# On-shell Renormalization with Vector-like Leptons, One-loop Muon–Higgs
Coupling and Muon $g-2$

Kilian Möhling<EMAIL_ADDRESS>Institut für Kern- und
Teilchenphysik, TU Dresden, Zellescher Weg 19, 01069 Dresden, Germany  Dominik
Stöckinger<EMAIL_ADDRESS>Institut für Kern- und
Teilchenphysik, TU Dresden, Zellescher Weg 19, 01069 Dresden, Germany  Hyejung
Stöckinger-Kim<EMAIL_ADDRESS>Institut für Kern- und
Teilchenphysik, TU Dresden, Zellescher Weg 19, 01069 Dresden, Germany

###### Abstract

Models with vector-like leptons can strongly modify the lepton mass generation
mechanism and lead to correlated effects in lepton–Higgs couplings and lepton
dipole moments. Here we begin an analysis of higher-order corrections in such
models by setting up a renormalization scheme with full on-shell conditions on
the lepton self energies, masses and fields. A minimal set of fundamental
parameters is renormalized in the $\overline{\text{MS}}$ scheme. We provide a
detailed discussion of lepton mixing and redundancies at higher orders, show
how the relevant counterterms can be obtained from the renormalization
conditions, and determine the $\beta$-functions corresponding to the scheme.
As a first application we calculate the one-loop effective muon–Higgs coupling
and analyse its correlation with the muon anomalous magnetic moment $\Delta
a_{\mu}^{\text{VLL}}$. In the interesting case of large masses and opposite-
sign coupling, the lowest-order correlation implies a fixed value of $\Delta
a_{\mu}^{\text{VLL}}$ around $22.5\times 10^{-10}$, while the higher-order
corrections significantly reduce this value to the interval $(10\ldots
18)\times 10^{-10}$.

###### Contents

  1. 1 Introduction
  2. 2 Model Definition
  3. 3 Renormalization
    1. 3.1 Gauge and Higgs Sector
    2. 3.2 Lepton Sector
    3. 3.3 Relation to $\overline{\text{MS}}$ and Renormalization Group Running
  4. 4 One-Loop Effective Higgs Coupling and Correlation with $a_{\mu}$
    1. 4.1 Definition and Vertex Renormalization
    2. 4.2 Cancellation of the Renormalization Scale Dependence
    3. 4.3 Qualitative Discussion and Muon $g-2$
    4. 4.4 Numerical Results
  5. 5 Conclusion
  6. A Feynman Rules
  7. B Renormalization Group Equations

##  1 Introduction

In recent years vector-like lepton (VLL) models have gained increasing
interest as a potential answer to some of the long standing open questions in
particle physics. We focus on a model where the Standard Model (SM) is
extended by additional vector-like leptons which can have fundamental gauge-
invariant Dirac masses and mix with the SM leptons. Such VLLs have been
considered in the context of the muon $g-2$ (also in combination with extended
scalar sectors) [1, 2, 3, 4, 5, 6, 7, 8, 9, 10], charged lepton flavor
violation (cLFV) [11, 12, 3], and electroweak precision observables [1, 2,
13].

The key feature of such VLL models lies in their change of the lepton mass
generation mechanism: The masses of the SM-like leptons result in part (or
even entirely) from mixing, rather than through their fundamental interactions
with the Higgs only. In general, SM fermion mass terms require a source of
chirality flips and a breaking of electroweak gauge invariance. These two
requirements are shared by other observables such as effective Higgs-lepton
couplings $\lambda_{ll^{\prime}}$ and lepton anomalous magnetic moments
$a_{l}=(g-2)_{l}/2$. The implications of these relations for $a_{\mu}$ are
discussed in Refs. [14, 15] as well as in Refs. [2, 16] with focus on the
correlation between $a_{\mu}$, $\lambda_{\mu\mu}$ and further observables in a
wide range of models. Many extensively studied models such as two-Higgs-
doublet (2HDM), supersymmetric (SUSY) and leptoquark (LQ) models introduce
modifications in the scalar sector, which lead to new chirality flips at the
loop level. In contrast, VLL models provide new sources of chirality flips
already at tree-level via the Dirac masses and Yukawa couplings of the VLL
fields and imply distinct correlations between the observables involving
chirality flips [2, 10, 17].

These correlations can be understood as a result of effective dimension-6
operators of the form $\bar{l}lHH^{\dagger}H$ that are generated at tree-level
in VLL models. Such operators can lead to $\mathcal{O}(1)$ tree-level
corrections to $\lambda_{\mu\mu}$ that turn out to be strongly correlated with
contributions to $a_{\mu}$ (for a general study of these operators see Ref.
[18]). On the other hand, experimental constrains restrict
$|\lambda_{\mu\mu}|$ to be close to the SM value. This allows either
$\lambda_{\mu\mu}\approx\lambda_{\mu\mu}^{\text{SM}}$ in which case $\Delta
a_{\mu}^{\text{VLL}}\approx 0$ or, intriguingly,
$\lambda_{\mu\mu}\approx-\lambda_{\mu\mu}^{\text{SM}}$ in which case the
correlation implies some fixed value $\Delta a_{\mu}^{\text{VLL}}\neq 0$. For
example, in the case of one generation of heavy VLLs coupling to the muon
discussed in Refs. [1, 2], the tree-level correlation predicts

$\displaystyle
a_{\mu}^{\text{VLL}}|_{(\lambda_{\mu\mu}=-\lambda_{\mu\mu}^{\text{SM}})}$
$\displaystyle\simeq 22.5\times 10^{-10}\,.$ (1)

This is intriguingly close to the deviation between the Fermilab Run 1,2,3
measurements [19, 20] and the 2020 white paper SM theory value based on [21,
22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40,
41], $a_{\mu}^{\text{exp}}-a_{\mu}^{\text{SM}}=(24.9\pm 4.8)\times 10^{-10}$.
Clearly, scrutiny of SM predictions (see Refs. [21, 42] for an overview of
results and prospects) and Fermilab Run 4,5,6 results might change this value.
Hence it is an open question whether the deviation in Eq. (1) remains a viable
value.

The aim of the present paper is to provide a starting point for higher-order
studies of VLL models. Here, we focus on the SM extended by one generation of
VLLs that mix with all of the SM leptons. We develop an on-shell
renormalization scheme and discuss the treatment of the lepton mixing at
higher orders. As a first application of the scheme, we determine the
modification of Eq. (1) from including one-loop corrections to the effective
muon-Higgs coupling $\lambda_{\mu\mu}$.

On-shell renormalization has already been studied in a variety of scenarios
beyond the Standard Model (BSM) and has often turned out to be highly
nontrivial. This is in contrast to the electroweak SM where there is a one-to-
one correspondence between fundamental parameters and masses or similarly
simple on-shell observables. E.g. the on-shell renormalization of the Minimal
Supersymmetric Standard Model (MSSM) has been studied in Refs. [43, 44, 45]
with focus on the SUSY sectors; here, SUSY relations between masses imply that
only a subset of masses can be on-shell renormalized. In addition, the
renormalization of parameters such as $\tan\beta$ or mixing angles is required
and troublesome [46, 47]. The on-shell renormalization of the two-Higgs
doublet model (2HDM) has been studied in Refs. [48, 49, 50, 51, 52, 53, 54]
with increasing focus on mixing angles and similar parameters for which an on-
shell scheme is not suitable. For these parameters $\overline{\text{MS}}$
renormalization conditions can be advantageous, but in such mixed schemes the
renormalization of the vacuum expectation values/of tadpoles becomes critical
[46, 52, 50, 53, 54, 55, 56, 57, 58].

In the VLL model considered in this paper, the $5\times 5$ lepton mixing is
responsible for the intriguing phenomenology, but it also complicates the
discussion of higher-order corrections. Nonetheless, with the steady increase
in experimental precision as well as spectacular predictions like Eq. (1),
increasing the theoretical accuracy is desirable. This requires a proper
definition of a renormalization scheme, where an on-shell scheme allows to
study correlations between observables in a particularly transparent way.

We begin with a concise review of the model in Sec. 2, including the
definition, the conventions for the Dirac masses, the Yukawa interaction
parameters, and the unitary mixing matrix for lepton mass eigenstates. In Sec.
3 we develop a renormalization scheme for this model where all masses and
fields are on-shell renormalized and additional
$\overline{\text{MS}}$-conditions are imposed on original parameters of the
gauge-invariant Lagrangian. We discuss in detail the counterterm structure and
solution to the on-shell condition as well as the residual renormalization
scale dependence of the fundamental parameters. Afterwards, in Sec. 4 we
extend the definition of the effective muon-Higgs coupling to one-loop order
and apply the renormalization scheme to compute the corresponding renormalized
higher-order contributions. Finally, in Sec. 5 we summarize the results and
our conclusion.

##  2 Model Definition

| SM | VL
---|---|---
| $l_{L}^{i}=\begin{pmatrix}\nu^{i}_{L}\\\ e^{i}_{L}\end{pmatrix}~{}~{}~{}e_{R}^{i}~{}~{}~{}H=\begin{pmatrix}\phi^{+}\\\ v+\frac{1}{\sqrt{2}}(h+i\phi^{0})\end{pmatrix}$ | $L_{L/R}=\begin{pmatrix}L^{0}_{L/R}\\\ L^{-}_{L/R}\end{pmatrix}~{}~{}~{}E_{L/R}$
$\text{SU}(2)_{\text{L}}$ — |  2 1 2 |  2 1
$\text{U}(1)_{\text{Y}}$ — |  $-\frac{1}{2}$ $-1$ $\frac{1}{2}$ |  $-\frac{1}{2}$ $-1$
Table 1: Definition of the SM and VL lepton fields (with generation index
$i=1,2,3$) and their corresponding quantum numbers (where the electric charge
is related to SU(2)L and U(1)Y quantum numbers as $Q=T^{3}+Y$). The fields
$h,~{}\phi^{0},~{}\phi^{+}$ correspond to the physical Higgs field and the
neutral and charged Goldstone bosons, and $v\approx 174$ GeV. The VLL fields
are defined as $L_{L/R}=P_{L/R}L$ and $E_{L/R}=P_{L/R}E$, where $L$ are
$SU(2)$ doublets and $E$ $SU(2)$ singlets with the same quantum numbers as the
SM lepton doublets and singlets respectively.

The SM lepton sector consists of three generations of left-handed
$\text{SU(2)}_{L}$ doublets $l_{L}^{i}$ and right-handed singlets $e_{R}^{i}$.
In addition, we consider one generation of vector-like leptons (VLL) comprised
of both left- and right handed doublets $L_{L/R}$ and singlets $E_{L/R}$ with
gauge-quantum numbers identical to those of the SM fields (see Table 1). This
model has previously been studied in the muon-specific limit in Refs. [1, 2]
and in the context of charged lepton flavor violation in Ref. [3]. It is
useful to combine the SM and VL lepton fields with identical quantum numbers
into the 4-tuples

$\displaystyle l^{a}_{L}=(l^{i}_{L},L_{L})^{\text{T}},\qquad
e^{a}_{R}=(e^{i}_{R},E_{R})^{\text{T}}\qquad\text{for}\qquad a=1,2,3,4.$ (2)

Viewed in this way, the model contains four generations of the usual chiral
leptons $l^{a}_{L}$ and $e^{a}_{R}$ as well as one generation of opposite-
chirality fields $L_{R}$ and $E_{L}$. Using these definitions, the most
general renormalizable Lagrangian can be written as

$\displaystyle\begin{split}\mathcal{L}&\supset\bar{l}^{a}_{L}i\not{D}l^{a}_{L}+\bar{e}^{a}_{R}i\not{D}e^{a}_{R}+\bar{L}_{R}i\not{D}L_{R}+\bar{E}_{L}i\not{D}E_{L}\\\
&\quad-\big{(}\bar{l}^{a}_{L}M_{L}^{a}L_{R}+\bar{E}_{L}M_{E}^{a}e^{a}_{R}+\bar{l}^{a}_{L}Y_{ab}e^{b}_{R}H+\bar{E}_{L}\bar{\lambda}L_{R}H^{\dagger}+h.c.\big{)},\end{split}$
(3)

where the $\text{SU}(2)_{\text{L}}\times\text{U}(1)_{\text{Y}}$ covariant
derivative is given by
$D_{\mu}=\partial_{\mu}-ig_{W}\frac{\sigma^{i}}{2}W^{i}_{\mu}+ig_{Y}YB_{\mu}$.
In the present paper we assume CP invariance and consider only real
parameters. $M_{L}$ and $M_{E}$ are 4-tuple vectors of mass parameters, $Y$ is
a $4\times 4$ Yukawa matrix and $\bar{\lambda}$ a single Yukawa coupling.
Similar to the SM Yukawa sector, the Lagrangian Eq. (3) contains a redundancy
and not all parameters carry physical information. This redundancy results
from the invariance of the Lagrangian under the following simultaneous
transformation of the fields and parameters

$\displaystyle l_{L}^{a}\to(V_{L}l_{L})^{a},\quad
e^{a}_{R}\to(V_{R}e_{R})^{a},\quad M^{a}_{L}\to(V_{L}M_{L})^{a},\quad
M_{E}^{a}\to(M_{E}^{\text{T}}V_{R}^{\dagger})^{a},\quad
Y_{ab}\to(V_{L}YV_{R}^{\dagger})_{ab},$ (4)

where $V_{L/R}$ are arbitrary unitary $4\times 4$ matrices. As a result, 12 of
the Yukawa and mass parameters can be absorbed into a redefinition of the
lepton fields. In particular, it is convenient to absorb the off-diagonal
Dirac masses together with the off-diagonal SM Yukawa couplings, such that
w.l.o.g. the mass and coupling matrices can be written as

