pt]
\vec{\bm{\mathcal{C}}}_{T}\\\\[1.50002pt]
\vec{\bm{\mathcal{C}}}_{D}\\\\[1.50002pt]
\vec{\bm{\mathcal{C}}}_{\mathrm{Yuk}}\\\\[1.50002pt]
\vec{\bm{\mathcal{C}}}_{H_{\ell}}\end{pmatrix}_{\Lambda}\,,$ (30)

where we only kept the contributions proportional to the top and bottom quark
Yukawas, as well as the SM gauge couplings, which are taken at the scale
$\mu\simeq\mu_{\mathrm{ew}}$. In the leading-logarithm approximation, the
matrices $U_{IJ}$ can be obtained from the SMEFT anomalous dimensions
$\gamma_{JI}$ given in Ref. [20] via the analogous expression of Eq. (13). The
effective coefficients are combined in the above equation in the following
vectors,

$\displaystyle\bm{\vec{\mathcal{C}}}_{V}$
$\displaystyle=\Big{(}{\vec{\mathcal{C}}}_{\begin{subarray}{c}lq\\\
ij\end{subarray}}^{\,(1)}\,,\leavevmode\nobreak\
{\vec{\mathcal{C}}}_{\begin{subarray}{c}lq\\\
ij\end{subarray}}^{\,(3)}\,,\leavevmode\nobreak\
{\vec{\mathcal{C}}}_{\begin{subarray}{c}lu\\\
ij\end{subarray}}\,,\leavevmode\nobreak\
{\vec{\mathcal{C}}}_{\begin{subarray}{c}ld\\\
ij\end{subarray}}\,,\leavevmode\nobreak\
{\vec{\mathcal{C}}}_{\begin{subarray}{c}eq\\\
ij\end{subarray}}\,,\leavevmode\nobreak\
{\vec{\mathcal{C}}}_{\begin{subarray}{c}eu\\\
ij\end{subarray}}\,,\leavevmode\nobreak\
{\vec{\mathcal{C}}}_{\begin{subarray}{c}ed\\\
ij\end{subarray}}\,,\leavevmode\nobreak\
{\vec{\mathcal{C}}}_{\begin{subarray}{c}ll\\\
ij\end{subarray}}\,,\leavevmode\nobreak\
{\vec{\mathcal{C}}}_{\begin{subarray}{c}le\\\
ij\end{subarray}}\,,\leavevmode\nobreak\
{\vec{\mathcal{C}}}_{\begin{subarray}{c}ee\\\ ij\end{subarray}}\Big{)}\,,$
(31) $\displaystyle\bm{\vec{\mathcal{C}}}_{S}$
$\displaystyle=\Big{(}\vec{\mathcal{C}}_{\begin{subarray}{c}ledq\\\
ij\end{subarray}}\,,\leavevmode\nobreak\
\vec{\mathcal{C}}_{\begin{subarray}{c}lequ\\\
ij\end{subarray}}^{\,(1)}\Big{)}\,,$ (32)
$\displaystyle\bm{\vec{\mathcal{C}}}_{T}$
$\displaystyle=\Big{(}\vec{\mathcal{C}}_{\begin{subarray}{c}lequ\\\
ij\end{subarray}}^{\,(3)}\Big{)}\,,$ (33)
$\displaystyle\bm{\vec{\mathcal{C}}}_{D}$
$\displaystyle=\Big{(}\mathcal{C}_{\begin{subarray}{c}eB\\\
ij\end{subarray}}\,,\leavevmode\nobreak\ \mathcal{C}_{\begin{subarray}{c}eW\\\
ij\end{subarray}}\Big{)}\,,$ (34)
$\displaystyle\bm{\vec{\mathcal{C}}}_{\mathrm{Yuk}}$
$\displaystyle=\Big{(}\mathcal{C}_{\begin{subarray}{c}eH\\\
ij\end{subarray}}\Big{)}\,,$ (35)
$\displaystyle\bm{\vec{\mathcal{C}}}_{H_{\ell}}$
$\displaystyle=\Big{(}\mathcal{C}_{\begin{subarray}{c}H\ell\\\
ij\end{subarray}}^{(1)}\,,\leavevmode\nobreak\
\mathcal{C}_{\begin{subarray}{c}H\ell\\\
ij\end{subarray}}^{(3)}\,,\leavevmode\nobreak\
\mathcal{C}_{\begin{subarray}{c}He\\\ ij\end{subarray}}\Big{)}\,,$ (36)

where the four-fermion subvectors are defined in such a way as to comprise all
possible quark flavor indices, for fixed lepton flavor indices $\\{i,j\\}$,
e.g.,

${\vec{\mathcal{C}}}_{\begin{subarray}{c}lq\\\
ij\end{subarray}}^{\,(1)}=\Big{(}\mathcal{C}_{\begin{subarray}{c}lq\\\
ij11\end{subarray}}^{(1)}\,,\leavevmode\nobreak\
\mathcal{C}_{\begin{subarray}{c}lq\\\
ij12\end{subarray}}^{(1)}\,,\leavevmode\nobreak\
\mathcal{C}_{\begin{subarray}{c}lq\\\
ij13\end{subarray}}^{(1)}\,,\leavevmode\nobreak\
\mathcal{C}_{\begin{subarray}{c}lq\\\
ij21\end{subarray}}^{(1)}\,,\leavevmode\nobreak\
\mathcal{C}_{\begin{subarray}{c}lq\\\
ij22\end{subarray}}^{(1)}\,,\leavevmode\nobreak\
\mathcal{C}_{\begin{subarray}{c}lq\\\
ij23\end{subarray}}^{(1)}\,,\leavevmode\nobreak\
\mathcal{C}_{\begin{subarray}{c}lq\\\
ij31\end{subarray}}^{(1)}\,,\leavevmode\nobreak\
\mathcal{C}_{\begin{subarray}{c}lq\\\
ij32\end{subarray}}^{(1)}\,,\leavevmode\nobreak\
\mathcal{C}_{\begin{subarray}{c}lq\\\ ij33\end{subarray}}^{(1)}\Big{)}\,,$
(37)

with similar definitions for the other operators.

Figure 3: Schematic representation of one-loop diagrams that can induce non-
diagonal operator-mixing via the $SU(2)_{L}\times U(1)_{Y}$ gauge and Yukawa
interactions above the electroweak scale [20]. The blue dots represent
insertions of specific $d=6$ operators appearing in Eq. (30).

Besides the leading-logarithm contributions discussed above, we stress once
again that two-loop RGE effects can be relevant if they introduce a new source
of mixing that does not appear at the one-loop level. A relevant example in
the SMEFT is the double-logarithm mixing of the scalar operator
$\mathcal{C}_{lequ}^{(1)}$ into the tensor $\mathcal{C}_{lequ}^{(3)}$, which
then mixes into the dipoles $\mathcal{C}_{eW}$ and $\mathcal{C}_{eB}$, via the
product $\mathcal{U}_{DT}\times\mathcal{U}_{TS}$ in Eq. (30). This mixing
induces a chirality-enhancement similar to Eq. (23), which provides the most
stringent constraint on the scalar coefficient
$\smash{\mathcal{C}_{lequ}^{(1)}}$ with third-generation quarks, as will be
shown in Sec. 5.

Finally, we note that Eq. (30) was obtained by neglecting the lepton Yukawas
($y_{e}$). However, we found that one particular type of contribution
proportional to the lepton Yukawas is relevant to phenomenology, namely the
Yukawa-induced mixing of the vector coefficients into tensors [19],

$\displaystyle\begin{split}\mu\,\dfrac{\mathrm{d}}{\mathrm{d}\mu}\mathcal{C}_{\begin{subarray}{c}lequ\\\
ijkl\end{subarray}}^{(3)}\overset{\mathrm{yuk.}}{=}&-\dfrac{[Y_{u}^{\dagger}]_{kw}[Y_{e}^{\dagger}]_{iv}}{32\pi^{2}}\,\Big{(}\mathcal{C}_{\begin{subarray}{c}eu\\\
vjwl\end{subarray}}+\mathcal{C}_{\begin{subarray}{c}lu\\\
vjwl\end{subarray}}\Big{)}-\dfrac{[Y_{u}^{\dagger}]_{wl}[Y_{e}^{\dagger}]_{vj}}{32\pi^{2}}\,\Big{(}\mathcal{C}_{\begin{subarray}{c}lq\\\
ivkw\end{subarray}}^{(1)}-3\,\mathcal{C}_{\begin{subarray}{c}lq\\\
ivkw\end{subarray}}^{(3)}\Big{)}\\\
&-\dfrac{[Y_{u}^{\dagger}]_{vl}[Y_{e}^{\dagger}]_{iw}}{32\pi^{2}}\mathcal{C}_{\begin{subarray}{c}eq\\\
wjkv\end{subarray}}+\dots\,,\end{split}$ (38)

where $Y_{u}=\mathrm{diag}(y_{u},y_{c},y_{t})\cdot V$ and
$Y_{e}=\mathrm{diag}(y_{e},y_{\mu},y_{\tau})$. This effect is sizable for
coefficients third-generation quarks since their contribution is proportional
to $y_{t}$. The tensor coefficient then mixes into dipoles [19], inducing
contributions scaling such as

$\displaystyle\mathcal{C}_{\begin{subarray}{c}eW\\\
ij\end{subarray}}(\mu_{\mathrm{ew}})\propto\dfrac{g_{2}\,N_{c}\,y_{t}^{2}\,y_{\ell}}{(16\pi^{2})^{2}}\Big{[}\log\Big{(}\frac{\mu_{\text{ew}}}{\Lambda}\Big{)}\Big{]}^{2}\mathcal{C}_{\begin{subarray}{c}V\\\
ij33\end{subarray}}(\Lambda)\,,$ (39)

where $\mathcal{C}_{V}$ denotes one of the coefficients appearing in the
right-hand-side of Eq. (38), and $m_{\ell}$ can be either $m_{\ell_{i}}$ or
$m_{\ell_{j}}$. Similar expressions apply for $C_{eB}$ by replacing $g_{2}\to
g_{1}$. We will include these effects in our analysis in Sec. 5, which allow
us to derive stringent limits on vector semileptonic coefficients, with third-
generation quarks, by using $\ell_{j}\to\ell_{i}\gamma$ (see also Ref. [17]).

###  2.3 Summary

Finally, we briefly summarize the loop effects that will be considered in our
phenomenological analysis in Sec. 5, which are classified in terms of a
logarithmic expansion. Firstly, for the low-energy EFT, we will consider

  * •

The one-loop mixing of the low-energy LFV operators from Ref. [16], which are
proportional to

$\dfrac{1}{16\pi^{2}}\log\bigg{(}\dfrac{\mu_{\mathrm{low}}}{\mu_{\mathrm{ew}}}\bigg{)}\,,$
(40)

as described in Eq. (14) (cf. also Ref. [19]).

  * •

The two-loop single logarithm mixing of semileptonic vector and operators into
leptonic dipoles calculated in Ref. [16],

$\dfrac{1}{(16\pi^{2})^{2}}\log\Big{(}\frac{\mu_{\text{low}}}{\mu_{\text{ew}}}\Big{)}\,,$
(41)

as illustrated in Eqs. (22) and (23).

  * •

The two-loop double logarithm mixing
$C_{S_{LL}}^{(q)}\overset{\mathrm{1-loop}}{\longrightarrow}C_{T_{LL}}^{(q)}\overset{\mathrm{1-loop}}{\longrightarrow}C_{D_{L}}^{(\ell)}$,
which induces contributions to dipoles proportional to

$\dfrac{1}{(16\pi^{2})^{2}}\bigg{[}\log\Big{(}\frac{\mu_{\text{low}}}{\mu_{\text{ew}}}\Big{)}\bigg{]}^{2}\,,$
(42)

which are chirality-enhanced for heavy-quarks.

  * •

The finite matching contributions of scalar semileptonic operators to the
gluonic ones [25], as given in Eq. (11).

For the SMEFT, we will consider

  * •

The one-loop mixing of the SMEFT operators taken from Ref. [20], which are
proportional to

$\dfrac{1}{16\pi^{2}}\log\Big{(}\frac{\mu_{\text{ew}}}{\Lambda}\Big{)}\,,$
(43)

as shown in Eq. (30).

  * •

The double-logarithm two-loop mixing
$\mathcal{O}_{lequ}^{(1)}\overset{\mathrm{1-loop}}{\longrightarrow}\mathcal{O}_{lequ}^{(3)}\overset{\mathrm{1-loop}}{\longrightarrow}\mathcal{O}_{eV}$
into dipoles, which is induced by the product of gauge and Yukawa running, and
the Yukawa-induced
${\\{\mathcal{O}_{lq}^{(1,3)},\mathcal{O}_{lu},\mathcal{O}_{eq},\mathcal{O}_{eu}\\}\overset{\mathrm{1-loop}}{\longrightarrow}\mathcal{O}_{lequ}^{(3)}\overset{\mathrm{1-loop}}{\longrightarrow}\mathcal{O}_{eV}}$
(with $V=W,B$), which is suppressed by lepton masses, but which remains
relevant for phenomenology. These effects are proportional to

$\dfrac{1}{(16\pi^{2})^{2}}\bigg{[}\log\Big{(}\frac{\mu_{\text{ew}}}{\Lambda}\Big{)}\bigg{]}^{2}\,.$
(44)

as described e.g. in Eq. (39).

The expressions for the relevant anomalous dimensions can be found in the
references given above. Finally, we stress once again that our analysis of
two-loop contributions is not entirely self-consistent and should be rather
understood as a qualitative indication of the dominant effects, which must be
refined in the future with a precise two-loop calculation.

##  3 Low-energy probes

In this Section, we provide the general expressions for the low-energy LFV
observables in terms of the EFT defined in Sec. 2.1. Firstly, we will discuss
direct probes of semileptonic operators at tree-level in LFV meson and
$\tau$-lepton decays in Sec. 3.1 and 3.2, respectively. 666In principle, we
could also consider baryon decays such as
$\Lambda_{b}\to\Lambda\ell_{i}\ell_{j}$ [36, 37], but there are no
experimental limits available for these decays yet. In Sec. 3.3, we will the
purely leptonic LFV decays, which are sensitive to semileptonic operators at
one-loop, cf. Fig. 2 and 3.

Process | Exp. limit | Ref. | Process | Exp. limit | Ref. | Process | Exp. limit | Ref.
---|---|---|---|---|---|---|---|---
$\pi^{0}\to e\mu$ | $4.8\times 10^{-10}$ | [30] | $\tau\to e\pi^{0}$ | $1.1\times 10^{-7}$ | [30] | $\tau\to\mu\pi^{0}$ | $1.5\times 10^{-7}$ | [30]
$\eta\to e\mu$ | $8.1\times 10^{-6}$ | [30] | $\tau\to e\eta$ | $1.2\times 10^{-7}$ | [30] | $\tau\to\mu\eta$ | $8.7\times 10^{-8}$ | [30]
$\eta^{\prime}\to e\mu$ | $6.3\times 10^{-4}$ | [30] | $\tau\to e\eta^{\prime}$ | $2.1\times 10^{-7}$ | [30] | $\tau\to\mu\eta^{\prime}$ | $1.8\times 10^{-7}$ | [30]
|  |  | $\tau\to e\rho^{0}$ | $2.4\times 10^{-8}$ | [30] | $\tau\to\mu\rho^{0}$ | $1.6\times 10^{-8}$ | [30]
$K_{L}\to e\mu$ | $6.3\times 10^{-12}$ | [30] | $\tau\to eK_{S}$ | $3.5\times 10^{-8}$ | [30] | $\tau\to\mu K_{S}$ | $3.1\times 10^{-8}$ | [30]
$K^{+}\to\pi^{+}\mu^{-}e^{+}$ | $1.1\times 10^{-10}$ | [30] | $\tau\to eK^{\ast 0}$ | $4.3\times 10^{-8}$ | [30] | $\tau\to\mu K^{\ast 0}$ | $7.9\times 10^{-8}$ | [30]
$K_{L}\to\pi^{0}e\mu$ | $1.0\times 10^{-10}$ | [30] |  |  |  |  |  |
$D^{0}\to e\mu$ | $1.7\times 10^{-8}$ | [30] | $\tau\to eD^{0}$ | – |  |  |  |
$D^{+}\to\pi^{+}e\mu$ | $4.5\times 10^{-7}$ | [38] |  |  |  |  |  |
$D_{s}\to K^{+}e\mu$ | $1.5\times 10^{-6}$ | [38] |  |  |  |  |  |
$B^{0}\to e\mu$ | $1.3\times 10^{-9}$ | [30] | $B^{0}\to e\tau$ | $2.1\times 10^{-5}$ | [30] | $B^{0}\to\mu\tau$ | $1.4\times 10^{-5}$ | [30]
$B^{+}\to\pi^{+}e\mu$ | $1.2\times 10^{-7}$ | [39] | $B^{+}\to\pi^{+}e\tau$ | $1.0\times 10^{-4}$ | [30] | $B^{+}\to\pi^{+}\mu\tau$ | $9.7\times 10^{-5}$ | [30]
$B^{0}\to\rho^{0}e\mu$ | – |  | $B^{0}\to\rho^{0}e\tau$ | – |  | $B^{0}\to\rho^{0}\mu\tau$ | – |
$B_{s}\to K^{0}e\mu$ | – |  | $B_{s}\to K^{0}e\tau$ | – |  | $B_{s}\to K^{0}\mu\tau$ | – |
$B_{s}\to e\mu$ | $7.2\times 10^{-9}$ | [30] | $B_{s}\to e\tau$ | $1.9\times 10^{-3}$ | [10] | $B_{s}\to\mu\tau$ | $4.2\times 10^{-5}$ | [30]
$B^{+}\to K^{+}e\mu$ | $1.8\times 10^{-8}$ | [9] | $B^{+}\to K^{+}e\tau$ | $4.1\times 10^{-5}$ | [10] | $B^{+}\to K^{+}\mu\tau$ | $4.1\times 10^{-5}$ | [10]
$B^{0}\to K^{\ast}e\mu$ | $1.2\times 10^{-8}$ | [9] | $B^{0}\to K^{\ast}e\tau$ | – |  | $B^{0}\to K^{\ast}\mu\tau$ | $2.2\times 10^{-5}$ | [9]
$B_{s}\to\phi e\mu$ | $2.0\times 10^{-8}$ | [9] | $B_{s}\to\phi e\tau$ | – |  | $B_{s}\to\phi\mu\tau$ | – |
$\phi\to e\mu$ | $2.7\times 10^{-6}$ | [30] | $\tau\to e\phi$ | $4.1\times 10^{-8}$ | [30] | $\tau\to\mu\phi$ | $1.1\times 10^{-7}$ | [30]
$J/\psi\to e\mu$ | $6.1\times 10^{-9}$ | [40] | $J/\psi\to e\tau$ | $1.0\times 10^{-7}$ | [40] | $J/\psi\to\mu\tau$ | $2.7\times 10^{-6}$ | [30]
$\Upsilon\to e\mu$ | $5.2\times 10^{-7}$ | [30] | $\Upsilon\to e\tau$ | $3.6\times 10^{-6}$ | [30] | $\Upsilon\to\mu\tau$ | $3.6\times 10^{-6}$ | [30]

Table 2: Experimental limits on LFV decays considered in our analysis (95%
CL.). For the decays with two leptons in the final state, we quote the limits
on the sum of the branching fractions with different lepton charges, i.e.
$\ell_{i}\ell_{j}\equiv\ell_{i}^{-}\ell_{j}^{+}+\ell_{i}^{+}\ell_{j}^{-}$ .
The dash symbol represent decays that are kinematically allowed, but for which
there is not an experimental limit yet. Experimental results provided at (90%
CL.) are converted to (95% CL.) following Ref. [3].

