#  HUPD-2105 A model with light and heavy scalars
in view of the effective theory

Apriadi Salim Adam1111E-mail address<EMAIL_ADDRESS>Yuta
Kawamura2222E-mail address<EMAIL_ADDRESS>and Takuya
Morozumi3,4333E-mail address<EMAIL_ADDRESS>

1Research Center for Physics, National Research and Innovation Agency (BRIN),
Serpong PUSPIPTEK Area, Tangerang Selatan 15314, Indonesia 2Graduate School of
Science, Hiroshima University,
Higashi-Hiroshima 739-8526, Japan 3Physics Program, Graduate School of
Advanced Science and Engineering, Hiroshima University,
Higashi-Hiroshima 739-8526, Japan 4Core of Research for the Energetic
Universe, Hiroshima University,
Higashi-Hiroshima 739-8526, Japan

( Abstract
The low energy effective potential for the model with a light scalar and a
heavy scalar is derived. We perform the path integration for both heavy and
light scalars and derive the low energy effective potential in terms of only
the light scalar. The effective potential is independent of the
renormalization scale approximately. By setting the renormalization scale
equal to the mass of the heavy scalar, one finds the corrections with the
logarithm of the ratio of the two scalar masses. The large logarithm is summed
with the renormalization group (RG) and the RG improved effective potential is
derived. The improved effective potential includes the one-loop correction of
the heavy scalar and the leading logarithmic corrections due to the light
scalar. We study the correction to the vacuum expectation value of the light
scalar and the dependence on the mass of the heavy scalar.  )

##  1 Introduction

Among various models beyond the standard model, the models including particles
with large mass hierarchies lead to two types of the corrections to the low
energy effective potential which is obtained by integrating the heavy
particles. One is large logarithmic corrections to the dimensionless coupling
constants and they have the form of the logarithm of the large energy scale
ratios. Another type of the corrections occurs for the mass parameters and the
cosmological constants. The correction to the light scalar mass often has the
form proportional to the heavy scalar mass. This type of correction
destabilizes the initial mass scale set in the tree level approximation. The
correction to the cosmological constants is also sensitive to the masses
included in the model.

In this paper, we study the model with two scalars and derive the low energy
effective potential which contains afore-mentioned corrections. The model
which we consider is a toy model which explains the origin of the small Yukawa
coupling to neutrinos 444In the framework of the standard model gauge group,
the two Higgs doublet model with the spirit similar to the present model has
been studied in [1, 2]. The constraints from electroweak precision data to
these models are discussed in [3].. In the present model, the light scalar
which corresponds to the standard model-like Higgs does not directly couple to
the neutrino. The light and heavy scalars have a mass mixing term and the
neutrino Yukawa coupling to the light scalar is generated after the heavy
scalar is integrated out. Then the Yukawa coupling is naturally suppressed by
a factor; the small mass mixing term between heavy and light scalars divided
by heavy scalar mass.

To suppress Yukawa coupling of the neutrinos, we need to increase the heavy
scalar mass provided that the small mass mixing term is fixed. By increasing
the heavy scalar mass, the couplings, the masses and the cosmological
constants in the low energy are changed. To quantify the change, we derive the
low energy effective potential for the model. With the effective theory
approach, one can show how the parameters at the low energy are related to
those of the full theory[4]. The vacuum expectation value (vev) of the lighter
scalar is also sensitive to the full theory parameters through the low energy
parameters of the effective theory. We study the dependence of the vev on the
heavy scalar mass. If one requires the correction to the vev lies within some
small range, one can set the upper bound on the heavy scalar mass.

To derive the effective potential, we introduce the source for the light
scalar only. One first integrates both light scalar and heavy scalar under the
constant expectation value of the light scalar. With this method, one particle
irreducibility of the light scalar is maintained while the heavy scalar
exchanged diagram that is no longer one-particle irreducible is also included.
Then we employ the approximation of the effective potential by keeping the
first few series of the small expansion parameters. Then the obtained
effective potential is also independent of the renormalization scale
approximately.

When there is a large mass difference between the light scalar and the heavy
scalar, the logarithm of their mass ratio is large and they should be
resummed. For this purpose, we derive the low energy effective theory for the
light scalar and apply the renormalization group (RG) equation in the leading
logarithmic approximation. Then we derive the RG improved effective potential.

The rest of this paper is organized as follows. In section 2, we present the
action for the model with heavy and light scalars. The definition of the
effective action for the light scalar is given in section 3. In section 4, we
integrate out the light and the heavy scalars. In section 5, the counter terms
and the renormalized effective potential are derived. In Section 6, the RG
improved effective potential is obtained and Section 7 is devoted to summary
and discussion.

##  2 The action for the model in terms of the renormalized quantities and
fields

We present a simple toy model which leads to a tiny Yukawa coupling to
neutrinos with the light scalar. The light scalar does not couple to the
neutrino directly and it couples through the second scalar with the mixing
term. The mass of the second scalar is large and it directly couples to the
neutrino. We denote the light scalar as $\rho_{1}$ and the heavy scalar as
$\rho_{2}$ respectively . The neutrino is denoted as $n$. The action is
written in terms of bare fields $\rho_{0i}$ , bare masses $m_{0i}$ and bare
couplings $\lambda_{0i}$ as,

$\displaystyle S=$ $\displaystyle\int
d^{d}x\left(-\frac{1}{2}\sum_{i=1}^{2}\rho_{0i}\left(\Box+m_{0i}^{2}+\frac{\lambda_{0i}}{2}\rho_{0i}^{2}\right)\rho_{0i}-\frac{\lambda_{03}}{4}\rho_{01}^{2}\rho_{02}^{2}-(y_{0}\bar{n_{0}}n_{0}+m_{012}^{2}\rho_{01})\rho_{02}\right.$
(1)
$\displaystyle\left.+\sum_{i=1}^{2}h_{0i}m^{4}_{0i}+h_{012}{m_{0}}_{12}^{4}+2h_{03}m^{2}_{01}m^{2}_{02}\right),$

where $m_{012}^{2}$ is the bare mixing mass. $y_{0}$ is the Yukawa coupling
between the neutrino and the heavy scalar. The last lines of Eq.(1) correspond
to the cosmological constants which consist of the mass parameters of the
model [5]. We do not include the kinetic term for the neutrino and its quantum
correction is not considered.

The model is renormalizable by imposing the following two $Z_{2}$ symmetries.
One of them is an exact symmetry called as $Z_{2}$ and another is a softly
broken symmetry called as $Z^{\prime}_{2}$. The charge assignment is shown in
Table 1. Two scalar fields $\rho_{1},\rho_{2}$ and right-handed neutrino
$n_{R}$ have odd $Z_{2}$ parity and the left-handed neutrino $n_{L}$ has even
$Z_{2}$ parity. Under the softly broken $Z^{\prime}_{2}$, only the scalar
field $\rho_{1}$ has odd parity. With this assignment, only the heavy scalar
$\rho_{2}$ has the Dirac type Yukawa coupling to the neutrino. In the scalar
potential, the cubic interactions of scalars are forbidden by $Z_{2}$
symmetry. About quartic couplings of the scalar field, one must have even
number of each scalar field due to the $Z^{\prime}_{2}$ symmetry. About the
quadratic part, the mixing term which is proportional to $\rho_{1}\rho_{2}$ is
allowed, and it softly breaks $Z^{\prime}_{2}$ symmetry. Concerning the Yukawa
interaction between neutrinos and scalar fields, only the Dirac type Yukawa
coupling with the type $\rho_{2}\overline{n_{L}}{n_{R}}$ is allowed. All the
other Yukawa couplings, such as $\rho_{1}\overline{n_{L}}{n_{R}}$ ,
$\rho_{i}\overline{(n_{L})^{c}}{n_{L}}$ and
$\rho_{i}\overline{(n_{R})^{c}}{n_{R}}$ are forbidden by considering both
$Z_{2}$ and $Z^{\prime}_{2}$ symmetries. About the neutrino mass term with the
dimension three, Dirac mass term $m_{\nu}\overline{n_{L}}{n_{R}}$ is forbidden
due to $Z_{2}$ symmetry. To forbid Majorana mass terms such as
$\overline{(n_{R})^{c}}{n_{R}}$ and $\overline{(n_{L})^{c}}{n_{L}}$, we impose
the symmetry under the transformation, $(n_{L},n_{R})$ into
$e^{i\frac{\pi}{2}}(n_{L},n_{R})$.

Symmetry | $\rho_{1}$ | $\rho_{2}$ | $n_{L}$ | $n_{R}$
---|---|---|---|---
$Z_{2}$ | – | – | + | –
$Z_{2}^{\prime}$ | – | + | + | +
Table 1: The charge assignment under $Z_{2}$ and $Z_{2}^{\prime}$ symmetries.

Next we rewrite the action in Eq.(1) in terms of the renormalized fields,
renormalized couplings and masses. The relations between bare quantities and
renomalized ones are given as follows;

$\displaystyle\rho_{0i}$ $\displaystyle=$ $\displaystyle\sqrt{Z_{i}}\rho_{i},$
(2) $\displaystyle n_{0}$ $\displaystyle=$ $\displaystyle\sqrt{Z_{n}}n,$ (3)
$\displaystyle m_{0i}^{2}Z_{i}$ $\displaystyle=$
$\displaystyle\sum_{j=1}^{2}Z_{mij}m_{j}^{2},$ (4) $\displaystyle
m_{012}^{2}\sqrt{Z_{1}Z_{2}}$ $\displaystyle=$ $\displaystyle
m^{2}_{12}Z_{12},$ (5) $\displaystyle\lambda_{0i}Z_{i}^{2}$ $\displaystyle=$
$\displaystyle\sum_{I=1}^{3}Z_{\lambda_{iI}}\lambda_{I}\mu^{2\eta},$ (6)
$\displaystyle\lambda_{03}Z_{1}Z_{2}$ $\displaystyle=$
$\displaystyle\sum_{I=1}^{3}Z_{\lambda_{3I}}\lambda_{I}\mu^{2\eta},$ (7)
$\displaystyle y_{0}Z_{n}\sqrt{Z_{2}}$ $\displaystyle=$ $\displaystyle
Z_{y}y\mu^{\eta},$ (8)

where the index $i$ ($i=1,2$) is not summed, $\mu$ denotes the renormalization
scale and $\eta$ is $2-\frac{d}{2}$. As for the cosmological constants, the
relations between the renormalized parameters and the bare ones are given as
follows,

$\displaystyle\sum_{i=1}^{2}h_{0i}m^{4}_{0i}+2h_{03}m^{2}_{01}m^{2}_{02}$
$\displaystyle=$
$\displaystyle\mu^{-2\eta}\left(\sum_{i=1}^{2}Z_{hi}h_{i}{m_{i}}^{4}+2Z_{h3}h_{3}m^{2}_{1}m^{2}_{2}\right),$
(9) $\displaystyle h_{012}{m_{0}}_{12}^{4}$ $\displaystyle=$
$\displaystyle\mu^{-2\eta}Z_{h12}h_{12}m_{12}^{4}.$ (10)

Using Eqs.(2)-(10), the action in Eq.(1) can be written in terms of
renormalized quantities as,

$\displaystyle S[\rho_{1},\rho_{2},n]=$ $\displaystyle-\frac{1}{2}\int
d^{d}x\sum_{i=1}^{2}\left(Z_{i}\rho_{i}\Box\rho_{i}+\rho_{i}^{2}Z_{mij}m_{j}^{2}+\frac{\mu^{2\eta}}{2}\sum_{I=1}^{3}\left(\rho_{i}^{4}Z_{\lambda_{iI}}\lambda_{I}+\rho_{1}^{2}\rho_{2}^{2}Z_{\lambda_{3}I}\lambda_{I}\right)\right)$
$\displaystyle-\int
d^{d}x\left(Z_{y}y\mu^{\eta}\bar{n}n+Z_{12}m_{12}^{2}\rho_{1}\right)\rho_{2}$
$\displaystyle+\int
d^{d}x\mu^{-2\eta}\left(\sum_{i=1}^{2}Z_{hi}h_{i}{m_{i}}^{4}+2Z_{h3}h_{3}m^{2}_{1}m^{2}_{2}+Z_{h12}h_{12}m_{12}^{4}\right).$
(11)

##  3 Definition of the effective action

In this section, we give a definition of the effective action for the light
scalar $\rho_{1}$ by integrating the heavy scalar $\rho_{2}$ as well as the
quantum fluctuation of the light scalar. To begin with, we define the
generating functional $W[J_{1},n]$,

$\displaystyle e^{iW[J_{1},n]}=\int d\rho_{1}\int
d\Delta_{2}e^{iS[\rho_{1},\Delta_{2},n]+i\int\rho_{1}J_{1}d^{4}x},$ (12)

where we have introduced the source term $J_{1}$ for the lighter field
$\rho_{1}$. About the heavy scalar field, we do not introduce any source term.
Instead, we integrate it by setting $\rho_{2}=\Delta_{2}$ in the action
Eq.(11) where we expand the heavy scalar field around the vanishing VEV.
$\bar{\rho}_{1}$ which is the expectation value of $\rho_{1}$ is defined by,

$\displaystyle\bar{\rho}_{1}|_{J_{1}}=\frac{\delta W[J_{1},n]}{\delta
J_{1}}=\frac{\int d\rho_{1}\int
d\Delta_{2}\rho_{1}e^{iS[\rho_{1},\Delta_{2},n]+i\int\rho_{1}J_{1}d^{4}x}}{\int
d\rho_{1}\int
d\Delta_{2}e^{iS[\rho_{1},\Delta_{2}]+i\int\rho_{1}J_{1}d^{4}x}}.$ (13)

Then one can define the effective action $\Gamma_{\operatorname{eff}}$, a
functional of $\bar{\rho}_{1}$ by Legendre transformation of $W[J_{1},n]$ as,

$\displaystyle\Gamma_{\operatorname{eff}}[\bar{\rho}_{1},n]$
$\displaystyle=W[J_{1},n]-\int J_{1}\bar{\rho}_{1}d^{4}x$
$\displaystyle=-i\log\int d\Delta_{1}\int
d\Delta_{2}e^{iS[\bar{\rho}_{1}+\Delta_{1},\Delta_{2},n]-i\int\Delta_{1}\frac{\delta\Gamma_{\operatorname{eff}}[\bar{\rho}_{1},n]}{\delta\bar{\rho}_{1}(x)}d^{4}x},$
(14) $\displaystyle\Delta_{1}\equiv\rho_{1}-\bar{\rho}_{1},$ (15)

where we substitute the relation
$J_{1}(x)=-\frac{\delta\Gamma_{\operatorname{eff}}[\bar{\rho}_{1},n]}{\delta\bar{\rho}_{1}(x)}$.
In Eq.(14), We change the path integral variable from $\rho_{1}$ to
$\Delta_{1}$ as defined in Eq.(15). $\Delta_{1}$ is the quantum fluctuation
from the expectation value $\bar{\rho}_{1}$ . Using Eq.(13), one can show
that,

$\displaystyle\int d\Delta_{1}\Delta_{1}\int
d\Delta_{2}e^{iS[\bar{\rho}_{1}+\Delta_{1},\Delta_{2},n]-i\int\Delta_{1}\frac{\delta\Gamma_{\operatorname{eff}}[\bar{\rho}_{1},n]}{\delta\bar{\rho}_{1}(x)}d^{4}x}$
$\displaystyle=$ $\displaystyle 0.$ (16)

The tadpole vanishing condition for $\Delta_{1}$ leads to the one particle
irreducibility of the effective action
$\Gamma_{\operatorname{eff}}[\bar{\rho}_{1},n]$. Concerned with the heavier
field $\Delta_{2}$, the one particle reducible diagram is included.

##  4 Integrating the scalar fields

In this section, we will integrate the scalar fields out. We assume the
following hierarchy for the mass parameters.

$\displaystyle m_{2}^{2}\gg-m_{1}^{2}\simeq\epsilon
m_{12}^{2}>0,\quad\epsilon=\frac{m_{12}^{2}}{m_{2}^{2}}\ll 1$ (17)

We first integrate the heavy scalar field $\Delta_{2}$ in Eq.(14). Next, we
integrate the lighter scalar field $\Delta_{1}$ which denotes the quantum
fluctuation around the background field $\bar{\rho}_{1}$. Concerning the
corrections, we keep them up to the second order of the coupling constants
within 1 loop approximation. Below we show the steps to integrate the heavy
scalar field $\Delta_{2}$ and the light scalar field $\Delta_{1}$.

