Cosmological constant as a finite temperature effect

I. Y. Park

†Department of Applied Mathematics, Philander Smith College
Little Rock, AR 72223, USA
<EMAIL_ADDRESS>
###### Abstract

We reexamine the cosmological constant problem in a finite temperature setup
and propose an intriguing possibility of carrying out perturbative analysis by
employing a renormalization scheme in which the renormalized Higgs mass (or
resummed mass, to be more precise) taken to be on the order of the CMB
temperature. Finite-temperature-induced complexification of the effective
potential is observed and its interpretation is given. It is shown that the
cosmological constant problem is avoided. A peculiarity of the perturbation
series in the new scheme is noted.

keywords: cosmological constant problem, finite temperature, quantization of
gravity, bare and renormalized perturbation theories

##  1 Introduction

The cosmological constant (CC) problem [1] (see, e.g., [2, 3, 4, 5] for
reviews111The inspiring review [5] is largely based on the previous works by
the author and his collaborators (see, e.g., [6, 7, 8]) that were further
developed in [9, 10, 11]. One of the main themes of these works is the so-
called running vacuum. These works are further commented on in the conclusion
and [12].) arises from the loop effects of Standard Model (SM) particles, such
as the Higgs particle. Since the electroweak scale is much higher than, say,
the temperature of the cosmic microwave background (CMB), it has been a
standard practice to apply the zero-temperature setup to formulate and tackle
the problem. However, since the CC, as vacuum energy, is an infrared effect
(in the sense explained below) and thus governed by the low energy sector, the
infrared structure of the theory must be important and its meticulous
description is desirable. The fact that the CC is a vacuum energy also implies
that quantization and renormalization of gravity must be involved in its
systematic treatment. In particular, the resolution of the problem would
require renormalization of the vacuum energy. In this work we show that the
finite temperature effects allow, when properly taken into account in the
quantum gravitational setup, one to avoid the CC fine-tuning problem.

The vacuum energy is defined as a minimum of the effective potential. In
vacuum energy computation both the ultraviolate (UV) and infrared (IR)
structures play roles. To some extent the UV and IR contributions to the
vacuum energy are intertwined. The relevance of the UV structure is evident
since renormalization procedure is involved. The relevance of the infrared
structure is subtler. It is well demonstrated in Casimir energy analysis (see,
e.g., the account in [13]). The point is that for proper evaluation of vacuum
energy it is necessary to employ an infrared regulator, such as a finite-size
box in the Casimir case in momentum cutoff regularization. In the body we
employ dimensional regularization with additional finite renormalization, and
pay a close attention to the infrared structure introduced by the presence of
the temperature.

Are there indications that the finite temperature effects may a priori be
important for CC analysis, despite the fact that the electroweak scale is much
higher than the CMB temperature? First of all, it should definitely be
possible, and is natural, to obtain zero-temperature results as a vanishing-
temperature limit of the corresponding finite-temperature results. A hint of
an indication comes from energy scalings in zero- and finite temperature loop
analyses. In zero temperature, a loop analysis typically yields logarithmic
factors such as $\ln\frac{m}{\mu}$, where $m$ is the renormalized mass of the
field and $\mu$ the renormalization scale. For the benefit of convergence, it
is necessary to choose $\mu\sim m$. (Or one may consider renormalization group
equations to conduct certain resummations.) By the same token it will be
desirable to take $\mu\sim m\sim T$ once the temperature enters. The question
is whether there is an additional, more quantitative rationale for enforcing
the scaling. The answer is affirmative: in the present work this scaling is
achieved in the course of improving the perturbative analysis by optimal
perturbation theory (OPT)222OPT is based on the variational principle. Other
thermal physics techniques based on the variational principle include screened
perturbation theory [14][15] and 2PI formalism [16]. [17] as well as standard
thermal resummation: we show that there is an OPT procedure that enforces the
scaling.

In the body we reformulate the CC problem as a zero-temperature limit of the
finite temperature counterpart. A potential obstruction to any finite-
temperature perturbative analysis is the well-known infrared problem. (Reviews
can be found in [18, 19, 20, 21, 22].) For a high temperature, say, that of
the QCD era, a well-known example is the ‘Linde problem’ [23], which has been
an active topic of research; see [24][25] and references therein. An effective
field theory approach combined with lattice computation was proposed to deal
with the problem [26]. The focus of the present work is a low temperature, the
temperature of CMB. (Nevertheless, some of the textbook results obtained for a
high temperature can be borrowed for reasons to be explained.) We show that
the finite-temperature effects are in fact crucial: in particular, once the
convergence property of the perturbation series is improved through a variant
OPT, the optimized renormalized mass of the Higgs turns out to be essentially
the temperature,333It has been revealed in the recent works of [27, 28, 29,
30] that quantum corrections can qualitatively change the classical solution.
In the present work we see a similar novelty: the finite temperature non-
perturbative effects dictate, small temperatures notwithstanding, the
renormalized mass. thus removing the root of the CC fine-tuning problem.

In essence, what is being proposed is an intriguing possibility of carrying
out perturbative analysis by employing - with proper justifications, namely,
the scaling and our variant OPT - a renormalization scheme in which the
renormalized Higgs mass is taken to be on the order of the CMB temperature. In
a general renormalization scheme, there is hardly anything that stops one from
adopting a renormalized mass of an arbitrary value - other than the question
of why one would make such a choice. Variant OPT is introduced to address this
question. Also, our point of contention is that such a choice accommodates a
wider range of physics and at the same time leads to interesting predictions
in cosmology. More on this in section 2 and the conclusion.

A cautionary remark is in order. The issue is that of renormalization schemes.
There are two widely-used renormalization schemes: onshell and “offshell”
scheme, such as $\overline{\mbox{MS}}$. The former was predominantly used in
the past whereas more recent literature employs the latter. (Among textbooks,
the former is used in [31]; many other textbooks, such as [32][33][13], employ
the latter.) In the onshell scheme, the mass appearing in the Lagrangian is
taken to be the physical mass whose more general definition is the pole of the
2-point Green’s function. In contrast, the mass in the $\overline{\mbox{MS}}$
scheme (or any scheme of renormalized perturbation theory,444The perturbative
analysis employing an “offshell” scheme often goes under the name of
renormalized perturbation theory. more generally) is a renormalized mass. The
renormalized mass is initially viewed arbitrary: it gets determined at the end
by requiring the pole of the Green’s function to match with the physical
value, 125 GeV in the case of Higgs field. We take the renormalized mass (or
the resummed mass, more precisely; see the body) of the Higgs to be on the
order of the temperature, with the value of the physical mass intact as 125
GeV. We do so by making use of the freedom associated with choice of the value
of the renormalized mass in renormalized perturbation theory. While the
scaling argument above is a qualitative justification for the temperature-
order renormalized mass, our OPT procedure provides a quantitative
justification.