###  3.1 $q_{l}\to q_{k}\ell_{i}\ell_{j}$

Firstly, we consider the leptonic and semileptonic decays of pseudoscalar
mesons. For convenience, we define the following combinations of
(axial-)vector coefficients that will allow us to express the branching
fractions in a more compact form,

$\displaystyle C_{VV}^{(q)}$
$\displaystyle={C_{V_{RR}}^{(q)}+C_{V_{RL}}^{(q)}+(L\leftrightarrow R)}\,,$
$\displaystyle C_{AA}^{(q)}$
$\displaystyle={C_{V_{RR}}^{(q)}-C_{V_{RL}}^{(q)}+(L\leftrightarrow R)}\,,$
(45) $\displaystyle C_{AV}^{(q)}$
$\displaystyle={C_{V_{RR}}^{(q)}+C_{V_{RL}}^{(q)}-(L\leftrightarrow R)}\,,$
$\displaystyle C_{VA}^{(q)}$
$\displaystyle={C_{V_{RR}}^{(q)}-C_{V_{RL}}^{(q)}-(L\leftrightarrow R)}\,,$
(46)

and similarly, for the (pseudo)scalar coefficients,

$\displaystyle C_{SS}^{(q)}$
$\displaystyle={C_{S_{RR}}^{(q)}+C_{S_{RL}}^{(q)}+(L\leftrightarrow R)}\,,$
$\displaystyle C_{PP}^{(q)}$
$\displaystyle={C_{S_{RR}}^{(q)}-C_{S_{RL}}^{(q)}+(L\leftrightarrow R)}\,,$
(47) $\displaystyle C_{PS}^{(q)}$
$\displaystyle={C_{S_{RR}}^{(q)}+C_{S_{RL}}^{(q)}-(L\leftrightarrow R)}\,,$
$\displaystyle C_{SP}^{(q)}$
$\displaystyle={C_{S_{RR}}^{(q)}-C_{S_{RL}}^{(q)}-(L\leftrightarrow R)}\,,$
(48)

and for the tensor ones,

$\displaystyle C_{T}^{(q)}$
$\displaystyle=C_{T_{R}}^{(q)}+C_{T_{L}}^{(q)}\,,\qquad\quad\;\;C_{T_{5}}^{(q)}=C_{T_{R}}^{(q)}-C_{T_{L}}^{(q)}\,,$
(49)

where, for simplicity, flavor indices are omitted. The current experimental
limits on the most relevant processes of this type are collected in Table 2.

####  3.1.1 Leptonic decays: $P\to\ell_{i}\ell_{j}$

One of the simplest probes of LFV at low-energies are leptonic decays of
pseudoscalar mesons. Firstly, we consider decays of flavored mesons of the
type $P=\bar{q}_{k}q_{l}$, with $k\neq l$, such as $K$-, $D$\- and $B$-mesons.
The hadronic matrix element needed to compute the decay rates reads

$\displaystyle\langle
0|\bar{q}_{k}\gamma^{\mu}\gamma_{5}q_{l}|P\rangle=if_{P}p^{\mu}\,,$ (50)

where $f_{P}$ denotes the $P$-meson decay constant, which has been computed
for the most relevant transition by means of numerical simulations of QCD on
the lattice, cf. Table 3 [41]. In terms of the low-energy EFT defined in Eq.
(1), the $P\to\ell_{i}\ell_{j}$ branching fraction then reads

$\displaystyle\begin{split}\mathcal{B}(P\to\ell_{i}^{-}\ell_{j}^{+})&=\tau_{P}\,\dfrac{f_{P}^{2}m_{P}m_{\ell_{j}}^{2}}{128\pi
v^{4}}\left(1-\dfrac{m_{\ell_{j}}^{2}}{m_{P}^{2}}\right)^{2}\\\
&\times\bigg{\\{}\bigg{|}C_{\begin{subarray}{c}VA\\\
ijkl\end{subarray}}^{(q)}+\dfrac{m_{P}^{2}}{m_{\ell_{j}}(m_{q_{k}}+m_{q_{l}})}\,C_{\begin{subarray}{c}SP\\\
ijkl\end{subarray}}^{(q)}\bigg{|}^{2}+\bigg{|}C_{\begin{subarray}{c}AA\\\
ijkl\end{subarray}}^{(q)}-\dfrac{m_{P}^{2}}{m_{\ell_{j}}(m_{q_{k}}+m_{q_{l}})}\,C_{\begin{subarray}{c}PP\\\
ijkl\end{subarray}}^{(q)}\bigg{|}^{2}\bigg{\\}}\,,\end{split}$ (51)

where $m_{P}$ denotes the $P$-meson mass, and we have assumed $i<j$ and
neglected the mass of the lightest lepton ($m_{\ell_{i}}$). 777See e.g. Ref.
[42] for the expression without this approximation. Note, also, that the sign
of the interference term between ${C_{VA}^{(q)}}$ and ${C_{SP}^{(q)}}$ changes
sign for the decay $P\to\ell_{i}^{+}\ell_{j}^{-}$, with opposite electric
charges for the leptons, as it is proportional to the difference of lepton
masses, see e.g. Ref. [42]. The above expressions can be used for the decays
of $D$\- and $B_{(s)}$-mesons with the appropriate replacement of the flavor
indices. Moreover, for the kaon decays $K_{L(S)}\to e\mu$, we have to take
into account that
$|K_{L(S)}\rangle\simeq(|K^{0}\rangle\pm|\overline{K^{0}}\rangle)/\sqrt{2}$,
which implies that these decays probe both $K_{0}\to e\mu$ and
$\overline{K}_{0}\to e\mu$. In this case, the above expression can be used
after making the following substitution [43],

$C_{\begin{subarray}{c}I\\\
ijkl\end{subarray}}^{(q)}\to\frac{1}{\sqrt{2}}\Big{(}C^{(d)}_{\begin{subarray}{c}I\\\
1212\end{subarray}}\pm C^{(d)}_{\begin{subarray}{c}I\\\
1221\end{subarray}}\Big{)}\;,$ (52)

where $I\in\\{VA,AA,SP,PP\\}$, and the upper (lower) sign corresponds to the
$K_{L}$ ($K_{S}$) decay.

####  3.1.2 Leptonic decays: $V\to\ell_{i}\ell_{j}$

Next, we provide the expressions for the leptonic decays
$V\to\ell_{i}^{-}\ell_{j}^{+}$ of vector mesons of the type
$V=\bar{q}_{k}q_{k}$, such as $V\in\\{\rho,\phi,J/\psi,\Upsilon\\}$. These
processes are typically much less constraining than the other probes discussed
in this paper, since vector mesons typically have a large total-width
($\tau_{V}$) that suppresses the LFV branching fractions [44, 45], but we also
discuss them for completeness. The relevant hadronic matrix elements can be
defined as,

$\displaystyle\begin{split}\langle
0|\bar{q}_{k}\gamma^{\mu}q_{k}|V(p,\lambda)\rangle&=f_{V}m_{V}\varepsilon_{\lambda}^{\mu}(p)\,,\\\\[4.49997pt]
\langle
0|\bar{q_{k}}\sigma^{\mu\nu}q_{k}|V(p,\lambda)\rangle&=if_{V}^{T}(\mu)\big{[}\varepsilon_{\lambda}^{\mu}(p)p^{\nu}-\varepsilon_{\lambda}^{\nu}(p)p^{\mu}\big{]}\,,\end{split}$
(53)

where $m_{V}$ is $V$-meson mass, $\varepsilon_{\lambda}^{\mu}$ denotes its
polarization-vector, and $f_{V}$ and $f_{V}^{T}$ stand for the vector and
tensor decay constants, respectively. See Ref. [46, 47, 49, 50, 48, 51] for
lattice QCD calculations of the relevant vector-meson decay constants. The
branching fraction can then be written in terms of Eq. (1) as follows,

$\displaystyle\mathcal{B}(V\to$
$\displaystyle\ell_{i}^{-}\ell_{j}^{+})=\tau_{V}\,\dfrac{f_{V}^{2}m_{V}^{3}}{192\pi
v^{4}}\left(1-\dfrac{m_{\ell_{j}}^{2}}{m_{V}^{2}}\right)^{2}$ (54)
$\displaystyle\times\bigg{\\{}\Big{[}|C_{\begin{subarray}{c}VV\\\
ijkk\end{subarray}}^{(q)}|^{2}+|C_{\begin{subarray}{c}AV\\\
ijkk\end{subarray}}^{(q)}|^{2}\Big{]}\bigg{(}1+\dfrac{m_{\ell_{j}}^{2}}{2m_{V}^{2}}\bigg{)}+8\bigg{(}\dfrac{f_{V}^{T}}{f_{V}}\bigg{)}^{2}\Big{[}|{C}_{\begin{subarray}{c}T\\\
ijkk\end{subarray}}^{(q)}|^{2}+|{C}_{\begin{subarray}{c}T_{5}\\\
ijkk\end{subarray}}^{(q)}|^{2}\Big{]}\bigg{(}1+\dfrac{2m_{\ell_{j}}^{2}}{m_{V}^{2}}\bigg{)}$
$\displaystyle+12\dfrac{m_{\ell_{j}}}{m_{V}}\dfrac{f_{V}^{T}}{f_{V}}\,\mathrm{Re}\big{[}{C}_{\begin{subarray}{c}T\\\
ijkk\end{subarray}}^{(q)}\,C_{\begin{subarray}{c}VV\\\
ijkk\end{subarray}}^{(q)\ast}-{C}_{\begin{subarray}{c}T_{5}\\\
ijkk\end{subarray}}^{(q)}\,C_{\begin{subarray}{c}AV\\\
ijkk\end{subarray}}^{(q)\ast}\big{]}\bigg{\\}}\,,$

where we have once again assumed $i<j$ and neglected the light-lepton mass,
$m_{\ell_{i}}$. Note, in particular, that scalar operators do not contribute
to these decays, since the hadronic-matrix element with a scalar density
vanishes in this case. Furthermore, we have not included the contributions
from the leptonic dipoles, as they are already tightly constrained by
$\ell_{j}\to\ell_{i}\gamma$.

$P$ | $f_{P}\leavevmode\nobreak\ [\mathrm{MeV}]$ | Ref. | $V$ | $f_{V}\leavevmode\nobreak\ [\mathrm{MeV}]$ | Ref.
---|---|---|---|---|---
$f_{\pi}$ | $130.2(8)$ | [41] | $f_{\rho}$ | $209.4(1.5)$ | [30]
$f_{K}$ | $155.7(3)$ | [41] | $f_{\phi}$ | $241(18)$ | [46]
$f_{D}$ | $212.0(7)$ | [41] | $f_{J/\psi}$ | $418(10)$ | [47]
$f_{B}$ | $190.0(1.3)$ | [41] | $f_{\Upsilon}$ | $649(31)$ | [48]
$f_{B_{s}}$ | $230.3(1.3)$ | [41] |  |  |
Table 3: Decay constants considered in our analysis.

####  3.1.3 Semileptonic decays: $P\to P^{\prime}\ell_{i}^{-}\ell_{j}^{+}$

The $P\to P^{\prime}\ell_{i}^{-}\ell_{j}^{+}$ decays (with $i<j$) can be
generally written in terms of the effective coefficients defined in Eq. (1) as

$\displaystyle\mathcal{B}(P\to P^{\prime}\ell_{i}^{-}\ell_{j}^{+})$
$\displaystyle=a_{VV}\,|C_{VV}^{(q)}|^{2}+a_{AV}\,|C_{AV}^{(q)}|^{2}+a_{TT}\,(|C_{T}^{(q)}|^{2}+|C_{T_{5}}^{(q)}|^{2})$
(55)
$\displaystyle+a_{SS}\,|C_{SS}^{(q)}|^{2}+a_{PS}\,|C_{PS}^{(q)}|^{2}+a_{VS}\,\mathrm{Re}[C_{VV}^{(q)}C_{SS}^{(q)\,\ast}]+a_{AP}\,\mathrm{Re}[C_{AV}^{(q)}C_{PS}^{(q)\,\ast}]$
$\displaystyle+a_{VT}\,\mathrm{Re}[C_{VV}^{(q)}\,C_{T}^{(q)\,\ast}]+a_{AT_{5}}\,\mathrm{Re}[C_{AV}^{(q)}\,C_{T_{5}}^{(q)\,\ast}]\,,$

where $C_{I}$ are the effective coefficients, evaluated at the relevant low-
energy scale, and $a_{I}$ stand for numeric coefficients that depend on the
form factors. Flavor indices are omitted in the above equation, for
simplicity, and should be replaced for the $q_{k}\to
q_{l}\ell_{i}^{-}\ell_{j}^{+}$ transition as follows,

$\displaystyle C_{I}^{(q)}\to C_{\begin{subarray}{c}I\\\
ijkl\end{subarray}}^{(q)}\,,$ (56)

where $I$ labels the Lorentz structure of the Wilson coefficients. In the
limit where the light-lepton mass ($m_{\ell_{i}}$) is neglected, we find that

$\displaystyle a_{VV}$ $\displaystyle\simeq a_{AV}\,,$ $\displaystyle a_{SS}$
$\displaystyle\simeq a_{PS}\,,$ $\displaystyle a_{VS}$ $\displaystyle\simeq-
a_{AP}\,,$ $\displaystyle a_{VT}$ $\displaystyle\simeq-a_{AT_{5}}\,,$ (57)

which are valid up to $\mathcal{O}(m_{\ell_{i}}/m_{\ell_{j}})$ corrections. In
the following, we compute these numerical coefficients for the most relevant
decays based on down- and up-type quark transitions, using the analytical
expressions from Appendix D and the $P\to P^{\prime}$ form factors described
below:

  * $\bullet$

$d_{l}\to d_{k}\ell_{i}^{-}\ell_{j}^{+}$: Our numerical coefficients for the
down-type quark transitions are collected in Table 5, where flavor indices are
omitted for simplicity and should be replaced following Eq. (56). For these
decays, the tensor coefficients $C_{T_{X}}$ are highly suppressed since they
only appear at $d=8$ once the $SU(2)_{L}\times U(1)_{Y}$ symmetry is imposed
[34], cf. also Appendix C. For this reason, we will not quote the numerical
values for the coefficients involving tensor operators. The numeric
coefficients are computed using lattice QCD form-factors for the $K\to\pi$
[52], $B\to K$ [53] (see Ref. [54]) and $B_{s}\to K$ [41] transitions. For
$B\to\pi$ decays, we have used the combined fit of experimental [55] and
lattice QCD [56] data made in Ref. [41] to have better control of the
uncertainties associated with the extrapolation of the lattice form-factors to
the entire physical region. 888Note, also, that we have not included effects
from $B_{s}-\overline{B}_{s}$ mixing in the above expressions, but these could
be easily implemented following Ref. [57].