$\displaystyle M^{a}_{L}=(0,0,0,m_{L})^{\text{T}},\qquad
M^{a}_{E}=(0,0,0,m_{E})^{\text{T}},\qquad
Y_{ab}=\begin{pmatrix}y_{i}\delta_{ij}&\lambda^{E}_{i}\\\
\lambda^{L}_{j}&\lambda\end{pmatrix}.$ (5)

The remaining 12 physical parameters in this sector are thus taken as the 3 SM
Yukawa couplings $y_{i}$, the 6 new Yukawa couplings
$\lambda^{L}_{i},\lambda^{E}_{i}$ which are responsible for the interactions
between the SM and VLL fields, the Yukawa coupling $\lambda$ involving the VLL
fields, and the 2 Dirac masses $m_{L}$ and $m_{E}$. We note that the 13th new
physical parameter $\bar{\lambda}$ cannot be absorbed by any field
redefinition. The vacuum expectation value (vev) of the Higgs field
spontaneously breaks the $\text{SU}(2)_{\text{L}}\times\text{U}(1)_{\text{Y}}$
symmetry and generates additional lepton mass terms in the Lagrangian. After
electroweak symmetry breaking (EWSB), the $5\times 5$ mass matrices for the
charged leptons and neutrinos are given by

$\displaystyle(\bar{e}_{L}^{a},\bar{E}_{L})\begin{pmatrix}(Yv)_{ab}&M_{L}^{a}\\\
M_{E}^{b}&\bar{\lambda}v\end{pmatrix}\begin{pmatrix}e_{R}^{b}\\\
L^{-}_{R}\end{pmatrix},\qquad(\bar{\nu}_{L}^{a},0)\begin{pmatrix}0&M_{L}^{a}\\\
0&0\end{pmatrix}\begin{pmatrix}0\\\ L^{0}_{R}\end{pmatrix}.$ (6)

We introduce the lepton mass eigenstate fields as the 5-tuples

$\hat{e}_{L}=U^{\dagger}_{L}\begin{pmatrix}e^{a}_{L}\\\
E_{L}\end{pmatrix},\qquad\hat{e}_{R}=U^{\dagger}_{R}\begin{pmatrix}e^{a}_{R}\\\
L^{-}_{R}\end{pmatrix},\qquad\hat{\nu}_{L}=U_{L}^{\nu\dagger}\begin{pmatrix}\nu^{a}_{L}\\\
0\end{pmatrix},\qquad\hat{\nu}_{R}=U_{R}^{\nu\dagger}\begin{pmatrix}0\\\
L^{0}_{R}\end{pmatrix},$ (7)

where the unitary $5\times 5$ matrices $U_{L/R}$ and $U^{\nu}_{L/R}$ are
chosen such that the mass matrices are diagonalized as

$\displaystyle\left[U_{L}^{\dagger}\begin{pmatrix}Yv&M_{L}\\\
M_{E}^{\text{T}}&\bar{\lambda}v\end{pmatrix}U_{R}\right]_{ab}:=m_{a}\delta_{ab},\qquad\left[U_{L}^{\nu\dagger}\begin{pmatrix}0&M_{L}\\\
0&0\end{pmatrix}U^{\nu}_{R}\right]_{ab}:=m_{L}\delta_{a4}\delta_{b4}$ (8)

Note that in our convention the only massive neutrino is $\hat{\nu}^{4}$ with
mass $m_{L}$, while the SM neutrinos $\hat{\nu}_{i}$ remain massless and
$\hat{\nu}_{5}=0$. The masses of the charged leptons are given approximately
by

$\displaystyle m_{i}$
$\displaystyle=y_{i}v+\frac{\bar{\lambda}\lambda^{E}_{i}\lambda^{L}_{i}v^{3}}{m_{E}m_{L}}+\mathcal{O}(v^{4})$
(9a) $\displaystyle m_{4,5}$ $\displaystyle=m_{L,E}+\mathcal{O}(v^{2})\,,$
(9b)

where $y_{i}$ is defined in Eq. (5). In the mass eigenstate basis the lepton
mass terms and physical Higgs-lepton Yukawa interactions are given by the
following terms in the classical Lagrangian

$\displaystyle\begin{split}\mathcal{L}&\supset-
m_{a}\bar{\hat{e}}^{a}_{L}\hat{e}^{a}_{R}-m_{L}\bar{\hat{\nu}}^{4}_{L}\hat{\nu}^{4}_{R}-\tfrac{1}{\sqrt{2}}\bar{\hat{e}}^{a}_{L}Y^{h}_{ab}\hat{e}_{R}^{b}h+h.c.,\end{split}$
(10)

where we have introduced the effective $5\times 5$ Yukawa coupling matrix

$\displaystyle Y^{h}=U_{L}^{\dagger}\begin{pmatrix}Y&0\\\
0&\bar{\lambda}\end{pmatrix}U_{R}\,.$ (11)

Our convention differs slightly from the one used in Refs. [1, 2] where
$U^{(\nu)}_{L/R}=\mathbb{1}$ in the limit $v\to 0$, but which results in less
symmetric expressions for the coupling and counterterm matrices. All coupling
matrices together with the Feynman rules are listed in appendix A.

##  3 Renormalization

In this section we present a renormalization scheme developed for the VLL
model with full on-shell conditions on the lepton, Higgs- and gauge-boson
self-energies. These on-shell conditions allow for a transparent computation
of higher-order corrections to observables in terms of the physical pole
masses and well-defined input parameters of the VLL model which satisfy Eq.
(8) at all orders. Typically, difficulties arise when the on-shell scheme is
applied to the renormalization of BSM scenarios since there is often no one-
to-one correspondence between masses and parameters. For example in the
chargino and neutralino sector of the MSSM a simultaneous on-shell
renormalization of all masses is impossible, as 3 of the 6 masses are fixed by
the remaining input parameters [44]. In the 2HDM a complete on-shell
renormalization is possible, but is made difficult by the appearance of
additional parameters such as mixing angles and several possible schemes have
been studied extensively in Refs. [48, 52, 50, 53, 54]. Our approach to the
renormalization of the VLL model is similar to the treatment of the chargino
and neutralino sector discussed in Refs. [43, 44]. The important difference is
that in our case a complete on-shell renormalization of all lepton masses is
possible.

###  Gauge and Higgs Sector

Here, we briefly list our notation for the one-loop on-shell (OS)
renormalization of the gauge and Higgs sector. It is chosen to be identical to
the one of the SM; hence we adopt the conventions and results from Refs. [59,
55]. The renormalization transformation of the bare fields and parameters is
done on the level of mass-eigenstate quantities,

$\displaystyle\quad\,h$ $\displaystyle\to\big{(}1+\tfrac{1}{2}\delta
Z_{h}\big{)}h,\qquad$ $\displaystyle M_{h}^{2}\to M_{h}^{2}+\delta M_{h}^{2},$
(12a) $\displaystyle\begin{pmatrix}A_{\mu}\\\ Z_{\mu}\end{pmatrix}$
$\displaystyle\to\begin{pmatrix}1+\tfrac{1}{2}\delta Z_{AA}&\tfrac{1}{2}\delta
Z_{AZ}\\\ \tfrac{1}{2}\delta Z_{ZA}&1+\tfrac{1}{2}\delta
Z_{ZZ}\end{pmatrix}\begin{pmatrix}A_{\mu}\\\ Z_{\mu}\end{pmatrix},\qquad$
$\displaystyle M^{2}_{Z}\to M_{Z}^{2}+\delta M_{Z}^{2},$ (12b)
$\displaystyle~{}\,W^{\pm}_{\mu}$ $\displaystyle\to\big{(}1+\tfrac{1}{2}\delta
Z_{WW}\big{)}W^{\pm}_{\mu},$ $\displaystyle M_{W}^{2}\to M_{W}^{2}+\delta
M_{W}^{2},$ (12c)

in terms of the renormalized quantities and renormalization constants $\delta
Z_{h}$, $\delta Z_{VV^{\prime}}$ (with $V=A,Z,W$) as well as $\delta M_{h}$,
$\delta M_{Z}$ and $\delta M_{W}$. Denoting the unrenormalized Higgs and
gauge-boson self-energies by

$\displaystyle\Sigma_{h}(p^{2}),\qquad\Sigma^{\mu\nu}_{VV^{\prime}}(q)=\Sigma^{T}_{VV^{\prime}}(q^{2})\big{(}\eta^{\mu\nu}-\tfrac{q^{\mu}q^{\nu}}{q^{2}}\big{)}+\Sigma^{L}_{VV^{\prime}}(q^{2})\tfrac{q^{\mu}q^{\nu}}{q^{2}},$
(13)

the OS conditions on the renormalized Higgs and gauge-boson self-energies
result in the following expressions for the renormalization constants,

$\displaystyle\delta Z_{h}$
$\displaystyle=-\widetilde{\real}\tfrac{d}{dp^{2}}\Sigma_{h}(p^{2})\big{|}_{p^{2}=M_{h}^{2}},\qquad$
$\displaystyle\delta M_{h}^{2}$
$\displaystyle=\widetilde{\real}\Sigma_{h}(M_{h}^{2}),$ (14a)
$\displaystyle\delta Z_{VV}$
$\displaystyle=-\widetilde{\real}\tfrac{d}{dq^{2}}\Sigma^{T}_{VV}(q^{2})\big{|}_{p^{2}=M_{V}^{2}},\qquad$
$\displaystyle\delta M_{V}^{2}$
$\displaystyle=\widetilde{\real}\Sigma^{T}_{VV}(M_{V}^{2}),$ (14b)
$\displaystyle\delta Z_{AZ}$
$\displaystyle=-\tfrac{2}{M_{Z}^{2}}\widetilde{\real}\Sigma^{T}_{AZ}(M_{Z}^{2}),\qquad$
$\displaystyle\delta Z_{ZA}$
$\displaystyle=\tfrac{2}{M_{Z}^{2}}\widetilde{\real}\Sigma^{T}_{AZ}(0),$ (14c)

where $\widetilde{\real}$ removes the imaginary parts of any appearing loop
integrals. From charge universality it follows that the renormalization of the
electric charge is given by

$\displaystyle e\to e+\delta e,\qquad\tfrac{\delta
e}{e}=-\tfrac{1}{2}\big{(}\delta Z_{AA}+\tfrac{s_{W}}{c_{W}}\delta
Z_{ZA}\big{)}.$ (15)

In this setup the original parameters of the Lagrangian, i.e. in particular
the Higgs vev $v$, are treated as a combination of the boson masses and
electric charge $e$. Consequently, the corresponding renormalization constant
is fixed by the above counterterms as

$\displaystyle v\to v+\delta v,\qquad\tfrac{\delta v}{v}=\tfrac{\delta
M_{W}^{2}}{2M_{W}^{2}}+\tfrac{c_{W}^{2}}{s_{W}^{2}}\Big{(}\tfrac{\delta
M_{Z}^{2}}{2M_{Z}^{2}}-\tfrac{\delta
M_{W}^{2}}{2M_{W}^{2}}\Big{)}-\tfrac{\delta e}{e}.$ (16)

Finally, we discuss the renormalization of tadpole diagrams and the
implication for the renormalization of $v$. All previous equations and
relations hold independently of the specific choice for the tadpole
counterterm $\delta t$, provided that the contribution of 1-particle reducible
tadpoles $T$ and the tadpole counterterm to the self-energies in Eq. (14) are
included. However, different definitions of $\delta t$ lead to different
values of the on-shell counterterms and subsequently $\delta v$. While this is
inconsequential in the pure SM, this choice can become important in BSM
scenarios where not all parameters are necessarily renormalized on-shell. A
discussion of this issue for the MSSM can be found in Ref. [46] and further
details along with several different schemes are given in Refs. [52, 53, 58,
56]. Here we choose the parameter-renormalized tadpole scheme (PRTS) in which
the renormalized tadpole diagrams $T$ are set to zero,

$\displaystyle T+\delta t^{\text{PRTS}}=0.$ (17)

With this requirement $v$ corresponds to the true minimum of the loop-
corrected effective potential, but $\delta v$ is gauge-dependent. This scheme
was found particularly advantageous in Ref. [46] in conjunction with the
$\overline{\text{MS}}$ renormalization of other parameters. Further
discussions of the vev renormalization in general theories up to the two-loop
level can be found in Refs. [60, 61].

###  Lepton Sector

The renormalization of the lepton sector of the VLL model at one-loop order is
the central part of our work. We begin by generating the counterterm
Lagrangian with the correct structure of renormalization constants needed to
fully absorb all appearing divergences and fulfil complete OS conditions.
Afterwards, we proceed to set up and evaluate the OS conditions on the lepton
self-energies and finally discuss the renormalization of the remaining
parameters that are defined in the $\overline{\text{MS}}$-scheme.

####  3.2.1 Renormalization Transformation and Counterterm Lagrangian

A complete counterterm Lagrangian reflecting the symmetries of the theory is
most easily obtained by a renormalization transformation of the fields and
parameters in the classical Lagrangian. To determine the appropriate
renormalization transformations, one must first choose a specific basis for
the fields and independent parameters like e.g. the gauge-eigenstate basis Eq.
(3) (for the SM this was worked out in Ref. [62]) or the mass-eigenstate basis
Eq. (10) (for the SM this was worked out in Refs. [63, 59]). As mentioned in
the Introduction, in many BSM theories, setting up an OS renormalization
scheme is significantly less straightforward than in the SM. In our case, the
main complication is the $5\times 5$ lepton mixing. There is enough freedom to
treat all masses as free parameters but the remaining parameters cannot be
written as simple mixing angles (in contrast to e.g. the 2HDM) and are
affected by the redundancy Eq. (4) between fields and parameters.

