#  One-loop Matching of Scotogenic Model onto Standard Model Effective Field
Theory up to Dimension 7

Yi Liao<EMAIL_ADDRESS>and Xiao-Dong Ma<EMAIL_ADDRESS>
a Guangdong Provincial Key Laboratory of Nuclear Science, Institute of Quantum
Matter,
South China Normal University, Guangzhou 510006, China
b Guangdong-Hong Kong Joint Laboratory of Quantum Matter,
Southern Nuclear Science Computing Center,
South China Normal University, Guangzhou 510006, China
c Tsung-Dao Lee Institute & School of Physics and Astronomy,
Shanghai Jiao Tong University, Shanghai 200240, China

###### Abstract

The scotogenic neutrino seesaw model is a minimal extension of the standard
model with three $\mathbb{Z}_{2}$-odd right-handed singlet fermions $N$ and
one $\mathbb{Z}_{2}$-odd Higgs doublet $\eta$ that can accommodate the tiny
neutrino mass and provide a dark matter candidate in a unified picture. Due to
lack of experimental signatures for electroweak scale new physics, it is
appealing to assume these new particles are well above the electroweak scale
and take the effective field theory approach to study their effects on low
energy observables. In this work we apply the recently developed functional
matching formalism to the one-loop matching of the model onto the standard
model effective field theory up to dimension seven for the case when all new
states $N$ and $\eta$ are heavy to be integrated out. This is a realistic
example which has no tree-level matching due to the $\mathbb{Z}_{2}$ symmetry.
Using the matching results, we analyze their phenomenological implications for
several physical processes, including the lepton number violating effect, the
CDF $W$ mass excess, and the lepton flavor violating decays $\mu\to e\gamma$
and $\mu\to 3e$.

###### Contents

  1. 1 Introduction
  2. 2 Basics of Functional Matching
  3. 3 Review of Scotogenic Model
  4. 4 One-loop Matching onto the SMEFT
    1. 4.1 Matching result in a Green basis
    2. 4.2 Matching result in the standard basis
  5. 5 Phenomenology
    1. 5.1 Dim-5 and dim-7 LNV processes
    2. 5.2 LNC processes due to dim-6 operators
  6. 6 Conclusion
  7. Acknowledgement
  8. A Dim-5, dim-6, and dim-7 operator bases in the SMEFT
  9. B Calculation of a power-type supertrace with double insertions of $\boldsymbol{X}_{N\eta}$ and $\boldsymbol{X}_{\eta N}$
  10. C Reduction into the standard basis

##  1 Introduction

The origin of neutrino mass and the nature of dark matter are two pieces of
well-established evidence for physics beyond the standard model (SM). Among
various beyond SM scenarios an attractive route is to correlate these two
seemingly separate problems and solve them within the same theoretical
framework. In this respect the scotogenic model [1] provides a nice example
that can explain the origin of neutrino mass and dark matter (DM) in a single
simple framework, with the introduction only of three $\mathbb{Z}_{2}$-odd
right-handed singlet fermions $N$ and one $\mathbb{Z}_{2}$-odd scalar doublet
$\eta$ on top of the SM content. The main merit of the model is that neutrinos
gain radiative mass from interactions with particles in the dark sector. The
lightest of the latter is stable due to the presumed exact $\mathbb{Z}_{2}$
symmetry, and thus potentially serves as a DM candidate. Since it was first
proposed in 2006, the model has attracted a lot of attentions from various
phenomenological aspects including the lepton flavor violating (LFV) golden
modes $\mu\to e\gamma,~{}3e$ and $\mu$-$e$ conversion in nuclei [2, 3, 4, 5,
6, 7], the scalar DM scenario [8, 9, 10, 11], the fermionic WIMP and FIMP DM
cases [12, 13, 14, 15, 16], the running effect of neutrino masses [17] and the
neutrino mass matrix textures [18], low scale leptogenesis [16, 19, 20], the
signals at hadron and lepton colliders [14, 21, 22] and in gravitational waves
[23], and the LFV $Z$ and Higgs boson decays [24], etc.

Nevertheless, there is no definite signature so far for the predicted
particles on the observational side. It then looks natural to assume that the
new particles lie well above the SM electroweak scale to be inaccessible to
current high energy colliders. This motivates us to study their indirect
effects on low energy observations by working with an effective field theory
(EFT) in which they are integrated out. The standard model effective field
theory (SMEFT) is tailored exactly for this purpose. It is a low energy
effective field theory for SM particles below some new physics scale. In this
framework the SM interactions appear as the leading term in a systematic
series expansion, and the effects from new physics at a high scale are
incorporated as suppressed high-dimensional operators and modifications to the
SM parameters. The most appealing feature of SMEFT is perhaps its
universality. It contains exclusively the SM fields which are governed by the
SM gauge symmetries but is otherwise not constrained. Different high scale new
physics will be reflected in Wilson coefficients and their interrelations. In
the past years, the bases of complete and independent high dimensional
operators have been built for the SMEFT up to dimension nine (dim-9) [25, 26,
27, 28, 29, 30, 31, 32, 33]. To apply the SMEFT to low energy phenomenology,
one evolves it to the electroweak scale with the help of SMEFT renormalization
group equations (RGE), matches it with the low energy effective field theory
(LEFT) at the electroweak scale, and further evolves the latter to the
experimental scale via the LEFT RGEs where one finally calculates the physical
observables. In this way, low energy experimental data can be employed
complementarily to constrain physics in the ultraviolet (UV).

An important task in this approach is the matching of a UV theory onto the
SMEFT at the UV scale. Assuming the UV theory is perturbative, the matching is
a double expansion, one in the number of loops and the other in the inverse
power of the heavy scale. The tree-level matching can be easily done by
solving the classical equations of motion (EoMs) for the heavy fields followed
by a low energy expansion to the desired order. However, in some cases
interesting phenomenology (like flavor changing neutral currents) arises as a
loop effect or precision data demands an improved theoretical analysis, so
that one-loop matching becomes more and more relevant. Confronted with this,
the recently developed functional matching via the effective action is a
tailor-made method to achieve this goal [34, 35, 36, 37, 38, 39, 40, 41, 42,
43, 44]. Unlike the diagrammatic approach, the matching is done by calculating
some functional supertraces without computing Feynman diagrams one by one for
a designed set of amplitudes. Some (semi-)automatic tools have been developed
to facilitate this job for the tree-level matching [45, 46, 47] and one-loop
matching [43, 48, 49, 47]. One-loop matching has recently been practiced for
several UV models, such as the three tree-level seesaws [50, 51, 52, 53, 54],
the Zee model [51], leptoquark models [55, 51, 56], and others [57, 58, 59,
60, 61, 62].

Considering rich phenomenology of the scotogenic model and null experimental
searches for new heavy states, it is plausible to take the above EFT approach
to investigate its low energy effects by treating both $N$ and $\eta$ as heavy
states. It turns out that the existence of $\mathbb{Z}_{2}$ symmetry implies
no tree-level matching to the SMEFT. Then we investigate its one-loop matching
up to dim-7 operators. The direct result of functional evaluation is organized
in the so-called Green basis with a handful of operators whose origin can be
relatively easily tracked from the diagrammatic picture. To recast the result
in terms of the standard dim-5 Weinberg operator [25], the dim-6 Warsaw basis
[26] and the dim-7 basis [29] (further improved in [63]), some manipulations
have to be made of the SM equations of motion (EoM), integration by parts
(IBP) and the Fierz identities (FI).

The rest of the paper is organized as follows. In section 2 we review the
basic ingredients of the functional matching in effective field theory.
Section 3 is a brief introduction of the scotogenic model as well as our
notational convention. In section 4 we consider the one-loop matching for the
scotogenic model to the SMEFT, and show our main result in both Green and
standard bases. The phenomenological analysis together with brief comparisons
to the literature is included in section 5. In section 6, we draw our
conclusions. Supplementary materials presented in the appendices include the
collection of the SMEFT operator bases up to dimension 7 in appendix A, the
calculation of the supertrace for four-lepton operators in appendix B, and the
reduction of operators from the Green basis to the standard basis in appendix
C.

##  2 Basics of Functional Matching

For a UV field theory, whether fundamental or effective, with a hierarchical
field spectrum, we collectively denote the heavy and light (scalar, fermion,
or vector) fields as $\Phi$ and $\phi$, respectively, with a mass hierarchy
$m_{\Phi}\gg m_{\phi}$. The low energy dynamics for light particles can be
calculated either from the UV Lagrangian ${\cal L}_{\tt UV}(\Phi,\phi)$
consisting of both heavy and light fields, or from the EFT Lagrangian ${\cal
L}_{\tt EFT}(\phi)$ consisting only of light fields. In matching calculation
${\cal L}_{\tt UV}(\Phi,\phi)$ is supposed to be known while ${\cal L}_{\tt
EFT}(\phi)$ is searched for. To reproduce the low-energy physics of ${\cal
L}_{\tt UV}(\Phi,\phi)$, ${\cal L}_{\tt EFT}(\phi)$ has to be carefully
determined from ${\cal L}_{\tt UV}(\Phi,\phi)$ by integrating out $\Phi$ and
performing a matching calculation. Conventionally, this matching is done by
designing judiciously a complete set of amplitudes, computing them in both
theories and equating them to determine the Wilson coefficients in ${\cal
L}_{\tt EFT}(\phi)$ which depend on the parameters associated with $\Phi$. The
main drawback of this diagrammatical matching is that one has to first
determine the correct basis of operators at each dimension for the sought
${\cal L}_{\tt EFT}(\phi)$ and compute amplitudes twice. The procedure
necessarily involves infrared physics of light particles which however
eventually does not enter ${\cal L}_{\tt EFT}(\phi)$ itself, causing
unnecessary complications.

A more elegant approach is the functional matching in the path integral
formalism [34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44]. The starting point is
the identification of one-particle-irreducible (1PI) generating functionals
$\Gamma_{\tt EFT}[\phi]=\Gamma_{\tt UV}^{\tt L}[\phi]$ at the matching scale
$m_{\Phi}$. Here $\Gamma_{\tt UV}^{\tt L}[\phi]$ is computed in the UV theory
and irreducible only to the light field $\phi$ while $\Gamma_{\tt EFT}[\phi]$
is computed in the EFT. This identification is made in a double expansion, one
in the inverse power expansion of $m_{\Phi}$ and the other in the number of
loops. At the tree order, this is easy: one solves in the UV theory the
classical EoM for $\Phi=\Phi_{c}[\phi]$ in terms of the light field $\phi$
which is a functional, and makes the inverse power expansion in $m_{\Phi}$ to
turn it into an infinite series of local functions $\Phi=\Phi_{c}(\phi)$.
Substituting it into ${\cal L}_{\tt UV}(\Phi,\phi)$ yields the answer:

$\displaystyle{\cal L}_{\tt EFT}^{\tt tree}(\phi)={\cal L}_{\tt
UV}(\Phi_{c}(\phi),\phi).$ (2.1)

To perform one-loop matching, let us start on the EFT side whose Lagrangian is

$\displaystyle{\cal L}_{\tt EFT}(\phi)={\cal L}_{\tt EFT}^{\rm
tree}(\phi)+{\cal L}_{\tt EFT}^{\tt 1\mbox{-}loop}(\phi)+\cdots,$ (2.2)

where the dots stand for higher-loop contributions and ${\cal L}_{\tt
EFT}^{\tt 1\mbox{-}loop}(\phi)$ is what we are seeking. The above contributes
to the one-loop 1PI generating functional in two manners,

$\displaystyle\Gamma_{\tt EFT}^{\tt
1\mbox{-}loop}[\phi]=\int\mbox{d}^{d}x~{}{\cal L}_{\tt EFT}^{\rm
1\mbox{-}loop}(\phi)+\frac{i}{2}{\rm STr}[\ln(\boldsymbol{\rm H}_{\tt EFT})],$
(2.3)

where the second term arises from one-loop diagrams formed with interactions
defined in ${\cal L}_{\tt EFT}^{\rm tree}(\phi)$. The supertrace (STr)
includes a minus sign for fields that are quantized as a Grassmanian field. In
the path integral formalism, it results from the Gaussian integral with the
Hessian matrix being

$\displaystyle\boldsymbol{\rm H}_{\tt EFT}(x,y)=\frac{\delta^{2}S_{\tt
EFT}^{\tt tree}[\phi]}{\delta\bar{\phi}(x)\delta\phi(y)},$ (2.4)

where $S_{\tt a}^{\tt i}[\varphi]=\displaystyle\int\mbox{d}^{d}z~{}{\cal
L}_{\tt a}^{\tt i}(\varphi(z))$ denotes the action in $d$ dimensional
spacetime at the ${\tt i}$ order in the ${\tt a}$ theory for the field
$\varphi$. (A global factor of $\mu^{d-4}$ has been suppressed for brevity
where $\mu$ is the usual renormalization scale.) To count correctly
independent degrees of freedom, the fields have been arranged in a self-
conjugate form up to a rotation $R$, i.e., the conjugate pair of fields
$\bar{\varphi}$ and $\varphi$ is related by
$\bar{\varphi}=\varphi^{\texttt{T}}R$ with $|{\rm det}(R)|=1$ [43, 48]. The
supertrace term in Eq. (2.3) contains all infrared physics associated with the
light field $\phi$.

A similar 1-loop manipulation can be made for the generating functional
$\Gamma_{\tt UV}^{\tt L}[\phi]$ in the UV theory. An important difference from
the usual case is that it is a generating functional only for and irreducible
only to the light field $\phi$ although the UV theory contains the heavy field
$\Phi$ as well. Therefore when we make Legendre transform from the generating
functional for connected Green’s functions of $\phi$ to that for 1PI Green’s
functions $\phi$, we have to implement the classical EoM for the $\Phi$ field
simply because no external source has been introduced to it. With this point
in mind, we have,

$\displaystyle\Gamma_{\tt UV}^{\tt L,1\mbox{-}loop}[\phi]=\frac{i}{2}{\rm
STr}[\ln(\boldsymbol{\rm H}_{\tt UV})],$ (2.5)

where

$\displaystyle\boldsymbol{\rm H}_{\tt UV}(x,y)=\left.\frac{\delta^{2}S_{\tt
UV}[\varphi]}{\delta\bar{\varphi}(x)\delta\varphi(y)}\right|_{\Phi=\Phi(\phi)}.$
(2.6)

We notice that the Hessian matrix is defined for the whole field space
$\varphi=(\phi,\Phi)$ and the substitution $\Phi=\Phi(\phi)$ is made only
after the functional derivatives have been finished.

The identification at the 1-loop order of $\Gamma_{\tt EFT}^{\tt
1\mbox{-}loop}[\phi]=\Gamma_{\tt UV}^{\tt L,1\mbox{-}loop}[\phi]$ then implies

$\displaystyle\int\mbox{d}^{d}x~{}{\cal L}_{\tt EFT}^{\rm
1\mbox{-}loop}(\phi)=\frac{i}{2}{\rm STr}[\ln(\boldsymbol{\rm H}_{\tt
UV})]-\frac{i}{2}{\rm STr}[\ln(\boldsymbol{\rm H}_{\tt EFT})].$ (2.7)

The recent development in functional matching is based on the following
crucial realization [36, 43, 38, 39]. When the loop integrals in the UV theory
is calculated by integration by regions [64, 65],

$\displaystyle\frac{i}{2}{\rm STr}[\ln(\boldsymbol{\rm H}_{\tt
UV})]=\left.\frac{i}{2}{\rm STr}[\ln(\boldsymbol{\rm H}_{\tt UV})]\right|_{\tt
hard}+\left.\frac{i}{2}{\rm STr}[\ln(\boldsymbol{\rm H}_{\tt UV})]\right|_{\tt
soft},$ (2.8)

the soft term exactly cancels out the EFT term in Eq. (2.7). The calculation
of one-loop matching then boils down to the calculation of the hard part in
the UV term:

$\displaystyle\int\mbox{d}^{d}x~{}{\cal L}_{\tt EFT}^{\rm
1\mbox{-}loop}(\phi)=\left.\frac{i}{2}{\rm STr}[\ln(\boldsymbol{\rm H}_{\tt
UV})]\right|_{\tt hard},$ (2.9)

where the subscript hard means that the loop integrands are first Taylor-
expanded for $q\sim m_{\Phi}\gg m_{\phi},k$ where $q,~{}k$ stand for the loop
and external momenta respectively, and then evaluated for the whole $q$ space
in $d$ dimensions.

The second functional derivative is split into two parts, $\boldsymbol{\rm
H}_{\tt UV}=\boldsymbol{K}-\boldsymbol{X}$, where $\boldsymbol{K}$ contains
only kinetic and mass terms and $\boldsymbol{X}$ includes all remaining
interactions. The matrix $\boldsymbol{K}$ is in a block-diagonal form:

$K_{i}=\begin{cases}P^{2}-m_{i}^{2}&\mbox{for spin-0 fields}\\\
\not{P}-m_{i}&\mbox{for spin-$1/2$ fields}\\\
-g^{\mu\nu}(P^{2}-m_{i}^{2})&\mbox{for spin-1 fields in Feynman
gauge}\end{cases},$ (2.10)

where $P_{\mu}=iD_{\mu}$ with $D_{\mu}$ being the covariant derivative with
respect to background gauge fields whose operators are under consideration. To
proceed further, we make the Taylor expansion,

$\ln(\boldsymbol{K}-\boldsymbol{X})=\ln(\boldsymbol{K})+\sum_{n=0}^{\infty}\frac{1}{n}(\boldsymbol{K}^{-1}\boldsymbol{X})^{n}.$
(2.11)

Since $\boldsymbol{X}$ contributes at least a mass dimension 1 ($3/2$) to
operators in search when it involves a bosonic (fermionic) field and
$\boldsymbol{K}^{-1}$ contributes a nonnegative mass dimension upon finishing
loop integrals, the expansion actually terminates for the sought operators
with a given mass dimension. The evaluation of supertraces is then classified
into a log-type and a power-type,

$\int\mbox{d}^{d}x{\cal L}_{\tt EFT}^{\tt
1\mbox{-}loop}[\phi_{c}]=\frac{i}{2}{\rm
STr}\left[\ln(\boldsymbol{K})\right]\Big{|}_{\tt
hard}-\frac{i}{2}\sum_{n=1}^{\infty}\frac{1}{n}{\rm
STr}\left[\left(\boldsymbol{K}^{-1}\boldsymbol{X}\right)^{n}\right]\Big{|}_{\tt
hard}.$ (2.12)

The log-type supertrace depends only on the representation in a gauge group
and is thus universal. The evaluation of the supertraces is done by the
technique of covariant derivative expansion (CDE) [66, 67, 68], which
automatically leads to gauge invariant operators. These methods have been
implemented into semi-automatic tools like STrEAM [43, 48] and SuperTracer
[49]. In this work, we will calculate both manually and with the help of
SuperTracer to achieve identical results.

