# Rectification of Random Walkers Induced by Energy Flow at Boundaries

Vidyesh Rao Anisetti1, Sharath Ananthamurthy2, J. M. Schwarz1,3 1 Physics
Department, Syracuse University, Syracuse, NY 13244 USA
2 School of Physics, University of Hyderabad, Hyderabad 500046, India
3 Indian Creek Farm, Ithaca, NY 14850 USA

###### Abstract

We explore rectification phenomena in a system where two-dimensional random
walkers interact with a funnel-shaped ratchet under two distinct classes of
reflection rules. The two classes include the angle of reflection exceeding
the angle of incidence ($\theta_{reflect}>\theta_{incident}$), or vice versa
($\theta_{reflect}<\theta_{incident}$). These generalized boundary reflection
rules are indicative of non-equilibrium conditions due to the introduction of
energy flows at the boundary. Our findings reveal that the nature of such
particle-wall interactions dictates the system’s behavior: the funnel either
acts as a pump, directing flow, or as a collector, demonstrating a ratchet
reversal. Importantly, we provide a geometric proof elucidating the underlying
mechanism of rectification, thereby offering insights into why certain
interactions lead to directed motion, while others do not.

##  I Introduction

Systems in thermal equilibrium do not show rectification as demonstrated by
the Feynman–Smoluchowski ratchetfeynman1963feynman ;
smoluchowski1912experimentell . However, non-equilibrium systems with an
underlying spatial asymmetry do exhibit sustained motion rectification, or
motor-like behavior hanggi2009artificial ; feynman1963feynman . An
experimental realization of what is now known as a Brownian motor consists of
four vanes that can freely rotate and surrounded by a vibrated granular gas
eshuis2010experimental . This system is out-of-equilibrium as the collisions
are inelastic and granular particles are driven by external vibrations (and
not thermal fluctuations). Of course, this example naturally leads one to the
field of active matter, in which each system constituent consumes energy to
lead to self-generated motion ramaswamy2010mechanics ;
marchetti2013hydrodynamics . Indeed, creating molecular motors/engines using
active Brownian particles, be it living or nonliving, has been explored by
many costantini2007granular ; Tailleur2009 ; Sekimoto1998 . One such
experimental example consists of active bacterial baths being used to operate
asymmetric gears Sokolov2010 ; Leonardo .

One of the simplest forms of an active engine is motion rectification of
active matter in the presence of funnel-shaped ratchets galajda ; wan .
Interestingly, it has been shown that active, or self-propelled, particles
show rectification in presence of funnel-shaped ratchets because of breaking
of detailed balance which occurs when particles slide along the boundary after
encountering it wan . However, for other types of particle-boundary
interactions, for example, pure reflection, rectification is lost, despite the
particles being active Tailleur2009 ; 10.1117/12.897424 . Therefore, we go
back to a simpler system of non-active Brownian particles and ask the
question: What types of particle-boundary interaction rules lead to motion
rectification?

By investigating this question, we show that there exists a class of particle-
boundary interactions that gives rise to rectification, and sliding along the
boundary is simply a special case of this class. Additionally, we provide a
physical understanding behind this effect, by using particle kinematics and
geometry of the boundary we provide a geometric proof on why, rectification
occurs for this class of interaction and not for pure reflection. Our approach
simplifies the system to its core processes by emphasizing only the essential
properties responsible for rectification. This primary simplification entails
simulating particle kinematics independently of the forces guiding their
trajectories, which is a departure from conventional methods wan . Our model
system consists of a two-dimensional rectangular chamber with a single funnel-
shaped ratchet in between (Fig. 2). Within this chamber, we introduce non-
interacting random walkers. While these walkers obey the reflection rule upon
contacting the rectangular boundary, they exhibit modified reflection behavior
when interacting with the funnel.

We define two classes of this modified reflection:-

$\displaystyle(i)~{}\theta_{r}=\theta_{i}+\alpha\left(\dfrac{\pi}{2}-\theta_{i}\right),$
(1)
$\displaystyle(ii)~{}\theta_{r}=\theta_{i}-\alpha\theta_{i}~{};\alpha\in[0,1]$
(2)

Here, $\theta_{r}$ and $\theta_{i}$ denote the angles of reflection and
incidence, respectively. Rule (i) results in $\theta_{r}>\theta_{i}$ while
Rule (ii) leads to $\theta_{r}<\theta_{i}$. Each value of $\alpha$ corresponds
to a specific reflection rule. The parameter $\alpha$ modulates the extent of
deviation from the standard law of reflection. When $\alpha=0$, the modified
rule reverts to $\theta_{r}=\theta_{i}$. Notably, for $\alpha=1$ in Rule (i),
the condition simulates particle sliding post-collision with the funnel, a
scenario explored in prior research with active particles wan ;
10.1117/12.897424 ; Tailleur2009 .

The rest of the manuscript is organized as follows. We detail the simulation
methodology, then present our simulation results. A simple, geometric proof
helps to interpret our simulation results. We conclude with a discussion of
the implications of our findings.

Figure 1: The left image shows an example of Rule (i) and right image shows an
example of Rule (ii). The blue outgoing arrow shows the path the particle
would have followed in case of perfect reflection rule. The green outgoing
arrow shows the path which particle takes in case of modified reflection rule.
Notice that in Rule (i) the particle is deviated towards the wall and in Rule
(ii) the particle is deviated towards the normal.  Figure 2: The shaded areas
specify region A and B respectively. Coordinates of orange dots are shown.

##  II Simulations

The system consists of a rectangular box of dimensions 1000 $\times$ 200, with
a single funnel in the middle (Fig. 2). This is different from the geometry
studied in Ref. wan  galajda , which used multiple funnels. In this system, we
study 5000 non-interacting random walkers following the iterative equation :-

$\displaystyle x_{t+dt}=x_{t}+\lambda\sin(2\pi\zeta(t))$ (3) $\displaystyle
y_{t+dt}=y_{t}+\lambda\cos(2\pi\zeta(t)),$ (4)

where $(x_{t},y_{t})$ define the position of each particle at time $t$ and
$\zeta$ is a function that outputs a random number $\zeta(t)\in[0,1]$ with a
uniform distribution. Moreover, $\lambda$ is the persistence length of these
random walkers. In the simulations, $dt=1$. If a particle encounters the
rectangular boundary during a time step, it reflects off it according to the
standard law of reflection. However, upon interacting with the funnel, it
follows one of the modified reflection rules detailed earlier. If a particle,
transitioning from $(x_{t},y_{t})$ to $(x_{t+1},y_{t+1})$, intersects the
funnel before completing its step, the point $(x_{t+1},y_{t+1})$ is reflected
about the funnel. This point is then rotated about the collision point based
on the reflection rule and $\alpha$ value, as shown in Fig. 1. It is crucial
to note that each particle traverses a distance equal to $\lambda$ in each
time step, even during a collision.

During the simulations, we observed that certain particles tend to collide
asymptotically with the corners in region B, where the funnel merges with the
rectangle. These particles generally follow Rule (i) with a high $\alpha$
value. When such a particle approaches a corner within a distance of less than
0.1 units, it receives a directional “kick”. This kick ensures that the
particle covers a distance $\lambda$ in one time step, while also ensuring
that the particle’s direction lies within the apical angle of the corner.

##  III Results

Figure 3: Snapshot of the simulation at t=1000 for for rules (i) (left column)
(ii) (right column) respectively for different $\alpha$s.

