# First-order Electroweak phase transition at finite density

Renhui Qin<EMAIL_ADDRESS>Department of Physics and Chongqing Key
Laboratory for Strongly Coupled Physics, Chongqing University, Chongqing
401331, P. R. China  Ligong Bian111Corresponding Author<EMAIL_ADDRESS>Department of Physics and Chongqing Key Laboratory for Strongly Coupled
Physics, Chongqing University, Chongqing 401331, P. R. China  Center for High
Energy Physics, Peking University, Beijing 100871, China

###### Abstract

We study the Electroweak phase transition with the Standard Model effective
field theory at finite temperature and finite density. Utilizing the
dimensional reduction approach, we construct the tree dimensional thermal
effective field theory at finite density and investigate the phase transition
dynamics. We evaluate how the results depend on the renormalization scale and
the chemical potential. Our results show that, with the tree dimensional
thermal effective potential at 2-loop level, we can effectively reduce the
theoretical uncertainty in the calculations of the phase transition parameters
due to the renormalization scale dependence, and the new physics scale is
restricted to be $\Lambda\lesssim(770-800)$ GeV by the baryon number washout
avoidance condition. Meanwhile, the presence of the chemical potential would
affect the phase transition parameter and make the constraints from the baryon
number washout avoidance condition more strict, especially for weaker first-
order phase transition scenarios at higher new physics scales.

###### Contents

  1. I Introduction
  2. II Thermal Standard model effective field theory at finite density
    1. II.1 The Model
    2. II.2 Dimensional Reduction
    3. II.3 The $3d$ thermal effective potential
    4. II.4 Phase transition dynamics
    5. II.5 Electroweak sphaleron
  3. III First-order Phase transition parameters
    1. III.1 Phase transition temperatures
    2. III.2 Phase transition parameters for gravitational wave prediction
    3. III.3 Baryon number washout avoidance condition
  4. IV Conclusion
  5. V Acknowledgements
  6. A The realtions between $\overline{MS}$ parameters and physical observables
  7. B Parameters of dimensional reduction
  8. C Effective potential at the two-loop level
  9. D Sphaleron energy

August 30, 2024

##  I Introduction

First-order phase transition (PT) is a common prediction of many particle
physics models beyond the Standard Model [1, 2, 3], such transitions are a
crucial ingredient for an explanation of the baryon asymmetry of the Universe
through electroweak baryogenesis (EWBG)[4, 5, 6, 7], and can produce
gravitational wave (GW) signatures to be probed by space-based
interferometers[8, 9, 10]. When new physics is integrated out, one arrives at
the standard model effective field theory (SMEFT), where the potential barrier
between symmetric and broken phase can appear at tree-level potential, which
is mostly captured by the dimension-six operator of
$(\Phi^{\dagger}\Phi)^{3}/\Lambda^{2}$ with the $\Lambda$ indicating the new
physics (NP) scale. Previously, there are many studies on the electroweak PT
in the framework of SMEFT based on four-dimensional finite temperature thermal
field theory [11, 12, 13, 14, 15, 16, 17, 18, 19]. These works show that a lot
of parameter spaces of SMEFT can predict strong first-order PT to be probed at
near future GW detectors.

To analyze the electroweak PT dynamics with four-dimensional finite
temperature thermal field theory, daisy resummation is usually applied to
improve the power like infrared divergence (IR) problem [20, 21, 22]. Another
way to study the property of PT, the 3d dimensional reduction (DR) method has
been developed, which applies to the non-perturbative lattice simulation and
suffer less theoretical uncertainty from the IR problem [22, 20, 21, 23].
Previous studies show that the DR suffers the uncertainty of the
renormalization scale dependence which is reflected in the convergence of
perturbation theory [24, 25, 12, 26]. Meanwhile, the EWBG and detectable GW in
the mHz to Hz range require a strong first-order electroweak PT caused by a
strong coupling which will reduce the convergence, and its correction is more
pronounced at 2-loop level and higher orders. Therefore, in the procedure of
calculating electroweak PT with the 3d DR method, at least 2-loop level is
needed to reduce the theoretical uncertainty from the renormalization scale
dependence.

In this paper, we adopt the DR approach to study the electroweak PT within the
framework of the SMEFT at finite density. To ensure a positive-definite value
for the number of bosons, the chemical potential needs less than the mass
$\mu_{i}\leq m_{i}$[27]. The DR approach usually needs to integrate heavy mode
to build a light theory $(g^{2}T)$, so the chemical potential in this work
requires less than light scale $\mu_{i}\leq g^{2}T$. In this work, we consider
theoretical uncertainties caused by renormalization scale dependence and the
effect of the chemical potential on the dynamics of strong first-order
electroweak PT relevant for the GW prediction and EWBG. Since there is only a
single parameter (the NP scale $\Lambda$) to characterize the phase transition
behavior and the electroweak sphaleron energy, we can easily consider the
effect of both the energy scale and chemical potential simultaneously.

The paper is organized as follows. In Section.II, we give a brief introduction
to the procedure of the DR approach and construct the thermal SMEFT at finite
density, where we elaborate on the definition of the PT parameter and the
electroweak sphaleron. In Section.III, we study the dependence of the
renormalization scale and the effect of chemical potential on the PT in
detail, including the PT parameters relevant for GW prediction and the baryon
number washout avoidance condition. In the end, we conclude with the
Section.IV. Appendix A presents the Beta functions and the input parameters of
our calculation. Appendix B gives the final result of the parameters of the DR
approach. Appendix C gives the effective potential at 2-loop level. Appendix B
shows the effective potential at different $\Lambda$ with critical temperature
and nucleation temperature. Appendix D gives the detail of our calculation for
the electroweak sphaleron energy.

##  II Thermal Standard model effective field theory at finite density

In this section, we construct the thermal SMEFT at finite density to study the
PT dynamics around electroweak scales.

###  II.1 The Model

We use the SMEFT with only a single dimension-six operator to study the first-
order PT property. The Euclidean action is defined by

$S=\int_{0}^{\beta}\mathrm{d}\,\tau\int\mathcal{L}\mathrm{d}^{3}x\;,$ (1)

with

$\displaystyle\mathcal{L}=$
$\displaystyle-\frac{1}{4}G_{\mu\nu}^{a}G_{\mu\nu}^{a}-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+(D_{\mu}\Phi)^{\dagger}(D_{\mu}\Phi)-V(\Phi)+\overline{q}_{L}\not{D}q_{L}$
(2)
$\displaystyle+\overline{l}_{L}\not{D}l_{L}+\overline{e}_{R}\not{D}e_{R}+\overline{u}_{R}\not{D}u_{R}+\overline{d}_{R}\not{D}d_{R}+g_{Y}(\overline{q}_{L}\tilde{\Phi}t_{R}+\overline{t}_{R}u\tilde{\Phi}^{\dagger}q_{L})\;,$

and

$V(\Phi)=\mu_{h}^{2}\Phi^{\dagger}\Phi+\lambda(\Phi^{\dagger}\Phi)^{2}+c_{6}(\Phi^{\dagger}\Phi)^{3},\quad
c_{6}=\Lambda^{-2}\;.$ (3)

Here, the $\Lambda$ is the cut-off scale describing where the NP enters. The
covariant derivative is defined as
$D_{\mu}=\partial_{\mu}-igIA_{\mu}^{i}\tau^{i}-ig^{\prime}YB_{\mu}$, where $I$
and $Y$ are the weak isospin and weak hypercharge of the corresponding
doublet/singlet. The $l_{L}$ and $q_{L}$ denote the left-hand lepton and quark
doublet, and $e_{R},u_{R},d_{R}$ denote the right-hand lepton, up-type and
down-type quarks. We consider the top quark has non-zero mass with Yukawa
coupling $g_{Y}$, and $\tilde{\Phi}=i\sigma_{2}\Phi^{*}$, where $\Phi$ is the
SM Higgs doublet.

To keep the charge conservation in this system, one can choose the hypercharge
and the third component of the isospin. The corresponding currents are given
as [23, 28]:

$\displaystyle j_{\mu}^{Y}=$
$\displaystyle\frac{1}{2}\sum\left(\frac{1}{3}\overline{q}_{L}\gamma_{\mu}q_{L}+\frac{4}{3}\overline{u}_{R}\gamma_{\mu}u_{R}-\frac{2}{3}\overline{d}_{R}\gamma_{\mu}d_{R}-\overline{l}_{L}\gamma_{\mu}l_{L}-2\overline{e}_{R}\gamma_{\mu}e_{R}\right)-\frac{i}{2}[(D_{\mu}\Phi)^{\dagger}\Phi-\Phi^{\dagger}D_{\mu}\Phi],$
(4) $\displaystyle j_{\mu}^{3}=$
$\displaystyle\frac{1}{2}\sum(\overline{q}_{L}\gamma_{\mu}\tau^{3}q_{L}+\overline{l}_{L}\gamma_{\mu}\tau^{3}l_{L})-\frac{i}{2}[(D_{\mu}\Phi)^{\dagger}\tau^{3}\Phi-\Phi^{\dagger}\tau^{3}D_{\mu}\Phi]-\epsilon^{3bc}A_{\nu}^{b}F_{\mu\nu}^{c}\;.$

Then, the electromagnetic and weak-neutral currents are

$\displaystyle N_{1}=$
$\displaystyle\int\mathrm{d}^{3}x(j_{0}^{Y}+j_{0}^{3})\;,$ (5) $\displaystyle
N_{3}=$ $\displaystyle\int\mathrm{d}^{3}x(\sin^{2}\theta
j_{0}^{Y}-\cos^{2}\theta j_{0}^{3})\frac{2}{\cos 2\theta}\;,$

where the $\theta$ is the Weinberg angle. In addition, the lepton number
currents and baryon number currents are not conserved due to the triangle
anomaly[29]. But one can make the $N_{f}$ conserved linear combinations of the
currents for baryon and lepton by define[23]

$N_{2,i}=\frac{1}{N_{f}}B-L_{i},\quad i=1,...,N_{f}$ (6)

with

$\displaystyle B$
$\displaystyle=\frac{1}{3}\sum_{f,c}\int\mathrm{d}^{3}x\overline{q}_{f,c}\gamma_{0}q_{f,c}\;,$
(7) $\displaystyle L_{i}$
$\displaystyle=\int\mathrm{d}^{3}x\left(\overline{e}_{i}\gamma_{0}e_{i}+\frac{1}{2}\overline{\nu}_{i}\gamma_{0}(1-\gamma_{5})\nu_{i}\right).$

The $f$ and $c$ are flavor and color for quark fields. Then, the partition
function with chemical potential is given by

$\displaystyle\mathcal{Z}=$
$\displaystyle\mathrm{Tr}e^{-\beta(H-\mu_{k}N_{k})}$ (8) $\displaystyle=$
$\displaystyle\mathrm{Tr}e^{-\beta(H-\mu_{i}N_{2,i}-\mu_{Y}^{\prime}N_{1}-\mu_{T_{3}}N_{3})}$
$\displaystyle=$ $\displaystyle\int\mathcal{D}\varphi
e^{-S+\int_{0}^{\beta}\mathrm{d}\tau\sum_{i}^{N_{f}}\mu_{i}N_{2,i}}\;,$

with

$\sum_{i}^{N_{f}}\mu_{i}N_{2,i}=\frac{1}{N_{f}}(\mu_{1}+\mu_{2}+\mu_{3})B-(\mu_{1}L_{1}+\mu_{2}L_{2}+\mu_{3}L_{3})\;.$
(9)

Here, the $S$ contains the gauge fixing and ghost terms. After redefining the
temporal components of the gauge fields

$g^{\prime}B_{0}\rightarrow g^{\prime}B_{0}-i\mu_{Y},\quad
gA_{0}^{3}\rightarrow gA_{0}^{3}-i\mu_{T^{3}},$ (10)

the action given above becomes the standard electroweak action and
$\mu_{Y},\mu_{T^{3}}$ are not explicit in the path integral anymore, which
means that the chemical potentials and the static modes of the temporal
components of gauge fields enter the path integral in the same way.

