Renormalization group improved implications of semileptonic operators in SMEFT

Jacky Kumar
TUM Institute for Advanced Study, Lichtenbergstr. 2a, D-85747 Garching,
Germany

E-Mail<EMAIL_ADDRESS>

###### Abstract

We study implications of the four-fermion semileptonic operators at the low-
energy and at electroweak (EW) scale in the framework of Standard Model
Effective Field Theory (SMEFT). We show how the renormalization group (RG)
running effects can play an important role in probing the _generic flavour
structure_ of such operators. It is shown that at the 1-loop level, through RG
running, depending upon the flavour structure, these operators can give rise
to sizable effects at low energy in the electroweak precision (EWP)
observables, the leptonic, quark, as well as the $Z$ boson flavour violating
decays. To this end, we isolate the phenomenologically relevant terms in the
full anomalous dimension matrices (ADMs) and discuss the impact of the QED+QCD
running in the Weak effective field theory (WET) and the SMEFT running due to
gauge and Yukawa interactions on the dim-4 and dim-6 operators at the low
energy. Considering all the relevant processes, we derive lower bounds on new
physics (NP) scale $\Lambda$ for each semileptonic operator, keeping a generic
flavour structure. In addition, we also report the allowed ranges for the
Wilson coefficients at a fixed value of $\Lambda=3$ TeV.

###### Contents

  1. 1 Introduction
  2. 2 General Strategy
  3. 3 Renormalization Group Running
    1. 3.1 Evolution of $\Delta F=(0,0)$ operators
      1. 3.1.1 Operator mixing due to Gauge interactions
      2. 3.1.2 Operator mixing due to top-Yukawa interactions
    2. 3.2 Evolution of $\Delta F=(1,0)$ operators
  4. 4 Observables induced at 1-loop level
    1. 4.1 Observables for $\Delta F=(0,0)$ operators
      1. 4.1.1 Electroweak Precision Observables
      2. 4.1.2 $B$ Meson Decays
      3. 4.1.3 Charged Current Decays
        1. 4.1.3.1 $K$ Meson Decays
        2. 4.1.3.2 $\tau\to K\nu$ and $\tau\to\pi\nu$ Decays
        3. 4.1.3.3 $\pi\to e\nu$ and Nuclear $\beta$ Decays
      4. 4.1.4 Correlations
    2. 4.2 Observables for $\Delta F=(1,0)$ operators
      1. 4.2.1 $Z\to\bar{\ell}_{i}\ell_{j}$ Decays
      2. 4.2.2 $\tau\to 3\ell$ and $\mu\to 3e$ Decays
      3. 4.2.3 $\tau\to\rho\ell$, $\tau\to P(\phi,\pi)\ell$ Decays
      4. 4.2.4 Correlations
  5. 5 Sensitivities to the NP scale $\Lambda$
  6. 6 Conclusions and Outlook
  7. A Tree-level shifts in the dim-4 gauge boson couplings
  8. B RG induced shifts in the dim-4 and dim-6 operators
    1. B.1 Dim-4 $Z$ boson couplings
    2. B.2 Dim-4 $W$ boson couplings
    3. B.3 Leptonic dim-6 WET operators
    4. B.4 Semileptonic dim-6 WET operators
  9. C Bounds on the Wilson coefficients

##  1 Introduction

In the absence of the discoveries of new particles at the Large Hadron
Collider (LHC), the SMEFT provides an elegant framework to parameterize and
quantify the effects of NP in terms of $SU(3)_{c}\times SU(2)_{L}\times
U(1)_{Y}$ gauge invariant higher dimensional ($d\geq 5$) operators[1, 2]. In
SMEFT, the operators are constructed using the field content of the Standard
Model (SM). Excluding the flavour structures, there are in total 59 operators
which conserve the baryon number[1].

An interesting aspect of the SMEFT is its built-in gauge symmetry. This
feature often leads to an intriguing pattern of correlations among low energy
observables due to enforcement of the model independent relations between the
Weak effective theory (WET) operators at the EW scale, on matching with SMEFT
[3, 4, 5, 6]. At the 1-loop level, through RG running, new effective operators
can also be generated at the EW scale as a result of operator mixing. For
instance, the four-fermion SMEFT operators can mix with the
$\psi^{2}\phi^{2}D$ type operators which can contribute to the observables of
different kinds. In this manner, the running from $\Lambda$ to the EW scale
can induce additional correlations [7, 8, 9, 10, 11, 12]. Therefore, in an
SMEFT analysis, considering such effects is very important in order to
correctly predict the low energy implications of new interactions introduced
at the NP scale.

By now, the complete ADMs for SMEFT as well as WET are known at the 1-loop
level[13, 14, 15, 16, 17]. The recent results for ADMs in the SMEFT extended
with right-handed neutrino fields can be found in Refs. [18, 19, 20]. Based on
these calculations, the tools such as wilson [21] and DsixTools [22, 23] have
been developed. (See also Ref. [24] for a discussion on the analytic solutions
to the SMEFT RGEs). Using these codes, it is possible to include the RG
running effects in the theoretical predictions which can be obtained using
flavio [25]. Several other packages [26, 27, 28] exist to facilitate the
different kinds of tasks for studying the phenomenology in the SMEFT
framework. As far as the present work is concerned, we have used flavio for
the theoretical predictions and the RG running effects have been taken care by
using the wilson package.

In this work, we focus on a subset of SMEFT operators which contain both
lepton and quark fields currents:

$(\overline{L}_{i}\gamma_{\mu}L_{j})(\overline{Q}_{k}\gamma^{\mu}Q_{l})\,,$

here, _i,j,k,l_ denote the flavour indices. Such operators are known as the
semileptonic operators. It is well-known that, at tree-level, these operators
enter into various semileptonic decays of mesons. In this context, the
operators which violate the quark flavour have been extensively studied in the
literature [8, 29, 9, 30, 31, 32, 33, 34, 35, 36]. On the other hand, a
generic flavour structure of these operators is not yet fully explored. In
particular, the quark flavour conserving counterparts deserve more attention.
A given NP model can generate both flavour violating as well as conserving
operators, therefore, it is essential to know what kind of constraints apply
on the later ones. One of the goals of the present work is to fill this gap by
identifying all possible low energy and EW scale observables, which can be
used to probe a generic flavour structure of the semileptonic operators.
Concerning this matter, the Ref. [37] discusses the contribution of
semileptonic operators to the EWP observables, assuming flavour universality.
Similarly, the Refs. [38, 7], pointed out the importance of EWP and lepton
flavour violating (LFV) constraints on the semileptonic operators needed to
resolve the $B$-anomalies. For more recent studies on this topic, see also
Refs. [39, 32, 30]. Due to the reasons outlined above, we will restrict
ourself to the flavour structures such that there is no quark-flavour
violation, to begin with, at the scale $\Lambda$. In other words, the
operators in which are interested are either flavour conserving in both
currents or at most they violate the lepton flavour at the NP scale. We will
emphasize the importance RG running from $\Lambda$ to the EW scale, through
which these operators can contribute to a verity of observables at lower
scales. In this regard, we will first isolate the most important terms in the
ADMs due to gauge couplings as well as Yukawas, _i.e._ , the ones which are
phenomenologically relevant, given the current precision of the measurements.

The remainder of the paper is structured as follows. In Sec. 2, we will
discuss our strategy, and in Sec. 3, we discuss the SMEFT RG running of
semileptonic operators. In Sec. 4, we will identify various observables which
are relevant for the different flavour structures of the operators under
consideration. In Sec. 5, we will discuss the sensitivities to the NP scales
$\Lambda$ for various operators. Finally, we move on to the conclusions in
Sec. 6. Additional material is collected in Appendices A, B, and C.

##  2 General Strategy

In SMEFT, the SM is extended with a series of higher dimensional effective
operators invariant under the full gauge symmetry of the SM. In general, the
SMEFT Lagrangian can be written as

$\displaystyle\mathcal{L}_{\rm SMEFT}^{d\geq
5}=\sum_{\mathcal{O}_{a}^{\dagger}=\mathcal{O}_{a}}\mathcal{C}_{a}\mathcal{O}_{a}+\sum_{\mathcal{O}_{a}^{\dagger}\neq\mathcal{O}_{a}}\left(\mathcal{C}_{a}\mathcal{O}_{a}+\mathcal{C}_{a}^{*}\mathcal{O}_{a}^{\dagger}\right)\,,$
(1)

here, $\mathcal{C}_{a}$ are known to be the Wilson coefficients. A complete
list of SMEFT operators can be found in Refs. [2, 1]. In this work, we will
focus on a subset, the four-fermion semileptonic operators:

$\displaystyle[{\cal O}_{\ell q}^{(1)}]_{ijkl}$ $\displaystyle=$
$\displaystyle(\bar{\ell}_{i}\gamma_{\mu}\ell_{j})(\bar{q}_{k}\gamma^{\mu}q_{l})\,,$
(2) $\displaystyle[{\cal O}_{\ell q}^{(3)}]_{ijkl}$ $\displaystyle=$
$\displaystyle(\bar{\ell}_{i}\gamma_{\mu}\tau^{a}\ell_{j})(\bar{q}_{k}\gamma^{\mu}\tau^{a}q_{l})\,,$
(3) $\displaystyle[{\cal O}_{\ell u}]_{ijkl}$ $\displaystyle=$
$\displaystyle(\bar{\ell}_{i}\gamma_{\mu}\ell_{j})(\bar{u}_{k}\gamma^{\mu}u_{l})\,,$
(4) $\displaystyle[{\cal O}_{\ell d}]_{ijkl}$ $\displaystyle=$
$\displaystyle(\bar{\ell}_{i}\gamma_{\mu}\ell_{j})(\bar{d}_{k}\gamma^{\mu}d_{l})\,,$
(5) $\displaystyle[{\cal O}_{ed}]_{ijkl}$ $\displaystyle=$
$\displaystyle(\bar{e}_{i}\gamma_{\mu}e_{j})(\bar{d}_{k}\gamma^{\mu}d_{l})\,,$
(6) $\displaystyle[{\cal O}_{eu}]_{ijkl}$ $\displaystyle=$
$\displaystyle(\bar{e}_{i}\gamma_{\mu}e_{j})(\bar{u}_{k}\gamma^{\mu}u_{l})\,,$
(7) $\displaystyle[{\cal O}_{qe}]_{ijkl}$ $\displaystyle=$
$\displaystyle(\bar{q}_{i}\gamma_{\mu}q_{j})(\bar{e}_{k}\gamma^{\mu}e_{l})\,,$
(8)

here, the flavour indices $i,j,k,l$ can take values from 1 to 3 and
$q,u,d,\ell,e$ represent the quark doublet, right-handed up-type quark, right-
handed down-type quark, lepton doublet and right-handed lepton fields,
respectively. The down-type quarks are chosen to be in the mass basis at the
scale $\Lambda$. This corresponds to the Warsaw-down basis of the SMEFT[40].
For convenience, we divide the operators into three classes based on their
flavour structure at the scale $\Lambda$:

  * •

$\Delta F=(0,0)$: the operators which do not violate quark111Since we work in
the Warsaw-down basis, the Cabibbo–Kobayashi–Maskawa (CKM) rotations can give
rise to quark flavour violating operators in the up-sector, even at the scale
$\Lambda$. Note that the CKM matrix itself can be affected by NP [41, 39],
however such effects are beyond the scope of present analyses. In our analyses
for numerics the CKM is obtained with wilson program using the input
$V_{us}=0.2243,V_{ub}=3.62\times 10^{-3},V_{cb}=4.221\times
10^{-2},\delta=1.27$ at the $Z$-mass scale.  and lepton flavours,

  * •

$\Delta F=(1,0)$: the operators which do violate lepton flavour by one unit
but do not violate the quark flavour,

  * •

$\Delta F=(i,1)$: the operators which do violate quark flavour by one unit but
may or may not violate the lepton flavour, _i.e_ , $i=0$ or 1.

Since, the $\Delta F=(i,1)$ type operators are already well studied in the
literature, we will not consider them here. On integrating out the new degrees
of freedom at $\Lambda$, a unique tower of SMEFT operators is generated.
However, a priori it is not obvious to which observables these operators can
enter into at the lower energies. This is because of three reasons. Firstly,
the $SU(2)_{L}$ invariance can lead to correlations between different type of
observables. Secondly, the pattern of mixing between different operators due
to running from $\Lambda$ to the EW scale is very complex in nature. Often
this leads to the appearance of new operators at the EW scale, which can give
rise to unpredictable correlations among low-energy observables. Finally, the
choice of the flavour basis at the NP scale is not invariant with respect to
the RG evolution. Therefore, it is extremely important to systematically
analyze these effects for all flavour structures of the operators of our
interest. This will allow us to identify all possible observables which are
sensitive to these operators.

With these motivations, first we will identify the most sensitive observables
which can be used to probe the operators listed in (2)-(8) for the following
flavour combinations:

$\displaystyle\Delta F=(0,0)$ $\displaystyle:$ $\displaystyle
1111,1122,1133,2211,2222,2233,3311,3322,3333\,,$ (9) $\displaystyle\Delta
F=(1,0)$ $\displaystyle:$ $\displaystyle
1211,1222,1233,1311,1322,1333,2311,2322,2333.$ (10)

Note that none of these flavour combinations are quark flavour violating.
Next, we will proceed in three steps: To begin with, we will study the
operator mixing pattern due RG evolution for the two classes of the flavour
structures as shown in Eqs. (9)-(10). Based on this, we will then identify all
possible observables which can be used to constrain a given operator directly
(at tree-level) or through the operators to which it mixes into, through the
RG effects (at 1-loop level). Using this information, we will finally derive
the lower bounds on the scale of each semileptonic operator assuming the
presence of a single operator at the scale $\Lambda$.

##  3 Renormalization Group Running

In this section we discuss the RG running of the SMEFT semileptonic Wilson
coefficients. In general, the running is governed by the coupled differential
equations

$\mathcal{\dot{C}}(\mu)\equiv
16\pi^{2}\frac{d\mathcal{C}(\mu)}{d\ln\mu}=\hat{\gamma}(\mu)~{}\mathcal{C}(\mu)\,,$
(11)

with

$\mathcal{C}(\mu)=(\mathcal{C}_{1}(\mu),\mathcal{C}_{2}(\mu),...)^{T}.$ (12)

Here, $\mu$ is the renormalization scale and $\hat{\gamma}$ is the anomalous
dimension matrix which is function of the SM parameters such as gauge and
Yukawa couplings. In the leading-log (LL) approximation, the solution to these
equations for running from scale $\Lambda$ and $\mu$ reads

$\mathcal{C}(\mu)=\left[\hat{1}-\frac{\hat{\gamma}}{16\pi^{2}}\ln\left(\frac{\Lambda}{\mu}\right)\right]\mathcal{C}(\Lambda).$
(13)

In the following we analyze the RG running and operator mixing of various
quark-flavour conserving semileptonic operators (shown in Eqs. (2)-(8)) of our
interest to various other operators. These effects can potentially relate them
to new observables generated at 1-loop level and hence allow us to put
additional constraints. Using the ADMs calculated in Refs. [13, 14, 15, 16] we
find that, depending upon the flavour structures, the quark flavour conserving
semileptonic operators can mix with two types of operators. The first category
is the $\psi^{2}\phi^{2}D$ type operators:

$\displaystyle[{\cal O}_{\phi\ell}^{(1)}]_{ij}$ $\displaystyle=$
$\displaystyle(\phi^{\dagger}i\,{\overleftrightarrow{D_{\mu}}}\,\phi)(\bar{\ell}_{i}\,\gamma^{\mu}\,\ell_{j})\,,$
(14) $\displaystyle[{\cal O}_{\phi\ell}^{(3)}]_{ij}$ $\displaystyle=$
$\displaystyle(\phi^{\dagger}i\,{\overleftrightarrow{D_{\mu}^{I}}}\,\phi)(\bar{\ell}_{i}\tau^{I}\gamma^{\mu}\ell_{j})\,,$
(15) $\displaystyle[{\cal O}_{\phi e}]_{ij}$ $\displaystyle=$
$\displaystyle(\phi^{\dagger}i\,{\overleftrightarrow{D_{\mu}}}\,\phi)(\bar{e}_{i}\gamma^{\mu}e_{j})\,,$
(16) $\displaystyle[{\cal O}_{\phi q}^{(1)}]_{ij}$ $\displaystyle=$
$\displaystyle(\phi^{\dagger}i\,{\overleftrightarrow{D_{\mu}}}\,\phi)(\bar{q}_{i}\gamma^{\mu}q_{j})\,,$
(17) $\displaystyle[{\cal O}_{\phi q}^{(3)}]_{ij}$ $\displaystyle=$
$\displaystyle(\phi^{\dagger}i\,{\overleftrightarrow{D_{\mu}^{I}}}\,\phi)(\bar{q}_{i}\tau^{I}\gamma^{\mu}q_{j})\,,$
(18) $\displaystyle[{\cal O}_{\phi u}]_{ij}$ $\displaystyle=$
$\displaystyle(\phi^{\dagger}i\,{\overleftrightarrow{D_{\mu}}}\,\phi)(\bar{u}_{i}\gamma^{\mu}u_{j})\,,$
(19) $\displaystyle[{\cal O}_{\phi d}]_{ij}$ $\displaystyle=$
$\displaystyle(\phi^{\dagger}i\,{\overleftrightarrow{D_{\mu}}}\,\phi)(\bar{d}_{i}\gamma^{\mu}d_{j}).$
(20)

Here, $I$ is the $SU(2)$ index and $D_{\mu}$ stands for the covariant
derivative. Secondly, they also mix with the purely leptonic operators given
by

$\displaystyle[{\cal O}_{\ell\ell}]_{ijkl}$ $\displaystyle=$
$\displaystyle(\bar{\ell}_{i}\gamma_{\mu}\ell_{j})(\bar{\ell}_{k}\gamma^{\mu}\ell_{l})\,,$
(21) $\displaystyle[{\cal O}_{\ell e}]_{ijkl}$ $\displaystyle=$
$\displaystyle(\bar{\ell}_{i}\gamma_{\mu}\ell_{j})(\bar{e}_{k}\gamma^{\mu}e_{l})\,,$
(22) $\displaystyle[{\cal O}_{ee}]_{ijkl}$ $\displaystyle=$
$\displaystyle(\bar{e}_{i}\gamma_{\mu}e_{j})(\bar{e}_{k}\gamma^{\mu}e_{l}).$
(23)

In addition, the $\Delta F=(0,0)$ type semileptonic operators can also mix
with the four-quark operators, however we do not find any significant
constraints due to such an operator mixing. In order to get the general
picture, for our qualitative discussion, we will use the LL solutions to the
corresponding RG equations as given by Eq. (13). However, for the numerics, we
sum the logs using full numerical solutions which have been obtained using the
wilson program [21].