The system is initialized with a uniform number density of random walkers
throughout the system. Gradually, we see number density increase in one of the
chambers depending on the reflection rule (see Fig. 3). We observe that an
increase in $\alpha$ leads to a higher left-right number density asymmetry in
the system at steady state. Additionally, within each chamber, a higher
$\alpha$ value results in a more heterogeneous steady-state number density
distribution. For $\alpha=0$, signifying perfect reflection, the particles
exhibit a uniform distribution across the system. The particles in the system
are initialized with a uniform number density.

For Rule (i), we observe a higher particle accumulation in chamber B, with the
number density ratio escalating with increasing $\alpha$. This suggests that
the funnel acts as a pump, directing particles into chamber B (see Fig. 3).
Conversely, for Rule (ii), more particles gather in chamber A. This indicates
a net movement of particles against the funnel’s “easy” direction, signifying
a ratchet reversal. We further analyzed how the ratio of particle number
density across the funnel evolves over time (Fig. 4). For the same $\alpha$
value, Rule (i) induces a greater asymmetry at steady state than Rule (ii).

Figure 4: Plots shows asymmetry defined as
$\dfrac{max(n_{a},n_{b})}{min(n_{a},n_{b})}$ with time for different $\alpha$
where plot (a) and (b) show Rule (i) and (ii) respectively.

To understand this phenomenon fundamentally we need to understand why there is
no rectification in case of pure reflection. In the next section we provide a
geometric proof of this.

Figure 5: The dotted lines show the steady state number density ratio found
from simulation and the solid line is
$\dfrac{P_{A\xrightarrow{}B}}{P_{B\xrightarrow{}A}}$.  Figure 6: Understanding
the Absence of Rectification in Purely Reflecting Particles:(a)In region A,
particle at $\vec{r}$ can transition directly into region B in one step, or
hit the funnel between points R and C to subsequently enter region B. This
range of angles over which this transfer can happen is $\theta_{eff}$ ( shaded
in orange ). Conversely, a point s in region B exhibits a smaller range of
directions (depicted in dark blue). The effective areas, $A_{eff}$ (in yellow)
and $B_{eff}$ (in light blue), demarcate the regions where $p(\vec{r})$ is
non-zero in regions A and B, respectively. It’s important to note that
$B_{eff}$ is larger than $A_{eff}$, showing that the funnel has restricted
some particles that could go in region B.  (b) : For a point $\vec{r}\in
A_{eff}$, the effective angle, $\theta_{eff}(\vec{r})$, is the sum of
$\theta(\vec{r})$ (in red) and $\theta_{ref}(\vec{r})$ (in green). (c) :
Consider a point $\vec{r}^{~{}\prime}$, which is reflection of point $\vec{r}$
about the funnel’s lower arm. This point subtends an angle,
$\theta(\vec{r^{\prime}})$, with respect to opening CD. As we are dealing with
the case of pure reflection, this angle corresponds to ,
$\theta(\vec{r}^{~{}\prime})=\theta_{ref}(\vec{r})$. This implies that a
particle originating from $\vec{r}$ and reflecting off RC to pass through CD
has the same probability as a hypothetical particle starting at
$\vec{r}^{~{}\prime}$ and moving through CD, ignoring the funnel’s presence.
This reasoning is extendable to cases of multiple reflections.  (d) : So now
instead of adding $\theta_{ref}(\vec{r})$ to account for increase in
probability we can ignore the funnel and rather extend the domain of
integration to include the points beyond the funnel. Consequently, $A_{eff}$
and $B_{eff}$ are symmetrical with respect to the line through CD. This
symmetry ensures that the integrals $P_{A\xrightarrow{}B}$ and
$P_{B\xrightarrow{}A}$ (referenced in Eq. 6) are equivalent.

##  IV A Geometric Proof

###  IV.1 Why $\alpha=0$ shows no rectification

For pure reflection, where $\alpha=0$, we evaluate the one-step transfer
probabilities, $P_{A\xrightarrow{}B}$ and $P_{B\xrightarrow{}A}$, of particles
transitioning between chambers A and B. Through a simple geometric proof, we
show why purely reflecting random walkers do not show rectification in the
presence of asymmetric boundaries. In the preceding subsection we explain why
we see rectification along the easy direction of funnel in Rule (i) and a
ratchet reversal in Rule (ii).

The probability of a particle at $\vec{r}$ transitioning to the adjacent
chamber in a single step is:

$p(\vec{r})=\begin{cases}\text{$\dfrac{\theta_{\text{eff}}(\vec{r})}{2\pi}$}&\text{if
}\lambda\geq R\\\ \text{0}&\text{if }\lambda<R,\\\ \end{cases}$ (5)

where $R$ represents the distance between $\vec{r}$ and the nearest point on
the opening, and $\theta_{\text{eff}}$ designates the angular range
facilitating chamber transition. This effective angle,
$\theta_{\text{eff}}(\vec{r})$, comprises contributions from both direct and
reflective transfers, as visualized in Fig. 6(a). The one-step transfer
probabilities are then:

$P_{A\xrightarrow{}B}=\int_{A}\dfrac{\theta_{eff}(\vec{r})}{2\pi}\,d^{2}r~{}~{}~{};~{}~{}~{}P_{B\xrightarrow{}A}=\int_{B}\dfrac{\theta_{eff}(\vec{r})}{2\pi}\,d^{2}r$
(6)

Upon examining Fig. 6(a), it looks like the transfer probabilities
$P_{A\xrightarrow{}B}$ and $P_{B\xrightarrow{}A}$ aren’t inherently equal. In
region A, the value of $p(\vec{r})$ is elevated because reflections expand the
angular range, augmenting particle transitions to the adjacent chamber. In
contrast, region B has a larger area with non-zero $p(\vec{r})$. Intriguingly,
the enhanced probability in $P_{A\xrightarrow{}B}$ (attributable to
reflection) is precisely offset by the increase in $P_{B\xrightarrow{}A}$
owing to the larger effective area $B_{\text{eff}}$ (Fig. 6(c-d)) .

This observation underscores a pivotal point: irrespective of the funnel’s
geometry, regions A and B will maintain equal particle number densities. This
is solely possible due to perfect reflection.

###  IV.2 Why $\alpha\neq 0$ is necessary for rectification

To understand what happens in case of asymmetric reflection we have to go back
to Fig. 6(c). For Rule (i), $\theta_{r}>\theta_{i}$, this means now the
particles with smaller incident angles can go into region B, this will result
in larger $\theta_{ref}(\vec{r})$ as particles hitting the funnel to the left
of point R can now enter region B. Therefore now
$\theta_{ref}(\vec{r})>\theta(\vec{r^{\prime}})$ which leads to
$P_{A\xrightarrow{}B}>P_{B\xrightarrow{}A}$. On the flip side, for Rule (ii),
we get $\theta_{ref}(\vec{r})<\theta(\vec{r^{\prime}})$ which leads to
$P_{A\xrightarrow{}B}<P_{B\xrightarrow{}A}$.

To validate these deductions, we conducted numerical calculations for
$P_{A\xrightarrow{}B}$ and $P_{B\xrightarrow{}A}$, juxtaposing them with the
steady-state number density ratios derived from our simulations, as depicted
in Fig. 5. In equilibrium, particles traversing the funnel from either side
are balanced, which translates to the relation
$n_{A}P_{A\xrightarrow{}B}=n_{B}P_{B\xrightarrow{}A}$. While our observations
align well for $\alpha>0$, discrepancies arise for $\alpha<0$. This divergence
might stem from the prolonged time systems with $\alpha<0$ require to reach a
steady state, as shown in Fig. 4. Hence, using one-step probabilities might
not effectively capture the number density ratios in these cases.