With the renormalized mass determined, the following task still remains: the
zero-temperature theories, such as the zero-temperature Standard Model, have
been quite successful. There, the renormalized masses turn out to be close to
the pole masses, usually within a few percent. In the case of the SM Higgs
field, for instance, the renormalized mass is close to 125 GeV, the pole mass
value. If one now wants to take the renormalized mass to be around the CMB
temperature, which is much smaller than the pole mass, one must yet maintain
compatibility with the zero-temperature analysis: the resulting perturbation
theory should preserve the success of the original zero-temperature theory.
This is the main task that we undertake in section 2; briefly, what underpins
the compatibility at a fundamental level is the fact that the theory is
defined by the bare Lagrangian. In section 3, after examining vast freedom in
choosing renormalization conditions, we invoke renormalization group
invariance of physical quantities to affirm this. We note that a certain
resummation is behind the invariance. We also note that the small renormalized
mass introduces a peculiarity in the renormalized perturbative series worth
examining further.

To be specific, the cosmological constant problem is examined by taking an
Einstein-scalar with a Higgs-type potential. A variant optimal perturbation
theory is implemented in the recently proposed quantum-gravitational
framework. The optimized renormalized mass, i.e., the renormalized mass
determined by the variant optimal perturbation theory, of the scalar field
turns out to be on the order of the temperature. Since the CMB temperature is
on the same order of magnitude as the measured value of the cosmological
constant in today’s Universe, the temperature-order CC implies that the
cosmological constant problem is avoided. More rigorously, the temperature-
order CC shifts the cosmological constant problem to compatibility of the
consequent perturbative analysis. The compatibility is guaranteed essentially
by invariance of physical quantities under renormalization scheme change.
(Several explicit checks are suggested in the conclusion.) We point out the
resummation behind the invariance.

In our OPT finite-temperature-induced complexification of the effective
potential is observed. In other words this effect is introduced through our
variant OPT. It is thus additional to the well-known presence of the imaginary
part in the zero-temperature effective potential. In particular, the vacuum
expectation value of the scalar field itself becomes complex. We interpret the
complexity to be instability - associated with the finite temperature - toward
zero temperature.

With the renormalized mass being of the order of the temperature, the CC
problem is avoided: the CMB temperature in terms of eV is $6.6\times 10^{-4}$
eV. Approximating this as $10^{-12}$ GeV, the vacuum energy contribution
associated with the thermal mass of a Higgs-type field is $\sim
10^{-48}\;\mbox{GeV}^{\,4}$. This is on the same order as the observed CC
value $\sim 10^{-48}\;\mbox{GeV}^{\,4}$.

The rest of the paper is organized as follows. Invariance of physical
quantities under renormalization scheme change (as well as renormalization
group flow) originates from the fact that the theory is defined by the bare
Lagrangian. The invariance will play a central role in the proposed resolution
of the CC problem. In section 2 we explore the freedom in adopting a
renormalization scheme and do so for a zero-T setup to avoid inessential
complications. We take a massive scalar theory with a quartic potential and
discuss the perspectives of both bare and renormalized perturbation theories.
In section 3 we carry out one-loop computation of the effective potential in a
finite temperature setup. We do this for a Higgs-type scalar in a flat
spacetime first, and subsequently extend the analysis to the graviton sector.
By employing an OPT-induced renormalization scheme, we obtain the main result
of the one-loop effective potential given in (31). In section 4 we end with
summary, comments on potential implications of the present result for
cosmology, and future directions.

##  2 Freedom in renormalization scheme

The renormalization scheme that we are employing in the main analysis is such
that the Higgs renormalized mass is taken to be far smaller than the physical
(i.e., pole) mass. This is not necessarily a problem (and with the help of
bare perturbation theory one will see that it indeed is not). It will
nevertheless be useful to examine the potential consistency issue with the new
scheme.

The gist of the conventional analysis is as follows. After UV regularization
one subtracts out the infinite part and fix the finite part of the vacuum
energy. In $\overline{\mbox{MS}}$ scheme one removes essentially the
$\frac{1}{\varepsilon}$ part, and this fixes the finite part. At this point
the renormalized mass is still undetermined; it is determined by matching the
pole expression of the 2-point function with the physical value of the field,
125 GeV for a Higgs field. In section 3, we put forth an OPT that leads to the
renormalized mass of the order of the temperature. With this one is to conduct
the consequent perturbative analysis. In the proposed new scheme it is the
value of the renormalized mass, instead of the finite part, that is fixed
first, through our OPT, to be on the order of the temperature. Matching with
the physical mass determines the finite part. This may raise a question on
consistency of the resulting perturbative analysis, since the new renormalized
mass is much smaller than the actual physical mass.555We stress that if one’s
theory is such that the renormalized mass happens to turn out to be much
smaller, unlike in the SM Higgs sector, than the physical value, the new
scheme would hardly require any justification. Although in nature the
modification amounts to finite renormalization and thus must not affect the
physics, it will be useful to take a close look at how the perturbative
analysis modifies in the new scheme.

The best formal framework for assuring the consistency is bare perturbation
theory. However, it will be useful to have a qualitative discussion of the
reasoning behind the new scheme and its potential ramifications before
quantitatively proving the consistency of the new scheme. Let us play with
informal and intuitive modifications of the standard $\overline{\mbox{MS}}$
scheme of renormalized perturbation theory. We mainly consider a zero-T setup,
since the issue at hand is already present at zero temperature. In the
standard renormalized perturbation theory, say, with dimensional
regularization and $\overline{\mbox{MS}}$ scheme, the finite part is pre-fixed
and then the renormalized mass is determined at the end by matching the pole
expression with the physical value of, say, 125 GeV. The renormalized mass
turns out to be within a few percent of 125 GeV. For the sake of argument, let
us suppose that the renormalized mass is determined to be 120 GeV. The precise
value doesn’t matter for the point to be made. Let us conduct the following
‘Gedankenexperiment’ with the new scheme, i.e., the scheme in which the
renormalized mass is fixed prior to the finite part denoted by $c_{m}$. (One
of the purposes of the Gedankenexperiment is to show without the
technicalities of bare perturbation theory that the new scheme points toward
interesting predictions in the case of a finite-T setup; see below). One may
take the renormalized mass $m$, say, $m=110$ GeV in the new scheme, and repeat
the renormalization analysis. Since the difference between 120 GeV and 110 GeV
is minor, the perturbation series must remain valid and the finite constant
$c_{m}$ will turn out to be close to the value prefixed in the standard
approach. (Since $c_{m}$ is zero in MS scheme, its value in the new scheme
will be small.) Now one keeps lowering the value of the renormalized mass $m$.
One can be certain that the series will definitely be valid at least for a
certain range of the renormalized mass values. The real question is whether or
not one can modify the renormalized mass to a degree required to solve the CC
problem. By employing bare perturbation theory we will see below that this is
obviously the case. One can conduct a similar Gedankenexperiment involving a
finite-T setup. Here one can play a similar game with two parameters (or
three, including $c_{m}$), the renormalized mass and temperature. Let us
consider the temperature to be around that of the electroweak (EW) era. The
scheme leads to potentially very interesting predictions: the CC will turn out
to be determined by the temperature, just the way it is determined in section
3. Then since the temperature should be time-dependent (checks are necessary
but the time-dependence of the temperature in the expanding Universe should be
generic), the result will lead to time-dependence of the CC.666Section 4.2 of
[35] demonstrated the time-dependence of the CC in the zero-T setup. More
generally, one will have cosmology substantially different [34] from the
standard one.