We remind the reader that the $K_{L(S)}\to\pi^{0}e^{-}\mu^{+}$ decays have to
be treated separately, since
$|K_{L(S)}\rangle\simeq(|K^{0}\rangle\pm|\overline{K^{0}}\rangle)/\sqrt{2}$
[43]. Therefore, we have to amend Eq. (55) via the replacements,

$\displaystyle
C_{V_{XY}}^{(q)}\rightarrow\frac{1}{2}\Big{(}C_{\begin{subarray}{c}V_{XY}\\\
1221\end{subarray}}^{(q)}\mp C_{\begin{subarray}{c}V_{XY}\\\
1212\end{subarray}}^{(q)}\Big{)}\,,\qquad\quad
C_{S_{LL}}^{(q)}\rightarrow\frac{1}{2}\Big{(}C_{\begin{subarray}{c}S_{XY}\\\
1221\end{subarray}}^{(q)}\pm C_{\begin{subarray}{c}S_{XY}\\\
1212\end{subarray}}^{(q)}\Big{)}\,,$ (58)

for $X,Y\in\\{L,R\\}$, where the upper (lower) sign corresponds to the $K_{L}$
($K_{S}$) decays. Similar replacements must be made for the coefficients with
$L\leftrightarrow R$.

  * $\bullet$

$u_{l}\to u_{k}\ell_{i}^{-}\ell_{j}^{+}$: The only up-type transition relevant
for our study is $c\to ue\mu$, which can induce e.g. the decays $D\to\pi e\mu$
and $D_{s}\to Ke\mu$. Processes with the $\tau$-lepton are phase-space
forbidden. For these decays, we consider the $D\to\pi$ lattice QCD form-
factors from Ref. [58] and the $D_{s}\to K$ ones from Ref. [59], 999See also
the recent lattice QCD results for the $D\to\pi$ from Ref. [59]. which allow
us to determine the numerical coefficients collected in Table 5 by neglecting
the electron mass.

$P\to P^{\prime}\ell_{i}^{-}\ell_{j}^{+}$ | $a_{VV}$ | $a_{AV}$ | $a_{SS}$ | $a_{PS}$ | $a_{VS}$ | $a_{AP}$
---|---|---|---|---|---|---
$K^{+}\to\pi^{+}e^{-}\mu^{+}$ | $0.1570(11)$ | $0.1578(11)$ | $2.71(3)$ | $2.72(3)$ | $-0.723(7)$ | $0.735(7)$
$K_{L}\to\pi^{0}e^{-}\mu^{+}$ | $0.690(5)$ | $0.693(5)$ | $12.18(14)$ | $12.23(14)$ | $-3.19(3)$ | $3.24(3)$
$K_{S}\to\pi^{0}e^{-}\mu^{+}$ | $0.001208(9)$ | $0.001213(9)$ | $0.0213(2)$ | $0.0214(2)$ | $-0.00558(5)$ | $0.00567(5)$
$B^{+}\to\pi^{+}e^{-}\mu^{+}$ | $1.46(12)$ | $1.46(12)$ | $2.37(14)$ | $2.37(14)$ | $-0.154(10)$ | $0.156(10)$
$B^{+}\to\pi^{+}e^{-}\tau^{+}$ | $1.01(7)$ | $1.01(7)$ | $1.49(9)$ | $1.49(9)$ | $-1.29(8)$ | $1.29(8)$
$B^{+}\to\pi^{+}\mu^{-}\tau^{+}$ | $1.00(7)$ | $1.03(7)$ | $1.45(8)$ | $1.53(9)$ | $-1.17(7)$ | $1.42(8)$
$B_{s}\to K_{S}e^{-}\mu^{+}$ | $0.42(9)$ | $0.42(9)$ | $0.69(8)$ | $0.69(8)$ | $-0.043(8)$ | $0.043(8)$
$B_{s}\to K_{S}e^{-}\tau^{+}$ | $0.31(4)$ | $0.31(4)$ | $0.43(4)$ | $0.43(4)$ | $-0.39(4)$ | $0.39(5)$
$B_{s}\to K_{S}\mu^{-}\tau^{+}$ | $0.31(4)$ | $0.32(4)$ | $0.42(4)$ | $0.44(4)$ | $-0.35(4)$ | $0.42(5)$
$B^{+}\to K^{+}e^{-}\mu^{+}$ | $1.92(6)$ | $1.92(6)$ | $2.72(7)$ | $2.72(7)$ | $-0.209(6)$ | $0.211(6)$
$B^{+}\to K^{+}e^{-}\tau^{+}$ | $1.20(3)$ | $1.20(3)$ | $1.55(3)$ | $1.55(3)$ | $-1.53(4)$ | $1.53(4)$
$B^{+}\to K^{+}\mu^{-}\tau^{+}$ | $1.18(3)$ | $1.22(3)$ | $1.49(3)$ | $1.60(4)$ | $-1.37(3)$ | $1.68(4)$

Table 4: Numerical coefficients defined in Eq. (55) for the decays $P\to
P^{\prime}\ell_{i}^{-}\ell_{j}^{+}$ based on the down-type transition
$\smash{d_{k}\to d_{l}\ell_{i}^{-}\ell_{j}^{+}}$ (cf. Eq. (56)). Decays with
opposite lepton electric charges in the final state can be obtained via the
replacement $a_{VS}\to-a_{VS}$. Moreover, the expressions for the
$\smash{K_{L(S)}\to\pi^{0}e^{+}\mu^{-}}$ decays are obtained through the
replacement specified in Eq. (58). See text for details on the hadronic inputs
considered.

$P\to P^{\prime}\ell_{i}^{-}\ell_{j}^{+}$ | $a_{VV}=a_{AV}$ | $a_{SS}=a_{PS}$ | $a_{VS}=-a_{AP}$ | $a_{TT}$ | $a_{VT}=a_{AT_{5}}$
---|---|---|---|---|---
$D^{+}\to\pi^{+}e^{-}\mu^{+}$ | $0.0166(10)$ | $0.0337(11)$ | $-0.0066(3)$ | $0.019(3)$ | $0.0081(7)$
$D^{0}\to\pi^{0}e^{-}\mu^{+}$ | $0.00326(19)$ | $0.0066(2)$ | $-0.00130(5)$ | $0.0038(6)$ | $0.00160(14)$
$D_{s}\to K^{+}e^{-}\mu^{+}$ | $0.00622(6)$ | $0.00893(3)$ | $-0.002364(8)$ | $0.00377(4)$ | $0.00250(2)$

Table 5: Numerical coefficients defined in Eq. (55) for the decays $P\to
P^{\prime}e^{-}\mu^{+}$ based on the up-type transition $\smash{c\to
ue^{-}\mu^{+}}$. These expressions have been obtained by neglecting the
electron mass, cf. Eq. (60). See caption of Fig. 5 for details.

$P\to V\ell\ell$ | $a_{VV}$ | $a_{VA}$ | $a_{AV}$ | $a_{AA}$ | $a_{PP}$ | $a_{SP}$ | $a_{AP}$ | $a_{AS}$
---|---|---|---|---|---|---|---|---
$B^{0}\to\rho e^{-}\mu^{+}$ | $0.62(10)$ | $2.9(6)$ | $0.62(10)$ | $2.9(6)$ | $1.1(2)$ | $1.1(2)$ | $0.12(2)$ | $-0.12(2)$
$B^{0}\to\rho e^{-}\tau^{+}$ | $0.32(5)$ | $1.6(3)$ | $0.32(5)$ | $1.6(3)$ | $0.49(9)$ | $0.49(9)$ | $0.63(12)$ | $-0.63(12)$
$B^{0}\to\rho\mu^{-}\tau^{+}$ | $0.35(6)$ | $1.6(3)$ | $0.31(5)$ | $1.5(3)$ | $0.52(10)$ | $0.47(9)$ | $0.56(11)$ | $-0.71(13)$
$B^{0}\to K^{\ast\,0}e^{-}\mu^{+}$ | $0.68(11)$ | $3.4(5)$ | $0.68(11)$ | $3.4(5)$ | $1.06(15)$ | $1.06(15)$ | $0.127(17)$ | $-0.128(18)$
$B^{0}\to K^{\ast\,0}e^{-}\tau^{+}$ | $0.34(5)$ | $1.8(3)$ | $0.33(5)$ | $1.8(3)$ | $0.46(7)$ | $0.46(7)$ | $0.62(9)$ | $-0.62(9)$
$B^{0}\to K^{\ast\,0}\mu^{-}\tau^{+}$ | $0.35(6)$ | $1.9(3)$ | $0.31(5)$ | $1.8(3)$ | $0.48(7)$ | $0.43(7)$ | $0.54(8)$ | $-0.69(10)$
$B_{s}\to\phi e^{-}\mu^{+}$ | $0.58(5)$ | $3.7(5)$ | $0.58(5)$ | $3.7(5)$ | $1.21(17)$ | $1.21(17)$ | $0.15(2)$ | $-0.15(2)$
$B_{s}\to\phi e^{-}\tau^{+}$ | $0.29(2)$ | $1.9(2)$ | $0.29(2)$ | $1.9(2)$ | $0.51(7)$ | $0.51(7)$ | $0.71(10)$ | $-0.71(10)$
$B_{s}\to\phi\mu^{-}\tau^{+}$ | $0.30(3)$ | $1.9(3)$ | $0.27(2)$ | $1.8(2)$ | $0.53(8)$ | $0.48(7)$ | $0.62(9)$ | $-0.79(11)$

Table 6: Numerical coefficients defined in Eq. (55) for the decays $P\to
V\ell_{i}^{-}\ell_{j}^{+}$ based on the down-type transition $\smash{d_{k}\to
d_{l}\ell_{i}^{-}\ell_{j}^{+}}$. Flavor indices are omitted and should be
replace as in Eq. (56). Decays with opposite lepton electric charges in the
final state can be obtained via the replacement $a_{AP}\to-a_{AP}$. See text
for details on the hadronic inputs considered.

####  3.1.4 Semileptonic decays: $P\to V\ell_{i}^{-}\ell_{j}^{+}$

We turn now our attention to the semileptonic decays $P\to
V\ell_{i}^{-}\ell_{j}^{+}$ with a vector meson $V$ in the final state decays.
In this case, the branching fraction can be expressed in terms of Eq. (1) as
follows,

$\displaystyle\mathcal{B}(P\to V$
$\displaystyle\ell_{i}^{-}\ell_{j}^{+})=a_{VV}\,|C_{VV}|^{2}+a_{VA}\,|C_{VA}|^{2}+a_{AV}\,|C_{AV}|^{2}+a_{AA}\,|C_{AA}|^{2}$
$\displaystyle+a_{PP}\,|C_{PP}|^{2}+a_{SP}\,|C_{SP}|^{2}+a_{AP}\,\mathrm{Re}[C_{VA}C_{SP}^{\ast}]+a_{AS}\,\mathrm{Re}[C_{AA}C_{PP}^{\ast}]$
$\displaystyle+a_{TT}\,\big{(}|C_{T}|^{2}+|C_{T_{5}}|^{2}\big{)}+a_{ST}\,\text{Re}[C_{SP}C_{T}^{\ast}]+a_{PT}\,\text{Re}[C_{PP}C_{T}^{\ast}]$
$\displaystyle+a_{ST_{5}}\,\text{Re}[C_{SP}C_{T_{5}}^{\ast}]+a_{PT_{5}}\,\text{Re}[C_{PP}C_{T_{5}}^{\ast}]\,,$
(59)

with $C_{I}$ are the relevant Wilson coefficients, defined at the relevant
low-energy scale, and $a_{I}$ are numerical coefficients related to the
hadronic matrix elements. Flavor indices are omitted and should be replaced as
in Eq. (55). In the limit where the light-lepton mass ($m_{\ell_{i}}$) is
neglected, we find the relations

$\displaystyle a_{VV}$ $\displaystyle\simeq a_{AV}\,,$ $\displaystyle a_{AP}$
$\displaystyle\simeq a_{AS}\,,$ $\displaystyle a_{PT}$ $\displaystyle\simeq-
a_{ST_{5}}\,,$ (60) $\displaystyle a_{VA}$ $\displaystyle\simeq a_{AA}\,,$
$\displaystyle a_{PP}$ $\displaystyle\simeq a_{SP}\,,$ $\displaystyle a_{ST}$
$\displaystyle\simeq-a_{PT_{5}}\,,$

which are once again valid up to corrections of the order
$m_{\ell_{i}}/m_{\ell_{j}}$.

For the $P\to V$ based on the $d_{k}\to d_{l}\ell_{i}\ell_{j}$ transition, we
use the general analytical expressions provided in Ref. [42], which are
summarized in the Appendix D.2, and the Light-Cone-Sum-Rules form-factors from
Ref. [60] to obtain the $a_{i}$ values collected in Table 6 for the
$B\to\rho$, $B\to K^{\ast}$ and $B_{s}\to\phi$ decays (see also Ref. [61]).
Once again, we do not quote the numerical coefficients corresponding to tensor
coefficients since the corresponding effective coefficients are necessarily
suppressed once $SU(2)_{L}\times U(1)_{Y}$ gauge symmetry is imposed [34]. The
only up-type decays that would appear in this case are
$D^{+}\to\rho^{+}\mu^{-}e^{+}$ and $D_{s}\to K^{\ast+}\mu^{-}e^{+}$, which are
experimentally challenging, and for which the form-factors have not yet been
determined on the lattice. For these reasons, we only report numerical results
for the $P\to V$ based on down-type quark transitions.

###  3.2 $\tau\to\ell M$

The simplest hadronic $\tau$-decays that can probe LFV are $\tau\to\ell M$,
where $M=\bar{q}_{k}q_{l}$ can be a light pseudoscalar or vector meson. Here,
it is important to distinguish light pseudoscalar mesons $P$ with open flavor
($k\neq l$), such as $K_{S}$, from the unflavored ones ($k=l$), such as
$P\in\\{\pi^{0},\eta,\eta^{\prime}\\}$ [62]. This is because the latter decays
are also sensitive to the CP-odd gluonic operators defined in Eq. (7) in
addition to the semileptonic ones. We also briefly discuss the $\tau$ decays
into vector mesons $V$, such as $K^{\ast}$, and $\rho$ and $\omega$.

####  3.2.1 Pseudoscalar mesons: $\tau\to\ell P$

Firstly, we consider the $\tau\to\ell P$ process, where $P=\bar{q}_{k}q_{l}$
denotes a generic pseudoscalar meson. We discuss separately the case where
$k\neq l$ from the one where $k=l$:

##### Flavored mesons

In the first case, it is sufficient to consider the decay constant $f_{P}$
defined in Eq. (50), which allows us to write e.g. for $P=K_{S}$,

$\displaystyle\begin{split}\mathcal{B}(\tau\to\ell_{i}K_{S})=\tau_{\tau}\,&\dfrac{f_{P}^{2}m_{\tau}^{3}}{256\pi
v^{4}}\left(1-\dfrac{m_{P}^{2}}{m_{\tau}^{2}}\right)^{2}\\\
&\times\bigg{\\{}\bigg{|}C_{VA}^{(q)}+\dfrac{m_{P}^{2}\,{C}_{SP}^{(q)}}{m_{\tau}(m_{Q}+m_{q})}\bigg{|}^{2}+\bigg{|}C_{AA}^{(q)}-\dfrac{m_{P}^{2}\,{C}_{PP}^{(q)}}{m_{\tau}(m_{Q}+m_{q})}\bigg{|}^{2}\bigg{\\}}\,,\end{split}$
(61)

where we have used that
$|K_{S}\rangle\simeq(|K^{0}\rangle-|\overline{K^{0}}\rangle)/\sqrt{2}$ and we
have neglected the light-lepton mass ($m_{\ell_{i}}$), as before. Flavor
indices in the above expressions are to be replaced as follows,

$C_{I}^{(q)}\to\frac{1}{\sqrt{2}}\Big{(}C_{\begin{subarray}{c}I\\\
i312\end{subarray}}^{(d)}-C_{\begin{subarray}{c}I\\\
i321\end{subarray}}^{(d)}\Big{)}\,,$ (62)

for $I\in\\{VA,AA,SP,PP\\}$. Similar expressions can be obtained for
$\tau\to\ell_{i}K_{L}$ by changing the relative sign of the coefficients in
the above equation. However, these decay modes are experimentally challenging
due to the $K_{L}$ lifetime.