###  3.1 Evolution of $\Delta F=(0,0)$ operators

The ADMs in the SMEFT depend on the gauge couplings as well the Yukawas. In
this section, we identify the phenomenologically important terms in the ADMs
of $\Delta F=(0,0)$ semileptonic operators.

####  3.1.1 Operator mixing due to Gauge interactions

First, we discuss the operator mixing of the $\Delta F=(0,0)$ semileptonic
operators due the gauge couplings. The flavour combinations for these
operators are specified in Eq. (9). Therefore, as an initial condition, at NP
scale, only the Wilson coefficients of the operators given in (2)-(8) are
assumed to be non-zero. It turns out that, these operators can mix with the
$\psi^{2}\phi^{2}D$-type operators, which are listed in Eqs. (14)-(20),
through EW interactions. Now we use the LL approximation to relate the Wilson
coefficients of the semileptonic operators at the scale $\Lambda$ to the
Wilson coefficients of $\psi^{2}\phi^{2}D$ operators which get generated at
the EW scale due to the operator mixing. We find

$\begin{pmatrix}\big{[}{\cal C}_{\phi\ell}^{(1)}\big{]}_{ii}\\\ \big{[}{\cal
C}_{\phi\ell}^{(3)}\big{]}_{ii}\\\ \big{[}{\cal C}_{\phi e}\big{]}_{ii}\\\
\big{[}{\cal C}_{\phi q}^{(1)}\big{]}_{kk}\\\ \big{[}{\cal C}_{\phi
q}^{(3)}\big{]}_{kk}\\\ \big{[}{\cal C}_{\phi d}\big{]}_{kk}\\\ \big{[}{\cal
C}_{\phi
u}\big{]}_{kk}\end{pmatrix}=L\begin{pmatrix}\frac{2}{3}$${g_{1}}^{2}$$&0&\frac{4}{3}$${g_{1}}^{2}$$&-\frac{2}{3}$${g_{1}}^{2}$$&0&0&0\\\
0&2$$g_{2}^{2}$$&0&&0&0&0\\\
0&0&0&0&-\frac{2}{3}$${g_{1}}^{2}$$&\frac{4}{3}$${g_{1}}^{2}$$&\frac{2}{3}$${g_{1}}^{2}$$\\\
-\frac{2}{3}$${g_{1}}^{2}$$&0&0&0&0&0&-\frac{2}{3}$${g_{1}}^{2}$$\\\
0&\frac{2}{3}$$g_{2}^{2}$$&0&0&0&0&0\\\
0&0&0&-\frac{2}{3}$${g_{1}}^{2}$$&-\frac{2}{3}$${g_{1}}^{2}$$&0&0\\\
0&0&-\frac{2}{3}$${g_{1}}^{2}$$&0&0&-\frac{2}{3}$${g_{1}}^{2}$$&0\\\
\end{pmatrix}\begin{pmatrix}\big{[}{\cal C}_{\ell q}^{(1)}\big{]}_{iikk}\\\
\big{[}{\cal C}_{\ell q}^{(3)}\big{]}_{iikk}\\\ \big{[}{\cal C}_{\ell
u}\big{]}_{iikk}\\\ \big{[}{\cal C}_{\ell d}\big{]}_{iikk}\\\ \big{[}{\cal
C}_{ed}\big{]}_{iikk}\\\ \big{[}{\cal C}_{eu}\big{]}_{iikk}\\\ \big{[}{\cal
C}_{qe}\big{]}_{kkii}\end{pmatrix}.$ (24)

Strictly speaking on the l.h.s. we should have written the difference between
the Wilson coefficients at the EW scale and their corresponding values at the
scale $\Lambda$, _i.e.,_

$\delta\mathcal{C}({{\mu_{\text{ew}}}})=\mathcal{C}({\mu_{\text{ew}}})-\mathcal{C}(\Lambda).$
(25)

However, since at the scale $\Lambda$ we assume only the semileptonic
operators to be non-zero, we have $\mathcal{C}(\Lambda)=0$ and therefore
$\delta\mathcal{C}({{\mu_{\text{ew}}}})=\mathcal{C}({\mu_{\text{ew}}})$. Also,
we have expressed the loop suppression factor and the log in terms of the
quantity $L$, given by

$L=\frac{1}{16\pi^{2}}\ln\left(\frac{{\mu_{\text{ew}}}}{\Lambda}\right).$ (26)

In Eq. (24), the $g_{1}$ and $g_{2}$ are the EW gauge couplings at the scale
$\Lambda$. It is important to note that for the first three (last four)
elements on the l.h.s., only the repeated index $k$ ($i$) is summed over on
the r.h.s. However, on the l.h.s. indices $k$ and $i$ are not summed over and
can take values in the range 1-3. For simplicity, we do not show the self-
mixing of the operators. Interestingly, after the EW symmetry breaking, the
$\psi^{2}\phi^{2}D$ operators are known to give corrections to the $W$ and $Z$
boson couplings with quarks and leptons [42, 43, 44, 45] (see also more recent
Refs. [39, 46, 37, 47, 48, 49, 50, 51]). Therefore, the quark-flavour
conserving semileptonic operators can be indirectly probed through EWP data.

In this context, one should also consider the operator $\big{[}{\cal
C}_{\ell\ell}\big{]}_{1221}$ which enters into the muon decay, _i.e._ ,
$\mu\to e\nu\bar{\nu}$ and hence affects the extraction of Fermi constant
$G_{F}$. At the EW scale the Wilson coefficient $\big{[}{\cal
C}_{\ell\ell}\big{]}_{1221}$ can be written as

$\big{[}{\cal C}_{\ell\ell}\big{]}_{1221}=4g_{2}^{2}L\big{[}{\cal C}_{\ell
q}^{(3)}\big{]}_{iikk}\,,$ (27)

here the indices on the r.h.s. are summed over $ii=11,22$ and $kk=11,22,33$.

In addition, we find the mixing between $\big{[}{\cal C}_{\ell
q}^{(1)}\big{]}$ and $\big{[}{\cal C}_{\ell q}^{(3)}\big{]}$ to be
phenomenologically relevant. At the 1-loop level, this goes through the EW
corrections. Therefore, the strength of mixing is driven by the gauge coupling
$g_{2}$. As before, solving the corresponding RGEs in LL approximation, one
finds

$\begin{pmatrix}\big{[}{\cal C}_{\ell q}^{(1)}\big{]}_{iikk}\\\ \big{[}{\cal
C}_{\ell
q}^{(3)}\big{]}_{iikk}\end{pmatrix}=L\begin{pmatrix}$--$&9$$g_{2}^{2}$$\\\
3$$g_{2}^{2}$$&$--$\\\ \end{pmatrix}\begin{pmatrix}\big{[}{\cal C}_{\ell
q}^{(1)}\big{]}_{iikk}\\\ \big{[}{\cal C}_{\ell
q}^{(3)}\big{]}_{iikk}\end{pmatrix}\,,$ (28)

here, the repeated indices $k$ and $i$ on the l.h.s. as well as on the r.h.s.
are not summed over. Once again, we have suppressed the self-mixing (indicated
by the entries with dashes) of these operators and log term $L$ is given by
(26). The mixing of $\big{[}{\cal C}_{\ell q}^{(1)}\big{]}_{iikk}$ with
$\big{[}{\cal C}_{\ell q}^{(3)}\big{]}_{iikk}$ can induce new charged current
transitions after EW symmetry breaking.

####  3.1.2 Operator mixing due to top-Yukawa interactions

In this subsection, we isolate phenomenologically important terms in the ADMs
which depend on the Yukawa interactions. In this regard, we will keep only the
largest terms involving the top-Yukawa coupling. Solving the RGEs in the LL
approximation, we find

$\begin{pmatrix}\big{[}{\cal C}_{\phi\ell}^{(1)}\big{]}_{ij}\\\ \big{[}{\cal
C}_{\phi\ell}^{(3)}\big{]}_{ij}\\\ \big{[}{\cal C}_{\phi
e}\big{]}_{ij}\end{pmatrix}\simeq
6y_{t}^{2}L\begin{pmatrix}1~{}&0~{}&-1~{}&0~{}&0~{}\\\
0~{}&-1~{}&0~{}&0~{}&0~{}\\\ 0~{}&0~{}&0~{}&1~{}&-1~{}\\\
\end{pmatrix}\begin{pmatrix}\big{[}{\cal C}_{\ell q}^{(1)}\big{]}_{ij33}\\\
\big{[}{\cal C}_{\ell q}^{(3)}\big{]}_{ij33}\\\ \big{[}{\cal C}_{\ell
u}\big{]}_{ij33}\\\ \big{[}{\cal C}_{qe}\big{]}_{33ij}\\\ \big{[}{\cal
C}_{eu}\big{]}_{ij33}\end{pmatrix}\,,$ (29)

here, $y_{t}$ is the Yukawa coupling for the top-quark. In Eq. (29), for
$\Delta F=(0,0)$ operators the flavour indices $i$ and $j$ can be set equal.
It is worth mentioning that for simplicity, we have not included the non-
standard effects due to running of the Yukawa couplings themselves, which can
lead to additional contributions to the quark-flavour violating semileptonic
operators at the EW scale. We will return to this point in Sec. 4.1.2.

###  3.2 Evolution of $\Delta F=(1,0)$ operators

Now we consider the $\Delta F=(1,0)$ type semileptonic operators which violate
the leptonic flavour by one unit but conserve the quark flavour (see Eq.
(10)). Apart from the self-mixing, these operators mix with the lepton flavour
violating leptonic and $\psi^{2}\phi^{2}D$ type operators, $\big{[}{\cal
C}_{\phi\ell}^{(1)}\big{]}_{ij},\big{[}{\cal C}_{\phi\ell}^{(1)}\big{]}_{ij}$
and $\big{[}{\cal C}_{\phi e}\big{]}_{ij}$ with $i\neq j$, through the EW
interactions. Once again, solving the RG equations in the LL approximation, we
find,

$\begin{pmatrix}\big{[}{\cal C}_{\ell\ell}\big{]}_{ijll}\\\ \big{[}{\cal
C}_{\ell e}\big{]}_{ijll}\\\ \big{[}{\cal C}_{\ell e}\big{]}_{llij}\\\
\big{[}{\cal C}_{ee}\big{]}_{ijll}\\\ \big{[}{\cal
C}_{\phi\ell}^{(1)}\big{]}_{ij}\\\ \big{[}{\cal
C}_{\phi\ell}^{(3)}\big{]}_{ij}\\\ \big{[}{\cal C}_{\phi e}\big{]}_{ij}\\\
\end{pmatrix}=L\begin{pmatrix}-\frac{1}{3}$${g_{1}}^{2}$$&$$g_{2}^{2}$$&-\frac{2}{3}$${g_{1}}^{2}$$&\frac{1}{3}$${g_{1}}^{2}$$&0&0&0\\\
-\frac{4}{3}$${g_{1}}^{2}$$&0&-\frac{8}{3}$${g_{1}}^{2}$$&\frac{4}{3}$${g_{1}}^{2}$$&0&0&0\\\
0&0&0&0&\frac{2}{3}$${g_{1}}^{2}$$&-\frac{4}{3}$${g_{1}}^{2}$$&-\frac{2}{3}$${g_{1}}^{2}$$\\\
0&0&0&0&\frac{2}{3}$${g_{1}}^{2}$$&-\frac{2}{3}$${g_{1}}^{2}$$&-\frac{1}{3}$${g_{1}}^{2}$$\\\
\frac{2}{3}$${g_{1}}^{2}$$&0&\frac{4}{3}$${g_{1}}^{2}$$&-\frac{2}{3}$${g_{1}}^{2}$$&0&0&0\\\
0&2$$g_{2}^{2}$$&0&&0&0&0\\\
0&0&0&0&-\frac{2}{3}$${g_{1}}^{2}$$&\frac{4}{3}$${g_{1}}^{2}$$&\frac{2}{3}$${g_{1}}^{2}$$\\\
\end{pmatrix}\begin{pmatrix}\big{[}{\cal C}_{\ell q}^{(1)}\big{]}_{ijkk}\\\
\big{[}{\cal C}_{\ell q}^{(3)}\big{]}_{ijkk}\\\ \big{[}{\cal C}_{\ell
u}\big{]}_{ijkk}\\\ \big{[}{\cal C}_{\ell d}\big{]}_{ijkk}\\\ \big{[}{\cal
C}_{ed}\big{]}_{ijkk}\\\ \big{[}{\cal C}_{eu}\big{]}_{ijkk}\\\ \big{[}{\cal
C}_{qe}\big{]}_{kkij}\\\ \end{pmatrix}\,,$ (30)

here the repeated indices $k$ on the r.h.s. is summed over the values $1-3$,
and $i\neq j$ holds on the both sides. Also, the repeated index $l$ on the
l.h.s. can take values in the range $1-3$ but it is not summed over. The
logarithmic piece $L$ is defined in equation (26). The operators on the l.h.s.
contribute to the LFV $W$, $Z$ boson couplings and leptonic decays. We will
come to this point later with more details. Note that, below the EW scale, the
semileptonic operators can mix into purely leptonic operators also through QED
interaction [52, 53].

##  4 Observables induced at 1-loop level

Based on our discussions in the previous section, we are now in position to
identify the 1-loop induced observables to which the semileptonic operators
can contribute. We will see how the SMEFT RG running can play an important
role in this respect. As discussed before, because of the complicated operator
mixing pattern in SMEFT, the semileptonic operators can contribute to
observables of very different nature depending upon the operators with which
they mix, which can be read out from the l.h.s. of Eqs. (24), (27), (28), (29)
and (30). In order to get a general picture, we will present the expressions
for the relevant low energy dim-4 and dim-6 couplings in terms of semileptonic
Wilson coefficients at the high scale by employing the LL approximation.
Eventually, we will sum the logs with the help of numerical solutions.

###  4.1 Observables for $\Delta F=(0,0)$ operators

We have identified three categories of the observables which are relevant for
$\Delta F=(0,0)$ type operators. Those are the EWP observables, flavour
violating $B$-decays and charged current decays. In the following, we discuss
how various semileptonic operators can contribute to them through operator
mixing.

####  4.1.1 Electroweak Precision Observables

As indicated by Eq. (24), the $\Delta F=(0,0)$ semileptonic operators can mix
with the $\psi^{2}\phi^{2}D$ type operators due to electroweak interactions.
In addition, this mixing also depends on the top-Yukawa interactions, see e.g.
Eq. (29). Interestingly, after EW symmetry breaking the latter operators give
corrections to the $Z$ and $W$ boson couplings with the fermions. Using the LL
solutions to the RGEs, as presented before, we can express the NP
contributions to these couplings directly in terms of the Wilson coefficients
of semileptonic operators at the high scale $\Lambda$. In general, NP shifts
in the neutral $Z$ boson couplings with the fermions can be parameterized as

$\mathcal{L}_{Z}\supset
g_{Z}\sum_{\psi=u,d,e,\nu_{L}}\bar{\psi}_{i}\gamma^{\mu}\left[\delta(g_{L}^{\psi})_{ij}P_{L}+\delta(g_{R}^{\psi})_{ij}P_{R}\right]\psi_{j}Z_{\mu}\,,$
(31)

here $g_{Z}=-g_{2}/\cos{\theta_{W}}$ and $\theta_{W}$ represents the weak-
mixing angle. NP can enter into $\delta g_{X}^{\psi}$ with $X=L,R$, through
three difference sources. This can be understood from the equation[49]

$\delta(g_{X}^{\psi})_{ij}=\delta g_{Z}~{}(g_{X}^{\psi,{\rm
SM}})_{ij}-Q_{\psi}~{}\delta\sin^{2}\theta_{W}~{}\delta_{ij}+\delta(g_{X}^{\psi})_{ij}^{\rm
dir}\,,$ (32)

here, the contributions in the first two and the last term can be thought of
as indirect and direct shifts to the $Z$ boson couplings, respectively. In
SMEFT, the tree-level expressions for the quantities $\delta g_{Z}$,
$\delta\sin^{2}\theta_{W}$ and $\delta(g_{X}^{\psi})_{ij}$ can be found in
Appendix A. Also, $Q_{\psi}$ represents the electric charge and
$g_{L}^{\psi,{\rm SM}}=T_{3}-Q_{\psi}\sin^{2}\theta_{W}$, $g_{R}^{\psi,{\rm
SM}}=-Q_{\psi}\sin^{2}\theta_{W}$. Below we present the 1-loop contributions
to all these quantities due to semileptonic operators in the LL approximation.
Using the Eqs. (24) and (78)-(84) one can obtain the NP shifts in the $Z$
couplings with quarks:

$\displaystyle\delta(g_{L}^{u})_{kk}^{\rm dir}$ $\displaystyle=$
$\displaystyle-\frac{g_{1}^{2}}{3}v^{2}LV_{km}\left(-\big{[}{\cal C}_{\ell
q}^{(1)}\big{]}_{iimn}-\frac{g_{2}^{2}}{g_{1}^{2}}\big{[}{\cal C}_{\ell
q}^{(3)}\big{]}_{iimn}-\big{[}{\cal
C}_{qe}\big{]}_{mnii}\right)V_{nk}^{\dagger}\,,$ (33)
$\displaystyle\delta(g_{R}^{u})_{kk}^{\rm dir}$ $\displaystyle=$
$\displaystyle-\frac{g_{1}^{2}}{3}v^{2}L\left(-\big{[}{\cal C}_{\ell
u}^{(1)}\big{]}_{iikk}-\big{[}{\cal C}_{eu}^{(3)}\big{]}_{iikk}\right)\,,$
(34) $\displaystyle\delta(g_{L}^{d})_{kk}^{\rm dir}$ $\displaystyle=$
$\displaystyle-\frac{g_{1}^{2}}{3}v^{2}L\left(-\big{[}{\cal C}_{\ell
q}^{(1)}\big{]}_{iikk}+\frac{g_{2}^{2}}{g_{1}^{2}}\big{[}{\cal C}_{\ell
q}^{(3)}\big{]}_{iikk}-\big{[}{\cal C}_{qe}\big{]}_{kkii}\right)\,,$ (35)
$\displaystyle\delta(g_{R}^{d})_{kk}^{\rm dir}$ $\displaystyle=$
$\displaystyle-\frac{g_{1}^{2}}{3}v^{2}L\left(-\big{[}{\cal C}_{\ell
d}\big{]}_{iikk}-\big{[}{\cal C}_{ed}\big{]}_{iikk}\right).$ (36)

