#  Constraints on scalar $q^{2}l^{2}$ operators from $\pi^{0}\rightarrow\mu e$

Mathew Thomas Arun<EMAIL_ADDRESS>School of Physics, Indian
Institute of Science Education and Research, Thiruvananthapuram, India
Priyanka Lamba<EMAIL_ADDRESS>Institute of Theoretical Physics,
Faculty of Physics, University of Warsaw, Poland  Sudhir K. Vempati
<EMAIL_ADDRESS>Centre for High Energy Physics, Indian Institute of
Science, Bangalore, India

#  Bounds or Limits on $q^{2}l^{2}$ operators from $\pi^{0}\rightarrow\mu e$

Mathew Thomas Arun<EMAIL_ADDRESS>School of Physics, Indian
Institute of Science Education and Research, Thiruvananthapuram, India
Priyanka Lamba<EMAIL_ADDRESS>Institute of Theoretical Physics,
Faculty of Physics, University of Warsaw, Poland  Sudhir K. Vempati
<EMAIL_ADDRESS>Centre for High Energy Physics, Indian Institute of
Science, Bangalore, India

#  Restricting $q^{2}l^{2}$ operators from $\pi^{0}\rightarrow\mu e$

Mathew Thomas Arun<EMAIL_ADDRESS>School of Physics, Indian
Institute of Science Education and Research, Thiruvananthapuram, India
Priyanka Lamba<EMAIL_ADDRESS>Institute of Theoretical Physics,
Faculty of Physics, University of Warsaw, Poland  Sudhir K. Vempati
<EMAIL_ADDRESS>Centre for High Energy Physics, Indian Institute of
Science, Bangalore, India

###### Abstract

In this paper we consider semileptonic lepton flavor violating operators of
the type $q^{2}l^{2}$ in low energy effective theory (LEFT). At the chiral
scale, we match these operators to chiral perturbation theory ($\chi$PT) and
place constraints from the process $\pi^{0}\rightarrow\mu^{+}\,\,e^{-}$. These
bounds are shown to depend on the chiral nature of the operators. The scalar
operators are significantly more constrained compared to the vector operators.
We then compare the limits from $\mu\to e$ conversion in Nuclei and show that
the limits on scalar operators are within an order of magnitude of the
corresponding limits from $\mu\to e$ conversion in Ti. On the other hand, the
limits on vector operators are however much weaker. Towards the end, we evolve
the LEFT operators to W-boson mass scale using RGE, and match them to the
Standard Model effective field theory (SMEFT) operators. We, then, derive the
constraints on the parameter space of Leptoquark models that could generate
these SMEFT operators at tree level.

##  I Introduction

Neutral pion decays of the type $\pi^{0}\to\mu e$ are excellent probes for
studying lepton number and flavour violation. To our knowledge the first
searches for this process constrains the Branching fraction of neutral pions
decaying to the combination ($\mu^{+}e^{-}+~{}\mu^{-}e^{+}$) to $1.72\times
10^{-8}$ ParticleDataGroup:1998hll . Whereas, later experiments performed at
Brookhaven AGS, BNL E777PhysRevLett.64.2450 , BNL E865BNL:1998apv , searching
for the decays of $K_{L}^{0}\rightarrow\mu^{\pm}e^{\mp}$ BNL:1998apv ,
$K_{L}^{0}\rightarrow\pi^{0}\mu^{\pm}e^{\mp}$ Arisaka:1998uf ; Appel:2000tc
and $K^{+}\rightarrow\pi^{+}\mu^{+}e^{-}$ ($K_{\pi\mu e}$)Appel:2000wg , in
particular, the one looking for Lepton Flavour Violating
$K^{+}\rightarrow\pi^{+}\ell\ell^{\prime}$ decays, restricts the process
$\pi^{0}\rightarrow\mu e$ through the subprocess
$K^{+}\rightarrow\pi^{+}\pi^{0}(\rightarrow\mu e)$. The Brookhaven
experiments, using them, place an upper limit on the Branching Fraction for
the $\pi^{0}\to\mu^{+}e^{-}$ process as $BR(\pi^{0}\to\mu^{+}e^{-})<3.8\times
10^{-10}$ and $BR(\pi^{0}\to\mu^{-}e^{+})~{}<3.4\times 10^{-9}$. Later, the
KTeV collaboration at the Fermilab has improved the limits on the combination
to BR($\pi^{0}\to\mu^{\mp}e^{\pm}$)$<3.59\times 10^{-10}$ KTeV:2007cvy  and
recently, the NA62 experiment at CERN constraint them further to
BR($\pi^{0}\to\mu^{-}e^{+}$)$<3.2\times 10^{-10}$ NA62:2021zxl . A summary of
the present limits is presented in Table 1.

Process | Experiment | Limit
---|---|---
$\pi^{0}\to\mu^{+}e^{-}$ | E865 BNL |  $3.4\times 10^{-10}$ Appel:2000wg
$\pi^{0}\to\mu^{-}e^{+}$ | NA62 CERN |  $3.2\times 10^{-10}$ NA62:2021zxl
$\pi^{0}\to\mu^{\mp}e^{\pm}$ | KTeV Fermilab |  $3.6\times 10^{-10}$ KTeV:2007cvy
Table 1: A list of the present limits on the Lepton number violating process
$\pi^{0}\to\mu e$.

It would be interesting to study the limits from these experiments on low
energy effective operators. For this, one needs to match the effective
operators to chiral perturbation theory. We do that in this article and derive
the limits. To our knowledge this is first time this process is used to obtain
the limits 111While finishing this paper, Ref.Hoferichter:2022mna  appeared on
the arxiv, addressing the same issue but in a way complementary to our work..
In general, due to the presence of large hadronic uncertainties, pion decays
(mostly charged ones) are more suitable for studying ratios like lepton flavor
universality etc, where purely leptonic processes like $\mu\to e\gamma$ are
good to study the lepton number/flavour violation in the leptonic sector.
However, there exists other decay modes like $K_{L}\to\mu e$ which provide
strong constraints on the effective theory operators.

Large flavor violation in the neutral lepton sector is very well established
through neutrino oscillation phenomenology. The (large) neutrino transitions
should typically lead to charged lepton flavor violation(cLFV). If neutrinos
attain masses through interactions with Higgs of the Standard Model, the
resultant cLFV is understood to be very small, leading to the Branching
Fraction of the most prominent channel $BR(\mu\rightarrow e\gamma)\sim
10^{-48}$. However, in the presence of new particles, with Weak scale masses,
we expect the cLFV rates to be within the range of present and future upcoming
experiments. The cLFV is a highly model dependent phenomena, such that, the
predicted rates for various cLFV processes vary significantly from one UV
model to another and with the parameter space of the model. For example in
supersymmetric models the rates can be large for supersymmetric particles
between masses of a few hundred GeV and several TeV. For a review please see
Calibbi et. al Calibbi:2006nq ; Calibbi:2017uvl . For the kind of the
operators we are interested in this work, the relevant leptonic number
violating process would be $\mu~{}\rightarrow~{}e$ conversion in Nuclei. This
process has been well studied and the limits are reasonably well known (for a
review, please see Kuno:1999jp ), though uncertainties concerning form factors
in various Nuclei persist. The best limit on $\mu\rightarrow e$ comes from the
SINDRUM II collaboration, which puts the limits on the Branching fraction to
be $\textbf{BR}(\mu\rightarrow e)$ in Ti (Au) is $6.1(7)\times 10^{-13}$
SINDRUMII:1998mwd ; SINDRUMII:2006dvw .

In the following we would like to compare the limits from
$\pi^{0}\to\mu^{+}e^{-}$ to those from $\mu\to e$ conversion in Nuclei. We do
the comparision in the set of low energy operators determined by the LEFT. In
this approach, the cLFV would appear as a contact interaction, four Fermi
operators at the scale far below the scale of new physics. The operators can
be written at the scale of the experiments which is typically far below the
scale of the Standard Model. The full effective theory at this scale is called
the low energy effective theory (LEFT). This theory is obtained from the
Standard Model lagrangian after integrating out the top, Higgs, $W$ and $Z$
particles. The resultant theory contains several operators which could be
classified as follows:

Scalar | Vector
---|---
$\mathcal{O}_{eu}^{SRR}=\displaystyle(\bar{e}_{Li}e_{Rj})(\bar{u}_{Lw}u_{Rt})$ | $\mathcal{O}_{eu}^{V\ LL}=\displaystyle(\bar{e}_{Li}\gamma_{\mu}e_{Lj})(\bar{u}_{Lw}\gamma^{\mu}u_{Lt})\quad\quad\mathcal{O}_{eu}^{V\ LR}=\displaystyle(\bar{e}_{Li}\gamma_{\mu}e_{Lj})(\bar{u}_{Rw}\gamma^{\mu}u_{Rt})$
$\mathcal{O}_{eu}^{SRL}=\displaystyle(\bar{e}_{Li}e_{Rj})(\bar{u}_{Rw}u_{Lt})$ | $\mathcal{O}_{eu}^{V\ RR}=\displaystyle(\bar{e}_{Ri}\gamma_{\mu}e_{Rj})(\bar{u}_{Rw}\gamma^{\mu}u_{Rt})\quad\quad\mathcal{O}_{eu}^{V\ RL}=\displaystyle(\bar{e}_{Ri}\gamma_{\mu}e_{Rj})(\bar{u}_{Lw}\gamma^{\mu}u_{Lt})$
$\mathcal{O}_{ed}^{SRR}=\displaystyle(\bar{e}_{Li}e_{Rj})(\bar{d}_{Lw}d_{Rt})$ | $\mathcal{O}_{ed}^{V\ LL}=\displaystyle(\bar{e}_{Li}\gamma_{\mu}e_{Lj})(\bar{d}_{Lw}\gamma^{\mu}d_{Lt})\quad\quad\mathcal{O}_{ed}^{V\ LR}=\displaystyle(\bar{e}_{Li}\gamma_{\mu}e_{Lj})(\bar{d}_{Rw}\gamma^{\mu}d_{Rt})$
$\mathcal{O}_{ed}^{SRL}=\displaystyle(\bar{e}_{Li}e_{Rj})(\bar{d}_{Rw}d_{Lt})$ | $\mathcal{O}_{ed}^{V\ RR}=\displaystyle(\bar{e}_{Ri}\gamma_{\mu}e_{Rj})(\bar{d}_{Rw}\gamma^{\mu}d_{Rt})\quad\quad\mathcal{O}_{ed}^{V\ RL}=\displaystyle(\bar{e}_{Ri}\gamma_{\mu}e_{Rj})(\bar{d}_{Lw}\gamma^{\mu}d_{Lt})$
Table 2: LEFT semileptonic scalar and vector operators. Notation: $e,u\;\&\;d$
are used for lepton sector, up quark sector and down quark sector
respectively. And $i,j,w,t$ are generation indices. For
$\pi^{0}\rightarrow\mu^{+}\,\,e^{-}$ value of i=$\mu$, j=e
,($w,\,\,t$)=(u,u),(d,d)

At the scale of LEFT, these operators can be considered independent. Of
course, they could be matched with SMEFT at some high scale and there could be
correlations within the operators. In the present work, we assume these
operators are independent and derive the limits on them. We find that for the
scalar operators, the limits obtained from $\pi^{0}\to\mu e$ can be comparable
with those obtained from $\mu\to e$ conversion in Ti. For the vector
operators, however, the limits from $\mu\to e$ conversion dominate
significantly. The ratio of the matrix element with the scalar operators to
the vector operators is roughly enhanced by a factor $B_{0}$ which is a low
energy constant appearing in the scalar densities ( in the scalar condensates
) of the chiral perturbation theory. The constant is measured on the lattice
and it is typically large $\sim GeV$ Aoki:2021kgd .

The rest of the paper is organised as follows. In the next section, we recap
the low energy effective theory and the chiral perturbation theory. In section
II, we do the matching of the chiral perturbation theory to that of the low
energy effective theory at tree level and derive the limits from
$BR(\pi^{0}\to\mu^{+}\,e^{-})$ in section III. We also derive the limits from
$\mu\to e$ conversion. In section IV, we discuss the Leptoquark completion at
high scale through a SMEFT matching with LEFT. Thus constraining the parameter
space of the Leptoquark models we have considered. We then close with a
summary in the last section.

##  II Low-Energy Effective Field Theory

Low Energy Effective Field Theory (LEFT) Jenkins:2017dyc ; Jenkins:2017jig  is
modeled to lie between the Standard Model Theory (SM) scale and the QCD non-
perturbative scale. The top quark, the Higgs and the massive gauge bosons are
integrated out, and hence LEFT has light quarks, photon and gluon as the
dynamical degrees of freedom. Within this framework, running and mixing of
effective operators can be computed using perturbative renormalization-group
(RG) equations up to the $\Lambda_{QCD}$ scale below which QCD becomes non-
perturbative. At which scale the theory must be matched with the $\chi$PT.
There are a total of 80 dimension-6 operators out of which 2 are pure gauge
($X^{3}$) and 78 are four-fermion contact operators ($\psi^{4}$). And there
exists 6 dimension-5 operators which have mixed fermion gauge structure
($\psi^{2}X$). These operators are further divided by their chirality. In
general they could be vector, scalar, and tensor like $\psi^{4}$ operators and
are denoted by $V,S,$ and $T$ superscripts.