Let us turn to bare perturbation theory and assure that the key to preserving
the success of the zero-temperature lies in vast freedom in choosing
subtraction schemes and/or renormalization conditions. The system that we
employ for this task is a real scalar theory with a quartic potential:

$\displaystyle S=-\int
d^{4}x\;\Big{[}\frac{1}{2}\partial_{\mu}\zeta\partial^{\mu}\zeta+\frac{1}{2}m^{2}\zeta^{2}\Big{]}-\int
d^{4}x\;\frac{\lambda}{4!}\zeta^{4}.$ (1)

Invariance of the physical quantities under renormalization scheme change is
based on the following well-known relation

$\displaystyle{\cal L}_{R}(m,g\mu^{\epsilon},c_{m})={\cal
L}_{cl}(m_{0},g_{0})$ (2)

where ${\cal L}_{R}$ (${\cal L}_{cl}$) denotes renormalized (classical)
Lagrangian. The theory is defined by the bare Lagrangian, i.e., the Lagrangian
on the right-hand side. Let us again focus on the mass sector. Although one
introduces $m$ and $c_{m}$, it is only a convenient “splitting” of a single
parameter $m_{0}$. One can obtain the pole mass expression to the desired
order in bare perturbation theory. For example, let us consider

$\displaystyle G_{0}(k,m_{0})=\frac{1}{k^{2}+m_{0}^{2}-\Sigma(k,m_{0})}$ (3)

where $\Sigma(k,m_{0})$ is a series in $\hbar$. The series is convergent; it
is convergent irrespective of the aforementioned splitting of $m_{0}$.777In
general, a perturbation series in quantum field theory is an asymptotic one.
However, the point to be stressed here is that whether the series is
asymptotic or genuinely convergent does not depend on one’s renormalization
scheme, e.g., how one splits $m_{0}$: if the series under consideration is
convergent for one renormalization scheme it will be so for other
renormalization schemes as well.

To be entirely heuristic, let us consider the self-energy at one-loop; one
finds

$\displaystyle\Sigma^{{\scriptscriptstyle(1)}}=-\frac{\lambda}{32\pi^{2}}m_{0}^{2}\Big{(}\frac{m_{0}^{2}}{4\pi\mu^{2}}\Big{)}^{-\varepsilon}\Gamma(1-D/2).$
(4)

where $\varepsilon\equiv\frac{4-D}{2}$ and $D$ denotes the spacetime
dimension. To remove the divergence at this point, one can split $m_{0}$ into
$m$ and the finite part $c_{m}$ according to one’s choice of renormalization
scheme. One can compute the series to a desired loop-order and subsequently
split $m_{0}$. As this example clearly demonstrates, the splitting can be
(quite arbitrarily) done after obtaining the series to a desired order without
having to worry about the convergence. How the splitting is done is part of
one’s renormalization scheme, which amounts to redistributions of various
values in the convergence series, i.e., certain resummation.

Considering higher loops, the corresponding set of finite constants $c^{(n)}$
where $n$ denotes the loop order can be introduced as a matter of principle.
Of course, it would be pleasant if things could work with a ‘minimal’ set of
$c^{(n)}$, and this is what happens, e.g., in the SM analysis with the usual
renormalization schemes. As we further comment below, one of the lessons of
the present work is that one can cover a wider range of physics with a
possibly slightly extended set of $c^{(n)}$ that are not prefixed.

$\fmffile{smcors2}\scalebox{1.3}{$\begin{gathered}\fmfgraph*(60,60)\fmfleft{i}\fmfright{o}\fmftop{m}\fmf{plain,tension=1}{i,v1}\fmf{plain,tension=1}{v1,o}\fmf{plain,left,tension=0}{v1,m,v1}\end{gathered}$}\quad\begin{gathered}\fmffile{olctr}\scalebox{1.3}{$\fmfgraph*(40,60)\fmfleft{i}\fmfright{o}\fmftop{m}\fmfv{decor.shape=cross,decor.size=10}{v1}\fmf{plain,tension=1}{i,v1}\fmf{plain,tension=1}{v1,o}\fmf{phantom,left,tension=0}{v1,m,v1}$}\end{gathered}$
$(a)\hskip 72.26999pt\;(b)$

Figure 1: Diagrams for one-loop self-energy: (a) one-loop correction for
propagator (b) its counter-term

Although the bare perturbation theory component above assures the validity of
our renormalization scheme, renormalized perturbation theory has its
advantages in practical computations, thus it will be useful to examine the
convergence issue in that framework. We do this by considering two-loop
renormalization of the propagator. The two-point proper vertex is defined as

$\Gamma^{[2]}\equiv k^{2}+m^{2}-\Sigma(k,m,c_{m})$ (5)

where $\Sigma$ again denotes self-energy. At one-loop, $\Sigma$ can be
computed by considering the diagrams in Fig. 1: the one-loop two-point
divergence introduces the counter-term:

$\displaystyle\begin{gathered}\fmffile{olctr}\scalebox{0.8}{$\fmfgraph*(40,60)\fmfleft{i}\fmfright{o}\fmftop{m}\fmfv{decor.shape=cross,decor.size=10}{v1}\fmf{plain,tension=1}{i,v1}\fmf{plain,tension=1}{v1,o}\fmf{phantom,left,tension=0}{v1,m,v1}$}\end{gathered}\;:\;-\frac{m^{2}}{4}\frac{\lambda}{(4\pi)^{2}}\;\Big{(}\frac{1}{\varepsilon}+c_{m}\Big{)}\zeta^{2}$
(7)

where $\varepsilon\equiv\frac{4-D}{2}$ as before and $D$ denotes the spacetime
dimension; $c_{m}$ is a constant to be determined by one’s subtraction scheme.
For instance, the modified minimal subtraction ($\overline{\mbox{MS}}$)
corresponds to setting $c_{m}=0$. Fixing the renormalized mass according to
the OPT principle of minimal sensitivity (PMS) (as we will in section 3) is in
contrast to the usual practice in zero-temperature: there, one fixes $c_{m}$
by one’s subtraction scheme, and then the renormalized mass by the pole mass
condition,

$k^{2}+m^{2}-\Sigma(k)\,\Big{|}_{k^{2}=-m_{P}^{2}}=0$ (8)

where $m_{P}$ denotes the physical pole mass. In the new scheme, it is the
coefficient $c_{m}$ that is determined by the pole mass condition, while the
renormalized mass is prefixed (by variant OPT in section 3).

