# PeV-scale leptogenesis, gravity waves and black holes
from a SUSY-breaking phase transition

James M. Cline  Benoit Laurent  Jean-Samuel Roux  McGill University,
Department of Physics, 3600 University St., Montréal, QC H3A2T8 Canada  Stuart
Raby  The Ohio State University, 191 W. Woodruff Ave., Columbus, OH 43210

###### Abstract

Supersymmetry is a highly motivated theoretical framework, whose scale of
breaking may be at PeV energies, to explain null searches at the Large Hadron
Collider. SUSY breaking through a first order phase transition may have
occurred in the early universe, leading to potential gravitational wave
signals. Constructing a realistic model for gauge-mediated supersymmetry
breaking, we show that such a transition can also induce masses for heavy
right-handed neutrinos and sneutrinos, whose CP-violating decays give
leptogenesis at the PeV scale, and a novel mechanism of neutrino mass
generation at one loop. For the same models we predict the possible gravity
wave signals, and we study the possibility of production of primordial black
holes during the phase transition.

##  I Introduction

First order phase transitions (FOPTs) in the early universe have taken on
renewed interest, since the sound waves produced during the transition may
produce primordial gravitational waves (GWs) that could be observed by
upcoming experiments, including LISA Audley:2017drz ; Caprini:2015zlo ;
Caprini:2019egz , DECIGO Seto:2001qf ; Sato:2017dkf  and BBO Crowder:2005nr ;
Harry:2006fi . Moreover FOPTs can produce nonequilibrium effects such as are
needed for baryogenesis, notably when the electroweak phase transition is
involved. Possible correlations between the gravity wave signals and
successful electroweak baryogenesis have been widely studied Dorsch:2016nrg ;
Vaskonen:2016yiu ; Cline:2021iff . However it is also possible to have
analogous scenarios for other kinds of phase transitions, notably where lepton
number is spontaneously broken Long:2017rdo ; Huang:2022vkf ; Dasgupta:2022isg
.

It is interesting to consider supersymmetric (SUSY) generalizations of this
framework. Although there is not yet experimental evidence for SUSY, it
figures prominently in string-theoretic completions of the standard model,
which may be the most promising means of reconciling quantum mechanics with
gravity. Current constraints from ATLAS and CMS suggest that SUSY is broken
well above the TeV scale Adam:2021rrw , disappointing hopes that naturalness
of the weak scale would imply a low scale of SUSY breaking. Nevertheless, it
has been argued that SUSY breaking even at the PeV scale need not be severely
fine tuned Wells:2004di . Moreover, it has been shown that hidden-sector SUSY
breaking at the PeV scale, through a FOPT, can readily produce observable
gravity waves, with novel features in the high-temperature dynamics of the
phase transition Craig:2020jfv .

In the present work, we propose an explicit model of gauge-mediated SUSY
breaking that can yield successful leptogenesis, through the out-of-
equilibrium decays of heavy right-handed neutrinos and their superpartners.
Usually the scale of leptogenesis is decoupled from that of SUSY breaking, but
in this work we consider a model where the two are tied together; we assume
that the same field that breaks SUSY also produces a large lepton-violating
mass for heavy right-handed neutrinos. We take advantage of the near-
degeneracy of heavy right-handed neutrinos in a minimal flavor violation (MFV)
hypothesis to resonantly enhance the CP asymmetry of their decays, a regime
known as resonant leptogenesis Pilaftsis:2003gt , and we extend it to the
decays of sneutrinos. The resonance allows the model to yield the observed
baryon asymmetry despite strong washout effects due to inverse decays.

We found that O’Refeartaigh-like models of SUSY breaking require at least two
right-handed neutrino superfields per lepton flavor to have a nonvanishing
vacuum energy in this scenario, in contrast to the usual assumption of just a
single family. The level repulsion between these two fields pushes the scale
of leptogenesis down to $\mathcal{O}(10^{5}-10^{6}{\rm\ GeV})$, while
generating light neutrino masses from one-loop effects at the
$\mathcal{O}(10^{8}{\rm\ GeV})$ scale.

We start with a description of the general framework of gauge-mediated hidden-
sector SUSY breaking in Section II, and there introduce a specific model. In
Section III we construct the finite-temperature effective potential needed for
analyzing the first-order SUSY-breaking phase transition that produces
gravitational waves (GWs). Section IV describes the generation of the lepton
asymmetry that gives rise to baryogenesis, via resonant leptogenesis, as well
as the one-loop mechanism for neutrino mass generation. In Section V we
describe two additional cosmological consequences: the production of
potentially observable gravity waves, and constraints arising from production
of primordial black holes at the end of the phase transition. Conclusions are
given in Section VI. Details of the one-loop effective potential including
finite temperature are given in the appendix.

##  II Framework

Ref. Craig:2020jfv  considered scenarios of gauge-mediated SUSY breaking from
a hidden sector, in which a chiral pseudomodulus superfield $X$ gets a vacuum
expectation value (VEV) in its scalar component $x$, through a first order
phase transition. The desired shape of the potential $V(x)$, with a barrier
separating the true and false vacua, is generated by four vectorlike messenger
superfields, $\Phi$, $\Phi^{\prime}$, $\overline{\Phi}$ and
$\overline{\Phi}^{\prime}$, which also communicate SUSY breaking to the
visible sector, as in the O’Raifeartaigh model ORaifeartaigh:1975nky . They
are SM singlets, but they carry charges $+1$, $+1$, $-1$, $-1$ respectively,
under a U(1)D gauge symmetry, which allows for a Fayet-Iliopoulos (FI)
$D$-term that can enhance the gravity wave signal.

This minimal particle content is sufficient to produce a FOPT, but to
realistically embed the standard model (in the guise of the minimal
supersymmetric standard model, MSSM), one can unify the SM gauge group in
SU(5), and include vectorlike messengers analogous to the $\Phi$ fields but in
the fundamental representation of SU(5), which we denote by $5_{M}$,
$5^{\prime}_{M}$, $\overline{5}_{M}$, $\overline{5}^{\prime}_{M}$. In
addition, we introduce right-handed neutrino superfields $N_{i}$,
$N_{i}^{\prime}$ whose coupling to $X$ leads to lepton number violating
interactions after SUSY breaking, enabling leptogenesis. We note that it is
not strictly necessary to unify the SM gauge group to SU(5), but the $5_{M}$
mediators provide a simple way of communicating SUSY breaking to the SM. We
can then consider SU(5) as an approximate global symmetry, broken by the gauge
couplings of the SM. It enlarges to SU(6) by ignoring the U(1)D coupling and
appropriately combining the $\Phi$ and $5_{M}$ fields:

$\displaystyle\Psi$ $\displaystyle=$
$\displaystyle(\Phi,\,5_{M})^{T},\quad\Psi^{\prime}=(\Phi^{\prime},\,5^{\prime}_{M})\,,$
$\displaystyle\overline{\Psi}$ $\displaystyle=$
$\displaystyle(\overline{\Phi},\,\overline{5}_{M})^{T},\quad\overline{\Psi}^{\prime}=(\overline{\Phi}^{\prime},\,\overline{5}^{\prime}_{M})\,.$
(1)

The various fields and their transformation properties are summarized in Table
1.

Field | $R$ | U(1)D | SU(5) | SU(2)L | U(1)y | $L$ | $\mathbb{Z}^{L}_{2}$
---|---|---|---|---|---|---|---
$X$ | $+2$ | 0 | 1 | 1 | 0 | 0 | $+1$
$\Phi$ | 0 | $+1$ | 1 | 1 | 0 | 0 | $+1$
$\overline{\Phi}$ | $+2$ | $-1$ | 1 | 1 | 0 | 0 | $+1$
$\Phi^{\prime}$ | $+2$ | $+1$ | 1 | 1 | 0 | 0 | $+1$
$\overline{\Phi}^{\prime}$ | $0$ | $-1$ | 1 | 1 | 0 | 0 | $+1$
$5_{M}$ | $0$ | $0$ | 5 | * | $+y$ | 0 | $+1$
$\overline{5}_{M}$ | $+2$ | $0$ | $\overline{5}$ | * | $-y$ | 0 | $+1$
$5^{\prime}_{M}$ | $+2$ | $0$ | 5 | * | $+y$ | 0 | $+1$
$\overline{5}^{\prime}_{M}$ | $0$ | $0$ | $\overline{5}$ | * | $-y$ | 0 | $+1$
$N_{i}$ | $0$ | 0 | 1 | 1 | 0 | $-1$ | $-1$
$N^{\prime}_{i}$ | $+2$ | 0 | 1 | 1 | 0 | $+1$ | $-1$
$L_{\alpha}$ | $2$ | 0 | 1 | 2 | $-1$ | $+1$ | $-1$
$H_{u}$ | $0$ | 0 | 1 | 2 | $+1$ | 0 | $+1$
Table 1: Superfield content and their charges. $*$ denotes that each $5$-plet
contains an SU(2)L doublet (as well as an SU(3)c triplet). The hypercharge
$\pm y$ of the $5_{M}$ mediators is undetermined.

We hypothesize that heavy sterile neutrinos get a signficant contribution to
their masses from spontaneous SUSY breaking, to provide a link between the
phase transition and leptogenesis. In our model, this comes from a
superpotential term $XN_{i}N_{j}$. However, as discussed below, this by itself
would lead to the sneutrino fields $\tilde{N}_{i}$ getting VEVs and yielding a
SUSY-preserving vacuum. Such an undesired outcome can be avoided by the
inclusion of two types of heavy neutrino fields $N_{i}$ and $N^{\prime}_{i}$,
carrying opposite lepton number, allowing us to generate light neutrino masses
at one loop, by $Z$ and $\tilde{Z}$ exchange.111Qualitatively similar one-loop
mechanisms were considered in Refs. Ma:2006uv ; Mohapatra:2022tgb .

In addition to the fields shown, the MSSM contains the quark doublets
$Q_{\alpha}$, the down-type Higgs $H_{d}$, and the SU(2) singlet quarks and
leptons, $U_{\alpha}$, $D_{\alpha}$, $E_{\alpha}$. Although they do not play
an immediate role in the present study, for completeness we suggest a possible
set of $R$-symmetry charges for them, that would be compatible with SU(5)
gauge coupling unification. The fields $Q_{\alpha}$, $U_{\alpha}$,
$E_{\alpha}$ in the 10 dimensional representation have $R(10)=1$, while
$D_{\alpha}$ and $L_{\alpha}$ in the $\bar{5}$ have $R(\bar{5})=2$, and
$R(H_{d})=-1$. The $\mu$ term ($\mu H_{u}H_{d}$) would be nonrenormalizable,
requiring additional fields with total $R=3$.

The resulting superpotential, which can realize both spontaneous SUSY breaking
and leptogenesis, is

$\displaystyle W$ $\displaystyle=$ $\displaystyle W_{MSSM}-FX+\lambda
X\Psi\overline{\Psi}^{\prime}+m(\Psi\overline{\Psi}+\Psi^{\prime}\overline{\Psi}^{\prime})$
(2) $\displaystyle+$
$\displaystyle\frac{\lambda^{\prime}_{ij}}{2}XN_{i}N_{j}+M_{ij}N_{i}N^{\prime}_{j}+Y_{i\alpha}\epsilon_{ab}N_{i}L_{\alpha}^{a}H_{u}^{b}\,,$

where $W_{MSSM}$ is the superpotential of the MSSM, $H_{u}$ is the up-type
Higgs doublet, $\alpha$ is the flavor index for the SM fields, and $i$ is that
for the right-handed neutrino superfields. At least two flavors of $N_{i}$ are
needed for CP-violation leading to leptogenesis and to explain light neutrino
oscillations; for simplicity we will assume only two such flavors. Terms of
the form $N^{\prime}NN$ and $F^{\prime}N^{\prime}$ are forbidden by a discrete
symmetry $\mathbb{Z}_{2}^{L}$ under which all fields acquire a sign
$(-1)^{L}$, where $L$ is the lepton number. This “leptonic parity” could be
considered as a remnant of the broken $U(1)_{L}$ global symmetry.

To construct a predictive and economical model, we will assume a version of
MFV, i.e., that SO($2$) symmetry of the sterile neutrino flavors (labeled by
$i$ for both $N_{i}$ and $N^{\prime}_{i}$) in the superpotential is broken
only by the Yukawa couplings $Y_{i\alpha}$, so that at leading order
$\lambda^{\prime}_{ij}=\lambda^{\prime}\delta_{ij}$ and
$M_{ij}=M\delta_{ij}+iM^{\prime}\epsilon_{ij}$, using the invariant tensors of
SO(2). At one loop, corrections are generated involving $\delta
Y^{2}_{ij}\equiv Y_{i\alpha}Y^{\dagger}_{\alpha j}/(16\pi^{2})$ Martin:1993zk
:

$\displaystyle\lambda^{\prime}_{ij}$ $\displaystyle=$
$\displaystyle\lambda^{\prime}\left[\delta_{ij}+c_{1}\,\delta
Y^{2}_{ij}\right]\,,$ $\displaystyle M_{ij}$ $\displaystyle=$
$\displaystyle\left(\delta_{ik}+c_{2}\,\delta
Y^{2}_{ik}\right)\left[M\delta_{kj}+iM^{\prime}\epsilon_{kj}\right]\,,$ (3)

where $c_{1,2}$ are constants of order unity. Ignoring the small corrections
of order $Y^{2}/16\pi^{2}$, the eigenvalues of $M_{ij}$ are $M\pm M^{\prime}$.

As we will show, the model requires $\delta Y^{2}_{ij}\sim 10^{-8}$ (that is,
$Y^{2}\sim 10^{-6}$) to radiatively generate neutrino mass of order $\sim
0.05$ eV. Therefore, Yukawa corrections to the mass eigenvalues are negligible
for $M^{\prime}\gtrsim 10^{-7}M$ only. It is technically natural to assume
that $M^{\prime}\ll M$ so that the two states, with masses $M\pm M^{\prime}$,
are nearly degenerate. It will be shown in Section IV that this leads to the
quasi-resonant enhancement of the $CP$ asymmetry of neutrino and sneutrino
decays. In this work, we will fix $M^{\prime}=10^{-7}M$ to maximize the effect
of this resonance. On the other hand, the Yukawa interactions contain CP-
violating phases, that can lead to the CP asymmetries needed for leptogenesis,
as we will show.

SUSY is spontaneously broken when the scalar component of $X$ gets its vacuum
expectation value (VEV). The scale

$\mu=\lambda^{\prime}\langle\tilde{X}\rangle\equiv{\lambda^{\prime}\over\sqrt{2}}\langle
x\rangle$ (4)

will play an important role in the following. It determines the mass of the
heavy sterile neutrinos $N_{i}$, in the regime $M\ll\mu$ where their mixing
with $N^{\prime}_{i}$ is suppressed. Then $N^{\prime}_{i}$ is relatively
light, with mass $\sim M^{2}/\mu$. We will see that this is the favorable
regime for obtaining successful leptogenesis. $\langle x\rangle$ is determined
by the microscopic model parameters as estimated in Eq. (27) below.