Delving deeper, when $\alpha>0$, particles experience a force directing them
towards the boundary, leading to accumulation of particles in the corners of
the box. Conversely, for $\alpha<0$, particles are repelled from the boundary,
moving away from the corners, an effect visible in Fig. 3. The scenario for
$\alpha=0$ is distinct, as the density distribution of purely reflecting
particles remains uniform, irrespective of boundary geometry.

##  V Discussion

Purely reflecting random walkers do not show rectification even for funnel-
shaped boundary. To observe rectification, we introduce a non-equilibrium
effect in the system by modifying the reflection rules. When the boundary
reflection rule deviates from pure reflection, time-reversal symmetry is
broken, particle-boundary interactions become non-reciprocal, which results in
the ratchet acting as a Maxwell demon.

Our result helps connects the different rectification phenomenon that has been
observed in self-propelled particles, ballistic chains, flexible vesicles ,
granular gases, and even thermal systems with nontrivial interactions galajda
; wan2013directed ; gandikota2023rectification ; solorzano2017thermal ;
costantini2007granular . Self- propelled particles that slide along the
boundary after collision follow Rule (i) with $\alpha=1$ galajda , and
therefore show rectification along the easy direction of funnel. Ballistic
chains when interacting elastically with the boundary show an effective
reflection law that looks like Rule (ii), which explains why a ratchet
reversal was observed wan2013directed . Deviations from pure reflection can be
experimentally achieved by introducing a temperature difference between the
gas molecules and the collision surface molecular_beam1966 , resulting in an
exchange of energy during collision and making the particle-wall interaction
“active”. Such experimental realizations have the potential to pave the way
for innovative engines that harness these non-equilibrium phenomena,
potentially leading to the creation of highly efficient engines reminiscent of
molecular motors.

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# First Order Electroweak Phase Transition from Weakly Coupled sub-GeV Physics
and
Possible Connection to Fermion Flavor

Hooman Davoudiasl<EMAIL_ADDRESS>High Energy Theory Group, Physics Department
Brookhaven National Laboratory, Upton, NY 11973, USA

###### Abstract

We propose that the dynamics of a scalar $\phi$ of mass $\mathcal{O}{(10)}$
MeV that is weakly coupled to the Higgs can lead to a first order electroweak
phase transition, fulfilling a key requirement for baryogenesis. Stability of
the model near the weak scale requires a suppressed - possibly vanishing - top
Yukawa coupling to the Higgs before the transition which rises to the Standard
Model value afterwards. This can be accomplished through the dynamics of
$\phi$ via a dimension-5 operator. We conjecture that the entire Standard
Model flavor structure could turn on, mutatis mutandis, after the electroweak
phase transition, via dimension-5 interactions of $\phi$ suppressed by scales
ranging from $\mathcal{O}{(10^{3})}$ TeV to near Planck mass. Due to its
suppressed couplings, $\phi$ is long-lived and can lead to missing energy
signals in rare kaon decays, which can be probed by the KOTO experiment.

##  I Introduction

It is widely accepted that the Standard Model (SM) does not contain sufficient
ingredients to generate the observed baryon asymmetry of the universe (BAU)
[1]. This motivates introduction of new interactions that can contribute to a
successful baryogenesis mechanism. Among the outstanding problems of particle
physics and cosmology, the origin of the BAU could be a good target for
laboratory experiments, since it concerns the visible world.

One of the necessary conditions for viable baryogenesis is departure from
equilibrium, in order to avoid erasure of the generated BAU [2]. This
condition could have been provided through a first order electroweak phase
transition in the SM [3]. However, the measured mass of the Higgs boson,
$m_{H}\approx 125$ GeV [1], rules out this possibility. The basic reason is
that at this mass, given the well-established vacuum expectation of the Higgs
$\langle{H}\rangle=v/\sqrt{2}\approx 174$ GeV, its deduced self-coupling
$\lambda_{H}\approx 0.13$ is too large to accommodate a first order phase
transition (FOPT).

To see the reason for the above situation, note that a FOPT is typically
assumed to be realized if at the transition critical temperature $T_{c}$ we
have

$\frac{v(T_{c})}{T_{c}}\gtrsim 1\,.$ (1)

In the SM, the above quantity is governed by the thermal contributions of the
$\\{W^{\pm},Z\\}$ gauge bosons and is given by [4, 5]

$\frac{v(T_{c})}{T_{c}}=\frac{2m_{W}^{3}+m_{Z}^{3}}{3\pi\lambda_{H}\,v^{3}}\approx
0.1\,,$ (2)

where $m_{W}\approx 80.4$ GeV and $m_{Z}\approx 91.2$ GeV [1] are the masses
of the $W^{\pm}$ and $Z$, respectively. Hence, the SM electroweak transition
seems to be far from being first order and thus fails to provide the necessary
non-equilibrium condition for successful baryogenesis.

The above circumstance motivates extensions of the SM that could lead to a
FOPT. Generally speaking, such extensions require new physics close to the
weak scale that has non-negligible interactions with the SM. For example,
additional bosons with $\mathcal{O}{(1)}$ coupling strength to the Higgs can
enhance the value of the transition parameter in Eq. (2) [6, 7, 8, 9, 10, 11,
12, 13, 14, 15, 16, 17, 18]. In that case, their masses could not be much
larger than the Higgs mass, so that their thermal population is not Boltzmann-
suppressed at $T_{c}\sim m_{H}$. However, see Refs. [19, 20] for mechanisms
that effect a FOPT through scalars of intermediate mass $\gtrsim 10$ GeV and
weak coupling with the Higgs. A model that achieves an electroweak FOPT from a
light axion-like-particle is presented in Ref. [21]. Ref. [5] examines a model
that employs TeV scale fermions. See Ref. [22] for recent work using a dark
gauge sector leading to a FOPT.

Another class of solutions for achieving a FOPT entails addition of higher
dimension operators $(H^{\dagger}H)^{n}$, with integer $n>2$. Such
interactions would allow deviations from the SM value of $\lambda_{H}$ in
order to satisfy the condition (1) and obtain a first order transition [23,
24]. In this class of models, it is generally required that the higher
dimension operators are suppressed by scales $\Lambda\sim 1$ TeV, which
implies the presence of new physics not far above $m_{H}$. For a realization
of such effective theories using weak scale fermions see Ref. [25].

We see that, in general, one is led to introduce new physics near the weak
scale that couples to the Higgs with un-suppressed strength. This situation
can cause tension with a growing body of data that typically does not seem to
favor the presence of the required states.

In this work, we propose an alternative approach to generating an electroweak
FOPT, where a light new scalar $\phi$, with suppressed couplings to the Higgs
boson and other SM particles, appears at energies well below the weak scale.
The main role of the scalar $\phi$ is to allow for a small Higgs self-coupling
before electroweak symmetry breaking, so that the ensuing transition is first
order. After the electroweak symmetry is broken, we arrange for the Higgs self
coupling to attain its low temperature value, corresponding to the value
deduced from experiment. This is achieved by the vacuum expectation value
(vev) of $\phi$ that becomes non-zero after the FOPT and drives the vev of the
Higgs to $v\approx 246$ GeV, observed at low energies.