Similarly, the shifts in the $Z$ boson couplings with leptons are found to be

$\displaystyle\delta(g_{L}^{\nu})_{ii}^{\rm dir}$ $\displaystyle=$
$\displaystyle-\frac{g_{1}^{2}}{3}v^{2}L\left(\big{[}{\cal C}_{\ell
q}^{(1)}\big{]}_{iikk}-\frac{3g_{2}^{2}}{g_{1}^{2}}\big{[}{\cal C}_{\ell
q}^{(3)}\big{]}_{iikk}+2\big{[}{\cal C}_{\ell u}\big{]}_{iikk}-\big{[}{\cal
C}_{\ell d}\big{]}_{iikk}\right)$ (37) $\displaystyle-$ $\displaystyle
6v^{2}y_{t}^{2}L\left(\big{[}{\cal C}_{\ell
q}^{(1)}\big{]}_{ii33}+\big{[}{\cal C}_{\ell
q}^{(3)}\big{]}_{ii33}-\big{[}{\cal C}_{\ell u}\big{]}_{ii33}\right)\,,$
$\displaystyle\delta(g_{L}^{e})_{ii}^{\rm dir}$ $\displaystyle=$
$\displaystyle-\frac{g_{1}^{2}}{3}v^{2}L\left(\big{[}{\cal C}_{\ell
q}^{(1)}\big{]}_{iikk}+\frac{3g_{2}^{2}}{g_{1}^{2}}\big{[}{\cal C}_{\ell
q}^{(3)}\big{]}_{iikk}+2\big{[}{\cal C}_{\ell u}\big{]}_{iikk}-\big{[}{\cal
C}_{\ell d}\big{]}_{iikk}\right)$ (38) $\displaystyle-$ $\displaystyle
6v^{2}y_{t}^{2}L\left(\big{[}{\cal C}_{\ell
q}^{(1)}\big{]}_{ii33}-\big{[}{\cal C}_{\ell
q}^{(3)}\big{]}_{ii33}-\big{[}{\cal C}_{\ell u}\big{]}_{ii33}\right)\,,$
$\displaystyle\delta(g_{R}^{e})_{ii}^{\rm dir}$ $\displaystyle=$
$\displaystyle-\frac{g_{1}^{2}}{3}v^{2}L\left(-\big{[}{\cal
C}_{ed}\big{]}_{iikk}+2\big{[}{\cal C}_{eu}\big{]}_{iikk}+\big{[}{\cal
C}_{qe}\big{]}_{kkii}\right)$ (39) $\displaystyle-$ $\displaystyle
6v^{2}y_{t}^{2}L\left(\big{[}{\cal C}_{qe}\big{]}_{33ii}-\big{[}{\cal
C}_{eu}\big{]}_{ii33}\right).$

Here, $v=246$ GeV is the vacuum expectation value and $L$ is given by (26).
Similarly, for $\delta g_{Z}$ and $\delta\sin^{2}\theta_{W}$ are found to be

$\displaystyle\delta g_{Z}$ $\displaystyle=$ $\displaystyle
3v^{2}y_{t}^{2}L\left(\big{[}{\cal C}_{\ell
q}^{(3)}\big{]}_{1133}+\big{[}{\cal C}_{\ell q}^{(3)}\big{]}_{2233}\right)\,,$
(40) $\displaystyle\delta\sin^{2}\theta_{W}$ $\displaystyle=$
$\displaystyle-\frac{\sin^{2}2\theta_{W}}{4\cos
2\theta_{W}}6v^{2}y_{t}^{2}L\left(\big{[}{\cal C}_{\ell
q}^{(3)}\big{]}_{1133}+\big{[}{\cal C}_{\ell q}^{(3)}\big{]}_{2233}\right).$
(41)

On the basis of the above discussions, now we point out a few important
observations:

  * •

First, note that the RG induced EW corrections to $\delta g_{Z}$ and
$\delta\sin^{2}\theta_{W}$ due to $\big{[}{\cal C}_{\ell
q}^{(3)}\big{]}_{iikk}$ for $ii=11,22$ and $kk=11,22,33$, get canceled among
$\big{[}{\cal C}_{\phi\ell}^{(3)}\big{]}_{11},\big{[}{\cal
C}_{\phi\ell}^{(3)}\big{]}_{22}$ and $\big{[}{\cal
C}_{\ell\ell}\big{]}_{1221}$. This can be seen by inserting the LL
contributions of $\big{[}{\cal C}_{\ell q}^{(3)}\big{]}_{iikk}$ from Eqs. (27)
and (24) to the latter Wilson coefficients in Eqs. (85)-(86). However, this is
no longer true once the contribution due to the Yukawa couplings, as shown in
Eq. (29), is included.

  * •

Next, in general, the top-Yukawa dependent contributions, given in
Eqs.(40)-(41), are larger in size as compared to the direct EW corrections to
the $Z$-couplings shown in Eqs. (33)-(39). So, for the cases in which both
effects exist simultaneously, the former has a greater impact.

  * •

Furthermore, it is evident from Eq. (32) that the impact of the $\big{[}{\cal
C}_{\ell q}^{(3)}\big{]}_{ii33}$ for $ii=11$ or $22$ on
$\delta(g_{X}^{Y})_{ii}$ through $\delta g_{Z}$ and $\delta\sin^{2}\theta_{W}$
is universal for all three families of leptons and quarks. On the other hand
the shifts $\delta(g_{L}^{e})_{ii}^{\rm dir}$, $\delta(g_{R}^{e})_{ii}^{\rm
dir}$ and $\delta(g_{L}^{\nu})_{ii}^{\rm dir}$ also experience effects due to
the top-Yukawa interactions, which however are lepton flavour dependent.

  * •

Finally, the contributions of the semileptonic operators due to top-Yukawa
effects do not affect the $Z$ boson couplings to quarks directly. This is
however still possible through $\delta g_{Z}$ and $\delta\sin^{2}\theta_{W}$,
but only for the case of $\big{[}{\cal C}_{\ell q}^{(3)}\big{]}_{ii33}$ with
$ii=11,22$.

Next, we look at the impact of semileptonic operators on the $W$-boson
couplings which can be parameterized as

$\mathcal{L}_{W}\supset\frac{-e}{\sqrt{2}\sin\theta_{W}}\left(\delta(\varepsilon_{L}^{\ell})_{ij}^{\rm
dir}~{}\bar{\nu}_{i}\gamma_{\mu}P_{L}e_{j}+\delta(\varepsilon_{L}^{q})_{ij}^{\rm
dir}~{}\bar{u}_{i}\gamma_{\mu}P_{L}d_{j}\right)W^{\mu}_{+}+h.c.~{}\,,$ (42)

here the NP shifts are given as[49]

$\displaystyle\delta(\varepsilon_{L}^{\psi}){ii}$ $\displaystyle=$
$\displaystyle\delta(\varepsilon_{L}^{\psi})_{ii}^{\rm
dir}-\frac{1}{2}\frac{\delta\sin^{2}\theta_{W}}{\sin^{2}\theta_{W}}.$ (43)

Using the LL solutions, we find the shifts
$\delta(\varepsilon_{L}^{\psi})_{ii}^{\rm dir}$ in terms of the semileptonic
operators to be

$\displaystyle\delta(\varepsilon_{L}^{\ell})_{ii}^{\rm dir}$ $\displaystyle=$
$\displaystyle 2g_{2}^{2}L\big{[}{\cal C}_{\ell
q}^{(3)}\big{]}_{iikk}-6v^{2}y_{t}^{2}L\big{[}{\cal C}_{\ell
q}^{(3)}\big{]}_{ii33}\,,$ (44)
$\displaystyle\delta(\varepsilon_{L}^{q})_{kk}^{\rm dir}$ $\displaystyle=$
$\displaystyle\frac{2}{3}g_{2}^{2}LV_{km}\big{[}{\cal C}_{\ell
q}^{(3)}\big{]}_{iimk}.$ (45)

We make the following observations for the RG induced shifts in the $W$
couplings:

  * •

The $W$ couplings to both quarks and leptons are universally affected for all
three families by $\big{[}{\cal C}_{\ell q}^{(3)}\big{]}_{ii33}$ for
$ii=11,22$ through $\delta\sin^{2}\theta_{W}$.

  * •

The top-Yukawa effects do not give direct contributions to the $W$ couplings.

  * •

The leptonic couplings can be directly affected by $\big{[}{\cal C}_{\ell
q}^{(3)}\big{]}_{ii33}$ for $ii=11,22,33$ through top-Yukawa interactions.
This effect is however flavour dependent.

In order to quantify these effects, in Appendix B we report tables for the
numerical values of the RG induced shifts in the $Z$ and $W$ couplings at the
EW scale due to semileptonic operators present at the NP scale $\Lambda$. To
understand the relative importance of the top-Yukawa and gauge interactions,
we present two sets of numbers with and without Yukawa RG running effects. It
is evident that, whenever present, the top-Yukawa effects always dominate.

Now, given that the $Z$ and $W$ boson couplings are strongly constrained by
the EWP observables, this implies that the flavour conserving semileptonic
$\Delta F=(0,0)$ type operators can be indirectly constrained by the EWP
measurements. The following list of operators can be constrained through this
mechanism:

$\big{[}{\cal C}_{\ell q}^{(1)}\big{]}_{ijkl}\,,\big{[}{\cal C}_{\ell
q}^{(3)}\big{]}_{ijkl}\,,\big{[}{\cal C}_{ed}\big{]}_{ijkl}\,,\big{[}{\cal
C}_{eu}\big{]}_{ijkl}\,,\big{[}{\cal C}_{\ell u}\big{]}_{ijkl}\,,\big{[}{\cal
C}_{\ell d}\big{]}_{ijkl}\,,\big{[}{\cal C}_{qe}\big{]}_{ijkl}\,,$ (46)

with

$ijkl\equiv 1111,1122,1133,2211,2222,2233,3311,3322,3333.$ (47)

In the next section, we will use the EWP measurements as constraints to derive
the lower bounds on the NP scales of the flavour conserving semileptonic
operators in given Eq. (46). To this end, the list of EWP observables [54, 55,
56, 57, 58, 59, 60] used in our analysis are taken from Tab. 13 of Ref. [39].
In addition, we have also included a recent measurement of the ratio
$R_{\tau\mu}(W\to\ell\bar{\nu})=\mathcal{B}(W\to\tau\bar{\nu})/\mathcal{B}(W\to\mu\bar{\nu})$[61].

####  4.1.2 $B$ Meson Decays

Throughout the analyses, we have used Warsaw-down basis at the high scale. In
this basis, to begin with the down-type quark and lepton mass matrices are
diagonal, whereas, the up-type quark matrix takes the form $V^{\dagger}{\rm
diag}(y_{u},y_{c},y_{t})$. Here, $V$ represents the CKM matrix. However, due
to the RG running of the Yukawa matrices, this choice of basis is not
preserved with respect to running from $\Lambda$ to the EW scale. As a result,
one has to perform back-rotation to the original (Warsaw-down) basis at the EW
scale. This process can generate flavour violating semileptonic operators from
their flavour conserving counterparts at the scale $\Lambda$ [10, 62, 9, 21].
The part of the WET Lagrangian that describes the $b\to s\ell^{+}\ell^{-}$
processes can be written as

$\displaystyle\mathcal{L}_{\rm eff}$ $\displaystyle=$
$\displaystyle\frac{\alpha
G_{F}}{\sqrt{2}\pi}V_{tb}V_{ts}^{*}\sum_{a=9,10}(C_{a}O_{a}+C^{\prime}_{a}O^{\prime}_{a})~{},$
$\displaystyle O_{9(10)}^{ij}$ $\displaystyle=$
$\displaystyle[{\bar{s}}\gamma_{\mu}P_{L}b][{\bar{e}_{i}}\gamma^{\mu}(\gamma_{5})e_{j}]\,,\,\
O_{9(10)}^{\prime
ij}=[{\bar{s}}\gamma_{\mu}P_{R}b][{\bar{e}_{i}}\gamma^{\mu}(\gamma_{5})e_{j}].$
(48)

We can match them to the corresponding SMEFT Wilson coefficients as[63] :

$\displaystyle C_{9,{\rm NP}}^{ij}$ $\displaystyle=$
$\displaystyle\frac{\pi}{\alpha}\frac{v^{2}}{V_{tb}V_{ts}^{*}}\left(\big{[}{\cal
C}_{\ell q}^{(1)}\big{]}_{ij23}+\big{[}{\cal C}_{\ell
q}^{(3)}\big{]}_{ij23}+\big{[}{\cal C}_{qe}\big{]}_{23ij}\right)\,,$
$\displaystyle C_{10,{\rm NP}}^{ij}$ $\displaystyle=$
$\displaystyle\frac{\pi}{\alpha}\frac{v^{2}}{V_{tb}V_{ts}^{*}}\left(\big{[}{\cal
C}_{qe}\big{]}_{23ij}-\big{[}{\cal C}_{\ell
q}^{(1)}\big{]}_{ij23}-\big{[}{\cal C}_{\ell q}^{(3)}\big{]}_{ij23}\right)\,,$
$\displaystyle C_{9,{\rm NP}}^{\prime ij}$ $\displaystyle=$
$\displaystyle\frac{\pi}{\alpha}\frac{v^{2}}{V_{tb}V_{ts}^{*}}\left(\big{[}{\cal
C}_{\ell d}\big{]}_{ij23}+\big{[}{\cal C}_{ed}\big{]}_{ij23}\right)\,,$
$\displaystyle C_{10,{\rm NP}}^{\prime ij}$ $\displaystyle=$
$\displaystyle\frac{\pi}{\alpha}\frac{v^{2}}{V_{tb}V_{ts}^{*}}\left(\big{[}{\cal
C}_{ed}\big{]}_{ij23}-\big{[}{\cal C}_{\ell d}\big{]}_{ij23}\right)~{}.$ (49)

Here, $ij=\mu\mu,ee$. We find that, through back-rotation, the following
semileptonic operators can contribute to $b\to s\ell^{+}\ell^{-}$ observables:

$\big{[}{\cal C}_{\ell q}^{(1)}\big{]}_{ijkl}\,,\big{[}{\cal
C}_{ed}\big{]}_{ijkl}\,,\big{[}{\cal C}_{\ell d}\big{]}_{ijkl}\,,\big{[}{\cal
C}_{qe}\big{]}_{klij}\,,$ (50)

with

$i,j,k,l\equiv 1122,1133,2222,2233.$ (51)

To constrain these operators we have used the measurements for LFUV observable
such as $R_{K^{(*)}}$ [64, 65, 66, 67, 68, 69, 70] and
$Q_{4,5}=P_{4,5}^{\mu\prime}-P_{4,5}^{e\prime}$ [71, 72]. However, the full
$b\to s\ell^{+}\ell^{-}$ data set can give rise to stronger constraints[73,
74].

####  4.1.3 Charged Current Decays

The $\rm(V-A)\times(V-A)$ type charged-current (c.c) interactions in low
energy WET are given by effective Lagrangian

$\mathcal{L}_{\rm eff}^{\rm c.c}=[C_{\nu
edu}^{V,LL}]_{ijkl}~{}(\bar{\nu}_{i}\gamma_{\mu}P_{L}e_{j})(\bar{d}_{k}\gamma_{\mu}P_{L}u_{l})+h.c.~{}.$
(52)

After the EW symmetry breaking, $[C_{\nu edu}^{V,LL}]_{ijkl}$ can be matched
with the SMEFT operator $\big{[}{\cal C}_{\ell q}^{(3)}\big{]}_{ijkl}$. Now at
1-loop level, using (28), one can express $[C_{\nu edu}^{V,LL}]_{ijkl}$ in
terms of high scale Wilson coefficient $\big{[}{\cal C}_{\ell
q}^{(1)}\big{]}_{ijkl}$ due to its mixing with $\big{[}{\cal C}_{\ell
q}^{(3)}\big{]}_{ijkl}$. We find

$[C_{\nu edu}^{V,LL}]_{ijkl}={6g_{2}^{2}}L\big{[}{\cal C}_{\ell
q}^{(1)}\big{]}_{ijkm}(V^{\dagger})_{ml}~{}.$ (53)

Here, the CKM elements $(V^{\dagger})_{ml}$ are needed to rotate the up-quark
states to the mass basis. The above equation indicates that through EW
corrections the singlet operator $\big{[}{\cal C}_{\ell
q}^{(1)}\big{]}_{ijkl}$ can lead to charged current transitions at the low
energy. In the following we will look at several charged current processes
which can be affected through this mechanism.

#####  4.1.3.1 $K$ Meson Decays

At the quark level, the charged-current $K$ decays involve the transition
$s\to ue\bar{\nu}$. Therefore, these decays are driven by the WET operator
$(\bar{\nu}_{1}\gamma_{\mu}P_{L}e_{1})(\bar{d}_{2}\gamma^{\mu}P_{L}u_{1})$.
Now, using Eq. (53), we note that this operator can be generated at the 1-loop
level from the high scale SMEFT Wilson coefficient $\big{[}{\cal C}_{\ell
q}^{(1)}\big{]}_{1122}$, _i.e._ ,

$[C_{\nu edu}^{V,LL}]_{1121}\propto\left(V_{ud}^{*}\big{[}{\cal C}_{\ell
q}^{(1)}\big{]}_{1121}+V_{us}^{*}\big{[}{\cal C}_{\ell
q}^{(1)}\big{]}_{1122}+V_{ub}^{*}\big{[}{\cal C}_{\ell
q}^{(1)}\big{]}_{1123}\right).$ (54)

Therefore, at the low-energy, $\big{[}{\cal C}_{\ell q}^{(1)}\big{]}_{1122}$
can lead to effects in the $K_{L}\to\pi e\bar{\nu}$, $K_{S}\to\pi e\bar{\nu}$,
$K^{+}\to\pi e\bar{\nu}$, and $K^{+}\to\ell\bar{\nu}$ decays.

#####  4.1.3.2 $\tau\to K\nu$ and $\tau\to\pi\nu$ Decays

The $\tau\to K\nu$ and $\tau\to\pi\nu$ decays involve $\tau\to\bar{u}s\nu$ and
$\tau\to\bar{u}d\nu$ transitions, respectively. At the low energy these are
governed by the WET operators
$(\bar{\nu}_{3}\gamma_{\mu}P_{L}e_{3})(\bar{d}_{2}\gamma^{\mu}P_{L}u_{1})$ and
$(\bar{\nu}_{3}\gamma_{\mu}P_{L}e_{3})(\bar{d}_{1}\gamma^{\mu}P_{L}u_{1})$.
However, at the EW scale these operators can be generated from high scale
Wilson coefficients $\big{[}{\cal C}_{\ell q}^{(1)}\big{]}_{3322}$ and
$\big{[}{\cal C}_{\ell q}^{(1)}\big{]}_{3311}$ respectively (see (53)),
because

$\displaystyle{[C_{\nu edu}^{V,LL}]}_{3321}$ $\displaystyle\propto$
$\displaystyle\left(V_{ud}^{*}\big{[}{\cal C}_{\ell
q}^{(1)}\big{]}_{3321}+V_{us}^{*}\big{[}{\cal C}_{\ell
q}^{(1)}\big{]}_{3322}+V_{ub}^{*}\big{[}{\cal C}_{\ell
q}^{(1)}\big{]}_{3323}\right)\,,$ (55) $\displaystyle{[C_{\nu
edu}^{V,LL}]}_{3311}$ $\displaystyle\propto$
$\displaystyle\left(V_{ud}^{*}\big{[}{\cal C}_{\ell
q}^{(1)}\big{]}_{3311}+V_{us}^{*}\big{[}{\cal C}_{\ell
q}^{(1)}\big{]}_{3312}+V_{ub}^{*}\big{[}{\cal C}_{\ell
q}^{(1)}\big{]}_{3313}\right).$ (56)

Therefore, one can use hadronic $\tau$ decays to constrain the SMEFT operators
on the r.h.s.

