#  OU-HET/906 KEK-TH-1926 MAD-TH-16-09 Meaning of the field dependence of the
renormalization scale in Higgs inflation

Yuta Hamada, 1,2 Hikaru Kawai, 3 Yukari Nakanishi, 4 and Kin-ya Oda 4

1KEK Theory Center, IPNS, KEK, Tsukuba, Ibaraki 305-0801, Japan
2Department of Physics, University of Wisconsin, Madison, WI 53706, USA
3Department of Physics, Kyoto University, Kyoto 606-8502, Japan
4Department of Physics, Osaka University, Osaka 560-0043, Japan

yhamada<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We consider the prescription dependence of the Higgs effective potential under
the presence of general nonminimal couplings. We evaluate the fermion loop
correction to the effective action in a simplified Higgs-Yukawa model whose
path integral measure takes simple form either in the Jordan or Einstein
frame. The resultant effective action becomes identical in both cases when we
properly take into account the quartically divergent term coming from the
change of measure. Working in the counterterm formalism, we clarify that the
difference between the prescriptions I and II comes from the counter term to
cancel the logarithmic divergence. This difference can be absorbed into the
choice of tree-level potential from the infinitely many possibilities,
including all the higher-dimensional terms. We also present another mechanism
to obtain a flat potential by freezing the running of the effective quartic
coupling for large field values, using the nonminimal coupling in the gauge
kinetic function.

##  1 Introduction

The Higgs inflation [1, 2] is one of the closest to the best fit point in the
tensor-to-scalar ratio vs spectral-index plane among various inflation models
[3]. The model requires rather a large nonminimal coupling $\xi\sim
10^{5\text{--}6}$ between the Higgs-squared $H^{\dagger}H$ and the Ricci
scalar $\mathcal{R}$.111 The earlier model [1, 4, 5, 6], without the Einstein-
Hilbert action at the tree level, requires $\xi\sim 10^{34}$ [2]. In Ref. [1],
the authors have also studied the Higgs inflation model in Ref. [2] with
essentially the same parameter $\xi\sim 10^{4}$ and
$\lambda\sim\left(\xi/10^{5}\right)^{2}\sim 10^{-2}$. See also Refs. [7, 8]
for inflation with the nonminimal coupling, and Ref. [9] for a possible issue
with the large nonminimal coupling.  On the other hand, the observed value of
the Higgs mass $m_{H}=125.09\pm 0.24\,\text{GeV}$ [10] indicates that the
Standard Model (SM) is at the _criticality_ , that is, the Higgs potential
becomes small and nearly flat when the Higgs-field value is close to the
Planck-scale; see e.g. Refs. [11, 12].

In the flat spacetime, the renormalized Higgs potential $V$ can be computed as
a sum of the tree-level potential $V_{\text{R}}$ and the loop correction
$\Delta V_{\text{R}}$, both of them being finite but depending on the
renormalization scale $\mu$, in the counterterm formalism. Since $V$ is
independent of $\mu$, we may choose $\mu$ arbitrarily. A convenient choice is
$\mu\sim\varphi$, where $\varphi$ is the Higgs field value. This choice
minimizes $\Delta V_{\text{R}}$, and then $V$ can be approximated by the tree-
level potential: $V\simeq\left.V_{\text{R}}\right|_{\mu=\varphi}$.

When we couple this system with gravity, in general, there arise corrections
from the non-renormalizable couplings such as $\xi$. Under the presence of
$\xi$, it has been said that there are two different “prescriptions” in which
the renormalized Higgs potential is approximated by the tree-level one with
[13, 14, 15]

$\displaystyle\mu$
$\displaystyle\sim\begin{cases}\displaystyle{\varphi\over\sqrt{1+\xi{\varphi^{2}\over
M_{\text{P}}^{2}}}}&\text{in prescription~{}I,}\\\ \varphi&\text{in
prescription~{}II,}\end{cases}$ (1)

where $M_{\text{P}}=1/\sqrt{8\pi G}=2.4\times 10^{18}\,\text{GeV}$ is the
reduced Planck scale. The prescriptions I and II are claimed to correspond to
a $\varphi$-independent ultraviolet (UV) cutoff in the Einstein and Jordan
frames, respectively [13, 14]. Here the Jordan frame refers to the original
action with a non-vanishing $\xi$, while the Einstein frame is the one without
$\xi$, obtained by the field re-definition of the metric $g_{\mu\nu}$. Note
that if we introduce a UV cutoff as a $\varphi$-independent constant in either
frame, then it becomes dependent on $\varphi$ in the other frame.  If we
accept Eq. (1) literally, the value of $\xi$ can be as low as of order 10 in
the prescription I [16, 17, 12] and $10^{2}$ in II [16, 12], under the SM
criticality. The physical difference comes from the different large $\varphi$
limit of Eq. (1).

In this paper, we will revisit the relation between the UV cutoff and the
renormalization scale. We clarify that the different choice of the cutoffs
does not directly lead to the difference in Eq. (1). We argue that, in the
counterterm formalism, it may be regarded as the choice of the counter term to
cancel the logarithmic divergence, and can be absorbed into the choice of the
tree-level potential from infinitely many possibilities.

For that purpose, we consider the fermion loop correction to the effective
action in a simplified Higgs-Yukawa model. This toy model captures the
essential features necessary to grasp what is going on in the realistic Higgs
inflation model: As in the real world, we neglect the Higgs mass term which is
much smaller than the one from quartic coupling $\lambda_{4}$ at the large
field values under consideration; the renormalization group (RG) running of
$\lambda_{4}$ is governed by the loop of top-quark, which is represented by
$\psi$.222 In reality, the loop of gauge bosons also contributes to the
running of $\lambda_{4}$. However, the $\varphi$-dependent effective mass of
the canonically normalized gauge boson, $g\varphi$, has the same
$F_{\mathcal{R}}\\!\left(\varphi\right)$ dependence in the Einstein frame,
$g\varphi/\sqrt{F_{\mathcal{R}}\\!\left(\varphi\right)}$, as the effective
mass of fermion $y\varphi$ which becomes
$y\varphi/\sqrt{F_{\mathcal{R}}\\!\left(\varphi\right)}$; see e.g. Ref. [12].
Therefore the arguments for frame independence and for prescription dependence
should apply without modification after we include gauge boson loops.

This paper is organized as follows. In Sec. 2, we obtain the one-loop
effective action in the Higgs-Yukawa model both in the Jordan and Einstein
frames. We show that the effective action is independent of the frame if we
properly take into account the change of the path integral measure. This
change of measure affects only the quartically divergent term, and has nothing
to do with the difference between the prescriptions I and II that is related
to the logarithmic divergence. In Sec. 3, in the counterterm formalism, we
show that the difference between prescriptions I and II comes from the choice
of the counter term to cancel the logarithmic divergence. We point out that
this difference can be absorbed into the choice of the tree-level potential,
including higher dimensional terms, from the infinite possibilities. In Sec.
4, we present a mechanism that uses the gauge kinetic function to stop the
running of effective quartic coupling for large $\varphi$ as in the
prescription I in Eq. (1), which helps to further flatten the Higgs potential
at high scales. In the last section, we summarize our result.

##  2 Quantum correction from fermion loop

###  2.1 Frames at classical level

We first review the transformation from Jordan to Einstein frames at the
classical level. Our starting action in Jordan frame is

$\displaystyle S$
$\displaystyle=\int\text{d}^{4}x\sqrt{-g}\Bigg{[}\frac{M_{\text{P}}^{2}}{2}F_{\mathcal{R}}\\!\left(\varphi\right)\mathcal{R}-\frac{1}{2}F_{\Phi}\\!\left(\varphi\right)g^{\mu\nu}\,\partial_{\mu}\varphi\,\partial_{\nu}\varphi-V\\!\left(\varphi\right)$
$\displaystyle\phantom{=\int\text{d}^{4}x\sqrt{-g}\Bigg{[}}-F_{\Psi}\\!\left(\varphi\right)\overline{\psi}\gamma^{\mu}D_{\mu}\psi-
F_{\text{Y}}\\!\left(\varphi\right)y\varphi\overline{\psi}\psi\Bigg{]},$ (2)

where $y$ is the Yukawa coupling333 This $y$ is related to the SM top Yukawa
coupling $y_{t}$ by $y=y_{t}/\sqrt{2}$.  and
$D_{\mu}=\partial_{\mu}+\Omega_{\mu}$ is the general covariant derivative on
spinor, with $\Omega_{\mu}$ being the spin-connection. We assume that we may
take a weak-field limit $\varphi\to 0$ so that we can expand the action around
$\varphi=0$. That is,