As stated in the introduction, the advantage of the new scheme is obvious: it
realizes the scaling mentioned in the introduction and thereby allows one to
avoid the CC fine-tuning problem. To see the disadvantage, let us note that
the pole mass condition in the new scheme yields

$c_{m}\,m^{2}\sim m_{P}^{2}-m^{2}$ (9)

and thus implies a larger value of $c_{m}$, compared with the standard
approach where the pole mass condition typically leads to $m^{2}\simeq
m_{P}^{2}$. Note, however, that it is the combination $m^{2}c_{m}\sim
m_{p}^{2}$, but not $c_{m}$ alone, that appears in the counter-term (7). In
general, for mass-related quantities it will not be possible at tree-level to
achieve suitable agreement with experimental values, since the renormalized
mass is pre-fixed: it will be necessary to go to one-loop where one has the
freedom of adjusting the finite parts. The two-loop-relevant diagrams are
given in Fig. 2. The circle in the last diagram in Fig. 2 represents the
counter-term for the one-loop four-point diagram (not explicitly shown).
Adding all up, the total two-loop self-energy
$\Sigma^{{\scriptscriptstyle(2)}}$ is

$\displaystyle\Sigma^{{\scriptscriptstyle(2)}}=m^{2}\frac{\lambda^{2}\mu^{4\varepsilon}}{(4\pi)^{4}}\Big{[}\frac{1}{2\varepsilon^{2}}+\frac{1}{4\varepsilon}(-1+c_{m}+3c_{\lambda})-\frac{p^{2}}{24m^{2}\varepsilon}+\cdots\Big{]}$
(10)

where $c_{\lambda}$888The new scheme concerns only the renormalized mass: for
the other quantities, such as the coupling constants, one can just employ the
standard schemes. denotes the finite part of the counter-term of the one-loop
four-point amplitude and $p$ the momentum entering through one end of the
diagrams. The $\varepsilon$-pole terms will have to be removed by two-loop
counter-terms. Let us focus on the finite parts. The constant $c_{m}$ \-
$m^{2}c_{m}$, more precisely - will also appear in the finite parts
represented by the ellipses.999Since the issue under consideration belongs to
the mass, one may well set $c_{\lambda}=0$ in the spirit of the
$\overline{\mbox{MS}}$. Similarly, the two-loop analogues of
$c_{m},c_{\lambda}$ (and the finite part associated with the wavefunction
renormalization) may be set to zero. In other words, too much freedom in
choosing the finite parts can be a burden: the freedom remaining after
determining $c_{m}$ can be fixed just as in a convenient subtraction scheme,
such as the $\overline{\mbox{MS}}$, to facilitate ‘next’ steps, e.g., solving
the renormalization group equations. By imposing the pole mass condition and
solving it, say, interactively for $c_{m}$, it will be possible to determine
its two-loop correction. Since $\Sigma^{{\scriptscriptstyle(2)}}$ has an
additional $\lambda$ and $\hbar$ compared with one-loop, the presence of
$m^{2}c_{m}$ cannot disturb the series in any significant way. (It is bare
perturbation theory that ultimately confirms this.) After all, the new
perturbation can be viewed as finite renormalization. We will point out later
that a certain resummation is behind this finite renormalization.

$\fmffile{twopointloop}\scalebox{0.9}{$\begin{gathered}\fmfgraph(40,60)\fmfleft{i1,i2,i3}\fmfright{o1,o2,o3}\fmf{plain}{i1,v1,o1}\fmffreeze\fmf{phantom}{i2,v2,o2}\fmf{phantom}{i3,v3,o3}\fmf{plain,left,tension=.3}{v1,v2,v1}\fmf{plain,left,tension=.2}{v2,v3,v2}\end{gathered}$}\quad\quad\fmffile{sunset}\scalebox{1.2}{$\begin{gathered}\fmfgraph(40,40)\fmfleft{i}\fmfright{o}\fmf{plain,tension=5}{i,v1}\fmf{plain,tension=5}{v2,o}\fmf{plain,left,tension=0.4}{v1,v2,v1}\fmf{plain}{v1,v2}\end{gathered}$}\quad\quad\fmffile{tlwc3}\scalebox{1.2}{$\begin{gathered}\fmfgraph*(40,60)\fmfleft{i}\fmfright{o}\fmftop{m}\fmfv{decor.shape=cross,decor.size=10}{m}\fmf{plain,tension=1}{i,v1}\fmf{plain,tension=1}{v1,o}\fmf{plain,left,tension=0}{v1,m,v1}\end{gathered}$}\quad\quad\fmffile{tlwc4}\scalebox{1.2}{$\begin{gathered}\fmfgraph*(40,60)\fmfleft{i}\fmfright{o}\fmftop{m}\fmfv{decor.shape=circle,decor.size=5}{v1}\fmf{plain,tension=1}{i,v1}\fmf{plain,tension=1}{v1,o}\fmf{plain,left,tension=0}{v1,m,v1}\end{gathered}$}$

Figure 2: Diagrams for two-loop self-energy

To be thorough, let us take the counter-term, $\sim
c_{m}^{{\scriptscriptstyle(1)}}m^{2}\zeta^{2}$ and examine its feedback to the
CC in renormalized perturbation theory to see how things are organized in this
sector. (Note that we have now denoted $c_{m}$ by
$c_{m}^{{\scriptscriptstyle(1)}}$ to distinguish the two-loop finite parts
from those at one-loop.) The counter-term
$c_{m}^{{\scriptscriptstyle(1)}}m^{2}\zeta^{2}$ will yield, upon self-
contracting the $\zeta$’s, a contribution to the CC at two-loop. Such a
contribution does not cause any harm to the two-loop analysis in [12], as we
now show. The contraction, being divergent, necessitates the corresponding CC
renormalization at two-loop by introducing a CC counter-term $\delta\Lambda$.
The renormalization procedure introduces
$c_{\Lambda}^{{\scriptscriptstyle(2)}}$. It is natural to take
$c_{\Lambda}^{{\scriptscriptstyle(2)}}$ to be small. Needless to say that an
additional factor of $\hbar$ dampens the two-loop result. One must choose its
value in a manner consistent with the one-loop result and known CC value. The
presence of the counter-term implies that it will be useful to carry out the
resummation analogous to the one invoked when treating the mass term of a
massive theory as a vertex. As well known, one gets the usual massive
propagator after resumming the mass vertex corrections to the massless
propagator. The resummation needed for the present case will of course be a
finite-T analogue.