##  3 Review of Scotogenic Model

We start first with the convention for the SM part. We denote the SM left-
handed lepton and quark doublet fields as $L(1,2,1/2)$ and $Q(3,2,1/6)$, the
right-handed up-type quark, down-type quark, and charged lepton singlet fields
as $u(3,1,2/3)$, $d(3,1,-1/3)$, and $e(1,1,-1)$, and the Higgs doublet as
$H(1,2,1/2)$, respectively. Here the numbers in brackets are the corresponding
representations under the SM gauge group $SU(3)_{C}\times SU(2)_{L}\times
U(1)_{Y}$, and the flavor index of the fermions is suppressed for simplicity.
Dropping the gauge-fixing related terms and topological terms, the SM
Lagrangian is,

$\displaystyle\mathcal{L}_{\tt SM}$ $\displaystyle=$
$\displaystyle-\frac{1}{4}G^{A}_{\mu\nu}G^{A\mu\nu}-\frac{1}{4}W^{I}_{\mu\nu}W^{I\mu\nu}-\frac{1}{4}B_{\mu\nu}B^{\mu\nu}$
(3.1)
$\displaystyle+|D_{\mu}H|^{2}+\mu_{H}^{2}(H^{\dagger}H)-\lambda_{H}(H^{\dagger}H)^{2}$
$\displaystyle+\sum_{\Psi=Q,L,u,d,e}\bar{\Psi}i\not{D}\Psi-\left(\bar{Q}Y_{u}u\tilde{H}+\bar{Q}Y_{d}dH+\bar{L}Y_{e}eH+\mbox{h.c.}\right),$

where $\mu_{H}^{2}$ is related to the vacuum expectation value (vev) $v$ of
the Higgs field via the relation, $\mu_{H}^{2}=\lambda_{H}v^{2}$, with
$\lambda_{H}$ being the Higgs self-coupling consant. The superscripts $A$ and
$I$ count the generators of the groups $SU(3)_{C}$ and $SU(2)_{L}$,
respectively. $Y_{u},~{}Y_{d},~{}Y_{e}$ are the Yukawa couplings which are
complex matrices in flavor space, and $\tilde{H}_{i}=\epsilon_{ij}H^{*}_{j}$.
The covariant derivative is defined by

$\displaystyle
D_{\mu}=\partial_{\mu}-ig_{3}T^{A}G^{A}_{\mu}-ig_{2}T^{I}W^{I}_{\mu}-ig_{1}YB_{\mu},$
(3.2)

where $g_{1},g_{2}$ and $g_{3}$ are the corresponding coupling constants for
$U(1)_{Y},SU(2)_{L}$, and $SU(3)_{C}$, and $T^{A},~{}T^{I},~{}Y$ are the
generator matrices appropriate for the fields to be acted on.

The scotogenic model proposed by E. Ma [1] is the extension of the SM with
three generations of the right-handed singlet fermion $N(1,1,0)$ and a second
scalar doublet $\eta(1,2,1/2)$. A discrete $\mathbb{Z}_{2}$ symmetry is
imposed under which these new fields are odd and the SM fields even. The
complete Lagrangian of the scotogenic model takes the form [1],

$\displaystyle{\cal L}$ $\displaystyle=$ $\displaystyle{\cal L}_{\tt SM}+{\cal
L}_{N,\eta},$ $\displaystyle{\cal L}_{N,\eta}$ $\displaystyle=$
$\displaystyle\bar{N}i\not{\partial}N-\left(\frac{1}{2}\overline{N}m_{N}N^{\texttt{C}}+\bar{L}\tilde{\eta}Y_{\eta}N+{\rm
h.c.}\right)+|D_{\mu}\eta|^{2}-V(H,\eta),$ $\displaystyle V(H,\eta)$
$\displaystyle=$ $\displaystyle
m_{\eta}^{2}\eta^{\dagger}\eta+\frac{1}{2}\lambda_{2}(\eta^{\dagger}\eta)^{2}+\lambda_{3}(H^{\dagger}H)(\eta^{\dagger}\eta)$
(3.3)
$\displaystyle+\lambda_{4}(H^{\dagger}\eta)(\eta^{\dagger}H)+\frac{1}{2}\lambda_{5}[(H^{\dagger}\eta)^{2}+(\eta^{\dagger}H)^{2}].$

The charge conjugation field is defined as $N^{\texttt{C}}\equiv
C\overline{N}^{\texttt{T}}$ with $(N^{\texttt{C}})^{\texttt{C}}=N$, where the
charge conjugation matrix $C$ satisfies the relations
$C^{\texttt{T}}=C^{\dagger}=-C$ and $C^{2}=-1$. $m_{N}$ is the Majorana mass
matrix and assumed without loss of generality to be diagonal with real
positive elements $m_{N_{i}}$, and $Y_{\eta}$ is the new Yukawa coupling that
will enter the generation of neutrino mass. While the couplings
$\lambda_{2,3,4}$ are real by themselves, the coupling $\lambda_{5}$ can be
chosen real by a phase redefinition of $\eta$, which only modifies the global
phase in the Yukawa coupling $Y_{\eta}$. Assuming $\mu_{H}^{2}>0$ and
$m_{\eta}^{2}>0$, when $H^{0}$ develops a vev as in the SM, $\eta^{0}$ is
guaranteed not to because of the exact $\mathbb{Z}_{2}$ symmetry. The mixing
between $H$ and $\eta$ is also forbidden by the symmetry.

In this work we assume both $\eta$ and $N$ particles are much heavier than the
electroweak scale and we match the scotogenic model to the SMEFT at the scale
$m_{\eta,N}$. Thus we will not shift the vev from the Higgs field $H$ to
maintain the complete SM gauge symmetry. In particular, the whole
$\lambda_{3,4,5}$ terms which would correct the mass of or even lift the
degeneracy of the $\eta$ components upon including the vev, will be treated as
perturbative interactions. This is indeed in accord with the EFT approach.
Because of $\mathbb{Z}_{2}$ symmetry, the terms in ${\cal L}$ are of even
powers in $N$ and $\eta$, so that their classical EoMs are of odd powers. This
means that they cannot be solved in terms of pure SM fields. Therefore, no
higher dimensional operators can appear from tree-level matching in this case.
This is obvious diagrammatically: pure $\mathbb{Z}_{2}$-even external lines
cannot be connected at tree level with pure $\mathbb{Z}_{2}$-odd internal
lines. The non-trivial matching then starts to appear at the one-loop level.
For the other two possible cases in which one of $N$ and $\eta$ is treated as
light and the other as heavy, tree-level matching indeed exists. For the case
with a heavy $\eta$ and light $N$s, the model will match onto the sterile
neutrino extended SMEFT ($\nu$SMEFT) [69], with the lightest $N$ being DM. For
the opposite case, it will match onto the DM EFT with a $\mathbb{Z}_{2}$-odd
scalar doublet [70], of which the lightest neutral scalar could act as DM.
These latter two cases have very different phenomenology and we defer the
study of these possibilities to a future publication.

##  4 One-loop Matching onto the SMEFT

To perform the one-loop matching using the functional method, we follow the
notations in refs [43, 48, 49] and introduce the pairs of fields that are
self-conjugate up to a rotation matrix:

$\displaystyle\varphi_{N}\equiv
N+N^{\texttt{C}},\quad\varphi_{\eta}\equiv\begin{pmatrix}\eta\\\
\eta^{*}\end{pmatrix}.$ (4.1)
$\displaystyle\bar{\varphi}_{N}=\varphi_{N}^{\texttt{T}}C,\quad\bar{\varphi}_{\eta}=\varphi_{\eta}^{\texttt{T}}\begin{pmatrix}0&1\\\
1&0\end{pmatrix}=(\eta^{\dagger},\eta^{T}).$ (4.2)

This facilitates their path integral formulation as they appear like real
variables. Let us first determine the interaction kernels
$\boldsymbol{X}_{ij}$ entering the power-type supertraces. The
$\mathbb{Z}_{2}$ symmetry and that both $\eta$ and $N$s are heavy have the
consequences: only the $\eta,N$ entries of $\boldsymbol{X}$ and only the
$\eta,N$-independent terms in those entries contribute to matching. The former
is because $\eta,N$ are odd and SM fields even under $\mathbb{Z}_{2}$, and the
latter is due to absence of relevant terms from EoMs of $\eta,N$. Thus we only
have the following three:

$\displaystyle\boldsymbol{X}_{N\eta}^{[3/2]}$ $\displaystyle=$
$\displaystyle\begin{pmatrix}Y_{\eta}^{\dagger}P_{L}(\epsilon^{\texttt{T}}L)&Y_{\eta}^{\texttt{T}}P_{R}(\epsilon^{\texttt{T}}L^{\texttt{C}})\end{pmatrix},$
(4.3a) $\displaystyle\boldsymbol{X}_{\eta N}^{[3/2]}$ $\displaystyle=$
$\displaystyle\begin{pmatrix}(\bar{L}\epsilon)P_{R}Y_{\eta}\\\
(\overline{L^{\texttt{C}}}\epsilon)P_{L}Y_{\eta}^{*}\end{pmatrix},$ (4.3b)
$\displaystyle\boldsymbol{X}_{\eta\eta}^{[2],ij}$ $\displaystyle=$
$\displaystyle\begin{pmatrix}\lambda_{3}(H^{\dagger}H)\delta^{ij}+\lambda_{4}H^{i}H^{j*}&\lambda_{5}H^{i}H^{j}\\\
\lambda_{5}H^{i*}H^{j*}&\lambda_{3}(H^{\dagger}H)\delta^{ij}+\lambda_{4}H^{i*}H^{j}\end{pmatrix}+\cdots,$
(4.3c)

where the superscript number in square brackets indicates the minimal
canonical dimension of $\boldsymbol{X}$ and the dots stand for irrelevant
$\eta$-dependent terms thus stated above.

###  4.1 Matching result in a Green basis

With the kernels Eq. (4.3) at hand, we form the supertraces that induce the
SMEFT operators up to dim 7:

$\displaystyle\int d^{d}x{\cal L}_{\tt EFT}^{\rm 1L,\,dim\leq 7}$
$\displaystyle=$ $\displaystyle\left\\{{i\over 2}{\rm
STr}\log\boldsymbol{K}-{i\over 2}{\rm
STr}\left(\boldsymbol{K}_{\eta}^{-1}\boldsymbol{X}_{\eta\eta}^{[2]}\right)-{1\over
2}{i\over 2}{\rm
STr}\left(\boldsymbol{K}_{\eta}^{-1}\boldsymbol{X}_{\eta\eta}^{[2]}\right)^{2}-{1\over
3}{i\over 2}{\rm
STr}\left(\boldsymbol{K}_{\eta}^{-1}\boldsymbol{X}_{\eta\eta}^{[2]}\right)^{3}\right.$
(4.4) $\displaystyle-$ $\displaystyle{i\over 2}{\rm
STr}\left(\boldsymbol{K}_{\eta}^{-1}\boldsymbol{X}_{\eta
N}^{[3/2]}\boldsymbol{K}_{N}^{-1}\boldsymbol{X}_{N\eta}^{[3/2]}\right)-{1\over
2}{i\over 2}{\rm STr}\left(\boldsymbol{K}_{\eta}^{-1}\boldsymbol{X}_{\eta
N}^{[3/2]}\boldsymbol{K}_{N}^{-1}\boldsymbol{X}_{N\eta}^{[3/2]}\right)^{2}$
$\displaystyle-$ $\displaystyle{i\over 2}{\rm
STr}\left(\boldsymbol{K}_{\eta}^{-1}\boldsymbol{X}_{\eta\eta}^{[2]}\boldsymbol{K}_{\eta}^{-1}\boldsymbol{X}_{\eta
N}^{[3/2]}\boldsymbol{K}_{N}^{-1}\boldsymbol{X}_{N\eta}^{[3/2]}\right)$
$\displaystyle-$ $\displaystyle\left.{i\over 2}{\rm
STr}\left[\left(\boldsymbol{K}_{\eta}^{-1}\boldsymbol{X}_{\eta\eta}^{[2]}\right)^{2}\boldsymbol{K}_{\eta}^{-1}\boldsymbol{X}_{\eta
N}^{[3/2]}\boldsymbol{K}_{N}^{-1}\boldsymbol{X}_{N\eta}^{[3/2]}\right]\right\\}\Big{|}_{\tt
hard}.$

Note that symmetry factors have been included. An inspection of
$\boldsymbol{X}$ shows that the first six terms only contribute to the lepton-
number-conserving (LNC) operators with an even mass dimension, the last but
one term contributes to both LNC and lepton-number-violating (LNV) operators,
and the last one only yields LNV dim-7 operators.

dim | Operator |  WCs [$1/(16\pi^{2})$]
---|---|---
2 | $H^{\dagger}H$ | $(1+L_{\eta})(2\lambda_{3}+\lambda_{4})m_{\eta}^{2}$
4 | $B_{\mu\nu}B^{\mu\nu}\,(\blacksquare)$ | $-{L_{\eta}\over 24}g_{1}^{2}$
$W^{I\mu\nu}W^{I}_{\mu\nu}\,(\blacksquare)$ | $-{L_{\eta}\over 24}g_{2}^{2}$
$\overline{L_{p}}i\not{D}L_{r}$ | $\left[Y_{\eta}{(1+2L_{\eta})m_{\eta}^{4}-4(1+L_{\eta})m_{N}^{2}m_{\eta}^{2}+(3+2L_{N})m_{N}^{4}\over 4(m_{\eta}^{2}-m_{N}^{2})^{2}}Y_{\eta}^{\dagger}\right]_{pr}$
$(H^{\dagger}H)^{2}$ | ${L_{\eta}\over 2}(2\lambda_{3}^{2}+2\lambda_{3}\lambda_{4}+\lambda_{4}^{2}+\lambda_{5}^{2})$
5 | $\epsilon_{im}\epsilon_{in}(\overline{L^{i\texttt{C}}_{p}}L_{r}^{j})H^{m}H^{n}+{\rm h.c.}$ | $-\lambda_{5}\left[Y_{\eta}^{*}m_{N}{m_{\eta}^{2}-(1-L_{\eta}+L_{N})m_{N}^{2}\over 2(m_{\eta}^{2}-m_{N}^{2})^{2}}Y_{\eta}^{\dagger}\right]_{pr}$
6 | $\partial_{\mu}B^{\mu\nu}\partial^{\rho}B_{\rho\nu}\,(\blacksquare)$ | $-{g_{1}^{2}\over 120m_{\eta}^{2}}$
$D_{\mu}W^{I\mu\nu}D^{\rho}W^{I}_{\rho\nu}\,(\blacksquare)({\color[rgb]{0,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,1}\pgfsys@color@cmyk@stroke{1}{0}{0}{0}\pgfsys@color@cmyk@fill{1}{0}{0}{0}\divideontimes})$ | $-{g_{2}^{2}\over 120m_{\eta}^{2}}$
$\epsilon^{IJK}W_{\mu}^{I\nu}W_{\nu}^{J\rho}W_{\rho}^{K\mu}\,(\blacksquare)$ | ${g_{2}^{3}\over 360m_{\eta}^{2}}$
$(H^{\dagger}H)^{3}$ | $-{2\lambda_{3}^{3}+\lambda_{4}^{3}+3(\lambda_{3}+\lambda_{4})(\lambda_{3}\lambda_{4}+\lambda_{5}^{2})\over 6m_{\eta}^{2}}$
$(H^{\dagger}H)\partial^{2}(H^{\dagger}H)$ | $-{2\lambda_{3}^{2}+2\lambda_{3}\lambda_{4}-\lambda_{5}^{2}\over 12m_{\eta}^{2}}$
$|H^{\dagger}D_{\mu}H|^{2}$ | $-{\lambda_{4}^{2}-\lambda_{5}^{2}\over 6m_{\eta}^{2}}$
$(H^{\dagger}H)(H^{\dagger}D^{2}H)+{\rm h.c.}({\color[rgb]{0,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,1}\pgfsys@color@cmyk@stroke{1}{0}{0}{0}\pgfsys@color@cmyk@fill{1}{0}{0}{0}\divideontimes})$ | $-{\lambda_{4}^{2}+\lambda_{5}^{2}\over 12m_{\eta}^{2}}$
$(H^{\dagger}H)B^{\mu\nu}B_{\mu\nu}$ | ${g_{1}^{2}(2\lambda_{3}+\lambda_{4})\over 48m_{\eta}^{2}}$
$(H^{\dagger}H)W^{I\mu\nu}W^{I}_{\mu\nu}$ | ${g_{2}^{2}(2\lambda_{3}+\lambda_{4})\over 48m_{\eta}^{2}}$
$(H^{\dagger}\sigma^{I}H)W^{I\mu\nu}B_{\mu\nu}$ | ${g_{1}g_{2}\lambda_{4}\over 24m_{\eta}^{2}}$
$(H^{\dagger}H)(\overline{L_{p}}i\overleftrightarrow{\not{D}}L_{r})$ | $-(2\lambda_{3}+\lambda_{4})\left[Y_{\eta}{m_{\eta}^{4}-4m_{N}^{2}m_{\eta}^{2}+(3-2L_{\eta}+2L_{N})m_{N}^{4}\over 8(m_{\eta}^{2}-m_{N}^{2})^{3}}Y_{\eta}^{\dagger}\right]_{pr}$
$(H^{\dagger}\sigma^{I}H)(\overline{L_{p}}i\overleftrightarrow{\not{D}}^{I}L_{r})$ | $\lambda_{4}\left[Y_{\eta}{m_{\eta}^{4}-4m_{N}^{2}m_{\eta}^{2}+(3-2L_{\eta}+2L_{N})m_{N}^{4}\over 8(m_{\eta}^{2}-m_{N}^{2})^{3}}Y_{\eta}^{\dagger}\right]_{pr}$
$\overline{L_{p}}i\overleftarrow{\not{D}}\not{D}\not{D}L_{r}$ | $\left[Y_{\eta}{m_{\eta}^{6}-6m_{N}^{2}m_{\eta}^{4}+3(1-2L_{\eta}+2L_{N})m_{N}^{4}m_{\eta}^{2}+2m_{N}^{6}\over 6(m_{\eta}^{2}-m_{N}^{2})^{4}}Y_{\eta}^{\dagger}\right]_{pr}$
$B^{\mu\nu}\overline{L_{p}}\sigma_{\mu\nu}i\not{D}L_{r}+{\rm h.c.}$ | $g_{1}\left[Y_{\eta}{m_{\eta}^{6}-6m_{N}^{2}m_{\eta}^{4}+3(1-2L_{\eta}+2L_{N})m_{N}^{4}m_{\eta}^{2}+2m_{N}^{6}\over 48(m_{\eta}^{2}-m_{N}^{2})^{4}}Y_{\eta}^{\dagger}\right]_{pr}$
$D_{\nu}B^{\mu\nu}\overline{L_{p}}\gamma_{\mu}L_{r}$ | $-g_{1}\left[Y_{\eta}{2m_{\eta}^{6}-9m_{N}^{2}m_{\eta}^{4}+18m_{N}^{4}m_{\eta}^{2}-(11-6L_{\eta}+6L_{N})m_{N}^{6}\over 72(m_{\eta}^{2}-m_{N}^{2})^{4}}Y_{\eta}^{\dagger}\right]_{pr}$
$W^{I\mu\nu}\overline{L_{p}}\sigma^{I}\sigma_{\mu\nu}i\not{D}L_{r}+{\rm h.c.}$ | $-g_{2}\left[Y_{\eta}{m_{\eta}^{6}-6m_{N}^{2}m_{\eta}^{4}+3(1-2L_{\eta}+2L_{N})m_{N}^{4}m_{\eta}^{2}+2m_{N}^{6}\over 48(m_{\eta}^{2}-m_{N}^{2})^{4}}Y_{\eta}^{\dagger}\right]_{pr}$
$D_{\nu}W^{I\mu\nu}\overline{L_{p}}\sigma^{I}\gamma_{\mu}L_{r}$ | $g_{2}\left[Y_{\eta}{2m_{\eta}^{6}-9m_{N}^{2}m_{\eta}^{4}+18m_{N}^{4}m_{\eta}^{2}-(11-6L_{\eta}+6L_{N})m_{N}^{6}\over 72(m_{\eta}^{2}-m_{N}^{2})^{4}}Y_{\eta}^{\dagger}\right]_{pr}$
$(\overline{L_{p}}\gamma_{\mu}L_{r})(\overline{L_{s}}\gamma^{\mu}L_{t})$ | $\,\,\,-{1\over 4}\left\\{\left[{(L_{\eta}-L_{N_{v}})m_{N_{v}}^{2}\over(m_{N_{v}}^{2}-m_{N_{w}}^{2})(m_{\eta}^{2}-m_{N_{v}}^{2})^{2}}+v\leftrightarrow w\right]+{1\over(m_{\eta}^{2}-m_{N_{v}}^{2})(m_{\eta}^{2}-m_{N_{w}}^{2})}\right\\}$
$\times
m_{N_{v}}m_{N_{w}}(Y_{\eta})_{pw}(Y_{\eta}^{\texttt{T}})_{ws}(Y_{\eta}^{*})_{rv}(Y_{\eta}^{\dagger})_{vt}$
$-{1\over
8}\left\\{\left[{(L_{\eta}-L_{N_{v}})m_{N_{v}}^{4}\over(m_{N_{v}}^{2}-m_{N_{w}}^{2})(m_{\eta}^{2}-m_{N_{v}}^{2})^{2}}+v\leftrightarrow
w\right]+{m_{\eta}^{2}\over(m_{\eta}^{2}-m_{N_{v}}^{2})(m_{\eta}^{2}-m_{N_{w}}^{2})}\right\\}$
$\times(Y_{\eta})_{pw}(Y_{\eta}^{\dagger})_{wt}(Y_{\eta})_{sv}(Y_{\eta}^{\dagger})_{vr}$
7 | $\epsilon_{im}\epsilon_{jn}(\overline{L^{i\texttt{C}}_{p}}L_{r}^{j})H^{m}H^{n}(H^{\dagger}H)+{\rm h.c.}$ | $(\lambda_{3}+\lambda_{4})\lambda_{5}\left[Y_{\eta}^{*}m_{N}{m_{\eta}^{4}+2(L_{\eta}-L_{N})m_{N}^{2}m_{\eta}^{2}-m_{N}^{4}\over 2m_{\eta}^{2}(m_{\eta}^{2}-m_{N}^{2})^{3}}Y_{\eta}^{\dagger}\right]_{pr}$
$\epsilon_{im}\epsilon_{jn}(\overline{L^{i\texttt{C}}_{p}}D^{2}L_{r}^{j})H^{m}H^{n}+{\rm h.c.}$ | $\lambda_{5}\left[Y_{\eta}^{*}m_{N}{m_{\eta}^{4}+4(1+L_{\eta}-L_{N})m_{N}^{2}m_{\eta}^{2}-(5-2L_{\eta}+2L_{N})m_{N}^{4}\over 4(m_{\eta}^{2}-m_{N}^{2})^{4}}Y_{\eta}^{\dagger}\right]_{pr}$
$\epsilon_{im}\epsilon_{jn}(\overline{L^{i\texttt{C}}_{p}}L_{r}^{j})(D_{\mu}H^{m}D^{\mu}H^{n})+{\rm h.c.}$ | $\lambda_{5}\left[Y_{\eta}^{*}m_{N}{m_{\eta}^{6}-6m_{N}^{2}m_{\eta}^{4}+3(1-2L_{\eta}+2L_{N})m_{N}^{4}m_{\eta}^{2}+2m_{N}^{6}\over 12m_{\eta}^{2}(m_{\eta}^{2}-m_{N}^{2})^{4}}Y_{\eta}^{\dagger}\right]_{pr}$
$\epsilon_{im}\epsilon_{jn}(\overline{L^{i\texttt{C}}_{p}}L_{r}^{j})H^{m}D^{2}H^{n}+{\rm h.c.}$ | $\lambda_{5}\left[Y_{\eta}^{*}m_{N}{m_{\eta}^{6}-6m_{N}^{2}m_{\eta}^{4}+3(1-2L_{\eta}+2L_{N})m_{N}^{4}m_{\eta}^{2}+2m_{N}^{6}\over 12m_{\eta}^{2}(m_{\eta}^{2}-m_{N}^{2})^{4}}Y_{\eta}^{\dagger}\right]_{pr}$