Assuming that all the chemical potentials of the leptons have the same value

$\mu_{B}=\frac{1}{N_{f}}\sum_{i=1}^{N_{f}}\mu_{i},\quad\mu_{L_{i}}=-\mu_{i}\;,$
(11)

which results in $N_{f}\mu_{B}+\sum_{i}\mu_{i}=0$ which can remove the Chern-
Simons terms that appear due to B+L nonconservation. We then have

$\mathcal{Z}=\int\mathcal{D}\varphi
e^{-S+\int_{0}^{\beta}\mathrm{d}\tau\left(\mu_{B}B+\sum_{i=1}^{n_{f}}\mu_{L_{i}}L_{i}\right)}\;.$
(12)

###  II.2 Dimensional Reduction

The path integral given by Eq.(12) can not be evaluated with perturbation
theory because of the IR divergence caused by bosonic zero modes. The other
modes are massive with $m\sim\pi T$, so they can be integrated out safely. For
this reason, we use the DR approach to include the contribution of the bosonic
zero modes. DR can be described as equilibrium thermodynamics of a weakly
interacting quantum field theory in the Matsubara formalism of

$\phi(x)=T\sum_{n=-\infty}^{\infty}\int\frac{\mathrm{d}^{3}\,p}{(2\pi)^{3}}\phi(\omega_{n},\mathbf{p})e^{i\omega_{n}\tau}e^{-i\mathbf{p\cdot
x}}$ (13)

where $\omega_{n}=2n\pi T$ for bosons and $\omega_{n}=(2n+1)\pi T$ for
fermions. This theory works when the coupling constant is small $g<1$, and the
couplings are dimensionful. The structure of perturbative expansion is
modified by resummations. To build the theory, one needs a matching between 3d
and 4d fields and parameters. The matching in DR needs to integrate out
heavier Matsubara mode. Thus we have to define the energy scales in this
theory. There are three different energy scales $\pi T$,$gT^{2}$, and
$g^{2}T^{2}$, named superheavy, heavy, and light mode (or heavy, soft, and
ultrasoft mode, but we don’t use it in this paper). The superheavy modes are
fermions and non-zero boson Matsubara modes with mass $\sim\pi T$, the heavy
mode is the Debye mass with mass $\sim gT$, and the light mode is spatial
gauge and scalar fields. The purpose of DR is to build a model with only light
mode, thus avoiding the IR problem that appeared in finite temperature field
theory. In general, the DR needs two steps to obtain the thermal effective
field theory containing only the light mode.

The original Lagrangian in this paper is Eq.(2). The first step is to build a
theory with heavy and light modes, that should integrate out fermion and non-
zero boson modes. With the accuracy $\mathcal{O}(g^{4})$, the power counting
we used in this paper are $g^{\prime}\sim g,g_{Y}\sim g,\lambda\sim
g^{2},c_{6}\sim g^{4}/\Lambda^{2}$. The $c_{6}$ term is non-renormalizable and
contains the NP scale. After integrating out superheavy mode, the effective
theory has the form

$S^{{\mbox{\tiny\rm{heavy}}}}_{{\mbox{\tiny\rm{3d}}}}=\int{\rm
d}^{3}x\biggl{[}\frac{1}{4}G_{ij}^{a}G_{ij}^{a}+\frac{1}{4}F_{ij}F_{ij}+\frac{1}{2}(D_{i}A_{0}^{a})^{2}+\frac{1}{2}(\partial_{i}B_{0})^{2}+\frac{1}{2}(D_{i}C_{0}^{\alpha})^{2}+(D_{i}\Phi)^{\dagger}(D_{i}\Phi)+V^{{\mbox{\tiny\rm{heavy}}}}_{{\mbox{\tiny\rm{3d}}}}\biggr{]}\;,$
(14)

where
$G^{a}_{ij}=\partial_{i}A^{a}_{j}-\partial_{j}A^{a}_{i}+g_{3}\epsilon^{abc}A_{i}^{b}A_{j}^{c}$,
$F_{ij}=\partial_{i}B_{j}-\partial_{j}B_{i}$ and
$D_{i}\Phi=(\partial_{i}-ig_{3}\tau^{a}A_{i}^{a}/2-ig^{\prime}_{3}B_{i}/2)\Phi$
with $\tau^{a}$ being the Pauli matrices. The heavy scalar potential has the
form

$\displaystyle V^{{\mbox{\tiny\rm{heavy}}}}_{{\mbox{\tiny\rm{3d}}}}=$
$\displaystyle\mu_{h,3}^{2}\Phi^{\dagger}\Phi+\lambda_{3}(\Phi^{\dagger}\Phi)^{2}+c_{6,3}(\Phi^{\dagger}\Phi)^{3}$
(15)
$\displaystyle+\frac{1}{2}m_{\mbox{\tiny\rm{D}}}^{2}\,A^{a}_{0}A^{a}_{0}+\frac{1}{2}m_{\mbox{\tiny\rm{D}}}^{\prime
2}\,B_{0}^{2}+\frac{1}{2}m_{\mbox{\tiny\rm{D}}}^{\prime\prime
2}\,C^{\alpha}_{0}C^{\alpha}_{0}$
$\displaystyle+\frac{1}{4}\kappa_{1}(A^{a}_{0}A^{a}_{0})^{2}+\frac{1}{4}\kappa_{2}B_{0}^{4}+\frac{1}{4}\kappa_{3}A^{a}_{0}A^{a}_{0}B_{0}^{2}$
$\displaystyle+h_{1}\Phi^{\dagger}\Phi
A^{a}_{0}A^{a}_{0}+h_{2}\Phi^{\dagger}\Phi
B_{0}^{2}+h_{3}B_{0}\Phi^{\dagger}{A}_{0}^{a}\tau^{a}\Phi+h_{4}\Phi^{\dagger}\Phi
C^{\alpha}_{0}C^{\alpha}_{0}$
$\displaystyle+\alpha_{0}\epsilon_{ijk}\left(A_{i}^{a}G_{jk}^{a}-\frac{i}{3}g_{3}\epsilon^{abc}A_{i}^{a}A_{i}^{b}A_{k}^{c}\right)+\alpha^{\prime}\epsilon_{ijk}B_{i}F_{jk}$
$\displaystyle+\kappa_{0}B_{0}+\rho\Phi^{\dagger}A_{0}^{a}\tau^{a}\Phi+\rho^{\prime}\Phi^{\dagger}\Phi
B_{0}+\rho_{G}B_{0}A_{0}^{a}A_{0}^{a}\;,$

where
$m_{\mbox{\tiny\rm{D}}},m_{\mbox{\tiny\rm{D}}}^{\prime},m_{\mbox{\tiny\rm{D}}}^{\prime\prime}$
are the Debye masses of the time component of gauge fields. The matching
between 3d and 4d fields can be obtained by considering the momentum-dependent
contribution of the superheavy modes to the 2-point Green function, and have
the form

$\displaystyle\phi_{3d}^{2}$
$\displaystyle=\frac{1}{T}Z_{\phi}\phi_{4d}^{2}\;,$ (16) $\displaystyle
A_{i,3d}^{2}$ $\displaystyle=\frac{1}{T}Z_{A_{i}}A_{i,4d}^{2}\;,$
$\displaystyle A_{0,3d}^{2}$
$\displaystyle=\frac{1}{T}Z_{A_{0}}A_{0,4d}^{2}\;,$

with

$\displaystyle Z_{\phi}^{\prime}$
$\displaystyle=1+\frac{1}{16\pi^{2}}\left(-\frac{3}{4}(3g^{2}+(g^{\prime})^{2})L_{b}+3g_{Y}^{2}L_{f}\right)\;,$
(17) $\displaystyle Z_{\phi}$
$\displaystyle=Z_{\phi}^{\prime}\left(1-\frac{3g_{Y}^{2}}{16\pi^{2}}\mathcal{A}\left(\frac{\mu_{B}}{3}\right)\right)\;,$
$\displaystyle Z_{A_{i}}^{\prime}$
$\displaystyle=1+\frac{g^{2}}{16\pi^{2}}\left(-\frac{25}{6}L_{b}-\frac{2}{3}+\frac{4}{3}N_{F}L_{f}\right)\;,$
$\displaystyle Z_{A_{i}}$
$\displaystyle=Z_{A_{i}}^{\prime}\left(1-\frac{g^{2}}{48\pi^{2}}\left[9\mathcal{A}\left(\frac{\mu_{B}}{3}\right)+\sum_{i=1}^{3}\mathcal{A}(\mu_{L_{i}})\right]\right)\;,$
$\displaystyle Z_{A_{0}}^{\prime}$
$\displaystyle=1+\frac{g^{2}}{16\pi^{2}}\left(-\frac{25}{6}L_{b}+3+\frac{4}{3}N_{F}(L_{f}-1)\right)\;,$
$\displaystyle Z_{A_{0}}$
$\displaystyle=Z_{A_{0}}^{\prime}\left(1-\frac{g^{2}}{48\pi^{2}}\left[9\mathcal{A}\left(\frac{\mu_{B}}{3}\right)+\sum_{i=1}^{3}\mathcal{A}(\mu_{L_{i}})\right]\right)\;,$

where the functions of $\mathcal{A}(x)$ describe the contribution of the
chemical potential defined in the Eq.(49)[23]. Then, the relation between the
3d and 4d couplings can be obtained by vanishing external momenta contribution
from superheavy mode to Green functions, the Debye masses can be obtained in
the same way. Here, the Chern-Simons term $\alpha_{0}$ and $\alpha^{\prime}$
vanished by the assumption of Eq. 11. See Appendix B for these parameters
formula.

The second step is to build a theory with only light mode. At the light scale,
the fields $A_{0,3d},B_{0,3d}$ and $C_{0,3d}$ are heavier and need to be
integrated out as well. The action of light scale reads

$S^{{\mbox{\tiny\rm{light}}}}_{{\mbox{\tiny\rm{3d}}}}=\int{\rm
d}^{3}x\biggl{[}\frac{1}{4}G_{ij}^{a}G_{ij}^{a}+\frac{1}{4}F_{ij}F_{ij}+(D_{i}\Phi)^{\dagger}(D_{i}\Phi)+V^{{\mbox{\tiny\rm{light}}}}_{{\mbox{\tiny\rm{3d}}}}\biggr{]}\;,$
(18)

with

$V^{{\mbox{\tiny\rm{light}}}}_{{\mbox{\tiny\rm{3d}}}}=\overline{\mu}_{h,3}^{2}\Phi^{\dagger}\Phi+\overline{\lambda}_{3}(\Phi^{\dagger}\Phi)^{2}+\overline{c}_{6,3}(\Phi^{\dagger}\Phi)^{3}$
(19)

and this theory is valid up to momenta $g^{2}T<p<gT$. There is no IR problem
due to these fields are massive. We list the couplings of heavy scale eq.(14)
and light scale eq.(18) in Appendix B.

Once we obtain the final light theory, the advantage of DR will be revealed.
The light theory is simpler than the 4d theory with fermions and bosons whose
masses are larger than $gT$ being integrated out. And all sum integrals have
been evaluated. The temperature only enters the parameters of the light
theory.

###  II.3 The $3d$ thermal effective potential

We construct the thermal 3d effective potential $V_{eff}$ using the light
scale theory given by the Eq.(18), where the non-zero Matsubara modes and the
temporal components of the gauge fields have been integrated out. Through the
process, the renormalization scale and the chemical potential enter the mass
parameters of $\bar{\mu}_{h,3}$ and the 3d couplings of
$\bar{g}_{3},\bar{g}^{\prime}_{3}$,$\bar{\lambda}_{3}$ and $\bar{c}_{6}$, See
Appendix.B for details. The effective potential up to the 2-loop order level
has the same form as the SM, and the $(\Phi^{\dagger}\Phi)^{3}/\Lambda^{2}$
term leads to corrections to the parameters in the effective potential. We use
the $\hbar-$expansion to expand the thermal effective potential

$V_{eff}=V_{3(0)}+\hbar V_{3(1)}+\hbar^{2}V_{3(2)}.$ (20)

After insert the background field to eq.(19), the tree-level potential is

$\displaystyle V_{3(0)}$
$\displaystyle=\frac{1}{2}\bar{\mu}_{h,3}^{2}\phi_{3}^{2}+\frac{1}{4}\bar{\lambda}_{3}\phi_{3}^{4}+\frac{1}{8}\bar{c}_{6,3}\phi_{3}^{6}\;.$
(21)

The 1-loop level potential in the Landau gauge is

$\displaystyle
V_{3(1)}=J_{{\mbox{\tiny\rm{soft}}}}(m_{\phi,3})+3J_{{\mbox{\tiny\rm{soft}}}}(m_{\chi,3})+(d-1)\Big{(}2J_{{\mbox{\tiny\rm{soft}}}}(m_{W,3})+J_{{\mbox{\tiny\rm{soft}}}}(m_{Z,3})\Big{)}\;,$

with

$\displaystyle
J_{{\mbox{\tiny\rm{soft}}}}(m)=-\frac{1}{12\pi}(m^{2})^{\frac{3}{2}}\;.$ (22)

Therein, the field dependent masses are

$\displaystyle m^{2}_{\phi,3}$
$\displaystyle=\bar{\mu}_{h,3}^{2}+3\bar{\lambda}_{3}\phi^{2}_{3}+\frac{15}{4}\bar{c}_{6,3}\phi^{4}_{3}\;,$
(23) $\displaystyle m^{2}_{\chi,3}$
$\displaystyle=\bar{\mu}_{h,3}^{2}+\bar{\lambda}_{3}\phi^{2}_{3}+\frac{3}{4}\bar{c}_{6,3}\phi^{4}_{3}\;,$
(24) $\displaystyle m^{2}_{W,3}$
$\displaystyle=\frac{1}{4}\bar{g}_{3}^{2}\phi^{2}_{3}\;,$ (25) $\displaystyle
m^{2}_{Z,3}$
$\displaystyle=\frac{1}{4}\Big{(}\bar{g}_{3}^{2}+\bar{g}_{3}^{\prime
2}\Big{)}\phi^{2}_{3}\;.$ (26)

Since contribution from $\bar{c}_{6,3}$ appear through the scalar mass
$m_{\chi,3}$ and $m_{\phi,3}$ and scalar self-interaction, the $\bar{c}_{6,3}$
term only affects (SS) and (SSS) part in the 2-loop potential of

$\displaystyle
V_{3(2)}=-\Big{(}\text{(SSS)}+\text{(VSS)}+\text{(VVS)}+\text{(VVV)}+\text{(VGG)}+\text{(SS)}+\text{(VS)}+\text{(VV)}\Big{)}\;,$
(27)

Different parts’ contributions to the 2-loop effective potential are listed in
Appendix.C.