Another frequently occurring scale is the effective $F$ term

$F_{X}\equiv
F-\lambda\langle\widetilde{\Phi}\widetilde{\overline{\Phi}}^{\prime}\rangle$
(5)

at the minimum of the potential, where the fields $\widetilde{\Phi}$,
$\widetilde{\overline{\Phi}}^{\prime}$ obtain VEVs, as described below. It
quantifies the spontaneous breaking of SUSY, and appears in the gluino mass
arising from gauge-mediated SUSY breaking,

$m_{\tilde{g}}=N_{m}\frac{g_{s}^{2}}{(4\pi)^{2}}\frac{F_{X}}{\langle\widetilde{X}\rangle}s_{M}\,C_{RG}\,,$
(6)

where $g_{s}$ is the strong coupling at the scale
$\langle\widetilde{X}\rangle$, and $N_{m}=2$ is the number of pairs of
messenger $SU(3)$ triplets. The factor $s_{M}\equiv\frac{1}{6}[\lambda
F/m^{2}]^{2}$ expresses the effect of gaugino screening Cheung:2007es ;
Komargodski:2009jf ; Cohen:2011aa .222The screening suppression comes from the
first subleading term in Eq. (7) of Ref. Martin:1996zb , where the leading
term is subtracted by the gaugino mass counterterm. The correction $C_{RG}$
accounts for the renormalization group (RG) relation of the soft mass to the
pole mass, from Eq. (8.3.3) of Ref. Martin:1997ns . The current experimental
limit of $m_{\tilde{g}}>2.3\,$TeV ATLAS:2020syg  will provide a significant
constraint on the parameter space favorable for gravity wave production, in
the following. Although we only consider the minimal model with $N_{m}=2$, one
could always consider a model with more sets of messenger fields, which would
increase gaugino masses with little impact on the FOPT dynamics or
leptogenesis.

###  II.1 Scalar potential

The $F$-term potential arising from the superpotential (2) includes the terms

$\displaystyle V_{F}$ $\displaystyle\supset$
$\displaystyle\absolutevalue{F-\frac{\lambda^{\prime}}{2}{\widetilde{N}}_{i}{\widetilde{N}}_{i}-\lambda\widetilde{\Psi}\widetilde{\overline{\Psi}}^{\prime}}^{2}+\left|m\right|^{2}\left(|{\widetilde{\Psi}}|^{2}+|{\widetilde{\overline{\Psi}}^{\prime}}|^{2}\right)$
$\displaystyle+$
$\displaystyle\absolutevalue{\lambda\widetilde{X}\widetilde{\Psi}+m\widetilde{\Psi}^{\prime}}^{2}+\absolutevalue{\lambda\widetilde{X}\widetilde{\overline{\Psi}}^{\prime}+m\widetilde{\overline{\Psi}}}^{2}$
$\displaystyle+$
$\displaystyle\absolutevalue{\lambda^{\prime}\widetilde{X}{\widetilde{N}}_{i}+M_{i}{\widetilde{N}}^{\prime}_{i}+\epsilon_{ab}Y_{i\alpha}\widetilde{L}_{\alpha}^{a}H_{u}^{b}}^{2}+\absolutevalue{M_{i}{\widetilde{N}}_{i}}^{2}$

where tilde denotes the scalar component of the superfield (except later for
$\widetilde{H}_{u}$, where it is the higgsino).

In addition, there is a $D$-term potential from the U(1)D interaction,
including a Fayet-Iliopoulos term,

$V_{D}={g^{2}\over 2}\left({D\over
g}+|\widetilde{\Phi}|^{2}+|\widetilde{\Phi}^{\prime}|^{2}-|\widetilde{\overline{\Phi}}|^{2}-|\widetilde{\overline{\Phi}}^{\prime}|^{2}\right)^{2}$
(8)

where $g$ is the U(1)D gauge coupling. We do not write the $D$ terms of the
$5_{M}$ fields associated with their SM charges, which only ensure that these
fields have vanishing VEVs at the minimum of the potential. Therefore the
$\Psi$ fields in Eq. (II.1) can be replaced by their corresponding $\Phi$
components, for the purposes of minimizing the potential.

At zero temperature, there is a metastable minimum of $V=V_{F}+V_{D}$ at
${\widetilde{N}}={\widetilde{N}}^{\prime}=\widetilde{\Psi}=\widetilde{\Psi}^{\prime}=0$,
provided the conditions $M_{i}^{2}>\lambda^{\prime}F$ and
$m^{4}>\lambda^{2}F^{2}+g^{2}D^{2}$ are satisfied. At tree level,
$\widetilde{X}$ is a flat direction near this local minimum. However, one-loop
corrections to the scalar potential slightly lift the flat direction, making
$\widetilde{X}=0$ the local minimum.

If $D>F$, the true minimum of the tree-level potential is a runaway solution,
$\widetilde{X}\to\infty$ and
$\widetilde{\Phi},\widetilde{\overline{\Phi}}^{\prime}\to 0$, with
$\widetilde{\Phi}^{\prime}=-(\lambda/m)\widetilde{X}\widetilde{\Phi}$,
$\widetilde{\overline{\Phi}}=-(\lambda/m)\widetilde{X}\widetilde{\overline{\Phi}}^{\prime}$
taking finite values that satisfy

$\big{|}\widetilde{\Phi}^{\prime}\big{|}^{2}-\big{|}\widetilde{\overline{\Phi}}\big{|}^{2}+\frac{D}{g}=0\,.$
(9)

Therefore at the minimum of $V$, $V_{D}$ vanishes while
$V_{F}=\absolutevalue{F}^{2}$. However, as we will show, loop corrections
shift the minimum to finite values of $\widetilde{X},\ \widetilde{\Phi}$ and
$\widetilde{\overline{\Phi}}^{\prime}$.

At one loop, $V$ receives a Coleman-Weinberg correction through the field-
dependent masses $m^{2}_{i}(x,\phi_{1},\phi_{2})$, where for brevity we define
$x=\sqrt{2}\big{|}\widetilde{X}\big{|}$,
$\phi_{1}=\sqrt{2}\big{|}\widetilde{\overline{\Phi}}\big{|}$ and
$\phi_{2}=\sqrt{2}\big{|}\widetilde{\Phi}^{\prime}\big{|}$. Expressions for
the masses and $V_{CW}$ are given in Appendix A. We define an effective
potential for the modulus alone, $V_{\rm eff}(x)$, by setting $\phi_{1}$ and
$\phi_{2}$ to the values that minimize $V$ for fixed $x$. Typical behaviors
for $V_{\rm eff}(x)$ at different temperatures are illustrated in Fig. 1. More
details about $V_{\rm eff}$ and its properties are derived in Sec. III.2.

Figure 1: Typical profiles of $V_{\rm eff}(x)$ at $T=0,\,T_{n}\ \mathrm{and}\
T_{c}$. The points show the position of the true vacuum. Model parameters are
fixed at $\lambda=4$, $D=8.6F$, $g=0.1$ and $m=\sqrt{5.3F}$.

###  II.2 Finite temperature corrections

Thermal corrections to the potential decrease the value of the potential near
the origin, but they have negligible effects at large field values, thus
making the $x=0$ vacuum the true minimum above a certain critical temperature
$T_{c}$. The thermal potential and its high-temperature expansion are given in
the Appendix A.

###  II.3 Interactions of fermions

The fermionic part of the potential, following from the superpotential,
contains the terms

$\displaystyle V_{f}$ $\displaystyle\supset$ $\displaystyle
M_{i}\overline{N_{i}^{C}}N^{\prime}_{i}+{\lambda^{\prime}\over
2}\widetilde{X}\overline{N_{i}^{C}}N_{i}+Y_{i\alpha}\epsilon_{ab}\left({\widetilde{N}}_{i}\overline{{\widetilde{H}}_{u}^{C,b}}P_{L}L_{\alpha}^{a}\right.$
(10) $\displaystyle+$
$\displaystyle\left.{\widetilde{L}}^{a}_{\alpha}\overline{{\widetilde{H}}_{u}^{C,b}}P_{L}N_{i}+H_{u}^{b}\overline{N_{i}^{C}}P_{L}L_{\alpha}^{a}\right)+\mathrm{h.c}.$

The second term on the first line violates lepton number conservation and
gives rise to a Majorana mass
$\mu=\lambda^{\prime}\langle\widetilde{X}\rangle$ for $N_{i}$ after
$\widetilde{X}$ acquires a VEV. In the limit $\mu\gg M$, $N_{i}$ and
$N_{i}^{\prime}$ split into two distinct eigenstates with masses $\sim\mu$ and
$M^{2}/\mu$ (see Eqs (33)).

As was the case for scalar decays, the Yukawa couplings are the only source of
CP violation for the fermionic decay channels of heavy neutrinos and
sneutrinos.

##  III SUSY-breaking first-order phase transition

The leptogenesis mechanism presented in this paper (see Section IV) does not
require the SUSY-breaking phase transition to be first order. Nevertheless,
FOPTs provide an interesting opportunity for probing high-energy physics by
producing primordial black holes and a stochastic background of GWs. Such
cosmological observables will be explored in Section V for the model presented
in this paper. In this section, we present our methodology for finding and
studying SUSY-breaking FOPTs and discuss their properties.

###  III.1 Bubble nucleation

FOPTs proceed through bubble nucleations of the true vacuum. It is triggered
by quantum tunnelling or thermal fluctuations, which can both be
quantitatively described by an instanton or a bounce solution interpolating
between the false and true vacua. In most cases of interest, the phase
transition happens at high temperatures where thermal fluctuations are much
more efficient than quantum tunnelling; hence we only describe the former
here.

The nucleation rate, which can be derived from a semi-classical calculation,
is given by Linde:1980tt

$\Gamma(T)\simeq T^{4}\left(\frac{S_{3}}{2\pi T}\right)^{3/2}e^{-S_{3}/T}\,,$
(11)

where $S_{3}$ is the O(3)-symmetric Euclidean action

$S_{3}(T)=4\pi\\!\\!\int\\!dr\,r^{2}\\!\left[\frac{1}{2}\left(\frac{d\phi_{i}}{dr}\right)^{2}+V(\phi_{i},T)\right],$
(12)

and $\phi_{i}$ denotes a scalar field whose VEV changes during the phase
transition. The tunneling path $\phi_{i}(r)$ is found by requiring $S_{3}$ to
be stationary, which leads to the equations of motion

$\frac{d^{2}\phi_{i}}{dr^{2}}+\frac{2}{r}\frac{d\phi_{i}}{dr}=\frac{\partial
V}{\partial\phi_{i}}\,,$ (13)

with the boundary conditions $\phi_{i}(r\rightarrow\infty)=\phi_{i}^{\rm
false}$ and $\left.\frac{d\phi_{i}}{dr}\right|_{r=0}=0$.

The phase transition begins at the nucleation temperature $T_{n}$, when there
is an average of one bubble per Hubble volume Huber:2007vva :

$1=\int_{T_{n}}^{T_{c}}\frac{dT}{T}\frac{\Gamma}{H^{4}}=\int_{T_{n}}^{T_{c}}dT\,\frac{T^{3}}{H^{4}}\left(\frac{S_{3}}{2\pi
T}\right)^{3/2}e^{-S_{3}/T}\,.$ (14)

$T_{c}$ is the critical temperature, defined as the temperature where the true
and false vacua are degenerate,

$\Delta V(T_{c})=0\,;$ (15)

$\Delta V$ is the potential difference between the two vacua. To obtain an
approximate solution of Eq. (14), one can expand the argument of the
exponential linearly in $T$, and fix $T=T_{n}$ elsewhere in the integrand;
this is justified because the exponential varies much more rapidly than the
other coefficients. This procedure yields the simpler condition

$\displaystyle\left.\frac{S_{3}}{T}\right|_{T=T_{n}}\simeq
92.5+\frac{3}{2}\log\left(\frac{S_{3}}{2\pi
T_{n}}\right)-4\log\left(\frac{T_{n}}{10\ \mathrm{PeV}}\right)\qquad$
$\displaystyle\quad-\log\left(\frac{\beta_{H}}{100(1-e^{-\beta_{H}(T_{c}/T_{n}-1)})}\right)-2\log\left(\frac{g}{300}\right)\
$ (16)

where $g$ is the effective number of degrees of freedom and

$\beta_{H}=\frac{\beta}{H}=T_{n}\left.\frac{d}{dT}\left(\frac{S_{3}}{T}\right)\right|_{T=T_{n}}$
(17)

quantifies the inverse duration of the phase transition. Using the fact that
the action is stationary under small variations of the tunneling path, one can
write $\beta_{H}$ in terms of $\phi_{i}(r)$ and $S_{3}$ as

$\beta_{H}=4\pi\\!\\!\int\\!dr\,r^{2}\,\frac{\partial V}{\partial
T}-\left.\frac{S_{3}}{T}\right|_{T=T_{n}},$ (18)

which is numerically much more efficient than computing Eq. (17) by finite
difference.

Another important quantity for characterizing the strength of the phase
transition is the ratio of vacuum energy released in the phase transition
compared to the radiation energy, given by

$\alpha=\frac{1}{\rho_{\gamma}}\left(\Delta
V-\frac{T_{n}}{4}\Delta\frac{dV}{dT}\right),$ (19)

where $\Delta V$ is the difference of potential energy between the true and
false vacua and $\rho_{\gamma}=g\pi^{2}T_{n}^{4}/30$. Larger $\alpha$
corresponds to more energy being released into the plasma, which yields a
stronger phase transition. As we will see in Section V, this quantity,
together with $\beta$, is important for determining the spectrum of GWs
produced during the FOPT. In general, the GW amplitude is enhanced at large
$\alpha$ and small $\beta$.

###  III.2 Effective bounce scalar potential

In order to study the dynamics of the phase transition, one must consider the
scalar potential as a function of the fields whose VEVs vary across the bubble
walls. As previously explained, these are the fields field $X$ and the four
vectorlike messengers $\Phi$, $\Phi^{\prime}$, $\overline{\Phi}$ and
$\overline{\Phi}^{\prime}$. The scalar potential appearing in the bounce
action thereby reduces to

$\displaystyle V$ $\displaystyle=$
$\displaystyle\absolutevalue{F-\lambda\widetilde{\Phi}\widetilde{\overline{\Phi}}^{\prime}}^{2}+\left|m\right|^{2}\left(|{\widetilde{\Phi}}|^{2}+|{\widetilde{\overline{\Phi}}^{\prime}}|^{2}\right)$
$\displaystyle+$
$\displaystyle\absolutevalue{\lambda\widetilde{X}\widetilde{\Phi}+m\widetilde{\Phi}^{\prime}}^{2}+\absolutevalue{\lambda\widetilde{X}\widetilde{\overline{\Phi}}^{\prime}+m\widetilde{\overline{\Phi}}}^{2}$
$\displaystyle+$ $\displaystyle{g^{2}\over 2}\left({D\over
g}+|\widetilde{\Phi}|^{2}+|\widetilde{\Phi}^{\prime}|^{2}-|\widetilde{\overline{\Phi}}|^{2}-|\widetilde{\overline{\Phi}}^{\prime}|^{2}\right)^{2}$
$\displaystyle+$ $\displaystyle V_{CW}+V_{T}\,,$ (21)

where $V_{CW}$ and $V_{T}$ are the 1-loop vacuum and thermal potentials,
respectively (see Appendix A for more details).