Table 1: The one-loop matching result for the scotogenic model in the heavy
$N$ and $\eta$ case in a Green basis. The pink sector contributes to the
threshold correction, while the blue sector will be reduced by using EoMs. See
the text for the notations $\blacksquare$ and
${\color[rgb]{0,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,1}\pgfsys@color@cmyk@stroke{1}{0}{0}{0}\pgfsys@color@cmyk@fill{1}{0}{0}{0}\divideontimes}$.

The supertraces are calculated in a gauge invariant way by the covariant
derivative expansion method [34, 36]. It has been incorporated into partially
automatic Mathmatica packages like STrEAM [48] and SuperTracer [49]. For
practical purposes, we have computed these supertraces both with the help of
the SuperTracer package and manually as a crosscheck. In Tab. 1, we first
provide the final matching result up to dim 7 in a minimal Green basis without
implementing the field redefinitions or EoMs to go back into the standard
basis [26, 63]. Such a basis are helpful in that one can relatively easily
track the origin of the matched operators from the diagrammatic viewpoint. In
addition, a handful of operators appear for the matching at each dimension.
For the result in Tab. 1, several comments are in order:

  1. 1.

We have defined
$L_{\eta}\equiv\epsilon^{-1}+\ln(4\pi\mu^{2}/m_{\eta}^{2})-\gamma_{\rm E}$ and
$L_{N}\equiv\epsilon^{-1}+\ln(4\pi\mu^{2}/m_{N}^{2})-\gamma_{\rm E}$, which
are related to the UV divergence in the modified minimal subtraction scheme in
$d=4-2\epsilon$ dimensions;

  2. 2.

The log-type supertrace is only for the scalar $\eta$ due to its non-trivial
$SU(2)_{L}\times U(1)_{Y}$ representation, and it generates the operators
consisting exclusively of gauge fields that are marked by a $\blacksquare$;

  3. 3.

The IBP relations and group $SU(2)_{L}$ and four-fermion Fierz identities are
extensively used to achieve the operators in the table. For instance, for the
dim-6 operators with pure Higgs fields and a pair of derivatives, we have used
the following transformations to reach the operators in the table,

$\displaystyle(H^{\dagger}H)|D_{\mu}H|^{2}$
$\displaystyle\overset{\texttt{IBP}}{\Rightarrow}$ $\displaystyle{1\over
2}(H^{\dagger}H)\partial^{2}(H^{\dagger}H)-{1\over
2}[(H^{\dagger}H)(H^{\dagger}D^{2}H)+{\rm h.c.}],$ (4.5a)
$\displaystyle(H^{\dagger}D_{\mu}H)(H^{\dagger}D^{\mu}H)$
$\displaystyle\overset{\texttt{IBP}}{\Rightarrow}$ $\displaystyle-{1\over
2}(H^{\dagger}H)\partial^{2}(H^{\dagger}H)-|H^{\dagger}D_{\mu}H|^{2}$ (4.5b)
$\displaystyle-{1\over 2}[(H^{\dagger}H)(H^{\dagger}D^{2}H)-{\rm
h.c.}],\quad\quad$
$\displaystyle(H^{\dagger}i\overleftrightarrow{D_{\mu}}H)(H^{\dagger}i\overleftrightarrow{D^{\mu}}H)$
$\displaystyle\overset{\texttt{IBP}}{\Rightarrow}$
$\displaystyle(H^{\dagger}H)\partial^{2}(H^{\dagger}H)+4|H^{\dagger}D_{\mu}H|^{2},$
(4.5c)
$\displaystyle(H^{\dagger}i\overleftrightarrow{D_{\mu}}^{I}H)(H^{\dagger}i\overleftrightarrow{D^{\mu}}^{I}H)$
$\displaystyle\overset{\texttt{IBP}}{\Rightarrow}$ $\displaystyle
3(H^{\dagger}H)\partial^{2}(H^{\dagger}H)-2[(H^{\dagger}H)(H^{\dagger}D^{2}H)+{\rm
h.c.}].$ (4.5d)

  4. 4.

A WC in the form, $[Y_{\eta}f(m_{N})Y_{\eta}^{\dagger}]_{pr}$, should be
understood as a matrix multiplication,
$(Y_{\eta})_{px}f(m_{N_{x}})(Y_{\eta}^{\dagger})_{xr}$ with the dummy index
$x$ of $N_{x}$ being summed over. In other words, the factor sandwiched
between the two Yuwaka matrices is understood as a diagonal matrix with the
$x$-th diagonal element evaluated at $m_{N_{x}}$. The same notation is used
throughout the paper.

  5. 5.

The four-lepton operators in the last row of dim-6 sector are generated by the
5th power-type supertrace in Eq. (4.4). Since the current version of
SuperTracer package cannot deal with the non-degenerate fermion case in a
fully automatic way, we demonstrate our manual calculation in appendix B. In
the degenerate mass limit, we find our result is consistent with the output
from using SuperTracer.

  6. 6.

The dim-4 or less operators in the pink sector give the so-called threshold
corrections to the SM, and they lead to the renormalization of the SM
parameters. The dim-6 and dim-7 operators in blue are not yet in the standard
basis [26, 63] and will be carefully dealt with in the following subsection to
obtain the final matching result in the standard basis.

  7. 7.

An interesting feature for the matching result in Tab. 1 is that UV divergence
only appears in the WCs of dim$\leq 4$ operators (characterized by the
$L_{\eta}$) and cancels out for dim$\geq 5$ operators in a combination form,
$L_{\eta}-L_{N}=\ln(m_{N}^{2}/m_{\eta}^{2})$. This is as expected, since a
correct implementation of renormalization should guarantee that all UV
divergence in a renormalizable theory like the scotogenic model can always be
absorbed into its parameters associated with dim$\leq 4$ operators.

###  4.2 Matching result in the standard basis

To translate the matching result in the Green basis in Tab. 1 into one in the
standard dim-6 [26] and dim-7 basis [63], we need to tackle both the threshold
correction to the SM terms in Eq. (3.1) and the reduction of those higher
dimensional operators not in the standard basis by field or coupling
redefinition. For this purpose one first renormalizes the SM terms, derives
EoMs from the renormalized SM Lagrangian and applies them to the reduction of
higher dimensional operators. From the pink sector in Tab. 1 the modified SM
terms are,

$\displaystyle{\cal L}_{\rm SM}$ $\displaystyle\supset$ $\displaystyle-{1\over
4}(1+\delta Z_{B})B_{\mu\nu}B^{\mu\nu}-{1\over 4}(1+\delta
Z_{W})W_{\mu\nu}^{I}W^{I\mu\nu}$ (4.6) $\displaystyle+\overline{L}(1+\delta
Z_{L})i\not{D}L+\hat{\mu}_{H}^{2}H^{\dagger}H-\hat{\lambda}_{H}(H^{\dagger}H)^{2},$

where the field renormalization constants and modified parameters in the Higgs
potential due to one-loop matching are given by

$\displaystyle\delta Z_{B}$ $\displaystyle=$ $\displaystyle{1\over
16\pi^{2}}{L_{\eta}\over 6}g_{1}^{2},$ (4.7a) $\displaystyle\delta Z_{W}$
$\displaystyle=$ $\displaystyle{1\over 16\pi^{2}}{L_{\eta}\over 6}g_{2}^{2},$
(4.7b) $\displaystyle(\delta Z_{L})_{pr}$ $\displaystyle=$
$\displaystyle{1\over
16\pi^{2}}\left[Y_{\eta}{(3+2L_{N})m_{N}^{4}-4(1+L_{\eta})m_{N}^{2}m_{\eta}^{2}+(1+2L_{\eta})m_{\eta}^{4}\over
4(m_{N}^{2}-m_{\eta}^{2})^{2}}Y_{\eta}^{\dagger}\right]_{pr},$ (4.7c)
$\displaystyle\hat{\mu}_{H}^{2}$ $\displaystyle=$
$\displaystyle\mu_{H}^{2}+{1\over
16\pi^{2}}(1+L_{\eta})(2\lambda_{3}+\lambda_{4})m_{\eta}^{2},$ (4.7d)
$\displaystyle\hat{\lambda}_{H}$ $\displaystyle=$
$\displaystyle\lambda_{H}-{1\over 16\pi^{2}}{L_{\eta}\over
2}(2\lambda_{3}^{2}+2\lambda_{3}\lambda_{4}+\lambda_{4}^{2}+\lambda_{5}^{2}),$
(4.7e)

with $\delta Z_{L}^{\dagger}=\delta Z_{L}$. To bring kinetic terms into the
canonical form, we make the following field and coupling redefinitions,

$\displaystyle B_{\mu}\to\left(1-{1\over 2}\delta
Z_{B}\right)B_{\mu},\,W^{I}_{\mu}\to\left(1-{1\over 2}\delta
Z_{B}\right)W^{I}_{\mu},\,L_{p}\to\left(1-{1\over 2}\delta
Z_{L}\right)_{pr}L_{r},$ (4.8a) $\displaystyle\,\,g_{1}\to\left(1+{1\over
2}\delta Z_{B}\right)g_{1},\,\quad g_{2}\to\left(1+{1\over 2}\delta
Z_{B}\right)g_{2}.$ (4.8b)

The gauge coupling redefinition ensures the covariant derivative is unchanged
under gauge field redefinition. These redefinitions restore the SM Lagrangian
in its canonical form but with modified lepton Yukawa coupling and Higgs
potential parameters,

$\displaystyle\mathcal{L}_{\rm SM}^{\rm R}$ $\displaystyle=$
$\displaystyle-\frac{1}{4}G^{A}_{\mu\nu}G^{A\mu\nu}-\frac{1}{4}W^{I}_{\mu\nu}W^{I\mu\nu}-\frac{1}{4}B_{\mu\nu}B^{\mu\nu}$
(4.9)
$\displaystyle+|D_{\mu}H|^{2}+\hat{\mu}_{H}^{2}(H^{\dagger}H)-\hat{\lambda}_{H}(H^{\dagger}H)^{2}$
$\displaystyle+\sum_{\Psi}\bar{\Psi}i\not{D}\Psi-\left[\bar{Q}Y_{u}u\tilde{H}+\bar{Q}Y_{d}dH+\bar{L}\hat{Y}_{e}eH+\mbox{h.c.}\right],$

where the renormalized $\hat{\mu}_{H}^{2}$ and $\hat{\lambda}_{H}$ are given
in Eq. (4.7) and the renormalized lepton Yuwaka coupling is

$\displaystyle(\hat{Y}_{e})_{pr}=(Y_{e})_{pr}-{1\over 2}(\delta
Z_{L})_{ps}(Y_{e})_{sr}.$ (4.10)

The above redefinition or renormalization also modifies the one-loop generated
higher dimensional operators, but the effect is a two-loop correction and can
thus be neglected in our one-loop matching.

We can now derive the EoMs from the renormalized SM Lagrangian
$\mathcal{L}_{\rm SM}^{\rm R}$ which are the usual ones with the
substitutions,
$(\mu^{2}_{H},~{}\lambda_{H},~{}Y_{e})\to(\hat{\mu}^{2}_{H},~{}\hat{\lambda}_{H},~{}\hat{Y}_{e})$.
For the reduction of operators in the blue sector in Tab. 1, only the
following ones are required,

$\displaystyle\partial^{\nu}B_{\mu\nu}$ $\displaystyle=$ $\displaystyle
g_{1}\left[{1\over 6}\bar{Q}\gamma_{\mu}Q+{2\over
3}\bar{u}\gamma_{\mu}u-{1\over 3}\bar{d}\gamma_{\mu}d-{1\over
2}\bar{L}\gamma_{\mu}L-\bar{e}\gamma_{\mu}e+{1\over
2}H^{\dagger}i\overleftrightarrow{D_{\mu}}H\right],$ (4.11a) $\displaystyle
D^{\nu}W^{I}_{\mu\nu}$ $\displaystyle=$ $\displaystyle{g_{2}\over
2}\left[\overline{Q}\sigma^{I}\gamma_{\mu}Q+\overline{L}\sigma^{I}\gamma_{\mu}L+H^{\dagger}i\overleftrightarrow{D_{\mu}}^{I}H\right],$
(4.11b) $\displaystyle D^{2}H$ $\displaystyle=$
$\displaystyle\hat{\mu}_{H}^{2}H-2\hat{\lambda}_{H}(H^{\dagger}H)H-\epsilon^{\texttt{T}}\bar{Q}Y_{u}u-\bar{d}Y^{\dagger}_{d}Q-\bar{e}Y^{\dagger}_{e}L,$
(4.11c) $\displaystyle(D^{2}H)^{\dagger}$ $\displaystyle=$
$\displaystyle\hat{\mu}_{H}^{2}H^{\dagger}-2\hat{\lambda}_{H}(H^{\dagger}H)H^{\dagger}-\bar{u}Y_{u}^{\dagger}Q\epsilon-\bar{Q}Y_{d}d-\bar{L}\hat{Y}_{e}e,$
(4.11d) $\displaystyle i\not{D}L$ $\displaystyle=$
$\displaystyle\hat{Y}_{e}eH,$ (4.11e)
$\displaystyle-i\bar{L}\overleftarrow{\not{D}}$ $\displaystyle=$
$\displaystyle H^{\dagger}\bar{e}Y_{e}^{\dagger},$ (4.11f) $\displaystyle
i\not{D}e$ $\displaystyle=$
$\displaystyle\hat{Y}^{\dagger}_{e}{H}^{\dagger}L,$ (4.11g)
$\displaystyle-i\bar{e}\overleftarrow{\not{D}}$ $\displaystyle=$
$\displaystyle\bar{L}\hat{Y}_{e}H.$ (4.11h)

Once again since the higher dimensional operators are generated at one-loop
order, the hat in the three parameters can be dropped causing a negligible
two-loop error in the result.