In our setup we follow the approach outlined in Secs. 4.2 and 4.3 of Ref. [43]
and used for the renormalization of the chargino and neutralino sector of the
MSSM in Ref. [44]. This approach is advantageous because it does not require a
renormalization of the mixing matrices and allows for a straight forward
combination of OS and $\overline{\text{MS}}$ renormalization conditions.

The approach proceeds in two steps. First, the fields and parameters in the
gauge-eigenstate basis are renormalized. Secondly, an additional finite
transformation of only the fields into mass-eigenstate fields is performed. In
combination, a redundancy-free counterterm Lagrangian is obtained which
contains renormalization constants of fundamental parameters and matrix-valued
field renormalization constants for mass-eigenstate fields, that allow to
fulfil complete OS conditions. In detail, starting from the original gauge-
eigenstate Lagrangian in Eq. (3) we first introduce the renormalization
transformation for the field multiplets as

$\displaystyle l^{a}_{L}$
$\displaystyle\to(Z^{l}_{L})^{\frac{1}{2}}_{ab}\,l^{b}_{L},\qquad$
$\displaystyle L_{R}\to(Z_{R}^{L})^{\frac{1}{2}}L_{R},$ (18a) $\displaystyle
e^{a}_{R}$ $\displaystyle\to(Z^{e}_{R})^{\frac{1}{2}}_{ab}\,e^{b}_{R},\qquad$
$\displaystyle E_{L}\to(Z_{L}^{E})^{\frac{1}{2}}E_{L},$ (18b)

where matrix-valued field renormalization constants allow mixing between
fields with identical quantum numbers. The Dirac mass and Yukawa parameters in
Eq. (3) are renormalized as

$\displaystyle M_{L}^{a}\to M_{L}^{a}+\delta M_{L}^{a},\qquad$ $\displaystyle
M_{E}^{a}\to M_{E}^{a}+\delta M_{E}^{a},$ (19a) $\displaystyle
Y_{ab}\,\to\,Y_{ab}+\delta Y_{ab},\qquad$
$\displaystyle~{}\bar{\lambda}~{}~{}\to\bar{\lambda}+\delta\bar{\lambda}.$
(19b)

At this point, as a consequence of the redundancy Eq. (4), there is an
ambiguity between the field and parameter counterterms. Despite assuming the
tree-level diagonalization Eq. (5), the same symmetry still implies an
invariance of the bare Lagrangian under the following transformation of the
renormalization constants

$\displaystyle(Z_{L}^{l})^{\frac{1}{2}}$ $\displaystyle\to(\mathbb{1}+\delta
V_{L})(Z_{L}^{l})^{\frac{1}{2}},$ (20a)
$\displaystyle(Z_{R}^{e})^{\frac{1}{2}}$ $\displaystyle\to(\mathbb{1}+\delta
V_{R})(Z_{R}^{e})^{\frac{1}{2}},$ (20b) $\displaystyle M_{L}+\delta M_{L}$
$\displaystyle\to(\mathbb{1}+\delta V_{L})(M_{L}+\delta M_{L}),$ (20c)
$\displaystyle M^{T}_{E}+\delta M^{T}_{E}$ $\displaystyle\to(M^{T}_{E}+\delta
M^{T}_{E})(\mathbb{1}+\delta V_{R}^{\dagger}),$ (20d) $\displaystyle Y+\delta
Y$ $\displaystyle\to(\mathbb{1}+\delta V_{L})(Y+\delta Y)(\mathbb{1}+\delta
V_{R}^{\dagger}),$ (20e)

where $\mathbb{1}+\delta V_{L/R}$ can be arbitrary unitary matrices. We make
use of this transformation to bring the counterterm matrices into the same
form as the tree-level couplings in Eq. (5), i.e. we assume that the 13
independent parameter renormalization constants in the sector have the form

$\displaystyle\delta M_{L}^{a}=(0,0,0,\delta m_{L})^{T},\qquad\delta
M^{a}_{E}=(0,0,0,\delta m_{E})^{T},\qquad\delta Y_{ab}=\begin{pmatrix}\delta
y_{i}\delta_{ij}&\delta\lambda^{E}_{i}\\\
\delta\lambda^{L}_{j}&\delta\lambda\end{pmatrix},\qquad\delta\bar{\lambda}.$
(21)

This removes the ambiguity and disentangles field and parameter counterterms.
However, it is important to keep this choice in mind when comparing to results
in the literature. For example, for the calculation of the $\beta$-functions
of general gauge theories in Refs. [64, 65] this ambiguity was removed by
assuming hermitian field renormalization matrices, which comes at the cost of
off-diagonal counterterms $\delta M_{L/E}^{i}$ and $\delta y_{ij}$. We will
discuss the consequences and our treatment in more detail in Sec. 3.3.

This structure of the renormalization constants corresponding to the gauge-
eigenstate basis is sufficient to cancel all UV divergences but not sufficient
to fulfil full OS conditions. Hence, as a second step we perform a finite
transformation to mass-eigenstate fields. We require that the diagonalization
condition in Eq. (8) holds exactly, i.e. at all orders for the renormalized
mass matrix. This provides an exact definition of the unitary matrices
$U_{L,R}$, $U^{\nu}_{L,R}$ as well as of the mass eigenvalues $m_{a}$ as
functions of the fundamental renormalized parameters. In contrast, Eq. (7)
cannot be required at all orders if full OS conditions are desired. Instead,
the mass-eigenstate fields are defined via the following higher-order
generalization of Eq. (7),

$\begin{pmatrix}e^{a}_{L}\\\
E_{L}\end{pmatrix}=U_{L}\mathcal{R}_{L}\hat{e}_{L},\quad\begin{pmatrix}e^{a}_{R}\\\
L^{-}_{R}\end{pmatrix}=U_{R}\mathcal{R}_{R}\hat{e}_{R},\quad\begin{pmatrix}\nu^{a}_{L}\\\
0\end{pmatrix}=U_{L}^{\nu}\mathcal{R}_{L}^{\nu}\hat{\nu}_{L},\quad\begin{pmatrix}0\\\
L^{0}_{R}\end{pmatrix}=U_{R}^{\nu}\mathcal{R}_{R}^{\nu}\hat{\nu}_{R}.$ (22)

Here, we have introduced additional finite and invertible matrices
$\mathcal{R}^{(\nu)}_{L/R}=\mathbb{1}+\delta\mathcal{R}^{(\nu)}_{L/R}$ (see
also Sec. 4.2 of Ref. [43]). In total, the combination of the field
transformations (18) and (22) amounts to a renormalization transformation from
bare gauge-eigenstate fields to renormalized mass-eigenstate fields in terms
of products of three matrices. Like in Refs. [43, 44] it is useful rewrite
these products in terms of final field renormalization constants which
effectively renormalize mass-eigenstate fields. Thus, the ultimate form of the
renormalization transformation from gauge-eigenstate fields to mass-eigenstate
fields is written as

$\displaystyle\begin{pmatrix}e^{a}_{L}\\\ E_{L}\end{pmatrix}$
$\displaystyle\to\begin{pmatrix}(Z^{l}_{L})^{\frac{1}{2}}&0\\\
0&(Z_{L}^{E})^{\frac{1}{2}}\end{pmatrix}U_{L}\mathcal{R}_{L}\hat{e}_{L}$
$\displaystyle\equiv U_{L}(Z_{L})^{\frac{1}{2}}\hat{e}_{L},$ (23a)
$\displaystyle\begin{pmatrix}e^{a}_{R}\\\ L^{-}_{R}\end{pmatrix}$
$\displaystyle\to\begin{pmatrix}(Z^{e}_{R})^{\frac{1}{2}}&0\\\
0&(Z_{R}^{L})^{\frac{1}{2}}\end{pmatrix}U_{R}\mathcal{R}_{R}\hat{e}_{R}$
$\displaystyle\equiv U_{R}(Z_{R})^{\frac{1}{2}}\hat{e}_{R},$ (23b)
$\displaystyle\begin{pmatrix}\nu^{a}_{L}\\\ 0\end{pmatrix}$
$\displaystyle\to\hskip
12.51927pt\begin{pmatrix}(Z^{l}_{L})^{\frac{1}{2}}&0~{}\\\
0&1~{}\end{pmatrix}\hskip
12.51927ptU_{L}^{\nu}\mathcal{R}_{L}^{\nu}\hat{\nu}_{L}$ $\displaystyle\equiv
U_{L}^{\nu}(Z_{L}^{\nu})^{\frac{1}{2}}\hat{\nu}_{L},$ (23c)
$\displaystyle\begin{pmatrix}0\\\ L^{0}_{R}\end{pmatrix}$
$\displaystyle\to\hskip 9.95863pt\begin{pmatrix}~{}\mathbb{1}&0\\\
~{}0&(Z_{R}^{L})^{\frac{1}{2}}\end{pmatrix}\hskip
8.5359ptU_{R}^{\nu}\mathcal{R}_{R}^{\nu}\hat{\nu}_{R}$ $\displaystyle\equiv
U_{R}^{\nu}(Z_{R}^{\nu})^{\frac{1}{2}}\hat{\nu}_{R},$ (23d)

where we have introduced the four $5\times 5$ matrices $Z_{L/R}$,
$Z_{L/R}^{\nu}$ which effectively implement the matrix-valued field
renormalization of the mass-eigenstate fields. We reiterate that the
diagonalization matrices $U_{L/R}$, $U_{L/R}^{\nu}$ split off on the left
always remain of tree-level order.

This completes the setup of the renormalization transformation. In summary,
the transformations are given by Eq. (19) for the fundamental parameters (with
the constraints (4), (21)) and by Eq. (23) (far right) for the fields. This
renormalization transformation is applied to the gauge-eigenstate basis
Lagrangian Eq. (3), using the definition of the diagonalization matrices
$U_{L/R}^{(\nu)}$ in Eq. (8). The result is the correct structure of the bare
Lagrangian expressed in terms of renormalized fundamental parameters and mass-
eigenstate fields. It can be written as

$\displaystyle\begin{split}\mathcal{L}&\supset\bar{\hat{e}}^{a}\bigg{\\{}i\not{\partial}\Big{[}Z_{L}^{\frac{1}{2}\dagger}Z_{L}^{\frac{1}{2}}\text{P}_{\text{L}}+Z_{R}^{\frac{1}{2}\dagger}Z_{R}^{\frac{1}{2}}\text{P}_{\text{R}}\Big{]}_{ab}-\Big{[}Z_{R}^{\frac{1}{2}\dagger}(m+\delta
m)^{\dagger}Z_{L}^{\frac{1}{2}}\text{P}_{\text{L}}+Z_{L}^{\frac{1}{2}\dagger}(m+\delta
m)Z_{R}^{\frac{1}{2}}\text{P}_{\text{R}}\Big{]}_{ab}\\\ &\hskip
15.6491pt-\tfrac{1}{\sqrt{2}}\Big{[}Z_{R}^{\frac{1}{2}\dagger}(Y^{h}+\delta
Y^{h})^{\dagger}Z_{L}^{\frac{1}{2}}\text{P}_{\text{L}}+Z_{L}^{\frac{1}{2}\dagger}(Y^{h}+\delta
Y^{h})Z^{\frac{1}{2}}_{R}\Big{]}_{ab}Z_{h}h\bigg{\\}}\hat{e}^{b}\\\
&+\bar{\hat{\nu}}^{a}\bigg{\\{}i\not{\partial}\Big{[}Z_{L}^{\nu\frac{1}{2}\dagger}Z_{L}^{\nu\frac{1}{2}}\text{P}_{\text{L}}+Z_{R}^{\nu\frac{1}{2}\dagger}Z_{R}^{\nu\frac{1}{2}}\text{P}_{\text{R}}\Big{]}_{ab}-(m_{L}+\delta
m_{L})\Big{[}Z^{\nu\frac{1}{2}\dagger}_{R,a4}Z^{\nu\frac{1}{2}}_{L,4b}\text{P}_{\text{L}}+Z^{\nu\frac{1}{2}\dagger}_{L,a4}Z^{\nu\frac{1}{2}}_{R,4b}\text{P}_{\text{R}}\Big{]}\bigg{\\}}\hat{\nu}^{b},\end{split}$
(24)

which includes a bare counterpart of the tree-level Lagrangian Eq. (10), where
the mass and Yukawa matrices effectively renormalize as

$\displaystyle m_{a}\delta_{ab}+\delta m_{ab}$
$\displaystyle=m_{a}\delta_{ab}+\left[U_{L}^{\dagger}~{}\begin{pmatrix}\delta(Yv)&\delta
M_{L}\\\ \delta
M^{T}_{E}&\delta(\bar{\lambda}v)\end{pmatrix}U_{R}\right]_{ab}$ (25a)
$\displaystyle Y^{h}_{ab}+\delta Y^{h}_{ab}~{}~{}$
$\displaystyle=~{}\,Y^{h}_{ab}~{}~{}+\left[U_{L}^{\dagger}~{}\begin{pmatrix}\delta
Y&0~{}\\\ 0&\delta\bar{\lambda}~{}\end{pmatrix}U_{R}\right]_{ab}$ (25b)

We reiterate that the fundamental parameters are taken as the ones of Eq. (3)
and the fundamental renormalization constants are the ones of Eq. (19) and Eq.
(23) (far right). Other quantities, in particular $U_{L/R}$ and $m_{a}$ as
well as the renormalization constants $\delta m_{ab}$ and $\delta Y^{h}$ are
derived from these definitions and are given by Eq. (8) and Eq. (25),
respectively.