Finding the bounce action for 10 real degrees of freedom is numerically
expensive, so it is advantageous to simplify the scalar potential to reduce
the effective number of degrees of freedom. This can be achieved with the
methodology of Ref. Craig:2020jfv . One first integrates out the
$\Phi^{\prime}$ and $\overline{\Phi}$ fields by solving for the $F$-terms of
$\Phi$ and $\overline{\Phi}^{\prime}$, which yields

$\widetilde{\Phi}^{\prime}=-\frac{\lambda}{m}\widetilde{X}\widetilde{\Phi},\quad\widetilde{\overline{\Phi}}=-\frac{\lambda}{m}\widetilde{X}\widetilde{\overline{\Phi}^{\prime}}\,.$
(22)

Substituting Eqs. (22) into Eq. III.2 gives the effective tunneling potential

$\displaystyle V_{\rm eff}$ $\displaystyle=$
$\displaystyle\absolutevalue{F-\lambda\widetilde{\Phi}\widetilde{\overline{\Phi}}^{\prime}}^{2}+\left|m\right|^{2}\left(|{\widetilde{\Phi}}|^{2}+|{\widetilde{\overline{\Phi}}^{\prime}}|^{2}\right)$
(23) $\displaystyle+$
$\displaystyle\frac{g^{2}}{2}\left[\frac{D}{g}+\left(\absolutevalue{\widetilde{\Phi}}^{2}-\absolutevalue{\widetilde{\overline{\Phi}}^{\prime}}^{2}\right)\left(1+\frac{\lambda^{2}}{m^{2}}\absolutevalue{\widetilde{X}}^{2}\right)\right]^{2}\quad$
$\displaystyle+$ $\displaystyle V_{CW}+V_{T}\,.$

$V_{\rm eff}$ depends on just four fields: the magnitude of $x$,
$\widetilde{\Phi}$, $\widetilde{\overline{\Phi}}^{\prime}$, and the relative
phase between the latter. Furthermore, the $F$-term of $X$ is minimized when
this relative phase vanishes, so without loss of generality, we can choose the
three fields to be real, leaving three degrees of freedom. The bounce action
Eq. (12) can then be rewritten in terms of the fields
$x=\sqrt{2}|\widetilde{X}|$, $\phi_{1}=\sqrt{2}|\widetilde{\Phi}|$ and
$\phi_{2}=\sqrt{2}|\widetilde{\overline{\Phi}}^{\prime}|$ as

$S_{3}^{\rm
eff}(T)=4\pi\\!\\!\int\\!\\!dr\,r^{2}\\!\left[\frac{\dot{x}^{2}}{2}+\frac{\dot{\phi}_{1}^{2}}{2}+\frac{\dot{\phi}_{2}^{2}}{2}+V_{\rm
eff}(x,\phi_{1},\phi_{2};T)\right].$ (24)

To get accurate results, one must minimize the potential (III.2) numerically
to find the true vacuum. We compute the bounce solution
$(x,\,\phi_{1},\,\phi_{2})(r)$, and the tunneling action numerically using the
package CosmoTransitions Wainwright:2011kj  which also calculates the
nucleation temperature $T_{n}$ as described in the last subsection (see Ref.
Craig:2020jfv  for an analytical estimate of the action).

One can get analytic insight into the phase transition dynamics by estimating
the position of the true vacuum. As argued in Section II, the tree-level true
vacuum is at $x\rightarrow\infty$, and it is rendered finite, though still
large, by small radiative contributions to $V_{\rm eff}$. Consequently, Eq.
(9) implies that $\langle\phi_{1}\rangle$ and $\langle\phi_{2}\rangle$ are
small. Moreover, to minimize the $F$ and $D$-terms, $\phi_{1}$ must be
suppressed with respect to $\phi_{2}$ by a factor of $gF/(\lambda D)$, which
is assumed to be small. Therefore, $\langle\phi_{1}\rangle$ is neglected in
the following.

We integrate out $\phi_{2}$ by partially minimizing $V_{\rm eff}$ with respect
to it, with the solution at large $x$

$\displaystyle\langle\phi_{2}\rangle_{x\rightarrow\infty}$
$\displaystyle\cong$ $\displaystyle\frac{2m}{\lambda x}\sqrt{\frac{D}{g}}\,,$
(25) $\displaystyle V_{\rm eff}(x\rightarrow\infty;T)$ $\displaystyle\cong$
$\displaystyle
F^{2}+\frac{2m^{4}D}{g\lambda^{2}x^{2}}+V_{CW}(x)+V_{T}(x;T)\,,$

where we neglected the one-loop contributions to compute
$\langle\phi_{2}\rangle$. As expected, the tree-level potential does not have
a minimum at finite $x$. Therefore, one must estimate the one-loop
contributions to obtain a finite VEV. We show in Appendix A that, for large
$x$,

$\displaystyle V_{CW}(x\rightarrow\infty)$ $\displaystyle\cong$
$\displaystyle\frac{3\lambda^{2}F^{2}}{8\pi^{2}}\log\left(\frac{\lambda^{2}x^{2}}{2m^{2}}\right),$
$\displaystyle V_{T}(x\rightarrow\infty;T)$ $\displaystyle\cong$
$\displaystyle\frac{3m^{4}T^{2}}{\lambda^{2}x^{2}}\,,$ (26)

neglecting constant terms that have no effect on the vacuum’s position. Adding
the one-loop corrections to the tree-level potential, one finds that the true
vacuum is located at

$\langle x\rangle\cong\frac{4\pi
m^{2}}{\sqrt{3}\lambda^{2}F}\sqrt{\frac{D}{g}+\frac{3T^{2}}{2}}\,.$ (27)

$\langle x\rangle$ depends on negative powers of the coupling constants,
confirming the hypothesis that $\langle x\rangle_{\rm true}$ is large when the
interactions are weak.

The fact that $x$ gains a large VEV has important consequences for the FOPT’s
dynamics. It implies that to get from one vacuum to the other, a large amount
of kinetic energy must be expended, which increases the tunneling action.
Therefore tunneling will only be possible at low temperature relative to
$T_{c}$, which leads to greater supercooling and consequently a stronger FOPT.
This has the potential to enhance GW production and the probability of
detection. In Section V, we will compute GW spectra for a range of parameters
and show that this is indeed the case.

##  IV Resonant leptogenesis

Figure 2: One-loop self energy diagrams for $\bm{{\widetilde{N}}^{\prime}}$
(top) and $\bm{N^{\prime}}$ (bottom) decays. Dotted vertical lines indicate
the intermediate states that go on-shell when evaluating the absorptive parts
of the diagrams, in accordance with the Cutkosky rules. Cutkosky:1960sp

Next we describe the mechanism of leptogenesis in the SUSY model, arising from
the out-of-equilibrium decays of the sterile neutrinos $N_{i}$,
$N^{\prime}_{i}$ and their superpartners ${\widetilde{N}}_{i}$,
${\widetilde{N}}^{\prime}_{i}$. CP and lepton number violation in the decays
generate a lepton asymmetry $Y_{L}^{\rm
tot}=\sum_{\alpha}Y_{L_{\alpha}}+Y_{{{\widetilde{L}}}_{\alpha}}$, where
$Y_{i}=n_{i}/s$ is the comoving abundance of a given species. Assuming $B-L$
is conserved, sphalerons partially convert the lepton asymmetry into a baryon
asymmetry. In the MSSM, the conversion efficiency is Harvey:1990qw

$Y_{B}=\frac{n_{B}-n_{\overline{B}}}{s}=-\frac{8}{23}Y_{L}^{\rm tot}.$ (28)

BBN and CMB data constrain the baryon asymmetry of the universe to
Workman:2022ynf ; Planck:2018vyg

$\displaystyle Y_{B}^{BBN}$ $\displaystyle=(8.7\pm 0.5)\times 10^{-11}$
$\displaystyle Y_{B}^{CMB}$ $\displaystyle=(8.69\pm 0.06)\times 10^{-11}$ (29)

at 95% C.L.

In our model, all lepton-number-violating interactions are related to the
breaking of SUSY via the nonvanishing $F$-term
$F_{X}=F-\lambda\langle\widetilde{\Phi}\widetilde{\overline{\Phi}}^{\prime}\rangle$
or the pseudomodulus $\widetilde{X}$,

$\displaystyle-\mathcal{L}_{\not L}$ $\displaystyle=$
$\displaystyle-\frac{\lambda^{\prime}F_{X}^{*}}{2}{\widetilde{N}}_{i}{\widetilde{N}}_{i}+\frac{\lambda^{\prime}\widetilde{X}}{2}\overline{N_{i}^{C}}N_{i}$
$\displaystyle+\lambda^{\prime}\widetilde{X}\,{\widetilde{N}}_{i}\left(M_{i}^{*}{\widetilde{N}}_{i}^{\prime*}+\epsilon_{ab}Y_{i\alpha}^{*}{\widetilde{L}}_{\alpha}^{a*}H_{u}^{b*}\right).$

One may recognize the first term of this expression as a soft SUSY-breaking
$B$-term, which mixes sneutrinos with their conjugates even before the phase
transition, allowing their decays to violate the lepton number outside the
bubble wall. When $\widetilde{X}$ gets a VEV, neutrinos acquire a Majorana
mass $\mu=\lambda^{\prime}\expectationvalue*{\widetilde{X}}$, opening their
decays to lepton number violation as well.

A priori, all parameters appearing in the superpotential (2) are complex. One
can rephase the fields $N_{i}$, $N^{\prime}_{i}$, $L_{\alpha}$ and $H_{u}$ to
remove the complex phases in the mass matrix and in three Yukawa couplings,
for example $Y_{1\alpha}$. If there are several right-handed neutrino flavors,
in general one cannot simultaneously remove the phases of $Y_{1\alpha}$ and
$Y_{j\alpha}$ ($j>1$), allowing one to define a set of invariant CP-violating
phases

$\theta_{j\alpha}=\arg\left(Y_{j\alpha}Y^{*}_{1\alpha}\right)\qquad\text{(no
sum on}\ \alpha;\ j>1),$ (31)

which can be assigned to $Y_{j\alpha}$. Hence at least two right-handed $N$
and $N^{\prime}$ species are needed for leptogenesis. Additional interactions
such as $A$-terms $A_{i}{\widetilde{N}}_{i}{\widetilde{L}}H_{u}$, which arise
from RG running of SUSY-breaking interactions and which are essential to
standard soft leptogenesis DAmbrosio:2003nfv ; Fong:2011yx ; Adhikari:2015ysa
, are not required and can be neglected if leptogenesis occurs at a scale
close to $\sqrt{F}$. In the following we will consider two heavy neutrino
flavors, so that only two SM neutrinos are massive.

There are two ways in which a lepton asymmetry can arise, depending on whether
$\mu\ll M$ or $\mu\gg M$. In the former case, (s)neutrinos decay in the
thermal bath, possibly even before the SUSY-breaking phase transition if
$M\gg\sqrt{F}$ (in which case only sneutrinos produce a lepton asymmetry due
to their $B$-term), and the CP asymmetry is resonantly enhanced by the small
mass splitting between flavors. Strong washout is then required to achieve the
observed baryon asymmetry. However, we have found that in this scenario, the
SM neutrino masses induced at one loop are too small to match observations.

Consequently, we will focus on the case $\mu\gg M$. After the phase
transition, $N$ and $N^{\prime}$ are weakly mixed with mixing angle $\sim
M/\mu\ll 1$, and we will refer to the corresponding mass eigenstates as
$\bm{N},\bm{N^{\prime}}$. Going to the basis where $M_{ij}=\left({M\atop-
iM^{\prime}}{iM^{\prime}\atop M}\right)$ becomes diagonal with eigenvalues
$M_{i}=M+s_{1}M^{\prime}$, where $s_{1}=\pm 1$, the four scalar masses are
given by

$\displaystyle m^{2}_{\bm{{\widetilde{N}}^{\prime}}_{i,\pm}}$ $\displaystyle=$
$\displaystyle
M_{i}^{2}\left(\frac{M_{i}^{2}+s_{2}\lambda^{\prime}F_{X}}{\mu^{2}}\right)\cong\frac{M_{i}^{4}}{\mu^{2}},$
$\displaystyle m^{2}_{\bm{{\widetilde{N}}}_{i,\pm}}$ $\displaystyle=$
$\displaystyle\mu^{2}+2M_{i}^{2}+s_{2}\lambda^{\prime}F_{X}\cong\mu^{2},$ (32)

with $s_{2}=\pm 1$. In the approximation
$m^{2}_{\bm{{\widetilde{N}}^{\prime}}_{i,\pm}}\cong M_{i}^{4}/\mu^{2}$, we
assumed that $\lambda^{\prime}F_{X}/\mu^{2}\ll M^{\prime}/M$. The fermionic
mass eigenvalues are

$\displaystyle m_{\bm{N^{\prime}}_{i}}$ $\displaystyle=$
$\displaystyle-{M_{i}^{2}\over\mu},$ $\displaystyle m_{\bm{N}_{i}}$
$\displaystyle=$
$\displaystyle\mu\left(1+{M_{i}^{2}\over\mu^{2}}\right)\cong\mu\,.$ (33)

It is useful to rewrite the Lagrangian interactions relevant for leptogenesis
in the mass eigenbasis. Keeping only the leading terms, one finds

$\displaystyle-\mathcal{L}_{s}$
$\displaystyle=\frac{\epsilon_{ab}Y_{i\alpha}}{\sqrt{2}}\times$
$\displaystyle\left\\{\bm{{\widetilde{N}}^{\prime}}_{i,-}\left(\frac{M_{i}^{3}}{\mu^{2}}{\widetilde{L}}^{a}_{\alpha}H_{u}^{b}-\frac{M_{i}}{\mu}\overline{{\widetilde{H}}^{C,b}_{u}}P_{L}L_{\alpha}^{a}\right)\right.$
$\displaystyle+i\bm{{\widetilde{N}}^{\prime}}_{i,+}\left(\frac{M_{i}^{3}}{\mu^{2}}{\widetilde{L}}^{a}_{\alpha}H_{u}^{b}+\frac{M_{i}}{\mu}\overline{{\widetilde{H}}^{C,b}_{u}}P_{L}L_{\alpha}^{a}\right)$
$\displaystyle+\bm{{\widetilde{N}}}_{i,-}\left(\mu{\widetilde{L}}^{a}_{\alpha}H_{u}^{b}+\overline{{\widetilde{H}}^{C,b}_{u}}P_{L}L_{\alpha}^{a}\right)$
$\displaystyle\left.+i\bm{{\widetilde{N}}}_{i,+}\left(\mu{\widetilde{L}}^{a}_{\alpha}H_{u}^{b}-\overline{{\widetilde{H}}^{C,b}_{u}}P_{L}L_{\alpha}^{a}\right)\right\\}+\rm{h.c.}$
(34)

for sneutrinos and

$\displaystyle-\mathcal{L}_{f}$ $\displaystyle=\epsilon_{ab}Y_{i\alpha}\times$
$\displaystyle\left\\{\bm{\overline{N_{i}^{\prime
C}}}\left(-\frac{M_{i}}{\mu}{\widetilde{L}}_{\alpha}^{a}P_{L}{\widetilde{H}}_{u}^{b}-\frac{M_{i}}{\mu}H_{u}^{b}P_{L}L_{\alpha}^{a}\right)\right.$
$\displaystyle+\bm{\overline{N_{i}^{C}}}\left({\widetilde{L}}_{\alpha}^{a}P_{L}{\widetilde{H}}_{u}^{b}+H_{u}^{b}P_{L}L_{\alpha}^{a}\right)\bigg{\\}}+\rm{h.c.}$
(35)

for neutrinos.