###  4.1 Integrating heavy scalar field

Using the relation $\Delta_{1}$ in Eq.(15) and Eq.(11), the action in Eq.(14)
is written as,

$\displaystyle S[\bar{\rho}_{1}+\Delta_{1},\Delta_{2},n]$ $\displaystyle=$
$\displaystyle S[\bar{\rho}_{1},0,0]+\int d^{d}x\Delta_{i}(x)\frac{\delta
S[\rho_{1},\rho_{2,}n]}{\delta\rho_{i}(x)}|_{\rho_{1}=\bar{\rho}_{1},\rho_{2}=0}$
(18) $\displaystyle+$ $\displaystyle\frac{1}{2}\int d^{d}x\int
d^{d}y\Delta_{i}(x)\frac{\delta
S[\rho_{1},\rho_{2,}n]}{\delta\rho_{i}(x)\delta\rho_{j}(y)}|_{\rho_{1}=\bar{\rho}_{1},\rho_{2}=0}\Delta_{j}(y)$
$\displaystyle+$ $\displaystyle
S_{\operatorname{int}}(\Delta_{i},\bar{\rho}_{1}).$

Each term in Eq(18) is given as follows,

$\displaystyle S[\bar{\rho}_{1},0,0]$ $\displaystyle=$ $\displaystyle\int
d^{d}x\left(\frac{Z_{1}}{2}\partial^{\mu}\bar{\rho}_{1}\partial_{\mu}\bar{\rho}_{1}-\frac{Z_{m11}}{2}m_{1}^{2}\bar{\rho}_{1}^{2}-\frac{\lambda_{01}Z_{1}^{2}}{4}\bar{\rho}_{1}^{4}\right)$
(19) $\displaystyle+$ $\displaystyle\int
d^{d}x\mu^{-2\eta}\left(\sum_{i=1}^{2}Z_{hi}h_{i}{m_{i}}^{4}+2Z_{h3}h_{3}m^{2}_{1}m^{2}_{2}+Z_{h12}h_{12}m_{12}^{4}\right)$
$\displaystyle\frac{\delta
S[\rho_{1},\rho_{2,}n]}{\delta\rho_{i}(x)}|_{\rho_{1}=\bar{\rho}_{1},\rho_{2}=0}$
$\displaystyle=$
$\displaystyle-\left(\begin{array}[]{c}\\{Z_{1}(\Box+m_{01}^{2})+\lambda_{01}Z_{1}^{2}\bar{\rho}_{1}^{2}\\}\bar{\rho}_{1}\\\
\mu^{\eta}yZ_{y}\bar{n}n+Z_{12}m_{12}^{2}\bar{\rho}_{1}\end{array}\right),$
(22) $\displaystyle\frac{\delta
S[\rho_{1},\rho_{2,}n]}{\delta\rho_{i}(x)\delta\rho_{j}(y)}|_{\rho_{1}=\bar{\rho}_{1},\rho_{2}=0}$
$\displaystyle=$
$\displaystyle-\left(\begin{array}[]{cc}Z_{1}(\Box+m_{01}^{2})+3\lambda_{01}Z_{1}^{2}\bar{\rho}_{1}^{2}&Z_{12}m_{12}^{2}\\\
Z_{12}m_{12}^{2}&Z_{2}(\Box_{x}+m_{02}^{2})+\frac{\lambda_{03}}{2}Z_{1}Z_{2}\bar{\rho}_{1}^{2}\end{array}\right)$
(26) $\displaystyle\times\delta^{d}(x-y),$ $\displaystyle
S_{\operatorname{int}}(\Delta_{i},\bar{\rho}_{1})$ $\displaystyle=$
$\displaystyle
S_{1\operatorname{int}}(\Delta_{1},\bar{\rho}_{1})+S_{2\operatorname{int}}(\Delta_{2})+S_{12\operatorname{int}}(\Delta_{1},\Delta_{2},\bar{\rho}_{1}),$
(27) $\displaystyle S_{1\operatorname{int}}(\Delta_{1},\bar{\rho}_{1})$
$\displaystyle=$ $\displaystyle-\int
d^{d}x\lambda_{01}Z_{1}^{2}\Delta_{1}^{3}\bar{\rho}_{1}-\int
d^{d}x\frac{1}{4}\lambda_{01}Z_{1}^{2}\Delta_{1}^{4},$ (28) $\displaystyle
S_{2\operatorname{int}}(\Delta_{2})$ $\displaystyle=$ $\displaystyle-\int
d^{d}x\frac{1}{4}\lambda_{02}Z_{2}^{2}\Delta_{2}^{4},$ (29) $\displaystyle
S_{12\operatorname{int}}(\Delta_{1},\Delta_{2},\bar{\rho}_{1})$
$\displaystyle=$ $\displaystyle-\int
d^{d}x\frac{1}{2}\lambda_{03}Z_{1}Z_{2}\Delta_{1}\Delta_{2}^{2}\bar{\rho}_{1}-\int
d^{d}x\frac{1}{4}\lambda_{03}Z_{1}Z_{2}\Delta_{1}^{2}\Delta_{2}^{2}.$ (30)

We define the following quantity by subtracting the classical action from the
effective action,

$\displaystyle\tilde{\Gamma}_{\operatorname{eff}}[\bar{\rho}_{1},n]$
$\displaystyle=$
$\displaystyle\Gamma_{\operatorname{eff}}[\bar{\rho}_{1},n]-S[\bar{\rho}_{1},0,0],$
(31)

with $\tilde{\Gamma}_{\operatorname{eff}}[\bar{\rho}_{1},n]$ is written as
follows,

$\displaystyle e^{i\tilde{\Gamma}_{\operatorname{eff}}[\bar{\rho}_{1},n]}$
$\displaystyle=\int d\Delta_{1}e^{i\left\\{\frac{1}{2}\int d^{d}x\int
d^{d}y\Delta_{1}(x)\frac{\delta^{2}S[\bar{\rho}_{1},0,n]}{\delta\bar{\rho}_{1}(x)\delta\bar{\rho}_{1}(y)}\Delta_{1}(y)+S_{1\operatorname{int}}(\Delta_{1},\bar{\rho}_{1})-\int
d^{d}x\Delta_{1}(x)\left(\frac{\delta\tilde{\Gamma}_{\operatorname{eff}}[\bar{\rho}_{1},n]}{\delta\bar{\rho}_{1}(x)}\right)\right\\}}e^{iW_{2}[J_{2},\Delta_{1},n]}.$
(32)

The last factor of Eq(32) summarizes the contribution from the heavy field to
the effective action. It is written in terms of the path integral with respect
to $\Delta_{2}$ as follows,

$\displaystyle e^{iW_{2}[J_{2},\bar{\rho}_{1},\Delta_{1}]}$ $\displaystyle=$
$\displaystyle\int d\Delta_{2}e^{\frac{i}{2}\int d^{d}x\int
d^{d}y\Delta_{2}(x)\frac{\delta^{2}S[\rho_{1},\rho_{2,}n]}{\delta\rho_{2}(x)\delta\rho_{2}(y)}|_{\rho_{1}=\bar{\rho}_{1},\rho_{2}=0}\Delta_{2}(y)+iS_{2\operatorname{int}}(\Delta_{2})+iS_{12\operatorname{int}}(\Delta_{1},\Delta_{2},\bar{\rho}_{1})+i\int
d^{d}x\Delta_{2}(x)J_{2}(x)},$ (33)

where $J_{2}$ is given by,

$\displaystyle J_{2}(x)=\left(\frac{\delta
S[\rho_{1},\rho_{2,}n]}{\delta\rho_{2}(x)}|_{\rho_{1}=\bar{\rho}_{1},\rho_{2}=0}-Z_{12}m_{12}^{2}\Delta_{1}\right)=-Z_{12}m^{2}_{12}(\Delta_{1}+\bar{\rho}_{1})-\mu^{\eta}yZ_{y}\bar{n}n.$
(34)

$J_{2}$ consists of the fields which linearly couple to the heavy scalar
$\Delta_{2}$. We note that $W_{2}$ in Eq.(33) is the generating functional for
Green functions for the heavy scalar $\Delta_{2}$. We calculate
$W_{2}[J_{2},\bar{\rho}_{1},\Delta_{1}]$ by integrating the heavy scalar field
$\Delta_{2}$. The result can be compactly written by introducing the
exponentiated functional differentiation and the notation
$\langle\cdot\cdot\cdot\rangle_{0}$ as,

$\displaystyle\braket{F[\Delta_{2}]}_{0}\equiv\exp\left[\frac{1}{2}\frac{\delta}{\delta\Delta_{2}}\cdot
D_{F22}\frac{\delta}{\delta\Delta_{2}}\right]F[\Delta_{2}]\Big{|}_{\Delta_{2}=0},$
(35)

where

$\displaystyle\frac{\delta}{\delta\Delta_{2}}\cdot
D_{F22}\frac{\delta}{\delta\Delta_{2}}\equiv\int d^{d}x\int
d^{d}y\frac{\delta}{\delta\Delta_{2}(x)}D_{F22}(x,y)\frac{\delta}{\delta\Delta_{2}(y)},$
(36)

$D_{Fii}(x,y)$ denotes the Feynman propagator for $\Delta_{i}$. Using the
notation, the generating functional is given as follows;

$\displaystyle e^{iW_{2}[J_{2},\bar{\rho}_{1},\Delta_{1}]}$
$\displaystyle=\left(\det\frac{\delta^{2}S[\rho_{1},\rho_{2,}n]}{\delta\rho_{2}(x)\delta\rho_{2}(y)}\Bigr{|}_{\rho_{1}=\bar{\rho}_{1},\rho_{2}=0}\right)^{-\frac{1}{2}}\langle
e^{iS_{2\operatorname{int}}(\Delta_{2})+iS_{12\operatorname{int}}(\Delta_{1},\Delta_{2},\bar{\rho}_{1})}\rangle_{0}\
e^{iW_{2}^{c}[J_{2},\bar{\rho}_{1},\Delta_{1}]},$ (37)

where $e^{iW_{2}^{c}[J_{2},\bar{\rho}_{1},\Delta_{1}]}$ is given as,

$\displaystyle e^{iW_{2}^{c}[J_{2},\bar{\rho}_{1},\Delta_{1}]}$
$\displaystyle=$ $\displaystyle\frac{\left\langle
e^{iS_{2\operatorname{int}}(\Delta_{2})+iS_{12\operatorname{int}}(\Delta_{1},\Delta_{2},\bar{\rho}_{1})+i\int
d^{d}x\Delta_{2}(x)J_{2}(x)}\right\rangle_{0}}{\langle
e^{iS_{2\operatorname{int}}(\Delta_{2})+iS_{12\operatorname{int}}(\Delta_{1},\Delta_{2},\bar{\rho}_{1})}\rangle_{0}}.$
(38)

In Eq.(37), the first factor is the vacuum graph contribution from the
quadratic part of $\Delta_{2}$ and the second part corresponds to that from
the interaction. The third factor is the connected Green function contribution
of $\Delta_{2}$ with the source term $J_{2}$ as given in Eq.(34). Next we
define $D^{-1}_{11}(x,y)$ and $D^{-1}_{22}(x,y)$ as follows,

$\displaystyle-D^{-1}_{11}(x,y)\equiv-\\{Z_{1}(\Box_{x}+m_{01}^{2})+3\lambda_{01}Z_{1}^{2}\bar{\rho}_{1}^{2}(x)\\}\delta^{d}(x-y),$
(39)
$\displaystyle-D^{-1}_{22}(x,y)\equiv-\left\\{Z_{2}(\Box_{x}+m_{02}^{2})+\frac{\lambda_{03}}{2}Z_{1}Z_{2}\bar{\rho}_{1}^{2}(x)\right\\}\delta^{d}(x-y),$
(40)

where $D^{-1}_{ii}(x,y)$ depends on the background field $\bar{\rho}_{1}$. The
propagators $D_{F11}(x,y)$ and $D_{F22}(x,y)$ are determined so that they
satisfy the following equations,

$\displaystyle\int d^{d}yiD^{-1}_{ii}(x,y)D_{Fii}(y,z)$ $\displaystyle=$
$\displaystyle\delta^{d}(x-z)\ \ (i=1,2).$ (41)

The propagators are symbolically written as follows,

$\displaystyle D_{F11}(x,y)$ $\displaystyle=$
$\displaystyle-\frac{i}{Z_{1}(\Box_{x}+m_{01}^{2})+3\lambda_{01}Z_{1}^{2}\bar{\rho}_{1}^{2}(x)},$
(42) $\displaystyle D_{F22}(x,y)$ $\displaystyle=$
$\displaystyle-\frac{i}{Z_{2}(\Box_{x}+m_{02}^{2})+\frac{\lambda_{03}}{2}Z_{1}Z_{2}\bar{\rho}_{1}^{2}(x)}.$
(43)

Note that $D_{Fii}(x,y)$ also depends on the background field
$\bar{\rho}_{1}$. One can write $W_{2}^{c}[J_{2},\Delta_{1}]$ as the sum of
the contribution from the tree diagrams and that of one loop diagrams,

$\displaystyle iW_{2}^{c}[J_{2},\bar{\rho}_{1},\Delta_{1}]$
$\displaystyle=iW_{2}^{\operatorname{c(tree)}}[J_{2},\bar{\rho}_{1},\Delta_{1}]+i\bar{W_{2}}^{c(1\operatorname{loop})}[\bar{\rho}_{1},n].$
(44)

$iW_{2}^{c}$ is obtained from the definition of Eq.(38) by using the
differentiation in Eq.(35). In Eq.(44), $iW_{2}^{\operatorname{c(tree)}}$ is
given as,

$\displaystyle
iW_{2}^{\operatorname{c(tree)}}[J_{2},\bar{\rho}_{1},\Delta_{1}]$
$\displaystyle=-\frac{1}{2}\int d^{d}x\int
d^{d}yJ_{2}(x)D_{F22}(x,y)J_{2}(y)-i\frac{\lambda_{02}}{4}\int
d^{d}xD_{J}(x)^{4}$ $\displaystyle-i\frac{\lambda_{3}\mu^{2\eta}}{4}\int
d^{d}x(\Delta_{1}^{2}(x)+2\Delta_{1}(x)\bar{\rho}_{1}(x))D_{J}(x)^{2}-\frac{\lambda_{02}^{2}}{2}\int
d^{d}x\int d^{d}yD_{F22}(x,y)D_{J}(x)^{3}D_{J}(y)^{3}$
$\displaystyle-\frac{\lambda_{3}^{2}\mu^{4\eta}}{8}\int d^{d}x\int
d^{d}y(\Delta_{1}^{2}(x)+2\Delta_{1}(x)\bar{\rho}_{1}(x))(\Delta_{1}^{2}(y)+2\Delta_{1}(y)\bar{\rho}_{1}(y))D_{F22}(x,y)D_{J}(x)D_{J}(y)$
$\displaystyle-\frac{\lambda_{2}\lambda_{3}\mu^{4\eta}}{2}\int d^{d}x\int
d^{d}y(\Delta_{1}^{2}(x)+2\Delta_{1}(x)\bar{\rho}_{1}(x))D_{F22}(x,y)D_{J}(x)D_{J}(y)^{3},$
(45)

where we have used the following definition,

$\displaystyle D_{J}(x)^{n}=\prod_{i=1}^{n}\int
d^{d}x_{i}D_{F22}(x,x_{i})iJ_{2}(x_{i}),\quad(n=1,2,3...).$ (46)

|  |
---|---|---
Figure 1: The tree level diagrams which contribute to
$i\bar{W}_{2}^{c(\operatorname{tree})}[\bar{\rho}_{1},n]$. All the propagators
are heavy scalars and the external source terms denoted by $O$ correspond to
the background field $\bar{\rho}_{1}$ and the neutrino bilinear
$\overline{n}n$. From the figure of the left to that of the right, each
Feynman diagram corresponds to the terms of Eq.(47) from the first term to the
last term respectively.

The tree contribution is obtained by setting $\Delta_{1}$ to zero in Eq.(45),

$\displaystyle i\bar{W}_{2}^{c(\operatorname{tree})}[\bar{\rho}_{1},n]$
$\displaystyle\equiv$ $\displaystyle
iW_{2}^{c(\operatorname{tree})}[J_{2}(\Delta_{1}=0),\bar{\rho}_{1},0]$ (47)
$\displaystyle=$ $\displaystyle-\frac{1}{2}\int d^{d}x\int
d^{d}yO(x)D_{F22}(x,y)O(y)-i\frac{\lambda_{02}}{4}\int d^{d}xD_{O}(x)^{4}$
$\displaystyle-\frac{\lambda_{02}^{2}}{2}\int d^{d}x\int
d^{d}yD_{F22}(x,y)D_{O}(x)^{3}D_{O}(y)^{3},$

where we have defined $D_{O}(x)$ as follows,

$\displaystyle D_{O}(x)=\int d^{d}x_{i}D_{F22}(x,x_{i})iO(x_{i}),$ (48)

where
$-J_{2}(\Delta_{1}=0)=O(x)=Z_{12}m^{2}_{12}\bar{\rho}_{1}+\mu^{\eta}yZ_{y}\bar{n}n$.
The diagrams for Eq.(47) are shown in Fig.1. All the propagators in the
diagrams are the heavy scalars and they are one particle reducible diagrams.
In Eq.(47), the bare coupling constant $\lambda_{02}$ is substituted because
these tree level contribution includes the counter terms which subtract the
divergences of one-loop graphs. The terms with $\Delta_{1}$ in Eq.(45)
contribute to the effective action beyond the tree level and the renormalized
coupling constants are substituted for their interactions.

In Fig.2, we have shown the diagrams for the one-loop contribution of
$\Delta_{2}$ defined by $i\bar{W}_{2}^{c(1\operatorname{loop})}$. The
contribution is summarized as,

$\displaystyle i\bar{W}_{2}^{c(1\operatorname{loop})}[\bar{\rho}_{1},n]\equiv
iW_{2}^{c(1\operatorname{loop})}[-O(x),\bar{\rho}_{1},0]$
$\displaystyle=-\frac{3i\lambda_{2}\mu^{2\eta}}{2}\int
d^{d}xD_{F22}(x,x)D_{O}(x)^{2}$
$\displaystyle-\frac{\lambda_{2}^{2}\mu^{4\eta}}{2}\int d^{d}x\int
d^{d}y\left(\frac{9}{2}D_{F22}(x,y)^{2}D_{O}(x)^{2}D_{O}(y)^{2}+6D_{F22}(x,y)D_{F22}(x,x)D_{O}(x)D_{O}(y)^{3}\right).$
(49)

In this contribution, one sets $\Delta_{1}$ equal to zero, since the
contribution of $\Delta_{1}$ generates either another loop effect or one-
particle reducible contribution which is excluded from
$\tilde{\Gamma}_{\mathrm{eff}}[\bar{\rho}_{1},n]$ in Eq.(32). With Eq.(45) and
Eq.(49) , the calculation of $W_{2}^{c}$ is completed.

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Figure 2: One loop diagrams of the heavy scalar $\Delta_{2}$ which contribute
to the $i\bar{W}_{2}^{c(1\operatorname{loop})}[\bar{\rho}_{1},n]$. From the
figure of the left to that of the right, each Feynman diagram corresponds to
terms of Eq.(49) from the first term to the last term respectively.

Next we move to the vacuum graph contribution; i.e., the second factor of
Eq.(37) and it is computed as,

$\displaystyle\langle
e^{iS_{2\operatorname{int}}(\Delta_{2})+iS_{12\operatorname{int}}(\Delta_{1},\Delta_{2},\bar{\rho}_{1})}\rangle_{0}$
$\displaystyle=\exp\left[-\frac{\lambda_{3}\mu^{2\eta}}{4}i\int
d^{d}x(\Delta_{1}^{2}(x)+2\Delta_{1}(x)\bar{\rho}_{1}(x))D_{F22}(x,x)\right.$
$\displaystyle\left.-\left(\frac{\lambda_{3}}{4}\right)^{2}\mu^{4\eta}\int
d^{d}x\int
d^{d}y(\Delta_{1}^{2}(x)+2\Delta_{1}(x)\bar{\rho}_{1}(x))(\Delta_{1}^{2}(y)+2\Delta_{1}(y)\bar{\rho}_{1}(y))D^{2}_{F22}(x,y)\right]$
$\displaystyle=1.$ (50)

We note in the contribution, one loop contribution of heavy scalar is present.
Therefore another loop effect of the light scalar leads to two loop
contribution. From the same reason as that of
$iW_{2}^{c(1\operatorname{loop})}$, we set $\Delta_{1}$ to be zero in the last
line of Eq.(50). Combining Eq.(45) and Eq.(49), one can summarize the
expression for Eq.(33).