$\displaystyle F_{X}\\!\left(\varphi\right)$
$\displaystyle=1+\xi_{X}{\varphi^{2}\over M^{2}}+\cdots$ (3)

for $X=\mathcal{R}$, $\Phi$, $\Psi$ and Y, where $\xi_{X}$ is the first-order
nonminimal coupling and $M$ is the typical scale of UV theory, such as the
string scale.444 The nonminimal coupling $\xi$ between Higgs and Ricci scalar
in the ordinary notation reads $\xi=\xi_{\mathcal{R}}{M_{\text{P}}^{2}/M^{2}}$
[2]. See also Ref. [18] for more arbitrary extension with large nonminimal
couplings.  We have also assumed for simplicity that the action is invariant
under a chiral $Z_{2}$ symmetry

$\displaystyle\varphi$ $\displaystyle\to-\varphi,$ $\displaystyle\psi$
$\displaystyle\to\gamma_{5}\psi.$ (4)

In general, the potential contains all the higher dimensional terms:

$\displaystyle V\\!\left(\varphi\right)$ $\displaystyle=\sum_{\text{$n$: even,
$n\geq 0$}}{\lambda_{n}}{\varphi^{n}\over M^{n-4}}.$ (5)

By the field redefinition

$\displaystyle
g^{\text{E}}_{\mu\nu}=F_{\mathcal{R}}\\!\left(\varphi\right)g_{\mu\nu},$ (6)

we obtain the Einstein-frame action555 In Ref. [12], the factor in front of
the scalar kinetic term has a typo and should read $\left[{B\over A}+{3\over
2}{M_{P}^{2}A^{\prime 2}\over A^{2}}\right]$.

$\displaystyle S$
$\displaystyle=\int\text{d}^{4}x\sqrt{-g_{\text{E}}}\Bigg{[}\frac{M_{\text{P}}^{2}}{2}\mathcal{R}_{\text{E}}-{1\over
2}\left({F_{\Phi}\\!\left(\varphi\right)\over
F_{\mathcal{R}}\\!\left(\varphi\right)}+{3\over
2}\left(M_{\text{P}}F_{\mathcal{R}}^{\prime}\\!\left(\varphi\right)\over
F_{\mathcal{R}}\\!\left(\varphi\right)\right)^{2}\right)g_{\text{E}}^{\mu\nu}\,\partial_{\mu}\varphi\,\partial_{\nu}\varphi-{V\\!\left(\varphi\right)\over\left(F_{\mathcal{R}}\\!\left(\varphi\right)\right)^{2}}$
$\displaystyle\phantom{=\int\text{d}^{4}x\sqrt{-g_{\text{E}}}\Bigg{[}}-{F_{\Psi}\\!\left(\varphi\right)\over\left(F_{\mathcal{R}}\\!\left(\varphi\right)\right)^{3/2}}\overline{\psi}\gamma_{\text{E}}^{\mu}D_{\mu}^{\text{E}}\psi-{F_{\text{Y}}\\!\left(\varphi\right)\over\left(F_{\mathcal{R}}\\!\left(\varphi\right)\right)^{2}}y\varphi\overline{\psi}\psi\Bigg{]}.$
(7)

Here and hereafter, we put either sub- or superscript “E” and “J” on
quantities in the Einstein and Jordan frames, respectively, when it is
preferable; the ones without such sub- or superscript are given in the Jordan
frame unless otherwise stated.

The original Higgs inflation [2] assumes that the potential (5) can be
approximated by

$\displaystyle V\\!\left(\varphi\right)$
$\displaystyle={\lambda_{4}}\varphi^{4},$ (8)

namely $\lambda_{n}\ll 1$ for $n\neq 4$, at around the scale $\varphi\sim
M\lesssim M_{\text{P}}$.666 For the Higgs inflation under SM criticality [16,
12], we further assume the flatness $\lambda_{4}\ll 1$ at high scales. In
terms of the quartic coupling $\lambda$ in Refs. [16, 12], the quartic
coupling in this paper is written as $\lambda_{4}=\lambda/4$.  That is, one
assumes that all the higher order terms are small at $\varphi\sim M$ so that
$V\ll M^{4}\lesssim M_{\text{P}}^{4}$. Combined with the assumption
$F_{\mathcal{R}}=1+\xi{\varphi^{2}/M_{\text{P}}^{2}}$, the potential in Eq.
(7),

$\displaystyle U:={V\over F_{\mathcal{R}}^{2}},$ (9)

becomes constant in the large $\varphi$ limit:

$\displaystyle U$ $\displaystyle\to{\lambda_{4}\over\xi^{2}}M_{\text{P}}^{4},$
(10)

leading to the inflation. The field $\varphi$ is not canonically normalized in
the Einstein frame, and the change of equation of motion must be taken into
account; see Appendix A. Note that it is important to terminate the expansion
of $V$ and $F_{\mathcal{R}}$ at $\varphi^{4}$ and $\varphi^{2}$, respectively,
in order to obtain this constant potential.777 The other option is to
terminate them at the $2n$th and $n$th orders, respectively; see the last
point in Appendix B.

###  2.2 Frame dependence of UV cutoffs

Now we take into account the quantum corrections. In this paper, we evaluate
the one-loop correction to the effective action from the fermion loop, leaving
those from the graviton and $\phi$ loops.888 The correction from $\phi$ loop
is proportional to $\lambda_{n}$, which are assumed to be small here; see
Refs. [11, 19, 20] for arguments in support of the smallness of $\lambda_{n}$
at high scales.  We compute the corrections to $\lambda_{n}$ only. That is, we
neglect all the corrections to other couplings $y$, $\xi_{X}$, etc., and hence
do not distinguish the bare and renormalized couplings for them.

In a given frame, short-distance cutoff $\ell$ is given by

$\displaystyle\ell^{2}=g_{\mu\nu}\,\Delta x^{\mu}\,\Delta x^{\nu}.$ (11)

Then the metric redefinition (6) relates the cutoff lengths in two frames by

$\displaystyle\ell_{\text{J}}^{2}$
$\displaystyle=g_{\mu\nu}^{\text{J}}\,\Delta x^{\mu}\,\Delta
x^{\nu}={g_{\mu\nu}^{\text{E}}\over
F_{\mathcal{R}}\\!\left(\varphi\right)}\Delta x^{\mu}\,\Delta
x^{\nu}={\ell_{\text{E}}^{2}\over F_{\mathcal{R}}\\!\left(\varphi\right)}.$
(12)

That is, the UV cutoff scales are related by [13]

$\displaystyle\Lambda_{\text{E}}^{2}$
$\displaystyle={\Lambda_{\text{J}}^{2}\over
F_{\mathcal{R}}\\!\left(\varphi\right)}.$ (13)

As pointed out in Refs. [13], we may choose either $\Lambda_{\text{J}}$ or
$\Lambda_{\text{E}}$ to be independent of $\varphi$, but not both.999 We note
that such a field-dependent cutoff itself does not lead to any logical
inconsistency. For example, a position-dependent momentum cutoff follows from
the Pauli-Villars regularization in warped space even if we start from a
position-independent bulk mass for the regulator [21].  In the prescriptions I
and II in the original sense [13], we set $\Lambda_{\text{E}}$ and
$\Lambda_{\text{J}}$ to be a constant, respectively.

###  2.3 Frame independence of effective action up to quartic divergence

In general, the effective action should not depend on the choice of frame if
we properly take into account the change of the path integral measure as well
as that of the cutoff (13). We demonstrate it at the one-loop level under the
simplifying assumption given above.

The one-loop effective action induced by the fermion loop in the Jordan frame
is given by

$\displaystyle e^{i\Delta S^{\text{J}}_{\text{eff}}}$
$\displaystyle:=\int\mathcal{D}_{g_{\text{J}}}\psi\
\mathcal{D}_{g_{\text{J}}}\overline{\psi}\
\exp\left[i\int\text{d}^{4}x\sqrt{-g_{\text{J}}}\,\overline{\psi}\left(-F_{\Psi}\not{D}_{g_{\text{J}}}-F_{\text{Y}}y\varphi\right)\psi\right].$
(14)

There is no unique definition of the path integral measure
$\mathcal{D}_{g_{\text{J}}}\psi$. Here we take a simple measure that is
induced from the following distance in the functional space

$\displaystyle\left\lVert\delta\psi\right\rVert_{g_{\text{J}}}^{2}=\int\text{d}^{4}x\sqrt{-g_{\text{J}}}\,\delta\overline{\psi}\,\delta\psi,$
(15)

which is invariant under the diffeomorphism.