When $\widetilde{X}$ acquires a VEV, the $\bm{N}$ states become heavy and
quickly decay, giving rise to an initial lepton asymmetry, due to the CP-
violating phases from the Yukawa couplings. In contrast, $\bm{N^{\prime}}$ is
typically lighter than the SUSY-breaking scale. As we will show, because of
its small mass and strong interaction with the thermal bath, $\bm{N^{\prime}}$
remains in equilibrium and its inverse decays exponentially wash out the
lepton asymmetry produced by $\bm{N}$ decays. Therefore, we can ignore the
$\bm{N}$ contribution to the asymmetry and focus on that of $\bm{N^{\prime}}$.

When $T$ drops below $m_{\bm{N^{\prime}}}\cong M^{2}/\mu$, the
$\bm{N^{\prime}}$ states fall out of equilibrium and a net lepton asymmetry
results. Within the MFV hypothesis, CP violation in (s)neutrino decays comes
from the one-loop mixing between $\bm{N^{\prime}}$
($\bm{{\widetilde{N}}^{\prime}}$) flavors. The one-loop diagrams are
illustrated in Fig. 2. The detailled calculation of the CP asymmetry will be
presented in section IV.2.

Ignoring corrections to the mass eigenvalues, $\bm{N^{\prime}_{i}}$ and
$\bm{{\widetilde{N}}^{\prime}_{i}}$ states have the same decay rates into
leptons, sleptons and their conjugates. The total decay rate is given by

$\Gamma_{i}=\frac{1}{4\pi}\left(YY^{\dagger}\right)_{ii}\left(\frac{M^{4}}{\mu^{3}}\right).$
(36)

Figure 3: Mass generation of the light neutrinos.

###  IV.1 Loop-induced neutrino masses

To fully determine the parameters consistent with successful leptogenesis, we
must relate the Yukawa couplings to the observed light neutrino masses. The
mass matrix in the basis $(\nu,\,N^{\prime},\,N)$ takes the form

$\left(\begin{array}[]{ccc}0&0&m_{D}\\\ 0&0&M\\\
m_{D}^{T}&M&\mu\end{array}\right)\,,$ (37)

where $m_{D,i\alpha}=Y_{i\alpha}\langle H_{u}\rangle\equiv Y_{i\alpha}v_{u}$,
$\mu_{ij}=\lambda^{\prime}\langle\widetilde{X}\rangle\delta_{ij}$ and $M$
denotes the matrix $M_{ij}$ from Eq. (2). We recall our assumption that
$|\mu|\gg|M|$. The lightest neutrinos remain massless with this matrix.
However at one loop, a direct Majorana mass $m_{\nu}$ is generated for the
$\nu$ states by the diagrams shown in Fig. 3. If $M\gg m_{D}$, mixing between
$\nu$ and $N^{\prime}$ can be neglected, and the light neutrino mass matrix is
given by

$\displaystyle m_{\nu}$ $\displaystyle=$
$\displaystyle{g_{2}^{2}\,Y^{T}Y\,(v_{u}^{2}/\mu)\over
8\pi^{2}\cos^{2}\theta_{W}}\left({\ln(\mu/m_{Z})\over
1-m_{Z}^{2}/\mu^{2}}+{\lambda^{\prime}F_{X}\,\over
16\,\mu\,m_{\tilde{Z}}}f(s_{m})\right)\,,$ $\displaystyle f(x)$
$\displaystyle=$ $\displaystyle
x^{2}(1-x^{2}+x^{2}\ln(x^{2}))\over(1-x^{2})^{2}\,,$ (38)

where $g_{2}$ is the SU(2) gauge coupling, $m_{\tilde{Z}}$ is the Zino mass,

$m_{\tilde{Z}}=N_{m}\frac{g_{2}^{2}}{(4\pi)^{2}}\frac{F_{X}}{\langle\widetilde{X}\rangle}s_{m}C_{RG}\,,$
(39)

including gaugino screening by the factor $s_{m}$, $N_{m}=2$ sets of messenger
fields and RG running correction $C_{RG}$ as in Eq. (6). We have approximated
$m_{\tilde{Z}}\cong m_{\tilde{\nu}}\,s_{m}$, and that $M$ is proportional to
the unit matrix in the $N$-$N^{\prime}$ flavor space. We find that $m_{\nu}$
is dominated by the first diagram, with the second making a correction of
$\sim 8\,\%$ for a screening factor of $s_{m}\sim 0.1$.

In addition to the previous assumptions, and the near-degeneracy of the heavy
RH neutrino $\bm{N_{i}}$ we consider the SM neutrinos to be hierarchical. To
relate light neutrino masses with the parameters of our model, we introduce a
Casas-Ibarra parametrization of the Yukawa matrix Casas:2001sr ; Ibarra:2003up
; Garbrecht:2014aga . Because our model only contains two sterile neutrino
flavors, the lightest neutrino is exactly massless. We will consider the
normal (NH) and inverted (IH) hierarchies, for which $m_{1}\cong 0$ and
$m_{3}\cong 0$ respectively. One obtains

$\displaystyle
Y_{i\alpha}=\sqrt{C\frac{\mu}{v_{u}^{2}}}\,\mathcal{R}\,D_{\sqrt{m_{\nu}}}\,U_{\nu}^{\dagger}$
(40)

where we have regrouped all numerical coefficients into $C\approx 9.7$,
$U_{\nu}$ is the PMNS matrix and $D_{\sqrt{m_{\nu}}}$ is the diagonal matrix

$\displaystyle D_{\sqrt{m_{\nu}}}$ $\displaystyle\,={\rm
diag}(0,\sqrt{m_{2}},\sqrt{m_{3}})\qquad{\rm(NH)},$ $\displaystyle
D_{\sqrt{m_{\nu}}}$ $\displaystyle\,={\rm
diag}(\sqrt{m_{1}},\sqrt{m_{2}},0)\qquad{\rm(IH)}.$ (41)

The $2\times 3$ matrix $\mathcal{R}$ contains a $2\times 2$ complex orthogonal
submatrix,

$\displaystyle\mathcal{R}$
$\displaystyle\,=\left(\begin{array}[]{ccc}\phantom{-}0&\phantom{-}\cos\hat{\varrho}&\sin\hat{\varrho}\\\
\phantom{-}0&-\sin\hat{\varrho}&\cos\hat{\varrho}\end{array}\right),\qquad{\rm(NH)}$
(44) $\displaystyle\mathcal{R}$
$\displaystyle\,=\left(\begin{array}[]{ccc}\phantom{-}\cos\hat{\varrho}&\sin\hat{\varrho}&\phantom{-}0\\\
-\sin\hat{\varrho}&\cos\hat{\varrho}&\phantom{-}0\end{array}\right),\qquad{\rm(IH)},$
(47)

where $\hat{\varrho}\equiv a+ib$ is a complex parameter. Using Eq. (IV.1), one
can estimate the numerical values of the Yukawa couplings required to yield
neutrino masses in agreement with observations,

$\displaystyle Y^{T}Y$ $\displaystyle=$ $\displaystyle 1.6\times
10^{-6}\left(\frac{\mu}{100\ \rm PeV}\right)$ (48) $\displaystyle\times$
$\displaystyle\left(\frac{m_{\nu}}{0.05\ \rm
eV}\right)\left(\frac{1}{\sin^{2}\beta}\right)$

where $\tan\beta=v_{u}/v_{d}$ and we used $v_{u}^{2}+v_{d}^{2}=(174\
\rm{GeV})^{2}$.

The matrix $YY^{\dagger}$ will be central in our analysis of leptogenesis, so
it is convenient to express it using the Casas-Ibarra parametrization,

$\displaystyle YY^{\dagger}$ $\displaystyle\equiv$ $\displaystyle
C\frac{\mu}{2v_{u}^{2}}\,\mathcal{M}$ (49) $\displaystyle=$ $\displaystyle
C\frac{\mu}{2v_{u}^{2}}\left(\begin{array}[]{cc}\phantom{-}\delta\,{\rm
c}_{2a}+\sigma\,{\rm ch}_{2b}&-\delta\,{\rm s}_{2a}+i\sigma\,{\rm sh}_{2b}\\\
-\delta\,{\rm s}_{2a}-i\sigma\,{\rm s}_{2b}&-\delta\,{\rm c}_{2a}+\sigma\,{\rm
ch}_{2b}\\\ \end{array}\right),$ (52)

where $\sigma=m_{2}+m_{3}$ ($m_{1}+m_{2})$ and $\delta=m_{2}-m_{3}$
($m_{1}-m_{2})$ for NH (IH). In this expression, $a$ and $b$ are the real and
imaginary parts of the complex angle $\hat{\varrho}$ in Eq. (47), and ${\rm
c}_{x}=\cos x,{\rm s}_{x}=\sin x,{\rm ch}_{x}=\cosh x,{\rm sh}_{x}=\sinh x$.
Note that the PMNS matrix doesn’t enter this expression and therefore has no
direct impact on leptogenesis.

###  IV.2 CP asymmetry

Because of the tiny mass splitting $\sim M^{\prime}$ between sterile neutrino
flavors, one-loop flavor mixing is resonantly enhanced. This makes the self-
energy diagrams of Fig. 2 the dominant contribution to CP asymmetry in
$\bm{N^{\prime}}$ and $\bm{{\widetilde{N}}^{\prime}}$ decays, which is often
referred to as $\varepsilon$-type CP violation. Vertex diagrams, or
$\varepsilon^{\prime}$-type CP violation, can therefore be neglected in our
model. Fong:2011yx ; Adhikari:2015ysa ; Pilaftsis:2003gt ; Pilaftsis:1997jf

Two types of sneutrino mixing can lead to CP violation:
$\bm{{\widetilde{N}}^{\prime}_{i,\pm}}$ with
$\bm{{\widetilde{N}}^{\prime}_{j,\pm}}$ (same sign), and
$\bm{{\widetilde{N}}^{\prime}_{i,\pm}}$ with
$\bm{{\widetilde{N}}^{\prime}_{j,\mp}}$ (opposite sign). In the former case,
the mass squared difference is of order (_cf._ Eq. (32)

$\delta m_{ij}^{2}\cong 8M^{2}\left(M\over\mu\right)^{2}\left(M^{\prime}\over
M\right),$ (53)

which is much smaller than the splitting of the second mixing,
$\sim\lambda^{\prime}F_{X}(M/\mu)^{2}$. Therefore, the second mixing is not as
resonant as the first one, and we can ignore CP violation coming from the
latter.

Using the resummation approach for unstable particle propagators described in
DAmbrosio:2003nfv ; Pilaftsis:1997jf ; Fong:2011yx ; Pilaftsis:2003gt , the
one-loop amplitude for the sneutrino decays
$\bm{{\widetilde{N}}_{i}^{\prime}}\to a_{\alpha}$, with
$a_{\alpha}={\widetilde{L}}_{\alpha}H_{u}$ or $L_{\alpha}{\widetilde{H}}_{u}$
is

$\hat{\mathcal{A}}^{a_{\alpha}}_{i}=A_{i}^{a_{\alpha}}-i\sum_{j\neq
i}A_{j}^{a_{\alpha}}\frac{\widetilde{\Sigma}^{\rm(abs)}_{ji}}{p^{2}-M_{j}^{2}+i\widetilde{\Sigma}^{\rm(abs)}_{jj}}$
(54)

In this expression, $A_{i}^{a_{\alpha}}$ is the tree-level amplitude of the
$\bm{{\widetilde{N}}_{i}^{\prime}}\to a_{\alpha}$ decay and
$\widetilde{\Sigma}_{ji}^{\rm(abs)}$ is the absorptive part of the
$\bm{{\widetilde{N}}_{i}^{\prime}}\to\bm{{\widetilde{N}}_{j}^{\prime}}$ self-
energy, which we evaluated with the Cutkosky rules Cutkosky:1960sp .

For neutrino decays $\bm{N_{i}^{\prime}}\to b_{\alpha}$, where
$b_{\alpha}=L_{\alpha}H_{u}$ or ${\widetilde{L}}_{\alpha}{\widetilde{H}}_{u}$,
the one-loop amplitude is given by a similar expression,

$\hat{\mathcal{A}}^{a_{\alpha}}_{i}=\overline{u}_{b_{\alpha}}P_{R}\left\\{h_{i\alpha}-i\sum_{j\neq
i}\frac{h_{j\alpha}}{\not{p}-M_{j}+i\Sigma^{\rm(abs)}_{jj}}\Sigma_{ji}^{\rm(abs)}\right\\}u_{\bm{N^{\prime}_{i}}}.$
(55)

Here, $\Sigma_{ji}^{\rm(abs)}$ is the absorptive part of the
$\bm{N_{i}^{\prime}}\to\bm{N_{j}^{\prime}}$ self-energy and $h_{i\alpha}$ is
the tree-level coupling between $\bm{N^{\prime}_{i}}$ and the final state
$b_{\alpha}$, _cf._ Eq. (35).

One can define the CP asymmetry parameters as

$\epsilon_{i}=\frac{\sum_{a_{\alpha},\alpha}\Gamma(\bm{N^{\prime}_{i}}\to
a_{\alpha})-\Gamma(\bm{N^{\prime}_{i}}\to
a_{\alpha}^{*})}{\sum_{a_{\alpha},\alpha}\Gamma(\bm{N^{\prime}_{i}}\to
a_{\alpha})+\Gamma(\bm{N^{\prime}_{i}}\to a_{\alpha}^{*})}$ (56)

for neutrinos and

$\widetilde{\epsilon}_{i}=\frac{\sum_{b_{\alpha},\alpha}\Gamma(\bm{{\widetilde{N}}^{\prime}_{i}}\to
b_{\alpha})-\Gamma(\bm{{\widetilde{N}}^{\prime}_{i}}\to
b_{\alpha}^{*})}{\sum_{b_{\alpha},\alpha}\Gamma(\bm{{\widetilde{N}}^{\prime}_{i}}\to
b_{\alpha})+\Gamma(\bm{{\widetilde{N}}^{\prime}_{i}}\to b_{\alpha}^{*})}$ (57)

for sneutrinos. Here, $a_{\alpha}^{*}$ and $b_{\alpha}^{*}$ are the CP
conjugates of the final states $a_{\alpha}$, $b_{\alpha}$.