#####  4.1.3.3 $\pi\to e\nu$ and Nuclear $\beta$ Decays

The nuclear $\beta$-decay and $\pi\to e\nu$ are controlled by the WET operator
$(\bar{\nu}_{1}\gamma_{\mu}P_{L}e_{1})(\bar{d}_{1}\gamma^{\mu}P_{L}u_{1})$.
From (53), clearly at the EW scale this operator can be generated from
$\big{[}{\cal C}_{\ell q}^{(1)}\big{]}_{1111}$ because

$[C_{\nu edu}^{V,LL}]_{1111}\propto\left(V_{ud}^{*}\big{[}{\cal C}_{\ell
q}^{(1)}\big{]}_{1111}+V_{us}^{*}\big{[}{\cal C}_{\ell
q}^{(1)}\big{]}_{1112}+V_{ub}^{*}\big{[}{\cal C}_{\ell
q}^{(1)}\big{]}_{1113}\right).$ (57)

To summarize, we have found that the following list of high scale coefficients
can contribute to various charged current decays at low energy:

$\big{[}{\cal C}_{\ell q}^{(1)}\big{]}_{1122}\,,\big{[}{\cal C}_{\ell
q}^{(1)}\big{]}_{3322}\,,\big{[}{\cal C}_{\ell
q}^{(1)}\big{]}_{3311}\,,\big{[}{\cal C}_{\ell q}^{(1)}\big{]}_{1111}.$ (58)

Note that the Wilson coefficient $\big{[}{\cal C}_{\ell
q}^{(3)}\big{]}_{ijkl}$ also give rise to tree-level contributions to the
charged current decays. The experimental measurements for various charged
current decays are shown in Tab. 1. In addition, the measurements for the
$\beta$-decays are taken from the Ref. [75].

Observable Experimental value Observable Experimental value
$\mathcal{B}(K_{L}\to\pi e\bar{\nu})$ $(40.55\pm 0.11)\times 10^{-2}$
$\mathcal{B}(K_{S}\to\pi e\bar{\nu})$ $(7.04\pm 0.08)\times 10^{-4}$
$\mathcal{B}(K^{+}\to\pi e\bar{\nu})$ $(5.07\pm 0.04)\times 10^{-2}$
$R_{e/\mu}({K^{+}\to\ell\bar{\nu}})$ $(2.488\pm 0.009)\times 10^{-5}$
$\mathcal{B}(\tau\to K\nu)$ $(6.96\pm 0.10)\times 10^{-3}$
$\mathcal{B}(\tau\to\pi\nu)$ $(10.82\pm 0.05)\times 10^{-2}$
$\mathcal{B}(\pi\to e\nu)$ $(1.2344\pm 0.0023\pm 0.0019)\times 10^{-4}$

Table 1: The experimental measured values for the charged current $K$, $\pi$
and $\tau$ decays [76, 77].

Note that for the charged current operators, in principle the three-body
$\tau$ decays such as the Belle spectrum for the process $\tau^{-}\to
K_{S}\pi^{-}\nu_{\tau}$ also apply [78, 79]. But as shown in Ref. [80] these
decays give rise to similar constraints as $\tau^{-}\to K\nu$ decay.

####  4.1.4 Correlations

Since a single operator can contribute to several different kinds of
observables through RG running, it would be interesting to see how these are
correlated to each other. For instance, the operator $\big{[}{\cal C}_{\ell
q}^{(1)}\big{]}_{1111}$ can contribute to $\beta$ decay as well as EWP
observables. Similarly, $\big{[}{\cal C}_{\ell q}^{(1)}\big{]}_{1122}$
contributes to EWP observables, $b\to s\ell^{+}\ell^{-}$ , as well as charged
current $K$ processes. Take another example of the operator $\big{[}{\cal
C}_{\ell q}^{(3)}\big{]}_{1111}$, which in addition to $\beta$ decay also
contributes to $\pi\to e\bar{\nu}$ process at tree level. In Fig. 1, we show
correlations between constraints due to various observables in the
$\big{[}{\cal C}_{\ell q}^{(1)}\big{]}_{1111}$ \- $\big{[}{\cal C}_{\ell
q}^{(1)}\big{]}_{1122}$ and $\big{[}{\cal C}_{\ell q}^{(1)}\big{]}_{1111}$ \-
$\big{[}{\cal C}_{\ell q}^{(3)}\big{]}_{1111}$ planes in left and right panels
respectively. Clearly, in order to get a complete picture about the constrains
on a given operator, it is very important to take into account all RG induced
observables.

Figure 1: The correlations between various observables in the $\big{[}{\cal
C}_{\ell q}^{(1)}\big{]}_{1111}$ \- $\big{[}{\cal C}_{\ell
q}^{(1)}\big{]}_{1122}$ plane (left) and $\big{[}{\cal C}_{\ell
q}^{(1)}\big{]}_{1111}$ \- $\big{[}{\cal C}_{\ell q}^{(3)}\big{]}_{1111}$
plane (right) .

###  4.2 Observables for $\Delta F=(1,0)$ operators

Next, we move on to the semileptonic operators which violate the lepton
flavour by one unit and conserve the quark flavour at the scale $\Lambda$. In
this case, depending upon the Dirac and flavour structures, both tree-level as
well as the 1-loop generated LFV observables are found to be important. The
1-loop generated processes are $\ell_{i}\to\ell_{j}\bar{\ell}\ell$ and
$Z\to\bar{\ell}_{i}\ell_{j}$ LFV decays. These processes are found to be
relevant for all $\Delta F=(1,0)$ operators under consideration. In addition,
the $\Delta F=(1,0)$ semileptonic operators can also contribute to $\tau\to
P\ell$ for $P=\pi,\phi$ and $\tau\to\rho\ell$ processes at the tree-level.

####  4.2.1 $Z\to\bar{\ell}_{i}\ell_{j}$ Decays

The tree-level SMEFT contributions to the LFV $Z$ boson couplings due to
$\psi^{2}\phi^{2}D$ operators are given in Eqs. (78)-(84). At the 1-loop
level, the semileptonic $\Delta F=(1,0)$ operators can mix with these
$\psi^{2}\phi^{2}D$-type operators, as shown in the Eqs. (30) and (29). As a
result the $\Delta F=(1,0)$ semileptonic operators can be constrained by the
$Z\to\bar{\ell}_{i}\ell_{j}$ LFV decays. The current experimental limits on
these decays are given in Tab. 2.

####  4.2.2 $\tau\to 3\ell$ and $\mu\to 3e$ Decays

In WET, the LFV processes such as $\tau\to 3\ell$ and $\mu\to 3e$ are governed
by purely leptonic operators, as given by the effective Lagrangian:

$\displaystyle\mathcal{L}_{\rm eff}^{\rm 1loop}$ $\displaystyle=$
$\displaystyle[C_{ee}^{V,LL}]_{ijll}~{}(\bar{e}_{i}\gamma^{\mu}P_{L}e_{j})(\bar{e}_{l}P_{L}\gamma_{\mu}e_{l})$
(59) $\displaystyle+$
$\displaystyle[C_{ee}^{V,LR}]_{ijll}~{}(\bar{e}_{i}\gamma^{\mu}P_{L}e_{j})(\bar{e}_{l}\gamma_{\mu}P_{R}e_{l})$
$\displaystyle+$
$\displaystyle[C_{ee}^{V,LR}]_{llij}~{}(\bar{e}_{l}\gamma^{\mu}P_{L}e_{l})(\bar{e}_{i}\gamma_{\mu}P_{R}e_{j})$
$\displaystyle+$
$\displaystyle[C_{ee}^{V,RR}]_{ijll}~{}(\bar{e}_{i}\gamma^{\mu}P_{R}e_{j})(\bar{e}_{l}\gamma_{\mu}P_{R}e_{l})+h.c.~{}.$

Here, the superscript indicates that such operators are generated only at the
1-loop level and are set to zero at the NP scale. In SMEFT, at tree-level the
corresponding four fermion leptonic operators are $\big{[}{\cal
C}_{\ell\ell}\big{]}_{ijkl}$, $\big{[}{\cal C}_{\ell e}\big{]}_{ijkl}$, and
$\big{[}{\cal C}_{ee}\big{]}_{ijkl}$ which are defined in Eqs. (21)-(23). In
addition to this, the SMEFT can also contribute through $\psi^{2}\phi^{2}D$
type operators after integrating out the $Z$ boson. Such contributions arise
by combining the LFV effective $Z$ boson vertices due to SMEFT with the
flavour conserving $Z$ boson interactions in the SM[81]. The $\Delta F=(1,0)$
semileptonic operators can give rise to both effects through operator mixing.
From Eq. (30), one can find that, at the 1-loop level through the EW
interactions, the contributing four-fermion SMEFT operators can be directly
generated from the semileptonic operators. In addition, the
$\psi^{2}\phi^{2}D$ operators get contributions through both the gauge (30) as
well as the top-Yukawa interactions (29). In the LL approximation, combining
the four fermion and $\psi^{2}\phi^{2}D$ type contributions, we can express
the contributing WET Wilson coefficients at the EW scale directly in terms of
SMEFT Wilson coefficients of the semileptonic operators at $\Lambda$:

$\displaystyle{[C_{ee}^{V,LL}]}_{ijll}$ $\displaystyle=$
$\displaystyle\frac{g_{1}^{2}}{3}L\left(-\big{[}{\cal C}_{\ell
q}^{(1)}\big{]}_{ijkk}+\frac{3g_{2}^{2}}{g_{1}^{2}}\big{[}{\cal C}_{\ell
q}^{(3)}\big{]}_{ijkk}-2\big{[}{\cal C}_{\ell u}\big{]}_{ijkk}+\big{[}{\cal
C}_{\ell d}\big{]}_{ijkk}\right)$ (60) $\displaystyle+$
$\displaystyle\frac{\mathcal{Z}}{4}\left(\big{[}{\cal C}_{\ell
q}^{(1)}\big{]}_{ij33}-\big{[}{\cal C}_{\ell
q}^{(3)}\big{]}_{ij33}-\big{[}{\cal C}_{\ell
u}\big{]}_{ij33}\right)g_{L}^{e,{\rm SM}}\,,$
$\displaystyle{[C_{ee}^{V,LR}]}_{ijll}$ $\displaystyle=$
$\displaystyle\frac{4g_{1}^{2}}{3}L\left(-\big{[}{\cal C}_{\ell
q}^{(1)}\big{]}_{ijkk}-2\big{[}{\cal C}_{\ell u}\big{]}_{ijkk}+\big{[}{\cal
C}_{\ell d}\big{]}_{ijkk}\right)$ (61) $\displaystyle+$
$\displaystyle\mathcal{Z}\left(\big{[}{\cal C}_{\ell
q}^{(1)}\big{]}_{ij33}-\big{[}{\cal C}_{\ell
q}^{(3)}\big{]}_{ij33}-\big{[}{\cal C}_{\ell
u}\big{]}_{ij33}\right)g_{R}^{e,{\rm SM}}\,,$
$\displaystyle{[C_{ee}^{V,LR}]}_{llij}$ $\displaystyle=$
$\displaystyle\frac{2g_{1}^{2}}{3}L\left(\big{[}{\cal
C}_{ed}^{(1)}\big{]}_{ijkk}-2\big{[}{\cal C}_{eu}\big{]}_{ijkk}-\big{[}{\cal
C}_{qe}\big{]}_{kkij}\right)$ (62) $\displaystyle+$
$\displaystyle\mathcal{Z}\left(\big{[}{\cal
C}_{qe}^{(1)}\big{]}_{33ij}-\big{[}{\cal
C}_{eu}\big{]}_{ij33}\right)g_{L}^{e,{\rm SM}}\,,$
$\displaystyle{[C_{ee}^{V,RR}]}_{ijll}$ $\displaystyle=$
$\displaystyle\frac{2g_{1}^{2}}{3}L\left(\big{[}{\cal
C}_{ed}^{(1)}\big{]}_{ijkk}-\big{[}{\cal C}_{eu}\big{]}_{ijkk}-\big{[}{\cal
C}_{qe}\big{]}_{kkij}\right)$ (63) $\displaystyle+$
$\displaystyle\frac{\mathcal{Z}}{4}\left(\big{[}{\cal
C}_{qe}^{(1)}\big{]}_{33ij}-\big{[}{\cal
C}_{eu}\big{]}_{ij33}\right)g_{R}^{e,{\rm SM}}\,,$

with the factor
$\mathcal{Z}=\left[3v^{2}y_{t}^{2}L\right]\left[\frac{g_{Z}^{2}}{M_{Z}^{2}}\right]$.
Here $L$ is the log term defined in (26). Numerically, the comparison of the
influence of the gauge and Yukawa interactions on the WET Wilson coefficients
on the l.h.s. is presented in Tab. 13 in Appendix B.4 . We find that the
following set of Wilson coefficients can be constrained through this
mechanism:

$\big{[}{\cal C}_{\ell q}^{(1)}\big{]}_{ijkl}\,,\big{[}{\cal C}_{\ell
q}^{(3)}\big{]}_{ijkl}\,,\big{[}{\cal C}_{ed}\big{]}_{ijkl}\,,\big{[}{\cal
C}_{eu}\big{]}_{ijkl}\,,\big{[}{\cal C}_{\ell u}\big{]}_{ijkl}\,,\big{[}{\cal
C}_{\ell d}\big{]}_{ijkl}\,,\big{[}{\cal C}_{qe}\big{]}_{klij}\,,$ (64)

with

$ijkl\equiv 1211,1222,1233,1311,1322,1333,2311,2322,2333.$ (65)

The experimental limits used for the LFV decays of $\tau$ and $\mu$ leptons
are collected in Tab. 2.

####  4.2.3 $\tau\to\rho\ell$, $\tau\to P(\phi,\pi)\ell$ Decays

In the WET, the hadronic decays $\tau\to P\ell$ for $P=\pi,\phi$ and
$\tau\to\rho\ell$ can be described by the effective Lagrangian

$\displaystyle\mathcal{L}_{\rm eff}^{\rm tree+1loop}$ $\displaystyle=$
$\displaystyle\big{[}{\cal
C}_{ed}^{V,LL}\big{]}_{ijkk}(\bar{e}_{i}\gamma^{\mu}P_{L}e_{j})(\bar{d}_{k}\gamma_{\mu}P_{L}d_{k})+\big{[}{\cal
C}_{eu}^{V,LL}\big{]}_{ij11}(\bar{e}_{i}\gamma^{\mu}P_{L}e_{j})(\bar{u}_{1}\gamma_{\mu}P_{L}u_{1})$
$\displaystyle+$ $\displaystyle\big{[}{\cal
C}_{ed}^{V,LR}\big{]}_{ijkk}(\bar{e}_{i}\gamma^{\mu}P_{L}e_{j})(\bar{d}_{k}\gamma_{\mu}P_{R}d_{k})+\big{[}{\cal
C}_{eu}^{V,LR}\big{]}_{ij11}(\bar{e}_{i}\gamma^{\mu}P_{L}e_{j})(\bar{u}_{1}\gamma_{\mu}P_{R}u_{1})$
$\displaystyle+$ $\displaystyle\big{[}{\cal
C}_{de}^{V,LR}\big{]}_{kkij}(\bar{d}_{k}\gamma^{\mu}P_{L}d_{k})(\bar{e}_{i}\gamma_{\mu}P_{R}e_{j})+\big{[}{\cal
C}_{ue}^{V,LR}\big{]}_{11ij}(\bar{u}_{1}\gamma^{\mu}P_{L}u_{1})(\bar{e}_{i}\gamma_{\mu}P_{R}e_{j})$
$\displaystyle+$ $\displaystyle\big{[}{\cal
C}_{ed}^{V,RR}\big{]}_{ijkk}(\bar{e}_{i}\gamma^{\mu}P_{R}e_{j})(\bar{d}_{k}\gamma_{\mu}P_{R}d_{k})+\big{[}{\cal
C}_{eu}^{V,RR}\big{]}_{ij11}(\bar{e}_{i}\gamma^{\mu}P_{R}e_{j})(\bar{u}_{1}\gamma_{\mu}P_{R}u_{1}).$

Here, the indices $kk=11$ for $\tau\to\pi\ell$, $\tau\to\rho\ell$ and $22$ for
$\tau\to\phi\ell$. In addition to the tree-level contributions, the
semileptonic operators in SMEFT can also contribute to these operators through
top-Yukawa loops. Adding the both contributions, in the LL approximation, at
the EW scale we obtain:

$\displaystyle\big{[}{\cal C}_{ed}^{V,LL}\big{]}_{ijkk}$ $\displaystyle=$
$\displaystyle\big{[}{\cal C}_{\ell q}^{(1)}\big{]}_{ijkk}+\big{[}{\cal
C}_{\ell q}^{(3)}\big{]}_{ijkk}$ (67) $\displaystyle+$
$\displaystyle\mathcal{Z}\left[\big{[}{\cal C}_{\ell
q}^{(1)}\big{]}_{ij33}-\big{[}{\cal C}_{\ell
q}^{(3)}\big{]}_{ij33}-\big{[}{\cal C}_{\ell
u}\big{]}_{ij33}\right]g_{L}^{d,{\rm SM}}\,,$ $\displaystyle\big{[}{\cal
C}_{eu}^{V,LL}\big{]}_{ij11}$ $\displaystyle=$ $\displaystyle
V_{1m}\left[\big{[}{\cal C}_{\ell q}^{(1)}\big{]}_{ijmn}-\big{[}{\cal C}_{\ell
q}^{(3)}\big{]}_{ijmn}\right]V^{\dagger}_{n1}$ (68) $\displaystyle+$
$\displaystyle\mathcal{Z}\left[\big{[}{\cal C}_{\ell
q}^{(1)}\big{]}_{ij33}-\big{[}{\cal C}_{\ell
q}^{(3)}\big{]}_{ij33}-\big{[}{\cal C}_{\ell
u}\big{]}_{ij33}\right]g_{L}^{u,{\rm SM}}\,,$ $\displaystyle\big{[}{\cal
C}_{ed}^{V,LR}\big{]}_{ijkk}$ $\displaystyle=$ $\displaystyle\big{[}{\cal
C}_{\ell d}\big{]}_{ijkk}~{}+~{}\mathcal{Z}\left[\big{[}{\cal C}_{\ell
q}^{(1)}\big{]}_{ij33}-\big{[}{\cal C}_{\ell
q}^{(3)}\big{]}_{ij33}-\big{[}{\cal C}_{\ell
u}\big{]}_{ij33}\right]g_{R}^{d,{\rm SM}}\,,$ (69) $\displaystyle\big{[}{\cal
C}_{eu}^{V,LR}\big{]}_{ij11}$ $\displaystyle=$ $\displaystyle\big{[}{\cal
C}_{\ell u}\big{]}_{ij11}~{}+~{}\mathcal{Z}\left[\big{[}{\cal C}_{\ell
q}^{(1)}\big{]}_{ij33}-\big{[}{\cal C}_{\ell
q}^{(3)}\big{]}_{ij33}-\big{[}{\cal C}_{\ell
u}\big{]}_{ij33}\right]g_{R}^{u,{\rm SM}}\,,$ (70) $\displaystyle\big{[}{\cal
C}_{de}^{V,LR}\big{]}_{kkij}$ $\displaystyle=$ $\displaystyle\big{[}{\cal
C}_{qe}\big{]}_{kkij}~{}+~{}\mathcal{Z}\left[\big{[}{\cal
C}_{qe}\big{]}_{33ij}-\big{[}{\cal C}_{eu}\big{]}_{ij33}\right]g_{L}^{d,{\rm
SM}}\,,$ (71) $\displaystyle\big{[}{\cal C}_{ue}^{V,LR}\big{]}_{11ij}$
$\displaystyle=$ $\displaystyle V_{1m}\big{[}{\cal
C}_{qe}\big{]}_{mnij}V^{\dagger}_{n1}~{}+~{}\mathcal{Z}\left[\big{[}{\cal
C}_{qe}\big{]}_{33ij}-\big{[}{\cal C}_{eu}\big{]}_{ij33}\right]g_{L}^{u,{\rm
SM}}\,,$ (72) $\displaystyle\big{[}{\cal C}_{ed}^{V,RR}\big{]}_{ijkk}$
$\displaystyle=$ $\displaystyle\big{[}{\cal
C}_{ed}\big{]}_{ijkk}~{}+~{}\mathcal{Z}\left[\big{[}{\cal
C}_{qe}\big{]}_{33ij}-\big{[}{\cal C}_{eu}\big{]}_{ij33}\right]g_{R}^{d,{\rm
SM}}\,,$ (73) $\displaystyle\big{[}{\cal C}_{eu}^{V,RR}\big{]}_{ij11}$
$\displaystyle=$ $\displaystyle\big{[}{\cal
C}_{eu}\big{]}_{ij11}~{}+~{}\mathcal{Z}\left[\big{[}{\cal
C}_{qe}\big{]}_{33ij}-\big{[}{\cal C}_{eu}\big{]}_{ij33}\right]g_{R}^{u,{\rm
SM}}.$ (74)

Here, the CKM elements $V_{ij}$ on the r.h.s. are needed to rotate the up-
quarks from the Warsaw-down to the mass basis for matching them with the WET
Wilson coefficients on the l.h.s. The quantity
$\mathcal{Z}=\left[3v^{2}y_{t}^{2}L\right]\left[\frac{g_{Z}^{2}}{M_{Z}^{2}}\right]$.
Numerically, the relative influence of the WET and SMEFT running can be found
in Tab. 14. The following set of Wilson coefficients can be constrained
through this mechanism:

$\big{[}{\cal C}_{\ell q}^{(1)}\big{]}_{ijkl}\,,\big{[}{\cal C}_{\ell
q}^{(3)}\big{]}_{ijkl}\,,\big{[}{\cal C}_{ed}\big{]}_{ijkl}\,,\big{[}{\cal
C}_{eu}\big{]}_{ijkl}\,,\big{[}{\cal C}_{\ell u}\big{]}_{ijkl}\,,\big{[}{\cal
C}_{\ell d}\big{]}_{ijkl}\,,\big{[}{\cal C}_{qe}\big{]}_{klij}\,,$ (75)

with

$ijkl\equiv 1211,1222,1233,1311,1322,1333,2311,2322,2333.$ (76)

Note that the operators involving $u\bar{u}$, $d\bar{d}$ and $s\bar{s}$
contribute at tree-level, whereas the operators involving the third generation
contribute to these processes only at the 1-loop level. Naively, the former
contributions are expected to dominate, but we will give a counterexample in
the next subsection. The experimental limits on the corresponding LFV decays
are shown in Tab. 2.

Figure 2: The correlations between various LFV observables in the $\big{[}{\cal C}_{\ell q}^{(1)}\big{]}_{1311}$ \- $\big{[}{\cal C}_{\ell q}^{(1)}\big{]}_{1333}$ plane (left) and in $\big{[}{\cal C}_{\ell q}^{(1)}\big{]}_{1311}$ \- $\big{[}{\cal C}_{\ell u}\big{]}_{1322}$ plane (right) . Observable | Experimental value | Observable | Experimental value
---|---|---|---
$\mathcal{B}(Z\to e^{\pm}\mu^{\mp})$ |  $<7.5\times 10^{-7}$[82] | $\mathcal{B}(Z\to e^{\pm}\tau^{\mp})$ |  $<5.0\times 10^{-6}$ [83]
$\mathcal{B}(Z\to\mu^{\pm}\tau^{\mp})$ |  $<6.5\times 10^{-6}$ [83] |  |
$\mathcal{B}(\tau^{-}\to e^{-}e^{+}e^{-})$ |  $<2.7\times 10^{-8}$ [84] | $\mathcal{B}(\tau^{-}\to\mu^{-}\mu^{+}\mu^{-})$ |  $<2.1\times 10^{-8}$ [84]
$\mathcal{B}(\tau^{-}\to e^{-}\mu^{+}\mu^{-})$ |  $<2.7\times 10^{-8}$ [84] | $\mathcal{B}(\tau^{-}\to\mu^{-}e^{+}e^{-})$ |  $<1.8\times 10^{-8}$ [84]
$\mathcal{B}(\tau^{-}\to\mu^{-}e^{+}\mu^{-})$ |  $<1.7\times 10^{-8}$ [84] | $\mathcal{B}(\tau^{-}\to e^{-}\mu^{+}e^{-})$ |  $<1.5\times 10^{-8}$ [84]
$\mathcal{B}(\mu^{-}\to e^{-}e^{+}e^{-})$ |  $<1.0\times 10^{-12}$ [85] |  |
$\mathcal{B}(\tau\to\phi\mu)$ |  $<8.4\times 10^{-8}$ [86] | $\mathcal{B}(\tau\to\phi e)$ |  $<3.1\times 10^{-8}$ [86]
$\mathcal{B}(\tau\to\rho\mu)$ |  $<1.2\times 10^{-8}$ [86] | $\mathcal{B}(\tau\to\rho e)$ |  $<1.8\times 10^{-8}$ [86]
$\mathcal{B}(\tau\to\pi\mu)$ |  $<1.1\times 10^{-7}$ [86] | $\mathcal{B}(\tau\to\pi e)$ |  $<8.8\times 10^{-8}$ [86]
Table 2: The experimental upper limits on the LFV decays of $Z$-boson, $\tau$
and $\mu$ leptons. Note only the strongest limits are shown and the upper
bounds correspond to $90\%$ CL for the $Z$ decays and at $90\%$ CL for all
other decays.

####  4.2.4 Correlations

Since, the semileptonic operators can contribute to the LFV processes at tree-
level as well as at the 1-loop, it would be interesting to see the relative
importance of the loop-level vs. tree-level LFV effects. For example, the
Wilson coefficient $\big{[}{\cal C}_{\ell q}^{(1)}\big{]}_{1311}$ could in
principle give a tree level contribution to $\tau\to Pe$ and $\tau\to\rho e$
processes. However, due to $SU(2)_{L}$ invariance, the WET Wilson coefficients
with $u\bar{u}$ and $d\bar{d}$ flavours get equal contributions on matching
with $\big{[}{\cal C}_{\ell q}^{(1)}\big{]}_{1311}$. As a result, its net
effect zero, because the $u\bar{u}$ and $d\bar{d}$ Wilson coefficients enter
with opposite sign in the branching ratios (see e.g. Eqs. (55) and (61) of
Ref. [39]). However, $\big{[}{\cal C}_{\ell q}^{(1)}\big{]}_{1311}$ can still
contribute at the 1-loop level through the EW corrections.

On the other hand, the Wilson coefficient $\big{[}{\cal C}_{\ell
q}^{(1)}\big{]}_{1333}$ can not give tree-level contribution to these LFV
processes, but it can contribute at 1-loop to $\tau\to 3e$, $\tau\to
e\mu^{+}\mu^{-}$ as well as $\tau\to\rho e$ and $\tau\to Pe$ processes.
Interestingly, this loop induced effect on the $u\bar{u}$ and $d\bar{d}$ WET
coefficients is not equal but depends on the SM couplings $g_{L}^{d,{\rm SM}}$
and $g_{L}^{u,{\rm SM}}$ and hence it is not canceled in the branching ratios
of our interest (see Eqs. (67) and (68)). In addition to VLL, the VLR (again
with unequal sizes for the $u\bar{u}$ and $d\bar{d}$ coefficients) WET
operators are also generated at the 1-loop level. In Fig. 2 (left), the loose
constraints on $\big{[}{\cal C}_{\ell q}^{(1)}\big{]}_{1311}$ as compared to
$\big{[}{\cal C}_{\ell q}^{(1)}\big{]}_{1333}$ confirm these findings. On the
right panel of the same figure, we show the impact of LFV constraints in the
plane of $\big{[}{\cal C}_{\ell q}^{(1)}\big{]}_{1311}$ and $\big{[}{\cal
C}_{\ell u}\big{]}_{1322}$ Wilson coefficients. In this case, since both
operators contribute at the 1-loop level, they are found to be constrained at
the similar level.

##  5 Sensitivities to the NP scale $\Lambda$

On the basis of our discussion about the RG running and the various low energy
and the EW scale observables identified in the previous sections, now we are
in position to look at the highest possible scales for each operator that can
be probed using these observables. In Eq. (1), the Wilson coefficients are
defined to be dimensionful quantities. However, these can be written in terms
of the dimensionless parameters ${[c_{X}]}_{ijkl}$ defined by

$\big{[}{\cal C}_{X}\big{]}_{ijkl}=\frac{[c_{X}]_{ijkl}}{\Lambda^{2}}.$ (77)

Assuming the presence of a single operator at the scale $\Lambda$, we perform
combined fits for each dimensionless parameter $[c_{X}]_{ijkl}$, using all
measurements relevant for a given operator. From this we obtain the quantity
${\Lambda}/{\sqrt{{[c_{X}]}_{ijkl}}}$ using the central values of
$[c_{X}]_{ijkl}$ from the fits. This gives us a rough estimate of the scales
that can be probed for each operator.

Figure 3: The figure shows the sensitivity of semileptonic operators to the
high scale $\Lambda$ normalized with dimensionless parameters ${[c_{\ell
q}^{(1)}]}_{ijkl}$ as defined in Eq. (77). Figure 4: Same as in Fig. 3 except
for ${[c_{\ell q}^{(3)}]}_{ijkl}$.  Figure 5: Same as in Fig. 3 except for
${[c_{\ell u}]}_{ijkl}$.  Figure 6: Same as in Fig. 3 except for ${[c_{\ell
d}]}_{ijkl}$.  Figure 7: Same as in Fig. 3 except for ${[c_{ed}]}_{ijkl}$.

Note that in a given NP model, more than one operators can be simultaneously
present which could change this simplified picture of single operator
dominance. However, the goal of present work is to systematically analyze the
low energy implications of each operator separately _i.e_ , to identify the
most sensitive observables and assess their potential to constrain the
individual operators. In Figs. 3-9, we show the central values for the lower
bounds on the quantity ${\Lambda}/{\sqrt{{[c_{X}]}_{ijkl}}}$ (expressed in the
units of TeV) for various Wilson coefficients under consideration. In the left
and right panels, the results for $\Delta F=(0,0)$ and $\Delta F=(1,0)$
operators are shown, respectively. Based on these results, we can now make
following important observations:

Figure 8: Same as in Fig. 3 except for ${[c_{eu}]}_{ijkl}$.

  * •

The lower bounds on $\Delta F=(0,0)$ operators vary between $\mathcal{O}(\rm
TeV)$ to $\mathcal{O}(10\rm TeV)$, whereas the lower bounds on the $\Delta
F=(1,0)$ operators go all the way up to $\mathcal{O}(100\rm TeV)$, much beyond
the present reach of the LHC.

  * •

Due to more stringent experimental limits on the branching ratio for the
process $\mu\to 3e$ as compared to the $\tau$ LFV modes, the $\Delta F=(1,0)$
operators involving the $12jj$ flavour indices are very strongly constrained.
As shown, the current lower bounds on such operators are always above the
ballpark of 10 TeV.

  * •

Among $12jj$ operators, the ones having $jj=33$ are more strongly constrained
as compared the operators with $jj=11$ or $22$. This can be attributed to the
fact that the former operators can mix strongly with the $\Delta F=(1,0)$
operators $\big{[}{\cal C}_{\phi\ell}^{(1)}\big{]}_{12}$, $\big{[}{\cal
C}_{\phi\ell}^{(3)}\big{]}_{12}$, and $\big{[}{\cal C}_{\phi e}\big{]}_{12}$
through top-Yukawa interactions. In this regard, the numerical impact of the
top-Yukawa is shown in Tab. 13 and Eq. (29). Clearly, the Wilson coefficients
$\big{[}{\cal C}_{\ell d}\big{]}_{ij33}$ and $\big{[}{\cal
C}_{ed}\big{]}_{1233}$ are exceptions here, because they do not mix with the
contributing operators through top-Yukawa interactions.

  * •

Since the operators with the indices $13jj$, $23jj$ contribute even at tree-
level to the LFV processes such as $\tau\to P\ell$ and $\tau\to\rho\ell$ for
$jj=11$ or $22$, as a result these operators are in general more strongly
constrained as compared to operators with $1333$ or $2333$ flavour indices.
This is due to the reason that the latter operators contribute only at the
1-loop to such processes or to purely leptonic LFV modes. However, in certain
cases like $\big{[}{\cal C}_{\ell q}^{(1)}\big{]}_{1333}$, $\big{[}{\cal
C}_{\ell q}^{(1)}\big{]}_{2333}$, $\big{[}{\cal C}_{\ell u}\big{]}_{1333}$ and
$\big{[}{\cal C}_{\ell u}\big{]}_{2333}$, the 1-loop effects due to the top-
Yukawa can dominate over the tree-level effects.

  * •

Except for $\big{[}{\cal C}_{\ell d}\big{]}$ and $\big{[}{\cal
C}_{ed}\big{]}$, the $\Delta F=(0,0)$ operators involving $ii33$ for
$ii=11,22$ or $33$ have the strongest bounds, because they contribute to the
$Z$ and $W$ boson couplings through the top-Yukawa.

  * •

Finally, among $\Delta F=(0,0)$ operators, $\big{[}{\cal C}_{\ell
d}\big{]}_{iijj}$ and $\big{[}{\cal C}_{ed}\big{]}_{iijj}$ are found to be
loosely constrained. In most cases, the lower bound on NP scale lies around
$\mathcal{O}(1{\rm TeV})$ or below. This is again due to the fact that these
operators do not exhibit the operator mixing due to large top-Yukawa coupling.

Figure 9: Same as in Fig. 3 except for ${[c_{qe}]}_{ijkl}$.

In Appendix C, we also provide the best-fit values along with $1\sigma$ errors
for the dimensionless Wilson coefficients.

##  6 Conclusions and Outlook

The SMEFT provides a convenient framework for parameterizing the NP effects
beyond the SM. It is well known that the semileptonic operators which violate
the quark flavour lead to effects in the flavour violating decays of $B$ and
$K$ mesons at low energy, which give rise to stringent constraints. In the
view of current anomalies in the $B$-decays, the semileptonic operators are of
great interest in general. However, it is important to probe the generic
flavour structure of such operators. In particular, often the quark and lepton
flavour conserving operators and the ones which violate only the lepton
flavour are also generated in the NP models. To probe such operators it is
important to know the type of observables to which these operators contribute.

In the present paper, we address this issue. We identify the low energy and
the EW scale observables which can be used to probe a generic flavour
structure of the semileptonic operators. However, in order to correctly
predict the low energy behaviour of these operators, it is necessary to know
the operator mixing pattern due to running from NP scale to the EW scale and
then below. To this end, by scrutinizing the ADMs due to the electroweak gauge
as well as the Yukawas interactions one can find that such operators can mix
with purely leptonic and $\psi^{2}\phi^{2}D$-type operators in the SMEFT. The
former operators can contribute to the LFV decays of leptons, whereas the
latter ones in addition also give corrections to the gauge boson couplings at
the EW scale.

Therefore, first we have identified the phenomenologically relevant terms in
the ADMs and then taking into account the WET and SMEFT RG running effects, we
identified a list of observables which can be used to constrain the
semileptonic operators having a generic flavour structure. We show that,
through SMEFT RG running effects at the 1-loop level, the semileptonic
operators can contribute to a variety of observables such as EWP observables,
flavour violating decays of $B$ and $K$-mesons $-$ involving neutral as well
as charged current transitions, the LFV decays of leptons as well as the LFV
$Z$-boson decays. The main findings of the present study can be summarized as
follows:

The semileptonic operators with flavour indices $iijj$ for $ii,jj=11,22$ or
$33$ contribute to the EWP observables at the 1-loop level. We have identified
two different types of contributions in this context (1) the operator mixing
with $\psi^{2}\phi^{2}D$ operators through gauge interactions affect the $W$
and $Z$ boson vertices at the EW scale. Then, through the top-Yukawa
interactions, there are additional contributions from the operators involving
third generation in the quark current, _i.e._ , $\big{[}{\cal C}_{\ell
q}^{(1)}\big{]}_{ii33}$, $\big{[}{\cal C}_{\ell q}^{(3)}\big{]}_{ii33}$,
$\big{[}{\cal C}_{\ell u}\big{]}_{ii33}$, $\big{[}{\cal C}_{eu}\big{]}_{ii33}$
and $\big{[}{\cal C}_{qe}\big{]}_{33ii}$. (2) The $W$ and $Z$ boson vertices
also receive corrections from $\big{[}{\cal C}_{\ell q}^{(3)}\big{]}_{1133}$
and $\big{[}{\cal C}_{\ell q}^{(3)}\big{]}_{2233}$ operators via shifts in the
$\delta g_{Z}$ and $\delta\sin^{2}\theta_{W}$. We present the relative impact
of the running due to the gauge and top-Yukawas on the $W/Z$ couplings at the
EW scale due to semileptonic operators at $\Lambda$. In addition, the quark
flavour conserving operators such as, $\big{[}{\cal C}_{\ell
q}^{(1)}\big{]}_{ijkl}$, $\big{[}{\cal C}_{ed}\big{]}_{ijkl}$, $\big{[}{\cal
C}_{\ell d}\big{]}_{ijkl}$, and $\big{[}{\cal C}_{qe}\big{]}_{ijkl}$ with
$ijkl=1122,1133,2222,2233$ can be constrained by $b\to s\ell^{+}\ell^{-}$
processes through the back-rotation effect. We have shown that the Wilson
coefficients $\big{[}{\cal C}_{\ell q}^{(1)}\big{]}_{ijkl}$ with $ijkl=1122$,
$3322$, $3311$, or $1111$ can contribute to various charged current processes
via operator mixing with $\big{[}{\cal C}_{\ell q}^{(3)}\big{]}_{ijkl}$. This
effects goes through the EW corrections.