###  II.4 Phase transition dynamics

The Universe first lives in the “symmetric” phase, and as the Universe cools
down, the “symmetric” and “broken” phases have the same free energy at the
critical temperature, and it can obtained by solving

$V_{eff}(\phi_{c},T_{c})=V_{eff}(0,T_{c}),\quad\left.\frac{\partial
V_{eff}(\phi,T)}{\partial\phi}\right|_{\phi=\phi_{c}}=0\;.$ (28)

In particular, when the Universe cools further, the vacuum bubbles of the true
phase nucleated in the symmetric phase. The nucleation temperature $T_{n}$ is
obtained when the bubble nucleation rate $\Gamma=A\exp[-S_{c}]$ is equal to
Hubble parameter $\Gamma\sim H$, i.e., $S_{c}\approx 140$ [30]. The Euclidean
action is

$S_{c}=\int\bigg{[}\frac{1}{2}(\partial_{i}\phi)^{2}+V_{eff}(\phi,T)\bigg{]}d^{3}x.$
(29)

The action can be obtained after the “bounce solution” acquired from the
equations of motion:

$\frac{d^{2}\phi}{d\rho^{2}}+\frac{2}{\rho}\frac{d\phi}{d\rho}=\frac{dV_{eff}(\phi,T)}{d\phi}\;,$
(30)

with the boundary condition

$\phi(\rho\rightarrow\infty)=0,\quad\left.\frac{d\phi}{d\rho}\right|_{\rho=0}=0\;.$
(31)

We use code “findbounce” to solve this equation and obtain the nucleation
temperature $T_{n}$ and the corresponding background field value $\phi_{n}$
[31]. The inverse PT duration is defined as:
$\beta/H_{n}=T_{n}(dS_{c}/dT)|_{T_{n}}$. The PT temperature and the duration
determine the peak frequency of the produced GW from PT [32, 33, 34], and the
trace anomaly ($\alpha$) usually determines the amplitude of the generated GW.
For the $4d$ theory, the $\alpha$ is defined as $\alpha=\Delta\rho/\rho_{rad}$
with

$\Delta\rho=\Delta V_{4d}(\phi_{n},T_{n})+\frac{1}{4}\left.T_{n}\frac{d\Delta
V_{4d}(\phi_{n},T)}{dT}\right|_{T=T_{n}}\;.\\\ $ (32)

After apply the relation between $4d$ and $3d$ potential $V_{4d}\approx
TV_{eff}$, we have $\alpha=T(\Delta\rho/\rho_{rad})$ with

$\Delta\rho=-\frac{3}{4}\Delta
V_{eff}(\phi_{n},T_{n})+\frac{1}{4}\left.T_{n}\frac{d\Delta
V_{eff}(\phi_{n},T)}{dT}\right|_{T=T_{n}}\;,$ (33)

where $\Delta V_{eff}(\phi,T)=V_{eff}(\phi,T)-V_{eff}(0,T)$ and
$\rho_{rad}=\pi^{2}g_{*}T_{n}^{4}/30$, $g_{*}=106.75$ is the effective number
of relativistic degrees of freedom [12].

###  II.5 Electroweak sphaleron

In the electroweak theory, the sphaleron is a static and unstable solution of
the field equations and corresponds to the top of the energy barrier between
two topologically distinct vacua [35, 36]. The electroweak sphaleron process
provides the baryon number violation condition (one of the three Sakharov
conditions [37]) to explain the baryon asymmetry of the Universe [4, 38, 7].
The electroweak sphaleron energy with $U(1)_{Y}$ contribution reads [39, 40]:

$\displaystyle E_{sph}^{EW}(\nu,T)=$ $\displaystyle\frac{4\pi
v}{g}\frac{v(T)}{v}\int_{0}^{\infty}d\xi\left\\{\sin^{2}{\nu}\left(\frac{8}{3}f^{\prime
2}+\frac{4}{3}f^{\prime
2}_{3}\right)+\sin^{4}{\nu}\frac{8}{\xi^{2}}\left[\frac{2}{3}f_{3}^{2}(1-f)^{2}\right.\right.$
(34)
$\displaystyle\left.+\frac{1}{3}(f(1-f)+f-f_{3})^{2}\right]+\frac{4}{3}\left(\frac{g}{g^{\prime}}\right)^{2}\left[\sin^{2}{\nu}f_{0}^{\prime
2}+\sin^{4}{\nu}\frac{2}{\xi^{2}}(1-f_{0})^{2}\right]$
$\displaystyle+\frac{1}{2}\xi^{2}h^{\prime
2}+\sin^{2}{\nu}h^{2}\left[\frac{1}{3}(f_{0}-f_{3})^{2}+\frac{2}{3}(1-f)^{2}\right]+\left.\frac{\xi^{2}}{g^{2}v(T)^{4}}V(h,T)\right\\}\;,$

where $\nu$ is related to Chern-Simons number through
$N_{CS}=\frac{2\nu-\sin{2\nu}}{2\pi}$. This sphaleron energy is valid in
$\nu\in[0,\pi]$ due to the $U(1)_{Y}$ gauge field will reduce the $SO(3)$
spherical symmetry to axial symmetry. Here, $h,f,f_{0},f_{3}$ are functions
that enter the Higgs and gauge field configurations, and $\xi$ is a
dimensionless parameter with $\xi=gvr$, for more details see Appendix. D. In
our work, we neglect the contribution of magnetic dipole moment. Here, we note
that the gauge invariant Higgs condensate $\langle\Phi^{\dagger}\Phi\rangle$
is related to the dimensionless quantity (the ratio between the Higgs vacuum
expectation value (v) and the temperature) during the PT through
$v/T=\sqrt{2\langle\Phi^{\dagger}\Phi\rangle}/\sqrt{T}$, in this work we
straightforwardly calculate the Higgs condensate as
$\langle\Phi^{\dagger}\Phi\rangle=dV_{eff}/d\overline{\mu}_{h,3}^{2}$ which
applies when the Coleman-Weinberg type radiative correction is important, the
spontaneous symmetry breaking is not necessarily to occur on the tree-level
potential [20].

##  III First-order Phase transition parameters

To study the renormalization scale dependence of the PT dynamics with the DR
approach, we consider three energy scales for $\overline{\mu}$:

$\displaystyle\pi T$ the lowest fermionic mode (35) $\displaystyle 2\pi T$ the
lowest nonzero bosonic mode $\displaystyle 4\pi e^{-\gamma_{E}}T$ the lowest
logarithmic contribution

To demonstrate the effects from the chemical potential, we take
$-\mu_{B}=\mu_{L_{i}}\equiv\mu_{ch}$ and consider two benchmark scenarios of
$\mu_{ch}=0.1$T and $\mu_{ch}=0.2$T.

###  III.1 Phase transition temperatures

Figure 1: The renormalization scale dependence and the effects of the chemical
potential in the critical temperature (upper panels) and nucleation
temperature $T_{n}$ (lower panels).

In Fig.1, we present the renormalization scale dependence and the effect of
the chemical potential on the critical temperature $T_{c}$ and the bubble
nucleation temperature $T_{n}$ at different $\Lambda$ of the first-order PT.
The renormalization scale dependence of $T_{c,n}$ at tree-level is very
significant, and this dependence is reduced at higher loop orders, This scale
dependence will lead to a large uncertainty, and make the result of tree-level
unreliable. Generally, the magnitude of the $T_{c}$ and $T_{n}$ decrease as
the calculation precision increases from one-loop to two-loop level. The
magnitudes of the $T_{n}$ and $T_{c}$ increase with the increase of the NP
scale, and decrease with the increase of precision of the calculation from
tree-level to two-loop level. The renormalization scale dependence have
$5\%$(one-loop) and $1\%$(two-loop) differences between the scenarios of
renormalization scale at $\pi T$ and $4\pi e^{-\gamma_{E}}T$. The effects of
the chemical potential on $T_{n}$ and $T_{c}$ are much smaller relative to the
renormalization scale dependence at tree- and one-loop levels. Since the
renormalization scale dependence is significantly suppressed at the two-loop
level, the effect of the chemical potential is relatively significant and
needs to be considered there, see the right two plots of Fig. 1 where we set
$\overline{\mu}=4\pi e^{-\gamma_{E}}T$ as an illustration. The nucleation
temperature $T_{n}$ ranges from $65$ GeV to $125$ GeV at the two-loop level
when the $\Lambda$ grows from 600 GeV to 900 GeV. The appearance of the
chemical potential reduces (amplifies) the magnitude of the $T_{c}$ and
$T_{n}$ at small (large) $\Lambda$. We observe around $1\%-2\%$ differences
between the cases of $\mu_{ch}=0$ and $\mu_{ch}=0.2$T at each order at large
$\Lambda$.

###  III.2 Phase transition parameters for gravitational wave prediction

Figure 2:  The renormalization scale dependence and chemical potential in the
trace anomaly $\alpha$ for left and right plots.

In Fig.2, we illustrate the behaviors of the trace anomaly $\alpha$ at the
tree-level and loop levels. When $\Lambda\sim 680$ GeV, the values of the
$\alpha$ at tree level and loop levels are the same. When the $\Lambda$ is
smaller, the magnitude of the $\alpha$ at tree level value will be larger than
that of loop levels, and vice versa when $\Lambda\gtrsim 680$ GeV. This
phenomenon is because the contributions of loop level grow to be more
important at a larger NP scale when $\Lambda\gtrsim 680$ GeV, see Figs.(11,12)
for illustration. Basically, this is because that the loop contributions of
$V_{3(1,2)}$ would cancel the tree-level’s contribution ($V_{3(0)}$) to the
potential barrier $\Delta V_{eff}(\phi,T)$. In the whole range of $\Lambda$,
the effect of the renormalization scale dependence decreases with the increase
of the precision of the calculation from the tree-level to the 2-loop level.
The chemical potential would affect the nucleation temperature and therefore
can affect the minimal of the thermal effective potential, and in turn,
affects the trace anomaly $\alpha$. This effect will be enlarged as the
increase of the NP scale $\Lambda$. Our results show that the trace anomaly
decreases with increasing chemical potential and that the effect of chemical
potential is more important for weak PT (with small trace anomaly) at large
$\Lambda$. The values of the trace anomaly ranges within the region of
$\alpha\sim[\mathcal{O}(10^{-3})-\mathcal{O}(10^{-1})]$ for
$\Lambda\sim[600,900]$ GeV, and the differences between the cases of
$\mu_{ch}=0$ and $\mu_{ch}=0.2\leavevmode\nobreak\ T$ at 2-loop level increase
from around $\sim 5\%$ to $\sim 35\%$ as the $\Lambda$ grows.

Figure 3: The magnitudes of $\beta/H_{n}$ as function of $\Lambda$. Left: The
renormalization scale dependence on $\beta/H_{n}$. Right: The effect of
chemical potential on $\beta/H_{n}$.

