aainstitutetext: Center for Theoretical Physics, Department of Physics and
Astronomy, Seoul National University, Seoul 08826, Korea

# Lepton-mediated electroweak baryogenesis, gravitational waves and the
$4\tau$ final state at the collider

Ke-Pan Xie<EMAIL_ADDRESS>
###### Abstract

An electroweak baryogenesis (EWBG) mechanism mediated by $\tau$ lepton
transport is proposed. We extend the Standard Model with a real singlet scalar
$S$ to trigger the strong first-order electroweak phase transition (SFOEWPT),
and with a set of leptophilic dimension-5 operators to provide sufficient CP
violating source. We demonstrate this model is able to generate the observed
baryon asymmetry of the universe. This scenario is experimentally testable via
either the SFOEWPT gravitational wave signals at the next-generation space-
based detectors, or the $pp\to h^{*}\to SS\to 4\tau$ process (where $h^{*}$ is
an off-shell Higgs) at the hadron colliders. A detailed collider simulation
shows that a considerable fraction of parameter space can be probed at the HL-
LHC, while almost the whole parameter space allowed by EWBG can be reached by
the 27 TeV HE-LHC.

##  1 Introduction

Cosmological observations show that there is an imbalance between the
abundance of matter and antimatter in our universe. This is also known as the
baryon asymmetry of the universe (BAU), which can be described by the baryon-
to-entropy ratio Tanabashi:2018oca

$\eta_{B}\equiv\frac{n_{B}}{s}=[0.82\sim 0.94]\times 10^{-10}.$ (1)

Creating the BAU requires three Sakharov conditions Sakharov:1967dj ;
Kuzmin:1970nx ; Ignatiev:1978uf : i) baryon number violation, ii) C and CP
violation, and iii) departure from equilibrium. The Standard Model (SM)
satisfies the first condition [via the electroweak (EW) sphaleron] but fails
in the last two: the CP violating (CPV) phase [from the CKM matrix] is too
tiny, and the SM EW phase transition (EWPT) is a smooth crossover
Rummukainen:1998as . Therefore, the observed BAU is a clear evidence for
physics beyond the SM (BSM).

There have been a lot of BSM mechanisms explaining BAU Bodeker:2020ghk , and
this paper will focus on EW baryogenesis (EWBG) Morrissey:2012db ;
Cline:2006ts ; Trodden:1998ym . In EWBG, the out-of-equilibrium environment is
provided by a strong first-order EWPT (SFOEWPT), during which bubbles
containing EW broken phase nucleate and expand in the EW symmetric background.
The CPV source on the bubble wall seeds a net density for the left-handed
fermions in front of the wall when the particles collide with the bubbles.
This “chiral asymmetry” then diffuses into the EW symmetric phase where it is
converted to a baryon asymmetry by the EW sphaleron process. The net baryon
number is swept into the expanding bubbles and then frozen, because the EW
sphaleron is strongly suppressed inside the bubble. The bubbles eventually
fill up the whole universe, completing the SFOEWPT and leaving the observed
BAU. EWBG is an attractive scenario as it typically involves BSM physics
around TeV scale, thus can be tested at current or future colliders Arkani-
Hamed:2015vfh . In addition, the phase transition gravitational waves (GWs)
might be detectable at the future space-based detectors Mazumdar:2018dfl .

Most literatures focus on EWBG mediated by quark transport, in particular by
top transport since the large top Yukawa ($y_{t}\approx 1$) enhances the CPV
source and hence the BAU Joyce:1994fu ; Joyce:1994zt ; Fromme:2006wx  (for
recent studies see Jiang:2015cwa ; Bruggisser:2018mus ; Chao:2019smr ;
Ellis:2019flb ; DeCurtis:2019rxl ; Cline:2020jre ; Xie:2020bkl )111There are
also EWBG mechanisms mediated by bottom Modak:2018csw ; Modak:2020uyq  or
charm Bruggisser:2018mrt  quark transport.. However, the top-mediated EWBG is
not as efficient as expected, due to three reasons summarized in Ref.
deVries:2018tgs  as follows. First, because of the strong QCD interaction with
the plasma, the diffusion of quark chiral asymmetry to the EW symmetric phase
is inefficient Joyce:1994bi . Second, the CPV in top sector has been
stringently constrained by the electron electric dipole moment (EDM)
experiments Cirigliano:2016nyn 222Actually, once the EDM measurements are
taken into account, the top-mediated EWBG within the SM effective field theory
(EFT) framework can only give a BAU that is 2 orders of magnitude smaller than
the observed value deVries:2017ncy . However this constraint can be relaxed if
we consider new light BSM degrees of freedom. For example, the SM plus a real
singlet scalar is able to generate the observed BAU and compatible with the
EDM experiments Espinosa:2011eu ; DeCurtis:2019rxl , especially when there is
a $\mathbb{Z}_{2}$ symmetry for the singlet Xie:2020bkl .. Third, the top
Yukawa as well as the strong sphaleron process tend to washout the chiral
asymmetry, significantly decreasing the generated BAU Giudice:1993bb ;
Tulin:2011wi .

Those three disadvantages may be overcome in a lepton-mediated EWBG
deVries:2018tgs : first, leptons are easier to diffuse in the plasma
Joyce:1994bi ; second, the EDM constraints for muon and tau are much weaker
Brod:2013cka ; third, the leptons are free from the washout effect of strong
sphaleron. Therefore, even though the CPV source from the lepton sector is
typically smaller than that from the top sector (for example the tau Yukawa
$y_{\tau}\approx 0.01\ll y_{t}$), a lepton-mediated EWBG is still possible to
account for the observed BAU. Indeed, Refs. Joyce:1994zn ; Chung:2008aya ;
Chung:2009cb ; Chiang:2016vgf ; Guo:2016ixx  have shown that a successful
$\tau$-mediated EWBG can be realized, if $y_{\tau}$ is enhanced in BSM models.
Recently, Refs. deVries:2018tgs ; Fuchs:2020uoc  study the CPV source from SM
EFT dimension-6 operators, and point out that even a SM-like $y_{\tau}$ is
able to realize EWBG.333See Ref. Fuchs:2019ore  for a $\mu$-mediated EWBG, and
Ref. Fernandez-Martinez:2020szk  for a neutrino-mediated EWBG.

In this work, we propose a novel $\tau$-mediated EWBG model based on the real
singlet scalar extension of SM (xSM). The CPV source comes from the
leptophilic dimension-5 operator $(1/\Lambda)S\bar{\ell}_{L}H\tau_{R}$, with
$\Lambda$ being the cutoff scale, $S$ the singlet,
$H=\left(G^{+},(h+iG^{0})/\sqrt{2}\right)^{T}$ the Higgs doublet,
$\ell_{L}=(\nu_{\tau L},\tau_{L})^{T}$ the third generation lepton doublet and
$\tau_{R}$ the right-handed tau. We will show that such a model can explain
today’s BAU with a SM-like $y_{\tau}$ in Section 2, where we also discuss the
possibility of detecting the phase transition GWs at the future space-based
detectors. Section 3 is devoted to collider phenomenology, in which we
demonstrate our scenario can be efficiently probed by the $pp\to h^{*}\to
SS\to\tau^{+}\tau^{-}\tau^{+}\tau^{-}$ process at the LHC and future 27 TeV
HE-LHC. Finally, we conclude in Section 4.

