# Global Symmetries and Effective Potential of 2HDM
in Orbit Space

Qing-Hong Cao<EMAIL_ADDRESS>School of Physics, Peking University,
Beijing 100871, China  Center for High Energy Physics, Peking University,
Beijing 100871, China  Kun Cheng<EMAIL_ADDRESS>School of Physics,
Peking University, Beijing 100871, China  Changlong Xu<EMAIL_ADDRESS>School of Physics, Peking University, Beijing 100871, China

###### Abstract

We extend the framework of analyzing the 2HDM in its orbit space to study the
one-loop effective potential before and after electroweak symmetry breaking.
In this framework, we present a comprehensive analysis of global symmetries of
the one-loop thermal effective potential in the 2HDM, demonstrating when the
global symmetries of the tree-level 2HDM potential are broken by loop
contributions. By introducing light-cone coordinates and generalizing the
bilinear notation around the vacuum, we present a geometric view of the scalar
mass matrix and on-shell renormalization conditions.

##  I Introduction

The Two-Higgs-Doublet Model (2HDM) is a simple extension of the SM Lee (1973).
It has received much attention for its potential to provide new sources of CP
violation and strong first-order phase transition Botella and Silva (1995);
Branco  _et al._ (2005); Gunion and Haber (2005); Trautner (2019); Cline  _et
al._ (2011); Basler  _et al._ (2017, 2020); Ferreira  _et al._ (2020); Branco
_et al._ (2012); Maniatis  _et al._ (2008). The most general tree-level 2HDM
scalar potential

$\begin{split}V=&m_{11}^{2}\Phi_{1}^{\dagger}\Phi_{1}+m_{22}^{2}\Phi_{2}^{\dagger}\Phi_{2}-\left(m_{12}^{2}\Phi_{1}^{\dagger}\Phi_{2}+h.c.\right)\\\
&+\frac{1}{2}\lambda_{1}\left(\Phi_{1}^{\dagger}\Phi_{1}\right)^{2}+\frac{1}{2}\lambda_{2}\left(\Phi_{2}^{\dagger}\Phi_{2}\right)^{2}+\lambda_{3}(\Phi_{2}^{\dagger}\Phi_{2})(\Phi_{1}^{\dagger}\Phi_{1})+\lambda_{4}(\Phi_{1}^{\dagger}\Phi_{2})(\Phi_{2}^{\dagger}\Phi_{1})\\\
&+\Big{(}\frac{1}{2}\lambda_{5}\left(\Phi_{1}^{\dagger}\Phi_{2}\right)^{2}+\lambda_{6}(\Phi_{1}^{\dagger}\Phi_{1})(\Phi_{1}^{\dagger}\Phi_{2})+\lambda_{7}(\Phi_{2}^{\dagger}\Phi_{2})(\Phi_{1}^{\dagger}\Phi_{2})+h.c.\Big{)}\\\
\end{split}$ (1)

is parameterized by 14 real parameters. Here,
($m_{12}^{2},\lambda_{5},\lambda_{6},\lambda_{7}$) are in general complex
while the others are real.

The CP conserving 2HDM, also called real 2HDM, require all the parameters in
Eq. (1) to be real with respective to a $U(2)_{\Phi}$ basis transformation
$\Phi_{i}^{\prime}=U_{ij}\Phi_{j}$. Due to the field redefinition, the CP
symmetry and other global symmetries of the potential are hard to determine
from the parameters in Eq. (1) directly, and one of the most efficient ways to
analyze these symmetries is to use the bilinear notation Maniatis  _et al._
(2006); Ivanov (2007, 2008); Nishi (2006) of the 2HDM. This method involves
expressing the tree-level 2HDM potential in terms of orbits of $SU(2)_{L}$
gauge transformations, which can be combined to form a four-vector,

$(K_{0},\vec{K})=K^{\mu}=\Phi_{i}^{\dagger}\sigma_{ij}^{\mu}\Phi_{j},\quad(\mu=0,1,2,3).$
(2)

In this notation, the $U(2)_{\Phi}$ basis transformation of the Higgs doublets
corresponds to a $SO(3)_{K}$ rotation of the three space-like components of
this four-vector, while CP transformations correspond to improper rotations in
these three dimensions Maniatis  _et al._ (2008).

The bilinear notation serves as a convenient tool for examining the symmetries
and vacuum conditions of 2HDM. However, its applications are usually
restricted to tree-level potential and global structures. In this work, we
establish a complete framework for discussing the 2HDM potential, by extending
the bilinear notation of the 2HDM to address the properties of physical fields
around the vacuum and one-loop effective potential including renormalization.
Recently, it is shown in Ref. Cao  _et al._ (2023); Sartore  _et al._ (2022)
that the bilinear notation can be extended to Yukawa couplings, making it
possible to express the 2HDM effective potential including fermion loop
contributions in the bilinear notation Cao  _et al._ (2023). With this
approach, we express the effective potential entirely as a function of gauge
orbits, and systematically analyze the possible global symmetries of the
effective potential. We generalize the bilinear notation to discuss physical
fields after electroweak symmetry breaking (EWSB), and provide a geometrical
description of scalar masses based on the light-cone coordinates in the orbit
space. We demonstrate that the scalar mass matrix can be viewed as a distance
matrix between two hyper-surfaces in the orbit space. Additionally, we
translate the renormalization conditions in the field space into a set of
geometrical conditions in the orbit space. Then numerous redundant
renormalization conditions Basler and Mühlleitner (2019) that depend on the
selection of background fields can be avoided. After the on-shell
renormalization, we give a comprehensive effective field theory description of
one-loop 2HDM effective potential for the first time.

In the rest of this paper, we first review the global symmetries of the tree-
level 2HDM potential in the bilinear notation in Section II, and then examine
whether these symmetries are preserved by one-loop corrections in Section III.
We explore the relationship between the orbit space and the field space around
the vacuum after EWSB in Section IV, and we demonstrate how to carry out the
on-shell renormalization in the orbit space in Section V. Finally, we conclude
in Section VI.

##  II Basis invariant description of global symmetry

In this section, we introduce the basis transformations and CP transformations
of the Higgs doublets, and the global symmetries of 2HDM potential. If a 2HDM
potential is invariant under some basis or CP transformations, then it
processes the corresponding symmetries. The bilinear notation is convenient to
discuss these global transformations, because the basis or CP transformations
simply corresponds to proper or improper rotations in the 3-dimensional space-
like part of the orbit space Branco  _et al._ (2012); Maniatis  _et al._
(2008); Ivanov (2007, 2008), and we refer this 3-dimensional subspace as
$\vec{K}$ space in the following.

###  II.1 Global transformations and symmetries in the bilinear notation

We first consider the $U(2)_{\Phi}$ basis transformations $\Phi_{i}\to
U_{ij}\Phi_{j}$. It is straightforward to see from Eq. (2) that an
$SU(2)_{\Phi}$ basis transformation corresponds to a rotation in the
$\vec{K}$-space,

$K_{0}\to K_{0},~{}K_{a}\to R_{ab}(U)K_{b},\quad
R_{ab}(U)=\frac{1}{2}\text{tr}\left[U^{\dagger}\sigma_{a}U\sigma_{b}\right],~{}a,b=1,2,3.$
(3)

Then we consider the CP transformation
$\Phi_{i}(t,\vec{x})\to\Phi_{i}^{*}(t,-\vec{x})$. Because the definition of
the standard CP transformation $\Phi_{i}\to\Phi_{i}^{*}$ will be changed if we
choose another set of basis to describe the scalar fields, e.g.
$\Phi_{i}^{\prime}=U_{ij}\Phi_{j}$, the CP transformations in the 2HDM are
extended as Ecker  _et al._ (1987); Lavoura and Silva (1994); Botella and
Silva (1995); Gunion and Haber (2005)

$\displaystyle{\rm GCP:}\quad\Phi_{i}\to X_{ij}\Phi_{j}^{*}.$ (4)

Here, $X_{ij}$ is an arbitrary unitary matrix, and such CP transformations are
called generalized CP (GCP) transformations. By plugging the GCP
transformation into Eq. (2), we find that $\vec{K}$ transforms in an improper
$O(3)_{K}$ rotation $\bar{R}(X)$.

$K_{0}\to K_{0},~{}K_{a}\to\bar{R}_{ab}(X)K_{b},\quad\bar{R}(X)\equiv
R(X)\operatorname{diag}(1,-1,1).$ (5)

Here, the $R(X)$ is defined in Eq. (3). Besides, for any GCP transformation,
one can always find a basis $\Phi_{i}$ so that $X_{ij}$ is a real rotation
matrix Ecker  _et al._ (1987). Therefore GCP transformations are often
classified into three cases Ferreira  _et al._ (2009, 2011):

$\displaystyle{\rm CP1:}$
$\displaystyle~{}\Phi_{1}\to\Phi_{1}^{*},~{}\Phi_{2}\to\Phi_{2}^{*},$ (6)
$\displaystyle{\rm CP2:}$
$\displaystyle~{}\Phi_{1}\to\Phi_{2}^{*},~{}\Phi_{2}\to-\Phi_{1}^{*},$ (7)
$\displaystyle{\rm CP3:}$
$\displaystyle~{}\begin{cases}\Phi_{1}\to\Phi_{1}^{*}\cos{\theta}+\Phi_{2}^{*}\sin{\theta}\\\
\Phi_{2}\to-\Phi_{1}^{*}\sin{\theta}+\Phi_{2}^{*}\cos{\theta}\end{cases},\quad
0<\theta<\pi/2,$ (8)

where CP3 is the most general CP transformation while CP1 and CP2 are some
special case with ${\rm CP1}^{2}={\rm CP2}^{2}=\mathbb{1}$, with respect to a
global sign.

After showing that the basis and GCP transformations correspond to $O(3)_{K}$
rotations in the $\vec{K}$-space, we examine the symmetries conserving
conditions on the 2HDM potential. The 2HDM potential in Eq. (1) can be written
as a function of gauge orbits Maniatis  _et al._ (2006); Ivanov (2007, 2008);
Nishi (2006),

$\begin{split}V&=\xi_{\mu}K^{\mu}+\eta_{\mu\nu}K^{\mu}K^{\nu}\\\
&=\xi_{0}K_{0}+\eta_{00}K_{0}^{2}+\vec{\xi}\cdot\vec{K}+2K_{0}\vec{\eta}\cdot\vec{K}+\vec{K}^{T}E\vec{K}.\end{split}$
(9)

Here, $\vec{\xi}$ parametrizes the scalar quadratic couplings while $E$ and
$\vec{\eta}$ parametrize the scalar quartic couplings. As discussed above, a
GCP or basis transformation corresponds to some (improper) rotation $R$ in the
$\vec{K}$-space. If a tree-level potential is invariant under a rotation $R$,
i.e., $V(K_{0},\vec{K})=V(K_{0},R\vec{K})$, its parameters should be invariant
under the rotation $R$,

$\vec{\xi}=R\vec{\xi},\quad\vec{\eta}=R\vec{\eta},\quad E=RER^{T}.$ (10)

A complete analysis of the symmetries of the 2HDM Lagrangian must include the
Yukawa interaction, which reads as

$-\mathcal{L}_{\rm
Yuk}=\bar{Q}_{L}y_{u,i}\tilde{\Phi}_{i}u_{R}+\bar{Q}_{L}y_{d,i}\Phi_{i}d_{R}+h.c.~{},\quad
i=1,2,$ (11)

for one generation of $u$-quark and $d$-quark. Note that $y_{d,i}$ and
$y_{u,i}^{*}$ transform like $\Phi_{i}$ under the $SU(2)_{\Phi}$ basis
redefinition. Although the Yukawa coupling terms in the Lagrangian cannot be
expressed in bilinear notation directly, it is shown that the bilinear
notation can still be extended to discuss whether the Yukawa couplings break
the global symmetries of the scalar potential Cao  _et al._ (2023). This is
done by projecting the Yukawa couplings into orbit space, and defining
covariant vectors in the dual space of $K^{\mu}$ in terms of the Yukawa
couplings as

$Y_{u}^{\mu}=y_{u,i}^{*}\sigma^{\mu}_{ij}y_{u,j},\quad
Y_{d}^{\mu}=y_{d,i}\sigma^{\mu}_{ij}y_{d,j}^{*}.$ (12)

In order to make sure that $\mathcal{L}_{\rm Yuk}$ is invariant under the
basis transformation $\Phi_{i}\to U_{ij}\Phi_{j}$ or the CP transformation in
the scalar sector $\Phi_{i}\to X_{ij}\Phi_{j}^{*}$, the vector $\vec{Y}$
projected by the Yukawa couplings should satisfy

$\vec{Y}=R(U)\vec{Y}\quad{\rm or}\quad\vec{Y}=\bar{R}(X)\vec{Y}.$ (13)

###  II.2 Examples of global symmetries

Next we show how to discuss some special symmetries that is widely considered
in the orbit space. We start with two characteristic examples, the CP1
symmetry and the $Z_{2}$ symmetry. The CP symmetry is often introduced to the
potential because large CP violations are prohibited by experiments. From Eq.
(5), the CP1 transformation $\Phi_{i}\to\Phi_{i}^{*}$ corresponds to a mirror
reflection in the $\vec{K}$-space. The $Z_{2}$ symmetry is introduced to
prevent the flavor changing neutral interactions,111For different types of
$Z_{2}$ charge assignments in the orbit space, see Table 2 in Appendix A.2.
and a softly broken $Z_{2}$ symmetry is often considered. From Eq. 3, the
$Z_{2}$ transformation $\Phi_{1}\to-\Phi_{1}$ corresponds to a 2-dimensional
rotation of $\pi$ in the $\vec{K}$ space.