Figure.3 shows how the $\beta/H_{n}$ depends on the renormalization scale
dependence and the chemical potential. As in the PT temperatures of
$T_{c},T_{n}$ and the trace anomaly $\alpha$, the renormalization scale
dependence of $\beta/H_{n}$ is magnificent at the tree-level, and the effect
of the chemical potential grows as the NP scale increases. The difference
between the value of $\beta/H_{n}$ at loop levels grows to be bigger and
bigger than that of the tree-level as the $\Lambda$ increases. At small
$\Lambda$ region, the values of one- and two-loop level of $\beta/H_{n}$ are
relatively close to each other around
$\mathcal{O}(10^{2})-\mathcal{O}(10^{4})$, while the difference between these
two values is significant at large $\Lambda$ region. The $\beta/H_{n}$ is
insensitive to renormalization scale dependence at loop levels, but there are
$\sim 37\%$ and $\sim 80\%$ differences between the renormalization scale at
$\pi T$ and $4\pi e^{-\gamma_{E}}T$ at tree-level for large $\Lambda$ regions.
The chemical potential has no significant effect on $\beta/H_{n}$ when
$\Lambda$ is small, but it has an impact of about $20\%$ differences between
the $\mu_{ch}=0$ and $\mu_{ch}=0.2T$ at large $\Lambda$ region at loop levels.

Recently, it was found that the $\alpha$ and $\beta/H$ are not independent
parameters of first-order PT [41, 42]. In the thin-wall limit, a polynomial
dependence of $\beta/H$ on $\alpha$ has been extracted as [41]:

$\beta/H=T\frac{d}{dT}\frac{S_{c}}{T}\propto\frac{S_{c}}{T}\propto\alpha^{-2}.$
(36)

Figure 4: $\beta/H$ as function of $\alpha$. Left: the effect of
renormalization scale dependence on the relation between $\beta/H$ and
$\alpha$. Right: the effect of chemical potential on the association between
$\beta/H$ and $\alpha$. The dashed lines correspond to the thin-wall
approximation with a constant set to $10^{0.6}$(green) and $10^{-1}$(purple).
The gradient color indicates the cut-off scale $\Lambda$.

In Figure.4, with the numerical results of mean bubble separation and the PT
strength at two-loop level, we present the relation between the $\beta/H_{n}$
and $\alpha$, which shows that the prediction of Eq. 36 is not really tenable
here. We examine the thin-wall limit in Fig.5. It shows that the potential
barrier height ($\Delta V_{eff}^{max}$) is smaller than the vacuum energy
difference ($\Delta V_{eff}^{min}$), which makes the thin-wall approximation
no longer applicable at the nucleation temperature $T_{n}$.

Figure 5: The ratio between the potential barrier height and the potential
energy difference as a function of $\Lambda$ at two-loop order. These figures
are plotted at the nucleation temperature $T_{n}$. Left: the renormalization
scale dependence of ratio without considering chemical potential. Right: the
effect of chemical potential on the ratio at $\overline{\mu}=4\pi
e^{-\gamma_{E}}T$.

###  III.3 Baryon number washout avoidance condition

The sphaleron process will reach equilibrium at high temperature, and it
should be strongly suppressed at low temperature to avoid washout of the
generated baryon number which corresponds to the requirement of a strongly-
first order electroweak PT [43], i.e.
$\Gamma_{sph}\varpropto\exp{(-E_{sph}^{EW}/T)}\ll H(T)$ with $E_{sph}^{EW}$
being the electroweak sphaleron energy, this requires that the sphaleron rate
in the broken phase is lower than the Hubble rate. After reorganizing the
relation among the sphaleron energy, PT temperature ($T_{n}$), and the scalar
VEV, the baryon number washout avoidance condition can be obtained to restrict
the baryon asymmetry explanation parameter spaces [44, 45]:

$PT_{sph}\equiv\frac{E_{sph}}{T}-7\ln\frac{v}{T}+\ln\frac{T_{n}}{100\text{GeV}}>(35.9-42.8)\;,$
(37)

Figure 6: The quantity $v/T$ as a function of $\Lambda$. Left: the
renormalization scale dependence of $v/T$ at $T_{n}$. Right: the effect of
chemical potential on $v/T$ with $\overline{\mu}=4\pi e^{-\gamma_{E}}T$ at
$T_{n}$.

In the two plots of Fig. 6, we present the values of $v/T$ at loop levels that
are dramatically different from that of the tree-level. The so-called strong
first-order PT condition in literatures $v/T\geq 1$ is stasified in all the
regions of $\Lambda$ under study, while $v/T\leq 1$ when $\Lambda\gtrsim 800$
GeV at loop levels. The bottom-left plot illustrates the renormalization scale
dependence at loop levels is weaker than at tree level. The right one
demonstrates the effect of the chemical potential is stronger at large
$\Lambda$ region. These two plots show that, at loop levels, the effect of the
chemical potential is slightly more pronounced compared to the renormalization
scale dependence in the large $\Lambda$ region.

Figure 7: The renormalization scale dependence and the effects of the chemical
potential in the sphaleron energy at different NP scales $\Lambda$.

In the figure.7, the renormalization scale dependence at 2-loop level is very
weak and can be ignored. The sphaleron energy at 2-loop grows to be larger
than that at 1-loop level after $\Lambda\gtrsim 700$ GeV. The chemical
potential reduces the sphaleron energy to be lower, and this effect at large
$\Lambda$ is stronger than the case with small $\Lambda$. This effect is
caused by the effect of chemical potential on the thermal effective potential,
see Figs.(11, 12) in Appendix C for illustration where the chemical potential
significantly changes the location of the minimum value of the potential
energy at large $\Lambda$. At the same time, we want to point out that the
effect of the chemical potential on the sphaleron solution is negligible, see
Fig. D in Appendix C for an illustration.

Figure 8: The quantity $PT_{sph}$ as a function of $\Lambda$. Left: the
renormalization scale dependence of $PT_{sph}$ with $\mu_{ch}=0$. Right: the
effect of chemical potential on $PT_{sph}$ with $\overline{\mu}=4\pi
e^{-\gamma_{E}}T$.

Fig.8 shows the renormalization scale dependence and the effect of chemical
potential on the behavior of the quantity $PT_{sph}$. The gray region
represents the uncertainty in Eq.(37). Above the gray region, the sphaleron
process is sufficiently suppressed to satisfy the requirement that the
sphaleron rate be lower than the Hubble rate. The two plots demonstrate that
the higher the NP scale $\Lambda$, the more pronounced the effects of the
renormalization scale and chemical potential dependences. As shown in the left
panel, the renormalization scale dependence of $PT_{sph}$ at loop levels is
weaker than that of the tree level. Comparatively, at the loop levels, the
effect of the chemical potential on the $PT_{sph}$ is stronger than the
theoretical uncertainty caused by the renormalization scale dependence, but
the opposite is true at the tree level. The increase of the chemical potential
yields the decrease of the value of $PT_{sph}$ since the magnitude of the
$E_{sph}/T$ decreases as $\mu_{ch}$ increases and $\ln(v/T)$ is not so
sensitive to $\mu_{ch}$. This leads to more tight constraints on the NP scale.

In this study, at the 2-loop level, the baryon number washout avoidance
condition requires the NP scale to be $\Lambda\lesssim(750-780)$ GeV for
$\mu_{ch}=0.2$ T and $\Lambda\lesssim(770-800)$ GeV for $\mu_{ch}=0$.
Meanwhile, the strongly first-order PT condition yields $\Lambda\lesssim
800(830)$ GeV for the scenario without (with) considering the effect of the
chemical potential. For the case without considering the chemical potential,
the washout avoidance condition yields slightly looser constraints on the NP
scale $\Lambda$ than that of Ref.[44]. This is mainly caused by the
corrections of loop levels.

##  IV Conclusion

In this paper, we have studied the renormalization scale dependence and the
chemical potential effect of phase transition, and the strong first-order PT
through the dimensional reduction approach. The results of this paper are that
the dimensional reduction at 2-loop level reduces the theoretical uncertainty
caused by renormalization scale dependence very well. The chemical potential
has a greater effect on the weak PT region than it does on the strong PT
region.

We find that when the NP scale $\Lambda\gtrsim 680$ GeV, the loop level
contributions become more and more important for the thermal effective
potential. The renormalization scale dependence affects the thermal effective
potential more significantly than that of PT temperature $T$, and the trace
anomaly (or the PT strength) $\alpha$ is very significantly affected by the
renormalization scale dependence. The effect of the chemical potential on the
PT temperature is different at different ranges of $\Lambda$. At small
$\Lambda$, the PT temperature increases with increasing chemical potential in
most first-order PT regions of $\Lambda$. The chemical potential will reduce
the PT strength and sphaleron energy to be lower. The chemical potential
affects the baryon number washout avoidance condition mainly through the
sphaleron energy, and its effect becomes evident in the weak PT region at
large NP scales. This phenomenon becomes more pronounced as $\Lambda$
increases due to the effect of the dimension-six operator becomes weaker and
the effect of chemical potential becomes more prominent.

Our calculations suggest that, at two-loop level, the nucleation temperature
$T_{n}$ would be lower than that of the tree-level around $\mathcal{O}(1-10)$
GeV, and the inverse PT duration grows several times than that of the tree-
level. Therefore, the peak frequency of the predicted gravitational wave would
be shifted to more higher frequency from tree-level to loop levels since the
peak frequency is proportional to the value of $\beta/H$. Our study also shows
that the PT strength would be greater (smaller) than that of the tree-level
when $\Lambda$ is smaller(greater) than $\sim 680$ GeV, therefore the
predicted GW signal would be suppressed (amplified) since the amplitude of the
GW from PT is mostly characterized by the PT strength. In comparison with the
strong first-order PT condition, the baryon number washout avoidance condition
yields more stringent constraints on the NP scale, with
$\Lambda\lesssim(770-800)$ GeV at two-loop level, and the constraints will
become more rigorous when the chemical potential is taken into account.

##  V Acknowledgements

This work is supported by the National Key Research and Development Program of
China under Grant No. 2021YFC2203004. L.B. is supported by the National
Natural Science Foundation of China (NSFC) under Grants Nos. 12075041,
12147102, 12322505. L.B. also acknowledges Chongqing Talents: Exceptional
Young Talents Project No. cstc2024ycjh-bgzxm0020..

##  Appendix A The realtions between $\overline{MS}$ parameters and physical
observables

We use the dimensional regularisation in the $\overline{MS}-$scheme in this
work. In the Landau gauge, the $\beta-$functions at one-loop order read

$\displaystyle\overline{\mu}\frac{d}{d\overline{\mu}}\mu_{h}^{2}$
$\displaystyle=\frac{1}{(4\pi)^{2}}\mu_{h}^{2}\left(-\frac{3}{4}(3g^{2}+g^{\prime
2})+12\lambda+6g_{Y}^{2}\right),$ (38)
$\displaystyle\overline{\mu}\frac{d}{d\overline{\mu}}\lambda$
$\displaystyle=\frac{1}{(4\pi)^{2}}\left(\frac{3}{8}(3g^{4}+g^{\prime
4}+2g^{2}g^{\prime
2})+24\lambda^{2}-6g_{Y}^{4}+24c_{6}\mu_{h}^{2}-3\lambda(3g^{2}+g^{\prime
2}-4g_{Y}^{2})\right),$ (39)
$\displaystyle\overline{\mu}\frac{d}{d\overline{\mu}}c_{6}$
$\displaystyle=\frac{1}{(4\pi)^{2}}c_{6}(108\lambda-\frac{9}{2}(3g^{2}+g^{\prime
2})+18g_{Y}^{2}).$ (40)

With these beta functions, one can obtain these couplings at different energy
scales that will appear in the 3d parameters after integrating out the heavy
and super-heavy modes. We relate the $\overline{MS}-$parameters to physical
observables by input Fermi constant $G_{F}=1.1663787\times
10^{-5}\text{GeV}^{-2}$ and pole mass[46]

$(M_{t},M_{W},M_{Z},M_{h})=(172.76,80.379,91.1876,125.1)\leavevmode\nobreak\
\text{GeV}.$ (41)

After use the shorthand notation $g_{0}^{2}=4\sqrt{2}G_{\mu}M_{W}^{2}$, the
tree level relation for gauge and Yukawa couplings($g_{Y}$) are read

$g^{2}=g_{0}^{2},\quad g^{\prime
2}=g_{0}^{2}\left(\frac{M_{Z}^{2}}{M_{W}^{2}}-1\right),\quad
g_{Y}=\sqrt{\frac{1}{2}g_{0}^{2}\frac{M_{t}^{2}}{m_{W}^{2}}}.$ (42)

and $v_{0}^{2}=4M_{W}^{2}/g_{0}^{2}=(246.22\text{GeV})^{2}$ for the tree level
VEV. We set the strong coupling constant $g_{s}^{2}=4\pi\alpha_{s}(m_{z})$
with $\alpha_{s}(m_{z})=0.1180$ and neglect its runing[46].

