# The milliQan Experiment: Search for milli-charged Particles at the LHC
(proceeding for ICHEP 2018111ICHEP 2018 SEOUL, International Conference on
High Energy Physics, 4-11 July 2018, SEOUL, KOREA )

Jae Hyeok Yoo (University of California, Santa Barbara)
On behalf of the milliQan Collaboration
E-mail<EMAIL_ADDRESS>
Recently, a search for milli-charged particles produced at the LHC has been
proposed. The experiment, named milliQan, is expected to obtain sensitivity to
charges of $10^{-1}-10^{-3}e$ for masses in the 0.1 - 100 GeV range. The
detector is composed of 3 stacks of 80 cm long plastic scintillator arrays
read out by PMTs. It will be installed in an existing tunnel 33 m from the CMS
interaction point at the LHC, with 17 m of rock shielding to suppress beam
backgrounds. In the fall of 2017 a 1% scale ”demonstrator” of the proposed
detector was installed at the planned site in order to study the feasibility
of the experiment, focusing on understanding various background sources such
as radioactivity of materials, PMT dark current, cosmic rays, and beam induced
backgrounds. The data from the demonstrator provides a unique opportunity to
understand the backgrounds and to optimize the design of the full detector.

##  1 Motivation of milliQan experiment

No evidence for new physics at the LHC has been found. This might mean that
there can be phase space that we are not exploring. The milliQan experiment
[1] searches for milli-charged particles (mCP) produced at the LHC. The
existing detectors miss such particles due to their small energy deposits and
the large background in a high luminosity environment.

One, well motivated, mechanism to predict such milli-charged particles is to
introduce a new U(1) in dark sector with a massless dark-photon ($A^{\prime}$)
and a massive dark-fermion ($\psi$),

$\mathcal{L}_{\textrm{dark
sector}}=-\frac{1}{4}A^{\prime}_{\mu\nu}A^{\prime\mu\nu}+i\bar{\psi}\left(\not{\partial}+ie^{\prime}\not{A}^{\prime}+i\textrm{M}_{\textrm{mCP}}\right)\psi-\frac{\kappa}{2}A^{\prime}_{\mu\nu}B^{\mu\nu}.$
(1)

where the last term shows that $A^{\prime}$ and $B$ kinetically mix with the
mixing parameter $\kappa$. After replacing $\not{A}^{\prime}$ with
$\not{A}^{\prime}+\kappa B_{\mu}$, the coupling between the dark fermion and
$B$ becomes $\kappa e^{\prime}$ ($~{}\kappa e^{\prime}\bar{\psi}\not{B}\psi$).
Therefore, the charge of $\psi$ can vary by the size of mixing.

Figure 1:  (a) Sensitivity of the existing experiments to mCP and (b) detector
concept.

Figure 1 shows the sensitivity of the existing experiments. Low mass and high
charge region is covered by the low energy and direct CMS/ATLAS searches, but
the $M_{mCP}>1$ GeV and $Q<0.3e$ region is unexplored. The milliQan experiment
provides a unique opportunity to explore this region.

##  2 Detector concept and demonstrator

The energy deposited by fractionally charged particles is proportional to
$Q^{2}$, meaning a particle with charge down to $10^{−3}e$ has $dE/dx$ of
$10^{−6}$ of a particle with $Q=1e$. Therefore, the detector should have a
large, active, and sensitive area to produce signals even as small as single
photon signals. As shown in Figure 1, The milliQan detector is composed of 3
layers of $80\times 5\times 5$ cm scintillator arrays pointing back to the CMS
interaction point (IP). The particles from the IP should go through all 3
layers; this feature is used to reduce random combinatoric backgrounds
significantly. The light from the scintillator bars is converted and amplified
by photomultiplier tube (PMT).

The detector is located in a underground tunnel at LHC P5. The distance to the
CMS IP is 33 m, of which 17 m is a rock that provides shielding for particles
from collisions. Since it is about 70m underground, the cosmic muons are
significantly reduced.

Figure 2: Design of the demonstrator and a photo of the demonstrator before
covering it with panels.

In order to verify the feasibility and optimize the design of the experiment
thoroughly, a 1 % of the detector was installed as a ”demonstrator” in the
fall of 2017. As shown in Figure 2, it is composed of 3 layers of $2\times 3$
scintillator+PMT units. In addition to the scintillator bars, additional
components were installed to reduce or understand certain types of
backgrounds; lead bricks were installed between the layers to shield radiation
between layers and scintillator slabs were installed to tag through-going
particles, get time info, and shield radiation, e.g., neutrons. Thin and large
scintillator panels were installed to cover the top and the sides and to
provide an ability to reject cosmic muons. Lastly, hodoscopes were installed
to get the tracks of beam and cosmic muons.

##  3 Results from demonstrator

The demonstrator has been running successfully since its installation, and
provided a valuable opportunity to understand the background and optimize the
final design of the full detector.

It is important to calibrate charge because it tells us how small charge the
milliQan detector is sensitive to. The calibration is done by calculating the
number of photo-electrons($\textrm{N}_{\textrm{PE}}$) for cosmic muons where
$\textrm{N}_{\textrm{PE}}$ is extracted by dividing pulse area of cosmic muons
by the pulse area of single photo-electrons (SPE). Then, the
$\textrm{N}_{\textrm{PE}}$ is extrapolated to fractional charges by $Q^{2}$.
The SPEs are taken from late pulses and measured in situ. SPE pulse area
measurement was also done on the bench as a validation. The typical value of
the measured $\textrm{N}_{\textrm{PE}}$ for $Q=1e$ is about $5k$. Taking the
difference in flight distance of cosmic muons and through-going muons in the
scintillator bars (80 cm/5 cm), the $\textrm{N}_{\textrm{PE}}$ for through-
going muon is approximately $5k\times 80/5=80k$. This gives
$\textrm{N}_{\textrm{PE}}=1$ for $Q\sim 3\times 10^{-3}e$, which is consistent
with the full Geant4 simulation results [2].

Figure 3:  (a) Number of through-going particles as a function of time [hour]
and (b) the time difference between the top and the bottom slabs.

Using muons from collisions we can understand the alignment, triggering, and
timing calibration of the demonstrator. Figure 3 shows the rate of beam muons
as a function of time for a few LHC fills. The rate decreases exponentially,
being consistent with the trend of the instantaneous luminosity. The time
constant from the exponential fit is $13\pm 2$ hours while that of the
instantaneous luminosity is $14$ hours. This is an evidence that the detector
”sees” the particles from collisions which indicates the trigger works and the
detector is pointing to CMS IP.

The distance between the top and the bottom slabs is 3.6 m. Therefore, the
time difference between the beam muons that hit the bottom slab first then the
top one and the cosmic muons that his the top slab then the bottom one is
$3.6\textrm{m}/0.3[\textrm{m}/\textrm{ns}]\times 2=24$ ns. Figure 3 shows the
time difference between two slabs. There are two populations peaking at -10 ns
and 12 ns. The time difference is consistent with the muons coming from the IP
and the sky; the peak around at -10 ns is due to cosmic muons and the peak
around at 12 ns is due to beam muons. This shows that the time is measured
correctly.

##  4 Summary

The milliQan experiment can probe the region of $Q=10^{-1}e-10^{-3}e$ for
masses in the range $0.1-100$ GeV, unexplored by previous experiments. The
detector is composed of 3 layers of long plastic scintillators that produce
enough photo-electrons for incoming charged particles with $Q=O(10^{-3}e)$ to
be observed. It will be installed in a tunnel at LHC P5, 33 m away from the
CMS interaction point. In order to understand the backgrounds more thoroughly,
a 1% version of the detector ”demonstrator” was installed and has been running
since fall 2017. The data shows that the muons from collisions go through the
demonstrator, the detector is sensitive to charges down to $3\times 10^{-3}e$,
and the cosmic and beam muons can be clearly distinguished using time
information. The full scale detector is expected to be installed during
2019-2020 and will provide an unprecedented sensitivity to the uncovered
region.

## Acknowledgement

This work was supported by the US Department of Energy under award number
DE‐SC0011702.

## References

  * [1] A. Haas, C. S. Hill, E. Izaguirre and I. Yavin, “Looking for milli-charged particles with a new experiment at the LHC”, Phys. Lett. B 746, 117 (2015), arXiv:1410.6816
  * [2] A. Ball et al., “A Letter of Intent to Install a milli-charged Particle Detector at LHC P5”, arXiv:1607.04669


#  OU-HET 1176 Quantum phase transition and absence of quadratic divergence in
generalized quantum field theories

Hikaru Kawai, Kiyoharu Kawana, Kin-ya Oda, and Kei Yagyu

∗Department of Physics and Center for Theoretical Physics,
National Taiwan University, Taipei 106, Taiwan
∗Physics Division, National Center for Theoretical Sciences, Taipei 10617,
Taiwan
†School of Physics, KIAS, Seoul 02455, Korea
‡Department of Mathematics, Tokyo Woman’s Christian University, Tokyo
167-8585, Japan
§Department of Physics, Osaka University, Toyonaka, Osaka 560-0043, Japan
E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

In ordinary thermodynamics, around first-order phase transitions, the
intensive parameters such as temperature and pressure are automatically fixed
to the phase transition point when one controls the extensive parameters such
as total volume and total energy. From the microscopic point of view, the
extensive parameters are more fundamental than the intensive parameters.
Analogously, in conventional quantum field theory (QFT), coupling constants
(including masses) in the path integral correspond to intensive parameters in
the partition function of the canonical formulation. Therefore, it is natural
to expect that in a more fundamental formulation of QFT, coupling constants
are dynamically fixed _a posteriori_ , just as the intensive parameter in the
micro-canonical formulation. Here, we demonstrate that the automatic tuning of
the coupling constants is realized at a quantum-phase-transition point at zero
temperature, even when the transition is of higher order, due to the
Lorentzian nature of the path integral. This naturally provides a basic
foundation for the multi-critical point principle. As a concrete toy model for
solving the Higgs hierarchy problem, we study how the mass parameter is fixed
in the $\phi^{4}$ theory at the one-loop level in the micro-canonical or
further generalized formulation of QFT. We find that there are two critical
points for the renormalized mass: zero and of the order of ultraviolet-cutoff.
In the former, the Higgs mass is automatically tuned to be zero and thus its
fine-tuning problem is solved. We also show that the quadratic divergence is
absent in a more realistic two-scalar model that realizes the dimensional
transmutation. Additionally, we explore the possibility of fixing quartic
coupling in $\phi^{4}$ theory and find that it can be fixed to a finite value.

###### Contents

  1. 1 Introduction and Outline
  2. 2 Generalized Quantum Field Theory
    1. 2.1 Statistical mechanics
    2. 2.2 Micro-canonical formulation of QFT
    3. 2.3 Further generalized QFT
  3. 3 Fixing mass in free scalar theory
    1. 3.1 Partition function of free scalar theory
    2. 3.2 Mass-squared in free scalar theory
    3. 3.3 Equivalence in large volume limit
  4. 4 Fixing mass in $\phi^{4}$ theory
    1. 4.1 Mass-squared at one-loop level
    2. 4.2 Mass-squared in Large-$N$ model
  5. 5 Fixing quartic coupling in $\phi^{4}$ theory
  6. 6 Dimensional transmutation in two-real-scalar model
  7. 7 Summary
  8. A Mathematical formulas
  9. B Fixing quartic coupling in large $N$ model
  10. C Details of two-real-scalar model

##  1 Introduction and Outline

By the discovery of the Higgs boson at LHC, it has been confirmed that the
electroweak symmetry breaking is triggered by the Higgs mechanism. However,
the question of why there is a huge hierarchy between the electroweak scale
$10^{2}$ GeV and the Planck scale $10^{18}$ GeV is not unveiled yet. In the
Standard Model (SM), the electroweak scale is merely obtained by fine-tuning
the mass parameter of the Higgs potential, which makes the Higgs boson mass
sensitive to ultraviolet (UV) scales such as the grand unification or Planck
scale. Although supersymmetry has been discussed as one of the most promising
new physics scenarios beyond the SM because of the stabilization of the Higgs
boson mass to be the electroweak scale, it does not explain the smallness of
the electroweak scale, nor is found near the expected TeV scale. In order to
confront this fine-tuning problem, more radical and fundamental approaches
beyond ordinary quantum field theory (QFT) appears to be required [1, 2, 3, 4,
5, 6, 7, 8, 9, 10, 11, 12, 13, 14].

