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# Ladder top-quark condensation imprints in supercooled electroweak phase
transition

Yuepeng Guan  Center for Theoretical Physics and College of Physics, Jilin
University, Changchun, 130012, China  Shinya Matsuzaki  Center for Theoretical
Physics and College of Physics, Jilin University, Changchun, 130012, China

###### Abstract

The electroweak (EW) phase transition in the early Universe might be
supercooled due to the presence of the classical scale invariance involving
Beyond the Standard Model (BSM) sectors and the supercooling could persist
down till a later epoch around which the QCD chiral phase transition is
supposed to take place. Since this supercooling period keeps masslessness for
all the six SM quarks, it has simply been argued that the QCD phase transition
is the first order, and so is the EW one. However, not only the QCD coupling
but also the top Yukawa and the Higgs quartic couplings get strong at around
the QCD scale due to the renormalization group running, hence this scenario is
potentially subject to a rigorous nonperturbative analysis. In this work, we
employ the ladder Schwinger-Dyson (LSD) analysis based on the Cornwall-Jackiw-
Tomboulis formalism at the two-loop level in such a gauge-Higgs-Yukawa system.
We show that the chiral broken QCD vacuum emerges with the nonperturbative top
condensate and the lightness of all six quarks is guaranteed due to the
accidental U(1) axial symmetry presented in the top-Higgs sector. We employ a
quark-meson model-like description in the mean field approximation to address
the impact on the EW phase transition arising due to the top quark
condensation at the QCD phase transition epoch. In the model, the LSD results
are encoded to constrain the model parameter space. We then observe the
cosmological phase transition of the first-order type and discuss the induced
gravitational wave (GW) productions. We find that in addition to the
conventional GW signals sourced from an expected BSM at around or over the TeV
scale, the dynamical topponium-Higgs system can yield another power spectrum
sensitive to the BBO, LISA, and DECIGO, etc.

##  I Introduction

The QCD phase transition is of particular importance to understand the origin
of mass and property for matter, i.e., nucleon. It has extensively been
explored so far based on lattice simulations as well as low-energy effective
field theories. However, in application to cosmology, the QCD phase transition
in the Hubble expanding universe cannot be observed in lattice QCD or
experiments today. Therefore, actually, we have little understood the QCD
cosmology in the sense of the thermal history of the universe.

The recent and prospected observations of nano hertz gravitational waves (GWs)
have the potential sensitivity to deeply clarify the thermal history around
the QCD phase transition epoch. The observation of a stochastic GW background
has been reported from the NANOGrav pulsar timing array (PTA) collaboration in
15 years of data NANOGrav:2023gor ; NANOGrav:2023hfp . Possible origins of the
detected nano Hz peak frequency in a view of the BSM have been investigated
also by the NANOGrav collaboration NANOGrav:2023hvm . Other recent PTA data,
such as those from the European PTA (EPTA)EPTA:2023sfo ; EPTA:2023akd ;
EPTA:2023fyk , Parkes PTA (PPTA)Reardon:2023gzh ; Reardon:2023zen , and
Chinese PTA (CPTA) Xu:2023wog  have also supported the presence of consistent
nano-hertz stochastic GWs. The tail of the GW spectra produced at the QCD
phase transition epoch still keeps having a high enough sensitivity also at
other interferometers designated aiming at the higher frequency spectra, like
the Laser Interferometer Space Antenna (LISA) LISA:2017pwj ; Caprini:2019egz ,
the Big Bang Observer (BBO) Corbin:2005ny ; Harry:2006fi , and Deci-hertz
Interferometer Gravitational Wave Observatory (DECIGO) Kawamura:2006up ;
Yagi:2011wg , etc. Thus, such GW evidence is anticipated to provide us with a
clue on the new aspect of the QCD cosmology in the thermal history of the
universe.

The QCD phase transition has been confirmed to be crossover by the lattice QCD
with 2 + 1 flavors at the physical point at the pseudocritical temperature
$T_{\rm pc}\sim 155$ MeV Aoki:2009sc ; Borsanyi:2011bn ; Ding:2015ona ;
HotQCD:2018pds ; Ding:2020rtq  for the chiral phase transition, and at the
same time the deconfinement-confinement transition is expected to take place
as well Bazavov:2016uvm ; Ding:2017giu . This is the current consensus of the
QCD phase transition to our best knowledge. In the Hubble evolutionary
universe with BSM, however, the cosmological QCD phase transition might be
more involved and richer.

One particular interest is in when the SM is extended to be scale-invariant
along with a dark sector, subsequent supercooling including the QCD phase
transition can be realized Iso:2017uuu ; Hambye:2018qjv ; vonHarling:2017yew .
In this class of the scenario context, the QCD phase transition also plays an
important role in triggering the electroweak (EW) phase transition, which can
be strong first-order at temperatures $T={\cal O}(T_{\rm pc})$, thus the EW
phase transition can be supercooled until then. In the literature several
phenomenological and cosmological consequences arising from this non-standard
QCD-scale cosmology have been discussed including the impact on the
baryogenesis Dichtl:2023xqd ; Ellis:2022lft ; Wang:2022ygk  and the
gravitational wave detection sensitivities mainly sourced from the scale-
invariant dark sector Iso:2017uuu ; Hambye:2018qjv ; Sagunski:2023ynd ;
Dichtl:2023xqd ; Frandsen:2022klh ; Ellis:2022lft ; Bodeker:2021mcj ;
vonHarling:2017yew

However, the quantum field theory in the thus supercooled epoch may not merely
be governed by QCD: the top Yukawa and the Higgs quartic couplings as well as
the QCD gauge coupling get strong at around the QCD scale due to the
renormalization group running, as plotted in Fig. 1. Hence this scenario is
potentially subject to a rigorous nonperturbative analysis, which has never
been addressed at this point.

In this paper, we make the first attempt to employ the nonperturbative
analysis of the QCD-induced EW phase transition scenario. The method that we
apply is the ladder Schwinger-Dyson (LSD) equation based on the Cornwall-
Jackiw-Tomboulis (CJT) formalism Cornwall:1974vz  at the two-loop level.
Working on the gauge-Higgs-Yukawa system including QCD, the Higgs-Yukawa, and
the Higgs quartic interactions, we show that the chiral broken QCD vacuum
emerges with the nonperturbative top condensate and the lightness of all six
quarks is guaranteed. The latter turns out to be due to the accidental U(1)
axial symmetry (approximately) present in the top-Higgs Yukawa sector (with
small enough EW gauge interactions neglected).

We discuss the impact on the EW phase transition in the thermal history
arising from the emergence of the top quark condensation in the QCD phase
transition epoch. We monitor the low-energy effective theory by a quark-meson
model-like description, for the top - Higgs hybrid system. The nonperturbative
scale anomaly induced from the dynamical top mass generation based on the LSD
analysis is encoded in the model by the anomaly matching procedure, to
constrain the model parameter space. We then observe the cosmological phase
transition of the first-order type in the mean field approximation and discuss
the associated GW productions. We find that in addition to the conventional GW
signals sourced from an expected BSM at around or over the TeV scale, the
dynamical topponium-Higgs system can yield another power spectrum sensitive to
the BBO, LISA, and DECIGO, etc.

This paper is structured as follows. In Sec. II we start with deriving the SD
equations in the gauge-Higgs-Yukawa system relevant to the presently concerned
supercooling scalegenesis, and introduce the ladder approximation. In Sec.
III, the LSD equations are reproduced in the framework of the CJT formalism at
the two-loop level, which also provides the nonperturbative scale anomaly
induced from the nontrivial solutions to the LSD equations including the top
and Higgs condensates. Then the phase structure with the scale-invariant SM
prediction is discussed in detail. In Sec. IV, we model a low-energy
description of the present gauge-Higgs-Yukawa theory to be a quark-meson
model-like, to discuss the top quark condensation on the effect on the
supercooled EW phase in the thermal history. Employing the mean-field
approximation, we evaluate the associated cosmological phase transition in the
top - Higgs hybrid system and explore the GW production and signals. The
summary of the present paper is presented in Section V, along with discussions
related to issues to be pursed in the future.

Figure 1: The perturbative two-loop running in the classically scale-invariant
SM for the Yukawa couplings (denoted as $y_{f}$ with $f$ being fermion flavor
label), the Higgs quartic coupling ($\lambda_{\phi}$), and the QCD gauge
coupling $g_{s}$. Here inputs have been made of $m_{h}=125.25$ GeV (the Higgs
mass); $v_{\text{EW}}=246$ GeV (the EW scale); $m_{Z}=91.1876$ GeV (the Z
boson mass); $\alpha_{s}(m_{Z})(\equiv g_{s}^{2}/(4\pi))=0.1184$ (the
$\overline{\text{MS}}$ QCD coupling renormalized at the $m_{Z}$ scale);
$m_{t}=172.5$ GeV (the top quark mass); $m_{b}=4.18$ GeV and $m_{c}=1.27$ GeV
(the $\overline{\text{MS}}$ masses renormalized at their
$\overline{\text{MS}}$ masses); $m_{s}=93.4\times 10^{-3}$ GeV,
$m_{d}=4.67\times 10^{-3}$ GeV, and $m_{u}=2.16\times 10^{-3}$ GeV (the
$\overline{\text{MS}}$ masses renormalized at 2 GeV). All the inputs above
follow the Particle Data Group ParticleDataGroup:2022pth .