###  II.1 $\chi$PT Lagrangian

In this subsection we give a brief outline of the chiral Lagrangian. For a
complete version we refer the reader to Gasser:1983yg ; Gasser:1984gg . In the
limit of vanishing quark masses the QCD Hamiltonian has an enhanced symmetry
under the chiral group $SU(N_{f})\times SU(N_{f})$, where $N_{f}$ is the
number of quark flavour. This symmetry is then assumed to be broken by the
ground state of the theory to give rise to $N_{f}^{2}-1$ Goldstone fields.
$\chi$PT is modeled through the addition of quark bilinears using spurion
fields which are $N_{f}\times N_{f}$ hermitian matrices in flavour space,
$v_{\mu}(x),a_{\mu}(x),s(x),p(x),t_{\mu\nu}(x)$ associated with vector, axial
vector, scalar, pseudo-scalar and tensor currents respectively. The generating
functional for the Greens function corresponding to the vacuum-to-vacuum
transition amplitude is then obtained from the Lagrangian,

$\displaystyle\mathcal{L}$ $\displaystyle=$
$\displaystyle\displaystyle\mathcal{L}_{\text{QCD}}^{0}+\bar{q}\gamma^{\mu}\Big{(}v_{\mu}(x)+\gamma^{5}a_{\mu}(x)\Big{)}q-\bar{q}\Big{(}s(x)-i\gamma^{5}p(x)\Big{)}q$
$\displaystyle+$
$\displaystyle\displaystyle\bar{q}\sigma^{\mu\nu}t_{\mu\nu}(x)q$
$\displaystyle\mathcal{L}_{\text{QCD}}^{0}$ $\displaystyle=$
$\displaystyle\displaystyle-\frac{1}{2g_{s}^{2}}G_{\mu\nu}G^{\mu\nu}+\bar{q}i\gamma^{\mu}\Big{(}\partial_{\mu}-iG_{\mu}\Big{)}q\
,$

where $G_{\mu\nu}$ is the field strength tensor of the gluon ($G_{\mu}$). For
future simplicity, it will be simpler to match with the LEFT operators if we
write the above Lagrangian in the chiral notation,

$\displaystyle\mathcal{L}$ $\displaystyle=$
$\displaystyle\displaystyle\mathcal{L}_{\text{QCD}}^{0}+\bar{q}_{L}\gamma^{\mu}l_{\mu}(x)q_{L}+\bar{q}_{R}\gamma^{\mu}r_{\mu}(x)q_{R}$
$\displaystyle+$
$\displaystyle\displaystyle\bar{q}_{L}S(x)q_{R}+\bar{q}_{R}S^{\dagger}(x)q_{L}$
$\displaystyle+$
$\displaystyle\displaystyle\bar{q}_{L}\sigma^{\mu\nu}t_{\mu\nu}(x)q_{R}+\bar{q}_{L}\sigma^{\mu\nu}t_{\mu\nu}^{\dagger}(x)q_{R}\
,$

where

$\displaystyle r_{\mu}(x)$ $\displaystyle=$ $\displaystyle\displaystyle
v_{\mu}(x)+a_{\mu}(x)\ ,$ (3) $\displaystyle l_{\mu}(x)$ $\displaystyle=$
$\displaystyle\displaystyle v_{\mu}(x)-a_{\mu}(x)\ ,$ $\displaystyle S(x)$
$\displaystyle=$ $\displaystyle\displaystyle s(x)-ip(x)\ .$

The three light quark mass matrix are contained in $S$ where are the heavy
quarks are integrated out and will not contribute to the low energy effective
chiral Lagrangian.

Considering three flavour, $N_{f}=3$, the 8 Goldstone fields corresponding to
the spontaneous symmetry breaking of $SU(3)\times SU(3)$ to $SU(3)$ is
collected in the unitary $3\times 3$ matrix $U(x)$. This matrix transforms as

$U^{\prime}(x)=V_{R}U(x)V^{\dagger}_{L}(x),$

under the chiral $SU(3)\times SU(3)$. In accordance with the above
transformation relation we can write,
$U(x)=exp\Big{(}i\frac{\lambda^{a}\phi^{a}(x)}{F_{0}}\Big{)}$, where
$\phi^{a}$ are the 8 Goldstone fields, $\lambda^{a}$ the Gell-Mann matrices
and $F_{0}$ the pion decay constant. In the non-linear representation in
chiral symmetry $U(x)$ can be written as Alves:2017avw

$U(x)\equiv\exp\Bigg{[}i\frac{\sqrt{2}}{F_{0}}\begin{pmatrix}\frac{\pi^{0}}{\sqrt{2}}+\frac{\eta}{\sqrt{6}}&\pi^{+}&K^{+}\\\
\pi^{-}&-\frac{\pi^{0}}{\sqrt{2}}+\frac{\eta}{\sqrt{6}}&K^{0}\\\
K^{-}&\bar{K}^{0}&-\frac{2\eta}{\sqrt{6}}\end{pmatrix}\Bigg{]}$ (4)

The Chiral Lagrangian could be written in terms of these fields as,

$\displaystyle\mathcal{L}$ $\displaystyle=$
$\displaystyle\displaystyle\mathcal{L}_{1}+\mathcal{L}_{2}$ (5)
$\displaystyle\mathcal{L}_{1}$ $\displaystyle=$
$\displaystyle\displaystyle\frac{1}{4}F_{0}^{2}\\{tr\Big{(}D_{\mu}U^{\dagger}D^{\mu}U\Big{)}+tr\Big{(}\chi^{\dagger}U+\chi
U^{\dagger}\Big{)}\\}\ ,$ $\displaystyle\mathcal{L}_{2}$ $\displaystyle=$
$\displaystyle\displaystyle L_{1}\langle
D^{\mu}U^{\dagger}D_{\mu}U\rangle^{2}+L_{2}\langle
D_{\mu}U^{\dagger}D_{\nu}U\rangle\langle D^{\mu}U^{\dagger}D^{\nu}U\rangle$
$\displaystyle+$ $\displaystyle L_{3}\langle
D^{\mu}U^{\dagger}D_{\mu}U\Big{(}\chi^{\dagger}U+U^{\dagger}\chi\Big{)}\rangle$
$\displaystyle+$ $\displaystyle L_{6}\langle\chi^{\dagger}U+\chi
U^{\dagger}\rangle^{2}+L_{7}\langle\chi^{\dagger}U-\chi U^{\dagger}\rangle$
$\displaystyle+$ $\displaystyle
L_{8}\langle\chi^{\dagger}U\chi^{\dagger}U+\chi U^{\dagger}\chi
U^{\dagger}\rangle$ $\displaystyle-$ $\displaystyle iL_{9}\langle
F_{R\mu\nu}D^{\mu}UD^{\nu}U^{\dagger}+F_{L\mu\nu}D^{\mu}U^{\dagger}D^{\nu}U\rangle$
$\displaystyle+$ $\displaystyle L_{10}\langle
U^{\dagger}F_{R\mu\nu}UF_{L}^{\mu\nu}\rangle+H_{1}\langle
F_{R\mu\nu}F_{R}^{\mu\nu}+F_{L\mu\nu}F_{L}^{\mu\nu}\rangle+H_{2}\langle\chi^{\dagger}\chi\rangle,$

where $\langle\,\rangle$ represents the trace of the matrix within and
$F^{\mu\nu}_{R}=\partial_{\mu}r_{\nu}-\partial_{\nu}r_{\mu}-i[r_{\mu},r_{\nu}]$
and
$F^{\mu\nu}_{L}=\partial_{\mu}l_{\nu}-\partial_{\nu}l_{\mu}-i[l_{\mu},l_{\nu}]$
are the field strength tensors with $r_{\mu}$ and $l_{\mu}$ as defined in
Eq.(3).

Finally, the low energy constants ($L_{i}$) are fixed from experimental
information on D-wave $\pi\pi$ scattering lengths, electro-magnetic charge
radius of the pion, decay of $\pi\rightarrow e\nu\gamma$, $K\pi$ scattering
and large-$N_{c}$ arguments (Zweig rule). These are presented in Table 3. The
contact term $H_{2}$ has no physical significance, but are needed as counter-
terms for renormalization. A more complete expression for a general Lagrangian
up to order $p^{4}$ along with the values computed for the low-energy
constants are given in Gasser:1984gg .

$L_{1}$ | $(0.9\pm 0.3)\times 10^{-3}\times m_{\pi}^{+}$ | $L_{2}$ | $(1.7\pm 0.7)\times 10^{-3}\times m_{\pi}^{+}$
---|---|---|---
$L_{3}$ | $(-4.4\pm 2.5)\times 10^{-3}\times m_{\pi}^{+}$ | $L_{4}$ | $(0\pm 0.5)\times 10^{-3}\times m_{\pi}^{+}$
$L_{5}$ | $(2.2\pm 0.5)\times 10^{-3}\times m_{\pi}^{+}$ | $L_{6}$ | $(0\pm 0.3)\times 10^{-3}\times m_{\pi}^{+}$
$L_{7}$ | $(-0.4\pm 0.15)\times 10^{-3}\times m_{\pi}^{+}$ | $L_{8}$ | $(1.1\pm 0.3)\times 10^{-3}\times m_{\pi}^{+}$
$L_{9}$ | $(7.4\pm 0.7)\times 10^{-3}\times m_{\pi}^{+}$ | $L_{10}$ | $(-6.0\pm 0.7)\times 10^{-3}\times m_{\pi}^{+}$
Table 3: All values for the low energy coupling constants are taken from
Gasser:1984gg .

##  III Matching and Lepton flavour violating Pion decay

For momentum transfers below $\Lambda_{QCD}$, the quark bilinears in scalar
and vector LEFT operators should be matched to $\chi$PT Lagrangian given in
Eq.(5). Since we are interested in the Lepton flavour violating decays of
neutral pion, lets consider the Hamiltonian matrix element

$\langle\ell_{i}\ell_{j}|\mathcal{O}|\pi^{0}\rangle\ ,$ (6)

where the matrix element $\mathcal{O}$ corresponds to the LEFT operators that
contributes to the process. In principle, for a UV complete model, all the
operators should contribute with the appropriate UV scale, but for simplicity
lets consider the scalar and vectors independently.

Before proceeding further, a note is in order. Not all of the operators
contribute to the decay. At tree level only 6 $\psi^{4}$ type of operators
will generate the Lepton flavour violating neutral pion decay. These operators
are given in Table 2. Regarding tensor operators, for quark bilinears that
transforms as a tensor the $\chi$PT replacement rule is given as,

$\bar{q}_{Lw}\sigma^{\mu\nu}q_{Rt}=F_{1}\langle
U_{wt}F_{L}^{\mu\nu}+F_{R}^{\mu\nu}U_{wt}\rangle+iF_{2}\langle
D^{\mu}U_{wr}U^{\dagger}_{rs}D^{\nu}U_{st}\rangle+\mathcal{O}(p^{6})\ .$ (7)

On matching with $\chi$PT it can be seen that tensor operators will always
have a photon in the final state. and hence will contribute in the process
$\pi^{0}\rightarrow e^{+}e^{-}\gamma$, for which the world average for the
decay fraction, $\frac{\Gamma(\pi^{0}\rightarrow
e^{+}e^{-}\gamma)}{\Gamma(\pi^{0}\rightarrow\gamma\gamma)}=1.188\%$refId0 , is
measured and matched with the theoretical value to $-0.3\sigma$ precision. And
hence, tensor operators will not be further discussed here.

###  III.1 Scalar operator

From Table 2, there are two scalar operators that contribute to the lepton
flavour violating pion decay process. Using these operators, the Hamiltonian
given in Eq.(6) could be written as,

$\langle\mu e|\Big{(}L^{S,RL}_{\begin{subarray}{c}eu_{i}\\\ \mu
eu_{i}u_{i}\end{subarray}}\mathcal{O}_{eu_{i}}^{SRL}+L^{S,RR}_{\begin{subarray}{c}eu_{i}\\\
\mu
eu_{i}u_{i}\end{subarray}}\mathcal{O}_{eu_{i}}^{SRR}\Big{)}|\pi^{0}\rangle$
(8)

where $L^{S,RL}_{\begin{subarray}{c}eu_{i}\\\ \mu
eu_{i}u_{i}\end{subarray}},\;\&\;L^{S,RR}_{\begin{subarray}{c}eu_{i}\\\ \mu
eu_{i}u_{i}\end{subarray}}$ are the Wilson Coefficients. Here, for brevity we
have used the notation $u_{i}=u,\;d$.

At low energies the $\bar{u_{i}}u_{i}$ condenses and at order $p^{4}$ in
chiral counting, the quark bilinear in the scalar operators are matched with
the chiral Lagrangian. The replacement rule as given in Dekens:2018pbu ;
Gasser:1983yg  is,

$\displaystyle\bar{u}_{iLw}u_{iRt}$ $\displaystyle=$
$\displaystyle-2B_{0}\left\\{\frac{1}{4}F_{0}^{2}U_{tw}+L_{4}\langle
D_{\mu}U^{\dagger}D^{\mu}U\rangle U_{tw}\right.$ (9) $\displaystyle+$
$\displaystyle\left.\displaystyle
L_{5}(UD_{\mu}U^{\dagger}D^{\mu}U)_{tw}+2L_{6}\langle
U^{\dagger}\chi+\chi^{\dagger}U\rangle U_{tw}-2L_{7}\langle
U^{\dagger}\chi-\chi^{\dagger}U\rangle U_{tw}\right.$ $\displaystyle+$
$\displaystyle\left.\displaystyle
2L_{8}(U\,\chi^{\dagger}\,U)_{tw}+H_{2}\,\chi_{tw}\right\\}+\mathcal{O}(p^{6})\
,$

where $F_{0}$ is the pion decay constant in the chiral limit, $B_{0},L_{i}$
and $H_{i}$ are the non-perturbative low energy contants and $u_{i}=u,d$ in
our case. $\chi=-2B_{0}S^{\dagger}$ where ‘S’ is mass matrix
$M_{u_{i}}=diag(m_{u},m_{d},m_{s})$. At leading order $F_{0}$ and $B_{0}$ is
sufficient to determine the low-energy behaviour.