Figure 1: Illustration figure of parameter vectors and tensor of the 2HDM
potential that obeys CP1 symmetry (left) or softly broken $Z_{2}$ symmetry
(right). The ellipsoid denotes the tensor $E$ and black dashed lines denote
its three principal axes. Red and blue arrows denote the directions of
$\vec{\eta}$ and $\vec{\xi}$ respectively.

Whether a 2HDM is invariant under the CP1 or $Z_{2}$ transformation can be
understood from the geometrical profile of parameter tensor and vectors, as
shown in Eqs. (10) and (13). Without loss of generality, we use an ellipsoid
to visualize the $3\times 3$ real symmetric tensor $E$ which possesses at
least three $C_{2}$ axis (principal axis) and three symmetry planes, and we
illustrate these two examples in Fig. 1. The CP1 symmetric 2HDM potential
satisfies a mirror reflection symmetry in the $\vec{K}$ space, requiring all
the parameter vectors to lie on the same reflection plane of $E$. The $Z_{2}$
symmetric potential is invariant under a rotation of $\pi$ in the $\vec{K}$
space. Hence the parameter vectors should point to the same principal axis of
$E$. As for the softly broken $Z_{2}$ symmetry, the quadratic term $\vec{\xi}$
is allowed to break the $Z_{2}$ symmetry as in Fig. 1.

Following Refs. Ferreira  _et al._ (2009, 2011), we list other global
symmetries in scalar family space by different geometric profile of the scalar
potential in Table 1. The $U(1)_{a}$ transformation $\Phi_{1}\to
e^{i\theta}\Phi_{1}$ corresponds to a rotation along a certain axis, the CP2
transformation corresponds to a point reflection, and the CP3 transformation
corresponds to a point reflection followed by an additional rotation of
$0\sim\pi$. The geometric profiles show the hierarchy chain of those global
symmetries clearly,

${\rm CP1}<Z_{2}<\left\\{\begin{matrix}{\rm CP2}\\\
U(1)_{a}\end{matrix}\right\\}<{\rm CP3}<U(2),$ (14)

i.e., a $Z_{2}$ symmetric tree-level 2HDM scalar potential must satisfy CP1
symmetry and likewise. For GCP properties of tree-level 2HDM scalar potential,
CP2 and CP3 symmetric conditions are more strict than CP1.222This situation is
different in $N$-Higgs doublet model for $N>2$. A potential that does not
conserve CP1 symmetry may satisfy some higher order GCP symmetries Ivanov and
Silva (2016); Ivanov  _et al._ (2019a). For detailed discussion of high order
GCP symmetries, see Ref. Ivanov (2017); Ivanov and Laletin (2018); Ivanov  _et
al._ (2019a, b) Besides, neither CP2 nor CP3 symmetry can be still preserved
after the Higgs field developed a non-vanishing vacuum expectation value.
Therefore, we will only discuss CP1 conserving (CPC) conditions and denote CP1
as CP in the following.

Table 1: Global symmetry and their geometric profile of parameter vectors and tensor $E$. $\vec{e}_{i}$ are direction of three eigenvectors of $E$, and $e_{i}$ denotes the three corresponding eigenvalues.  Symmetry | Transformation | Vector $\vec{\eta}$, $\vec{\xi}$ and $\vec{Y}$ | Tensor
---|---|---|---
U(2) | $\Phi_{i}\to U_{ij}\Phi_{j}$ | 0 | spherical
CP3 | Eq. (8) | 0 | $e_{1}=e_{2}$
CP2 | Eq. (7) | 0 | -
$U(1)_{a}$ | $\Phi_{1}\to e^{i\theta}\Phi_{1}$ | collinear with $\vec{e}_{3}$ | $e_{1}=e_{2}$
$Z_{2}$ | $\Phi_{1,2}\to\pm\Phi_{1,2}$ | collinear with an axes $\vec{e}_{i}$ | -
CP1 | $\Phi_{i}\to\Phi_{i}^{*}$ | orthogonal to an axes $\vec{e}_{i}$ | -

##  III Effective potential and thermal correction

The global symmetries of thermal effective potential are important in the
study of the vacuum structure and CP violation in the early universe. The use
of bilinear notation simplifies, from a geometrical perspective, the analysis
of the global symmetries of the effective potential. In this section, we
employ the bilinear notation to evaluate the effective potential and discuss
its global symmetries.

The thermal effective potential of the 2HDM is written as

$V_{\rm eff}(T)=V_{\rm tree}+V_{\rm CW}+V_{T}+V_{\rm daisy},$ (15)

where $V_{\rm tree}$ is the tree-level potential, $V_{\rm CW}$ is the one-loop
Coleman-Weinberg potential at zero temperature Coleman and Weinberg (1973),
and $V_{T}+V_{\rm daisy}$ are the thermal corrections at finite temperature.
Using the background field method, the one-loop Coleman-Weinberg potential
calculated in Landau gauge under the $\overline{\rm MS}$ scheme is

$\displaystyle V_{\rm CW}(\phi_{c})$
$\displaystyle=\frac{1}{2}\mathbf{Tr}\int\frac{d^{4}p_{E}}{2\pi^{4}}\ln\left[p^{2}_{E}+\mathbf{M}^{2}(\phi_{c})\right]$
(16)
$\displaystyle=\frac{1}{64\pi^{2}}\sum_{i}n_{i}m_{i}^{4}(\phi_{c})\left[\ln\frac{m_{i}^{2}(\phi_{c})}{\mu^{2}}-c_{i}\right].$

Here, $p_{E}=\left(-ip^{0},\vec{p}\right)$, $\mathbf{M}^{2}$ is the mass
matrix of scalar or fermion in the loop and $\mathbf{Tr}$ traces over the
dimension of mass matrix, $m_{i}^{2}$ is the eigenvalue of the
$\mathbf{M}^{2}$ for the field $i$, and $n_{i}$ is the degree of freedom of
the field $i$. The constant $c_{i}$ equals to 5/6 for gauge bosons and 3/2 for
others.

The effective potential of the 2HDM has been extensively studied in the
literature Cline  _et al._ (2011); Basler  _et al._ (2020, 2017); Ferreira
_et al._ (2020); Bernon  _et al._ (2018). Typically, only the neutral or CP-
even components of the Higgs boson doublets are treated as background fields,
which breaks the $SU(2)_{L}$ invariance explicitly. Consequently, the bilinear
notation cannot be applied to study $V_{\rm eff}(\phi_{c})$ directly. In order
to analyze the global symmetries of the effective potential using bilinear
notation, a global $SU(2)_{L}$ invariance must be preserved in the calculation
Cao  _et al._ (2023), which means that the masses in Eq. (16) need to be
evaluated in a $SU(2)_{L}$ invariant way. To achieve this, we treat all the
components of the Higgs boson doublets $\Phi_{i}$’s,

$\Phi_{i}=\begin{pmatrix}\phi_{i\uparrow}\\\
\phi_{i\downarrow}\end{pmatrix},\quad i=1,2,$ (17)

as background fields, and $K^{\mu}$ should be understood as bilinear forms of
background fields in this section.

###  III.1 Symmetries of Coleman-Weinberg potential

We first consider the zero temperature effective potential by calculating the
contributions from gauge boson loop, fermion loop and scalar loop to the
Coleman-Weinberg potential respectively.

#### Contributions from gauge boson loop.

The masses of gauge bosons arise from the kinetic term $|D_{\mu}\Phi_{i}|^{2}$
with
$D^{\mu}\Phi_{i}=(\partial^{\mu}+i\frac{g}{2}\sigma_{a}W_{a}^{\mu}+i\frac{g^{\prime}}{2}B^{\mu})\Phi_{i}$,
where $i=1,2$. Expanding the covariant derivative term directly yields the
gauge boson mass term

$\begin{split}\frac{1}{4}\Phi_{i}^{\dagger}(g\sigma_{a}W_{a}+g^{\prime}B)^{2}\Phi_{i}=&\frac{1}{4}\Phi_{i}^{\dagger}(g^{\prime
2}B^{2}+2gg^{\prime}BW_{a}\sigma_{a}+g^{2}\sigma_{a}\sigma_{b}W_{a}W_{b})\Phi_{i}\\\
=&\frac{1}{4}\Phi_{i}^{\dagger}(g^{\prime
2}B^{2}+2gg^{\prime}BW_{a}\sigma_{a}+g^{2}\sigma_{\\{a}\sigma_{b\\}}W_{a}W_{b})\Phi_{i}\\\
=&\frac{\Phi_{i}^{\dagger}\Phi_{i}}{4}(g^{\prime
2}B^{2}+g^{2}W_{a}W_{a})+\frac{\Phi_{i}^{\dagger}\sigma_{a}\Phi_{i}}{2}gg^{\prime}BW_{a}.\end{split}$
(18)

Then the gauge boson mass matrix in basis $\vec{G}=(W_{1},W_{2},W_{3},B)$ is

$\mathbf{M}^{2}_{G}(\Phi_{i})=\frac{\partial^{2}L}{\partial\vec{G}\partial\vec{G}}=\frac{g^{2}}{4}\begin{pmatrix}\Phi_{i}^{\dagger}\Phi_{i}&0&0&t_{W}\Phi_{i}^{\dagger}\sigma_{1}\Phi_{i}\\\
0&\Phi_{i}^{\dagger}\Phi_{i}&0&t_{W}\Phi_{i}^{\dagger}\sigma_{2}\Phi_{i}\\\
0&0&\Phi_{i}^{\dagger}\Phi_{i}&t_{W}\Phi_{i}^{\dagger}\sigma_{3}\Phi_{i}\\\
t_{W}\Phi_{i}^{\dagger}\sigma_{1}\Phi_{i}&t_{W}\Phi_{i}^{\dagger}\sigma_{2}\Phi_{i}&t_{W}\Phi_{i}^{\dagger}\sigma_{3}\Phi_{i}&t_{W}^{2}\Phi_{i}^{\dagger}\Phi_{i}\end{pmatrix}$
(19)

where $t_{W}=\tan\theta_{W}=g^{\prime}/g$. For a matrix with the shape of Eq.
(19), its eigenvalues are

$\text{Eigenvalues}\begin{pmatrix}e&&&a\\\ &e&&b\\\ &&e&c\\\
a&b&c&d\end{pmatrix}=(e,e,\frac{d+e\pm\sqrt{4(a^{2}+b^{2}+c^{2})+(d-e)^{2}}}{2}).$
(20)

With the help of the Fierz identities,

$\displaystyle(\Phi_{i}^{\dagger}\sigma_{a}\Phi_{i})(\Phi_{j}^{\dagger}\sigma_{a}\Phi_{j})$
$\displaystyle=(\Phi_{1}^{\dagger}\Phi_{1}-\Phi_{2}^{\dagger}\Phi_{2})^{2}+4(\Phi_{1}^{\dagger}\Phi_{2})(\Phi_{2}^{\dagger}\Phi_{1})=|\vec{K}|^{2},$
(21)

we present four eigenvalues of the gauge boson mass matrix,

$\displaystyle m^{2}_{W^{\pm}}$ $\displaystyle=\frac{g^{2}}{4}K_{0},$ (22)
$\displaystyle m^{2}_{Z}$
$\displaystyle=\frac{g^{2}}{8}\left((1+t_{W}^{2})K_{0}+\sqrt{4t^{2}_{W}|\vec{K}|^{2}+(t_{W}^{2}-1)^{2}K_{0}^{2}}\right),$
$\displaystyle m^{2}_{\gamma}$
$\displaystyle=\frac{g^{2}}{8}\left((1+t_{W}^{2})K_{0}-\sqrt{4t^{2}_{W}|\vec{K}|^{2}+(t_{W}^{2}-1)^{2}K_{0}^{2}}\right).$

Notice that there is a massless photon when the vacuum is neutral, i.e.,
$K_{0}=|\vec{K}|$. By plugging Eq. (22) into Eq. (16), we find that the gauge
boson loop contributions to the Coleman-Weinberg potential, $V_{\rm
CW}^{(G)}=V_{\rm CW}^{(G)}(K_{0},|\vec{K}|)$, is spherically symmetric and
preserve any rotational symmetry in the $\vec{K}$ space, i.e.,

$V_{\rm CW}^{(G)}(K_{0},\vec{K})=V_{\rm CW}^{(G)}(K_{0},R\vec{K}),\qquad R\in
O(3).$

#### Contributions from the quark loop.

Typically, only the contribution from the heaviest quark needs to be included
in the effective potential. However, we include both the top and bottom quarks
in our calculation to ensure an explicit $SU(2)_{L}$ invariance. The top and
bottom quark masses mix due to the presence of charged background fields, and
the fermion mass matrix given by
$-\partial^{2}\mathcal{L}/\partial\bar{\psi}^{i}_{L}\partial\psi^{j}_{R}$ is

$(\bar{t}_{L},\bar{b}_{L})\mathbf{M}_{F}\begin{pmatrix}t_{R}\\\
b_{R}\end{pmatrix},\quad\mathbf{M}_{F}=\begin{pmatrix}y_{it}\phi_{i\downarrow}^{*}&y_{ib}\phi_{i\uparrow}\\\
-y_{it}\phi_{i\uparrow}^{*}&y_{ib}\phi_{i\downarrow}\end{pmatrix}.$ (23)

We obtain the fermion masses after singular decomposition,

$L^{-1}\mathbf{M}_{F}R=\begin{pmatrix}m_{t}&\\\ &m_{b}\end{pmatrix},\quad
m_{t/b}^{2}=\dfrac{B\pm\sqrt{B^{2}+C}}{2},$ (24)

where, with the help of vector $\vec{Y}$ defined in Eq. (12), $B$ and $C$ can
be written as $SO(3)_{K}$ basis invariant forms as follows:

$\begin{split}B=&\frac{1}{2}(Y_{t0}+Y_{b0})K_{0}+\frac{1}{2}(\vec{Y}_{t}+\vec{Y}_{b})\cdot\vec{K},\\\
C=&-\frac{1}{2}(Y_{t}\cdot
Y_{b})K_{0}^{2}-K_{0}(Y_{t0}\vec{Y}_{b}+Y_{b0}\vec{Y}_{t})\cdot\vec{K}\\\
&+\frac{1}{2}\vec{K}\cdot(\vec{Y}_{t}\cdot\vec{Y}_{b}-Y_{t0}Y_{b0}-\vec{Y}_{t}\otimes\vec{Y}_{b}-\vec{Y}_{b}\otimes\vec{Y}_{t})\cdot\vec{K}.\end{split}$
(25)

The masses can be simplified in the case that the Yukawa couplings exhibit a
large hierarchy; for example, when $y_{t}\gg y_{b}$, only the top quark mass
$m_{t}^{2}(K)=(Y_{t0}K_{0}+\vec{Y}_{t}\cdot\vec{K})/4$ needs to be considered.
Equations (24) and (25) show that the symmetry of $V_{\rm CW}^{(F)}$ is
completely determined by the direction of vector $\vec{Y}$. When the vector
$\vec{Y}$ is invariant under the rotation, i.e.,
$\vec{Y}_{t/b}=R\vec{Y}_{t/b}$ for $R\in O(3)$,

$V_{\rm CW}^{(F)}(K_{0},\vec{K})=V_{\rm CW}^{(F)}(K_{0},R\vec{K}).$

When $\vec{Y}_{t/b}\neq R\vec{Y}_{t/b}$ ,

$V_{\rm CW}^{(F)}(K_{0},\vec{K})\neq V_{\rm CW}^{(F)}(K_{0},R\vec{K}).$

Therefore, whether the fermion loop contribution to $V_{\rm CW}$ breaks the
global symmetry of the tree-level potential depends on the pattern of Yukawa
couplings.

#### Contributions from the scalar loop.

The calculation of $V^{(S)}_{\rm CW}(K_{0},\vec{K})$ can be performed
straightforwards from Eq. (16), in which the mass matrix of scalars is given
by

$\mathbf{M}_{S}^{2}(\varphi)_{ab}=\frac{\delta^{2}V}{\delta\varphi_{a}\delta\varphi_{b}},$
(26)

where $\varphi_{a}$ are real vectors in the 8-dimensional field space. Though
$\mathbf{M}_{S}^{2}$ cannot be diagonalized analytically, we still find a way
to investigate the global symmetries of $V^{(G)}_{\rm CW}$. We firstly employ
the notations in Ref. Degee and Ivanov (2010), where the components of
$\varphi_{a}$ are ordered as

$\varphi_{a}^{T}=\left(\operatorname{Re}\phi_{1\uparrow},\operatorname{Im}\phi_{1\uparrow},\operatorname{Re}\phi_{2\uparrow},\operatorname{Im}\phi_{2\uparrow},\operatorname{Re}\phi_{1\downarrow},\operatorname{Im}\phi_{1\downarrow},\operatorname{Re}\phi_{2\downarrow},\operatorname{Im}\phi_{2\downarrow}\right),$
(27)

and $\varphi_{a}$ is related to the bilinear form by
$K^{\mu}=\varphi_{a}\Sigma_{ab}^{\mu}\varphi_{b}$. The $8\times 8$ matrices
$\Sigma^{\mu}$ are defined as

$\Sigma^{\mu}=\Sigma^{\mu}_{4}\oplus\Sigma^{\mu}_{4},\quad\Sigma_{4}^{0}=\mathbb{1}_{4},\quad\Sigma_{4}^{1}=\begin{pmatrix}0&\mathbb{1}_{2}\\\
\mathbb{1}_{2}&0\\\
\end{pmatrix},\quad\Sigma_{4}^{2}=\begin{pmatrix}0&\mathbb{i}_{2}\\\
-\mathbb{i}_{2}&0\\\
\end{pmatrix},\quad\Sigma_{4}^{3}=\begin{pmatrix}\mathbb{1}_{2}&0\\\
0&-\mathbb{1}_{2}\\\ \end{pmatrix},$ (28)

where $\mathbb{1}_{d}$ is the $d\times d$ identity matrix and
$\mathbb{i}_{2}\equiv\left(\begin{smallmatrix}0&1\\\
-1&0\end{smallmatrix}\right).$

The $V_{\rm CW}^{(S)}$ can be expanded in the powers of $\mathbf{M}_{S}$
Quiros (1999),

$\displaystyle V_{\rm CW}^{(S)}$
$\displaystyle=\frac{1}{2}\mathbf{Tr}\int\frac{d^{4}p_{E}}{2\pi^{4}}\ln\left[p_{E}^{2}+\mathbf{M}^{2}_{S}\right]$
(29)
$\displaystyle=\frac{1}{2}\int\frac{d^{4}p_{E}}{2\pi^{4}}\left[\mathbf{Tr}\sum_{n=1}^{\infty}\frac{1}{n}\left(-\frac{\mathbf{M}_{S}^{2}}{p^{2}_{E}}\right)^{n}+\ln
p^{2}_{E}\right],$

where $\mathbf{Tr}$ stands for taking a trace over the 8-dimensional field
space. For example, the leading power is

$\mathbf{Tr}(\mathbf{M}^{2}_{S})=\left(20\eta_{00}+4\operatorname{tr}(E)\right)K_{0}+24\vec{K}\cdot\vec{\eta}+8\xi_{0},$
(30)

which is consistent with Ref. Degee and Ivanov (2010). We show that all the
traces $\mathbf{Tr}(\mathbf{M}_{S}^{2n})$ in Eq. (29) are functions of gauge
orbits $K^{\mu}$, and the complete calculations are deferred to Appendix B.
Here, we present the final calculation result expressed in the bilinear
notation as

$V_{\rm CW}^{(S)}=\mathcal{F}\left(S_{p}^{\mu\nu},\eta^{\mu\nu}\right).$ (31)

The function $\mathcal{F}$ only depends on the trace of the inner products of
$S_{p}^{\mu\nu}$ and $\eta^{\mu\nu}$, and $S_{p}^{\mu\nu}$ is defined as

$S_{p}^{\mu\nu}=F(p)^{\mu}K^{\nu}+F(p)^{\nu}K^{\mu}-g^{\mu\nu}(F(p)K).$ (32)

Here, $F(p)_{\mu}$ is a function of $K^{\mu}$ that depends on the integer $p$,

$\displaystyle F(p)_{0}$
$\displaystyle\equiv\sum_{k=0}^{p/2}C_{p}^{2k}(A_{0})^{p-2k}|\vec{A}|^{2k},$
(33) $\displaystyle\vec{F}(p)$ $\displaystyle\equiv\left\\{\begin{aligned}
-&\sum_{k=0}^{(p-1)/2}C_{p}^{2k+1}(A_{0})^{p-2k-1}|\vec{A}|^{2k}\vec{A}\qquad&(p\neq
0),\\\ &0\qquad&(p=0),\end{aligned}\right.$ (34)

where $(A_{0},\vec{A})=A_{\mu}=2\eta_{\mu\nu}K^{\nu}+\xi_{\mu}$ and
$C_{p}^{k}$ is the binomial coefficient. Notice that the global symmetries are
determined only by the 3-dimension vector $\vec{A}$.

Upon expressing $V_{\rm CW}^{(S)}$ as a function in the orbit space, we find
that the tensor structures in $V_{\rm CW}^{(S)}$ are constructed entirely by
tree level parameter tensors $\eta^{\mu\nu}$ and $\xi^{\mu}$, i.e., no new
tensor structure appears, therefore, the rotation symmetries of $V_{\rm
CW}^{(S)}$ in the $\vec{K}$-space are determined by the tree-level parameter
tensors. If the tree-level potential is invariant under a rotation $R\in O(3)$
in the $\vec{K}$-space, i.e., $V_{\rm tree}(K_{0},\vec{K})=V_{\rm
tree}(K_{0},R\vec{K})$, the scalar loop contribution $V_{\rm CW}^{(S)}$ also
preserves the rotation invariance,

$V_{\rm CW}^{(S)}(K_{0},\vec{K})=V_{\rm CW}^{(S)}(K_{0},R\vec{K}).$

###  III.2 Symmetries of thermal potential

As for the finite temperature corrections in Eq. (15), $V_{T}$ stands for the
contribution from one-loop diagrams, and $V_{\rm daisy}$ denotes the
correction from higher loop Daisy diagrams Quiros (1999). The one-loop
correction $V_{T}$ is given by

$V_{T}=\sum_{i}n_{i}\frac{T^{4}}{2\pi^{2}}J_{B/F}\left(m_{i}^{2}/T^{2}\right),$
(35)

where the thermal bosonic function $J_{B}$ and fermionic function $J_{F}$ are

$\displaystyle J_{B}\left(m^{2}/T^{2}\right)=$
$\displaystyle-\frac{\pi^{4}}{45}+\frac{\pi^{2}}{12}\frac{m^{2}}{T^{2}}-\frac{\pi}{6}\left(\frac{m^{2}}{T^{2}}\right)^{3/2}-\frac{1}{32}\frac{m^{4}}{T^{4}}\log\frac{m^{2}}{a_{b}T^{2}}$
$\displaystyle-2\pi^{7/2}\sum_{\ell=1}^{\infty}(-1)^{\ell}\frac{\zeta(2\ell+1)}{(\ell+1)!}\Gamma\left(\ell+\frac{1}{2}\right)\left(\frac{m^{2}}{4\pi^{2}T^{2}}\right)^{\ell+2},$
(36) $\displaystyle J_{F}\left(m^{2}/T^{2}\right)=$
$\displaystyle\frac{7\pi^{4}}{360}-\frac{\pi^{2}}{24}\frac{m^{2}}{T^{2}}-\frac{1}{32}\frac{m^{4}}{T^{4}}\log\frac{m^{2}}{a_{f}T^{2}}$
$\displaystyle-\frac{\pi^{7/2}}{4}\sum_{\ell=1}^{\infty}(-1)^{\ell}\frac{\zeta(2\ell+1)}{(\ell+1)!}\left(1-2^{-2\ell-1}\right)\Gamma\left(\ell+\frac{1}{2}\right)\left(\frac{m^{2}}{\pi^{2}T^{2}}\right)^{\ell+2}.$
(37)

Here, $a_{b}=16a_{f}=16\pi^{2}e^{3/2-2\gamma_{E}}$, and $\zeta$ is the
Riemann-$\zeta$ function. The leading $T$-dependent terms of $J_{B/F}$ are
given by the mass-square terms,

$J_{B}=\frac{\pi^{2}}{12}\frac{m^{2}}{T^{2}}+O(T^{-4}),\quad
J_{F}=-\frac{\pi^{2}}{24}\frac{m^{2}}{T^{2}}+O(T^{-4}),$ (38)

where the background-field-independent terms are dropped. By collecting the
results in Eqs. (22), (24) and (30), we obtain the leading contributions from
gauge boson loops, fermion loops, and scalar loops to $V_{T}$ as follows:

$\displaystyle V_{T}^{(G)}$
$\displaystyle\approx\frac{g^{2}T^{2}}{32}(3+t_{W}^{2})K_{0},$ (39)
$\displaystyle V_{T}^{(F)}$
$\displaystyle\approx-\frac{T^{2}}{8}\left[(Y_{t0}+Y_{b0})K_{0}+(\vec{Y}_{t}+\vec{Y}_{b})\cdot\vec{K})\right],$
(40) $\displaystyle V_{T}^{(S)}$
$\displaystyle\approx\frac{T^{2}}{6}\left[\left(5\eta_{00}+\operatorname{tr}(E)\right)K_{0}+6\vec{K}\cdot\vec{\eta}+2\xi_{0}\right].$
(41)

We find that the corrections from Eqs. (39)-(41) to the tree-level potential
is equivalent to shifting the quadratic couplings $\xi_{\mu}$ in the orbit
space, i.e.,

$\displaystyle\xi_{0}$ $\displaystyle\to\xi_{0}+T^{2}c_{T0},$
$\displaystyle\vec{\xi}$ $\displaystyle\to\vec{\xi}+T^{2}\vec{c}_{T},$ (42)

where

$\displaystyle c_{T0}$
$\displaystyle=\frac{g^{2}}{32}(3+t_{W}^{2})-\frac{Y_{t0}+Y_{b0}}{8}+\frac{5\eta_{00}+\operatorname{tr}(E)}{6},$
$\displaystyle\vec{c}_{T}$
$\displaystyle=\frac{1}{8}\left[8\vec{\eta}-\vec{Y}_{t}-\vec{Y}_{b}\right].$
(43)

The direction of $\vec{\xi}$ is shifted by the quartic couplings $\vec{\eta}$
and Yukawa interactions $\vec{Y}$ from thermal corrections. At a sufficient
high temperature with $T^{2}\gg|\vec{\xi}|/|\vec{c}_{T}|$, the direction of
shifted $\vec{\xi}$ is aligned along the direction of $\vec{c}_{T}$. As a
result, the symmetries of thermal effective potential under the basis
transformation and CP transformation are determined by $\vec{c}_{T}$.