The above picture can generally lead to an unstable scalar potential just
above the weak scale. This is mostly due to the quantum effects of the top
quark Yukawa coupling, $y_{t}\approx 1$ in the SM, that drive the Higgs
quartic to negative values with increasing energy scale. We hence additionally
propose that the top Yukawa coupling can be small or vanishing before the
transition, ensuring a quantum mechanically stable potential at energies well
above the weak scale. The non-zero $\phi$ vev attained after the transition is
then postulated to set $y_{t}$ to its SM value, through a dimension-5
operator. Motivated by symmetry considerations, we are led to conjecture that
all SM flavor may be established in this fashion, through dimension-5
operators suppressed by mass scales ranging from $\sim 10^{3}$ TeV, for the
top, to near the Planck mass $M_{P}\sim 10^{19}$ GeV, for neutrinos.

The end result of our mechanism is that the Higgs potential and couplings
after the FOPT are to a very good approximation those of the SM. The
suppressed interactions of $\phi$ imply that other new physical states are
well above the weak scale or else have small couplings to the SM. The only
other field that we explicitly introduce is a scalar with a mass $\sim 10$ TeV
after the transition and a small coupling to $\phi$, which is therefore
largely inaccessible. This typically leads to the only observable new effects
being from $\phi$, which in our setup is a weakly coupled sub-GeV state. See
also Ref. [26] for a model that contains a light scalar and relates flavor
dynamics to an electroweak FOPT.

Our proposal would then generically be consistent with the lack of significant
deviations at or near the weak scale, which is currently probed at the LHC and
various precision experiments. However, low energy probes of rare phenomena,
such as certain kaon decay modes, could in principle provide experimental
tests of our scenario. Here, we note that if the FOPT generated in our
proposal is to be part of a larger framework for baryogenesis, there would
likely be other new states that couple to the SM and could gives rise to
additional signals. Such signals will depend on the specific features of
baryogenesis models; we will come back to this point later.

We will next introduce an explicit model that realizes our FOPT scenario.

##  II Scalar Potential

Consider the following potential

$\displaystyle V(\phi,H,\eta)$ $\displaystyle=$
$\displaystyle\frac{m_{0\phi}^{2}}{2}\phi^{2}-(\mu_{0}^{2}+2\mu\,\phi)H^{\dagger}H$
(3) $\displaystyle+$
$\displaystyle(\lambda_{0}+2\frac{\phi^{2}}{M^{2}})(H^{\dagger}H)^{2}+\frac{\kappa}{4}\phi^{2}\eta^{2}\,,$

where $m_{0\phi}^{2}>0$ is the initial mass of $\phi$, positive constants
$\mu_{0}^{2}$ and $\lambda_{0}$ are the initial values of the Higgs mass
parameter and self-interactions for $\langle{\phi}\rangle=0$, respectively,
and $0<\mu\ll m_{H}$. The scale $M\gg m_{H}$ descends from some ultraviolet
(UV) dynamics that we do not specify here. We have included an additional
scalar $\eta$ coupled to $\phi$ with positive strength $\kappa\ll 1$ whose
role is to allow a thermal “slow roll” for $\phi$, as will be described below.
The above potential is consistent with a softly broken ${\mathbb{Z}}_{2}$
symmetry acting on $\phi$.

There can be other possible terms in the potential $V(\phi,H,\eta)$ that we
may invoke later, but the above interactions suffice to elucidate key aspects
of our mechanism, for now. We will assume that other interactions not invoked
in this work are suppressed compared to those that we explicitly write down
here and below. Also, we will only examine if the chosen parameters can yield
a consistent phenomenology, regardless of whether they are considered tuned or
not. In particular, a more complete model leading to our effective theory may
be required to avoid potentially large quantum corrections to $\phi$ mass.
While we do not offer any concrete examples here, we point out that $\phi$
only needs to emerge in our theory below scales of
$\Lambda\sim\mathcal{O}{(100~{}\text{GeV})}$ to affect the FOPT. In that case,
loop corrections to its mass could in principle be cutoff at $\Lambda$. The
largest contribution to $m_{\phi}\sim$ few MeV would be from interactions with
$\eta$ in our effective description. Assuming $\kappa\sim 10^{-4}$, as we will
later, one can show the required tuning is $\mathcal{O}{(10^{-3})}$, for
$\Lambda\sim 100$ GeV.

In what follows, we will assume that a FOPT can be achieved for
$v(T_{c})/T_{c}\sim 1$, in agreement with condition (1). Over the range of
parameters that we will consider, we find that the values of the SM
$SU(2)_{L}$ and $U(1)_{Y}$ hypercharge couplings, respectively $g$ and
$g^{\prime}$, would not run significantly and stay nearly constant. As
$m_{W},m_{Z}\propto v$, Eq. (2) then implies that we will need
$\lambda_{0}\lesssim 0.1\,\lambda_{H}$ at the onset of the transition, to
satisfy relation (1).

To find the requisite values of various parameters, we obtain the vacuum
solutions for the scalars, corresponding to
$\partial_{h}V=\partial_{\phi}V=\partial_{\eta}V=0$, where $h$ is the
background value of the Higgs boson, with $\langle{h}\rangle=v=246$ GeV. We
will set $\langle{\eta}\rangle=0$. Let $\bar{\phi}$ denote the background
value of $\phi$ at $T=0$. We find

$\bar{\phi}=\frac{\mu\,v^{2}\,M^{2}}{m^{2}_{0\phi}M^{2}+v^{4}}\quad\quad(T=0).$
(4)

Using Eq. (3), the SM Higgs mass parameter $\mu_{H}\approx 89$ GeV and self-
coupling $\lambda_{H}\approx 0.13$ – which we will assume as the effective low
energy values – are given by

$\mu_{H}^{2}=\mu_{0}^{2}+2\mu\,\bar{\phi}\;\;\;\text{and}\;\;\;\lambda_{H}=\lambda_{0}+2\frac{{\bar{\phi}}^{2}}{M^{2}}\,.$
(5)

Let us parameterize $\lambda_{0}=\varepsilon\lambda_{H}$, with
$\varepsilon\lesssim 0.1$ a small quantity. In what follows, we will choose
$\varepsilon$, $\mu$, and $M$ as our input parameters that will determine the
required values of $\mu_{0}$ and $m_{0\phi}$, setting $h=v$ at $T=0$. We
demand $m_{0\phi}^{2}>0$ and $\mu_{0}^{2}>0$, so that the potential is stable
and that electroweak symmetry can be spontaneously broken, respectively. Then,
using Eqs.(4) and (5), we obtain

$\frac{\bar{\phi}\,v^{2}}{M^{2}}<\mu<\frac{\mu_{H}^{2}}{2\bar{\phi}}\,.$ (6)

##  III Mixing with the Higgs

The mixing between the Higgs and $\phi$ is governed by the angle

$\theta\approx\frac{2v^{2}}{m_{H}^{2}}\left(\frac{2\bar{\phi}\,v}{M^{2}}-\frac{\mu}{v}\right)\,.$
(7)

We will work in the regime $|\theta|\ll 1$ and hence the mixing effect on the
Higgs can be ignored, which yields $v=\mu_{H}/\sqrt{\lambda_{H}}$. After
diagonalizing the mass matrix, to a good approximation, we will have
$m_{H}\approx\sqrt{2}\,\mu_{H}$, as in the SM. In that limit, the lightest
scalar eigenstate, which we will continue to denote by $\phi$, will end up
having a mass given by

$m_{\phi}^{2}\approx
m_{0\phi}^{2}+\frac{v^{4}}{M^{2}}-\theta^{2}\,m_{H}^{2}\,,$ (8)

that we will require to be positive, to ensure a valid solution for $\phi$.