##  2 Lepton-mediated EWBG

###  2.1 SFOEWPT and GWs

The SFOEWPT of xSM has been extensively studied McDonald:1993ey ;
Profumo:2007wc ; Espinosa:2011ax ; Cline:2012hg ; Alanne:2014bra ;
Vaskonen:2016yiu ; Huang:2018aja ; Cheng:2018ajh ; Alanne:2019bsm ;
Gould:2019qek ; Carena:2019une , and we needn’t to repeat all those results
here. For simplicity, we assume the following tree level potential

$V_{0}=\mu_{H}^{2}|H|^{2}+\frac{\mu_{S}^{2}}{2}S^{2}+\lambda_{H}|H|^{4}+\frac{\lambda_{S}}{4}S^{4}+\lambda_{HS}|H|^{2}S^{2},$
(2)

which is symmetric under the $\mathbb{Z}_{2}$ transformation $S\to-S$. Under
the unitary gauge,

$V_{0}=\frac{\mu_{H}^{2}}{2}h^{2}+\frac{\mu_{S}^{2}}{2}S^{2}+\frac{\lambda_{H}}{4}h^{4}+\frac{\lambda_{S}}{4}S^{4}+\frac{\lambda_{HS}}{2}h^{2}S^{2}.$
(3)

The bounded below condition for this potential is $\lambda_{H}>0$,
$\lambda_{S}>0$, and $\sqrt{\lambda_{H}\lambda_{S}}+\lambda_{HS}>0$. We are
interested in the parameter space with Bian:2019kmg

$\mu_{H}^{2}<0,\quad\mu_{S}^{2}<0,\quad\lambda_{H}\mu_{S}^{2}>\lambda_{HS}\mu_{H}^{2},\quad\lambda_{S}\mu_{H}^{2}>\lambda_{HS}\mu_{S}^{2},\quad-\frac{\mu_{H}^{4}}{\lambda_{H}}<-\frac{\mu_{S}^{4}}{\lambda_{S}},$
(4)

in which $V_{0}$ has two local minima $(v,0)$ and $(0,w)$ with

$v=\sqrt{-\mu_{H}^{2}/\lambda_{H}},\quad w=\sqrt{-\mu_{S}^{2}/\lambda_{S}},$
(5)

and the former is the global minimum, i.e. the vacuum. Expanding potential
around the vacuum gives physical masses of the scalars

$M_{h}^{2}=-2\mu_{H}^{2},\quad M_{S}^{2}=\mu_{S}^{2}+\lambda_{HS}v^{2}.$ (6)

In other words, the measured values $M_{h}=125$ GeV and $v=246$ GeV have fixed
$\mu_{H}^{2}$ and $\lambda_{H}$, and we only have three free parameters in Eq.
(3).

At one-loop level, $V_{0}$ becomes

$V_{1}=V_{0}+V_{\rm CW}+V_{\rm CT},$ (7)

where $V_{\rm CW}$ is the Coleman-Weinberg potential, and $V_{\rm CT}$ is the
counter term. At finite temperature, the scalar potential receives thermal
corrections,

$V_{T}=V_{1}+\tilde{V}_{T}+V_{\rm daisy},$ (8)

where $\tilde{V}_{T}$ is the one-loop thermal integral correction, while
$V_{\rm daisy}$ is the daisy resummation term. The detailed expressions for
$V_{\rm CW}$, $V_{\rm CT}$, $\tilde{V}_{T}$ and $V_{\rm daisy}$ are given in
Appendix A. Taking only the leading $T^{2}$ terms Dolan:1973qd , $V_{T}$ can
be approximated as

$V_{T}\approx\frac{1}{2}\left(\mu_{H}^{2}+c_{H}T^{2}\right)h^{2}+\frac{1}{2}\left(\mu_{S}^{2}+c_{S}T^{2}\right)S^{2}+\frac{\lambda_{H}}{4}S^{4}+\frac{\lambda_{S}}{4}S^{4}+\frac{\lambda_{HS}}{2}h^{2}S^{2},$
(9)

where

$c_{H}=\frac{3g^{2}+g^{\prime
2}}{16}+\frac{y_{t}^{2}}{4}+\frac{\lambda_{H}}{2}+\frac{\lambda_{HS}}{12},\quad
c_{S}=\frac{\lambda_{S}}{4}+\frac{\lambda_{HS}}{3},$ (10)

with $g^{(\prime)}$ and $y_{t}$ being the EW gauge couplings and top Yukawa,
respectively.

The thermal potential in Eq. (9) can realize a two-step phase transition in
which a second-order phase transition first occurs along the $S$-axis, and
then a first-order EWPT happens through the vacuum decay between the $S$\- and
$h$\- axes, i.e.

$\left(\left\langle h\right\rangle=0,\left\langle
S\right\rangle=0\right)\xrightarrow[\rm step]{\rm First}\left(\left\langle
h\right\rangle=0,\left\langle S\right\rangle\neq 0\right)\xrightarrow[\rm
step]{\rm Second}\left(\left\langle h\right\rangle\neq 0,\left\langle
S\right\rangle=0\right),$ (11)

as the temperature decreases. For the first step, if there were an exact
$\mathbb{Z}_{2}$ symmetry for $S$ in $V_{T}$, then the probability of
transition to $+S$ and $-S$ directions would be equal, leaving the domain wall
problem. To avoid this issue, we assume a small $\mathbb{Z}_{2}$-breaking is
present, such that the transition along the $+S$ direction is energetically
preferred Espinosa:2011eu .

The second step is a first-order EWPT, whose necessary condition is two
degenerate vacua separated by a barrier at the critical temperature $T_{c}$.
For the polynomial potential in Eq. (9), this condition is equivalent to
Bian:2019kmg

$\frac{c_{S}}{c_{H}}<\frac{\mu_{S}^{2}}{\mu_{H}^{2}}<\frac{\sqrt{\lambda_{S}}}{\sqrt{\lambda_{H}}}<\frac{\lambda_{HS}}{\lambda_{H}}.$
(12)

The sufficient condition for a first-order EWPT is the onset of nucleation,
which is resolved from the equality of vacuum decay rate and the universe
expansion rate at the nucleation temperature $T_{n}$,

$T_{n}^{4}e^{-S_{3}(T_{n})/T_{n}}\approx H^{4}(T_{n}),$ (13)

with $S_{3}(T)$ being the Euclidean action of the $O(3)$-symmetric bounce
solution Linde:1981zj , and $H(T)$ the Hubble constant. For a radiation-
dominated universe, Eq. (13) is approximately

$\frac{S_{3}(T_{n})}{T_{n}}\sim 140.$ (14)

for a $T_{n}$ around the EW scale Quiros:1999jp . For the EW sphaleron process
to be suppressed inside the bubble, we further require Moore:1998swa ;
Zhou:2019uzq

$v_{n}/T_{n}\gtrsim 1,$ (15)

which is the definition for a strong transition, namely a SFOEWPT.