$\displaystyle
e^{iW_{2}[J_{2},\bar{\rho}_{1},\Delta_{1}]}=e^{-\frac{1}{2}\operatorname{TrLn}D^{-1}_{22}(\bar{\rho}_{1})+i\bar{W}_{2}^{c(\operatorname{tree})}[\bar{\rho}_{1},n]+i\bar{W}_{2}^{c(1\operatorname{loop})}[\bar{\rho}_{1},n]}e^{iW_{2}^{c(\operatorname{tree})}[J_{2},\bar{\rho}_{1},\Delta_{1}]|_{\mathrm{rest}}},$
(51)

where $iW_{2}^{c(\operatorname{tree})}|_{\mathrm{rest}}$ is defined as the
difference of Eq.(45) and Eq.(47),

$\displaystyle
iW_{2}^{c(\operatorname{tree})}[J_{2},\bar{\rho}_{1},\Delta_{1}]|_{\mathrm{rest}}\equiv
iW_{2}^{\operatorname{c(tree)}}[J_{2},\bar{\rho}_{1},\Delta_{1}]-i\bar{W}_{2}^{c(\operatorname{tree})}[\bar{\rho}_{1},n].$
(52)

###  4.2 Integrating lighter scalar field

In the following, we integrate the lighter scalar field $\Delta_{1}$. We
include the corrections up to one loop level. Using the result of previous
section, one loop part of the effective action is written by,

$\displaystyle
e^{i\tilde{\Gamma}_{\operatorname{eff}}[\bar{\rho}_{1},n]}=e^{-\frac{1}{2}\operatorname{TrLn}D^{-1}_{11}(\bar{\rho}_{1})}e^{-\frac{1}{2}\operatorname{Tr}\operatorname{Ln}D_{22}^{-1}(\bar{\rho}_{1})}e^{i\bar{W}_{2}^{c(\operatorname{tree})}[\bar{\rho}_{1},n]+i\bar{W}_{2}^{c(1\operatorname{loop})}[\bar{\rho}_{1},n]}$
$\displaystyle\times\frac{\int d\Delta_{1}e^{i\left\\{\frac{1}{2}\int
d^{d}x\int
d^{d}y\Delta_{1}(x)\frac{\delta^{2}S[\bar{\rho}_{1},0,n]}{\delta\bar{\rho}_{1}(x)\delta\bar{\rho}_{1}(y)}\Delta_{1}(y)+S_{1\operatorname{int}}\left(\Delta_{1},\bar{\rho}_{1}\right)+W_{2}^{c(\operatorname{tree})}[J_{2},\bar{\rho}_{1},\Delta_{1}]|_{\mathrm{rest}}-\int
d^{d}x\Delta_{1}(x)\left(\frac{\delta\tilde{\Gamma}_{\operatorname{eff}}[\bar{\rho}_{1},n]}{\delta\bar{\rho}_{1}(x)}\right)\right\\}}}{\int
d\Delta_{1}e^{i\left\\{\frac{1}{2}\int d^{d}x\int
d^{d}y\Delta_{1}(x)\frac{\delta^{2}S[\bar{\rho}_{1},0,n]}{\delta\bar{\rho}_{1}(x)\delta\bar{\rho}_{1}(y)}\Delta_{1}(y)\right\\}}}.$
(53)

From Eq.(53), one obtains the effective action as.

$\displaystyle i\tilde{\Gamma}_{\operatorname{eff}}[\bar{\rho}_{1},n]$
$\displaystyle=$
$\displaystyle-\frac{1}{2}\operatorname{TrLn}D^{-1}_{22}(\bar{\rho}_{1})-\frac{1}{2}\operatorname{Tr}\operatorname{Ln}\\{(D_{11}^{-1})^{\prime}(\bar{\rho}_{1})\\}+i\bar{W}_{2}^{c(\operatorname{tree})}[\bar{\rho}_{1},n]+i\bar{W}_{2}^{c(1\operatorname{loop})}[\bar{\rho}_{1},n]$
$\displaystyle+$ $\displaystyle\log\left[\frac{\int
d\Delta_{1}e^{i\left\\{-\frac{1}{2}\int d^{d}x\int
d^{d}y\Delta_{1}(x)\\{(D_{11}^{-1})^{\prime}(\bar{\rho}_{1})\\}\Delta_{1}(y)-\int
d^{d}x\Delta_{1}(x)\left(\frac{\delta\tilde{\Gamma}_{\operatorname{eff}}[\bar{\rho}_{1},n]}{\delta\bar{\rho}_{1}(x)}\right)\right\\}}e^{\mathcal{L}_{\operatorname{int}}(\Delta_{1})}}{\int
d\Delta_{1}e^{i\left\\{-\frac{1}{2}\int d^{d}x\int
d^{d}y\Delta_{1}(x)\\{(D_{11}^{-1})^{\prime}(\bar{\rho}_{1})\\}\Delta_{1}(y)\right\\}}}\right].$

In Eq.(LABEL:eq:_iGamma), we define
$\mathcal{L}_{\operatorname{int}}(\Delta_{1})$ by;

$\displaystyle\mathcal{L}_{\operatorname{int}}(\Delta_{1})$
$\displaystyle\equiv
iS_{1\operatorname{int}}(\Delta_{1},\bar{\rho}_{1})+iW_{2}^{c\operatorname{tree}}[J_{2},\Delta_{1},\bar{\rho}_{1}]|_{\mathrm{rest}}+\frac{1}{2}m_{12}^{4}\int
d^{d}x\int d^{d}y\Delta_{1}(x)D_{F22}(x,y)\Delta_{1}(y)$ $\displaystyle\simeq
iW_{2}^{c\operatorname{tree}}[J_{2},\Delta_{1},\bar{\rho}_{1}]|_{\mathrm{rest}}^{\mbox{\tiny
quadratic part of $\Delta_{1}$}}+\frac{1}{2}m_{12}^{4}\int d^{d}x\int
d^{d}y\Delta_{1}(x)D_{F22}(x,y)\Delta_{1}(y),$ (55)

To derive Eq.(LABEL:eq:_iGamma) from Eq.(53), we absorb the mixing effect of
the heavy scalar into the propagator of the light scalar. The corresponding
inverse propagator is given as,

$\displaystyle-
D_{11}^{-1^{\prime}}(x,y)=-D^{-1}_{11}(x,y)+im_{12}^{4}D_{F22}(x,y).$ (56)

The mixing of the heavy scalar is the second term of Eq.(56) and the modified
propagator for the light scalar is defined by the following equation,

$\displaystyle i\int
d^{d}zD_{11}^{-1^{\prime}}(x,z)D^{\prime}_{F11}(z,y)=\delta^{d}(x-y).$ (57)

Because of this change, the quadratic term with respect to $\Delta_{1}$ in
$iW_{2}^{c\operatorname{tree}}[J_{2},\Delta_{1},\bar{\rho}_{1}]|_{\mathrm{rest}}$;

$\displaystyle-\frac{1}{2}m_{12}^{4}\int\int
d^{d}{x}d^{d}{y}\Delta_{1}(x)D_{F22}(x,y)\Delta_{1}(y),$ (58)

should be subtracted. The second term of the last line of Eq.(55) is added for
this purpose. As for the other parts of Eq.(55), from the second line to the
third line of Eq.(55), the quartic interaction term with respect to
$\Delta_{1}$ is ignored because it contributes to beyond the one loop order.
The contribution to the tadpole diagram from the cubic interaction of
$\Delta_{1}$ is absent in 1 PI effective action and the second order
contribution from the cubic interaction is also ignored because it contributes
to beyond the one loop order. Therefore within one-loop approximation , we
keep only the quadratic terms with respect to $\Delta_{1}$. The explicit
expression is given as follows,

$\displaystyle\mathcal{L}_{\operatorname{int}}(\Delta_{1})\equiv-\frac{3im_{12}^{4}\lambda_{2}\mu^{2\eta}}{2}\int
d^{d}x\left(\prod_{i=1}^{4}\int
d^{d}x_{i}D_{F22}(x,x_{i})\right)O(x_{3})O(x_{4})\Delta_{1}(x_{1})\Delta_{1}(x_{2})$
$\displaystyle+\frac{i\lambda_{3}\mu^{2\eta}}{4}\int
d^{d}x\Delta_{1}^{2}(x)\left(\prod_{i=1}^{2}\int
d^{d}x_{i}D_{F22}(x,x_{i})O(x_{i})\right)$
$\displaystyle+im_{12}^{2}\lambda_{3}\mu^{2\eta}\int
d^{d}x\Delta_{1}(x)\bar{\rho}_{1}(x)\left(\prod_{i=1}^{2}\int
d^{d}x_{i}D_{F22}(x,x_{i})\right)\Delta_{1}(x_{1})O(x_{2})$
$\displaystyle+\frac{\lambda_{3}^{2}\mu^{4\eta}}{2}\int d^{d}x\int
d^{d}y\Delta_{1}(x)\bar{\rho}_{1}(x)\Delta_{1}(y)\bar{\rho}_{1}(y)D_{F22}(x,y)\int
d^{d}x_{1}\int d^{d}y_{1}D_{F22}(x,x_{1})D_{F22}(y,y_{1})O(x_{1})O(y_{1})$
$\displaystyle-\frac{\lambda_{2}\lambda_{3}\mu^{4\eta}}{4}\int d^{d}x\int
d^{d}y\Delta_{1}^{2}(x)D_{F22}(x,y)\int
d^{d}x_{1}D_{F22}(x,x_{1})O(x_{1})\left(\prod_{i=1}^{3}\int
d^{d}y_{i}D_{F22}(y,y_{i})O(y_{i})\right)$
$\displaystyle-\frac{m_{12}^{2}\lambda_{2}\lambda_{3}\mu^{4\eta}}{2}\int
d^{d}x\int d^{d}y\Delta_{1}(x)\bar{\rho}_{1}(x)D_{F22}(x,y)\int
d^{d}x_{1}D_{F22}(x,x_{1})\Delta_{1}(x_{1})\left(\prod_{i=1}^{3}\int
d^{d}y_{i}D_{F22}(y,y_{i})O(y_{i})\right)$
$\displaystyle-\frac{3}{2}m_{12}^{2}\lambda_{2}\lambda_{3}\mu^{4\eta}\int
d^{d}x\int d^{d}y\Delta_{1}(x)\bar{\rho}_{1}(x)D_{F22}(x,y)\int
d^{d}x_{1}D_{F22}(x,x_{1})O(x_{1})\left(\prod_{i=1}^{3}\int
d^{d}y_{i}D_{F22}(y,y_{i})\right)\Delta_{1}(y_{1})O(y_{2})O(y_{3})$
$\displaystyle+3m_{12}^{4}\lambda_{2}^{2}\mu^{4\eta}\int d^{d}x\int
d^{d}yD_{F22}(x,y)\left(\prod_{i=1}^{3}\int
d^{d}x_{i}D_{F22}(x,x_{i})\right)\Delta_{1}(x_{1})\Delta_{1}(x_{2})O(x_{3})\left(\prod_{j=1}^{3}\int
d^{d}y_{j}D_{F22}(y,y_{j})O(y_{j})\right)$
$\displaystyle+\frac{9m_{12}^{4}\lambda_{2}^{2}\mu^{4\eta}}{2}\int d^{d}x\int
d^{d}yD_{F22}(x,y)\left(\prod_{i=1}^{3}\int d^{d}x_{i}\int
d^{d}y_{i}D_{F22}(x,x_{i})D_{F22}(y,y_{j})\right)\Delta_{1}(x_{1})O(x_{2})O(x_{3})\Delta_{1}(y_{1})O(y_{2})O(y_{3}).$
(59)

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Figure 3: From the figure of the upper left to that of the lower right, each
Feynman diagram corresponds to each line of Eq.(59) from the top line to the
bottom. The cross mark denotes an insertion of the mass mixing term
$m_{12}^{2}$. All the propagators are heavy scalars.

In Fig.3, we have shown the diagrams for Eq.(59). Below we calculate the
contribution from $\mathcal{L}_{\operatorname{int}}(\Delta_{1})$ in Eq.(59).
In the last term of Eq.(LABEL:eq:_iGamma), the expression inside the
logarithmic function is defined as follows,

$\displaystyle F[\bar{\rho}_{1},n]$ $\displaystyle\equiv\frac{\int
d\Delta_{1}e^{i\left\\{-\frac{1}{2}\int d^{d}x\int
d^{d}y\Delta_{1}(x)(D^{-1}_{11})^{\prime}(\bar{\rho}_{1})\Delta_{1}(y)\right\\}}e^{\left(\mathfrak{L}_{\operatorname{int}}(\Delta_{1})-i\int
d^{d}x\Delta_{1}(x)\left(\frac{\delta\tilde{\Gamma}_{\operatorname{eff}}[\bar{\rho}_{1},n]}{\delta\bar{\rho}_{1}(x)}\right)\right)}}{\int
d\Delta_{1}e^{i\left\\{\frac{1}{2}\int d^{d}x\int
d^{d}y\Delta_{1}(x)(D^{-1}_{11})^{\prime}(\bar{\rho}_{1})\Delta_{1}(y)\right\\}}}.$
(60)

Note that the effect of $\int
d^{d}x\Delta_{1}(x)\left(\frac{\delta\tilde{\Gamma}_{\operatorname{eff}}[\bar{\rho}_{1},n]}{\delta\bar{\rho}_{1}(x)}\right)$
removes the contribution from one particle reducible graphs. We expand
$F[\bar{\rho}_{1},n]$ with respect to the series of the coupling constants. Up
to the second order, it is given by,

$\displaystyle F[\bar{\rho}_{1},n]$ $\displaystyle\approx$ $\displaystyle
1+\frac{i}{4}\lambda_{3}\mu^{2\eta}\int d^{d}x\left(\prod_{i=1}^{2}\int
d^{d}x_{i}D^{(0)}_{F22}(x-x_{i})O(x_{i})\right)D^{\prime}_{F11}(x,x)$ (61)
$\displaystyle+$ $\displaystyle i\lambda_{3}m_{12}^{2}\mu^{2\eta}\int
d^{d}x\bar{\rho}_{1}(x)\left(\prod_{i=1}^{2}\int
d^{d}x_{i}D^{(0)}_{F22}(x-x_{i})\right)O(x_{2})D^{\prime}_{F11}(x,x_{1})$
$\displaystyle+$ $\displaystyle i\lambda_{3}m_{12}^{2}\mu^{2\eta}\int
d^{d}x\bar{\rho}_{1}(x)\int d^{d}x_{1}\int
d^{d}x_{2}D^{(1)}_{F22}(x,x_{1})D^{(0)}_{F22}(x-x_{2})O(x_{2})D^{\prime}_{F11}(x,x_{1})$
$\displaystyle+$ $\displaystyle i\lambda_{3}m_{12}^{2}\mu^{2\eta}\int
d^{d}x\bar{\rho}_{1}(x)\int d^{d}x_{1}\int
d^{d}x_{2}D^{(0)}_{F22}(x-x_{1})D^{(1)}_{F22}(x,x_{2})O(x_{2})D^{\prime}_{F11}(x,x_{1})$
$\displaystyle+$ $\displaystyle\frac{\lambda_{3}^{2}\mu^{4\eta}}{2}\int
d^{d}x\int d^{d}y\bar{\rho}_{1}(x)\bar{\rho}_{1}(y)D^{(0)}_{F22}(x-y)$
$\displaystyle\times$ $\displaystyle\int d^{d}x_{1}\int
d^{d}y_{1}D^{(0)}_{F22}(x-x_{1})D^{(0)}_{F22}(y-y_{1})O(x_{1})O(y_{1})D^{\prime}_{F11}(x,y).$

To derive Eq.(61), we approximate the exact expression by expanding the result
with the small parameter $\epsilon=\frac{m^{2}_{12}}{m^{2}_{2}}\ll 1$. About
the quadratic terms for the background field $\bar{\rho}_{1}$ , we keep the
the terms suppressed up to the order of
$m^{2}_{12}\epsilon\bar{\rho}_{1}^{2}$. About the quartic terms, we keep those
suppressed by $\epsilon^{2}\bar{\rho}_{1}^{4}$. In Eq.(61) ,
$D_{F22}^{(n)}(x,y),(n=0,1,2)$ are obtained by solving the following integral
equation with iteration.

$\displaystyle D_{F22}(x,y)$ $\displaystyle=D_{F22}^{(0)}(x-y)-i\int
d^{d}zD_{F22}^{(0)}(x-z)\frac{\lambda_{3}}{2}\bar{\rho}_{1}^{2}(z)D_{F22}(z,y),$
(62)

where the leading contribution $D_{F22}^{(0)}(x-y)$ is given as follows,

$\displaystyle D_{F22}^{(0)}(x-y)$
$\displaystyle=\int\frac{d^{d}k}{(2\pi)^{d}i}\frac{1}{m_{2}^{2}-k^{2}}e^{-ik(x-y)}.$
(63)

One also finds the first order correction and the second order correction. The
same approximation is also used when we expand $D_{F22}$ in Eq.(47),

$\displaystyle D_{F22}^{(1)}(x,y)$ $\displaystyle=-i\int
d^{d}zD_{F22}^{(0)}(x-z)\frac{\lambda_{3}}{2}\bar{\rho}_{1}^{2}(z)D^{(0)}_{F22}(z-y),$
(64) $\displaystyle D_{F22}^{(2)}(x,z)$ $\displaystyle=-\int d^{d}z\int
d^{d}\omega
D_{F22}^{(0)}(x-z)\frac{\lambda_{3}}{2}\bar{\rho}_{1}^{2}(z)D_{F22}^{(0)}(z-\omega)\frac{\lambda_{3}}{2}\bar{\rho}_{1}^{2}(\omega)D^{(0)}_{F22}(\omega-y).$
(65)

The propagator $D^{\prime}_{F11}(x,y)$ is expanded with respect to $m_{12}$.
When $\bar{\rho}_{1}$ is independent of the space time,
$D^{\prime}_{F11}(x,y)$ up to the fourth power of $m_{12}$ is given as
follows.

$\displaystyle D^{\prime}_{F11}(x-y)$ $\displaystyle=$
$\displaystyle\int\frac{d^{d}k}{(2\pi)^{d}}e^{-ik\cdot(x-y)}\frac{-i}{m_{1}^{2}-k^{2}+3\lambda_{1}\bar{\rho}_{1}^{2}}$
(66) $\displaystyle+$
$\displaystyle\int\frac{d^{d}k}{(2\pi)^{d}}e^{-ik\cdot(x-y)}\frac{-im_{12}^{4}}{(m_{1}^{2}-k^{2}+3\lambda_{1}\bar{\rho}_{1}^{2})^{2}\left(m_{2}^{2}-k^{2}+\frac{\lambda_{3}}{2}\bar{\rho}_{1}^{2}\right)}.$

Hereafter, we consider the case for the space time independent field
$\bar{\rho}_{1}$ and the fermion bilinear of $\bar{n}n$ for the neutrino. We
study the effective potential for the scalar and the bilinear. In order to
study the scalar loop effect on the neutrinos Yukawa coupling, it is
sufficient to consider the space time independent mode of the bilinear
$\bar{n}n$. We summarize the tree level action and one loop contribution for
the constant background fields.