Class | Operator |  WCs [$1/(16\pi^{2}m_{\eta}^{2})$]
---|---|---
$X^{3}$ | ${\cal O}_{W}$ | ${1\over 360}g_{2}^{3}$
$H^{6}$ | ${\cal O}_{H}$ | $-{1\over 6}[2\lambda_{3}^{3}+\lambda_{4}^{3}+3(\lambda_{3}+\lambda_{4})(\lambda_{3}\lambda_{4}+\lambda_{5}^{2})]+{1\over 60}[20(\lambda_{4}^{2}+\lambda_{5}^{2})-g_{2}^{4}]\lambda_{H}$
$H^{4}D^{2}$ | ${\cal O}_{H\Box}$ | $-{1\over 12}(2\lambda_{3}^{2}+2\lambda_{3}\lambda_{4}-\lambda_{5}^{2})-{1\over 480}(g_{1}^{4}+3g_{2}^{4})$
${\cal O}_{HD}$ | $-{1\over 6}(\lambda_{4}^{2}-\lambda_{5}^{2})-{1\over 120}g_{1}^{4}$
$\psi^{2}H^{3}+{\rm h.c.}$ | ${\cal O}_{eH}^{pr}$ | ${1\over 240}[20(\lambda_{4}^{2}+\lambda_{5}^{2})-g_{2}^{4}](Y_{e})_{pr}-{1\over 4}\lambda_{3}\left[Y_{\eta}F_{1}(x)Y_{\eta}^{\dagger}Y_{e}\right]_{pr}+{1\over 12}\left[Y_{e}Y_{e}^{\dagger}Y_{\eta}F_{2}(x)Y_{\eta}^{\dagger}Y_{e}\right]_{pr}$
${\cal O}_{uH}^{pr}$ | ${1\over 240}[20(\lambda_{4}^{2}+\lambda_{5}^{2})-g_{2}^{4}](Y_{u})_{pr}$
${\cal O}_{dH}^{pr}$ | ${1\over 240}[20(\lambda_{4}^{2}+\lambda_{5}^{2})-g_{2}^{4}](Y_{d})_{pr}$
$X^{2}H^{2}$ | ${\cal O}_{HW}$ | ${1\over 48}g_{2}^{2}(2\lambda_{3}+\lambda_{4})$
${\cal O}_{HB}$ | ${1\over 48}g_{1}^{2}(2\lambda_{3}+\lambda_{4})$
${\cal O}_{HWB}$ | ${1\over 24}g_{1}g_{2}\lambda_{4}$
$\psi^{2}XH+{\rm h.c.}$ | ${\cal O}_{eW}^{pr}$ | $-{1\over 48}g_{2}\left[Y_{\eta}F_{2}(x)Y_{\eta}^{\dagger}Y_{e}\right]_{pr}$
${\cal O}_{eB}^{pr}$ | $~{}~{}{1\over 48}g_{1}\left[Y_{\eta}F_{2}(x)Y_{\eta}^{\dagger}Y_{e}\right]_{pr}$
$\psi^{2}H^{2}D$ | ${\cal O}_{Hl}^{(1),pr}$ | $~{}~{}\,{1\over 240}g_{1}^{4}\delta_{pr}-{1\over 72}g_{1}^{2}\left[Y_{\eta}F_{3}(x)Y_{\eta}^{\dagger}\right]_{pr}$
${\cal O}_{Hl}^{(3),pr}$ | $-{1\over 240}g_{2}^{4}\delta_{pr}+{1\over 72}g_{2}^{2}\left[Y_{\eta}F_{3}(x)Y_{\eta}^{\dagger}\right]_{pr}$
${\cal O}_{He}^{pr}$ | $~{}~{}~{}~{}~{}~{}{1\over 120}g_{1}^{4}\delta_{pr}+{1\over 12}\left[Y_{e}^{\dagger}Y_{\eta}F_{2}(x)Y_{\eta}^{\dagger}Y_{e}\right]_{pr}$
${\cal O}_{Hq}^{(1),pr}$ | $-{1\over 720}g_{1}^{4}\delta_{pr}$
${\cal O}_{Hq}^{(3),pr}$ | $-{1\over 240}g_{2}^{4}\delta_{pr}$
${\cal O}_{Hu}^{pr}$ | $-{1\over 180}g_{1}^{4}\delta_{pr}$
${\cal O}_{Hd}^{pr}$ | $~{}~{}{1\over 360}g_{1}^{4}\delta_{pr}$
$(\bar{L}L)(\bar{L}L)$ | ${\cal O}_{ll}^{prst}$ | $F_{4,1}(x_{v},x_{w})(Y_{\eta})_{pw}(Y_{\eta}^{\texttt{T}})_{ws}(Y_{\eta}^{*})_{rv}(Y_{\eta}^{\dagger})_{vt}-F_{4,2}(x_{v},x_{w})(Y_{\eta})_{pw}(Y_{\eta}^{\dagger})_{wt}(Y_{\eta})_{sv}(Y_{\eta}^{\dagger})_{vr}$
$+{1\over
144}\left\\{(g_{1}^{2}-g_{2}^{2})\left[Y_{\eta}F_{3}(x)Y_{\eta}^{\dagger}\right]_{pr}\delta_{st}+2g_{2}^{2}\left[Y_{\eta}F_{3}(x)Y_{\eta}^{\dagger}\right]_{pt}\delta_{rs}+(pr)\leftrightarrow(st)\right\\}$
$-{1\over
480}[(g_{1}^{4}-g_{2}^{4})\delta_{pr}\delta_{st}+2g_{2}^{4}\delta_{pt}\delta_{rs}]$
${\cal O}_{qq}^{(1),prst}$ | $-{1\over 4320}g_{1}^{4}\delta_{pr}\delta_{st}$
${\cal O}_{qq}^{(3),prst}$ | $\,\,-{1\over 480}g_{2}^{4}\delta_{pr}\delta_{st}$
${\cal O}_{lq}^{(1),prst}$ | $~{}~{}~{}~{}{1\over 720}g_{1}^{4}\delta_{pr}\delta_{st}-{1\over 216}g_{1}^{2}\left[Y_{\eta}F_{3}(x)Y_{\eta}^{\dagger}\right]_{pr}\delta_{st}$
${\cal O}_{lq}^{(3),prst}$ | $-{1\over 240}g_{2}^{4}\delta_{pr}\delta_{st}+{1\over 72}g_{2}^{2}\left[Y_{\eta}F_{3}(x)Y_{\eta}^{\dagger}\right]_{pr}\delta_{st}$
$(\bar{R}R)(\bar{R}R)$ | ${\cal O}_{ee}^{prst}$ | $\,\,\,-{1\over 120}g_{1}^{4}\delta_{pr}\delta_{st}$
${\cal O}_{uu}^{prst}$ | $\,\,\,-{1\over 270}g_{1}^{4}\delta_{pr}\delta_{st}$
${\cal O}_{dd}^{prst}$ | $-{1\over 1080}g_{1}^{4}\delta_{pr}\delta_{st}$
${\cal O}_{eu}^{prst}$ | $~{}~{}~{}~{}~{}{1\over 90}g_{1}^{4}\delta_{pr}\delta_{st}$
${\cal O}_{ed}^{prst}$ | $\,\,\,-{1\over 180}g_{1}^{4}\delta_{pr}\delta_{st}$
${\cal O}_{ud}^{(1),prst}$ | $~{}~{}~{}~{}{1\over 270}g_{1}^{4}\delta_{pr}\delta_{st}$
$(\bar{L}L)(\bar{R}R)$ | ${\cal O}_{le}^{prst}$ | $\,\,\,-{1\over 120}g_{1}^{4}\delta_{pr}\delta_{st}+{1\over 36}g_{1}^{2}\left[Y_{\eta}F_{3}(x)Y_{\eta}^{\dagger}\right]_{pr}\delta_{st}$
${\cal O}_{lu}^{prst}$ | $\,\,\,\,\,\,\,{1\over 180}g_{1}^{4}\delta_{pr}\delta_{st}-{1\over 54}g_{1}^{2}\left[Y_{\eta}F_{3}(x)Y_{\eta}^{\dagger}\right]_{pr}\delta_{st}$
${\cal O}_{ld}^{prst}$ | $\,\,\,\,-{1\over 360}g_{1}^{4}\delta_{pr}\delta_{st}+{1\over 108}g_{1}^{2}\left[Y_{\eta}F_{3}(x)Y_{\eta}^{\dagger}\right]_{pr}\delta_{st}$
${\cal O}_{qe}^{prst}$ | $~{}~{}{1\over 360}g_{1}^{4}\delta_{pr}\delta_{st}$
${\cal O}_{qu}^{(1),prst}$ | $-{1\over 540}g_{1}^{4}\delta_{pr}\delta_{st}$
${\cal O}_{qd}^{(1),prst}$ | $~{}~{}{1\over 1080}g_{1}^{4}\delta_{pr}\delta_{st}$

Table 2: The matching result of dim-6 operators in the standard basis [26].

Class | Operator |  WCs [$1/(16\pi^{2})$]
---|---|---
$\psi^{2}H^{2}$ | ${\cal O}_{LH,5}^{pr}$ | $-{\lambda_{5}\over 2m_{\eta}^{2}}\left\\{\left[Y_{\eta}^{*}m_{N}G_{1}(x)Y_{\eta}^{\dagger}\right]_{pr}-{\mu_{H}^{2}\over 6m_{\eta}^{2}}\left[Y_{\eta}^{*}m_{N}G_{4}(x)Y_{\eta}^{\dagger}\right]_{pr}\right\\}$
$\psi^{2}H^{4}$ | ${\cal O}_{LH}^{pr}$ | ${\lambda_{5}\over 2m_{\eta}^{4}}\left\\{(\lambda_{3}+\lambda_{4})\left[Y_{\eta}^{*}m_{N}G_{2}(x)Y_{\eta}^{\dagger}\right]_{pr}-{\lambda_{H}\over 3}\left[Y_{\eta}^{*}m_{N}G_{4}(x)Y_{\eta}^{\dagger}\right]_{pr}\right\\}$
$\psi^{2}H^{3}D$ | ${\cal O}_{LeHD}^{pr}$ | $\,{\lambda_{5}\over 4m_{\eta}^{4}}\left[Y_{\eta}^{*}m_{N}G_{3}(x)Y_{\eta}^{\dagger}Y_{e}\right]_{pr}$
$\psi^{2}H^{2}X$ | ${\cal O}_{LHW}^{pr}$ | $-{\lambda_{5}\over 16m_{\eta}^{4}}\left[Y_{\eta}^{*}m_{N}G_{3}(x)Y_{\eta}^{\dagger}\right]_{pr}$
$\psi^{2}H^{2}D^{2}$ | ${\cal O}_{LDH2}^{pr}$ |  ${\lambda_{5}\over 12m_{\eta}^{4}}\left[Y_{\eta}^{*}m_{N}G_{4}(x)Y_{\eta}^{\dagger}\right]_{pr}$
$\psi^{4}H$ | ${\cal O}_{\bar{e}LLLH}^{prst}$ |  ${\lambda_{5}\over 12m_{\eta}^{4}}(Y_{e}^{\dagger})_{pr}\left[Y_{\eta}^{*}m_{N}G_{4}(x)Y_{\eta}^{\dagger}\right]_{st}$
${\cal O}_{\bar{d}QLLH1}^{prst}$ |  ${\lambda_{5}\over 12m_{\eta}^{4}}(Y_{d}^{\dagger})_{pr}\left[Y_{\eta}^{*}m_{N}G_{4}(x)Y_{\eta}^{\dagger}\right]_{st}$
${\cal O}_{\bar{Q}uLLH}^{prst}$ | $-{\lambda_{5}\over 12m_{\eta}^{4}}(Y_{u})_{pr}\left[Y_{\eta}^{*}m_{N}G_{4}(x)Y_{\eta}^{\dagger}\right]_{st}$

Table 3: The matching result of dim-5 and dim-7 operators in the standard
bases [25, 63]. Seven out of twelve dim-7 LNV but baryon-number-conserving
operators are generated.

Using the above EoMs together with IBP and Fierz relations, we can reduce the
operators mentioned above to the standard basis. The final matching results
for the LNC dim-6 and LNV dim-5 and dim-7 operators are tabulated in Tab. 2
and Tab. 3 respectively. We provide some details of reduction in appendix C.
In the tables we introduced some loop functions appearing in Tab. 1 for
brevity. Denoting the ratio of the two masses squared $x\equiv
m_{N}^{2}/m_{\eta}^{2}$ where the flavor index of $N$ is not shown for
brevity, we define the five loop functions associated with the dim-6 LNC
operators,

$\displaystyle F_{1}(x)$ $\displaystyle\equiv$ $\displaystyle{1-4x+3(1-2\ln
x)x^{2}\over(1-x)^{3}},$ (4.12a) $\displaystyle F_{2}(x)$
$\displaystyle\equiv$ $\displaystyle{1-6x+3(1-2\ln
x)x^{2}+2x^{3}\over(1-x)^{4}},$ (4.12b) $\displaystyle F_{3}(x)$
$\displaystyle\equiv$ $\displaystyle{2-9x+18x^{2}-(11-6\ln{x})x^{3}\over
2(1-x)^{4}},$ (4.12c) $\displaystyle F_{4,1}(x_{v},x_{w})$
$\displaystyle\equiv$ $\displaystyle-{\sqrt{x_{v}x_{w}}\over
4}\left\\{\left[{x_{v}\ln{x_{v}}\over(x_{v}-x_{w})(1-x_{v})^{2}}+v\leftrightarrow
w\right]+{1\over(1-x_{v})(1-x_{w})}\right\\},$ (4.12d) $\displaystyle
F_{4,2}(x_{v},x_{w})$ $\displaystyle\equiv$ $\displaystyle{1\over
8}\left\\{\left[{x_{v}^{2}\ln{x_{v}}\over(x_{v}-x_{w})(1-x_{v})^{2}}+v\leftrightarrow
w\right]+{1\over(1-x_{v})(1-x_{w})}\right\\}.$ (4.12e)

Note that the functions $F_{4,1}(x_{v},x_{w})$ and $F_{4,2}(x_{v},x_{w})$
depend on two masses of $N_{v}$ and $N_{w}$. In the degenerate limit of
$x_{v}=x_{w}=x$, they become $F_{4,i}(x,x)=F_{4,i}(x)$ where

$\displaystyle F_{4,1}(x)$ $\displaystyle\equiv$
$\displaystyle-{(2+\ln{x})x-(2-\ln{x})x^{2}\over 4(1-x)^{3}},$ (4.13a)
$\displaystyle F_{4,2}(x)$ $\displaystyle\equiv$
$\displaystyle{1+2x\ln{x}-x^{2}\over 8(1-x)^{3}}.$ (4.13b)

The four loop functions associated with the LNV dim-5 and dim-7 operators are,

$\displaystyle G_{1}(x)$ $\displaystyle\equiv$ $\displaystyle{1-(1-\ln
x)x\over(1-x)^{2}},$ (4.14a) $\displaystyle G_{2}(x)$ $\displaystyle\equiv$
$\displaystyle{1+2x\ln x-x^{2}\over(1-x)^{3}},$ (4.14b) $\displaystyle
G_{3}(x)$ $\displaystyle\equiv$ $\displaystyle{1+4(1+\ln x)x-(5-2\ln
x)x^{2}\over(1-x)^{4}},$ (4.14c) $\displaystyle G_{4}(x)$
$\displaystyle\equiv$ $\displaystyle{1-6x+3(1-2\ln
x)x^{2}+2x^{3}\over(1-x)^{4}}.$ (4.14d)

Except for $F_{4,i}$, all functions are normalized to unity at $x=0$. We make
a few comments concerning the matching result. First, the operators in blue in
Tab. 2 and Tab. 3 are completely generated through the EoM operators in blue
in Tab. 1. Second, application of the EoMs also causes significant changes in
the WCs for the operators already in the standard form, except for a few
operators, i.e., ${\cal O}_{W,HW,HB,HWB}$ in Tab. 2, and ${\cal O}_{LDH2}$ in
Tab. 3. And finally, application of EoMs for the two operators with a
${\color[rgb]{0,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,1}\pgfsys@color@cmyk@stroke{1}{0}{0}{0}\pgfsys@color@cmyk@fill{1}{0}{0}{0}\divideontimes}$
in Tab. 1 results in a further shift in the Higgs self-coupling
$\hat{\lambda}_{H}$ in Eq. (4.7) with the final answer being

$\displaystyle\hat{\lambda}_{H}\to\tilde{\lambda}_{H}$ $\displaystyle\equiv$
$\displaystyle\hat{\lambda}_{H}+{1\over 16\pi^{2}}{1\over
120}{\mu_{H}^{2}\over
m_{\eta}^{2}}[20(\lambda_{4}^{2}+\lambda_{5}^{2})-g_{2}^{4}]$ (4.15)
$\displaystyle=$ $\displaystyle\lambda_{H}-{1\over
16\pi^{2}}\left\\{{L_{\eta}\over
2}(2\lambda_{3}^{2}+2\lambda_{3}\lambda_{4}+\lambda_{4}^{2}+\lambda_{5}^{2})-{1\over
120}{\mu_{H}^{2}\over
m_{\eta}^{2}}[20(\lambda_{4}^{2}+\lambda_{5}^{2})-g_{2}^{4}]\right\\}.$

##  5 Phenomenology

From the SMEFT matching result in Tab. 2 and Tab. 3, together with the
matching results onto the LEFT given in [71, 72], one can readily consider
interesting physical processes to explore the parameter space of the model.
Differently from all previous studies on the scotogeneric model [2, 3, 4, 5,
6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24], we
here use the SMEFT approach to consider the heavy new physics implications for
low energy physics, with both $m_{\eta}$ and $m_{N}$ being above the TeV
scale. We will also make comparisons with the study in literature when
necessary.