For a numerical study, we adopt the complete one-loop potential Eq. (8) as the
input. Using Eq. (6), we choose the three free parameter of the scalar
potential as $M_{S}$, $\lambda_{HS}$ and $\lambda_{S}$. For a fixed
$\lambda_{S}$, we scan over $M_{S}$ and $\lambda_{HS}$ and derive the SFOEWPT
parameter space by solving Eq. (13) with the CosmoTransitions package
Wainwright:2011kj  and checking Eq. (14). The results are shown as shaded
regions in Fig. 1. We found that the shape of those regions match the
analytical condition in Eq. (12) very well. This is because in the two-step
phase transition paradigm, the SFOEWPT is induced by the tree level potential
barrier dominated by the $\lambda_{HS}h^{2}S^{2}/2$ term in the potential,
thus a leading $T^{2}$ analysis already provides very good approximation. Due
to the same reason, a sizable $\lambda_{HS}$ is needed for a successful
SFOEWPT. The correlation between SFOEWPT and the $\lambda_{HS}$ parameter
provides an excellent channel for probing our mechanism at the collider, as
will be shown in Section 3.

Figure 1: Left: parameter space allowed by a successful SFOEWPT. The colored
shaded regions are for individual $\lambda_{S}$ values, except the light blue
shaded area is for the non-perturbative region that $\lambda_{S}>4\pi$. The
colored scatter points are the parameter regions that are detectable at the
LISA. For $\lambda_{S}=8$, to avoid the plot to be too messy we omit the
general SFOEWPT region and only show the LISA detectable points. The star-
symbols highlight the two BPs in Eq. (2.1). Right: the GW spectrum for the two
BPs selected from the left panel.

A SFOEWPT in the early universe may show detectable cosmological signals
today, as the phase transition GWs typically peak around millihertz, within
the sensitive region of a few next-generation space-based detectors, such as
LISA Audley:2017drz , BBO Crowder:2005nr , TianQin Luo:2015ght ; Hu:2017yoc ,
Taiji Hu:2017mde ; Guo:2018npi  and DECIGO Kawamura:2011zz ; Kawamura:2006up .
GWs are produced by three sources: bubble collision, sound waves and
turbulence. For a SFOEWPT with a non-luminal terminal bubble velocity, the
bubble collision contribution is negligible and the GWs are dominated by sound
waves Ellis:2018mja . The GW spectrum today is described by

$\Omega_{\rm GW}(f)=\frac{1}{\rho_{c}}\frac{\rho_{\rm GW}}{d\ln f},$ (16)

where $f$ is the frequency, $\rho_{\rm GW}$ is the GW energy density and
$\rho_{c}$ is the critical energy density of the present universe. For a given
SFOEWPT, Eq. (16) can be expressed as numerical functions of the following
three physical parameters Grojean:2006bp ; Caprini:2015zlo ; Caprini:2019egz :

  1. 1.

$v_{b}$, the wall velocity of the expanding bubbles.

  2. 2.

$\alpha$, the ratio of the SFOEWPT latent heat to the cosmic radiative energy
density

$\alpha=\frac{1}{\rho_{R}(T_{n})}\left(-\Delta V_{T}+T\frac{\partial\Delta
V_{T}}{\partial T}\right)\Big{|}_{T_{n}},$ (17)

where $\rho_{R}(T_{n})=\pi^{2}g(T_{n})T_{n}^{4}/30$ with $g(T_{n})$ being the
number of relativistic degrees of freedom, and $\Delta V_{T}$ is the free
energy difference between the true and false vaca.

  3. 3.

$\beta/H$, the inverse ratio of the time scale of the SFOEWPT and the universe
expansion,

$\beta/H=T_{n}\frac{d}{dT}\left(\frac{S_{3}(T)}{T}\right)\Big{|}_{T_{n}}.$
(18)

In general, $\beta/H$ is related to the duration of the SFOEWPT hence the peak
frequency of the GWs, while $\alpha$ is relevant to the signal strength.

We obtain the GWs spectrum using the formulae from the references mentioned
above, with the energy budget from Ref. Espinosa:2010hh 444See Ref.
Wang:2020nzm  for a recent study about the energy budget beyond the bag model
equation of state.. The bubble velocity is adopted as $v_{b}=0.6$ as a
benchmark. Strictly speaking, the $\alpha$ and $\beta/H$ parameters should be
calculated at the percolation temperature $T_{p}$ Megevand:2016lpr ;
Kobakhidze:2017mru ; Ellis:2018mja ; Ellis:2020awk ; Wang:2020jrd ; however
the supercooling effect in our scenario is not strong – we have checked that
$\alpha\lesssim 1$ for our parameter space – thus $T_{p}\approx T_{n}$ is a
good approximation. We have taken into account the suppression factor
$H\tau_{\rm sw}$ for the sound wave contribution due to the shortness of its
duration compared to the Hubble time Ellis:2018mja 555We adopt the suppression
factor as $\Upsilon=1-(1+2H\tau_{\rm sw})^{-1/2}$ Guo:2020grp .. For the
detectability of GWs signals we take the LISA as an example and evaluate the
signal-to-noise-ratio (SNR) Caprini:2015zlo

${\rm SNR}=\sqrt{\mathcal{T}\int_{f_{\rm min}}^{f_{\rm
max}}df\left(\frac{\Omega_{\rm GW}(f)}{\Omega_{\rm LISA}(f)}\right)^{2}},$
(19)

with $\Omega_{\rm LISA}$ being the sensitive curve of the LISA detector
Caprini:2015zlo , and $\mathcal{T}=9.46\times 10^{7}$ s the data-taking
duration Caprini:2019egz . With ${\rm SNR}=10$ as the detectable threshold, we
find that for a fixed $\lambda_{S}$, there is a narrow band in the
$M_{S}-\lambda_{HS}$ plane that can be probed by the LISA, as shown in the
left panel of Fig. 1. Fixing $\lambda_{S}=1$, we choose two benchmark points
(BPs) as

$\displaystyle{\rm BP1}:\quad M_{S}=100~{}{\rm
GeV},\quad\lambda_{HS}=0.46;\quad\alpha=0.13,\quad\beta/H=406,$
$\displaystyle{\rm BP2}:\quad M_{S}=200~{}{\rm
GeV},\quad\lambda_{HS}=0.88;\quad\alpha=0.05,\quad\beta/H=109,$ (20)

and plot their GW spectra in the right panel of Fig. 1 as an illustration.

###  2.2 Transport equations and the generation of BAU

Consider a SFOEWPT triggered by the vacuum decay $(0,w_{n})\to(v_{n},0)$ at
$T_{n}$. Since EWBG happens in the vicinity of the bubble wall, it’s
convenient to work in the wall rest frame. Under the planar wall
approximation, the scalars can be treated as background fields $\hat{h}$ and
$\hat{S}$ that depend only on the spatial distance $z$ from the center of
wall. Assuming $z\to\pm\infty$ is the broken/symmetric phase, the scalar
backgrounds can be parametrized as

$\hat{h}(z)=\frac{v_{n}}{2}\left(1+\tanh\frac{z}{L_{w}}\right),\quad\hat{S}(z)=\frac{w_{n}}{2}\left(1-\tanh\frac{z}{L_{w}}\right),$
(21)

with $L_{w}$ being the wall thickness.