The tree-level Higgs mass parameter and self-interaction can be obtained by
solving the

$\left.\frac{\partial^{2}V_{0}}{\partial\phi^{2}}\right|_{\phi=v_{0}}=M_{h}^{2},\quad\left.\frac{\partial
V_{0}}{\partial\phi}\right|_{\phi=v_{0}}=0$ (43)

where $V_{0}$ is defined in eq.(21). Then we can obtain

$\mu_{h}^{2}=-\frac{1}{2}M_{h}^{2}+\frac{3}{4}c_{6}v_{0}^{4},\quad\lambda=\frac{1}{2}\frac{M_{h}^{2}}{v_{0}^{2}}-\frac{3}{2}c_{6}v_{0}^{2}.$
(44)

Figure 9: The scalar quartic coupling and the Higgs mass parameter as
functions of the NP scale $\Lambda$.

The tree-level relation is at $\mathcal{O}(g^{2})$ accuracy, for the complete
$\mathcal{O}(g^{4})$ accuracy of the 2-loop order, the tree level relations
need to be improved by their one-loop corrections. These corrections have been
done in Ref.[12], and we show the result in Fig.9 for the tree-level relation
and its one-loop correction. We would like to mention that, in our
calculations, the $\lambda$ is different from that of Ref.[12] which directly
affects the thermal effective potential, and the PT temperature will be
increased when the scalar quartic coupling $\lambda$ is decreased. For
example, at tree level, the $\lambda=-0.12$ with
$\Lambda=600\leavevmode\nobreak\ \text{GeV}$ while $\lambda=-0.16$ with
$\Lambda=600\leavevmode\nobreak\ \text{GeV}$ in Fig.11 of Ref.[12]. Meanwhile,
the behavior of $\mu_{h}^{2}$ is the same as that of Ref.[12]. Our results of
$T_{c},T_{n},\alpha,\beta/H$ therefore differ from that of Ref.[12]. However,
our results also show that the renormalization scale dependence of these
physical quantities will be lower at the loop level than the tree level, which
is the same as the results in Ref.[12].

##  Appendix B Parameters of dimensional reduction

We use the code “DRalgo” to obtain the parameters of dimensional reduction
[47]. The tree and one-loop result of “DRalgo” can be directly matched with
literature, but for the two-loop level result, the “DRalgo” uses the Glaisher-
Kinkelin constant. The relation between the Glaisher-Kinkelin constant $A$ and
Riemann zeta function $\zeta(x)$ is

$\ln(2\pi)-(\ln(\zeta(2)))^{\prime}=1-{\gamma_{\mbox{\tiny\rm{E}}}}+(\ln(\zeta(-1))^{\prime},\quad
1+(\ln(\zeta(-1))^{\prime}=12\ln[A]\;,$ (45)

where
$\ln[A]=(1/12)\left({\gamma_{\mbox{\tiny\rm{E}}}}-L_{b}-2\left(c+\log\left[3T/\bar{\mu}\right]\right)\right)\;$,
with

$\displaystyle
c\equiv\frac{1}{2}\bigg{(}\ln\Big{(}\frac{8\pi}{9}\Big{)}+\frac{\zeta^{\prime}(2)}{\zeta(2)}-2{\gamma_{\mbox{\tiny\rm{E}}}}\bigg{)}\;,$
(46)

and

$\displaystyle L_{b}$ $\displaystyle\equiv
2\ln\Big{(}\frac{\bar{\mu}}{T}\Big{)}-2[\ln(4\pi)-{\gamma_{\mbox{\tiny\rm{E}}}}]\;,$
(47) $\displaystyle L_{f}$ $\displaystyle\equiv L_{b}+4\ln 2\;,$ (48)

${\gamma_{\mbox{\tiny\rm{E}}}}$ is the Euler-Mascheroni constant. It should be
noticed that “DRalgo” is worked in Landau gauge. The chemical potential part
of the parameters has been calculated in Ref.[23], we add the chemical
potential part contributions to the corresponding parameters. After defining
the function[23]

$\mathcal{A}(\mu_{ch})=\psi(\frac{1}{2}+\frac{i\mu_{ch}}{2\pi
T})+\psi(\frac{1}{2}-\frac{i\mu_{ch}}{2\pi
T})+2{\gamma_{\mbox{\tiny\rm{E}}}}+2\ln{4},\quad\psi(z)=\partial_{z}\ln{\Gamma(z)}$
(49)

the final results in Landau gauge ($\xi=0$) for the 3d parameters at the heavy
scale read

$\displaystyle\lambda_{3}=$ $\displaystyle
T\Big{(}\lambda(\bar{\mu})+\frac{1}{(4\pi)^{2}}\bigg{[}\frac{1}{8}\Big{(}3g^{4}+{g^{\prime}}^{4}+2g^{2}{g^{\prime}}^{2}\Big{)}+3L_{f}\Big{(}g_{Y}^{4}-2\lambda
g_{Y}^{2}\Big{)}$ $\displaystyle-
L_{b}\bigg{(}\frac{3}{16}\Big{(}3g^{4}+{g^{\prime}}^{4}+2g^{2}{g^{\prime}}^{2}\Big{)}-\frac{3}{2}\Big{(}3g^{2}+{g^{\prime}}^{2}-8\lambda\Big{)}\lambda\bigg{)}\bigg{]}\Big{)}$
$\displaystyle+T^{3}c_{6}-\frac{\mu_{h}^{2}}{(4\pi)^{2}}12Tc_{6}L_{b}-\frac{3g_{Y}^{2}T}{16\pi^{2}}(g_{Y}^{2}-2\lambda)\mathcal{A}\left(\frac{\mu_{B}}{3}\right)\;,$
(50) $\displaystyle c_{6,3}=$ $\displaystyle
T^{2}c_{6}(\bar{\mu})\bigg{(}1+\frac{1}{(4\pi)^{2}}\bigg{[}\Big{(}-54\lambda+\frac{9}{4}(3g^{2}+{g^{\prime}}^{2})\Big{)}L_{b}-9g_{Y}^{2}L_{f}\bigg{]}\bigg{)}$
$\displaystyle-\frac{\zeta(3)}{768\pi^{4}}\bigg{(}-\frac{3}{8}\Big{(}3g^{6}+3g^{4}{g^{\prime}}^{2}+3g^{2}{g^{\prime}}^{4}+{g^{\prime}}^{6}\Big{)}-240\lambda^{3}+84g_{Y}^{6}\bigg{)}\;,$
(51) $\displaystyle g_{3}^{2}=$ $\displaystyle
g^{2}(\bar{\mu})T\bigg{[}1+\frac{g^{2}}{(4\pi)^{2}}\bigg{(}\frac{43}{6}L_{b}+\frac{2}{3}-\frac{4N_{\rm
f}}{3}L_{f}\bigg{)}\bigg{]}+\frac{g^{4}T}{48\pi^{2}}\left[9\mathcal{A}\left(\frac{\mu_{B}}{3}\right)+\sum_{i=1}^{3}\mathcal{A}(\mu_{L_{i}})\right]\;,$
(52) $\displaystyle{g^{\prime}_{3}}^{2}=$
$\displaystyle{g^{\prime}}^{2}(\bar{\mu})T\bigg{[}1+\frac{{g^{\prime}}^{2}}{(4\pi)^{2}}\bigg{(}-\frac{1}{6}L_{b}-\frac{20N_{\rm
f}}{9}L_{f}\bigg{)}\bigg{]}\;,$ (53)

$\displaystyle h_{1}=$
$\displaystyle\frac{g^{2}(\bar{\mu})T}{4}\bigg{(}1+\frac{1}{(4\pi)^{2}}\bigg{\\{}\bigg{[}\frac{43}{6}L_{b}+\frac{17}{2}-\frac{4N_{\rm
f}}{3}(L_{f}-1)\bigg{]}g^{2}+\frac{{g^{\prime}}^{2}}{2}-6g_{Y}^{2}+12\lambda\bigg{\\}}\bigg{)}$
$\displaystyle+\frac{g^{4}T}{192\pi^{2}}\left[9\mathcal{A}\left(\frac{\mu_{B}}{3}\right)+\sum_{i=1}^{3}\mathcal{A}(\mu_{L_{i}})\right]\;,$
(54) $\displaystyle h_{2}=$
$\displaystyle\frac{{g^{\prime}}^{2}(\bar{\mu})T}{4}\bigg{(}1+\frac{1}{(4\pi)^{2}}\bigg{\\{}\frac{3g^{2}}{2}-\bigg{[}\frac{(L_{b}-1)}{6}+\frac{20N_{\rm
f}(L_{f}-1)}{9}\bigg{]}{g^{\prime}}^{2}-\frac{34}{3}g_{Y}^{2}+12\lambda\bigg{\\}}\bigg{)}\;,$
(55) $\displaystyle h_{3}=$
$\displaystyle\frac{g(\bar{\mu}){g^{\prime}}(\bar{\mu})T}{2}\bigg{\\{}1+\frac{1}{(4\pi)^{2}}\bigg{[}-g^{2}+\frac{1}{3}{g^{\prime}}^{2}+L_{b}\bigg{(}\frac{43}{12}g^{2}-\frac{1}{12}{g^{\prime}}^{2}\bigg{)}$
$\displaystyle-N_{\rm
f}(L_{f}-1)\bigg{(}\frac{2}{3}g^{2}+\frac{10}{9}{g^{\prime}}^{2}\bigg{)}+4\lambda+2g_{Y}^{2}\bigg{]}\bigg{\\}}\;,$
(56) $\displaystyle h_{4}=$
$\displaystyle-T\frac{1}{(4\pi)^{2}}2g_{\mbox{\scriptsize s}}^{2}g_{Y}^{2}\;,$
(57) $\displaystyle\kappa_{1}=$ $\displaystyle
T\frac{g^{4}}{(4\pi)^{2}}\bigg{(}\frac{17-4N_{\rm f}}{3}\bigg{)}\;,$ (58)
$\displaystyle\kappa_{2}=$ $\displaystyle
T\frac{{g^{\prime}}^{4}}{(4\pi)^{2}}\bigg{(}\frac{1}{3}-\frac{380}{81}N_{\rm
f}\bigg{)}\;,$ (59) $\displaystyle\kappa_{3}=$ $\displaystyle
T\frac{g^{2}{g^{\prime}}^{2}}{(4\pi)^{2}}\bigg{(}2-\frac{8}{3}N_{\rm
f}\bigg{)}\;,$ (60) $\displaystyle m_{\mbox{\tiny\rm{D}}}^{2}=$ $\displaystyle
g^{2}(\bar{\mu})T^{2}\bigg{(}\frac{5}{6}+\frac{N_{\rm
f}}{3}\bigg{)}+\frac{g^{2}}{4\pi^{2}}\left(\mu_{B}^{2}+\sum_{i=1}^{3}\mu_{L_{i}}^{2}\right)\;,$
(61) $\displaystyle m_{\mbox{\tiny\rm{D}}}^{\prime 2}=$
$\displaystyle{g^{\prime}}^{2}(\bar{\mu})T^{2}\bigg{(}\frac{1}{6}+\frac{5N_{\rm
f}}{9}\bigg{)}+\frac{g^{\prime
2}}{4\pi^{2}}\left(\frac{11}{9}\mu_{B}^{2}+3\sum_{i=1}^{3}\mu_{L_{i}}^{2}\right)\;,$
(62) $\displaystyle m_{\mbox{\tiny\rm{D}}}^{\prime\prime 2}=$ $\displaystyle
g_{\mbox{\scriptsize s}}^{2}T^{2}\bigg{(}1+\frac{N_{\rm f}}{3}\bigg{)}\;,$
(63) $\displaystyle\mu_{h,3}^{2}=$
$\displaystyle\mu^{2}_{h}(\bar{\mu})+\frac{T^{2}}{16}\Big{(}3g^{2}(\bar{\mu})+{g^{\prime}}^{2}(\bar{\mu})+4g_{Y}^{2}(\bar{\mu})+8\lambda(\bar{\mu})\Big{)}+\frac{1}{4}T^{4}c_{6}$
$\displaystyle+\frac{1}{(4\pi)^{2}}\bigg{\\{}\mu_{h}^{2}\bigg{[}\Big{(}\frac{3}{4}(3g^{2}+{g^{\prime}}^{2})-6\lambda\Big{)}L_{b}-3g_{Y}^{2}L_{f}\bigg{]}$
$\displaystyle+T^{2}\bigg{[}\frac{167}{96}g^{4}+\frac{1}{288}{g^{\prime}}^{4}-\frac{3}{16}g^{2}{g^{\prime}}^{2}+\frac{1}{4}\lambda(3g^{2}+{g^{\prime}}^{2})$
$\displaystyle+L_{b}\Big{(}\frac{17}{16}g^{4}-\frac{5}{48}{g^{\prime}}^{4}-\frac{3}{16}g^{2}{g^{\prime}}^{2}+\frac{3}{4}\lambda(3g^{2}+{g^{\prime}}^{2})-6\lambda^{2}\Big{)}$
$\displaystyle+\Big{(}c+\ln\left(\frac{3T}{\bar{\mu}_{3}}\right)\Big{)}\Big{(}\frac{81}{16}g^{4}+3\lambda(3g^{2}+{g^{\prime}}^{2})-12\lambda^{2}$
$\displaystyle-\frac{7}{16}{g^{\prime}}^{4}-\frac{15}{8}g^{2}{g^{\prime}}^{2}\Big{)}$
$\displaystyle-
g_{Y}^{2}\Big{(}\frac{3}{16}g^{2}+\frac{11}{48}{g^{\prime}}^{2}+2g_{\mbox{\scriptsize
s}}^{2}\Big{)}+\Big{(}\frac{1}{12}g^{4}+\frac{5}{108}{g^{\prime}}^{4}\Big{)}N_{\rm
f}$
$\displaystyle+L_{f}\Big{(}g_{Y}^{2}\Big{(}\frac{9}{16}g^{2}+\frac{17}{48}{g^{\prime}}^{2}+2g_{\mbox{\scriptsize
s}}^{2}-3\lambda\Big{)}+\frac{3}{8}g_{Y}^{4}$
$\displaystyle-\Big{(}\frac{1}{4}g^{4}+\frac{5}{36}{g^{\prime}}^{4}\Big{)}N_{\rm
f}\Big{)}$
$\displaystyle+\ln(2)\Big{(}g_{Y}^{2}\Big{(}-\frac{21}{8}g^{2}-\frac{47}{72}{g^{\prime}}^{2}+\frac{8}{3}g_{\mbox{\scriptsize
s}}^{2}+9\lambda\Big{)}-\frac{3}{2}g_{Y}^{4}$
$\displaystyle+\Big{(}\frac{3}{2}g^{4}+\frac{5}{6}{g^{\prime}}^{4}\Big{)}N_{\rm
f}\Big{)}\bigg{]}\bigg{\\}}+\Delta\mu_{h,3}^{2}\;.$ (64)