Nature might already give us some important clues to tackle this problem. Let
us recall a basic notion of statistical mechanics: there are several different
formulations depending on which parameters to use as control parameters, and
they are equivalent in the thermodynamic limit $V\rightarrow\infty$. In
particular, the most fundamental formulation is obtained with the micro-
canonical ensemble where all the extensive parameters, e.g., energy $E$,
volume $V$, number of particles $N$ are chosen as control parameters, while
intensive parameters, e.g., temperature $T$, pressure $p$, and chemical
potential $\mu$, are determined as functions of these extensive parameters in
the thermodynamic limit. Here, the important point is that this correspondence
is not injective when a system undergoes phase transitions. For example, in a
first-order phase transition, the temperature stays at the critical
temperature $T_{\rm cri}(E)$ until the system releases or absorbs all the
latent heat, which means that the critical point spans the finite region of
$E$ in the micro-canonical formulation.111Although the terminology “critical
point” is often used to express the end point of a phase equilibrium curve
such as a vapor-liquid critical point, we do not use the term in this sense
here, but use it in the sense that a phase transition of any order occurs.
More generally, with a finite probability, intensive parameters are fixed at
the point where the extensive quantities become discontinuous.

In this paper, we explore a correspondence in QFT similar to that observed in
statistical mechanics. Our objective is to address the fine-tuning problem by
determining parameters in the canonical partition function of QFT from the
micro-canonical partition function, thereby eliminating the need for tuning.
In essence, the fine-tuning problem is resolved when a finite region in the
parameter space of micro-canonical (or further generalized) QFT corresponds to
a specific point in the canonical-QFT parameter space that appears fine-tuned.
We also acknowledge various attempts to construct micro-canonical QFT, as
discussed in previous studies [1, 2, 3, 11, 12, 13, 14] and the references
therein.

The concept of generalized QFT is both fascinating and intriguing; however,
the actual fine-tuning of couplings remains unclear, necessitating further
research. As a first step, we investigate the free scalar theory to explore
how the bare mass-squared parameter $m_{\text{B}}^{2}$ is fixed in the
generalized QFT. We calculate the generalized partition function and find two
critical points in the large volume limit: $m_{\text{B}}^{2}=0$ and ${\cal
O}(\Lambda^{2})$, where $\Lambda$ denotes the cut-off scale. In particular,
while the latter corresponds to a saddle point of the vacuum energy and
depends on the regularization schemes in general, the former corresponds to
its discontinuity and does not depend on the regularization schemes. In this
regard, one might find $m_{\text{B}}^{2}=0$ physically more preferable,
meaning that massless theory is naturally realized in the micro-canonical (or
further generalized) picture. This can be also interpreted as a theoretical
explanation of the origin of the so-called _classical scale invariance_ or
_classical conformality_ in the literature [15, 16, 17, 18, 19, 20, 21, 22,
23, 24, 25, 26, 27, 28, 29, 30], which is also the basic assumption behind the
Coleman-Weinberg mechanism [31].

We proceed to examine the $\phi^{4}$ theory at the one-loop level and
generalize it to a large-$N$ model. In both cases, the mass term receives UV
divergent corrections $\delta m_{\rm UV}^{2}\sim\Lambda^{2}$, prompting us to
investigate at which value the renormalized mass
$m^{2}:=m_{\text{B}}^{2}+\delta m_{\rm UV}^{2}$ is fixed in the generalized
QFT. A key distinction from the above free theory lies in the vacuum
transition at $m^{2}=0$; For $m^{2}>0$, we observe a trivial vacuum
expectation value (VEV) $\langle\phi\rangle=0$, while it becomes nonzero for
$m^{2}<0$. This behavior corresponds to the discontinuity of the (second)
derivative of the vacuum energy at $m^{2}=0$, which remains a critical point
in both cases. Although our analysis is limited to the one-loop level or the
large $N$ limit, we anticipate that the conclusions will remain unaffected by
higher order corrections as long as the theory is in the perturbative region.

In our next non-trivial example, we examine a two-real-scalar model at the
one-loop level. This model has been extensively studied in a phenomenological
context [32, 33, 34, 35, 36, 37] because it can realize the Coleman-Weinberg
mechanism [31] under the assumption of classical conformality, i.e.,
$m^{2}=0$. However, in the generalized QFT, this is not an assumption but a
point of discussion for determining where and how renormalized masses are
fixed. To streamline our analysis, we assume
$\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ symmetry and focus on the coupling space
where one real scalar $\phi$ can develop a nonzero VEV $\langle\phi\rangle\neq
0$ due to the radiative potential. We discover that the classical conformal
point $m_{\phi}^{2}=m_{S}^{2}=0$ is not a critical point in the generalized
partition function. Instead, we identify another critical point with
$m_{\phi}^{2}\neq 0$ and $m_{S}^{2}=0$, which corresponds to a quantum first-
order phase transition point. From a phenomenological perspective, this
critical point may serve as an alternative possibility for a dimensional
transmutation mechanism [36, 37] compared to the conventional classical
conformal point; see also Ref. [35] for other possible critical points without
$\mathbb{Z}_{2}$ symmetry.

The organization of this paper is as follows: In Sec. 2, we introduce the
generalized QFT and explain how the fine-tuning of coupling constants can be
automatically achieved within this framework. We also review the standard
discussion on the equivalence between different ensembles in statistical
mechanics to clarify the underlying concept. In Sec. 3, we investigate the
free scalar theory in the context of generalized QFT, focusing on how the bare
mass-squared parameter is fixed in the large volume limit. In Sec. 4, we delve
into a more non-trivial example by considering the $\phi^{4}$ theory,
discussing how the renormalized mass-squared is fixed at the one-loop level,
as well as in the large-$N$ model. In Sec. 5, we explore the possibility of
automatically tuning the quartic coupling constant. In Sec. 6, we analyze the
two-real-scalar model within a simple parameter space where only one scalar
can develop a VEV. Finally, in Sec. 7, we summarize our result.

Throughout the paper, we work in natural units $\hbar=c=1$ and employ the
metric convention $\eta_{\mu\nu}={\rm diag}(-,+,\cdots,+)$.

##  2 Generalized Quantum Field Theory

In this section, we briefly review the basics of statistical mechanics and
introduce a generalized partition function in the micro-canonical formulation
of QFT, and illustrate how the tuning of the coupling constants becomes
possible. See e.g. Ref. [3] and Appendix D in Ref. [38] for other reviews.

###  2.1 Statistical mechanics

To clarify the idea, let us briefly recall statistical mechanics. When control
parameters are temperature $T$ and volume $V$, the system is described by the
canonical partition function

$\displaystyle Z_{V}(T)=e^{-F_{V}(T)/T}=\sum_{n}e^{-E_{n}/T},$ (1)

where $E_{n}$ denotes the energy eigenvalue and $F_{V}(T)=-T\log Z_{V}(T)$ is
the Helmhortz free energy.

On the other hand, we can also consider the micro-canonical formulation where
$E$ is a control parameter instead of $T$. The corresponding partition
function, the number of states, is given by

$\displaystyle\Omega_{V}(E)=\Delta E\sum_{n}\delta(E_{n}-E)=e^{S_{V}(E)},$ (2)

where $S_{V}(E)$ denotes the entropy and $\Delta E$ denotes a sufficiently
small energy interval whose effects on observables vanish in the thermodynamic
limit $V\rightarrow\infty$. In the following, we omit the subscript $V$ for
simplicity.

The equivalence between the canonical and micro-canonical formulations can be
shown as

$\displaystyle e^{-F(T)/T}$ $\displaystyle=\sum_{n}e^{-E_{n}/T}=\int
dE\sum_{n}\delta(E-E_{n})e^{-E/T}=\frac{1}{\Delta E}\int dE\,e^{S(E)-E/T}$
$\displaystyle=\frac{V}{\Delta E}\int
d\varepsilon\,e^{V\left(s(\varepsilon)-\varepsilon/T\right)},$ (3)

where $s=S/V$ ($\varepsilon=E/V$) represents the entropy (energy) density. The
integration is dominated by the saddle point in the thermodynamic limit
$V\rightarrow\infty$ as

$\displaystyle Z_{V}(T)=e^{-F(T)/T}$ $\displaystyle\approx\frac{T(2\pi
C)^{1/2}}{\Delta E}e^{V(s(\varepsilon_{*})-\varepsilon_{*}/T)},$ (4)
$\displaystyle\therefore\ \frac{F(T)}{T}$
$\displaystyle=\frac{E_{*}}{T}-S(E_{*})+{\cal O}(\log V),$ (5)

where $C=\partial E/\partial T$ is the specific heat and
$E_{*}=\varepsilon_{*}V$ is the solution of

$\displaystyle\frac{\partial S}{\partial E}=\frac{1}{T}.$ (6)

Eqs. (5) and (6) are nothing but the Legendre transformation between the free
energy and the entropy in thermodynamics. It is also straightforward to check
the equivalence of the ensemble averages of a general observable $\hat{x}$.
The canonical ensemble average is given by

$\displaystyle\langle\hat{x}\rangle_{\rm
can}=\frac{1}{Z_{V}(T)}\sum_{n}x_{n}e^{-E_{n}/T},$ (7)

where $x_{n}=\langle E_{n}|\hat{x}|E_{n}\rangle$. This can be written as

$\displaystyle\langle\hat{x}\rangle^{\rm can}_{T}$
$\displaystyle=\frac{1}{Z_{V}(T)}\sum_{n}x_{n}\int dEe^{-E/T}\delta(E-E_{n})$
(8) $\displaystyle=\frac{1}{Z_{V}(T)\Delta E}\int
dEe^{S(E)-E/T}\langle\hat{x}\rangle^{\rm mic}_{E}~{},$ (9)

where

$\displaystyle\langle\hat{x}\rangle^{\rm
mic}_{E}=\frac{\sum_{n}x_{n}\delta(E-E_{n})}{\Omega_{V}(E)}$ (10)

is the micro-canonical ensemble average. Because the integrand in Eq. (9) has
a strong peak at $E=E_{*}$ as in Eq. (4), we obtain

$\displaystyle\langle\hat{x}\rangle_{T}^{\rm can}=\langle\hat{x}\rangle^{\rm
mic}_{E}$ (11)

in the thermodynamic limit $V\rightarrow\infty$. In particular, during the
first-order phase transition, $T$ is equal to the critical temperature
$T_{c}(E)$, but $E$ varies between the two phases. This means that the
coexisting phases are described in the finite parameter region of the micro-
canonical scheme. In this sense, a fine-tuning to a first-order phase
transition point $T=T_{c}(E)$ is automatically realized as long as $E$ is the
finite region. The correspondence is summarized in Table 1.

| Parameter | Partition function | Thermodynamic function
---|---|---|---
Canonical | $T$ | $Z(T)=\sum_{n}e^{-E_{n}/T}$ | $F(T)=-T\log Z(T)$
Micro-canonical | $E$ | $\Omega(E)=\Delta E\sum_{n}\delta(E_{n}-E)$ | $S(E)=\log\Omega(E)$
Table 1: Relation between canonical and micro-canonical formulations in
statistical mechanics

###  2.2 Micro-canonical formulation of QFT

Now let us return to QFT. Conventionally we start from “canonical” partition
function with the bare coupling constants
$\set{\lambda}:=\set{\lambda_{j}}_{j=0,1,2,\dots}$:

$\displaystyle Z\left(\set{\lambda}\right)=\int{\cal
D}\phi\exp\left(i\sum_{j}\lambda_{j}O_{j}[\phi]\right),$ (12)

where $O_{j}[\phi]$ ($j=0,1,2,\dots$) denotes a spacetime integral of a local
operator such as $\int d^{d}x{1\over
2}\left(\partial_{\mu}\phi(x)\right)^{2}$, $\int d^{d}x{1\over
2}\left(\phi(x)\right)^{2}$, $\int d^{d}x{1\over 4!}\left(\phi(x)\right)^{4}$,
etc. After the renormalization procedure, we can calculate physical
observables finitely as functions of (renormalized) couplings. However, the
problem is that there is no principle to pick up specific values of
$\set{\lambda}$ theoretically, and this is the very origin of the fine-tuning
problems. Here comes the analogy between QFT and statistical mechanics into
play: What if we start from the “micro-canonical” picture in QFT?