To constraint the Wilson Coefficients responsible for the pion decay process,
the terms $\mathcal{O}(p^{4})$ in chiral counting are subdominant
Gasser:1984gg . Hence, keeping the expansion to order $p^{2}$ in Eq.(9), we
get

$\displaystyle\mathcal{O}_{eu_{i}}^{SRR}$ $\displaystyle=$
$\displaystyle\displaystyle(\bar{\mu}_{L}e_{R})(\bar{u}_{iLw}u_{iRt})$ (10)
$\displaystyle=$
$\displaystyle\displaystyle(\bar{\mu}_{L}e_{R})\Big{[}-2B_{0}\frac{1}{4}F_{0}^{2}U_{tw}\Big{]}+\mathcal{O}(p^{4}),$
$\displaystyle\mathcal{O}_{eu_{i}}^{SRL}$ $\displaystyle=$
$\displaystyle\displaystyle(\bar{\mu}_{L}e_{R})(\bar{u}_{iRw}u_{iLt})$ (11)
$\displaystyle=$
$\displaystyle\displaystyle(\bar{\mu}_{L}e_{R})\Big{[}-2B_{0}\frac{1}{4}F_{0}^{2}U^{\dagger}_{tw}\Big{]}+\mathcal{O}(p^{4}),$

in $\chi$PT.

To study the effect of operators Eq.(10) and Eq.(11) in $\pi^{0}\rightarrow\mu
e$, we need to expand the matrix in Eq.(4) to the leading order,
$1+i\frac{\lambda^{a}\phi^{a}}{F_{0}}$, and the corresponding Hamiltonian
element then becomes,

$\displaystyle\langle e\mu|\Big{(}L^{S,RL}_{\begin{subarray}{c}eu_{i}\\\ \mu
eu_{i}u_{i}\end{subarray}}\mathcal{O}_{eu_{i}}^{SRL}+L^{S,RR}_{\begin{subarray}{c}eu_{i}\\\
\mu
eu_{i}u_{i}\end{subarray}}\mathcal{O}_{eu_{i}}^{SRR}\Big{)}|\pi^{0}\rangle$
$\displaystyle=$
$\displaystyle-\Big{(}\frac{iB_{0}F_{0}}{2}\Big{)}(L^{S,RR}_{\begin{subarray}{c}eu\\\
\mu euu\end{subarray}}-L^{S,RR}_{\begin{subarray}{c}ed\\\ \mu
edd\end{subarray}}-L^{S,RL}_{\begin{subarray}{c}eu\\\ \mu
euu\end{subarray}}+L^{S,RL}_{\begin{subarray}{c}ed\\\ \mu edd\end{subarray}})$
(12) $\displaystyle\times$ $\displaystyle\displaystyle\langle
e\mu|\displaystyle(\bar{\mu}_{L}e_{R})\pi^{0}|\pi^{0}\rangle$

The amplitude, $\mathcal{M}_{\pi^{0}\mu^{+}e^{-}}$, for the process computed
from the matrix element above, is given as

$\displaystyle\mathcal{M}(\pi^{0}(p^{\pi^{0}})\rightarrow\mu^{+}(p_{\mu})e^{-}(p_{e}))$
$\displaystyle=$
$\displaystyle\displaystyle-\Big{(}\frac{iB_{0}F_{0}}{2}\Big{)}(L^{S,RR}_{\begin{subarray}{c}eu\\\
\mu euu\end{subarray}}-L^{S,RR}_{\begin{subarray}{c}ed\\\ \mu
edd\end{subarray}}-L^{S,RL}_{\begin{subarray}{c}eu\\\ \mu
euu\end{subarray}}+L^{S,RL}_{\begin{subarray}{c}ed\\\ \mu edd\end{subarray}})$
(13) $\displaystyle\times$
$\displaystyle\displaystyle\bar{u}(p_{e})\frac{1}{2}(1-\gamma^{5})v(p_{\mu})\
.$

For the rate, the square of the amplitude is given by,

$\displaystyle\sum_{spin}|\mathcal{M}_{\pi^{0}\mu^{+}e^{-}}|^{2}$
$\displaystyle=$
$\displaystyle\Big{|}-\Big{(}\frac{iB_{0}F_{0}}{2}\Big{)}\Big{(}L^{S,RR}_{\begin{subarray}{c}eu\\\
\mu euu\end{subarray}}-L^{S,RR}_{\begin{subarray}{c}ed\\\ \mu
edd\end{subarray}}-L^{S,RL}_{\begin{subarray}{c}eu\\\ \mu
euu\end{subarray}}+L^{S,RL}_{\begin{subarray}{c}ed\\\ \mu
edd\end{subarray}}\Big{)}\Big{|}^{2}$ (14)
$\displaystyle\displaystyle\sum_{spin}\Big{(}\bar{u}(p_{e})\frac{1}{2}(1-\gamma^{5})v(p_{\mu})\Big{)}\Big{(}\bar{u}(p_{e})\frac{1}{2}(1-\gamma^{5})v(p_{\mu})\Big{)}^{\dagger}$
$\displaystyle=$
$\displaystyle\displaystyle\frac{B_{0}^{2}}{4}F_{0}^{2}\Big{|}\Big{(}L^{S,RR}_{\begin{subarray}{c}eu\\\
\mu euu\end{subarray}}-L^{S,RR}_{\begin{subarray}{c}ed\\\ \mu
edd\end{subarray}}-L^{S,RL}_{\begin{subarray}{c}eu\\\ \mu
euu\end{subarray}}+L^{S,RL}_{\begin{subarray}{c}ed\\\ \mu
edd\end{subarray}}\Big{)}\Big{|}^{2}\Big{(}m_{\pi}^{2}-(m_{e}+m_{\mu})^{2}\Big{)}$

| In GeV |  | In GeV
---|---|---|---
$m_{\pi}^{0}$ | 134.97$\times 10^{-3}$ | $F_{0}$ | 92.3$\times 10^{-3}$Dekens:2018pbu
$m_{\mu}$ | 105.65$\times 10^{-3}$ | $m_{e}$ | 0.51$\times 10^{-3}$
$m_{u}$ | 2.16$\times 10^{-3}$ | $m_{d}$ | 4.67 $\times 10^{-3}$
$\Gamma(\pi^{0}\rightarrow\mu^{+}e^{-})$ | 2.97$\times 10^{-18}$ | $B_{0}=\frac{m_{\pi^{0}}^{2}}{m_{u}+m_{d}}$ | 2.667
Table 4: Input values in GeV. $m_{u},m_{d}$ are MS bar masses at 2 GeV. All
values are taken from PDGZyla:2020zbs .

Using the masses given in Table 4 and Eq.(13), the decay width of
$\pi^{0}\rightarrow\mu^{+}e^{-}$ could be directly computed to be,

$\displaystyle\Gamma(\pi^{0}\rightarrow\mu^{+}e^{-})$ $\displaystyle=$
$\displaystyle\displaystyle\frac{|p_{\mu}|}{8\pi
m_{\pi}^{2}}|\mathcal{M}_{\pi^{0}\mu^{+}e^{-}}|^{2}$ $\displaystyle=$
$\displaystyle\Big{|}\Big{(}L^{S,RR}_{\begin{subarray}{c}eu\\\ \mu
euu\end{subarray}}-L^{S,RR}_{\begin{subarray}{c}ed\\\ \mu
edd\end{subarray}}-L^{S,RL}_{\begin{subarray}{c}eu\\\ \mu
euu\end{subarray}}+L^{S,RL}_{\begin{subarray}{c}ed\\\ \mu
edd\end{subarray}}\Big{)}\Big{|}^{2}(6.007\times 10^{-6})\,\mbox{GeV}^{5}$
$\displaystyle\leq$ $\displaystyle 2.97\times 10^{-18}$

And the constraint on Wilson Coefficients becomes,

$\Big{(}L^{S,RR}_{\begin{subarray}{c}eu_{i}\\\ \mu
eu_{i}u_{i}\end{subarray}}-L^{S,RL}_{\begin{subarray}{c}eu_{i}\\\ \mu
eu_{i}u_{i}\end{subarray}}\Big{)}\leq 7.03\times 10^{-7}\quad\mbox{GeV}^{-2}$
(16)

where the definitions $L^{S,RR}_{\begin{subarray}{c}eu_{i}\\\ \mu
eu_{i}u_{i}\end{subarray}}=L^{S,RR}_{\begin{subarray}{c}eu\\\ \mu
euu\end{subarray}}-L^{S,RR}_{\begin{subarray}{c}ed\\\ \mu edd\end{subarray}}$
and $L^{S,RL}_{\begin{subarray}{c}eu_{i}\\\ \mu
eu_{i}u_{i}\end{subarray}}=L^{S,RL}_{\begin{subarray}{c}eu\\\ \mu
euu\end{subarray}}-L^{S,RL}_{\begin{subarray}{c}ed\\\ \mu edd\end{subarray}}$
have been used.

The bounds derived so far are at the scale close to mass of the pion and are
thus close to QCD scale $\Lambda_{QCD}\sim 2~{}\text{GeV}$. If one would like
to see the value of the coupling close to the electro-weak scale, then, one
needs to utilise the RGE of the low energy effective theory. These RGE take
into consideration the running effects on the couplings between the scales
$\sim 2$ GeV to $100$ GeV. The relevant anomalous dimensions are presented in
Jenkins:2017dyc  and reproduced in the Appendix V.1. It is understood that
these couplings mix under the anomalous dimension matrix, as an example the
scalar couplings $L^{S,RR}_{\begin{subarray}{c}ed\\\ 1211\end{subarray}}$ are
mixed with $L^{S,RR}_{\begin{subarray}{c}ed\\\ 1222\end{subarray}}$ and
$L^{S,RR}_{\begin{subarray}{c}ed\\\ 1233\end{subarray}}$. If all the three
couplings $L^{S,RR}_{\begin{subarray}{c}ed\\\
1211\end{subarray}},L^{S,RR}_{\begin{subarray}{c}ed\\\
1222\end{subarray}},L^{S,RR}_{\begin{subarray}{c}ed\\\ 1233\end{subarray}}$
are of similar order, then the considered bounds from $\pi\rightarrow\mu e$ on
$L^{S,RR}_{\begin{subarray}{c}ed\\\ 1211\end{subarray}}$ are not significantly
affected by the running of the RGE to the high scale. Similar argument also
holds for mixing between tensor operators though we have not explicitly given
examples of them here. In summary RGE effects do not play an important role
and can be safely neglected.

The result could be seen in Fig. 1, where we have plotted on the left hand
side the couplings $L^{SRL}_{\mu eu_{i}u_{i}}$ vs $L^{SRR}_{\mu eu_{i}u_{i}}$.
The cut-off scale has been chosen to be 100 GeV. RGE effects have been taken
in to consideration while doing this plot. As can be seen, the limits on these
couplings are $\sim\text{few}\times 10^{-7}$ (boundary of the shaded region)
and weakens when there are cancellations, along the diagonal of the plot.

Figure 1: On the left hand side, we plot $L^{SRL}_{\mu eu_{i}u_{i}}$ vs
$L^{SRR}_{\mu eu_{i}u_{i}}$ with the cut-off scale $\Lambda=100$ GeV. In the
shaded regions experimental limits from BR($\pi^{0}\to\mu e$) are satisfied.
On the right hand side, the couplings $A_{RR}$ vs $A_{LL}$ are plotted with
the shaded region showing the limits from BR($\pi^{0}\to\mu e$).

We now proceed to compare the limits from $\mu\to e$ conversion in Nuclei. At
the leading order the Br($\mu N\to eN$) is given for scalar operators as (in
the notation of Kitano:2002mt ):

$Br(\mu N\to
eN)=\frac{2G_{F}^{2}(G_{S}^{(d,p)}S^{(p)}+G_{S}^{(d,n)}S^{(n)})^{2}(|g_{LS(d)}|^{2}+|g_{RS(d)}|^{2})}{w_{capt}},$
(17)

where $G_{F}$ stands for the Fermi constant and
$G_{S}^{(d,p)},S^{(p)},G_{S}^{(d,n)},S^{(n)}$ correspond to various form
factors and number densities represented in terms of overlap integrals.
$w_{capt}$ is the capture rate. Using the values quoted for $Ti$ we have the
limit on the scalar couplings to be:

$\sqrt{(|g_{LS(d)}|^{2}+|g_{RS(d)}|^{2})}\leq 7.8\times 10^{-7}.$ (18)

The limit is of similar order of magnitude as one gets from Pion LFV decay.

###  III.2 Vector operators

The NP can also contribute to the process $\pi^{0}\rightarrow\mu^{+}e^{-}$
through effective vector operators.The corresponding Lepton Flavour violating
Hamiltonian element is,

$\langle\ell_{i}\ell_{j}|\mathcal{O}|\pi^{0}\rangle\ ,$ (19)

where $\mathcal{O}$ is now the vector operators given in Table 2. Using these
operators the above matrix element could be written as,

$\langle e\mu|\Big{(}L^{V,LL}_{\begin{subarray}{c}eu_{i}\\\ \mu
eu_{i}u_{i}\end{subarray}}\mathcal{O}_{eu_{i}}^{VLL}+L^{V,LR}_{\begin{subarray}{c}eu_{i}\\\
\mu
eu_{i}u_{i}\end{subarray}}\mathcal{O}_{eu_{i}}^{VLR}+L^{V,RR}_{\begin{subarray}{c}eu_{i}\\\
\mu
eu_{i}u_{i}\end{subarray}}\mathcal{O}_{eu_{i}}^{VRR}+L^{V,RL}_{\begin{subarray}{c}eu_{i}\\\
\mu
eu_{i}u_{i}\end{subarray}}\mathcal{O}_{eu_{i}}^{VRL}\Big{)}|\pi^{0}\rangle$
(20)

where $L^{V,LL}_{\begin{subarray}{c}eu_{i}\\\ \mu
eu_{i}u_{i}\end{subarray}},L^{V,LR}_{\begin{subarray}{c}eu_{i}\\\ \mu
eu_{i}u_{i}\end{subarray}}$, $L^{V,RR}_{\begin{subarray}{c}eu_{i}\\\ \mu
eu_{i}u_{i}\end{subarray}}$ and $L^{V,RL}_{\begin{subarray}{c}eu_{i}\\\ \mu
eu_{i}u_{i}\end{subarray}}$ are the Wilson Coefficient. And as before
$u_{i}=u,\;d$.