###  5.1 Dim-5 and dim-7 LNV processes

Neutrino mass: The neutrino mass is generated from both the dim-5 Weinberg
operator ${\cal O}_{LH,5}$ and the dim-7 operator ${\cal O}_{LH}$. By going to
the Higgs phase $H\to v/\sqrt{2}$, we obtain the symmetric Majorana neutrino
mass matrix as

$\displaystyle M^{\nu}_{pr}={\lambda_{5}v^{2}\over
32\pi^{2}m_{\eta}^{2}}\left\\{Y_{\eta}^{*}m_{N}\left[G_{1}(x)-{(\lambda_{3}+\lambda_{4})v^{2}\over
2m_{\eta}^{2}}G_{2}(x)\right]Y_{\eta}^{\dagger}\right\\}_{pr},$ (5.1)

where the $G_{1}(x)$ part comes from the dim-5 Weinberg operator and the
$G_{2}(x)$ part is due to the dim-7 contribution. We recall that a sum over
$N_{w}$ is implied in the above, where the subscript enters
$m_{N_{w}}(Y^{*}_{\eta})_{pw}(Y^{*}_{\eta})_{rw}$ and
$x_{w}=m_{N_{w}}^{2}/m_{\eta}^{2}$. In the limit of $\lambda_{5}\to 0$, our
result for the dim-5 contribution differs from the original one given in Eq.
(12) in [1] by a factor of 2, but is consistent with the result provided in
[17]. The dim-7 contribution is new and has not been considered before in the
literature. To assess roughly its potential relevance, we consider the ratio
of their contributions:

$\displaystyle{(\lambda_{3}+\lambda_{4})v^{2}\over
2m_{\eta}^{2}}{G_{2}(x)\over G_{1}(x)}\sim 0.3{\lambda_{3}+\lambda_{4}\over
10}\left(1\,$\mathrm{T}\mathrm{e}\mathrm{V}$\over
m_{\eta}\right)^{2}G_{2/1}(x),$ (5.2)

where $G_{2/1}(x)\equiv G_{2}(x)/G_{1}(x)$ deceases monotonically from 1 at
$x=0$ to $0.17$ at $x=10^{3}$ (Fig. 1). Thus, for a relatively large
$\lambda_{3,4}\sim{\cal O}(1)-{\cal O}(10)$ and $m_{\eta}\sim{\cal
O}($\mathrm{T}\mathrm{e}\mathrm{V}$)$, the dim-7 contribution could be non-
negligible and should be included in a refined analysis.

Neutrino transition moment: From the matching results in Tab. 3 one would
expect a contribution to the Majorana neutrino electromagnetic moment from the
operator ${\cal O}_{LHW}^{pr}$. However, this potential contribution vanishes
since its WC $C_{LHW}^{pr}$ is symmetric in the flavor indices while the
neutrino transition magnetic moment is antisymmetric. It is evident from
Feynman diagrams as there is no contributing diagram due to the
$\mathbb{Z}_{2}$ symmetry and charge conservation.

Figure 1: Ratios of loop functions associated with LNV dim-5 and dim-7
matching coefficients as a function of the mass squared ratio $x$.

Effects of remaining dim-7 operators: To see the relative size of LNV signals
due to the matching results of other dim-7 operators, it is helpful to first
compare the loop functions $G_{2,3,4}(x)$ in dim-7 WCs with $G_{1}(x)$ for the
dim-5 Weinberg operator. In Fig. 1 we show the ratios of the three functions,
$G_{i/1}(x)=G_{i}(x)/G_{1}(x)$ for $i=2,3,4$. As can be seen in the figure,
$G_{2,3,4}(x)$ are almost always smaller than $G_{1}(x)$ in the whole range,
with the only exception of $G_{4}(x)$ for $0\leq x\leq 1$ where the two
functions are comparable. Thus to explore their potential largest effects, we
can approximately replace $G_{2,3,4}(x)$ by $G_{1}(x)$ and
$\lambda_{5}v^{2}/(32\pi^{2}m_{\eta}^{2})[Y_{\eta}^{*}m_{N}G_{1}(x)Y_{\eta}^{\dagger}]_{pr}$
by the neutrino mass matrix $M^{\nu}_{pr}$. This results in the estimation:

$\displaystyle|C_{LeHD}^{pr}|\lesssim{(M^{\nu}Y_{e})_{pr}\over
2v^{2}m_{\eta}^{2}};\quad|C_{LHW}^{pr}|,\,|C_{LDH2}^{pr}|\lesssim{M^{\nu}_{pr}\over
6v^{2}m_{\eta}^{2}};$
$\displaystyle|C_{\bar{e}LLLH}^{prst}|,\,|C_{\bar{d}QLLH1}^{prst}|,\,|C_{\bar{Q}uLLH}^{prst}|\lesssim{(Y_{x})_{pr}M^{\nu}_{st}\over
6v^{2}m_{\eta}^{2}},$ (5.3)

where in the last line $Y_{x}=Y^{\dagger}_{e},~{}Y^{\dagger}_{d},~{}Y_{u}$ for
the three different WCs in question. Clearly, these WCs are suppressed by the
neutrino mass as well as the heavy scale $m_{\eta}$, and in some cases further
suppressed by the SM Yukawa couplings. If we take $M_{\nu}\sim{\cal
O}(0.1\,$\mathrm{e}\mathrm{V}$)$ and $m_{\eta}\sim{\cal
O}(1\,$\mathrm{T}\mathrm{e}\mathrm{V}$)$, the largest WC is
$|C_{\bar{Q}uLLH}^{prst}|\lesssim
1/(10^{4}\,$\mathrm{T}\mathrm{e}\mathrm{V}$)^{3}($\mathrm{T}\mathrm{e}\mathrm{V}$/m_{\eta})^{2}$,
far below the current sensitivity of nuclear neutrinoless double-$\beta$ decay
($0\nu\beta\beta$) [73, 74]. We therefore conclude that the most interesting
LNV processes like $0\nu\beta\beta$ are dominantly mediated by the neutrino
mass term through the contribution of dim-5 and dim-7 Weinberg operators
${\cal O}_{LH,5}$ and ${\cal O}_{LH}$, and the effects from the remaining
dim-7 operators can be safely neglected. This pattern of LNV signals being
bound to the tiny neutrino mass is understandable from the fact that lepton
number violation can only happen for
$\lambda_{5}Y_{\eta}^{*}Y_{\eta}^{\dagger}\neq 0$, to which the neutrino mass
matrix is proportional.

###  5.2 LNC processes due to dim-6 operators

CDF $W$-mass anomaly: The CDF Collaboration recently reported a measurement
for the $W$ boson mass $m_{W}^{\tt CDF}=80433.5\pm
9.4\,$\mathrm{M}\mathrm{e}\mathrm{V}$$ [75], which is a 7$\sigma$ deviation
from the SM prediction, $m_{W}^{\tt SM}=80357\pm
4\,$\mathrm{M}\mathrm{e}\mathrm{V}$$ [76]. Combining the result with all
previous measurements yields a new world average, $m_{W}^{\tt
new\,ave}=80417\pm 18\,$\mathrm{M}\mathrm{e}\mathrm{V}$$ [76], which is still
far away from the SM prediction. Assuming Gaussian error propagation, the CDF
result implies the following relative size of the excess,

$\displaystyle{\delta m_{W}^{2}\over m_{W}^{2}}\Big{|}_{\tt
CDF}\equiv{m_{W,\tt CDF}^{2}-m_{W,\tt SM}^{2}\over m_{W,\tt CDF}^{2}}=(0.95\pm
0.13)\times 10^{-3},$ $\displaystyle{\delta m_{W}^{2}\over
m_{W}^{2}}\Big{|}_{\tt new\,ave}=(0.75\pm 0.23)\times 10^{-3}.$ (5.4)

The anomaly has stimulated many studies, see for instance [77, 80, 79, 78] and
references therein. An investigation of the scotogenic model to explain the
anomaly has been given in [81]. We will work with the SMEFT approach by using
the matching result obtained in the previous section to examine its
implications.

To study the correction to $m_{W}^{2}$ in SMEFT we proceed as in [80] and
employ the following most precisely known parameters as the input:

$\displaystyle\alpha_{\rm em}^{-1}(m_{Z})=127.95,\quad G_{F}=1.16638\times
10^{-5}\,$\mathrm{G}\mathrm{e}\mathrm{V}$^{-2},\quad
m_{Z}=91.1876\,$\mathrm{G}\mathrm{e}\mathrm{V}$.$ (5.5)

Then the correction to $m_{W}^{2}$ enters through the modifications to the
expressions of $m_{Z}$ and the Fermi constant $G_{F}$ measured in muon decay,
which are mainly induced by the four dim-6 SMEFT operators,

$\displaystyle{\cal
O}_{HWB}=H^{\dagger}\sigma^{I}HW_{\mu\nu}^{I}B^{\mu\nu},\quad{\cal
O}_{HD}=|H^{\dagger}D_{\mu}H|^{2},$ $\displaystyle{\cal
O}_{ll}=(\overline{L}\gamma_{\mu}L)(\overline{L}\gamma^{\mu}L),\quad{\cal
O}_{Hl}^{(3)}=(H^{\dagger}i\overleftrightarrow{D_{\mu}^{I}}H)(\overline{L}\sigma^{I}\gamma^{\mu}L).$
(5.6)

The first two affects the SM prediction on $m_{Z}$ through field
diagonalization while the latter two lead to a correction to $G_{F}$. Defining
the effective weak mixing angle by the input parameters as,

$\displaystyle\cos^{2}\theta_{W}\equiv{1\over
2}\left(1+\sqrt{1-{4\pi\alpha_{\rm em}\over\sqrt{2}G_{F}m_{Z}^{2}}}\right),$
(5.7)

the correction to the pole mass $m_{W}$ relative to the SM prediction
($m_{W}^{\tt SM}=m_{Z}^{2}\cos^{2}\theta_{W}$) takes the form, to the linear
order in the WCs associated with the above operators,

$\displaystyle{\delta m_{W}^{2}\over m_{W}^{2}}$ $\displaystyle=$
$\displaystyle-{\sin 2\theta_{W}\over\cos 2\theta_{W}}{1\over
8G_{F}}\left[{\cos\theta_{W}\over\sin\theta_{W}}C_{HD}+4C_{HWB}\right.$ (5.8)
$\displaystyle\left.+{\sin\theta_{W}\over\cos\theta_{W}}\left(2C_{Hl}^{(3),ee}+2C_{Hl}^{(3),\mu\mu}-C_{ll}^{\mu
ee\mu}-C_{ll}^{e\mu\mu e}\right)\right].$

If the $SU(3)^{5}$ global flavor symmetry is assumed, the above expression
reduces to Eq. (2.3) given in [80]. Note that the usual electroweak oblique
parameters $S$ and $T$ [83] are related to $C_{HWB}$ and $C_{HD}$ via the
relations $S=(4\sin\theta_{W}\cos\theta_{W}/\alpha_{\rm
em})C_{WB}/(\sqrt{2}G_{F})$ and $T=-(1/2\alpha_{\rm
em})C_{HD}/(\sqrt{2}G_{F})$. From the matching result in Tab. 2, we obtain,

$\displaystyle{\delta m_{W}^{2}\over m_{W}^{2}}={\tan 2\theta_{W}\over
128\pi^{2}G_{F}m_{\eta}^{2}}\left[{(4\pi)^{2}\alpha_{\rm em}^{2}\over
15\sin^{3}2\theta_{W}}+{\cos\theta_{W}\over\sin\theta_{W}}{\lambda_{4}^{2}-\lambda_{5}^{2}\over
6}-{4\pi\alpha_{\rm em}\lambda_{4}\over 3\sin
2\theta_{W}}+2{\sin\theta_{W}\over\cos\theta_{W}}\tilde{C}_{ll}^{e\mu\mu
e}\right],$ (5.9)

where we have used the approximate relations $g_{1}=e/\cos\theta_{W}$,
$g_{2}=e/\sin\theta_{W}$, and $\alpha_{\rm em}=e^{2}/(4\pi)$, as well as the
abbreviation

$\displaystyle\tilde{C}_{ll}^{e\mu\mu e}$ $\displaystyle=$ $\displaystyle
F_{4,1}(x_{w},x_{v})(Y_{\eta})_{1w}(Y_{\eta}^{\texttt{T}})_{w2}(Y_{\eta}^{*})_{1v}(Y_{\eta}^{\dagger})_{v2}$
(5.10) $\displaystyle-
F_{4,2}(x_{w},x_{v})(Y_{\eta})_{1w}(Y_{\eta}^{\dagger})_{w1}(Y_{\eta})_{2v}(Y_{\eta}^{\dagger})_{v2},$

where summation over $N_{v},~{}N_{w}$ is implied.

Since $\lambda_{5}$ is related to the Majorana neutrino mass, it is naturally
much smaller than unity to yield an ${\cal O}(0.1\,$\mathrm{e}\mathrm{V}$)$
neutrino mass for ${\cal O}(\,$\mathrm{T}\mathrm{e}\mathrm{V}$)$ scale heavy
masses. Dropping the $\lambda_{5}$ term and using the values in Eq. (5.5), we
obtain

$\displaystyle{\delta m_{W}^{2}\over
m_{W}^{2}}=10^{-3}\left($\mathrm{T}\mathrm{e}\mathrm{V}$\over
m_{\eta}\right)^{2}\left(1.6\times
10^{-4}+0.046\lambda_{4}^{2}-0.006\lambda_{4}+0.168\tilde{C}_{ll}^{e\mu\mu
e}\right).$ (5.11)

To reproduce the amount of excess in Eq. (5.4) for ${\cal
O}($\mathrm{T}\mathrm{e}\mathrm{V}$)$ scale $m_{\eta}$, it requires either a
relatively large $\lambda_{4}$ or $\tilde{C}_{ll}^{e\mu\mu e}$. Let us first
consider the case with negligible $\tilde{C}_{ll}^{e\mu\mu e}$. To explain the
CDF anomaly, the allowed parameter space in the $m_{\eta}$-$\lambda_{4}$ plane
is shown in Fig. 2 for the two cases: the CDF-only and the new world average.
It can be seen that $\lambda_{4}\gtrsim 4$ is necessary to match the excess
for $m_{\eta}\geq 1\,$\mathrm{T}\mathrm{e}\mathrm{V}$$. This result basically
agrees with the estimation given in [78]. On the other hand, the excess could
be accommodated by $\tilde{C}_{ll}^{e\mu\mu e}\in[4.9,\,6.4]~{}([3.1,\,5.8])$
for the CDF-only (new world average) case as well. Since the loop functions
are bound by $F_{4,1}\leq 1/24$ and $F_{4,2}\leq 1/8$, this would require
${\cal O}({\rm few})$ Yukawa couplings to achieve a large enough
$\tilde{C}_{ll}^{e\mu\mu e}$. This may cause some tension with the constraints
from the LFV processes ($\mu\to e\gamma$ and $\mu\to 3e$), and needs a careful
analysis of the Yukawa sector.

Figure 2: The allowed region in the $m_{\eta}$-$\lambda_{4}$ plane for
explaining the $W$ boson mass anomaly at the 1$\sigma$ level.

Lepton $\bm{g-2}$ and LFV decay $\bm{\ell_{i}\to\ell_{j}\gamma}$: These
observables are connected to the dim-5 dipole operator in the LEFT,

$\displaystyle{\cal L}_{e\gamma}=C_{e\gamma}^{pr}{\cal O}_{e\gamma}^{pr}+{\rm
h.c.}=C_{e\gamma}^{pr}\overline{e_{Lp}}\sigma_{\mu\nu}e_{Rr}F^{\mu\nu}+{\rm
h.c.},$ (5.12)

where the WC $C_{e\gamma}^{pr}$ in LEFT are related to those in Tab. 2 by

$\displaystyle
C_{e\gamma}^{pr}={v\over\sqrt{2}}\left(-\sin\theta_{w}C_{eW}^{pr}+\cos\theta_{w}C_{eB}^{pr}\right)={e\,m_{r}\over
384\pi^{2}}{1\over
m_{\eta}^{2}}\left[Y_{\eta}F_{2}(x)Y_{\eta}^{\dagger}\right]_{pr}.$ (5.13)

We have assumed that the SM Yukawa coupling matrix for charged leptons has
been diagonalized so that $M_{e}=vY_{e}/\sqrt{2}={\rm
diag}(m_{e},m_{\mu},m_{\tau})$. The decay width for the LFV process
$\ell_{i}\to\ell_{j}\gamma$ is given by

$\displaystyle\Gamma_{\ell_{i}\to\ell_{j}\gamma}={m_{\ell_{i}}^{3}\over
4\pi}\left(|C_{e\gamma}^{ji}|^{2}+|C_{e\gamma}^{ij}|^{2}\right)={\alpha_{\rm
em}m_{\ell_{i}}^{5}\over(384\pi^{2})^{2}m_{\eta}^{4}}\left|\left[Y_{\eta}F_{2}(x)Y_{\eta}^{\dagger}\right]_{ji}\right|^{2},$
(5.14)

where in the second step we have neglected the small correction proportional
to ${m_{\ell_{j}}^{2}/m_{\ell_{i}}^{2}}$ from $|C_{e\gamma}^{ij}|^{2}$. The
above leads to the branching ratio

$\displaystyle{\cal B}_{\ell_{i}\to\ell_{j}\gamma}$ $\displaystyle=$
$\displaystyle{\Gamma_{\ell_{i}\to\ell_{j}\gamma}\over\Gamma_{\ell_{i}\to\ell_{j}\bar{\nu}_{j}\nu_{i}}^{\tt
SM}}={\alpha_{\rm em}\over 768\pi
G_{F}^{2}m_{\eta}^{4}}\left|\left[Y_{\eta}F_{2}(x)Y_{\eta}^{\dagger}\right]_{ji}\right|^{2}$
(5.15) $\displaystyle=$ $\displaystyle 2.2\times
10^{-8}\left($\mathrm{T}\mathrm{e}\mathrm{V}$\over
m_{\eta}\right)^{4}\left|\left[Y_{\eta}F_{2}(x)Y_{\eta}^{\dagger}\right]_{ji}\right|^{2},$

where the SM prediction for the dominant decay width has been used,
$\Gamma_{\ell_{i}\to\ell_{j}\bar{\nu}_{j}\nu_{i}}^{\tt
SM}=G_{F}^{2}m_{\ell_{i}}^{5}/(192\pi^{3})$. In the heavy mass limit, our
result agrees with those given in [2, 7]. The current experimental upper
bounds on these LFV processes are, ${\cal B}_{\mu\to e\gamma}^{\tt
exp}\lesssim 4.2\times 10^{-13}$ and ${\cal B}_{\tau\to e(\mu)\gamma}^{\tt
exp}\lesssim 3.3(4.4)\times 10^{-8}$ [76], which implies

$\displaystyle{{\cal B}_{\mu\to e\gamma}/{\cal B}_{\mu\to
e\bar{\nu}_{e}\nu_{\tau}}}\lesssim 4.2\times 10^{-13},\quad{{\cal B}_{\tau\to
e\gamma}/{\cal B}_{\tau\to e\bar{\nu}_{e}\nu_{\tau}}}\lesssim 1.9\times
10^{-7},\quad{{\cal B}_{\tau\to\mu\gamma}/{\cal
B}_{\tau\to\mu\bar{\nu}_{\mu}\nu_{\tau}}}\lesssim 2.5\times 10^{-7}.$ (5.16)

As can be seen from the above, for $m_{\eta,N}\sim{\cal
O}($\mathrm{T}\mathrm{e}\mathrm{V}$)$ and ${\cal O}(1)$ Yukawa couplings, the
branching ratios of the LFV processes $\tau\to e(\mu)\gamma$ are below the
current experimental bounds by an order of magnitude, whereas the $\mu\to
e\gamma$ can probe the new physics scale up to tens of TeV for ${\cal O}(1)$
Yukawa couplings.