Let us take $\varepsilon=0.1$ and $M=4.0\times 10^{3}$ TeV as benchmark
values, for concreteness. Using Eqs. (7) and (8), one can then obtain a range
of values for $m_{\phi}$ and $\theta$ as a function of $\mu$. By numerical
inspection of Eq. (8), we find that $\phi$ mass varies by a factor of $\sim 2$
over the range (6) and we have

$5~{}\text{MeV}\lesssim m_{\phi}\lesssim 10~{}\text{MeV}\,,$ (9)

given our benchmark parameters.

For the region of parameters typical of our benchmark choices, the lifetime
$\tau_{\phi}$ of $\phi$ corresponds to a macroscopic decay length

$c\,\tau_{\phi}\approx\frac{8\pi}{\theta^{2}y_{e}^{2}m_{\phi}}\approx
8.4\times
10^{3}~{}\text{km}\left(\frac{10^{-8}}{\theta^{2}}\right)\\!\\!\left(\frac{7~{}\text{MeV}}{m_{\phi}}\right)\,,$
(10)

where $c$ is the speed of light and $y_{e}\approx 2.9\times 10^{-6}$ is the SM
Yukawa coupling of the electron. The above formula is valid for $\phi$ masses
below the muon pair threshold. Later, we will introduce an additional $\phi$
coupling to the electron, which however will not drive the $\phi$ lifetime
away from the above order of magnitude, set by $\theta y_{e}$.

##  IV Evolution with Temperature

Here, we will provide an estimate for the expected critical temperature
$T_{c}$ of the FOPT. One could obtain this quantity from the following
expression (see, for example, Refs. [4, 5])

$T_{c}\approx\frac{T_{0}}{\sqrt{1-E^{2}/(\lambda_{0}\,D)}}\,$ (11)

where

$\displaystyle
T_{0}^{2}=\frac{m_{0H}^{2}-8Bv_{0}^{2}}{4D}\;;\;B=\frac{3(2m_{W}^{4}+m_{Z}^{4}-4m_{t}^{4})}{64\pi^{2}v^{4}}$
$\displaystyle
D=\frac{2m_{W}^{2}+m_{Z}^{2}+2m_{t}^{2}}{8v^{2}}\;;\;E=\frac{2m_{W}^{3}+m_{Z}^{3}}{6\pi
v^{3}}\,.$ (12)

In the above, $T_{c}$ corresponds to the temperature when there are two
degenerate minima, which allows a transition to begin. Also,
$m_{0H}=\sqrt{2}\,\mu_{0}$, and $v_{0}^{2}=\mu_{0}^{2}/\lambda_{0}$,
corresponding to the initial Higgs mass and vev, respectively, at the onset of
the transition ($\langle{\phi}\rangle=0$). The value of the field at the new
minimum, which will eventually become the new vacuum, is given by
$h(T_{c})\approx 2ET_{c}/\lambda_{0}$. At temperature $T_{0}$, the barrier
separating the two minima disappears and the origin at $h=0$ becomes a local
maximum. Hence, the transition is complete between $T_{c}$ and $T_{0}$. The
value of the Higgs field at this minimum is given by $h(T_{0})\approx
3ET_{0}/\lambda_{0}$.

For the benchmark values of this work, and with $\mu$ in the range (6), we
find that $0\leq T_{c}\leq 95$ GeV, with $T_{c}=0$ at the upper limit of the
range $\mu=\mu_{H}^{2}/(2\bar{\phi})\approx 4.1$ MeV. We do not consider this
endpoint to yield a sensible cosmology and hence will only consider

$3.7~{}\text{MeV}\leq\mu\leq 4.0~{}\text{MeV}\,,$ (13)

as a possible range, corresponding to $38~{}\text{GeV}\leq T_{c}\leq 90$ GeV
and $37~{}\text{GeV}\leq T_{0}\leq 87$ GeV. A narrower range can also be
considered, if desirable. Note that in the above, $m_{t}$ is the top mass at
the onset of electroweak symmetry breaking, which we set $m_{t}=0$. Later, we
will discuss the reason for this choice, which is motivated by vacuum
stability of the Higgs potential above the weak scale. A mechanism for
achieving the SM value after the FOPT will be provided below. Thus, here we
use $m_{t}=0$ to obtain $T_{c}$ from the above expressions.

Figure 1: Values of $\phi/M$ at $T_{c}$ (solid) and $T_{0}$ (dashed), assuming
the benchmark parameters, from Eq. (15). Here, $\kappa=10^{-4}$ has been
assumed, for $\phi$-$\eta$ coupling in $V(\phi,H,\eta)$. The smallness of
$\phi/M$ implies that the Higgs parameters in Eq. (5) remain close to their
initial values during the transition.

In order to make sure that the jump in the order parameter of the FOPT remains
effective during the transition, the evolution of $\phi$ away from zero cannot
be too prompt; this has been an implicit assumption in the preceding
discussion. In particular, if $\phi$ becomes large once $h$ tunnels to the new
minimum, the Higgs quartic coupling could approach $\mathcal{O}{(1)}$ values
and the condition in Eq. (1) may no longer hold. To see why this could be a
problem, note that the Hubble scale around the time of the phase transition is
roughly $T^{2}/M_{\rm P}\sim 10^{-12}$ MeV, whereas the initial mass of $\phi$
is $\mathcal{O}{(\text{MeV})}$. Hence, in the absence of some plasma effect,
the evolution of $\phi$ could in principle be much faster than the relevant
time scales for the completion of the transition. To avoid this possibility,
we will employ thermal effects from the interactions of $\phi$ with $\eta$,
introduced earlier in Eq. (3). The scalar $\eta$ generates a temperature
dependence $\propto\kappa$ for $\phi$ mass [4]

$m_{\phi}^{2}(T)=m_{0\phi}^{2}+\frac{h^{4}(T)}{M^{2}}+\frac{\kappa}{24}T^{2}\,,$
(14)

where $h(T)$ is the Higgs background value at temperature $T$.

By inspecting the expression in Eq. (4) for $\bar{\phi}$, we see that a
sufficiently large thermal mass for $\phi$ will keep it pinned near $\phi=0$.
Once $T$ is low enough, $\phi$ can reach its late time values. We have plotted
values of

$\frac{\phi(T)}{M}=\frac{\mu\,h^{2}(T)}{m_{\phi}^{2}(T)M}\,,$ (15)

at $T_{c}$ (solid) and $T_{0}$ (dashed), in Fig.1. We have taken
$\kappa=10^{-4}$ as a reference value. As one can see, over the range (13) of
$\mu$ considered in this work, $\phi/M$ remains sufficiently small that one
could assume, as implied by Eq. (5), the Higgs parameters remain near their
initial values $(\mu_{0},\lambda_{0})$ during the transition, to a good
approximation. Numerical inspection shows that $\phi/M$ remains small compared
to unity down to $T\sim 10$ GeV. For these temperatures, we have checked that
$\kappa\lesssim\text{few}\times 10^{-5}$ yields $\phi/M\gtrsim 0.1$ where the
Higgs quartic starts to become large compared to $\lambda_{0}\sim 0.01$ and
hence we do not consider $\kappa\ll 10^{-4}$, to be conservative.