$\displaystyle\Gamma_{\operatorname{eff}}[\bar{\rho}_{1},n]$ $\displaystyle=$
$\displaystyle
S[\bar{\rho}_{1},0,0]+\tilde{\Gamma}_{\operatorname{eff}}[\bar{\rho}_{1},n],$
(67) $\displaystyle S[\bar{\rho}_{1},0,0]$ $\displaystyle=$ $\displaystyle\int
d^{d}x\left[-\frac{Z_{m11}}{2}m_{1}^{2}\bar{\rho}_{1}^{2}-\frac{\lambda_{01}{Z_{1}}^{2}}{4}\bar{\rho}_{1}^{4}\right]$
(68) $\displaystyle+$ $\displaystyle\int
d^{d}x\left[\mu^{-2\eta}\left(\sum_{i=1}^{2}Z_{hi}h_{i}{m_{i}}^{4}+2Z_{h3}h_{3}m^{2}_{1}m^{2}_{2}+Z_{h12}h_{12}m_{12}^{4}\right)\right],$
$\displaystyle\tilde{\Gamma}_{\operatorname{eff}}[\bar{\rho}_{1},n]$
$\displaystyle=$
$\displaystyle\tilde{\Gamma}_{\operatorname{eff}}[\bar{\rho}_{1},n]^{\operatorname{tree}}+\tilde{\Gamma}_{\operatorname{eff}}[\bar{\rho}_{1},n]^{1\operatorname{loop}}_{\Delta_{2}}+\tilde{\Gamma}_{\operatorname{eff}}[\bar{\rho}_{1},n]^{1\operatorname{loop}}_{\Delta_{1}}+\tilde{\Gamma}_{\operatorname{eff}}[\bar{\rho}_{1},n]^{\operatorname{TrLn}},$
(69)
$\displaystyle\tilde{\Gamma}_{\operatorname{eff}}[\bar{\rho}_{1},n]^{\operatorname{tree}}$
$\displaystyle=$ $\displaystyle\int
d^{d}x\left[\frac{m_{012}^{2}}{2}\epsilon_{0}\bar{\rho}_{01}^{2}+y_{0}\epsilon_{0}\bar{n_{0}}n_{0}\bar{\rho}_{01}-\frac{\lambda_{03}}{4}\epsilon_{0}^{2}\bar{\rho}_{01}^{4}+\frac{1}{2m_{2}^{2}}\left(\frac{\lambda_{3}\mu^{2\eta}}{2}\epsilon\bar{\rho}_{1}^{3}-y\mu^{\eta}\bar{n}n\right)^{2}\right],$
(70)
$\displaystyle\tilde{\Gamma}_{\operatorname{eff}}[\bar{\rho}_{1},n]^{1\operatorname{loop}}_{\Delta_{2}}$
$\displaystyle=$ $\displaystyle\int
d^{d}x\left[\frac{3\lambda_{2}\mu^{2\eta}}{16\pi^{2}}(\Gamma(\eta)+1+\log
4\pi-\log
m_{2}^{2})\epsilon(y\mu^{\eta}\bar{n}n\bar{\rho}_{1}+\frac{m_{12}^{2}}{2}\bar{\rho}_{1}^{2})\right.$
(71)
$\displaystyle+\left.\frac{3\lambda_{2}\lambda_{3}\mu^{4\eta}}{64\pi^{2}}(-\Gamma(\eta)-2-\log
4\pi+\log m_{2}^{2})\epsilon^{2}\bar{\rho}_{1}^{4}\right],$
$\displaystyle\tilde{\Gamma}_{\operatorname{eff}}[\bar{\rho}_{1},n]^{1\operatorname{loop}}_{\Delta_{1}}$
$\displaystyle=$ $\displaystyle\int
d^{d}x\left[-\frac{\lambda_{3}\mu^{2\eta}}{16\pi^{2}}(\Gamma(\eta)+\log
4\pi+1-\log
m_{2}^{2})\epsilon(y\mu^{\eta}\bar{n}n\bar{\rho}_{1}+m_{12}^{2}\bar{\rho}_{1}^{2})\right.$
(72) $\displaystyle+$
$\displaystyle\frac{\lambda_{3}\mu^{2\eta}}{16\pi^{2}}\left\\{\frac{3\lambda_{1}\mu^{2\eta}}{4}\left(\Gamma(\eta)+1+\log
4\pi-\log(m_{1}^{2}+3\lambda_{1}\mu^{2\eta}\bar{\rho}_{1}^{2})-4\log\left(\frac{m_{1}^{2}+3\lambda_{1}\mu^{2\eta}\bar{\rho}_{1}^{2}}{m_{2}^{2}}\right)\right)\right.$
$\displaystyle+$
$\displaystyle\left.\left.\lambda_{3}\mu^{2\eta}\left(\Gamma(\eta)+\log
4\pi+\frac{3}{2}-\log
m_{2}^{2}\right)\right\\}\epsilon^{2}\bar{\rho}_{1}^{4}\right],$
$\displaystyle\tilde{\Gamma}_{\operatorname{eff}}[\bar{\rho}_{1},n]^{\operatorname{TrLn}}$
$\displaystyle=$ $\displaystyle\int
d^{d}x\left[\frac{m_{12}^{4}}{32\pi^{2}}(\Gamma[\eta]+\log 4\pi+1-\log
m_{2}^{2})+\frac{\epsilon^{2}m_{1}^{2}m_{2}^{2}}{32\pi^{2}}\log\frac{m_{1}^{2}+3\lambda_{1}\mu^{2\eta}\bar{\rho}_{1}^{2}}{m_{2}^{2}}\right.$
$\displaystyle+$
$\displaystyle\frac{m_{1}^{4}}{64\pi^{2}}\left(\Gamma[\eta]+\log
4\pi+\frac{3}{2}-\log(m_{1}^{2}+3\lambda_{1}\bar{\rho}_{1}^{2}\mu^{2\eta})\right)+\frac{m_{2}^{4}}{64\pi^{2}}\left(\Gamma[\eta]+\log
4\pi+\frac{3}{2}-\log m_{2}^{2}\right)$ $\displaystyle+$
$\displaystyle\frac{m_{12}^{2}}{2}\left(\frac{3\lambda_{1}\mu^{2\eta}}{16\pi^{2}}\log\frac{m_{1}^{2}+3\lambda_{1}\mu^{2\eta}\bar{\rho}_{1}^{2}}{m_{2}^{2}}-\frac{\lambda_{3}\mu^{2\eta}}{32\pi^{2}}\right)\epsilon\bar{\rho}_{1}^{2}$
$\displaystyle+$
$\displaystyle\left\\{\frac{3\lambda_{1}\mu^{2\eta}m_{1}^{2}}{32\pi^{2}}\left(\Gamma[\eta]+\frac{3}{2}+\log
4\pi-\log(m_{1}^{2}+3\lambda_{1}\bar{\rho}_{1}^{2}\mu^{2\eta})\right)\right.$
$\displaystyle+$
$\displaystyle\left.\frac{\lambda_{3}\mu^{2\eta}m_{2}^{2}}{64\pi^{2}}\left(\Gamma[\eta]+1+\log
4\pi-\log m_{2}^{2}\right)\right\\}\bar{\rho}_{1}^{2}$ $\displaystyle+$
$\displaystyle\left\\{\frac{\lambda_{3}^{2}\mu^{2\eta}}{256\pi^{2}}\left(\Gamma[\eta]+\log
4\pi-\log m_{2}^{2}\right)\right.$ $\displaystyle+$
$\displaystyle\left.\frac{9\lambda_{1}^{2}\mu^{4\eta}}{64\pi^{2}}\left(\Gamma[\eta]+\frac{3}{2}+\log
4\pi-\log(m_{1}^{2}+3\lambda_{1}\bar{\rho}_{1}^{2}\mu^{2\eta})\right)\right\\}\bar{\rho}_{1}^{4}$
$\displaystyle+$
$\displaystyle\left.\left\\{\frac{9\lambda_{1}^{2}\mu^{2\eta}}{32\pi^{2}}\log\frac{m_{1}^{2}+3\lambda_{1}\mu^{2\eta}\bar{\rho}_{1}^{2}}{m_{2}^{2}}-\frac{3\lambda_{1}\lambda_{3}\mu^{4\eta}}{64\pi^{2}}\left(1+\log\frac{m_{1}^{2}+3\lambda_{1}\mu^{2\eta}\bar{\rho}_{1}^{2}}{m_{2}^{2}}\right)+\frac{\lambda_{3}^{2}\mu^{4\eta}}{256\pi^{2}}\right\\}\epsilon^{2}\bar{\rho}_{1}^{4}\right],$

where $\epsilon_{0}=\frac{m_{012}^{2}}{m_{02}^{2}}$. $\Gamma[\eta]$ is a gamma
function which represents divergence in the dimensional regularization.

##  5 The counter terms and the effective potential

One loop contribution in Eqs.(71-LABEL:eq:TrLn) includes the divergent terms.
We show that the counter terms of $S[\bar{\rho_{1}},0,0]$ in Eq.(68) and
$\tilde{\Gamma}_{\operatorname{eff}}^{\operatorname{tree}}$ in Eq.(70) are
determined so that the divergences are canceled. We replace the bare mass
terms and the bare coupling constants with the renormalized ones using the
relations from Eqs.(2-8). We also use the fact that there is no wave function
renormalization for scalars from their one-loop diagrams. Since we do not take
into account of the fermion loop contribution, one can set $Z_{i}=1(i=1,2)$.
Since $m_{1}^{2}m_{2}^{2}$ term does not include divergence, one can set
$Z_{h_{3}}=1$. The other counter terms are generated by splitting the Z
factors as ,

$\displaystyle Z_{12}$ $\displaystyle=$ $\displaystyle 1+(Z_{12}-1),$ (74)
$\displaystyle Z_{\lambda_{ii}}$ $\displaystyle=$ $\displaystyle
1+(Z_{\lambda_{ii}}-1)(i=1,2,3),$ (75) $\displaystyle Z_{mii}$
$\displaystyle=$ $\displaystyle 1+(Z_{mii}-1)(i=1,2),$ (76) $\displaystyle
Z_{h_{i}}$ $\displaystyle=$ $\displaystyle 1+(Z_{h_{i}}-1)(i=1,2),\quad
Z_{h_{12}}=1+(Z_{h_{12}}-1).$ (77)

As the result, the tree part of the effective action and the counter terms are
obtained as follows.

$\displaystyle S_{\operatorname{tree}}[\bar{\rho}_{1},n]$ $\displaystyle=$
$\displaystyle
S[\bar{\rho}_{1},0,0]+\tilde{\Gamma}_{\operatorname{eff}}[\bar{\rho}_{1},n]^{\operatorname{tree}}=S_{\operatorname{tree}}^{\operatorname{4dim}}[\bar{\rho}_{1},n]+S_{\operatorname{tree}}^{\operatorname{6dim}}[\bar{\rho}_{1},n]+S_{\operatorname{C}}[\bar{\rho}_{1},n],$
(78) $\displaystyle
S_{\operatorname{tree}}^{\operatorname{4dim}}[\bar{\rho}_{1},n]$
$\displaystyle=$ $\displaystyle\int
d^{d}x\left[-\frac{1}{2}m_{1}^{2}\bar{\rho}_{1}^{2}-\frac{\lambda_{1}}{4}\bar{\rho}_{1}^{4}+\frac{m_{12}^{2}}{2}\epsilon\bar{\rho}_{1}^{2}+y\mu^{\eta}\epsilon\bar{n}n\bar{\rho}_{1}-\frac{\lambda_{3}\mu^{2\eta}}{4}\epsilon^{2}\bar{\rho}_{1}^{4}\right]$
(79) $\displaystyle+$ $\displaystyle\int
d^{d}x\mu^{-2\eta}\left[\sum_{i=1}^{2}h_{i}{m_{i}}^{4}+2h_{3}m^{2}_{1}m^{2}_{2}+h_{12}m_{12}^{4}\right]$
$\displaystyle
S_{\operatorname{tree}}^{\operatorname{6dim}}[\bar{\rho}_{1},n]$
$\displaystyle=$ $\displaystyle\int
d^{d}x\left[\frac{1}{2m_{2}^{2}}\left(\frac{\lambda_{3}\mu^{2\eta}}{2}\epsilon\bar{\rho}_{1}^{3}-y\mu^{\eta}\bar{n}n\right)^{2}\right],$
(80) $\displaystyle S_{\operatorname{C}}[\bar{\rho}_{1},n]$ $\displaystyle=$
$\displaystyle\int
d^{d}x\left[\left(-\frac{(Z_{m11}-1)m_{1}^{2}+Z_{m12}m_{2}^{2}}{2}\bar{\rho}_{1}^{2}-\frac{\\{(Z_{\lambda_{11}}-1)\lambda_{1}+Z_{\lambda_{12}}\lambda_{2}+Z_{\lambda_{13}}\lambda_{3}\\}\mu^{2\eta}}{4}\bar{\rho}_{1}^{4}\right)\right.$
(81) $\displaystyle+$
$\displaystyle\frac{1}{2}m_{12}^{2}\left(2(Z_{12}-1)-\left\\{(Z_{m22}-1)\right\\}\right)\epsilon\bar{\rho}_{1}^{2}+\left((Z_{12}-1)-\left\\{(Z_{m22}-1)\right\\}\right)y\mu^{\eta}\epsilon\bar{n}n\bar{\rho}_{1}$
$\displaystyle-$
$\displaystyle\frac{\mu^{2\eta}}{4}\left[Z_{\lambda_{31}}\lambda_{1}+Z_{\lambda_{32}}\lambda_{2}+(Z_{\lambda_{33}}-1)\lambda_{3}+2\lambda_{3}\left\\{(Z_{12}-1)-\left((Z_{m22}-1)\right)\right\\}\right]\epsilon^{2}\bar{\rho}_{1}^{4}$
$\displaystyle+$
$\displaystyle\left.\mu^{-2\eta}\left(\sum_{i=1}^{2}(Z_{hi}-1)h_{i}{m_{i}}^{4}+(Z_{h12}-1)h_{12}m_{12}^{4}\right)\right].$

We keep the terms up to those suppressed as $\epsilon\bar{\rho}_{1}^{2}$ ,
$\epsilon^{2}\bar{\rho}_{1}^{4}$ and $\epsilon\bar{n}n\bar{\rho}_{1}$ and
ignore the terms with the further suppression factor of
$\frac{m_{1}^{2}}{m_{2}^{2}}$. Correspondingly, the counter terms with the
same suppression factor such as $Z_{m21}\frac{m_{1}^{2}}{m_{2}^{2}}$ are also
ignored in Eq.(81). The $Z$ factors in Eq.(81) are determined in the full
theory and the derivation is given in appendix A. By substituting the $Z$
factors in Eqs.(123-127), all the divergences in Eqs.(71-LABEL:eq:TrLn) are
canceled. We define the tree level part and the one loop part of the effective
action respectively as,

$\displaystyle S_{\operatorname{eff}}^{\operatorname{tree}}$ $\displaystyle=$
$\displaystyle
S_{\operatorname{tree}}^{\operatorname{4dim}}[\bar{\rho}_{1},n]+S_{\operatorname{tree}}^{\operatorname{6dim}}[\bar{\rho}_{1},n],$
(82) $\displaystyle S_{\operatorname{eff}}^{\operatorname{loop}}$
$\displaystyle=$
$\displaystyle\tilde{\Gamma}_{\operatorname{eff}}[\bar{\rho}_{1},n]^{1\operatorname{loop}}_{\Delta_{2}}+\tilde{\Gamma}_{\operatorname{eff}}[\bar{\rho}_{1},n]^{1\operatorname{loop}}_{\Delta_{1}}+\tilde{\Gamma}_{\operatorname{eff}}[\bar{\rho}_{1},n]^{\operatorname{TrLn}}+S_{\operatorname{C}}[\bar{\rho}_{1},n],$
(83) $\displaystyle S_{\operatorname{eff}}$ $\displaystyle=$ $\displaystyle
S_{\operatorname{eff}}^{\operatorname{tree}}+S_{\operatorname{eff}}^{\operatorname{loop}}$
(84) $\displaystyle=$ $\displaystyle\int
d^{4}x\Bigl{[}m_{1}^{4}\left\\{h_{1}+\frac{1}{64\pi^{2}}\left(\frac{3}{2}-\log\frac{m_{1}^{2}+3\lambda_{1}\bar{\rho}_{1}^{2}}{\mu^{2}}\right)\right\\}+m_{2}^{4}\left\\{h_{2}+\frac{1}{64\pi^{2}}\left(\frac{3}{2}-\log\frac{m_{2}^{2}}{\mu^{2}}\right)\right\\}$
$\displaystyle+$ $\displaystyle
m_{12}^{4}\left\\{h_{12}+\frac{1}{32\pi^{2}}\left(1-\log\frac{m_{2}^{2}}{\mu^{2}}\right)\right\\}+2m_{1}^{2}m_{2}^{2}\left\\{h_{3}+\frac{\epsilon^{2}}{32\pi^{2}}\log\frac{m_{1}^{2}+3\lambda_{1}\bar{\rho}_{1}^{2}}{\mu^{2}}\right\\}$
$\displaystyle-$
$\displaystyle\frac{1}{2}\left\\{m_{1}^{2}\left(1-\frac{3\lambda_{1}}{16\pi^{2}}\left(\frac{3}{2}-\log\frac{m_{1}^{2}+3\lambda_{1}\bar{\rho}_{1}^{2}}{\mu^{2}}\right)\right)-\frac{\lambda_{3}m_{2}^{2}}{32\pi^{2}}\left(1-\log\frac{m_{2}^{2}}{\mu^{2}}\right)\right\\}\bar{\rho}_{1}^{2}$
$\displaystyle+$
$\displaystyle\frac{m_{12}^{2}}{2}\left\\{1+\frac{3\lambda_{1}}{16\pi^{2}}\log\frac{m_{1}^{2}+3\lambda_{1}\bar{\rho}_{1}^{2}}{m_{2}^{2}}+\frac{3\lambda_{2}}{16\pi^{2}}\left(1-\log\frac{m_{2}^{2}}{\mu^{2}}\right)-\frac{\lambda_{3}}{8\pi^{2}}\left(\frac{5}{4}-\log\frac{m_{2}^{2}}{\mu^{2}}\right)\right\\}\epsilon\bar{\rho}_{1}^{2}$
$\displaystyle-$
$\displaystyle\frac{\lambda_{1}}{4}\left\\{1-\frac{9\lambda_{1}}{16\pi^{2}}\left(\frac{3}{2}-\log\frac{m_{1}^{2}+3\lambda_{1}\bar{\rho}_{1}^{2}}{\mu^{2}}\right)+\frac{1}{64\pi^{2}}\frac{\lambda_{3}^{2}}{\lambda_{1}}\log\frac{m_{2}^{2}}{\mu^{2}}\right\\}\bar{\rho}_{1}^{4}$
$\displaystyle-$
$\displaystyle\frac{\lambda_{3}}{4}\left\\{1+\frac{3\lambda_{2}}{16\pi^{2}}\left(2-\log\frac{m_{2}^{2}}{\mu^{2}}\right)\right.-\frac{\lambda_{3}}{4\pi^{2}}\left(\frac{25}{16}-\log\frac{m_{2}^{2}}{\mu^{2}}\right)$
$\displaystyle+$
$\displaystyle\left.\frac{3\lambda_{1}}{16\pi^{2}}\left(\log\frac{m_{1}^{2}+3\lambda_{1}\bar{\rho}_{1}^{2}}{\mu^{2}}+5\log\frac{m_{1}^{2}+3\lambda_{1}\bar{\rho}_{1}^{2}}{m_{2}^{2}}-6\frac{\lambda_{1}}{\lambda_{3}}\log\frac{m_{1}^{2}+3\lambda_{1}\bar{\rho}_{1}^{2}}{m_{2}^{2}}\right)\right\\}\epsilon^{2}\bar{\rho}_{1}^{4}$
$\displaystyle+$
$\displaystyle\left.\left\\{1+\frac{3\lambda_{2}-\lambda_{3}}{16\pi^{2}}\left(1-\log\frac{m_{2}^{2}}{\mu^{2}}\right)\right\\}y\epsilon\bar{n}n\bar{\rho}_{1}+\frac{\lambda_{3}^{2}}{8m_{2}^{2}}\epsilon^{2}\bar{\rho}_{1}^{6}-\frac{\lambda_{3}y}{2m_{2}^{2}}\epsilon\bar{n}n\bar{\rho}_{1}^{3}+\frac{y^{2}}{2m_{2}^{2}}(\bar{n}n)^{2}\right],$

where the limit $d\to 4$ is taken. By substituting the vacuum expectation
value $v_{1}$ for $\bar{\rho}_{1}$, one obtains the effective potential as;