Let us rewrite Eq. (14) into the path integral in Einstein frame. Because we
have

$\displaystyle\left\lVert\delta\psi\right\rVert_{g_{\text{J}}}^{2}$
$\displaystyle=\int\text{d}^{4}x\sqrt{-{g_{\text{J}}}}\,\delta\overline{\psi}\,\delta\psi$
$\displaystyle=\int\text{d}^{4}x\sqrt{-g_{\text{E}}}\,F_{\mathcal{R}}^{-2}\,\delta\overline{\psi}\,\delta\psi,$
(16)

the functional measure satisfies

$\displaystyle\mathcal{D}_{g_{\text{J}}}\psi\
\mathcal{D}_{g_{\text{J}}}\overline{\psi}$
$\displaystyle=\mathcal{D}_{g_{\text{E}}}\psi\
\mathcal{D}_{g_{\text{E}}}\overline{\psi}\
\left(\prod_{x}F_{\mathcal{R}}^{-2}\right)^{-4}$
$\displaystyle=\mathcal{D}_{g_{\text{E}}}\psi\
\mathcal{D}_{g_{\text{E}}}\overline{\psi}\
\exp\left[-4\,\underset{g_{\text{E}},\Lambda_{\text{E}}}{\operatorname{Tr}}\ln
F_{\mathcal{R}}^{-2}\right],$ (17)

where the measure $\mathcal{D}_{g_{\text{E}}}\psi$ is the one induced from the
following distance in the functional space

$\displaystyle\left\lVert\delta\psi\right\rVert_{g_{\text{E}}}^{2}=\int\text{d}^{4}x\sqrt{-g_{\text{E}}}\,\delta\overline{\psi}\,\delta\psi,$
(18)

and $\operatorname{Tr}_{g_{\text{E}},\Lambda_{\text{E}}}$ indicates that the
functional trace depends both on the metric $g_{\text{E}}$ and the cutoff
$\Lambda_{\text{E}}$; the more explicit form will be presented below.101010
The extra minus sign of $-4$ in Eq. (17) is from the Jacobian for fermionic
variables.

When we rewrite the functional measure $\mathcal{D}_{g_{\text{J}}}\psi\
\mathcal{D}_{g_{\text{J}}}\overline{\psi}$ in terms of
$\mathcal{D}_{g_{\text{E}}}\psi\ \mathcal{D}_{g_{\text{E}}}\overline{\psi}$,
there has appeared the extra contribution to the effective action:

$\displaystyle\exp\left[-4\,\underset{g_{\text{E}},\Lambda_{\text{E}}}{\operatorname{Tr}}\,\ln
F_{\mathcal{R}}^{-2}\right].$ (19)

As we will see below, this factor contains quartic divergence, and is absorbed
into the renormalized couplings including the coefficients of higher
dimensional terms. Finally, Eq. (14) becomes

$\displaystyle e^{i\Delta S^{\text{J}}_{\text{eff}}}$
$\displaystyle=\exp\left[-4\,\underset{g_{\text{E}},\Lambda_{\text{E}}}{\operatorname{Tr}}\,\ln
F_{\mathcal{R}}^{-2}\right]\int{\mathcal{D}_{g_{\text{E}}}\psi}\
{\mathcal{D}_{g_{\text{E}}}\overline{\psi}}\
\exp\left[i\int\text{d}^{4}x\sqrt{-g_{\text{E}}}\,\overline{\psi}\left(-{F_{\Psi}\over
F_{\mathcal{R}}^{3/2}}\not{D}_{g_{\text{E}}}-{F_{\text{Y}}\over
F_{\mathcal{R}}^{2}}y\varphi\right)\psi\right]$
$\displaystyle=:\exp\left[-4\,\underset{g_{\text{E}},\Lambda_{\text{E}}}{\operatorname{Tr}}\,\ln
F_{\mathcal{R}}^{-2}\right]e^{i\Delta S^{\text{E}}_{\text{eff}}},$ (20)

where we have defined the Einstein-frame effective action $\Delta
S_{\text{eff}}^{\text{E}}$, which is obtained from the path integral measure
$\mathcal{D}_{g_{\text{E}}}\psi$.111111 The same argument applies if we start
from a different theory defined with another measure induced from the distance
(18) instead of Eq. (16). Then the Jordan-frame effective action will receive
extra contribution from the change of measure,
$\exp\left[4\,\underset{g_{\text{J}},\Lambda_{\text{J}}}{\operatorname{Tr}}\,\ln
F_{\mathcal{R}}^{-2}\right]$, which again will make the difference only in the
renormalization conditions.  In this section hereafter, we will put the
superscript J (E) for the effective potential of the theory that is defined by
using the measure $\mathcal{D}_{g_{\text{J}}}\psi$
($\mathcal{D}_{g_{\text{E}}}\psi$), as well as the effective action.

###  2.4 Explicit computations

Now we verify the equality (28) through more explicit computations under the
assumption that $\varphi$ and $g_{\mu\nu}$ are slowly varying backgrounds so
that they may be treated as constants in the computation of the effective
action.

As a preparation, let us first compute the extra factor (19) coming from the
change of measure:

$\displaystyle\exp\left[-4\,\underset{g_{\text{E}},\Lambda_{\text{E}}}{\operatorname{Tr}}\,\ln
F_{\mathcal{R}}^{-2}\right]$
$\displaystyle=\exp\left[-4i\int\text{d}^{4}x\sqrt{-g_{\text{E}}}\Braket{x}{\ln
F_{\mathcal{R}}^{-2}}{x}_{\Lambda_{\text{E}}}\right]$
$\displaystyle=\exp\left[i\int\text{d}^{4}x\sqrt{-g_{\text{E}}}\left(-{\Lambda_{\text{E}}^{4}\over
8\pi^{2}}\ln F_{\mathcal{R}}^{-2}\right)\right],$ (21)

where we have used

$\displaystyle\Braket{x}{x}_{\Lambda_{\text{E}}}$
$\displaystyle=\int^{\Lambda_{\text{E}}}{\text{d}^{4}p\over\left(2\pi\right)^{4}}=\int_{0}^{\Lambda_{\text{E}}}{2\pi^{2}p^{3}\text{d}p\over\left(2\pi\right)^{4}}={\Lambda_{\text{E}}^{4}\over
32\pi^{2}}.$ (22)

Here and hereafter, the momentum integral is taken in the Euclidean space.

The effective action (14) reads

$\displaystyle e^{i\Delta S^{\text{J}}_{\text{eff}}}$
$\displaystyle=\underset{g_{\text{J}},\Lambda_{\text{J}}}{\operatorname{Det}}\\!\left(-{F_{\Psi}}\not{D}_{g_{\text{J}}}-F_{\text{Y}}y\varphi\over\mu_{0}\right)$
$\displaystyle=\exp\left[\underset{g_{\text{J}},\Lambda_{\text{J}}}{\operatorname{Tr}}\,\ln\\!\left(-F_{\Psi}\not{D}_{g_{\text{J}}}-F_{\text{Y}}y\varphi\over\mu_{0}\right)\right]$
$\displaystyle=\exp\left[4i\int\text{d}^{4}x\sqrt{-g_{\text{J}}}\int^{\Lambda_{\text{J}}}{\text{d}^{4}p\over\left(2\pi\right)^{4}}\,{{1\over
2}\ln\\!\left(F_{\Psi}^{2}p^{2}+F_{\text{Y}}^{2}\left(y\varphi\right)^{2}\over\mu_{0}^{2}\right)}\right]$
$\displaystyle=:\exp\left[i\int\text{d}^{4}x\sqrt{-g_{\text{J}}}\left(-\Delta
V^{\text{J}}_{\text{eff}}\right)\right],$ (23)

where $\mu_{0}$ is an arbitrary reference scale and we have defined the
correction to the Jordan-frame potential $\Delta V_{\text{eff}}^{\text{J}}$.
It may be computed as

$\displaystyle\Delta V^{\text{J}}_{\text{eff}}$
$\displaystyle=-2\int^{\Lambda_{\text{J}}}{\text{d}^{4}p\over\left(2\pi\right)^{4}}\,\ln\\!\left(F_{\Psi}^{2}p^{2}+F_{\text{Y}}^{2}\left(y\varphi\right)^{2}\over\mu_{0}^{2}\right)$
$\displaystyle=-{1\over
16\pi^{2}}\left\\{\Lambda_{\text{J}}^{4}\left[\ln\\!\left(F_{\Psi}^{2}{\Lambda_{\text{J}}^{2}+\mathcal{M}_{\text{J}}^{2}\over\mu_{0}^{2}}\right)-{1\over
2}\right]+\Lambda_{\text{J}}^{2}\mathcal{M}_{\text{J}}^{2}+\mathcal{M}_{\text{J}}^{4}\ln\\!\left(\mathcal{M}_{\text{J}}^{2}\over\Lambda_{\text{J}}^{2}+\mathcal{M}_{\text{J}}^{2}\right)\right\\},$
(24)

where

$\displaystyle\mathcal{M}_{\text{J}}\\!\left(\varphi\right):=y\varphi\,{F_{\text{Y}}\\!\left(\varphi\right)\over
F_{\Psi}\\!\left(\varphi\right)}$ (25)

is the field-dependent mass for canonically normalized fermion in the Jordan
frame.121212 For more realistic top quark loop, $\Delta V_{\text{eff}}$ is
multiplied by the color degrees of freedom $N=3$.  We may rewrite the
effective action (23) with the Einstein-frame metric:

$\displaystyle e^{i\Delta S_{\text{eff}}^{\text{J}}}$
$\displaystyle=\exp\left[i\int\text{d}^{4}x\sqrt{-g_{\text{E}}}\left(-{\Delta
V^{\text{J}}_{\text{eff}}\over
F_{\mathcal{R}}^{2}}\right)\right]=:\exp\left[i\int\text{d}^{4}x\sqrt{-g_{\text{E}}}\left(-\Delta
U^{\text{J}}_{\text{eff}}\right)\right],$ (26)

where we have defined the correction to the potential (9). That is,

$\displaystyle\Delta U^{\text{J}}_{\text{eff}}={\Delta
V^{\text{J}}_{\text{eff}}\over F_{\mathcal{R}}^{2}}$ $\displaystyle=-{1\over
16\pi^{2}F_{\mathcal{R}}^{2}}\left\\{\Lambda_{\text{J}}^{4}\left[\ln\\!\left(F_{\Psi}^{2}{\Lambda_{\text{J}}^{2}+\mathcal{M}_{\text{J}}^{2}\over\mu_{0}^{2}}\right)-{1\over
2}\right]+\Lambda_{\text{J}}^{2}\mathcal{M}_{\text{J}}^{2}+\mathcal{M}_{\text{J}}^{4}\ln\\!\left(\mathcal{M}_{\text{J}}^{2}\over\Lambda_{\text{J}}^{2}+\mathcal{M}_{\text{J}}^{2}\right)\right\\}.$
(27)

To summarize, we have started from the measure (15), and computed the one-loop
correction (24). One may worry that the change of path measure (17) might
introduce a trace anomaly in addition to Eq. (21). However, it is taken into
account as a form of the logarithmic UV cutoff dependence in Eq. (24). Indeed,
the constant shift of $\ln\Lambda_{\text{J}}$ correctly reproduces the trace
anomaly, as can be seen from Eq. (34) with Eq. (25), compared to Eq. (41).

We may instead perform the field redefinition (6) to the Einstein frame first,
and compute the Einstein-frame effective action in the right-hand side of Eq.
(28):

$\displaystyle e^{i\Delta S^{\text{E}}_{\text{eff}}}$
$\displaystyle=\exp\left[\underset{g_{\text{E}},\Lambda_{\text{E}}}{\operatorname{Tr}}\,\ln\left(-{F_{\Psi}\over
F_{\mathcal{R}}^{3/2}}\not{D}_{g_{\text{E}}}-{F_{\text{Y}}\over
F_{\mathcal{R}}^{2}}y\varphi\over\mu_{0}\right)\right]$
$\displaystyle=\exp\left[4i\int\text{d}^{4}x\sqrt{-g_{\text{E}}}\,\int^{\Lambda_{\text{E}}}{\text{d}^{4}p\over\left(2\pi\right)^{4}}\,{1\over
2}\ln\\!\left({F_{\Psi}^{2}\over
F_{\mathcal{R}}^{3}}p^{2}+{F_{\text{Y}}^{2}\over
F_{\mathcal{R}}^{4}}\left(y\varphi\right)^{2}\over\mu_{0}^{2}\right)\right]$
$\displaystyle=:\exp\left[i\int\text{d}^{4}x\sqrt{-g_{\text{E}}}\left(-\Delta
U^{\text{E}}_{\text{eff}}\right)\right],$ (28)

where $\Delta U_{\text{eff}}^{\text{E}}$ is the fermion loop correction to the
potential (9) that is obtained with the measure
$\mathcal{D}_{g_{\text{E}}}\psi$:

$\displaystyle\Delta U^{\text{E}}_{\text{eff}}$
$\displaystyle=-2\int_{0}^{\Lambda_{\text{E}}}{\text{d}^{4}p\over\left(2\pi\right)^{4}}\left[\ln\\!\left({F_{\Psi}^{2}\over
F_{\mathcal{R}}^{3}}p^{2}+{F_{\text{Y}}^{2}\over
F_{\mathcal{R}}^{4}}y^{2}\varphi^{2}\over\mu_{0}^{2}\right)\right]$
$\displaystyle=-{1\over
16\pi^{2}}\left\\{\Lambda_{\text{E}}^{4}\left[\ln\\!\left({F_{\Psi}^{2}\over
F_{\mathcal{R}}^{3}}{\Lambda_{\text{E}}^{2}+{\mathcal{M}_{\text{J}}^{2}\over
F_{\mathcal{R}}}\over\mu_{0}^{2}}\right)-{1\over
2}\right]+\Lambda_{\text{E}}^{2}{\mathcal{M}_{\text{J}}^{2}\over
F_{\mathcal{R}}}+{\mathcal{M}_{\text{J}}^{4}\over
F_{\mathcal{R}}^{2}}\ln\\!\left({\mathcal{M}_{\text{J}}^{2}\over
F_{\mathcal{R}}}\over\Lambda_{\text{E}}^{2}+{\mathcal{M}_{\text{J}}^{2}\over
F_{\mathcal{R}}}\right)\right\\}.$ (29)

With the identification of UV cutoff scales (13), we may rewrite

$\displaystyle\Delta U^{\text{E}}_{\text{eff}}$ $\displaystyle=-{1\over
16\pi^{2}F_{\mathcal{R}}^{2}}\left\\{{\Lambda_{\text{J}}^{4}}\left[\ln\\!\left({F_{\Psi}^{2}\over
F_{\mathcal{R}}^{4}}{\Lambda_{\text{J}}^{2}+\mathcal{M}_{\text{J}}^{2}\over\mu_{0}^{2}}\right)-{1\over
2}\right]+{\Lambda_{\text{J}}^{2}}{\mathcal{M}_{\text{J}}^{2}}+{\mathcal{M}_{\text{J}}^{4}}\ln\\!\left({\mathcal{M}_{\text{J}}^{2}}\over{\Lambda_{\text{J}}^{2}}+{\mathcal{M}_{\text{J}}^{2}}\right)\right\\}.$
(30)

We see that

$\displaystyle\Delta U^{\text{E}}_{\text{eff}}$ $\displaystyle=\Delta
U^{\text{J}}_{\text{eff}}-{1\over 16\pi^{2}}\Lambda_{\text{E}}^{4}\ln
F_{\mathcal{R}}^{-4}.$ (31)

Using Eq. (21), we see that Eq. (31) is equivalent to Eq. (30). We note that
the difference in (31) is quartically divergent, which will be subtracted by
the renormalization. In particular, this difference does not change the
running of couplings, as we will see.

To summarize, once we fix the path integral measure, say, to be
$\mathcal{D}_{g_{\text{J}}}\psi$, we obtain the same result $\Delta
U^{\text{J}}_{\text{eff}}$, no matter in which frame we compute it: When we
compute it in the Jordan frame, we obtain

$\displaystyle e^{i\Delta S_{\text{eff}}^{\text{J}}}$
$\displaystyle=e^{i\int\text{d}^{4}x\sqrt{-g_{\text{J}}}\left(-\Delta
V_{\text{eff}}^{\text{J}}\right)}$
$\displaystyle=e^{i\int\text{d}^{4}x\sqrt{-g_{\text{E}}}\left(-\Delta
U_{\text{eff}}^{\text{J}}\right)},$ (32)

while when we compute it in the Einstein frame,

$\displaystyle e^{i\Delta S_{\text{eff}}^{\text{J}}}$
$\displaystyle=e^{i\int\text{d}^{4}x\sqrt{-g_{\text{E}}}\left(-{\Lambda_{\text{E}}^{4}\over
8\pi^{2}}\ln
F_{\mathcal{R}}^{-2}\right)}e^{i\int\text{d}^{4}x\sqrt{-g_{\text{E}}}\left(-\left(\Delta
U_{\text{eff}}^{\text{J}}-{\Lambda_{\text{E}}^{4}\over 16\pi^{2}}\ln
F_{\mathcal{R}}^{-4}\right)\right)}$
$\displaystyle=e^{i\int\text{d}^{4}x\sqrt{-g_{\text{E}}}\left(-\Delta
U_{\text{eff}}^{\text{J}}\right)}.$ (33)

We have explicitly checked that these two agree.

The frame independence of the effective potential has been verified in various
ways: In Refs. [22, 23], the authors have obtained one-loop RG equations for
the tree level action in both the Jordan and Einstein frames, and have found
the agreement between both results; see also appendix of Ref. [24]. In Ref.
[25], the authors checked that both the tree-level actions are equivalent when
written in terms of dimensionless variables, as it should be. In Ref. [26],
the authors have computed the one-loop divergent part of the effective
potential in both frames, and have shown that both coincide at on-shell. In
Refs. [27, 28, 29], the authors have discussed frame independence of physical
observables.