While the electric dipole moment is not induced at one loop, there is a
contribution to the anomalous magnetic moment of charged leptons:

$\displaystyle\Delta a_{\ell}={m_{\ell}^{2}\over
96\pi^{2}m_{\eta}^{2}}\left[Y_{\eta}F_{2}(x)Y_{\eta}^{\dagger}\right]_{\ell\ell}.$
(5.17)

Considering $0<F_{2}(x)\leq 1$, the anomalous magnetic moment is approximately
bound by

$\displaystyle|\Delta a_{\ell}|\lesssim 1.2\times 10^{-11}{m_{\ell}^{2}\over
m_{\mu}^{2}}\left({1\,{\rm TeV}\over
m_{\eta}}\right)^{2}\left|\left[Y_{\eta}Y_{\eta}^{\dagger}\right]_{\ell\ell}\right|,$
(5.18)

which cannot fill the gap between the SM prediction and the observed values in
the Fermilab and BNL E821 experiments [84, 85], $\Delta a_{\mu}=a_{\mu}^{\tt
exp}-a_{\mu}^{\tt SM}=251(59)\times 10^{-11}$ for reasonably large Yukawa
couplings.

LFV decay $\mu\to 3e$: In the LEFT framework [71, 86], the dominant
contribution to $\mu\to 3e$ is from the long-distance dipole operator ${\cal
O}_{e\gamma}^{e\mu}$ through a virtual photon as well as the following dim-6
contact interactions,

$\displaystyle{\cal O}^{V,LL}_{e\mu ee}$ $\displaystyle=$
$\displaystyle(\overline{e_{L}}\gamma_{\mu}\mu_{L})(\overline{e_{L}}\gamma^{\mu}e_{L}),\quad\,{\cal
O}^{V,LR}_{e\mu
ee}=(\overline{e_{L}}\gamma_{\mu}\mu_{L})(\overline{e_{R}}\gamma^{\mu}e_{R}),\quad$
(5.19a) $\displaystyle{\cal O}^{V,LR}_{eee\mu}$ $\displaystyle=$
$\displaystyle(\overline{e_{R}}\gamma_{\mu}\mu_{R})(\overline{e_{L}}\gamma^{\mu}e_{L}),\quad{\cal
O}^{V,RR}_{e\mu
ee}=(\overline{e_{R}}\gamma_{\mu}\mu_{R})(\overline{e_{R}}\gamma^{\mu}e_{R}),\quad$
(5.19b) $\displaystyle{\cal O}^{S,LL}_{e\mu ee}$ $\displaystyle=$
$\displaystyle(\overline{e_{R}}\mu_{L})(\overline{e_{R}}e_{L}),\quad\quad\quad\,{\cal
O}^{S,RR}_{e\mu ee}=(\overline{e_{L}}\mu_{R})(\overline{e_{L}}e_{R}).$ (5.19c)

Based on the matching result in Tab. 2 as well as the LEFT-SMEFT matching
result in [71], we find only the first two dim-6 operators (${\cal
O}^{V,LL}_{e\mu ee},{\cal O}^{V,LR}_{eee\mu}$) have non-negligible
contributions,

$\displaystyle C_{e\mu ee}^{V,LL}={1\over
16\pi^{2}m_{\eta}^{2}}\left\\{2\tilde{C}_{ll}^{e\mu ee}+{e^{2}\over
18}\left[Y_{\eta}F_{3}(x)Y_{\eta}^{\dagger}\right]_{e\mu}\right\\},\quad
C_{e\mu ee}^{V,LR}={1\over 16\pi^{2}m_{\eta}^{2}}{e^{2}\over
18}\left[Y_{\eta}F_{3}(x)Y_{\eta}^{\dagger}\right]_{e\mu}.$ (5.20)

Rewriting the WC of the dipole operator in Eq. (5.13) as
$C_{e\gamma}^{e\mu}\equiv e\,m_{\mu}\tilde{C}_{e\gamma}^{e\mu}$, the decay
width of $\mu\to 3e$ takes the form [86],

$\displaystyle{\Gamma_{\mu\to 3e}}$ $\displaystyle=$
$\displaystyle{m_{\mu}^{5}\over
192\pi^{3}}\bigg{\\{}e^{4}|\tilde{C}_{e\gamma}^{e\mu}|^{2}\left(8\ln{m_{\mu}\over
m_{e}}-11\right)+{1\over 8}\left(2|C_{e\mu ee}^{V,LL}|^{2}+|C_{e\mu
ee}^{V,LR}|^{2}\right)$ (5.21)
$\displaystyle-e^{2}\Re\left[\tilde{C}_{e\gamma}^{e\mu}(2C_{e\mu
ee}^{V,LL}+C_{e\mu ee}^{V,LR})^{*}\right]\bigg{\\}},$

where again we have neglected the small correction proportional to
$m_{e}^{2}/m_{\mu}^{2}$ from $|\tilde{C}_{e\gamma}^{\mu e}|^{2}$. In terms of
branching ratios, we obtain

$\displaystyle{{\cal B}_{\mu\to 3e}\over{\cal B}_{\mu\to
e\bar{\nu}_{e}\nu_{\mu}}}$ $\displaystyle=$ $\displaystyle{3e^{4}\over
8G_{F}^{2}}\bigg{\\{}{8\over
3}|\tilde{C}_{e\gamma}^{e\mu}|^{2}\left(8\ln{m_{\mu}\over
m_{e}}-11\right)+{1\over 3e^{4}}\left(2|C_{e\mu ee}^{V,LL}|^{2}+|C_{e\mu
ee}^{V,LR}|^{2}\right)$ (5.22) $\displaystyle-{8\over
3e^{2}}\Re\left[\tilde{C}_{e\gamma}^{e\mu}(2C_{e\mu ee}^{V,LL}+C_{e\mu
ee}^{V,LR})^{*}\right]\bigg{\\}}.$

Once again in the heavy mass limit, our result agrees with that in [7] upon
the following correspondence of WCs between ours and theirs,

$\displaystyle\tilde{C}_{e\gamma}^{e\mu}={1\over 2}A_{D},\quad C_{e\mu
ee}^{V,LL}={1\over 2}e^{2}(B+2A_{ND}),\quad C_{e\mu ee}^{V,LR}=e^{2}A_{ND}.$
(5.23)

Besides the processes discussed above, our matching results can also be
applied to many other interesting processes, for instance, the LFV decays of
the Higgs and $Z$ bosons, the LFV decays of heavy mesons and baryons, the
$\mu$-$e$ conversion in nuclei, and non-standard neutrino interactions, etc.
This more complete phenomenological investigation deserves a future work.

##  6 Conclusion

The scotogenic model is an economical extension of the SM that generates tiny
neutrino mass at one-loop level, and at the same time naturally provides a
dark matter candidate. The one-loop origin of neutrino mass implies a lower
new physics scale than that of the three tree-level seesaws, which makes this
model potentially testable at the future high energy colliders. Confronting
with the current null experimental searches for weak scale new particles, it
is appropriate to assume these new particles are well above the electroweak
scale so that their indirect effects on low energy observables can be
incorporated in an effective field theory where they have been integrated out.
With this spirit, we employ the newly developed functional method to match the
scotogenic model onto the standard model effective field theory up to
dimension 7 for the case when both new particles $N$ and $\eta$ are heavy .
Because of the $\mathbb{Z}_{2}$ symmetry, non-trivial matching only starts to
appear at one loop. Our matching results are first organized in a Green basis
(see Tab. 1) in which the origin of the matched operators can be relatively
easily tracked. Then we use the SM equations of motion, the integration by
parts relations, and various algebraic identities to recast the results in the
standard bases for dim-5 [25], dim-6 [26], and dim-7 operators [63] (Tab. 2
and Tab. 3). Finally, we apply our results to study implications on several
interesting physical processes at low energy and make comparisons with those
in the literature to further confirm our results.

## Acknowledgement

This work was supported in part by the Grants No. NSFC-12035008, No.
NSFC-11975130, No. NSFC-12090064, by the Chung-Yao Chao Fellowship at Chinese
Academy of Sciences Center for Excellence in Particle Physics (CCEPP), by the
National Key Research and Development Program of China under Grant No.
2017YFA0402200, by the Guangdong Major Project of Basic and Applied Basic
Research No. 2020B0301030008 and Science and Technology Program of Guangzhou
No. 2019050001, and by Shanghai Pujiang Program (20PJ1407800).

##  Appendix A Dim-5, dim-6, and dim-7 operator bases in the SMEFT

$\color[rgb]{1,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,1}\pgfsys@color@cmyk@stroke{0}{1}{0}{0}\pgfsys@color@cmyk@fill{0}{1}{0}{0}X^{3}$ | $\color[rgb]{1,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,1}\pgfsys@color@cmyk@stroke{0}{1}{0}{0}\pgfsys@color@cmyk@fill{0}{1}{0}{0}\psi^{2}H^{3}+{\rm h.c.}$ | $\color[rgb]{1,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,1}\pgfsys@color@cmyk@stroke{0}{1}{0}{0}\pgfsys@color@cmyk@fill{0}{1}{0}{0}(\bar{L}L)(\bar{L}L)$
---|---|---
${\cal O}_{G}$ | $f^{ABC}G_{\mu}^{A\nu}G_{\nu}^{B\rho}G_{\rho}^{C\mu}$ | ${\cal O}_{eH}$ | $(H^{\dagger}H)(\overline{L}eH)$ | ${\cal O}_{ll}$ | $(\overline{L}\gamma_{\mu}L)(\overline{L}\gamma^{\mu}L)$
${\cal O}_{\tilde{G}}$ | $f^{ABC}\tilde{G}_{\mu}^{A\nu}G_{\nu}^{B\rho}G_{\rho}^{C\mu}$ | ${\cal O}_{uH}$ | $(H^{\dagger}H)(\overline{Q}u\tilde{H})$ | ${\cal O}_{qq}^{(1)}$ | $(\overline{Q}\gamma_{\mu}Q)(\overline{Q}\gamma^{\mu}Q)$
${\cal O}_{W}$ | $\epsilon^{IJK}W_{\mu}^{I\nu}W_{\nu}^{J\rho}W_{\rho}^{K\mu}$ | ${\cal O}_{dH}$ | $(H^{\dagger}H)(\overline{Q}dH)$ | ${\cal O}_{qq}^{(3)}$ | $(\overline{Q}\gamma_{\mu}\tau^{I}Q)(\overline{Q}\gamma^{\mu}\tau^{I}Q)$
${\cal O}_{\tilde{W}}$ | $\epsilon^{IJK}\tilde{W}_{\mu}^{I\nu}W_{\nu}^{J\rho}W_{\rho}^{K\mu}$ | $\color[rgb]{1,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,1}\pgfsys@color@cmyk@stroke{0}{1}{0}{0}\pgfsys@color@cmyk@fill{0}{1}{0}{0}\psi^{2}XH+{\rm h.c.}$ | ${\cal O}_{lq}^{(1)}$ | $(\overline{L}\gamma_{\mu}L)(\overline{Q}\gamma^{\mu}Q)$
$\color[rgb]{1,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,1}\pgfsys@color@cmyk@stroke{0}{1}{0}{0}\pgfsys@color@cmyk@fill{0}{1}{0}{0}H^{6}$ | ${\cal O}_{eW}$ | $(\overline{L}\sigma^{\mu\nu}e)\tau^{I}HW^{I}_{\mu\nu}$ | ${\cal O}_{lq}^{(3)}$ | $(\overline{L}\gamma_{\mu}\tau^{I}L)(\overline{Q}\gamma^{\mu}\tau^{I}Q)$
${\cal O}_{H}$ | $(H^{\dagger}H)^{3}$ | ${\cal O}_{eB}$ | $(\overline{L}\sigma^{\mu\nu}e)HB_{\mu\nu}$ | $\color[rgb]{1,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,1}\pgfsys@color@cmyk@stroke{0}{1}{0}{0}\pgfsys@color@cmyk@fill{0}{1}{0}{0}(\bar{R}R)(\bar{R}R)$
$\color[rgb]{1,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,1}\pgfsys@color@cmyk@stroke{0}{1}{0}{0}\pgfsys@color@cmyk@fill{0}{1}{0}{0}H^{4}D^{2}$ | ${\cal O}_{uG}$ | $(\overline{Q}\sigma^{\mu\nu}T^{A}u)\tilde{H}G^{A}_{\mu\nu}$ | ${\cal O}_{ee}$ | $(\overline{e}\gamma_{\mu}e)(\overline{e}\gamma^{\mu}e)$
${\cal O}_{H\Box}$ | $(H^{\dagger}H)\Box(H^{\dagger}H)$ | ${\cal O}_{uW}$ | $(\overline{Q}\sigma^{\mu\nu}u)\tau^{I}\tilde{H}W^{I}_{\mu\nu}$ | ${\cal O}_{uu}$ | $(\overline{u}\gamma_{\mu}u)(\overline{u}\gamma^{\mu}u)$
${\cal O}_{HD}$ | $(H^{\dagger}D_{\mu}H)^{*}(H^{\dagger}D_{\mu}H)$ | ${\cal O}_{uB}$ | $(\overline{Q}\sigma^{\mu\nu}u)\tilde{H}B_{\mu\nu}$ | ${\cal O}_{dd}$ | $(\overline{d}\gamma_{\mu}d)(\overline{d}\gamma^{\mu}d)$
$\color[rgb]{1,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,1}\pgfsys@color@cmyk@stroke{0}{1}{0}{0}\pgfsys@color@cmyk@fill{0}{1}{0}{0}X^{2}H^{2}$ | ${\cal O}_{dG}$ | $(\overline{Q}\sigma^{\mu\nu}T^{A}d)HG^{A}_{\mu\nu}$ | ${\cal O}_{eu}$ | $(\overline{e}\gamma_{\mu}e)(\overline{u}\gamma^{\mu}u)$
${\cal O}_{HG}$ | $H^{\dagger}HG^{A}_{\mu\nu}G^{A\mu\nu}$ | ${\cal O}_{dW}$ | $(\overline{Q}\sigma^{\mu\nu}d)\tau^{I}HW^{I}_{\mu\nu}$ | ${\cal O}_{ed}$ | $(\overline{e}\gamma_{\mu}e)(\overline{d}\gamma^{\mu}d)$
${\cal O}_{H\tilde{G}}$ | $H^{\dagger}H\tilde{G}^{A}_{\mu\nu}G^{A\mu\nu}$ | ${\cal O}_{dB}$ | $(\overline{Q}\sigma^{\mu\nu}d)HB_{\mu\nu}$ | ${\cal O}_{ud}^{(1)}$ | $(\overline{u}\gamma_{\mu}u)(\overline{d}\gamma^{\mu}d)$
${\cal O}_{HW}$ | $H^{\dagger}HW^{I}_{\mu\nu}W^{I\mu\nu}$ | $\color[rgb]{1,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,1}\pgfsys@color@cmyk@stroke{0}{1}{0}{0}\pgfsys@color@cmyk@fill{0}{1}{0}{0}\psi^{2}H^{2}D$ | ${\cal O}_{ud}^{(8)}$ | $(\overline{u}\gamma_{\mu}T^{A}u)(\overline{d}\gamma^{\mu}T^{A}d)$
${\cal O}_{H\tilde{W}}$ | $H^{\dagger}H\tilde{W}^{I}_{\mu\nu}W^{I\mu\nu}$ | ${\cal O}_{Hl}^{(1)}$ | $(H^{\dagger}i\overleftrightarrow{D_{\mu}}H)(\overline{L}\gamma^{\mu}L)$ | $\color[rgb]{1,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,1}\pgfsys@color@cmyk@stroke{0}{1}{0}{0}\pgfsys@color@cmyk@fill{0}{1}{0}{0}(\bar{L}L)(\bar{R}R)$
${\cal O}_{HB}$ | $H^{\dagger}HB_{\mu\nu}B^{\mu\nu}$ | ${\cal O}_{Hl}^{(3)}$ | $(H^{\dagger}i\overleftrightarrow{D_{\mu}^{I}}H)(\overline{L}\gamma^{\mu}\tau^{I}L)$ | ${\cal O}_{le}$ | $(\overline{L}\gamma_{\mu}L)(\overline{e}\gamma^{\mu}e)$
${\cal O}_{H\tilde{B}}$ | $H^{\dagger}H\tilde{B}_{\mu\nu}B^{\mu\nu}$ | ${\cal O}_{He}$ | $(H^{\dagger}i\overleftrightarrow{D_{\mu}}H)(\overline{e}\gamma^{\mu}e)$ | ${\cal O}_{lu}$ | $(\overline{L}\gamma_{\mu}L)(\overline{u}\gamma^{\mu}u)$
${\cal O}_{HWB}$ | $H^{\dagger}\tau^{I}HW^{I}_{\mu\nu}B^{\mu\nu}$ | ${\cal O}_{Hq}^{(1)}$ | $(H^{\dagger}i\overleftrightarrow{D_{\mu}}H)(\overline{Q}\gamma^{\mu}Q)$ | ${\cal O}_{ld}$ | $(\overline{L}\gamma_{\mu}L)(\overline{d}\gamma^{\mu}d)$
${\cal O}_{H\tilde{W}B}$ | $H^{\dagger}\tau^{I}H\tilde{W}^{I}_{\mu\nu}B^{\mu\nu}$ | ${\cal O}_{Hq}^{(3)}$ | $(H^{\dagger}i\overleftrightarrow{D_{\mu}^{I}}H)(\overline{Q}\gamma^{\mu}\tau^{I}Q)$ | ${\cal O}_{qe}$ | $(\overline{Q}\gamma_{\mu}Q)(\overline{e}\gamma^{\mu}e)$
|  | ${\cal O}_{Hu}$ | $(H^{\dagger}i\overleftrightarrow{D_{\mu}}H)(\overline{u}\gamma^{\mu}u)$ | ${\cal O}_{qu}^{(1)}$ | $(\overline{Q}\gamma_{\mu}Q)(\overline{u}\gamma^{\mu}u)$
|  | ${\cal O}_{Hd}$ | $(H^{\dagger}i\overleftrightarrow{D_{\mu}}H)(\overline{d}\gamma^{\mu}d)$ | ${\cal O}_{qu}^{(8)}$ | $(\overline{Q}\gamma_{\mu}T^{A}Q)(\overline{u}\gamma^{\mu}T^{A}u)$
|  | ${\cal O}_{Hud}+{\rm h.c.}$ | $(\tilde{H}^{\dagger}iD_{\mu}H)(\overline{u}\gamma^{\mu}d)$ | ${\cal O}_{qd}^{(1)}$ | $(\overline{Q}\gamma_{\mu}Q)(\overline{d}\gamma^{\mu}d)$
|  |  |  | ${\cal O}_{qd}^{(8)}$ | $(\overline{Q}\gamma_{\mu}T^{A}Q)(\overline{d}\gamma^{\mu}T^{A}d)$
|  |  |  | $\color[rgb]{1,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,1}\pgfsys@color@cmyk@stroke{0}{1}{0}{0}\pgfsys@color@cmyk@fill{0}{1}{0}{0}(\bar{L}R)(\bar{R}L)+{\rm h.c.}$
|  |  |  | ${\cal O}_{ledq}$ | $(\overline{L}e)(\overline{d}Q)$
|  |  |  | $\color[rgb]{1,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,1}\pgfsys@color@cmyk@stroke{0}{1}{0}{0}\pgfsys@color@cmyk@fill{0}{1}{0}{0}(\bar{L}R)(\bar{L}R)+{\rm h.c.}$
|  |  |  | ${\cal O}_{quqd}^{(1)}$ | $\epsilon_{ij}(\overline{Q^{i}}u)(\overline{Q^{j}}d)$
|  |  |  | ${\cal O}_{quqd}^{(8)}$ | $\epsilon_{ij}(\overline{Q^{i}}T^{A}u)(\overline{Q^{j}}T^{A}d)$
|  |  |  | ${\cal O}_{lequ}^{(1)}$ | $\epsilon_{ij}(\overline{L^{i}}e)(\overline{Q^{j}}u)$
|  |  |  | ${\cal O}_{lequ}^{(3)}$ |  $\epsilon_{ij}(\overline{L^{i}}\sigma_{\mu\nu}e)(\overline{Q^{j}}\sigma^{\mu\nu}u)$

Table 4: The Warsaw basis of dim-6 operators that conserve baryon and lepton
number in SMEFT [26]. Note that in our convention the capital symbols $Q$ and
$L$ are used to represent the quark and lepton doublets instead of $q$ and
$l$.