$\displaystyle V_{\mathrm{eff}}(v_{1})$ $\displaystyle=$ $\displaystyle
V_{\operatorname{cosmo}}+\frac{m_{1\mathrm{eff}}^{2}}{2}v_{1}^{2}-\frac{m_{12\mathrm{eff}}^{2}}{2}\epsilon
v_{1}^{2}+\frac{\lambda_{1\mathrm{eff}}}{4}v_{1}^{4}+\frac{\lambda_{3\mathrm{eff}}}{4}\epsilon^{2}v_{1}^{4}-y_{\mathrm{eff}}\epsilon\bar{n}nv_{1}$
(85) $\displaystyle-$
$\displaystyle\frac{\lambda_{3}^{2}}{8m_{2}^{2}}\epsilon^{2}v_{1}^{6}+\frac{\lambda_{3}}{2m_{2}^{2}}\epsilon(\bar{n}n)v_{1}^{3}-\frac{y^{2}}{2m_{2}^{2}}(\bar{n}n)^{2}.$

where the cosmological constant, the effective masses and couplings are
defined as;

$\displaystyle V_{\operatorname{cosmo}}$ $\displaystyle=$ $\displaystyle-
h_{1\mathrm{eff}}{m_{1}}^{4}-h_{2\mathrm{eff}}{m_{2}}^{4}-h_{12\mathrm{eff}}{m_{12}}^{4}-2h_{3\mathrm{eff}}{m_{1}}^{2}{m_{2}}^{2},$
(86) $\displaystyle m_{1\mathrm{eff}}^{2}$ $\displaystyle=$ $\displaystyle
m_{1}^{2}\left(1-\frac{3\lambda_{1}}{16\pi^{2}}\left(\frac{3}{2}-\log\frac{m_{1}^{2}+3\lambda_{1}v_{1}^{2}}{\mu^{2}}\right)\right)-\frac{\lambda_{3}m_{2}^{2}}{32\pi^{2}}\left(1-\log\frac{m_{2}^{2}}{\mu^{2}}\right),$
(87) $\displaystyle m_{12\mathrm{eff}}^{2}$ $\displaystyle=$ $\displaystyle
m_{12}^{2}\left\\{1+\frac{3\lambda_{1}}{16\pi^{2}}\log\frac{m_{1}^{2}+3\lambda_{1}v_{1}^{2}}{m_{2}^{2}}+\frac{3\lambda_{2}}{16\pi^{2}}\left(1-\log\frac{m_{2}^{2}}{\mu^{2}}\right)-\frac{\lambda_{3}}{8\pi^{2}}\left(\frac{5}{4}-\log\frac{m_{2}^{2}}{\mu^{2}}\right)\right\\},$
(88) $\displaystyle\frac{\lambda_{1\mathrm{eff}}}{4}$ $\displaystyle=$
$\displaystyle\frac{\lambda_{1}}{4}\left\\{1-\frac{9\lambda_{1}}{16\pi^{2}}\left(\frac{3}{2}-\log\frac{m_{1}^{2}+3\lambda_{1}v_{1}^{2}}{\mu^{2}}\right)+\frac{1}{64\pi^{2}}\frac{\lambda_{3}^{2}}{\lambda_{1}}\log\frac{m_{2}^{2}}{\mu^{2}}\right\\},$
(89) $\displaystyle\frac{\lambda_{3\mathrm{eff}}}{4}$ $\displaystyle=$
$\displaystyle\frac{\lambda_{3}}{4}\left\\{1+\frac{3\lambda_{2}}{16\pi^{2}}\left(2-\log\frac{m_{2}^{2}}{\mu^{2}}\right)\right.-\frac{\lambda_{3}}{4\pi^{2}}\left(\frac{25}{16}-\log\frac{m_{2}^{2}}{\mu^{2}}\right)$
(90) $\displaystyle+$
$\displaystyle\left.\frac{3\lambda_{1}}{16\pi^{2}}\left(\log\frac{m_{1}^{2}+3\lambda_{1}v_{1}^{2}}{\mu^{2}}+5\log\frac{m_{1}^{2}+3\lambda_{1}v_{1}^{2}}{m_{2}^{2}}-6\frac{\lambda_{1}}{\lambda_{3}}\log\frac{m_{1}^{2}+3\lambda_{1}v_{1}^{2}}{m_{2}^{2}}\right)\right\\},$
$\displaystyle h_{1\mathrm{eff}}$ $\displaystyle=$ $\displaystyle
h_{1}+\frac{1}{64\pi^{2}}\left(\frac{3}{2}-\log\frac{m_{1}^{2}+3\lambda_{1}v_{1}^{2}}{\mu^{2}}\right),$
(91) $\displaystyle h_{2\mathrm{eff}}$ $\displaystyle=$ $\displaystyle
h_{2}+\frac{1}{64\pi^{2}}\left(\frac{3}{2}-\log\frac{m_{2}^{2}}{\mu^{2}}\right),$
(92) $\displaystyle h_{3\mathrm{eff}}$ $\displaystyle=$ $\displaystyle
h_{3}+\frac{\epsilon^{2}}{32\pi^{2}}\log\frac{m_{1}^{2}+3\lambda_{1}\bar{\rho}_{1}^{2}}{\mu^{2}},$
(93) $\displaystyle h_{12\mathrm{eff}}$ $\displaystyle=$ $\displaystyle
h_{12}+\frac{1}{32\pi^{2}}\left(1-\log\frac{m_{2}^{2}}{\mu^{2}}\right),$ (94)
$\displaystyle y_{\mathrm{eff}}$ $\displaystyle=$ $\displaystyle
y\left(1+\frac{3\lambda_{2}-\lambda_{3}}{16\pi^{2}}\left(1-\log\frac{m_{2}^{2}}{\mu^{2}}\right)\right).$
(95)

The renormalization scale $\mu$ independence of the effective mass, coupling
constant and cosmological constant is studied in the appendix A. In the
appendix, we have shown,

$\displaystyle\mu\frac{dm_{1\operatorname{eff}}^{2}}{d\mu}=0,\quad\mu\frac{d\lambda_{1\operatorname{eff}}}{d\mu}=0,\quad\mu\frac{dV_{\mathrm{cosmo}}}{d\mu}=0.$
(96)

The other parameters are approximately scale independent. Within the leading
order of the expansion with respect to $\frac{m_{1}^{2}}{m_{2}^{2}}$, they
satisfy,

$\displaystyle\mu\frac{d(m_{12\operatorname{eff}}^{2}\epsilon)}{d\mu}=m_{12}^{2}\epsilon
O\left(\frac{m_{1}^{2}}{m_{2}^{2}}\right)\simeq 0,$ (97)
$\displaystyle\mu\frac{d(\lambda_{3\operatorname{eff}}\epsilon^{2})}{d\mu}=\epsilon^{2}O\left(\frac{m_{1}^{2}}{m_{2}^{2}}\right)\simeq
0,$ (98)
$\displaystyle\mu\frac{d(y_{\operatorname{eff}}\epsilon)}{d\mu}=\epsilon
yO\left(\frac{m_{1}^{2}}{m_{2}^{2}}\right)\simeq 0.$ (99)

##  6 RG improvement

In this section, we discuss the RG improvement of the effective potential of
Eqs.(85-95). The RG improved effective potential for the models with two
scalars is studied in [4, 6]. The RG improved effective potential with multi-
scale is also studied in [7].

By setting the renormalization scale $\mu$ equal to the heavy scalar mass
$m_{2}$, the obtained effective couplings and masses in Eqs.(87-91) include
the large logarithmic correction which is proportional to
$\log\frac{m_{2}^{2}}{m_{1}^{2}+3\lambda_{1}v_{1}^{2}}$. In this section, we
will resum this type of logarithmic corrections. Since their origin is the
loop correction of the light scalar whose virtual momentum ranges from the
heavy scalar mass $m_{2}$ down to the low energy, one can compute the
correction by using the effective low energy theory without the heavy scalar.
We derive the low energy effective Lagrangian by integrating the tree level
contribution of the heavy scalar in Eq.(47),

$\displaystyle S$ $\displaystyle=$ $\displaystyle
S[\bar{\rho}_{1}]+\bar{W}^{c\ \rm{tree}}_{2}(\bar{\rho}_{1},n)$ (100)
$\displaystyle=$ $\displaystyle\int
d^{d}x\left\\{\frac{1}{2}\partial_{\mu}\bar{\rho}_{1}\partial^{\mu}\bar{\rho}_{1}-\frac{m_{1}^{2}}{2}{\bar{\rho}_{1}}^{2}-\frac{\lambda_{1}}{4}{\bar{\rho}_{1}}^{4}+h_{1}{m_{1}}^{4}+h_{2}{m_{2}}^{4}+2h_{3}{m_{1}}^{2}{m_{2}}^{2}+{h_{12}}m_{12}^{4}\right\\}$
$\displaystyle+$ $\displaystyle\frac{i}{2}\int
d^{d}xd^{d}yO(x)D_{F22}(x,y)O(y),$

where $D_{F22}(x,y)$ has the following form of the low energy expansion,

$\displaystyle
D_{F22}(x,y)=\frac{1}{i}\left(\frac{1}{m_{2}^{2}}-\frac{\Box_{x}+\frac{\lambda_{3}}{2}\bar{\rho_{1}}^{2}}{m_{2}^{4}}+\frac{(\Box_{x}+\frac{\lambda_{3}}{2}\bar{\rho_{1}}^{2})^{2}}{m_{2}^{6}}\right)\delta^{d}(x-y).$
(101)

If we keep the terms up to the dimension six ($d=6$) operators, the effective
action is given as,

$\displaystyle S$ $\displaystyle=$ $\displaystyle\int
d^{d}x\left\\{\frac{1}{2}\partial_{\mu}\bar{\rho}_{1}\partial^{\mu}\bar{\rho}_{1}-\frac{m_{1}^{2}}{2}{\bar{\rho}_{1}}^{2}-\frac{\lambda_{1}}{4}{\bar{\rho}_{1}}^{4}+\frac{m_{12}^{4}}{2m_{2}^{2}}{\bar{\rho}_{1}}^{2}+y\frac{m^{2}_{12}}{m_{2}^{2}}\overline{n}{n}\bar{\rho}_{1}\right.$
$\displaystyle+$ $\displaystyle
h_{1}{m_{1}}^{4}+h_{2}{m_{2}}^{4}+2h_{3}{m_{1}}^{2}{m_{2}}^{2}+{h_{12}}m_{12}^{4}$
$\displaystyle+$
$\displaystyle\left.\frac{y^{2}}{2m_{2}^{2}}(\overline{n}{n})^{2}+\frac{m_{12}^{4}}{2m_{2}^{4}}\partial_{\mu}\bar{\rho}_{1}\partial^{\mu}\bar{\rho}_{1}-\frac{\lambda_{3}}{4}\frac{m_{12}^{4}}{m_{2}^{4}}\bar{\rho}_{1}^{4}-y\frac{m^{2}_{12}}{m_{2}^{2}}\frac{\Box\bar{\rho}_{1}+\frac{\lambda_{3}\bar{\rho}^{3}_{1}}{2}}{m_{2}^{2}}(\overline{n}n)+\frac{m_{12}^{4}(\Box\bar{\rho}_{1}+\frac{\lambda_{3}\bar{\rho}_{1}^{3}}{2})^{2}}{2m_{2}^{6}}\right\\}.$

One rewrites the action with $\epsilon=\frac{m^{2}_{12}}{m_{2}^{2}}$ and the
rescaled field $\rho_{1}^{\prime}=\sqrt{1+\epsilon^{2}}\rho_{1}$.

$\displaystyle S$ $\displaystyle=$ $\displaystyle\int
d^{d}x\left\\{\frac{1}{2}\partial_{\mu}\overline{\rho}^{\prime}_{1}\partial^{\mu}\overline{\rho}^{\prime}_{1}-\frac{m_{1}^{2}-\epsilon
m_{12}^{2}}{2}{\overline{\rho}^{\prime}_{1}}^{2}-\frac{\lambda_{1}+\epsilon^{2}(\lambda_{3}-2\lambda_{1})}{4}{\overline{\rho}^{\prime}_{1}}^{4}+\epsilon
y\bar{n}n{\overline{\rho}^{\prime}_{1}}\right.$ (103) $\displaystyle-$
$\displaystyle\left.\epsilon
y\bar{n}n\frac{\Box\overline{\rho}^{\prime}_{1}}{m_{2}^{2}}+\frac{1}{2m_{2}^{2}}\left(\frac{\epsilon\lambda_{3}{\overline{\rho}^{\prime}_{1}}^{3}}{2}-y\bar{n}n\right)^{2}+\frac{(\epsilon\Box{\overline{\rho}^{\prime}_{1}})^{2}}{2m_{2}^{2}}+\frac{(\epsilon^{2}\Box{\overline{\rho}^{\prime}_{1}})}{2m_{2}^{2}}\lambda_{3}{\overline{\rho}^{\prime}_{1}}^{3}\right.$
$\displaystyle+$
$\displaystyle\left.h_{1}{m_{1}}^{4}+h_{2}{m_{2}}^{4}+2h_{3}{m_{1}}^{2}{m_{2}}^{2}+{h_{12}}m_{12}^{4}\right\\}.$

Below we derive the effective potential including one loop effects of light
scalar and improve it with RG equation. As for $d=6$ operators, we estimate
the one loop contribution to the renormalizable terms in the effective
potential. As the result, the contribution turned out to be suppressed by the
higher powers of $\epsilon$ and $\frac{m_{1}^{2}}{m_{2}^{2}}$. Then ignoring
this contribution, we will obtain the effective potential within the following
accuracy: (1) For higher dimensional ($d=6$) terms, we calculate the
contribution within the tree level approximation. (2) For the renormalizable
part of the effective potential, the one-loop contribution is included. The
tree level effective potential is given by substituting the constant vacuum
expectation value $(1+\epsilon^{2})v_{1}^{2}$ to $\bar{\rho_{1}}^{\prime 2}$
in Eq.(103).