##### Unflavored mesons

The case of light unflavored mesons is rather different since there is not a
simple relation between the axial and pseudoscalar matrix elements, and since
the matrix element of the $G\widetilde{G}$ operator does not vanish in this
case. Following Ref. [63], we define

$\displaystyle\langle 0|\bar{q}\gamma^{\mu}\gamma_{5}q|P(p)\rangle$
$\displaystyle\equiv-ib_{q}f_{P}^{(q)}p^{\mu}\,,\qquad\quad 2m_{q}\langle
0|\bar{q}\gamma_{5}q|P\rangle\equiv ib_{q}h_{P}^{(q)}\,,$ (63)

and

$\displaystyle\langle
0|\dfrac{\alpha_{s}}{4\pi}G_{\mu\nu}\widetilde{G}^{\mu\nu}|P\rangle\equiv
a_{P}\,,$ (64)

where $q\in\\{u,d,s\\}$, $f_{P}^{(q)}$ and $h_{P}^{(q)}$ stand for the axial
and pseudoscalar densities, and the pre-factors $b_{q}$ are given by
$b_{u(d)}={1}/{\sqrt{2}}$ and $b_{s}=1$. In our numerical analysis, we will
compute the $\pi^{0}$ pseudoscalar density in the exact isospin limit, i.e.
$h_{\pi}^{(u)}=-h_{\pi}^{(d)}=f_{\pi}\,m_{\pi}^{2}$, and the anomaly
contribution by taking the divergence of the axial current [64], which gives

$\displaystyle
a_{\pi}=-\dfrac{1-z}{1+z}\dfrac{f_{\pi}m_{\pi}^{2}}{\sqrt{2}}\,,$ (65)

where $z=m_{u}/m_{d}$. For $\eta^{(\prime)}$, we rely on the computation of
$a_{\eta^{(\prime)}}$, $f_{\eta^{(\prime)}}^{(q)}$ and
$h_{\eta^{(\prime)}}^{(q)}$ from the so-called Feldmann-Kroll-Stech (FKS)
mixing scheme [65, 63], which leads to the phenomenological estimations
collected in Table 7. The additional contributions, in this case, can be
reabsorbed in the (pseudo)scalar effective coefficients,

$\displaystyle\overline{C}_{\begin{subarray}{c}SP\\\
i3kk\end{subarray}}^{(q)}$ $\displaystyle\equiv{C}_{\begin{subarray}{c}SP\\\
i3kk\end{subarray}}^{(q)}+\dfrac{4i}{b_{q}}\dfrac{m_{\tau}m_{q_{k}}}{v^{2}}\dfrac{a_{P}}{h_{P}}(C_{\begin{subarray}{c}\tilde{G}_{R}\\\
i3\end{subarray}}^{(\ell)}+C_{\begin{subarray}{c}\tilde{G}_{L}\\\
i3\end{subarray}}^{(\ell)})\,,$ (66)
$\displaystyle\overline{C}_{\begin{subarray}{c}PP\\\
i3kk\end{subarray}}^{(q)}$ $\displaystyle\equiv{C}_{\begin{subarray}{c}PP\\\
i3kk\end{subarray}}^{(q)}+\dfrac{4i}{b_{q}}\dfrac{m_{\tau}m_{q_{k}}}{v^{2}}\dfrac{a_{P}}{h_{P}}(C_{\begin{subarray}{c}\tilde{G}_{R}\\\
i3\end{subarray}}^{(\ell)}-C_{\begin{subarray}{c}\tilde{G}_{L}\\\
i3\end{subarray}}^{(\ell)})\,,$ (67)

$P$ | $f_{P}^{(q)}\leavevmode\nobreak\ [\mathrm{MeV}]$ | $f_{P}^{(s)}\leavevmode\nobreak\ [\mathrm{MeV}]$ | $h_{P}^{(q)}\leavevmode\nobreak\ [\mathrm{GeV}^{3}]$ | $h_{P}^{(s)}\leavevmode\nobreak\ [\mathrm{GeV}^{3}]$ | $a_{P}\leavevmode\nobreak\ [\mathrm{GeV}^{3}]$
---|---|---|---|---|---
$\eta$ | $108(3)$ | $-111(6)$ | $0.001(3)$ | $-0.055(3)$ | $-0.022(2)$
$\eta^{\prime}$ | $89(3)$ | $136(6)$ | $0.001(2)$ | $0.068(5)$ | $-0.057(2)$
Table 7: Hadronic inputs for $\eta^{(\prime)}$ obtained in Ref. [65, 63] by
using the FKS scheme.

where $m_{\tau}$ comes from the definition of the gluonic operators, $k$ spans
the light-quark flavors (i.e., $q_{k}\in\\{u,d,s\\}$) and the
$C_{{G}_{L}}^{(\ell)}$ and $C_{\tilde{G}_{L}}^{(\ell)}$ coefficients are
generated by heavy quarks at one loop, cf. Eq. (11). The final branching
fraction for $P\in\\{\pi^{0},\eta,\eta^{\prime}\\}$ thus reads

$\displaystyle\begin{split}\mathcal{B}(\tau\to\ell_{i}P)&=\tau_{\tau}\,\dfrac{m_{\tau}^{3}}{256\pi
v^{4}}\left(1-\dfrac{m_{P}^{2}}{m_{\tau}^{2}}\right)^{2}\times\Bigg{\\{}\bigg{|}\sum_{q_{k}=u,d,s}\bigg{[}b_{q_{k}}f_{P}^{(q_{k})}C_{\begin{subarray}{c}VA\\\
i3kk\end{subarray}}^{(q)}+\dfrac{b_{q_{k}}h_{P}^{(q_{k})}}{2m_{\tau}m_{q_{k}}}\,\overline{C}_{\begin{subarray}{c}SP\\\
i3kk\end{subarray}}^{(q)}\bigg{]}\bigg{|}^{2}\\\
&+\bigg{|}\sum_{q_{k}=u,d,s}\bigg{[}b_{q_{k}}f_{P}^{(q_{k})}C_{\begin{subarray}{c}AA\\\
i3kk\end{subarray}}^{(q)}-\dfrac{b_{q_{k}}h_{P}^{(q_{k})}}{2m_{\tau}m_{q_{k}}}\,\overline{C}_{\begin{subarray}{c}PP\\\
i3kk\end{subarray}}^{(q)}\bigg{]}\bigg{|}^{2}\Bigg{\\}}\,,\end{split}$ (68)

where we have neglected the light-lepton mass, $m_{\ell_{i}}$.

####  3.2.2 Vector mesons: $\tau\to\ell V$

Lastly, we consider the decays $\tau\to\ell V$, where $V=\bar{q}_{k}q_{l}$
denotes a generic vector meson such as $V\in\\{K^{\ast};\omega,\rho,\phi\\}$.
By using the decay constants defined in Eq. (53), we can show that

$\displaystyle\mathcal{B}(\tau\to\ell_{i}V)$
$\displaystyle=\tau_{\tau}\,\dfrac{f_{V}^{2}m_{\tau}^{3}}{256\pi
v^{4}}\left(1-\dfrac{m_{V}^{2}}{m_{\tau}^{2}}\right)^{2}$ (69)
$\displaystyle\times\bigg{\\{}\Big{[}|C_{\begin{subarray}{c}VV\\\
i3kl\end{subarray}}^{(q)}|^{2}+|C_{\begin{subarray}{c}AV\\\
i3kl\end{subarray}}^{(q)}|^{2}\Big{]}\bigg{(}1+\dfrac{2m_{V}^{2}}{m_{\tau}^{2}}\bigg{)}+32\bigg{(}\dfrac{f_{V}^{T}}{f_{V}}\bigg{)}^{2}\Big{[}|\bar{C}_{\begin{subarray}{c}T\\\
i3kl\end{subarray}}^{(q)}|^{2}+|\bar{C}_{\begin{subarray}{c}T_{5}\\\
i3kl\end{subarray}}^{(q)}|^{2}\Big{]}\bigg{(}1+\dfrac{m_{V}^{2}}{2m_{\tau}^{2}}\bigg{)}$
$\displaystyle+24\dfrac{m_{V}}{m_{\tau}}\dfrac{f_{V}^{T}}{f_{V}}\,\mathrm{Re}\big{[}\bar{C}_{\begin{subarray}{c}T\\\
i3kl\end{subarray}}^{(q)}\,C_{\begin{subarray}{c}VV\\\
i3kl\end{subarray}}^{(q)\ast}-\bar{C}_{\begin{subarray}{c}T_{5}\\\
i3kl\end{subarray}}^{(q)}\,C_{\begin{subarray}{c}AV\\\
i3kl\end{subarray}}^{(q)\ast}\big{]}\,,$

with

$\displaystyle\bar{C}_{\begin{subarray}{c}T\\\ i3kl\end{subarray}}^{(q)}$
$\displaystyle\equiv C_{\begin{subarray}{c}T\\\
i3kl\end{subarray}}^{(q)}-\delta_{kl}eQ_{q}\frac{m_{\tau}f_{V}}{m_{V}f_{V}^{T}}\Big{(}C_{\begin{subarray}{c}D_{R}\\\
i3\end{subarray}}^{(\ell)}+C_{\begin{subarray}{c}D_{L}\\\
i3\end{subarray}}^{(\ell)}\Big{)}\;,$ (70)
$\displaystyle\bar{C}_{\begin{subarray}{c}T_{5}\\\ i3kl\end{subarray}}^{(q)}$
$\displaystyle\equiv
C_{T_{5}}^{(q)}-\delta_{kl}eQ_{q}\frac{m_{\tau}f_{V}}{m_{V}f_{V}^{T}}\Big{(}C_{\begin{subarray}{c}D_{R}\\\
i3\end{subarray}}^{(\ell)}-C_{\begin{subarray}{c}D_{L}\\\
i3\end{subarray}}^{(\ell)}\Big{)}\;,$ (71)

where $Q_{q}$ denotes the $q_{k,l}$ electric charges, and we have neglected
again the light-lepton mass, $m_{\ell_{i}}$. For $V\in\\{\rho,\omega\\}$, in
the isospin limit, we have to perform the following trivial replacements in
the above expression,

$\displaystyle
C_{I}^{(q)}\overset{V=\rho}{\to}\frac{1}{\sqrt{2}}\Big{(}C_{\begin{subarray}{c}I\\\
i311\end{subarray}}^{(u)}-C_{\begin{subarray}{c}I\\\
i311\end{subarray}}^{(d)}\Big{)}\,,\qquad\quad
C_{I}^{(q)}\overset{V=\omega}{\to}\frac{1}{\sqrt{2}}\Big{(}C_{\begin{subarray}{c}I\\\
i311\end{subarray}}^{(u)}+C_{\begin{subarray}{c}I\\\
i311\end{subarray}}^{(d)}\Big{)}\,,$ (72)

where $I\in\\{VV,AV,T,T_{5}\\}$.

###  3.3 $\ell_{j}\to\ell_{i}\gamma$ and $\ell_{j}\to\ell_{i}\ell_{k}\ell_{k}$

We now turn our discussion to purely leptonic observables (see e.g. Ref. [15,
66, 16] for previous EFT studies). These processes receive contributions from
semileptonic operators at loop level, as illustrated in Fig. 3. The
experimental limits considered in our analysis are collected in Table 8.

##### Radiative decays

The simplest of these processes is the radiative decay
$\ell_{j}\to\ell_{i}\gamma$ with $i<j$, where the photon is on-shell. This
process is described by

$\displaystyle\mathcal{B}(\ell_{j}\to\ell_{i}\gamma)=\frac{\tau_{\ell_{j}}m_{\ell_{j}}^{5}}{4\pi
v^{4}}\Big{(}|C_{\begin{subarray}{c}D_{L}\\\
ij\end{subarray}}^{(\ell)}|^{2}+|C_{\begin{subarray}{c}D_{R}\\\
ij\end{subarray}}^{(\ell)}|^{2}\Big{)}\,.$ (73)

where we have neglected the lightest lepton mass, as before.

##### Three-body decays

The $\ell_{j}^{-}\to\ell_{i}^{-}\ell_{k}^{+}\ell_{k}^{-}$ decays receive
contributions from the dipole operators, in addition to operators with four
leptons. We distinguish the case where $j>i=k$ from $j>i>k$ which have
slightly different expressions. In the first case (i.e., $\tau^{-}\to
e^{-}e^{+}e^{+}$ and $\tau^{-}\to\mu^{-}\mu^{+}\mu^{+}$) [15],

$\displaystyle\mathcal{B}(\ell_{j}^{-}\to\ell_{i}^{-}\ell_{i}^{+}\ell_{i}^{-})=\frac{\tau_{\ell_{j}}m_{\ell_{j}}^{5}}{1536\pi^{3}v^{4}}$
$\displaystyle\bigg{\\{}2|C_{\begin{subarray}{c}V_{LL}\\\
ijii\end{subarray}}^{(\ell)}|^{2}+|C_{\begin{subarray}{c}V_{LR}\\\
ijii\end{subarray}}^{(\ell)}|^{2}+64e^{2}\Big{(}\log\frac{m_{\ell_{j}}}{m_{\ell_{k}}}-\frac{11}{8}\Big{)}|C_{\begin{subarray}{c}D_{L}\\\
ij\end{subarray}}^{(\ell)}|^{2}$ (74)
$\displaystyle+8e\text{Re}\Big{[}C_{\begin{subarray}{c}D_{R}\\\
ij\end{subarray}}^{(\ell)}(2C_{\begin{subarray}{c}V_{LL}\\\
ijii\end{subarray}}^{(\ell)}+C_{\begin{subarray}{c}V_{LR}\\\
ijii\end{subarray}}^{(\ell)})^{\ast}\Big{]}+(L\leftrightarrow R)\bigg{\\}}\,,$

where the light lepton massesare neglected in the above expression, except in
the dipole term, which would otherwise be infrared divergent. Note, also, that
we have not included the contributions from the scalar coefficients
$\smash{C_{{S_{XY}}}}$ (with $X,L\in\\{L,R\\}$), as they are not induced by
$d=6$ operators in the SMEFT, cf. Appendix C.

A similar expression can be obtained for the decay with $j>i>k$ (i.e.,
$\tau^{-}\to\mu^{-}e^{+}e^{-}$), which has no identical particles in the final
state [15],

$\displaystyle\mathcal{B}(\ell_{j}^{-}\to\ell_{i}^{-}\ell_{k}^{+}\ell_{k}^{-})=\frac{\tau_{\ell_{j}}m_{\ell_{j}}^{5}}{1536\pi^{3}v^{4}}$
$\displaystyle\bigg{\\{}|C_{\begin{subarray}{c}V_{LL}\\\
ijkk\end{subarray}}^{(\ell)}|^{2}+|C_{\begin{subarray}{c}V_{LR}\\\
ijkk\end{subarray}}^{(\ell)}|^{2}+64e^{2}\Big{(}\log\frac{m_{\ell_{j}}}{m_{\ell_{k}}}-\frac{3}{2}\Big{)}|C_{\begin{subarray}{c}D_{L}\\\
ij\end{subarray}}^{(\ell)}|^{2}$ (75)
$\displaystyle+8e\,\text{Re}\Big{[}C_{\begin{subarray}{c}D_{R}\\\
ij\end{subarray}}^{(\ell)}(C_{\begin{subarray}{c}V_{LL}\\\
ijkk\end{subarray}}^{(\ell)}+C_{\begin{subarray}{c}V_{LR}\\\
ijkk\end{subarray}}^{(\ell)})^{\ast}\Big{]}+(L\leftrightarrow R)\bigg{\\}}\,,$

where we have kept, once again, the light-lepton masses only in the
logarithmic term. We do not include the contributions from the scalar
$C_{S_{XY}}$ and tensor $C_{T_{X}}$ coefficients (with $X,L\in\\{L,R\\}$), as
before, since they do not appear in the tree-level matching to the SMEFT, as
shown in Appendix C.

Process | Exp. limit | Process | Exp. limit | Process | Exp. limit
---|---|---|---|---|---
$\mu\to e\gamma$ | $5.6\times 10^{-13}$ | $\tau\to e\gamma$ | $4.4\times 10^{-8}$ | $\tau\to\mu\gamma$ | $5.6\times 10^{-8}$
$\mu\to eee$ | $1.3\times 10^{-12}$ | $\tau\to eee$ | $3.6\times 10^{-8}$ | $\tau\to\mu\mu\mu$ | $2.8\times 10^{-8}$
|  | $\tau\to e\mu\mu$ | $3.6\times 10^{-8}$ | $\tau\to\mu ee$ | $2.4\times 10^{-8}$
Table 8: Experimental limits on purely leptonic LFV decays at 95$\%$ CL [30].

###  3.4 $\mu N\to eN$

Lastly, we discuss $\mu\to e$ conversion in nuclei, which has been thoroughly
studied within the EFT approach e.g. in Ref. [67]. The nucleon effective
coefficients are given in terms of our Wilson coefficients as follows [16],

$\displaystyle\begin{split}\widetilde{C}^{VX}_{(p)}&=\sum_{q_{k}=u,d,s}\sum_{Y=L,R}C^{(q)}_{\begin{subarray}{c}V_{XY}\\\
12kk\end{subarray}}\,f^{(q_{k})}_{Vp}\,,\\\\[4.49997pt]
\widetilde{C}^{SX}_{(p)}&=\sum_{q_{k}=u,d,s}\sum_{Y=L,R}\dfrac{m_{p}}{m_{q_{k}}}C^{(q)}_{\begin{subarray}{c}S_{XY}\\\
12kk\end{subarray}}\,f^{(q_{k})}_{S\,p}+\dfrac{m_{\mu}m_{p}}{4\pi
v^{2}}{C}^{(\ell)}_{\begin{subarray}{c}G_{X}\\\
12\end{subarray}}\,f_{G\,p}\,,\end{split}$ (76)

where $X=L,R$, with analogous expressions for $p\to n$. The effective
coefficients are evaluated at $\mu\approx 1$ GeV, thus including contributions
from scalar operators made of heavy quarks [25], as described in Eq. (11). For
the vector current, the nucleon form-factors are given by
$f^{(u)}_{Vp}=f^{(d)}_{Vn}=2$, $f^{(d)}_{Vp}=f^{(u)}_{Vn}=1$ and
$f^{(s)}_{Vp}=f^{(s)}_{Vn}=1$. We consider the numerical results for the
scalar form-factors from Ref. [16], which are based on Ref. [68],

$\displaystyle f_{S\,p}^{(u)}$ $\displaystyle=(20.8\pm 1.5)\times 10^{-3}\,,$
$\displaystyle f_{S\,n}^{(u)}$ $\displaystyle=(18.9\pm 1.4)\times 10^{-3}\,,$
(77) $\displaystyle f_{S\,p}^{(d)}$ $\displaystyle=(41.1\pm 2.8)\times
10^{-3}\,,$ $\displaystyle f_{S\,n}^{(u)}$ $\displaystyle=(45.1\pm 2.7)\times
10^{-3}\,,$ (78) $\displaystyle f_{S\,p}^{(s)}$
$\displaystyle=f_{Sn}^{(s)}=(53\pm 27)\times 10^{-3}\,,$ (79)

with the gluonic form-factors given by

$\displaystyle
f_{G\,p(n)}=-\dfrac{8\pi}{9}\Big{(}1-\sum_{q=u,d,s}f_{S\,p(n)}^{(q)}\Big{)}\,.$
(80)

The $\mu\to e$ conversion rate normalized by the muon capture rate
$(\Gamma_{\mathrm{capt}})$ is denoted by $\mathcal{B}^{(N)}_{\mu e}$ and it
can be written in terms of nucleon EFT Wilson coefficients as follows [16],

$\displaystyle\mathcal{B}^{(N)}_{\mu
e}=\frac{m_{\mu}^{5}}{4\upsilon^{4}\Gamma^{(N)}_{\text{capt}}}\Big{|}C^{(\ell)}_{D_{L}}\,D_{N}+4\Big{[}\widetilde{C}^{VR}_{(p)}\,V_{N}^{(p)}+\widetilde{C}^{SL}_{(p)}\,S^{(p)}_{N}+(p\to
n)\Big{]}\Big{|}^{2}+(L\leftrightarrow R)\,.$ (81)

Nuclei | $D$ | $V^{(p)}$ | $V^{(n)}$ | $S^{(p)}$ | $S^{(n)}$
---|---|---|---|---|---
Au | $0.189$ | $0.0974$ | $0.146$ | $0.0614$ | $0.0918$
Al | $0.0362$ | $0.0161$ | $0.0173$ | $0.0155$ | $0.0167$
Table 9: Overlap integrals for gold and aluminum atoms from Ref. [69].