The number of states $\Omega(E)$ in statistical mechanics can naturally be
promoted to the partition function in QFT as

$\displaystyle\Omega\\!\left(\set{A}\right)=\int{\cal
D}\phi\prod_{j}\delta\\!\left(O_{j}[\phi]-A_{j}\right),$ (13)

where we write $\set{A}:=\set{A_{j}}_{j=0,1,2,\dots}$ and each $A_{j}$ is an
“extensive parameter” corresponding to $O_{j}[\phi]$, which is proportional to
the spacetime volume $V_{d}$. By using the Fourier transform of the delta
function, Eq. (13) can be written as

$\displaystyle\Omega\\!\left(\set{A}\right)=\int\left(\prod_{j}d\lambda_{j}\right)\int{\cal
D}\phi\exp\left(i\sum_{l}\lambda_{l}\left(O_{l}[\phi]-A_{l}\right)\right)=\int\left(\prod_{j}d\lambda_{j}\right)Z\\!\left(\set{\lambda}\right)\,e^{-i\sum_{l}\lambda_{l}A_{l}}.$
(14)

Now, one can see that we have an ensemble average of various coupling
constants and their weight is proportional to the canonical partition function
(12).

###  2.3 Further generalized QFT

Further, we can generalize [3] the product of delta functions in Eq. (13) to
an arbitrary weight function $W\\!\left(\set{x}\right)$ as 222 Here, it is
understood that $\set{x}:=\set{x_{j}}_{j=0,1,2,\dots}$.

$\displaystyle\Omega\\!\left(\set{A}\right)$ $\displaystyle=\int{\cal
D}\phi\,W\\!\left(\set{O[\phi]-A}\right)$
$\displaystyle=\int\left(\prod_{j}d\lambda_{j}\right)Z\\!\left(\set{\lambda}\right)e^{-i\sum_{l}\lambda_{l}A_{l}}\omega\\!\left(\set{\lambda}\right),$
(15)

where $\omega\\!\left(\set{\lambda}\right)$ is the Fourier transform of
$W\\!\left(\set{x}\right)$:

$\displaystyle W\\!\left(\set{x}\right)$
$\displaystyle=\int\left(\prod_{j}d\lambda_{j}\,e^{i\lambda_{j}x_{j}}\right)\omega\\!\left(\set{\lambda}\right).$
(16)

In this generalization, we assume that $W\\!\left(\Set{x}\right)$ can depend
on extensive parameters solely through $x_{j}={\cal O}_{j}[\phi]-A_{j}$ so
that $\omega\\!\left(\Set{\lambda}\right)$ does not depend on any extensive
parameters.

If there exists a strong peak $\set{\lambda^{*}}$ of
$Z\\!\left(\set{\lambda}\right)$ in the infinite volume limit
$V\rightarrow\infty$, the generalized QFT is equivalent to the ordinary
canonical QFT whose (bare) coupling constants are fixed at
$\\{\lambda^{*}\\}$. In other words, the fine-tuning of coupling constants is
automatically realized in the generalized partition function. We will see that
only the behavior of the _canonical_ partition function
$Z\\!\left(\Set{\lambda}\right)$ matters for the realization of the Higgs
fine-tuning, regardless of whether it is in the micro-canonical QFT (13) or in
the generalized QFT (15). See Table 2 for the summary of naive correspondence.
In the following sections, we will verify this fine-tuning mechanism for the
(bare) mass term $m_{\text{B}}^{2}$ in scalar field theory.

| Parameter | Partition function | Generating function
---|---|---|---
Canonical QFT | $\lambda$ | $Z(\lambda)=\int{\cal D}\phi e^{i\lambda O[\phi]}$ | $F(\lambda)=-i^{-1}\log Z(\lambda)$
Micro-canonical QFT | $A$ | $\Omega(A)=\int{\cal D}\phi\,\delta\\!\left(O[\phi]-A\right)$ | $S(A)=\log\Omega(A)$
Generalized QFT | $A$ | $\Omega(A)=\int{\cal D}\phi\,W\\!\left(O[\phi]-A\right)$ | $S(A)=\log\Omega(A)$
Table 2: Naive correspondence in QFT

##  3 Fixing mass in free scalar theory

We study the free scalar theory in the generalized QFT and show how the bare
mass parameter is fixed.

###  3.1 Partition function of free scalar theory

We consider the free scalar theory in the $d$-dimensional spacetime:

$\displaystyle S_{0}=-\int
d^{d}x\frac{1}{2}\left(\partial_{\mu}\phi\right)^{2}.$ (17)

Here, we introduce the bare mass term according to the generalized QFT, while
leaving the kinetic term (17) as is, for simplicity. Then the generalized
partition function is defined by

$\displaystyle\Omega(A)$ $\displaystyle:=\int{\cal D}\phi
e^{iS_{0}}W\\!\left(A-\frac{1}{2}\int d^{d}x\phi^{2}(x)\right)$ (18)
$\displaystyle=\int_{-\infty}^{\infty}\frac{dm_{\text{B}}^{2}}{2\pi}\int{\cal
D}\phi\omega\\!\left(m_{\text{B}}^{2}\right)\exp\left\\{i\left[S_{0}+m_{\text{B}}^{2}\left(A-\frac{1}{2}\int
d^{d}x\phi^{2}(x)\right)\right]\right\\}$ (19)
$\displaystyle=\int_{-\infty}^{\infty}\frac{dm_{\text{B}}^{2}}{2\pi}\omega\\!\left(m_{\text{B}}^{2}\right)e^{im_{\text{B}}^{2}A}Z\\!\left(m_{\text{B}}^{2}\right),$
(20)

where $\omega\\!\left(m_{\text{B}}^{2}\right)$ is the Fourier transform of
$W(x)$ and $Z\\!\left(m_{\text{B}}^{2}\right)$ denotes the ordinary canonical
partition function

$\displaystyle Z\\!\left(m_{\text{B}}^{2}\right)=\int{\cal
D}\phi\exp\left[i\left(S_{0}-\frac{m_{\text{B}}^{2}}{2}\int
d^{d}x\phi^{2}(x)\right)\right].$ (21)

We assume that $\omega\\!\left(m_{\text{B}}^{2}\right)$ is an ordinary smooth
function. We can now perform the path-integral as

$\displaystyle\Omega(A)$
$\displaystyle\propto\int_{-\infty}^{\infty}\frac{dm_{\text{B}}^{2}}{2\pi}\omega\\!\left(m_{\text{B}}^{2}\right)\exp\left[im_{\text{B}}^{2}A-\frac{1}{2}\text{tr}\log\left[i\left(-\Box+m_{\text{B}}^{2}-i\varepsilon\right)\right]\right]$
(22)
$\displaystyle=\int_{-\infty}^{\infty}\frac{dm_{\text{B}}^{2}}{2\pi}\omega\\!\left(m_{\text{B}}^{2}\right)\exp\left[iV_{d}\left(am_{\text{B}}^{2}-F\\!\left(m_{\text{B}}^{2}\right)\right)\right],$
(23)

where $\Box=\eta^{\mu\nu}\partial_{\mu}\partial_{\nu}$, $a=A/V_{d}$,
$i\varepsilon$ is the Feynman’s prescription, and

$\displaystyle
F\\!\left(m_{\text{B}}^{2}\right)=\frac{1}{2}\int\frac{d^{d}p_{\text{E}}}{(2\pi)^{d}}\log\\!\left(p_{\text{E}}^{2}+m_{\text{B}}^{2}-i\varepsilon\right)+\rm
const.,$ (24)

with $p_{\text{E}}^{\mu}$ being the Euclidean momentum. This is apparently UV
divergent and we need a regularization.

###  3.2 Mass-squared in free scalar theory

To discuss how the mass is tuned, we employ cut-off and dimensional
regularizations.

#### Cut-off regularization

Figure 1: Left: Integration line. Right: $f(\xi)$ (blue) and
$32\pi^{2}a/(N\Lambda^{2})$ (orange).

First, let us study the cut-off regularization. As a function of
$m_{\text{B}}^{2}$, the integrand in Eq. (23) has a branch cut at

$\displaystyle-\Lambda^{2}+i\varepsilon\leq m_{\text{B}}^{2}\leq
i\varepsilon,$ (25)

and the integration line $-\infty<m_{\text{B}}^{2}<+\infty$ is located below
this branch cut as shown in the left panel in Fig. 1. Then, Eq. (24) can be
evaluated as

$\displaystyle F\\!\left(m_{\text{B}}^{2}\right)$
$\displaystyle=\frac{S_{d-1}}{2(2\pi)^{d}}\int_{0}^{\Lambda}dp_{\text{E}}p_{\text{E}}^{d-1}\log\\!\left(p_{\text{E}}^{2}+m_{\text{B}}^{2}-i\varepsilon\right)$
(26)
$\displaystyle=\frac{S_{d-1}\Lambda^{d}}{4(2\pi)^{d}}\int_{0}^{1}dxx^{\frac{d}{2}-1}\log\\!\left(x+\xi-i\varepsilon\right),\quad\xi=m_{\text{B}}^{2}/\Lambda^{2}$
(27)
$\displaystyle=\frac{S_{d-1}\Lambda^{d}}{4(2\pi)^{d}}\times\begin{cases}\int_{0}^{1}dxx^{\frac{d}{2}-1}\log(x+\xi)&\text{for
}\xi>0\\\
\int_{0}^{1}dxx^{\frac{d}{2}-1}\log|x+\xi|-i\frac{2\pi}{d}(-\xi)^{\frac{d}{2}}&\text{for
}0>\xi>-1\\\
\int_{0}^{1}dxx^{\frac{d}{2}-1}\log|x+\xi|-i\frac{2\pi}{d}&\text{for
}-1>\xi\end{cases},$ (28)

where $S_{d-1}$ denotes the area of a $d-1$ dimensional sphere.

We see that the function $F(m_{\text{B}}^{2})$ contains the imaginary part for
$m_{\text{B}}^{2}<0$, which gives a large suppression in the partition
function when $V_{d}$ is large. Qualitatively, the contribution from
$m_{\text{B}}^{2}<0$ is

$\displaystyle\int^{0}_{-\infty}d\xi e^{-V_{d}\Lambda^{d}(-\xi)^{d/2}}={\cal
O}\left(\frac{1}{(V_{d}\Lambda^{d})^{\frac{2}{d}}}\right).$ (29)

On the other hand, there is no such suppression for $m_{\text{B}}^{2}>0$, and
the integrand is a rapidly oscillating function of $m_{\text{B}}^{2}$. In this
case, the saddle point can exist at the point determined by

$\displaystyle a$
$\displaystyle=\frac{dF}{dm_{\text{B}}^{2}}=\frac{S_{d-1}\Lambda^{d-2}}{4(2\pi)^{d}}\int_{0}^{1}dx\frac{x^{\frac{d}{2}-1}}{x+\xi},$
(30)

When $d=4$, this equation becomes

$\displaystyle
a=\frac{S_{3}\Lambda^{2}}{4(2\pi)^{4}}\left(1-\xi\log\frac{1+\xi}{\xi}\right)=:\frac{\Lambda^{2}}{32\pi^{2}}f(\xi),$
(31)

where $f(\xi)$ is a monotonic function satisfying $f(0)=1$ and $f(\infty)=0$;
see the right panel in Fig. 1. Thus, there can be a saddle point when
$a/\Lambda^{2}\leq\left(32\pi^{2}\right)^{-1}$, while the exponent,
$am_{\text{B}}^{2}-F\\!\left(m_{\text{B}}^{2}\right)$, becomes a monotonic
function when $a/\Lambda^{2}>\left(32\pi^{2}\right)^{-1}$.