To realize another necessary condition for EWBG, i.e. the CP violation, we
shall assume the singlet has the following leptophilic interactions with the
SM fields via dimension-5 operators,

$\mathcal{L}_{5}=\sum_{i,j}\frac{c_{ij}}{\Lambda}S\bar{\ell}_{L}^{i}He_{R}^{j}+{\rm
h.c.},$ (22)

where $i$, $j$ are generation indices, $c_{ij}$ are the (complex) Wilson
coefficients, and $\Lambda$ is the cutoff scale of EFT. For simplicity, we
assume the Willson coefficients are flavor-diagonal, i.e. $c_{ij}={\rm
diag}\\{c_{e},c_{\mu},c_{\tau}\\}$. During a SFOEWPT, the effective mass of a
lepton reads

$\bar{m}_{i}(z)=\frac{y_{i}}{\sqrt{2}}\hat{h}(z)+\frac{c_{i}}{\sqrt{2}\Lambda}\hat{h}(z)\hat{S}(z),$
(23)

where $i=e$, $\mu$, $\tau$, and $y_{i}=\sqrt{2}M_{i}/v$ is the SM Yukawa
coupling, with $M_{i}$ being the physical lepton mass at zero temperature.
Since the effective mass is space-dependent, the complex phase of $c_{i}$
cannot be globally rotated away, resulting in physical CPV effects.

The Yukawa couplings for the first and second generation leptons are too small
to generate a considerable CPV source deVries:2018tgs , thus we neglect them
and focus on the $\tau$ lepton. In this case, the magnitude of $c_{\tau}$ can
be absorbed into the definition of $\Lambda$, and we can rewrite the relevant
part of Eq. (22) as

$\mathcal{L}_{5}\supset\frac{e^{i\phi_{\tau}}}{\Lambda_{\tau}}S\bar{\ell}_{L}H\tau_{R}+{\rm
h.c.},$ (24)

such that $\phi_{\tau}$ is a pure CPV phase. Note that this operator is rather
weakly constrained by the EDM experiments, especially in our scenario that
$V_{0}$ is $\mathbb{Z}_{2}$ symmetric and $\left\langle S\right\rangle=0$ so
that there is no tree level $h-S$ mixing at zero temperature.

In the plasma, the $\tau$ leptons participate in gauge, Yukawa, helicity-
flipping and EW sphaleron interactions. We deal with those interactions
following the standard two-step approach: in the first step we derive the
generation and diffusion of the (lepton) chiral asymmetry by solving the
Boltzmann equations of the “fast processes”, and then in the second step the
chiral asymmetry is converted into a baryon asymmetry via the EW sphaleron.
Here EW sphaleron is the “slow process” and all other interactions are treated
as “fast”.

First we establish and solve the Boltzmann equations. Denoting the net
particle density (i.e. the number density difference between particle and
antiparticle) for left-handed third generation leptons, right-handed taus, and
the Higgs bosons as $\ell=n_{\nu_{\tau L}}+n_{\tau_{L}}$, $\tau=n_{\tau_{R}}$
and $h=n_{G^{+}}+n_{H^{0}}$, respectively, the transport equations read
deVries:2018tgs

$\displaystyle v_{w}\ell^{\prime}-D_{\ell}\ell^{\prime\prime}$
$\displaystyle=$
$\displaystyle\Gamma_{M}\left(\frac{\tau}{k_{\tau}}-\frac{\ell}{k_{\ell}}\right)+\Gamma_{Y}\left(\frac{\tau}{k_{\tau}}-\frac{\ell}{k_{\ell}}+\frac{h}{k_{h}}\right)-S_{\tau},$
$\displaystyle v_{w}\tau^{\prime}-D_{\tau}\tau^{\prime\prime}$
$\displaystyle=$
$\displaystyle-\Gamma_{M}\left(\frac{\tau}{k_{\tau}}-\frac{\ell}{k_{\ell}}\right)-\Gamma_{Y}\left(\frac{\tau}{k_{\tau}}-\frac{\ell}{k_{\ell}}+\frac{h}{k_{h}}\right)+S_{\tau},$
(25)

where $\ell$, $\tau$ and $h$ are functions of the spatial coordinate $z$, and
“ $\prime$ ” represents the $d/dz$ operation. $v_{w}\in(0,1)$ is the bubble
wall velocity with respect to the plasma just in front of the wall. Note that
$v_{w}$ can differ from $v_{b}$ defined in Section 2.1, which is relative wall
velocity to the plasma at infinite distance No:2011fi . $D_{\ell}=100/T_{n}$
and $D_{\tau}=380/T_{n}$ are respectively the diffusion constants for left-
and right-handed leptons (note that they are much bigger than quark diffusion
constant $D_{q}=6/T_{n}$) Joyce:1994zn . $\Gamma_{M}$ and $\Gamma_{Y}$ denote
the helicity-flipping and Yukawa rates, respectively, and their detail
definitions as well as the temperature-dependent coefficients $k_{i}$ are
given in Appendix B. The EW gauge interactions are not written in Eq. (2.2)
because they are treated as in equilibrium, so that the two components of the
same $SU(2)_{L}$ doublet share a common chemical potential. $S_{\tau}$ is the
CPV source given by the closed time path method Lee:2004we

$S_{\tau}=\frac{v_{w}}{\pi^{2}}{\rm
Im}\left[\bar{m}_{\tau}^{\prime}m_{\tau}^{*}\right]J_{\tau},$ (26)

where $\bar{m}_{\tau}$ is the $z$-dependent effective mass in Eq. (23), while
$J_{\tau}$ is a numerical factor whose expression is presented in Appendix B.
Eq. (26) clearly reveals that the number density difference between $\ell_{L}$
and $\bar{\ell}_{L}$ is sourced by the space-dependent imaginary part of the
effective mass (namely the physical CPV phase) when the $\tau$ leptons are
passing across the bubble wall.

There are three unknown functions $\ell$, $\tau$ and $h$ in Eq. (2.2), but
only two equations (for leptons). To solve the equations self-consistently we
need one more equation about $h$. However, the transport equation of $h$
involves the heavy quarks such as top and bottom, because they interact
strongly with the Higgs. As a result, a complete treatment for lepton
transport should also include the equations for quarks as well. Thanks to the
small lepton Yukawa couplings, the impact of $h$ on lepton transport is
negligible, hence we can reasonably approximate $h\approx 0$ and avoid the
complexity of the quark transport equations. In addition, the diffusion
constant for left- and right-handed leptons can be approximated as the same,
i.e. $D_{\ell}\approx D_{\tau}\approx 100/T_{n}$, which provides a relation
$\ell=-\tau$ and therefore Eq. (2.2) is simplified into a single ordinary
differential equation about $\ell$

$-D_{\ell}\ell^{\prime\prime}+v_{w}\ell^{\prime}+(\Gamma_{M}+\Gamma_{Y})\left(\frac{1}{k_{\ell}}+\frac{1}{k_{\tau}}\right)\ell=-S_{\tau},$
(27)

which can be solved either semi-analytically or numerically. Ref.
deVries:2018tgs  has shown that Eq. (27) is a very good approximation to the
complete treatment which takes into account $D_{\ell}\neq D_{\tau}$ and the
heavy quark transport.