At high temperatures, the contribution from higher loop Daisy diagrams $V_{\rm
daisy}$ is comparable with $V_{T}$, and it is given by Quiros (1999)

$V_{\rm daisy}=-\frac{T}{12\pi}\sum_{i=\rm
bosons}n_{i}\left[\mathcal{M}_{i}^{3}(\phi_{c},T)-m_{i}^{3}(\phi_{c})\right].$
(44)

Here, the $\mathcal{M}_{i}(\phi_{c},T)$ are thermal corrected masses
calculated from $V_{\rm tree}+V_{T}$, which is obtained from the tree level
potential by parameter shifting $\xi^{\mu}\to\xi^{\mu}+T^{2}c_{T}^{\mu}$.
Therefore, the $T$-dependent terms in $\mathcal{M}_{i}(\phi_{c},T)$ are in the
form of $T^{2}c_{T}^{\mu}$. As $c_{T}^{0}$ plays no role in global
transformations, the behavior of $V_{\rm daisy}$ under the $O(3)_{K}$
transformation depends only on $\vec{c}_{T}$.

After understanding the behavior of $V_{\rm CW}$, $V_{T}$ and $V_{\rm daisy}$
under the $O(3)_{K}$ transformation, we are ready to discuss whether a global
symmetry preserved by the tree-level potential will be violated by the loop
corrections. Consider a tree-level potential that processes the symmetry of a
basis or CP transformation, then the potential is invariant under a rotation
$R$ in the $\vec{K}$-space, $V_{\rm tree}(K_{0},\vec{K})=V_{\rm
tree}(K_{0},R\vec{K})$, and its parameters satisfy

$\vec{\xi}=R\vec{\xi},\quad\vec{\eta}=R\vec{\eta},\quad E=RER^{T}.$ (45)

The only quantum correction that may violate the symmetry is the contribution
from fermion loops. The global symmetry is maintained in effective potential
if and only if all the Yukawa couplings are invariant under $R$, i.e.,
$\vec{Y}=R\vec{Y}$.

If the symmetry is softly broken at tree level, i.e., only the scalar
quadratic coupling $\vec{\xi}\neq R\vec{\xi}$ violates the symmetry while
other conditions in Eq. (45) are preserved, then the symmetry violation effect
from soft terms tend to be suppressed at high temperature. This is because the
leading thermal corrections shift the scalar quadratic couplings $\vec{\xi}$
with Yukawa coupling $\vec{Y}$ and scalar quartic couplings $\vec{\eta}$, and
both $\vec{Y}$ and $\vec{\eta}$ preserve the symmetry.

Another noteworthy example is the custodial symmetry. In the orbit space, the
custodial symmetry of the 2HDM does not correspond to a rotation symmetry but
a shift symmetry Grzadkowski  _et al._ (2011). As the effective potential is
not invariant under any shift symmetry in the orbit space, the custodial
symmetry of 2HDM is bound to be broken by the effective potential.

##  IV Bilinear notation after EWSB

In this section, we extend the bilinear notation to discuss EWSB and physical
fields. There are two reasons to discuss the EWSB in the bilinear notation.

Firstly, a global symmetry exhibited by the potential, as shown in Table 1,
can be broken spontaneously after the potential develops a vacuum. For
example, consider the CP symmetry. Even if the potential is explicitly CP
conserving,
$V(\Phi_{1},\Phi_{2})|_{\Phi_{i}\to\Phi_{i}^{*}}=V(\Phi_{1},\Phi_{2})$, the
physical fields, which are fluctuations around the vacuum, may still break CP
symmetry after EWSB if the vacuum has an irremovable CP phase as follows:

$\langle\Phi_{1}\rangle=\begin{pmatrix}0\\\
v_{1}\end{pmatrix},\quad\langle\Phi_{2}\rangle=\begin{pmatrix}0\\\
v_{2}e^{i\delta}\end{pmatrix}.$ (46)

This is called spontaneous CP violation (SCPV). In the bilinear notation, the
SCPV happens when the potential but not the vacuum is invariant under a mirror
reflection. In this case, there are two degenerate vacuums related by a CP
transformation as in Fig. 2. After analyzing the vacuum conditions in the
orbit space, we can easily determine whether the CP symmetry or other global
symmetry is spontaneously broken.

Figure 2: Illustration figures of tree-level parameters for an SCPV potential.
Here $\vec{K}_{v}$ and $\vec{K}_{v^{\prime}}$ are a pair of degenerated vacuum
expectation values that are related by a mirror reflection, and the reflection
plane spanned by $\vec{\xi}$ and $\vec{\eta}$.

Secondly, exploring the physical fields after the EWSB is necessary for
performing on-shell renormalization. The renormalized effective potential can
be expressed fully in the bilinear notation if we can perform the on-shell
renormalization in the orbit space. For that, we examine the vacuum structures
in the orbit space and investigate the relations between the field space and
orbit space. Furthermore, we demonstrate that the mass matrix of the physical
neutral scalars corresponds to a geometric structure in the orbit space,
making it convenient to handle the mass spectrum and on-shell renormalization.

###  IV.1 Vacuum condition

We start with the vacuum conditions of $V(K^{\mu})$, where $V(K^{\mu})$
represents the tree-level or effective potential in the orbit space. Figure 3
displays the light-cone in the orbit space, and the light-cone is a hyper-
surface defined by $K_{0}=|\vec{K}|$. The orbit space inside the forward
light-cone $LC^{+}$ is the physical region, satisfying $K_{0}\geq|\vec{K}|$
Maniatis  _et al._ (2006); Ivanov (2007, 2008); Nishi (2006). A neutral vacuum
expectation value requires the minimum of the potential, denoted as
$K_{v}^{\mu}$, to lie on the $LC^{+}$, i.e., $K_{v,0}=|\vec{K}_{v}|$ Maniatis
_et al._ (2006); Ivanov (2007, 2008); Nishi (2006). Therefore, $K_{v}^{\mu}$
is a conditional minimum of $V(K^{\mu})$ on the $LC^{+}$.

The vacuum of the potential $V(K^{\mu})$ is solved by minimizing the function
$V_{u}(K^{\mu})=V(K^{\mu})-uL(K^{\mu})$, where $u$ is a Lagrange multiplier
and $L(K^{\mu})=0$ is the light-cone condition with $L(K^{\mu})$ defined as

$L(K_{0},\vec{K})=K_{0}^{2}-|\vec{K}|^{2}=4K_{+}K_{-}-|\vec{K}_{T}|^{2}.$ (47)

Here, for the convenience, we introduce the light-cone coordinates

$K_{\pm}\equiv\frac{(K_{0}\pm K_{3})}{2},\qquad\vec{K}_{\rm
T}\equiv(K_{1},K_{2}),$ (48)

which are defined after rotating the vacuum along the $K_{3}$ direction, i.e.,
$K_{v}^{\mu}=\frac{v^{2}}{2}(1,0,0,1)^{T}$. The solution of the conditional
minimum satisfies

$\displaystyle\frac{\partial V}{\partial
K_{-}}\bigg{|}_{K_{v}}=2v^{2}u>0,\quad\frac{\partial V}{\partial
K_{+}}\bigg{|}_{K_{v}}=0,\quad\frac{\partial V}{\partial\vec{K}_{\rm
T}}\bigg{|}_{K_{v}}=0.$ (49)

Note that we require $\frac{\partial V}{\partial K_{-}}>0$ to ensure no global
minimum inside the light-cone to avoid a charged vacuum.

Figure 3: Vacuum expectation value and light-cone coordinates in the orbit
space. The yellow surface denotes $LC^{+}$ and the green denotes the
equipotential surface $\mathcal{M}_{\rm vev}$. $K_{v}$ is the tangent point of
these 3-dimensional hyper-surfaces.

In addition to the conditions in Eq. (49), we need to make sure that $K_{v}$
is a minimal point rather than a saddle point. In the 4-dimensional orbit
space, $K_{v}$ is the tangent point of $LC^{+}$ and an equipotential surface
$\mathcal{M}_{\rm vev}$ defined by $V(K^{\mu})=V(K^{\mu}_{v})$ Ivanov (2008),
and the normal direction of their tangent space is $K_{-}$, as shown in Fig.
3. Therefore, the requirement that $K_{v}$ is not a saddle point indicates
that $\mathcal{M}_{\rm vev}$ must be outside of the $LC^{+}$. Equivalently,
the distance between $LC^{+}$ and $\mathcal{M}_{\rm vev}$ is non-negative.
Expanding the distance between $LC^{+}$ and $\mathcal{M}_{\rm vev}$ at their
tangent point yields

$\delta h=(\delta K_{+},\delta\vec{K}_{\rm T})~{}\mathbf{M}_{\rm
dist}^{2}\begin{pmatrix}\delta K_{+}\\\ \delta\vec{K}_{\rm
T}\end{pmatrix},\quad\mathbf{M}_{\rm dist}^{2}=\frac{1}{\partial V/\partial
K_{-}}\begin{pmatrix}\frac{\partial^{2}V_{u}}{\partial
K_{+}^{2}}&\frac{\partial^{2}V_{u}}{\partial K_{+}\partial\vec{K}_{\rm T}}\\\
\frac{\partial^{2}V_{u}}{\partial K_{+}\partial\vec{K}_{\rm
T}}&\frac{\partial^{2}V_{u}}{\partial\vec{K}_{\rm T}^{2}}\end{pmatrix}.$ (50)

Therefore, the matrix $\mathbf{M}_{\rm dist}^{2}$ must be positive definite.
Here, the distance $\delta h$ is measured in the coordinate $K_{-}$. As to be
shown later, the distance matrix $\mathbf{M}_{\rm dist}^{2}$ between the two
hyper-surfaces directly yields the neutral scalar mass matrix.

Now we have introduced the vacuum conditions fully in the orbit space. These
conditions apply to both the tree-level and the effective potentials.
Specifically, the tree-level potential in Eq. (9) can be written in terms of
the light-cone coordinates as follows,

$\displaystyle V_{\rm tree}=$
$\displaystyle\xi_{+}K_{+}+\xi_{-}K_{-}+\vec{\xi}_{\rm T}\cdot\vec{K}_{\rm
T}+(K_{+},K_{-},\vec{K}_{\rm
T})\begin{pmatrix}\eta_{++}&\eta_{+-}&\vec{\eta}_{\rm T+}\\\
\eta_{+-}&\eta_{--}&\vec{\eta}_{\rm T-}\\\ \vec{\eta}_{\rm T+}&\vec{\eta}_{\rm
T-}&{\eta}_{\rm TT}\end{pmatrix}\begin{pmatrix}K_{+}\\\ K_{-}\\\ \vec{K}_{\rm
T}\end{pmatrix}.$ (51)

Then the minimal conditions for the tree-level potential from Eq. (49) are

$\displaystyle\frac{\partial V_{\rm tree}}{\partial
K_{-}}\bigg{|}_{K_{v}}=\xi_{-}+v^{2}\eta_{+-}=2v^{2}u>0,$
$\displaystyle\frac{\partial V_{\rm tree}}{\partial
K_{+}}\bigg{|}_{K_{v}}=\xi_{+}+v^{2}\eta_{++}=0,$ $\displaystyle\frac{\partial
V_{\rm tree}}{\partial\vec{K}_{\rm T}}\bigg{|}_{K_{v}}=\vec{\xi}_{\rm
T}+v^{2}\vec{\eta}_{\rm T+}=0,$ (52)

which are equivalent to the minimal conditions given in Ref. Maniatis  _et
al._ (2006).

###  IV.2 A geometrical view of the scalar mass matrix

After the potential develops a vacuum expectation value, the scalar fields
become massive. The field components after the EWSB, which are fluctuations
around the vacuum. Without loss of generality, we use the Higgs basis in which
the vacuum $v$ is rotated to the first doublet, and the field components are

$H_{1}=\begin{pmatrix}G^{+}\\\
\dfrac{v+\phi+iG^{0}}{\sqrt{2}}\end{pmatrix},~{}~{}~{}~{}H_{2}=\begin{pmatrix}H^{+}\\\
\dfrac{R+iI}{\sqrt{2}}\end{pmatrix},$ (53)

where $\phi,R,I$ and $H^{\pm}$ are physical fields while $G_{0}$ and $G^{\pm}$
are Goldstone fields. By substituting the field components of $H_{i}$ into Eq.
(2), and rewriting them in terms of the light-cone coordinates, we have

$\begin{pmatrix}K_{+}\\\ K_{1}\\\ K_{2}\\\
K_{-}\end{pmatrix}=\frac{v^{2}}{2}\begin{pmatrix}1\\\ 0\\\ 0\\\
0\end{pmatrix}+v\begin{pmatrix}\phi\\\ R\\\ I\\\
0\end{pmatrix}+\begin{pmatrix}\frac{\phi^{2}}{2}+\frac{G_{0}^{2}}{2}+G^{+}G^{-}\\\
\phi R+IG_{0}+G^{+}H^{-}+G^{-}H^{+}\\\ \phi
I-RG_{0}+i(G^{+}H^{-}-G^{-}H^{+})\\\
\frac{I^{2}}{2}+\frac{R^{2}}{2}+H^{-}H^{+}\end{pmatrix}.$ (54)

The charged Higgs boson mass is given by

$m_{H^{\pm}}^{2}=\left.\frac{\partial V}{\partial H^{-}H^{+}}\right|_{\rm
vev}=\left.\frac{\partial V}{\partial
K_{-}}\right|_{K_{v}}\left.\frac{\partial K_{-}}{\partial
H^{-}H^{+}}\right|_{\rm vev}=\left.\frac{\partial V}{\partial
K_{-}}\right|_{K_{v}}.$ (55)

As for the neutral physical scalars $\phi,R$ and $I$, their mass matrix is
calculated by expanding the potential in the field space as follows,

$\delta V=(\delta\phi,\delta R,\delta I)~{}\mathbf{M}^{2}_{\rm
neutral}~{}\begin{pmatrix}\delta\phi\\\ \delta R\\\ \delta I\end{pmatrix},$
(56)

where $\delta\phi,\delta R$ and $\delta I$ are small expansions of the fields
around the vacuum. Equation (54) shows that the three directions
$(K_{+},\vec{K}_{\rm T})$, which span the tangent space of $LC^{+}$ and
$\mathcal{M}_{\rm vev}$, are linearly related to the three neutral scalar
fields $(\phi,R,I)$ around the vacuum. The linear relationship between field
space and orbit space directly links the scalar mass matrix and the distance
matrix between $LC^{+}$ and $\mathcal{M}_{\rm vev}$. By combining Eq. (56)
with Eqs. (50) and (54), we obtain

$\mathbf{M}^{2}_{\rm
neutral}=v^{2}\begin{pmatrix}\frac{\partial^{2}V_{u}}{\partial
K_{+}^{2}}&\frac{\partial^{2}V_{u}}{\partial K_{+}\partial\vec{K}_{\rm T}}\\\
\frac{\partial^{2}V_{u}}{\partial K_{+}\partial\vec{K}_{\rm
T}}&\frac{\partial^{2}V_{u}}{\partial\vec{K}_{\rm
T}^{2}}\end{pmatrix}=v^{2}\frac{\partial V}{\partial K_{-}}\mathbf{M}^{2}_{\rm
dist}.$ (57)

Therefore, the neutral mass matrix is simply proportional to the distance
matrix between the two hyper-surfaces $LC^{+}$ and $\mathcal{M}_{\rm vev}$.