#### Dimensional regularization

Let us also calculate the partition function in the dimensional
regularization. We define $\epsilon=4-d$. In this case, the free energy is
calculated as

$\displaystyle F\\!\left(m_{\text{B}}^{2}\right)$
$\displaystyle=-\frac{\left(m_{\text{B}}^{2}\right)^{2}}{2(4\pi)^{2}}\left(\frac{1}{\epsilon}-\frac{\gamma}{2}+\frac{3}{4}+\frac{1}{2}\log
4\pi-\frac{1}{2}\log\left(\frac{m_{\text{B}}^{2}-i{\varepsilon}}{\mu^{2}}\right)\right)$
$\displaystyle=:-\frac{\left(m_{\text{B}}^{2}\right)^{2}}{2(4\pi)^{2}}\left(c_{\overline{\rm
MS}}-\frac{1}{2}\log\left(\frac{m_{\text{B}}^{2}-i{\varepsilon}}{\mu^{2}e^{3/2}}\right)\right),$
(32)

where $\mu$ is the renormalization scale and $c_{\overline{\rm MS}}$ contains
both the finite and divergent terms.

For $m_{\text{B}}^{2}\leq 0$, we again have the imaginary part from the
logarithmic term, and the partition function is highly suppressed. On the
other hand, for $m_{\text{B}}^{2}>0$, the saddle point is determined by

$\displaystyle
a=\frac{dF}{dm_{\text{B}}^{2}}=\frac{m_{\text{B}}^{2}}{32\pi^{2}}\log\left(\frac{m_{\text{B}}^{2}}{\mu^{2}e^{1+2c_{\overline{\rm
MS}}}}\right),$ (33)

which always has a solution unlike the cut-off case. If we identify $\mu
e^{c_{\overline{\rm MS}}}$ as the cut-off scale, the location of the saddle
points is

$\displaystyle m_{\text{B}}^{2}\sim\Lambda^{2}\sim\mu^{2}e^{2c_{\overline{\rm
MS}}}~{}$ (34)

in both of the regularizations.

#### Mass tuning in free scalar theory

As we have seen, the existence of a saddle point depends on the regularization
scheme. But in any case, $m_{\text{B}}^{2}=0$ holds over a wide range of the
parameter space. First, if there is no saddle point, the $m_{\text{B}}^{2}$
integration is dominated by the boundary $m_{\text{B}}^{2}=0$ by the
mathematical formula (88) for any smooth weight function
$\omega(m_{\text{B}}^{2})$. On the other hand, if there is a saddle point,
$am_{\text{B}}^{2}-F(m_{\text{B}}^{2})$ is monotonic below the saddle point
$m_{\text{B}}^{2}\lesssim\Lambda^{2}$. Thus, when the weight function
$\omega\\!\left(m_{\text{B}}^{2}\right)$ has a finite support for
$m_{\text{B}}^{2}\lesssim\Lambda^{2}$, the free energy
$am_{\text{B}}^{2}-F\\!\left(m_{\text{B}}^{2}\right)$ is monotonic, and the
boundary $m_{\text{B}}^{2}=0$ is always dominant due to the mathematical
formula (88). In conclusion, $m_{\text{B}}^{2}=0$ appears to be a unique
critical point that is physically reasonable in the generalized partition
function (23).

###  3.3 Equivalence in large volume limit

Finally, let us confirm the equivalence between generalized QFT and canonical
QFT in the large volume limit. When $A\neq 0$, the first derivative of
$m_{\text{B}}^{2}A/V_{d}-F(m_{\text{B}}^{2})$ is continuous and nonzero at
$m_{\text{B}}^{2}=0$ while the second derivative is discontinuous. Then, we
can use the mathematical formula (92) and $\Omega(A)$ is evaluated as

$\displaystyle\lim_{V_{d}\rightarrow\infty}\Omega(A)$
$\displaystyle=\lim_{V_{d}\rightarrow\infty}\int_{-\infty}^{\infty}\frac{dm_{\text{B}}^{2}}{2\pi}\omega(m_{\text{B}}^{2})e^{im_{\text{B}}^{2}A}Z(m_{\text{B}}^{2})={\cal
N}\lim_{V_{d}\rightarrow\infty}\frac{Z(m_{\text{B}}^{2}=0)}{V_{d}^{2}}~{},$
(35)

where ${\cal N}$ is an unimportant numerical factor. That is,

$\displaystyle\lim_{V_{d}\rightarrow\infty}\log\Omega(A)=\lim_{V_{d}\rightarrow\infty}\log
Z(m_{\text{B}}^{2}=0)+{\cal O}(\log V_{d}).$ (36)

We apparently see the equivalence in the large volume limit.

We can also check the equivalence of correlation functions. We introduce a
source term as

$\displaystyle\Omega[A;J]$ $\displaystyle=\int{\cal D}\phi e^{iS_{0}-i\int
d^{d}xJ(x)\phi(x)}W\\!\left(A-\frac{1}{2}\int d^{d}x\phi^{2}(x)\right)$
$\displaystyle=\int_{-\infty}^{\infty}\frac{dm_{\text{B}}^{2}}{2\pi}\omega\\!\left(m_{\text{B}}^{2}\right)e^{iV_{d}\left(m_{\text{B}}^{2}a-F(m^{2})\right)}\exp\left(\frac{i}{2}\int
d^{d}xJ(x)(-\Box+m_{\text{B}}^{2}-i\epsilon)^{-1}J(x)\right)$
$\displaystyle=\int_{-\infty}^{\infty}\frac{dm_{\text{B}}^{2}}{2\pi}\omega\\!\left(m_{\text{B}}^{2}\right)e^{iAm_{\text{B}}^{2}}Z\\!\left(m_{\text{B}}^{2}\right)G(J;m_{\text{B}}^{2})~{}.$
(37)

As long as $J(x)$ is a finite supported function, i.e., when it has no volume
dependence, the factor $G(J;m_{\text{B}}^{2})$ does not have exponentially
large volume dependence, and the $m_{\text{B}}^{2}$ integral in Eq. (37) is
dominated by the critical point of $Z\\!\left(m_{\text{B}}^{2}\right)$.
Namely, we have

$\displaystyle\lim_{V_{d}\rightarrow\infty}\Omega[A;J]={\rm const.}\times
G(J;m_{\text{B}}^{2}=0).$ (38)

Then, by taking the functional derivatives with respect to $J(x)$, we obtain

$\displaystyle\left\langle
T\left\\{\phi(x_{1})\cdots\phi(x_{n})\right\\}\right\rangle_{\rm
mic}=\left\langle
T\left\\{\phi(x_{1})\cdots\phi(x_{n})\right\\}\right\rangle_{\rm
can}\bigg{|}_{m_{\text{B}}^{2}=0},$ (39)

which corresponds to Eq. (11) in statistical mechanics.

##  4 Fixing mass in $\phi^{4}$ theory

In this section, we analyze the $\phi^{4}$ theory in the scope of generalized
QFT. In particular, we study how the mass term is fixed by studying its
critical point at the one-loop level, and also in the large-$N$ extended
model. Our investigation reveals that the renormalized mass settles at zero in
the one-loop analysis as well as in the large-$N$ model, due to the
discontinuity of the derivative of the vacuum energy.

In this section, we assume that the quartic coupling is prefixed at a certain
value, as in the ordinary QFT. The possibility of its tuning will be discussed
in the next section.

###  4.1 Mass-squared at one-loop level

We first introduce a bare action without the bare mass term:

$\displaystyle S_{\text{B}}[\phi]=\int
d^{d}x\left(-\frac{1}{2}(\partial_{\mu}\phi)^{2}-\frac{\lambda_{\text{B}}}{4!}\phi^{4}-\Lambda_{\rm
B}\right),$ (40)

where we have also included the bare cosmological constant term for later
convenience. For simplicity, we have imposed the $\mathbb{Z}_{2}$ symmetry
$\phi\rightarrow-\phi$, which leaves only the quadratic and quartic terms in
the renormalizable potential.

Our starting point is the generalized partition function $\Omega(A)$ which can
be expressed in terms of the canonical partition function
$Z\\!\left(m_{\text{B}}^{2}\right)$ by the same procedure as in Sec. 3.1:

$\displaystyle\Omega(A)$
$\displaystyle=\int_{-\infty}^{\infty}\frac{dm_{\text{B}}^{2}}{2\pi}\omega\\!\left(m_{\text{B}}^{2}\right)e^{im_{\text{B}}^{2}A}\,Z\\!\left(m_{\text{B}}^{2}\right),$
(41)

where

$\displaystyle Z\\!\left(m_{\text{B}}^{2}\right)$ $\displaystyle=\int{\cal
D}\phi e^{iS[\phi]},$ $\displaystyle S[\phi]$
$\displaystyle=S_{\text{B}}[\phi]-\frac{m_{\text{B}}^{2}}{2}\int
d^{d}x\phi^{2}.$ (42)

The canonical partition function $Z$ can be obtained from the ordinary
effective action $\Gamma[\phi]$ via $Z=e^{i\min_{\phi}\Gamma[\phi]}$. At the
one-loop level, we obtain

$\displaystyle\Gamma[\phi]$ $\displaystyle=S[\phi]-\frac{1}{2i}{\rm
tr}\log\left(-\Box+m_{\text{B}}^{2}+\frac{\lambda_{\text{B}}}{2}\phi^{2}-i\varepsilon\right)$
$\displaystyle=S[\phi]+\frac{V_{d}}{2}\int\frac{d^{d}p_{\text{E}}}{(2\pi)^{d}}\log\left(p_{\text{E}}^{2}+m_{\text{B}}^{2}+\frac{\lambda_{\text{B}}}{2}\phi^{2}-i\varepsilon\right)$
$\displaystyle=S[\phi]+\frac{V_{d}}{2}\frac{(M_{\phi}^{2}(\phi))^{2}}{(4\pi)^{2}}\left[c_{\overline{\rm
MS}}-\frac{1}{2}\log\left(\frac{M_{\phi}^{2}(\phi)-i\varepsilon}{\mu^{2}e^{3/2}}\right)\right],$
(43)

where
$M_{\phi}^{2}(\phi)=m_{\text{B}}^{2}+\frac{\lambda_{\text{B}}}{2}\phi^{2}$.

Now the one-loop effective potential is given by

$\displaystyle V_{\rm
eff}(\phi):=-\frac{\Gamma[\phi]}{V_{d}}\bigg{|}_{\phi={\rm const}}$
$\displaystyle=\Lambda_{\text{B}}+\frac{m_{\text{B}}^{2}}{2}\phi^{2}+\frac{\lambda_{\text{B}}}{4!}\phi^{4}-\frac{(M_{\phi}^{2}(\phi))^{2}}{32\pi^{2}}\left[c_{\overline{\rm
MS}}-\frac{1}{2}\log\left(\frac{M_{\phi}^{2}(\phi)-i\varepsilon}{\mu^{2}e^{3/2}}\right)\right]$
$\displaystyle=\Lambda_{\text{R}}+\frac{m_{\text{R}}^{2}}{2}\phi^{2}+\frac{\lambda_{\text{R}}}{4!}\phi^{4}+\frac{(M_{\phi}^{2}(\phi))^{2}}{64\pi^{2}}\log\left(\frac{M_{\phi}^{2}(\phi)-i\varepsilon}{\mu^{2}e^{3/2}}\right),$
(44)

where

$\displaystyle\Lambda_{\text{R}}$
$\displaystyle=\Lambda_{\text{B}}-c_{\overline{\rm
MS}}\frac{m_{\text{B}}^{4}}{32\pi^{2}},$ $\displaystyle m_{\text{R}}^{2}$
$\displaystyle=m_{\text{B}}^{2}-\frac{c_{\overline{\rm
MS}}}{16\pi^{2}}\lambda_{\text{B}}m_{\text{B}}^{2},$
$\displaystyle\lambda_{\text{R}}$
$\displaystyle=\lambda_{\text{B}}-\frac{3c_{\overline{\rm
MS}}}{16\pi^{2}}\lambda_{\text{B}}^{2},$ (45)

denote the renormalized parameters. We can regard these renormalized couplings
as our parameters instead of the bare couplings
$(m_{\text{B}}^{2},\lambda_{\text{B}})$.