For the nuclear-physics inputs, we consider the results of Ref. [69], which
are summarized in Table 9, and the nuclear capture rates determined
experimentally [70],

$\displaystyle\Gamma_{\mathrm{capt}}^{(\mathrm{Al})}$
$\displaystyle\simeq{6.99\times 10^{-7}\leavevmode\nobreak\
\mathrm{ps}^{-1}}\,,\qquad\qquad\Gamma_{\mathrm{capt}}^{(\mathrm{Au})}\simeq{1.32\times
10^{-5}\leavevmode\nobreak\ \mathrm{ps}^{-1}}\,.$ (82)

Currently, the most stringent experimental limit is
$\mathcal{B}^{(\mathrm{Au})}_{\mu e}<7\times 10^{-13}$ (90% CL.), which was
set by the SINDRUM-II experiment [71]. The experimental sensitivity is planned
to be considerably improved in the near future by the Mu2e experiment at
Fermilab [5] and COMET [7] at J-PARC, which are expected to reach a
sensitivity of $\mathcal{O}(10^{-17})$ with aluminum atoms.

##  4 High-energy probes

In this Section, we describe the high-energy probes of LFV in terms of the SM
EFT Lagrangian defined in Eq. (24). We will consider the decays of the
$Z$-boson, the Higgs boson, and the top quark, which are subject to the
experimental limits collected in Table 10. Furthermore, we will briefly
discuss the Drell-Yan processes $pp\to\ell_{i}\ell_{j}$ at the LHC that are
sensitive to non-resonant contributions from the SM EFT operators.

###  4.1 $Z\to\ell_{i}\ell_{j}$

The $Z$-boson LFV decays receive contributions at tree level from the
effective coefficients $\smash{\mathcal{C}_{Hl}^{(1)}}$ and
$\smash{\mathcal{C}_{Hl}^{(3)}}$, as well as the dipoles $\mathcal{C}_{eB}$
and $\mathcal{C}_{eW}$, which are both sensitive to one-loop contributions
from semileptonic operators (see e.g. Ref.[78]), 101010We note that the
orthogonal combination $\mathcal{C}_{e\gamma}\equiv-
s_{W}\,\mathcal{C}_{eW}+c_{W}\,\mathcal{C}_{eB}$ of the dipole coefficients is
already tightly constrained by $\ell_{i}\to\ell_{j}\gamma$.

$\displaystyle\mathcal{B}(Z\to\ell_{i}^{-}\ell_{j}^{+})=\dfrac{\tau_{Z}m_{Z}^{3}v^{2}}{24\pi\Lambda^{4}}\Big{[}\big{|}\mathcal{C}_{\begin{subarray}{c}Hl\\\
ij\end{subarray}}^{(1+3)}\big{|}^{2}+\big{|}\mathcal{C}_{\begin{subarray}{c}He\\\
ij\end{subarray}}\big{|}^{2}+\big{|}\mathcal{C}_{\begin{subarray}{c}eZ\\\
ij\end{subarray}}\big{|}^{2}+\big{|}\mathcal{C}_{\begin{subarray}{c}eZ\\\
ji\end{subarray}}\big{|}^{2}\Big{]}\,,$ (83)

where we define
$\mathcal{C}_{eZ}\equiv-\cos{\theta_{W}}\,\mathcal{C}_{eW}-\sin{\theta_{W}}\,\mathcal{C}_{eB}$,
$\theta_{W}$ is the Weinberg angle, and $\mathcal{C}_{Hl}^{(1\pm
3)}\equiv\mathcal{C}_{Hl}^{(1)}\pm\mathcal{C}_{Hl}^{(3)}$. The $Z$-boson mass
and lifetime are denoted by $m_{Z}$ and $\tau_{Z}$, respectively, and we
neglect lepton masses in the above expression. We notice that far more
stringent constraints on $\mathcal{C}_{eW}$ and $\mathcal{C}_{eB}$ can be
obtained by the $\ell_{i}\to\ell_{j}\gamma$ processes. However, these
observables still provided useful constraints on
$\smash{\mathcal{C}_{Hl}^{(1,3)}}$ and $\mathcal{C}_{He}$. The latter
coefficients are related to $\smash{\mathcal{C}_{lq}^{(1,3)}}$ and
$\mathcal{C}_{eu}$, respectively, with third-generation quarks, through the
RGE effects depicted in Fig. 3, cf. e.g. Ref. [22].

###  4.2 $h\to\ell_{i}\ell_{j}$

The Higgs boson decays are also efficient probes of LFV [79] (see also Ref.
[80]). The only SM EFT $d=6$ operator that contributes at tree level to this
process is $\mathcal{O}_{eH}$, which can receive sizable one-loop
contributions from the scalar operator $\mathcal{C}_{ledq}$ through RGEs [29].
After electroweak-symmetry breaking, this contributes not only to the Higgs
coupling to leptons, but it also induces a non-diagonal contribution to the
fermion masses,

$\displaystyle\begin{split}\mathcal{L}_{\mathrm{SMEFT}}&\supset\dfrac{1}{\Lambda^{2}}\mathcal{C}_{\begin{subarray}{c}eH\\\
ij\end{subarray}}\big{(}\bar{l}_{i}e_{j}H\big{)}\big{(}H^{\dagger}H\big{)}+\mathrm{h.c.}\\\\[3.50006pt]
&\overset{\langle H\rangle\neq
0}{\rightarrow}\dfrac{v^{3}}{2\sqrt{2}\Lambda^{2}}\mathcal{C}_{\begin{subarray}{c}eH\\\
ij\end{subarray}}\big{(}\bar{\ell}_{i}P_{R}\ell_{j}\big{)}\Big{(}1+3\frac{h}{v}+3\frac{h^{2}}{v^{2}}+\frac{h^{3}}{v^{3}}\Big{)}+\mathrm{h.c.}\,,\end{split}$
(84)

Process | Exp. limit | Ref. | Process | Exp. limit | Ref. | Process | Exp. limit | Ref.
---|---|---|---|---|---|---|---|---
$Z\to e\mu$ | $2.6\times 10^{-7}$ | [72] | $Z\to e\tau$ | $5.0\times 10^{-6}$ | [30] | $Z\to\mu\tau$ | $6.5\times 10^{-6}$ | [30]
$h\to e\mu$ | $4.4\times 10^{-5}$ | [73] | $h\to e\tau$ | $2.0\times 10^{-3}$ | [74] | $h\to\mu\tau$ | $1.5\times 10^{-3}$ | [30]
$t\to ce\mu$ | $2.6\times 10^{-6}$ | [75] | $t\to ce\tau$ | $1.9\times 10^{-5}$ | [76] | $t\to c\mu\tau$ | $1.1\times 10^{-6}$ | [77]

Table 10: Experimental limits on LFV decays of the $Z$-boson, Higgs and top-
quark at 95$\%$ CL. Note that the decays with opposite lepton charges are
combined, i.e.
$\ell_{i}\ell_{j}\equiv\ell_{j}^{+}\ell_{i}^{-}+\ell_{j}^{-}\ell_{i}^{+}$.

which must be re-absorbed by a rotation of the lepton fields. The effective
Higgs coupling to leptons is then given by 111111See Appendix E for a
discussion of Yukawa coupling running in the SM.

$\displaystyle y_{\begin{subarray}{c}e\\\
ij\end{subarray}}^{\mathrm{eff}}\equiv\sqrt{2}\,\dfrac{m_{\ell_{i}}}{v}\delta_{ij}-\dfrac{v^{2}}{\Lambda^{2}}\mathcal{C}_{\begin{subarray}{c}eH\\\
ji\end{subarray}}^{\ast}\,,$ (85)

which implies that

$\displaystyle\mathcal{B}(H\to\ell_{i}^{-}\ell_{j}^{+})=\frac{\tau_{h}v^{4}m_{h}}{32\pi\Lambda^{4}}\Big{[}|\mathcal{C}_{\begin{subarray}{c}eH\\\
ij\end{subarray}}|^{2}+|\mathcal{C}_{\begin{subarray}{c}eH\\\
ji\end{subarray}}|^{2}\Big{]}\,,$ (86)

where the lepton masses have been neglected, and $m_{h}$ and $\tau_{h}$ denote
the Higgs-boson mass and lifetime, respectively.

###  4.3 $t\to c\ell_{i}\ell_{j}$

The top-quark decays $t\to c\ell_{i}\ell_{j}$ can also be used to probe LFV at
high-energies [81]. The current experimental precision only allows us to
consistently probe tree-level contributions, cf. Table 10. The branching
fractions for these decays are given by,

$\displaystyle\begin{split}\mathcal{B}(t\to
c\ell_{i}^{-}\ell_{j}^{+})=\frac{\tau_{t}m_{t}^{5}}{1536\pi^{3}\Lambda^{4}}&\bigg{[}\big{|}\mathcal{C}_{\begin{subarray}{c}lq\\\
ij23\end{subarray}}^{\prime(1-3)}\big{|}^{2}+\big{|}\mathcal{C}^{\prime}_{\begin{subarray}{c}eq\\\
ij23\end{subarray}}\big{|}^{2}+\big{|}\mathcal{C}_{\begin{subarray}{c}eu\\\
ij23\end{subarray}}\big{|}^{2}+\big{|}\mathcal{C}_{\begin{subarray}{c}lu\\\
ij23\end{subarray}}\big{|}^{2}\\\\[3.00003pt]
&+\frac{1}{4}\Big{(}\big{|}\mathcal{C}_{\begin{subarray}{c}lequ\\\
ij23\end{subarray}}^{\prime(1)}\big{|}^{2}+\big{|}\mathcal{C}_{\begin{subarray}{c}lequ\\\
ji32\end{subarray}}^{\prime(1)}\big{|}^{2}\Big{)}+12\Big{(}\big{|}\mathcal{C}_{\begin{subarray}{c}lequ\\\
ij23\end{subarray}}^{\prime(3)}\big{|}^{2}+\big{|}\mathcal{C}_{\begin{subarray}{c}lequ\\\
ji32\end{subarray}}^{\prime(3)}\big{|}^{2}\Big{)}\bigg{]}\,,\end{split}$ (87)

where lepton and charm-quark masses have been neglected, and $m_{t}$ and
$\tau_{t}$ denote the top-quark mass and lifetime, respectively. We use once
again the shorthand notation $\mathcal{C}_{lq}^{(1\pm
3)}=\mathcal{C}_{lq}^{(1)}\pm\mathcal{C}_{lq}^{(3)}$, and we define the primed
Wilson coefficients as follows,

$\displaystyle\begin{split}\mathcal{C}_{lq}^{\prime\,(1-3)}&\equiv
V\mathcal{C}_{lq}^{(1-3)}V^{\dagger}\,,\\\\[3.99994pt]
\mathcal{C}_{eq}^{\prime}&\equiv V\mathcal{C}_{eq}^{\dagger}\,,\\\\[3.99994pt]
\mathcal{C}_{lequ}^{\prime\,(1)}&\equiv
V\mathcal{C}_{lequ}^{\,(1)}\,,\\\\[3.99994pt]
\mathcal{C}_{lequ}^{\prime\,(3)}&\equiv
V\mathcal{C}_{lequ}^{\,(3)}\,,\end{split}$ (88)

where the CKM matrix ($V$) acts on quark-flavor indices, e.g.
$\smash{\mathcal{C}^{\prime\,(1)}_{\begin{subarray}{c}lequ\\\
ijkl\end{subarray}}\equiv\sum_{k^{\prime}}V_{kk^{\prime}}\,\mathcal{C}_{\begin{subarray}{c}lequ\\\
ijk^{\prime}l\end{subarray}}^{(1)}}$ . Analogous expressions apply to the
$t\to u\ell_{i}\ell_{j}$ decays.

###  4.4 $pp\to\ell_{i}\ell_{j}$

The study of $pp\to\ell_{i}^{-}\ell_{j}^{+}$ at high-$p_{T}$ can also provide
useful probes of LFV in semileptonic operators since these EFT contributions
are energy-enhanced at the tails of the distributions [82]. The
$q_{l}\bar{q}_{k}\to\ell_{i}^{-}\ell_{j}^{+}$ partonic cross-section (with
$q=u,d$) can be written as [83]

$\hat{\sigma}(q_{k}\bar{q}_{l}\to\ell_{i}^{-}\ell_{j}^{+})=\dfrac{\hat{s}}{144\pi
v^{4}}\sum_{IJ}\mathcal{C}_{I}^{(q)\ast}\,\Omega_{IJ}\,\mathcal{C}_{J}^{(q)}\,,$
(89)

where $\hat{s}$ is the partonic center-of-mass energy, and $\mathcal{C}^{(u)}$
and $\mathcal{C}^{(d)}$ are vectors of effective coefficients [44, 45],
121212The zeros in the $\mathcal{C}^{(d)}$ reflect the fact there are no
tensor effective coefficients for the $d_{i}\to d_{j}\ell\ell$ transition at
$d=6$ in the SM EFT [34].

$\displaystyle\vec{\mathcal{C}}^{\,(u)}$
$\displaystyle=\Big{[}\mathcal{C}_{\begin{subarray}{c}lq\\\
ijkl\end{subarray}}^{\prime\,(1-3)}\,,\;\mathcal{C}_{\begin{subarray}{c}lu\\\
ijkl\end{subarray}}\,,\;\mathcal{C}_{\begin{subarray}{c}eq\\\
ijkl\end{subarray}}^{\prime}\,,\;\mathcal{C}_{\begin{subarray}{c}eu\\\
ijkl\end{subarray}}\,,\;\mathcal{C}_{\begin{subarray}{c}lequ\\\
ijkl\end{subarray}}^{\prime\,(1)}\,,\;\mathcal{C}_{\begin{subarray}{c}lequ\\\
jilk\end{subarray}}^{\prime\,(1)\,\ast}\,,\;\mathcal{C}_{\begin{subarray}{c}lequ\\\
ijkl\end{subarray}}^{\prime\,(3)}\,,\;\mathcal{C}_{\begin{subarray}{c}lequ\\\
jilk\end{subarray}}^{\prime\,(3)\,\ast}\Big{]}\,,$ (90)
$\displaystyle\vec{\mathcal{C}}^{\,(d)}$
$\displaystyle=\Big{[}\mathcal{C}_{\begin{subarray}{c}lq\\\
ijkl\end{subarray}}^{(1+3)}\,,\;\mathcal{C}_{\begin{subarray}{c}ld\\\
ijkl\end{subarray}}\,,\;\mathcal{C}_{\begin{subarray}{c}eq\\\
ijkl\end{subarray}}\,,\;\mathcal{C}_{\begin{subarray}{c}ed\\\
ijkl\end{subarray}}\,,\;\mathcal{C}_{\begin{subarray}{c}ledq\\\
ijkl\end{subarray}}\,,\;\mathcal{C}_{\begin{subarray}{c}ledq\\\
jilk\end{subarray}}^{\ast}\,,\;0\,,\;0\Big{]}\,,$ (91)

where the primed effective-coefficients appearing in $\mathcal{\vec{C}}^{(u)}$
are defined in Eq. (88). Moreover, $\Omega$ is a $8\times 8$ matrix that takes
a diagonal form $\Omega_{IJ}=\Omega_{I}\,\delta_{IJ}$ for the full partonic
cross-section,

$\Omega=\mathrm{diag}\big{(}1\,,\;1\,,\;1\,,\;1\,,\;3/4\,,\;3/4\,,\;4\,,\;4\big{)}\,.$
(92)

The Drell-Yan cross-section is given by the convolution of (89) with the
parton luminosity functions,

$\displaystyle{\sigma}(pp\to\ell_{i}^{-}\ell_{j}^{+})=\sum_{k,l}\int\dfrac{\mathrm{d}\hat{s}}{s}\mathcal{L}_{q_{k}\bar{q}_{l}}\,\hat{\sigma}(q_{k}\bar{q}_{l}\to\ell_{i}^{-}\ell_{j}^{+})\,,$
(93)

where

$\displaystyle\mathcal{L}_{q_{k}\bar{q}_{l}}(\hat{s})\equiv\int_{\hat{s}/s}^{1}\dfrac{\mathrm{d}x}{x}\Big{[}f_{{q}_{k}}(x,\mu_{F})f_{\bar{q}_{l}}(\frac{\hat{s}}{sx},\mu_{F})+({q}_{k}\leftrightarrow\bar{q}_{l})\Big{]}\,,$
(94)

where $f_{{q}_{k}}$ and $f_{\bar{q}_{l}}$ are the Parton Distribution
Functions (PDFs) of $q_{k}$ and $\bar{q}_{l}$, and $\mu_{F}$ denotes the
factorization scale. In this paper, we consider the constraints on SMEFT
operators derived in the HighPT package [84] through an appropriate recast of
the latest CMS search (with $140\leavevmode\nobreak\ \mathrm{fb}^{-1}$) for
heavy resonances decaying into LFV lepton pairs [85].