The semileptonic operators which violate the lepton flavour, while conserving
the quark flavour can contribute to the LFV decays of $\tau$ and $\mu$
leptons. Some of the semileptonic operators contribute at tree-level to
$\tau\to P\ell$ and $\tau\to\rho\ell$ processes. On the other hand, purely
leptonic LFV decays such as $\tau\to 3\mu$ and $\mu\to 3e$ etc., are generated
only at the 1-loop level. Again this happens through operator mixing which
depends on the gauge as well as Yukawa interactions. In this regard, we find
that depending upon the flavour structures the Yukawas play an important in
constraining such operators. We have also studied the relative impact of RG
running due to various sources on the contributing WET operators at the low
scale. These results are presented in Tab. 13 for purely leptonic WET
operators and in Tabs. 14 for semileptonic WET operators.

Finally, using the latest measurements, we derived lower bounds on the cut-off
scale $\Lambda$ for each semileptonic operator under consideration. We observe
that depending upon the flavour structure, the RG induced constraints can lead
to sensitivities to very high NP scales varying between $\mathcal{O}({\rm
1TeV})$ to $\mathcal{O}(100{\rm TeV})$. Finally, for a fixed value of
$\Lambda=3$ TeV, we also provide allowed ranges for the semileptonic Wilson
coefficients.

## Note Added

While finalizing our paper we notice Ref. [87] on arXiv which presented a
study of the LFV decays through $Z$-flavour violation in the context of future
colliders. The discussion on the LFV decays partly overlaps with the present
work. However, the aim of the current study is not limited to the LFV modes
but is to analyze both flavour conserving as well as the flavour violating
processes to which semileptonic operators can contribute at tree-level or
through operator mixing.

## Acknowledgments

J. K. thanks Andrzej Buras and Jason Aebischer for carefully reading the
manuscript and for the useful comments. J.K. is financially supported by the
Alexander von Humboldt Foundation’s postdoctoral research fellowship.

##  Appendix A Tree-level shifts in the dim-4 gauge boson couplings

For completeness, in this section we collect formulae for the tree-level
shifts due to SMEFT in the dim-4 $Z$ and $W$ boson couplings at the EW scale.
For this purpose we closely follow Ref. [49]. The shifts in the $Z$ boson
couplings $\delta(g_{X}^{Y})_{ij}^{\rm dir}$ (see Eq. (31) for definition) get
tree-level contributions due to $\psi^{2}\phi^{2}D$-type operators in the
SMEFT. At the EW scale for the quarks these are given by

$\displaystyle\delta(g_{L}^{u})_{ij}^{\rm dir}$ $\displaystyle=$
$\displaystyle-\frac{v^{2}}{2}V_{im}\left(\big{[}{\cal C}_{\phi
q}^{(1)}\big{]}_{mn}-\big{[}{\cal C}_{\phi
q}^{(3)}\big{]}_{mn}\right)V^{\dagger}_{nj}\,,$ (78)
$\displaystyle\quad\delta(g_{R}^{u})_{ij}^{\rm dir}$ $\displaystyle=$
$\displaystyle-\frac{v^{2}}{2}\big{[}{\cal C}_{\phi u}\big{]}_{ij}\,,$ (79)
$\displaystyle\delta(g_{L}^{d})_{ij}^{\rm dir}$ $\displaystyle=$
$\displaystyle-\frac{v^{2}}{2}\left(\big{[}{\cal C}_{\phi
q}^{(1)}\big{]}_{ij}+\big{[}{\cal C}_{\phi q}^{(3)}\big{]}_{ij}\right)\,,$
(80) $\displaystyle\quad\delta(g_{R}^{d})_{ij}^{\rm dir}$ $\displaystyle=$
$\displaystyle-\frac{v^{2}}{2}\big{[}{\cal C}_{\phi d}\big{]}_{ij}.$ (81)

Similarly, for the leptons we have

$\displaystyle\delta(g_{L}^{\nu})_{ij}^{\rm dir}$ $\displaystyle=$
$\displaystyle-\frac{v^{2}}{2}\left(\big{[}{\cal
C}_{\phi\ell}^{(1)}\big{]}_{ij}-\big{[}{\cal
C}_{\phi\ell}^{(3)}\big{]}_{ij}\right)\,,$ (82)
$\displaystyle\delta(g_{L}^{e})_{ij}^{\rm dir}$ $\displaystyle=$
$\displaystyle-\frac{v^{2}}{2}\left(\big{[}{\cal
C}_{\phi\ell}^{(1)}\big{]}_{ij}+\big{[}{\cal
C}_{\phi\ell}^{(3)}\big{]}_{ij}\right)\,,$ (83)
$\displaystyle\quad\delta(g_{R}^{e})_{ij}^{\rm dir}$ $\displaystyle=$
$\displaystyle-\frac{v^{2}}{2}\big{[}{\cal C}_{\phi e}\big{]}_{ij}.$ (84)

The CKM elements appearing in the expressions for $\delta(g_{L}^{u})_{ij}^{\rm
dir}$ are needed due to our Warsaw-down basis choice for the SMEFT operators.
In addition, the shifts in the parameters $g_{Z}$ and $\sin\theta_{W}$ are
given by

$\delta g_{Z}=-\frac{v^{2}}{2}\left(\big{[}{\cal
C}_{\phi\ell}^{(3)}\big{]}_{11}+\big{[}{\cal
C}_{\phi\ell}^{(3)}\big{]}_{22}-\frac{\big{[}{\cal
C}_{\ell\ell}\big{]}_{1221}}{2}\right)\,,$ (85)

and

$\delta\sin^{2}{\theta_{W}}=\frac{v^{2}\sin^{2}{2\theta_{W}}}{4\cos{2\theta_{W}}}\left(\big{[}{\cal
C}_{\phi\ell}^{(3)}\big{]}_{11}+\big{[}{\cal
C}_{\phi\ell}^{(3)}\big{]}_{22}-\frac{\big{[}{\cal
C}_{\ell\ell}\big{]}_{1221}}{2}\right).$ (86)

The $W$ boson couplings can be analogously parameterized as

$\displaystyle\delta(\varepsilon_{L}^{\ell})_{ij}^{\rm dir}$ $\displaystyle=$
$\displaystyle v^{2}\big{[}{\cal C}_{\phi\ell}^{(3)}\big{]}_{ij}\,,$ (87)
$\displaystyle\delta(\varepsilon_{L}^{q})_{ij}^{\rm dir}$ $\displaystyle=$
$\displaystyle v^{2}V_{im}\big{[}{\cal C}_{\phi q}^{(3)}\big{]}_{mj}.$ (88)

Since, we are interested to study the effects of only the semileptonic
operators on the EWP observables, we have ignored additional corrections due
to $\big{[}{\cal C}_{\phi D}\big{]}$ and $\big{[}{\cal C}_{\phi WB}\big{]}$
operators[49]. In this regard, we have checked that these operators can not be
generated from semileptonic operators via operator mixing.

##  Appendix B RG induced shifts in the dim-4 and dim-6 operators

Depending upon the scale and the interactions involved, there are three types
of RGEs, _i.e._ , due to the gauge and Yukawa interactions in the SMEFT and
due to QCD+QED interactions in the WET. In this section, we analyze the
relative impact of three different types of RG runnings on the dim-4 and dim-6
operators which contribute to the EWP observables and the LFV processes at the
low energy.

###  B.1 Dim-4 $Z$ boson couplings

In Tab. 3-10, we show the $Z$ and $W$ boson gauge couplings at the EW scale
for a given semileptonic operator introduced at the high scale.

${\rm Coupling}(m_{Z})$ Gauge Couplings Yukawa Couplings $\big{[}{\cal
C}_{\ell q}^{(1)}\big{]}_{ii33}(\Lambda)=1$ $\delta(g_{L}^{u})_{11}$ - 2.28
$\delta(g_{L}^{u})_{22}$ - 2.28 $\delta(g_{L}^{d})_{11}$ - -1.68
$\delta(g_{L}^{d})_{22}$ - -1.68 $\delta(g_{L}^{d})_{33}$ -4.02 -5.67
$\delta(g_{L}^{e})_{ii}$ 3.58 220.63 $\delta(g_{L}^{e})_{jj}$ - -2.87
$\delta(g_{R}^{u})_{11}$ - 1.19 $\delta(g_{R}^{u})_{22}$ - 1.19
$\delta(g_{R}^{e})_{kk}$ - -1.79

Table 3: The shifts in the $Z$ boson couplings with fermions _i.e.,_
$\delta(g_{X}^{Y})_{ii}\times 10^{5}$ at $\mu\simeq 91$ GeV due to
semileptonic Wilson coefficient $\big{[}{\cal C}_{\ell
q}^{(1)}\big{]}_{ii33}(\Lambda)=1{\rm TeV^{-2}}$ at $\Lambda=1{\rm TeV}$. Here
$ii=11$ or $22$, $jj=11,22,33\neq ii$ and $kk=11,22,33$. In the second and
third column, the RG running due to gauge interactions only and gauge + Yukawa
interactions is shown, respectively. Note, only the non-zero entries are
shown.

${\rm Coupling}(m_{Z})$ Gauge Couplings Yukawa Couplings $\big{[}{\cal
C}_{\ell q}^{(1)}\big{]}_{3333}(\Lambda)=1$ $\delta(g_{L}^{d})_{33}$ -4.04
-4.27 $\delta(g_{L}^{e})_{33}$ 3.54 222.69

Table 4: Same as in Tab. 3 except for $\big{[}{\cal C}_{\ell
q}^{(1)}\big{]}_{3333}(\Lambda)$.

${\rm Coupling}(m_{Z})$ Gauge Couplings Yukawa Couplings $\big{[}{\cal
C}_{\ell q}^{(3)}\big{]}_{ii33}(\Lambda)=1$ $\delta(g_{L}^{u})_{11}$ - -181.11
$\delta(g_{L}^{u})_{22}$ - -181.11 $\delta(g_{L}^{d})_{11}$ - 147.73
$\delta(g_{L}^{d})_{22}$ - 147.73 $\delta(g_{L}^{d})_{33}$ 12.86 160.61
$\delta(g_{L}^{e})_{ii}$ 38.66 17.72 $\delta(g_{L}^{e})_{jj}$ - 214.52
$\delta(g_{R}^{u})_{11}$ - -66.79 $\delta(g_{R}^{u})_{22}$ - -66.79
$\delta(g_{R}^{d})_{kk}$ - 33.40 $\delta(g_{R}^{e})_{kk}$ - 100.19

Table 5: Same as in Tab. 3 except for $\big{[}{\cal C}_{\ell
q}^{(3)}\big{]}_{ii33}(\Lambda)$ with $ii=11$ or $22$.

${\rm Coupling}(m_{Z})$ Gauge Couplings Yukawa Couplings $\big{[}{\cal
C}_{\ell q}^{(3)}\big{]}_{3333}(\Lambda)=1$ $\delta(g_{L}^{d})_{33}$ 12.93
12.61 $\delta(g_{L}^{e})_{33}$ 38.76 -197.61

Table 6: Same as in Tab. 3 except for $\big{[}{\cal C}_{\ell
q}^{(3)}\big{]}_{3333}(\Lambda)$.

${\rm Coupling}(m_{Z})$ Gauge Couplings Yukawa Couplings $\big{[}{\cal
C}_{\ell u}\big{]}_{ii33}(\Lambda)=1$ $\delta(g_{L}^{e})_{ii}$ 7.85 -208.36

Table 7: Same as in Tab. 3 except for $\big{[}{\cal C}_{\ell
u}\big{]}_{ii33}(\Lambda)$ with $ii=11,22$ or $33$.

${\rm Coupling}(m_{Z})$ Gauge Couplings Yukawa Couplings $\big{[}{\cal
C}_{\ell d}\big{]}_{ii33}(\Lambda)=1$ $\delta(g_{L}^{e})_{ii}$ -3.95 -3.89
$\delta(g_{R}^{d})_{33}$ -3.92 -3.92

Table 8: Same as in Tab. 3 except for $\big{[}{\cal C}_{\ell
d}\big{]}_{ii33}(\Lambda)$ with $ii=11,22$ or $33$.

${\rm Coupling}(m_{Z})$ Gauge Couplings Yukawa Couplings $\big{[}{\cal
C}_{eu}\big{]}_{ii33}(\Lambda)=1$ $\delta(g_{R}^{e})_{ii}$ 7.81 -211.10

Table 9: Same as in Tab. 3 except for $\big{[}{\cal
C}_{eu}\big{]}_{ii33}(\Lambda)$ with $ii=11,22$ or $33$.

${\rm Coupling}(m_{Z})$ Gauge Couplings Yukawa Couplings $\big{[}{\cal
C}_{qe}^{(1)}\big{]}_{33ii}(\Lambda)=1$ $\delta(g_{L}^{d})_{33}$ -3.91 -4.13
$\delta(g_{R}^{e})_{ii}$ 3.87 220.04

Table 10: Same as in Tab. 3 except for $\big{[}{\cal
C}_{qe}\big{]}_{33ii}(\Lambda)$ with $ii=11,22$ or $33$.

###  B.2 Dim-4 $W$ boson couplings

In Tab. 11 and 12 we show impact of RG running due to Yukawas on the $W$
couplings to fermions at the EW scale.

${\rm Coupling}(m_{W})$ Gauge Couplings Yukawa Couplings $\big{[}{\cal
C}_{\ell q}^{(3)}\big{]}_{ii33}(\Lambda)=1$ $\delta(\varepsilon_{L}^{q})_{11}$
- -328.37 $\delta(\varepsilon_{L}^{q})_{22}$ - -328.37
$\delta(\varepsilon_{L}^{\ell})_{ii}$ -77.31 52.11
$\delta(\varepsilon_{L}^{\ell})_{jj}$ - -328.37

Table 11: The shifts in the $W$ boson couplings _i.e.,_
$\delta(\varepsilon_{L}^{Y})_{ii}\times 10^{5}$ to fermions at $\mu\simeq 91$
GeV due to semileptonic operator $\big{[}{\cal C}_{\ell
q}^{(3)}\big{]}_{ii33}(\Lambda)=1{\rm TeV^{-2}}$ at $\Lambda=1{\rm TeV}$. Here
$ii=11$ or $22$, and $jj=11,22,33\neq ii$. In the second and third column the
RG running due to gauge only and gauge + Yukawa interactions is included,
respectively. Note, only the non-zero entries are shown.

${\rm Coupling}(m_{W})$ Gauge Couplings Yukawa Couplings $\big{[}{\cal
C}_{\ell q}^{(3)}\big{]}_{3333}(\Lambda)=1$
$\delta(\varepsilon_{L}^{\ell})_{33}$ -77.46 381.44

Table 12: Same as in Tab. 11 except for $\big{[}{\cal C}_{\ell
q}^{(3)}\big{]}_{3333}(\Lambda)$.

###  B.3 Leptonic dim-6 WET operators

In Tab. 13, we present the numerical values for the leptonic WET Wilson
coefficients by setting the SMEFT Wilson coefficients to 1 ${\rm TeV^{-2}}$ at
$\Lambda=1{\rm TeV}$. In order to study the roles played by different kinds of
RG effects, we present three kind of numbers, (1) with only WET (QED+QCD)
running, (2) WET + SMEFT running due to gauge interactions, and (3) the full
WET+ SMEFT gauge + Yukawa running. As we can see, except for $\big{[}{\cal
C}_{\ell d}\big{]}_{1333}$ and $\big{[}{\cal C}_{ed}\big{]}_{1333}$, the top-
Yukawa effects always play an important role for the operators involving third
generation quarks.