As a consistency check, let us consider the free theory limit:
$\lambda_{\text{B}}=\lambda_{\text{R}}=0$. In this case, there is no
difference between the bare mass and the renormalized mass, and the effective
potential is exactly given by

$\displaystyle V_{\rm
eff}(\phi)\bigg{|}_{\lambda_{\text{B}}=0}=\Lambda_{\text{R}}+\frac{m_{\text{B}}^{2}}{2}\phi^{2}+\frac{m_{\text{B}}^{4}}{64\pi^{2}}\log\left(\frac{m_{\text{B}}^{2}-i{\varepsilon}}{\mu^{2}e^{3/2}}\right),$
(46)

which has a trivial minimum at $\phi=0$ and the corresponding vacuum energy is

$\displaystyle V_{\rm
min}=\Lambda_{\text{B}}-\frac{\left(m_{\text{B}}^{2}\right)^{2}}{32\pi^{2}}\left(c_{\overline{\rm
MS}}-\frac{1}{2}\log\left(\frac{m_{\text{B}}^{2}-i{\varepsilon}}{\mu^{2}e^{3/2}}\right)\right),$
(47)

which is nothing but Eq. (32).

Let us come back to the interacting scalar theory. We assume
$\lambda_{\text{R}}>0$ to ensure the stability of the effective potential. As
usual, we can obtain the RG improved effective potential by choosing the
renormalization scale $\mu$ appropriately. For $m_{\text{R}}^{2}\geq 0$, the
VEV is trivial $v=0$, and we take $\mu=m_{R}e^{-3/4}$, which will give a
simple expression of the vacuum energy (as the first line in Eq. (50) below).
On the other hand, for $m_{\text{R}}^{2}<0$, the field $\phi$ develops a
nonzero VEV, which is determined by

$\displaystyle\frac{\partial V_{\rm eff}(\phi)}{\partial\phi}=0$
$\displaystyle\quad\Leftrightarrow\quad
m_{\text{R}}^{2}v+\frac{\lambda_{\text{R}}}{6}v^{3}+\frac{\lambda_{\text{R}}v}{32\pi^{2}}\left(M_{\phi}^{2}(v)\log\left(\frac{M_{\phi}^{2}(v)}{\mu^{2}e}\right)\right)=0~{}.$
(48)

By choosing the renormalization scale at $\mu^{2}=M_{\phi}^{2}(v)e^{-1}$, the
VEV is given by

$\displaystyle v^{2}=-\frac{6m_{\text{R}}^{2}}{\lambda_{\text{R}}}~{}.$ (49)

Now the vacuum energy as a function of $m_{\text{R}}^{2}$ is given by

$\displaystyle V_{\rm
min}(m_{\text{R}}^{2})=\Lambda_{\text{B}}-\frac{c_{\overline{\rm
MS}}}{32\pi^{2}}(m_{\text{R}}^{2})^{2}+\begin{cases}0&\text{for
}m_{\text{R}}^{2}\geq 0\\\
-\frac{3(m_{\text{R}}^{2})^{2}}{2\lambda_{\text{R}}}-\frac{\left(2m_{\text{R}}^{2}\right)^{2}}{128\pi^{2}}&\text{for
}m_{\text{R}}^{2}<0\end{cases},$ (50)

which shows that the second derivative of $V_{\rm min}(m_{\text{R}}^{2})$ is
discontinuous at $m_{\text{R}}^{2}=0$ as

$\displaystyle\frac{\partial^{2}V_{\rm
min}}{\partial(m_{\text{R}}^{2})^{2}}\bigg{|}_{m_{\text{R}}^{2}=0+}$
$\displaystyle=-\frac{c_{\overline{\rm MS}}}{16\pi^{2}},$
$\displaystyle\frac{\partial^{2}V_{\rm
min}}{\partial(m_{\text{R}}^{2})^{2}}\bigg{|}_{m_{\text{R}}^{2}=0-}$
$\displaystyle\simeq-\frac{c_{\overline{\rm
MS}}}{16\pi^{2}}-\frac{3}{\lambda_{\text{R}}}+\frac{1}{16\pi^{2}}~{}.$ (51)

Note that the first derivative of $m_{\text{B}}^{2}A/V_{d}-V_{\rm
min}(m_{\text{R}}^{2})$ is continuous and nonzero at $m_{\text{R}}^{2}=0$ as
long as $A\neq 0$, which means that we can use the mathematical formula Eq.
(92). Then, the partition function is evaluated as

$\displaystyle\lim_{V_{d}\rightarrow\infty}e^{im_{B}^{2}A+\log
Z(m_{\text{R}}^{2})}=\lim_{V_{d}\rightarrow\infty}e^{im_{B}^{2}A-iV_{d}V_{\rm
min}(m_{\text{R}}^{2})}\propto\lim_{V_{d}\rightarrow\infty}\frac{e^{-iV_{d}V_{\rm
min}(m_{\text{R}}^{2}=0)}}{V_{d}^{2}}\delta(m_{\text{R}}^{2}),$ (52)

which leads to

$\displaystyle\lim_{V_{d}\rightarrow\infty}\Omega(A)$
$\displaystyle=\int_{-\infty}^{\infty}\frac{dm_{\text{R}}^{2}}{2\pi}\left(\frac{\partial
m_{\text{B}}^{2}}{\partial
m_{\text{R}}^{2}}\right)\omega(m_{\text{B}}^{2})e^{im_{\text{B}}^{2}A+\log
Z(m_{\text{R}}^{2})}=\frac{{\cal N}}{V_{d}^{2}}e^{\log
Z(m_{\text{R}}^{2}=0)},$ (53)

where ${\cal N}$ is an unimportant factor. That is,

$\displaystyle\lim_{V_{d}\rightarrow\infty}\log\Omega(A)$
$\displaystyle=\lim_{V_{d}\rightarrow\infty}\log Z(m_{\text{R}}^{2}=0)+{\cal
O}\\!\left(\log V_{d}\right),$ (54)

We see that the equivalence between two formulations still holds in the
$\phi^{4}$ theory at the one-loop level in the large volume limit. In
particular, the renormalized mass parameter $m_{\text{R}}^{2}$ is fixed at
zero because of the discontinuity of the second derivatives of the vacuum
energy. 333Precisely speaking, there is also a saddle point solution
$m_{\text{R}}^{2}\sim-\frac{A}{V_{d}c_{\overline{\rm MS}}}$. The value of
which depends on the regularization schemes and the way the large volume limit
$A/(c_{\overline{\rm MS}}V_{d})=$fixed is taken. On the other hand, the
critical point $m_{\text{R}}^{2}=0$ has no such uncertainty and is uniquely
determined as in the free case.  This result implies that the quadratic
divergence problem is absent in the micro-canonical formulation. Moreover, in
the simple $\phi^{4}$ theory, the conclusion is not affected by higher-loop
corrections as long as $\lambda_{\text{R}}\lesssim 1$. However, the situation
can be different in more general theories with more than one field because
another mass scale can be radiatively generated by other fields. See Sec. 6
for a concrete two-scalar model.

###  4.2 Mass-squared in Large-$N$ model

In this section, we turn to the $O(N+1)$ symmetric scalar theory with
$\phi_{i}$ ($i=0,1,\dots,N$), and discuss how the mass parameter is fixed in
the generalized partition function in the large-$N$ limit. The bare action is
given by

$\displaystyle S_{\text{B}}=\int
d^{d}x\left(-\frac{1}{2}(\partial_{\mu}\phi_{i})^{2}-\frac{\lambda_{\text{B}}}{4!}(\phi_{i}^{2})^{2}\right),$
(55)

where we have omitted the bare mass term as before and apply the Einstein
summation convention for the field index $i$.

The generalized partition function now becomes

$\displaystyle\Omega(A)=\int_{-\infty}^{\infty}\frac{dm_{\text{B}}^{2}}{2\pi}\omega\\!\left(m_{\text{B}}^{2}\right)e^{im_{\text{B}}^{2}NA}\int{\cal
D}\phi e^{iS}$
$\displaystyle=\int_{-\infty}^{\infty}\frac{dm_{\text{B}}^{2}}{2\pi}\omega\\!\left(m_{\text{B}}^{2}\right)e^{im_{\text{B}}^{2}NA}Z\\!\left(m_{\text{B}}^{2}\right),$
(56)

where

$\displaystyle S=\int
d^{d}x\left(-\frac{1}{2}(\partial_{\mu}\phi_{i})^{2}-\frac{1}{2}m_{\text{B}}^{2}\phi_{i}^{2}-\frac{\lambda_{\text{B}}}{4!}(\phi_{i}^{2})^{2}\right)~{}.$
(57)

We can generally separate the original field $\phi_{i}$ as

$\displaystyle\phi_{i}=\begin{cases}\sqrt{N}s&i=0\\\
\pi_{i}&i=1,2,\cdots,N\end{cases}\quad,$ (58)

where $s$ is the field that may acquire a VEV. The Lagrangian now becomes

$\displaystyle{\cal
L}=N\left(-\frac{1}{2}(\partial_{\mu}s)^{2}-\frac{m_{\text{B}}^{2}}{2}s^{2}\right)-\frac{1}{2}(\partial_{\mu}\pi_{i})^{2}-\frac{m_{\text{B}}^{2}}{2}\pi_{i}^{2}-\frac{\lambda_{\text{B}}}{4!}(Ns^{2}+\pi_{i}^{2})^{2}.$
(59)

We can introduce an auxiliary scalar field $c$ in the Lagrangian such that the
partition function does not change after performing the path-integral over
$c$:

$\displaystyle{\cal L}$
$\displaystyle=N\left(-\frac{1}{2}(\partial_{\mu}s)^{2}-\frac{1}{2}(m_{\text{B}}^{2}+c)s^{2}\right)-\frac{1}{2}(\partial_{\mu}\pi_{i})^{2}-\frac{1}{2}(m_{\text{B}}^{2}+c)\pi_{i}^{2}+\frac{3}{2\lambda_{\text{B}}}c^{2}.$
(60)

By performing the path-integral of $\pi_{i}$, the partition function is given
as

$\displaystyle Z(m_{\text{B}}^{2})=\int{\cal D}\phi e^{iS}=\int{\cal
D}c\int{\cal D}s\exp\bigg{[}iN\int d^{d}x\bigg{(}$
$\displaystyle-\frac{1}{2}(\partial_{\mu}s)^{2}-\frac{1}{2}(m_{\text{B}}^{2}+c)s^{2}+\frac{1}{4\tilde{\lambda}_{\text{B}}}c^{2}$
$\displaystyle-\frac{1}{2}\int\frac{d^{d}p_{\text{E}}}{(2\pi)^{d}}\log(p_{\text{E}}^{2}+m_{\text{B}}^{2}+c-i\epsilon)\bigg{)}\bigg{]}.$
(61)

where $\tilde{\lambda}_{\text{B}}:=\lambda_{\text{B}}N/6$. The variation of
$c$ gives

$\displaystyle-s^{2}+\frac{1}{\tilde{\lambda}_{\text{B}}}c-\int\frac{d^{d}p_{\text{E}}}{(2\pi)^{d}}\frac{1}{p_{\text{E}}^{2}+m_{\text{B}}^{2}+c-i\epsilon}=0,$
(62)

while the variation of $s$ gives

$\displaystyle(-\Box+m_{\text{B}}^{2}+c)s=0.$ (63)

As long as we focus on the ground state, we can omit $\Box s$. For a given
value of $\tilde{\lambda}_{\text{B}}$, the parameter space of
$m_{\text{B}}^{2}$ is divided into two regions as shown in Fig. 2, which
correspond to the broken phase (blue) and the unbroken phase (uncolored)
respectively. Now let us discuss the behavior of the partition function in
each region.