####  3.2.2 One-loop On-Shell Renormalization

Having established the structure of the counterterm Lagrangian we set out to
solve the OS renormalization conditions on the lepton self-energies for the
renormalization constants at one-loop order. In terms of the renormalized
charged lepton self-energy matrix $\hat{\Sigma}_{ab}$ these conditions read

$\widetilde{\real}\hat{\Sigma}_{ab}(\not{p})u_{b}(p)\big{|}_{p^{2}=m_{b}^{2}}=0,\qquad\quad\lim_{p^{2}\to
m_{a}^{2}}\tfrac{1}{\not{p}-m_{a}}\widetilde{\real}\hat{\Sigma}_{aa}(\not{p})u_{a}(p)=0,\qquad
a,b=1,...,5$ (26)

where $\widetilde{\real}$ discards the imaginary parts resulting from the loop
integration. To solve for the renormalization constants we first write the
renormalized self-energy in terms of the covariant decomposition

$\displaystyle
i\hat{\Sigma}_{ab}(\not{p})=i\not{p}\Big{(}\hat{\Sigma}^{R}_{ab}(p^{2})\text{P}_{\text{R}}+\hat{\Sigma}_{ab}^{L}(p^{2})\text{P}_{\text{L}}\Big{)}+i\Big{(}\hat{\Sigma}_{ab}^{SR}(p^{2})\text{P}_{\text{R}}+\hat{\Sigma}_{ab}^{SL}(p^{2})\text{P}_{\text{L}}\Big{)},$
(27)

where hermiticity of the (counterterm) Lagrangian relates the coefficient
functions by

$\displaystyle\widetilde{\real}\hat{\Sigma}^{R}_{ab}(p^{2})=\widetilde{\real}\hat{\Sigma}^{R*}_{ba}(p^{2}),\quad\widetilde{\real}\hat{\Sigma}^{L}_{ab}(p^{2})=\widetilde{\real}\hat{\Sigma}^{L*}_{ba}(p^{2}),\quad\widetilde{\real}\hat{\Sigma}^{SR}_{ab}(p^{2})=\widetilde{\real}\hat{\Sigma}^{SL*}_{ba}(p^{2}).$
(28)

Next, we expand the charged lepton field renormalization matrices in Eq. (24)
to one-loop order

$\displaystyle Z^{\frac{1}{2}}_{L/R}\approx\mathbb{1}+\tfrac{1}{2}\delta
Z_{L/R}.$ (29)

The covariant coefficients in Eq. (27) can now be expressed in terms of the
corresponding coefficients of the unrenormalized one-loop self-energy matrix
$\Sigma_{ab}$ and the one-loop counterterm contributions

$\displaystyle\hat{\Sigma}^{R}_{ab}(p^{2})$
$\displaystyle=\Sigma^{R}_{ab}(p^{2})+\tfrac{1}{2}(\delta
Z_{R})_{ab}+\tfrac{1}{2}(\delta Z_{R}^{\dagger})_{ab}$ (30a)
$\displaystyle\hat{\Sigma}^{L}_{ab}(p^{2})$
$\displaystyle=\Sigma^{L}_{ab}(p^{2})+\tfrac{1}{2}(\delta
Z_{L})_{ab}+\tfrac{1}{2}(\delta Z_{L}^{\dagger})_{ab}$ (30b)
$\displaystyle\hat{\Sigma}^{SR}_{ab}(p^{2})$
$\displaystyle=\Sigma^{SR}_{ab}(p^{2})-\tfrac{1}{2}m_{a}(\delta
Z_{R})_{ab}-\tfrac{1}{2}(\delta Z_{L}^{\dagger})_{ab}m_{b}-(\delta m)_{ab}$
(30c) $\displaystyle\hat{\Sigma}^{SL}_{ab}(p^{2})$
$\displaystyle=\Sigma^{SL}_{ab}(p^{2})-\tfrac{1}{2}m_{a}(\delta
Z_{L})_{ab}-\tfrac{1}{2}(\delta Z_{R}^{\dagger})_{ab}m_{b}-(\delta
m^{\dagger})_{ab}.$ (30d)

Finally, the OS conditions Eq. (26) can be solved and imply the following
expression for the renormalization constants of the charged leptons

$\displaystyle\text{for~{}}a=b\quad\begin{cases}(\delta
m)_{aa}&=\frac{1}{2}\,\widetilde{\real}[m_{a}\Sigma^{R}_{aa}+m_{a}\Sigma^{L}_{aa}+\Sigma^{SR}_{aa}+\Sigma^{SL}_{aa}\bigg{]}\Big{|}_{p^{2}=m_{a}^{2}}\\\\[8.5359pt]
(\delta
Z_{R})_{aa}&=-\widetilde{\real}[\Sigma^{R}_{aa}+m_{a}\Big{(}\Sigma^{SR\prime}_{aa}+\Sigma^{SL\prime}_{aa}\Big{)}+m_{a}^{2}\Big{(}\Sigma^{R\prime}_{aa}+\Sigma^{L\prime}_{aa}\Big{)}\bigg{]}\Big{|}_{p^{2}=m_{a}^{2}}\\\\[8.5359pt]
(\delta
Z_{L})_{aa}&=-\widetilde{\real}[\Sigma^{L}_{aa}+m_{a}\Big{(}\Sigma^{SR\prime}_{aa}+\Sigma^{SL\prime}_{aa}\Big{)}+m_{a}^{2}\Big{(}\Sigma^{R\prime}_{aa}+\Sigma^{L\prime}_{aa}\Big{)}\bigg{]}\Big{|}_{p^{2}=m_{a}^{2}}\end{cases}$
(31a) $\displaystyle\text{for~{}}a\neq b\quad\begin{cases}(\delta
Z_{R})_{ab}&=\frac{2}{m_{a}^{2}-m_{b}^{2}}\bigg{[}m_{b}^{2}\widetilde{\real}\Sigma^{R}_{ab}+m_{a}m_{b}\widetilde{\real}\Sigma^{L}_{ab}+m_{a}\widetilde{\real}\Sigma^{SR}_{ab}\\\
&\hskip 59.75095pt+m_{b}\widetilde{\real}\Sigma^{SL}_{ab}-m_{a}(\delta
m)_{ab}-m_{b}(\delta
m^{\dagger})_{ab}\bigg{]}\Big{|}_{p^{2}=m_{b}^{2}}\\\\[8.5359pt] (\delta
Z_{L})_{ab}&=\frac{2}{m_{a}^{2}-m_{b}^{2}}\bigg{[}m_{b}^{2}\widetilde{\real}\Sigma^{L}_{ab}+m_{a}m_{b}\widetilde{\real}\Sigma^{R}_{ab}+m_{a}\widetilde{\real}\Sigma^{SL}_{ab}\\\
&\hskip 59.75095pt+m_{b}\widetilde{\real}\Sigma^{SR}_{ab}-m_{a}(\delta
m^{\dagger})_{ab}-m_{b}(\delta
m)_{ab}\bigg{]}\Big{|}_{p^{2}=m_{b}^{2}}\end{cases}$ (31b)

The neutral lepton sector can be treated analogously. The main difference is
that the structure of the theory guarantees that there are three massless and
one massive neutrino and the mixing is trivial. Hence, the The corresponding
conditions on the neutrino self-energies result in

$\displaystyle\text{for~{}}a=b$ $\displaystyle\begin{cases}\delta
m_{L}&=\frac{1}{2}\,\widetilde{\real}[m_{L}\Sigma^{\nu
R}_{44}+m_{L}\Sigma^{\nu L}_{44}+\Sigma^{\nu SR}_{44}+\Sigma^{\nu
SL}_{44}\bigg{]}\Big{|}_{p^{2}=m_{L}^{2}}\\\\[8.5359pt] (\delta
Z^{\nu}_{R})_{aa}&=-\widetilde{\real}[\Sigma^{\nu
R}_{aa}+\delta_{a4}m_{L}\Big{(}\Sigma^{\nu SR\prime}_{aa}+\Sigma^{\nu
SL\prime}_{aa}+m_{L}\Sigma^{\nu R\prime}_{aa}+m_{L}\Sigma^{\nu
L\prime}_{aa}\Big{)}\bigg{]}\Big{|}_{p^{2}=\delta_{a4}m_{L}^{2}}\\\\[8.5359pt]
(\delta Z^{\nu}_{L})_{aa}&=-\widetilde{\real}[\Sigma^{\nu
L}_{aa}+\delta_{a4}m_{L}\Big{(}\Sigma^{\nu SR\prime}_{aa}+\Sigma^{\nu
SL\prime}_{aa}+m_{L}\Sigma^{\nu R\prime}_{aa}+m_{L}\Sigma^{\nu
L\prime}_{aa}\Big{)}\bigg{]}\Big{|}_{p^{2}=\delta_{a4}m_{L}^{2}}\end{cases}$
(32a) $\displaystyle\text{for~{}}a,b\neq 4$ $\displaystyle\begin{cases}(\delta
Z^{\nu}_{L/R})_{a4}&=-\frac{2}{m_{L}}\widetilde{\real}[m_{L}\Sigma^{\nu
L/R}_{a4}(m_{L}^{2})+\Sigma^{\nu SR/SL}_{a4}(m_{L}^{2})\bigg{]}\\\\[8.5359pt]
(\delta
Z^{\nu}_{L/R})_{4b}&=\hphantom{-}\frac{2}{m_{L}}\widetilde{\real}\Sigma^{\nu
SL/SR}_{4b}(0)\end{cases}.$ (32b)

Note that the OS conditions on the off-diagonal neutrino self-energies for
$a,b\neq 4$ are fulfilled automatically and do not provide any non-trivial
constraints for renormalization constants.

####  3.2.3 Renormalization of the Fundamental Parameters

The OS conditions listed above do not fix all fundamental renormalization
constants of Eq. (19) and Eq. (23) unambiguously. The remaining freedom has to
be fixed by imposing additional conditions. In order to motivate our choice we
start by writing the relation Eq. (25) for the diagonal mass renormalization
constants as

$\displaystyle\begin{split}\delta
m_{aa}&=\sum_{b,c}(U_{L}^{\dagger})_{ab}(U_{R})_{ca}\delta M_{bc},\qquad
a=1,...,5,\qquad\delta M=\begin{pmatrix}\delta(Yv)&\delta M_{L}\\\ \delta
M_{E}^{\text{T}}&\delta(\bar{\lambda}v)\end{pmatrix}.\end{split}$ (33)

The OS conditions fix the left-hand side for all $a=1\ldots 5$ via Eq. (31a).
In addition, the neutral lepton OS condition (32a) directly fixes $\delta
m_{L}$. Hence, 6 of the 13 renormalization constants $\delta m_{L}$, $\delta
m_{E}$, $\delta Y_{ab}$ and $\delta\bar{\lambda}$ can be determined from OS
conditions but further conditions need to be imposed on the remaining
renormalization constants. Some care has to be taken in order to avoid
absorbing large corrections of order of the VLL masses $m_{4,5}\gg v$ into
e.g. $\delta(\lambda v)$ or $\delta(\bar{\lambda}v)$. Such corrections are
instead naturally absorbed by $\delta m_{E}$ and $\delta m_{L}$. In principle,
this implies we should solve Eq. (33) for $\delta m_{L}$ and $\delta m_{E}$ in
addition to 3 of the Yukawa counterterms. However, since $\delta m_{L}$ is
already fixed by the neutrino OS conditions we are forced to choose a fourth
Yukawa counterterm instead. Fortunately, the difference between $\delta m_{L}$
obtained from the neutral lepton OS conditions and from solving Eq. (33), like
the difference between $m_{4}$ and $m_{L}$, is generated by EWSB effects. It
is therefore of order $v$ and can safely be absorbed into e.g. $\delta(\lambda
v)$ or $\delta(\bar{\lambda}v)$. An obvious choice for the four Yukawa
counterterms is $\delta y_{i}$ (to allow for the OS renormalized SM limit) and
either $\delta\lambda$ or $\delta\bar{\lambda}$. Here we choose
$\delta\lambda$, such that Eq. (33) has to be solved for $\delta(Y_{aa}v)$ and
$\delta m_{E}$. For simplicity, we fix the remaining renormalization constants
in the $\overline{\text{MS}}$ scheme. To summarize, the parameter counterterms
are determined in the following way:

  * •

$\delta m_{E}$, $\delta(y_{i}v)$ and $\delta(\lambda v)$ are fixed via the
charged lepton masses by Eq. (33) together with Eq. (31a)

  * •

$\delta m_{L}$ is fixed via the neutrino mass by Eq. (32a)

  * •

$\delta\lambda^{L}_{i},\delta\lambda^{E}_{i}$ and $\delta\bar{\lambda}$ are
fixed by the $\overline{\text{MS}}$ scheme

  * •

$\delta v$ is fixed by Eq. (16) and the tadpole is fixed by Eq. (17).