##  3 Prescriptions I and II

We now discuss the meaning of the prescriptions I and II. We first clarify how
the difference of the prescriptions in Eq. (1) arises in the ordinary context.
Then we will show, in the counterterm formalism, that this difference can be
absorbed into the choice from infinitely many possibilities of the
coefficients of higher dimensional terms in the tree-level potential.

We consider the cutoff theory containing infinite number of higher dimensional
terms. We tune the infinite number of bare couplings in the large cutoff limit
$\Lambda_{\text{J}},\Lambda_{\text{E}}\to\infty$ such that the renormalized
effective potential becomes a function of $\varphi/M$, where $M$ is the
physical mass scale; see e.g. Ref. [30]. We work in the counterterm formalism
so that $\mathcal{M}_{\text{J}}$ and $F_{\Psi}$ are treated as finite
renormalized quantities. To be concrete, we consider the theory defined by the
path integral measure $\mathcal{D}_{g_{\text{J}}}\psi$, and we omit the
superscript J from the potentials $V^{\text{J}}$, $\Delta
V_{\text{eff}}^{\text{J}}$, …, etc. hereafter.131313 Exactly the same argument
applies if we consider the theory defined by the measure
$\mathcal{D}_{g_{\text{E}}}\psi$.

###  3.1 Prescription II in ordinary context

We clarify how the prescription II in Eq. (1) appears in the ordinary context.
The contribution from fermion to the effective potential (24) contains the
quartic, quadratic, and logarithmic divergences:

$\displaystyle\Delta V_{\text{eff}}$ $\displaystyle=-{1\over
16\pi^{2}}\left\\{\Lambda_{\text{J}}^{4}\left[\ln\\!\left(F_{\Psi}^{2}{\Lambda_{\text{J}}^{2}\over\mu_{0}^{2}}\right)-{1\over
2}\right]+2\Lambda_{\text{J}}^{2}\mathcal{M}_{\text{J}}^{2}+\mathcal{M}_{\text{J}}^{4}\left(\ln{\mathcal{M}_{\text{J}}^{2}\over\Lambda_{\text{J}}^{2}}-{1\over
2}\right)\right\\}+\mathcal{O}\\!\left(\Lambda_{\text{J}}^{-2}\right).$ (34)

Then in the full effective potential

$\displaystyle V=V_{\text{B}}+\Delta V_{\text{eff}},$ (35)

we cancel the divergences in Eq. (34) by the bare couplings
$\lambda_{n\text{B}}$ in the bare potential

$\displaystyle V_{\text{B}}:=\sum_{n}\lambda_{n\text{B}}{\varphi^{n}\over
M^{n-4}}.$ (36)

The quartic and quadratic divergences in Eq. (34) can be simply subtracted by
the counter term

$\displaystyle V^{\text{c.t.}}_{\text{power}}$ $\displaystyle={1\over
16\pi^{2}}\left\\{\Lambda_{\text{J}}^{4}\left[\ln\\!\left(F_{\Psi}^{2}{\Lambda_{\text{J}}^{2}\over\mu_{0}^{2}}\right)-{1\over
2}\right]+2\Lambda_{\text{J}}^{2}\mathcal{M}_{\text{J}}^{2}\right\\}.$ (37)

However, we need a special care in subtracting the logarithmic divergence in
Eq. (34) because the counter term should be analytic around $\varphi=0$: A
counter term having $\ln\mathcal{M}_{\text{J}}=\ln\varphi+\cdots$ breaks the
analyticity around $\varphi=0$. In particular, the $(n+1)$th derivative of the
term $\varphi^{n}\ln\varphi$ is singular, and the $(n+1)$-point function
becomes ill-defined in the weak field limit $\varphi\to 0$ if the bare action
has such a term.141414 In contrast, the singular behavior of $(n+1)$th
derivative of the effective action represents the infrared singularity of the
$(n+1)$-point scattering of massless scalar, which is cured by taking into
account the Higgs mass in reality and/or by concentrating on the infrared-safe
physical quantities.  Because we employ the analytic tree-level potential, the
counter term should then be analytic too.

A natural choice of the counter term that is analytic around $\varphi=0$ would
be

$\displaystyle V^{\text{c.t.II}}_{\text{log}}$
$\displaystyle={\mathcal{M}_{\text{J}}^{4}\over
16\pi^{2}}\ln{\mu^{2}\over\Lambda_{\text{J}}^{2}},$ (38)

where $\mu$ is the renormalization scale. The resultant bare potential is

$\displaystyle V_{\text{B}}^{\text{II}}$
$\displaystyle=V_{\text{R}}^{\text{{\color[rgb]{0,0,0}II}}}+V^{\text{c.t.}}_{\text{power}}+V^{\text{c.t.II}}_{\text{log}}$
$\displaystyle=V_{\text{R}}^{\text{{\color[rgb]{0,0,0}II}}}+{1\over
16\pi^{2}}\left\\{\Lambda_{\text{J}}^{4}\left[\ln\\!\left(F_{\Psi}^{2}{\Lambda_{\text{J}}^{2}\over\mu_{0}^{2}}\right)-{1\over
2}\right]+2\Lambda_{\text{J}}^{2}\mathcal{M}_{\text{J}}^{2}+\mathcal{M}_{\text{J}}^{4}\ln{\mu^{2}\over\Lambda_{\text{J}}^{2}}\right\\},$
(39)

where $V_{\text{R}}^{\text{{\color[rgb]{0,0,0}II}}}$ is the tree-level
potential in the counterterm formalism. We note that
$V_{\text{R}}^{\text{{\color[rgb]{0,0,0}II}}}$ is $\mu$-dependent:

$\displaystyle
V_{\text{R}}^{\text{{\color[rgb]{0,0,0}II}}}\\!\left(\varphi,\mu\right)$
$\displaystyle=\sum_{n}\lambda_{n\text{R}}\\!\left(\mu\right){\varphi^{n}\over
M^{n-4}},$ (40)

and the $\mu$-independence of $V_{\text{B}}$ determines the running of
$\lambda_{n\text{R}}\\!\left(\mu\right)$ via Eq. (39). In particular, because
$\mathcal{M}_{\text{J}}=y\varphi+\mathcal{O}\\!\left(\varphi^{3}\right)$, we
obtain the ordinary running of the quartic coupling:

$\displaystyle{\text{d}\lambda_{4\text{R}}\\!\left(\mu\right)\over\text{d}\ln\mu}$
$\displaystyle=-{y^{4}\over 8\pi^{2}}.$ (41)

Substituting the bare potential (39) into $V_{\text{B}}$ in Eq. (35), we
obtain

$\displaystyle V\\!\left(\varphi\right)$
$\displaystyle=V_{\text{B}}^{\text{II}}+\Delta V_{\text{eff}}$
$\displaystyle=V_{\text{R}}^{\text{{\color[rgb]{0,0,0}II}}}\\!\left(\varphi,\mu\right)+{\color[rgb]{0,0,0}\Delta
V^{\text{II}}_{\text{R}}\\!\left(\varphi,\mu\right),}$ (42)

where

$\displaystyle{\color[rgb]{0,0,0}\Delta
V_{\text{R}}^{\text{II}}\\!\left(\varphi,\mu\right)}:=-{\left[\mathcal{M}_{\text{J}}\\!\left(\varphi\right)\right]^{4}\over
16\pi^{2}}\left(\ln{\left[\mathcal{M}_{\text{J}}\\!\left(\varphi\right)\right]^{2}\over\mu^{2}}-{1\over
2}\right)$ (43)

is the one-loop correction in the counterterm formalism in the prescription
II. Now both $V_{\text{R}}^{\text{II}}$ and $\Delta V_{\text{R}}^{\text{II}}$
are finite.

When we want to minimize the correction (43), we may choose the
renormalization scale151515 The constant $-1/2$ is scheme-dependent and does
not affect our argument here.

$\displaystyle\mu\sim\mathcal{M}_{\text{J}}.$ (44)

This result reproduces the prescription II in the sense of Eq. (1), namely
$\mu\sim\varphi$ for $F_{\Psi}=F_{\text{Y}}=1$.

###  3.2 Prescription I in ordinary context

We clarify how the prescription I in Eq. (1) appears in the ordinary context.
We can rewrite Eq. (34) by using Eq. (13):161616 This may also be verified by
substituting Eq. (29) into Eq. (31), dividing both-hand sides by
$F_{\mathcal{R}}^{2}$, and expanding it in terms of $\Lambda_{\text{E}}$.