Broken phase
When $s\neq 0$, we have $m_{\text{B}}^{2}+c=0$. Eq. (62) then becomes

$\displaystyle
s^{2}=-\frac{1}{\tilde{\lambda}_{\text{B}}}m_{\text{B}}^{2}-\int^{\Lambda}\frac{d^{d}p_{\text{E}}}{(2\pi)^{d}}\frac{1}{p_{\text{E}}^{2}}>0,$
(64)

which indicates the parameter space of broken phase. This is shown by the blue
region in Fig. 2.

Figure 2: Left: Parameter space in the large $N$ limit.

We represent the critical value of $m_{\text{B}}^{2}$ as $-m_{\Lambda}^{2}$.
For $d=4$, it is

$\displaystyle
m_{\Lambda}^{2}=\frac{\tilde{\lambda}_{\text{B}}\Lambda^{2}}{16\pi^{2}}~{}.$
(65)

For the parameters obeying Eq. (64), the canonical partition function Eq. (61)
becomes very simple in the large $N$ analysis. In fact,

$\displaystyle e^{im_{B}^{2}NA}Z\left(m_{\text{B}}^{2}\right)$
$\displaystyle\sim\exp\left(N\left\\{{\rm
const}.+im_{\text{B}}^{2}A+\frac{i}{4\tilde{\lambda}_{\text{B}}}\int
d^{d}xc^{2}\right\\}\right)$ (66) $\displaystyle=\exp\left[N\left\\{{\rm
const}.+i\frac{V_{d}}{4\tilde{\lambda}_{\text{B}}}\left(m_{\text{B}}^{2}+\frac{2\tilde{\lambda}_{\text{B}}A}{V_{d}}\right)^{2}\right\\}\right]~{}.$
(67)

One can see that there is a saddle point at
$m_{\text{B}}^{2}=-2\tilde{\lambda}_{\text{B}}A/V_{d}$ if it is smaller than
the critical value $-m_{\Lambda}^{2}$. Otherwise, the exponent in Eq. (67) is
a monotonic function for $m_{\text{B}}^{2}\leq-m_{\Lambda}^{2}$. In
particular, the second derivative is given by

$\displaystyle\frac{\partial^{2}\log
Z(m_{\text{B}}^{2})}{\partial(m_{\text{B}}^{2})^{2}}=i\frac{NV_{d}}{2\tilde{\lambda}_{\text{B}}}~{}.$
(68)

Unbroken phase
When $s=0$, the system is in the unbroken phase. In this case, $c$ is
determined by the gap equation

$\displaystyle
c=\tilde{\lambda}_{\text{B}}\int\frac{d^{d}p_{\text{E}}}{(2\pi)^{d}}\frac{1}{p_{\text{E}}^{2}+m_{\text{B}}^{2}+c-i\varepsilon}.$
(69)

This equation has a solution only for

$\displaystyle-\frac{1}{\tilde{\lambda}_{\text{B}}}m_{\text{B}}^{2}-\int_{0}^{\Lambda}\frac{d^{d}p_{\text{E}}}{(2\pi)^{d}}\frac{1}{p_{\text{E}}^{2}}<0,$
(70)

which is consistent with Eq. (64). In this case, the canonical partition
function becomes

$\displaystyle
Z\\!\left(m_{\text{B}}^{2}\right)\sim\exp\left[iNV_{d}\left\\{-\frac{1}{2}\int\frac{d^{d}p_{\text{E}}}{(2\pi)^{d}}\log(p_{\text{E}}^{2}+m_{\text{B}}^{2}+c-i\varepsilon)+\frac{1}{4\tilde{\lambda}_{\text{B}}}c^{2}\right\\}\right],$
(71)

where $c$ depends on $m_{\text{B}}^{2}$ via the gap equation (69). The
exponent in Eq. (71) is a monotonically decreasing function of
$m_{\text{B}}^{2}$ and the second derivative with respect to
$m_{\text{B}}^{2}$ at the critical point $-m_{\Lambda}^{2}$ is

$\displaystyle\frac{\partial^{2}\log
Z(m_{\text{B}}^{2})}{\partial(m_{\text{B}}^{2})^{2}}\bigg{|}_{m_{\text{B}}^{2}=-m_{\Lambda}^{2}}$
$\displaystyle=\frac{iNV_{d}}{2}\left[\int\frac{d^{d}p_{\text{E}}}{(2\pi)^{d}}\frac{1}{(p_{\text{E}}^{2}+m_{\text{B}}^{2}+c)^{2}}\left(1+\frac{dc}{dm_{\text{B}}^{2}}\right)^{2}+\frac{1}{\tilde{\lambda}_{\text{B}}}\left(\frac{dc}{dm_{\text{B}}^{2}}\right)^{2}\right]\bigg{|}_{m_{\text{B}}^{2}=-m_{\Lambda}^{2}}$
$\displaystyle=\frac{iNV_{d}}{2\tilde{\lambda}_{\text{B}}}\frac{\tilde{\lambda}_{\text{B}}\int\frac{d^{d}p_{\text{E}}}{(2\pi)^{d}}\frac{1}{(p_{\text{E}}^{2})^{2}}}{1+\tilde{\lambda}_{\text{B}}\int\frac{d^{d}p_{\text{E}}}{(2\pi)^{d}}\frac{1}{(p_{\text{E}}^{2})^{2}}}~{}.$
(72)

By comparing Eq. (68) and Eq. (72), one can see that the second derivative is
discontinuous at the critical point $m_{\text{B}}^{2}=-m_{\Lambda}^{2}$. Thus,
we conclude that $m_{\text{B}}^{2}$ is fixed at $-m_{\Lambda}^{2}$ by the
mathematical formula (92) (as long as $A\neq 0$) at which the renormalized
mass $m_{\text{R}}^{2}=m_{\text{B}}^{2}+c$ is zero.

##  5 Fixing quartic coupling in $\phi^{4}$ theory

We briefly comment on the possibility of fixing the quartic coupling. As well
as the mass term, we can also consider the generalized partition function for
the quartic coupling

$\displaystyle\int dm_{\text{R}}^{2}\int
d\lambda_{\text{R}}\omega(m_{\text{R}}^{2},\lambda_{\text{R}})e^{im_{\text{R}}^{2}A_{m^{2}}+i\lambda_{\text{R}}A_{\lambda}}Z(m_{\text{R}}^{2},\lambda_{\text{R}}),$
(73)

where we take the renormalized couplings as the integration parameters instead
of the bare couplings for simplicity.444 The Jacobian from the change of
variables has been absorbed by the redefinition of $\omega$.  Note that
$A_{\lambda}$ is another extensive parameter proportional to the spacetime
volume $V$. For simplicity, let us restrict the parameter space to
$\lambda_{\text{R}}\geq 0$ to guarantee the stability of the system.

At the one-loop level of the $\phi^{4}$ theory, the vacuum energy (50)
vanishes at the critical point of the mass-squared $m_{\text{R}}^{2}=0$ for
any $\lambda_{\text{R}}\geq 0$, which means that the above partition function
becomes

$\displaystyle\sim\int_{0}^{\infty}d\lambda_{\text{R}}\omega(\lambda_{\text{R}})e^{i\lambda_{\text{R}}A_{\lambda}}.$
(74)

As long as $A_{\lambda}\neq 0$, the exponent is a linear function of
$\lambda_{\text{R}}$, and this integration is strongly dominated by
$\lambda_{\text{R}}=0$ by the formula Eq. (88) in the large volume limit.
Thus, the free scalar theory seems to be realized in the generalized QFT in
the $\phi^{4}$ theory at least at one-loop level. As long as the renormalized
mass $m_{\text{R}}^{2}$ is fixed at zero and we focus on the cut-off
independent part of the vacuum energy, this conclusion will not be
significantly altered by the higher-loop effects because the cut-off
independent part should be proportional to $(m_{\text{R}}^{2})^{2}$ by
dimensional analysis and they vanish at the critical point
$m_{\text{R}}^{2}=0$.

On the other hand, non-trivial saddle point can appear if we also include the
cut-off dependent parts. For example, at the three-loop level, we have the
following contributions to the vacuum energy:

$\displaystyle\lambda_{\rm R}\frac{A_{\lambda}}{V_{4}}+(c_{1}\lambda_{\rm
R}-c_{2}\lambda_{\rm R}^{2})\Lambda^{4}~{},$ (75)

where $c_{i}>0$ are just constants containing the loop suppression factors.
Thus, one can see that there exists a saddle point at

$\displaystyle\lambda_{\text{R}}\sim\left(\frac{A_{\lambda}}{V_{4}}+c_{1}\right)\bigg{/}c_{2}~{}.$
(76)

This is a very interesting possibility but its physical meaning is subtle as
in the mass parameter case. More detailed studies are left for future
investigations. See also Appendix B for the possibility of fixing
$\lambda_{\text{R}}$ in the large $N$ limit.

##  6 Dimensional transmutation in two-real-scalar model

As a next non-trivial example, we study a two-real-scalar model [32, 33, 34,
35, 36, 37] at the one-loop level. We show that an automatic tuning of the
mass-squared parameter is realized so that the dimensional transmutation is
successfully achieved.

For simplicity, we again impose the $\mathbb{Z}_{2}\times\mathbb{Z}_{2}$
symmetry, which leads to the following Lagrangian

$\displaystyle{\cal L}=-\frac{1}{2}(\partial\phi)^{2}-\frac{1}{2}(\partial
S)^{2}-\frac{m_{\phi}^{2}}{2}\phi^{2}-\frac{m_{S}^{2}}{2}S^{2}-\frac{\lambda_{\phi
S}}{4}\phi^{2}S^{2}-\frac{\lambda_{\phi}}{4!}\phi^{4}-\frac{\lambda_{S}}{4!}S^{4}.$
(77)

To simplify the discussion further, we focus on the parameter space
$\lambda_{\phi}\ll\lambda_{S}\sim 1$ and $\lambda_{\phi S}>0$, which means
that the $\phi$ direction is almost flat for $m_{\phi}^{2}\sim 0$. $\phi$ and
$S$ play the roles of the scalar and gauge fields in the original Coleman-
Weinberg mechanism. On the other hand, the $S$ direction is always dominated
by the tree-level potential, that is, its VEV is well determined by
$m_{S}^{2}S^{2}+\frac{\lambda_{S}}{4!}S^{4}$.

Under these conditions, we will show that there is a critical point at
$m_{S}^{2}=0$ and $m_{\phi}^{2}\neq 0$ which corresponds to a quantum first-
order phase transition point. Although the conventional classical conformal
point $m_{S}^{2}=m_{\phi}^{2}=0$ is not realized in the current formulation,
our result serves as a concrete way of dimensional transmutation without
assuming an artificial symmetry such as classical conformality.

Now let us discuss the details. The one-loop effective potential in the
$\overline{\rm MS}$ scheme is given by

$\displaystyle V_{\rm eff}(\phi,S)=\frac{m_{\phi}^{2}}{2}\phi^{2}$
$\displaystyle+\frac{m_{S}^{2}}{2}S^{2}+\frac{\lambda_{\phi
S}}{4}\phi^{2}S^{2}+\frac{\lambda_{\phi}}{4!}\phi^{4}+\frac{\lambda_{S}}{4!}S^{4}$
$\displaystyle+\frac{(M_{+}^{2}(\phi,S))^{2}}{64\pi^{2}}\log\left(\frac{M_{+}^{2}(\phi,S)}{\mu^{2}e^{3/2}}\right)+\frac{(M_{-}^{2}(\phi,S))^{2}}{64\pi^{2}}\log\left(\frac{M_{-}^{2}(\phi,S)}{\mu^{2}e^{3/2}}\right),$
(78)

where

$\displaystyle
M_{\pm}^{2}(\phi,S)=\frac{1}{2}\left(M_{\phi}^{2}+M_{S}^{2}\pm\sqrt{(M_{\phi}^{2}-M_{S}^{2})^{2}+4\lambda_{\phi
S}^{2}\phi^{2}S^{2}}\right)$ (79)

with

$\displaystyle M_{\phi}^{2}=m_{\phi}^{2}+\frac{\lambda_{\phi
S}}{2}S^{2}+\frac{\lambda_{\phi}}{2}\phi^{2},\quad
M_{S}^{2}=m_{S}^{2}+\frac{\lambda_{\phi
S}}{2}\phi^{2}+\frac{\lambda_{S}}{2}S^{2}.$ (80)

Here, all the coupling constants are the renormalized ones. As usual, we can
take the renormalization scale at the point $\mu=M$ where $\lambda_{\phi}$
vanishes. In the following, we first examine how $m_{\phi}^{2}$ is fixed for a
given value of $m_{S}^{2}$ and then discuss the fixing of $m_{S}^{2}$.