$\displaystyle V_{\operatorname{eff}}^{\operatorname{tree}}$ $\displaystyle=$
$\displaystyle\frac{m_{1}^{2}-\epsilon
m_{12}^{2}}{2}{v_{1}}^{2}+\frac{\lambda_{1}+\epsilon^{2}\lambda_{3}}{4}{v_{1}}^{4}-\epsilon
y\bar{n}n{v_{1}}$ (104) $\displaystyle-$ $\displaystyle
h_{1}{m_{1}}^{4}-h_{2}{m_{2}}^{4}-2h_{3}{m_{1}}^{2}{m_{2}}^{2}-{h_{12}}m_{12}^{4}$
$\displaystyle-$
$\displaystyle\frac{1}{2m_{2}^{2}}\left(\frac{\epsilon\lambda_{3}{v_{1}}^{3}}{2}-y\bar{n}n\right)^{2}.$

The total contribution including the one-loop correction and its counter terms
is given as;

$\displaystyle V^{\mathrm{Low}}_{\operatorname{eff}}$ $\displaystyle=$
$\displaystyle
V_{\operatorname{eff}}^{\operatorname{tree}}+V_{\operatorname{eff}}^{\operatorname{1loop}}+V^{c}_{\operatorname{eff}}$
(105) $\displaystyle=$
$\displaystyle\frac{m_{1}^{2}}{2}\left\\{1-\frac{3\lambda_{1}}{16\pi^{2}}\left(\frac{3}{2}-\log\left(\frac{m_{1}^{2}+3\lambda_{1}v_{1}^{2}}{\mu^{2}}\right)\right)\right\\}v_{1}^{2}$
$\displaystyle-\frac{m_{12}^{2}}{2}\left\\{1-\frac{3\lambda_{1}}{16\pi^{2}}\left(1-\log\left(\frac{m_{1}^{2}+3\lambda_{1}v_{1}^{2}}{\mu^{2}}\right)\right)\right\\}\epsilon
v_{1}^{2}-\epsilon
y\bar{n}nv_{1}-\frac{1}{2m_{2}^{2}}\left(\frac{\lambda_{3}\epsilon}{2}v_{1}^{3}-y\bar{n}n\right)^{2}$
$\displaystyle+\frac{\lambda_{1}}{4}\left\\{1-\frac{9\lambda_{1}}{16\pi^{2}}\left(\frac{3}{2}-\log\left(\frac{m_{1}^{2}+3\lambda_{1}v_{1}^{2}}{\mu^{2}}\right)\right)\right\\}v_{1}^{4}$
$\displaystyle+\frac{\lambda_{3}}{4}\left\\{1+\frac{9}{8\pi^{2}}\frac{\lambda_{1}(\lambda_{1}-\lambda_{3})}{\lambda_{3}}+\frac{9}{8\pi^{2}}\frac{\lambda_{1}(\lambda_{3}-\lambda_{1})}{\lambda_{3}}\log\left(\frac{m_{1}^{2}+3\lambda_{1}v_{1}^{2}}{\mu^{2}}\right)\right\\}\epsilon^{2}v_{1}^{4}$
$\displaystyle-
m_{1}^{4}\left\\{h_{1}+\frac{1}{64\pi^{2}}\left(\frac{3}{2}-\log\left(\frac{m_{1}^{2}+3\lambda_{1}v_{1}^{2}}{\mu^{2}}\right)\right)\right\\}$
$\displaystyle+2\epsilon^{2}m_{1}^{2}m_{2}^{2}\left\\{-\frac{h_{3}}{\epsilon^{2}}+\frac{1}{64\pi^{2}}\left(1-\log\left(\frac{m_{1}^{2}+3\lambda_{1}v_{1}^{2}}{\mu^{2}}\right)\right)\right\\}$
$\displaystyle-\epsilon^{2}m_{12}^{4}\left\\{\frac{h_{12}}{\epsilon^{2}}-\frac{1}{64\pi^{2}}\log\left(\frac{m_{1}^{2}+3\lambda_{1}v_{1}^{2}}{\mu^{2}}\right)\right\\}-h_{2}{m_{2}}^{4}.$

The derivation of Eq.(105) is given in appendix B.
$V_{\operatorname{eff}}^{\operatorname{1loop}}$ and
$V^{c}_{\operatorname{eff}}$ can be found in Eq.(144) and in
Eq.(LABEL:eq:countertermforveff), respectively. To obtain the effective
potential Eq.(105) from Eq.(150), we keep the terms such as
$m_{1}^{2}v_{1}^{2}\epsilon^{0}$, $m_{12}^{2}v_{1}^{2}\epsilon$ , $v_{1}^{4}$
and $\epsilon^{2}v_{1}^{4}$. About the cosmological constant terms, we keep
the terms of the forms $m_{1}^{4}\epsilon^{0}$, $m_{1}^{2}m^{2}_{12}\epsilon$
and $m_{12}^{4}\epsilon^{2}$. We drop the other terms with the extra
suppression factors.

We compare the low energy effective potential in Eq.(105) to the effective
potential Eq.(85). The latter includes both heavy and light scalar loop effect
. We set the renormalization scale $\mu$ equal to $m_{2}$ in both effective
potentials. For $\mu=m_{2}$, the effective potential in Eq.(85) becomes,

$\displaystyle V_{\mathrm{eff}}$ $\displaystyle=$
$\displaystyle\frac{m_{1}^{2}}{2}\left[1-\frac{3\lambda_{1}}{16\pi^{2}}\frac{3}{2}-\frac{\lambda_{3}}{32\pi^{2}}\frac{m_{2}^{2}}{m_{1}^{2}}+\frac{3\lambda_{1}}{16\pi^{2}}\log\left(\frac{m_{1}^{2}+3\lambda_{1}v_{1}^{2}}{m_{2}^{2}}\right)\right]v_{1}^{2}$
(106)
$\displaystyle-\frac{m_{12}^{2}}{2}\left[1-\frac{5\lambda_{3}}{32\pi^{2}}+\frac{3\lambda_{2}}{16\pi^{2}}+\frac{3\lambda_{1}}{16\pi^{2}}\log\left(\frac{m_{1}^{2}+3\lambda_{1}v_{1}^{2}}{m_{2}^{2}}\right)\right]\epsilon
v_{1}^{2}$
$\displaystyle+\frac{\lambda_{1}}{4}\left[1-\frac{9\lambda_{1}}{16\pi^{2}}\frac{3}{2}+\frac{9\lambda_{1}}{16\pi^{2}}\log\frac{m_{1}^{2}+3\lambda_{1}v_{1}^{2}}{m_{2}^{2}}\right]v_{1}^{4}$
$\displaystyle+\frac{\lambda_{3}}{4}\left[1+\frac{6\lambda_{2}}{16\pi^{2}}-\frac{25\lambda_{3}}{64\pi^{2}}\frac{9\lambda_{1}}{8\pi^{2}}\left(1-\frac{\lambda_{1}}{\lambda_{3}}\right)\log\left(\frac{m_{1}^{2}+3\lambda_{1}v_{1}^{2}}{m_{2}^{2}}\right)\right]\epsilon^{2}v_{1}^{4}$
$\displaystyle-\left(1+\frac{3\lambda_{2}-\lambda_{3}}{16\pi^{2}}\right)y\epsilon\bar{n}nv_{1}$
$\displaystyle-
m_{1}^{4}\left\\{h_{1}+\frac{1}{64\pi^{2}}\left(\frac{3}{2}-\log\frac{m_{1}^{2}+3\lambda_{1}v_{1}^{2}}{m_{2}^{2}}\right)\right\\}$
$\displaystyle-
m_{2}^{4}\left(h_{2}+\frac{3}{128\pi^{2}}\right)-m_{12}^{4}\left(h_{12}+\frac{1}{32\pi^{2}}\right)$
$\displaystyle-2m_{1}^{2}m_{2}^{2}\left\\{h_{3}+\frac{\epsilon^{2}}{64\pi^{2}}\log\left(\frac{m_{1}^{2}+3\lambda_{1}v_{1}^{2}}{m_{2}^{2}}\right)\right\\}-\frac{1}{2m_{2}^{2}}\left(\frac{\lambda_{3}}{2}\epsilon
v_{1}^{3}-y\bar{n}n\right)^{2}.$

We observe that the coefficients of the logarithmic term
$\log\frac{m_{1}^{2}+3\lambda_{1}v_{1}^{2}}{m_{2}^{2}}$ in both expressions
are identical to each other. This implies that the low energy effective
potential of Eq.(105) correctly incorporates the loop effect of the light
scalar. Since one can improve the low energy effective potential Eq.(105) by
RG, we are able to obtain the RG improved effective potential defined by,

$\displaystyle V^{\operatorname{Improved}}_{\operatorname{eff}}$
$\displaystyle=$ $\displaystyle
V_{\operatorname{eff}}-V_{\operatorname{eff}}^{\operatorname{Low}}+V_{\operatorname{eff}}^{\operatorname{Low\
RGimproved}},$ (107)

where we set the renormalization scale $\mu$ equal to $m_{2}$ in
$V_{\operatorname{eff}}$ and $V_{\operatorname{eff}}^{\operatorname{Low}}$. On
the right-hand side, the first two terms
$V_{\operatorname{eff}}-V_{\operatorname{eff}}^{\operatorname{Low}}$ include
the loop effect of the heavy scalar. Using Eqs.(156-158), the solutions of the
RG equations, the RG improved effective potential is obtained,

$\displaystyle V_{\operatorname{eff}}^{\operatorname{Low\ RGimproved}}$
$\displaystyle=$
$\displaystyle\frac{1}{2}\frac{m_{1}^{2}(m_{2})}{\left(1+\frac{9\lambda_{1}(m_{2})}{16\pi^{2}}\log\left(\frac{m_{2}^{2}}{m_{1}^{2}+3\lambda_{1}v_{1}^{2}}\right)\right)^{\frac{1}{3}}}\left(1-\frac{9\lambda_{1}(m_{2})}{32\pi^{2}}\right)v_{1}^{2}$
(108)
$\displaystyle-\frac{1}{2}\frac{m_{12}^{2}(m_{2})}{\left(1+\frac{9\lambda_{1}(m_{2})}{16\pi^{2}}\log\left(\frac{m_{2}^{2}}{m_{1}^{2}+3\lambda_{1}v_{1}^{2}}\right)\right)^{\frac{1}{3}}}\left(1-\frac{3\lambda_{1}(m_{2})}{16\pi^{2}}\right)\epsilon
v_{1}^{2}$
$\displaystyle+\frac{1}{4}\frac{\lambda_{1}(m_{2})}{1+\frac{9\lambda_{1}(m_{2})}{16\pi^{2}}\log\left(\frac{m_{2}^{2}}{m_{1}^{2}+3\lambda_{1}v_{1}^{2}}\right)}\left(1-\frac{27\lambda_{1}(m_{2})}{32\pi^{2}}\right)v_{1}^{4}$
$\displaystyle+\frac{1}{4}\frac{\lambda_{3}(m_{2})}{1+\frac{9\lambda_{1}(m_{2})}{8\pi^{2}}\left(1-\frac{\lambda_{1}(m_{2})}{\lambda_{3}(m_{2})}\right)\log\left(\frac{m_{2}^{2}}{m_{1}^{2}+3\lambda_{1}v_{1}^{2}}\right)}\left(1-\frac{9\lambda_{1}(m_{2})}{8\pi^{2}}\left(1-\frac{\lambda_{1}(m_{2})}{\lambda_{3}(m_{2})}\right)\right)\epsilon^{2}v_{1}^{4}$
$\displaystyle-
m_{1}^{4}\left\\{\bar{h}\left(\sqrt{m_{1}^{2}+3\lambda_{1}v_{1}^{2}}\right)+\frac{3}{128\pi^{2}}\right\\}+2\epsilon
m_{1}^{2}m_{12}^{2}\left\\{\bar{h}\left(\sqrt{m_{1}^{2}+3\lambda_{1}v_{1}^{2}}\right)+\frac{1}{64\pi^{2}}\right\\}$
$\displaystyle-\epsilon^{2}m_{12}^{4}\left\\{\bar{h}\left(\sqrt{m_{1}^{2}+3\lambda_{1}v_{1}^{2}}\right)\right\\}-\epsilon
y\bar{n}nv_{1}-\frac{1}{2m_{2}^{2}}\left(\frac{\lambda_{3}\epsilon}{2}v_{1}^{3}-y\bar{n}n\right)^{2}.$

where the leading logarithmic corrections are summed. $\bar{h}$ denotes the
running cosmological constant defined in Eq.(158) in the appendix C. By adding
Eq.(108) to
$V_{\operatorname{eff}}-V_{\operatorname{eff}}^{\operatorname{Low}}$, one
finally obtains the renormalization group improved effective potential,

$\displaystyle V^{\operatorname{Improved}}_{\operatorname{eff}}$
$\displaystyle=$
$\displaystyle\frac{m_{1}^{2}(m_{2})}{2}\left[-\frac{\lambda_{3}}{32\pi^{2}}\frac{m_{2}^{2}}{m_{1}^{2}}+\frac{1-\frac{9\lambda_{1}(m_{2})}{32\pi^{2}}}{\left(1+\frac{9\lambda_{1}(m_{2})}{16\pi^{2}}\log\left(\frac{m_{2}^{2}}{m_{1}^{2}+3\lambda_{1}v_{1}^{2}}\right)\right)^{\frac{1}{3}}}\right]v_{1}^{2}$
(109)
$\displaystyle-\frac{m_{12}^{2}(m_{2})}{2}\left[-\frac{5\lambda_{3}}{32\pi^{2}}+\frac{3\lambda_{2}}{16\pi^{2}}+\frac{3\lambda_{1}}{16\pi^{2}}+\frac{1-\frac{3\lambda_{1}(m_{2})}{16\pi^{2}}}{\left(1+\frac{9\lambda_{1}(m_{2})}{16\pi^{2}}\log\left(\frac{m_{2}^{2}}{m_{1}^{2}+3\lambda_{1}v_{1}^{2}}\right)\right)^{\frac{1}{3}}}\right]\epsilon
v_{1}^{2}$
$\displaystyle+\frac{\lambda_{1}(m_{2})}{4}\left[\frac{1-\frac{27\lambda_{1}(m_{2})}{32\pi^{2}}}{1+\frac{9\lambda_{1}(m_{2})}{16\pi^{2}}\log\left(\frac{m_{2}^{2}}{m_{1}^{2}+3\lambda_{1}v_{1}^{2}}\right)}\right]v_{1}^{4}$
$\displaystyle+\frac{\lambda_{3}(m_{2})}{4}\left[\frac{6\lambda_{2}}{16\pi^{2}}-\frac{25\lambda_{3}}{64\pi^{2}}+\frac{9\lambda_{1}}{8\pi^{2}}\left(1-\frac{\lambda_{1}}{\lambda_{3}}\right)+\frac{1-\frac{9\lambda_{1}(m_{2})}{8\pi^{2}}\left(1-\frac{\lambda_{1}(m_{2})}{\lambda_{3}(m_{2})}\right)}{1+\frac{9\lambda_{1}(m_{2})}{8\pi^{2}}\left(1-\frac{\lambda_{1}(m_{2})}{\lambda_{3}(m_{2})}\right)\log\left(\frac{m_{2}^{2}}{m_{1}^{2}+3\lambda_{1}v_{1}^{2}}\right)}\right]\epsilon^{2}v_{1}^{4}$
$\displaystyle-\left(1+\frac{3\lambda_{2}-\lambda_{3}}{16\pi^{2}}\right)y\epsilon\bar{n}nv_{1}-m_{1}^{4}\left\\{h_{1}(m_{2})+\frac{1}{64\pi^{2}}\left(\frac{3}{2}-\log\frac{m_{1}^{2}+3\lambda_{1}v_{1}^{2}}{m_{2}^{2}}\right)\right\\}$
$\displaystyle-
m_{2}^{4}\left(h_{2}(m_{2})+\frac{3}{128\pi^{2}}\right)-m_{12}^{4}\left(h_{12}(m_{2})+\frac{1}{32\pi^{2}}\right)$
$\displaystyle-2m_{1}^{2}m_{2}^{2}\left\\{h_{3}(m_{2})+\frac{\epsilon^{2}}{64\pi^{2}}\log\left(\frac{m_{1}^{2}+3\lambda_{1}v_{1}^{2}}{m_{2}^{2}}\right)\right\\}-\frac{1}{2m_{2}^{2}}\left(\frac{\lambda_{3}}{2}\epsilon
v_{1}^{3}-y\bar{n}n\right)^{2}.$

In Eq.(109), we find the large logarithmic correction to the cosmological
constant terms which are proportional to $m_{1}^{4}$ and $m_{1}^{2}m_{2}^{2}$
. The logarithmic correction to the term $m_{1}^{4}$ can be interpreted as the
running of the coefficient for the cosmological constant of the low energy
effective theory (see Eq.(111) and Eq.(158)).

$\displaystyle
m_{1}^{4}\left\\{h_{1}(m_{2})+\frac{1}{64\pi^{2}}\left(\frac{3}{2}-\log\frac{m_{1}^{2}+3\lambda_{1}v_{1}^{2}}{m_{2}^{2}}\right)\right\\}$
(110) $\displaystyle=$ $\displaystyle
m_{1}^{4}\left\\{h_{1}(m_{2})+\frac{3}{128\pi^{2}}-\left(\bar{h}(m_{2})-\bar{h}\left(\sqrt{m_{1}^{2}+3\lambda_{1}v_{1}^{2}}\right)\right)\right\\},$

where,

$\displaystyle\frac{1}{64\pi^{2}}\log\frac{m_{1}^{2}+3\lambda_{1}v_{1}^{2}}{m_{2}^{2}}=\bar{h}(m_{2})-\bar{h}\left(\sqrt{m_{1}^{2}+3\lambda_{1}v_{1}^{2}}\right).$
(111)

The running of another cosmological constant term which is proportional to
$m_{1}^{2}m_{2}^{2}$ in Eq.(109) can be also related to that of the
cosmological constant term in the low energy effective potential in Eq.(105).
The corresponding term in Eq.(109) is,

$\displaystyle-2m_{1}^{2}m_{2}^{2}\left\\{h_{3}(m_{2})+\frac{\epsilon^{2}}{64\pi^{2}}\log\left(\frac{m_{1}^{2}+3\lambda_{1}v_{1}^{2}}{m_{2}^{2}}\right)\right\\}$
(112) $\displaystyle=$ $\displaystyle
2m_{1}^{2}m_{12}^{2}\epsilon\left\\{-\frac{h_{3}(m_{2})}{\epsilon^{2}}-\left(\bar{h}(m_{2})-\bar{h}\left(\sqrt{m_{1}^{2}+3\lambda_{1}v_{1}^{2}}\right)\right)\right\\}.$

In the second line of Eq.(112), we factor out the coefficient $\epsilon
m_{1}^{2}m_{12}^{2}$ so that the comparison to the corresponding term in the
low energy effective potential in Eq.(105) is feasible.