The delayed evolution of $\phi$ could possibly lead to a period of inflation
which could result in unwanted effects, such as dilution of any BAU generated
during or before the FOPT. To examine this question, we note that any such
inflation may be caused if the potential energy stored in $\phi$ \- that we
estimate to be at most $\delta V\sim m_{0\phi}^{2}\,{\bar{\phi}}^{2}\sim
10^{6}$ GeV4 for our choice of reference parameters - dominates the cosmic
energy budget. The radiation energy density is given by $\rho_{R}\sim
g_{*}T^{4}$, where $g_{*}\sim 100$ is the number of relativistic degrees of
freedom. For $T\sim 10$ GeV, below which - as mentioned above - $\phi$ starts
to evolve significantly towards the minimum of the potential, we find
$\rho_{R}\sim 10^{6}$ GeV4. Thus, we see that such an inflationary period is
not a typical expectation for the regime of parameters considered here.

An implicit assumption in the above discussion is that $\phi$ and $\eta$ are
in thermal equilibrium during the phase transition. To see why this is the
case, note that the $\phi$-Higgs coupling leads to $HH^{\dagger}\to\phi$ at a
rate $\Gamma\sim\mu^{2}/T$, with $\mu\sim$ MeV. The Hubble rate is roughly
given by ${\cal H}\sim T^{2}/M_{P}$. Demanding $\Gamma\gtrsim{\cal H}$, in
order to populate a thermal bath of $\phi$ states, we obtain the condition
$T\lesssim(\mu^{2}M_{P})^{1/3}$. For $\mu\sim$ MeV, we then find $T\lesssim
10^{4}$ GeV. Since $\eta$ is initially massless, its production via
$\phi\phi\to\eta\eta$ has a rate $\sim\kappa^{2}T$. Hence, it can be in
thermal equilibrium for $\kappa=10^{-4}$, used as a reference value above, as
long as $T\lesssim 10^{11}$ GeV. We thus conclude that these considerations do
not pose any restrictions on the FOPT - which only begins at $T_{c}\lesssim
100$ GeV - and we can have $\phi$ and $\eta$ in thermal equilibrium with the
SM.

We note that once the final values of the fields have been achieved, $\eta$
obtains a mass $m_{\eta}\sim\kappa^{1/2}\bar{\phi}\sim 10$ TeV, through its
coupling to $\phi$. Since we are assuming that $\eta$ has a thermal population
in the early Universe, we need to make sure that its number density is
sufficiently depleted, so that it would not lead to excessive contributions to
dark matter. One could in principle entertain the possibility that $\eta$ is
dark matter, which would require arranging for suitable interactions that
would yield an acceptable relic density for it. We do not pursue that option
here, though it would be an interesting complement to our model. However, in
lieu of such model building, one may simply postulate a dimension-5 operator,
say $\eta F_{\mu\nu}F^{\mu\nu}/M_{\eta}$, with $F_{\mu\nu}$ the photon field
strength tensor and $M_{\eta}$ some UV scale, to deplete the $\eta$ number
density. This operator would lead to $\eta\to\gamma\gamma$ sufficiently fast -
that is before Big Bang Nucleosynthesis (BBN) at $T\sim$ MeV - as long as
$M_{\eta}\lesssim 10^{18}$ GeV, and would remove $\eta$ from the cosmic energy
budget.

##  V UV Stability of the Potential

So far, we have shown that the proposed model can in principle yield a FOPT.
However, the central feature of our setup, namely a small initial quartic
coupling $\lambda_{0}$ for the Higgs, can lead to an unstable potential, due
to running which would drive $\lambda_{0}$ negative at large energy scales.
Denoting the top Yukawa coupling by $y_{t}$, the running of the initial Higgs
quartic coupling is given by (see, for example, Ref. [27]; note that this
reference defines the Higgs quartic with an extra factor of 1/2 compared to
our convention)

$\displaystyle 16\pi^{2}\frac{d\lambda_{0}}{dt}$ $\displaystyle=$
$\displaystyle 24\lambda_{0}^{2}+\lambda_{0}(12y_{t}^{2}-9g^{2}-3g^{\prime
2})-6y_{t}^{4}$ (16) $\displaystyle+$
$\displaystyle\frac{9}{8}g^{4}+\frac{3}{8}g^{\prime
4}+\frac{3}{4}g^{2}g^{\prime 2}\,,$

where $t\equiv\ln Q/Q_{0}$ and $Q$ is the renormalization scale; $Q_{0}$ is a
reference scale. We will set $Q_{0}=100$ GeV as the typical scale of the
electroweak symmetry breaking. As was discussed before, within the regime of
parameters in this work, the critical temperature $T_{c}\lesssim 100$ GeV and
hence this choice of $Q_{0}$ is reasonable.

Figure 2: Running of $\lambda_{0}$, for $\lambda_{0}(Q_{0})=0.1\lambda_{H}$,
corresponding to a top Yukawa coupling that is initially vanishing (solid) or
at the SM value (dashed); for the latter choice the Higgs self-coupling goes
negative at an energy scale $Q\gtrsim 170$ GeV. The dotted curve corresponds
to the running in the SM, with $\lambda_{0}(Q_{0})=\lambda_{H}=0.13$. The
reference scale is chosen to be $Q_{0}=100$ GeV.

Let us set $y_{t}(Q_{0})=\sqrt{2}\,160\,\text{GeV}/v\approx 0.92$ and the
gauge couplings to their SM values near the weak scale. For our benchmark
choice, $\lambda_{0}(Q_{0})\approx 0.013$, using Eq. (16) we find that this
parameter goes negative at energies above $\sim 170$ GeV, not far from the
electroweak symmetry breaking temperature, as shown in Fig. 2 by the dashed
line. This is largely due to the quantum effects of the top quark that couples
to the Higgs with $\mathcal{O}{(1)}$ strength and drives $\lambda_{0}$ to
negative values. Hence, as it is, the validity of the model above such energy
scales requires the introduction new bosonic degrees of freedom that would
stabilize the Higgs potential. Such a solution, however, would lessen the
motivation for our mechanism, since new weak scale bosons that couple to the
Higgs with $\mathcal{O}{(1)}$ stregngth could also provide the necessary
ingredients for a FOPT, as has been studied extensively before. As we will
discuss below, the dynamics of $\phi$ itself offers a way to stabilize the
Higgs potential without introducing additional weak scale bosons with
significant coupling to the SM.

To see how $\phi$ can help with the stability of the potential, first note
that it is the large top Yukawa coupling that is the origin of the problem, as
mentioned before. Thus, if we set the initial value of this coupling to be
smaller than the SM value, we can in principle avoid having a negative Higgs
self-coupling just above $T_{c}$. However, we still would need a mechanism to
restore $y_{t}$ to its SM value at the weak scale. Since this is what we have
achieved with $\lambda_{H}$ through the dynamics of $\phi$, we can extend the
mechanism to $y_{t}$. To do this, we introduce a dimension-5 operator

$O_{t}=\frac{\phi}{M_{t}}H^{*}\epsilon\,{\bar{Q}_{L}}t_{R}+\text{\small
H.C.}\,,$ (17)

where $M_{t}$ is a new UV scale and $\epsilon$ is the two-dimensional Levi-
Civita symbol for isospin indices. In Eq. (17), $Q_{L}$ is the third
generation quark doublet and $t_{R}$ is the right-handed top quark, which we
assume has odd $\mathbb{Z}_{2}$ parity. Once $\langle{\phi}\rangle\neq 0$,
this operator contributes to the low energy Yukawa coupling of the top quark
$y^{\rm SM}_{t}(Q_{0})$ which we assume has the SM value, as the notation
suggests.