##  II The SD formalism in the gauge - Higgs - Yukawa system: gauge fixing in
the ladder approximation

In this section, we derive the exact SD equations in the currently concerned
gauge - Higgs - Yukawa system and discuss the ladder truncation. The gauge-
Higgs Yukawa system, which dominates the low-energy description of the
classically scale-invariant SM, is described by the following Lagrangian:

$\mathcal{L}=\mathcal{L}_{\text{gauge}}+\mathcal{L}_{\text{quarks}}+\mathcal{L}_{\text{Higgs}}+\mathcal{L}_{\text{Yukawa}},$
(1)

where

$\displaystyle\mathcal{L}_{\text{quark}}=i\bar{q_{i}}\left(\not{\partial}+ig_{s}\not{A}\right)q_{i},$
(2)
$\displaystyle\mathcal{L}_{\text{Higgs}}=(\partial_{\mu}H)^{\dagger}\partial^{\mu}H-\lambda_{\phi}(H^{\dagger}H)^{2},$
$\displaystyle\mathcal{L}_{\text{Yukawa}}=-y_{t}\bar{q_{3}}_{L}\tilde{H}t_{R}-y_{b}\bar{q_{3}}_{L}Hb_{R}+\text{h.c.}\,,$

and ${\cal L}_{\rm gauge}$ includes the QCD gauge kinetic and Lorentz-
covariant gauge fixing terms
($-\frac{1}{2\xi}(\partial_{\mu}A^{a}_{\mu})^{2}$) with the gluon fields
$A_{\mu}^{a}$ ($a=1,\cdots,8$) and the gauge fixing parameter $\xi$. In Eq.(1)
${q_{i}}_{L}$ $(i=1,2,3)$ denotes the $i$th generation-$SU(2)_{L}$ quark
doublet; $H$ denotes the Higgs doublet parameterized as
$H=\left(\begin{array}[]{cc}\phi^{+}\\\
\phi^{0}\end{array}\right)=\frac{1}{\sqrt{2}}\left(\begin{array}[]{cc}\pi_{2}-i\pi_{1}\\\
\sigma+i\pi_{3}\end{array}\right)$, and its charge conjugate field $\tilde{H}$
is defined as $\tilde{H}\equiv i\tau_{2}H^{*}$ with $\tau_{2}$ being the Pauli
matrix. The Higgs (vacuum expectation value) VEV $\sigma_{h}$ arises as
$\sigma=\sigma_{h}+\widetilde{\sigma}$, with
$\sigma_{h}\equiv\sqrt{2\braket{H^{\dagger}H}}$. Only the third-generation
quarks (top and bottom) become strongly coupled to the Higgs as well as to
QCD, as seen from the renormalization group evolution in Fig. 1, while other
lighter quarks strongly couple only to QCD. The present analysis extends the
earlier work in Kondo:1993ty  in the sense that the isospin breaking in the
Yukawa sector will be taken into account.

###  II.1 The exact SD equations

From the Lagrangian 1 the exact SD equations for the inverse-full propagators
for $t$ and $b$ quark fields, $S_{t}^{-1}$ and $S_{b}^{-1}$, are read off as
(for Feynman graphical interpretations, see also Fig. 2)

$\displaystyle iS_{\text{t}}^{-1}$
$\displaystyle=\not{p}-\frac{y_{t}}{\sqrt{2}}\sigma_{h}-i\int\frac{d^{4}q}{(2\pi)^{4}}\,\frac{-iy_{t}}{\sqrt{2}}S_{\text{t}}(q)\Gamma^{(3)}_{\bar{t}\sigma
t}(p,q)\Delta_{\sigma}(p-q)$ $\displaystyle\hskip
20.0pt-i\int\frac{d^{4}q}{(2\pi)^{4}}\,\frac{-\gamma_{5}y_{t}}{\sqrt{2}}S_{\text{t}}(q)\Gamma^{(3)}_{\bar{t}\pi_{3}t}(p,q)\Delta_{\pi_{3}}(p-q)$
$\displaystyle\hskip
20.0pt-i\int\frac{d^{4}q}{(2\pi)^{4}}\,\frac{-(y_{b}P_{R}-y_{t}P_{L})}{\sqrt{2}}S_{\text{b}}(q)\left[\Gamma^{(3)}_{\bar{b}\pi_{1}t,R}(p,q)+\Gamma^{(3)}_{\bar{b}\pi_{1}t,L}(p,q)\right]\Delta_{\pi_{1}}(p-q)$
$\displaystyle\hskip
20.0pt-i\int\frac{d^{4}q}{(2\pi)^{4}}\,\frac{-i(y_{b}P_{R}-y_{t}P_{L})}{\sqrt{2}}S_{\text{b}}(q)\left[\Gamma^{(3)}_{\bar{b}\pi_{2}t,R}(p,q)+\Gamma^{(3)}_{\bar{b}\pi_{2}t,L}(p,q)\right]\Delta_{\pi_{2}}(p-q)$
$\displaystyle\hskip
20.0pt-i\int\frac{d^{4}q}{(2\pi)^{4}}\,(-ig_{s}\gamma^{\mu}T^{a})S_{\text{t}}(q)\Gamma^{(3),\nu}_{\bar{t}At,a}(p,q)\Delta_{g,\mu\nu}(p-q)\,,$
(3)

and

$\displaystyle iS_{\text{b}}^{-1}$
$\displaystyle=\not{p}-\frac{y_{b}}{\sqrt{2}}\sigma_{h}-i\int\frac{d^{4}q}{(2\pi)^{4}}\,\frac{-iy_{b}}{\sqrt{2}}S_{\text{b}}(q)\Gamma^{(3)}_{\bar{b}\sigma
b}(p,q)\Delta_{\sigma}(p-q)$ $\displaystyle\hskip
20.0pt-i\int\frac{d^{4}q}{(2\pi)^{4}}\,\frac{\gamma_{5}y_{b}}{\sqrt{2}}S_{\text{b}}(q)\Gamma^{(3)}_{\bar{b}\pi_{3}b}(p,q)\Delta_{\pi_{3}}(p-q)$
$\displaystyle\hskip
20.0pt-i\int\frac{d^{4}q}{(2\pi)^{4}}\,\frac{-(y_{t}P_{R}-y_{b}P_{L})}{\sqrt{2}}S_{\text{t}}(q)\left[\Gamma^{(3)}_{\bar{t}\pi_{1}b,R}(p,q)+\Gamma^{(3)}_{\bar{t}\pi_{1}b,L}(p,q)\right]\Delta_{\pi_{1}}(p-q)$
$\displaystyle\hskip
20.0pt-i\int\frac{d^{4}q}{(2\pi)^{4}}\,\frac{i(y_{t}P_{R}-y_{b}P_{L})}{\sqrt{2}}S_{\text{t}}(q)\left[\Gamma^{(3)}_{\bar{t}\pi_{2}b,R}(p,q)+\Gamma^{(3)}_{\bar{t}\pi_{2}b,L}(p,q)\right]\Delta_{\pi_{2}}(p-q)$
$\displaystyle\hskip
20.0pt-i\int\frac{d^{4}q}{(2\pi)^{4}}\,(-ig_{s}\gamma^{\mu}T_{a})S_{\text{b}}(q)\Gamma^{(3),\nu}_{\bar{b}Ab,b}(p,q)\Delta_{g,\mu\nu}^{ab}(p-q)\,,$
(4)

where $\Gamma^{(3)}_{\bar{\psi_{i}}\phi\psi_{j}}(p,q)$ represent the three-
point vertex functions in the momentum space among the fermion field pair
$\bar{\psi_{i}}$, $\psi_{j}$, and the boson field $\phi$; $T_{a}$ accompanied
with the QCD vertex stands for the generator of $SU(3)$; the labels $L$ and
$R$ denote the chirality projection involved in the vertex functions, defined
like $\Gamma^{(3)}_{L/R}\equiv\Gamma^{(3)}P_{L/R}$ with
$P_{L/R}\equiv(1\mp\gamma_{5})/2$; $\Delta_{\phi,g}$ denote the full
propagators of boson fields. These equations with $y_{t}=y_{b}$ taken are
smoothly reduced to the global chiral $U(2)_{L}\times U(2)_{R}$ invariant case
which has been addressed in the literature Kondo:1993ty .

The SD equation for the Higgs VEV is evaluated at the leading order of the
Yukawa interactions, which is dominated by the top-quark contribution, to be

$\lambda_{\phi}\braket{\sigma(\sigma^{2}+\vec{\pi}^{2})}=-\frac{y_{t}}{\sqrt{2}}\braket{\bar{t}t}\,,$
(5)

where $\vec{\pi}=(\pi_{1},\pi_{2},\pi_{3})$.