Like for the scalar operators, at low energies the $\bar{u_{i}}u_{i}$
condenses and then quark bilinear in the vector operators should be matched
with the chiral Lagrangian. The replacement rule for vector LEFT quark
bilinear could be written as,

$\displaystyle\bar{u_{i}}_{L}\gamma^{\mu}u_{iL}$ $\displaystyle=$
$\displaystyle\frac{i}{2}F_{0}^{2}\langle
D_{\mu}UU^{\dagger}\rangle+4iL_{1}\left\langle
D_{\nu}U^{\dagger}D^{\nu}U\right\rangle\left\langle
D_{\mu}UU^{\dagger}\right\rangle+4iL_{2}\left\langle
D^{\mu}U^{\dagger}D^{\nu}U\right\rangle\left\langle
D_{\nu}UU^{\dagger}\right\rangle$ (21) $\displaystyle+$
$\displaystyle\displaystyle
2iL_{3}\left\langle\left(U^{\dagger}D^{\mu}U-D^{\mu}U^{\dagger}U\right)D_{\nu}U^{\dagger}D^{\nu}U\right\rangle+2iL_{4}\left\langle
D_{\mu}UU^{\dagger}\right\rangle\left\langle
U^{\dagger}\chi+\chi^{\dagger}U\right\rangle$ $\displaystyle+$
$\displaystyle\displaystyle
iL_{5}\left\langle\left(U^{\dagger}D^{\mu}U-D^{\mu}U^{\dagger}U\right)(U^{\dagger}\chi+\chi^{\dagger}U)\right\rangle$
$\displaystyle+$ $\displaystyle\displaystyle L_{9}\left[-\left\langle
F_{R}^{\mu\nu}D_{\nu}UU^{\dagger}\right\rangle-\left\langle
UD_{\nu}U^{\dagger}F_{R}^{\mu\nu}\right\rangle+\left\langle
D_{\nu}UF_{L}^{\mu\nu}U^{\dagger}\right\rangle+\left\langle
UF_{L}^{\mu\nu}D_{\nu}U^{\dagger}\right\rangle\right]$ $\displaystyle-$
$\displaystyle\displaystyle iL_{9}\left\langle
D^{\nu}(D_{\mu}UD_{\nu}U^{\dagger}-D_{\nu}UD_{\mu}U^{\dagger}\right\rangle+2L_{10}\left\langle
D_{\nu}(UF_{L}^{\mu\nu}U^{\dagger})\right\rangle$ $\displaystyle+$
$\displaystyle\displaystyle 4H_{1}\left\langle
D_{\nu}F_{R}^{\mu\nu}\right\rangle+\epsilon\text{ terms}+\mathcal{O}(p^{6})\
,$

where $F^{\mu\nu}_{R}$ and $F^{\mu\nu}_{L}$ are the field strength tensors as
defined previously. Similarly $\bar{u}_{iR}\gamma^{\mu}u_{iR}$ matching is
same as in Eq.(21) by replacing $U\rightarrow U^{\dagger}$,
$\chi\rightarrow\chi^{\dagger}$.

To the most dominant term, contributing to the process
$\pi^{0}\rightarrow\mu^{+}e^{-}$, in the order in expansion of chiral
Lagrangian becomes,

$\displaystyle\mathcal{O}_{eu_{i}}^{VLL}$ $\displaystyle=$
$\displaystyle\displaystyle(\bar{\mu}_{L}\gamma_{\mu}e_{L})(\bar{u}_{iLw}\gamma^{\mu}u_{iLt})\quad\quad\quad\quad\mathcal{O}_{eu_{i}}^{VLR}=\displaystyle(\bar{\mu}_{L}\gamma_{\mu}e_{L})(\bar{u}_{iRw}\gamma^{\mu}u_{iRt})$
(22) $\displaystyle=$
$\displaystyle\displaystyle(\bar{\mu}_{L}\gamma^{\mu}e_{L})\Big{(}\frac{i}{2}F_{0}^{2}(D_{\mu}UU^{\dagger})_{tw}\Big{)}\quad\quad\quad\,\,\,=\displaystyle(\bar{\mu}_{L}\gamma^{\mu}e_{L})\Big{(}\frac{i}{2}F_{0}^{2}(D_{\mu}U^{\dagger}U)_{tw}\Big{)}$

and

$\displaystyle\mathcal{O}_{eu_{i}}^{VRL}$ $\displaystyle=$
$\displaystyle\displaystyle(\bar{\mu}_{R}\gamma_{\mu}e_{R})(\bar{u}_{iLw}\gamma^{\mu}u_{iLt})\quad\quad\quad\quad\mathcal{O}_{eu_{i}}^{VLR}=\displaystyle(\bar{\mu}_{R}\gamma_{\mu}e_{R})(\bar{u}_{iRw}\gamma^{\mu}u_{iRt})$
(23) $\displaystyle=$
$\displaystyle\displaystyle(\bar{\mu}_{R}\gamma^{\mu}e_{R})\Big{(}\frac{i}{2}F_{0}^{2}(D_{\mu}UU^{\dagger})_{tw}\Big{)}\quad\quad\quad\,\,\,=\displaystyle(\bar{\mu}_{R}\gamma^{\mu}e_{R})\Big{(}\frac{i}{2}F_{0}^{2}(D_{\mu}U^{\dagger}U)_{tw}\Big{)}$

where the neutral pion arises from
$(D_{\mu}UU^{\dagger})_{uu}\rightarrow\frac{i}{F_{0}}\partial_{\mu}\pi^{0},\quad(D_{\mu}UU^{\dagger})_{dd}\rightarrow-\frac{i}{F_{0}}\partial_{\mu}\pi^{0}$
and
$(D_{\mu}U^{\dagger}U)_{uu}\rightarrow-\frac{i}{F_{0}}\partial_{\mu}\pi^{0},\
\text{and}\
(D_{\mu}U^{\dagger}U)_{dd}\rightarrow\frac{i}{F_{0}}\partial_{\mu}\pi^{0}$, in
the expansion of $U(x)$ given in Eq.(3) up to first order in
$\frac{1}{F_{0}}$.

Then, to study the effect of the operators in Eq.(22) and Eq.(23) in the
Lepton flavour violating neutral pion decay process, the matrix element could
be derived as,

$\displaystyle\langle e\mu|\Big{(}L^{V,LL}_{\begin{subarray}{c}eu_{i}\\\ \mu
eu_{i}u_{i}\end{subarray}}\mathcal{O}_{eu_{i}}^{VLL}+L^{V,LR}_{\begin{subarray}{c}eu_{i}\\\
\mu
eu_{i}u_{i}\end{subarray}}\mathcal{O}_{eu_{i}}^{VLR}+L^{V,RR}_{\begin{subarray}{c}eu_{i}\\\
\mu
eu_{i}u_{i}\end{subarray}}\mathcal{O}_{eu_{i}}^{VRR}+L^{V,RL}_{\begin{subarray}{c}eu_{i}\\\
\mu
eu_{i}u_{i}\end{subarray}}\mathcal{O}_{eu_{i}}^{VRL}\Big{)}|\pi^{0}\rangle$
$\displaystyle=$ (24) $\displaystyle\langle
e\mu|\Big{(}L^{V,LR}_{\begin{subarray}{c}eu_{i}\\\ \mu
eu_{i}u_{i}\end{subarray}}-L^{V,LL}_{\begin{subarray}{c}eu_{i}\\\ \mu
eu_{i}u_{i}\end{subarray}}\Big{)}(\bar{\mu}_{L}\gamma^{\mu}e_{L})\frac{F_{0}}{2}\partial_{\mu}\pi^{0}|\pi^{0}\rangle$
$\displaystyle+$ $\displaystyle\langle
e\mu|\Big{(}L^{V,RR}_{\begin{subarray}{c}eu_{i}\\\ \mu
eu_{i}u_{i}\end{subarray}}-L^{V,RL}_{\begin{subarray}{c}eu_{i}\\\ \mu
eu_{i}u_{i}\end{subarray}}\Big{)}(\bar{\mu}_{R}\gamma^{\mu}e_{R})\frac{F_{0}}{2}\partial_{\mu}\pi^{0}|\pi^{0}\rangle$

where $L^{V,AB}_{\begin{subarray}{c}eu_{i}\\\ \mu
eu_{i}u_{i}\end{subarray}}=L^{V,AB}_{\begin{subarray}{c}eu\\\ \mu
euu\end{subarray}}-L^{V,AB}_{\begin{subarray}{c}ed\\\ \mu edd\end{subarray}}$.
For simplicity, lets define

$\displaystyle A_{LL}=L^{V,LR}_{\begin{subarray}{c}eu_{i}\\\ \mu
eu_{i}u_{i}\end{subarray}}-L^{V,LL}_{\begin{subarray}{c}eu_{i}\\\ \mu
eu_{i}u_{i}\end{subarray}}$ (25) $\displaystyle
A_{RR}=L^{V,RR}_{\begin{subarray}{c}eu_{i}\\\ \mu
eu_{i}u_{i}\end{subarray}}-L^{V,RL}_{\begin{subarray}{c}eu_{i}\\\ \mu
eu_{i}u_{i}\end{subarray}}$

The corresponding amplitude can be written as

$\displaystyle\mathcal{M}(\pi^{0}(p^{\pi^{0}})\rightarrow\mu^{+}(p_{\mu})e^{-}(p_{e}))$
$\displaystyle=$
$\displaystyle\displaystyle\frac{F_{0}}{2}\Big{(}A_{LL}(\bar{u}(p_{e})\gamma^{\mu}\frac{(1-\gamma^{5})}{2}v(p_{\mu}))p_{\mu}^{\pi_{0}}$
(26) $\displaystyle+$ $\displaystyle
A_{RR}(\bar{u}(p_{e})\gamma^{\mu}\frac{(1+\gamma^{5})}{2}v(p_{\mu}))p_{\mu}^{\pi_{0}}\Big{)}\
.$

Squaring this amplitude we get,

$\displaystyle\sum_{spin}|\mathcal{M}_{\pi\mu e}|^{2}$ $\displaystyle=$
$\displaystyle\displaystyle\frac{F_{0}^{2}}{4}\Big{[}2(|A_{LL}|^{2}+|A_{RR}|^{2})\frac{1}{2}\Big{(}m_{\pi^{0}}^{2}(m_{\mu}^{2}+m_{e}^{2})-(m_{\mu}^{2}-m_{e}^{2})^{2}\Big{)}\Big{]}$
(27)

In limit $m_{e}\rightarrow 0$, using masses in Table 4 and Eq.(26), the decay
width of neutral pion decay process could be computed to be,

$\displaystyle\Gamma(\pi^{0}\rightarrow\mu^{+}e^{-})$ $\displaystyle=$
$\displaystyle\displaystyle\frac{|\vec{p}_{\mu}|}{8\pi
m_{\pi^{0}}^{2}}\sum_{spin}|\mathcal{M}_{\pi^{0}\mu^{+}e^{-}}|^{2}$ (28)
$\displaystyle=$
$\displaystyle\displaystyle(|A_{LL}|^{2}+|A_{RR}|^{2})\times(9.57\times
10^{-9})\,\,(\mbox{GeV})^{5}$

This puts a constraint on the combination of the Wilson Coefficients as

$\displaystyle(|A_{LL}|^{2}+|A_{RR}|^{2})$ $\displaystyle\leq$ $\displaystyle
3.104\times 10^{-10}\,\,(\mbox{GeV})^{-4}$ (29)

We now compare the limits for the same from $\mu\to e$ conversion in Ti.
Following Kitano:2002mt  we have:

$Br(\mu N\to
eN)=\frac{2G_{F}^{2}(V^{(p)})^{2}(|\bar{g}_{LV}^{(p)}|^{2}+|\bar{g}_{RV}^{(p)}|^{2})}{w_{capt}},$
(30)

where $V^{(p)}$ represents vectorial form factors. For Ti, we find the limits
to be

$(|\bar{g}_{LV}^{(p)}|^{2}+|\bar{g}_{RV}^{(p)}|^{2})<4.0\times 10^{-14},$ (31)

which is far stronger than the one from Pion LFV decay. In Fig. 1 we show
limit (shaded region) in the plane of $A_{RR}$ and $A_{RL}$, as can be seen
the limit is of the order of $\sim 10^{-4}$ which becomes weaker along the
diagonal due to cancellations. As mentioned before RGE corrections are taken
in to consideration while plotting this graph.

##  IV Discussion

Though the results are be adaptable for any BSM NP model, in this section we
relie on Lepto-Quarks to generate the flavour violating coupling.
Renormalisable Lagrangian of scalar and vector Lepto-Quarks that couple to SM
without generating tree level proton decay operatorsAssad:2017iib  could be
written as

$\displaystyle\mathcal{L}_{\Delta}$ $\displaystyle=$
$\displaystyle\displaystyle\Big{(}\lambda^{ij}_{Lu}\bar{L}_{L}^{i}u^{j}_{R}+\lambda^{ij}_{eQ}\bar{e}^{i}_{R}i\tau_{2}Q^{j}_{L}\Big{)}\Delta_{-\frac{7}{6}}$
(32) $\displaystyle\mathcal{L}_{V}$ $\displaystyle=$
$\displaystyle\displaystyle\Big{(}g_{LQ}\bar{L}_{L}\gamma_{\mu}Q_{L}+g_{ed}\bar{e}_{R}\gamma_{\mu}d_{R}\Big{)}V_{-2/3}^{\mu}+g_{eu}\bar{e}_{R}\gamma_{\mu}u_{R}V_{-5/3}^{\mu}+g^{\prime}_{LQ}\bar{L}_{L}\gamma_{\mu}\tau^{a}Q_{L}V_{-2/3}^{\mu\
a}$ (33)

where $\Delta_{-7/6}$ is the scalar and $V^{\mu}_{-2/3},V^{\mu}_{-5/6},V^{\mu\
a}_{-2/3},$ are the vector Lepto-Quarks.