If we denote the initial value of the top quark Yukawa coupling at $T\gg
T_{c}$ by $y_{0t}$ we then have

$y^{\rm SM}_{t}(Q_{0})=y_{0t}(Q_{0})+\frac{\bar{\phi}}{M_{t}}\,.$ (18)

We will choose $y_{0t}=0$, so that the Yukawa coupling for the top vanishes
before the FOPT when $\langle{\phi}\rangle=0$, consistent with the assumed
$\mathbb{Z}_{2}$ symmetry. With $y^{\rm SM}_{t}(Q_{0})\approx 0.92$, we find
$M_{t}\approx 1.1\times 10^{3}$ TeV. Hence, before electroweak symmetry
breaking, the running of $\lambda_{0}$ in Eq. (16) will be governed by
$y_{t}=y_{0t}=0$, which is shown in Fig. 2 as the solid curve. As one can see,
the initial Higgs potential remains stable up to very high scales, beyond the
scale $M$ of our effective theory. In the figure, we have also shown the
running of $\lambda_{H}$ in the SM, given by the dotted curve, which is what
we would obtain after electroweak symmetry breaking.

Note that due to the soft-breaking of the $\mathbb{Z}_{2}$, one could in
principle generate a Yukawa coupling for the top at 1-loop order, before the
FOPT. However, this contribution is estimated to be

$\delta y_{t}(Q_{0})\sim\frac{\mu}{16\pi^{2}\,M_{t}}\ln(M_{t}/Q_{0})\,,$ (19)

which for our reference values is $\mathcal{O}{(10^{-10})}$ and hence
completely negligible. The Feynman diagram of the above 1-loop process, for a
fermion $f$, is shown in Fig.3.

Figure 3: One-loop process for generating fermion $f$ Yukawa coupling. The
vertices marked by $\oplus$ and $\blacksquare$ refer to the $\mathbb{Z}_{2}$
soft-breaking interaction $\propto\mu$ and a dimension-5 operator of the type
in Eq. (20), respectively.

##  VI Flavor from $\phi$ Dynamics

Here, we will examine whether we can generate all Yukawa couplings $y_{f}$,
where $f$ is a fermion in the SM, from the dynamics of $\phi$. In the context
of our model, the motivation for this would be the $\mathbb{Z}_{2}$ symmetry
that is so far assumed to be only softly broken. That is, since we have
assumed that $t_{R}$ is odd under this parity, we cannot write down a generic
tree-level Yukawa matrix for quarks without violating the $\mathbb{Z}_{2}$.
Also, there is no obvious reason in our model that $t_{R}$ should be
fundamentally different from other spin-1/2 fields. The extension of the same
charge assignment to all fermions in the SM, while not rigorously required,
can nonetheless be a natural assumption.

Let us consider the operator

$O_{f}=\frac{\phi}{M_{f}}H\,{\bar{F}_{L}}f_{R}+\text{\small H.C.}\,,$ (20)

where $F$ denotes an $SU(2)_{L}$ doublet containing the $f$ flavor; this
expression is a schematic representation and the proper contraction of gauge
indices is assumed depending on the fermion.111Similar operators have been
considered in other contexts, see e.g. Refs. [28, 29]. In the above, the scale
$M_{f}$ needs to be chosen so that the correct SM Yukawa coupling value is
obtained via

$y^{\rm SM}_{f}(Q_{0})=\frac{\bar{\phi}}{M_{f}}\,,$ (21)

implying

$M_{f}=\frac{y^{\rm SM}_{t}(Q_{0})}{y^{\rm SM}_{f}(Q_{0})}M_{t}.$ (22)

The above interaction generates a direct coupling of the fermion $f$ to $\phi$
given by

$\xi_{f}=\langle{H}\rangle/M_{f}\,.$ (23)

which can potentially lead to phenomenological problems if it is too large. We
will show that if the mixing angle $\theta$ is at an acceptable level, the
direct coupling $\xi_{f}$ is also generically allowed. To see this, first note
that in our model $(\bar{\phi}/M)^{2}\sim 0.1$ in order to attain
$\lambda_{H}$ after the FOPT. Then, Eq. (6) implies $\mu\sim
0.1v^{2}/{\bar{\phi}}$ and hence one can deduce $\theta\sim v/{\bar{\phi}}$.
Using Eqs. (21) and (23), we obtain

$\theta\,y_{f}\sim\xi_{f}\,.$ (24)

Since we have both mixing-induced and direct coupling for $\phi$, its total
coupling to fermions is given by

$\bar{\xi}_{f}\equiv\xi_{f}+\theta\,y_{f}^{\rm SM}\,.$ (25)

Given the range of allowed values for $\theta$ [30], which covers about an
order of magnitude for our reference values of $m_{\phi}$, $\bar{\xi}_{f}$
couplings are also acceptable, since they are within a factor of $\sim 2$ of
$\theta y_{f}$, as implied by Eq. (24).

Figure 4: Values of $\phi$ coupling to electrons. The solid line corresponds
to the total coupling $\bar{\xi}_{e}$ and the dot-dashed line is only
accounting for $\phi$-Higgs mixing, for $\varepsilon=0.1$ and $M=4\times
10^{3}$ TeV, as a function of $\mu$ in MeV. The shaded region between dashed
lines approximately corresponds to the allowed values from Ref. [30] and the
shaded region extended to the dotted line is set by demanding that $\phi$
decay by BBN [30], based on a revised supernova constraint from Ref. [31].
Here, we have assumed $m_{\phi}=7$ MeV, as a typical value from the range (9),
for our benchmark parameters.

Using Eq. (22), we find $M_{e}\approx 3.3\times 10^{11}$ GeV. The interaction
(20) generates a direct coupling $\xi_{e}\phi\,\bar{e}_{L}e_{R}+\text{\small
H.C.}$. Let us choose $m_{\phi}=7$ MeV, typical of the range (9), as a
representative choice, for concreteness and illustrative purposes. From the
results of Ref. [30], we find that the value of $\theta y_{e}$ for
$m_{\phi}\approx 7$ MeV must lie approximately in the dark shaded region
between the two dashed lines, in Fig.4. The lower bound is set by supernova
physics constraints which were revisited and found to be less severe in Ref.
[31]. In that case, the new lower bound moves to the lightly shaded region
marked by the dotted line, which is set by requiring a sufficiently short
lifetime for $\phi$, i.e. below $\sim 1$ s, from BBN requirements.

As we can see from Fig. 4, our benchmark parameters yield acceptable values
for $\theta y_{e}$. However, we must also include the effect of $\xi_{e}$ if
we assume the flavor scenario presented above. Accounting for both
$\phi$-Higgs mixing and $\xi_{e}>0$, we have plotted the value of
$\bar{\xi}_{e}$ in Fig. 4, which as one can see lies inside the allowed region
of $\phi$-$e$ coupling, in agreement with the preceding general discussion.
Also, note that the overall magnitude of the coupling remains close to that
from mixing alone and hence $\phi$ remains long-lived, as deduced from Eq.
(10), via the replacement $\theta y_{e}\to\bar{\xi}_{e}$.