$\displaystyle\Delta V_{\text{eff}}$ $\displaystyle=-{F_{\mathcal{R}}^{2}\over
16\pi^{2}}\left\\{\Lambda_{\text{E}}^{4}\left[\ln\\!\left(F_{\Psi}^{2}{F_{\mathcal{R}}\Lambda_{\text{E}}^{2}\over\mu_{0}^{2}}\right)-{1\over
2}\right]+2\Lambda_{\text{E}}^{2}{\mathcal{M}_{\text{J}}^{2}\over
F_{\mathcal{R}}}+{\mathcal{M}_{\text{J}}^{4}\over
F_{\mathcal{R}}^{2}}\left(\ln{\mathcal{M}_{\text{J}}^{2}/F_{\mathcal{R}}\over\Lambda_{\text{E}}^{2}}-{1\over
2}\right)\right\\}+\mathcal{O}\\!\left(\Lambda_{\text{E}}^{-2}\right).$ (45)

The quartic and quadratic divergences are canceled by the same counter term
(37). This time, a natural choice to cancel the logarithmic divergence would
be, instead of Eq. (38),

$\displaystyle V^{\text{c.t.I}}_{\text{log}}$
$\displaystyle={\mathcal{M}_{\text{J}}^{4}\over
16\pi^{2}}\ln{\mu^{2}\over\Lambda_{\text{E}}^{2}}.$ (46)

Then the bare potential becomes

$\displaystyle V_{\text{B}}^{\text{I}}$
$\displaystyle=V_{\text{R}}^{\text{{\color[rgb]{0,0,0}I}}}+V_{\text{power}}^{\text{c.t.}}+V_{\text{log}}^{\text{c.t.I}}$
$\displaystyle=V_{\text{R}}^{\text{{\color[rgb]{0,0,0}I}}}+{1\over
16\pi^{2}}\left\\{\Lambda_{\text{E}}^{4}F_{\mathcal{R}}^{2}\left[\ln\\!\left(F_{\Psi}^{2}{F_{\mathcal{R}}\Lambda_{\text{E}}^{2}\over\mu_{0}^{2}}\right)-{1\over
2}\right]+2\Lambda_{\text{E}}^{2}F_{\mathcal{R}}{\mathcal{M}_{\text{J}}^{2}}+{\mathcal{M}_{\text{J}}^{4}}\ln{\mu^{2}\over\Lambda_{\text{E}}^{2}}\right\\},$
(47)

and we obtain

$\displaystyle V\\!\left(\varphi\right)$
$\displaystyle=V_{\text{B}}^{\text{I}}+\Delta V_{\text{eff}}$
$\displaystyle=V_{\text{R}}^{\text{{\color[rgb]{0,0,0}I}}}\\!\left(\varphi,\mu\right)+{\color[rgb]{0,0,0}\Delta
V_{\text{R}}^{\text{{\color[rgb]{0,0,0}I}}}\\!\left(\varphi,\mu\right),}$ (48)

where

$\displaystyle{\color[rgb]{0,0,0}\Delta
V_{\text{R}}^{\text{{\color[rgb]{0,0,0}I}}}\\!\left(\varphi,\mu\right)}:=-{\left[\mathcal{M}_{\text{J}}\\!\left(\varphi\right)\right]^{4}\over
16\pi^{2}}\left(\ln{\left[\mathcal{M}_{\text{J}}\\!\left(\varphi\right)\right]^{2}/F_{\mathcal{R}}\over\mu^{2}}-{1\over
2}\right).$ (49)

Again the $\mu$-independence of $V$ fixes the running of the couplings. The
running of quartic coupling becomes the same as in Eq. (41) because
$F_{\mathcal{R}}=1+\mathcal{O}\\!\left(\varphi^{2}\right)$ and hence the
$\varphi^{4}$ term is not affected by $\ln F_{\mathcal{R}}$; see the
discussion below.

When we want to minimize the second term in Eq. (48), we may choose the
renormalization scale

$\displaystyle\mu\sim{\mathcal{M}_{\text{J}}\over\sqrt{F_{\mathcal{R}}}}.$
(50)

This result reproduces the prescription I in the sense of Eq. (1), namely
$\mu\sim{\varphi/\sqrt{1+\xi{\varphi^{2}/M_{\text{P}}^{2}}}}$ for
$F_{\Psi}=F_{\text{Y}}=1$; see also footnote 4.

###  3.3 Where the difference comes from

Let us summarize the difference between the prescriptions I and II. The
difference of prescriptions I and II comes from that of the subtractions of
logarithmic divergence in Eqs. (38) and (46):

$\displaystyle
V_{\text{log}}^{\text{c.t.I}}-V_{\text{log}}^{\text{c.t.II}}={\mathcal{M}_{\text{J}}^{4}\over
16\pi^{2}}\ln{\Lambda_{\text{J}}^{2}\over\Lambda_{\text{E}}^{2}}={\mathcal{M}_{\text{J}}^{4}\over
16\pi^{2}}\ln F_{\mathcal{R}}={\xi_{\mathcal{R}}y^{4}\over
16\pi^{2}}{\varphi^{6}\over M^{2}}+\cdots,$ (51)

where we used Eq. (13). This difference amounts to the finite renormalization
of $V_{\text{R}}$. Note that the difference (51) is analytic around
$\varphi=0$ and that it has only higher order terms with $n\geq 6$.

Originally the prescription has been introduced as a choice of frame in which
the theory is defined, and it was believed that the radiative correction to
the effective potential is minimized by the choice of the renormalization
scale as in Eq. (1). As we have shown, however, the physics does not depend on
the choice of the frame in which the theory is defined. Instead, for a given
renormalized tree-level action, the difference of the prescriptions (1) can be
understood as that of the logarithmic counter terms (51): The different
counter terms lead to the different scales (44) and (50) that minimize the
radiative corrections.

###  3.4 Renormalized potential

Theoretically, the potential $V_{\text{R}}$ in Eq. (42) or (48),

$\displaystyle V_{\text{R}}$ $\displaystyle=\sum_{\text{$n$: even, $n\geq
0$}}\lambda_{n\text{R}}\\!\left(\mu\right){\varphi^{n}\over M^{n-4}},$ (52)

may take arbitrary form, so long as it is analytic around $\varphi=0$. How do
we determine its form?

We may reproduce the ordinary Higgs inflation [2] that does not assume the
criticality, by tuning the infinite number of bare couplings such that
$V_{\text{R}}$ becomes

$\displaystyle\left.V_{\text{R}}\\!\left(\varphi,\mu\right)\right|_{\mu\sim
M}$ $\displaystyle\simeq\lambda_{4\text{R}}\\!\left(\mu\right)\varphi^{4},$
(53)

where all the couplings $\lambda_{n\text{R}}\\!\left(\mu\right)$ with $n\neq
4$ are suppressed at $\mu=M$. When the form (53) is put into Eqs. (42) and
(48), which result from the counter terms (38) and (46), we obtain the Higgs
potential in the prescriptions II and I in the ordinary context, respectively:

$\displaystyle V^{\text{II}}$
$\displaystyle=\lambda_{4\text{R}}\\!\left(\mu\right)\varphi^{4}-{\mathcal{M}_{\text{J}}^{4}\over
16\pi^{2}}\left(\ln{\mathcal{M}_{\text{J}}^{2}\over\mu^{2}}-{1\over
2}\right),$ (54) $\displaystyle V^{\text{I}}$
$\displaystyle=\lambda_{4\text{R}}\\!\left(\mu\right)\varphi^{4}-{\mathcal{M}_{\text{J}}^{4}\over
16\pi^{2}}\left(\ln{\mathcal{M}_{\text{J}}^{2}/F_{\mathcal{R}}\over\mu^{2}}-{1\over
2}\right).$ (55)

However, we may as well obtain the potential of the form of $V^{\text{I}}$ in
Eq. (55) even when we employ the counter term $V_{\text{log}}^{\text{c.t.II}}$
in Eq. (38) if we choose the following form of the tree-level potential
$V_{\text{R}}$ in Eq. (42),

$\displaystyle V_{\text{R}}\\!\left(\varphi,\mu\right)$
$\displaystyle=\lambda_{4\text{R}}\\!\left(\mu\right)\varphi^{4}+{\left[\mathcal{M}_{\text{J}}\\!\left(\varphi\right)\right]^{4}\over
16\pi^{2}}\ln F_{\mathcal{R}}\\!\left(\varphi\right),$ (56)

instead of the form (53). From the same counter term
$V_{\text{log}}^{\text{c.t.II}}$ in Eq. (38), we may obtain the forms (54) and
(55) by assuming the tree-level potentials (53) and (56), respectively. Recall
that the second term in the right-hand side of Eq. (56) modifies only the
higher dimensional terms of $\mathcal{O}\\!\left(\varphi^{6}\right)$. There
are infinitely many possibilities for the tree-level potential. Therefore
there is no reason to suppose one (or any) of the above two as the proper
tree-level potential if we restrict ourselves to the low-energy effective
field theory.171717 The tree-level potential is determined once we fix the
underlying UV-finite theory; see e.g. Refs. [19, 20] for discussions based on
string theory.