This provides sufficient conditions to fix all $\delta M_{ab}$ and in turn all
$\delta m_{ab}$ and $\delta Y^{h}_{ab}$ unambiguously. To solve for the set of
renormalization constants, we first write Eq. (33) in matrix notation

$\displaystyle\bm{\delta}\mathbf{m}=\bm{\kappa}\cdot\bm{\delta}\mathbf{M}+\bm{\delta}\mathbf{u},\qquad\text{where}\quad\bm{\delta}\mathbf{M}=\\{\delta(y_{1}v),\delta(y_{2}v),\delta(y_{3}v),\delta(\lambda
v),\delta m_{E}\\},$ (34)

with the coefficient matrix $\bm{\kappa}$ and constant vector
$\bm{\delta}\mathbf{u}$ given by

$\displaystyle\bm{\kappa}_{ab}=\begin{cases}(U_{L}^{\dagger})_{ab}(U_{R})_{ba}&b=1,...,4\\\
(U_{L}^{\dagger})_{a5}(U_{R})_{4a}&b=5\end{cases},\qquad\bm{\delta}\mathbf{u}_{a}=(U_{L}^{\dagger})_{a5}(U_{R})_{5a}\delta
M_{55}+\sum_{\begin{subarray}{c}b\neq c\\\ b\neq
5\end{subarray}}(U_{L}^{\dagger})_{ab}(U_{R})_{ca}\delta M_{bc}.$ (35)

This allows us to write the solution in the following compact form

$\displaystyle\bm{\delta}\mathbf{M}=\bm{\kappa}^{-1}(\bm{\delta}\mathbf{m}-\bm{\delta}\mathbf{u}),\qquad\delta
Y_{aa}=\tfrac{1}{v}\big{(}\delta M_{aa}-Y_{aa}\delta v\big{)}.$ (36)

These equations hold separately for the finite and UV divergent parts and can
be evaluated numerically or using perturbative expressions for the rotation
matrices such as those given in [2]. The remaining $\overline{\text{MS}}$
parameter counterterms are taken from the literature [64, 65].

###  Relation to $\overline{\text{MS}}$ and Renormalization Group Running

In the renormalization scheme defined by the renormalization transformations
Eq. (19) and Eq. (23) (right), the on-shell conditions Eq. (26) and by the
relation Eq. (36), the mass-eigenstate fields are on-shell renormalized and
the diagonalized mass matrices Eq. (8) correspond to their respective physical
pole masses. However, in this way only a subset of the fundamental parameters
are determined by physical on-shell conditions while the rest are defined in
the $\overline{\text{MS}}$ scheme with the constraint (21). Here we describe
how to determine the values of the $\overline{\text{MS}}$ counterterms and the
resulting running of all renormalized parameters. We begin with the well-known
$\beta$-functions derived for general gauge theories in the
$\overline{\text{MS}}$ scheme, see e.g. [65] and references therein. As
mentioned in the previous section, these $\beta$-functions are derived
assuming hermitian field renormalization matrices which resolves the
redundancy between field and parameter renormalization. We instead removed
this redundancy by imposing Eq. (21) on the parameter counterterms, allowing
for general field renormalization matrices. As a result, the renormalization
constants including the $\frac{1}{\epsilon}$ poles, and consequently the
$\beta$-functions, change. We have explicitly computed the required matrices
$\delta V_{L/R}$ of Eq. (20) that absorb the off-diagonal parameter
counterterms into the field renormalization matrices. The resulting
$\beta$-functions corresponding to $\overline{\text{MS}}$ with condition Eq.
(21) are listed in App. B. These $\beta$-functions unambiguously determine the
$\overline{\text{MS}}$ renormalization constants $\delta\lambda^{E}_{i}$,
$\delta\lambda^{L}_{j}$, $\delta\bar{\lambda}$ required in our scheme.

The running of parameters in our (partial) OS scheme is further modified by
the presence of finite contributions $\delta\Theta_{\alpha}^{[0]}$ to the
renormalization constants
$\delta\Theta_{\alpha}=\frac{1}{\epsilon}\delta\Theta^{[1]}_{\alpha}+\delta\Theta^{[0]}_{\alpha}$
for some of the fundamental model parameters collectively denoted by
$\Theta_{\alpha}$. Specifically, for our choices in the previous section,
these finite contributions relate the model parameters in our scheme to the
ones in the pure $\overline{\text{MS}}$ scheme by

$\displaystyle m_{L}^{\overline{\text{MS}}}=m_{L}+\delta m^{[0]}_{L},\qquad
m_{E}^{\overline{\text{MS}}}=m_{E}+\delta m^{[0]}_{E},\qquad
Y_{aa}^{\overline{\text{MS}}}=Y_{aa}+\delta Y_{aa}^{[0]}$ (37)

The resulting running of the parameters $\Theta_{\alpha}$ in our scheme is
therefore given by

$\displaystyle\mu\frac{d\Theta_{\alpha}}{d\mu}\equiv\beta^{\text{eff}}(\Theta_{\alpha})\approx\beta(\Theta_{\alpha})-\mu\frac{\partial\delta\Theta_{\alpha}^{[0]}}{\partial\mu}+\mathcal{O}(2L)$
(38)

For purely OS renormalized parameters, like $m_{L}$, the effective
$\beta$-function vanishes. However, because of the mixing between OS and
$\overline{\text{MS}}$ parameters when going to the mass basis the parameters,
$Y_{aa}$ and $m_{E}$ necessarily retain some residual $\mu$ dependence. We
also note that due to the mixing between $\overline{\text{MS}}$ and OS
counterterms in Eq. (36) the usual (one-loop) OS relation

$\displaystyle\mu\frac{\partial\delta\Theta_{\alpha}^{[0]}}{\partial\mu}=2\delta\Theta_{\alpha}^{[1]}$
(39)

does not hold for these parameters.

##  4 One-Loop Effective Higgs Coupling and Correlation with $a_{\mu}$

Having established a renormalization scheme that addresses the higher-order
mixing between the lepton fields in the VLL model and establishes OS
conditions on masses and fields, in this section we test its validity and
discuss a first application to the phenomenological impact of higher-order
corrections.

In particular, we focus on the effective muon–Higgs coupling
$\lambda_{\mu\mu}$ in the muon-specific case, where the VLL couplings to the
first and third generation of SM leptons are zero,
$\lambda^{E}_{1,3}=\lambda^{L}_{1,3}=0$. The effective coupling
$\lambda_{\mu\mu}$ is one of the most interesting observables in the VLL model
as it receives large chirally enhanced tree-level corrections from the mixing
between SM and VL leptons. Furthermore, these corrections are directly
correlated with the one-loop VLL contributions to the anomalous magnetic
moment $a_{\mu}$. For this reason the tree-level effective coupling has been
studied extensively in the muon-specific VLL model in search for an
explanation of the tentative experimental deviation of $\Delta a_{\mu}$ from
the SM prediction [1, 2, 6, 10]. Here, we calculate the one-loop corrections
to $\lambda_{\mu\mu}$, and we show how these higher-order contributions can
significantly modify the correlation between $\lambda_{\mu\mu}$ and $\Delta
a_{\mu}^{\text{VLL}}$ found at tree-level, see Eq. (1).

###  Definition and Vertex Renormalization

$h$$\mu^{-}$$\mu^{+}$$\hat{e}_{c}$$\hat{e}_{d}$$h,\phi^{0}$ (a)

$h$$\mu^{-}$$\mu^{+}$$\hat{e}_{c}$$\hat{e}_{d}$$\gamma,Z$ (b)

$h$$\mu^{-}$$\mu^{+}$$h,\phi^{0,\pm}$$h,\phi^{0,\pm}$$\hat{e}_{c},\nu_{c}$ (c)

$h$$\mu^{-}$$\mu^{+}$$Z,W^{\pm}$$Z,W^{\pm}$$\hat{e}_{c}$ (d)

$h$$\mu^{-}$$\mu^{+}$$\phi^{0},\phi^{\pm}$$Z,W^{\pm}$$\hat{e}_{c},\nu_{c}$ (e)

$h$$\mu^{-}$$\mu^{+}$$Z,W^{\pm}$$\phi^{0},\phi^{\pm}$$\hat{e}_{c},\nu_{c}$ (f)

Figure 1: One-loop virtual contributions to $h\to\mu^{+}\mu^{-}$ in the VLL
model.

At tree-level, the effective muon–Higgs coupling $\lambda_{\mu\mu}$ can be
directly defined by the partial decay width of an on-shell Higgs into a
muon/anti-muon pair

$\displaystyle\Gamma(h\to\mu^{-}\mu^{+})\approx\frac{M_{h}}{8\pi^{2}}|\lambda_{\mu\mu}^{\text{tree}}|^{2}+\mathcal{O}(1L).$
(40)

Neglecting the muon mass, it is related to the tree-level matrix element by

$\displaystyle|\lambda^{\text{tree}}_{\mu\mu}|^{2}=\tfrac{1}{M_{h}^{2}}\overline{|\mathcal{M}|^{2}}(h\to\mu^{-}\mu^{+})_{\text{tree}}=(Y^{h}_{22})^{2}.$
(41)

This definition can be extended to the one-loop level in the following simple
way. In general, the loop-corrected on-shell matrix element can be written as

$\displaystyle\mathcal{M}\big{(}h(q)\to\mu^{-}(p)\mu^{+}(p^{\prime})\big{)}=-i\bar{u}(p)\Big{[}\zeta(q^{2},p^{2},p^{\prime
2})^{*}\text{P}_{\text{L}}+\zeta(q^{2},p^{2},p^{\prime
2})\text{P}_{\text{R}}\Big{]}v(p^{\prime}).$ (42)

At next-to-leading order (NLO) the coefficient function $\zeta$ develops an
infrared (IR) divergence due to the vertex and self-energy corrections from
soft photon loops. This divergence can be factored out into a universal gauge-
invariant quantity $\alpha B$ [66],

$\displaystyle\zeta=\exp(\alpha B)\zeta_{0},$ (43)

such that $\zeta_{0}$ is IR finite, and where (in dimensional regularization)

$\displaystyle\begin{split}\alpha
B=\frac{e^{2}}{16\pi^{2}}\Big{\\{}4&+B_{0}(0,m_{\mu}^{2},m_{\mu}^{2})-B_{0}(M_{h}^{2},m_{\mu}^{2},m_{\mu}^{2})+4m_{\mu}^{2}B^{\prime}_{0}(m_{\mu}^{2},0,m_{\mu}^{2})\\\
&-2(M_{h}^{2}-2m_{\mu}^{2})C_{0}(M_{h}^{2},m_{\mu}^{2},m_{\mu}^{2},m_{\mu}^{2},m_{\mu}^{2},0)\Big{\\}}.\end{split}$
(44)

We can thus define an IR finite effective muon–Higgs coupling at the loop-
level by111It would be straightforward to extend this definition to include
the real radiation corrections that remove the IR divergences from the
physical decay width. However, like the soft singularities, these
contributions cancel in the ratio
$\lambda_{\mu\mu}/\lambda_{\mu\mu}^{\text{SM}}$ and have no impact on our
further analysis.

$\displaystyle\lambda_{\mu\mu}\equiv\sqrt{2}\zeta_{0}(M_{h}^{2},m_{\mu}^{2},m_{\mu}^{2})=\sqrt{2}\exp(-\alpha
B)\zeta(M_{h}^{2},m_{\mu}^{2},m_{\mu}^{2}).$ (45)

At one-loop order this expression can be written explicitly as the tree-level
value plus the virtual corrections $\zeta^{1l}$ from the diagrams depicted in
Fig. 1 and the counterterm contribution generated by the Lagrangian Eq. (24).

$\displaystyle\lambda^{\text{tree}+1l}_{\mu\mu}=Y^{h}_{22}(1-\alpha
B)+\sqrt{2}\zeta^{1l}(M_{h}^{2},m_{\mu}^{2},m_{\mu}^{2})+\delta
Y^{h}_{22}+\tfrac{1}{2}(\delta Z_{L}^{\dagger}Y^{h}+Y^{h}\delta Z_{R}+\delta
Z_{h}Y^{h})_{22}.$ (46)

To compute the numerical results, the amplitudes for the diagrams in Fig. 1
were obtained using FeynArts [67] and FeynCalc [68, 69, 70, 71] and the
appearing loop function were evaluated using LoopTools [72]. In LoopTools IR
divergent loop functions are automatically regulated using a finite photon
mass. We have checked that our results are independent of the artificial
photon mass within the numerical precision.

Importantly, we have also confirmed numerically that the UV divergences in Eq.
(46) cancel. This cancellation is non-trivial since the renormalization
constants are defined using a combination of self-energy diagrams and explicit
${1}/{\epsilon}$ poles for the $\overline{\text{MS}}$ counterterms derived
from the modified $\beta$-functions in App. B, while the UV divergences of
$\zeta^{1l}$ are obtained from the genuine 3-point loop diagrams in Fig. 1.
The cancellation thus constitutes a significant check of the renormalization
scheme.

###  Cancellation of the Renormalization Scale Dependence

Figure 2: Example of cancellation between the explicit scale dependence of the
one-loop contributions to the effective muon–Higgs coupling
$\lambda_{\mu\mu}^{1l}(\mu)$ and the (linearized) running of the tree-level
effective coupling $Y^{h}_{22}(\mu)$.

A second important check for the validity of the scheme is the cancellation of
the renormalization scale dependence in physical observables. As discussed in
Sec. 3.3, due to the combination of OS and $\overline{\text{MS}}$ conditions,
the fundamental model parameters retain a residual renormalization scale
$\mu$-dependence. At the same time the one-loop contributions to the effective
Higgs coupling in Eq. (45) depend on the renormalization scale $\mu$
explicitly. In total, the explicit and implicit $\mu$-dependences must cancel
exactly at the one-loop level.