##  3 Variant OPT-induced renormalized mass

The crux of the variant optimal perturbation theory (OPT) can be captured by
considering a scalar system in a flat spacetime. Since the UV divergences
originate locally from a short distance, they are insensitive to global
geometry. For this reason, the zero-temperature UV regularization can be
employed in a finite-temperature theory. As for quantities depending on the
infrared structure, the prime example of which is vacuum energy, one must
consider in principle the actual background. The difference between using the
curved background and the flat one lies in the finite parts. (However, the
finite parts are adjusted by the renormalization conditions anyway; we refer
to [36] for further discussion.)

In thermal field theory, convergence of perturbative analysis is improved by
resummation. The convergence can be further enhanced with a touch of non-
perturbative techniques, such as OPT. The OPT implemented in this work is a
relatively minor, but nonetheless crucial, variation of the widely-studied
one. In the widely-used OPT, an artificial mass term is subtracted out after
adding. This is one way of ensuring artificial-mass independence of the full
closed results. In our case, the renormalized mass itself serves as the OPT
parameter to be fixed by the OPT principle of minimal sensitivity. As well
known in the context of the variational principle in quantum mechanics, there
is no unique way of implementing the principle. For instance, the more
variational parameters one introduces, the more accurate the approximation
generally becomes. Our OPT is one that has an advantage of achieving the goal
of avoiding the CC problem through improved infrared convergence.

The gravity-scalar system that we consider is

$\displaystyle S=\frac{1}{\kappa^{2}}\int d^{4}x\sqrt{-g}\;R-\int
d^{4}x\sqrt{-g}\;\Big{(}\frac{1}{2}g^{\mu\nu}\partial_{\mu}\zeta\partial_{\nu}\zeta+V(\zeta)\Big{)}$
(11)

where $\kappa^{2}=16\pi G$ with $G$ being Newton’s constant. The potential
$V(\zeta)$ is

$\displaystyle
V(\zeta)=\frac{\lambda}{4!}\Big{(}\zeta^{2}+\frac{6}{\lambda}\nu^{2}\Big{)}^{2}.$
(12)

Note the notation change as compared with section 2: the mass has been denoted
by $\nu$. One conceptual hurdle is the justification of the complete-square
form of the potential instead of the usual
$V=\frac{1}{2}\nu^{2}\zeta^{2}+\frac{1}{4!}\lambda\zeta^{4}$. The value of CC
depends, of course, on whether one uses the complete-square form or the form
without the constant piece. A shift of potential by a constant is immaterial
in flat spacetime quantum field theory. The same is not true, however, in the
quantum gravitational context. Whether one should use the complete-square form
or the more usual form is not part of the CC problem. A closely related
question, whose answer is not currently known [31], is why the minimum value
of the classical Higgs potential should be taken to be zero. It is an
independent problem that must ultimately be answered experimentally. Our goal
here is to show that in the setup dictated largely by convergence of thermal
perturbation theory, the fine tuning-problem is not present; this goal can be
achieved more conveniently with the complete-square form.

To set the stage for the refined BFM [37, 35, 36, 38], we shift the fields as

$\displaystyle g_{\mu\nu}\rightarrow
h_{\mu\nu}+\tilde{g}_{\mu\nu}\quad,\quad\zeta\rightarrow\hat{\zeta}+\tilde{\zeta}$
(13)

with

$\tilde{g}_{\mu\nu}\equiv
g_{c\,\mu\nu}+\varphi_{\mu\nu}\quad,\quad\tilde{\zeta}\equiv\zeta_{c}+\xi$
(14)

where $g_{c\,\mu\nu},\zeta_{c}$ denote the classical solutions,
$\varphi_{\mu\nu},\xi$ the background fields, and $h_{\mu\nu},\hat{\zeta}$ the
fluctuation fields. The zero-temperature loop analysis is based on the
following two-point functions (see [38] for the conventions). For the metric,

$\displaystyle<h_{\mu\nu}(x_{1})h_{\rho\sigma}(x_{2})>$ $\displaystyle=$
$\displaystyle\tilde{P}_{\mu\nu\rho\sigma}\,\tilde{\Delta}(x_{1}-x_{2})$ (15)

where the tensor $\tilde{P}_{\mu\nu\rho\sigma}$ is given by

$\displaystyle\tilde{P}_{\mu\nu\rho\sigma}$ $\displaystyle\equiv$
$\displaystyle\frac{\bar{\kappa}^{2}}{2}\Big{(}\tilde{g}_{\mu\rho}\tilde{g}_{\nu\sigma}+\tilde{g}_{\mu\sigma}\tilde{g}_{\nu\rho}-\frac{1}{2}\tilde{g}_{\mu\nu}\tilde{g}_{\rho\sigma}\Big{)};$
(16)

where $\bar{\kappa}^{2}\equiv 2\kappa^{2}$ and satisfies

$\displaystyle\tilde{P}_{\mu\nu\kappa_{1}\kappa_{2}}\tilde{P}^{\kappa_{1}\kappa_{2}}{}_{\rho\sigma}=\bar{\kappa}^{2}\tilde{P}_{\mu\nu\rho\sigma}.$
(17)

$\tilde{\Delta}(x_{1}-x_{2})$ is Green’s function for a (massless) scalar
theory in the background metric $\tilde{g}_{\mu\nu}$:

$\displaystyle<\hat{\zeta}(x_{1})\hat{\zeta}(x_{2})>$ $\displaystyle=$
$\displaystyle\tilde{\Delta}(x_{1}-x_{2}).$ (18)

The explicit form of $\tilde{\Delta}$ for a massive scalar theory will be
given and utilized in section 3.2 where the curved space analysis of the
matter-involving sector is conducted. With these the finite temperature can be
computed in the standard way.

In passing, the following can be said for breaking of gauge invariance in a
finite temperature setup. In gravity theory, introducing a finite temperature
should be viewed as part of specifying the background. Once temperature
enters, the diffeomorphism is lost in a manner that can be dubbed as
generalized spontaneous symmetry breaking. The roles played by this concept of
generalized spontaneous symmetry breaking can be found, e.g., in [39][40][41].

###  3.1 Flat spacetime analysis

As stated in the beginning, the crux of our OPT is captured by considering a
scalar system in a flat spacetime. We employ the $\overline{\mbox{MS}}$
subtraction scheme in the present section. We will come back to the deviation
from the $\overline{\mbox{MS}}$ scheme in section 3.3.