The $\mu_{h,3}^{2}$ represents the contribution of the chemical potential to
$\mu_{h,3}^{2}$. After defining the function

$\mathcal{B}(\mu)=\zeta^{\prime}\left(-1,\frac{1}{2}+\frac{i\mu}{2\pi
T}\right)+\zeta^{\prime}\left(-1,\frac{1}{2}-\frac{i\mu}{2\pi
T}\right)-2\zeta^{\prime}\left(-1,\frac{1}{2}\right),$ (65)

the $\mu_{h,3}^{2}$ becomes

$\displaystyle\Delta\mu_{h,3}^{2}=$
$\displaystyle\frac{g_{Y}^{2}(\overline{\mu})}{12\pi^{2}}\mu_{B}^{2}-\frac{3g_{Y}^{2}}{16\pi^{2}}\left(\mu_{h}^{2}-\frac{\lambda
T^{2}}{2}-\frac{3g^{2}T^{2}}{16}-\frac{g_{Y}^{2}T^{2}}{4}\right)\mathcal{A}\left(\frac{\mu_{B}}{3}\right)+\frac{g_{Y}^{2}\mu_{B}^{2}}{64\pi^{4}}\left\\{\frac{3}{4}g^{2}L_{b}(\overline{\mu})\right.$
(66)
$\displaystyle\left.-g_{Y}^{2}\left[L_{f}(\overline{\mu})-\mathcal{A}\left(\frac{\mu_{B}}{3}\right)\right]\right\\}-\left[\left(9g_{Y}^{4}+\frac{9}{2}g_{Y}^{2}g^{2}+16g_{Y}^{2}g_{s}^{2}\right)\mathcal{A}\left(\frac{\mu_{B}}{3}\right)\right.$
$\displaystyle\left.-\left(9g_{Y}^{2}+\frac{9}{4}g_{Y}^{2}g^{2}-18\lambda
g_{Y}^{2}-16g_{Y}^{2}g_{s}^{2}-\frac{27}{4}g^{4}\right)16\mathcal{B}\left(\frac{\mu_{B}}{3}\right)\right]\frac{T^{2}}{128\pi^{2}}$
$\displaystyle+\left(9g_{Y}^{4}+\frac{9}{4}g_{Y}^{2}g^{2}-18\lambda
g_{Y}^{2}-16g_{Y}^{2}g_{s}^{2}-\frac{27}{4}g^{4}\right)\frac{i\mu_{B}T}{48\pi^{3}}\ln{\left(\frac{\Gamma\left(\frac{1}{2}-\frac{i\mu_{B}}{6\pi
T}\right)}{\Gamma\left(\frac{1}{2}+\frac{i\mu_{B}}{6\pi T}\right)}\right)}$
$\displaystyle+\left[9g_{Y}^{2}g^{2}-6(3g_{Y}^{4}-8g_{Y}^{2}g_{s}^{2})L_{b}(\overline{\mu})+9g_{Y}^{4}L_{f}(\overline{\mu})+\left(\frac{9}{2}g_{Y}^{2}g^{2}+16g_{Y}^{2}g_{s}^{2}\right)(4\ln{2}-1)\right.$
$\displaystyle+\left(9g_{Y}^{4}+\frac{9}{4}g_{Y}^{2}g^{2}-18\lambda
g_{Y}^{2}-16g_{Y}^{2}g_{s}^{2}-\frac{27}{4}g^{4}\right)4{\gamma_{\mbox{\tiny\rm{E}}}}-\left(9g_{Y}^{4}+\frac{9}{2}g_{Y}^{2}g^{2}+16g_{Y}^{2}g_{s}^{2}\right)$
$\displaystyle\left.\mathcal{A}\left(\frac{\mu_{B}}{3}\right)\right]\frac{\mu_{B}^{2}}{1152\pi^{4}}-\frac{3}{4}g^{4}\sum_{i=1}^{3}\left[\frac{T^{2}}{8\pi^{2}}\mathcal{B}(\mu_{L_{i}})+\frac{i\mu_{L_{i}}T}{16\pi^{3}}\ln{\left(\frac{\Gamma\left(\frac{1}{2}-\frac{i\mu_{L_{i}}}{2\pi
T}\right)}{\Gamma\left(\frac{1}{2}+\frac{i\mu_{L_{i}}}{2\pi
T}\right)}\right)}+\frac{\mu_{L_{i}}^{2}}{32\pi^{4}}{\gamma_{\mbox{\tiny\rm{E}}}}\right]$

These relations show that the tree level part runs terms of 4d renormalization
scale $\overline{\mu}$. After applying the corresponding $\beta-$functions,
this running is canceled out by the logarithmic term in the $L_{f}$ and
$L_{b}$ terms. This leads to a large difference in scale dependence between
the tree level and the loop level. However, the scale dependence is not
completely removed, but is present in the Debye mass $m_{D}^{2},m_{D}^{\prime
2}$ and $m_{D}^{\prime\prime 2}$ at $\mathcal{O}(g^{4})$. This scale
dependence can be removed by the 2-loop Debye mass at a higher order than
$\mathcal{O}(g^{4})$ but is beyond the scope of this article.

The chemical potential also includes new terms for effective theories. These
terms are read

$\displaystyle\kappa_{0}=$
$\displaystyle-\frac{i\pi}{3}g^{\prime}T^{\frac{5}{2}}\left\\{\left(1-\frac{9g^{2}}{64\pi^{2}}\right)\sum_{i=1}^{3}\frac{\mu_{L_{i}}}{\pi
T}\left[1+\left(\frac{\mu_{L_{i}}}{\pi T}\right)^{2}\right]-\right.$
$\displaystyle\left.\left(1-\frac{5g_{Y}^{2}}{32\pi^{2}}-\frac{9g^{2}}{64\pi^{2}}-\frac{g_{s}^{2}}{2\pi^{2}}\right)\frac{\mu_{B}}{\pi
T}\left[1+\frac{1}{9}\left(\frac{\mu_{B}}{\pi
T}\right)^{2}\right]\right\\}\;,$ (67) $\displaystyle\rho=$
$\displaystyle\frac{i}{8\pi}gg_{Y}^{2}T^{3/2}\frac{\mu_{B}}{\pi T}\;,$ (68)
$\displaystyle\rho^{\prime}=$
$\displaystyle-\frac{5i}{24\pi}g^{\prime}g_{Y}^{2}T^{3/2}\frac{\mu_{B}}{\pi
T}\;,$ (69) $\displaystyle\rho_{G}=$
$\displaystyle-\frac{i}{8\pi}g^{\prime}g^{2}T^{3/2}\left(\frac{\mu_{B}}{\pi
T}-\sum_{i=1}^{3}\frac{\mu_{L_{i}}}{\pi T}\right)\;,$ (70)
$\displaystyle\alpha_{0}=$
$\displaystyle\frac{g^{2}}{32\pi^{2}}(N_{f}\mu_{B}+\sum_{i=1}^{3}\mu_{L_{i}})=0\;,$
(71) $\displaystyle\alpha^{\prime}=$ $\displaystyle-\frac{g^{\prime
2}}{32\pi^{2}}(N_{f}\mu_{B}+\sum_{i=1}^{3}\mu_{L_{i}})=0\;.$ (72)

Here, the Chern-Simons terms of $\alpha_{0}$ and $\alpha^{\prime}$ are removed
due to the chemical potential arrangement.

Finally, when the heavy temporal scalar scalars are integrated out in the
second step of dimensional reduction. The parameters of this light EFT read

$\displaystyle\bar{g}^{2}_{3}=$ $\displaystyle
g_{3}^{2}\Big{(}1-\frac{g_{3}^{2}}{6(4\pi)m_{\mbox{\tiny\rm{D}}}}\Big{)}\;,$
(73) $\displaystyle\bar{g}^{\prime 2}_{3}=$
$\displaystyle{g^{\prime}_{3}}^{2}\;,$ (74)
$\displaystyle\bar{\mu}^{2}_{h,3}=$
$\displaystyle\mu^{2}_{h,3}-\frac{1}{4\pi}\Big{(}3h_{1}m_{\mbox{\tiny\rm{D}}}+h_{2}m_{\mbox{\tiny\rm{D}}}^{\prime}+8h_{4}m_{\mbox{\tiny\rm{D}}}^{\prime\prime}\Big{)}+\frac{1}{(4\pi)^{2}}\bigg{(}3g_{3}^{2}h_{1}-3h_{1}^{2}-h_{2}^{2}-\frac{3}{2}h_{3}^{2}$
$\displaystyle+\Big{(}-\frac{3}{4}g_{3}^{4}+12g_{3}^{2}h_{1}\Big{)}\ln\Big{(}\frac{\bar{\mu}_{3}}{2m_{\mbox{\tiny\rm{D}}}}\Big{)}-6h_{1}^{2}\ln\Big{(}\frac{\bar{\mu}_{3}}{2m_{\mbox{\tiny\rm{D}}}}\Big{)}-2h_{2}^{2}\ln\Big{(}\frac{\bar{\mu}_{3}}{2m_{\mbox{\tiny\rm{D}}}^{\prime}}\Big{)}$
$\displaystyle-3h_{3}^{2}\ln\Big{(}\frac{\bar{\mu}_{3}}{m_{\mbox{\tiny\rm{D}}}+m_{\mbox{\tiny\rm{D}}}^{\prime}}\Big{)}\bigg{)}+\Delta\overline{\mu}_{h}^{2}\;,$
(75) $\displaystyle\bar{\lambda}_{3}=$
$\displaystyle\lambda_{3}-\frac{1}{2(4\pi)}\Big{(}\frac{3h_{1}^{2}}{m_{\mbox{\tiny\rm{D}}}}+\frac{h_{2}^{2}}{m_{\mbox{\tiny\rm{D}}}^{\prime}}+\frac{h_{3}^{2}}{m_{\mbox{\tiny\rm{D}}}+m_{\mbox{\tiny\rm{D}}}^{\prime}}\Big{)}+\Delta\overline{\lambda}\;,$
(76) $\displaystyle\bar{c}_{6,3}=$ $\displaystyle
c_{6,3}+\frac{1}{2(4\pi)}\frac{h_{1}^{3}}{m_{\mbox{\tiny\rm{D}}}^{3}}\;,$ (77)

where

$\displaystyle\Delta\overline{\mu}_{h}^{2}$
$\displaystyle=\frac{h_{3}\kappa_{0}^{2}}{m_{D}^{\prime
4}}-\frac{\rho^{\prime}\kappa_{0}}{m_{D}^{\prime
2}}-\frac{1}{4\pi}\left[(\frac{3g_{3}^{2}g_{3}^{\prime
2}}{4m_{D}}+\frac{4h_{2}^{2}}{m_{D}^{\prime}})\frac{\kappa_{0}^{2}}{m_{D}^{\prime
4}}-(\frac{3g_{3}g_{3}^{\prime}\rho}{m_{D}}+\frac{4h_{2}\rho^{\prime}}{m_{D}^{\prime}})\frac{\kappa_{0}}{m_{D}^{\prime
2}}+\frac{3\rho^{2}}{m_{D}}+\frac{\rho^{\prime 2}}{m_{D}^{\prime}}\right]$
(78) $\displaystyle\Delta\overline{\lambda}$
$\displaystyle=-\frac{1}{2}\left[(\frac{g_{3}^{2}g_{3}^{\prime
2}}{4m_{D}^{2}}+\frac{4h_{2}^{\prime 2}}{m_{D}^{\prime
2}})\frac{\kappa_{0}^{2}}{m_{D}^{\prime
4}}-(\frac{g_{3}g_{3}^{\prime}\rho}{m_{D}^{2}}+\frac{4h_{2}\rho^{\prime}}{m_{D}^{\prime
2}})\frac{\kappa_{0}}{m_{D}^{\prime
2}}+\frac{\rho^{2}}{m_{D}^{2}}+\frac{\rho^{\prime 2}}{m_{D}^{\prime
2}}\right]$ (79)

The light theory depends on an additional renormalization scale
$\overline{\mu}_{3}$ at Eq.(73). This scale is unphysical as well as
$\overline{\mu}$. We do not consider the dependence of $\overline{\mu}_{3}$
since $\overline{\mu}_{3}$ has a very insignificant effect on the results, and
take $\overline{\mu}_{3}=g^{2}T$ is this paper[12, 25].