By numerically solving Eq. (27), we get the non-zero chiral asymmetry $\ell$.
This asymmetry is then converted into a baryon-to-entropy ratio
deVries:2018tgs

$\eta_{B}=\frac{n_{B}}{s}=-\frac{3\Gamma_{\rm
ws}}{2sD_{q}\alpha_{+}}\int_{-\infty}^{0}dz\ell(z)\exp\\{-\alpha_{-}z\\},$
(28)

where $D_{q}=6/T_{n}$ is the quark diffusion constant Joyce:1994zn ,
$\Gamma_{\rm ws}=6\kappa\alpha_{W}^{5}T_{n}$ is the EW sphaleron rate with
$\kappa\approx 18$ DOnofrio:2014rug , and

$\alpha_{\pm}=\frac{v_{w}\pm\sqrt{15D_{q}\Gamma_{\rm ws}+v_{w}^{2}}}{2D_{q}}.$
(29)

The integration region of Eq. (28) is limited within the EW symmetric phase
where the EW sphaleron is active. The generated baryon number survives until
today, becoming the observed BAU. Note that the dimension-5 operator in Eq.
(24) is odd under the $\mathbb{Z}_{2}$ transformation, so does the CPV source
$S_{\tau}$ in Eq. (26). Consequently, if there were an exact $\mathbb{Z}_{2}$
for $S$ in the potential, the $+S$ and $-S$ transitions equally distributed in
different patches of the universe finally give a zero net BAU on average. This
consideration gives another motivation for the small $\mathbb{Z}_{2}$-breaking
term Espinosa:2011eu .

Figure 2: Left: the chiral asymmetry $\ell(z)\equiv
n_{\ell}(z)-n_{\bar{\ell}}(z)$ solved from the transport equation (27). Right:
the BAU as a function of wall velocity $v_{w}$. The gray band represents the
BBN fitted result of $\eta_{B}$. The CPV phase is $\phi_{\tau}=0.02$ for both
panels.

For illustration, we consider the two SFOEWPT BPs in Eq. (2.1) of Section 2.1.
Their SFOEWPT profiles are

$\displaystyle{\rm BP1}:\quad T_{n}=64.44~{}{\rm GeV},\quad
v_{n}=239.06~{}{\rm GeV},\quad w_{n}=132.13~{}{\rm GeV};$ $\displaystyle{\rm
BP2}:\quad T_{n}=94.32~{}{\rm GeV},\quad v_{n}=220.55~{}{\rm GeV},\quad
w_{n}=108.97~{}{\rm GeV},$ (30)

and we take the wall thickness $L_{w}=15/T_{n}$ according to Ref.
Konstandin:2014zta . The two BPs are fed into the Boltzmann equation (27) to
get the chiral asymmetry and then we use Eq. (28) to evaluate the BAU. The CPV
phase and cutoff scale are chosen to be $\phi_{\tau}=0.02$ and
$\Lambda_{\tau}=10$ TeV,666The choice of $\Lambda_{\tau}=10$ TeV seems much
larger than the scale adopted in Ref. deVries:2018tgs , which studies the CPV
dimension-6 operators in SM EFT and uses $\Lambda_{\tau}=1$ TeV. However, the
normalization schemes are different for these two researches: in Ref.
deVries:2018tgs , the CPV operator is
$(iy_{\tau}/\Lambda_{\tau}^{2})|H|^{2}\bar{\ell}_{L}H\tau_{R}$, which has an
additional suppression factor $y_{\tau}\approx 0.01$; while in this article,
the relevant operator is
$(e^{i\phi_{\tau}}/\Lambda_{\tau})S\bar{\ell}_{L}H\tau_{R}$. Once taken into
account this difference, the amounts of CPV in our scheme and Ref.
deVries:2018tgs  are of the same order (or more specifically, for the selected
benchmarks, the CPV effects in our article are several times larger).
respectively. The chiral asymmetry $\ell(z)\equiv
n_{\ell}(z)-n_{\bar{\ell}}(z)$ is shown in the left panel of Fig. 2, where we
can recognize that the asymmetry is generated on the bubble wall, and then
diffuses to both the EW symmetric and broken phases. Only the chiral asymmetry
in the symmetric phase is responsible for generating BAU, and we plot the
resultant BAU as a function of $v_{w}$ in the right panel of Fig. 2. The
observed BAU is shown in gray band and we find it can be achieved for
$v_{w}\sim 0.01$ or 0.1 in the figure, where $\phi_{\tau}=0.02$ is fixed.
Since $\phi_{\tau}$ is actually a free parameter, we conclude that our
mechanism can explain the BAU within a vast parameter space of $\phi_{\tau}$
and $v_{w}$.

##  3 Collider search: the $4\tau$ final state

Complementary to the space-based GW detectors, the terrestrial collider
experiments also serve as a powerful probe for the SFOEWPT by either measuring
the Higgs potential shape or directly detecting the relevant BSM physics
around TeV scale Profumo:2014opa ; Cao:2017oez ; Alves:2018oct ; Zhou:2020idp
; Chen:2019ebq ; Huang:2016cjm ; Kozaczuk:2019pet ; Papaefstathiou:2020iag ;
Alves:2020bpi . In this section we assess the possibility of detecting the
lepton-mediated EWBG at the 14 TeV LHC and the 27 TeV HE-LHC Abada:2019ono .
At the collider, the real singlet in the xSM can be probed by the resonant or
non-resonant di-Higgs production, the modified Higgs-fermion or Higgs-gauge
couplings, etc. However, most of the proposed channels require the mixing
between $S$ and $h$. If unfortunately the $S-h$ mixing is negligible, e.g. in
the scenario considered in Section 2.1 that $S$ has an approximate
$\mathbb{Z}_{2}$ symmetry in the potential and $\left\langle
S\right\rangle|_{T=0}=0$, then it would be very challenging to test the
SFOEWPT at the collider Ashoorioon:2009nf ; Curtin:2014jma . However, in our
scenario, there exists the following process at a $pp$ collider,

$gg\to h^{*}\to SS\to\tau^{+}\tau^{-}\tau^{+}\tau^{-},$ (31)

i.e. the pair production of $S$ through an off-shell Higgs (mediated by the
$\lambda_{HS}v\,hS^{2}\subset\lambda_{HS}|H|^{2}S^{2}$ vertex), followed by
the $S\to\tau^{+}\tau^{-}$ decay (mediated by the dimension-5 operator
$(c_{\tau}/\Lambda_{\tau})S\bar{\ell}_{L}H\tau_{R}$). This robust channel is
present even in the absence of a $h-S$ mixing. Since the $\lambda_{HS}$
coupling is required by a SFOEWPT, while the dimension-5 operator is required
by the CPV, both two necessary ingredients of our model is tested by the
single process in Eq. (31).

Figure 3: Production rates of $pp\to h^{*}\to
SS\to\tau^{+}\tau^{-}\tau^{+}\tau^{-}$ at the 14 TeV LHC and 27 TeV HE-LHC. We
have assumed $\lambda_{HS}=0.5$ and $\text{Br}(S\to\tau^{+}\tau^{-})=100\%$.