At one-loop order, we find that neutrinos and sneutrinos have the same CP
asymmetry:

$\epsilon_{i}=\widetilde{\epsilon}_{i}=\frac{1}{2}\sum_{i\neq
j}\frac{\imaginary\left[(YY^{\dagger})^{2}_{ij}\right]}{(YY^{\dagger})_{ii}(YY^{\dagger})_{jj}}\frac{(m_{i}\Gamma_{j})\,(\delta
m^{2}_{ij})}{(\delta m^{2}_{ij})^{2}+(m_{i}\Gamma_{j})^{2}}\,.$ (58)

This expression is the same as in resonant leptogenesis Pilaftsis:2003gt ;
Garbrecht:2014aga . Significantly, because all decay channels of a given state
contribute with the same sign to the CP asymmetry, thermal factors of the
decay product cancel out in the expression for $\epsilon_{i}$, which implies
the CP asymmetry survives even in the $T\to 0$ limit. This is to be contrasted
with the standard case of soft leptogenesis where bosonic and fermionic decay
channels have opposite CP asymmetry and leptogenesis requires thermal effects
to avoid exact cancellation between them DAmbrosio:2003nfv ; Fong:2011yx ;
Adhikari:2015ysa .

In the absence of SUSY breaking, the resonance condition
$m_{i}\Gamma_{j}\sim\delta m_{ij}^{2}$ would be satisfied assuming $Y^{2}\sim
M^{\prime}/M$, as can by seen by comparing Eqs. (32), (33), (36) and (53). In
that case, the fraction $(m_{i}\Gamma_{j})\delta m^{2}_{ij}/[(\delta
m^{2}_{ij})^{2}+(m_{j}\Gamma_{j})^{2}]$ is maximized, leading to
$\epsilon_{i}\sim 1$. This would be an example of resonant leptogenesis.

However, in our model the breaking of SUSY suppresses the decay width
$m_{i}\Gamma_{j}$ by $(M/\mu)^{4}$ (cf., Eq. (36)) while $\delta m^{2}$ is
suppressed by $(M/\mu)^{2}$ only (cf., Eq. (53)).

With $M^{\prime}/M=10^{-7}$, we find that $\delta m_{ij}\gg m_{i}\Gamma_{j}$
and the resonant enhancement is not maximal, but it is still sufficient to
yield successful leptogenesis at the PeV scale, as we will show. We can
rewrite Eq. (58) in terms of the leptogenesis scale, $M^{2}/\mu$,

$\epsilon_{i}\simeq\frac{C}{64\pi\,v_{u}^{2}}\left({M\over
M^{\prime}}\right)\left(\frac{M^{2}}{\mu}\right)\sum_{j\neq
i}\frac{\absolutevalue{\mathcal{M}}^{2}_{ij}\sin(2\phi_{ij})}{(\mathcal{M})_{ii}},$
(59)

where $\phi_{ij}=\arg(\mathcal{M}_{ij})$. We recall that $C\approx 9.7$ and
$\mathcal{M}$ is the matrix introduced in the Casas-Ibarra parametrization of
$YY^{\dagger}$, Eq. (49). Assuming $M^{2}/\mu\approx 1$ PeV,
$\mathcal{M}\approx 0.05$ eV and $M^{\prime}/M=10^{-7}$, and barring a strong
hierarchy between on- and off-diagonal entries of the matrix $\mathcal{M}$ (or
equivalently, of $YY^{\dagger}$), one could therefore expect the CP asymmetry
to be of order $\epsilon_{i}\sim 10^{-3}$. Although this is not as large as
what standard resonant leptogenesis models can achieve, this is sufficient to
produce the observed baryon asymmetry as we will now show.

###  IV.3 Evolution of lepton asymmetry

(a) (b)

Figure 4: (a) Evolution of the (s)neutrino abundance $Y_{N}$
($Y_{\widetilde{N}}$) and the baryon asymmetry as a function of
$z=m_{i}/T=(M^{2}/\mu)/T$ for a benchmark model with washout parameter
$K=1000$. (b) Final baryon asymmetry for varying washout parameter $K$ and CP
asymmetry $\epsilon$. The light (dark) blue horizontal band illustrates the 95
% C.L. limits set by BBN (CMB) data., cf. Eq. (29).

The initial asymmetry generated by the decays is partially washed out by
scattering and inverse decay processes. The strength of the latter is
characterized by a washout parameter, defined as

$K_{i}=\frac{\Gamma_{i}}{H_{i}}\,,$ (60)

where $\Gamma_{i}$ is given in Eq. (36) and
$H_{i}=1.66\sqrt{g_{*}}\,m_{i}^{2}/M_{\rm pl}$ is the Hubble rate evaluated at
$T=m_{i}$, with Planck mass $M_{\rm pl}=1.22\times 10^{19}$ GeV. Numerically
estimating the value of $K_{i}$ yields

$K_{i}=1000\left(\frac{(YY^{\dagger})_{ii}}{1.6\times
10^{-6}}\right)\left(\frac{100\ {\rm
PeV}}{\mu}\right)\left(\frac{230}{g_{*}}\right)^{1/2},$ (61)

which is in the strong washout regime, $K_{i}\gg 1$.

In the strong washout regime, inverse decays of right-handed neutrinos remain
in equilibrium down to temperatures well below their masses. This implies
significant suppression of the final lepton asymmetry, but it also allows for
several simplifying assumptions:

  * •

the final lepton asymmetry is independent of the initial right-handed neutrino
abundances because they rapidly come into thermal equilibrium;

  * •

any lepton asymmetry coming from the decays of heavy $\bm{N}$ and
$\bm{{\widetilde{N}}}$ (whose mass is $\mu\gg M,\sqrt{F}$) is exponentially
suppressed and can be ignored relative to those of the lighter
$\bm{N^{\prime}}$ and $\bm{{\widetilde{N}}^{\prime}}$;

  * •

thermal effects and CP violation from $2\to 2$ scattering processes can be
ignored, as those are significant only at large temperature Buchmuller:2004nz
.

The coupled Boltzmann equations for the evolution of the right-handed
neutrino, sneutrino and lepton number abundances, $Y_{X}=n_{X}/s$, are

$\displaystyle\frac{dY_{\bm{N^{\prime}_{i}}}}{dz}$
$\displaystyle=-K_{i}z\frac{\mathcal{K}_{1}(z)}{\mathcal{K}_{2}(z)}\left(Y_{\bm{N^{\prime}_{i}}}-Y_{\bm{N^{\prime}_{i}}}^{\rm
eq}\right)$ (62)
$\displaystyle\frac{dY_{\bm{{\widetilde{N}}}^{\prime}_{i}}}{dz}$
$\displaystyle=-K_{i}z\frac{\mathcal{K}_{1}(z)}{\mathcal{K}_{2}(z)}\left(Y_{\bm{{\widetilde{N}}}^{\prime}_{i}}-Y_{\bm{{\widetilde{N}}}^{\prime}_{i}}^{\rm
eq}\right)$ (63) $\displaystyle\frac{dY_{L}}{dz}$
$\displaystyle=\sum_{i}\left[\epsilon_{i}K_{i}z\frac{\mathcal{K}_{1}(z)}{\mathcal{K}_{2}(z)}\left(Y_{\bm{N^{\prime}_{i}}}-Y_{\bm{N^{\prime}_{i}}}^{\rm
eq}+Y_{\bm{{\widetilde{N}}}^{\prime}_{i}}-Y_{\bm{{\widetilde{N}}}^{\prime}_{i}}^{\rm
eq}\right)\right.$ (64) $\displaystyle\left.-2\
\frac{K_{i}z^{3}}{4}\mathcal{K}_{1}(z)Y_{L}\right]$

where $z=m/T=(M^{2}/\mu)/T$, indicating our approximation that all states have
the same mass, ${\cal K}_{i}(x)$ are the modified Bessel functions of the
second kind, and the equilibrium abundances in the Maxwell-Boltzmann
approximation are given by $Y^{\rm
eq}=45/(4\pi^{4}g_{*})z^{2}\mathcal{K}_{2}$.

In Fig. 4(a) we show the numerical solutions of the Boltzmann equations
assuming the two right handed neutrino flavors have the same washout parameter
$K_{i}=1000$ and the same CP asymmetry $\epsilon_{i}$. The initial conditions
we used were
$Y_{\bm{N}^{\prime}_{i}}=Y_{\bm{{\widetilde{N}}}^{\prime}_{i}}=Y^{\rm eq}$ and
$Y_{L}=10^{-12}$, from the initial decays of the heavy states. The final
values relevant for the observed baryon asymmetry are however quite
insensitive to the initial conditions, which is characteristic of the strong
washout regime. In Fig. 4(b), the resulting baryon asymmetry as a function of
$K$ is plotted for several values of $\epsilon$. Numerically, we obtain that
the solution $Y_{L}$ scales as

$Y_{L}\sim 10^{-4}\frac{\epsilon}{K}$ (65)

up to a logarithmic dependence on $K$, which agrees with the analytical
estimate found in Ref. Buchmuller:2004nz  for the strong washout regime. These
estimates show that we can obtain the observed asymmetry with $\epsilon\sim
10^{-3}$.

(a) (b)

Figure 5: Contour plot of the leptogenesis scale $M^{2}/\mu$ that yields the
observed baryon asymmetry for a given complex angle $\hat{\varrho}=a+ib$ in
the $\mathcal{R}$ matrix of Eq. 47. The left and right panels show scenarios
with normal and inverted light neutrino hierarchy, respectively. Black dashed
curves show the contour lines of some benchmark values. White dotted curves
show parameters corresponding to models 1 (left) and 2 (right) of Table 2.

Combining the Casas-Ibarra parametrization of $YY^{\dagger}$ (49) with the
expression for the washout parameter (61) and the CP asymmetry (59), one finds
that the final lepton asymmetry depends on a handful of parameters, namely the
light neutrino masses, the leptogenesis scale $M^{2}/\mu$, the mass splitting
parameter $M^{\prime}/M$ and the complex angle $\hat{\varrho}=a+ib$ that
enters the matrix $\mathcal{R}$.

The contour plots of Fig. 5 show the leptogenesis scale $M^{2}/\mu$ that
yields the experimental baryon asymmetry, $Y_{B}=8.7\times 10^{-11}$, for
given values of $a$ and $b$, and in the NH (a) and IH (b) scenarios. For the
NH case, a large region of the parameter space yields successful leptogenesis
at the PeV scale. The IH scenario requires a slightly larger scale,
$M^{2}/\mu\gtrsim 6$ PeV. This is because the imaginary part of
$YY^{\dagger}_{12}$ scales with the light neutrino mass difference, which is
smaller for IH.

Eq. (6) and the experimental bound $m_{\tilde{g}}\gtrsim 2.3$ TeV put a lower
bound on the ratio $F_{X}/\widetilde{X}$. We also recall that stability of the
${\widetilde{N}}={\widetilde{N}}^{\prime}=0$ vacuum requires
$\lambda^{\prime}F_{X}/M^{2}\approx\lambda^{\prime}F/M^{2}\leq 1$. Combining
these constraints, one obtains a lower bound on the mass scale of the
$\bm{N}^{\prime}_{i}$ states,

$\frac{M^{2}}{\mu}\gtrsim 700\ {\rm TeV}\ \left(0.1\over s_{M}\right)\
\left(\frac{M^{2}}{\lambda^{\prime}F_{X}}\right)$ (66)

where we used $g_{s}^{2}/4\pi\approx 0.12$. Eq. (66) can be understood as an
absolute lower bound for the scale of leptogenesis in our model, which is
orders of magnitude below the $\sim 10^{9}$ GeV Davidson-Ibarra bound in
standard leptogenesis Davidson:2002qv  and the $\sim 10^{7}$ GeV limit seen in
soft leptogenesis Fong:2011yx  and other models with radiatively-induced light
neutrino masses Ma:2006ci .

###  IV.4 Charged lepton flavor violation

SUSY-breaking models typically open the way to charged lepton flavor violating
processes which are highly constrained, such as $\mu\to e\gamma$ Casas:2001sr
; Hisano:1995cp ; Kashti:2004vj . These interactions are allowed by off-
diagonal entries in the slepton soft mass matrix,

$(m^{2}_{\widetilde{L}})_{\alpha\beta}\approx-\frac{3m_{0}^{2}}{8\pi^{2}}\,(Y^{\dagger}Y)_{\alpha\beta}\,\ln\left(M_{\rm
GUT}\over M\right),$ (67)

where $m_{0}$ is the universal slepton mass. The branching ratio of
$L_{\alpha}\to L_{\beta}\gamma$ decays is approximately given by

$\mathrm{BR}\left(L_{\alpha}\to
L_{\beta}\gamma\right)\approx\frac{\alpha^{3}}{G_{F}^{2}}\frac{|(m_{\widetilde{L}}^{2})_{\alpha\beta}|^{2}}{m_{\mathrm{SUSY}}^{8}}\tan^{2}\beta,$
(68)

where $\alpha$ is the fine structure constant, $G_{F}$ is the Fermi constant
and $m_{\mathrm{SUSY}}$ is the superpartner mass scale.

Assuming $m_{0}\approx m_{\mathrm{SUSY}}$, we can numerically estimate this
branching ratio in our leptogenesis model,

$\displaystyle\mathrm{BR}\left(L_{\alpha}\to L_{\beta}\gamma\right)$
$\displaystyle\approx$ $\displaystyle 4.5\times
10^{-19}\left(\frac{(Y^{\dagger}Y)_{\alpha\beta}}{1.6\times
10^{-6}}\right)^{2}$ (69) $\displaystyle\times\left(\frac{\mathrm{1\
TeV}}{m_{\rm SUSY}}\right)^{4}\left(\frac{\tan\beta}{10}\right)^{2},$

where we used $M_{\mathrm{GUT}}\approx 10^{16}$ GeV and $M\approx 10$ PeV.
This estimate is orders of magnitude below the current bound for lepton flavor
violation in muon decays, $\mathrm{BR}(\mu\to e\gamma)\lesssim 4.2\times
10^{-13}$ MEG:2016leq , and well below the $\sim 6\times 10^{-14}$ predicted
sensitivity for the MEG II experiment Meucci:2022qbh ; MEGII:2021fah . Charged
lepton flavor violation in our model is therefore too small to be observed in
the near future, which is a general consequence of the flavor blindness of
gauge-mediated SUSY breaking models Martin:1997ns ; Giudice:1998bp .

Model | $\sqrt{F}$ | $m$ | $\lambda$ | $g$ | $\sqrt{D}$ | $T_{n}$ | $\langle x\rangle_{T=T_{n}}$ | $\alpha$ | $\beta_{H}$ | $\sqrt{F_{X}}$ |  $m_{\tilde{g}}$ (TeV) | $\epsilon_{R}$ | $\lambda^{\prime}$ | $M$ | $\mu$ | $M^{2}/\mu$
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
1 | 30 | 50.7 | 2.14 | 0.1 | 84.0 | 26.9 | 769 | 0.429 | 76.8 | 24.6 | $2.49$ | $1.90\times 10^{-3}$ | 1.58 | 40 | 890 | 1.80
2 | 30 | 59.7 | 2.97 | 0.117 | 78.3 | 26.9 | 555 | 0.317 | 74.4 | 21.5 | $2.58$ | $2.69\times 10^{-3}$ | 0.623 | 40 | 258 | 6.20
Table 2: Parameters for two benchmark models with successful leptogenesis and
high GW production. All the dimensionful quantities are expressed in PeV
except for the gluino mass.