$\displaystyle 2\epsilon
m_{1}^{2}m_{12}^{2}\left\\{\bar{h}(m_{2})+\frac{1}{64\pi^{2}}\left(1-\log\left(\frac{m_{1}^{2}+3\lambda_{1}v_{1}^{2}}{m_{2}^{2}}\right)\right)\right\\}$
(113) $\displaystyle=$ $\displaystyle 2\epsilon
m_{1}^{2}m_{12}^{2}\left\\{\frac{1}{64\pi^{2}}+\bar{h}(m_{2})-\left(\bar{h}(m_{2})-\bar{h}\left(\sqrt{m_{1}^{2}+3\lambda_{1}v_{1}^{2}}\right)\right)\right\\}.$

The running of the coefficient for the cosmological constant
$\bar{h}(m_{2})-\bar{h}\left(\sqrt{m_{1}^{2}+3\lambda_{1}v_{1}^{2}}\right)$ in
Eq.(112) and Eq.(113) is the same. Using Eqs.(111-113), we rewrite the
renormalization improved effective potential, Eq.(109) as,

$\displaystyle V^{\operatorname{Improved}}_{\operatorname{eff}}$
$\displaystyle=\frac{m_{1}^{2}(m_{2})}{2}\left[-\frac{\lambda_{3}}{32\pi^{2}}\frac{m_{2}^{2}}{m_{1}^{2}}+\frac{1-\frac{9\lambda_{1}(m_{2})}{32\pi^{2}}}{\left(1+\frac{9\lambda_{1}(m_{2})}{16\pi^{2}}\log\left(\frac{m_{2}^{2}}{m_{1}^{2}+3\lambda_{1}v_{1}^{2}}\right)\right)^{\frac{1}{3}}}\right]v_{1}^{2}$
$\displaystyle-\frac{m_{12}^{2}(m_{2})}{2}\left[-\frac{5\lambda_{3}}{32\pi^{2}}+\frac{3\lambda_{2}}{16\pi^{2}}+\frac{3\lambda_{1}}{16\pi^{2}}+\frac{1-\frac{3\lambda_{1}(m_{2})}{16\pi^{2}}}{\left(1+\frac{9\lambda_{1}(m_{2})}{16\pi^{2}}\log\left(\frac{m_{2}^{2}}{m_{1}^{2}+3\lambda_{1}v_{1}^{2}}\right)\right)^{\frac{1}{3}}}\right]\epsilon
v_{1}^{2}$
$\displaystyle+\frac{\lambda_{1}(m_{2})}{4}\left[\frac{1-\frac{27\lambda_{1}(m_{2})}{32\pi^{2}}}{1+\frac{9\lambda_{1}(m_{2})}{16\pi^{2}}\log\left(\frac{m_{2}^{2}}{m_{1}^{2}+3\lambda_{1}v_{1}^{2}}\right)}\right]v_{1}^{4}$
$\displaystyle+\frac{\lambda_{3}(m_{2})}{4}\left[\frac{6\lambda_{2}}{16\pi^{2}}-\frac{25\lambda_{3}}{64\pi^{2}}+\frac{9\lambda_{1}}{8\pi^{2}}\left(1-\frac{\lambda_{1}}{\lambda_{3}}\right)+\frac{1-\frac{9\lambda_{1}(m_{2})}{8\pi^{2}}\left(1-\frac{\lambda_{1}(m_{2})}{\lambda_{3}(m_{2})}\right)}{1+\frac{9\lambda_{1}(m_{2})}{8\pi^{2}}\left(1-\frac{\lambda_{1}(m_{2})}{\lambda_{3}(m_{2})}\right)\log\left(\frac{m_{2}^{2}}{m_{1}^{2}+3\lambda_{1}v_{1}^{2}}\right)}\right]\epsilon^{2}v_{1}^{4}$
$\displaystyle-\left(1+\frac{3\lambda_{2}-\lambda_{3}}{16\pi^{2}}\right)y\epsilon\bar{n}nv_{1}-m_{1}^{4}\left\\{h_{1}(m_{2})+\frac{3}{128\pi^{2}}-\bar{h}(m_{2})+\bar{h}\left(\sqrt{m_{1}^{2}+3\lambda_{1}v_{1}^{2}}\right)\right\\}$
$\displaystyle-
m_{2}^{4}\left(h_{2}(m_{2})+\frac{3}{128\pi^{2}}\right)-m_{12}^{4}\left(h_{12}(m_{2})+\frac{1}{32\pi^{2}}\right)$
$\displaystyle-2m_{1}^{2}m_{12}^{2}\epsilon\left\\{\frac{1}{\epsilon^{2}}h_{3}(m_{2})+\bar{h}(m_{2})-\bar{h}\left(\sqrt{m_{1}^{2}+3\lambda_{1}v_{1}^{2}}\right)\right\\}-\frac{1}{2m_{2}^{2}}\left(\frac{\lambda_{3}}{2}\epsilon
v_{1}^{3}-y\bar{n}n\right)^{2}.$ (114)

This completes the derivation of the RG improved effective potential.

To illustrate how the vev depends on the heavy scalar mass, we study the
stationary condition of the effective potential,

$\displaystyle\frac{\partial
V^{\operatorname{Improved}}_{\operatorname{eff}}}{\partial v_{1}^{2}}=0.$
(115)

The solution satisfies,

$\displaystyle\frac{v_{1}}{v_{10}}=\sqrt{\Biggl{[}-\frac{\lambda_{3}m_{2}^{2}}{32\pi^{2}m_{1}^{2}}+\frac{1}{\Bigl{\\{}1-\frac{9\lambda_{1}}{16\pi^{2}}\log\left(\frac{m_{1}^{2}+3\lambda_{1}v_{1}^{2}}{m_{2}^{2}}\right)\Bigr{\\}}^{\frac{1}{3}}}\Biggr{]}\Biggl{[}1-\frac{9\lambda_{1}}{16\pi^{2}}\log\left(\frac{m_{1}^{2}+3\lambda_{1}v_{1}^{2}}{m_{2}^{2}}\right)\Biggr{]}},$
$\displaystyle v_{10}=\sqrt{-\frac{m_{1}^{2}}{\lambda_{1}}},$ (116)

where we keep the leading logarithmic correction and the correction
proportional to the heavy scalar mass squared. The other small corrections
such as those suppressed by powers of $\epsilon$ are ignored. To study the
dependence on the heavy scalar mass of the vev, we solve the vev with
radiative correction in Eq.(116). In Fig.(4), the ratio in Eq.(116) is plotted
as a function of the heavy scalar mass $m_{2}$. We fix the parameters as
$m_{1}^{2}=-(100)^{2}$(GeV)2 and $\lambda_{1}=\lambda_{3}=1$. This corresponds
to $v_{10}=100$ (GeV). The correction to the vev increases as the heavy scalar
mass increases. If we require the correction is within $20\%$ compared to the
vev without the radiative correction, the upper bound on the heavy scalar mass
is about $1000$ (GeV).

Figure 4: The vevs’ ratio in Eq.(116) as a function of heavy scalar mass
$m_{2}$. The dashed line corresponds to $\frac{v_{1}}{v_{10}}=1.2$.

##  7 Summary and Discussion

In this paper, we have studied a model consists of a light scalar and a heavy
scalar. In the model, we introduce the Yukawa interaction for neutrinos
coupled to the heavy scalar. We also add the cosmological constant terms which
are related to the mass parameters of the model. To derive the low energy
effective potential, we integrate both light and heavy scalars out with the
constant expectation value of the light scalar. This is realized by
introducing generating functional with the source only for the light scalar.
Then performing the Legendre transformation, the effective potential of the
light scalar is obtained. In this way, one particle irreducibility for the
light scalar is maintained and the effects of the diagrams where the heavy
scalar is exchanged can be also included.

We have found that the effective potential is independent of the
renormalization scale approximately. The large logarithmic correction for the
heavy scalar loop is suppressed by setting the renormalization scale equal to
the heavy scalar mass. With this choice of the renormalization scale, the
large logarithmic corrections in the effective potential originate from the
loop effect of the light scalar. We derive the low energy effective potential
with the tree level matching and one-loop calculation of the light scalar. As
a result, we find that the logarithmic corrections in the both effective
potential are the same. As for the low energy effective potential, we perform
the RG improvement by summing the leading logarithmic corrections. Since the
loop correction due to the heavy scalar is absent in the low energy effective
potential, we add this correction as the difference of the two effective
potentials.

We found that the effective Yukawa coupling $y_{\rm{eff}}$ between the light
scalar and the Dirac neutrino is inversely proportional to the heavy scalar
mass squared as $y_{\rm{eff}}\simeq y\frac{m_{12}^{2}}{m_{2}^{2}}$ and is
naturally suppressed. As for the dimension six operators, the six point
interaction for the light scalars and the four Fermi interaction for the
neutrinos are also generated. A part of the radiative corrections to the light
scalar mass squared is proportional to the heavy scalar mass squared. We
numerically study the effect to the vev using the stationary condition of the
potential. The vev depends on the heavy scalar mass and one can set the upper
bound on the mass by demanding the radiative correction to the vev should be
limited within a certain range.

The cosmological constants also suffer from the large contribution
proportional to the fourth power of the heavy scalar mass. As for the
renormalizable coupling such as the quartic interaction of the light scalar,
the contribution suppressed by a factor of $\epsilon^{2}$ appear. This implies
the renormalizable coupling constant of the low energy effective potential is
also sensitive to the coupling and the mass of the full theory.

For the future work, we will extend the present analysis to the realistic
model based upon the standard model gauge group such as Davidson and Logan
Model [1], and derive the low energy effective theory [8, 9]. With the
realistic model, one can constrain the parameters of the full theory such as
the heavy scalar mass and the small mass mixing term for two scalars from the
correction to the vev of the light scalar and the effective Yukawa coupling to
neutrinos. One can also obtain the constrains on the coefficients of the
cosmological constants.

Acknowledgement
The work of T.M. is supported by Japan Society for the Promotion of Science
(JSPS) KAKENHI Grant Number JP17K05418 .

##  Appendix A The counter terms for the full theory

In this appendix, we determine the counter terms for the full theory. We also
derive the RG equation and examine the renormalization point independence of
the effective couplings and masses. Since there is no wave function
renormalization from one-loop contribution of scalar fields, it is sufficient
to know the counter terms for the effective potential. The Lagrangian density
for the full theory is;

$\displaystyle\mathcal{L}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\left(\sum_{i=1}^{2}Z_{i}\rho_{i}\Box\rho_{i}+\sum_{i,j=1}^{2}\rho_{i}^{2}Z_{\operatorname{mij}}m_{j}^{2}+\frac{\mu^{2\eta}}{2}\sum_{I=1}^{3}\left(\sum_{i=1}^{2}\rho_{i}^{4}Z_{\lambda\operatorname{iI}}\lambda_{I}+\rho_{1}^{2}\rho_{2}^{2}Z_{\lambda
3I}\lambda_{I}\right)\right)$ (117) $\displaystyle-$
$\displaystyle(Z_{y}y\mu^{\eta}\bar{n}n+Z_{12}m_{12}^{2}\rho_{1})\rho_{2}+\bar{n}i\not{\partial}nZ_{n}+\mu^{d-4}Z_{h_{1}}h_{1}m_{1}^{4}+\mu^{d-4}Z_{h_{2}}h_{2}m_{2}^{4}+\mu^{d-4}Z_{h_{12}}h_{12}m_{12}^{4}$
$\displaystyle=$
$\displaystyle-\frac{1}{2}\left(\sum_{i=1}^{2}\rho_{0i}\Box\rho_{0i}+\sum_{i=1}^{2}\rho_{0i}^{2}m_{0i}^{2}+\frac{1}{2}\left(\sum_{i=1}^{2}\rho_{0i}^{4}\lambda_{i}+\rho_{01}^{2}\rho_{02}^{2}\lambda_{03}\right)\right)$
$\displaystyle-$
$\displaystyle(y_{0}\bar{n}_{0}n_{0}+m_{012}^{2}\rho_{01})\rho_{02}+\mu^{d-4}Z_{h_{1}}h_{1}m_{1}^{4}+\mu^{d-4}Z_{h_{2}}h_{2}m_{2}^{4}+\mu^{d-4}Z_{h_{12}}h_{12}m_{12}^{4}.$

The counter terms are given by,

$\displaystyle\mathcal{L}_{c}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\sum_{i=1}^{2}(Z_{i}-1)\rho_{i}\Box\rho_{i}-\frac{1}{2}\sum_{i,j=1}^{2}\rho_{i}^{2}(Z_{\operatorname{mij}}-\delta_{\operatorname{ij}})m_{j}^{2}-((Z_{y}-1)y\mu^{\eta}\bar{n}n+(Z_{12}-1)m_{12}^{2}\rho_{1})\rho_{2}$
$\displaystyle-$
$\displaystyle\frac{\mu^{2\eta}}{4}\sum_{I=1}^{3}\left(\sum_{i=1}^{2}\rho_{i}^{4}(Z_{\lambda\operatorname{iI}}-\delta_{\operatorname{iI}})\lambda_{I}+\rho_{1}^{2}\rho_{2}^{2}(Z_{\lambda
3I}-\delta_{3I})\lambda_{I}\right)$ $\displaystyle+$ $\displaystyle
i\bar{n}\not{\partial}n(Z_{n}-1)-\mu^{d-4}(Z_{h_{1}}-1)h_{1}m_{1}^{4}-\mu^{d-4}(Z_{h_{2}}-1)h_{2}m_{2}^{4}-\mu^{d-4}(Z_{h_{12}}-1)h_{12}m_{12}^{4}.$

Next we compute the one-loop effective potential.

$\displaystyle V_{\operatorname{eff}}^{1\operatorname{loop}}$ $\displaystyle=$
$\displaystyle-i\frac{1}{2V^{d-1}T}\log\det\left(\left.\frac{-\delta^{2}S_{\operatorname{tree}}}{\delta\rho_{i}(x)\delta\rho_{j}(y)}\right|_{\rho_{i}=v_{i}\mu^{-\eta}}\right)+V_{c}$
(118) $\displaystyle=$
$\displaystyle\frac{1}{2}\int\frac{d^{d}k}{(2\pi)^{d}i}\log\Bigl{[}\left(m_{1}^{2}+\left(3\lambda_{1}v^{2}_{1}+\frac{\lambda_{3}}{2}v_{2}^{2}\right)-k^{2}\right)\left(m_{2}^{2}+\left(3\lambda_{2}v_{2}^{2}+\frac{\lambda_{3}}{2}v_{1}^{2}\right)-k^{2}\right)$
$\displaystyle-$
$\displaystyle(m_{12}^{2}+v_{1}v_{2}\lambda_{3})^{2}\Bigr{]}+V_{c}.$

where $V_{c}$ is the countertems for effective potential given by,

$\displaystyle V_{c}$ $\displaystyle=$
$\displaystyle\frac{\mu^{-2\eta}}{2}\Bigl{[}\\{(Z_{m11}-1)m_{1}^{2}+Z_{m12}m_{2}^{2}\\}v_{1}^{2}+\\{(Z_{m22}-1)m_{2}^{2}+Z_{m21}m_{1}^{2}\\}v_{2}^{2}+2(Z_{12}-1)m_{12}^{2}v_{1}v_{2}\Bigr{]}$
(119) $\displaystyle+$ $\displaystyle\mu^{-2\eta}\frac{(Z_{\lambda
11}-1)\lambda_{1}+Z_{\lambda 12}\lambda_{2}+Z_{\lambda
13}\lambda_{3}}{4}v_{1}^{4}+\mu^{-2\eta}\frac{Z_{\lambda
21}\lambda_{1}+(Z_{\lambda 22}-1)\lambda_{2}+Z_{\lambda
23}\lambda_{3}}{4}v_{2}^{4}$ $\displaystyle+$
$\displaystyle\mu^{-2\eta}\frac{Z_{\lambda 31}\lambda_{1}+Z_{\lambda
32}\lambda_{2}+(Z_{\lambda
33}-1)\lambda_{3}}{4}v_{1}^{2}v_{2}^{2}+(Z_{y}-1)y\bar{n}nv_{2}$
$\displaystyle-$
$\displaystyle\mu^{-2\eta}\\{(Z_{h_{1}}-1)h_{1}m_{1}^{4}+(Z_{h_{2}}-1)h_{2}m_{2}^{4}+(Z_{h_{12}}-1)h_{12}m_{12}^{4}\\}.$

The divergent parts can be extracted by expanding the logarithmic terms up to
$(m_{12}^{2}+v_{1}v_{2}\lambda_{3})^{2}$.

$\displaystyle V_{\operatorname{eff}}^{1\operatorname{loop}}$
$\displaystyle\simeq$
$\displaystyle\frac{1}{2}\int\frac{d^{d}k}{(2\pi)^{d}i}\left\\{\log(m_{1}^{2}+3\lambda_{1}v^{2}_{1}+\frac{\lambda_{3}}{2}v_{2}^{2}-k^{2})+\log(m_{2}^{2}+3\lambda_{2}v_{2}^{2}+\frac{\lambda_{3}}{2}v_{1}^{2}-k^{2})\right\\}$
$\displaystyle-$
$\displaystyle\frac{1}{2}\int\frac{d^{d}k}{(2\pi)^{d}i}\frac{(m_{12}^{2}+v_{1}v_{2}\lambda_{3})^{2}}{(m_{1}^{2}+3\lambda_{1}v^{2}_{1}+\frac{\lambda_{3}}{2}v_{2}^{2}-k^{2})(m_{2}^{2}+3\lambda_{2}v_{2}^{2}+\frac{\lambda_{3}}{2}v_{1}^{2}-k^{2})}+V_{c}+...$

We collect only the one loop divergences and the counter terms as follows.