The implicit $\mu$-dependence is governed by the modified renormalization
group equation Eq. (38) which can, in principle, be solved to obtain the one-
loop running of the fundamental gauge-eigenstate parameters $Y_{ab}(\mu)$ and
$m_{E}(\mu)$. To check the cancellation of the scale dependence in the
physical observable, i.e. in Eq. (45), we are interested in the effect of this
running on the effective coupling matrix $Y^{h}_{22}(\mu)$ in the mass-
eigenstate basis. By construction, the diagonalized lepton mass matrices
represent the physical pole masses and are independent of $\mu$. However, the
effective Yukawa coupling matrix inherits a scale dependence from both the
running couplings and the scale dependence of the rotation matrices $U_{L/R}$.
In general, solving the RGE and switching to the mass basis results in a
complicated non-linear dependence of the coupling matrices on the
renormalization scale. Since the cancellation between the implicit and the
explicit $\mu$-dependence can only be seen up to the considered loop order, it
is useful to truncate the higher-order terms analytically. Working at one-loop
order this means we should only consider the linear dependence on $\ln(\mu)$
in both the running couplings and loop corrections. In this approximation, the
renormalization scale dependence of the fundamental parameters can be obtained
directly from Eq. (38) by integration

$\displaystyle\Theta_{\alpha}(\mu)\approx\Theta_{\alpha}(\mu_{0})+\beta^{\text{eff}}(\Theta_{\alpha}(\mu_{0}))\ln(\tfrac{\mu}{\mu_{0}})+\mathcal{O}(2L).$
(47)

From this, the linearized running of the effective couplings in the mass basis
can be obtained analytically using the perturbative expressions of the
diagonalization matrices given in Ref. [2]

$\displaystyle Y^{h}_{22}(\mu)\approx
Y^{h}_{22}(\mu_{0})+\frac{2v^{2}\bar{\lambda}\lambda^{E}_{2}\lambda^{L}_{2}}{m_{4}m_{5}}\bigg{(}\frac{\beta^{\text{eff}}(\bar{\lambda})}{\bar{\lambda}}+\frac{\beta^{\text{eff}}(\lambda^{E}_{2})}{\lambda^{E}_{2}}+\frac{\beta^{\text{eff}}(\lambda^{L}_{2})}{\lambda^{L}_{2}}\bigg{)}\ln(\tfrac{\mu}{\mu_{0}})+\mathcal{O}(v^{4}).$
(48)

Fig. 2 shows an example of how this linearized running of the tree-level
coupling exactly cancels the explicit one-loop logarithms appearing in the
renormalized (IR-subtracted) vertex corrections.

###  Qualitative Discussion and Muon $g-2$

Having confirmed the validity of the renormalization scheme, we move on to
discuss the qualitative impact of the one-loop corrections. In particular we
focus on the correlation between $\lambda_{\mu\mu}$ and the VLL contributions
to the muon anomalous magnetic moment $\Delta a_{\mu}^{\text{VLL}}$. For the
calculation of $\Delta a_{\mu}^{\text{VLL}}$, our implementation agrees with
the formula presented e.g. in Ref. [2]. For simplicity, we summarize here only
the main behaviour, see Eq. (39) in Ref. [2],

$\displaystyle\Delta
a_{\mu}^{\text{VLL}}\approx-c(m_{L,E})\frac{\bar{\lambda}\lambda^{E}_{2}\lambda^{L}_{2}v^{3}}{m_{L}m_{E}m_{\mu}}\times
22.5\times 10^{-10}\,,$ (49)

where in our convention the coefficient $c(m_{L,E})$ is approximately $+1$
$(-1)$ for heavy (EW-scale) VLL masses. Unlike in most other models where the
chirally enhanced contributions to the effective Higgs coupling and $a_{\mu}$
arise simultaneously at the one-loop level (cf. Ref. [16]), in the VLL model,
new contributions to $m_{\mu}$ and to $\lambda_{\mu\mu}$ are already generated
at tree-level by the lepton mixing [1, 2, 3, 17]. Hence, there is a very
peculiar correlation between the effective Higgs coupling $\lambda_{\mu\mu}$
and $\Delta a_{\mu}^{\text{VLL}}$. For heavy VLL masses $m_{L,E}\gg v$ it can
be written as

$\displaystyle\bigg{|}\frac{\lambda_{\mu\mu}}{\lambda_{\mu\mu}^{\text{SM}}}\bigg{|}^{\text{tree}}\approx\quad\bigg{|}1-\frac{2\Delta
a_{\mu}^{\text{VLL}}}{22.5\times 10^{-10}}\bigg{|}.$ (50)

This equation is the origin of Eq. (1) quoted in the Introduction.

$\displaystyle\leavevmode\hbox to0pt{\vbox
to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}} \feynman[small] \vertex(m1) at
(-2,0) {$\mu_{R}$}; \vertex(m2) at (2,0) {$\mu_{L}$}; \vertex(mh1) at (-1,0);
\vertex(mh2) at (0,0); \vertex(mh3) at (1,0); \vertex(h1) at (-1,1) {$H$};
\vertex(h2) at (0,1) {$H$}; \vertex(h3) at (1,1) {$H$}; \diagram*{ (m1) --
[fermion] (mh1) -- [fermion, thick, edge label'=$E_{L}$] (mh2) -- [fermion,
thick, edge label'=$L_{R}$] (mh3) -- [fermion] (m2), (mh1) -- [scalar] (h1),
(mh2) -- [scalar] (h2), (mh3) -- [scalar] (h3), };
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\sim\frac{\lambda^{E}_{2}\bar{\lambda}\lambda^{L}_{2}}{m_{L}m_{E}}\hskip
56.9055pt\leavevmode\hbox to0pt{\vbox
to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}} \feynman[small] \vertex(h) at
(-1.5,0) {$H$}; \vertex(m1) at (1.5,1) {$\mu_{L}$}; \vertex(m2) at (1.5,-1)
{$\mu_{R}$}; \vertex(a) at (-.5,0); \vertex(b) at (.5,.5); \vertex(c) at
(.5,-.5); \diagram*{ (h) -- [scalar] (a) -- [fermion, thick, edge
label=$L_{R}$] (b) -- [fermion] (m1), (m2) -- [fermion] (c) -- [fermion,
thick, edge label=$E_{L}$] (a), (c) -- [scalar] (b) };
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\sim\frac{\lambda^{E}_{2}\bar{\lambda}\lambda^{L}_{2}}{16\pi^{2}}$
Figure 3: _Left:_ matching diagram for $C_{\mu H}$ at tree-level. _Right:_
matching diagram for $y_{\mu}^{\text{EFT}}$ at one-loop order. The thick lines
represent the heavy internal VLL that are integrated out.

Previous studies [1, 2] focused in particular on the region of small VLL
masses $m_{L,E}\sim v$ where $c(m_{L,E})\approx-1$ and the sign in the
correlation changes, such that $\Delta a_{\mu}^{\text{VLL}}$ of order of the
Fermilab result [20, 21] would require a large deviation
$|\lambda_{\mu\mu}/\lambda_{\mu\mu}^{\text{SM}}|\sim 3$. After these studies,
LHC constraints on the Higgs coupling have become much more stringent,
restricting $|\lambda_{\mu\mu}/\lambda_{\mu\mu}^{\text{SM}}|\simeq 1$. This
has been used as a motivation for extended VLL models, where the correlation
Eq. (50) is modified e.g. by couplings to additional scalar particles [5, 4,
6]. On the other hand, for large VLL masses the Fermilab result and LHC data
are perfectly compatible with Eq. (50) if
$\lambda_{\mu\mu}/\lambda_{\mu\mu}^{\text{SM}}\simeq-1$. It is therefore of
interest to better understand how one-loop corrections to the effective Higgs
coupling affect the above correlation.

The effective field theory (EFT) approach used in Refs. [1, 2] to study the
correlation due to the lepton mixing also nicely extends to the one-loop
level. After integrating out the VLLs, the resulting Lagrangian contains two
chirality-flipping terms, a SM-like Yukawa interaction and a dimension-6
operator with three Higgs fields,

$\displaystyle\mathcal{L}_{EFT}$
$\displaystyle\supset-y^{\text{EFT}}_{\mu}\bar{l}^{2}_{L}\mu_{R}H-C_{\mu
H}~{}\bar{l}^{2}_{L}\mu_{R}H(H^{\dagger}H)+h.c.\,.$ (51)

Both $y^{\text{EFT}}_{\mu}$ and the Wilson coefficient $C_{\mu H}$ are
determined by matching and admit an expansion into tree-level and loop
contributions (see Fig. 3)

$\displaystyle
y_{\mu}^{\text{EFT}}=y_{\mu}+\mathcal{O}\Big{(}\frac{\bar{\lambda}\lambda^{E}_{2}\lambda^{L}_{2}}{16\pi^{2}}\Big{)},\qquad
C_{\mu
H}=\frac{\bar{\lambda}\lambda^{E}_{2}\lambda^{L}_{2}}{m_{L}m_{E}}\Big{(}1+\mathcal{O}(1L)\Big{)}.$
(52)

Both the tree-level matching contribution to $C_{\mu H}$ as well as the one-
loop matching contribution to $y_{\mu}^{\text{EFT}}$ are chirally enhanced
compared to the fundamental Yukawa coupling $y_{\mu}$, with the difference
that the tree-level contribution is suppressed by $1/(m_{L}m_{E})$ while the
one-loop contribution is suppressed by $1/(16\pi^{2})$. In the EFT, the muon
pole mass $m_{\mu}$ and the physical muon–Higgs coupling $\lambda_{\mu\mu}$
are given by

$\displaystyle m_{\mu}$ $\displaystyle=y_{\mu}^{\text{EFT}}v+~{}~{}C_{\mu
H}v^{3}+\Delta m_{\mu}^{\text{1L,EFT}},$ (53a) $\displaystyle\lambda_{\mu\mu}$
$\displaystyle=y_{\mu}^{\text{EFT}}~{}~{}+3C_{\mu
H}v^{2}+\Delta\lambda_{\mu\mu}^{\text{1L,EFT}}.$ (53b)

The important factor $3$ arises from combinatorics and the last terms
correspond to loop corrections within the EFT. Those EFT loop corrections can
arise e.g. from loops with gauge or Higgs bosons and cannot contain additional
chiral enhancements. Chiral enhancements only appear in the matching
coefficients as discussed above. Interestingly, for VLL masses above the TeV
scale the one-loop corrections entering via $y_{\mu}^{\text{EFT}}$ can be
larger than the tree-level corrections entering via $C_{\mu H}$, which
provides further motivation for a one-loop analysis.

Eliminating $y_{\mu}^{\text{EFT}}$ between the previous two equations, using
Eqs. (49) and (52) and neglecting higher orders leads to the lowest-order
correlation formula (50). Taking into account higher orders, we find that no
new chiral enhancements beyond those present at tree-level are introduced by
the loop corrections and the one-loop corrected correlation for large VLL
masses is of the form222If other dimension-6 operators besides $\bar{l}^{2}\mu
H^{3}$ are included, additional mass-suppressed terms can appear independently
of $\Delta a_{\mu}^{\text{VLL}}$. In our case none of the possible operators
introduce new chiral enhancements and are therefore neglected.

$\displaystyle\bigg{|}\frac{\lambda_{\mu\mu}}{\lambda_{\mu\mu}^{\text{SM}}}\bigg{|}^{\text{tree+1L}}$
$\displaystyle\approx\quad\bigg{|}1-\frac{2\Delta
a_{\mu}^{\text{VLL}}}{22.5\times
10^{-10}}\times\Big{(}1+\mathcal{O}(\text{1L})\Big{)}\bigg{|}.$ (54)

This can also be compared with the analysis of Ref. [16], where generic models
without tree-level mixing were considered, such that $C_{\mu H}$ arises only
at the one-loop level. Going through similar steps, a correlation like Eq.
(54) can be derived, but $\Delta a_{\mu}$ would only appear multiplied with
$\mathcal{O}(\text{1L})$ effects (i.e. the “$1+$” term in the round brackets
would be absent in Eq. (54)).

Figure 4: _Left:_ Plot of $\delta v/v$ as a function of $C_{\mu H}$ for
several fixed values of the VLL masses. _Right:_ Plot of the corresponding
contribution to $\lambda_{\mu\mu}/\lambda_{\mu\mu}^{\text{SM}}$.

Eq. (54) defines our expectation for the behaviour of the explicit one-loop
results discussed below. Despite the absence of new chiral enhancements at
one-loop order, it turns out that the magnitude of the one-loop corrections
can be expected to be large, due to logarithms of the VLL masses and mixing
effects. To provide a qualitative estimate, we mention that one particular
contribution that exhibits both of these effects is the correction due to the
renormalization constants $\delta v$ and $\delta Z_{h}$. These are obtained
from gauge- and Higgs-boson self-energy diagrams, where loops with heavy
fermions typically result in large logarithmically enhanced contributions
which enter the effective Higgs coupling through the Yukawa counterterm and
field renormalization. For example, the leading VLL Yukawa contribution to
$\delta v$ in the limit $m_{L}\simeq m_{E}\gg v$ is given by

$\displaystyle\frac{\delta
v}{v}\Big{|}_{\text{VLL}}\simeq\frac{1}{32\pi^{2}}\Big{(}\lambda^{2}+\bar{\lambda}^{2}+(\lambda^{E}_{2})^{2}+(\lambda^{L}_{2})^{2}\Big{)}\ln(\frac{m_{L}^{2}}{\bar{\mu}^{2}})$
(55)

Similarly, the contribution due to the top quark reads

$\displaystyle\frac{\delta
v}{v}\Big{|}_{m_{t}}\approx\frac{3y_{t}^{2}}{32\pi^{2}}\bigg{[}\ln(\frac{m_{t}^{2}}{\bar{\mu}^{2}})+\frac{1-2s_{W}^{2}}{2s_{W}^{2}}\bigg{]}.$
(56)

Both contributions involve potentially large coefficients and a logarithmic
scale dependence, such that $\delta v/v$ will always result in large
contributions to the effective Higgs coupling, regardless of the choice of
renormalization scale.