The cross sections of Eq. (31) are shown as functions of $M_{S}$ in Fig. 3,
where we have set $\lambda_{HS}=0.5$ and assumed $S$ decays exclusively to
$\tau^{+}\tau^{-}$. Note that the production rate is proportional to
$\lambda_{HS}^{2}$, thus this channel is sensitive to the SFOEWPT parameter
space, which requires a sizable $\lambda_{HS}$ (see the left panel of Fig. 1).
The decay branching ratio for a $\tau$ lepton is $\sim 35\%$ and $\sim 65\%$
for leptonic and hadronic channels, respectively. The leptonic decay yields a
charged lepton ($e^{\pm}$ or $\mu^{\pm}$) and missing transverse momentum,
while the hadronic decay typically results in a $\tau$-jet Bagliesi:2007qx .
Therefore, the $4\tau$ final state can be categorized into the following final
states directly available at the detector:

$4\tau_{j}~{}(17.9\%),\quad 1\ell 3\tau_{j}~{}(38.4\%),\quad 2\ell
2\tau_{j}~{}(31.1\%),\quad 3\ell 1\tau_{j}~{}(11.1\%),\quad 4\ell~{}(1.5\%),$
(32)

where $\ell$ denotes the charged leptons $e^{\pm}$ and $\mu^{\pm}$, $\tau_{j}$
denotes the $\tau$-jets, and the numbers in the brackets are the corresponding
probabilities. We find that the $1\ell 3\tau_{j}$ and $2\ell 2\tau_{j}$
channels are hopefully to be probed, as they enjoy considerable cross section
fractions and have charged leptons to trigger the events. While the $2\ell
2\tau_{j}$ final state is searched at the LHC in supersymmetry-relevant
experiments Aaboud:2018zeb , the $1\ell 3\tau_{j}$ final state has not yet
been dedicatedly searched at the LHC.

Unit: fb |  | Signal
---
BP1
| Signal
---
BP2
$W^{\pm}+{\rm jets}$ | $Z+{\rm jets}$ | $t\bar{t}$ | $W^{\pm}\tau^{+}\tau^{-}j$ | $\tau^{+}\tau^{-}+{\rm jets}$ | $\tau^{+}\tau^{-}\tau^{+}\tau^{-}$
14 TeV LHC
Before | $12.3$ | $1.19$ | $1.45\times 10^{6}$ | $6.18\times 10^{5}$ | $1.21\times 10^{5}$ | $129$ | $1.49\times 10^{5}$ | $7.15$
Cut I | $1.76$ | $0.352$ | $2.43\times 10^{5}$ | $5.91\times 10^{4}$ | $6.73\times 10^{4}$ | $34.5$ | $6.35\times 10^{3}$ | $0.511$
Cut II | $0.0733$ | $0.0269$ | $0.832$ | $3.28$ | $3.41$ | $0.152$ | $0.841$ | $0.0378$
Cut III | $0.0661$ | $0.0245$ | $0.681$ | $2.64$ | $0.243$ | $0.134$ | $0.762$ | $0.0356$
27 TeV HE-LHC
Before | $42.7$ | $5.30$ | $4.10\times 10^{6}$ | $1.59\times 10^{6}$ | $1.06\times 10^{6}$ | $321$ | $3.34\times 10^{5}$ | $13.4$
Cut I | $6.74$ | $1.64$ | $6.66\times 10^{5}$ | $1.69\times 10^{5}$ | $5.55\times 10^{5}$ | $95.8$ | $1.72\times 10^{4}$ | $1.19$
Cut II | $0.267$ | $0.115$ | $2.54$ | $13.9$ | $45.7$ | $0.369$ | $2.23$ | $0.0724$
Cut III | $0.245$ | $0.103$ | $2.05$ | $10.9$ | $9.14$ | $0.315$ | $1.87$ | $0.0635$
Table 1: Cut flows for backgrounds and the two signal BPs at the 14 TeV LHC
and 27 TeV HE-LHC. The two signal BPs are from Eq. (2.1) in Section 2.1.

In this work, we focus on the $1\ell 3\tau_{j}$ channel, whose main
backgrounds are the SM $W^{\pm}+{\rm jets}$, $Z+{\rm jets}$, $t\bar{t}$,
$W^{\pm}\tau^{+}\tau^{-}j$, $\tau^{+}\tau^{-}+{\rm jets}$ and
$\tau^{+}\tau^{-}\tau^{+}\tau^{-}$ processes. We perform a collider simulation
using the packages FeynRules Alloul:2013bka  (to write the UFO model file
Degrande:2011ua  for our scenario), MadGraph5_aMC@NLO Alwall:2014hca  (to
generate parton level events for signal and backgrounds) and Pythia8
Sjostrand:2007gs  and Delphes deFavereau:2013fsa  (for parton shower and fast
detector simulation, respectively). The $W^{\pm}+{\rm jets}$, $Z+{\rm jets}$
and $\tau^{+}\tau^{-}+{\rm jets}$ backgrounds are realized by matching
$W^{\pm}jjj$ to $W^{\pm}jj$ process, matching $Zjj$ to $Zj$ process, matching
$\tau^{+}\tau^{-}jj$ to $\tau^{+}\tau^{-}j$ process, respectively. The
$t\bar{t}$ is also matched to $+1~{}{\rm jet}$ final state. The inclusive
decay of $\tau$ lepton is implemented by the Pythia8 package.

To suppress the backgrounds, we apply the following cuts to the events:

  1. 1.

Exactly 1 charged lepton with $p_{T}^{\ell}>25$ GeV and $|\eta_{\ell}|<2.5$,
and at least 3 jets with $p_{T}^{j}>20$ GeV and $|\eta_{j}|<5$.

  2. 2.

At least 3 $\tau$-tagged jets. The tagging and mistag rates for a $\tau$-jet
are set to be 60% and 1%, respectively.

  3. 3.

Veto any event with $b$-tagged jets. The $b$-tagging efficiency is chosen as
77%, and the mistag rate for $c$ and other light quarks are 17% and 0.75%,
respectively Aaboud:2018xpj .

The cut flows for the backgrounds and the two signal BPs are listed in Table
1. We can see that the requirement of three $\tau$-jets significantly
suppresses the backgrounds, while the $b$-veto cut works very well in reducing
the $t\bar{t}$ background.

Given the cut flow data, now we can derive the signal significance for a given
set of $(M_{S},\lambda_{HS})$ and integrated luminosity $\mathcal{L}$:

${\rm
Significance}=\frac{\sigma_{S}\epsilon_{S}}{\sqrt{\sigma_{B}\epsilon_{B}}}\sqrt{\mathcal{L}},$
(33)

where $\sigma_{S,B}$ and $\epsilon_{S,B}$ denote the cross sections and cut
efficiencies for signal and total background, respectively. We define the
significance equal to $5\sigma$ and $2\sigma$ as the experimental discovery
and exclusion limits of the collider at a given integrated luminosity,
respectively. Note that the collider phenomenology is irrelevant to the
singlet quartic coupling $\lambda_{S}$, while the SFOEWPT parameter space
depends on $\lambda_{S}$, see the left panel of Fig. 1. In Fig. 4, we obtain
the whole SFOEWPT parameter space at the $\lambda_{HS}-M_{S}$ plane by varying
$\lambda_{S}$ from 0 to the non-perturbative limit $4\pi$, and taking the
envelop of the SFOEWPT regions. The expected reach of different collider
setups is shown in the figure with solid or dashed lines with different
colors. We can see that those reach curves have similar shapes with the
SFOEWPT parameter space (gray shaded region), because both the $gg\to h^{*}\to
SS$ process and the SFOEWPT prefer a large $\lambda_{HS}$. A considerable
fraction of the SFOEWPT space can be probed at the HL-LHC (namely the LHC with
an integrated luminosity of 3 ab-1), while almost all the SFOEWPT parameter
space can be covered by the 27 HE-LHC with an integrated luminosity of 15
ab-1.