We note that the interaction in Eq. (17), analogous to the case of the
electron, induces a coupling of the top quark to $\phi$ given by
$\xi_{t}\phi\,\bar{t}_{L}t_{R}+\text{\small H.C.}$ For our benchmark choice of
parameters, we find $\xi_{t}\approx 1.6\times 10^{-4}$. This is of order a
$\phi$-top coupling induced by the $\phi$-Higgs mixing angle $\theta$, because
the top Yukawa coupling attains its SM value $y_{t}^{\rm SM}\approx 1$ after
electroweak symmetry breaking in our model. Assuming no cancellation between
the two effects, we find $\bar{\xi}_{t}\approx 2.6\times 10^{-4}$ for our
benchmark values, which can accommodate known constraints [32].

For Dirac neutrinos, we have $y^{\rm SM}_{\nu}(Q_{0})\sim 10^{-12}$, which
suggests $M_{\nu}\sim 4\times 10^{18}$ GeV close to, but below, the Planck
mass. We hence see that it is in principle possible that the physics of flavor
in the SM is governed by the dynamics of $\phi$, with UV scales ranging from
$\mathcal{O}{(10^{3})}$ TeV all the way up to near Planck mass. Alternatively,
this could signify an exponential hierarchy of flavor dependent couplings
between $\phi$ and some UV physics responsible for mediation of the SM flavor
structure at scales of $\mathcal{O}{(10^{3})}$ TeV.

Before closing the discussion of flavor in our model, we will briefly address
how transport of chiral asymmetries invoked in electroweak baryogenesis may be
realized, if the SM Yukawa couplings arise only after the FOPT. Nonetheless,
we first emphasize that the viability of our FOPT scenario only requires that
the top Yukawa coupling, and perhaps by extension quark Yukawa couplings,
remain small until after the transition. This could still leave lepton Yukawa
couplings as a possible avenue for generation of the BAU; see for example
Refs. [33, 34, 35] for work along this direction, in different contexts.
However, even if we adopt the view that all SM fermion Yukawa couplings are
generated only after the FOPT, other interactions may be present, as outlined
below.

For example, consider a singlet Dirac fermion $\chi$ that carries a unit of
lepton number and whose right (left)-handed component is even (odd) under the
assumed $\mathbb{Z}_{2}$. One can write down a Yukawa coupling $\lambda
H\bar{\chi}_{R}L+$ H.C. that involves the SM lepton doublet $L$, providing a
potential route for generating chiral asymmetries. We may also include an
interaction $\lambda^{\prime}\phi\bar{\chi}_{L}\chi_{R}+$ H.C. that would
allow $\chi$ to become more massive than the Higgs once
$\langle{\phi}\rangle\neq 0$, to avoid inducing a large deviation, governed by
$\lambda$, in the Higgs width. Note that these couplings would not lead to
large masses for the observed neutrinos, even if $\lambda\gtrsim 0.1$. The
above interactions point to possibilities that may allow our proposed
$\phi$-induced SM flavor structure to be consistent with viable electroweak
baryogenesis. A fuller treatment - which is beyond the scope of this work - is
required for quantitative conclusions.

We have collected the benchmark parameters used in the preceding discussions,
in Table 1.

$\varepsilon$ | $\kappa$ |  $M$ (TeV) |  $M_{t}$ (TeV)
---|---|---|---
0.1 | $10^{-4}$ | $4.0\times 10^{3}$ | $1.1\times 10^{3}$
Table 1: Benchmark parameters $\\{\varepsilon,\kappa,M,M_{t}\\}$, treated as
input choices.

##  VII Low Energy Tests and Other Signals

From Eq. (10), we see that in our scenario $\phi$ will be typically identified
as missing energy in laboratory experiments. This provides a possible avenue
for testing our proposal in rare meson decay experiments, such as KOTO. This
experiment aims to measure the branching fraction of the rare kaon decay
$K_{L}\to\pi^{0}\bar{\nu}\nu$, which is predicted to be $3.4\pm 0.6\times
10^{-11}$ in the SM [36, 37].

The KOTO experiment had observed a few events [38] that appeared to point to
an excess, but at low statistical significance [37]. The study in Ref. [37]
considers an interaction of the type in Eq. (17) and the results of their
analysis suggests that with a $\phi$-Higgs mixing parameter or a $\phi$-top
coupling $\sim 2\times 10^{-4}$ and $m_{\phi}\lesssim 50$ MeV, one could
obtain a 2$\sigma$ explanation of the seeming KOTO excess. This results from
the coupling of $\phi$ to the top quark in the 1-loop penguin diagram that
mediates $K_{L}\to\pi^{0}\phi$, with the ‘invisible’ $\phi$ mimicking the
missing energy from $\bar{\nu}\nu$ (see also [39]). Hence, parameters quite
close to our benchmark region could explain that excess with
$\bar{\xi}_{t}\sim 2\times 10^{-4}$, at around 2$\sigma$.

The above measurement has recently been further studied by the KOTO
collaboration, leading to identification of new sources of background [40].
The experiment has now determined that the observed events are statistically
consistent with the level of background. Thus, one may not need to invoke new
physics to explain the data. Nonetheless, the target precision of the KOTO
experiment for the branching fraction of $K_{L}\to\pi^{0}\bar{\nu}\nu$, which
is at the 10% level222See the experimental proposal at
https://koto.kek.jp/pub/p14.pdf., suggests that it could potentially probe
values of $\bar{\xi}_{t}\lesssim 10^{-5}$ in the future, providing an
experimental test of our model and its variants over a significant region of
parameter space. We note that in our model the penguin diagrams that mediate
$K_{L}\to\pi^{0}\phi$ [37] are dominated by the top quark, since the couplings
$\bar{\xi}_{f}$ scale with fermion mass.

Before closing, we would like to point out that it can be typically expected
that the FOPT in this scenario would lead to primordial gravitational waves,
corresponding to temperatures $\sim T_{c}$. This signal may be detectable by
future observatories like LISA (see, for example, Ref. [41]). We leave a
quantitative examination of this potential signal of our model to other work.

##  VIII Summary

In this work, we have introduced a model for a first order electroweak phase
transition which employs a light sub-GeV scalar $\phi$ that is weakly coupled
to the Higgs. Such a phase transition provides a necessary ingredient for a
successful baryogenesis mechanism. The consistency of the model requires new
interactions between $\phi$ and the top quark, for stability at scales
$\gtrsim 100$ GeV. This interaction can be consistent with a softly broken
parity of the scalar potential. We proposed that such interactions could be
the source of the entire SM flavor structure, via the dynamics of $\phi$. The
low energy effective theory in our proposal is to a good approximation that of
the SM. Nonetheless, precision low energy data, such as from measurements of
rare kaon decays, can provide tests of our model. We expect the KOTO
experiment measurements of $K_{L}\to\pi^{0}+$ ‘missing energy’ to have
significant reach for the regime of parameters discussed in this work. General
considerations regarding a strong first order phase transition at the weak
scale also suggest that primordial gravitational wave signals of this scenario
could be detected by a future observatory, such as LISA. In that case, our
model provides an interesting scenario where primordial gravitational wave
signals are related to rare phenomena at low energies. Definite quantitative
conclusions on that possibility, however, require further analysis that we
leave to other work.

###### Acknowledgements.

Work supported by the US Department of Energy under Grant Contract DE-
SC0012704.

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