If the lepton masses were generated purely by the Higgs mechanism as in the
SM, only the combination $\delta Z_{h}/2-\delta v/v$ would appear in the one-
loop effective Higgs coupling (this combination corresponds to the quantity
studied in Refs. [60, 61]). In this combination the logarithmically enhanced
terms cancel and the remaining correction is at most $\mathcal{O}(1\%)$, even
if heavy new particles are present. However, in the VLL model additional terms
$\propto\delta v$ exist due to the tree-level lepton mixing. This is most
directly seen by applying the transformation $v\to v+\delta v$ and the Higgs
field renormalization in the tree-level result,

$\displaystyle\lambda^{\text{tree}}_{\mu\mu}=\tfrac{m_{\mu}}{v}+2C_{\mu
H}v^{2}\quad\implies\quad\lambda_{\mu\mu}^{\text{1L}}\big{|}_{\delta
v}=\big{(}4C_{\mu H}v^{2}-\tfrac{m_{\mu}}{v}\big{)}\tfrac{\delta
v}{v}+\lambda_{\mu\mu}^{\text{tree}}\tfrac{1}{2}\delta Z_{h}.$ (57)

Simplifying, this results in the following contribution to the one-loop
effective muon–Higgs coupling,

$\displaystyle\lambda^{\text{1L}}_{\mu\mu}\big{|}_{\delta
v}=\lambda_{\mu\mu}^{\text{tree}}(\tfrac{1}{2}\delta Z_{h}-\tfrac{\delta
v}{v})+6C_{\mu H}v^{2}\tfrac{\delta v}{v},$ (58)

where the first term is small but the second term is doubly enhanced by the
prefactor $6$ and the large contributions within $\delta v$. In particular in
the region where $\lambda_{\mu\mu}/\lambda_{\mu\mu}^{\text{SM}}\approx-1$ and
the VLL masses are $\gtrsim 1~{}\text{TeV}$, the contribution of $\delta v/v$
can account for several $\mathcal{O}(10\%)$ corrections to the tree-level
value. This effect is illustrated in Fig. 4 where the value of $\delta v$
(left) and corresponding correction to $\lambda_{\mu\mu}$ (right) are plotted
for several different VLL masses.

Including the contributions from all loop and counterterm diagrams also beyond
$\delta v$, the leading logarithmic one-loop VLL Yukawa contribution to the
effective Higgs coupling is given by

$\displaystyle\frac{\lambda_{\mu\mu}}{\lambda_{\mu\mu}^{\text{SM}}}\Big{|}_{\text{1l,VLL}}\simeq\frac{\lambda^{E}_{2}\lambda^{L}_{2}\bar{\lambda}v^{3}}{32\pi^{2}m_{L}^{2}m_{\mu}}\bigg{[}9\Big{(}\lambda^{2}+\bar{\lambda}^{2}+(\lambda^{E}_{2})^{2}+(\lambda^{L}_{2})^{2}\Big{)}-12\lambda\bar{\lambda}\bigg{]}\ln(\frac{m_{L}^{2}}{M^{2}_{h}}).$
(59)

Note that the terms in the square brackets are always positive and thus lead
to an increase of $|\lambda_{\mu\mu}|$ compared to the tree-level result.

###  Numerical Results

Figure 5: Comparison between the exact result (solid) and fit (dashed) of the
one-loop results for $\lambda_{\mu\mu}/\lambda_{\mu\mu}^{\text{SM}}$ as a
function of $m_{L}$. We have set $m_{E}=m_{L}$ and
$\lambda^{E}_{2}=\lambda^{L}_{2}=0.02\frac{m_{L}}{v}$. The colours correspond
to different values of $\bar{\lambda}$ as indicated, while $\lambda=0$ (left)
and $\lambda=1$ (right).

Finally we present numerical results on the complete one-loop corrections to
$\lambda_{\mu\mu}$ in our renormalization scheme. We find good agreement with
the qualitative discussion above and determine the numerical impact on the
correlation Eq. (54) between $\lambda_{\mu\mu}$ and $\Delta
a_{\mu}^{\text{VLL}}$. We note that the updated analysis of Ref. [73] on the
EW precision constrains stemming from Z-pole observables translates into the
following bounds on $\lambda^{E}_{2}$ and $\lambda^{L}_{2}$

$\displaystyle\bigg{|}\frac{\lambda^{L}_{2}v}{m_{L}}\bigg{|}\lesssim
0.02,\qquad\bigg{(}\frac{\lambda^{E}_{2}v}{m_{E}}\bigg{)}^{2}+1.16\bigg{(}\frac{\lambda^{L}_{2}v}{m_{L}}\bigg{)}^{2}\lesssim
0.00054.$ (60)

In particular, in order to achieve
$\lambda_{\mu\mu}^{\text{tree}}=-\lambda_{\mu\mu}^{\text{SM}}$ these bounds
require $|\bar{\lambda}|\gtrsim 1.5$.333These constraints are potentially
slightly modified by inclusion of the one-loop corrections to the muon–Z
coupling which, however, are beyond the scope of this paper.

Figure 6: Scatter plot of the correlation between $\Delta
a_{\mu}^{\text{VLL}}$ and $m_{L}$ for
$\lambda_{\mu\mu}/\lambda^{\text{SM}}_{\mu\mu}=-1\pm 0.1$. The red points show
the results using the tree-level value of $\lambda_{\mu\mu}$ while the one-
loop results are colour-coded to highlight the dependence on $\bar{\lambda}$.
The greyed-out points violate the EW constraints Eq. (60). The data was
obtained from a random scan with $m_{L}=m_{E}\in[250,10000]$ GeV and
$\lambda=0$ (left), $\lambda=1$ (right).

We have performed a grid and a random scan for $m_{L}=m_{E}\in[250,10000]$
GeV, $\bar{\lambda}\in[-4,0]$, $\lambda\in[-1,1]$ and $\lambda^{L/E}_{2}$ in
the range allowed by the above constraints. Using the grid scan data together
with an informed ansatz following the discussion in the previous section, we
obtain the following approximate numerical formula for the one-loop corrected
normalized effective Higgs-coupling

$\displaystyle\begin{split}\frac{\lambda_{\mu\mu}}{\lambda^{SM}_{\mu\mu}}\approx
1+\frac{\lambda^{E}_{2}\lambda^{L}_{2}\bar{\lambda}v^{3}}{m_{L}^{2}m_{\mu}}&\bigg{\\{}2+0.056-0.032\ln(\tfrac{m^{2}_{L}}{M^{2}_{h}})+\frac{v^{2}}{m_{L}^{2}}\Big{[}4\bar{\lambda}\lambda-(\lambda^{E}_{2})^{2}-(\lambda^{L}_{2})^{2}\Big{]}\\\
&+0.028\Big{[}\bar{\lambda}^{2}+\lambda^{2}+(\lambda^{L}_{2})^{2}+(\lambda^{E}_{2})^{2}\Big{]}\ln(\tfrac{m_{L}^{2}}{M_{h}^{2}})-0.038\lambda\bar{\lambda}\ln(\tfrac{m_{L}^{2}}{M_{h}^{2}})\\\
&-0.035\Big{[}\bar{\lambda}^{2}+\lambda^{2}+(\lambda^{L}_{2})^{2}\Big{]}+0.033(\lambda^{E}_{2})^{2}+0.013\lambda\bar{\lambda}\bigg{\\}}.\end{split}$
(61)

The “$2$” in the brackets corresponds to lowest order, all other terms are
one-loop corrections. The logarithmically enhanced terms correspond to the
analytical result (59). In the fit we have also included
$\mathcal{O}(v^{2}/m_{L}^{2})$ corrections corresponding to the tree-level
expansion of $Y^{h}_{22}$ to improve the convergence at small VLL masses. In
Fig. 5 we have plotted a comparison between the numerical fit (dashed) and the
results obtained from the full loop calculation and tree-level couplings from
the exact diagonalization (solid) as a function of $m_{L}$. The plot shows
good agreement at large VLL masses $\gtrsim 1000$ GeV, but a deviation at
small masses (in particular for $\lambda\neq 0$) which can be attributed to
the missing terms of even higher orders in $v/m_{L}$.

Finally, in Fig. 6 we have plotted points from the random scan in the
$m_{L}$–$\Delta a_{\mu}^{\text{VLL}}$ plane, for which
$\lambda_{\mu\mu}/\lambda_{\mu\mu}^{\text{SM}}=-1\pm 0.1$. The red points show
the results using $\lambda_{\mu\mu}^{\text{tree}}$ and thus reflect the
correlation Eq. (50) leading to the asymptotic value $\Delta
a_{\mu}^{\text{VLL}}\approx 22.5\times 10^{-10}$. In contrast, including the
one-loop results (color-coded according to the value of $\bar{\lambda}$) leads
to a significantly lower asymptotic value of $\Delta a_{\mu}^{\text{VLL}}$.
This can be understood as a consequence of the logarithmic enhancement of the
one-loop corrections. According to Eq. (61), in combination the logarithmic
terms can easily increase the effective Higgs coupling by $10$–$50\%$ compared
to the tree-level value, which in turn requires smaller couplings to fulfil
$\lambda_{\mu\mu}/\lambda_{\mu\mu}^{\text{SM}}\approx-1$ at one-loop order.
Consequently, with the asymptotic behavior of $\Delta a_{\mu}^{\text{VLL}}$
determined by Eq. (49), the one-loop corrections to the correlation result in
an overall decrease of the predicted value of the deviation.

##  5 Conclusion

In models with vector-like leptons, the lepton mass generation mechanism can
be strongly modified. Similar to the neutrino seesaw mechanism, SM-like lepton
masses can receive significant contributions from mixing with VLL with
fundamental gauge-invariant Dirac mass terms. The modified mass generation
mechanism is accompanied by potentially large effects in the effective
lepton–Higgs couplings and in the anomalous magnetic moment $a_{\mu}$, which
are strongly correlated in a peculiar way.

Here we have started an analysis of higher-order effects in a VLL model where
one VLL generation can mix with all SM leptons via additional Yukawa
couplings. As the first theoretical step we have developed a renormalization
scheme. Our scheme addresses the mixing between the SM and vector-like leptons
and implements full on-shell conditions on the lepton self-energies. It thus
establishes a direct physical interpretation of mass parameters and fields and
allows a transparent computation of observables. The remaining freedom is
fixed by imposing $\overline{\text{MS}}$ conditions on a judiciously chosen
set of fundamental parameters of the gauge-eigenstate Lagrangian. Here we use
the redundancy Eq. (4) between parameters and fields to minimize the number of
such parameters.

Because of the combination of on-shell and $\overline{\text{MS}}$ conditions,
there is a residual renormalization-scale dependence of the fundamental model
parameters. The corresponding $\beta$ functions differ from standard ones
because of the finite parts of counterterms and because of the use of the
redundancy. Both changes have been evaluated. The cancellation of divergences
as well as the cancellation of the renormalization-scale dependence in the
computation of observables provides a strong check of the scheme and its
implementation.

As the first phenomenological application we have calculated the one-loop
corrections to the effective muon-Higgs coupling $\lambda_{\mu\mu}$ and its
correlation with the VLL contribution to the muon magnetic moment $\Delta
a_{\mu}^{\text{VLL}}$. We focused in particular on the interesting region
$\lambda_{\mu\mu}\simeq-\lambda_{\mu\mu}^{\text{SM}}$, where the tree-level
correlation predicts $\Delta a_{\mu}^{\text{VLL}}$ roughly of order of the
current tentative experimental deviation. We found that, despite not
introducing new chiral enhancements, the one-loop corrections can
significantly modify the results found at tree-level and in particular lead to
a smaller predicted value of $\Delta a_{\mu}^{\text{VLL}}$. This is mainly due
to the logarithmic enhancements present in the one-loop results for
$\lambda_{\mu\mu}$, together with the large couplings required to achieve
$\lambda_{\mu\mu}\simeq-\lambda_{\mu\mu}^{\text{SM}}$.

Quantitatively, the lowest-order correlation Eq. (1) predicts a rigidly fixed,
large value for $\Delta a_{\mu}^{\text{VLL}}$ of around $22.5\times 10^{-10}$,
but the loop-corrected correlation is incompatible with such large values of
$\Delta a_{\mu}^{\text{VLL}}$ and predicts instead a range between around
$(10\ldots 18)\times 10^{-10}$ for masses in the multi-TeV region. It will be
interesting to compare this range to future updates of the Fermilab $a_{\mu}$
measurement and the SM theory prediction.

As a further outlook, given the large corrections found here it will be
motivated to explore the phenomenological impact of one-loop corrections and
the properties of the renormalization scheme in a broader range of
observables. The renormalization scheme can be used to evaluate further
observables in the lepton/Higgs/EW sector at the loop level, for example to
obtain more accurate EW constraints on the model from Z-pole measurements.
Moreover, the scheme is not restricted to the flavour-conserving case but also
allows to compute e.g. lepton-flavour violating Higgs-, Z-, or muon decays.

## Acknowledgments

We acknowledge financial support by the German Research Foundation (DFG) under
grant number STO 876/7-2.