The experimentally preferred Higgs alignment limit can be read out from
Eq.(57) directly. In the alignment limit, the neutral scalar $\phi$ in Eq.
(53) corresponds to the SM-like Higgs boson, and all of its properties are
very close to the SM Higgs boson, including mass, gauge couplings, Yukawa
couplings, and CP property. Technically, the alignment limit is reached when
the neutral scalar $\phi$ in Eq. (53) is approximately the 125 GeV mass
eigenstate and does not mix with other neutral scalars, therefore, we obtain
the following relations from Eq. (57),

$\left.\frac{\partial^{2}V_{u}}{\partial K_{+}\partial\vec{K}_{\rm
T}}\right|_{K_{v}}=\left.\frac{\partial^{2}V}{\partial
K_{+}\partial\vec{K}_{\rm T}}\right|_{K_{v}}\approx 0,$ (58)

where $K_{+}$ and $\vec{K}_{\rm T}$ are light-cone coordinates in orbit space.
At tree-level, this condition yields $\vec{\eta}_{\rm T+}\approx 0$
straightforwards.

Another demonstration is to discuss the ultra-light CP-odd particle, which is
also known as the axion-like particle (ALP). The ALP is of widespread interest
for its rich phenomenology, and the 2HDM is a simple model that can provide
the ALP. From the geometric relations in the orbit space, a massless scalar
appears when the two hyper-surfaces $LC^{+}$ and $\mathcal{M}_{\rm vev}$
osculate at $K_{v}$ along a certain direction. There are two possibilities in
the 2HDM to produce an ALP naturally, due to symmetries rather than accidental
parameter choice. One possibility is the 2HDM potential with an approximately
$U(1)_{a}$ symmetry. An exact $U(1)_{a}$ symmetry in the 2HDM potential
results in an additional Goldstone boson, and the Goldstone boson will develop
a small mass if the $U(1)_{a}$ symmetry is slightly broken as shown in Fig.
4(a). In this case, the ALP is a pseudo-Goldston boson as in the
Dine–Fischler–Srednicki–Zhitnitsky axion model Zhitnitsky (1980); Dine  _et
al._ (1981). Another possibility is the 2HDM potential with a CP symmetry that
is spontaneously broken. When the SCPV phase $\delta$ is very small, the two
degenerate vacuums turn to merge, and the two hyper-surfaces $LC^{+}$ and
$\mathcal{M}_{\rm vev}$ turn to osculate with each other at $K_{v}$, as shown
in Fig. 4(b), therefore, a massless boson appears when the SCPV phase $\delta$
goes to zero Mao and Zhu (2014). In this case, the ALP is not a pseudo-
Goldston boson.

Figure 4: A two dimensional slice of Fig. 3 with $K_{0}=K_{v,0}$, viewed from
the $K_{0}$ direction. The symbol $\odot$ denotes the $K_{0}$ axis. The yellow
line denotes $LC^{+}$ and the green denotes $\mathcal{M}_{\rm vev}$. There are
two scenarios with an ultra-light scalar: (a) potential with a slightly broken
$U(1)_{a}$ symmetry; (b) the SCPV potential with a small CP phase $\delta$.

##  V On-shell renormalization in the orbit space

The masses and mixing angles of physical states derived from the one-loop CW
potential in the $\overline{\rm MS}$ renormalization scheme differ from their
tree-level values. To directly use the loop-corrected masses and mixing angles
as inputs, the on-shell renormalization scheme is often preferred. This is
achieved by adding the counterterm potential $V_{\rm CT}$ to the zero
temperature effective potential

$V_{\rm eff}=V_{\rm tree}+V_{\rm CW}+V_{\rm CT},$ (59)

and then enforcing the loop-corrected vacuum and masses to be the same as the
tree-level values. Consequently, the renormalization conditions in the field
space are given by

$\displaystyle\partial_{\varphi_{a}}(V_{\rm CT}+V_{\rm
CW})\big{|}_{\varphi_{a}=\langle\varphi_{a}\rangle_{\rm tree}}=0,$ (60)
$\displaystyle\partial_{\varphi_{a}}\partial_{\varphi_{b}}(V_{\rm CT}+V_{\rm
CW})\big{|}_{\varphi_{a}=\langle\varphi_{a}\rangle_{\rm tree}}=0,$ (61)

where $\varphi_{a}~{}(a=1\cdots 8)$ denote the eight scalar field components
in the two Higgs doublets.

However, most of the renormalization conditions are redundant due to
unphysical fields and quite a few identities, and it is convenient to deal
with the renormalization condition in orbit space.333A detailed analysis of
the number of renormalization conditions in the field space and their
equivalence with the conditions in the orbit space is presented in Appendix C.
To achieve this, we express the counterterm potential in the bilinear notation
as $V_{\rm CT}=\delta\xi_{\mu}\,K^{\mu}+\delta\eta_{\mu\nu}\,K^{\mu}K^{\nu}$.
Based on the vacuum conditions in Eq. (49) and the scalar masses given in Eqs.
(55) and (57), we obtain ten independent renormalization conditions that are
related to the physical fields as follows:

$\displaystyle 0=\partial_{K_{+}}(V_{\rm CT}+V_{\rm CW})\big{|}_{K_{v}},$ (62)
$\displaystyle 0=\partial_{\vec{K}_{T}}(V_{\rm CT}+V_{\rm
CW})\big{|}_{K_{v}},$ (63) $\displaystyle 0=\partial_{K_{-}}(V_{\rm CT}+V_{\rm
CW})\big{|}_{K_{v}},$ (64) $\displaystyle 0=\partial_{K_{+}}^{2}(V_{\rm
CT}+V_{\rm CW})\big{|}_{K_{v}},$ (65) $\displaystyle
0=\partial_{\vec{K}_{T}}^{2}(V_{\rm CT}+V_{\rm CW})\big{|}_{K_{v}},$ (66)
$\displaystyle 0=\partial_{K_{+}}\partial_{\vec{K}_{T}}(V_{\rm CT}+V_{\rm
CW})\big{|}_{K_{v}}.$ (67)

Here the light-cone coordinates are defined still by the tree-level vacuum
$K_{v}$, and the derivatives are evaluated around $K_{v}$. Note that only part
of the first and second derivatives $\partial_{K^{\mu}}(V_{\rm CT}+V_{\rm
CW})\big{|}_{K_{v}}$ and $\partial_{K^{\mu}K^{\nu}}(V_{\rm CT}+V_{\rm
CW})\big{|}_{K_{v}}$ are related to the vacuum conditions and scalar masses
and should be included in the renormalization conditions, while the others are
irrelevant to physical quantities. Specifically, four conditions from the
first derivative in Eqs. (62)-(64) ensure that the loop-corrected vacuum
expectation value is the same as the tree-level case, and Eq. (64) also
ensures that the charged scalar mass is the same as the tree-level value. The
other six conditions involving the second derivatives in Eqs. (65)-(67) ensure
that the neutral scalar masses and mixing angles are the same as those of the
tree-level potential.

The counterterms $\delta\xi_{\mu}$ and $\delta\eta_{\mu\nu}$ can be determined
from the renormalization conditions in Eqs. (62)-(67). For a general 2HDM
without any constrains on the parameters, there are fourteen free parameters,
four in $\delta\xi_{\mu}$ and ten in $\delta\eta_{\mu\nu}$, to be determined
by the renormalization conditions. After expressing $\delta\xi_{\mu}$ and
$\delta\eta_{\mu\nu}$ in terms of the light-cone coordinates, the
renormalization conditions are

$\displaystyle\delta\eta_{++}$ $\displaystyle=-\partial_{K_{+}}^{2}V_{\rm
CW}\big{|}_{K_{v}},$ (68) $\displaystyle\delta\vec{\eta}_{T+}$
$\displaystyle=-\partial_{\vec{K}_{T}}\partial_{\vec{K}_{+}}V_{\rm
CW}\big{|}_{K_{v}},$ (69) $\displaystyle\delta\eta_{TT}$
$\displaystyle=-\partial_{\vec{K}_{T}}^{2}V_{\rm CW}\big{|}_{K_{v}},$ (70)
$\displaystyle\delta\xi_{+}$ $\displaystyle=-\partial_{K_{+}}V_{\rm
CW}\big{|}_{K_{v}}-v^{2}\delta\eta_{++},$ (71)
$\displaystyle\delta\vec{\xi}_{T}$
$\displaystyle=-\partial_{\vec{K}_{T}}V_{\rm
CW}\big{|}_{K_{v}}-v^{2}\delta\vec{\eta}_{T+},$ (72)
$\displaystyle\delta\xi_{-}$ $\displaystyle=-\partial_{K_{-}}V_{\rm
CW}\big{|}_{K_{v}}-v^{2}\delta\eta_{+-}.$ (73)

Note that neither the vacuum condition nor the scalar mass matrix depends on
the counterterms $\delta\eta_{--}$, $\delta\eta_{+-}$ and
$\delta\vec{\eta}_{T-}$, therefore, these four parameters are up to free
choices.

In addition, our convention is to set the tadpole terms to zero whenever
possible. Generally, one can allow the development of vacuum in the field
space and introduce the tadpole terms in $V_{\rm CT}$ as done in Refs. Basler
_et al._ (2018); Basler and Mühlleitner (2019). However, for the most general
2HDM potential, there will be more parameters than renormalization conditions
and we can always set the tadpole terms to zero. Tadpole terms may be
necessary if we require the counterterms to satisfy some specific constraints
such that the remaining parameters cannot satisfy the renormalization
conditions.

For the 2HDM with some specific parameter constraints required by symmetries
or alignment, it is a common practice to demand the counterterms
$\delta\xi_{\mu}$ and $\delta\eta_{\mu\nu}$ satisfying the same constraints as
the tree-level parameters $\xi_{\mu}$ and $\eta_{\mu\nu}$. Then the number of
parameters in $\delta\xi_{\mu}$ and $\delta\eta_{\mu\nu}$ is less than
fourteen as in the general 2HDM, and the renormalization conditions need to be
dealt with case-by-case. For illustration, we discuss the renormalization
conditions used in three 2HDMs below.

#### Softly broken $Z_{2}$ symmetric potential.

Imposing a softly broken $Z_{2}$ symmetry on the 2HDM Lagrangian is the most
popular way to prevent flavor-changing neutral interactions. For a complex
2HDM with softly broken $Z_{2}$ symmetry, the $Z_{2}$ symmetry gives four
additional constraints on $\delta\eta_{\mu\nu}$, and the remaining six
counterterms can be fixed by the six conditions in Eqs. (68)-(70). The soft
quadratic couplings are not constrained, and four parameters in
$\delta\xi^{\mu}$ can be fixed by the four conditions in Eqs. (71)-(73).

#### Real 2HDM with softly broken $Z_{2}$ symmetry.

In addition to the softly broken $Z_{2}$ symmetry, a CP symmetry is often
imposed on the potential. The tree-level potential is invariant under a mirror
reflection $\bar{R}$ in the orbit space, $V_{\rm tree}(K_{0},\vec{K})=V_{\rm
tree}(K_{0},\bar{R}\vec{K})$. Note that the CP symmetry does not impose any
additional constraint on the quartic counterterms $\delta\eta^{\mu\nu}$, as
the $Z_{2}$ symmetry provides stronger constraints than the CP symmetry. On
the other hand, the softly broken terms are constrained by the CP symmetry.
Say that the mirror reflection is along the second direction
$\bar{R}:K_{2}\to-K_{2}$, then $\delta\xi_{2}$ should be set to zero, leaving
three free parameters in $\delta\xi^{\mu}$.