Using the above Lagrangian, let us discuss the tree level matching of the LEFT
Wilson coefficients that contribute to the $\pi^{0}\rightarrow\mu e$ process
with the Standard Model Effective Field Theory (SMEFT) operators generated
from the model.

Scalar operators :

At tree level,

$\displaystyle L^{S,RR}_{\begin{subarray}{c}eu\\\ \mu
euu\end{subarray}}=-C^{(1)}_{lequ}(\mu
euu)=\frac{\lambda^{ij}_{Lu}\lambda^{*ij}_{eQ}}{M_{\Delta_{7/6}}^{2}}\
,L^{S,RR}_{\begin{subarray}{c}ed\\\ \mu edd\end{subarray}}=0$ (34)

At tree level,

$\displaystyle L^{S,RL}_{\begin{subarray}{c}ed\\\ \mu
edd\end{subarray}}=\sum_{i=d,s,b}V_{di}C_{ledq}(\mu edi)=0\
,L^{S,RL}_{\begin{subarray}{c}eu\\\ \mu euu\end{subarray}}=0$ (35)

Then, from the constraint equation

$\Big{(}L^{S,RR}_{\begin{subarray}{c}eu_{i}\\\ \mu
eu_{i}u_{i}\end{subarray}}-L^{S,RL}_{\begin{subarray}{c}eu_{i}\\\ \mu
eu_{i}u_{i}\end{subarray}}\Big{)}\leq 7.03\times 10^{-7}\quad\mbox{GeV}^{-2}$
(36)

we get

$\frac{\lambda^{ij}_{Lu}\lambda^{*ij}_{eQ}}{M_{\Delta_{7/6}}^{2}}\leq
7.03\times 10^{-7}\quad\mbox{GeV}^{-2}$ (37)

Vector Fields :

At tree level,

$\displaystyle L^{V,LL}_{\begin{subarray}{c}eu\\\ \mu
euu\end{subarray}}=C^{(1)}_{lq}(\mu eq_{1}q_{1})-C^{(3)}_{lq}(\mu
eq_{1}q_{1})=0$ (38)

At tree level, assuming Lepto-Quark coupling to be universal,

$\displaystyle L^{V,LL}_{\begin{subarray}{c}ed\\\ \mu
edd\end{subarray}}=\sum_{i,j=d,s,b}\Big{(}C^{(1)}_{lq}(\mu
eij)+C^{(3)}_{lq}(\mu
eij)\Big{)}V_{jd}V^{\dagger}_{di}=\frac{g_{LQ}g^{*}_{ed}}{M_{V_{2/3}}^{2}}$
(39)

At tree level,

$\displaystyle L^{V,LR}_{\begin{subarray}{c}eu\\\ \mu
euu\end{subarray}}=C_{lu}(\mu euu)=0\ ,C_{\mu
edd}^{VLR}=L^{V,LL}_{\begin{subarray}{c}ed\\\ \mu edd\end{subarray}}=0$ (40)

At tree level,

$\displaystyle L^{V,RR}_{\begin{subarray}{c}eu\\\ \mu
euu\end{subarray}}=C_{eu}(\mu euu)=\frac{g_{eu}g^{*}_{eu}}{M_{V_{5/3}}^{2}}\
,L^{V,RR}_{\begin{subarray}{c}ed\\\ \mu edd\end{subarray}}=C_{ed}(\mu
edd)=\frac{g_{ed}g^{*}_{ed}}{M_{V_{2/3}}^{2}}$ (41)

At tree level,

$\displaystyle L^{V,RL}_{\begin{subarray}{c}eu\\\ \mu
euu\end{subarray}}=C_{qe}(\mu euu)=0\ ,L^{V,RL}_{\begin{subarray}{c}ed\\\ \mu
edd\end{subarray}}=C_{qe}(\mu edd)=0$ (42)

From the constraint on the vector LEFT operator’s Wilson coefficients

$\displaystyle(|A_{LL}|^{2}+|A_{RR}|^{2})$ $\displaystyle\leq$ $\displaystyle
3.104\times 10^{-10}\,\,(\mbox{GeV})^{-4},$ (43)

where

$\displaystyle A_{LL}=L^{V,LR}_{\begin{subarray}{c}eu_{i}\\\ \mu
eu_{i}u_{i}\end{subarray}}-L^{V,LL}_{\begin{subarray}{c}eu_{i}\\\ \mu
eu_{i}u_{i}\end{subarray}}$ (44) $\displaystyle
A_{RR}=L^{V,RR}_{\begin{subarray}{c}eu_{i}\\\ \mu
eu_{i}u_{i}\end{subarray}}-L^{V,RL}_{\begin{subarray}{c}eu_{i}\\\ \mu
eu_{i}u_{i}\end{subarray}}$

we get,

$\Big{|}\frac{g_{LQ}g^{*}_{ed}}{M_{V_{2/3}}^{2}}\Big{|}^{2}+\Big{|}\frac{g_{eu}g^{*}_{eu}}{M_{V_{5/3}}^{2}}-\frac{g_{ed}g^{*}_{ed}}{M_{V_{2/3}}^{2}}\Big{|}^{2}\leq
3.104\times 10^{-10}\,\,(\mbox{GeV})^{-4}$ (45)

In Table 5, we give the summary of the matching done at $M_{W}$ scale, for
detailed calculation refer to the Appendix V.2.

LeptoQuark Model | Coupling | Limit
---|---|---
Scalar LQ ($\Delta_{-\frac{7}{6}}$) | $\frac{\lambda^{ij}_{Lu}\lambda^{*ij}_{eQ}}{M_{\Delta_{7/6}}^{2}}$ | $\leq 7.03\times 10^{-7}\quad\mbox{GeV}^{-2}$
Vector LQ ($V^{\mu}_{-\frac{2}{3}}$, $V^{\mu}_{-\frac{5}{3}}$) | $\Big{|}\frac{g_{LQ}g^{*}_{ed}}{M_{V_{2/3}}^{2}}\Big{|}^{2}+\Big{|}\frac{g_{eu}g^{*}_{eu}}{M_{V_{5/3}}^{2}}-\frac{g_{ed}g^{*}_{ed}}{M_{V_{2/3}}^{2}}\Big{|}^{2}$ | $\leq 3.104\times 10^{-10}\,\,(\mbox{GeV})^{-4}$
Table 5: Limits on first two generation flavour violating couplings of Scalar
and Vector LeptoQuarks

###  IV.1 Summary

In this paper we derived the model independent bounds on Lepton flavour
violating LEFT operators in neutral pion decay process which involves only the
first generation quarks. By using LEFT-$\chi$PT matching it is clear that only
scalar and vector operators contribute to the process. Tensor operators, as
shown in Eq.(7), will always result in a photon emission along with the
dileptons. We have also shown that constraint on scalar operators are much
stringent than the one on vector operators. This is because of the scalar
amplitudes are enhanced by the condesation parameter $B_{0}$ compared to the
vector ones.

It is generally believed that $\mu\to e$ conversion in Nuclei gives the strong
bounds on the first two generational lepton number/flavour violating operators
of the $q^{2}l^{2}$ type. While this is generally true, the limits on the
process $\pi^{0}\to\mu e$ also give comparable constraints on scalar
$q^{2}l^{2}$ operators for vector operators, the limits are weaker than those
from $\mu\to e$ conversion.

###### Acknowledgements.

We are thankful to B. Ananthanarayan and G. D’Ambrosio for extensive
discussions. M.T.A. acknowledges financial support of DST through INSPIRE
Faculty grant [DST/INSPIRE/04/2019/002507]. SKV acknowledges support from the
Institute of Excellence grants from IISc. P.L. partially supported by the
National Science Centre, Poland, under research grant 2017/26/E/ST2/00135.

##  V Appendix

###  V.1 RGE

The needed LEFT Renormalization Group Equations Jenkins:2017dyc  for the
operators given in Table 2 is given as,