##  5 Numerical results

###  5.1 LFV from top-quark loops

In this Section, we illustrate our results by considering EFT scenarios with
semileptonic operators only involving third-generation quarks at high
energies. This choice is motivated by Minimal Flavor Violation [86] or the
$U(2)^{5}$ flavor symmetry [87] (see also Ref. [88]), in which the couplings
to quarks are hierarchical. This example is also convenient from a pragmatic
point of view since the various quark-level transitions become related through
RGE effects induced e.g. by the top-quark Yukawa, allowing us to compare the
sensitivity of different observables with a minimal set of Wilson
coefficients.

With the assumption that only the effective operators made of third-generation
quark are present at the scale $\Lambda\gg v$, we have derived constraints on
$\Lambda$ (for $\mathcal{C}=1$) for each semileptonic operator appearing in
Table 1, by using the various processes that receive contributions at tree-
and loop-level, cf. Figs. 2 and 3. We collect these results in Table 11, 13
and 13 for the $\mu\to e$, $\tau\to e$ and $\tau\to\mu$ transitions,
respectively. We find that the different low-energy processes are
complementary in probing these operators through loop-level contributions. The
only relevant constraints at tree level for such scenarios are the
high-$p_{T}$ processes $pp\to\ell_{i}\ell_{j}$ [44, 83, 84, 45]. By inspecting
Table 11–13, we arrive at the following conclusions:

  * $\bullet$

$\mu\to e$: For the $\mu\to e$ transition, the dominant bounds come from the
stringent experimental limit on the purely leptonic process $\mu\to e\gamma$
and $\mu\to eee$, as well as on the $\mu\to e$ conversion in nuclei (see Ref.
[16, 17]). More specifically, we find that the best constraints on the vector-
type operators
$\big{\\{}\mathcal{C}_{lq}^{(1)},\,\mathcal{C}_{lq}^{(3)},\,\mathcal{C}_{eq},\,\mathcal{C}_{eu},\,\mathcal{C}_{lu},\,\mathcal{C}_{ld}\big{\\}}$
come from the one-loop penguin diagrams that induce
$(\bar{e}\gamma^{\mu}\mu)(\bar{q}\gamma_{\mu}q)$ (with $q=u,d$), which are
precisely bounded by $\mu N\to eN$, cf. Figs. 2 and 3. For the scalar operator
$\mathcal{C}_{ledq}$, we find that the most stringent constraint comes from
the one-loop finite contribution to the gluonic operators
${O}_{G_{X}}^{(\ell)}$ (cf. Fig. 1), which is again strictly constrained by
$\mu N\to eN$. Lastly, the tensor $\mathcal{C}_{lequ}^{(3)}$ and the scalar
$\mathcal{C}_{lequ}^{(1)}$ are better constrained by $\mu\to e\gamma$ via the
one-loop $(T\to D_{\ell})$ and the two-loop double-logarithm mixing $(S\to
T\to D_{\ell})$, which supersede the constraints from $\mu N\to eN$ and give
the most stringent constraint in Table 11, probing scales as large as
$\mathcal{O}(10^{4}\leavevmode\nobreak\ \mathrm{TeV})$. This can be understood
from the chirality enhancement ($\propto m_{t}/m_{\mu}$) of the one-loop
tensor contribution to the dipole operators entering $\mu\to e\gamma$ [29].

  * $\bullet$

$\tau\to l$ (with $l=e,\mu$): For the $\tau\to l$ transitions (with
$l=e,\mu$), there is an even more pronounced complementarity between the
various experimental probes that are available. We find that the most
stringent constraints come from the $\tau\to lP$ and $\tau\to lV$ decays,
which are induced at loop level, or from the LHC searches for $pp\to l\tau$ at
high-$p_{T}$, which receives tree-level contributions in our setup. The
$\tau$-lepton decays are particularly useful to probe the vector operators
containing the top-quark (i.e.,
$\mathcal{C}_{lq}^{(1-3)}\,,\mathcal{C}_{eq}\,,\mathcal{C}_{eu}\,,\mathcal{C}_{lu}$),
which mixes into the $(\bar{\mu}\gamma^{\mu}\tau)(\bar{q}\gamma_{\mu}q)$
operators (with $q=u,d,s$) via penguins induced by the gauge couplings, as
well as the Yukawa running depicted in Fig. 3. Instead, for the vector
operators containing the $b$-quark (i.e.,
$\mathcal{C}_{lq}^{(1+3)},\,\mathcal{C}_{ld},\,\mathcal{C}_{ed}$), these
contributions are smaller, giving constraints that are weaker than the tree-
level ones arising from Drell-Yan processes. Finally, we find once again that
the tensor $\mathcal{C}_{lequ}^{(3)}$ and the scalar
$\smash{\mathcal{C}_{lequ}^{(1)}}$ are tightly constrained thanks to their
chirality-enhanced contributions to $\tau\to l\gamma$ at one- and two-loops,
respectively.

Lower limits on $\Lambda$ (for $\mathcal{C}=1$)
---
Coeff. | $s\to de\mu$ | $b\to de\mu$ | $b\to se\mu$ | $\mu\to e\gamma$ | $\mu\to eee$ | $\mu N\to eN$ | $Z\to e\mu$ | $h\to e\mu$ | $pp\to e\mu$
$\mathcal{C}_{\begin{subarray}{c}lq\\\ 1233\end{subarray}}^{(1+3)}$ | – | $\ast$ |  $1.0$ TeV |  $69$ TeV |  $19$ TeV |  $68$ TeV | $\ast$ | – |  $2.5$ TeV
$\mathcal{C}_{\begin{subarray}{c}lq\\\ 1233\end{subarray}}^{(1-3)}$ | – | – | – |  $103$ TeV |  $68$ TeV | $340$ TeV |  $2.0$ TeV | – | –
$\mathcal{C}_{\begin{subarray}{c}eq\\\ 1233\end{subarray}}$ | – | $\ast$ |  $1.0$ TeV |  $69$ TeV |  $70$ TeV |  $250$ TeV |  $2.1$ TeV | – |  $2.5$ TeV
$\mathcal{C}_{\begin{subarray}{c}eu\\\ 1233\end{subarray}}$ | $\ast$ | $\ast$ |  $0.6$ TeV |  $69$ TeV |  $71$ TeV | $240$ TeV |  $2.1$ TeV | – | –
$\mathcal{C}_{\begin{subarray}{c}lu\\\ 1233\end{subarray}}$ | $\ast$ | $\ast$ |  $0.6$ TeV |  $69$ TeV |  $69$ TeV |  $240$ TeV |  $2.1$ TeV | – | –
$\mathcal{C}_{\begin{subarray}{c}ld\\\ 1233\end{subarray}}$ | – | – | – |  $3.6$ TeV |  $19$ TeV |  $26$ TeV | $\ast$ | – |  $2.5$ TeV
$\mathcal{C}_{\begin{subarray}{c}ed\\\ 1233\end{subarray}}$ | – | – | – |  $4.6$ TeV |  $16$ TeV |  $26$ TeV | $\ast$ | – |  $2.5$ TeV
$\mathcal{C}_{\begin{subarray}{c}lequ\\\ 1233\end{subarray}}^{(1)}$ | – | $\ast$ | $\ast$ |  $3.2\times 10^{3}$ TeV |  $600$ TeV |  $580$ TeV | – |  $7.6$ TeV | –
$\mathcal{C}_{\begin{subarray}{c}lequ\\\ 1233\end{subarray}}^{(3)}$ | – | – | – |  $4.0\times 10^{4}$ TeV |  $8.4\times 10^{3}$ TeV |  $8.2\times 10^{3}$ TeV |  $1$ TeV | – | –
$\mathcal{C}_{\begin{subarray}{c}ledq\\\ 1233\end{subarray}}$ | – |  $2.2$ TeV |  $3.7$ TeV |  $21$ TeV |  $4.8$ TeV |  $82$ TeV | – | $\ast$ |  $2.4$ TeV

Table 11: Lower limits on $\Lambda$ in TeV units (for $\mathcal{C}=1$) derived
from LFV observables for each semileptonic operator coupled to third-
generation couplings that contributes to the $\mu\to e$ transition. The cells
highlighted in blue correspond to the most stringent limits on each operator.
The dashes denote processes that are not sensitive to a given coefficient at
one-loop order within the leading logarithmic approximation. The asterisks
denote upper limits for which the SM EFT is not the valid description, i.e.
with $\Lambda/\sqrt{|\mathcal{C}|}\lesssim v/\sqrt{4\pi}$.

Lower limits on $\Lambda$ (for $\mathcal{C}=1$)
---
Coeff. | $b\to de\tau$ | $b\to se\tau$ | $\tau\to e\gamma$ | $\tau\to e\ell\ell$ | $\tau\to eP$ | $\tau\to eV$ | $Z\to e\tau$ | $h\to e\tau$ | $pp\to e\tau$
$\mathcal{C}_{\begin{subarray}{c}lq\\\ 1333\end{subarray}}^{(1+3)}$ | $\ast$ | $\ast$ |  $0.9$ TeV | $\ast$ | $\ast$ |  $2.3$ TeV | $\ast$ | – |  $1.8$ TeV
$\mathcal{C}_{\begin{subarray}{c}lq\\\ 1333\end{subarray}}^{(1-3)}$ | – | – |  $1.7$ TeV |  $2.7$ TeV |  $2.2$ TeV | $3.6$ TeV |  $0.8$ TeV | – | –
$\mathcal{C}_{\begin{subarray}{c}eq\\\ 1333\end{subarray}}$ | $\ast$ | $\ast$ |  $0.9$ TeV |  $2.8$ TeV |  $2.2$ TeV |  $3.2$ TeV |  $0.9$ TeV | – |  $1.8$ TeV
$\mathcal{C}_{\begin{subarray}{c}eu\\\ 1333\end{subarray}}$ | $\ast$ | $\ast$ |  $0.9$ TeV |  $2.8$ TeV |  $2.2$ TeV | $3.3$ TeV |  $0.8$ TeV | – | –
$\mathcal{C}_{\begin{subarray}{c}lu\\\ 1333\end{subarray}}$ | $\ast$ | $\ast$ |  $1.0$ TeV |  $2.9$ TeV |  $2.2$ TeV |  $3.3$ TeV |  $0.8$ TeV | – | –
$\mathcal{C}_{\begin{subarray}{c}ld\\\ 1333\end{subarray}}$ | – | – | $\ast$ |  $0.6$ TeV | $\ast$ |  $2.3$ TeV | $\ast$ | – |  $1.8$ TeV
$\mathcal{C}_{\begin{subarray}{c}ed\\\ 1333\end{subarray}}$ | – | – | $\ast$ |  $0.7$ TeV | $\ast$ |  $2.3$ TeV | $\ast$ | – |  $1.8$ TeV
$\mathcal{C}_{\begin{subarray}{c}lequ\\\ 1333\end{subarray}}^{(1)}$ | $\ast$ | $\ast$ |  $15$ TeV |  $3.6$ TeV | $\ast$ |  $1.9$ TeV | – |  $2.5$ TeV | –
$\mathcal{C}_{\begin{subarray}{c}lequ\\\ 1333\end{subarray}}^{(3)}$ | – | – |  $300$ TeV |  $93$ TeV | – |  $75$ TeV | $\ast$ | – | –
$\mathcal{C}_{\begin{subarray}{c}ledq\\\ 1333\end{subarray}}$ | $\ast$ | $\ast$ | $\ast$ | $\ast$ |  $1.3$ TeV | – | – | $\ast$ |  $1.7$ TeV

Table 12: Same as Table 11 for the $\tau\to e$ transition..

Lower limits on $\Lambda$ (for $\mathcal{C}=1$)
---
Coeff. | $b\to d\mu\tau$ | $b\to s\mu\tau$ | $\tau\to\mu\gamma$ | $\tau\to\mu\ell\ell$ | $\tau\to\mu P$ | $\tau\to\mu V$ | $Z\to\mu\tau$ | $h\to\mu\tau$ | $pp\to\mu\tau$
$\mathcal{C}_{\begin{subarray}{c}lq\\\ 2333\end{subarray}}^{(1+3)}$ | $\ast$ | $\ast$ |  $1.1$ TeV | $\ast$ | $\ast$ |  $1.8$ TeV | $\ast$ | – |  $2.3$ TeV
$\mathcal{C}_{\begin{subarray}{c}lq\\\ 2333\end{subarray}}^{(1-3)}$ | – | – |  $2.0$ TeV |  $3.0$ TeV |  $2.1$ TeV | $3.9$ TeV |  $0.7$ TeV | – | –
$\mathcal{C}_{\begin{subarray}{c}eq\\\ 2333\end{subarray}}$ | $\ast$ | $\ast$ |  $0.8$ TeV |  $3.0$ TeV |  $2.1$ TeV |  $3.6$ TeV |  $0.8$ TeV | – |  $2.4$ TeV
$\mathcal{C}_{\begin{subarray}{c}eu\\\ 2333\end{subarray}}$ | $\ast$ | $\ast$ |  $0.8$ TeV |  $3.1$ TeV |  $2.1$ TeV | $3.5$ TeV |  $0.8$ TeV | – | –
$\mathcal{C}_{\begin{subarray}{c}lu\\\ 2333\end{subarray}}$ | $\ast$ | $\ast$ |  $0.9$ TeV |  $3.0$ TeV |  $2.1$ TeV |  $3.5$ TeV |  $0.8$ TeV | – | –
$\mathcal{C}_{\begin{subarray}{c}ld\\\ 2333\end{subarray}}$ | – | – | $\ast$ |  $0.7$ TeV | $\ast$ |  $1.8$ TeV | $\ast$ | – |  $2.4$ TeV
$\mathcal{C}_{\begin{subarray}{c}ed\\\ 2333\end{subarray}}$ | – | – | $\ast$ |  $0.8$ TeV | $\ast$ |  $1.8$ TeV | $\ast$ | – |  $2.4$ TeV
$\mathcal{C}_{\begin{subarray}{c}lequ\\\ 2333\end{subarray}}^{(1)}$ | $\ast$ | $\ast$ |  $13$ TeV |  $4.0$ TeV | $\ast$ |  $2.2$ TeV | – |  $2.7$ TeV | –
$\mathcal{C}_{\begin{subarray}{c}lequ\\\ 2333\end{subarray}}^{(3)}$ | – | – |  $280$ TeV |  $100$ TeV | – |  $83$ TeV | $\ast$ | – | –
$\mathcal{C}_{\begin{subarray}{c}ledq\\\ 2333\end{subarray}}$ | $\ast$ | $\ast$ | $\ast$ | $\ast$ |  $1.3$ TeV | – | – | $\ast$ |  $2.2$ TeV

Table 13: Same as Table 11 for the $\tau\to\mu$ transition.

It is important to stress that $K$\- and $B$-meson decays are not competitive
with the other constraints depicted in Tables 11–13 because of our assumption
on the quark-flavor content of the operators appearing at $\mu=\Lambda$. Since
the only source of quark-flavor violation that we introduce is the top Yukawa,
we find that the $s\to d$, $b\to d$ and $b\to s$ decays are suppressed by
$V_{ts}V_{td}^{\ast}$, $V_{tb}V_{td}^{\ast}$ and $V_{tb}V_{ts}^{\ast}$,
respectively. For this reason, we are not able to obtain meaningful
constraints from these processes. However, it is clear that these observables
can be useful to probe scenarios with different flavor assumptions, with
several concrete examples recently studied in the context of leptoquark models
[89, 90, 36, 91].

Figure 4: Constraints on selected $\mu\to e$ semileptonic Wilson coefficients,
with of third generation quarks, derived to $95\%$ accuracy from $\mu N\to eN$
(blue), $\mu\to eee$ (orange) and $\mu\to e\gamma$ (magenta), cf. Table 11.
The combined constraint is shown in green in the upper left and bottom right
panels. The EFT cutoff is fixed to $\Lambda=1$.