$\displaystyle
V_{\operatorname{eff}}^{1\operatorname{loop}\operatorname{div}}+V_{c}$ (121)
$\displaystyle=$
$\displaystyle-\frac{C_{\operatorname{UV}}}{64\pi^{2}}\\{m_{1}^{4}+m_{2}^{4}+2m_{12}^{4}\\}-\frac{C_{\operatorname{UV}}}{32\pi^{2}}\left(3m_{1}^{2}\lambda_{1}+m_{2}^{2}\frac{\lambda_{3}}{2}\right)v_{1}^{2}$
$\displaystyle-$
$\displaystyle\frac{C_{\operatorname{UV}}}{32\pi^{2}}\left(3m_{2}^{2}\lambda_{2}+m_{1}^{2}\frac{\lambda_{3}}{2}\right)v_{2}^{2}-\frac{C_{\operatorname{UV}}}{16\pi^{2}}m_{12}^{2}\lambda_{3}v_{1}v_{2}$
$\displaystyle-$
$\displaystyle\frac{C_{\operatorname{UV}}}{64\pi^{2}}\left\\{\left(9\lambda_{1}^{2}+\frac{\lambda_{3}^{2}}{4}\right)v_{1}^{4}+\left(9\lambda_{2}^{2}+\frac{\lambda_{3}^{2}}{4}\right)v_{2}^{4}+3(\lambda_{1}+\lambda_{2})\lambda_{3}v^{2}_{1}v_{2}^{2}+2\lambda_{3}^{2}v_{1}^{2}v_{2}^{2}\right\\}$
$\displaystyle+$
$\displaystyle\frac{\mu^{-2\eta}}{2}\Bigl{[}\\{(Z_{m11}-1)m_{1}^{2}+Z_{m12}m_{2}^{2}\\}v_{1}^{2}+\\{(Z_{m22}-1)m_{2}^{2}+Z_{m21}m_{1}^{2}\\}v_{2}^{2}+2(Z_{12}-1)m_{12}^{2}v_{1}v_{2}\Bigr{]}$
$\displaystyle+$ $\displaystyle\mu^{-2\eta}\frac{(Z_{\lambda
11}-1)\lambda_{1}+Z_{\lambda 12}\lambda_{2}+Z_{\lambda
13}\lambda_{3}}{4}v_{1}^{4}+\mu^{-2\eta}\frac{Z_{\lambda
21}\lambda_{1}+(Z_{\lambda 22}-1)\lambda_{2}+Z_{\lambda
23}\lambda_{3}}{4}v_{2}^{4}$ $\displaystyle+$
$\displaystyle\mu^{-2\eta}\frac{Z_{\lambda 31}\lambda_{1}+Z_{\lambda
32}\lambda_{2}+(Z_{\lambda
33}-1)\lambda_{3}}{4}v_{1}^{2}v_{2}^{2}+(Z_{y}-1)y\bar{n}nv_{2}$
$\displaystyle-$
$\displaystyle\mu^{-2\eta}\\{(Z_{h_{1}}-1)h_{1}m_{1}^{4}+(Z_{h_{2}}-1)h_{2}m_{2}^{4}+(Z_{h_{12}}-1)h_{12}m_{12}^{4}\\}.$

We require that the counter terms subtract all the divergences,

$\displaystyle V_{c}$ $\displaystyle=$ $\displaystyle-
V_{\operatorname{eff}}^{1\operatorname{loop}\operatorname{div}}.$ (122)

Then one can determine the Z factors of the counter terms as follows,

$\displaystyle Z_{m\operatorname{ii}}$ $\displaystyle=$ $\displaystyle
1+\frac{3C_{\operatorname{UV}}}{16\pi^{2}}\lambda_{i},\quad
Z_{m12}=Z_{m21}=\frac{C_{\operatorname{UV}}}{32\pi^{2}}\lambda_{3},$ (123)
$\displaystyle Z_{12}$ $\displaystyle=$ $\displaystyle
1+\frac{C_{\operatorname{UV}}}{16\pi^{2}}\lambda_{3},\quad
Z_{\lambda\operatorname{ii}}=1+\frac{C_{\operatorname{UV}}}{16\pi^{2}}9\lambda_{i},$
(124) $\displaystyle Z_{\lambda i3}$ $\displaystyle=$
$\displaystyle\frac{C_{\operatorname{UV}}}{64\pi^{2}}\lambda_{3},\quad
Z_{\lambda 3i}=\frac{C_{\operatorname{UV}}}{16\pi^{2}}3\lambda_{3},\quad
Z_{\lambda 33}=1+\frac{C_{\operatorname{UV}}}{8\pi^{2}}\lambda_{3},$ (125)
$\displaystyle Z_{y}$ $\displaystyle=$ $\displaystyle 1,\quad Z_{\lambda
12}=Z_{\lambda 21}=0,$ (126) $\displaystyle(Z_{h_{i}}-1)h_{i}$
$\displaystyle=$
$\displaystyle-\frac{C_{\operatorname{UV}}}{64\pi^{2}},\quad(Z_{h_{12}}-1)h_{12}=-\frac{C_{\operatorname{UV}}}{32\pi^{2}},$
(127)

where $i=1,2$. This completes the derivation of the Z factors in the counter
terms for the full theory. The RG equations for coupling constants can be also
derived as,

$\displaystyle\mu\frac{d\lambda_{3}}{d\mu}$ $\displaystyle=$
$\displaystyle\frac{1}{8\pi^{2}}(2\lambda_{3}+3\lambda_{2}+3\lambda_{1})\lambda_{3},$
(128) $\displaystyle\mu\frac{d\lambda_{i}}{d\mu}$ $\displaystyle=$
$\displaystyle\frac{1}{8\pi^{2}}9\lambda_{i}^{2}+\frac{1}{32\pi^{2}}\lambda_{3}^{2},\quad(i=1,2).$
(129)

The coefficients of the cosmological constants and the mass parameters satisfy
the following RG equations.

$\displaystyle\mu\frac{dh_{1}}{d\mu}$ $\displaystyle=$
$\displaystyle\frac{-1}{16\pi^{2}}\left(\frac{1}{2}+12\lambda_{1}h_{1}+2\lambda_{3}h_{3}\right),$
(130) $\displaystyle\mu\frac{dh_{2}}{d\mu}$ $\displaystyle=$
$\displaystyle\frac{-1}{16\pi^{2}}\left(\frac{1}{2}+12\lambda_{2}h_{2}+2\lambda_{3}h_{3}\right),$
(131) $\displaystyle\mu\frac{dh_{3}}{d\mu}$ $\displaystyle=$
$\displaystyle\frac{-1}{16\pi^{2}}(\lambda_{3}(h_{1}+h_{2})+6(\lambda_{2}+\lambda_{1})h_{3}),$
(132) $\displaystyle\mu\frac{dh_{12}}{d\mu}$ $\displaystyle=$
$\displaystyle-\frac{1}{16\pi^{2}}(1+4\lambda_{3}h_{12}),$ (133)
$\displaystyle\mu\frac{dm_{1}^{2}}{d\mu}$ $\displaystyle=$
$\displaystyle\frac{3}{8\pi^{2}}\lambda_{1}m_{1}^{2}+\frac{1}{16\pi^{2}}\lambda_{3}m_{2}^{2},$
(134) $\displaystyle\mu\frac{dm_{2}^{2}}{d\mu}$ $\displaystyle=$
$\displaystyle\frac{3}{8\pi^{2}}\lambda_{2}m_{2}^{2}+\frac{1}{16\pi^{2}}\lambda_{3}m_{1}^{2},$
(135) $\displaystyle\mu\frac{dm_{12}^{2}}{d\mu}$ $\displaystyle=$
$\displaystyle\frac{\lambda_{3}}{8\pi^{2}}m_{12}^{2}.$ (136)

Using the RG equations from Eq.(128) to Eq.(136), one can study the
renormalization point independence of the effective masses and the effective
coupling constants in the one-loop effective potential of Eq.(85).

$\displaystyle\mu\frac{dm_{1\operatorname{eff}}^{2}}{d\mu}$ $\displaystyle=$
$\displaystyle
m_{1}^{2}\frac{3\lambda_{1}}{8\pi^{2}}+m_{2}^{2}\frac{\lambda_{3}}{16\pi^{2}}-m_{1}^{2}\frac{3\lambda_{1}}{8\pi^{2}}-\frac{\lambda_{3}m_{2}^{2}}{16\pi^{2}}=0,$
(137) $\displaystyle\mu\frac{d(m_{12\operatorname{eff}}^{2}\epsilon)}{d\mu}$
$\displaystyle=$
$\displaystyle\frac{m_{12}^{4}}{m_{2}^{2}}\left\\{\frac{\lambda_{3}}{4\pi^{2}}-\frac{3\lambda_{2}}{8\pi^{2}}+\frac{3\lambda_{2}}{8\pi^{2}}-\frac{\lambda_{3}}{4\pi^{2}}+\frac{\lambda_{3}}{16\pi^{2}}\frac{m_{1}^{2}}{m_{2}^{2}}\right\\}=\frac{\lambda_{3}m_{12}^{2}\epsilon}{16\pi^{2}}\left(\frac{m_{1}^{2}}{m_{2}^{2}}\right),$
(138) $\displaystyle\mu\frac{d\lambda_{1\operatorname{eff}}}{d\mu}$
$\displaystyle=$
$\displaystyle\frac{9\lambda_{1}^{2}}{8\pi^{2}}+\frac{\lambda_{3}^{2}}{32\pi^{2}}+\lambda_{1}\left\\{-\frac{9\lambda_{1}}{8\pi^{2}}-\frac{1}{32\pi^{2}}\frac{\lambda_{3}^{2}}{\lambda_{1}}\right\\}=0,$
(139)
$\displaystyle\mu\frac{d(\lambda_{3\operatorname{eff}}\epsilon^{2})}{d\mu}$
$\displaystyle=$
$\displaystyle\epsilon^{2}\left\\{\frac{\lambda_{3}(2\lambda_{3}+3\lambda_{2}+3\lambda_{1}+3\lambda_{2}-4\lambda_{3}-3\lambda_{1}+2\lambda_{3}-6\lambda_{2})}{8\pi^{2}}\right\\}-\frac{\lambda_{3}}{8\pi^{2}}\frac{m_{1}^{2}}{m_{2}^{2}}$
(140) $\displaystyle=$
$\displaystyle-\frac{\lambda_{3}\epsilon^{2}}{8\pi^{2}}\left(\frac{m_{1}^{2}}{m_{2}^{2}}\right),$
$\displaystyle\mu\frac{d(y_{\operatorname{eff}}\epsilon)}{d\mu}$
$\displaystyle=$ $\displaystyle\epsilon
y\left\\{\frac{3\lambda_{2}-\lambda_{3}+\lambda_{3}-3\lambda_{2}}{8\pi^{2}}\right\\}-\frac{\epsilon
y\lambda_{3}}{16\pi^{2}}\frac{m_{1}^{2}}{m_{2}^{2}}=-\frac{\lambda_{3}y\epsilon}{16\pi^{2}}\left(\frac{m_{1}^{2}}{m_{2}^{2}}\right).$
(141)

One finds all of the effective masses and the couplings are renormalization
point independent by ignoring the sub-leading corrections
$O(\frac{m_{1}^{2}}{m_{2}^{2}})$. The remaining dependence is due to the
truncation of these suppressed contribution in the derivation of the effective
potential. The cosmological constants of Eq.$(\ref{Vcosmo})$ is written as
follows,

$\displaystyle V_{\operatorname{cosmo}}$ $\displaystyle=$
$\displaystyle-\frac{m_{12}^{4}}{32\pi^{2}}\left(1-\log\frac{m_{2}^{2}}{\mu^{2}}\right)-\frac{m_{1}^{4}}{64\pi^{2}}\left(\frac{3}{2}-\log\left(\frac{m_{1}^{2}+3\lambda_{1}v_{1}^{2}}{\mu^{2}}\right)\right)$
(142)
$\displaystyle-\frac{m_{2}^{4}}{64\pi^{2}}\left(\frac{3}{2}-\log\frac{m_{2}^{2}}{\mu^{2}}\right)-h_{1}m_{1}^{4}-h_{2}m_{2}^{4}-h_{12}m_{12}^{4}-2h_{3}m_{1}^{2}m_{2}^{2}.$

The renormalization point independence of $V_{\operatorname{cosmo}}$ is
explicitly shown below.

$\displaystyle\mu\frac{dV_{\operatorname{cosmo}}}{d\mu}$ $\displaystyle=$
$\displaystyle\frac{2\lambda_{3}h_{3}}{16\pi^{2}}(m_{1}^{4}+m_{2}^{4})-2(h_{1}+h_{2})m_{1}^{2}\frac{1}{16\pi^{2}}\lambda_{3}m_{2}^{2}$
(143)
$\displaystyle+\frac{2}{16\pi^{2}}(\lambda_{3}(h_{1}+h_{2}))m_{1}^{2}m_{2}^{2}-h_{3}\frac{2}{16\pi^{2}}\lambda_{3}(m_{2}^{4}+m_{1}^{4})$
$\displaystyle=$ $\displaystyle 0.$

##  Appendix B Derivation of $V_{\rm{eff}}^{\rm{Low}}$ in Eq.(105)

In this appendix, we give an outline of the derivation of Eq.(105). In
Eq.(105), $V_{\operatorname{eff}}^{1\operatorname{loop}}$ is given by,

$\displaystyle V_{\operatorname{eff}}^{1\operatorname{loop}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\int\frac{d^{d}k}{(2\pi)^{d}i}\log(m_{1}^{\prime
2}+3\lambda_{1}^{\prime}v_{1}^{2}-k^{2})$ (144) $\displaystyle=$
$\displaystyle-\frac{(m_{1}^{\prime
2}+3\lambda_{1}^{\prime}v_{1}^{2})^{2}}{64\pi^{2}}\left(C_{UV}+\frac{3}{2}-\log(m_{1}^{\prime
2}+3\lambda_{1}^{\prime}v_{1}^{2})\right),$

where the divergence is denoted by $C_{UV}=\frac{1}{\eta}-\gamma+\log 4\pi$
and the coupling constant and the mass are defined as,

$\displaystyle m_{1}^{\prime 2}=m_{1}^{2}-\epsilon
m_{12}^{2},\quad\lambda_{1}^{\prime}=\lambda_{1}+\epsilon^{2}(\lambda_{3}-\lambda_{1}).$
(145)

The divergent part is extracted from Eq.(144) and is given as,

$\displaystyle V_{\operatorname{eff}}^{1\operatorname{loop\ div.}}$
$\displaystyle=$ $\displaystyle-\frac{(m_{1}^{\prime
2}+3\lambda_{1}^{\prime}v_{1}^{2})^{2}}{64\pi^{2}}C_{UV}$ $\displaystyle=$
$\displaystyle-\frac{(m_{1}^{2}-\epsilon m_{12}^{2})^{2}+6(m_{1}^{2}-\epsilon
m_{12}^{2})(\lambda_{1}+\epsilon^{2}(\lambda_{3}-\lambda_{1}))v_{1}^{2}+9(\lambda_{1}+\epsilon^{2}(\lambda_{3}-\lambda_{1}))^{2}v_{1}^{4}}{64\pi^{2}}C_{UV}.$

Then the divergence can be subtracted by the counter terms,

$\displaystyle V^{c}_{\operatorname{eff}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}(Z_{m^{\prime}_{1}}-1)(m_{1}^{2}-\epsilon
m^{2}_{12})\mu^{-2\eta}v_{1}^{2}+\frac{\lambda_{1}^{\prime}}{4}(Z_{\lambda_{1}^{\prime}}-1)v_{1}^{4}\mu^{-2\eta}-(Z_{\bar{h}}-1)\mu^{-2\eta}\bar{h}(m_{1}^{2}-\epsilon
m^{2}_{12})^{2}.$

where the $Z$ factors are determined so that the counter terms satisfy
$V^{c}_{\operatorname{eff}}=-V_{\operatorname{eff}}^{1\operatorname{loop\
div.}}$.

$\displaystyle
Z_{m^{\prime}_{1}}-1=\frac{3\lambda^{\prime}_{1}C_{\operatorname{UV}}}{16\pi^{2}},\quad
Z_{\lambda_{1}^{\prime}}-1=\frac{9C_{\operatorname{UV}}}{16\pi^{2}}\lambda_{1}^{\prime}.$
(148) $\displaystyle(Z_{\bar{h}}-1)\bar{h}=-\frac{1}{64\pi^{2}}C_{UV}.$ (149)

By adding the counter terms to the tree and one-loop contribution, the finite
effective potential is obtained,

$\displaystyle
V_{\rm{eff}}+V_{\operatorname{eff}}^{1\operatorname{loop}}+V^{c}_{\operatorname{eff}}$
$\displaystyle=-h_{1}{m_{1}}^{4}-h_{2}{m_{2}}^{4}-2h_{3}{m_{1}}^{2}{m_{2}}^{2}-{h_{12}}m_{12}^{4}-\frac{1}{2m_{2}^{2}}\left(\frac{\epsilon\lambda_{3}{v_{1}}^{3}}{2}-y\bar{n}n\right)^{2}$
$\displaystyle+\frac{m_{1}^{2}-\epsilon
m_{12}^{2}}{2}{v_{1}}^{2}+\frac{\lambda_{1}+\epsilon^{2}\lambda_{3}}{4}{v_{1}}^{4}-\epsilon
y\bar{n}n{v_{1}}$ $\displaystyle+\frac{(m_{1}^{2}-\epsilon
m_{12}^{2}+3(\lambda_{1}+\epsilon^{2}(\lambda_{3}-\lambda_{1}))v_{1}^{2})^{2}}{64\pi^{2}}\left(\log\frac{m_{1}^{2}-\epsilon
m_{12}^{2}+3(\lambda_{1}+\epsilon^{2}(\lambda_{3}-\lambda_{1}))v_{1}^{2}}{\mu^{2}}-\frac{3}{2}\right).$
(150)

To obtain the final form of the effective potential Eq.(105) from Eq.(150), we
keep the terms such as $m_{1}^{2}v_{1}^{2}\epsilon^{0}$,
$m_{12}^{2}v_{1}^{2}\epsilon$ , $v_{1}^{4}$ and $\epsilon^{2}v_{1}^{4}$. About
the cosmological constant terms, we keep the terms of the forms
$m_{1}^{4}\epsilon^{0}$, $m_{1}^{2}m^{2}_{12}\epsilon$ and
$m_{12}^{4}\epsilon^{2}$.

##  Appendix C  RG equation and its solutions for low energy effective theory

In this appendix, the RG equations for the coupling ,mass, and cosmological
constant of the low energy effective action in Eq.(103) are obtained. The
solutions in the leading logarithmic approximation are also derived. The
relations between the renormalized quantities and the bare ones are given as
follows;

$\displaystyle\lambda^{\prime}_{10}=Z_{\lambda^{\prime}_{1}}\lambda^{\prime}_{1}\mu^{2\eta};\quad
m_{10}^{\prime 2}=Z_{m_{1}^{\prime}}m_{1}^{\prime 2};\quad\bar{h}_{0}m^{\prime
4}_{10}=Z_{\bar{h}}\bar{h}m_{1}^{\prime 4}\mu^{-2\eta}$ (151)

where $Z$ factors are given by,

$\displaystyle
Z_{\lambda_{1}^{\prime}}=1+\frac{9C_{\operatorname{UV}}}{16\pi^{2}}\lambda_{1}^{\prime},\quad
Z_{m_{1}^{\prime}}=1+\frac{3\lambda_{1}^{\prime}C_{\operatorname{UV}}}{16\pi^{2}},\quad(Z_{\bar{h}}-1)\bar{h}=-\frac{C_{\operatorname{UV}}}{64\pi^{2}}.$
(152)

Using the relations, the RG equations are derived as,

$\displaystyle\mu\frac{dm^{\prime 2}_{1}(\mu)}{d\mu}-m^{\prime
2}_{1}(\mu)\frac{3\lambda_{1}^{\prime}}{8\pi^{2}}=0,$ (153)
$\displaystyle\mu\frac{d\lambda_{1}^{\prime}(\mu)}{d\mu}-\frac{9\lambda_{1}^{\prime
2}}{8\pi^{2}}=0,$ (154)
$\displaystyle\mu\frac{d\bar{h}}{d\mu}+\frac{1}{32\pi^{2}}=0.$ (155)

By solving the RG equations, we obtain the RG improved couplings and masses.

$\displaystyle\lambda_{1}^{\prime}(\mu_{0})$ $\displaystyle=$
$\displaystyle\frac{\lambda_{1}^{\prime}(\mu)}{1+\frac{9\lambda_{1}^{\prime}(\mu)}{16\pi^{2}}\log\frac{\mu^{2}}{\mu_{0}^{2}}},$
(156) $\displaystyle m_{1}^{\prime 2}(\mu_{0})$ $\displaystyle=$
$\displaystyle\frac{m^{\prime
2}_{1}(\mu)}{\left(1+\frac{9\lambda_{1}^{\prime}(\mu)}{16\pi^{2}}\log\frac{\mu^{2}}{\mu_{0}^{2}}\right)^{\frac{1}{3}}},$
(157) $\displaystyle\bar{h}(\mu_{0})$ $\displaystyle=$
$\displaystyle\bar{h}(\mu)-\frac{1}{64\pi^{2}}\log\frac{\mu_{0}^{2}}{\mu^{2}}.$
(158)

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