$\displaystyle{\dot{L}}^{S,RR}_{\begin{subarray}{c}eu\\\ prst\end{subarray}}$
$\displaystyle=-\left[6e^{2}(\mathsf{q}_{e}^{2}+\mathsf{q}_{u}^{2})+6g^{2}C_{F}\right]L^{S,RR}_{\begin{subarray}{c}eu\\\
prst\end{subarray}}-96e^{2}\mathsf{q}_{e}\mathsf{q}_{u}L^{T,RR}_{\begin{subarray}{c}eu\\\
prst\end{subarray}}-192e^{2}\mathsf{q}_{e}\mathsf{q}_{u}L_{\begin{subarray}{c}e\gamma\\\
pr\end{subarray}}L_{\begin{subarray}{c}u\gamma\\\ st\end{subarray}}\,,$ (46)
$\displaystyle{\dot{L}}^{S,RR}_{\begin{subarray}{c}ed\\\ prst\end{subarray}}$
$\displaystyle=-\left[6e^{2}(\mathsf{q}_{e}^{2}+\mathsf{q}_{d}^{2})+6g^{2}C_{F}\right]L^{S,RR}_{\begin{subarray}{c}ed\\\
prst\end{subarray}}-96e^{2}\mathsf{q}_{e}\mathsf{q}_{d}L^{T,RR}_{\begin{subarray}{c}ed\\\
prst\end{subarray}}-192e^{2}\mathsf{q}_{e}\mathsf{q}_{d}L_{\begin{subarray}{c}e\gamma\\\
pr\end{subarray}}L_{\begin{subarray}{c}d\gamma\\\ st\end{subarray}}\,,$ (47)
$\displaystyle{\dot{L}}^{S,RL}_{\begin{subarray}{c}eu\\\ prst\end{subarray}}$
$\displaystyle=-\left[6e^{2}(\mathsf{q}_{e}^{2}+\mathsf{q}_{u}^{2})+6g^{2}C_{F}\right]L^{S,RL}_{\begin{subarray}{c}eu\\\
prst\end{subarray}}-192e^{2}\mathsf{q}_{e}\mathsf{q}_{u}L_{\begin{subarray}{c}e\gamma\\\
pr\end{subarray}}L_{\begin{subarray}{c}u\gamma\\\ st\end{subarray}}\,,$ (48)
$\displaystyle{\dot{L}}^{S,RL}_{\begin{subarray}{c}ed\\\ prst\end{subarray}}$
$\displaystyle=-\left[6e^{2}(\mathsf{q}_{e}^{2}+\mathsf{q}_{d}^{2})+6g^{2}C_{F}\right]L^{S,RL}_{\begin{subarray}{c}ed\\\
prst\end{subarray}}-192e^{2}\mathsf{q}_{e}\mathsf{q}_{d}L_{\begin{subarray}{c}e\gamma\\\
pr\end{subarray}}L_{\begin{subarray}{c}d\gamma\\\ st\end{subarray}}\,,$ (49)
$\displaystyle{\dot{L}}^{V,LL}_{\begin{subarray}{c}eu\\\ prst\end{subarray}}$
$\displaystyle=\frac{4}{3}e^{2}\mathsf{q}_{e}\delta_{pr}\begin{aligned}
&\bigg{[}N_{c}\mathsf{q}_{d}\Bigl{(}L^{V,LL}_{\begin{subarray}{c}ud\\\
stww\end{subarray}}+L^{V,LR}_{\begin{subarray}{c}ud\\\
stww\end{subarray}}\Bigr{)}+N_{c}\mathsf{q}_{u}\Bigl{(}2L^{V,LL}_{\begin{subarray}{c}uu\\\
stww\end{subarray}}+L^{V,LR}_{\begin{subarray}{c}uu\\\
stww\end{subarray}}\Bigr{)}\\\
&+\mathsf{q}_{e}\Bigl{(}L^{V,LL}_{\begin{subarray}{c}eu\\\
wwst\end{subarray}}+L^{V,LR}_{\begin{subarray}{c}ue\\\
stww\end{subarray}}\Bigr{)}+2\mathsf{q}_{u}L^{V,LL}_{\begin{subarray}{c}uu\\\
swwt\end{subarray}}\bigg{]}\end{aligned}$ (50)
$\displaystyle\quad+\frac{4}{3}e^{2}\mathsf{q}_{u}\delta_{st}\begin{aligned}
&\bigg{[}N_{c}\mathsf{q}_{d}\Bigl{(}L^{V,LL}_{\begin{subarray}{c}ed\\\
prww\end{subarray}}+L^{V,LR}_{\begin{subarray}{c}ed\\\
prww\end{subarray}}\Bigr{)}+N_{c}\mathsf{q}_{u}\Bigl{(}L^{V,LL}_{\begin{subarray}{c}eu\\\
prww\end{subarray}}+L^{V,LR}_{\begin{subarray}{c}eu\\\
prww\end{subarray}}\Bigr{)}\\\
&+\mathsf{q}_{e}\Bigl{(}4L^{V,LL}_{\begin{subarray}{c}ee\\\
prww\end{subarray}}+L^{V,LR}_{\begin{subarray}{c}ee\\\
prww\end{subarray}}\Bigr{)}\bigg{]}\end{aligned}$
$\displaystyle\quad+12e^{2}\mathsf{q}_{e}\mathsf{q}_{u}L^{V,LL}_{\begin{subarray}{c}eu\\\
prst\end{subarray}}-\frac{16}{3}C_{F}e^{2}\mathsf{q}_{e}\mathsf{q}_{u}L_{\begin{subarray}{c}uG\\\
sw\end{subarray}}L^{*}_{\begin{subarray}{c}uG\\\
tw\end{subarray}}\delta_{pr}+8C_{F}eg\mathsf{q}_{e}\Bigl{(}L_{\begin{subarray}{c}uG\\\
sw\end{subarray}}L^{*}_{\begin{subarray}{c}u\gamma\\\
tw\end{subarray}}+L_{\begin{subarray}{c}u\gamma\\\
sw\end{subarray}}L^{*}_{\begin{subarray}{c}uG\\\
tw\end{subarray}}\Bigr{)}\delta_{pr}$
$\displaystyle\quad+e^{2}\Bigl{(}24\mathsf{q}_{u}^{2}+\frac{32}{3}\mathsf{q}_{e}\mathsf{q}_{u}\Bigr{)}L_{\begin{subarray}{c}e\gamma\\\
pw\end{subarray}}L^{*}_{\begin{subarray}{c}e\gamma\\\
rw\end{subarray}}\delta_{st}+e^{2}\Bigl{(}24\mathsf{q}_{e}^{2}+\frac{32}{3}\mathsf{q}_{e}\mathsf{q}_{u}\Bigr{)}L_{\begin{subarray}{c}u\gamma\\\
sw\end{subarray}}L^{*}_{\begin{subarray}{c}u\gamma\\\
tw\end{subarray}}\delta_{pr}$
$\displaystyle\quad+2e^{2}\mathsf{q}_{e}\mathsf{q}_{u}\zeta_{e}\delta_{pr}\delta_{st}\,,$
$\displaystyle{\dot{L}}^{V,LL}_{\begin{subarray}{c}ed\\\ prst\end{subarray}}$
$\displaystyle=\frac{4}{3}e^{2}\mathsf{q}_{e}\delta_{pr}\begin{aligned}
&\bigg{[}N_{c}\mathsf{q}_{u}\Bigl{(}L^{V,LL}_{\begin{subarray}{c}ud\\\
wwst\end{subarray}}+L^{V,LR}_{\begin{subarray}{c}du\\\
stww\end{subarray}}\Bigr{)}+N_{c}\mathsf{q}_{d}\Bigl{(}2L^{V,LL}_{\begin{subarray}{c}dd\\\
stww\end{subarray}}+L^{V,LR}_{\begin{subarray}{c}dd\\\
stww\end{subarray}}\Bigr{)}\\\
&+\mathsf{q}_{e}\Bigl{(}L^{V,LL}_{\begin{subarray}{c}ed\\\
wwst\end{subarray}}+L^{V,LR}_{\begin{subarray}{c}de\\\
stww\end{subarray}}\Bigr{)}+2\mathsf{q}_{d}L^{V,LL}_{\begin{subarray}{c}dd\\\
swwt\end{subarray}}\bigg{]}\end{aligned}$
$\displaystyle\quad+\frac{4}{3}e^{2}\mathsf{q}_{d}\delta_{st}\begin{aligned}
&\bigg{[}N_{c}\mathsf{q}_{d}\Bigl{(}L^{V,LL}_{\begin{subarray}{c}ed\\\
prww\end{subarray}}+L^{V,LR}_{\begin{subarray}{c}ed\\\
prww\end{subarray}}\Bigr{)}+N_{c}\mathsf{q}_{u}\Bigl{(}L^{V,LL}_{\begin{subarray}{c}eu\\\
prww\end{subarray}}+L^{V,LR}_{\begin{subarray}{c}eu\\\
prww\end{subarray}}\Bigr{)}\\\
&+\mathsf{q}_{e}\Bigl{(}4L^{V,LL}_{\begin{subarray}{c}ee\\\
prww\end{subarray}}+L^{V,LR}_{\begin{subarray}{c}ee\\\
prww\end{subarray}}\Bigr{)}\bigg{]}\end{aligned}$
$\displaystyle\quad+12e^{2}\mathsf{q}_{d}\mathsf{q}_{e}L^{V,LL}_{\begin{subarray}{c}ed\\\
prst\end{subarray}}-\frac{16}{3}C_{F}e^{2}\mathsf{q}_{e}\mathsf{q}_{d}L_{\begin{subarray}{c}dG\\\
sw\end{subarray}}L^{*}_{\begin{subarray}{c}dG\\\
tw\end{subarray}}\delta_{pr}+8C_{F}eg\mathsf{q}_{e}\Bigl{(}L_{\begin{subarray}{c}dG\\\
sw\end{subarray}}L^{*}_{\begin{subarray}{c}d\gamma\\\
tw\end{subarray}}+L_{\begin{subarray}{c}d\gamma\\\
sw\end{subarray}}L^{*}_{\begin{subarray}{c}dG\\\
tw\end{subarray}}\Bigr{)}\delta_{pr}$
$\displaystyle\quad+e^{2}\Bigl{(}24\mathsf{q}_{d}^{2}+\frac{32}{3}\mathsf{q}_{e}\mathsf{q}_{d}\Bigr{)}L_{\begin{subarray}{c}e\gamma\\\
pw\end{subarray}}L^{*}_{\begin{subarray}{c}e\gamma\\\
rw\end{subarray}}\delta_{st}+e^{2}\Bigl{(}24\mathsf{q}_{e}^{2}+\frac{32}{3}\mathsf{q}_{e}\mathsf{q}_{d}\Bigr{)}L_{\begin{subarray}{c}d\gamma\\\
sw\end{subarray}}L^{*}_{\begin{subarray}{c}d\gamma\\\
tw\end{subarray}}\delta_{pr}$
$\displaystyle\quad+2e^{2}\mathsf{q}_{e}\mathsf{q}_{d}\zeta_{e}\delta_{pr}\delta_{st}\,,$
$\displaystyle{\dot{L}}^{V,LR}_{\begin{subarray}{c}eu\\\ prst\end{subarray}}$
$\displaystyle=\frac{4}{3}e^{2}\mathsf{q}_{e}\delta_{pr}\begin{aligned}
&\bigg{[}N_{c}\mathsf{q}_{d}\Bigl{(}L^{V1,LR}_{\begin{subarray}{c}du\\\
wwst\end{subarray}}+L^{V1,RR}_{\begin{subarray}{c}ud\\\
stww\end{subarray}}\Bigr{)}+N_{c}\mathsf{q}_{u}\Bigl{(}L^{V1,LR}_{\begin{subarray}{c}uu\\\
wwst\end{subarray}}+2L^{V,RR}_{\begin{subarray}{c}uu\\\
stww\end{subarray}}\Bigr{)}\\\
&+\mathsf{q}_{e}\Bigl{(}L^{V,LR}_{\begin{subarray}{c}eu\\\
wwst\end{subarray}}+L^{V,RR}_{\begin{subarray}{c}eu\\\
wwst\end{subarray}}\Bigr{)}+2\mathsf{q}_{u}L^{V,RR}_{\begin{subarray}{c}uu\\\
swwt\end{subarray}}\bigg{]}\end{aligned}$ (52)
$\displaystyle\quad+\frac{4}{3}e^{2}\mathsf{q}_{u}\delta_{st}\begin{aligned}
&\bigg{[}N_{c}\mathsf{q}_{d}\Bigl{(}L^{V,LL}_{\begin{subarray}{c}ed\\\
prww\end{subarray}}+L^{V,LR}_{\begin{subarray}{c}ed\\\
prww\end{subarray}}\Bigr{)}+N_{c}\mathsf{q}_{u}\Bigl{(}L^{V,LL}_{\begin{subarray}{c}eu\\\
prww\end{subarray}}+L^{V,LR}_{\begin{subarray}{c}eu\\\
prww\end{subarray}}\Bigr{)}\\\
&+\mathsf{q}_{e}\Bigl{(}4L^{V,LL}_{\begin{subarray}{c}ee\\\
prww\end{subarray}}+L^{V,LR}_{\begin{subarray}{c}ee\\\
prww\end{subarray}}\Bigr{)}\bigg{]}\end{aligned}$
$\displaystyle\quad-12e^{2}\mathsf{q}_{e}\mathsf{q}_{u}L^{V,LR}_{\begin{subarray}{c}eu\\\
prst\end{subarray}}-\frac{16}{3}C_{F}e^{2}\mathsf{q}_{e}\mathsf{q}_{u}L^{*}_{\begin{subarray}{c}uG\\\
ws\end{subarray}}L_{\begin{subarray}{c}uG\\\
wt\end{subarray}}\delta_{pr}+8C_{F}eg\mathsf{q}_{e}\Bigl{(}L^{*}_{\begin{subarray}{c}uG\\\
ws\end{subarray}}L_{\begin{subarray}{c}u\gamma\\\
wt\end{subarray}}+L^{*}_{\begin{subarray}{c}u\gamma\\\
ws\end{subarray}}L_{\begin{subarray}{c}uG\\\
wt\end{subarray}}\Bigr{)}\delta_{pr}$
$\displaystyle\quad+e^{2}\Bigl{(}\frac{32}{3}\mathsf{q}_{e}\mathsf{q}_{u}-24\mathsf{q}_{u}^{2}\Bigr{)}L_{\begin{subarray}{c}e\gamma\\\
pw\end{subarray}}L^{*}_{\begin{subarray}{c}e\gamma\\\
rw\end{subarray}}\delta_{st}+e^{2}\Bigl{(}\frac{32}{3}\mathsf{q}_{e}\mathsf{q}_{u}-24\mathsf{q}_{e}^{2}\Bigr{)}L^{*}_{\begin{subarray}{c}u\gamma\\\
ws\end{subarray}}L_{\begin{subarray}{c}u\gamma\\\
wt\end{subarray}}\delta_{pr}$
$\displaystyle\quad+2e^{2}\mathsf{q}_{u}\mathsf{q}_{e}\zeta_{e}\delta_{pr}\delta_{st}\,,$
$\displaystyle{\dot{L}}^{V,LR}_{\begin{subarray}{c}ed\\\ prst\end{subarray}}$
$\displaystyle=\frac{4}{3}e^{2}\mathsf{q}_{e}\delta_{pr}\begin{aligned}
&\bigg{[}N_{c}\mathsf{q}_{d}\Bigl{(}L^{V1,LR}_{\begin{subarray}{c}dd\\\
wwst\end{subarray}}+2L^{V,RR}_{\begin{subarray}{c}dd\\\
stww\end{subarray}}\Bigr{)}+N_{c}\mathsf{q}_{u}\Bigl{(}L^{V1,LR}_{\begin{subarray}{c}ud\\\
wwst\end{subarray}}+L^{V1,RR}_{\begin{subarray}{c}ud\\\
wwst\end{subarray}}\Bigr{)}\\\
&+\mathsf{q}_{e}\Bigl{(}L^{V,LR}_{\begin{subarray}{c}ed\\\
wwst\end{subarray}}+L^{V,RR}_{\begin{subarray}{c}ed\\\
wwst\end{subarray}}\Bigr{)}+2\mathsf{q}_{d}L^{V,RR}_{\begin{subarray}{c}dd\\\
swwt\end{subarray}}\bigg{]}\end{aligned}$ (53)
$\displaystyle\quad+\frac{4}{3}e^{2}\mathsf{q}_{d}\delta_{st}\begin{aligned}
&\bigg{[}N_{c}\mathsf{q}_{d}\Bigl{(}L^{V,LL}_{\begin{subarray}{c}ed\\\
prww\end{subarray}}+L^{V,LR}_{\begin{subarray}{c}ed\\\
prww\end{subarray}}\Bigr{)}+N_{c}\mathsf{q}_{u}\Bigl{(}L^{V,LL}_{\begin{subarray}{c}eu\\\
prww\end{subarray}}+L^{V,LR}_{\begin{subarray}{c}eu\\\
prww\end{subarray}}\Bigr{)}\\\
&+\mathsf{q}_{e}\Bigl{(}4L^{V,LL}_{\begin{subarray}{c}ee\\\
prww\end{subarray}}+L^{V,LR}_{\begin{subarray}{c}ee\\\
prww\end{subarray}}\Bigr{)}\bigg{]}\end{aligned}$
$\displaystyle\quad-12e^{2}\mathsf{q}_{d}\mathsf{q}_{e}L^{V,LR}_{\begin{subarray}{c}ed\\\
prst\end{subarray}}-\frac{16}{3}C_{F}e^{2}\mathsf{q}_{e}\mathsf{q}_{d}L^{*}_{\begin{subarray}{c}dG\\\