$iS^{-1}$ | = | $i{S_{0}}^{-1}$ | + |  | + |
---|---|---|---|---|---|---
Figure 2: Feynman diagrams of the exact SD equations for the $t$ and $b$ quark
propagators corresponding to Eqs.(11) and (12). $S_{0}^{-1}$ in the first term
of the right-hand side stands for the tree-level inverse propagators,
$iS_{0}^{-1}(p)|_{t,b}=\not{p}-\frac{y_{t,b}}{\sqrt{2}}\sigma_{h}$. The blobs
include all possible quantum corrections in the present gauge-Higgs-Yukawa
system.

###  II.2 The ladder truncation and gauge fixing

We apply the ladder approximation for QCD, such that the gauge-fermion-fermion
vertex functions are taken to be of tree level:

$\Gamma^{(3),\nu}_{\bar{t}At,a}(p,q)=\Gamma^{(3),\nu}_{\bar{b}Ab,a}(p,q)=-ig_{s}\gamma^{\nu}T_{a}.$
(6)

The abelian-type Ward-Takahashi identity with this truncation constrains the
fermion full propagators as Miransky:1994vk

$iS_{t/b}^{-1}=A_{t/b}(-p^{2})\not{p}-B_{t/b}(-p^{2})\,,\qquad{\rm with}\qquad
A_{t/b}(-p^{2})=1\,.$ (7)

It is well-known that in the pure QCD case, this condition can be fulfilled by
taking the Landau gauge: $\xi=0$. The present case is, however, more involved,
which also includes the Yukawa and Higgs quartic interactions. First of all,
we therefore examine the consistency of the Landau gauge choice for the
present system to satisfy Eq.(7). As it will turn out right below, the
extended ladder approximation with the bare propagators for both scalar and
gauge fields in the Landau gauge is shown to be numerically consistent well
with Eq.(7).

To prove this point, we take the tree-level propagators for the scalar and
gauge fields as

$\displaystyle\Delta_{\sigma}(k)=\frac{i}{k^{2}-3\lambda_{\phi}\sigma_{h}^{2}},$
(8)
$\displaystyle\Delta_{\pi_{i}}(k)=\frac{i}{k^{2}-\lambda_{\phi}\sigma_{h}^{2}},$
$\displaystyle\Delta_{g,\mu\nu}^{ab}(k)=\frac{-i\delta^{ab}}{k^{2}}\left(g_{\mu\nu}-(1-\xi)\frac{k_{\mu}k_{\nu}}{k^{2}}\right),$

and all the relevant vertices to be bare ones:

$\displaystyle\Gamma^{(3)}_{\bar{t}\sigma
t}(p,q)=\frac{-iy_{t}}{\sqrt{2}},\quad\Gamma^{(3)}_{\bar{t}\pi_{3}t}(p,q)=\frac{-\gamma_{5}y_{t}}{\sqrt{2}},$
(9)
$\displaystyle\Gamma^{(3)}_{\bar{t}\pi_{1}b,R}(p,q)=\frac{-y_{b}P_{R}}{\sqrt{2}},\quad\Gamma^{(3)}_{\bar{t}\pi_{2}b,R}(p,q)=\frac{-iy_{b}P_{R}}{\sqrt{2}},$
$\displaystyle\Gamma^{(3)}_{\bar{t}\pi_{1}b,L}(p,q)=\frac{y_{t}P_{L}}{\sqrt{2}},\quad\Gamma^{(3)}_{\bar{t}\pi_{2}b,L}(p,q)=\frac{iy_{t}P_{L}}{\sqrt{2}},$
$\displaystyle\Gamma^{(3)}_{\bar{b}\sigma
b}(p,q)=\frac{-iy_{b}}{\sqrt{2}},\quad\Gamma^{(3)}_{\bar{b}\pi_{3}b}(p,q)=\frac{\gamma_{5}y_{b}}{\sqrt{2}},$
(10)
$\displaystyle\Gamma^{(3)}_{\bar{b}\pi_{1}t,R}(p,q)=\frac{-y_{t}P_{R}}{\sqrt{2}},\quad\Gamma^{(3)}_{\bar{b}\pi_{2}t,R}(p,q)=\frac{iy_{t}P_{R}}{\sqrt{2}},$
$\displaystyle\Gamma^{(3)}_{\bar{b}\pi_{1}t,L}(p,q)=\frac{y_{b}P_{L}}{\sqrt{2}},\quad\Gamma^{(3)}_{\bar{b}\pi_{2}t,L}(p,q)=\frac{-iy_{b}P_{L}}{\sqrt{2}}.$

With this prescription, after a tedious computation, the $A_{t/b}(p_{E}^{2})$
and $B_{t/b}(p_{E}^{2})$ functions (with $p_{E}^{2}\equiv-p^{2}$) take the
form

$\displaystyle B_{t}(p_{E}^{2})=$
$\displaystyle\frac{y_{t}}{\sqrt{2}}\sigma_{h}+\int^{\Lambda^{2}}dq_{E}^{2}\mathcal{K}_{B_{t}}(p_{E}^{2},q_{E}^{2})\frac{q_{E}^{2}B_{t}(q_{E}^{2})}{A_{t}^{2}(q_{E}^{2})q_{E}^{2}+B_{t}^{2}(q_{E}^{2})}$
(11)
$\displaystyle+\frac{y_{t}y_{b}}{16\pi^{2}}\int^{\Lambda^{2}}dq_{E}^{2}K_{B}(p_{E}^{2},q_{E}^{2};\lambda_{\phi}\sigma_{h}^{2})\frac{q_{E}^{2}B_{b}(q_{E}^{2})}{A_{b}^{2}(q_{E}^{2})q_{E}^{2}+B_{b}^{2}(q_{E}^{2})},$
$\displaystyle A_{t}(p_{E}^{2})=$ $\displaystyle
1+\int^{\Lambda^{2}}dq_{E}^{2}\mathcal{K}_{A_{t}}(p_{E}^{2},q_{E}^{2})\frac{A_{t}(q_{E}^{2})}{A_{t}^{2}(q_{E}^{2})q_{E}^{2}+B_{t}^{2}(q_{E}^{2})}$
$\displaystyle+\frac{{y_{t}}^{2}+{y_{b}}^{2}}{32\pi^{2}}\int^{\Lambda^{2}}dq_{E}^{2}K_{A}(p_{E}^{2},q_{E}^{2};\lambda_{\phi}\sigma_{h}^{2})\frac{A_{b}(q_{E}^{2})}{A_{b}^{2}(q_{E}^{2})q_{E}^{2}+B_{b}^{2}(q_{E}^{2})},$
$\displaystyle B_{b}(p_{E}^{2})=$
$\displaystyle\frac{y_{b}}{\sqrt{2}}\sigma_{h}+\int^{\Lambda^{2}}dq_{E}^{2}\mathcal{K}_{B_{b}}(p_{E}^{2},q_{E}^{2})\frac{q_{E}^{2}B_{t}(q_{E}^{2})}{A_{b}^{2}(q_{E}^{2})q_{E}^{2}+B_{b}^{2}(q_{E}^{2})}$
(12)
$\displaystyle+\frac{y_{t}y_{b}}{16\pi^{2}}\int^{\Lambda^{2}}dq_{E}^{2}K_{B}(p_{E}^{2},q_{E}^{2};\lambda_{\phi}\sigma_{h}^{2})\frac{q_{E}^{2}B_{t}(q_{E}^{2})}{A_{t}^{2}(q_{E}^{2})q_{E}^{2}+B_{t}^{2}(q_{E}^{2})},$
$\displaystyle A_{b}(p_{E}^{2})=$ $\displaystyle
1+\int^{\Lambda^{2}}dq_{E}^{2}\mathcal{K}_{A_{b}}(p_{E}^{2},q_{E}^{2})\frac{A_{b}(q_{E}^{2})}{A_{b}^{2}(q_{E}^{2})q_{E}^{2}+B_{b}^{2}(q_{E}^{2})}$
$\displaystyle+\frac{{y_{t}}^{2}+{y_{b}}^{2}}{32\pi^{2}}\int^{\Lambda^{2}}dq_{E}^{2}K_{A}(p_{E}^{2},q_{E}^{2};\lambda_{\phi}\sigma_{h}^{2})\frac{A_{t}(q_{E}^{2})}{A_{t}^{2}(q_{E}^{2})q_{E}^{2}+B_{t}^{2}(q_{E}^{2})}\,,$

where the integral kernels are given by

$\displaystyle\mathcal{K}_{B_{t}}(x,y)=(1+\xi/3)\frac{3{g_{s}}^{2}C_{F}}{16\pi^{2}}K_{B}(x,y;0)+\frac{{y_{t}}^{2}}{32\pi^{2}}\left[K_{B}(x,y;\lambda_{\phi}\sigma_{h}^{2})-K_{B}(x,y;3\lambda_{\phi}\sigma_{h}^{2})\right],$
(13)
$\displaystyle\mathcal{K}_{A_{t}}(x,y)=\frac{y}{x}\left\\{\xi\frac{4{g_{s}}^{2}C_{F}}{16\pi^{2}}K_{A}(x,y;0)+\frac{{y_{t}}^{2}}{32\pi^{2}}\left[K_{A}(x,y;3\lambda_{\phi}\sigma_{h}^{2})+K_{A}(x,y;\lambda_{\phi}\sigma_{h}^{2})\right]\right\\}$

and

$\displaystyle K_{B}(x,y;M^{2})$
$\displaystyle=\frac{2}{x+y+M^{2}+\sqrt{(x+y+M^{2})^{2}-4xy}},$ (14)
$\displaystyle K_{A}(x,y;M^{2})$
$\displaystyle=\frac{2xy}{\left[x+y+M^{2}+\sqrt{(x+y+M^{2})^{2}-4xy}\right]^{2}},$

with $C_{F}=T^{a}T_{a}=({N_{c}}^{2}-1)/(2N_{c})=4/3$. We have regularized the
momentum integrals by the momentum cutoff $\Lambda$. This cutoff scale
$\Lambda$ acts as the ultraviolet regulator in the LSD equations, and is
identified as infrared scales for the perturbative renormalization group
running. This is in the later section to coincide with a matching scale
between the quark-meson model-like description at low energy and the present
gauge-Higgs-Yukawa system.