${\rm Coefficent}$ QCD+QED Gauge Couplings Yukawa Couplings
$[\mathcal{C}_{\ell q}^{(1)}]_{1311}(\Lambda)=1$ $[C_{ee}^{V,LL}]_{\ell\ell
13}$ 3.41 5.48 5.48 $[C_{ee}^{V,LR}]_{13\ell\ell}$ 3.41 5.64 5.63
$[\mathcal{C}_{\ell q}^{(1)}]_{1333}(\Lambda)=1$ $[C_{ee}^{V,LL}]_{\ell\ell
13}$ -3.40 -1.49 -35.25 $[C_{ee}^{V,LR}]_{13\ell\ell}$ -3.40 -1.33 36.98
$[\mathcal{C}_{\ell q}^{(3)}]_{1311}(\Lambda)=1$ $[C_{ee}^{V,LL}]_{\ell\ell
13}$ -10.24 -15.91 15.91 $[C_{ee}^{V,LR}]_{13\ell\ell}$ -10.24 -16.66 16.66
$[\mathcal{C}_{\ell q}^{(3)}]_{1333}(\Lambda)=1$ $[C_{ee}^{V,LL}]_{\ell\ell
13}$ -3.42 -8.42 47.68 $[C_{ee}^{V,LR}]_{13\ell\ell}$ -3.42 -9.17 -29.10
$[\mathcal{C}_{\ell u}]_{1311}(\Lambda)=1$ $[C_{ee}^{V,LL}]_{\ell\ell 13}$
6.83 10.37 10.37 $[C_{ee}^{V,LR}]_{13\ell\ell}$ 6.83 10.67 10.67
$[\mathcal{C}_{\ell u}]_{1333}(\Lambda)=1$ $[C_{ee}^{V,LL}]_{\ell\ell 13}$ -
3.60 -39.64 $[C_{ee}^{V,LR}]_{13\ell\ell}$ - 3.90 -39.08 $[\mathcal{C}_{\ell
d}]_{1311}(\Lambda)=1$ $[C_{ee}^{V,LL}]_{\ell\ell 13}$ -3.14 -5.24 5.24
$[C_{ee}^{V,LR}]_{13\ell\ell}$ -3.14 -5.39 5.39 $[\mathcal{C}_{\ell
d}]_{1333}(\Lambda)=1$ $[C_{ee}^{V,LL}]_{\ell\ell 13}$ -3.14 -5.24 5.23
$[C_{ee}^{V,LR}]_{13\ell\ell}$ -3.14 -5.39 5.40
$[\mathcal{C}_{eu}]_{1311}(\Lambda)=1$ $[C_{ee}^{V,LR}]_{\ell\ell 13}$ 6.83
10.88 -10.88 $[C_{ee}^{V,RR}]_{\ell\ell 13}$ 6.83 10.88 -10.88
$[\mathcal{C}_{eu}]_{1333}(\Lambda)=1$ $[C_{ee}^{V,LR}]_{\ell\ell 13}$ - 3.94
-35.65 $[C_{ee}^{V,RR}]_{\ell\ell 13}$ - 3.94 -35.65
$[\mathcal{C}_{ed}]_{1311}(\Lambda)=1$ $[C_{ee}^{V,LR}]_{\ell\ell 13}$ -3.14
-5.33 5.33 $[C_{ee}^{V,RR}]_{\ell\ell 13}$ -3.14 -5.33 5.33
$[\mathcal{C}_{ed}]_{1333}(\Lambda)=1$ $[C_{ee}^{V,LR}]_{\ell\ell 13}$ -3.14
-5.33 -5.32 $[C_{ee}^{V,RR}]_{\ell\ell 13}$ -3.14 -5.33 -5.32
$[\mathcal{C}_{qe}]_{1311}(\Lambda)=1$ $[C_{ee}^{V,LR}]_{\ell\ell 13}$ 3.41
5.35 -5.35 $[C_{ee}^{V,RR}]_{\ell\ell 13}$ 3.41 5.35 -5.35
$[\mathcal{C}_{qe}]_{1333}(\Lambda)=1$ $[C_{ee}^{V,LR}]_{\ell\ell 13}$ -3.4
-1.44 -37.66 $[C_{ee}^{V,RR}]_{\ell\ell 13}$ -3.400 -1.44 -37.66

Table 13: The impact of semileptonic operators on the low energy purely
leptonic $\Delta F=(1,0)$ Wilson coefficients (in $10^{-9}~{}{\rm TeV^{-2}}$
units) at $\mu_{\rm low}=2$GeV is shown. The second column refers to the
values with only WET (QED+QCD) running, the third column refers to the WET +
SMEFT running due to gauge interactions, and the fourth column refers to the
full WET+ SMEFT gauge + Yukawa running. Here, the indices $\ell\ell=11$,$22$,
or $33$ and $\Lambda=1$TeV. The SMEFT Wilson coefficients are set to 1 (in
$\rm TeV^{-2}$ units) at $\Lambda$.

###  B.4 Semileptonic dim-6 WET operators

In Tab. 14, we present the low energy semileptonic WET Wilson coefficients by
setting various semileptonic SMEFT Wilson coefficients, involving the third
generation of quarks, equal to 1 ${\rm TeV^{-2}}$ at $\Lambda=1{\rm TeV}$.
Again, in order to study the roles played by different kinds of RG running, we
have presented three kind of numbers. As found in the case of WET leptonic
operators, except for the Wilson coefficients $\big{[}{\cal C}_{\ell
d}\big{]}_{1333}$ and $\big{[}{\cal C}_{ed}\big{]}_{1333}$, the top-Yukawa
always plays an important role.

${\rm Coefficent}$ QCD+QED Gauge Couplings Yukawa Couplings
$[\mathcal{C}_{\ell q}^{(1)}]_{1333}(\Lambda)=1$ $[C_{ed}^{V,LL}]_{13jj}$
-1.13 -0.47 -59.92 $[C_{ed}^{V,LR}]_{13jj}$ -1.13 -0.46 12.11
$[C_{eu}^{V,LL}]_{1311}$ 2.28 0.92 49.59 $[C_{eu}^{V,LR}]_{1311}$ 2.27 0.97
-23.60 $[\mathcal{C}_{\ell q}^{(3)}]_{1333}(\Lambda)=1$
$[C_{ed}^{V,LL}]_{13jj}$ -1.14 -2.63 67.29 $[C_{ed}^{V,LR}]_{13jj}$ -1.14
-3.02 -9.50 $[C_{eu}^{V,LL}]_{1311}$ 2.27 6.44 -60.48 $[C_{eu}^{V,LR}]_{1311}$
2.28 5.91 18.38 $[\mathcal{C}_{\ell u}]_{1333}(\Lambda)=1$
$[C_{ed}^{V,LL}]_{13jj}$ - 1.24 -58.33 $[C_{ed}^{V,LR}]_{13jj}$ - 1.27 12.77
$[C_{eu}^{V,LL}]_{1311}$ - -2.56 47.26 $[C_{eu}^{V,LR}]_{1311}$ - -2.43 -24.76
$[\mathcal{C}_{\ell d}]_{1333}(\Lambda)=1$ $[C_{ed}^{V,LL}]_{13jj}$ -1.14
-1.77 1.75 $[C_{ed}^{V,LR}]_{13jj}$ -1.14 -1.78 1.78 $[C_{eu}^{V,LL}]_{1311}$
2.28 3.57 -3.56 $[C_{eu}^{V,LR}]_{1311}$ 2.28 3.51 -3.51
$[\mathcal{C}_{eu}]_{1333}(\Lambda)=1$ $[C_{de}^{V,LR}]_{jj13}$ - 1.28 -60.47
$[C_{ed}^{V,RR}]_{13jj}$ - 1.25 12.50 $[C_{ue}^{V,LR}]_{1113}$ - -2.48 46.25
$[C_{eu}^{V,RR}]_{1311}$ - -2.62 -25.80 $[\mathcal{C}_{ed}]_{1333}(\Lambda)=1$
$[C_{de}^{V,LR}]_{jj13}$ -1.14 -1.76 -1.75 $[C_{ed}^{V,RR}]_{13jj}$ -1.14
-1.75 -1.75 $[C_{ue}^{V,LR}]_{1113}$ 2.28 3.48 3.47 $[C_{eu}^{V,RR}]_{1311}$
2.28 3.55 3.56 $[\mathcal{C}_{qe}]_{1333}(\Lambda)=1$ $[C_{de}^{V,LR}]_{jj13}$
-1.13 -0.49 -60.48 $[C_{ed}^{V,RR}]_{13jj}$ -1.13 -0.51 11.62
$[C_{ue}^{V,LR}]_{1113}$ 2.28 1.04 47.08 $[C_{eu}^{V,RR}]_{1311}$ 2.27 0.96
-23.86

Table 14: The impact of semileptonic operators on the low energy semileptonic
$\Delta F=(1,0)$ Wilson coefficients (in $10^{-9}~{}{\rm TeV^{-2}}$ units) at
$\mu_{\rm low}=2$GeV is shown. The second column refers to the values with
only WET (QED+QCD) running, the third column refers to WET + SMEFT running due
to gauge interactions, and the fourth column refers to the full WET+ SMEFT
gauge + Yukawa running. Here the indices $jj=11$ or $22$ and $\Lambda=1$TeV.
The SMEFT Wilson coefficients are set to 1 (in $\rm TeV^{-2}$ units) at the
high scale.

##  Appendix C Bounds on the Wilson coefficients

In this section, we report the allowed ranges for the semileptonic Wilson
coefficients $[c_{X}]_{ijkl}$ which are defined in (77). Considering various
1-loop induced constraints, as discussed in Sec. 4, the allowed values of the
dimensionless parameters $[c_{X}]_{ijkl}$ are obtained by performing the fits.
The results are presented in Tables 15-21. The uncertainties are also
indicated at $1\sigma$ level. In all cases, the cut-off scale $\Lambda$ is set
to 3 TeV.

$\bar{\ell}_{i}\gamma^{\mu}\ell_{j}\bar{q}_{k}\gamma_{\mu}q_{l}$
---
$\Delta F=(0,0)$ | $\Delta F=(1,0)$
1111 | $(-1.37\pm 1.15).10^{0}$ | 1211 | ($1.64\pm 0.62).10^{-2}$
1122 | $(6.94\pm 1.68).10^{0}$ | 1222 | ($1.63\pm 0.62).10^{-2}$
1133 | $(-7.42\pm 3.71).10^{-1}$ | 1233 | ($2.42\pm 0.91).10^{-3}$
2211 | $(8.77\pm 6.07).10^{1}$ | 1311 | $(3.99\pm 1.03).10^{0}$
2222 | $(-6.32\pm 1.46).10^{0}$ | 1322 | $(7.48\pm 2.74).10^{-2}$
2233 | ($2.85\pm 0.62).10^{0}$ | 1333 | $(8.64\pm 1.73).10^{-1}$
3311 | $(-2.68\pm 1.82).10^{1}$ | 2311 | ($3.16\pm 0.91).10^{0}$
3322 | $(-4.91\pm 3.36).10^{1}$ | 2322 | ($1.19\pm 0.46).10^{-1}$
3333 | $(-0.19\pm 1.36).10^{0}$ | 2333 | $(6.93\pm 1.64).10^{-1}$
Table 15: Allowed values of the Wilson coefficients
$\big{[}{c}_{lq}^{(1)}\big{]}_{ijkl}$ based on the constraints discussed in
the main text.
$\bar{\ell}_{i}\gamma^{\mu}\tau^{a}\ell_{j}\bar{q}_{k}\gamma_{\mu}\tau^{a}q_{l}$
---
$\Delta F=(0,0)$ | $\Delta F=(1,0)$
1111 | $(1.84\pm 3.44).10^{-2}$ | 1211 | $(5.75\pm 2.15).10^{-3}$
1122 | $(2.08\pm 1.85).10^{-1}$ | 1222 | $(5.75\pm 2.14).10^{-3}$
1133 | ($-1.49\pm 0.39).10^{0}$ | 1233 | ($2.07\pm 0.77).10^{-3}$
2211 | $(7.41\pm 5.86).10^{0}$ | 1311 | $(3.99\pm 1.31).10^{-2}$
2222 | $(-2.17\pm 1.76).10^{-1}$ | 1322 | $(7.56\pm 2.74).10^{-2}$
2233 | $(4.01\pm 2.53).10^{-1}$ | 1333 | $(6.86\pm 1.43).10^{-1}$
3311 | $(8.73\pm 5.04).10^{-1}$ | 2311 | $(3.13\pm 1.09).10^{-2}$
3322 | $(1.71\pm 1.01).10^{0}$ | 2322 | ($1.28\pm 0.44).10^{-1}$
3333 | $(-0.10\pm 1.07).10^{0}$ | 2333 | $(5.40\pm 1.30).10^{-1}$
Table 16: Same as in Tab. 15, except for Wilson coefficient
$\big{[}{c}_{lq}^{(3)}\big{]}_{ijkl}$.
$\bar{\ell}_{i}\gamma^{\mu}\ell_{j}\bar{d}_{k}\gamma_{\mu}d_{l}$
---
$\Delta F=(0,0)$ | $\Delta F=(1,0)$
1111 | $(-1.49\pm 2.26).10^{1}$ | 1211 | ($1.76\pm 0.67).10^{-2}$
1122 | $(2.47\pm 1.59).10^{1}$ | 1222 | ($1.76\pm 0.67).10^{-2}$
1133 | $(-4.09\pm 1.63).10^{1}$ | 1233 | ($2.24\pm 0.84).10^{-2}$
2211 | $(-8.99\pm 5.58).10^{1}$ | 1311 | $(7.80\pm 2.73).10^{-2}$
2222 | $(-7.04\pm 2.16).10^{1}$ | 1322 | $(6.94\pm 2.59).10^{-2}$
2233 | $(4.34\pm 2.12).10^{1}$ | 1333 | $(7.77\pm 1.94).10^{0}$
3311 | $(-0.22\pm 6.83).10^{1}$ | 2311 | $(6.36\pm 2.25).10^{-2}$
3322 | $(-0.24\pm 6.83).10^{1}$ | 2322 | ($1.18\pm 0.43).10^{-1}$
3333 | $(-1.54\pm 6.89).10^{1}$ | 2333 | $(6.29\pm 1.65).10^{0}$
Table 17: Same as in Tab. 15, except for Wilson coefficient
$\big{[}{c}_{ld}\big{]}_{ijkl}$.
$\bar{\ell}_{i}\gamma^{\mu}\ell_{j}\bar{u}_{k}\gamma_{\mu}u_{l}$
---
$\Delta F=(0,0)$ | $\Delta F=(1,0)$
1111 | $(0.60\pm 1.15).10^{1}$ | 1211 | $(9.01\pm 3.38).10^{-3}$
1122 | $(0.60\pm 1.15).10^{1}$ | 1222 | $(9.01\pm 3.38).10^{-3}$
1133 | $(4.23\pm 4.46).10^{-1}$ | 1233 | ($2.53\pm 0.95).10^{-3}$
2211 | $(4.02\pm 2.80).10^{1}$ | 1311 | $(8.28\pm 2.89).10^{-2}$
2222 | $(4.05\pm 2.80).10^{1}$ | 1322 | ($3.42\pm 0.84).10^{0}$
2233 | ($-3.70\pm 0.93).10^{0}$ | 1333 | $(9.54\pm 1.88).10^{-1}$
3311 | $(-0.52\pm 3.44).10^{1}$ | 2311 | $(6.45\pm 2.50).10^{-2}$
3322 | $(-0.48\pm 3.44).10^{1}$ | 2322 | ($2.72\pm 0.74).10^{0}$
3333 | $(0.39\pm 1.44).10^{0}$ | 2333 | $(7.69\pm 1.79).10^{-1}$
Table 18: Same as in Tab. 15, except for Wilson coefficient
$\big{[}{c}_{lu}\big{]}_{ijkl}$.
$\bar{e}_{i}\gamma^{\mu}e_{j}\bar{d}_{k}\gamma_{\mu}d_{l}$
---
$\Delta F=(0,0)$ | $\Delta F=(1,0)$
1111 | $(1.75\pm 2.83).10^{1}$ | 1211 | ($1.80\pm 0.68).10^{-2}$
1122 | $(1.55\pm 3.00).10^{1}$ | 1222 | ($1.80\pm 0.68).10^{-2}$
1133 | $(2.15\pm 3.03).10^{1}$ | 1233 | ($2.27\pm 0.86).10^{-2}$
2211 | $(9.50\pm 7.27).10^{1}$ | 1311 | $(7.89\pm 2.90).10^{-2}$
2222 | $(6.65\pm 5.43).10^{1}$ | 1322 | $(7.18\pm 2.77).10^{-2}$
2233 | $(3.04\pm 5.43).10^{1}$ | 1333 | $(7.69\pm 2.05).10^{0}$
3311 | $(-7.88\pm 8.08).10^{1}$ | 2311 | $(6.57\pm 2.35).10^{-2}$
3322 | $(-7.92\pm 8.08).10^{1}$ | 2322 | ($1.19\pm 0.46).10^{-1}$
3333 | $(-9.74\pm 8.16).10^{1}$ | 2333 | $(6.15\pm 1.56).10^{0}$
Table 19: Same as in Tab. 15, except for Wilson coefficient
$\big{[}{c}_{ed}\big{]}_{ijkl}$.
$\bar{e}_{i}\gamma^{\mu}e_{j}\bar{u}_{k}\gamma_{\mu}u_{l}$
---
$\Delta F=(0,0)$ | $\Delta F=(1,0)$
1111 | $(-0.78\pm 1.35).10^{1}$ | 1211 | $(8.71\pm 3.31).10^{-3}$
1122 | $(-0.78\pm 1.35).10^{1}$ | 1222 | $(8.71\pm 3.31).10^{-3}$
1133 | $(2.63\pm 5.32).10^{-1}$ | 1233 | ($2.42\pm 0.91).10^{-3}$
2211 | $(-4.43\pm 3.51).10^{1}$ | 1311 | $(7.73\pm 2.61).10^{-2}$
2222 | $(-4.39\pm 3.50).10^{1}$ | 1322 | ($3.30\pm 0.84).10^{0}$
2233 | $(2.06\pm 1.30).10^{0}$ | 1333 | $(9.41\pm 1.84).10^{-1}$
3311 | $(4.26\pm 4.02).10^{1}$ | 2311 | $(6.39\pm 2.18).10^{-2}$
3322 | $(4.31\pm 4.01).10^{1}$ | 2322 | ($2.65\pm 0.66).10^{0}$
3333 | $(-1.44\pm 1.57).10^{0}$ | 2333 | $(7.76\pm 1.67).10^{-1}$
Table 20: Same as in Tab. 15, except for Wilson coefficient
$\big{[}{c}_{eu}\big{]}_{ijkl}$.
$\bar{q}_{i}\gamma^{\mu}q_{j}\bar{e}_{k}\gamma_{\mu}e_{l}$
---
$\Delta F=(0,0)$ | $\Delta F=(1,0)$
1111 | $(-1.98\pm 2.76).10^{1}$ | 1112 | ($1.82\pm 0.68).10^{-2}$
2211 | ($-2.93\pm 0.44).10^{1}$ | 2212 | ($1.83\pm 0.68).10^{-2}$
3311 | $(-2.86\pm 5.26).10^{-1}$ | 3312 | ($2.53\pm 0.94).10^{-3}$
1122 | $(-8.69\pm 7.11).10^{1}$ | 1113 | $(8.26\pm 2.96).10^{-1}$
2222 | $(5.22\pm 4.87).10^{0}$ | 2213 | $(6.90\pm 2.68).10^{-2}$
3322 | $(-2.04\pm 1.16).10^{0}$ | 3313 | $(8.85\pm 1.78).10^{-1}$
1133 | $(8.87\pm 8.13).10^{1}$ | 1123 | $(6.82\pm 2.47).10^{-1}$
2233 | $(7.40\pm 7.39).10^{1}$ | 2223 | ($1.16\pm 0.44).10^{-1}$
3333 | $(1.32\pm 1.52).10^{0}$ | 3323 | $(7.30\pm 1.67).10^{-1}$
Table 21: Same as in Tab. 15, except for Wilson coefficient
$\big{[}{c}_{qe}\big{]}_{ijkl}$.