##  Appendix A Feynman Rules

Here we list Feynman rules of the VLL model in the mass-eigenstate basis
defined in Eq. (7). The SM vertices are taken from Ref. [74]. For the gauge
sector we adapt the sign convention from [59] who define the gauge-boson mass
eigenstates by

$\displaystyle\begin{pmatrix}~{}W^{+}_{\mu}~{}\\\
W^{-}_{\mu}\end{pmatrix}=\begin{pmatrix}\frac{1}{\sqrt{2}}&-\frac{i}{\sqrt{2}}\\\
\frac{1}{\sqrt{2}}&\hphantom{-}\frac{i}{\sqrt{2}}\end{pmatrix}\begin{pmatrix}~{}W^{1}_{\mu}~{}\\\
W^{2}_{\mu}\end{pmatrix},\qquad\begin{pmatrix}~{}A_{\mu}~{}\\\
Z_{\mu}\end{pmatrix}=\begin{pmatrix}c_{W}&-s_{W}\\\
s_{W}&c_{W}\end{pmatrix}\begin{pmatrix}~{}B_{\mu}~{}\\\
W^{3}_{\mu}\end{pmatrix},$ (62)

and the covariant derivative as

$\displaystyle
D_{\mu}=\partial_{\mu}-i\tfrac{g_{W}}{\sqrt{2}}\left(t^{+}W^{+}_{\mu}+t^{-}W^{-}_{\mu}\right)-i\tfrac{g_{W}}{c_{W}}(t^{3}-Qs_{W}^{2})Z_{\mu}+ieQA_{\mu},$
(63)

where $t^{\pm}=(t^{1}\pm it^{2})$ (for doublets,
$t^{i}=\frac{\sigma^{i}}{2}$), $Q=t^{3}+Y$ and $e=g_{Y}c_{W}=g_{W}s_{W}$.

### Gauge Interactions

$\leavevmode\hbox to0pt{\vbox to0pt{\pgfpicture\makeatletter\hbox{\hskip
0.0pt\lower 0.0pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}} \feynman[small] \vertex(a) at
(0,0); \vertex(q) at (1.75, 1) {$f_{a}$}; \vertex(v1) at (-1.75,0)
{$V_{\mu}$}; \vertex(v2) at (1.75,-1) {$\bar{f}_{b}$}; \diagram* { (v1) --
[photon] (a) -- [anti fermion] (v2), (a) -- [fermion] (q) };
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}=i\gamma_{\mu}\Big{(}C_{R}\text{P}_{\text{R}}+C_{L}\text{P}_{\text{L}}\Big{)}_{ab}$

$V\bar{f}f$ | $A\,\bar{\hat{e}}^{b}\hat{e}^{a}$ | $Z\,\bar{\hat{e}}^{b}\hat{e}^{a}$ | $Z\,\bar{\hat{\nu}}^{b}\hat{\nu}^{a}$ | $W\bar{\hat{e}}^{b}\hat{\nu}^{a}$ | $W\bar{\hat{\nu}}^{b}\hat{e}^{a}$
---|---|---|---|---|---
$C^{V}_{R}$ | $e\delta_{ab}$ | $(g^{Z}_{R})_{ab}$ | $(g^{Z\nu}_{R})_{ab}$ | $(g^{W}_{R})_{ab}$ | $(g^{W}_{R})^{*}_{ba}$
$C^{V}_{L}$ | $e\delta_{ab}$ | $(g^{Z}_{L})_{ab}$ | $(g^{Z\nu}_{L})_{ab}$ | $(g^{W}_{L})_{ab}$ | $(g^{W}_{L})^{*}_{ba}$

Figure 7: Feynman rules for gauge interactions

where

$\displaystyle g^{Z}_{L/R}$
$\displaystyle=\tfrac{e}{c_{W}s_{W}}U_{L/R}^{\dagger}\cdot\big{(}s_{W}^{2}-\tfrac{1}{2}T_{L/R}\big{)}\cdot
U_{L/R},$ (64a) $\displaystyle g^{Z\nu}_{L/R}$
$\displaystyle=\tfrac{e}{c_{W}s_{W}}U^{\nu\dagger}_{L/R}\cdot\big{(}\tfrac{1}{2}T_{L/R}\big{)}\cdot
U^{\nu}_{L/R}$ (64b) $\displaystyle g^{W}_{L/R}$
$\displaystyle=\tfrac{e}{\sqrt{2}s_{W}}U^{\nu\dagger}_{L/R}\cdot\big{(}T_{L/R}\big{)}\cdot
U_{L/R}$ (64c)

with $T_{R}=\text{diag}(0,0,0,0,1)$ and $T_{L}=\text{diag}(1,1,1,1,0)$.

### Yukawa Interactions

$\leavevmode\hbox to0pt{\vbox to0pt{\pgfpicture\makeatletter\hbox{\hskip
0.0pt\lower 0.0pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}} \feynman[small] \vertex(a) at
(0,0); \vertex(q) at (1.75, 1) {$f_{a}$}; \vertex(v1) at (-1.75,0) {$S$};
\vertex(v2) at (1.75,-1) {$\bar{f}_{b}$}; \diagram* { (v1) -- [scalar] (a) --
[anti fermion] (v2), (a) -- [fermion] (q) }; \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}=-\tfrac{i}{\sqrt{2}}\Big{(}C^{S}_{R}\text{P}_{\text{R}}+C^{S}_{L}\text{P}_{\text{L}}\Big{)}_{ab}$

$S\bar{f}f$ | $h\,\bar{\hat{e}}^{b}\hat{e}^{a}$ | $\phi^{0}\,\bar{\hat{e}}^{b}\hat{e}^{a}$ | $\phi^{+}\bar{\hat{e}}^{b}\hat{\nu}^{a}$ | $\phi^{-}\bar{\hat{\nu}}^{b}\hat{e}^{a}$
---|---|---|---|---
$C^{S}_{R}$ | $Y^{h}_{ab}$ | $iY^{\phi^{0}}_{ab}$ | $\sqrt{2}Y^{\phi^{+}}_{ab}$ | $\sqrt{2}Y^{\phi^{-}}_{ab}$
$C^{S}_{L}$ | $Y^{h*}_{ba}$ | $-iY^{\phi^{0}*}_{ba}$ | $\sqrt{2}Y^{\phi^{-}*}_{ba}$ | $\sqrt{2}Y^{\phi^{+}*}_{ba}$

Figure 8: Feynman rules for Yukawa interactions

where

$\displaystyle Y^{h}$ $\displaystyle=U_{L}^{\dagger}$
$\displaystyle\begin{pmatrix}\,Y_{ab}&0~{}\\\
0&\bar{\lambda}~{}\end{pmatrix}U_{R},$ (65a) $\displaystyle Y^{\phi^{0}}$
$\displaystyle=U_{L}^{\dagger}$ $\displaystyle\begin{pmatrix}\,Y_{ab}&~{}0\\\
0&-\bar{\lambda}~{}\end{pmatrix}U_{R},$ (65b) $\displaystyle Y^{\phi^{+}}$
$\displaystyle=U^{\nu\dagger}_{L}$
$\displaystyle\begin{pmatrix}\,Y_{ab}&0~{}\\\ 0&0~{}\end{pmatrix}U_{R},$ (65c)
$\displaystyle Y^{\phi^{-}}$ $\displaystyle=U_{L}^{\dagger}$
$\displaystyle\begin{pmatrix}~{}0&~{}0~{}\\\
~{}0&\bar{\lambda}\end{pmatrix}U_{R}^{\nu}$ (65d)

##  Appendix B Renormalization Group Equations

Here we list the one-loop $\beta$-functions of the model assuming the off-
diagonal counterterms $\delta M_{L}^{i},\delta M_{E}^{i}$ and $\delta y_{ij}$
have been absorbed into $\delta V_{L}$ and $\delta V_{R}$, see Eq. (21). In
general, if we consider a theory with a set of bare parameters
$\Theta^{0}_{\alpha}=\mu^{\xi_{\alpha}\epsilon}(\Theta_{\alpha}+\delta\Theta_{\alpha})$,
where the $\mu$-dependent prefactor is introduced to generate a
$D$-dimensional bare Lagrangian while keeping parameters integer-dimensional,
the $\beta$-functions in the $\overline{\text{MS}}$-scheme are given by

$\displaystyle\beta(\Theta_{\alpha})=-\xi_{\alpha}\delta\Theta^{[1]}_{\alpha}+\sum_{\beta}\frac{\partial\delta\Theta_{\alpha}^{[1]}}{\partial\Theta_{\beta}}\xi_{\beta}\Theta_{\beta}$
(66)

where $\delta\Theta_{\alpha}^{[1]}$ denotes the coefficient of the
$1/\epsilon$ pole in $\delta\Theta_{\alpha}$. Using this relationship and
using the abbreviations $g_{W}=e/s_{W}$, $g_{Y}=e/c_{W}$ and

$\displaystyle\delta\Upsilon=2\bar{\lambda}^{2}+2\lambda^{2}+2\sum_{j}\Big{(}(\lambda_{j}^{E})^{2}+(\lambda_{j}^{L})^{2}+y_{j}^{2}+3(y_{j}^{u})^{2}+3(y_{j}^{d})^{2}\Big{)}-\frac{9}{2}g_{W}^{2}-\frac{15}{2}g_{Y}^{2},$
(67)

the Yukawa $\beta$-functions are given by

$\displaystyle(32\pi^{2})\,\beta(y_{i})=~{}y_{i}\,\bigg{[}\delta\Upsilon+3y_{i}^{2}+2(\lambda^{L}_{i})^{2}+(\lambda^{E}_{i})^{2}\bigg{]}-4\lambda^{E}_{i}\lambda^{L}_{i}\bar{\lambda}\tfrac{m_{E}^{2}+2m_{L}^{2}}{m_{E}m_{L}}$
(68a)
$\displaystyle\begin{split}&(32\pi^{2})\beta(\lambda^{L}_{i})=\lambda^{L}_{i}\bigg{[}d\Upsilon+3(\lambda^{L}_{i})^{2}+4y_{i}^{2}+\lambda\Big{(}\lambda-\bar{\lambda}\tfrac{8m_{L}}{m_{E}}\Big{)}\bigg{]}+2\lambda^{E}_{i}y_{i}\bigg{[}\lambda+\bar{\lambda}\tfrac{2m_{E}}{m_{L}}\bigg{]}\\\
&\qquad+\sum_{k\neq
i}\tfrac{\lambda^{L}_{k}}{y_{i}^{2}-y_{k}^{2}}\bigg{[}\lambda^{L}_{i}\lambda^{L}_{k}\Big{(}5y_{i}^{2}-y_{k}^{2}\Big{)}+2\lambda^{E}_{i}\lambda^{E}_{k}y_{i}y_{k}-4\bar{\lambda}\Big{(}\lambda^{L}_{i}\lambda^{E}_{k}y_{k}+\lambda^{E}_{i}\lambda^{L}_{k}y_{i}\Big{)}\tfrac{m_{E}^{2}+2m_{L}^{2}}{m_{E}m_{L}}\bigg{]}\end{split}$
(68b)
$\displaystyle\begin{split}&(32\pi^{2})\beta(\lambda^{E}_{i})=\lambda^{E}_{i}\bigg{[}d\Upsilon+3(\lambda^{E}_{i})^{2}+5y_{i}^{2}+2\lambda\Big{(}\lambda-\bar{\lambda}\tfrac{2m_{E}}{m_{L}}\Big{)}\bigg{]}+4\lambda^{L}_{i}y_{i}\bigg{[}\lambda+\bar{\lambda}\tfrac{2m_{L}}{m_{E}}\bigg{]}\\\
&\qquad+\sum_{k\neq
i}\tfrac{\lambda^{E}_{k}}{y_{i}^{2}-y_{k}^{2}}\bigg{[}\lambda^{E}_{i}\lambda^{E}_{k}\Big{(}4y_{i}^{2}-2y_{k}^{2}\Big{)}+4\lambda^{L}_{i}\lambda^{L}_{k}y_{i}y_{k}-4\bar{\lambda}\Big{(}\lambda^{L}_{i}\lambda^{E}_{k}y_{i}+\lambda^{E}_{i}\lambda^{L}_{k}y_{k}\Big{)}\tfrac{m_{E}^{2}+2m_{L}^{2}}{m_{E}m_{L}}\bigg{]}\end{split}$
(68c)
$\displaystyle\begin{split}&(32\pi^{2})~{}\beta(\lambda)=\lambda~{}\bigg{[}\delta\Upsilon+3\lambda^{2}+\sum_{i}\Big{(}4(\lambda^{E}_{i})^{2}+5(\lambda^{L}_{i})^{2}\Big{)}\bigg{]}+\sum_{i}\bigg{[}6y_{i}\lambda^{E}_{i}\lambda^{L}_{i}+4\bar{\lambda}\tfrac{m_{E}^{2}(\lambda^{E}_{i})^{2}+2m_{L}^{2}(\lambda^{L}_{i})^{2}}{m_{E}m_{L}}\bigg{]}\end{split}$
(68d)
$\displaystyle(32\pi^{2})~{}\beta(\bar{\lambda})=\bar{\lambda}~{}\bigg{[}\delta\Upsilon+3\bar{\lambda}^{2}\bigg{]}.$
(68e)

Similarly, the $\beta$-functions of the Dirac masses are given by

$\displaystyle(32\pi^{2})\beta(m_{L})$ $\displaystyle=4\bar{\lambda}\lambda
m_{E}+\Big{(}\lambda^{2}+\bar{\lambda}^{2}+\sum_{j}(\lambda^{L}_{j})^{2}-9g_{W}^{2}-3g_{Y}^{2}\Big{)}m_{L}$
(69a) $\displaystyle(16\pi^{2})\beta(m_{E})$
$\displaystyle=4\bar{\lambda}\lambda
m_{L}+\Big{(}\lambda^{2}+\bar{\lambda}^{2}+\sum_{j}(\lambda^{E}_{j})^{2}-6g_{Y}^{2}\Big{)}m_{E}.$
(69b)

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