First, let us consider the parameter space $m_{S}^{2}\geq 0$. In this case,
$\langle S\rangle=0$ as long as the tree-level potential dominates in the $S$
direction. The effective potential for $\phi$ is then

$\displaystyle V_{\rm eff}(0,\phi)$
$\displaystyle=\frac{(m_{\phi}^{2})^{2}}{64\pi^{2}}\log\left(\frac{m_{\phi}^{2}}{M^{2}e^{3/2}}\right)+\frac{m_{\phi}^{2}}{2}\phi^{2}+\frac{(m_{S}^{2}+\frac{\lambda_{\phi
S}}{2}\phi^{2})^{2}}{64\pi^{2}}\log\left(\frac{m_{S}^{2}+\frac{\lambda_{\phi
S}}{2}\phi^{2}}{M^{2}e^{3/2}}\right)~{}.$ (81)

In Fig. 3, we show the plots of $V_{\rm eff}(0,\phi)$ where the different
colors correspond to the different values of $m_{\phi}^{2}$.

Figure 3: One loop effective potential of $\tilde{\phi}$. Different colors
correspond to the different values of $m_{\phi}^{2}$.

In particular, one can see that there exists a quantum first-order phase
transition at

$\displaystyle m_{\phi}^{2}=\frac{\lambda_{\phi
S}e^{1/2}}{64\pi^{2}}M^{2}=:m_{c}^{2}~{},$ (82)

meaning that $\phi$ achieves a nonzero VEV, $\langle\phi\rangle=v_{\phi}$, for
$m_{\phi}^{2}<m_{c}^{2}$. See Appendix C for the derivation of Eq. (82).
Correspondingly, the VEV and the vacuum energy are given by

$\displaystyle\langle\phi\rangle$ $\displaystyle=\begin{cases}0&{\rm
for}~{}m_{\phi}^{2}\geq m_{c}^{2}\\\ v_{\phi}&{\rm
for}~{}m_{\phi}^{2}<m_{c}^{2}\end{cases}~{},$ (83) $\displaystyle V_{\rm
eff}(0,\langle\phi\rangle)-\frac{(m_{\phi}^{2})^{2}}{64\pi^{2}}\log\left(\frac{m_{\phi}^{2}}{M^{2}e^{3/2}}\right)$
$\displaystyle=\begin{cases}\frac{(m_{S}^{2})^{2}}{64\pi^{2}}\log\left(\frac{m_{S}^{2}}{M^{2}e^{3/2}}\right)&{\rm
for}~{}m_{\phi}^{2}\geq m_{c}^{2}\\\
\frac{m_{\phi}^{2}}{2}v_{\phi}^{2}+\frac{(m_{S}^{2}+\frac{\lambda_{\phi
S}}{2}v_{\phi}^{2})^{2}}{64\pi^{2}}\log\left(\frac{m_{S}^{2}+\frac{\lambda_{\phi
S}}{2}v_{\phi}^{2}}{M^{2}e^{3/2}}\right)&{\rm
for}~{}m_{\phi}^{2}<m_{c}^{2}\end{cases}~{}.$ (84)

As in the simple $\phi^{4}$ theory, the additional contribution in the second
line gives the discontinuity of the second derivative of the vacuum energy at
$m_{\phi}^{2}=m_{c}^{2}$, and this point is dominant in the generalized
partition function.

Second, we turn to the case $m_{S}^{2}<0$. In this case, $S$ already has a VEV
at $\langle S\rangle^{2}=-6m_{S}^{2}/\lambda_{S}$ at tree level, and there
exists the tree-level vacuum energy $-3(m_{S}^{2})^{2}/(2\lambda_{S})$ as in
the simple $\phi^{4}$ theory discussed in the previous sections. As for the
$\phi$ potential, nonzero VEV of $S$ just changes the effective mass in Eq.
(80) as

$\displaystyle M_{S}^{2}=m_{S}^{2}+\frac{\lambda_{S}}{2}\langle
S\rangle^{2}+\frac{\lambda_{\phi
S}}{2}\phi^{2}=2|m_{S}^{2}|+\frac{\lambda_{\phi S}}{2}\phi^{2}.$ (85)

Note that $M_{\phi}^{2}$ is just a constant because of $\lambda_{\phi}=0$.
Thus, we can repeat the same discussion as above and conclude that
$m_{\phi}^{2}$ is still fixed at the critical point $m_{\phi}^{2}=m_{c}^{2}$.

Finally, at such a critical point of $m_{\phi}^{2}$, the vacuum energy as a
function of $m_{S}^{2}$ is given by

$\displaystyle V_{\rm
min}\bigg{|}_{m_{\phi}^{2}=m_{c}^{2}}\simeq\begin{cases}\frac{(m_{S}^{2})^{2}}{64\pi^{2}}\log\left(\frac{m_{S}^{2}}{M^{2}e^{3/2}}\right)&\text{for
$m_{S}\geq 0$}\\\
-\frac{3(m_{S}^{2})^{2}}{2\lambda_{S}}+\frac{(m_{S}^{2}+\frac{\lambda_{\phi
S}}{2}v_{\phi}^{2})^{2}}{64\pi^{2}}\log\left(\frac{m_{S}^{2}+\frac{\lambda_{\phi
S}}{2}v_{\phi}^{2}}{M^{2}e^{3/2}}\right)&\text{for $m_{S}<0$}\end{cases}~{}.$
(86)

As in the simple $\phi^{4}$ theory, $m_{S}^{2}$ is fixed at zero because the
second derivative of this equation is discontinuous at $m_{S}^{2}=0$. The
conventional Coleman-Weinberg point $m_{S}^{2}=m_{\phi}^{2}=0$ is not realized
in the current formation because such a point $m_{S}^{2}=m_{\phi}^{2}=0$
corresponds to the degeneracy of false vacua as discussed in Ref. [35]. We
leave the discussion on the criticality of general extrema for future
investigation.

##  7 Summary

In this paper, we have investigated the Higgs hierarchy problem in the
generalized QFT. We first studied the free scalar theory and found that there
are two critical points in the large volume limit: One is $m_{\text{B}}^{2}=0$
and the other is $m_{\text{B}}^{2}={\cal O}(\Lambda^{2})$. While the former
does not depend on the regularization methods, the latter does, implying that
$m_{\text{B}}^{2}=0$ is a physically reasonable critical point. This can be a
theoretical origin/explanation of classical conformality which is implicitly
assumed in many literatures.

We then studied the $\phi^{4}$ theory at the one-loop level, as well as the
large $N$ model. In this case, we found that a critical point exists at the
point where the renormalized mass $m^{2}=m_{\text{B}}^{2}+\delta m_{\rm
UV}^{2}$ vanishes due to the discontinuity of the vacuum energy’s derivative.
This further backs up the fine-tuning of the (Higgs) mass-squared being
automatically accomplished in the generalized QFT.

We have also discussed the possibility that the quartic coupling is
automatically fixed, with a positive result.

As a next non-trivial example, we examined the $\mathbb{Z}_{2}$ invariant two-
real-scalar model at the one-loop level by focussing on a simple parameter
space where only one real scalar $\phi$ can develop a nonzero VEV
$\langle\phi\rangle\neq 0$. Under these conditions, we found that there exists
a critical point where $m_{\phi}^{2}=m_{c}^{2}>0$ and $m_{S}^{2}=0$, which
corresponds to a quantum first-order phase transition point of the theory.
This result gives a complete realization of dimensional transmutation without
assuming classical conformality which has been implicitly assumed in many
literatures.

## Acknowledgements

The work of KK is supported by KIAS Individual Grants, Grant No. 090901 The
work of KO is partly supported by the JSPS Kakenhi Grant Nos. 19H01899 and
21H01107. The work of KY is supported in part by the Grant-in-Aid for Early-
Career Scientists, No. 19K14714. H.K. thanks Prof. Shin-Nan Yang and his
family for their kind support through the Chin-Yu chair professorship. H.K. is
partially supported by JSPS (Grants-in-Aid for Scientific Research Grants
No.20K03970), by the Ministry of Science and Technology, R.O.C. (MOST
111-2811-M-002-016), and by National Taiwan University

## Appendix

##  Appendix A Mathematical formulas

In the following, we often use mathematical formulas of generalized functions.
We summarize them here. The first one is well known as the saddle point
approximation:

$\displaystyle\lim_{V\rightarrow\infty}e^{-Vg(x)}\simeq
e^{-Vg(x_{0})}\sqrt{\frac{2\pi}{Vg^{\prime\prime}(x_{0})}}\times\delta(x).$
(87)

where $g(x)$ is a smooth function and $x_{0}$ is a saddle point,
$g^{\prime}(x_{0})=0$. The proof of this equation is textbook level and we
will not repeat it here. The second formula is

$\displaystyle\lim_{V\rightarrow\infty}e^{iVg(x)}\theta(x)\simeq\frac{i}{V}\left|\frac{dg}{dx}\right|^{-1}e^{iVg(0)}\times\delta(x),$
(88)

where $g(x)$ is now a real, smooth and monotonic function for $x\geq 0$ and
satisfies $g^{\prime}(0)\neq 0$. The proof is as follows. By multiplying a
test function $f(x)$ with a finite support and integrating from $0$ to
$\infty$, we have

$\displaystyle\int_{0}^{\infty}dxe^{iVg(x)}f(x)$
$\displaystyle=\int_{g(0)}^{g(\infty)}dg\left|\frac{dg}{dx}\right|^{-1}e^{iVg}f(x=x(g))$
(89)
$\displaystyle=\left[\frac{e^{iVg}}{iV}\left|\frac{dg}{dx}\right|^{-1}f(x(g))\right]_{g(0)}^{g(\infty)}-\frac{1}{iV}\int_{g(0)}^{g(\infty)}dg\frac{d}{dg}\left(\left|\frac{dg}{dx}\right|^{-1}f(x(g))\right)e^{iVg}$
$\displaystyle=i\frac{e^{iVg(0)}}{V}\left|\frac{dg}{dx}\right|^{-1}f(x)\bigg{|}_{x=0}-\frac{1}{(iV)^{2}}\left[\frac{d}{dg}\left(\left|\frac{dg}{dx}\right|^{-1}f(x(g))\right)e^{iVg}\right]_{g(0)}^{g(\infty)}+\cdots$
$\displaystyle=i\frac{e^{iVg(0)}}{V}\left|\frac{dg}{dx}\right|^{-1}f(x)\bigg{|}_{x=0}+{\cal
O}(V^{-2}),$ (90)

which confirms Eq. (88). Note that the contribution from $x=\infty$ is zero
because we have assumed that $f(x)$ has a finite support. More generally, when
$g(x)$ is a monotonic and smooth function for both sides $x>0$ and $x<0$, but
its derivative $g^{\prime}(x)$ is discontinuous at $x=0$, we have

$\displaystyle\lim_{V\rightarrow\infty}e^{iVg(x)}=\lim_{V\rightarrow\infty}\frac{i}{V}\left[\left|\frac{dg}{dx}\right|^{-1}_{0+}-\left|\frac{dg}{dx}\right|^{-1}_{0-}\right]e^{iVg(0)}\times\delta(x).$
(91)

Note that when $g^{\prime}(x)$ is also continuous at $x=0$, the right-hand
side vanishes and higher order terms dominate. In general, when the
derivatives of $g(x)$ are continuous and nonzero up to $(n-1)$-th order but
the $n$-th derivative of $g(x)$ is discontinuous, we have

$\displaystyle\lim_{V\rightarrow\infty}e^{iVg(x)}=c\lim_{V\rightarrow\infty}\frac{i^{n}}{V^{n}}e^{iVg(0)}\times\delta(x),$
(92)

where $c$ is a coefficient determined by
$g^{\prime}(0),g^{\prime\prime}(0),\cdots,d^{(n-1)}g(0)/dx^{n-1},dg^{(n)}(0+)/dx^{n},dg^{(n)}(0-)/dx^{n}$.