The starting point of the OPT-improved thermal resummation can be taken as the
following renormalized action

$\displaystyle S(\zeta)$ $\displaystyle=$ $\displaystyle-\int
d^{4}x\;\frac{1}{2}\partial_{\mu}\zeta\partial^{\mu}\zeta-\int
d^{4}x\;\Big{(}\frac{1}{2}M^{2}\zeta^{2}+\frac{\lambda}{4!}\zeta^{4}\Big{)}-\int
d^{4}x\;\frac{3\nu^{4}}{2\lambda}$

with

$M^{2}(T)\equiv\nu^{2}+\frac{\lambda}{24}T^{2}$ (20)

We take the expression $M^{2}(T)\equiv\nu^{2}+\frac{\lambda}{24}T^{2}$ for the
purpose of computing the loop contributions, namely, to take the thermal
resummation into account. Later, we use $M^{2}\equiv\nu^{2}$ for the classical
part of the one-loop effective potential eq. (26).101010One may consider using
the expression $M^{2}(T)\equiv\nu^{2}+\frac{\lambda}{24}T^{2}$ even for the
classical part of the potential in eq. (26). This would amount to finite
renormalization of the mass term. This possibility will be further examined in
[12].  Shift the field

$\zeta\rightarrow\hat{\zeta}+\tilde{\zeta}$ (21)

where $\hat{\zeta},\tilde{\zeta}$ denote the fluctuation field and background
field, respectively. Since we are interested in the potential as opposed to
the action, the background field $\tilde{\zeta}$ can be treated as a constant.
Then the potential can be effectively computed by considering the field-
dependent mass term

$M^{2}(T)\rightarrow\tilde{M}^{2}(T,\tilde{\zeta})=\nu^{2}+\frac{\lambda}{24}T^{2}+\frac{\lambda}{2}\tilde{\zeta}^{2}$
(22)

and integrating out the fluctuation field $\hat{\zeta}$. A remark is in order
before proceeding to determination of the OPT-induced renormalized mass. We
stated earlier that although we consider a CMB-order temperature, the high-
energy expansion can be utilized. The temperature being high or low is
relative to the mass and we will show below that our OPT implies
$\tilde{M}\sim\hbar^{1/2}T$ ($\hbar$ will be kept implicit). One can now tell
why the high-energy expansion is justified: since the auxiliary mass
$\tilde{M}$ satisfies $\tilde{M}/T\sim\hbar^{1/2}$, the intermediate analysis
corresponds to that of high temperature: $\frac{\tilde{M}}{T}<<1$.

The two-loop calculation of the effective potential was conducted long ago,
e.g., in [42], [43], and [44]. For our goal, it is necessary to keep tract of
the field-independent terms as well. Also, the $M$\- and $\tilde{M}$\-
dependence is important. Let us focus on the one-loop potential; after
carefully following these terms, one gets

$\displaystyle V_{\mbox{opt}}(\tilde{\zeta})$ $\displaystyle=$
$\displaystyle\frac{3\nu^{4}}{2\lambda}-\frac{\pi^{2}T^{4}}{90}-\frac{\tilde{M}^{4}}{32\pi^{2}}\ln\frac{\bar{\mu}e^{\gamma_{E}}}{4\pi
T}+\frac{1}{24}\tilde{M}^{2}T^{2}$ (23)
$\displaystyle+\frac{1}{2}\nu^{2}\tilde{\zeta}^{2}-\frac{1}{12\pi}\tilde{M}^{3}T+\frac{1}{4!}\lambda\tilde{\zeta}^{4}+{\cal
O}\Big{(}\frac{\tilde{M}^{6}}{T^{2}}\Big{)}$

where $\bar{\mu}\equiv\mu\Big{(}\frac{4\pi}{e^{\gamma_{E}}}\Big{)}^{1/2}$ is a
scaling parameter of dimensional regularization (with the
$\overline{\mbox{MS}}$ scheme). The field equation associated with (26),
$\frac{\partial}{\partial\tilde{\zeta}}V_{\mbox{opt}}=0$, yields

$\displaystyle\frac{\lambda}{6}\tilde{\zeta}^{2}+\nu^{2}-\frac{\lambda}{(4\pi)^{2}}\tilde{M}^{2}\ln\frac{\bar{\mu}e^{\gamma_{E}}}{4\pi
T}+\frac{1}{24}\lambda T^{2}-\frac{\lambda}{8\pi}\tilde{M}T=0$ (24)

up to terms of two-loop order. The solution is

$\displaystyle\tilde{\zeta}^{2}(M)$ $\displaystyle\simeq$
$\displaystyle-\frac{6\nu^{2}}{\lambda}-\frac{T^{2}}{4}+\frac{3}{8\pi^{2}}\Big{(}-3\nu^{2}+M^{2}\Big{)}\ln\Big{(}\frac{\bar{\mu}e^{\gamma_{E}}}{4\pi
T}\Big{)}+\frac{3T\sqrt{\left(M^{2}-3\nu^{2}\right)}}{4\pi}+\cdots.$

With this, one gets the following onshell potential:

$\displaystyle V_{\mbox{opt}}$ $\displaystyle=$
$\displaystyle\frac{3\nu^{4}}{2\lambda}-\frac{\pi^{2}T^{4}}{90}-\frac{\tilde{M}^{4}}{32\pi^{2}}\ln\frac{\bar{\mu}e^{\gamma_{E}}}{4\pi
T}+\frac{1}{24}\tilde{M}^{2}T^{2}$ (26)
$\displaystyle+\frac{1}{2}\nu^{2}\tilde{\zeta}^{2}(M)-\frac{1}{12\pi}\tilde{M}^{3}T+\frac{1}{4!}\lambda\tilde{\zeta}^{4}(M)+{\cal
O}\Big{(}\frac{\tilde{M}^{6}}{T^{2}}\Big{)}.$

The following PMS condition for $\nu$

$\displaystyle\frac{\partial V_{\mbox{opt}}}{\partial\nu}=0$ (27)

admits111111Strictly speaking, one will have to keep $\nu$ approaching zero to
be compatible with renormalization group consideration. (This is a relatively
minor point and does not, of course, qualitatively change the conclusions.)
See further comments in the note added at the end of the Conclusion.

$\displaystyle\nu=0$ (28)

as a solution. This implies

$\displaystyle{M}^{2}=\frac{1}{24}\lambda T^{2}$ (29)

Interestingly, this is the same as the result obtained in [45] (see also [46]
and [47]) by considering renormalization group and choosing appropriate
renormalization conditions.121212Two cautionary remarks: firstly, although we
have been loosely speaking that the optimized renormalized mass is on the
order of the temperature, it is really ${M}^{2}$ that is on the order of the
temperature. (However, the other branches of the solutions of (27) have that
property, and in this sense one may say that the solution is generically on
the order of the temperature.) Secondly, in the earlier arXiv version (v1) of
this work, there was an algebraic error. Although they were relatively minor,
they rendered the coefficient in (29) different from $1/24$. The other
branches of the solutions have undesirable features. For instance, in those
branches the small $\tilde{M}$-expansion is not justified. The condition (28)
yields

$\displaystyle\tilde{\zeta}^{2}$ $\displaystyle=$
$\displaystyle-\frac{T^{2}}{4}+i\frac{\sqrt{3\lambda}}{8\pi}T^{2}+\cdots$ (30)

Note the novelty that $\tilde{\zeta}^{2}$ takes a complex value. Once the
potential (26) is evaluated with this value, one gets

$\displaystyle V_{\mbox{opt}}=-\frac{1}{90}\pi^{2}T^{4}+\cdots$ (31)

which then allows one to avoid the fine-tuning problem.