At dim 5, there is a unique LNV operator related to the Majorana neutrino mass
[25]

$\displaystyle{\cal
O}_{LH,5}^{pr}=\epsilon_{ij}\epsilon_{mn}(\overline{L^{\texttt{C},i}_{p}}L^{m}_{r})H^{j}H^{n},$
(A.1)

plus its hermitian conjugate. Tab. 4 and Tab. 5 reproduce the bases of dim-6
and dim-7 operators conserving the baryon number in SMEFT [26, 27, 29],
respectively. The convention for fields is as follows: $L,~{}Q$ are the left-
handed lepton and quark doublet fields, $u,~{}d,~{}e$ are the right-handed up-
type quark, down-type quark, and charged lepton singlet fields, and $H$
denotes the Higgs doublet. For the dim-7 operators, they were first
systematically studied in [27], and corrected by [29]. In Tab. 5 we use the
further improved basis in [63] in which lepton flavor symmetry is more
manifest and the operators containing both quark and lepton fields have a
factorized quark-lepton current form.

$\color[rgb]{1,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,1}\pgfsys@color@cmyk@stroke{0}{1}{0}{0}\pgfsys@color@cmyk@fill{0}{1}{0}{0}\psi^{2}H^{4}+{\rm h.c.}$ | $\color[rgb]{1,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,1}\pgfsys@color@cmyk@stroke{0}{1}{0}{0}\pgfsys@color@cmyk@fill{0}{1}{0}{0}\psi^{2}H^{3}D+{\rm h.c.}$
---|---
${\cal O}_{LH}$ | $\epsilon_{ij}\epsilon_{mn}(\overline{L^{\texttt{C},i}}L^{m})H^{j}H^{n}(H^{\dagger}H)$ | ${\cal O}_{LeHD}$ | $\epsilon_{ij}\epsilon_{mn}(\overline{L^{\texttt{C},i}}\gamma_{\mu}e)H^{j}(H^{m}iD^{\mu}H^{n})$
$\color[rgb]{1,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,1}\pgfsys@color@cmyk@stroke{0}{1}{0}{0}\pgfsys@color@cmyk@fill{0}{1}{0}{0}\psi^{2}H^{2}D^{2}+{\rm h.c.}$ | $\color[rgb]{1,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,1}\pgfsys@color@cmyk@stroke{0}{1}{0}{0}\pgfsys@color@cmyk@fill{0}{1}{0}{0}\psi^{2}H^{2}X+{\rm h.c.}$
${\cal O}_{LDH1}$ | $\epsilon_{ij}\epsilon_{mn}(\overline{L^{\texttt{C},i}}\overleftrightarrow{D_{\mu}}L^{j})(H^{m}D^{\mu}H^{n})$ | ${\cal O}_{LHB}$ | $g_{1}\epsilon_{ij}\epsilon_{mn}(\overline{L^{\texttt{C},i}}\sigma_{\mu\nu}L^{m})H^{j}H^{n}B^{\mu\nu}$
${\cal O}_{LDH2}$ | $\epsilon_{im}\epsilon_{jn}(\overline{L^{\texttt{C},i}}L^{j})(D_{\mu}H^{m}D^{\mu}H^{n})$ | ${\cal O}_{LHW}$ | $g_{2}\epsilon_{ij}(\epsilon\tau^{I})_{mn}(\overline{L^{\texttt{C},i}}\sigma_{\mu\nu}L^{m})H^{j}H^{n}W^{I\mu\nu}$
$\color[rgb]{1,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,1}\pgfsys@color@cmyk@stroke{0}{1}{0}{0}\pgfsys@color@cmyk@fill{0}{1}{0}{0}\psi^{4}D+{\rm h.c.}$ | $\color[rgb]{1,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,1}\pgfsys@color@cmyk@stroke{0}{1}{0}{0}\pgfsys@color@cmyk@fill{0}{1}{0}{0}\psi^{4}H+{\rm h.c.}$
${\cal O}_{\overline{d}uLDL}$ | $\epsilon_{ij}(\overline{d}\gamma_{\mu}u)(\overline{L^{\texttt{C},i}}i\overleftrightarrow{D}^{\mu}L^{j})$ | ${\cal O}_{\overline{e}LLLH}$ | $\epsilon_{ij}\epsilon_{mn}(\overline{e}L^{i})(\overline{L^{\texttt{C},j}}L^{m})H^{n}$
|  | ${\cal O}_{\overline{d}QLLH1}$ | $\epsilon_{ij}\epsilon_{mn}(\overline{d}Q^{i})(\overline{L^{\texttt{C},j}}L^{m})H^{n}$
|  | ${\cal O}_{\overline{d}QLLH2}$ | $\epsilon_{ij}\epsilon_{mn}(\overline{d}\sigma_{\mu\nu}Q^{i})(\overline{L^{\texttt{C},j}}\sigma^{\mu\nu}L^{m})H^{n}$
|  | ${\cal O}_{\overline{d}uLeH}$ | $\epsilon_{ij}(\overline{d}\gamma_{\mu}u)(\overline{L^{\texttt{C},i}}\gamma^{\mu}e)H^{j}$
|  | ${\cal O}_{\overline{Q}uLLH}$ | $\epsilon_{ij}(\overline{Q}u)(\overline{L^{\texttt{C}}}L^{i})H^{j}$

Table 5: Basis of dim-7 lepton operators that conserve baryon number in SMEFT
[63]. Here $D_{\mu}H^{n}$ is understood as $(D_{\mu}H)^{n}$, etc.

##  Appendix B Calculation of a power-type supertrace with double insertions
of $\boldsymbol{X}_{N\eta}$ and $\boldsymbol{X}_{\eta N}$

The supertraces to calculate all have a general form ${\rm
STr}\left[f(P_{\mu},\\{U_{k}(x)\\})\right]$, where $P_{\mu}=iD_{\mu}$ with
$D_{\mu}$ being the usual covariant derivative operator and $U_{k}$ is a set
of $P_{\mu}$-independent functionals of classical fields and coupling
constants. With the use of completeness relation and Baker-Campbell-Hausdorff
formula, the CDE method leads to [48]

$\displaystyle{\rm
Tr}\left[f(P_{\mu},\\{U_{k}(x)\\})\right]=\int{\mbox{d}^{d}q\over(2\pi)^{d}}\left\langle
q\left|{\rm
tr}\left[f(P_{\mu},\\{U_{k}\\})\right]\right|q\right\rangle=\int\mbox{d}^{d}x\int{\mbox{d}^{d}q\over(2\pi)^{d}}{\rm
tr}\left[f(\bar{P}_{\mu},\\{\bar{U}_{k}\\})\right],$ (B.1)

where the CDE transformed quantities are represented by a bar and take the
following form,

$\displaystyle\bar{P}_{\mu}$ $\displaystyle=$
$\displaystyle\bar{G}_{\mu\nu}\tilde{\partial}^{\nu}-q_{\mu},$
$\displaystyle\bar{U}_{k}$ $\displaystyle=$
$\displaystyle\sum_{n=0}^{\infty}{1\over n!}(P_{\mu_{1}}\dotsi
P_{\mu_{n}}U_{k})\tilde{\partial}^{\mu_{1}}\dotsi\tilde{\partial}^{\mu_{n}};$
$\displaystyle i\bar{G}_{\mu\nu}$ $\displaystyle=$
$\displaystyle\sum_{n=0}{n+1\over(n+2)!}(P_{\mu_{1}}\dotsi
P_{\mu_{n}}F_{\mu\nu})\tilde{\partial}^{\mu_{1}}\dotsi\tilde{\partial}^{\mu_{n}},$
(B.2)

where $\tilde{\partial}^{\mu}\equiv\partial/\partial q_{\mu}$ stands for the
partial derivative with respect to the loop momentum.

With the above formula, the supertrace with double insertions of
$\boldsymbol{X}_{N\eta}$ and $\boldsymbol{X}_{\eta N}$ in Eq. (4.4) becomes,

$\displaystyle-{i\over 4}{\rm
STr}\left(\boldsymbol{K}_{N}^{-1}\boldsymbol{X}_{N\eta}^{[3/2]}\boldsymbol{K}_{\eta}^{-1}\boldsymbol{X}_{\eta
N}^{[3/2]}\right)^{2}\Big{|}_{\tt hard}=\int\mbox{d}^{4}x{i\over
4}\mu^{2\epsilon}\int{\mbox{d}^{d}q\over(2\pi)^{d}}{\rm
tr}\left(\bar{\boldsymbol{K}}_{N}^{-1}\bar{\boldsymbol{X}}_{N\eta}^{[3/2]}\bar{\boldsymbol{K}}_{\eta}^{-1}\bar{\boldsymbol{X}}_{\eta
N}^{[3/2]}\right)^{2}\Big{|}_{\tt hard},$ (B.3)

where a minus sign is included due to the fermionic nature of the first
propagator. Since the fermion $N$ is a gauge singlet, the CDE transformed
inverse propagator $\bar{\boldsymbol{K}}_{N}=-\not{q}+m_{N}$ with vanishing of
$\bar{G}_{\mu\nu}^{N}$ due to $F_{\mu\nu}^{N}=0$; while for the doublet
$\eta$,
$\bar{\boldsymbol{K}}_{\eta}=(\bar{G}_{\mu\nu}^{\eta}\tilde{\partial}^{\nu}-q_{\mu})^{2}-m_{\eta}^{2}$
with $F_{\mu\nu}^{\eta}=g_{2}W_{\mu\nu}^{I}\sigma^{I}/2+g_{1}B_{\mu\nu}/2$
entering into the definition of $\bar{G}_{\mu\nu}^{\eta}$ in Eq. (B.2).
Denoting the obtained effective Lagrangian by ${\cal L}_{4L}$ (with
integration over spacetime $x$ implied on both sides), we have

$\displaystyle{\cal L}_{4L}\overset{\tt CDE}{=}{i\over
4}\mu^{2\epsilon}\int{\mbox{d}^{d}q\over(2\pi)^{d}}{\rm tr}\left[{1\over
q^{2}-m_{N}^{2}}(-\not{q}+m_{N})\bar{\boldsymbol{X}}_{N\eta}^{[3/2]}{1\over(\bar{G}_{\mu\nu}^{\eta}\tilde{\partial}^{\nu}-q_{\mu})^{2}-m_{\eta}^{2}}\bar{\boldsymbol{X}}_{\eta
N}^{[3/2]}\right]^{2}\Big{|}_{\tt hard}.$ (B.4)

Since each $\boldsymbol{X}_{\eta N}^{[3/2]}$ contributes a lepton field
$L(L^{\texttt{C}})$ from Eq. (4.3), the supertrace will generate operators
containing four $L(L^{\texttt{C}})$s without any Higgs field. Thus it cannot
contribute to the LNV dim-7 operators but only LNC dim-6 ones involving four
leptons. Therefore, up to dim-7 matching, we can remove the bars associated
with CDE transformation and neglect the $\bar{G}_{\mu\nu}^{\eta}$ part in the
denominator, which leads to

$\displaystyle{\cal L}_{4L}^{\tt dim\leq 7}$ $\displaystyle=$
$\displaystyle{i\over 4}\mu^{2\epsilon}\int{d^{d}q\over(2\pi)^{d}}{\rm
tr}\left({-\not{q}+m_{N}\over
q^{2}-m_{N}^{2}}\boldsymbol{X}_{N\eta}^{[3/2]}{1\over
q^{2}-m_{\eta}^{2}}\boldsymbol{X}_{\eta N}^{[3/2]}\right)^{2}\Big{|}_{\tt
hard}$ (B.5) $\displaystyle=$ $\displaystyle{i\over
4}\mu^{2\epsilon}\int{\mbox{d}^{d}q\over(2\pi)^{d}}{{\rm
tr}\left[(-\not{q}+m_{N_{v}})\boldsymbol{X}_{N_{v}\eta}\boldsymbol{X}_{\eta
N_{w}}(-\not{q}+m_{N_{w}})\boldsymbol{X}_{N_{w}\eta}\boldsymbol{X}_{\eta
N_{v}}\right]\over(q^{2}-m_{\eta}^{2})^{2}(q^{2}-m_{N_{v}}^{2})(q^{2}-m_{N_{w}}^{2})},$

where in the second step we have dropped the subscript “hard” since the
integrand already lives in the hard momentum region, and some flavor labels
$(v,w)$ associated with the fermion $N$ are added. By taking into account the
$\boldsymbol{X}_{N\eta}$ and $\boldsymbol{X}_{\eta N}$ in Eq. (4.3), the trace
part in the above can be calculated step by step as follows,

$\displaystyle{\rm trace}$ $\displaystyle\equiv$ $\displaystyle{\rm
tr}\left[(-\not{q}+m_{N_{v}})\boldsymbol{X}_{N_{v}\eta}\boldsymbol{X}_{\eta
N_{w}}(-\not{q}+m_{N_{w}})\boldsymbol{X}_{N_{w}\eta}\boldsymbol{X}_{\eta
N_{v}}\right]$ (B.6) $\displaystyle\overset{(\ref{eq:newX})}{=}$
$\displaystyle{\rm
tr}\left\\{(-\not{q}+m_{N_{v}})[(Y_{\eta}^{\dagger})_{vp}P_{L}L_{p}^{i}\bar{L}_{r}^{i}P_{R}(Y_{\eta})_{rw}+(Y_{\eta}^{\texttt{T}})_{vp}P_{R}L_{p}^{i\texttt{C}}\overline{L_{r}^{i\texttt{C}}}P_{L}(Y_{\eta}^{*})_{rw}]\right.$
$\displaystyle\times\left.(-\not{q}+m_{N_{w}})[(Y_{\eta}^{\dagger})_{ws}P_{L}L_{s}^{j}\bar{L}_{t}^{j}P_{R}(Y_{\eta})_{tv}+(Y_{\eta}^{\texttt{T}})_{ws}P_{R}L_{s}^{j\texttt{C}}\overline{L_{t}^{j\texttt{C}}}P_{L}(Y_{\eta}^{*})_{tv}]\right\\},$

where $(i,j)$ stand for the $SU(2)_{L}$ doublet indices and $(p,r,s,t,v,w)$
for the fermion flavor indices. Using the chiral projection property
$P_{R}P_{L}=0$ and dropping the linear $q$ terms which vanish upon loop
integration, it splits into two terms,

$\displaystyle{\rm
trace}=m_{N_{v}}m_{N_{w}}T_{1}+q_{\alpha}q_{\beta}T_{2}^{\alpha\beta},$ (B.7)

where

$\displaystyle T_{1}$ $\displaystyle=$ $\displaystyle{\rm
tr}\Big{[}(Y_{\eta}^{\dagger})_{vp}L_{p}^{i}\bar{L}_{r}^{i}(Y_{\eta})_{rw}(Y_{\eta}^{\texttt{T}})_{ws}L_{s}^{j\texttt{C}}\overline{L_{t}^{j\texttt{C}}}(Y_{\eta}^{*})_{tv}$
(B.8a)
$\displaystyle~{}~{}+(Y_{\eta}^{\texttt{T}})_{vp}L_{p}^{i\texttt{C}}\overline{L_{r}^{i\texttt{C}}}(Y_{\eta}^{*})_{rw}(Y_{\eta}^{\dagger})_{ws}L_{s}^{j}\bar{L}_{t}^{j}(Y_{\eta})_{tv}\Big{]}$
$\displaystyle=$
$\displaystyle-2(Y_{\eta}^{\dagger})_{vp}(Y_{\eta})_{rw}(Y_{\eta}^{\texttt{T}})_{ws}(Y_{\eta}^{*})_{tv}\left[(\bar{L}_{r}^{i}L_{s}^{j\texttt{C}})(\overline{L_{t}^{j\texttt{C}}}L_{p}^{i})\right],$
$\displaystyle T_{2}^{\alpha\beta}$ $\displaystyle=$ $\displaystyle{\rm
tr}\Big{[}\gamma^{\beta}(Y_{\eta}^{\dagger})_{vp}L_{p}^{i}\bar{L}_{r}^{i}(Y_{\eta})_{rw}\gamma^{\alpha}(Y_{\eta}^{\dagger})_{ws}L_{s}^{j}\bar{L}_{t}^{j}(Y_{\eta})_{tv}$
(B.8b)
$\displaystyle~{}~{}+\gamma^{\beta}(Y_{\eta}^{\texttt{T}})_{vp}L_{p}^{i\texttt{C}}\overline{L_{r}^{i\texttt{C}}}(Y_{\eta}^{*})_{rw}\gamma^{\alpha}(Y_{\eta}^{\texttt{T}})_{ws}L_{s}^{j\texttt{C}}\overline{L_{t}^{j\texttt{C}}}(Y_{\eta}^{*})_{tv}\Big{]}$
$\displaystyle=$
$\displaystyle-2(Y_{\eta}^{\dagger})_{vp}(Y_{\eta})_{rw}(Y_{\eta}^{\dagger})_{ws}(Y_{\eta})_{tv}\left[(\bar{L}_{r}^{i}\gamma^{\alpha}L_{s}^{j})(\bar{L}_{t}^{j}\gamma^{\beta}L_{p}^{i})\right].$

We have removed chiral projectors since our $L(L^{\texttt{C}})$ is left-
handed, finished the trace in the spinor space, and reshuffled dummy indices
to make the expressions more compact. Note that a minus sign has been included
when interchanging a pair of fermion fields. Finally, we apply the FI in
$T_{1}$,
$2(\overline{\psi_{1L}}\psi^{\texttt{C}}_{3L})(\overline{\psi^{\texttt{C}}_{4L}}\psi_{2L})=(\overline{\psi_{1L}}\gamma^{\mu}\psi_{2L})(\overline{\psi_{3L}}\gamma_{\mu}\psi_{4L})$,
and make symmetric loop integration in
$q_{\alpha}q_{\beta}T_{2}^{\alpha\beta}\to(q^{2}/d)g_{\alpha\beta}T_{2}^{\alpha\beta}$,
to obtain

$\displaystyle T_{1}$ $\displaystyle=$
$\displaystyle-(Y_{\eta})_{pw}(Y_{\eta}^{\texttt{T}})_{ws}(Y_{\eta}^{*})_{tv}(Y_{\eta}^{\dagger})_{vr}(\bar{L}_{p}\gamma_{\mu}L_{r})(\bar{L}_{s}\gamma^{\mu}L_{t}),$
(B.9a) $\displaystyle q_{\alpha}q_{\beta}T_{2}^{\alpha\beta}$
$\displaystyle\Rightarrow$ $\displaystyle-{2q^{2}\over
d}(Y_{\eta})_{pw}(Y_{\eta}^{\dagger})_{wt}(Y_{\eta})_{sv}(Y_{\eta}^{\dagger})_{vr}(\bar{L}_{p}\gamma_{\mu}L_{r})(\bar{L}_{s}\gamma^{\mu}L_{t}).$
(B.9b)

The loop integrals in Eq. (B.5) can now be worked out to yield the loop
functions $F_{4,1}(x)$ and $F_{4,2}(x)$. The end result is that the WC for the
operator $(\bar{L}_{p}\gamma_{\mu}L_{r})(\bar{L}_{s}\gamma^{\mu}L_{t})$ is
$\tilde{C}_{ll}^{prst}/(16\pi^{2}m_{\eta}^{2})$, where

$\displaystyle\tilde{C}_{ll}^{prst}$ $\displaystyle=$ $\displaystyle
F_{4,1}(x_{v},x_{w})(Y_{\eta})_{pw}(Y_{\eta}^{\texttt{T}})_{ws}(Y_{\eta}^{*})_{rv}(Y_{\eta}^{\dagger})_{vt}$
(B.10) $\displaystyle-
F_{4,2}(x_{v},x_{w})(Y_{\eta})_{pw}(Y_{\eta}^{\dagger})_{wt}(Y_{\eta})_{sv}(Y_{\eta}^{\dagger})_{vr},$

which is given in the last row of the dim-6 sector in Tab. 1.