Usually, the three parameters in $\delta\xi^{\mu}$ are not enough to satisfy
the four equations in Eqs. (71)-(73). But when the vacuum is invariant under
the CP transformation, e.g., $K_{v}^{\mu}=\frac{v^{2}}{2}(1,0,0,1)$ and
$\vec{K}_{v}=\bar{R}\vec{K}_{v}$, there are only three independent conditions
in Eqs. (71)-(73), because the CW potential satisfies the CP symmetry, $V_{\rm
CW}(K_{0},\vec{K})=V_{\rm CW}(K_{0},\bar{R}\vec{K})$, and we have

$\displaystyle\partial_{K_{2}}V_{\rm
CW}\big{|}_{K_{v}}=0,\quad\partial_{K_{2}}\partial_{K^{\mu}}V_{\rm
CW}\big{|}_{K_{v}}=0.$ (74)

Then one renormalization condition $\delta\xi_{2}=-\partial_{K_{2}}V_{\rm
CW}-v^{2}\delta\vec{\eta}_{2+}=0$ automatically holds from Eq. (72), and we
end up with three parameters and three conditions.

However, if the vacuum develops an SCPV phase $\delta$, the CP symmetry is
broken spontaneously. The vacuum $\vec{K}_{v}$ is no longer invariant under
the CP transformation, e.g.,
$K_{v}^{\mu}=\frac{v^{2}}{2}(1,0,\sin\delta,\cos\delta)$ and
$\vec{K}_{v}\neq\bar{R}\vec{K}_{v}$. As a result, Eqs. (74) no longer hold.
The rest three parameters in $\delta\xi^{\mu}$ are not enough to satisfy the
renormalization conditions if we still require the counterterm
$\delta\xi_{2}=0$. The remaining renormalization condition, which is
equivalent to $\partial_{\delta}(V_{\rm CW}+V_{\rm CT})=0$, cannot be
fulfilled, and this corresponds to a change of the SCPV phase $\delta$. It
could be fixed with a tadpole counterterm of the CP-violating vacuum.

#### Exact aligned 2HDM.

In the 2HDM, the exact alignment condition requires that the neutral scalar
$\phi$ in Eq. (53) is the 125 GeV mass eigenstate, then the tree-level
parameters satisfy $\vec{\eta}_{T+}=0$ as shown in Eq. (58). However, the
alignment condition is not protected by any symmetry, and there is no
guarantee that the counterterms
$\delta\vec{\eta}_{T+}=-\partial_{\vec{K}_{T}}\partial_{\vec{K}_{+}}V_{\rm
CW}$ vanish. Therefore, the alignment condition is usually broken by quantum
corrections.

##  VI Conclusion and Discussion

We performed a complete analysis of the CP and basis transformation symmetries
of the 2HDM in the orbit space. We extended the study of the global symmetries
in orbit space to one-loop thermal effective potential. We demonstrated that
the global symmetries of the tree-level potential are preserved by quantum
corrections from boson loop contributions, but may be broken by fermion loop
contributions, depending on the Yukawa interactions.

In order to study the vacuum conditions and physical masses in the orbit
space, we introduced the light-cone coordinates and generalized the bilinear
notation to study the physical scalar fields around the vacuum. It provides a
geometric view of the scalar mass matrix and on-shell renormalization
conditions. By translating the on-shell renormalization conditions of the
vacuum and scalar mass into geometric conditions in the orbit space, we
calculated the renormalized one-loop effective potential completely.

We extend our study to the case after the EWSB. The geometrical view of scalar
masses can provide insight into special limits of the 2HDM mass spectrum, such
as alignment limit and ultra-light scalars, thereby simplifying the analysis.
The renormalization conditions are much simpler to be dealt with in the orbit
space, and there are at most 10 independent on-shell renormalization
conditions for a general 2HDM potential. Our work provides a foundation for
future study of the 2HDM effective potential and its implications in orbit
space.

###### Acknowledgements.

The work is supported in part by the National Science Foundation of China
under Grants No. 11725520, No. 11675002 and No. 12235001.

##  Appendix A Basis invariant notations of 2HDM potential

###  A.1 Explicit expression of bilinear notation

The explicit expression for each component of $K^{\mu}$ is

$K^{\mu}=\Phi_{i}^{\dagger}\sigma_{ij}^{\mu}\Phi_{j}=\begin{pmatrix}\Phi_{1}^{\dagger}\Phi_{1}+\Phi_{2}^{\dagger}\Phi_{2}\\\
\Phi_{1}^{\dagger}\Phi_{2}+\Phi_{2}^{\dagger}\Phi_{1}\\\
i(\Phi_{2}^{\dagger}\Phi_{1}-\Phi_{1}^{\dagger}\Phi_{2})\\\
\Phi_{1}^{\dagger}\Phi_{1}-\Phi_{2}^{\dagger}\Phi_{2}\\\ \end{pmatrix}.$ (75)

By comparing the potential in the blinear notation (Eq. 9) with the
tranditional notation (Eq. 1), we can explicitly relate these two sets of
parameters,

$\displaystyle\xi_{0}$ $\displaystyle\equiv$
$\displaystyle\frac{1}{2}(m_{11}^{2}+m_{22}^{2}),\qquad\eta_{00}=(\lambda_{1}+\lambda_{2}+2\lambda_{3})/8,$
$\displaystyle\vec{\xi}$ $\displaystyle=$
$\displaystyle\left(-\Re\left(m_{12}^{2}\right),\Im\left(m_{12}^{2}\right),\tfrac{1}{2}(m_{11}^{2}-m_{22}^{2})\right)^{T},$
$\displaystyle\vec{\eta}$ $\displaystyle=$
$\displaystyle\left(\Re(\lambda_{6}+\lambda_{7})/4,-\Im(\lambda_{6}+\lambda_{7})/4,(\lambda_{1}-\lambda_{2})/8\right)^{T},$
$\displaystyle E$ $\displaystyle=$
$\displaystyle\frac{1}{4}\left(\begin{array}[]{ccc}\lambda_{4}+\Re\left(\lambda_{5}\right)&-\Im\left(\lambda_{5}\right)&\Re\left(\lambda_{6}-\lambda_{7}\right)\\\
-\Im\left(\lambda_{5}\right)&\lambda_{4}-\Re\left(\lambda_{5}\right)&\Im\left(\lambda_{7}-\lambda_{6}\right)\\\
\Re\left(\lambda_{6}-\lambda_{7}\right)&\Im\left(\lambda_{7}-\lambda_{6}\right)&\left(\lambda_{1}+\lambda_{2}-2\lambda_{3}\right)/2\end{array}\right).$
(79)

In the 4-dimensional orbit space, the physical region is confined to the
interior of the forward light-cone, i.e., $K_{0}\geqslant|\vec{K}|$. Because
$K^{\mu}$ can be decomposed from
$\underline{K}_{ij}=\Phi_{i}^{\dagger}\Phi_{j}$, by definition:

$\underline{K}\equiv\begin{pmatrix}\Phi_{1}^{\dagger}\Phi_{1}&\Phi_{2}^{\dagger}\Phi_{1}\\\
\Phi_{1}^{\dagger}\Phi_{2}&\Phi_{2}^{\dagger}\Phi_{2}\\\
\end{pmatrix}\equiv\frac{1}{2}\begin{pmatrix}K_{0}+K_{3}&K_{1}-iK_{2}\\\
K_{1}+iK_{2}&K_{0}-K_{3}\\\ \end{pmatrix},$ (80)

and the matrix $\underline{K}$ is actually a semi-positive matrix when
$\Phi_{i}=(\phi_{i\uparrow},\phi_{i\downarrow})^{T}$ are $SU(2)_{L}$ doublets,

$\underline{K}=\underline{\phi}\underline{\phi}^{\dagger},\quad\underline{\phi}=\begin{pmatrix}\phi_{1\uparrow}&\phi_{1\downarrow}\\\
\phi_{2\downarrow}&\phi_{2\downarrow}\end{pmatrix},$ (81)

which directly leads to

$\begin{cases}~{}~{}\text{tr}\underline{K}=K_{0}\geqslant 0,\\\
\det\underline{K}=(K_{0}^{2}-|\vec{K}|^{2})/4\geqslant 0.\end{cases}$ (82)

Therefore in the bilinear notation, the tree-level 2HDM scalar potential is a
real quadratic function of $(K_{0},\vec{K})$, and the physical region is
defined inside the forward light-cone.

###  A.2 $Z_{2}$ symmetry in the bilinear notation

The $Z_{2}$ symmetry is imposed on the 2HDM by assigning $Z_{2}$ charges to
scalar and fermion fields. In Eq. (1), the two Higgs doublets $\Phi_{1}$ and
$\Phi_{2}$ carry the $Z_{2}$ charges of $-1$ and $+1$ respectively, forbidding
the $(\Phi_{1}^{\dagger}\Phi_{1})(\Phi_{1}^{\dagger}\Phi_{2})$ and
$(\Phi_{2}^{\dagger}\Phi_{2})(\Phi_{1}^{\dagger}\Phi_{2})$ terms in the
potential.

As for the Yukawa interactions, fermions are also assigned with negative or
positive $Z_{2}$ charges, then forced to interact with only $\Phi_{1}$ or
$\Phi_{2}$. Usually, the patterns of $Z_{2}$ charges assignments are divided
into four types Haber  _et al._ (1979); Donoghue and Li (1979); Hall and Wise
(1981); Barger  _et al._ (1990); Barnett  _et al._ (1984a, b): Type I, Type
II, Type X and Type Y, as listed in Table 2. For fermions with different
$Z_{2}$ charges, the vectors $\vec{Y}$’s projected by their Yukawa couplings
are opposite to each other. For example, in the orbit space of $Z_{2}$
eigenbasis $(\Phi_{1},\Phi_{2})$, the Yukawa coupling of fermion with positive
$Z_{2}$ charge yield $Y^{\mu}\propto(1,0,0,-1)$ and the Yukawa coupling of
fermion with negative $Z_{2}$ charge yield $Y^{\mu}\propto(1,0,0,1)$.

Table 2: $Z_{2}$ charge assignment for different types | $u_{R}$ | $d_{R}$ | $l_{R}$ | $\Phi_{1}$ | $\Phi_{2}$ | Directions of $\hat{Y}_{f}$
---|---|---|---|---|---|---
Type I | $+$ | $+$ | $+$ | $-$ | $+$ | $\hat{Y}_{u}=\hat{Y}_{d}=\hat{Y}_{l}$
Type II | $+$ | $-$ | $-$ | $-$ | $+$ | $-\hat{Y}_{u}=\hat{Y}_{d}=\hat{Y}_{l}$
Type X | $+$ | $+$ | $-$ | $-$ | $+$ | $\hat{Y}_{u}=\hat{Y}_{d}=-\hat{Y}_{l}$
Type Y | $+$ | $-$ | $+$ | $-$ | $+$ | $\hat{Y}_{u}=-\hat{Y}_{d}=\hat{Y}_{l}$

###  A.3 Tensor notation

For the completeness of this paper, here we reviewed another basis invariant
notation to analyze the 2HDM potential, the tensor notation Botella and Silva
(1995); Branco  _et al._ (2005); Gunion and Haber (2005). It is
straightforward to express the 2HDM scalar potential in an $U(2)_{\Phi}$ basis
invariant form,

$V=\mu_{ij}\Phi_{i}^{\dagger}\Phi_{j}+\lambda_{ij,kl}(\Phi_{i}^{\dagger}\Phi_{j})(\Phi_{k}^{\dagger}\Phi_{l}).$
(83)

As a result $\mu_{ij}$ and $\lambda_{ij,kl}$ transform covariantly with
$\Phi_{i}$ under the $U(2)_{\Phi}$ basis transformation,

$\mu^{\prime}_{ij}=U_{ik}\mu_{kl}U^{*}_{jl},\quad\lambda^{\prime}_{ij,kl}=U_{ip}U_{kr}\lambda_{pq,rs}U^{*}_{jq}U^{*}_{ls}.$
(84)

By definition, $\lambda_{ij,kl}=\lambda_{kl,ij}$, and hermiticity requires
that $\mu_{ij}=\mu_{ji}^{*},~{}\lambda_{kl,ij}=\lambda_{lk,ji}^{*}$. Under the
basis of Eq. (1), we have the following relations explicitly,

$\begin{array}[]{rl}\mu_{11}=m_{11}^{2},&\quad\mu_{22}=m_{22}^{2},\\\
\mu_{12}=-m_{12}^{2},&\quad\mu_{21}=-m_{12}^{2}{}^{*}\\\
\lambda_{11,11}=\lambda_{1},&\quad\lambda_{22,22}=\lambda_{2},\\\
\lambda_{11,22}=\lambda_{22,11}=\lambda_{3},&\quad\lambda_{12,21}=\lambda_{21,12}=\lambda_{4},\\\
\lambda_{12,12}=\lambda_{5},&\quad\lambda_{21,21}=\lambda_{5}^{*},\\\
\lambda_{11,12}=\lambda_{12,11}=\lambda_{6},&\quad\lambda_{11,21}=\lambda_{21,11}=\lambda_{6}^{*},\\\
\lambda_{22,12}=\lambda_{12,22}=\lambda_{7},&\quad\lambda_{22,21}=\lambda_{21,22}=\lambda_{7}^{*}.\end{array}$
(85)

The potential is invariant under the GCP symmetry Eq. (4) when $\mu_{ij}$ and
$\lambda_{ij,kl}$ satisfy

$\mu_{ij}=X_{ik}\mu_{kl}^{*}X_{lj}^{*},\quad\lambda_{ij,kl}=X_{im}X_{kn}\lambda_{mp,nq}^{*}X_{jp}^{*}X_{lq}^{*}.$
(86)

One can construct several CP invariants to determine whether a potential is
GCP invariant Botella and Silva (1995). Similar to the Jarlskog invariant
Jarlskog (1985), a $SU(3)_{L/R}$ invariant in quark family space, the CP
invariants of 2HDM scalar potential are constructed from tensor products of
$\mu_{ij}$ and $\lambda_{ij,kl}$ as $U(2)_{\Phi}$ invariants in scalar family
space. And tensor notation can also be used to construct CP invariants for
scalar fermion interaction after extending tensor structures to fermion family
space Botella and Silva (1995). In addition, a recent development in tensor
notation is using the Hilbert series to systematically construct all possible
CP invariants Trautner (2019), and similar procedures can also be used to
construct CP invariant in the lepton sector with Majorana terms Yu and Zhou
(2021).