Fig.10 shows the renormalization scale dependence and the chemical potential
effect on the parameters $\overline{g}_{3},\overline{\lambda}_{3}$ and
$\overline{m}_{3}^{2}$ at temperature $T_{c}$. The parameter
$\overline{m}_{2}^{3}$ at large $\Lambda$ is less than it at small $\Lambda$,
but the opposite is true for $\overline{g}_{3}^{2}$ and
$\overline{\lambda}_{3}$. The renormalization scale dependence of these
parameters at loop levels is weaker than it is at the tree level. The
parameter $\overline{g}_{3}^{2}$ is more susceptible to renormalization scale
at 1-loop level relative to $\overline{\lambda}_{3}$ and
$\overline{m}_{3}^{2}$. These parameters are not sensitive to the chemical
potential.

Figure 10: The renormalization scale dependence and effect of chemical
potential of $\overline{g}_{3},\overline{\lambda}_{3}$ and
$\overline{m}_{3}^{2}$ at $T_{c}$. Left: The renormalization scale dependence
of $\overline{g}_{3},\overline{\lambda}_{3}$ and $\overline{m}_{3}^{2}$.
Right: The effect of chemical potential on
$\overline{g}_{3},\overline{\lambda}_{3}$ and $\overline{m}_{3}^{2}$ with
$\overline{\mu}=4\pi e^{-\gamma-E}T$.

##  Appendix C Effective potential at the two-loop level

With the definitions of the master integrations of $\mathcal{D}$ and
$I_{1}^{3}$ being given in Ref. [48], different part contributions to the
effective potential $V_{eff}$ at 2-loop level are:

$\displaystyle{\rm(SSS)}=$
$\displaystyle\frac{3}{4}\phi^{2}_{3}\Big{(}2\bar{\lambda}_{3}+3\bar{c}_{6,3}\phi_{3}^{2}\Big{)}^{2}\mathcal{D}_{{\mbox{\tiny\rm{SSS}}}}(m_{\phi,3},m_{\chi,3},m_{\chi,3})$
$\displaystyle+\frac{3}{4}\phi^{2}_{3}\Big{(}2\bar{\lambda}_{3}+5\bar{c}_{6,3}\phi_{3}^{2}\Big{)}^{2}\mathcal{D}_{{\mbox{\tiny\rm{SSS}}}}(m_{\phi,3},m_{\phi,3},m_{\phi,3})\;,$
(80) $\displaystyle{\rm(VSS)}=$
$\displaystyle\frac{1}{4}\bar{g}_{3}^{2}\mathcal{D}_{{\mbox{\tiny\rm{VSS}}}}(m_{\chi,3},m_{\chi,3},m_{W,3})+\frac{1}{4}\bar{g}_{3}^{2}\mathcal{D}_{{\mbox{\tiny\rm{VSS}}}}(m_{\phi,3},m_{\chi,3},m_{W,3})$
$\displaystyle+\frac{1}{8}(\bar{g}_{3}^{2}+\bar{g}_{3}^{\prime
2})\mathcal{D}_{{\mbox{\tiny\rm{VSS}}}}(m_{\phi,3},m_{\chi,3},m_{Z,3})$
$\displaystyle+\frac{1}{8}\frac{(\bar{g}_{3}^{2}-\bar{g}_{3}^{\prime
2})^{2}}{\bar{g}_{3}^{2}+\bar{g}_{3}^{\prime
2}}\mathcal{D}_{{\mbox{\tiny\rm{VSS}}}}(m_{\chi,3},m_{\chi,3},m_{Z,3})+\frac{1}{2}\frac{\bar{g}_{3}^{2}\bar{g}_{3}^{\prime
2}}{\bar{g}_{3}^{2}+\bar{g}_{3}^{\prime
2}}\mathcal{D}_{{\mbox{\tiny\rm{VSS}}}}(m_{\chi,3},m_{\chi,3},0)\;,$ (81)
$\displaystyle{\rm(VVS)}=$
$\displaystyle\frac{1}{8}\bar{g}_{3}^{4}\phi_{3}^{2}\mathcal{D}_{{\mbox{\tiny\rm{VVS}}}}(m_{\phi,3},m_{W,3},m_{W,3})+\frac{1}{16}(\bar{g}_{3}^{2}+\bar{g}_{3}^{\prime
2})^{2}\phi_{3}^{2}\mathcal{D}_{{\mbox{\tiny\rm{VVS}}}}(m_{\phi,3},m_{Z,3},m_{Z,3})$
$\displaystyle+\frac{1}{4}\frac{\bar{g}_{3}^{4}\bar{g}_{3}^{\prime
2}\phi_{3}^{2}}{\bar{g}_{3}^{2}+\bar{g}_{3}^{\prime
2}}\mathcal{D}_{{\mbox{\tiny\rm{VVS}}}}(m_{\chi,3},m_{W},0)+\frac{1}{4}\frac{\bar{g}_{3}^{2}\bar{g}_{3}^{\prime
4}\phi_{3}^{2}}{\bar{g}_{3}^{2}+\bar{g}_{3}^{\prime
2}}\mathcal{D}_{{\mbox{\tiny\rm{VVS}}}}(m_{\chi,3},m_{W,3},m_{Z,3})\;,$ (82)
$\displaystyle{\rm(VVV)}=$
$\displaystyle-\frac{1}{2}\frac{\bar{g}_{3}^{4}}{\bar{g}_{3}^{2}+\bar{g}_{3}^{\prime
2}}\mathcal{D}_{{\mbox{\tiny\rm{VVV}}}}(m_{W,3},m_{W,3},m_{Z,3})-\frac{1}{2}\frac{\bar{g}_{3}^{2}\bar{g}_{3}^{\prime
2}}{\bar{g}_{3}^{2}+\bar{g}_{3}^{\prime
2}}\mathcal{D}_{{\mbox{\tiny\rm{VVV}}}}(m_{W,3},m_{W,3},0)\;,$ (83)
$\displaystyle{\rm(VGG)}=$
$\displaystyle-2\bar{g}_{3}^{2}\mathcal{D}_{{\mbox{\tiny\rm{VGG}}}}(m_{W,3})-\frac{\bar{g}_{3}^{4}}{\bar{g}_{3}^{2}+\bar{g}_{3}^{\prime
2}}\mathcal{D}_{{\mbox{\tiny\rm{VGG}}}}(m_{Z,3})\;,$ (84)
$\displaystyle{\rm(SS)}=$
$\displaystyle-\frac{15}{8}\Big{(}2\bar{\lambda}_{3}+3\bar{c}_{6,3}\phi_{3}^{2}\Big{)}\Big{(}I^{3}_{1}(m_{\chi,3})\Big{)}^{2}-\frac{3}{4}\Big{(}9\bar{c}_{6,3}\phi_{3}^{2}+2\bar{\lambda}_{3}\Big{)}I^{3}_{1}(m_{\chi,3})I^{3}_{1}(m_{\phi,3})$
$\displaystyle-\frac{3}{8}\Big{(}2\bar{\lambda}_{3}+15\bar{c}_{6,3}\phi_{3}^{2}\Big{)}\Big{(}I^{3}_{1}(m_{\phi,3})\Big{)}^{2}\;,$
(85) $\displaystyle{\rm(VS)}=$
$\displaystyle-\frac{3}{4}(d-1)\bar{g}_{3}^{2}I^{3}_{1}(m_{\chi,3})I^{3}_{1}(m_{W,3})-\frac{1}{4}(d-1)\frac{(\bar{g}_{3}^{2}-\bar{g}_{3}^{\prime
2})^{2}}{\bar{g}_{3}^{2}+\bar{g}_{3}^{\prime
2}}I^{3}_{1}(m_{\chi,3})I^{3}_{1}(m_{Z,3})$ (86)
$\displaystyle-\frac{1}{4}(d-1)\bar{g}_{3}^{2}I^{3}_{1}(m_{\phi,3})I^{3}_{1}(m_{W,3})-\frac{1}{8}(d-1)(\bar{g}_{3}^{2}+\bar{g}_{3}^{\prime
2})I^{3}_{1}(m_{\chi,3})I^{3}_{1}(m_{Z,3})$
$\displaystyle-\frac{1}{8}(d-1)(\bar{g}_{3}^{2}+\bar{g}_{3}^{\prime
2})I^{3}_{1}(m_{\phi,3})I^{3}_{1}(m_{Z,3})\;,$ (87) $\displaystyle{\rm(VV)}=$
$\displaystyle-\frac{1}{2}\bar{g}_{3}^{2}\mathcal{D}_{{\mbox{\tiny\rm{VV}}}}(m_{W,3},m_{W,3})-\frac{\bar{g}_{3}^{4}}{\bar{g}_{3}^{2}+\bar{g}_{3}^{\prime
2}}\mathcal{D}_{{\mbox{\tiny\rm{VV}}}}(m_{W,3},m_{Z,3})\;.$ (88)

Figure 11: The effective potential at $\Lambda=610$ GeV with different
renormalization scales and chemical potentials.

Figure 12: The effective potential at $\Lambda=910$ GeV with different
renormalization scales and chemical potentials.

The Figs.(11, 12) show the effective potential at $T_{c}$(red) and
$T_{n}$(green) with $\Lambda=610$ GeV and $910$ GeV. Where, we can find that
the renormalization scale dependence of the thermal effective potential
decreases from tree-level to two-loop level, and the chemical potential can
reduce the magnitude of the potential barrier $\bar{g}^{-6}_{3}\Delta
V_{eff}$. For the case of $\Lambda=610$ GeV, we can see that the magnitude of
$|\Delta V_{eff}|$ at loop levels is larger than that at tree level at the
nucleation temperature $T_{n}$, and the $|\Delta V_{eff}|$ at loop levels is
smaller than it at tree level with $\Lambda=910$GeV. This means that, in the
range of $\Lambda=590-910$GeV, there exists a value of $\Lambda$ where the
loop levels contribution cancels that from the tree level, i.e., $|\Delta
V_{eff}^{tree}|=|\Delta(V_{eff}^{1loop}+V_{eff}^{2loop})|$.