ws\end{subarray}}L_{\begin{subarray}{c}dG\\\
wt\end{subarray}}\delta_{pr}+8C_{F}eg\mathsf{q}_{e}\Bigl{(}L^{*}_{\begin{subarray}{c}dG\\\
ws\end{subarray}}L_{\begin{subarray}{c}d\gamma\\\
wt\end{subarray}}+L^{*}_{\begin{subarray}{c}d\gamma\\\
ws\end{subarray}}L_{\begin{subarray}{c}dG\\\
wt\end{subarray}}\Bigr{)}\delta_{pr}$
$\displaystyle\quad+e^{2}\Bigl{(}\frac{32}{3}\mathsf{q}_{e}\mathsf{q}_{d}-24\mathsf{q}_{d}^{2}\Bigr{)}L_{\begin{subarray}{c}e\gamma\\\
pw\end{subarray}}L^{*}_{\begin{subarray}{c}e\gamma\\\
rw\end{subarray}}\delta_{st}+e^{2}\Bigl{(}\frac{32}{3}\mathsf{q}_{e}\mathsf{q}_{d}-24\mathsf{q}_{e}^{2}\Bigr{)}L^{*}_{\begin{subarray}{c}d\gamma\\\
ws\end{subarray}}L_{\begin{subarray}{c}d\gamma\\\
wt\end{subarray}}\delta_{pr}$
$\displaystyle\quad+2e^{2}\mathsf{q}_{d}\mathsf{q}_{e}\zeta_{e}\delta_{pr}\delta_{st}\,,$
$\displaystyle{\dot{L}}^{V,RR}_{\begin{subarray}{c}eu\\\ prst\end{subarray}}$
$\displaystyle=\frac{4}{3}e^{2}\mathsf{q}_{e}\delta_{pr}\begin{aligned}
&\bigg{[}N_{c}\mathsf{q}_{d}\Bigl{(}L^{V,LR}_{\begin{subarray}{c}du\\\
wwst\end{subarray}}+L^{V,RR}_{\begin{subarray}{c}ud\\\
stww\end{subarray}})+N_{c}\mathsf{q}_{u}\Bigl{(}L^{V,LR}_{\begin{subarray}{c}uu\\\
wwst\end{subarray}}+2L^{V,RR}_{\begin{subarray}{c}uu\\\
stww\end{subarray}}\Bigr{)}\\\
&+\mathsf{q}_{e}\Bigl{(}L^{V,LR}_{\begin{subarray}{c}eu\\\
wwst\end{subarray}}+L^{V,RR}_{\begin{subarray}{c}eu\\\
wwst\end{subarray}}\Bigr{)}+2\mathsf{q}_{u}L^{V,RR}_{\begin{subarray}{c}uu\\\
swwt\end{subarray}}\bigg{]}\end{aligned}$
$\displaystyle\quad+\frac{4}{3}e^{2}\mathsf{q}_{u}\delta_{st}\begin{aligned}
&\bigg{[}N_{c}\mathsf{q}_{d}\Bigl{(}L^{V,LR}_{\begin{subarray}{c}de\\\
wwpr\end{subarray}}+L^{V,RR}_{\begin{subarray}{c}ed\\\
prww\end{subarray}}\Bigr{)}+N_{c}\mathsf{q}_{u}\Bigl{(}L^{V,LR}_{\begin{subarray}{c}ue\\\
wwpr\end{subarray}}+L^{V,RR}_{\begin{subarray}{c}eu\\\
prww\end{subarray}}\Bigr{)}\\\
&+\mathsf{q}_{e}\Bigl{(}L^{V,LR}_{\begin{subarray}{c}ee\\\
wwpr\end{subarray}}+4L^{V,RR}_{\begin{subarray}{c}ee\\\
prww\end{subarray}}\Bigr{)}\bigg{]}\end{aligned}$
$\displaystyle\quad+12e^{2}\mathsf{q}_{e}\mathsf{q}_{u}L^{V,RR}_{\begin{subarray}{c}eu\\\
prst\end{subarray}}-\frac{16}{3}C_{F}e^{2}\mathsf{q}_{e}\mathsf{q}_{u}L^{*}_{\begin{subarray}{c}uG\\\
ws\end{subarray}}L_{\begin{subarray}{c}uG\\\
wt\end{subarray}}\delta_{pr}+8C_{F}eg\mathsf{q}_{e}\Bigl{(}L^{*}_{\begin{subarray}{c}uG\\\
ws\end{subarray}}L_{\begin{subarray}{c}u\gamma\\\
wt\end{subarray}}+L^{*}_{\begin{subarray}{c}u\gamma\\\
ws\end{subarray}}L_{\begin{subarray}{c}uG\\\
wt\end{subarray}}\Bigr{)}\delta_{pr}$
$\displaystyle\quad+e^{2}\Bigl{(}24\mathsf{q}_{u}^{2}+\frac{32}{3}\mathsf{q}_{e}\mathsf{q}_{u}\Bigr{)}L^{*}_{\begin{subarray}{c}e\gamma\\\
wp\end{subarray}}L_{\begin{subarray}{c}e\gamma\\\
wr\end{subarray}}\delta_{st}+e^{2}\Bigl{(}24\mathsf{q}_{e}^{2}+\frac{32}{3}\mathsf{q}_{e}\mathsf{q}_{u}\Bigr{)}L^{*}_{\begin{subarray}{c}u\gamma\\\
ws\end{subarray}}L_{\begin{subarray}{c}u\gamma\\\
wt\end{subarray}}\delta_{pr}$
$\displaystyle\quad+2e^{2}\mathsf{q}_{e}\mathsf{q}_{u}\zeta_{e}\delta_{pr}\delta_{st}\,,$
$\displaystyle{\dot{L}}^{V,RR}_{\begin{subarray}{c}ed\\\ prst\end{subarray}}$
$\displaystyle=\frac{4}{3}e^{2}\mathsf{q}_{e}\delta_{pr}\begin{aligned}
&\bigg{[}N_{c}\mathsf{q}_{d}\Bigl{(}L^{V,LR}_{\begin{subarray}{c}dd\\\
wwst\end{subarray}}+2L^{V,RR}_{\begin{subarray}{c}dd\\\
stww\end{subarray}}\Bigr{)}+N_{c}\mathsf{q}_{u}\Bigl{(}L^{V,LR}_{\begin{subarray}{c}ud\\\
wwst\end{subarray}}+L^{V,RR}_{\begin{subarray}{c}ud\\\
wwst\end{subarray}}\Bigr{)}\\\
&+\mathsf{q}_{e}\Bigl{(}L^{V,LR}_{\begin{subarray}{c}ed\\\
wwst\end{subarray}}+L^{V,RR}_{\begin{subarray}{c}ed\\\
wwst\end{subarray}}\Bigr{)}+2\mathsf{q}_{d}L^{V,RR}_{\begin{subarray}{c}dd\\\
swwt\end{subarray}}\bigg{]}\end{aligned}$ (55)
$\displaystyle\quad+\frac{4}{3}e^{2}\mathsf{q}_{d}\delta_{st}\begin{aligned}
&\bigg{[}N_{c}\mathsf{q}_{d}\Bigl{(}L^{V,LR}_{\begin{subarray}{c}de\\\
wwpr\end{subarray}}+L^{V,RR}_{\begin{subarray}{c}ed\\\
prww\end{subarray}}\Bigr{)}+N_{c}\mathsf{q}_{u}\Bigl{(}L^{V,LR}_{\begin{subarray}{c}ue\\\
wwpr\end{subarray}}+L^{V,RR}_{\begin{subarray}{c}eu\\\
prww\end{subarray}}\Bigr{)}\\\
&+\mathsf{q}_{e}\Bigl{(}L^{V,LR}_{\begin{subarray}{c}ee\\\
wwpr\end{subarray}}+4L^{V,RR}_{\begin{subarray}{c}ee\\\
prww\end{subarray}}\Bigr{)}\bigg{]}\end{aligned}$
$\displaystyle\quad+12e^{2}\mathsf{q}_{d}\mathsf{q}_{e}L^{V,RR}_{\begin{subarray}{c}ed\\\
prst\end{subarray}}-\frac{16}{3}C_{F}e^{2}\mathsf{q}_{e}\mathsf{q}_{d}L^{*}_{\begin{subarray}{c}dG\\\
ws\end{subarray}}L_{\begin{subarray}{c}dG\\\
wt\end{subarray}}\delta_{pr}+8C_{F}eg\mathsf{q}_{e}\Bigl{(}L^{*}_{\begin{subarray}{c}dG\\\
ws\end{subarray}}L_{\begin{subarray}{c}d\gamma\\\
wt\end{subarray}}+L^{*}_{\begin{subarray}{c}d\gamma\\\
ws\end{subarray}}L_{\begin{subarray}{c}dG\\\
wt\end{subarray}}\Bigr{)}\delta_{pr}$
$\displaystyle\quad+e^{2}\Bigl{(}24\mathsf{q}_{d}^{2}+\frac{32}{3}\mathsf{q}_{e}\mathsf{q}_{d}\Bigr{)}L^{*}_{\begin{subarray}{c}e\gamma\\\
wp\end{subarray}}L_{\begin{subarray}{c}e\gamma\\\
wr\end{subarray}}\delta_{st}+e^{2}\Bigl{(}24\mathsf{q}_{e}^{2}+\frac{32}{3}\mathsf{q}_{e}\mathsf{q}_{d}\Bigr{)}L^{*}_{\begin{subarray}{c}d\gamma\\\
ws\end{subarray}}L_{\begin{subarray}{c}d\gamma\\\
wt\end{subarray}}\delta_{pr}$
$\displaystyle\quad+2e^{2}\mathsf{q}_{e}\mathsf{q}_{d}\zeta_{e}\delta_{pr}\delta_{st}\,,$
$\displaystyle{\dot{L}}^{V,LR}_{\begin{subarray}{c}ue\\\ prst\end{subarray}}$
$\displaystyle=\frac{4}{3}e^{2}\mathsf{q}_{u}\delta_{pr}\begin{aligned}
&\bigg{[}N_{c}\mathsf{q}_{d}\Bigl{(}L^{V,LR}_{\begin{subarray}{c}de\\\
wwst\end{subarray}}+L^{V,RR}_{\begin{subarray}{c}ed\\\
stww\end{subarray}}\Bigr{)}+N_{c}\mathsf{q}_{u}\Bigl{(}L^{V,LR}_{\begin{subarray}{c}ue\\\
wwst\end{subarray}}+L^{V,RR}_{\begin{subarray}{c}eu\\\
stww\end{subarray}}\Bigr{)}\\\
&+\mathsf{q}_{e}\Bigl{(}L^{V,LR}_{\begin{subarray}{c}ee\\\
wwst\end{subarray}}+4L^{V,RR}_{\begin{subarray}{c}ee\\\
stww\end{subarray}}\Bigr{)}\bigg{]}\end{aligned}$ (56)
$\displaystyle\quad+\frac{4}{3}e^{2}\mathsf{q}_{e}\delta_{st}\begin{aligned}
&\bigg{[}N_{c}\mathsf{q}_{d}\Bigl{(}L^{V1,LL}_{\begin{subarray}{c}ud\\\
prww\end{subarray}}+L^{V1,LR}_{\begin{subarray}{c}ud\\\
prww\end{subarray}}\Bigr{)}+N_{c}\mathsf{q}_{u}\Bigl{(}2L^{V,LL}_{\begin{subarray}{c}uu\\\
prww\end{subarray}}+L^{V1,LR}_{\begin{subarray}{c}uu\\\
prww\end{subarray}}\Bigr{)}\\\
&+\mathsf{q}_{e}\Bigl{(}L^{V,LL}_{\begin{subarray}{c}eu\\\
wwpr\end{subarray}}+L^{V,LR}_{\begin{subarray}{c}ue\\\
prww\end{subarray}}\Bigr{)}+2\mathsf{q}_{u}L^{V,LL}_{\begin{subarray}{c}uu\\\
pwwr\end{subarray}}\bigg{]}\end{aligned}$
$\displaystyle\quad-12e^{2}\mathsf{q}_{e}\mathsf{q}_{u}L^{V,LR}_{\begin{subarray}{c}ue\\\
prst\end{subarray}}-\frac{16}{3}C_{F}e^{2}\mathsf{q}_{e}\mathsf{q}_{u}L_{\begin{subarray}{c}uG\\\
pw\end{subarray}}L^{*}_{\begin{subarray}{c}uG\\\
rw\end{subarray}}\delta_{st}+8C_{F}eg\mathsf{q}_{e}\Bigl{(}L_{\begin{subarray}{c}uG\\\
pw\end{subarray}}L^{*}_{\begin{subarray}{c}u\gamma\\\
rw\end{subarray}}+L_{\begin{subarray}{c}u\gamma\\\
pw\end{subarray}}L^{*}_{\begin{subarray}{c}uG\\\
rw\end{subarray}}\Bigr{)}\delta_{st}$
$\displaystyle\quad+e^{2}\Bigl{(}\frac{32}{3}\mathsf{q}_{e}\mathsf{q}_{u}-24\mathsf{q}_{e}^{2}\Bigr{)}L_{\begin{subarray}{c}u\gamma\\\
pw\end{subarray}}L^{*}_{\begin{subarray}{c}u\gamma\\\
rw\end{subarray}}\delta_{st}+e^{2}\Bigl{(}\frac{32}{3}\mathsf{q}_{e}\mathsf{q}_{u}-24\mathsf{q}_{u}^{2}\Bigr{)}L^{*}_{\begin{subarray}{c}e\gamma\\\
ws\end{subarray}}L_{\begin{subarray}{c}e\gamma\\\
wt\end{subarray}}\delta_{pr}$
$\displaystyle\quad+2e^{2}\mathsf{q}_{u}\mathsf{q}_{e}\zeta_{e}\delta_{pr}\delta_{st}\,,$
$\displaystyle{\dot{L}}^{V,LR}_{\begin{subarray}{c}de\\\ prst\end{subarray}}$
$\displaystyle=\frac{4}{3}e^{2}\mathsf{q}_{d}\delta_{pr}\begin{aligned}
&\bigg{[}N_{c}\mathsf{q}_{d}\Bigl{(}L^{V,LR}_{\begin{subarray}{c}de\\\
wwst\end{subarray}}+L^{V,RR}_{\begin{subarray}{c}ed\\\
stww\end{subarray}}\Bigr{)}+N_{c}\mathsf{q}_{u}\Bigl{(}L^{V,LR}_{\begin{subarray}{c}ue\\\
wwst\end{subarray}}+L^{V,RR}_{\begin{subarray}{c}eu\\\
stww\end{subarray}}\Bigr{)}\\\
&+\mathsf{q}_{e}\Bigl{(}L^{V,LR}_{\begin{subarray}{c}ee\\\
wwst\end{subarray}}+4L^{V,RR}_{\begin{subarray}{c}ee\\\
stww\end{subarray}}\Bigr{)}\bigg{]}\end{aligned}$
$\displaystyle\quad+\frac{4}{3}e^{2}\mathsf{q}_{e}\delta_{st}\begin{aligned}
&\bigg{[}N_{c}\mathsf{q}_{d}\Bigl{(}2L^{V,LL}_{\begin{subarray}{c}dd\\\
prww\end{subarray}}+L^{V1,LR}_{\begin{subarray}{c}dd\\\
prww\end{subarray}}\Bigr{)}+N_{c}\mathsf{q}_{u}\Bigl{(}L^{V1,LL}_{\begin{subarray}{c}ud\\\
wwpr\end{subarray}}+L^{V1,LR}_{\begin{subarray}{c}du\\\
prww\end{subarray}}\Bigr{)}\\\
&+\mathsf{q}_{e}\Bigl{(}L^{V,LL}_{\begin{subarray}{c}ed\\\
wwpr\end{subarray}}+L^{V,LR}_{\begin{subarray}{c}de\\\
prww\end{subarray}}\Bigr{)}+2\mathsf{q}_{d}L^{V,LL}_{\begin{subarray}{c}dd\\\
pwwr\end{subarray}}\bigg{]}\end{aligned}$
$\displaystyle\quad-12e^{2}\mathsf{q}_{d}\mathsf{q}_{e}L^{V,LR}_{\begin{subarray}{c}de\\\
prst\end{subarray}}-\frac{16}{3}C_{F}e^{2}\mathsf{q}_{e}\mathsf{q}_{d}L_{\begin{subarray}{c}dG\\\
pw\end{subarray}}L^{*}_{\begin{subarray}{c}dG\\\
rw\end{subarray}}\delta_{st}+8C_{F}eg\mathsf{q}_{e}\Bigl{(}L_{\begin{subarray}{c}dG\\\
pw\end{subarray}}L^{*}_{\begin{subarray}{c}d\gamma\\\
rw\end{subarray}}+L_{\begin{subarray}{c}d\gamma\\\
pw\end{subarray}}L^{*}_{\begin{subarray}{c}dG\\\
rw\end{subarray}}\Bigr{)}\delta_{st}$
$\displaystyle\quad+e^{2}\Bigl{(}\frac{32}{3}\mathsf{q}_{e}\mathsf{q}_{d}-24\mathsf{q}_{e}^{2}\Bigr{)}L_{\begin{subarray}{c}d\gamma\\\
pw\end{subarray}}L^{*}_{\begin{subarray}{c}d\gamma\\\
rw\end{subarray}}\delta_{st}+e^{2}\Bigl{(}\frac{32}{3}\mathsf{q}_{e}\mathsf{q}_{d}-24\mathsf{q}_{d}^{2}\Bigr{)}L^{*}_{\begin{subarray}{c}e\gamma\\\
ws\end{subarray}}L_{\begin{subarray}{c}e\gamma\\\
wt\end{subarray}}\delta_{pr}$
$\displaystyle\quad+2e^{2}\mathsf{q}_{d}\mathsf{q}_{e}\zeta_{e}\delta_{pr}\delta_{st}\,,$