The complete two-loop offshell form of the potential will be presented in
[12]. There one again encounters the novelty: the potential develops an
imaginary part, signaling instability of the vacuum toward zero
temperature.131313Strictly speaking, the potential itself remains real even at
two-loop. However, this is because the source of the imaginary parts,
$\tilde{M}^{3}$, does not contribute even at two-loop. Due to the expected
contribution of $\tilde{M}^{3}$ term (and the terms with higher odd-integer
powers of $\tilde{M}$ that should appear in higher-loop computations), it is
expected that the complexity of the potential will become manifest at three-
loop and on. More on this in the conclusion.

###  3.2 Curved space analysis

Whereas what we referred to as the second-layer perturbation in [36] is
necessary for the pure gravity sector computation, there exists, for the
matter sector, a powerful shortcut based on the first-layer perturbation, the
“one-stroke” method. In the present work the first-layer perturbation will be
used exclusively. From the results obtained, it becomes evident that the
qualitative conclusion of the flat space analysis remains unchanged.

#### Graviton sector

Let us recall the zero-temperature case first. In [35] and [36], we conducted
the computation in a brute-force manner by employing the second-layer
perturbation and viewing the classical CC as the graviton mass term. As shown,
e.g., in [36], the result is a divergent CC term

$\sim\int\sqrt{-\tilde{g}}.$ (32)

Strictly speaking, for a flat background, the one-loop results vanish in
dimensional regularization in the absence of the classical CC treated as the
graviton mass term. For consistency with the observed value of the CC, the
classical (i.e., renormalized) CC will have to be set to $\sim T^{4}$ in the
renormalization program described below, and thus will not qualitatively
affect the proposed resolution of the CC problem.

The two-loop vanishes at zero temperature, due to the tracelessness of
$P_{\mu\nu\rho\sigma}$ and (17). The results of $n$-loop with $n>2$ will
presumably vanish in dimensional regularization due to the structures of the
vacuum diagrams. Let us examine things in more detail. An arbitrary $n$-loop
graph with $n\geq 2$ contains a product of vertices that can be written as

$\displaystyle<h_{\alpha_{1}\alpha_{2}}h_{\alpha_{3}\alpha_{4}}h_{\alpha_{5}\alpha_{6}}\cdots
h^{\beta_{1}\beta_{2}}h^{\beta_{3}\beta_{4}}h^{\beta_{5}\beta_{6}}\cdots>$
(33)

where the upper and lower indices are fully contracted. Contractions of the
fields lead to

$\displaystyle\tilde{P}_{\mu_{1}\nu_{1}\rho_{1}\sigma_{1}}\tilde{P}_{\mu_{2}\nu_{2}\rho_{2}\sigma_{2}}\cdots$
(34)

where again, all the indices are contracted one way or another with
$\tilde{g}^{\mu\nu}$’s. Whenever a pair of $\tilde{P}$’s have a pair of the
indices contracted, the explicit expression for $\tilde{P}$ given in (16) can
be used. One can also use (17) to reduce the total number of $\tilde{P}$’s
until only one $\tilde{P}$ remains at the end. Since all of the indices must
be contracted, the final expression must be
$\sim\tilde{P}_{\mu\nu}{}^{\mu\nu}$. Thus the overall tensor structure gets
simplified. As for the loop integrals, the two-loop diagrams, i.e., the
sunset- and “8” - types of diagrams vanish in dimensional regularization due
to masslessness of a graviton. We expect the same to be true for higher loops.
However, these diagrams will have non-vanishing contributions if either a CC
term is present and treated as a graviton mass term or temperature enters.
Further analysis of the finite temperature case will be presented in [12].

#### Matter-involving sector

The matter-sector diagrams can be subdivided, depending on whether or not they
involve a graviton loop. The matter-involving vertices do not, unlike the pure
graviton vertices, come with $\frac{1}{\kappa^{2}}$. Since the graviton
propagator comes with $\kappa^{2}$, the diagrams leading in $\kappa$ are those
with a matter loop, which are our focus.

There exists a highly effective “one-stroke” method of computing the matter-
involving part of the effective action, in which the flat spacetime analysis
can be entirely carried over. For this, note the explicit form of
$\tilde{\Delta}(x_{1}-x_{2})$ can be written as

$\displaystyle\tilde{\Delta}(x_{1}-x_{2})=\int\frac{d^{4}k}{(2\pi)^{4}}\frac{1}{\sqrt{-\tilde{g}(x_{1})}}\;\frac{e^{ik\cdot(x_{1}-x_{2})}}{i(k_{\mu}k_{\nu}\tilde{g}^{\mu\nu}(x_{1})+m^{2})}.$
(35)

Defining “flattened” momentum and coordinates as

$\displaystyle
K_{\underline{\alpha}}\equiv\tilde{e}_{\underline{\alpha}}^{\mu}\,k_{\mu}\quad,\quad
X^{\underline{\beta}}\equiv\tilde{e}^{\underline{\beta}}_{\nu}\,x^{\nu}\quad,\quad\tilde{e}_{\underline{\alpha}}^{\mu}\tilde{e}_{\underline{\beta}}^{\nu}\,\tilde{g}_{\mu\nu}=\eta_{\underline{\alpha}\underline{\beta}}$
(36)

where the underlined indices are flattened, one gets the flattened propagator:

$\displaystyle\tilde{\Delta}(X_{1}-X_{2})=\int\frac{d^{4}K}{(2\pi)^{4}}\frac{e^{iK_{\underline{\gamma}}(X_{1}-X_{2})^{\underline{\gamma}}}}{i(K_{\underline{\alpha}}K_{\underline{\beta}}\eta^{\underline{\alpha}\underline{\beta}}+m^{2})}.$
(37)

When computing a diagram, one can pull out all of the background fields and
contract the fluctuation fields. The propagators can then be transformed to
the above. Afterward, the steps become parallel to those corresponding to the
flat cases. The matter part of the effective action can thus be computed
exactly in the same manner in which it is computed in the flat case.