Figure 3 shows the momentum dependence of $A_{t/b}(p_{E}^{2})$ with $\xi=0$ in
the regime that is what is called the nonperturbative regime, relevant to the
present study with the input parameters to be addressed in details in the
later section. We find $A_{t/b}(p_{E}^{2})\simeq 1$ in the target momentum
range, thus the present extended ladder truncation works with $A_{t/b}=1$ in
the Landau gauge $\xi=0$, in a way similar to the literature Kondo:1993ty  in
the $U(2)_{L}\times U(2)_{R}$ invariant limit.

Figure 3:  The Euclidean momentum dependence of $A\equiv A_{t/b}$. The
parameter setup has been taken as $y_{t}\simeq 2.6$, $g_{s}\simeq 3.1$,
$y_{b}\simeq 0.063$, and $\lambda_{\phi}\simeq 0.93$ at the cutoff scale
$\Lambda\simeq 234$ MeV below which the nonperturbative regime in the chiral
broken phase governs.

##  III The CJT potential in the ladder approximation and phase structure of
the gauge-Higgs-Yukawa system

In this section, we work on the CJT formalism Cornwall:1974vz  to get the CJT
potential at the two-loop level in the ladder approximation and reproduce the
LSD equations, Eqs.(11), (12), and (5). This CJT formalism also makes a
preliminary setup to facilitate the matching procedure linked to a quark-meson
model-like description in the later section. We then discuss the phase
structure of the present gauge-Higgs-Yukawa system on the model parameter
space along with the SM benchmark point, as referred to in Figs. 1 and 3. The
SM quark masses at the benchmark infrared scale ($\Lambda\simeq 234$ MeV) are
also computed.

Figure 4: The two types of diagrams contributing to $V_{2}$ in Eq.(15) in the
ladder approximation, where the solid curve represents quark propagators, the
dashed one the scalars, and the wavy curve the one for the gluons.

###  III.1 The CJT effective potential

We begin by writing the CJT effective potential in the presently applied
ladder approximation up to the two-loop level,

$V_{\rm
eff}[\sigma_{h},S_{t},S_{b}]=V_{0}(\sigma_{h})+V_{\text{scalar}}^{\text{1-loop}}(\sigma_{h},\Delta_{\sigma,\pi_{i}})+V_{\text{quarks}}^{\text{1-loop}}(S_{t,b})+V_{\text{gauge}}^{\text{1-loop}}(\Delta_{g})+V_{2}(\phi_{c},\Delta_{\sigma,\pi_{i}},S_{t,b},\Delta_{g})+\text{const.},$
(15)

where $V_{0}(\sigma_{h})=\frac{\lambda_{\phi}}{4}\sigma_{h}^{4}$ is the tree-
level potential, and

$\displaystyle
V_{\text{scalar}}^{\text{1-loop}}(\sigma_{h},\Delta_{\sigma/\pi_{i}})$
$\displaystyle=-\frac{i}{2}\text{tr}\,\ln{i\Delta_{\sigma_{h}/\pi_{i}}^{-1}}\,,$
$\displaystyle V_{\text{quarks}}^{\text{1-loop}}(S_{t/b})$
$\displaystyle=i\,\text{tr}\,\ln{iS^{-1}_{t/b}}+i\,\text{tr}\left[S^{-1}_{0t/b}(\sigma_{h})S_{t/b}\right]\,,$
$\displaystyle V_{\text{gauge}}^{\text{1-loop}}(\Delta_{g})$
$\displaystyle=-\frac{i}{2}\text{tr}\,\ln{i\Delta_{g}^{-1}}\,,$ (16)

with the inverse-tree propagators for $t$ and $b$ quarks defined as
$S_{0t/b}^{-1}=\not{p}-\frac{y_{t/b}}{\sqrt{2}}\sigma_{h}$. The $V_{2}$ term
denotes the contribution from diagrams at the two-loop level, which in the
present ladder approximation includes the graphs depicted in Fig. 4. In the
first line of Eq.(16), $\Delta_{\sigma/\pi_{i}}^{-1}$ do not take into account
the nonperturbative effects on the scalar two-point functions, i.e.,
$V_{\text{scalar}}^{\text{1-loop}}(\sigma_{h},\Delta_{\sigma/\pi_{i}})$ should
be the one-particle irreducible potential for $\sigma$ and $\pi_{i}$ fields,
which depends only on the Higgs VEV. Taking into account $A_{t/b}=1$ in the
Landau gauge ($\xi=0$), one can compute $V_{\rm eff}$ to reach the form

$\displaystyle V_{\text{eff}}[\sigma_{h},B_{t},B_{b}]=$
$\displaystyle\frac{\lambda_{\phi}}{4}{\sigma_{h}}^{4}+\frac{1}{32\pi^{2}}\int^{\Lambda^{2}}dk_{E}^{2}\,k_{E}^{2}\ln{\frac{k_{E}^{2}+3\lambda_{\phi}\sigma_{h}^{2}}{k_{E}^{2}}}+\frac{3}{32\pi^{2}}\int^{\Lambda^{2}}dk_{E}^{2}\,k_{E}^{2}\ln{\frac{k_{E}^{2}+\lambda_{\phi}\sigma_{h}^{2}}{k_{E}^{2}}}$
$\displaystyle+\frac{N_{c}}{4\pi^{2}}\int^{\Lambda^{2}}dp_{E}^{2}\,p_{E}^{2}\frac{B_{t}^{2}(p_{E}^{2})-y_{t}\sigma_{h}{B_{t}}(p_{E}^{2})/\sqrt{2}}{p_{E}^{2}+B_{t}^{2}(p_{E}^{2})}$
$\displaystyle+\frac{N_{c}}{4\pi^{2}}\int^{\Lambda^{2}}dp_{E}^{2}\,p_{E}^{2}\frac{B_{b}^{2}(p_{E}^{2})-y_{b}\sigma_{h}{B_{b}}(p_{E}^{2})/\sqrt{2}}{p_{E}^{2}+B_{b}^{2}(p_{E}^{2})}\,.$

The stationary conditions with respect to $B_{t}$ and $B_{b}$ respectively
reproduce the SD equations for $B_{t}$ and $B_{b}$ in Eqs.(11) and Eq.(12)
with the present ladder approximation prescribed.

The tadpole condition for the Higgs VEV in Eq.(5) is also derived with respect
to $\sigma_{h}$, which, neglecting the small $b$ quark contribution, takes the
same form as in Eq.(5). The three-point function in the left-hand side of
Eq.(5) can be decomposed as

$\braket{\sigma(\sigma^{2}+\vec{\pi}^{2})}=\braket{\sigma}^{3}+3\braket{\sigma}\braket{\sigma\sigma}_{\text{conn.}}+\braket{\sigma}\braket{\pi_{i}\pi_{i}}_{\text{conn.}}+\braket{\sigma(\sigma^{2}+\pi_{i}^{2})}_{\text{conn.}}\,,$
(17)

where the subscript “conn.” stands for the amplitude constructed from
connected diagrams. The present ladder approximation the right-hand side of
Eq.(17) into only the first two terms with the bare bosonic propagators, so
that we have

$\displaystyle\braket{\sigma(\sigma^{2}+\vec{\pi}^{2})}$
$\displaystyle=\lambda_{\phi}\sigma_{h}^{3}+\lambda_{\phi}\sigma_{h}\int\frac{d^{4}q}{(2\pi)^{4}}\left(\frac{3i}{q^{2}-3\lambda_{\phi}\sigma_{h}^{2}}+\frac{3i}{q^{2}-\lambda_{\phi}\sigma_{h}^{2}}\right)$
(18)
$\displaystyle=\lambda_{\phi}\sigma_{h}^{3}+\frac{3\lambda_{\phi}\sigma_{h}}{16\pi^{2}}\left(2\Lambda^{2}-3\lambda_{\phi}\sigma_{h}^{2}\ln{\frac{\Lambda^{2}-3\lambda_{\phi}\sigma_{h}^{2}}{3\lambda_{\phi}\sigma_{h}^{2}}}-\lambda_{\phi}\sigma_{h}^{2}\ln{\frac{\Lambda^{2}-\lambda_{\phi}\sigma_{h}^{2}}{\lambda_{\phi}\sigma_{h}^{2}}}\right).$