##  Appendix C Reduction into the standard basis

Using the integration by parts relations in Eq. (4.5) and the EoMs in Eq.
(4.11), we reduce the dim-6 non-standard basis operators (blue ones) in Tab. 1
to the Warsaw basis as follows,

$\displaystyle\partial_{\mu}B^{\mu\nu}\partial^{\rho}B_{\rho\nu}$
$\displaystyle\overset{\texttt{EoM}}{\Rightarrow}$ $\displaystyle
g_{1}^{2}\left[{1\over 6}\bar{Q}\gamma_{\mu}Q+{2\over
3}\bar{u}\gamma_{\mu}u-{1\over 3}\bar{d}\gamma_{\mu}d-{1\over
2}\bar{L}\gamma_{\mu}L-\bar{e}\gamma_{\mu}e+{1\over
2}H^{\dagger}i\overleftrightarrow{D_{\mu}}H\right]^{2}$ (C.1a)
$\displaystyle\Rightarrow$ $\displaystyle{g_{1}^{2}\over 4}{\cal
O}_{H\Box}+g_{1}^{2}{\cal O}_{HD}-{g_{1}^{2}\over 2}\delta_{pr}{\cal
O}_{Hl}^{(1),pr}-g_{1}^{2}\delta_{pr}{\cal O}_{He}^{pr}$
$\displaystyle+{g_{1}^{2}\over 6}\delta_{pr}{\cal O}_{Hq}^{(1),pr}+{2\over
3}g_{1}^{2}\delta_{pr}{\cal O}_{Hu}^{pr}-{g_{1}^{2}\over 3}\delta_{pr}{\cal
O}_{Hd}^{pr}$ $\displaystyle+{g_{1}^{2}\over 4}\delta_{pr}\delta_{st}{\cal
O}_{ll}^{prst}+{g_{1}^{2}\over 36}\delta_{pr}\delta_{st}{\cal
O}_{qq}^{(1),prst}-{g_{1}^{2}\over 6}\delta_{pr}\delta_{st}{\cal
O}_{lq}^{(1),prst}$ $\displaystyle+g_{1}^{2}\delta_{pr}\delta_{st}{\cal
O}_{ee}+{4\over 9}g_{1}^{2}\delta_{pr}\delta_{st}{\cal
O}_{uu}^{prst}+{g_{1}^{2}\over 9}\delta_{pr}\delta_{st}{\cal O}_{dd}^{prst}$
$\displaystyle-{4\over 3}g_{1}^{2}\delta_{pr}\delta_{st}{\cal
O}_{eu}^{prst}+{2\over 3}g_{1}^{2}\delta_{pr}\delta_{st}{\cal
O}_{ed}^{prst}-{4\over 9}g_{1}^{2}\delta_{pr}\delta_{st}{\cal
O}_{ud}^{(1),prst}$ $\displaystyle+g_{1}^{2}\delta_{pr}\delta_{st}{\cal
O}_{le}^{prst}-{2\over 3}g_{1}^{2}\delta_{pr}\delta_{st}{\cal
O}_{lu}^{prst}+{g_{1}^{2}\over 3}\delta_{pr}\delta_{st}{\cal O}_{ld}^{prst}$
$\displaystyle-{g_{1}^{2}\over 3}\delta_{pr}\delta_{st}{\cal
O}_{qe}^{prst}+{2\over 9}g_{1}^{2}\delta_{pr}\delta_{st}{\cal
O}_{qu}^{(1),prst}-{g_{1}^{2}\over 9}\delta_{pr}\delta_{st}{\cal
O}_{qd}^{(1),prst},$ $\displaystyle D_{\mu}W^{I\mu\nu}D^{\rho}W^{I}_{\rho\nu}$
$\displaystyle\overset{\texttt{EoM}}{\Rightarrow}$
$\displaystyle{g_{2}^{2}\over
4}\left[\overline{Q}\sigma^{I}\gamma_{\mu}Q+\overline{L}\sigma^{I}\gamma_{\mu}L+H^{\dagger}i\overleftrightarrow{D_{\mu}}^{I}H\right]^{2}$
(C.1b) $\displaystyle\Rightarrow$
$\displaystyle\underline{\underline{-g_{2}^{2}\mu_{H}^{2}(H^{\dagger}H)^{2}}}+2g_{2}^{2}\lambda_{H}{\cal
O}_{H}+{3\over 4}g_{2}^{2}{\cal O}_{H\Box}$ $\displaystyle+{g_{2}^{2}\over
2}\delta_{pr}{\cal O}_{Hl}^{(3),pr}+{g_{2}^{2}\over 2}\delta_{pr}{\cal
O}_{Hq}^{(3),pr}$ $\displaystyle+{g_{2}^{2}\over 2}\big{[}(Y_{e})_{pr}{\cal
O}_{eH}^{pr}+(Y_{u})_{pr}{\cal O}_{uH}^{pr}+(Y_{d})_{pr}{\cal
O}_{dH}^{pr}+{\rm h.c.}\big{]}$ $\displaystyle+{g_{2}^{2}\over
4}(2\delta_{pt}\delta_{rs}-\delta_{pr}\delta_{st}){\cal O}_{ll}^{prst}$
$\displaystyle+{g_{2}^{2}\over 4}\delta_{pr}\delta_{st}{\cal
O}_{qq}^{(3),prst}+{g_{2}^{2}\over 2}\delta_{pr}\delta_{st}{\cal
O}_{lq}^{(3),prst},$ $\displaystyle(H^{\dagger}H)(H^{\dagger}D^{2}H)+{\rm
h.c.}$ $\displaystyle\overset{\texttt{EoM}}{\Rightarrow}$
$\displaystyle\underline{\underline{2\mu_{H}^{2}(H^{\dagger}H)^{2}}}-4\lambda_{H}{\cal
O}_{H}$ (C.1c) $\displaystyle-[(Y_{e})_{pr}{\cal
O}_{eH}^{pr}+(Y_{u})_{pr}{\cal O}_{uH}^{pr}+(Y_{d})_{pr}{\cal
O}_{dH}^{pr}+{\rm h.c.}],$ $\displaystyle
C_{pr}(H^{\dagger}H)(\overline{L_{p}}i\overleftrightarrow{\not{D}}L_{r})$
$\displaystyle\overset{\texttt{EoM}}{\Rightarrow}$
$\displaystyle(CY_{e})_{pr}{\cal O}_{eH}^{pr}+{\rm h.c.},$ (C.1d)
$\displaystyle
C_{pr}(H^{\dagger}\sigma^{I}H)(\overline{L_{p}}i\overleftrightarrow{\not{D}}^{I}L_{r})$
$\displaystyle\overset{\texttt{EoM}}{\Rightarrow}$
$\displaystyle(CY_{e})_{pr}{\cal O}_{eH}^{pr}+{\rm h.c.},$ (C.1e)
$\displaystyle-
C_{pr}\overline{L_{p}}i\overleftarrow{\not{D}}i\not{D}i\not{D}L_{r}$
$\displaystyle\overset{\texttt{EoM}}{\Rightarrow}$
$\displaystyle(Y_{e}^{\dagger}CY_{e})_{pr}(H^{\dagger}\overline{e_{p}})i\not{D}(e_{r}H)$
(C.1f) $\displaystyle=$
$\displaystyle(Y_{e}^{\dagger}CY_{e})_{pr}\left[(\overline{e_{p}}\gamma_{\mu}e_{r})(H^{\dagger}iD^{\mu}H)+(H^{\dagger}H)(\overline{e_{p}}i\not{D}e_{r})\right]$
$\displaystyle\overset{\texttt{IBP}}{\Rightarrow}$ $\displaystyle{1\over
2}(Y_{e}^{\dagger}CY_{e})_{pr}\left[(\overline{e_{p}}\gamma_{\mu}e_{r})(H^{\dagger}i\overleftrightarrow{D^{\mu}}H)+(H^{\dagger}H)(\overline{e_{p}}i\overleftrightarrow{\not{D}}e_{r})\right]$
$\displaystyle\overset{\texttt{EoM}}{\Rightarrow}$ $\displaystyle{1\over
2}(Y_{e}^{\dagger}CY_{e})_{pr}{\cal O}_{He}^{pr}+{1\over
2}\left[(Y_{e}Y_{e}^{\dagger}CY_{e})_{pr}{\cal O}_{eH}^{pr}+{\rm
h.c.}\right],$ $\displaystyle
C_{pr}B^{\mu\nu}\overline{L_{p}}\sigma_{\mu\nu}i\not{D}L_{r}$
$\displaystyle\overset{\texttt{EoM}}{\Rightarrow}$
$\displaystyle(CY_{e})_{pr}{\cal O}_{eB}^{pr},$ (C.1g) $\displaystyle
C_{pr}D_{\nu}B^{\mu\nu}\overline{L_{p}}\gamma_{\mu}L_{r}$
$\displaystyle\overset{\texttt{EoM}}{\Rightarrow}$ $\displaystyle
C_{pr}g_{1}^{2}(\overline{L_{p}}\gamma_{\mu}L_{r})\bigg{[}{1\over
6}\bar{Q}\gamma_{\mu}Q+{2\over 3}\bar{u}\gamma_{\mu}u-{1\over
3}\bar{d}\gamma_{\mu}d-{1\over 2}\bar{L}\gamma_{\mu}L-\bar{e}\gamma_{\mu}e$
(C.1h) $\displaystyle+{1\over
2}H^{\dagger}i\overleftrightarrow{D_{\mu}}H\bigg{]}$
$\displaystyle\Rightarrow$ $\displaystyle{g_{1}\over 2}C_{pr}{\cal
O}_{Hl}^{(1),pr}-{g_{1}\over 4}(C_{pr}\delta_{st}+C_{st}\delta_{pr}){\cal
O}_{ll}^{prst}+{g_{1}\over 6}C_{pr}\delta_{st}{\cal O}_{lq}^{(1),prst}$
$\displaystyle-$ $\displaystyle g_{1}C_{pr}\delta_{st}{\cal
O}_{le}^{prst}+{2g_{1}\over 3}C_{pr}\delta_{st}{\cal
O}_{lu}^{prst}-{g_{1}\over 3}C_{pr}\delta_{st}{\cal O}_{ld}^{prst},$
$\displaystyle
C_{pr}W^{I\mu\nu}\overline{L_{p}}\sigma^{I}\sigma_{\mu\nu}i\not{D}L_{r}$
$\displaystyle\overset{\texttt{EoM}}{\Rightarrow}$
$\displaystyle(CY_{e})_{pr}{\cal O}_{eW}^{pr},$ (C.1i) $\displaystyle
C_{pr}D_{\nu}W^{I\mu\nu}\overline{L_{p}}\sigma^{I}\gamma_{\mu}L_{r}$
$\displaystyle\overset{\texttt{EoM}}{=}$ $\displaystyle{g_{2}\over
2}C_{pr}(\overline{L_{p}}\sigma^{I}\gamma^{\mu}L_{r})\left[\overline{Q}\sigma^{I}\gamma_{\mu}Q+\overline{L}\sigma^{I}\gamma_{\mu}L+H^{\dagger}i\overleftrightarrow{D_{\mu}}^{I}H\right]$
(C.1j) $\displaystyle\Rightarrow$ $\displaystyle{g_{2}\over 2}C_{pr}{\cal
O}_{Hl}^{(3),pr}+{g_{2}\over
4}\big{[}2C_{pt}\delta_{rs}-C_{pr}\delta_{st}+(pr)\leftrightarrow(st)\big{]}{\cal
O}_{ll}^{prst}$ $\displaystyle+{g_{2}\over 2}C_{pr}\delta_{st}{\cal
O}_{lq}^{(3),prst},\quad$

where we have used a generic symbol $C_{pr}$ to stand for the WCs of the
operators involving fermion fields. The WC of the operator ${\cal
O}_{ll}^{prst}$ is made symmetric using the flavor symmetry ${\cal
O}_{ll}^{prst}={\cal O}_{ll}^{stpr}$. The doubly underlined terms contribute
to the dim-4 Higgs quartic interaction.

To reduce the two dim-7 EoM operators in Tab. 1, we need the identities,

$\displaystyle D^{2}=\not{D}\not{D}+{1\over
2}\sigma^{\mu\nu}F_{\mu\nu},\,F_{\mu\nu}=i[D_{\mu},D_{\nu}]=g_{1}YB_{\mu\nu}+g_{2}T^{I}W^{I}_{\mu\nu},$
(C.2)

where $Y$ is the hypercharge operator and $T^{I}$ the $SU(2)_{L}$ generators.
For the lepton doublet field $L$ being acted on by $D^{2}$, $YL=-{1\over 2}L$
and $T^{I}L={1\over 2}\sigma^{I}L$, we have

$\displaystyle\epsilon_{ik}\epsilon_{jl}(\overline{L_{p}^{i,\texttt{C}}}D^{2}L_{r}^{j})H^{k}H^{l}$
$\displaystyle=$
$\displaystyle\epsilon_{ik}\left[\epsilon_{jl}(\overline{L_{p}^{i,\texttt{C}}}\not{D}\not{D}L_{r}^{j})-{g_{1}\over
4}\epsilon_{jl}(\overline{L_{p}^{i,\texttt{C}}}\sigma^{\mu\nu}L_{r}^{j})B_{\mu\nu}-{g_{2}\over
4}(\epsilon\sigma^{I})_{jl}(\overline{L_{p}^{i,\texttt{C}}}\sigma^{\mu\nu}L_{r}^{j})W^{I}_{\mu\nu}\right]H^{k}H^{l}$
$\displaystyle\overset{\texttt{EoM}}{\Rightarrow}$
$\displaystyle(Y_{e})_{rs}{\cal O}^{ps}_{LeHD}-{1\over 4}{\cal
O}_{LHB}^{pr}-{1\over 4}{\cal O}_{LHW}^{pr},$
$\displaystyle\epsilon_{ik}\epsilon_{jl}(\overline{L_{p}^{i,\texttt{C}}}L_{r}^{j})H^{k}D^{2}H^{l}$
$\displaystyle\overset{\texttt{EoM}}{\Rightarrow}$
$\displaystyle\epsilon_{ik}\epsilon_{jl}(\overline{L_{p}^{i,\texttt{C}}}L_{r}^{j})H^{k}\left[\mu_{H}^{2}H^{l}-2\lambda_{H}(H^{\dagger}H)H^{l}+\epsilon_{lm}\overline{Q^{m}}Y_{u}u-\overline{d}Y^{\dagger}_{d}Q^{l}-\bar{e}Y^{\dagger}_{e}L^{l}\right]$
$\displaystyle=$ $\displaystyle\mu_{H}^{2}{\cal
O}_{LH,5}^{pr}-2\lambda_{H}{\cal O}_{LH}^{pr}-(Y_{u})_{vw}{\cal
O}_{\bar{Q}uLLH}^{vwrp}+(Y^{\dagger}_{d})_{vw}{\cal
O}_{\bar{d}QLLH1}^{vwrp}+(Y^{\dagger}_{e})_{vw}{\cal
O}_{\bar{e}LLLH}^{vwrp},\quad\quad$ (C.3b)

which lead to

$\displaystyle
S_{pr}\epsilon_{im}\epsilon_{jn}(\overline{L_{p}^{i,\texttt{C}}}D^{2}L_{r}^{j})H^{m}H^{n}$
$\displaystyle\overset{\texttt{EoM}}{\Rightarrow}$
$\displaystyle(SY_{e})_{pr}{\cal O}^{pr}_{LeHD}-{1\over 4}S_{pr}{\cal
O}_{LHW}^{pr},$ $\displaystyle
S_{pr}\epsilon_{im}\epsilon_{jn}(\overline{L_{p}^{i,\texttt{C}}}L_{r}^{j})H^{m}D^{2}H^{n}$
$\displaystyle\overset{\texttt{EoM}}{\Rightarrow}$
$\displaystyle\underline{\underline{\mu_{H}^{2}S_{pr}{\cal
O}_{LH,5}^{pr}}}-2\lambda_{H}S_{pr}{\cal O}_{LH}^{pr}$ $\displaystyle+$
$\displaystyle(Y^{\dagger}_{e})_{pr}S_{st}{\cal
O}_{\bar{e}LLLH}^{prst}+(Y^{\dagger}_{d})_{pr}S_{st}{\cal
O}_{\bar{d}QLLH1}^{prst}-(Y_{u})_{pr}S_{st}{\cal O}_{\bar{Q}uLLH}^{prst}.$
(C.5)

Here $S_{pr}$ stands for the corresponding WC which is flavor symmetric,
$S_{pr}=S_{rp}$, and the doubly underlined term contributes to the dim-5
Weinberg operator.

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