###  3.3 Consistency of new subtraction scheme

Figure 3: Resummation behind the new scheme: (a) a connected diagram in
$\overline{\mbox{MS}}$ scheme (b) resummation required in the new scheme

With the renormalized mass around the CMB temperature, one should make sure
that that framework preserves the success of the zero-temperature theory such
as that seen in the zero-temperature SM. The new perturbation has been
illustrated in section 2 by taking a simple scalar theory. In the case of the
SM, each sector in the SM, i.e., the Higgs, gauge, and fermion, the
renormalization process should be modified from that of the standard
renormalization scheme, e.g., $\overline{\mbox{MS}}$. More specifically, in
the modified scheme, the pole mass value must be realized by adjusting the
finite parts of the divergent integrals that are chosen differently from those
in the standard renormalization scheme: in analysis with
$\overline{\mbox{MS}}$, the pole mass condition is part of the
$\overline{\mbox{MS}}$ scheme, and the renormalized mass is not determined
prior to the pole mass condition. It is the pole mass condition that
determines the renormalized mass. In contrast, in the new scheme the
renormalized mass is determined, as demonstrated in section 3.1, by the OPT.
The finite parts are to be determined by the physical pole mass condition.
What it implies is that it is necessary to go to the one-loop level to achieve
the accuracy of amplitudes obtained in $\overline{\mbox{MS}}$. In order to
match the values of, e.g., the tree amplitudes computed in
$\overline{\mbox{MS}}$, one must insert one-particle-irreducible diagrams to
the internal lines. (We anticipated that one-loop insertion will normally be
sufficient. Since one-loop contributions will be dominant even over tree-
level, it will be useful to examine two-loop contributions to see how fast
two-loop, compared to one-loop, decreases. We suggest this as a future task in
the conclusion.) The insertions amount to a certain resummation. The situation
is generally illustrated in Fig. 3. In light of this, it is also worth noting
that the $T^{2}$-scaling of the renormalized mass was previously obtained in
[45] by choosing appropriate renormalization conditions.

##  4 Conclusion

Since some of the Standard Model particles, such as the Higgs, are massive,
the matter contributions to the CC are naively expected to be larger than that
of the graviton. The variant OPT reveals, however, that the ultimate
determining factor of the CC is the temperature. We believe that this
identifies the cosmological constant problem at its root. This leads to
necessity of employing a new renormalization scheme. In the new scheme, one
needs to go a few orders higher and perform a certain resummation to achieve
the same level of proximity to the values of the physical observables as in
the zero-temperature standard schemes. Given what it brings, this seems to a
relatively small price to pay.

The finding in the present work that a non-traditional renormalization scheme
is needed to tackle the cosmological constant problem is in line with the
observation, e.g., in [7]. The overarching umbrella for this and [6, 8, 9, 10,
11] is the so-called running vacuum with time-varying and dynamical energy, a
view that we find very natural. In these works, the cosmological constant
problem was coherently tackled from different angles and various cosmological
implications of such a vacuum were discussed. (Even the relevance of
temperature to the cosmological constant problem was discussed to some
extent.) Several different regularization methods and their pertinent
renormalization schemes were analyzed. We believe that the present work lends
support, by providing independent motivation to consider a non-standard
renormalization scheme, to those approaches. It is also interesting to note
that the present finite temperature contribution to the stress-energy tensor
seems to bear a certain resemblance to the ‘stress-energy tensor of cosmic
interaction’ obtained in [49].

Before commenting on potential implications of the new scheme for cosmology,
let us discuss some checks that are desirable for solidifying the new
renormalization scheme in the context of renormalized perturbation theory. One
peculiarity of the present scheme worth a closer look is that the one-loop
part should be bigger than the classical part. This is because the
renormalized mass is taken to be far smaller than the physical value. This is
the point illustrated in Fig. 3: since the pole mass in the new scheme will
match poorly with the physical value at the classical level, one should go to
at least one-loop, which must be large enough to offset the deficit of the
classical value. It should be useful to look into things in two directions:
Firstly, it will be illuminating to explicitly check how the new scheme will
play out in an explicit SM example. Secondly, it will be useful to examine in
the new scheme the rate at which the two-loop decreases compared to one-loop.
In the case of a scalar theory, one should be able to check this with
reasonable amount of effort. We will report on progress in [34].

We end with three ramifications of our results and further directions. The
results obtained in this work suggest that, earlier in the thermal history of
the Universe, the value of the CC should have been larger. In other words the
CC becomes time-dependent through the temperature, and the present small value
must be due to the age of the Universe[31]. More quantitatively, the following
is the basis for this anticipation. In [35] it was shown that there exists a
time-dependent solution that approaches the minimal value of the potential. By
using the one-parameter family of the potentials labeled by the temperature,
one can repeat the analysis with the present finite temperature potential. One
can then analyze the resulting quantum-level action, and it should be
possible, at the time-dependent solution level, to establish a CC that
decreases to a small value. One may introduce a renormalized CC,
$\Lambda_{ren}$. To be consistent with the fact that the observed value of the
CC is small, one will have to take $\Lambda_{ren}$ to be small. It will be of
interest to pursue this line of study. Another ramification, not unrelated to
the first, is that the large value of the CC becomes natural when the
temperature is on the order of the EW scale. (This may be something profound,
and have implications for the hierarchy problem in the SM.) Since the Universe
was at higher temperatures in the previous eras, it will be a meaningful
endeavor to explore whether one could come up with a streamlined description
covering the entire temperature range, say, from the near-Planck era to the
present. With the recent progress in the OPT literature, the present results
indicate toward an affirmative answer. Lastly, as we will report in [12], the
potential is expected to develop a imaginary part. We interpret this as an
indication of the vacuum decay from a finite temperature to zero temperature.
Just as an analogous decay induced by a bounce worm hole solution plays an
important role in black hole information [40][48], the present vacuum is
likely to have interesting cosmological implications that deserve further
study.

Note added: As a matter of fact, a novel renormalization condition had been
adopted (with $\overline{\mbox{MS}}$ scheme) in the work by Hamada et al. [50]
\- which I became aware of sometime after publication of the present work -
wherein bare perturbation theory was employed. Their scheme can be translated,
in renormalized perturbation theory, into one that is in line with the present
scheme. The renormalization condition employed therein amounts to setting the
renormalized mass approximately to zero. More specifically, their
renormalization condition in bare perturbation theory is spelled out in eqs.
(5) - (7). Let us focus on the one-loop case eq. (6). The momentum was
evaluated at $k=0$ as a way of narrowing down to the potential, as opposed to
kinetic, term. Let us consider a zero-temperature Higgs type scalar theory
with the renormalized mass denoted by $m$ (as in section 2). To one-loop
order, one gets

$\displaystyle\Sigma$ $\displaystyle=$ $\displaystyle-\frac{\lambda
m^{2}}{32\pi^{2}}\Big{(}\frac{2}{D-4}-1+\gamma_{E}+\cdots\Big{)}.$ (38)

On the other hand one can write, by employing $\overline{\mbox{MS}}$
subtraction scheme,

$\displaystyle m_{0}^{2}=m^{2}-\frac{\lambda
m^{2}}{2(4\pi)^{2}}\Big{(}\frac{1}{D-4}-1+\gamma_{E}\Big{)}.$ (39)

Combining these two equations one sees

$\displaystyle m_{0}^{2}-\Sigma=m^{2}.$ (40)

With eq. (5) in [50] - which is approximately valid, the renormalization
condition eq. (5) above implies

$m^{2}\simeq 0.$ (41)

The lesson of this and the present renormalization condition is that a vast
amount of the freedom in choosing renormalization conditions can (and must) be
utilized to accommodate a wider range of physics.

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