The right hand side of Eq.(5), i.e., the top quark condensate
$(-\frac{y_{t}}{\sqrt{2}}\langle\bar{t}t\rangle)$, is similarly simplified, in
the present ladder approximation, to

$\displaystyle-\frac{y_{t}}{\sqrt{2}}\langle\bar{t}t\rangle$
$\displaystyle=\frac{y_{t}N_{c}}{\sqrt{2}}\int\frac{d^{4}q}{(2\pi)^{4}}\text{tr}S_{t}(q)=\frac{y_{t}N_{c}}{4\sqrt{2}\pi^{2}}\int^{\Lambda^{2}}dp_{E}^{2}\frac{p_{E}^{2}B_{t}(p_{E}^{2})}{p_{E}^{2}+{B_{t}^{2}(p_{E}^{2})}}\,.$
(19)

The current quark masses are generated by the Higgs VEV $\sigma_{h}$ as the
solution of Eq.(5), coupled with the SD equations for $B_{t}$ and $B_{b}$ in
Eqs.(11) and (12), to be

$m_{q}=\frac{y_{q}}{\sqrt{2}}\sigma_{h},$ (20)

for $q$-quark. The full masses for $t$ and $b$ quarks are defined as

$m_{t/b,full}=B_{t/b}(p^{2}_{E}=m_{t/b,full}^{2})\,.$ (21)

###  III.2 The phase structure and implication to SM quark masses

Numerically solving the coupled LSD equations, Eqs.(11), (11), and (5), in
Fig. 5 we show a phase diagram in the space spanned by the dynamical top mass
normalized to the cutoff $m_{t,full}/\Lambda$, $y_{t}$, and $\lambda_{\phi}$,
with $g_{s}=y_{t}$ and $y_{b}=0.5y_{t}$. The critical line has been observed
on the $(y_{t},\lambda_{\phi})$ plane placed by $m_{t,full}=0$. In the limit
where $\lambda_{\phi}\to 0$, $m_{t,full}$ goes divergent, which reflects the
non-renormalizability of the Yukawa theory in the absence of the scalar
quartic coupling.

Here we have several remarks.

  * •

The bare parameter setup on $g_{s}=y_{t}$ well models the infrared feature of
the renormalization group running in the SM as seen from Fig. 1, when the
cutoff $\Lambda$ for the SD equations matches with a strong coupling scale of
$\alpha_{s}$, $\alpha_{s}(\Lambda)\sim\pi$.

  * •

Though the input isospin breaking by $y_{b}=0.5y_{t}$ is too small to be
realistic, the generic feature of the phase diagram does not substantially
alter for smaller $y_{b}$.

  * •

The chiral symmetric phase is still observed even with nonzero
$\lambda_{\phi}$ and $y_{t}$ as long as $y_{t}\lesssim 1$, because the tadpole
of the Higgs is generated via the top quark condensate following the SD
Eq.(5).

  * •

In the strong QED limit where $y_{t}=y_{b}=0$ and $\lambda_{\phi}\to\infty$
(i.e. the decoupling limit of the Higgs), we have observed the well-known
critical coupling $g_{s}^{\rm cr}=\pi$ ($\alpha_{s}=\pi/(3C_{F})=\pi/4)$)
Miransky:1994vk .

Figure 5: A mock-up phase diagram on $m_{t,full}/\Lambda$, $y_{t}$, and
$\lambda_{\phi}$, with $g_{s}=y_{t}$ and $y_{b}=0.5y_{t}$.  Figure 6:  The
phase diagram on $m_{t,full}/\Lambda$, $y_{t}$, and $y_{b}$, with
$g_{s}=\lambda_{\phi}=0$. The critical line separating the chiral symmetric
and broken phases is given as $y_{t}y_{b}=4\pi^{2}$.

In the Yukawa-theory limit with the massless scalars, we have found a critical
line $y_{t}y_{b}=4\pi^{2}$ as shown in Fig. 6. The seemingly dominated
$y_{t}^{2}$ terms in the LSD equation for $B_{t}$ in Eq.(11) are actually
canceled in the kernel function ${\cal K}_{B_{t}}$ in Eq.(14) (with $\xi=0$).
This feature is still approximately operative even with the massive scalar
exchange contributions: ${\cal K}_{B_{t}}(x,y;m^{2})\simeq{\cal
K}_{B_{t}}(x,y;3m^{2})$ as long as $m/\Lambda\ll 1$. This is essentially due
to an accidental $U(1)$ axial symmetry between $\sigma$ and $\pi^{3}$
exchanges (repulsive and attractive channels): that is, $U(1)_{A}\equiv
U(1)_{L,3}\times U(1)_{R,3}$, which is associated with the chiral rotation
with respect to the third component of the Pauli matrix $\tau_{3}$ and
transforms $\phi^{0}$ and the quark doublet field $q$ in the Yukawa terms, in
Eq.(1), as

$\phi^{0}\rightarrow e^{-i\theta}\phi^{0}\quad\text{and}\quad q\rightarrow
e^{i\frac{\theta}{2}\gamma_{5}\tau_{3}}q\,,$ (22)

making the coupling strengths of $\bar{q}\sigma q$ and $\bar{q}\pi_{3}q$
identical to each other.

Thus, the strong top Yukawa coupling $y_{t}$ cannot solely develop the
critical line or does not dominantly generate the chiral broken phase, or the
top quark condensate. Therefore, the dynamical chiral symmetry in the present
gauge-Higgs-Yukawa system is essentially triggered by the ladder QCD (i.e.
strong QED), and the top flavor characteristics on the top-quark full mass
$m_{t,full}$ is provided only through the overall top Yukawa coupling:
$m_{t,full}$ still lies in the QCD scale $={\cal O}$(100) MeV. This conclusion
favors the description of almost massless six-flavor QCD when the QCD phase
transition is addressed, which is to be clarified right below.

Figure 7 shows the current quark masses as a function of $\Lambda$, which
arise from the Higgs tadpole contribution in Eq.(23) together with the LSD
solutions. The selected range of $\Lambda$ includes a reference point
$\Lambda=234$ MeV where we have $g_{s}(\Lambda)\simeq 3.1\sim\pi$, i.e., close
to the critical coupling in strong QED, which corresponds to a boundary
between the perturbative nonperturbative coupling regimes as quoted in Fig. 3:
at $\Lambda=234$ MeV the SM perturbative two-loop running, with the EW scale
inputs as in Fig. 1, yields $y_{t}\simeq 2.6$, $g_{s}\simeq 3.1$, $y_{b}\simeq
0.063$, and $\lambda_{\phi}\simeq 0.93$. Figure 3 indeed implies that at
around the reference point $\Lambda=234$ MeV, we have current quark masses for
the SM heavier three quarks (the Higgs tadpole contribution in Eq.(23)),

$\displaystyle m_{t}(\Lambda=234\,{\rm MeV})$ $\displaystyle\simeq 140\,{\rm
MeV}\,,$ $\displaystyle m_{b}(\Lambda=234\,{\rm MeV})$ $\displaystyle\simeq
3.5\,{\rm MeV}\,,$ $\displaystyle m_{c}(\Lambda=234\,{\rm MeV})$
$\displaystyle\simeq 0.94\,{\rm MeV}\,.$ (23)

In addition, the top and bottom quarks get the pure ladder QCD and Yukawa
contributions due to the top condensation. At the same reference point of
$\Lambda$, we find the full masses for $t$ and $b$ quarks,

$\displaystyle m_{t,full}(\Lambda=234\,{\rm MeV})$ $\displaystyle\simeq
175\,{\rm MeV}\,,$ $\displaystyle m_{b,full}(\Lambda=234\,{\rm MeV})$
$\displaystyle\simeq 12\,{\rm MeV}\,.$ (24)

Thus all the six quarks can be light enough in the typical QCD phase
transition epoch at around the temperature $\sim 100-200$ MeV.

Figure 7: The plots of current quark masses versus $\Lambda$ including a
reference point $\Lambda=234$ MeV, at which we have $g_{s}(\Lambda)\simeq
3.1\sim\pi$, i.e., close to the critical coupling in strong QED, following the
SM perturbative renormalization evolution as in Fig. 3. The panel (b) excludes
the top quark, while the panel (a) includes it.

Figure 8: Quark full masses as a function of $\Lambda$ in the same range as in
Fig. 7.