##  Appendix B Fixing quartic coupling in large $N$ model

In this Appendix, we study a possibility of fixing the quartic coupling by
taking the $N^{-1}$ corrections. In the following, the mass term $m^{2}$ is
fixed at the critical point $m^{2}=m_{\text{B}}^{2}+c_{\rm cl}=0$ and we focus
on $d=4$.

Around the large $N$ solution $(c,s)=(c_{\rm cl},s_{\rm cl})$ studied in
Section 4.2, the effective action in Eq. (61) becomes

$\displaystyle
S=\frac{N\Lambda^{4}V_{4}}{4\tilde{\lambda}_{\text{B}}}\left(\frac{\tilde{\lambda}_{\text{B}}}{16\pi^{2}}\Lambda^{2}\right)^{2}+N\int
d^{4}x\left(-\frac{1}{2}(\partial_{\mu}\delta s)^{2}-\frac{1}{2}\delta c\delta
s^{2}+\frac{1}{4}\left(\frac{1}{\tilde{\lambda}_{\text{B}}}+\int\frac{d^{4}p_{\text{E}}}{(2\pi)^{4}}\frac{1}{(p_{\text{E}}^{2})^{2}}\right)\delta
c^{2}+\cdots\right),$ (93)

where $\delta s=s-s_{\rm cl},~{}\delta c=c-c_{\rm cl}$ and $\cdots$ represents
the higher order terms. Note that the first term is the leading vacuum energy
contribution and it is proportional to $\tilde{\lambda}_{\text{B}}$. Thus,
this can be absorbed into the definition of $A_{\lambda}$. In the following,
we represent

$\displaystyle\frac{1}{\tilde{\lambda}_{\text{B}}}+\int\frac{d^{4}p_{\text{E}}}{(2\pi)^{4}}\frac{1}{(p_{\text{E}}^{2})^{2}}=\frac{1}{\tilde{\lambda}_{\text{B}}}+\frac{1}{8\pi^{2}}\log\left(\frac{\Lambda}{\mu}\right):=\frac{1}{N\lambda(\mu)},$
(94)

where $\mu$ is some (IR) scale. We can check that $\lambda(\mu)$ corresponds
to the usual renormalized quartic coupling because Eq. (94) is nothing but the
solution of RGE in the large $N$ limit:

$\displaystyle\frac{d\lambda}{d\log\mu}\approx\frac{N}{8\pi^{2}}\lambda^{2}.$
(95)

In Eq. (93), $\delta c$ part can be rewritten as

$\displaystyle\frac{1}{4N\lambda(\mu)}\left(\delta c-N\lambda(\mu)\delta
s^{2}\right)^{2}-\frac{N\lambda(\mu)}{4}\delta s^{4}.$ (96)

By the field redefinitions $N^{1/2}\delta s\rightarrow\delta s$ and
$N^{1/2}(\delta c-N\lambda(\mu)\delta s^{2})\rightarrow\delta c$, the action
becomes

$\displaystyle S=\int d^{4}x\left(-\frac{1}{2}(\partial_{\mu}\delta
s)^{2}-\frac{\lambda(\mu)}{4}\delta s^{4}+\frac{1}{4N\lambda(\mu)}\delta
c^{2}+{\cal O}(N^{-1})\right),$ (97)

Now we can perform the one-loop integration of $\delta c$ as

$\displaystyle-\frac{V_{4}}{2}\int\frac{d^{4}q_{\text{E}}}{(2\pi)^{4}}\log(N\lambda(\mu)/2)^{-1}=-\frac{F}{2}\log\left(\frac{2}{N\lambda(\mu)}\right)=-\frac{F}{2}\log\left(\frac{1}{\lambda_{\text{B}}}+\frac{N}{8\pi^{2}}\log\left(\frac{\Lambda}{\mu}\right)\right)+{\rm
const},$ (98)

where $F=V_{4}\Lambda^{4}/(32\pi^{2})$. Note that we have singular points at

$\displaystyle\lambda(\mu)=0,\quad\pm\infty.$ (99)

Now the $\lambda_{\text{B}}$ integration in the micro-canonical partition
function is given by

$\displaystyle\Omega(A)$
$\displaystyle=\int_{-\infty}^{\infty}\frac{d\lambda_{\text{B}}}{2\pi}\exp\left(i\lambda_{\text{B}}NA_{\lambda}-i\frac{F}{2}\log\left(\frac{1}{\lambda_{\text{B}}}+\frac{N}{8\pi^{2}}\log\left(\frac{\Lambda}{\mu}\right)\right)\right)$
(100)
$\displaystyle=\int_{-\infty}^{\infty}\frac{d\lambda_{\text{B}}}{2\pi}e^{iNA_{\lambda}G(\lambda(\mu))},$
(101)

where

$\displaystyle
G(\lambda(\mu)):=\frac{\lambda(\mu)}{1-\frac{N\lambda(\mu)}{8\pi^{2}}\log\left(\frac{\Lambda}{\mu}\right)}+\frac{F}{2NA_{\lambda}}\log\lambda(\mu)~{}.$
(102)

One can check that there exist two saddle points:

$\displaystyle
N\lambda(\mu)=N\lambda_{\pm}=\frac{1}{B}\left(1-\frac{A_{\lambda}}{FB}\pm\sqrt{\left(1-\frac{A_{\lambda}}{FB}\right)^{2}-1}\right),$
(103)

where

$\displaystyle B=\frac{1}{8\pi^{2}}\log\left(\frac{\Lambda}{\mu}\right).$
(104)

We can see that $N\lambda_{\pm}$ is real and positive if we take the following
large $N$ limit.

$\displaystyle N\rightarrow\infty~{}\quad N\lambda_{\text{R}}(\mu)={\rm
fixed},\quad\frac{A_{\lambda}}{F}={\rm
fixed},\quad\bigg{|}1-\frac{A_{\lambda}}{FB}\bigg{|}\geq 1~{}.$ (105)

More detailed analysis is necessary to verify the validity of this limit
because we now have additional parameter $A_{\lambda}$ that is absent in the
usual large $N$ analysis.

##  Appendix C Details of two-real-scalar model

By putting $\tilde{\phi}^{2}=(m_{S}^{2}+\lambda_{\phi
S}\phi^{2}/2)/M^{2}e^{3/2}$, we can rewrite Eq. (81) as

$\displaystyle V_{\rm eff}(0,\phi)$
$\displaystyle=\Lambda(m_{\phi}^{2},m_{S}^{2})+\tilde{V}_{\phi}(\tilde{\phi}),$
(106)

where

$\displaystyle\Lambda(m_{\phi}^{2},m_{S}^{2})$
$\displaystyle:=\frac{(m_{\phi}^{2})^{2}}{64\pi^{2}}\log\left(\frac{m_{\phi}^{2}}{\mu^{2}e^{3/2}}\right)-\frac{m_{S}^{2}m_{\phi}^{2}}{\lambda_{\phi
S}},$ (107) $\displaystyle\tilde{V}_{\phi}(\tilde{\phi})$
$\displaystyle:=(M^{2}e^{3/2})^{2}\left(\frac{\tilde{m}_{\phi}^{2}}{2}\tilde{\phi}^{2}+\frac{\tilde{\phi}^{4}}{64\pi^{2}}\log\tilde{\phi}^{2}\right),$
(108)

in which

$\displaystyle\tilde{m}_{\phi}^{2}$
$\displaystyle:=\frac{2m_{\phi}^{2}}{\lambda_{\phi S}M^{2}e^{3/2}}.$ (109)

The effective potential has minima at $\phi=0$and

$\displaystyle\tilde{\phi}=\tilde{v}_{\phi}~{},$ (110)

where $\tilde{v}_{\phi}^{2}$ is a solution of

$\displaystyle\tilde{m}_{\phi}^{2}+\frac{\tilde{\phi}^{2}}{16\pi^{2}}\log(\tilde{\phi}^{2}e^{1/2})=0.$
(111)

Correspondingly, $v_{\phi}$ will denote the VEV of the original field $\phi$
below.

Note that $\langle\phi\rangle=0$ is always the true vacuum for
$m_{S}^{2}>\frac{\lambda_{\phi S}}{2}v_{\phi}^{2}$ because
$\tilde{\phi}^{2}>\tilde{v}_{\phi}^{2}$ for any values of $\phi^{2}>0$. On the
other hand, the minimum of $\tilde{V}_{\phi}(\tilde{\phi})$ depends on
$m_{\phi}^{2}$ for $\frac{\lambda_{\phi S}}{2}v_{\phi}^{2}\geq m_{S}^{2}\geq
0$. In particular, it has a critical point at

$\displaystyle
0=\tilde{V}_{\phi}(0)=\tilde{V}_{\phi}(\tilde{v}_{\phi})\quad\rightarrow\quad\tilde{m}_{\phi}^{2}=\frac{1}{32\pi^{2}e}=:\tilde{m}_{c}^{2}~{}.$
(112)

See Fig. 3 for the explicit plots of $\tilde{V}_{\phi}(\tilde{\phi})$.

As a consistency check of $\langle S\rangle=0$ for $m_{S}^{2}\geq 0$, let us
also check the positivity of the effective mass of $S$ at $S=0$:

$\displaystyle m_{{\rm eff},S}^{2}$ $\displaystyle=\frac{\partial V_{\rm
eff}}{\partial S^{2}}\bigg{|}_{S=0,\phi=\langle\phi\rangle}$ (113)
$\displaystyle=m_{S}^{2}+\frac{\lambda_{\phi
S}}{2}\langle\phi\rangle^{2}+\frac{\lambda_{S}}{32\pi^{2}}\left(m_{S}^{2}+\frac{\lambda_{\phi
S}}{2}\langle\phi\rangle^{2}\right)\log\left(\frac{m_{S}^{2}+\frac{\lambda_{\phi
S}}{2}\langle\phi\rangle^{2}}{M^{2}e}\right)+\frac{\lambda_{\phi
S}m_{\phi}^{2}}{32\pi^{2}}\log\left(\frac{m_{\phi}^{2}}{M^{2}e}\right).$ (114)

The last term is negligible at around the critical point
$m_{\phi}^{2}=m_{c}^{2}$ because

$\displaystyle\frac{\lambda_{\phi
S}m_{\phi}^{2}}{32\pi^{2}}\log\left(\frac{m_{\phi}^{2}}{M^{2}e}\right)\sim-\frac{\lambda_{\phi
S}^{2}}{(32\pi)^{2}}\langle\phi\rangle^{2}\ll\frac{\lambda_{\phi
S}^{2}}{2}\langle\phi\rangle^{2}.$ (115)

On the other hand, the second term in Eq. (114) seems to become negative at
around $m_{S}^{2}\sim 0$ for $\langle\phi\rangle=0$, but it is just an
illusion of looking at the first term of leading log corrections. By summing
up all the leading log terms (which corresponds to the RG improvement), we
have

$\displaystyle
1+\frac{\lambda_{S}}{32\pi^{2}}\log\left(\frac{m_{S}^{2}}{M^{2}e}\right)+\left(\frac{\lambda_{S}}{32\pi^{2}}\log\left(\frac{m_{S}^{2}}{M^{2}e}\right)\right)^{2}+\cdots=\frac{1}{1-\frac{\lambda_{S}}{32\pi^{2}}\log\left(\frac{m_{S}^{2}}{M^{2}e}\right)},$
(116)

which is always positive for $m_{S}^{2}\geq 0$.555This is nothing but the
well-known fact [31] that the Coleman-Weinberg mechanism does not work for a
single (real) scalar. Thus, $\langle S\rangle=0$ is justified at this one-loop
level calculations and $m_{\phi}^{2}$ is fixed at the critical point
$m_{c}^{2}$ by the discontinuity of the vacuum energy.

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