##  V Other cosmological signatures

In addition to providing a mechanism for leptogenesis, a first-order SUSY-
breaking phase transition may produce other cosmological signals that future
experiments can observe. In this section, we start by estimating the GW
spectrum generated by such a FOPT and compare it to the expected noise
spectrum of several proposed GW detectors. We then demonstrate that this SUSY-
breaking phase transition would be prone to producing primordial black holes.

###  V.1 Gravitational waves

(a) (b)

Figure 6: Intensity plot of maximal amplitude of the GW spectrum. The regions
(a) below or (b) left of the dashed lines would be detected by the indicated
experiment. We fixed (a) $\sqrt{F}=30\ \mathrm{PeV}$, $g=0.1$ and $\lambda
F/m^{2}=3/4$, and (b) $\sqrt{F}=30\ \mathrm{PeV}$, $\lambda F/m^{2}=3/4$ and
$gD/m^{2}=1/5$. The red signs $\times$ and $+$ show respectively the position
of the models 1 and 2 of Table 2.

To assess the observability of GWs produced in the FOPT, one considers the
spectrum $\Omega_{{\rm gw}}(f)$, which is the contribution per frequency
octave to the energy density in gravitational waves, i.e., $\int\Omega_{{\rm
gw}}(f)\ d(\log f)$, the fraction of energy density compared to the critical
density of the universe. In general, the spectrum can be separated into
contributions from the scalar fields, sound waves in the plasma and
magnetohydrodynamical turbulence. However, the scalar field contribution is
only important for runaway walls ($\gamma_{w}\rightarrow\infty$); Ref.
Bodeker:2017cim  showed that for ultrarelativistic walls, interactions with
gauge bosons create a pressure on the wall proportional to $\gamma_{w}$ which
prevents it from running away. Furthermore, the estimates for the
magnetohydrodynamical turbulence are uncertain and sensitive to the details of
the phase transition dynamics RoperPol:2019wvy , and are expected to be much
smaller than the contribution from sound waves. Hence, we only consider the
contribution from the latter.

The GW spectrum from sound waves observed today can be parameterized as
Hindmarsh:2020hop

$\Omega_{{\rm gw}}(f)=2.061F_{{\rm gw},0}\tilde{\Omega}_{\rm
gw}\frac{(HR)^{2}}{\sqrt{K}+HR}\,K^{2}\,C(f/f_{\rm p,0})\,,$ (70)

where $F_{{\rm gw},0}=3.57\times 10^{-5}\left({100}/{g_{*}}\right)^{1/3}$,
quantifies the decrease in GW energy from the expansion of the universe,
$R=(8\pi)^{1/3}v_{w}/{\beta}$ is the mean bubble radius at the time of
percolation,333We make the approximation $T_{*}\simeq T_{n}$, where $T_{*}$ is
the percolation temperature. $K=\kappa\alpha/({1+\alpha})$ is the kinetic
energy fraction, $C(s)=s^{3}\left({7}/({4+3s^{2}})\right)^{7/2}$ is a function
determined from simulations that approximate the spectrum’s shape, and the
peak frequency is

$f_{\rm p,0}=2.62\left(\frac{1}{HR}\right)\left(\frac{T_{n}}{100\
\mathrm{PeV}}\right)\left(\frac{g_{*}}{100}\right)^{1/6}\ \mathrm{Hz}\,.$ (71)

Furthermore, $g_{*}$ is the effective number of degrees of freedom, $\beta$
and $\alpha$ were given in Eqs. (17-19), $v_{w}$ is the wall
velocity,444Determining $v_{w}$ is a notoriously difficult problem; hence we
do not try to do a complete calculation here. Nevertheless, Refs.
Cline:2021iff ; Laurent:2022jrs  showed that for strong FOPTs ($\alpha\gtrsim
0.01$), the wall becomes ultrarelativistic. Since our model always yields
$\alpha\gtrsim 0.1$, $v_{w}\cong 1$ and we adopt this value.
$\tilde{\Omega}_{\rm gw}=0.012$ is a constant determined numerically which
represents the efficiency with which kinetic energy is converted into GWs, and
$\kappa\cong{\alpha}/({0.73+0.083\sqrt{\alpha}+\alpha})$ is the efficiency
with which vacuum energy is turned into kinetic energy.

Once the GW spectrum is known, it must be compared to the sensitivity of a
detector to assess its detectability. The signal-to-noise ratio (SNR) is
defined by

${\rm SNR}=\sqrt{\mathcal{T}\int_{f_{\rm min}}^{f_{\rm
max}}\\!df\left[\frac{\Omega_{\rm gw}(f)}{\Omega_{\rm sens}(f)}\right]^{2}},$
(72)

where $\Omega_{\rm sens}(f)$ denotes the sensitivity curve of the detector and
$\mathcal{T}$ is the duration of the mission. Whenever SNR is greater than a
given threshold ${\rm SNR}_{\rm thr}$, we conclude that the signal can be
detected. In general, the threshold depends upon the configuration of the
detector and can be complicated to compute, but the value ${\rm SNR}_{\rm
thr}=10$ is reasonable for most detectors. We compare the prediction (70) for
a range of parameters to several proposed GW detectors: the earth-based
detectors LIGO LIGOScientific:2014pky , ET Sathyaprakash:2012jk  and CE
LIGOScientific:2016wof , and the space-based detectors LISA Audley:2017drz ;
Robson:2018ifk , AEDGE AEDGE:2019nxb , DECIGO Seto:2001qf ; Yagi:2013du  and
BBO Crowder:2005nr ; Yagi:2013du . For each experiment, we assume ${\rm
SNR}_{\rm thr}=10$ and $\mathcal{T}=4\ \mathrm{years}$, and the sensitivity
curves can be found in the previous references.

We performed two scans of the parameter space at a scale $\sqrt{F}=30\
\mathrm{PeV}$ and computed the SNR for each detector. In both scans, $\lambda$
ranges from 1 to 7, with $D/F\in[6,13]$ in the first scan and $g\in[0.02,0.2]$
in the second. Fig. 6 shows the amplitude of the GW signal in the planes of
the varied parameters, and the regions of sensitivity of the future
experiments. It demonstrates that a large region of parameter space can be
probed by the proposed experiments, especially by the earth-based detectors ET
and CE. In contrast, no model is detectable by LIGO. Fig. 6 shows that the GW
amplitude is maximized at small couplings $\lambda\lesssim 2$, $g\lesssim 0.1$
and for $D/F\lesssim 8$. The amplitude is maximal at the boundary of the ‘No
nucleation’ region, where no solution to Eq. (III.1) exists. Close to this
boundary, the phase transitions have enhanced supercooling, which leads to a
stronger FOPT and consequently a larger GW amplitude.

Fig. 7 shows the peak-integrated sensitivity curves Schmitz:2020syl  (PISC) of
the detectors, along with the peak value of the GW spectrum for each model
appearing in Fig. 6. The PISC is defined in such a way that any GW signal
whose amplitude peaks above the PISC will be detected. It is therefore an
intuitive figure of merit for the sensitivity of a detector. At the scale
$\sqrt{F}=30\ \mathrm{PeV}$, the frequency range of the GW signal produced by
the FOPT coincides with the region of peak sensitivity of the earth-based
detectors. This allows them to probe a larger region of parameter space,
despite their lower sensitivity relative to space-based detectors. If we lower
the scale to $\sqrt{F}=3\ \mathrm{PeV}$, the peak frequency is also rescaled
by the same factor since all the dimensionful parameters are expressed as a
ratio of $\sqrt{F}$555Only the Hubble parameter $H$, which appears in Eq.
(14), has a different scaling relation $H\propto F/m_{P}$. However, $T_{n}$
only depends logarithmically on $H$, so it has a small effect and one still
approximately has $T_{n}\propto\sqrt{F}$., so we must have $f_{\rm
p,0}\propto\sqrt{F}$. The frequency range then becomes closer to the
sensitivity region of the space-based detectors. This highlights the
importance of having both types of detectors to cover a large range of
frequencies, and of the corresponding energy scales.

Figure 7: Peak-integrated sensitivity curves of the detectors (background) and
peak values of the GW signal (foreground) for the models used to generate Fig.
6. The signs $\times$ and $+$ show respectively the position of the models 1
and 2 of Table 2.

###  V.2 Primordial black holes

(a) (b)

Figure 8: $\alpha$ and $\beta_{H}$ intensity plots for the scans presented in
Fig. 6

Several recent studies suggest that primordial black holes (PBHs) can be
produced by collision of true-vacuum bubbles during a FOPT. One mechanism
relies on the energy stored in the wall to cause gravitational collapse when
neighboring bubbles collide Jung:2021mku . However, to reach the required
energy density, the bubbles must have a large radius, which implies
$\beta_{H}\ll 1$. In our model, obtaining such values of $\beta_{H}$ requires
significant fine tuning, since typically $\beta_{H}\sim 100$, and the smallest
value from the 2000 models represented in Fig. 6 was $\beta_{H}\sim 1$.

A more promising mechanism in the present context is mass gain by particles
across the bubble walls Baker:2021nyl ; Baker:2021sno ; Huang:2022him . It
requires a species that is initially light to acquire a large mass $\Delta
m\gg\gamma_{w}T$ during the FOPT. These particles do not have enough kinetic
energy to go through the wall and are trapped in the false vacuum. At the end
of the phase transition, they form false-vacuum bubbles that are compressed by
the vacuum pressure and can thereby lead to a gravitational collapse.

As argued in Section III, the pseudomodulus field $x$ naturally gets a large
VEV during the phase transition; hence it is generic to have large variation
of masses across the wall. For example, if $m\lesssim T$ and $\lambda\langle
x\rangle\gg\gamma_{w}T$, half of the messenger fields666For simplicity, we
only consider the contribution from the messengers, but the same mechanism
would work with the neutrinos $N_{i}$ if $M\lesssim T$ and
$\lambda^{\prime}\langle x\rangle\gg\gamma_{w}T$. would gain a large enough
mass to contribute to the collapse (the other half becoming light and not
trapped in the false-vacuum bubble).

Determining the final abundance and mass spectrum of the resulting PBHs is a
complicated task that is beyond the scope of this paper. Nevertheless, one can
follow the methodology of Ref. Baker:2021sno  to estimate that PBH production
should be efficient. For simplicity, we take the false-vacuum bubble at the
end of the phase transition to be spherical, with radius $r(t)$. One can show
that, for nonrelativistic walls,777We use the nonrelativistic limit to
simplify the analysis. In reality, the wall is ultrarelativistic and therefore
transmits more energy to the particles that reflect on it. Thus, the estimates
derived here are conservative and one should expect the actual PBH production
to be larger., the energy density of the heavy messengers inside the bubble
scales like $r(t)^{-4}$, assuming that none of them can escape the bubble.

For the bubble to collapse into a black hole, $r(t)$ must become smaller than
the Schwarzschild radius $r_{s}$:

$r(t)<r_{s}=2GE_{\rm tot},$ (73)

where $E_{\rm tot}$ is the total energy of the heavy messengers inside the
bubble. Using the previous scaling relation for the energy density, one can
show that this condition leads to

$\frac{r(t)}{r_{H}}<\sqrt{\frac{ag_{\Psi}}{g_{*}}}\left(\frac{r_{0}}{r_{H}}\right)^{2},$
(74)

where $g_{\Psi}=45$ ($g_{*}\cong 341.25$) is the heavy messenger (total)
effective number of degrees of freedom, $r_{0}=r(t_{0})$ is the initial bubble
radius and $r_{H}=1/H$ is the Hubble radius.

We have also introduced the parameter

$a\equiv\frac{\rho_{\Psi}(r_{0})}{\rho_{n}}\,,$ (75)

where $\rho_{n}=\pi^{2}g_{\Psi}T_{n}^{4}/30$ is the messenger’s thermal energy
density at the beginning of the phase transition and $\rho_{\Psi}(r_{0})$ is
the energy density when the false-vacuum bubbles form. In general, one expects
$a>1$ since when the bubbles form, there is already a nonnegligible fraction
of the universe in the true vacuum, which decreases the volume of the false
vacuum regions, and thereby increases their energy density. Assuming that the
energy scales like $V^{-4/3}$ and that the first false vacuum bubbles form at
the time of percolation, when the volume fraction remaining in the false
vacuum is $p_{f}\cong 0.71$ Rintoul_1997 , we estimate $a\cong
p_{f}^{-4/3}\cong 1.6$. For a typical initial size of $r_{0}=r_{H}$, the
bubble need only shrink by a factor of 2.2 to collapse.

One must still assess whether the bubble can shrink this much. A necessary
condition is that the net inward pressure remains positive while $r(t)>r_{s}$.
Assuming that the plasma remains in thermal equilibrium and that the bubble
shrinks adiabatically without losing particles, the thermal pressure which
opposes the contraction is given by

$P_{T}(r)=\frac{1}{3}\rho_{\Psi}(r)=\frac{a\rho_{n}}{3}\left(\frac{r_{0}}{r(t)}\right)^{4}.$
(76)

This must be compared to the vacuum pressure $\Delta V$ and the gravitational
pressure, both of which promote the collapse. For a uniform spherical bubble,
the gravitational energy is

$\displaystyle E_{G}$ $\displaystyle=$
$\displaystyle-\frac{3GE_{\Psi}^{2}}{5r}=-\frac{3G}{5r}\left(\frac{4}{3}\pi
r^{3}\right)^{2}(a\rho_{n})^{2}\left(\frac{r_{0}}{r}\right)^{8}$ (77)
$\displaystyle=$
$\displaystyle-\frac{2\pi}{5r^{3}}r_{0}^{8}H^{2}a^{2}\rho_{n}\left(\frac{g_{\Psi}}{g_{*}}\right)\,,$

yielding the gravitational pressure

$P_{G}=\frac{\partial E_{G}}{\partial
V}=\frac{3r_{0}^{8}}{10r_{H}^{2}r^{6}}\,a^{2}\rho_{n}\left(\frac{g_{\Psi}}{g_{*}}\right)\,.$
(78)

It follows that the net pressure $P_{\rm net}=\Delta V+P_{G}-P_{T}$ is
minimized for a false vacuum bubble of radius

$r_{\rm
min}=\frac{3}{2}\sqrt{\frac{3ag_{\Psi}}{5g_{*}}}\frac{r_{0}^{2}}{r_{H}}\cong
1.162\,r_{s}\,.$ (79)

Hence, the most stringent constraint does not arise from $r(t)$ reaching the
Schwarzschild radius $r_{s}$, but rather at the slightly larger radius $r_{\rm
min}$. Once a bubble becomes smaller than $r_{\rm min}$, the gravitational
pressure starts to dominate and collapse is inevitable. The condition on the
net inward pressure at $r_{\rm min}$ for collapse is

$P_{\rm net}(r_{\rm min})=\Delta
V-\frac{400}{6561\,a}\left(\frac{g_{*}}{g_{\Psi}}\right)^{2}\left(\frac{r_{H}}{r_{0}}\right)^{4}\rho_{n}>0\,.$
(80)

Neglecting the second term in the definition (19) of $\alpha$, one then
obtains the condition

$\alpha>\frac{400}{6561a}\left(\frac{g_{*}}{g_{\Psi}}\right)\left(\frac{r_{H}}{r_{0}}\right)^{4}\cong
0.29\left(\frac{r_{H}}{r_{0}}\right)^{4}$ (81)

for the formation of PBHs.