##  IV Cosmological implication of the top quark condensation in supercooled
EW phase transition

Since the dynamically generated top condensate couples to the SM Higgs, the
supercooled EW phase transition should get a significant correlation with the
thermal evolution of the top condensate. In this section, we address this
issue by employing a quark-meson model in the large $N_{c}$ limit. namely, the
mean-field approximation for the Higgs and QCD meson fields. We fix the model
parameters by matching them to the LSD results with the matching scale set at
$\Lambda=234$ MeV, which is identified as the LSD cutoff scale as has been
done in the previous section. We then discuss the cosmological phase
transition arising from the evidence of the top quark condensation and its
cosmological consequences such as the GW production.

###  IV.1 Quark meson model for the top-Higgs sector

Figure 9: The Feynman graph generating the $\sigma_{t}$ \- $\sigma_{h}$
mixing. The blob and (internal) top loop lines include the nonperturbative
ladder-type contributions as has been discussed in Sec.III.

We consider a quark-meson model-like model for the low-energy effective
description for the present gauge-Higgs-Yukawa system at the matching scale
$\Lambda$. Since the correlation with the SM Higgs arising only through the
Yukawa couplings to quarks, the mean-field associated with only the top quark
condensate becomes significant. The model Lagrangian is thus constructed from
the scale-invariant SM Higgs sector (only with neutral scalars) coupled to the
top quark:

$\displaystyle\mathcal{L}_{\text{QMM}}$
$\displaystyle=\bar{t}i\not{\partial}t-g\,\bar{t}_{L}M_{\bar{t}t}t_{R}-y_{t}\,\bar{t}_{L}M_{h}t_{R}+\text{h.c.}$
$\displaystyle+\partial_{\mu}M_{\bar{t}t}^{\dagger}\partial^{\mu}M_{\bar{t}t}+\partial_{\mu}M_{h}^{\dagger}\partial^{\mu}M_{h}-V_{\rm
QMM}\,,$ $\displaystyle V_{\rm QMM}$
$\displaystyle=\lambda_{\phi}\left(M_{h}^{\dagger}M_{h}\right)^{2}-g_{m}(M_{h}^{\dagger}M_{\bar{t}t}+{\rm
h.c.})+V(M_{\bar{t}t})\,,$ (25)

where $M_{\bar{t}t}=(\sigma_{t}+i\pi_{t})/\sqrt{2}$ for the $\bar{t}t$ scalar
meson, i.e., topponium, and $M_{h}=(\sigma_{h}+i\pi_{h})/\sqrt{2}$ for the
neutral Higgs scalar fields in the SM. Here several comments are in order:

  * •

The $g_{m}$ coupling term denotes the mixing between the topponium and the SM
Higgs mean fields, $-g_{m}\langle\sigma_{h}\rangle\langle\sigma_{t}\rangle$,
which corresponds to the disconnected graph contribution in the ladder
approximation. See Fig. 9 for the typical Feyman graph. The figure also
displays a squared blob, which denotes the Bethe-Salpeter amplitude to
describe the dynamical $\sigma_{t}$ \- $t$ \- $t$ vertex, including not merely
the static potential terms, but also the momentum dependence. The latter type
of quantum corrections are discarded in the current ladder approximation. The
momentum dependence will not give any significance on the static potential
analysis aiming at discussion on the thermal phase transition, which might,
however, be the issue when the dynamic cosmological phase transition is
addressed. We will come back to this point later in the Summary and Discussion
section.

  * •

In addition to the tadpole term, in Eq.(25) the top Yukawa coupling to $M_{h}$
has been taken into account, so that the top-quark full mass contributions are
shared by those two terms due to the presence of the dynamical Higgs field.
This is in contrast to the case of the ordinary QCD quark-meson model which
includes the current quark mass effect in the tadpole term and the dynamical
mass part in the Yukawa term.

  * •

One can also see that the second derivative of the CJT potential with respect
to $\sigma_{h}$ in Eq.(III.1) vanishes, which implies no Higgs portal coupling
to $\sigma_{t}$, such as $|M_{h}|^{2}|M_{\bar{t}t}|^{2}$, hence the $g_{m}$
coupling is the unique way to allow the mixing of $M_{\bar{t}t}$ with $M_{h}$
up to operators smaller than dimension four.

  * •

The topponium VEV at the vacuum, $\braket{\sigma_{t}}$, is set by the topped
pion decay constant $f_{t}$, which we nonperturbatively compute by using the
Pagels-Stokar formula Pagels:1979hd  with the solution of the coupled LSD
equations (Eqs.(11), (12), and (5)) as

$f_{t}^{2}=\frac{N_{c}}{4\pi^{2}}\int^{\Lambda^{2}}dp_{E}^{2}\cdot
p_{E}^{2}\frac{B_{t}^{2}(p_{E}^{2})-\frac{p_{E}^{2}}{4}\frac{d}{dp_{E}^{2}}B_{t}^{2}(p_{E}^{2})}{(p_{E}^{2}+B_{t}^{2}(p_{E}^{2}))^{2}}\,.$
(26)

The numerical estimate gives $f_{t}/\Lambda\simeq 0.091$ and for Higgs VEV
$\sigma_{h}/\Lambda\simeq 0.34$ at $\Lambda=234$ MeV with the same inputs as
in Eqs.(23) and (24).

Besides, we have the potential term for $M_{\bar{t}t}$ in Eq.(25). Taking
$\pi_{t}=0$ via the U(1) axial transformation, the mesonic potential can be
written as a function of only $\sigma_{t}$. At this point, a simple-minded
parametrization of the potential is of the Ginzburg-Landau polynomial form,

$V(M_{\bar{t}t})\Bigg{|}_{\rm
GL}=\mu^{2}\sigma_{t}^{2}+\lambda\sigma_{t}^{2}\,.$ (27)

The presence of the mass term $\mu^{2}$ would reflect the hard scale breaking
at the deeper infrared regime of QCD, simply following the perturbative
running of the gauge coupling (as seen also from Fig. 1). Still, we could have
another possibility featuring the deeply infrared QCD nature: in a recent
nonperturbative analysis based on the functional renormalization group, like
in the literature Deur:2023dzc , the QCD gauge coupling might possess a
nontrivial infrared fixed point, which implies an almost scale invariance (to
be eventually broken by the dynamical chiral symmetry breaking, though) #1#1#1
The almost scale-invariant QCD with six quarks has also been discussed in
works other than the functional renormalization group analysis, motivated by
addressing the QCD sigma meson as a dilaton associated with the spontaneous
breaking of the approximate scale invariance Li:2016uzn ; Cata:2019edh ;
Zwicky:2023bzk ; Zwicky:2023krx ; Crewther:2020tgd ; Crewther:2013vea . . From
this perspective, we might also consider the Coleman-Weinberg type
Coleman:1973jx ,

$V(M_{\bar{t}t})\Bigg{|}_{\rm
CW}=\lambda\sigma_{t}^{4}\left[a+b\ln\left(\frac{\sigma_{t}}{f_{t}}\right)\right]\,.$
(28)

Taking into account both two possibilities, thus we model the form of
$V(M_{\bar{t}t})$ as

$V_{M_{\bar{t}t}}=\lambda_{D}\,\left(\frac{\sigma_{t}}{f_{t}}\right)^{4-\gamma_{D}}{f_{t}}^{4}+\lambda\left(\frac{\sigma_{t}}{f_{t}}\right)^{4}{f_{t}}^{4}.$
(29)

Here we have introduced an anomalous dimension parameter $\gamma_{D}$
associated with the scale anomaly, which can be fixed by solving the
underlying theory. In the present study, we take it as a free parameter and
consider $\gamma_{D}$ in a range of $0<\gamma_{D}\leq 2$. The $\lambda_{D}$
term thus plays a role of interplay between the Ginzburg-Landau and almost
scale-invariant description. When $\gamma_{D}=2$ the $\lambda_{D}$ term is
reduced to the normal quadratic mass term as in Eq.(27), while for
$\gamma_{D}\simeq 0$ it provides a logarithmic potential term like
$\lambda_{D}\,\gamma_{D}\sigma_{t}^{4}\log(\sigma_{t}/f_{t})$:

$\displaystyle V(M_{\bar{t}t})\Bigg{|}_{\gamma_{D}\simeq 0}$
$\displaystyle=\sigma_{t}^{4}\left[(\lambda+\lambda_{D})-\lambda_{D}\gamma_{D}\ln\left(\frac{\sigma_{t}}{f_{t}}\right)+\mathcal{O}(\gamma_{D}^{2})\right]\equiv
V(M_{\bar{t}t})\Bigg{|}_{\rm CW}\quad{\rm in}\quad{\rm Eq.(\ref{CW-type})}\,.$
(30)

We fix the vacuum feature of the model including the Higgs VEV
$\langle\sigma_{h}\rangle$, the topponium VEV
$\langle\sigma_{t}\rangle=f_{t}$, and the potential energy i.e., the scale
anomaly, by matching with the nonperturbative results associated with the CJT
effective potential in Eq.(III.1). First, the stationary condition for the
potential terms, together with the LSD solutions to
$\langle\sigma_{t}\rangle=f_{t}$ (Eq.(26)) and $\langle\sigma_{h}\rangle$
(Eq.(5)), constrain $\lambda_{D}$, $\gamma_{D}$, and $g_{m}$, as follows:

$\displaystyle\left.\frac{\partial
V}{\partial\sigma_{h}}\right|_{\text{vacuum}}=\lambda_{\phi}\sigma_{h}^{3}-g_{m}f_{t}=0;$
(31) $\displaystyle\left.\frac{\partial
V}{\partial\sigma_{t}}\right|_{\text{vacuum}}=\lambda_{D}(4-\gamma_{D}){f_{t}}^{3}+4\lambda{f_{t}}^{3}-g_{m}\sigma_{h}=0.$

Then, matching with the vacuum energy gives a further constraint like

$V_{\rm CJT}\Bigg{|}_{\rm vacuum}\equiv V_{QMM}\Bigg{|}_{\rm
vacuum}=\frac{1}{4}\left(\lambda_{D}{f_{t}}^{4}-2g_{m}\langle\sigma_{h}\rangle
f_{t}\right)\,.$ (32)

Though the $g$ coupling between quark and meson is at this point just free
parameter, referring to the LSD results may fix it via the constituent quark
mass in the following two ways (I and II)

$\displaystyle
g|_{I}=\left(m_{t,full}-\frac{y_{t}}{\sqrt{2}}\langle\sigma_{h}\rangle\right)/f_{t},$
(33) $\displaystyle g|_{II}=m_{t,full}/f_{t}.$

In the former case ($g|_{I}$), $M_{\bar{t}t}$ couples to the top quark pair
only via the dynamical mass of the top quark, while keeping the tadpole term
coupled to the Higgs mean field. The latter case ($g|_{II}$) implies that the
top quark pair still couples to $M_{\bar{t}t}$ also through the current mass,
so that there are two current mass contributions to the model along with the
tadpole term. The LSD result at the reference point $\Lambda=234$ MeV gives

$\displaystyle g|_{I}$ $\displaystyle\simeq 1.3\,,$ $\displaystyle g|_{II}$
$\displaystyle\simeq 8.0\,.$ (34)

Thus the model parameters can completely be fixed once $\gamma_{D}$ is given
for $0<\gamma_{D}\leq 2$. Table 1 gives the model parameter sets fixed by
matching with the present LSD analysis as described above.

$\gamma_{D}$ | 2 | 0.5 | 0.1
---|---|---|---
$g$ | 1.3 | 8.0 | 1.3 | 8.0 | 1.3 | 8.0
$g_{m}/\Lambda^{2}$ | 0.39
$\lambda$ | 50 | 89 | 299
$\lambda_{D}$ | -13 | -52 | -262
Table 1: The parameter setting of the present quark-meson model fixed by the
matching with the LSD analysis at $\Lambda=234$ MeV, given a value of
$\gamma_{D}$ in a range of $0<\gamma_{D}\leq 2$.

###  IV.2 Thermal phase transition

We incorporate the thermal loop corrections arising from the top-Yukawa
interaction at the one-loop level of the quark-meson model in Eq.(25), based
on the imaginary time formalism, to derive the thermodynamic potential
($\Omega(\sigma_{t},\sigma_{h})$) for the mean fields $\sigma_{t}$ and
$\sigma_{h}$. The top loop corrections to the vacuum part are not included in
the thermodynamic potential, instead, which involves nonperturbative
corrections from the LSD analysis in the tree level as has been discussed
above.

Figure 10 shows the temperature ($T$) dependence (normalized to $\Lambda=234$
MeV) of $\sigma_{t}$ and $\sigma_{h}$ in the case of $\gamma_{D}=2$ with
$g|_{I}$ and $g|_{II}$ in Eq.(34). For larger $g$, the phase transition goes
like the type of the first-order with the critical temperature $T_{c}\simeq
65$ MeV, while for smaller $g$, it tends to be changed to crossover with the
pseudocritical temperature (defined as the inflection point of the order
parameter) $T_{\rm pc}\simeq 80$ MeV, which holds approximately for both
$\sigma_{t}$ and $\sigma_{h}$. We have checked that this trend does not
substantially change for smaller $\gamma_{D}$. The first-order phase
transition has been undergone due to the emergence of the thermal potential
barrier between the false and true vacua, which has been created along a
hybrid direction involving two nonzero $\sigma_{t}$ and $\sigma_{h}$, as
plotted in Fig. 11. This is the characteristic consequence of the presence of
the top quark condensate in the thermal history.

Figure 10: The temperature dependence of $\sigma_{t}$ and $\sigma_{h}$ in the
case of $\gamma_{D}=2$ for (a) $g|_{II}\simeq 8.0$ and (b) $g|_{I}\simeq 1.3$.

Figure 11: The thermodynamic potentials as a function of $\sigma_{t}$ and
$\sigma_{h}$ for $\gamma_{D}=2$ with $g|_{II}\simeq 8.0$ around the first
order phase transition point. The panel (a) corresponds to the case when
$T=0.5T_{c}$; (b) $T=1.0T_{c}$; (c) $T=1.5T_{c}$, where $T_{c}\simeq 65$ MeV
for the matching scale $\Lambda=234$ MeV.

###  IV.3 Cosmological phase transition

The first-order phase transition in the hybrid $\sigma_{t}$-$\sigma_{h}$
system at around $T_{c}\simeq 65$ MeV would be supercooled in the universe to
produce the bubble nucleation and percolation. We discuss this supercooling in
terms of the false vacuum decay and evaluate the false vacuum decay rate
approximately as Linde:1981zj

$\Gamma\simeq T^{4}\left(\frac{\mathcal{S}_{3}}{2\pi
T}\right)^{\frac{3}{2}}\exp{\left(-\frac{\mathcal{S}_{3}}{T}\right)}\,,$ (35)

where the $\mathcal{S}_{3}=\mathcal{S}_{3}[\phi_{\text{b}}(r)]$ is the three-
dimensional ${\cal O}(3)$ symmetric Euclidean action #2#2#2 As discussed in
Ref. Helmboldt:2019pan , the quark-meson model introduces the mesonic degrees
of freedom as fundamental, $\mathcal{S}_{3}$ takes the form having the
canonical kinetic term for the meson fields. ,

$\mathcal{S}_{3}=\int dr\,(4\pi
r^{2})\left[\frac{1}{2}\left(\frac{d\vec{\sigma}}{dr}+\Omega(\vec{\sigma},T)\right)\right]\,,$
(36)

with a set of the bounce solutions
$\vec{\sigma}(r)=(\sigma_{t},\sigma_{h})^{T}$. The bounce solutions are given
by solving the stationary condition of the action:

$\frac{d^{2}\vec{\sigma}}{dr^{2}}+\frac{2}{r}\frac{d\vec{\sigma}}{dr}=\nabla_{\sigma}\Omega(\vec{\sigma},r)\,,$
(37)

with the boundary conditions

$\displaystyle\left.\frac{d\vec{\sigma}}{dr}\right|_{r=0}=0,$ (38)
$\displaystyle\left.\vec{\sigma}\right|_{r\rightarrow\infty}=0.$

To get the multi-field bounce solutions, we have used the numerical source
pack CosmoTransitions Wainwright:2011kj .

Then the nucleation temperature $T_{n}$ can be determined by

$\frac{\Gamma(T_{n})}{H^{4}(T_{n})}\sim 1.$ (39)

where the Hubble parameter is parametrized as

$H^{2}(T)=\frac{\rho_{\text{rad}}(T)+\rho_{\text{vac}}(T)}{{3M_{\text{Pl}}}^{2}}\simeq\frac{1}{{3M_{\text{Pl}}}^{2}}\left(\frac{\pi^{2}}{30}g_{*}(T_{i})T_{i}^{4}+\Delta
V\right)\,,$ (40)

with $\Delta V$ being the vacuum energy density difference between the false
vacuum and the true vacuum. Here $g_{*}(T)$ denotes the effective degrees of
freedom at $T$, and $M_{\rm pl}$ is the reduced Planck scale $\simeq 1.2\times
10^{18}$ GeV. We have also taken into account the dark-sector induced
potential density, which is parametrized by a dark thermal density with
$T_{i}$ as done in the literature Sagunski:2023ynd . This term can be turned
off or on depending on whether the electroweak/dark sector phase transition is
triggered before the hybrid $\sigma_{t}$ \- $\sigma_{h}$ phase transition
takes place, or not.

We also evaluate the percolation temperature $T_{p}$ via the probability still
staying in the false vacuum Guth:1979bh ; Guth:1981uk , $P=e^{-I(T)}$, where
the exponent function is available in the literature Ellis:2018mja ;
Sagunski:2023ynd  We define $T_{p}$ at which $I(T_{p})=0.34$ yielding $P\simeq
0.7$ Guth:1979bh ; Guth:1981uk .