Figure 4: Probing the SFOEWPT parameter space at the LHC and HE-LHC. The left
and right panels show the $5\sigma$ discovery and $2\sigma$ exclusion limits,
respectively. The gray shaded region labeled as “SFOEWPT” is derived by
varying $\lambda_{S}\in(0,4\pi)$.

Note that the results in Fig. 4 are for the $1\ell 3\tau_{j}$ channel only. We
have also checked that the $2\ell 2\tau_{j}$ channel shows a similar reach by
using the backgrounds simulated in Ref. Aaboud:2018zeb . Therefore, the
combination of those two channels can provide an even better probe for our
mechanism. Finally, we emphasize that the collider phenomenology presented in
this section is mainly determined by the existence of the dimension-5 operator
$(c_{\tau}/\Lambda_{\tau})S\bar{\ell}_{L}H\tau_{R}$, but not sensitive to its
CPV phase. If in the future we really detected an excess in the $4\tau$
channel, then a further analysis should be performed to reveal its CP
structure.

##  4 Conclusion

In this article, we have proposed a novel EWBG mechanism mediated by $\tau$
lepton transport. The SM is extended with a real singlet to trigger the
SFOEWPT and with a set of dimension-5 operators to get sufficient CPV source.
For simplicity, we assume a scalar potential that is symmetric under $S\to-S$
at tree level, and demonstrate that it is able to realize the SFOEWPT with the
real scalar mass at $\mathcal{O}(100~{}{\rm GeV})$. By establishing and
solving the Boltzmann equations, we have shown that this model is able to
account for current BAU using the $\tau$-mediated EWBG.

Our model is testable in current or near-future experiments. On one hand, the
phase transition GWs can be probed by the near-future space-based detectors
(e.g. LISA); on the other hand, the pair production of singlets via the off-
shell Higgs leads to distinguishable $4\tau$ final state at hadron colliders
such as LHC and HE-LHC. This $pp\to h^{*}\to
SS\to\tau^{+}\tau^{-}\tau^{+}\tau^{-}$ process is a characteristic channel of
our mechanism, as the production of $S$ pair is induced by the $\lambda_{HS}$
term which is required by the SFOEWPT, while the decay of $S$ is induced by
the dimension-5 operator which is needed for CPV. A detailed collider
simulation shows that the EWBG parameter space can be efficiently explored
through the $1\ell 3\tau_{j}$ channel.

###### Acknowledgements.

I am grateful to Ligong Bian, Peisi Huang, Benoit Laurent, Zhen Liu and
Yehonatan Viernik for the useful discussions. I also thank Huai-Ke Guo for the
discussion about the sound wave period of GWs. This work is supported by the
Grant Korea NRF-2019R1C1C1010050.

##  Appendix A The complete one-loop scalar potential

The one-loop Coleman Weinberg potential in $\overline{\rm MS}$ scheme is

$V_{\rm
CW}=\sum_{j}n_{j}\frac{m_{j}^{4}}{64\pi^{2}}\left(\ln\frac{m_{j}^{2}}{\mu^{2}}-C_{j}\right),$
(34)

where the subscript $j$ runs over all particles of the model, and he field-
dependent masses are

$\displaystyle m_{h}^{2}=\mu_{H}^{2}+3\lambda_{H}h^{2}+\lambda_{HS}S^{2},\quad
m_{S}^{2}=\mu_{S}^{2}+3\lambda_{S}S^{2}+\lambda_{HS}h^{2},\quad
m_{hS}^{2}=2\lambda_{HS}hS,$ $\displaystyle
m_{G^{\pm,0}}^{2}=\mu_{H}^{2}+\lambda_{H}h^{2}+\lambda_{HS}S^{2},$ (35)

for the scalars (where $G^{\pm,0}$ denote the Goldstone modes of the Higgs
doublet) and

$m_{W}^{2}=\frac{g^{2}}{4}h^{2},\quad m_{Z}^{2}=\frac{g^{2}+g^{\prime
2}}{4}h^{2},\quad m_{t}^{2}=\frac{y_{t}^{2}}{2}h^{2}.$ (36)

for the gauge bosons and the top quark. The numerical factors are

$\begin{cases}~{}n_{j}=1,\quad C_{j}=3/2,\quad&\text{for scalar fields;}\\\
~{}n_{j}=3,\quad C_{j}=5/6,\quad&\text{for vector fields;}\\\
~{}n_{j}=-4N_{c},\quad C_{j}=3/2,\quad&\text{for Dirac fermions,}\end{cases}$
(37)

where $N_{c}$ is the color number, which is 3 for a quark and 1 for a lepton.
For our numerical study, the renormalization scale is chosen as the top mass,
i.e. $\mu=M_{t}=173.2$ GeV.

The counter term is defined as

$V_{\rm
CT}=\frac{\delta\mu_{H}^{2}}{2}h^{2}+\frac{\delta\mu_{S}^{2}}{2}S^{2}+\frac{\delta\lambda_{H}}{4}h^{4}+\frac{\delta\lambda_{S}}{4}S^{4}+\frac{\delta\lambda_{HS}}{2}h^{2}S^{2}.$
(38)

We choose the following renormalization conditions,

$\displaystyle(V_{\rm CW}+V_{\rm
CT})\big{|}_{(0,w)}=0,\quad\frac{\partial(V_{\rm CW}+V_{\rm CT})}{\partial
h}\Big{|}_{(v,0)}=\frac{\partial(V_{\rm CW}+V_{\rm CT})}{\partial
S}\Big{|}_{(0,w)}=0,$ $\displaystyle\frac{\partial^{2}(V_{\rm CW}+V_{\rm
CT})}{\partial h^{2}}\Big{|}_{(v,0)}=\frac{\partial^{2}(V_{\rm CW}+V_{\rm
CT})}{\partial S^{2}}\Big{|}_{(v,0)}=0,$ (39)

to determine the coefficients in Eq. (38), such that the tree-level relations
in Eq. (5) and Eq. (6) still hold at one-loop level.

The thermal integral correction is

$\tilde{V}_{T}=\sum_{j}n_{j}\frac{T^{4}}{2\pi^{2}}J_{j}\left(\frac{m_{j}^{2}}{T^{2}}\right),$
(40)

where $j$ again runs over all particles and $m_{j}$ again is the field-
dependent mass given in last subsection. The thermal integral are

$J_{B/F}(y)=\int_{0}^{\infty}x^{2}dx\ln\left(1\mp e^{-\sqrt{x^{2}+y}}\right),$
(41)

where $B/F$ is for bosons/fermions. Finally, the daisy resummation term is

$V_{\rm
daisy}=-\sum_{j^{\prime}}\frac{T}{12\pi}\left[\left(m_{j^{\prime}}^{T}\right)^{3}-m_{j^{\prime}}^{3}\right],$
(42)

where $j^{\prime}$ runs over the scalars and longitudinal modes of the vector
bosons, and $m_{j^{\prime}}^{T}$ is the Debye mass whose expression can be
found in Refs. Carrington:1991hz ; Espinosa:2011ax .