##  4 Flattening Higgs potential by gauge kinetic function

Let us turn to more realistic running of the quartic coupling in the SM:

$\displaystyle V_{\text{SM}}$
$\displaystyle=\lambda_{4\text{R}}\\!\left(\mu\right)\varphi^{4}+{\color[rgb]{0,0,0}\Delta
V_{\text{R}}\\!\left(\varphi,\mu\right)},$ (57)

where $\Delta V_{\text{R}}$ is the finite correction (43) or (49) in the
counterterm formalism; see also footnote 6. In the SM,
$\beta_{4\text{R}}:={\text{d}\over\text{d}\ln\mu}\lambda_{4\text{R}}$ turns
from negative to positive around the scale $\mu_{\text{min}}\sim
10^{17}\,\text{GeV}$, and we may approximate as [16, 12]

$\displaystyle\lambda_{4\text{R}}\\!\left(\mu\right)$
$\displaystyle\approx\lambda_{4\text{R}}^{\text{min}}+b_{4\text{R}}\,\left(\ln{\mu\over\mu_{\text{min}}}\right)^{2},$
(58)

where $b_{4\text{R}}$ can be computed within the SM as

$\displaystyle b_{4\text{R}}$
$\displaystyle\simeq{0.1\over\left(16\pi^{2}\right)^{2}}\simeq 5\times
10^{-6}.$ (59)

The negative $\beta_{4\text{R}}$ for $\mu<\mu_{\text{min}}$ is dominated by
top quark loop, while the positive $\beta_{4\text{R}}$ for
$\mu>\mu_{\text{min}}$ by the $U(1)_{Y}$ and $SU(2)_{L}$ gauge boson loops.

For top quark loop, the contribution is through the effective mass
$\mathcal{M}_{\text{J}}=y\varphi F_{\text{Y}}/F_{\Psi}$. In the prescription I
in the sense of Eq. (1), namely in Eq. (55) with the tree-level potential
(53), we get the constant $\mu$ in the large $\varphi$ limit,

$\displaystyle\mu\sim{\mathcal{M}_{\text{J}}\over\sqrt{F_{\mathcal{R}}}}\to{M_{\text{P}}\over\sqrt{\xi}},$
(60)

and the effective quartic coupling
$\left.\lambda_{4\text{R}}\\!\left(\mu\right)\right|_{\mu\sim\mathcal{M}_{\text{J}}/\sqrt{F_{\mathcal{R}}}}$
stops running for large $\varphi$ [15].181818 Here we have assumed
$F_{\text{Y}}=F_{\Psi}=1$. If $F_{\text{Y}}=1+\xi_{\text{Y}}\varphi^{2}/M^{2}$
and $F_{\Psi}=1+\xi_{\Psi}\varphi^{2}/M^{2}$, we obtain
$\displaystyle\mu\sim{\mathcal{M}_{\text{J}}\over\sqrt{F_{\mathcal{R}}}}\to{\xi_{\text{Y}}\over\sqrt{\xi}\xi_{\Psi}}M_{\text{P}}$
instead of Eq. (60); see also footnote 4.  This mechanism makes the potential
even flatter at the SM criticality and helps to earn a sufficiently large
$e$-folding number for smaller $\xi\sim 10$; in the prescription II we lack
this mechanism and need larger $\xi\sim 10^{2}$ [16, 12].

Similarly, the contribution of the gauge boson loop is through [12]

$\displaystyle\mathcal{M}^{\text{gauge}}_{\text{J}}$
$\displaystyle=g\varphi\sqrt{F_{\Phi}\over F_{\text{g}}},$ (61)

where $g$ is the gauge coupling and $F_{\text{g}}$ is the gauge kinetic
function, namely the function of $\varphi$ in front of the gauge kinetic term.

When we raise the scale beyond $\mu>\mu_{\text{min}}$ in the SM, the top
Yukawa coupling becomes smaller and smaller. To the first approximation, the
running at $\mu>\mu_{\text{min}}$ is governed by the gauge boson loop. Then in
the prescription II in the ordinary context, which corresponds to Eq. (54)
with the tree-level potential (53), the effective potential becomes

$\displaystyle
V_{\text{SM}}=\left.\lambda_{4\text{R}}\\!\left(\mu\right)\varphi^{4}\right|_{\mu=\mathcal{M}_{\text{J}}^{\text{gauge}}}.$
(62)

When we assume that $F_{\Phi}\simeq 1$, we obtain

$\displaystyle\mathcal{M}^{\text{gauge}}_{\text{J}}$
$\displaystyle={g\varphi\over\sqrt{1+\xi_{\text{g}}{\varphi^{2}\over
M^{2}}}}.$ (63)

In the large-$\varphi$ limit,

$\displaystyle\mathcal{M}^{\text{gauge}}_{\text{J}}\to{g\over\sqrt{\xi_{\text{g}}}}M.$
(64)

We propose that this can be used in the prescription II in the ordinary
context as an alternative mechanism to Eq. (60) in order to stop the running
of quartic coupling
$\left.\lambda_{4\text{R}}\\!\left(\mu\right)\right|_{\mu\sim\mathcal{M}_{\text{J}}^{\text{gauge}}}$
for large $\varphi$.191919 When the top Yukawa contribution is non-negligible,
one may further introduce e.g.  $\displaystyle
F_{\Psi}\\!\left(\varphi\right)$
$\displaystyle=\sqrt{1+2\xi_{\Psi}{\varphi^{2}\over
M^{2}}}=1+\xi_{\Psi}{\varphi^{2}\over M^{2}}-{\xi_{\Psi}^{2}\over
2}{\varphi^{4}\over M^{4}}+{\xi_{\Psi}^{3}\over 2}{\varphi^{6}\over
M^{6}}+\cdots$ together with $F_{\text{Y}}=1$, which stops running due to the
top contribution too:  $\displaystyle\mathcal{M}_{\text{J}}$
$\displaystyle={y\varphi\over\sqrt{1+2\xi_{\Psi}{\varphi^{2}\over
M^{2}}}}\to{y\over\sqrt{2\xi_{\Psi}}}M.$

The $\varphi$-dependent mass (63) takes the same form as the prescription I in
Eq. (1) if we neglect the gauge coupling $g$. Therefore, for example, we may
set $\xi_{\text{g}}=\xi_{\mathcal{R}}$ and $M=M_{\text{P}}$, then the
subsequent analysis becomes identical to those in Ref. [12]. This serves as an
explicit example of viable parameter set that realizes the above-mentioned
idea.

##  5 Summary

We have analyzed the one-loop effective action in the simplified Higgs-Yukawa
model, which captures essential features of the Higgs potential in the Higgs
inflation. We have shown that the effective actions obtained in the Jordan and
Einstein frames are exactly the same if we properly take into account the
change of path integral measure. We show that, in the counterterm formalism,
the prescriptions I and II are merely two specific choices of counter terms to
cancel the logarithmic divergence. We point out that the difference between I
and II can be absorbed into the choice of tree-level potential, including
higher dimensional terms, from infinitely many possibilities.

We have also proposed a mechanism to stop the running of the effective quartic
coupling in the prescription II in the ordinary context, using the gauge
kinetic function: $\mu\sim\varphi/\sqrt{F_{\text{g}}}\to
M/\sqrt{\xi_{\text{g}}}$. Detailed phenomenological study of this scenario
will be presented in a separate publication.

We briefly comment on the remaining points to be addressed: In this paper, we
have concentrated on the fermion loop. It would be worth including the scalar
loop, as in Ref. [26], and also the gauge boson loop. It is also worth
studying the issue of gauge dependence in these loops.

### Acknowledgment

We thank Seong Chan Park, Mikhail Shaposhnikov, and Satoshi Yamaguchi for
useful discussions. The work of Y.H. and K.O. are supported in part by JSPS
KAKENHI Grant Nos. 16J06151 and 23104009, 15K05053, respectively.

## Appendix

##  Appendix A Classical dynamics in terms of Jordan-frame field

When we consider the classical dynamics of scalar field under gravity, it is
convenient to define the canonically normalized scalar field

$\displaystyle\text{d}\chi$
$\displaystyle=\mathcal{G}\\!\left(\varphi\right)\,\text{d}\varphi,$ (65)

where

$\displaystyle\mathcal{G}\\!\left(\varphi\right)$
$\displaystyle:=\sqrt{{F_{\Phi}\\!\left(\varphi\right)\over
F_{\mathcal{R}}\\!\left(\varphi\right)}+{3\over
2}\left(M_{\text{P}}F_{\mathcal{R}}^{\prime}\\!\left(\varphi\right)\over
F_{\mathcal{R}}\\!\left(\varphi\right)\right)^{2}}.$ (66)

We use the Friedmann-Lemaître-Robertson-Walker ansatz202020 We neglect the
spatial curvature, but may recover it by
${\text{d}\boldsymbol{x}}^{2}\to{\text{d}\boldsymbol{x}}^{2}+K{\left(\boldsymbol{x}\cdot\text{d}\boldsymbol{x}\right)^{2}\over
1-K\boldsymbol{x}^{2}}$.