###  IV.4 GW signals

We now discuss the GW signals produced from the supercooled QCD -EW phase
transitions induced from the topponium - Higgs correlation. We evaluate the
two characteristic signal parameters, $\alpha$ and $\beta$. The normalized
latent heat $\alpha$ is defined as

$\alpha(T_{p})\equiv\frac{1}{\rho_{\text{rad}}}\left(\rho_{\rm
rad}(T_{i})+\Delta V-T\frac{d(\Delta V)}{dT}\right)_{T=T_{p}}\,,$ (41)

where $\rho_{\rm rad}(T_{i})=\frac{\pi^{2}}{30}g_{*}(T_{i})T_{i}^{4}$, which
is removed depending on the assumed dark-sector supercooling scenario
Sagunski:2023ynd . The other parameter $\beta$ measures the inverse duration
of the phase transition, and the normalized one ($\beta$) is defined as

$\tilde{\beta}\equiv\frac{\beta}{H(T_{p})}=T_{p}\frac{d}{dT}\left(\frac{\mathcal{S}_{3}(T)}{T}\right).$
(42)

Given the signal parameters as above, we calculate the gravitational wave
spectra sourced from the bubble collision. The power spectrum is given by
Brdar:2018num

$\Omega_{\text{coll}}h^{2}=e^{-4N_{e}}\left(\frac{\rho_{p}}{\rho_{r}}\right)\times
1.67\times
10^{-5}\left(\frac{\beta}{H(T_{p})}\right)^{-2}\left(\frac{\kappa_{\text{coll}}\alpha}{1+\alpha}\right)^{2}\left(\frac{100}{g_{*}}\right)^{1/3}\left(\frac{0.11{v_{w}}^{3}}{0.42+{v_{w}}^{2}}\right)\frac{3.8(f/f_{\text{coll}})^{2.8}}{1+2.8(f/f_{\text{coll}})^{3.8}},$
(43)

where $\kappa_{\text{coll}}$ characterizes the energy transfer between the
vacuum energy and the kinetic energy of the bubble wallZhang:2024vpp , which
we take $\kappa_{\text{coll}}\simeq 1$ with the velocity $v_{\omega}\simeq 1$;
$N_{e}$ denotes the e-folding number during the reheating and $\rho_{p,r}$
denotes the energy density at the percolation/reheating temperature. The peak
frequency is given by Zhang:2024vpp

$f_{\text{coll}}=e^{-N_{e}}\left(\frac{\rho_{p}}{\rho_{r}}\right)^{1/2}\times
1.65\times
10^{-5}\left(\frac{\beta}{H(T_{p})}\right)\left(\frac{0.62}{1.8-0.1v_{w}+{v_{w}}^{2}}\right)\left(\frac{T_{r}}{100\,\text{GeV}}\right)\left(\frac{g_{*}}{100}\right)^{1/6}\text{Hz}.$
(44)

In the present study, we simply assume the instantaneous reheating, such that
in terms of the literature Zhang:2024vpp , $\rho_{p}=\rho_{r}$ and $N_{e}=0$.

Figure 12: GW power spectra compared to the sensitivities of the prospected GW
interferometers Schmitz:2020syl . The black solid curves seen around the
centers of the plots draw the trajectories following the shift of the peak
frequency generated by varying $T_{i}$ from $1.5\times 10^{-2}$ MeV to
$1.0\times 10^{6}$ GeV. The dashed curves around the bottom of the figure
correspond to the case where the supercooling effect in the dark sector (to
supply the EW scale) is no longer left in the QCD phase transition epoch.

In Fig. 12 we show the GW spectra predicted from the supercooling phase
transition in the topponium - Higgs coupled system around the QCD phase
transition epoch. As seen from the figure, we have observed no significant
dependence of $\gamma_{D}$ in the range of $0<\gamma_{D}\leq 2$. The produced
GW spectra can be probed by the LISA, DECIGO, BBO, CE, and ET depending on the
dark-sector energy-density scale characterized by $T_{i}$. The signals will be
present in addition to those arising from the supercooling in the dark sector
(scalegenesis) phase transition as discussed in the literature Iso:2017uuu ;
Hambye:2018qjv ; Sagunski:2023ynd ; Dichtl:2023xqd ; Frandsen:2022klh ;
Ellis:2022lft ; Bodeker:2021mcj ; vonHarling:2017yew . Just as a reference,
also has been drawn the curves without the dark sector contribution (dashed
curves around the bottom of the figure).

##  V Summary and discussions

In summary, we have made the first attempt to employ the nonperturbative
analysis of the QCD-induced EW phase transition scenario, based on the LSD
equations and the CJT formalism in the gauge-Higgs-Yukawa system. We have
observed that the chiral broken QCD vacuum emerges with the nonperturbative
top quark condensate, which can essentially be generated by the QCD sector:
the EW phase transition at the QCD scale can be mainly triggered by QCD, even
taking into account the strong top Yukawa interaction. Thus all the six quarks
get masses small enough compared to the typical QCD critical temperature of
the order of 100 MeV, where even the top quark can merely be in the same order
(Fig.(7)). This is due to the big suppression of the top-Yukawa contributions
to the ladder kernel caused by an accidental U(1) axial symmetry present in
the top-Higgs Yukawa sector (Fig.3).

We then have discussed the impact on the top quark condensation in the QCD-EW
phase transition by employing a quark-meson model-like description as the low-
energy effective theory. There the SM Higgs couples with the mean field of the
top quark condensate and topponium-like scalar bound state. The
nonperturbative scale anomaly induced from the dynamical top quark
condensation based on the LSD analysis as well as the associated
nonperturbative results have been encoded in the quark meson model, which
highly constrains the model parameter space. we have observed the EW phase
transition of the first order type along the topponium - Higgs hybrid
direction in the field space (Figs. 10) and (11). We have discussed the
associated GW productions and found that in addition to the conventional GW
signals sourced from an expected BSM at around or over the TeV scale, the
dynamical topponium-Higgs system can yield another power spectrum sensitive to
the BBO and LISA, and DECIGO, etc (Fig. 12).

Thus the presence of the nonperturbative top quark condensation in the
supercooled EW phase transition would leave cosmologically probable footprints
in the thermal history of the universe as the prospected GW spectra. This
finding has presently been bench-marked based on the LSD method coupled with
the quark meson-like and mean-field approach, which will pave a way to
pursuing the QCD-origin cosmology coupled with possible BSM embedded in the
classically scale-invariant scenario. In closing, we give several comments
related to improvements from the present analysis, which is in more detail to
be addressed in another publication.

Figure 13: A Feynman diagram generating the kinetic term mixing between
$\sigma_{t}$ and $\sigma_{h}$ beyond the ladder approximation. The round blob
attached on the $\sigma_{h}$ \- $t$ \- $t$ vertex, includes non-ladder
corrections, in contrast to the ladder Fig. 9, generating the
$\langle\sigma_{t}\rangle$ \- $\langle\sigma_{h}\rangle$ mixing, constructed
only from the ladder graphs.

  * •

In the present work, the thermal corrections to the top-Higgs system have been
taken into account in the quark-meson model, where a couple of the results
from the LSD analysis have been used as inputs to fix the model parameters.
The straightforward incorporation of the thermal corrections in the LSD
analysis is possible and provides the temperature dependence of the top and
bottom quarks along with the Higgs VEV $\sigma_{h}$. Thus we could have
discussed both QCD and EW phase transitions solely in the LSD framework, not
invoking the mean field (or bosonization of $\bar{t}t$) $\sigma_{t}$ or
passing the quark meson model. However, the cosmological phase transition
issue currently seems to be challenging to argue without the bounce scalar
action or the false vacuum decay via the scalar field theory, namely, without
$\sigma_{t}$. This issue will be addressed in future work.

  * •

The LSD analysis can be improved by coupling to the functional renormalization
group method, as has been discussed in Gao:2020qsj . It would be noteworthy to
examine how non-ladder diagram corrections, incorporated into by the
functional renormalization group, can affect the criticality of the top quark
condensation and the thermal QCD and EW phase transitions.

  * •

The significant topponium effect on the EW phase transition arises from the
mixing with the SM Higgs as depicted in Fig. 9. This is essentially generated
via the condensate mixture; $\langle\sigma_{t}\rangle$ \-
$\langle\sigma_{h}\rangle$, at the ladder approximation level, which thus
contributes to the mean field potential as in Eq.(25). The kinetic term mixing
can also generically be generated in the full gauge-Higgs-Yukawa theory,
however, which goes beyond the ladder approximation because it is necessary to
include the Yukawa vertex corrections. See Fig. 13. This beyond-the-ladder
kinetic term mixing at finite temperature would modify the bounce action
Eq.(36), which might give a nontrivial correction to the nucleation and
percolation processes in the cosmological phase transition. Similar kinetic
term corrections have been addressed in the literature Sagunski:2023ynd ;
Aoki:2019mlt  based on the Nambu-Jona-Lasinio type model for a dark QCD
theory, in which, however, the kinetic mixing effect has been disregarded in
the bounce equation. This issue is intriguing to be discussed in detail
elsewhere.

## Acknowledgments

We are grateful to Fei Gao, and He-Xu Zhang for fruitful discussions. We would
like to present a special thanks to Masatoshi Yamada for enlightening
discussions and comments. This work was supported in part by the National
Science Foundation of China (NSFC) under Grant No.11747308, 11975108,
12047569, and the Seeds Funding of Jilin University.

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