##  Appendix D Sphaleron energy

The sphaleron energy has the form

$E_{sph}(T)=\int
d^{3}x\left[-\frac{1}{4}G_{ij}^{a}G_{ij}^{a}-\frac{1}{4}F_{ij}F_{ij}+(D_{i}\Phi)^{\dagger}(D_{i}\Phi)+V_{eff}\right]$
(89)

and $G_{ij},F_{ij},D_{i}$ are defined in Eq.(14). Then, we consider sphaleron
energy with $U(1)$ gauge field. For $\theta_{w}\neq 0$, the $U(1)$ gauge field
will contribute to sphaleron energy, and the spherical symmetry reduces to an
axial symmetry[35, 40, 49, 39]. The $SU(2)$ matrix is defined as

$U^{\infty}(\nu,\theta,\phi)=\begin{pmatrix}e^{i\nu}(\cos\nu-i\sin\nu\cos\theta)&e^{i\phi}\sin\nu\sin\theta\\\
-e^{-i\phi}\sin\nu\sin\theta&e^{-i\nu}(\cos\nu+i\sin\nu\cos\theta),\end{pmatrix}$
(90)

then we can define

$U^{\infty-1}dU^{\infty}=\sum_{a=1}^{3}F_{a}\frac{\tau^{a}}{2i},$ (91)

where

$\displaystyle F_{1}=$
$\displaystyle-2\sin\nu\left[\cos{\theta}\cos^{2}{\nu}\sin{\phi}-2\sin^{2}{(\theta/2)}\sin{\nu}\cos{\nu}\cos{\phi}+\sin^{2}{\nu}\sin{\phi}\right]d\theta$
(92)
$\displaystyle-2\sin{\nu}\left[\cos{\phi}\sin{\theta}-\cos{\phi}\sin{\theta}\sin^{2}{\nu}+\cos{\theta}\cos{\phi}\sin{\theta}\sin^{2}{\nu}\right.$
$\displaystyle\left.-2\cos{\nu}\sin^{2}{(\theta/2)}\sin{\theta}\sin{\nu}\sin{\phi}\right]d\phi,$
$\displaystyle F_{2}=$
$\displaystyle-2\sin{\nu}\left[\cos{\phi}(\cos{\theta}\cos^{2}{\nu}+\sin^{2}{\nu}+\sin^{2}{(\theta/2)}\sin{2\nu}\sin{\phi})\right]d\theta$
$\displaystyle+2\sin{\theta}\sin{\nu}\left[\cos{\phi}\sin^{2}{(\theta/2)}\sin{2\nu}+(1+(-1+\cos{\theta})\sin^{2}{\nu})\sin{\phi}\right]d\phi,$
$\displaystyle F_{3}=$ $\displaystyle 2(\sin\nu\cos\nu\sin\theta
d\theta+\sin^{2}\nu\sin^{2}\theta d\phi).$

After set $\nu=\pi/2$, the function $F_{i}$ becomes

$\displaystyle F_{1}$ $\displaystyle=-2\sin\phi d\theta-\sin 2\theta\cos\phi
d\phi,$ (93) $\displaystyle F_{2}$ $\displaystyle=-2\cos\phi d\theta+\sin
2\theta\sin\phi d\phi,$ $\displaystyle F_{3}$
$\displaystyle=2\sin^{2}{\theta}d\phi.$

The configuration of gauge and Higgs fields is expressed as

$\displaystyle g^{\prime}B_{i}dx^{i}$ $\displaystyle=(1-f_{0}(\xi))F_{3},$
(94) $\displaystyle gA_{i}^{a}\tau^{a}dx^{i}$
$\displaystyle=(1-f(\xi))(F_{1}\tau^{1}+F_{2}\tau^{2})+(1-f_{3}(\xi))F_{3}\tau^{3},$
$\displaystyle\phi$ $\displaystyle=\frac{v(T)}{\sqrt{2}}\begin{pmatrix}0\\\
h(\xi)\end{pmatrix},$

and the sphaleron energy read as

$\displaystyle E_{sph}^{EW}(\nu,\theta,\phi,T)=$
$\displaystyle\frac{v}{g}\int\xi^{2}\sin\theta d\xi d\theta
d\phi\left[\sin^{2}\nu\frac{1}{\xi^{2}}\\{4f^{\prime
2}+2\sin^{2}\theta(f_{3}^{\prime 2}-f^{\prime 2})\\}\right.$ (95)
$\displaystyle+\sin^{4}\nu\frac{8}{\xi^{4}}\left\\{\sin^{2}\theta
f_{3}^{2}(1-f)^{2}+\cos^{2}\theta(f(1-f)+f-f_{3})^{2}\right\\}$
$\displaystyle+\frac{g}{g^{\prime}}\left\\{\sin^{2}\nu\sin^{2}\theta\frac{2}{\xi^{2}}f_{0}^{\prime^{2}}+\sin^{4}\nu\cos^{2}\theta\frac{8}{\xi^{4}}(1-f_{0})^{2}\right\\}$
$\displaystyle+\frac{1}{2}h^{\prime
2}+\sin^{2}\nu\frac{h^{2}}{2\xi^{2}}\left\\{\sin^{2}\theta(f_{0}-f_{3})^{2}+(2-\sin^{2}\theta)(1-f)^{2}\right\\}$
$\displaystyle\left.+\frac{\xi^{2}}{g^{2}v(T)^{4}}V(h,T)\right].$

Figure 13: The behaviors of $f,h,f_{0},f_{3}$ as functions of variables
$\xi$,$\theta$ and $\nu$ with $\theta\in[0,\pi]$ and $\nu\in[-\pi,\pi]$, here
$\Lambda=610$ GeV and we take $\overline{\mu}=4\pi E^{-\gamma_{E}}T$ and
$\mu_{ch}=0$ as an expample.

After integrating over the variables $\theta$ and $\phi$, the sphaleron energy
become

$\displaystyle E_{sph}^{EW}(\nu,T)=$ $\displaystyle\frac{4\pi
v}{g}\frac{v(T)}{v}\int_{0}^{\infty}d\xi\left\\{\sin^{2}{\nu}\left(\frac{8}{3}f^{\prime
2}+\frac{4}{3}f^{\prime
2}_{3}\right)+\sin^{4}{\nu}\frac{8}{\xi^{2}}\left[\frac{2}{3}f_{3}^{2}(1-f)^{2}\right.\right.$
(96)
$\displaystyle\left.+\frac{1}{3}(f(1-f)+f-f_{3})^{2}\right]+\frac{4}{3}\left(\frac{g}{g^{\prime}}\right)^{2}\left[\sin^{2}{\nu}f_{0}^{\prime
2}+\sin^{4}{\nu}\frac{2}{\xi^{2}}(1-f_{0})^{2}\right]$
$\displaystyle+\frac{1}{2}\xi^{2}h^{\prime
2}+\sin^{2}{\nu}h^{2}\left[\frac{1}{3}(f_{0}-f_{3})^{2}+\frac{2}{3}(1-f)^{2}\right]+\left.\frac{\xi^{2}}{g^{2}v(T)^{4}}V(h,T)\right\\}.$

The function $f(\xi),h(\xi),f_{0}(\xi)$ and $f_{3}(\xi)$ in eq.(96) can be
obtained by solve equations(97)

$\displaystyle
f^{\prime\prime}+\frac{1-f}{4\xi^{2}}\left[8\sin^{2}\nu(f(f-2)+f_{3}+f_{3}^{2})+\xi^{2}h^{2}\right]=0,$
(97) $\displaystyle
f^{\prime\prime}_{3}-\frac{2}{\xi^{2}}\sin^{2}\nu\left[3f_{3}+f(f-2)(1+2f_{3})\right]-\frac{h^{2}}{4}(f_{3}-f_{0})=0,$
$\displaystyle f^{\prime\prime}_{0}+\sin^{2}\nu\frac{g^{\prime
2}}{4g^{2}}h^{2}(f_{3}-f_{0})+2\frac{1-f_{0}}{\xi^{2}}=0,$ $\displaystyle
h^{\prime\prime}+\frac{2}{\xi}h^{\prime}-\sin^{2}\nu\frac{2}{3\xi^{2}}h\left[2(f-1)^{2}+(f_{3}-f_{0})^{2}\right]-\frac{1}{g^{2}v(T)^{4}}\frac{\partial
V(h,T)}{\partial h}=0$

with boundary condition

$\displaystyle f(\xi)=0,h(\xi)=0,f_{3}(\xi)=0,f_{0}(\xi)=1$
$\displaystyle,\qquad\xi\rightarrow 0$ (98) $\displaystyle
f(\xi)=1,h(\xi)=1,f_{3}(\xi)=1,f_{0}(\xi)=1$
$\displaystyle,\qquad\xi\rightarrow\infty$

A solutions of Eq.(97) without integrating over $\theta$ are shown in Fig. 13,
one can find that all the configurations of $f,h,f_{0},f_{3}$ finally approach
to spherical at large $\xi$. As shown in Fig.14, there is a periodic structure
of the sphaleron energy with Chern-Simons number(left) and $\nu$(right). The
integral value of $N_{CS}$ need $\sin{2\nu}=0(x=n\pi)$, this condition will
remove lots of terms of sphaleron energy Eq.(34) and lead a periodic structure
to the sphaleron energy with $N_{CS}$ and $\nu$. In this study, we set
$\nu=\pi/2$ to maximize the sphaleron energy.

Figure 14: The sphaleron energy as a function of $\nu$ and $N_{CS}$. The left
plot is the sphaleron energy as a function of $N_{CS}$ and in the right plot
the sphaleron energy is shown as a function of $\nu$. The lowest value is the
vanishing sphaleron energy and each highest value in the barrier between two
adjacent vacua corresponds to a sphaleron solution.

Figure 15: Examples of solutions to eq.(97) at two-loop order with $\nu=\pi/2$
for the cases of $\Lambda=600$ GeV (left plot) and $\Lambda=900$ GeV (right
plot).

Fig.15 shows that the renormalization scale and chemical potential have a
negligible effect on the solution of Eq.(97) where both angles of $\theta$ and
$\phi$ are integrated out. The figure.16 shows that the decrease in sphaleron
energy with increasing temperature for two benchmarks of $\Lambda=610$ GeV and
$\Lambda=900$ GeV when $\nu=\pi/2$, that shows that the chemical potentials
contribution will be enlarged at high NP scale.

Figure 16: The sphaleron energys as functions of temperature with
$\Lambda=610$ GeV (left panel) and $910$ GeV (right panel) at 2-loop level.

When the $U(1)$ gauge field has contributed to sphaleron energy, a magnetic
dipole moment will arise. The dipole energy with a constant magnetic field $B$
has an additional contribution to sphaleron energy with

$E_{dip}=-\mu^{(1)}B,$ (99)

where the dipole moment $\mu^{(1)}$ was given by

$\mu^{(1)}=\frac{2\pi}{3}\frac{g^{\prime}}{g^{3}v}\int_{0}^{\infty}\mathrm{d}\xi\,\xi^{2}h^{2}(\xi)[1-f(\xi)]\;.$
(100)

use $x=\xi/(\xi+3)$, it can be rewritten as

$\mu^{(1)}=\frac{2\pi}{3}\frac{g^{\prime}}{g^{3}v}\int_{0}^{1}\mathrm{d}x\,\left(-\frac{3}{-1+x}+\frac{3x}{(-1+x)^{2}}\right)\left(-\frac{3x}{-1+x}\right)^{2}h^{2}(x)[1-f(x)]\;.$
(101)

Then the sphaleron energy with magnetic field becomes:
$E_{sph}=E^{EW}_{sph}+E_{dip}$. Therefore, the magnetic field would threaten
the EWBG by changing the electroweak sphaleron rate around the PT [40].

Figure 17: The effect of the dipole moment of $E_{sph}$ (left plot) and
$PT_{sph}$ (right plot) with magnetic field $B=bT^{2}$ at $T_{n}$. We set the
renormalization scale at $\overline{\mu}=4\pi Te^{-\gamma_{E}}T$ with null
chemical potential.

In Fig.17, we show the effect of dipole moment on $E_{sph}$ and $PT_{sph}$
with the magnetic field being $B=bT_{n}^{2}$. We take $b=0.1,0.01$ as an
illustration since the magnetic field generated at PT is found to be around
the magnitude [50, 51, 52, 53, 54]. The nucleation temperature $T_{n}$ is
obtained by Fig.1 and we do not consider the effects of the magnetic field on
the PT dynamics since it was found not to drive first-order PT [55] , while it
might yield W condensate see Refs.[56, 57] for example. The effect of the
dipole moment on the quantities of $E_{sph}$ and $PT_{sph}$ at small $\Lambda$
region is negligible, but it becomes very significant at large $\Lambda$
region. The magnetic field can modify the restriction on $\Lambda$ by changing
the washout avoidance condition. At the 2-loop level with $\overline{\mu}=4\pi
e^{-\gamma_{E}}T$ and $\mu_{ch}=0$, the magnetic field changes the restriction
on $\Lambda$ from $770-800$ GeV ($B=0$) to $750-780$ GeV ($B=0.1T^{2}$). This
result is similar to the scenario of $\mu_{ch}=0.2T$ in Fig.8, and when
$\Lambda\gtrsim 800$ GeV, the effect of magnetic field on the $PT_{sph}$
increases with $\Lambda$ and grows to be larger than that of the chemical
potential. This implies that the role of the magnetic field in the weak PT is
non-negligible relative to the chemical potential.

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