$\displaystyle g_{\mu\nu}^{\text{E}}\text{d}x^{\mu}\text{d}x^{\nu}$
$\displaystyle=-{\text{d}t_{\text{E}}}^{2}+\left(a_{\text{E}}\\!\left(t_{\text{E}}\right)\right)^{2}{\text{d}\boldsymbol{x}}^{2}.$
(67)

In the Einstein frame, the Einstein equations reduce to the ordinary Friedmann
equation:212121 We may recover the spatial curvature by $H_{\text{E}}^{2}\to
H_{\text{E}}^{2}+{K\over a^{2}}$.

$\displaystyle H_{\text{E}}^{2}$ $\displaystyle={\rho_{\text{E}}\over
3M_{\text{P}}^{2}},$ (68)
$\displaystyle{\text{d}\rho_{\text{E}}\over\text{d}t_{\text{E}}}$
$\displaystyle=-3\left(\rho_{\text{E}}+p_{\text{E}}\right)H_{\text{E}},$ (69)

where

$\displaystyle\rho_{\text{E}}$ $\displaystyle={1\over
2}\left(\text{d}\chi\over\text{d}t_{\text{E}}\right)^{2}+U_{\text{E}}\\!\left(\varphi\right),$
(70) $\displaystyle p_{\text{E}}$ $\displaystyle={1\over
2}\left(\text{d}\chi\over\text{d}t_{\text{E}}\right)^{2}-U_{\text{E}}\\!\left(\varphi\right),$
(71)

and $H_{\text{E}}:={1\over
a_{\text{E}}}{\text{d}a_{\text{E}}\over\text{d}t_{\text{E}}}$.

The Higgs-field equation reads

$\displaystyle{\text{d}^{2}\chi\over\text{d}t_{\text{E}}^{2}}+3H_{\text{E}}{\text{d}\chi\over\text{d}t_{\text{E}}}$
$\displaystyle=-{\text{d}U_{\text{E}}\over\text{d}\chi}.$ (72)

In terms of the Jordan-frame field,

$\displaystyle{\text{d}^{2}\varphi\over\text{d}t_{\text{E}}^{2}}+{\text{d}\varphi\over\text{d}t_{\text{E}}}\left(3H_{\text{E}}+{\text{d}\over\text{d}t_{\text{E}}}\ln\mathcal{G}\right)$
$\displaystyle=-{1\over\mathcal{G}^{2}}{\text{d}U_{\text{E}}\over\text{d}\varphi}.$
(73)

The universe expands with the rate $H_{\text{E}}$, whereas the Jordan-frame
field $\varphi$ receives extra friction
${\text{d}\over\text{d}t_{\text{E}}}\ln\mathcal{G}$, and rolls slower (faster)
than under its absence when it is positive (negative). This term will turn out
to be the same order as the slow-roll parameter under the slow-roll condition
shown below.

We assume the slow-roll inflation. The slow-roll parameters read

$\displaystyle\epsilon$ $\displaystyle:={M_{\text{P}}^{2}\over
2U_{\text{E}}^{2}}\left(\text{d}U_{\text{E}}\over\text{d}\chi\right)^{2}={M_{\text{P}}^{2}\over
2U_{\text{E}}^{2}\mathcal{G}^{2}}\left(\text{d}U_{\text{E}}\over\text{d}\varphi\right)^{2}\ll
1,$ (74) $\displaystyle\eta$ $\displaystyle:={M_{\text{P}}^{2}\over
U_{\text{E}}}{\text{d}^{2}U_{\text{E}}\over\text{d}\chi^{2}}={M_{\text{P}}^{2}\over
U_{\text{E}}\mathcal{G}}{\text{d}\over\text{d}\varphi}\left({1\over\mathcal{G}}{\text{d}U_{\text{E}}\over\text{d}\varphi}\right)\ll
1.$ (75)

The Friedmann and Higgs-field equations become, respectively,

$\displaystyle 3M_{\text{P}}^{2}H_{\text{E}}^{2}$
$\displaystyle=U_{\text{E}},$ (76) $\displaystyle
3H_{\text{E}}{\text{d}\chi\over\text{d}t_{\text{E}}}$
$\displaystyle=-{\text{d}U_{\text{E}}\over\text{d}\chi}.$ (77)

In terms of the Jordan-frame field, the latter reads

$\displaystyle 3H_{\text{E}}{\text{d}\varphi\over\text{d}t_{\text{E}}}$
$\displaystyle=-{1\over\mathcal{G}^{2}}{\text{d}U_{\text{E}}\over\text{d}\varphi}=:-{\text{d}\mathcal{U}_{\text{E}}\over\text{d}\varphi},$
(78)

where we have defined the effectual potential

$\displaystyle\mathcal{U}_{\text{E}}$
$\displaystyle=\int\text{d}\varphi{1\over\mathcal{G}^{2}}{\text{d}U_{\text{E}}\over\text{d}\varphi}+\text{const.},$
(79)

which takes into account the effect from $\mathcal{G}$. Using this potential,
the slow-roll parameters can be rewritten as

$\displaystyle\epsilon$ $\displaystyle={\mathcal{G}^{2}M_{\text{P}}^{2}\over
2U_{\text{E}}^{2}}\left(\text{d}\mathcal{U}_{\text{E}}\over\text{d}\varphi\right)^{2},$
(80) $\displaystyle\eta$ $\displaystyle={M_{\text{P}}^{2}\over
U_{\text{E}}}\left({\text{d}^{2}\mathcal{U}_{\text{E}}\over\text{d}\varphi^{2}}+{\text{d}\mathcal{U}_{\text{E}}\over\text{d}\varphi}{\text{d}\over\text{d}\varphi}\ln\mathcal{G}\right),$
(81)

though this expression may not be particularly convenient. In this paper, we
omit the subscript E from $U_{\text{E}}$ and $\mathcal{U}_{\text{E}}$ which
are always given in the Einstein frame.

##  Appendix B Examples that lead to various Higgs inflations

Even if we decide to take the simple form of the tree-level potential (53) in
Eq. (42) or (48), we still have freedom to choose any form of
$F_{X}\\!\left(\varphi\right)$s. We review several examples that lead to
viable cosmic inflations:

  * •

In Ref. [12], the authors have spelled out the result from prescriptions I and
II in the ordinary context, with the tree-level potential (53) in Eq. (48) and
(42), respectively, and with the function $F_{X}=1$ except for
$F_{\mathcal{R}}=1+\xi_{\mathcal{R}}{\varphi^{2}\over M^{2}}$. The former
prescription I allows smaller
$\xi:=\xi_{\mathcal{R}}M_{\text{P}}^{2}/M^{2}\sim 10$ because the coupling
stops running for $\varphi\gg M/\sqrt{\xi_{\mathcal{R}}}$:

$\displaystyle\mu\sim{\varphi\over\sqrt{F_{\mathcal{R}}}}\to{M\over\sqrt{\xi_{\mathcal{R}}}}.$
(82)

The latter prescription II can have a chaotic inflation for $\xi\sim 10^{2}$,
since the effectual potential (79) becomes

$\displaystyle\mathcal{U}$
$\displaystyle\sim\text{const.}+{\beta_{\lambda}M_{\text{P}}^{2}\over
48\xi^{2}}\varphi^{2},$ (83)

due to

$\displaystyle\mathcal{G}$
$\displaystyle\to{\sqrt{6}M_{\text{P}}\over\varphi}$ (84)

for large $\xi_{\mathcal{R}}$; see Appendix A.

  * •

When we have large $\xi_{\Phi}$ only, in particular with $F_{\mathcal{R}}=1$
which gives $U=V={\lambda\over 4}\varphi^{4}$, we get

$\displaystyle\mathcal{G}\to\sqrt{\xi_{\Phi}}\,{\varphi\over M}$ (85)

and hence

$\displaystyle\mathcal{U}$ $\displaystyle=\text{const.}+{\lambda M^{2}\over
2\xi_{\Phi}}\varphi^{2}.$ (86)

This can also lead to a chaotic inflation when $\lambda/\xi_{\Phi}\ll 1$ [31].

  * •

When we terminate the tree-level potential at the $2n$th order

$\displaystyle V=\sum_{\text{$n^{\prime}$: even, $4\leq n^{\prime}\leq
2n$}}\lambda_{n^{\prime}}{\varphi^{n^{\prime}}\over M^{n^{\prime}-4}},$ (87)

instead of Eq. (53), and the function at the $n$th order

$\displaystyle F_{\mathcal{R}}=\sum_{\text{$n^{\prime}$: even, $0\leq
n^{\prime}\leq n$}}\xi_{\mathcal{R},n^{\prime}}{\varphi^{n^{\prime}}\over
M^{n^{\prime}}},$ (88)

the resultant classical potential becomes constant [32]:

$\displaystyle U\to{\lambda_{2n}\over\xi_{\mathcal{R},n}^{2}}M^{4}.$ (89)

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