The criterion (81) implies that only very strong FOPTs can produce PBHs. For
weaker transitions, the thermal pressure grows too rapidly as the false-vacuum
bubble shrinks, and it eventually overcomes the vacuum and gravitational
pressures. It is also apparent that PBH production is favored in large initial
bubbles. Since one expects $r_{0}/r_{H}\sim 1/\beta_{H}$, $\beta_{H}$ should
not be too large. Intensity plots of $\alpha$ and $\beta_{H}$ are shown in
Fig. 8, that demonstrate the existence of extended regions with large
$\alpha$, close to the no-nucleation boundary. These correspond to the regions
of minimal $\beta_{H}$, which favors PBH production.

We emphasize that the heuristic analysis made here neglects several physical
effects. For example, for relativistic walls, the particles gain more energy
with each collision, making it easier to satisfy the Schwarzschild radius
criterion. On the other hand, several processes can reduce the number of
particles in the bubbles (e.g., decays into light messengers, annihilation
into $XX$ pairs, sufficiently energetic particles able to cross the wall),
which reduce the energy density. Such effects were investigated in Refs.
Baker:2021nyl ; Baker:2021sno . Furthermore, we made the approximation of
relativistic messengers, whereas they typically have a mass of $m/T_{n}\sim
1-2$. Recomputing the criterion (81) numerically with a finite mass, we find
that it scales approximately as $n^{-5/4}$, where $n\sim
g_{\Psi}T_{n}^{3}e^{-m/T_{n}}$ is the messenger number density. Therefore, PBH
production is strongly Boltzmann suppressed at high $m$, but the bound on
$\alpha$ remains reasonably low for $m/T_{n}\sim 1-2$. A quantitative
calculation of the final PBH abundance would require determining the
distribution of the $r_{0}$ and $a$ value, which is beyond the scope of this
paper.

Nevertheless, the criterion (81) is very general. As long as a few fundamental
conditions are satisfied—namely $\Delta m\gg\gamma_{w}T$, $v_{w}\ll 1$ and
that the number of particles stays roughly constant—efficient PBH production
should be predicted by a condition similar to (81), irrespective of the
details of the phase transition.

##  VI Conclusions

In this work we have developed the paradigm started in Ref. Craig:2020jfv ,
where the potential for first-order supersymmetry breaking phase transitions
to produce observable gravity waves was initially explored. A primary
challenge undertaken here was to extend the original framework to encompass
viable leptogenesis, to simultaneously explain the baryon asymmetry of the
universe. This proved to be more constraining than might be expected a priori,
due to our hypothesis that the asymmetry could be linked to the phase
transition.

In particular, we assumed that lepton number is broken by a single interaction
$\lambda^{\prime}XN_{i}N_{i}$ coupling heavy sterile neutrinos to the
pseudomodulus field in the superpotential $W$,888 The corresponding bare mass
term $MNN$ is forbidden in $W$ by $R$ symmetry. which leads to the heavy
neutrino mass $m_{N}=\lambda^{\prime}\langle\tilde{X}\rangle$ being correlated
with the scale of SUSY breaking, taken to be $\sim 10\,$PeV to get observable
gravity waves. This is too low for conventional leptogenesis. We found these
challenges could be overcome by introducing a second set of heavy neutrinos
$N^{\prime}_{i}$ that pair with $N_{i}$ to form Dirac states before SUSY
breaking (but become lighter than $N_{i}$ after SUSY breaking), and whose out-
of-equilibrium decays, along with those of the corresponding sneutrinos,
produce the lepton asymmetry. In this setup, light neutrino masses vanish at
tree level, but get generated at one loop via virtual $Z$-$N$ exchange. A
large enough lepton asymmetry is achieved by assuming minimal flavor violation
in the leptonic sector, which makes the heavy $N^{\prime}_{i}$ nearly
degenerate, leading to partially resonant leptogenesis. This occurs at the
scale $T\lesssim\sqrt{F}$, which is much lower than in conventional
leptogenesis.

A novel outcome is our proposal for neutrino mass generation at one loop, due
to the presence of additional right-handed neutrino species $N_{i}^{\prime}$
that cause the tree-level masses to vanish. The resulting neutrino mass
spectrum is similar to that provided by the seesaw mechanism, with an
effective right-handed neutrino mass that is parametrically larger than the
actual mass by a factor of $2\pi/\alpha_{w}$.

Ref. Craig:2020jfv  noted that the SUSY-breaking scale $\sqrt{F}$ is rather
narrowly constrained, since LHC limits on the gluino mass bound it from below,
while gravitino overproduction, combined with Big Bang Nucleosythesis, bounds
it from above. The LHC constraint is strengthened in our model, which predicts
a definite degree of gaugino screening, pushing the gluino mass close toward
its current limit. In this sense the model is quite predictive, requiring
$m_{\tilde{g}}$ to be not much higher than 2 TeV.

We have taken a preliminary step toward estimation of the primordial black
holes, by the mechanism of particles being trapped in the disappearing false
vacuum regions toward the end of the phase transition. A full study of the
spectrum of produced PBHs would require simulating the phase transition on a
lattice, which is beyond the scope of the present work. The criterion we
derived for which false vacuum bubbles would lead to PBH formation may be
useful in such a future investigation.

Acknowledgements. This work was supported by the Natural Sciences and
Engineering Research Council (NSERC) of Canada and the Fonds de recherche du
Québec Nature et technologies (FRQNT). We thank M. Baker, N. Craig, G.
Giudice, T. Han, E. Ma, R. Mohapatra and D. Redigolo for helpful
correspondence or feedback.

##  Appendix A One-loop potential and mass eigenvalues

One-loop corrections are given by the Coleman-Weinberg potential:

$V_{CW}(\phi_{i})=\sum_{j}(-1)^{F}\frac{g_{j}m_{j}^{4}(\phi_{i})}{64\pi^{2}}\left(\log(\frac{m_{j}^{2}(\phi_{i})}{Q^{2}})-c_{j}\right),$
(82)

where $F=0$ ($1$) for bosons (fermions), $g_{i}=1$ (2) for scalars (fermions),
$c_{i}$ = 3/2 for scalars and fermions in the $\overline{\rm MS}$
renormalization scheme, and $Q$ is the renormalization scale. We will take
$Q=m$. Here $\phi_{i}$ indicates the dependence of the mass eigenvalues on $x$
and the VEVs of the $U(1)_{D}$ messenger fields. The sum is taken over all
tree-level mass eigenvalues in our model.

Similarly, the thermal corrections to the potential are given by

$V_{T}(\phi_{i},T)=\frac{T^{4}}{2\pi^{2}}\sum_{j}(-1)^{F}g_{j}J_{B/F}\left(\frac{m_{j}^{2}(\phi_{i})}{T^{2}}\right),$
(83)

with the thermal functions

$J_{B/F}(z^{2})=\int_{0}^{\infty}dxx^{2}\log\left[1-(-1)^{F}\exp(-\sqrt{x^{2}+z^{2}})\right].$
(84)

At high temperature, these functions can be approximated to lowest order by
the following expansions:

$\displaystyle J_{B}(z^{2})$ $\displaystyle=$
$\displaystyle-\frac{\pi^{4}}{45}+\frac{\pi^{2}}{12}z^{2}+\cdots$
$\displaystyle J_{F}(z^{2})$ $\displaystyle=$
$\displaystyle\frac{7\pi^{4}}{360}-\frac{\pi^{2}}{24}z^{2}+\cdots$ (85)

To find the mass eigenvalues, we write the quadratic terms of the potential as

$\frac{1}{2}\begin{pmatrix}\phi^{*}&\phi\end{pmatrix}\mathbf{m_{s}^{2}}(\phi_{i})\begin{pmatrix}\phi\\\
\phi^{*}\end{pmatrix}$

and diagonalize the $x$-dependent matrix $\mathbf{m_{s}^{2}}(\phi_{i})$.

To get the fermionic mass eigenvalues, we must diagonalize the matrix

$(\mathbf{m_{f}})_{ij}=\left.\frac{\delta^{2}W}{\delta\Phi_{i}\delta\Phi_{j}}\right|_{\Phi_{i}\to\phi_{i}}.$

The notation $\Phi_{i}\to\phi_{i}$ means we replace the superfields with their
scalar components.

At tree-level, the components of $X$ and of all MSSM fields are massless, with
the exception of the Higgs doublets $H_{u},H_{d}$ that have a mass $\mu$ prior
to the breaking of SUSY. However, in the limit where $\mu$ is much smaller
than $m$ and $\sqrt{F}$, we can treat the Higgs doublets as massless.

Right-handed neutrinos. The sneutrino mass eigenvalues are given by:

$\displaystyle\left(m_{{\widetilde{N}}_{i},\pm}\right)^{2}$
$\displaystyle=M_{i}^{2}+\frac{1}{2}\left(\mu^{2}+\Delta_{i,\pm}^{2}\pm\lambda^{\prime}F_{X}\right),$
$\displaystyle\left(m_{{\widetilde{N}}^{\prime}_{i},\pm}\right)^{2}$
$\displaystyle=M_{i}^{2}+\frac{1}{2}\left(\mu^{2}-\Delta_{i,\pm}^{2}\pm\lambda^{\prime}F_{X}\right),$
(86)

where $\mu=\lambda^{\prime}x/\sqrt{2}$,
$F_{X}=|F-\lambda\phi_{2}\widetilde{\phi}_{2}|$ and

$\Delta_{i,\pm}^{2}=\sqrt{\mu^{4}+4\mu^{2}M_{i}^{2}\pm
2\mu^{2}(\lambda^{\prime}F_{X})+(\lambda^{\prime}F_{X})^{2}}.$ (87)

The neutrino masses are given by

$\displaystyle m_{N_{i}}$
$\displaystyle=\frac{1}{2}\left(\mu+\sqrt{\mu^{2}+4M_{i}^{2}}\right),$
$\displaystyle m_{N^{\prime}_{i}}$
$\displaystyle=\frac{1}{2}\left(\mu-\sqrt{\mu^{2}+4M_{i}^{2}}\right).$ (88)

In the limit $\mu\gg M_{i}$ the superfields $N_{i}$ and $N^{\prime}_{i}$ unmix
and split into heavy and light eigenstates, which are approximately given by

$\displaystyle\left(m_{{\widetilde{N}}_{i},\pm}\right)^{2}$
$\displaystyle\cong\mu^{2}+2M_{i}^{2}\pm\lambda^{\prime}F_{X}$
$\displaystyle\left(m_{{\widetilde{N}}^{\prime}_{i},\pm}\right)^{2}$
$\displaystyle\cong
M_{i}^{2}\left(\frac{M_{i}^{2}\pm\lambda^{\prime}F_{X}}{\mu^{2}}\right)$ (89)

for the scalars and

$\displaystyle m_{N_{i}}$
$\displaystyle\cong\mu\left(1+{M_{i}^{2}\over\mu^{2}}\right)\cong\mu$
$\displaystyle m_{N^{\prime}_{i}}$ $\displaystyle\cong-{M_{i}^{2}\over\mu}.$
(90)

for the fermions.

$SU(5)$ gauge mediators. The mass matrix of the $SU(5)$ gauge mediators is
very similar to that of the right-handed neutrinos. The scalar eigenvalues are

$\displaystyle\left(m_{\widetilde{5}_{M},\pm}\right)^{2}$
$\displaystyle=m^{2}+\frac{1}{2}\left[(\lambda
x)^{2}/2+\Delta_{5,\pm}^{2}\pm\lambda F_{X}\right],$
$\displaystyle\left(m_{\widetilde{\overline{5}}_{M},\pm}\right)^{2}$
$\displaystyle=m^{2}+\frac{1}{2}\left[(\lambda
x)^{2}/2-\Delta_{5,\pm}^{2}\pm\lambda F_{X}\right],$ (91)

where

$\Delta_{5,\pm}^{2}=\sqrt{(\lambda x)^{4}/4+2(\lambda x)^{2}m^{2}\pm(\lambda
x)^{2}(\lambda F_{X})+(\lambda F_{X})^{2}}.$ (92)

These are identical to the sneutrino eigenvalues with
$\lambda^{\prime}\to\lambda$ and $M_{i}\to m$.

Similarly the fermion eigenvalues are

$\displaystyle m_{{5}_{M}}$ $\displaystyle=\frac{1}{2^{3/2}}\left(\lambda
x+\sqrt{(\lambda x)^{2}+8m^{2}}\right),$ $\displaystyle
m_{{\overline{5}_{M}}}$ $\displaystyle=\frac{1}{2^{3/2}}\left(\lambda
x-\sqrt{(\lambda x)^{2}+8m^{2}}\right).$ (93)

$\bm{U(1)_{D}}$ gauge mediators. The mass matrix of the $\phi$ fields is
identical to that of the $5_{M}$ with the exception of the additional Fayet-
Iliopoulos contribution. This doesn’t affect the fermion mass matrix, so their
mass eigenvalues is equal to Eq. (A).

Unfortunately, solving for the mass squared eigenvalues of the scalars
requires finding the roots of a 4th order polynomial, which we cannot do
analytically. In the limit $g\ll 1$, those eigenvalues are also given by Eq.
(A). We shall use this approximation to estimate the loop corrections to the
scalar potential.

At leading order, we can simply shift the eigenvalues of the $SU(5)$
messengers by $\pm gD$. This is enough to see what happens in the limit of
large $x$: some eigenvalues which previously converged to $m^{2}\to 0$ will
instead converge to $m^{2}\to-gD$, that is, the model will have tachyons. At
this point the potential becomes unstable at $\phi=\tilde{\phi}=0$ and the
$U(1)$ messengers will get a VEV. This VEV will allow the scalar fields to
cancel out the $D$ term in the limit $x\to\infty$. In other words, the true
vacuum of the tree-level potential is the runaway solution $x\to\infty$. A
slightly more accurate estimate of the mass eigenvalues is given by Eqs.
(4.17)-(4.19) in Craig et al.

Large $x$ approximation. For large values of the pseudomodulus field $x$, most
of the masses become either very large ($\sim x$) or very small ($\sim 1/x$).
This allows one to derive simple approximations for the vacuum and thermal
corrections (82,83). To simplify the analysis, we will only consider the
contribution from the messengers. This is justified since they have a much
larger number of degrees of freedom than the neutrinos. Furthermore, we will
use the analytical formulas (A-A) for all the messenger masses, including the
$\phi$.