###  V.2 SMEFT matching

Scalar Operators | Matching
---|---
$\mathcal{O}^{S,RR}_{eu}$ | $-C^{(1)}_{\begin{subarray}{c}lequ\\\ prst\end{subarray}}$
$\mathcal{O}^{S,RR}_{ed}$ | 0
$\mathcal{O}^{S,RL}_{eu}$ | 0
$\mathcal{O}^{S,RL}_{ed}$ | $C_{\begin{subarray}{c}ledq\\\ prst\end{subarray}}$

Table 6: LEFT-SMEFT matching of Scalar operators at tree level Jenkins:2017jig
.

Vector Operators | Matching
---|---
$\mathcal{O}^{V,LL}_{eu}$ | $C^{(1)}_{\begin{subarray}{c}lq\\\ prst\end{subarray}}-C^{(3)}_{\begin{subarray}{c}lq\\\ prst\end{subarray}}-\frac{{\overline{g}_{Z}}^{2}}{M_{Z}^{2}}\left[Z_{e_{L}}\right]_{pr}\left[Z_{u_{L}}\right]_{st}$
$\mathcal{O}^{V,LL}_{ed}$ | $C^{(1)}_{\begin{subarray}{c}lq\\\ prst\end{subarray}}+C^{(3)}_{\begin{subarray}{c}lq\\\ prst\end{subarray}}-\frac{{\overline{g}_{Z}}^{2}}{M_{Z}^{2}}\left[Z_{e_{L}}\right]_{pr}\left[Z_{d_{L}}\right]_{st}$
$\mathcal{O}^{V,RR}_{eu}$ | $C_{\begin{subarray}{c}eu\\\ prst\end{subarray}}-\frac{{\overline{g}_{Z}}^{2}}{M_{Z}^{2}}\left[Z_{e_{R}}\right]_{pr}\left[Z_{u_{R}}\right]_{st}$
$\mathcal{O}^{V,RR}_{ed}$ | $C_{\begin{subarray}{c}ed\\\ prst\end{subarray}}-\frac{{\overline{g}_{Z}}^{2}}{M_{Z}^{2}}\left[Z_{e_{R}}\right]_{pr}\left[Z_{d_{R}}\right]_{st}$
$\mathcal{O}^{V,LR}_{eu}$ | $C_{\begin{subarray}{c}lu\\\ prst\end{subarray}}-\frac{{\overline{g}_{Z}}^{2}}{M_{Z}^{2}}\left[Z_{e_{L}}\right]_{pr}\left[Z_{u_{R}}\right]_{st}$
$\mathcal{O}^{V,LR}_{ed}$ | $C_{\begin{subarray}{c}ld\\\ prst\end{subarray}}-\frac{{\overline{g}_{Z}}^{2}}{M_{Z}^{2}}\left[Z_{e_{L}}\right]_{pr}\left[Z_{d_{R}}\right]_{st}$
$\mathcal{O}^{V,RL}_{eu}$ | $C_{\begin{subarray}{c}qe\\\ prst\end{subarray}}-\frac{{\overline{g}_{Z}}^{2}}{M_{Z}^{2}}\left[Z_{u_{L}}\right]_{pr}\left[Z_{e_{R}}\right]_{st}$
$\mathcal{O}^{V,RL}_{ed}$ | $C_{\begin{subarray}{c}qe\\\ prst\end{subarray}}-\frac{{\overline{g}_{Z}}^{2}}{M_{Z}^{2}}\left[Z_{d_{L}}\right]_{pr}\left[Z_{e_{R}}\right]_{st}$

Table 7: LEFT-SMEFT matching of Vector operators at tree level Jenkins:2017jig
.

where

$\displaystyle[Z_{e_{L}}]_{pr}$
$\displaystyle=\left[\delta_{pr}\left(-\frac{1}{2}+{\overline{s}}^{2}\right)-\frac{1}{2}v_{T}^{2}C^{(1)}_{\begin{subarray}{c}Hl\\\
pr\end{subarray}}-\frac{1}{2}v_{T}^{2}C^{(3)}_{\begin{subarray}{c}Hl\\\
pr\end{subarray}}\right],\quad[Z_{e_{R}}]_{pr}$
$\displaystyle=\left[\delta_{pr}\left(+{\overline{s}}^{2}\right)-\frac{1}{2}v_{T}^{2}C_{\begin{subarray}{c}He\\\
pr\end{subarray}}\right],$ (58) $\displaystyle[Z_{u_{L}}]_{pr}$
$\displaystyle=\left[\delta_{pr}\left(\frac{1}{2}-\frac{2}{3}{\overline{s}}^{2}\right)-\frac{1}{2}v_{T}^{2}C^{(1)}_{\begin{subarray}{c}Hq\\\
pr\end{subarray}}+\frac{1}{2}v_{T}^{2}C^{(3)}_{\begin{subarray}{c}Hq\\\
pr\end{subarray}}\right],\quad[Z_{u_{R}}]_{pr}$
$\displaystyle=\left[\delta_{pr}\left(-\frac{2}{3}{\overline{s}}^{2}\right)-\frac{1}{2}v_{T}^{2}C_{\begin{subarray}{c}Hu\\\
pr\end{subarray}}\right],$ (59) $\displaystyle[Z_{d_{L}}]_{pr}$
$\displaystyle=\left[\delta_{pr}\left(-\frac{1}{2}+\frac{1}{3}{\overline{s}}^{2}\right)-\frac{1}{2}v_{T}^{2}C^{(1)}_{\begin{subarray}{c}Hq\\\
pr\end{subarray}}-\frac{1}{2}v_{T}^{2}C^{(3)}_{\begin{subarray}{c}Hq\\\
pr\end{subarray}}\right],\quad[Z_{d_{R}}]_{pr}$
$\displaystyle=\left[\delta_{pr}\left(+\frac{1}{3}{\overline{s}}^{2}\right)-\frac{1}{2}v_{T}^{2}C_{\begin{subarray}{c}Hd\\\
pr\end{subarray}}\right].$ (60)

## References

  * (1) C. Caso et al., “Review of particle physics. Particle Data Group,” Eur. Phys. J. C, vol. 3, pp. 1–794, 1998.
  * (2) N. Nagaosa and P. A. Lee, “Normal-state properties of the uniform resonating-valence-bond state,” Phys. Rev. Lett., vol. 64, pp. 2450–2453, May 1990.
  * (3) D. Ambrose et al., “New limit on muon and electron lepton number violation from K0(L) —$>$ mu+- e-+ decay,” Phys. Rev. Lett., vol. 81, pp. 5734–5737, 1998.
  * (4) K. Arisaka et al., “Search for the lepton-family number violating decays K(L) –$>$ pi0 mu+- e-+,” Phys. Lett. B, vol. 432, pp. 230–234, 1998.
  * (5) R. Appel et al., “Search for lepton flavor violation in K+ decays,” Phys. Rev. Lett., vol. 85, pp. 2877–2880, 2000.
  * (6) R. Appel et al., “An Improved limit on the rate of decay K+ —$>$ pi+ muon+ e-,” Phys. Rev. Lett., vol. 85, pp. 2450–2453, 2000.
  * (7) E. Abouzaid et al., “Search for lepton flavor violating decays of the neutral kaon,” Phys. Rev. Lett., vol. 100, p. 131803, 2008.
  * (8) E. Cortina Gil et al., “Search for Lepton Number and Flavor Violation in $K^{+}$ and $\pi^{0}$ Decays,” Phys. Rev. Lett., vol. 127, no. 13, p. 131802, 2021.
  * (9) M. Hoferichter, J. Menéndez, and F. Noël, “Improved limits on lepton-flavor-violating decays of light pseudoscalars via spin-dependent $\mu\to e$ conversion in nuclei,” 4 2022.
  * (10) L. Calibbi, A. Faccia, A. Masiero, and S. K. Vempati, “Lepton flavour violation from SUSY-GUTs: Where do we stand for MEG, PRISM/PRIME and a super flavour factory,” Phys. Rev. D, vol. 74, p. 116002, 2006.
  * (11) L. Calibbi and G. Signorelli, “Charged Lepton Flavour Violation: An Experimental and Theoretical Introduction,” Riv. Nuovo Cim., vol. 41, no. 2, pp. 71–174, 2018.
  * (12) Y. Kuno and Y. Okada, “Muon decay and physics beyond the standard model,” Rev. Mod. Phys., vol. 73, pp. 151–202, 2001.
  * (13) J. Kaulard et al., “Improved limit on the branching ratio of mu- –$>$ e+ conversion on titanium,” Phys. Lett. B, vol. 422, pp. 334–338, 1998.
  * (14) W. H. Bertl et al., “A Search for muon to electron conversion in muonic gold,” Eur. Phys. J. C, vol. 47, pp. 337–346, 2006.
  * (15) Y. Aoki et al., “FLAG Review 2021,” 11 2021.
  * (16) E. E. Jenkins, A. V. Manohar, and P. Stoffer, “Low-Energy Effective Field Theory below the Electroweak Scale: Anomalous Dimensions,” JHEP, vol. 01, p. 084, 2018.
  * (17) E. E. Jenkins, A. V. Manohar, and P. Stoffer, “Low-Energy Effective Field Theory below the Electroweak Scale: Operators and Matching,” JHEP, vol. 03, p. 016, 2018.
  * (18) J. Gasser and H. Leutwyler, “Chiral Perturbation Theory to One Loop,” Annals Phys., vol. 158, p. 142, 1984.
  * (19) J. Gasser and H. Leutwyler, “Chiral Perturbation Theory: Expansions in the Mass of the Strange Quark,” Nucl. Phys. B, vol. 250, pp. 465–516, 1985\.
  * (20) D. S. M. Alves and N. Weiner, “A viable QCD axion in the MeV mass range,” JHEP, vol. 07, p. 092, 2018.
  * (21) Beddall, A. and Beddall, A., “Measurement of the dalitz-decay branching ratio of the,” Eur. Phys. J. C, vol. 54, no. 3, pp. 365–370, 2008.
  * (22) W. Dekens, E. E. Jenkins, A. V. Manohar, and P. Stoffer, “Non-perturbative effects in $\mu\to e\gamma$,” JHEP, vol. 01, p. 088, 2019.
  * (23) P. Zyla et al., “Review of Particle Physics,” PTEP, vol. 2020, no. 8, p. 083C01, 2020.
  * (24) R. Kitano, M. Koike, and Y. Okada, “Detailed calculation of lepton flavor violating muon electron conversion rate for various nuclei,” Phys. Rev. D, vol. 66, p. 096002, 2002.  [Erratum: Phys.Rev.D 76, 059902 (2007)].
  * (25) N. Assad, B. Fornal, and B. Grinstein, “Baryon Number and Lepton Universality Violation in Leptoquark and Diquark Models,” Phys. Lett. B, vol. 777, pp. 324–331, 2018.