## References

  * [1] B. Grzadkowski, M. Iskrzynski, M. Misiak, and J. Rosiek, Dimension-Six Terms in the Standard Model Lagrangian, JHEP 1010 (2010) 085, [arXiv:1008.4884].
  * [2] W. Buchmuller and D. Wyler, Effective Lagrangian Analysis of New Interactions and Flavor Conservation, Nucl. Phys. B268 (1986) 621–653.
  * [3] A. Buras, Gauge Theory of Weak Decays.  Cambridge University Press, 6, 2020.
  * [4] B. Bhattacharya, A. Datta, D. London, and S. Shivashankara, Simultaneous Explanation of the $R_{K}$ and $R(D^{(*)})$ Puzzles, Phys. Lett. B 742 (2015) 370–374, [arXiv:1412.7164].
  * [5] B. Capdevila, A. Crivellin, S. Descotes-Genon, L. Hofer, and J. Matias, Searching for New Physics with $b\to s\tau^{+}\tau^{-}$ processes, Phys. Rev. Lett. 120 (2018), no. 18 181802, [arXiv:1712.01919].
  * [6] A. J. Buras, J. Girrbach-Noe, C. Niehoff, and D. M. Straub, $B\to K^{(*)}\nu\bar{\nu}$ decays in the Standard Model and beyond, JHEP 1502 (2015) 184, [arXiv:1409.4557].
  * [7] F. Feruglio, P. Paradisi, and A. Pattori, On the Importance of Electroweak Corrections for B Anomalies, JHEP 09 (2017) 061, [arXiv:1705.00929].
  * [8] C. Cornella, F. Feruglio, and P. Paradisi, Low-energy Effects of Lepton Flavour Universality Violation, arXiv:1803.00945.
  * [9] J. Aebischer, A. J. Buras, and J. Kumar, Another SMEFT story: $Z^{\prime}$ facing new results on $\epsilon^{\prime}/\epsilon$, $\Delta M_{K}$ and $K\to\pi\nu\overline{\nu}$, JHEP 12 (2020) 097, [arXiv:2006.01138].
  * [10] J. Aebischer and J. Kumar, Flavour violating effects of Yukawa running in SMEFT, JHEP 09 (2020) 187, [arXiv:2005.12283].
  * [11] J. Aebischer, C. Bobeth, A. J. Buras, and J. Kumar, SMEFT ATLAS of $\Delta$F = 2 transitions, JHEP 12 (2020) 187, [arXiv:2009.07276].
  * [12] S. Bruggisser, R. Schäfer, D. van Dyk, and S. Westhoff, The Flavor of UV Physics, JHEP 05 (2021) 257, [arXiv:2101.07273].
  * [13] R. Alonso, E. E. Jenkins, A. V. Manohar, and M. Trott, Renormalization Group Evolution of the Standard Model Dimension Six Operators III: Gauge Coupling Dependence and Phenomenology, JHEP 04 (2014) 159, [arXiv:1312.2014].
  * [14] E. E. Jenkins, A. V. Manohar, and M. Trott, Renormalization Group Evolution of the Standard Model Dimension Six Operators II: Yukawa Dependence, JHEP 01 (2014) 035, [arXiv:1310.4838].
  * [15] E. E. Jenkins, A. V. Manohar, and M. Trott, Renormalization Group Evolution of the Standard Model Dimension Six Operators I: Formalism and lambda Dependence, JHEP 10 (2013) 087, [arXiv:1308.2627].
  * [16] E. E. Jenkins, A. V. Manohar, and P. Stoffer, Low-Energy Effective Field Theory below the Electroweak Scale: Anomalous Dimensions, JHEP 01 (2018) 084, [arXiv:1711.05270].
  * [17] J. Aebischer, M. Fael, C. Greub, and J. Virto, B physics Beyond the Standard Model at One Loop: Complete Renormalization Group Evolution below the Electroweak Scale, JHEP 09 (2017) 158, [arXiv:1704.06639].
  * [18] M. Chala and A. Titov, One-loop running of dimension-six Higgs-neutrino operators and implications of a large neutrino dipole moment, JHEP 09 (2020) 188, [arXiv:2006.14596].
  * [19] A. Datta, J. Kumar, H. Liu, and D. Marfatia, Anomalous dimensions from gauge couplings in SMEFT with right-handed neutrinos, JHEP 02 (2021) 015, [arXiv:2010.12109].
  * [20] A. Datta, J. Kumar, H. Liu, and D. Marfatia, Anomalous dimensions from Yukawa couplings in SMNEFT: four-fermion operators, JHEP 05 (2021) 037, [arXiv:2103.04441].
  * [21] J. Aebischer, J. Kumar, and D. M. Straub, Wilson: a Python package for the running and matching of Wilson coefficients above and below the electroweak scale, Eur. Phys. J. C 78 (2018), no. 12 1026, [arXiv:1804.05033].
  * [22] A. Celis, J. Fuentes-Martin, A. Vicente, and J. Virto, DsixTools: The Standard Model Effective Field Theory Toolkit, Eur. Phys. J. C77 (2017), no. 6 405, [arXiv:1704.04504].
  * [23] J. Fuentes-Martin, P. Ruiz-Femenia, A. Vicente, and J. Virto, DsixTools 2.0: The Effective Field Theory Toolkit, Eur. Phys. J. C 81 (2021), no. 2 167, [arXiv:2010.16341].
  * [24] A. J. Buras and M. Jung, Analytic inclusion of the scale dependence of the anomalous dimension matrix in Standard Model Effective Theory, JHEP 06 (2018) 067, [arXiv:1804.05852].
  * [25] D. M. Straub, flavio: a Python package for flavour and precision phenomenology in the Standard Model and beyond, arXiv:1810.08132.
  * [26] I. Brivio, Y. Jiang, and M. Trott, The SMEFTsim package, theory and tools, JHEP 12 (2017) 070, [arXiv:1709.06492].
  * [27] A. Dedes, M. Paraskevas, J. Rosiek, K. Suxho, and L. Trifyllis, SmeftFR – Feynman rules generator for the Standard Model Effective Field Theory, Comput. Phys. Commun. 247 (2020) 106931, [arXiv:1904.03204].
  * [28] J. Aebischer, M. Fael, A. Lenz, M. Spannowsky, and J. Virto, eds., Computing Tools for the SMEFT, 10, 2019.
  * [29] M. Endo, S. Mishima, and D. Ueda, Revisiting electroweak radiative corrections to $b\to s\ell\ell$ in SMEFT, JHEP 05 (2021) 050, [arXiv:2012.06197].
  * [30] L. Alasfar, A. Azatov, J. de Blas, A. Paul, and M. Valli, $B$ anomalies under the lens of electroweak precision, JHEP 12 (2020) 016, [arXiv:2007.04400].
  * [31] A. Celis, J. Fuentes-Martin, A. Vicente, and J. Virto, Gauge-invariant implications of the LHCb measurements on lepton-flavor nonuniversality, Phys. Rev. D 96 (2017), no. 3 035026, [arXiv:1704.05672].
  * [32] J. Kumar, D. London, and R. Watanabe, Combined Explanations of the $b\to s\mu^{+}\mu^{-}$ and $b\to c\tau^{-}{\bar{\nu}}$ Anomalies: a General Model Analysis, Phys. Rev. D 99 (2019), no. 1 015007, [arXiv:1806.07403].
  * [33] C. Bobeth and A. J. Buras, Leptoquarks meet $\varepsilon^{\prime}/\varepsilon$ and rare Kaon processes, JHEP 02 (2018) 101, [arXiv:1712.01295].
  * [34] D. Buttazzo, A. Greljo, G. Isidori, and D. Marzocca, B-physics anomalies: a guide to combined explanations, arXiv:1706.07808.
  * [35] L. Calibbi, A. Crivellin, and T. Ota, Effective Field Theory Approach to $b\to s\ell\ell^{(^{\prime})}$, $B\to K^{(*)}\nu\overline{\nu}$ and $B\to D^{(*)}\tau\nu$ with Third Generation Couplings, Phys. Rev. Lett. 115 (2015) 181801, [arXiv:1506.02661].
  * [36] R. Alonso, B. Grinstein, and J. Martin Camalich, Lepton universality violation and lepton flavor conservation in $B$-meson decays, JHEP 10 (2015) 184, [arXiv:1505.05164].
  * [37] S. Dawson and P. P. Giardino, Electroweak and QCD corrections to $Z$ and $W$ pole observables in the standard model EFT, Phys. Rev. D 101 (2020), no. 1 013001, [arXiv:1909.02000].
  * [38] F. Feruglio, P. Paradisi, and A. Pattori, Revisiting Lepton Flavor Universality in B Decays, Phys. Rev. Lett. 118 (2017), no. 1 011801, [arXiv:1606.00524].
  * [39] J. Aebischer, J. Kumar, P. Stangl, and D. M. Straub, A Global Likelihood for Precision Constraints and Flavour Anomalies, Eur. Phys. J. C 79 (2019), no. 6 509, [arXiv:1810.07698].
  * [40] J. Aebischer et al., WCxf: an exchange format for Wilson coefficients beyond the Standard Model, Comput. Phys. Commun. 232 (2018) 71–83, [arXiv:1712.05298].
  * [41] S. Descotes-Genon, A. Falkowski, M. Fedele, M. González-Alonso, and J. Virto, The CKM parameters in the SMEFT, JHEP 05 (2019) 172, [arXiv:1812.08163].
  * [42] A. Falkowski and F. Riva, Model-independent precision constraints on dimension-6 operators, JHEP 02 (2015) 039, [arXiv:1411.0669].
  * [43] J. Ellis, V. Sanz, and T. You, The Effective Standard Model after LHC Run I, JHEP 03 (2015) 157, [arXiv:1410.7703].
  * [44] A. Efrati, A. Falkowski, and Y. Soreq, Electroweak constraints on flavorful effective theories, JHEP 07 (2015) 018, [arXiv:1503.07872].
  * [45] L. Berthier and M. Trott, Towards consistent Electroweak Precision Data constraints in the SMEFT, JHEP 05 (2015) 024, [arXiv:1502.02570].
  * [46] J. Ellis, C. W. Murphy, V. Sanz, and T. You, Updated Global SMEFT Fit to Higgs, Diboson and Electroweak Data, JHEP 06 (2018) 146, [arXiv:1803.03252].
  * [47] V. Bresó-Pla, A. Falkowski, and M. González-Alonso, $A_{FB}$ in the SMEFT: precision Z physics at the LHC, arXiv:2103.12074.
  * [48] A. Falkowski and D. Straub, Flavourful SMEFT likelihood for Higgs and electroweak data, JHEP 04 (2020) 066, [arXiv:1911.07866].
  * [49] I. Brivio and M. Trott, The Standard Model as an Effective Field Theory, Phys. Rept. 793 (2019) 1–98, [arXiv:1706.08945].
  * [50] S. Dawson and P. P. Giardino, Higgs decays to $ZZ$ and $Z\gamma$ in the standard model effective field theory: An NLO analysis, Phys. Rev. D 97 (2018), no. 9 093003, [arXiv:1801.01136].
  * [51] Anisha, S. Das Bakshi, J. Chakrabortty, and S. K. Patra, Connecting electroweak-scale observables to BSM physics through EFT and Bayesian statistics, Phys. Rev. D 103 (2021), no. 7 076007, [arXiv:2010.04088].
  * [52] A. Crivellin, S. Davidson, G. M. Pruna, and A. Signer, Renormalisation-group improved analysis of $\mu\to e$ processes in a systematic effective-field-theory approach, JHEP 05 (2017) 117, [arXiv:1702.03020].
  * [53] V. Cirigliano, K. Fuyuto, C. Lee, E. Mereghetti, and B. Yan, Charged Lepton Flavor Violation at the EIC, JHEP 03 (2021) 256, [arXiv:2102.06176].
  * [54] D0 Collaboration, B. Abbott et al., A measurement of the $W\to\tau\nu$ production cross section in $p\bar{p}$ collisions at $\sqrt{s}=1.8$ TeV, Phys. Rev. Lett. 84 (2000) 5710–5715, [hep-ex/9912065].
  * [55] LHCb Collaboration, R. Aaij et al., Measurement of forward $W\to e\nu$ production in $pp$ collisions at $\sqrt{s}=8\,$TeV, JHEP 10 (2016) 030, [arXiv:1608.01484].
  * [56] ALEPH, DELPHI, L3, OPAL, LEP Electroweak Collaboration, S. Schael et al., Electroweak Measurements in Electron-Positron Collisions at W-Boson-Pair Energies at LEP, Phys. Rept. 532 (2013) 119–244, [arXiv:1302.3415].
  * [57] CDF, D0 Collaboration, T. A. Aaltonen et al., Combination of CDF and D0 $W$-Boson Mass Measurements, Phys. Rev. D 88 (2013), no. 5 052018, [arXiv:1307.7627].
  * [58] ATLAS Collaboration, M. Aaboud et al., Measurement of the $W$-boson mass in pp collisions at $\sqrt{s}=7$ TeV with the ATLAS detector, Eur. Phys. J. C 78 (2018), no. 2 110, [arXiv:1701.07240]. [Erratum: Eur.Phys.J.C 78, 898 (2018)].
  * [59] SLD Collaboration, K. Abe et al., First direct measurement of the parity violating coupling of the Z0 to the s quark, Phys. Rev. Lett. 85 (2000) 5059–5063, [hep-ex/0006019].
  * [60] ALEPH, DELPHI, L3, OPAL, SLD, LEP Electroweak Working Group, SLD Electroweak Group, SLD Heavy Flavour Group Collaboration, S. Schael et al., Precision electroweak measurements on the $Z$ resonance, Phys. Rept. 427 (2006) 257–454, [hep-ex/0509008].
  * [61] ATLAS Collaboration, G. Aad et al., Test of the universality of $\tau$ and $\mu$ lepton couplings in $W$-boson decays from $t\bar{t}$ events with the ATLAS detector, arXiv:2007.14040.
  * [62] R. Coy, M. Frigerio, F. Mescia, and O. Sumensari, New physics in $b\to s\ell\ell$ transitions at one loop, Eur. Phys. J. C 80 (2020), no. 1 52, [arXiv:1909.08567].
  * [63] J. Aebischer, A. Crivellin, M. Fael, and C. Greub, Matching of gauge invariant dimension-six operators for $b\to s$ and $b\to c$ transitions, JHEP 05 (2016) 037, [arXiv:1512.02830].
  * [64] G. Hiller and F. Kruger, More model-independent analysis of $b\to s$ processes, Phys. Rev. D 69 (2004) 074020, [hep-ph/0310219].
  * [65] LHCb Collaboration, R. Aaij et al., Test of lepton universality using $B^{+}\rightarrow K^{+}\ell^{+}\ell^{-}$ decays, Phys. Rev. Lett. 113 (2014) 151601, [arXiv:1406.6482].
  * [66] LHCb Collaboration, R. Aaij et al., Search for lepton-universality violation in $B^{+}\to K^{+}\ell^{+}\ell^{-}$ decays, Phys. Rev. Lett. 122 (2019), no. 19 191801, [arXiv:1903.09252].
  * [67] BELLE Collaboration, S. Choudhury et al., Test of lepton flavor universality and search for lepton flavor violation in $B\rightarrow K\ell\ell$ decays, JHEP 03 (2021) 105, [arXiv:1908.01848].
  * [68] LHCb Collaboration, R. Aaij et al., Test of lepton universality in beauty-quark decays, arXiv:2103.11769.
  * [69] LHCb Collaboration, R. Aaij et al., Test of lepton universality with $B^{0}\rightarrow K^{*0}\ell^{+}\ell^{-}$ decays, JHEP 08 (2017) 055, [arXiv:1705.05802].
  * [70] Belle Collaboration, A. Abdesselam et al., Test of Lepton-Flavor Universality in ${B\to K^{\ast}\ell^{+}\ell^{-}}$ Decays at Belle, Phys. Rev. Lett. 126 (2021), no. 16 161801, [arXiv:1904.02440].
  * [71] Belle Collaboration, S. Wehle et al., Lepton-Flavor-Dependent Angular Analysis of $B\to K^{\ast}\ell^{+}\ell^{-}$, Phys. Rev. Lett. 118 (2017), no. 11 111801, [arXiv:1612.05014].
  * [72] B. Capdevila, S. Descotes-Genon, J. Matias, and J. Virto, Assessing lepton-flavour non-universality from $B\to K^{*}\ell\ell$ angular analyses, JHEP 10 (2016) 075, [arXiv:1605.03156].
  * [73] A. Datta, J. Kumar, and D. London, The $B$ anomalies and new physics in $b\to se^{+}e^{-}$, Phys. Lett. B 797 (2019) 134858, [arXiv:1903.10086].
  * [74] W. Altmannshofer and P. Stangl, New Physics in Rare B Decays after Moriond 2021, arXiv:2103.13370.
  * [75] J. C. Hardy and I. S. Towner, Superallowed $0^{+}\to 0^{+}$ nuclear $\beta$ decays: 2014 critical survey, with precise results for $V_{ud}$ and CKM unitarity, Phys. Rev. C 91 (2015), no. 2 025501, [arXiv:1411.5987].
  * [76] Particle Data Group Collaboration, K. A. Olive et al., Review of Particle Physics, Chin. Phys. C 38 (2014) 090001.
  * [77] PiENu Collaboration, A. Aguilar-Arevalo et al., Improved Measurement of the $\pi\to\textrm{e}\nu$ Branching Ratio, Phys. Rev. Lett. 115 (2015), no. 7 071801, [arXiv:1506.05845].
  * [78] Belle Collaboration, D. Epifanov et al., Study of tau- —$>$ K(S) pi- nu(tau) decay at Belle, Phys. Lett. B 654 (2007) 65–73, [arXiv:0706.2231].
  * [79] A. Pich, Precision Tau Physics, Prog. Part. Nucl. Phys. 75 (2014) 41–85, [arXiv:1310.7922].
  * [80] S. Gonzàlez-Solís, A. Miranda, J. Rendón, and P. Roig, Exclusive hadronic tau decays as probes of non-SM interactions, Phys. Lett. B 804 (2020) 135371, [arXiv:1912.08725].
  * [81] E. E. Jenkins, A. V. Manohar, and P. Stoffer, Low-Energy Effective Field Theory below the Electroweak Scale: Operators and Matching, JHEP 03 (2018) 016, [arXiv:1709.04486].
  * [82] ATLAS Collaboration, G. Aad et al., Search for the lepton flavor violating decay $Z\rightarrow e\mu$ in pp collisions at $\sqrt{s}$ TeV with the ATLAS detector, Phys. Rev. D 90 (2014), no. 7 072010, [arXiv:1408.5774].
  * [83] ATLAS Collaboration, G. Aad et al., Search for charged-lepton-flavour violation in Z-boson decays with the ATLAS detector, Nature Phys. 17 (2021), no. 7 819–825, [arXiv:2105.12491].
  * [84] K. Hayasaka et al., Search for Lepton Flavor Violating Tau Decays into Three Leptons with 719 Million Produced Tau+Tau- Pairs, Phys. Lett. B 687 (2010) 139–143, [arXiv:1001.3221].
  * [85] SINDRUM Collaboration, U. Bellgardt et al., Search for the Decay mu+ —$>$ e+ e+ e-, Nucl. Phys. B 299 (1988) 1–6.
  * [86] Particle Data Group Collaboration, C. Patrignani et al., Review of Particle Physics, Chin. Phys. C 40 (2016), no. 10 100001.
  * [87] L. Calibbi, X. Marcano, and J. Roy, Z lepton flavour violation as a probe for new physics at future $e^{+}e^{-}$ colliders, arXiv:2107.10273.