##  Appendix B Effective potential from Scalar Loop Contribution

Here we show the calculation of the effective potential from scalar loop
contribution in detail. We employ the notations in Ref. Degee and Ivanov
(2010) to link the eight scalar fields $\varphi_{i}$ with the bilinear forms
$K^{\mu}$,

$\displaystyle\mathcal{L}=\Omega_{\mu}\left(\partial_{\alpha}\Phi_{i}\right)^{\dagger}\sigma_{ij}^{\mu}\left(\partial^{\alpha}\Phi_{j}\right)-V,\quad\Omega^{2}=1,$
(87) $\displaystyle V_{\rm
tree}=\xi_{\mu}K^{\mu}+\eta_{\mu\nu}K^{\mu}K^{\nu},$
$\displaystyle\varphi_{a}=\left(\operatorname{Re}\phi_{1,\uparrow},\operatorname{Im}\phi_{1,\uparrow},\operatorname{Re}\phi_{2,\uparrow},\operatorname{Im}\phi_{2,\uparrow},\operatorname{Re}\phi_{1,\downarrow},\operatorname{Im}\phi_{1,\downarrow},\operatorname{Re}\phi_{2,\downarrow},\operatorname{Im}\phi_{2,\downarrow}\right),$
$\displaystyle K^{\mu}=\varphi_{a}\Sigma_{ab}^{\mu}\varphi_{b},$
$\displaystyle(\Omega_{\rho}\Sigma^{\rho})^{-1}=\Omega_{\rho}\bar{\Sigma}^{\rho}.$

Note that $\Omega_{\mu}=(1,0,0,0)$ for the canonical kinetic term The matrix
$\bar{\Sigma}^{\mu}=(\Sigma^{0},-\Sigma^{i})$ and the $8\times 8$ symmetric
matrics $\Sigma^{\mu}$ defined in Eq. (28) are

$\Sigma^{\mu}=\Sigma^{\mu}_{4}\oplus\Sigma^{\mu}_{4},\quad\Sigma_{4}^{0}=\mathbb{1}_{4},\quad\Sigma_{4}^{1}=\begin{pmatrix}0&\mathbb{1}_{2}\\\
\mathbb{1}_{2}&0\\\
\end{pmatrix},\quad\Sigma_{4}^{2}=\begin{pmatrix}0&\mathbb{i}_{2}\\\
-\mathbb{i}_{2}&0\\\
\end{pmatrix},\quad\Sigma_{4}^{3}=\begin{pmatrix}\mathbb{1}_{2}&0\\\
0&-\mathbb{1}_{2}\\\ \end{pmatrix},$ (88)

where $\mathbb{1}_{d}$ is the $d\times d$ identity matrix and
$\mathbb{i}_{2}\equiv\left(\begin{smallmatrix}0&1\\\
-1&0\end{smallmatrix}\right)$. Because $(\mathbb{i}_{2})^{2}=-\mathbb{1}_{2}$,
the matrix $\Sigma^{\mu}$ share the same algebra with the pauli matrix
$\sigma^{\mu}$, e.g.,

$\displaystyle[\Sigma^{i},\Sigma^{j}]=2\mathbb{i}_{8}\epsilon^{ijk}\Sigma^{k}$
$\displaystyle,\quad(\vec{w}\cdot\vec{\Sigma})^{2}=|\vec{w}|^{2}\mathbb{1}_{8},$
(89)
$\displaystyle\frac{1}{2}(\bar{\Sigma}^{\mu}\Sigma^{\nu}+\bar{\Sigma}^{\nu}\Sigma^{\mu})$
$\displaystyle=g^{\mu\nu}\mathbb{1}_{8},$ (90)
$\displaystyle\Sigma^{\mu}\bar{\Sigma}^{\rho}\Sigma^{\nu}=g^{\mu\rho}\Sigma^{\nu}+g^{\rho\nu}\Sigma^{\mu}$
$\displaystyle-g^{\mu\nu}\Sigma^{\rho}+\mathbb{i}_{8}\epsilon^{\mu\rho\nu}_{\quad\lambda}\Sigma^{\lambda},$
(91)
$\displaystyle\bar{\Sigma}^{\mu}\Sigma^{\rho}\bar{\Sigma}^{\nu}=g^{\mu\rho}\bar{\Sigma}^{\nu}+g^{\rho\nu}\bar{\Sigma}^{\mu}$
$\displaystyle-g^{\mu\nu}\bar{\Sigma}^{\rho}-\mathbb{i}_{8}\epsilon^{\mu\rho\nu}_{\quad\lambda}\bar{\Sigma}^{\lambda}.$
(92)

Here $\mathbb{i}_{8}\equiv\mathbb{1}_{4}\otimes\mathbb{i}_{2}$ is an anti-
symmetric matrix who commutes with $\Sigma^{\mu}$ and satisfies
$(\mathbb{i}_{8})^{2}=-\mathbb{1}_{8}$, and $\vec{w}$ is an arbitrary vector.
These identities help to translate some expressions of $\varphi_{a}$ to
bilinear forms. For example,

$\displaystyle\varphi\mathbb{i}_{8}\Sigma^{\mu}\varphi=0,\quad\varphi\Sigma^{\mu}\bar{\Sigma}^{\rho}\Sigma^{\nu}\varphi=g^{\mu\rho}K^{\nu}+g^{\rho\nu}K^{\mu}-g^{\mu\nu}K^{\rho}.$
(93)

Then we evaluate the second derivative of $\mathcal{L}$

$-\frac{\delta^{2}\mathcal{L}}{\delta\varphi_{a}\delta\varphi_{b}}=\Omega_{\rho}\Sigma_{ab}^{\rho}\partial^{2}+\xi_{\mu}\Sigma_{ab}^{\mu}+2\eta_{\mu\nu}(\varphi_{c}\Sigma_{cd}^{\mu}\varphi_{d})\Sigma_{ab}^{\nu}+4\eta_{\mu\nu}\Sigma^{\mu}_{ac}(\varphi_{c}\varphi_{d})\Sigma^{\nu}_{db}.$
(94)

In the following, we work in the frame with the canonical kinetic term with
$\Omega_{\mu}=(1,0,0,0)$, and the scalar mass matrix is

$\displaystyle\mathbf{M}_{S}^{2}(\varphi)_{ab}=A_{ab}+B_{ab},$ (95)
$\displaystyle A_{ab}=A_{\mu}\Sigma_{ab}^{\mu},~{}\quad
A_{\mu}=2\eta_{\mu\nu}K^{\nu}+\xi_{\mu},$ $\displaystyle
B_{ab}=4\eta_{\mu\nu}\Sigma_{ac}^{\mu}\varphi_{c}\varphi_{d}\Sigma_{db}^{\nu}.$

To deal with $\mathbf{Tr}(\mathbf{M}_{S}^{2n})$ in Eq. (29), we expand the
binomial

$\mathbf{Tr}[(A_{ab}+B_{ab})^{n}]=\sum_{l=0}^{n}\sum_{\\{p_{i}\\}}^{\sum
p_{i}=n-l}N_{s}(\\{p_{i}\\})\mathbf{Tr}(A^{p_{1}}BA^{p_{2}}B\cdots
A^{p_{l}}B).$ (96)

And we need to evaluate $(A_{\mu}\Sigma^{\mu})^{p}$. Using the identities in
Eq. (89),

$\displaystyle(A_{\mu}\Sigma^{\mu})^{p}$
$\displaystyle=(A_{0}\mathbb{1}_{8}+\vec{A}\cdot\vec{\Sigma})^{p},$
$\displaystyle=\sum_{k=0}^{p}C_{p}^{k}(A_{0})^{p-k}(\vec{A}\cdot\vec{\Sigma})^{k},$
$\displaystyle=\sum_{k=0}^{p/2}C_{p}^{2k}(A_{0})^{p-2k}|\vec{A}|^{2k}\mathbb{1}_{8}+\sum_{k=0}^{(p-1)/2}C_{p}^{2k+1}(A_{0})^{p-2k-1}|\vec{A}|^{2k}(\vec{A}\cdot\vec{\Sigma}),$
(97)

where $C_{p}^{k}$ is the binomial coefficient and

$\displaystyle A_{0}$
$\displaystyle=2\eta_{00}K_{0}+2\vec{\eta}\cdot\vec{K}+\xi_{0},$ (98)
$\displaystyle\vec{A}$ $\displaystyle=2K_{0}\vec{\eta}+2E\vec{K}+\vec{\xi}.$
(99)

For simplicity, we define a new four-vector $F(p)_{\mu}$ from
$(A_{0},\vec{A})$

$\displaystyle F(p)_{0}$
$\displaystyle\equiv\sum_{k=0}^{p/2}C_{p}^{2k}(A_{0})^{p-2k}|\vec{A}|^{2k},$
(100) $\displaystyle\vec{F}(p)$ $\displaystyle\equiv\left\\{\begin{aligned}
-&\sum_{k=0}^{(p-1)/2}C_{p}^{2k+1}(A_{0})^{p-2k-1}|\vec{A}|^{2k}\vec{A}\qquad&(p\neq
0),\\\ &0\qquad&(p=0),\end{aligned}\right.$ (101)

and we have

$(A_{\mu}\Sigma^{\mu})^{p}=F(p)_{\mu}\bar{\Sigma}^{\mu}.$ (102)

The series in Eq. (96) are then calculated as

$\begin{split}\mathbf{Tr}(A^{p_{1}}BA^{p_{2}}B\cdots
A^{p_{l}}B)&=4^{l}\eta_{\mu_{1}\nu_{1}}\cdots\eta_{\mu_{l}\nu_{l}}\prod_{i=1}^{l}A(p_{i})_{\rho_{i}}\varphi\Sigma^{\nu_{i}}\bar{\Sigma}^{\rho_{i}}\Sigma^{\mu_{i+1}}\varphi\\\
&=4^{l}\eta_{\mu_{1}\nu_{1}}\cdots\eta_{\mu_{l}\nu_{l}}\prod_{i=1}^{l}S_{p_{i}}^{\nu_{i}\mu_{i+1}}\\\
&=4^{l}\mathbf{tr}\left(\eta\cdot S_{p_{1}}\cdots\eta\cdot
S_{p_{l}}\right),\\\ \end{split}$

where $\mu_{l+1}\equiv\mu_{1}$ and the trace $\mathbf{tr}$ is taken in the
orbit space. The symmetric tensor
$S_{p}^{\mu\nu}=F(p)_{\rho}\varphi\Sigma^{\mu}\bar{\Sigma}^{\rho}\Sigma^{\nu}\varphi=F(p)^{\mu}K^{\nu}+F(p)^{\nu}K^{\mu}-g^{\mu\nu}(F(p)K)$.
And the effective potential can be expressed as 444For simplicity, the $\ln
p_{E}$ is dropped here.

$\displaystyle V_{\rm CW}^{(S)}$
$\displaystyle=\frac{1}{2}\int\frac{d^{4}p_{E}}{2\pi^{4}}\left[\mathbf{Tr}\sum_{n=1}^{\infty}\frac{1}{n}\left(-\frac{\mathbf{M}_{S}^{2}}{p^{2}_{E}}\right)^{n}\right]$
$\displaystyle=\frac{1}{2}\int\frac{d^{4}p_{E}}{2\pi^{4}}\sum_{n}(-)^{n}\frac{1}{n(p_{E}^{2})^{n}}\sum_{l=0}^{n}\sum_{\\{p_{i}\\}}^{\sum
p_{i}=n-l}N_{s}(\\{p_{i}\\})\mathbf{Tr}(A^{p_{1}}BA^{p_{2}}B\cdots
A^{p_{l}}B)$
$\displaystyle=\frac{1}{2}\int\frac{d^{4}p_{E}}{2\pi^{4}}\sum_{n}(-)^{n}\frac{1}{n(p_{E}^{2})^{n}}\sum_{l=0}^{n}\sum_{\\{p_{i}\\}}^{\sum
p_{i}=n-l}N_{s}(\\{p_{i}\\})\mathbf{tr}\left(\eta\cdot
S_{p_{1}}\cdots\eta\cdot S_{p_{l}}\right)$ (103)

In the end, the $V_{\rm CW}^{(S)}$ is expressed as a series defined in the
orbit space. It is worth mentioning that the discussion of CP property is
independent of regularization. When the potential is CP-even, as we discussed,
we can apply the CP transformation before and after the regularization and
nothing will change. Finally, we can conclude that the CP property of the (CP
conserving) potential tree-level potential is not violated by the Coleman-
Weinberg potential from scalar loop contribution.