Figure 5: Constraints on selected $\tau\to\mu$ semileptonic Wilson
coefficients, with of third generation quarks, derived to $95\%$ accuracy from
$pp\to\mu\tau$ (red), $\tau\to\mu ll$ (orange), $\tau\to\mu V$ (purple),
$\tau\to\mu P$ (gray) and $\tau\to\mu\gamma$ (magenta), cf. Table 13. The
combined constraint is shown in green in the upper left, upper right and
bottom left panels. The EFT cutoff is fixed to $\Lambda=1$. Similar results
are obtained for the $\tau\to e$ decays, which are not shown above.

For the sake of comparison, we also quote the limits that we derive from the
$\Upsilon\to\ell_{i}\ell_{j}$ decays at tree level (cf. Table 2). For
instance, for the $\mathcal{C}_{ed}$ Wilson coefficients,

$\dfrac{\Lambda}{\sqrt{|\mathcal{C}_{\begin{subarray}{c}ed\\\
1233\end{subarray}}|}}\gtrsim 0.5\text{
TeV}\,,\qquad\qquad\dfrac{\Lambda}{\sqrt{|\mathcal{C}_{\begin{subarray}{c}ed\\\
1333\end{subarray}}|}}\gtrsim 0.3\text{
TeV}\,,\qquad\qquad\dfrac{\Lambda}{\sqrt{|\mathcal{C}_{\begin{subarray}{c}ed\\\
2333\end{subarray}}|}}\gtrsim 0.3\text{ TeV}\,,$ (95)

which are much weaker than the loop constraints collected in Tables 11–13.
Similar conclusions can be derived for the other operators that contribute to
quarkonium decays at tree level. This exercise demonstrates once again the
importance of accounting for the loop constraints discussed in this paper for
the quark-flavor conserving operators made of $c$\- and $b$-quarks, which
allow us to derive far better constraints e.g. via $\mu\to e$ conversion in
nuclei or the various $\tau$-lepton LFV decays. These limits are also
superseded by the Drell-Yan constraints, as shown in Tables 11–13 and as
previously discussed in Ref. [44, 45].

The complementarity between the different types of processes is more striking
when more than one Wilson coefficient is simultaneously present, as it is in
fact predicted in several concrete scenarios (cf. Sec. 5.2). In Fig. 4, we
plot constraints from $\mu\to e$ observables on selected pairs of effective
coefficients to illustrate this complementarity. In particular, we find that
it is necessary to consider more than one $\mu\to e$ observable to remove flat
directions that could appear e.g. in the $\mathcal{C}_{lq}^{(1)}$ vs.
$\mathcal{C}_{lq}^{(3)}$ (upper left) and $\mathcal{C}_{lequ}^{(1)}$ vs.
$\mathcal{C}_{ledq}$ (bottom right) planes. Other combinations of effective
couplings are dominated by a single observable, such as
$\mathcal{C}_{lequ}^{(1)}$ vs. $\mathcal{C}_{lequ}^{(3)}$ (upper right) and
$\mathcal{C}_{ed}$ vs. $\mathcal{C}_{ld}$ (bottom left). Similar conclusions
hold for the $\tau\to\mu$ observables, as shown in Fig. 5, where the
$pp\to\mu\tau$ bounds play an important role in constraining potential flat
directions. Notice that we do not show the analogous plot for the $\tau\to e$
transition since it has the same qualitative features as Fig. 5.

Lastly, we have also performed a numerical comparison of our results for the
observables that have been implemented in flavio [92], by using the RGEs from
the wilson package [93]. These observables are the LFV decays of the $K$\- and
$B$-mesons, the purely leptonic processes $\ell\to\ell^{\prime}\gamma$ and
$\ell\to 3\ell^{\prime}$, as well as $\mu N\to eN$ and LFV decays of the
$Z$-boson. These packages are based on numerical integration of the one-loop
RGEs, thus resumming the logarithms, which is a very useful cross-check of our
numerical results. We find an overall good agreement between our results, with
deviations smaller than $\approx 10\%$, but there are a two exceptions that we
fully understand and that we discuss now in detail. For the coefficient
$\mathcal{C}_{ledq}$, we find more stringent constraints from $\mu N\to eN$
and $\tau\to\ell P$ than flavio, which can be traced back to the higher-
dimensional gluonic operators that contribute to these processes via the
(finite) one-loop effects depicted in Fig. 1. The other discrepancy that we
have encountered concerns the contributions from $\mathcal{C}_{ed}$ and
$\mathcal{C}_{ld}$ from $\mu\to e\gamma$, which are a factor of
$\mathcal{O}(10)$ stronger for us. This disagreement can be traced back to the
two-loop single-logarithm mixing of vector operators into dipoles [16], which
is not implemented in flavio. 131313Note that similar discrepancies do not
appear for the vector operators containing the top-quark such as
$\smash{\mathcal{C}_{lq}^{(1-3)}}$, $\mathcal{C}_{eq}$, $\mathcal{C}_{eq}$ and
$\mathcal{C}_{lu}$, because there are two-loop double-logarithm contributions
of these coefficients in the SM EFT that are dominant over the two-loop single
logarithm ones. Since wilson integrates the RGEs numerically, the double
logarithm effects are included in their results.

Field | Spin | Quantum numbers | $\mathcal{C}_{lq}^{(1)}$ | $\mathcal{C}_{lq}^{(3)}$ | $\mathcal{C}_{lu}$ | $\mathcal{C}_{ld}$ | $\mathcal{C}_{eq}$ | $\mathcal{C}_{eu}$ | $\mathcal{C}_{ed}$ | $\mathcal{C}_{ledq}$ | $\mathcal{C}_{lequ}^{(1)}$ | $\mathcal{C}_{lequ}^{(3)}$
---|---|---|---|---|---|---|---|---|---|---|---|---
$\Phi^{\prime}$ | 0 | $(\mathbf{1},\,\mathbf{2},\,1/2)$ |  |  |  |  |  |  |  | ✓ | ✓ |
$Z^{\prime}$ | 1 | $(\mathbf{1},\,\mathbf{1},\,0)$ | ✓ |  | ✓ | ✓ | ✓ | ✓ | ✓ |  |  |
$V$ | 1 | $(\mathbf{1},\,\mathbf{3},\,0)$ |  | ✓ |  |  |  |  |  |  |  |
$S_{1}$ | 0 | $(\mathbf{\bar{3}},\,\mathbf{1},\,1/3)$ | ✓ | ✓ |  |  |  | ✓ |  |  | ✓ | ✓
$\widetilde{S}_{1}$ | 0 | $(\mathbf{\bar{3}},\,\mathbf{1},\,4/3)$ |  |  |  |  |  |  | ✓ |  |  |
$S_{3}$ | 0 | $(\mathbf{\bar{3}},\,\mathbf{3},\,1/3)$ | ✓ | ✓ |  |  |  |  |  |  |  |
$R_{2}$ | 0 | $(\mathbf{3},\,\mathbf{2},\,7/6)$ |  |  | ✓ |  | ✓ |  |  |  | ✓ | ✓
$\widetilde{R}_{2}$ | 0 | $(\mathbf{3},\,\mathbf{2},\,1/6)$ |  |  |  | ✓ |  |  |  |  |  |
$U_{1}$ | 1 | $(\mathbf{3},\,\mathbf{1},\,2/3)$ | ✓ | ✓ |  |  |  |  | ✓ | ✓ |  |
$\widetilde{U}_{1}$ | 1 | $(\mathbf{3},\,\mathbf{1},\,5/3)$ |  |  |  |  |  | ✓ |  |  |  |
$V_{2}$ | 1 | $(\mathbf{\bar{3}},\,\mathbf{2},\,5/6)$ |  |  |  | ✓ | ✓ |  |  | ✓ |  |
$\widetilde{V}_{2}$ | 1 | $(\mathbf{\bar{3}},\,\mathbf{2},\,-1/6)$ |  |  | ✓ |  |  |  |  |  |  |
$U_{3}$ | 1 | $(\mathbf{3},\,\mathbf{3},\,2/3)$ | ✓ | ✓ |  |  |  |  |  |  |  |

Table 14: Bosonic mediators that can induce the semileptonic LFV operators at
tree level are classified in terms of the SM quantum numbers,
$(SU(3)_{c},SU(2)_{L},Y)$, with $Q=Y+T_{3}$ as hypercharge convention. The
tree-level matching between these concrete models and the SM EFT is provided
in Appendix F.

###  5.2 Correlations in concrete models

To further demonstrate the relevance of our results, we consider the bosonic
mediators that can induce the semileptonic operators in Table 11–13 at tree
level and we explore the correlation between the LFV effective coefficients
arising in these models [94]. To this purpose, we assume a minimalistic flavor
structure, only considering couplings to third-generation quarks and leptons
with different flavors, in agreement with the assumption made in the previous
Section. These mediators are collected in Table 14. They can be a second Higgs
doublet [95, 96], a singlet or triplet $SU(2)_{L}$ vector-boson [97], or
various low-energy scalar and vector leptoquarks [42, 98]. 141414Note, in
particular, that we neglect di-quark couplings of leptoquarks since they would
make the proton unstable [98]. The Lagrangian of each scenario and the tree-
level matching to the SM EFT Lagrangian are given in detail in Appendix F.

Figure 6: Combined constraints on $\smash{\mathcal{C}_{lq}^{(1)}}$ vs.
$\smash{\mathcal{C}_{lq}^{(3)}}$, with third-generation quark flavor indices,
derived from the most constraining $\tau\to e$ (top left), $\tau\to\mu$ (top
right) and $\mu\to e$ (bottom) observables to $95\%$ CL. accuracy. The
correlations between these effective coefficients arising in the concrete
scenarios listed in Table 14 are depicted by the dashed lines. The masses of
the mediators is fixed to $\Lambda=1$ TeV.

In Fig. 6, we consider the specific example of operators coupled to left-
handed fermions, which are induced by several of the concrete models listed in
Table 14, and we confront them to the global constraints from the LFV
observables discussed above, by fixing the mediator masses to $\Lambda=1$ TeV.
We choose this example, since the correlations between
$\mathcal{C}_{lq}^{(1)}$ and $\mathcal{C}_{lq}^{(3)}$ are fixed by the quantum
numbers of the mediators in several cases, without requiring specific
assumptions about the New Physics couplings. For the $\mu\to e$ transition,
the least constrained scenario is $U_{1}\sim(\mathbf{3},\mathbf{1},2/3)$ with
left-handed couplings, which lies along the diagonal
$\mathcal{C}_{lq}^{(1)}=\mathcal{C}_{lq}^{(3)}$, since LFV operators
containing the top-quark do not appear at tree level in this scenario [90, 36,
91]. 151515See Ref. [99] for detailed one-loop calculation in ultraviolet-
complete models containing the $U_{1}$ leptoquark. For the other scenarios, we
find even more stringent constraints via the loop effects discussed above. For
the $\tau\to l$ transition, we obtain comparable bounds for the various models
that are shown in Fig. 6. Finally, we note that correlations between the
scalar $\smash{\mathcal{C}_{lequ}^{(1)}}$ and tensor
$\smash{\mathcal{C}_{lequ}^{(3)}}$ also can arise for the scenarios with
$R_{2}\sim(\mathbf{3},\mathbf{2},7/6)$ and
$S_{1}\sim(\bar{\mathbf{3}},\mathbf{1},1/3)$ [29], cf. Appendix F. In this
case, the constraints on the tensor operators are far more stringent than on
the scalar ones, thanks to the chirality-enhancement of the leptonic dipoles
operators, as shown in the Tables 11–13. These observables give the dominant
bounds on the Yukawa couplings appearing in these models, see e.g. Ref. [100].

##  6 Conclusion

In this paper, we have studied Lepton Flavor Violation (LFV) in semileptonic
observables using an Effective Field Theory (EFT) approach with a general
flavor structure. Besides the tree-level effects that have been extensively
studied in the literature, we have included the one-loop effects induced by
the Renormalization Group Equations (RGE) below [16, 19] and above [20] the
electroweak scale, as well as the relevant two-loop effects that are available
in the literature [16]. In this way, we provided general expressions for
several high-energy observables such as the decays $h\to\ell\ell^{\prime}$,
$Z\to\ell\ell^{\prime}$ and $t\to c\ell\ell^{\prime}$, in addition to the
high-energy tails of $pp\to\ell\ell^{\prime}$ that were studied in Ref. [44,
45]. At low energies, we have provided general expressions for the
semileptonic processes $\mu N\to eN$, $M\to\ell\ell^{\prime}$, $M\to
M^{\prime}\ell\ell^{\prime}$ and $\tau\to\ell M$, where $M^{(\prime)}$ stands
for a vector or pseudoscalar meson, and the purely leptonic processes
$\ell\to\ell^{\prime}\gamma$ and $\ell\to 3\ell^{\prime}$ that can constrain
semileptonic operators at the loop level. In particular, we have updated the
predictions for the $M\to M^{\prime}\ell\ell^{\prime}$ decays by using the
latest determination of the relevant form-factors on the lattice.

The loop effects that we consider induce important correlations between the
various LFV observables and, in several instances, they allow us to derive
even stronger constraints than those we would obtain at tree level. To
illustrate these results, we have considered the scenarios where only
semileptonic LFV Wilson coefficients with third-generation-quarks are non-zero
at the high scale $\Lambda$ and we have computed the various observables
induced either at tree- or loop-level. We have obtained the lower limits on
the EFT cutoff ($\Lambda$) that are given in Tables 11–13 by taking
$\mathcal{C}_{a}(\Lambda)=1$, where $\mathcal{C}_{a}$ denotes the possible
effective coefficient at the scale $\Lambda$. For the $\mu\to e$ transition,
we have found that the most stringent constraints arise from the searches for
$\mu\to e\gamma$, $\mu\to eee$, and $\mu N\to eN$, depending on the operator
one considers, see also Ref. [16]. For the $\tau\to\ell$ transitions (with
$\ell=e,\mu$), we highlight an interesting complementarity between the
processes $\tau\to\ell\gamma$, $\tau\to\ell\ell\ell$, $\tau\to\ell P$ and
$\tau\to\ell V$, which are induced at one-loop, with $pp\to\ell\tau$ that
receives tree-level contributions. The complementarity between the different
probes is even more evident in the two-dimensional plane, where pairs of
effective coefficients are considered, as shown in Fig. 4 and 5 for the
$\mu\to e$ and $\tau\to\mu$ transitions, respectively.

It is important to emphasize that going beyond the leading logarithm solution
is needed to probe several effective coefficients at the loop level (cf. Ref.
[16, 101, 17]). Among the effects that we have included, we accounted for the
finite matching of scalar to gluonic operators, which contributes to $\mu N\to
eN$ and to the $\tau\to\ell P$ decays, where
$P\in\\{\pi^{0},\eta,\eta^{\prime}\\}$ [26], as depicted in Fig. 1. We have
also accounted for the two-loop single-logarithm mixing between vector
operators and dipoles, which provides the most stringent constraint on several
$\mu\to e$ operators, as computed in Ref. [16] for the low-energy EFT, cf.
Table 11. Lastly, several two-loop double-logarithm effects are relevant to
the phenomenology of $\mu\to e\gamma$ and $\tau\to\ell\gamma$. These include
the mixing of a scalar into a tensor, which then mixes into dipoles, both in
the low-energy EFT and within the SM EFT. Another example is the mixing of
vector into tensor operators in the SM EFT, which mixes again into dipoles,
and which has also been included in our phenomenological analysis. Even though
these are only partial results, since the full two-loop anomalous dimension
for the SM EFT is not known yet, it shows the importance of computing these
effects that can be relevant given the current precision of the experimental
searches.

Finally, we have briefly discussed the implications of our results to concrete
scenarios that generate semileptonic operators at tree level. We have provided
the tree-level matching of these models to the SM EFT in Appendix C. Moreover,
we have shown that these scenarios induce different correlations of the
effective coefficients that can be tested experimentally, as depicted in Fig.
6 for the effective coefficients $\mathcal{C}_{lq}^{(1)}$ and
$\mathcal{C}_{lq}^{(3)}$. This example again shows the importance of
accounting for loop-induced effects to constrain the size of LFV contributions
to low-energy decays, with an interesting complementarity to high-energy
processes.

##  7 Acknowledgments

We thank Peter Stangl for useful exchanges on wilson [93] and flavio [92],
Lukas Allwicher for useful discussions, and Sébastien Descotes-Genon for
comments on the manuscript. This project has received support from the
European Union’s Horizon 2020 research and innovation programme under the
Marie Skłodowska-Curie grant agreement N∘ 860881-HIDDeN. I.P. receives funding
from “P2IO LabEx (ANR-10-LABX-0038)” in the framework “Investissements
d’Avenir” (ANR-11-IDEX-0003-01) managed by the Agence Nationale de la
Recherche (ANR), France.