For $m_{j}^{2}/T^{2}\lesssim 2$, the thermal integrals can be approximated as

$\frac{T^{4}}{2\pi^{2}}J_{B}\left(\frac{m_{j}^{2}}{T^{2}}\right)\approx\frac{T^{2}}{24}m_{j}^{2},\quad\frac{T^{4}}{2\pi^{2}}J_{F}\left(\frac{m_{j}^{2}}{T^{2}}\right)\approx-\frac{T^{2}}{48}m_{j}^{2}.$
(43)

Using this expansion, and ignoring the Coleman-Weinberg potential, the counter
terms and the daisy resummation, we can get the leading $T^{2}$ approximation
for $V_{T}$, i.e. Eq. (9).

##  Appendix B Coefficients in the Boltzmann equations

In this appendix we list the definitions of the coefficients exist in the
transport equation Eq. (2.2). The $k_{i}$ coefficients are given by
deVries:2017ncy ; Fuchs:2020pun

$k_{i}=\tilde{k}_{i}\frac{c_{B/F}}{\pi^{2}}\int_{a_{i}}^{\infty}dx\frac{xe^{x}}{(e^{x}\mp
1)^{2}}\sqrt{x^{2}-a_{i}^{2}},$ (44)

where the upper/lower sign is for bosons/fermions, $c_{B/F}=3$ or 6 for
bosons/fermions, $\tilde{k}_{i}$ is the physical degrees of freedom in a
multiplet ($\tilde{k}_{h}=4$, $\tilde{k}_{\ell}=2$ and $\tilde{k}_{\tau}=1$ in
our scenario), and $a_{i}=\sqrt{{\rm Re}[\delta m_{i}^{2}(T_{n})]}/T_{n}$,
with the real part of the thermal masses being Enqvist:1997ff

$\displaystyle{\rm Re}[\delta m_{h}^{2}(T)]$ $\displaystyle=$
$\displaystyle\left[\frac{3}{16}g^{2}+\frac{1}{16}g^{\prime
2}+\frac{1}{12}\sum_{j}\left(y_{e^{j}}^{2}+3y_{u^{j}}^{2}+3y_{d^{j}}^{2}\right)\right]T^{2},$
$\displaystyle{\rm Re}[\delta m_{\ell}^{2}(T)]$ $\displaystyle=$
$\displaystyle\left(\frac{3}{32}g^{2}+\frac{1}{32}g^{\prime
2}+\frac{1}{16}y_{\tau}^{2}\right)T^{2},$ $\displaystyle{\rm Re}[\delta
m_{\tau}^{2}(T)]$ $\displaystyle=$ $\displaystyle\left(\frac{1}{8}g^{\prime
2}+\frac{1}{8}y_{\tau}^{2}\right)T^{2}.$ (45)

The interactions rates in Eq. (2.2) are Fuchs:2020pun

$\Gamma_{M}=\frac{3}{\pi^{2}T_{n}^{3}}|\bar{m}_{\tau}|^{2}\int_{0}^{\infty}\frac{k^{2}dk}{\omega_{L}\omega_{R}}\times\\\
{\rm
Im}\left[\frac{h_{F}(\epsilon_{R})+h_{F}(\epsilon_{L})}{\epsilon_{R}+\epsilon_{L}}\left(\epsilon_{L}\epsilon_{R}+k^{2}\right)-\frac{h_{F}(\epsilon_{R}^{*})+h_{F}(\epsilon_{L})}{\epsilon_{R}^{*}-\epsilon_{L}}\left(\epsilon_{L}\epsilon_{R}^{*}-k^{2}\right)\right],$
(46)

and

$\displaystyle\Gamma_{Y}$ $\displaystyle=$
$\displaystyle\frac{3y_{\tau}^{2}}{4\pi^{3}T_{n}^{2}}(m_{h}^{2}-m_{\ell}^{2}-m_{\tau}^{2})\int_{m_{\tau}}^{\infty}d\omega_{R}h_{F}(\omega_{R})\ln\left(\frac{e^{\omega_{R}/T_{n}}+e^{\omega_{-}/T_{n}}}{e^{\omega_{R}/T_{n}}+e^{\omega_{+}/T_{n}}}\frac{e^{\omega_{+}/T_{n}}-1}{e^{\omega_{-}/T_{n}}-1}\right)$
(47)
$\displaystyle+\frac{3\zeta_{3}}{32\pi^{3}}g^{2}y_{\tau}^{2}T_{n}\ln\left(\frac{8T_{n}^{2}}{m_{\ell}^{2}}\right),$

where $\zeta_{3}\approx 1.202$, $\bar{m}_{\tau}$ is the effective mass in Eq.
(23), $m_{\ell,\tau,h}^{2}$ are short for the thermal masses in Eq. (B), and

$\omega_{\pm}=\frac{1}{2m_{\tau}^{2}}\Big{[}\omega_{R}|m_{h}^{2}+m_{\tau}^{2}-m_{\ell}^{2}|\\\
\pm\sqrt{(\omega_{R}^{2}-m_{\tau}^{2})(m_{\tau}^{2}-(m_{\ell}+m_{h})^{2})(m_{\tau}^{2}-(m_{\ell}-m_{h})^{2})}\Big{]}.$
(48)

The $\omega_{R/L}$ and $\epsilon_{R/L}$ functions are

$\omega_{R/L}(\textbf{k})=\sqrt{|\textbf{k}|^{2}+{\rm Re}[\delta
m_{\tau/\ell}^{2}(T)]},\quad\epsilon_{R/L}(\textbf{k})=\omega_{R/L}(\textbf{k})-i\Gamma_{\tau},$
(49)

where $\Gamma_{\tau}\approx 0.002\,T_{n}$ Elmfors:1998hh . The $n_{F}$ and
$h_{F}$ functions are

$n_{F}(k_{0})=\frac{1}{e^{k_{0}/T_{n}}+1},\quad
h_{F}(k_{0})=\frac{e^{k_{0}/T_{n}}}{(e^{k_{0}/T_{n}}+1)^{2}}.$ (50)

Finally, the factor $J_{\tau}$ in the CPV source of Eq. (26) is given by
Lee:2004we ; Cirigliano:2006wh ; Fuchs:2020pun

$J_{\tau}=\int_{0}^{\infty}\frac{k^{2}dk}{\omega_{L}\omega_{R}}{\rm
Im}\left[\frac{n_{F}(\epsilon_{L})-n_{F}(\epsilon_{R}^{*})}{(\epsilon_{L}-\epsilon_{R}^{*})^{2}}(\epsilon_{L}\epsilon_{R}^{*}-k^{2})+\frac{n_{F}(\epsilon_{L})+n_{F}(\epsilon_{R})}{(\epsilon_{L}+\epsilon_{R})^{2}}(\epsilon_{L}\epsilon_{R}+k^{2})\right].$
(51)

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