##  Appendix C Renormalization Conditions

To compare with the renormalization conditions in Ref. Basler and Mühlleitner
(2019), we follow their notations and the field expanded around the vacuum
$v_{1},v_{2}$ are

$\Phi_{1}=\frac{1}{\sqrt{2}}\left(\begin{array}[]{c}\rho_{1}+\mathrm{i}\eta_{1}\\\
v_{1}+\zeta_{1}+\mathrm{i}\psi_{1}\end{array}\right),\quad\Phi_{2}=\frac{1}{\sqrt{2}}\left(\begin{array}[]{c}\rho_{2}+\mathrm{i}\eta_{2}\\\
v_{2}+\zeta_{2}+\mathrm{i}\psi_{2}\end{array}\right).$ (104)

The renormalization conditions are

$\displaystyle\partial_{\varphi_{a}}(V_{\rm CT}+V_{\rm
CW})\big{|}_{\varphi_{a}=\langle\varphi_{a}\rangle_{\rm tree}}=0,$ (105)
$\displaystyle\partial_{\varphi_{a}}\partial_{\varphi_{b}}(V_{\rm CT}+V_{\rm
CW})\big{|}_{\varphi_{a}=\langle\varphi_{a}\rangle_{\rm tree}}=0.$ (106)
$\displaystyle\varphi_{a}\equiv\left\\{\rho_{1},\eta_{1},\zeta_{1},\psi_{1},\rho_{2},\eta_{2},\zeta_{2},\psi_{2}\right\\},~{}\langle\varphi_{a}\rangle_{\rm
tree}=\left\\{0,0,v_{1},0,0,0,v_{2},0\right\\}.$

Naively, there are 8+36 renormalization conditions from Eqs. (105) and (106).
However, for any function of the form $f(\Phi^{\dagger}_{i}\Phi_{j})$, its
first and second derivative satisfy some identities so that most of the
renormalization conditions are redundant.

We have the following 5 identities for the first derivatives,

$\displaystyle\partial_{\rho_{1}}=0,$ (107)
$\displaystyle\partial_{\rho_{2}}=0,$ (108)
$\displaystyle\partial_{\eta_{1}}=0,$ (109)
$\displaystyle\partial_{\eta_{2}}=0,$ (110) $\displaystyle
c_{\beta}\partial_{\psi_{1}}+s_{\beta}\partial_{\psi_{2}}=0,$ (111)

where $\partial_{\phi_{i}}=0$ denotes
$\partial_{\phi_{i}}f|_{\phi=\langle\phi\rangle_{\rm tree}}=0$ for any
function $f(\Phi^{\dagger}_{i}\Phi_{j})$ and $\tan\beta=v_{2}/v_{1}$.
Therefore, we are left with 3 independent renormalization conditions from Eq.
(105),

$\displaystyle\partial_{\zeta_{1}}(V_{\rm CT}+V_{\rm CW})=0,$ (112)
$\displaystyle\partial_{\zeta_{2}}(V_{\rm CT}+V_{\rm CW})=0,$ (113)
$\displaystyle(c_{\beta}\partial_{\psi_{2}}-s_{\beta}\partial_{\psi_{1}})(V_{\rm
CT}+V_{\rm CW})=0.$ (114)

We have the following 26 identities for the second derivatives,

$\displaystyle(c_{\beta}\partial_{\rho_{1}}+s_{\beta}\partial_{\rho_{2}})(c_{\beta}\partial_{\eta_{1}}+s_{\beta}\partial_{\eta_{2}})=0,$
(115)
$\displaystyle(c_{\beta}\partial_{\rho_{1}}+s_{\beta}\partial_{\rho_{2}})(c_{\beta}\partial_{\zeta_{1}}+s_{\beta}\partial_{\zeta_{2}})=0,$
(116)
$\displaystyle(c_{\beta}\partial_{\rho_{1}}+s_{\beta}\partial_{\rho_{2}})(c_{\beta}\partial_{\psi_{1}}+s_{\beta}\partial_{\psi_{2}})=0,$
(117)
$\displaystyle(c_{\beta}\partial_{\rho_{1}}+s_{\beta}\partial_{\rho_{2}})(c_{\beta}\partial_{\zeta_{2}}-s_{\beta}\partial_{\zeta_{1}})=0,$
(118)
$\displaystyle(c_{\beta}\partial_{\rho_{1}}+s_{\beta}\partial_{\rho_{2}})(c_{\beta}\partial_{\psi_{2}}-s_{\beta}\partial_{\psi_{1}})=0,$
(119)
$\displaystyle(c_{\beta}\partial_{\eta_{1}}+s_{\beta}\partial_{\eta_{2}})(c_{\beta}\partial_{\zeta_{1}}+s_{\beta}\partial_{\zeta_{2}})=0,$
(120)
$\displaystyle(c_{\beta}\partial_{\eta_{1}}+s_{\beta}\partial_{\eta_{2}})(c_{\beta}\partial_{\psi_{1}}+s_{\beta}\partial_{\psi_{2}})=0,$
(121)
$\displaystyle(c_{\beta}\partial_{\eta_{1}}+s_{\beta}\partial_{\eta_{2}})(c_{\beta}\partial_{\zeta_{2}}-s_{\beta}\partial_{\zeta_{1}})=0,$
(122)
$\displaystyle(c_{\beta}\partial_{\eta_{1}}+s_{\beta}\partial_{\eta_{2}})(c_{\beta}\partial_{\psi_{2}}-s_{\beta}\partial_{\psi_{1}})=0,$
(123)
$\displaystyle(c_{\beta}\partial_{\zeta_{1}}+s_{\beta}\partial_{\zeta_{2}})(c_{\beta}\partial_{\psi_{1}}+s_{\beta}\partial_{\psi_{2}})=0,$
(124)
$\displaystyle(c_{\beta}\partial_{\zeta_{1}}+s_{\beta}\partial_{\zeta_{2}})(c_{\beta}\partial_{\rho_{2}}-s_{\beta}\partial_{\rho_{1}})=0,$
(125)
$\displaystyle(c_{\beta}\partial_{\zeta_{1}}+s_{\beta}\partial_{\zeta_{2}})(c_{\beta}\partial_{\eta_{2}}-s_{\beta}\partial_{\eta_{1}})=0,$
(126)
$\displaystyle(c_{\beta}\partial_{\psi_{1}}+s_{\beta}\partial_{\psi_{2}})(c_{\beta}\partial_{\rho_{2}}-s_{\beta}\partial_{\rho_{1}})=0,$
(127)
$\displaystyle(c_{\beta}\partial_{\psi_{1}}+s_{\beta}\partial_{\psi_{2}})(c_{\beta}\partial_{\eta_{2}}-s_{\beta}\partial_{\eta_{1}})=0,$
(128)
$\displaystyle(c_{\beta}\partial_{\rho_{2}}-s_{\beta}\partial_{\rho_{1}})(c_{\beta}\partial_{\eta_{2}}-s_{\beta}\partial_{\eta_{1}})=0,$
(129)
$\displaystyle(c_{\beta}\partial_{\rho_{2}}-s_{\beta}\partial_{\rho_{1}})(c_{\beta}\partial_{\zeta_{2}}-s_{\beta}\partial_{\zeta_{1}})=0,$
(130)
$\displaystyle(c_{\beta}\partial_{\rho_{2}}-s_{\beta}\partial_{\rho_{1}})(c_{\beta}\partial_{\psi_{2}}-s_{\beta}\partial_{\psi_{1}})=0,$
(131)
$\displaystyle(c_{\beta}\partial_{\eta_{2}}-s_{\beta}\partial_{\eta_{1}})(c_{\beta}\partial_{\zeta_{2}}-s_{\beta}\partial_{\zeta_{1}})=0,$
(132)
$\displaystyle(c_{\beta}\partial_{\eta_{2}}-s_{\beta}\partial_{\eta_{1}})(c_{\beta}\partial_{\psi_{2}}-s_{\beta}\partial_{\psi_{1}})=0,$
(133)
$\displaystyle(c_{\beta}\partial_{\rho_{1}}+s_{\beta}\partial_{\rho_{2}})(c_{\beta}\partial_{\rho_{1}}+s_{\beta}\partial_{\rho_{2}})=(c_{\beta}\partial_{\eta_{1}}+s_{\beta}\partial_{\eta_{2}})(c_{\beta}\partial_{\eta_{1}}+s_{\beta}\partial_{\eta_{2}}),$
(134)
$\displaystyle(c_{\beta}\partial_{\rho_{1}}+s_{\beta}\partial_{\rho_{2}})(c_{\beta}\partial_{\rho_{1}}+s_{\beta}\partial_{\rho_{2}})=(c_{\beta}\partial_{\psi_{1}}+s_{\beta}\partial_{\psi_{2}})(c_{\beta}\partial_{\psi_{1}}+s_{\beta}\partial_{\psi_{2}}),$
(135)
$\displaystyle(c_{\beta}\partial_{\rho_{2}}-s_{\beta}\partial_{\rho_{1}})(c_{\beta}\partial_{\rho_{2}}-s_{\beta}\partial_{\rho_{1}})=(c_{\beta}\partial_{\eta_{2}}-s_{\beta}\partial_{\eta_{1}})(c_{\beta}\partial_{\eta_{2}}-s_{\beta}\partial_{\eta_{1}}),$
(136)
$\displaystyle(c_{\beta}\partial_{\rho_{1}}+s_{\beta}\partial_{\rho_{2}})(c_{\beta}\partial_{\rho_{2}}-s_{\beta}\partial_{\rho_{1}})=(c_{\beta}\partial_{\eta_{1}}+s_{\beta}\partial_{\eta_{2}})(c_{\beta}\partial_{\eta_{2}}-s_{\beta}\partial_{\eta_{1}}),$
(137)
$\displaystyle(c_{\beta}\partial_{\rho_{1}}+s_{\beta}\partial_{\rho_{2}})(c_{\beta}\partial_{\rho_{2}}-s_{\beta}\partial_{\rho_{1}})=(c_{\beta}\partial_{\psi_{1}}+s_{\beta}\partial_{\psi_{2}})(c_{\beta}\partial_{\psi_{2}}-s_{\beta}\partial_{\psi_{1}}),$
(138)
$\displaystyle(c_{\beta}\partial_{\eta_{1}}+s_{\beta}\partial_{\eta_{2}})(c_{\beta}\partial_{\rho_{2}}-s_{\beta}\partial_{\rho_{1}})=-(c_{\beta}\partial_{\rho_{1}}+s_{\beta}\partial_{\rho_{2}})(c_{\beta}\partial_{\eta_{2}}-s_{\beta}\partial_{\eta_{1}}),$
(139)
$\displaystyle(c_{\beta}\partial_{\eta_{1}}+s_{\beta}\partial_{\eta_{2}})(c_{\beta}\partial_{\rho_{2}}-s_{\beta}\partial_{\rho_{1}})=(c_{\beta}\partial_{\psi_{1}}+s_{\beta}\partial_{\psi_{2}})(c_{\beta}\partial_{\zeta_{2}}-s_{\beta}\partial_{\zeta_{1}}).$
(140)

Then, there are 10 independent renormalization conditions from the second
derivatives. However, three of them are satisfied automatically when the
renormalization conditions from the first derivatives are satisfied, because
of the following identities,

$\displaystyle(c_{\beta}\partial_{\rho_{1}}+s_{\beta}\partial_{\rho_{2}})^{2}=\frac{1}{2v}(c_{\beta}\partial_{\zeta_{1}}+s_{\beta}\partial_{\zeta_{2}}),$
(141)
$\displaystyle(c_{\beta}\partial_{\rho_{1}}+s_{\beta}\partial_{\rho_{2}})(c_{\beta}\partial_{\rho_{2}}-s_{\beta}\partial_{\rho_{1}})=\frac{1}{2v}(c_{\beta}\partial_{\zeta_{2}}-s_{\beta}\partial_{\zeta_{1}}),$
(142)
$\displaystyle(c_{\beta}\partial_{\rho_{1}}+s_{\beta}\partial_{\rho_{2}})(c_{\beta}\partial_{\eta_{2}}-s_{\beta}\partial_{\eta_{1}})=\frac{1}{2v}(c_{\beta}\partial_{\psi_{2}}-s_{\beta}\partial_{\psi_{1}}).$
(143)

Finally, we are left with only 7 independent renormalization conditions from
Eq. (106),

$\displaystyle(c_{\beta}\partial_{\rho_{2}}-s_{\beta}\partial_{\rho_{1}})^{2}(V_{\rm
CT}+V_{\rm CW})=0,$ (144)
$\displaystyle(c_{\beta}\partial_{\zeta_{1}}+s_{\beta}\partial_{\zeta_{2}})^{2}(V_{\rm
CT}+V_{\rm CW})=0,$ (145)
$\displaystyle(c_{\beta}\partial_{\zeta_{2}}-s_{\beta}\partial_{\zeta_{1}})^{2}(V_{\rm
CT}+V_{\rm CW})=0,$ (146)
$\displaystyle(c_{\beta}\partial_{\psi_{2}}-s_{\beta}\partial_{\psi_{1}})^{2}(V_{\rm
CT}+V_{\rm CW})=0,$ (147)
$\displaystyle(c_{\beta}\partial_{\zeta_{1}}+s_{\beta}\partial_{\zeta_{2}})(c_{\beta}\partial_{\zeta_{2}}-s_{\beta}\partial_{\zeta_{1}})(V_{\rm
CT}+V_{\rm CW})=0,$ (148)
$\displaystyle(c_{\beta}\partial_{\zeta_{1}}+s_{\beta}\partial_{\zeta_{2}})(c_{\beta}\partial_{\psi_{2}}-s_{\beta}\partial_{\psi_{1}})(V_{\rm
CT}+V_{\rm CW})=0,$ (149)
$\displaystyle(c_{\beta}\partial_{\zeta_{2}}-s_{\beta}\partial_{\zeta_{1}})(c_{\beta}\partial_{\psi_{2}}-s_{\beta}\partial_{\psi_{1}})(V_{\rm
CT}+V_{\rm CW})=0.$ (150)

And we have 10 independent renormalization conditions from Eqs. (112)-(114)
and Eqs. (144)-(150) in total.

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