##  Appendix A Conventions

We consider the same notation of Ref. [20] for the operators in the Warsaw
basis [14]. Quark and lepton weak-doublets are denoted by $q$ and $l$, while
up- and down-type quarks, and lepton singlets are denoted by $u$, $d$, and
$e$, respectively. The covariant derivative acting on a quark-doublet reads

$D_{\mu}=\partial_{\mu}+ig^{\prime}\,YB_{\mu}+ig\,\frac{\tau^{I}}{2}W^{I}_{\mu}+ig_{3}\,T^{A}G_{\mu}^{A}\,,$
(96)

where $T^{A}=\lambda^{A}/2$ are the $SU(3)_{c}$ generators, $\tau^{I}$ are the
Pauli matrices and $Y$ denotes the hypercharge. The Yukawa interactions are
defined in the flavour basis as follows,

$-\mathcal{L}_{\mathrm{yuk}}=H^{\dagger}\,\bar{d}\,y_{d}\,q+\widetilde{H}^{\dagger}\,\bar{u}\,y_{u}\,q+H^{\dagger}\,\bar{e}\,y_{e}\,l+\mathrm{h.c.}\,,$
(97)

where $y_{f}$ (with $f\in\\{u,d,e\\}$) stand for the Yukawa matrices and
flavor indices have been omitted. The SM Higgs doublet is denoted by $H$, with
the conjugate field defined as $\widetilde{H}\equiv\epsilon H^{\ast}$ and the
$SU(2)$ anti-symmetric tensor is $\epsilon\equiv i\tau^{2}$. We opt to work in
the basis where $y_{\ell}$ and $y_{d}$ are diagonal matrices, while $y_{u}$
depends on the CKM matrix, $V\equiv V_{\mathrm{CKM}}$ (i.e.
$y_{u}=\hat{y}_{u}V$).

##  Appendix B LEFT basis for $b\to s\ell_{i}\ell_{j}$

For the convenience of the reader, we provide the matching of our operator
basis for the $b\to s\ell_{i}\ell_{j}$ transition to the one that is usually
considered in the literature on $B$-meson LFV decays [42, 89],

$\displaystyle\mathcal{H}_{\mathrm{eff}}=-\frac{4G_{F}}{\sqrt{2}}V_{tb}V_{ts}^{\ast}$
$\displaystyle\sum_{a=9,10,S,P}\Big{[}C^{ij}_{a}(\mu)\mathcal{O}^{ij}_{a}(\mu)+C^{ij}_{a^{\prime}}(\mu)\mathcal{O}^{ij}_{a^{\prime}}(\mu)\Big{]}+\mathrm{h.c.}\,,$
(98)

where

$\displaystyle\mathcal{O}_{9}^{ij}$
$\displaystyle=\frac{e^{2}}{(4\pi)^{2}}(\bar{s}\gamma_{\mu}P_{L}b)(\bar{\ell}_{i}\gamma^{\mu}\ell_{j})\,,\qquad\qquad\hskip
11.38092pt\mathcal{O}_{S}^{ij}=\frac{e^{2}}{(4\pi)^{2}}(\bar{s}P_{R}b)(\bar{\ell}_{i}\ell_{j})\,,$
(99) $\displaystyle\mathcal{O}_{10}^{ij}$
$\displaystyle=\frac{e^{2}}{(4\pi)^{2}}(\bar{s}\gamma_{\mu}P_{L}b)(\bar{\ell}_{i}\gamma^{\mu}\gamma^{5}\ell_{j})\,,\qquad\qquad\mathcal{O}_{P}^{ij}=\frac{e^{2}}{(4\pi)^{2}}(\bar{s}P_{R}b)(\bar{\ell}_{i}\gamma^{5}\ell_{j})\,.$

The two operator bases are related via

$\displaystyle C_{9}^{\ell_{i}\ell_{j}}$
$\displaystyle=\frac{\pi}{\lambda_{t}\alpha_{em}}\Big{(}C^{(d)}_{\begin{subarray}{c}V_{RL}\\\
ij23\end{subarray}}+C^{(d)}_{\begin{subarray}{c}V_{LL}\\\
ij23\end{subarray}}\Big{)}\,,$ $\displaystyle
C_{9^{\prime}}^{\ell_{i}\ell_{j}}$
$\displaystyle=\frac{\pi}{\lambda_{t}\alpha_{em}}\Big{(}C^{(d)}_{\begin{subarray}{c}V_{RR}\\\
{ij23}\end{subarray}}+C^{(d)}_{\begin{subarray}{c}V_{LR}\\\
ij23\end{subarray}}\Big{)}\,,$ (100) $\displaystyle C_{10}^{\ell_{i}\ell_{j}}$
$\displaystyle=\frac{\pi}{\lambda_{t}\alpha_{em}}\Big{(}C^{(d)}_{\begin{subarray}{c}V_{RL}\\\
ij23\end{subarray}}-C^{(d)}_{\begin{subarray}{c}V_{LL}\\\
ij23\end{subarray}}\Big{)}\,,$ $\displaystyle
C_{10^{\prime}}^{\ell_{i}\ell_{j}}$
$\displaystyle=\frac{\pi}{\lambda_{t}\alpha_{em}}\Big{(}C^{(d)}_{\begin{subarray}{c}V_{RR}\\\
ij23\end{subarray}}-C^{(d)}_{\begin{subarray}{c}V_{LR}\\\
ij23\end{subarray}}\Big{)}\,,$ $\displaystyle C_{S}^{\ell_{i}\ell_{j}}$
$\displaystyle=\frac{\pi}{\lambda_{t}\alpha_{em}}\Big{(}C^{(d)}_{\begin{subarray}{c}S_{RR}\\\
ij23\end{subarray}}+C^{(d)}_{\begin{subarray}{c}S_{LR}\\\
ij23\end{subarray}}\Big{)}\,,$ $\displaystyle
C_{S^{\prime}}^{\ell_{i}\ell_{j}}$
$\displaystyle=\frac{\pi}{\lambda_{t}\alpha_{em}}\Big{(}C^{(d)}_{\begin{subarray}{c}S_{RL}\\\
ij23\end{subarray}}+C^{(d)}_{\begin{subarray}{c}S_{LL}\\\
ij23\end{subarray}}\Big{)}\,,$ $\displaystyle C_{P}^{\ell_{i}\ell_{j}}$
$\displaystyle=\frac{\pi}{\lambda_{t}\alpha_{em}}\Big{(}C^{(d)}_{\begin{subarray}{c}S_{RR}\\\
ij23\end{subarray}}-C^{(d)}_{\begin{subarray}{c}S_{LR}\\\
ij23\end{subarray}}\Big{)}\,,$ $\displaystyle
C_{P^{\prime}}^{\ell_{i}\ell_{j}}$
$\displaystyle=\frac{\pi}{\lambda_{t}\alpha_{em}}\Big{(}C^{(d)}_{\begin{subarray}{c}S_{RL}\\\
ij23\end{subarray}}-C^{(d)}_{\begin{subarray}{c}S_{LL}\\\
ij23\end{subarray}}\Big{)}\,,$

with $\lambda_{t}=V_{tb}V_{ts}^{\ast}$ . The results for the $b\to
d\ell_{i}\ell_{j}$ transition can be obtained by a trivial replacement of the
flavor indices.

##  Appendix C Matching to the SM EFT

The matching between LEFT coefficients defined above with the SM EFT operators
introduced in Sec. 4 reads:

### Vector Operators

$\displaystyle C_{\begin{subarray}{c}V_{LL}\\\ ijkl\end{subarray}}^{(u)}$
$\displaystyle=\dfrac{v^{2}}{\Lambda^{2}}\mathcal{C}_{\begin{subarray}{c}lq\\\
ijkl\end{subarray}}^{\prime(1-3)}+2g^{Z}_{u_{L}}\delta_{kl}\dfrac{v^{2}}{\Lambda^{2}}\,\mathcal{C}_{\begin{subarray}{c}Hl\\\
ij\end{subarray}}^{(1+3)}\,,\quad$ $\displaystyle
C_{\begin{subarray}{c}V_{RR}\\\ ijkl\end{subarray}}^{(u)}$
$\displaystyle=\dfrac{v^{2}}{\Lambda^{2}}\mathcal{C}_{\begin{subarray}{c}eu\\\
ijkl\end{subarray}}+2g^{Z}_{u_{R}}\delta_{kl}\dfrac{v^{2}}{\Lambda^{2}}\,\mathcal{C}_{\begin{subarray}{c}He\\\
ij\end{subarray}}\,,$ (101) $\displaystyle C_{\begin{subarray}{c}V_{LR}\\\
ijkl\end{subarray}}^{(u)}$
$\displaystyle=\dfrac{v^{2}}{\Lambda^{2}}\mathcal{C}_{\begin{subarray}{c}lu\\\
ijkl\end{subarray}}+2g^{Z}_{u_{R}}\delta_{kl}\dfrac{v^{2}}{\Lambda^{2}}\,\mathcal{C}_{\begin{subarray}{c}Hl\\\
ij\end{subarray}}^{(1+3)}\,,$ $\displaystyle C_{\begin{subarray}{c}V_{RL}\\\
ijkl\end{subarray}}^{(u)}$
$\displaystyle=\dfrac{v^{2}}{\Lambda^{2}}\mathcal{C}^{\prime}_{\begin{subarray}{c}eq\\\
ijkl\end{subarray}}+2g^{Z}_{u_{L}}\delta_{kl}\dfrac{v^{2}}{\Lambda^{2}}\,\mathcal{C}_{\begin{subarray}{c}He\\\
ij\end{subarray}}\,,$ $\displaystyle C_{\begin{subarray}{c}V_{LL}\\\
ijkl\end{subarray}}^{(d)}$
$\displaystyle=\dfrac{v^{2}}{\Lambda^{2}}\mathcal{C}_{\begin{subarray}{c}lq\\\
ijkl\end{subarray}}^{(1+3)}+2g^{Z}_{d_{L}}\delta_{kl}\dfrac{v^{2}}{\Lambda^{2}}\,\mathcal{C}_{\begin{subarray}{c}Hl\\\
ij\end{subarray}}^{(1+3)}\,,$ $\displaystyle C_{\begin{subarray}{c}V_{RR}\\\
ijkl\end{subarray}}^{(d)}$
$\displaystyle=\dfrac{v^{2}}{\Lambda^{2}}\mathcal{C}_{\begin{subarray}{c}ed\\\
ijkl\end{subarray}}+2g^{Z}_{d_{R}}\delta_{kl}\dfrac{v^{2}}{\Lambda^{2}}\,\mathcal{C}_{\begin{subarray}{c}He\\\
ij\end{subarray}}\,,$ $\displaystyle C_{\begin{subarray}{c}V_{LR}\\\
ijkl\end{subarray}}^{(d)}$
$\displaystyle=\dfrac{v^{2}}{\Lambda^{2}}\mathcal{C}_{\begin{subarray}{c}ld\\\
ijkl\end{subarray}}+2g^{Z}_{d_{R}}\delta_{kl}\dfrac{v^{2}}{\Lambda^{2}}\,\mathcal{C}_{\begin{subarray}{c}Hl\\\
ij\end{subarray}}^{(1+3)}\,,$ $\displaystyle C_{\begin{subarray}{c}V_{RL}\\\
ijkl\end{subarray}}^{(d)}$
$\displaystyle=\dfrac{v^{2}}{\Lambda^{2}}\mathcal{C}_{\begin{subarray}{c}eq\\\
ijkl\end{subarray}}+2g^{Z}_{d_{L}}\delta_{kl}\dfrac{v^{2}}{\Lambda^{2}}\,\mathcal{C}_{\begin{subarray}{c}He\\\
ij\end{subarray}}\,,$ $\displaystyle C_{\begin{subarray}{c}V_{LL}\\\
ijkk\end{subarray}}^{(\ell)}$
$\displaystyle=\frac{\upsilon^{2}}{\Lambda^{2}}\mathcal{C}_{\begin{subarray}{c}ll\\\
ijkk\end{subarray}}+2g^{Z}_{e_{L}}\dfrac{v^{2}}{\Lambda^{2}}\,\mathcal{C}_{\begin{subarray}{c}Hl\\\
ij\end{subarray}}^{(1+3)}\,,$ $\displaystyle C_{\begin{subarray}{c}V_{RR}\\\
ijkk\end{subarray}}^{(\ell)}$
$\displaystyle=\frac{\upsilon^{2}}{\Lambda^{2}}\mathcal{C}_{\begin{subarray}{c}ee\\\
ijkk\end{subarray}}+2g^{Z}_{e_{R}}\dfrac{v^{2}}{\Lambda^{2}}\,\mathcal{C}_{\begin{subarray}{c}He\\\
ij\end{subarray}}\,,$ $\displaystyle C_{\begin{subarray}{c}V_{LR}\\\
ijkk\end{subarray}}^{(\ell)}$
$\displaystyle=\frac{\upsilon^{2}}{\Lambda^{2}}\mathcal{C}_{\begin{subarray}{c}le\\\
ijkk\end{subarray}}++2g^{Z}_{e_{R}}\dfrac{v^{2}}{\Lambda^{2}}\,\mathcal{C}_{\begin{subarray}{c}Hl\\\
ij\end{subarray}}^{(1+3)}\,,$ $\displaystyle C_{\begin{subarray}{c}V_{RL}\\\
ijkk\end{subarray}}^{(\ell)}$
$\displaystyle=\frac{\upsilon^{2}}{\Lambda^{2}}\mathcal{C}_{\begin{subarray}{c}le\\\
kkij\end{subarray}}+2g^{Z}_{e_{L}}\dfrac{v^{2}}{\Lambda^{2}}\,\mathcal{C}_{\begin{subarray}{c}He\\\
ij\end{subarray}}\,,$

where the couplings of the $Z$-boson to the SM fermions are defined as
$\smash{g_{f_{L}}^{Z}=T_{3}^{f}-Q_{f}\sin^{2}\theta_{W}}$ and
$\smash{g_{f_{R}}^{Z}=-Q_{f}\sin^{2}\theta_{W}}$, where $T_{3}^{f}$ stands for
the third-component of the weak isospin and $Q_{f}$ is the electric charge of
the fermion $f$. The primed coefficients are defined in Eq. (88) and we use
again the shorthand notation $\mathcal{C}_{lq}^{(1\pm
3)}=\mathcal{C}_{lq}^{(1)}\pm\mathcal{C}_{lq}^{(3)}$ for the left-handed
operators.

### Scalar Operators

$\displaystyle C^{(u)}_{\begin{subarray}{c}S_{RL}\\\ ijkl\end{subarray}}$
$\displaystyle=-\delta_{kl}\frac{\upsilon^{3}m_{u_{k}}}{\sqrt{2}m_{h}^{2}\Lambda^{2}}\,\mathcal{C}_{\begin{subarray}{c}eH\\\
ij\end{subarray}}\,,\qquad$ $\displaystyle\qquad
C_{\begin{subarray}{c}S_{LR}\\\ ijkl\end{subarray}}^{(u)}$
$\displaystyle=-\delta_{kl}\frac{v^{2}}{\Lambda^{2}}\frac{vm_{u_{k}}}{\sqrt{2}m_{h}^{2}}\,\mathcal{C}^{\ast}_{\begin{subarray}{c}eH\\\
ji\end{subarray}}\,,$ (102) $\displaystyle
C^{(u)}_{\begin{subarray}{c}S_{RR}\\\ ijkl\end{subarray}}$
$\displaystyle=-\frac{\upsilon^{2}}{\Lambda^{2}}\,\mathcal{C}_{\begin{subarray}{c}lequ\\\
ijkl\end{subarray}}^{\prime(1)}-\frac{v^{2}}{\Lambda^{2}}\frac{vm_{u_{k}}}{\sqrt{2}m_{h}^{2}}\,\mathcal{C}_{\begin{subarray}{c}eH\\\
ij\end{subarray}}\,,$ $\displaystyle C_{\begin{subarray}{c}S_{LL}\\\
ijkl\end{subarray}}^{(u)}$
$\displaystyle=-\frac{\upsilon^{2}}{\Lambda^{2}}\mathcal{C}_{\begin{subarray}{c}lequ\\\
jilk\end{subarray}}^{\prime(1)\ast}-\delta_{kl}\frac{v^{2}}{\Lambda^{2}}\frac{vm_{u_{k}}}{\sqrt{2}m_{h}^{2}}\,\mathcal{C}^{\ast}_{\begin{subarray}{c}eH\\\
ji\end{subarray}}\,,$ $\displaystyle C^{(d)}_{\begin{subarray}{c}S_{RL}\\\
ijkl\end{subarray}}$
$\displaystyle=\frac{\upsilon^{2}}{\Lambda^{2}}\,\mathcal{C}_{\begin{subarray}{c}ledq\\\
ijkl\end{subarray}}-\delta_{kl}\frac{v^{2}}{\Lambda^{2}}\frac{vm_{d_{k}}}{\sqrt{2}m_{h}^{2}}\,\mathcal{C}_{\begin{subarray}{c}eH\\\
ij\end{subarray}}\,,$ $\displaystyle C^{(d)}_{\begin{subarray}{c}S_{LR}\\\
ijkl\end{subarray}}$
$\displaystyle=\frac{\upsilon^{2}}{\Lambda^{2}}\,\mathcal{C}_{\begin{subarray}{c}ledq\\\
jilk\end{subarray}}^{\ast}-\delta_{kl}\frac{v^{2}}{\Lambda^{2}}\frac{vm_{d_{k}}}{\sqrt{2}m_{h}^{2}}\,\mathcal{C}^{\ast}_{\begin{subarray}{c}eH\\\
ji\end{subarray}}\,,$ $\displaystyle C^{(d)}_{\begin{subarray}{c}S_{RR}\\\
ijkl\end{subarray}}$
$\displaystyle=-\delta_{kl}\frac{\upsilon^{3}m_{d_{k}}}{\sqrt{2}m_{h}^{2}\Lambda^{2}}\mathcal{C}_{\begin{subarray}{c}eH\\\
ij\end{subarray}}\,,$ $\displaystyle C^{(d)}_{\begin{subarray}{c}S_{LL}\\\
ijkl\end{subarray}}$