Summing Eq. (82) over all the mass eigenstates (A-A) and expanding to lowest
order in $1/x$, one obtains

$V_{CW}(x\rightarrow\infty)\cong\frac{3\lambda^{2}F^{2}}{8\pi^{2}}\log\left(\frac{\lambda^{2}x^{2}}{2m^{2}}\right),$
(94)

where we neglected constant terms. The calculation of the thermal potential is
slightly different, as large mass eigenstates are Boltzmann suppressed. One
can therefore only consider the light eigenstates with the high-temperature
approximation (A). The leading-order thermal correction is then

$V_{T}(x\rightarrow\infty,T)\cong\frac{3m^{4}T^{2}}{\lambda^{2}x^{2}}.$ (95)

## References

  * (1) LISA Collaboration, P. Amaro-Seoane et al., “Laser Interferometer Space Antenna,” arXiv:1702.00786 [astro-ph.IM].
  * (2) C. Caprini et al., “Science with the space-based interferometer eLISA. II: Gravitational waves from cosmological phase transitions,” JCAP 04 (2016) 001, arXiv:1512.06239 [astro-ph.CO].
  * (3) C. Caprini et al., “Detecting gravitational waves from cosmological phase transitions with LISA: an update,” JCAP 03 (2020) 024, arXiv:1910.13125 [astro-ph.CO].
  * (4) N. Seto, S. Kawamura, and T. Nakamura, “Possibility of direct measurement of the acceleration of the universe using 0.1-Hz band laser interferometer gravitational wave antenna in space,” Phys. Rev. Lett. 87 (2001) 221103, arXiv:astro-ph/0108011.
  * (5) S. Sato et al., “The status of DECIGO,” J. Phys. Conf. Ser. 840 no. 1, (2017) 012010.
  * (6) J. Crowder and N. J. Cornish, “Beyond LISA: Exploring future gravitational wave missions,” Phys. Rev. D 72 (2005) 083005, arXiv:gr-qc/0506015.
  * (7) G. M. Harry, P. Fritschel, D. A. Shaddock, W. Folkner, and E. S. Phinney, “Laser interferometry for the big bang observer,” Class. Quant. Grav. 23 (2006) 4887–4894. [Erratum: Class.Quant.Grav. 23, 7361 (2006)].
  * (8) G. C. Dorsch, S. J. Huber, T. Konstandin, and J. M. No, “A Second Higgs Doublet in the Early Universe: Baryogenesis and Gravitational Waves,” JCAP 05 (2017) 052, arXiv:1611.05874 [hep-ph].
  * (9) V. Vaskonen, “Electroweak baryogenesis and gravitational waves from a real scalar singlet,” Phys. Rev. D 95 no. 12, (2017) 123515, arXiv:1611.02073 [hep-ph].
  * (10) J. M. Cline, A. Friedlander, D.-M. He, K. Kainulainen, B. Laurent, and D. Tucker-Smith, “Baryogenesis and gravity waves from a UV-completed electroweak phase transition,” Phys. Rev. D 103 no. 12, (2021) 123529, arXiv:2102.12490 [hep-ph].
  * (11) A. J. Long, A. Tesi, and L.-T. Wang, “Baryogenesis at a Lepton-Number-Breaking Phase Transition,” JHEP 10 (2017) 095, arXiv:1703.04902 [hep-ph].
  * (12) P. Huang and K.-P. Xie, “Leptogenesis triggered by a first-order phase transition,” arXiv:2206.04691 [hep-ph].
  * (13) A. Dasgupta, P. S. B. Dev, A. Ghoshal, and A. Mazumdar, “Gravitational Wave Pathway to Testable Leptogenesis,” arXiv:2206.07032 [hep-ph].
  * (14) W. Adam and I. Vivarelli, “Status of searches for electroweak-scale supersymmetry after LHC Run 2,” Int. J. Mod. Phys. A 37 no. 02, (2022) 2130022, arXiv:2111.10180 [hep-ex].
  * (15) J. D. Wells, “PeV-scale supersymmetry,” Phys. Rev. D 71 (2005) 015013, arXiv:hep-ph/0411041.
  * (16) N. Craig, N. Levi, A. Mariotti, and D. Redigolo, “Ripples in Spacetime from Broken Supersymmetry,” JHEP 21 (2020) 184, arXiv:2011.13949 [hep-ph].
  * (17) A. Pilaftsis and T. E. J. Underwood, “Resonant leptogenesis,” Nucl. Phys. B 692 (2004) 303–345, arXiv:hep-ph/0309342.
  * (18) L. O’Raifeartaigh, “Spontaneous Symmetry Breaking for Chiral Scalar Superfields,” Nucl. Phys. B 96 (1975) 331–352.
  * (19) E. Ma, “Supersymmetric Model of Radiative Seesaw Majorana Neutrino Masses,” Annales Fond. Broglie 31 (2006) 285, arXiv:hep-ph/0607142.
  * (20) R. N. Mohapatra and N. Okada, “Affleck-Dine Leptogenesis with One Loop Neutrino Mass and strong CP,” arXiv:2207.10619 [hep-ph].
  * (21) S. P. Martin and M. T. Vaughn, “Two loop renormalization group equations for soft supersymmetry breaking couplings,” Phys. Rev. D 50 (1994) 2282, arXiv:hep-ph/9311340. [Erratum: Phys.Rev.D 78, 039903 (2008)].
  * (22) C. Cheung, A. L. Fitzpatrick, and D. Shih, “(Extra)ordinary gauge mediation,” JHEP 07 (2008) 054, arXiv:0710.3585 [hep-ph].
  * (23) Z. Komargodski and D. Shih, “Notes on SUSY and R-Symmetry Breaking in Wess-Zumino Models,” JHEP 04 (2009) 093, arXiv:0902.0030 [hep-th].
  * (24) T. Cohen, A. Hook, and B. Wecht, “Comments on Gaugino Screening,” Phys. Rev. D 85 (2012) 115004, arXiv:1112.1699 [hep-ph].
  * (25) S. P. Martin, “Generalized messengers of supersymmetry breaking and the sparticle mass spectrum,” Phys. Rev. D 55 (1997) 3177–3187, arXiv:hep-ph/9608224.
  * (26) S. P. Martin, “A Supersymmetry primer,” Adv. Ser. Direct. High Energy Phys. 18 (1998) 1–98, arXiv:hep-ph/9709356.
  * (27) ATLAS Collaboration, G. Aad et al., “Search for squarks and gluinos in final states with jets and missing transverse momentum using 139 fb-1 of $\sqrt{s}$ =13 TeV $pp$ collision data with the ATLAS detector,” JHEP 02 (2021) 143, arXiv:2010.14293 [hep-ex].
  * (28) A. D. Linde, “Fate of the False Vacuum at Finite Temperature: Theory and Applications,” Phys. Lett. B 100 (1981) 37–40.
  * (29) S. J. Huber and T. Konstandin, “Production of gravitational waves in the nMSSM,” JCAP 05 (2008) 017, arXiv:0709.2091 [hep-ph].
  * (30) C. L. Wainwright, “CosmoTransitions: Computing Cosmological Phase Transition Temperatures and Bubble Profiles with Multiple Fields,” Comput. Phys. Commun. 183 (2012) 2006–2013, arXiv:1109.4189 [hep-ph].
  * (31) R. E. Cutkosky, “Singularities and discontinuities of Feynman amplitudes,” J. Math. Phys. 1 (1960) 429–433.
  * (32) J. A. Harvey and M. S. Turner, “Cosmological baryon and lepton number in the presence of electroweak fermion number violation,” Phys. Rev. D 42 (1990) 3344–3349.
  * (33) Particle Data Group Collaboration, R. L. Workman, “Review of Particle Physics,” PTEP 2022 (2022) 083C01.
  * (34) Planck Collaboration, N. Aghanim et al., “Planck 2018 results. VI. Cosmological parameters,” Astron. Astrophys. 641 (2020) A6, arXiv:1807.06209 [astro-ph.CO]. [Erratum: Astron.Astrophys. 652, C4 (2021)].
  * (35) G. D’Ambrosio, G. F. Giudice, and M. Raidal, “Soft leptogenesis,” Phys. Lett. B 575 (2003) 75–84, arXiv:hep-ph/0308031.
  * (36) C. S. Fong, M. C. Gonzalez-Garcia, and E. Nardi, “Leptogenesis from Soft Supersymmetry Breaking (Soft Leptogenesis),” Int. J. Mod. Phys. A 26 (2011) 3491–3604, arXiv:1107.5312 [hep-ph].
  * (37) R. Adhikari, A. Dasgupta, C. S. Fong, and R. Rangarajan, “Nonthermal $CP$ violation in soft leptogenesis,” Phys. Rev. D 91 no. 9, (2015) 096001, arXiv:1501.06310 [hep-ph].
  * (38) J. A. Casas and A. Ibarra, “Oscillating neutrinos and $\mu\to e,\gamma$,” Nucl. Phys. B 618 (2001) 171–204, arXiv:hep-ph/0103065.
  * (39) A. Ibarra and G. G. Ross, “Neutrino phenomenology: The Case of two right-handed neutrinos,” Phys. Lett. B 591 (2004) 285–296, arXiv:hep-ph/0312138.
  * (40) B. Garbrecht, F. Gautier, and J. Klaric, “Strong Washout Approximation to Resonant Leptogenesis,” JCAP 09 (2014) 033, arXiv:1406.4190 [hep-ph].
  * (41) A. Pilaftsis, “CP violation and baryogenesis due to heavy Majorana neutrinos,” Phys. Rev. D 56 (1997) 5431–5451, arXiv:hep-ph/9707235.
  * (42) W. Buchmuller, P. Di Bari, and M. Plumacher, “Leptogenesis for pedestrians,” Annals Phys. 315 (2005) 305–351, arXiv:hep-ph/0401240.
  * (43) S. Davidson and A. Ibarra, “A Lower bound on the right-handed neutrino mass from leptogenesis,” Phys. Lett. B 535 (2002) 25–32, arXiv:hep-ph/0202239.
  * (44) E. Ma, N. Sahu, and U. Sarkar, “Leptogenesis below the Davidson-Ibarra bound,” J. Phys. G 32 (2006) L65–L68, arXiv:hep-ph/0603043.
  * (45) J. Hisano, T. Moroi, K. Tobe, and M. Yamaguchi, “Lepton flavor violation via right-handed neutrino Yukawa couplings in supersymmetric standard model,” Phys. Rev. D 53 (1996) 2442–2459, arXiv:hep-ph/9510309.
  * (46) T. Kashti, “Phenomenological consequences of soft leptogenesis,” Phys. Rev. D 71 (2005) 013008, arXiv:hep-ph/0410319.
  * (47) MEG Collaboration, A. M. Baldini et al., “Search for the lepton flavour violating decay $\mu^{+}\rightarrow\mathrm{e}^{+}\gamma$ with the full dataset of the MEG experiment,” Eur. Phys. J. C 76 no. 8, (2016) 434, arXiv:1605.05081 [hep-ex].
  * (48) MEG II Collaboration, M. Meucci, “MEG II experiment status and prospect,” PoS NuFact2021 (2022) 120, arXiv:2201.08200 [hep-ex].
  * (49) MEG II Collaboration, A. M. Baldini et al., “The Search for $\mu$\+ → e+$\gamma$ with 10–14 Sensitivity: The Upgrade of the MEG Experiment,” Symmetry 13 no. 9, (2021) 1591, arXiv:2107.10767 [hep-ex].
  * (50) G. F. Giudice and R. Rattazzi, “Theories with gauge mediated supersymmetry breaking,” Phys. Rept. 322 (1999) 419–499, arXiv:hep-ph/9801271.
  * (51) D. Bodeker and G. D. Moore, “Electroweak Bubble Wall Speed Limit,” JCAP 05 (2017) 025, arXiv:1703.08215 [hep-ph].
  * (52) A. Roper Pol, S. Mandal, A. Brandenburg, T. Kahniashvili, and A. Kosowsky, “Numerical simulations of gravitational waves from early-universe turbulence,” Phys. Rev. D 102 no. 8, (2020) 083512, arXiv:1903.08585 [astro-ph.CO].
  * (53) M. B. Hindmarsh, M. Lüben, J. Lumma, and M. Pauly, “Phase transitions in the early universe,” SciPost Phys. Lect. Notes 24 (2021) 1, arXiv:2008.09136 [astro-ph.CO].
  * (54) B. Laurent and J. M. Cline, “First principles determination of bubble wall velocity,” Phys. Rev. D 106 no. 2, (2022) 023501, arXiv:2204.13120 [hep-ph].
  * (55) LIGO Scientific Collaboration, J. Aasi et al., “Advanced LIGO,” Class. Quant. Grav. 32 (2015) 074001, arXiv:1411.4547 [gr-qc].
  * (56) B. Sathyaprakash et al., “Scientific Objectives of Einstein Telescope,” Class. Quant. Grav. 29 (2012) 124013, arXiv:1206.0331 [gr-qc]. [Erratum: Class.Quant.Grav. 30, 079501 (2013)].
  * (57) LIGO Scientific Collaboration, B. P. Abbott et al., “Exploring the Sensitivity of Next Generation Gravitational Wave Detectors,” Class. Quant. Grav. 34 no. 4, (2017) 044001, arXiv:1607.08697 [astro-ph.IM].
  * (58) T. Robson, N. J. Cornish, and C. Liu, “The construction and use of LISA sensitivity curves,” Class. Quant. Grav. 36 no. 10, (2019) 105011, arXiv:1803.01944 [astro-ph.HE].
  * (59) AEDGE Collaboration, Y. A. El-Neaj et al., “AEDGE: Atomic Experiment for Dark Matter and Gravity Exploration in Space,” EPJ Quant. Technol. 7 (2020) 6, arXiv:1908.00802 [gr-qc].
  * (60) K. Yagi, “Scientific Potential of DECIGO Pathfinder and Testing GR with Space-Borne Gravitational Wave Interferometers,” Int. J. Mod. Phys. D 22 (2013) 1341013, arXiv:1302.2388 [gr-qc].
  * (61) K. Schmitz, “New Sensitivity Curves for Gravitational-Wave Signals from Cosmological Phase Transitions,” JHEP 01 (2021) 097, arXiv:2002.04615 [hep-ph].
  * (62) T. H. Jung and T. Okui, “Primordial black holes from bubble collisions during a first-order phase transition,” arXiv:2110.04271 [hep-ph].
  * (63) M. J. Baker, M. Breitbach, J. Kopp, and L. Mittnacht, “Primordial Black Holes from First-Order Cosmological Phase Transitions,” arXiv:2105.07481 [astro-ph.CO].
  * (64) M. J. Baker, M. Breitbach, J. Kopp, and L. Mittnacht, “Detailed Calculation of Primordial Black Hole Formation During First-Order Cosmological Phase Transitions,” arXiv:2110.00005 [astro-ph.CO].
  * (65) P. Huang and K.-P. Xie, “Primordial black holes from an electroweak phase transition,” Phys. Rev. D 105 no. 11, (2022) 115033, arXiv:2201.07243 [hep-ph].
  * (66) M. D. Rintoul and S. Torquato, “Precise determination of the critical threshold and exponents in a three-dimensional continuum percolation model,” Journal of Physics A: Mathematical and General 30 no. 16, (Aug, 1997) L585–L592.