# Gravitational Wave Signatures of Gauged Baryon and Lepton Number

Jessica Bosch  Department of Chemistry and Physics, Barry University, Miami
Shores, Florida 33161, USA  Zoraida Delgado  Department of Chemistry and
Physics, Barry University, Miami Shores, Florida 33161, USA  Bartosz Fornal
Department of Chemistry and Physics, Barry University, Miami Shores, Florida
33161, USA  Alejandra Leon  Department of Chemistry and Physics, Barry
University, Miami Shores, Florida 33161, USA

###### Abstract

We demonstrate that novel types of gravitational wave signatures arise in
theories with new gauge symmetries broken at high energy scales. For
concreteness, we focus on models with gauged baryon number and lepton number,
in which neutrino masses are generated via the type I seesaw mechanism,
leptogenesis occurs through the decay of a heavy right-handed neutrino, and
one of the new baryonic fields is a good dark matter candidate. Depending on
the scalar content of the theory, the gravitational wave spectrum consists of
contributions from cosmic strings, domain walls, and first order phase
transitions. We show that a characteristic double-peaked signal from domain
walls or a sharp domain wall peak over a flat cosmic string background may be
generated. Those new signatures are within the reach of future experiments,
such as Cosmic Explorer, Einstein Telescope, DECIGO, Big Bang Observer, and
LISA.

##  I Introduction

In the Standard Model Glashow (1961); Higgs (1964); Englert and Brout (1964);
Weinberg (1967); Salam (1968); Fritzsch  _et al._ (1973); Gross and Wilczek
(1973); Politzer (1973), baryon number and lepton number are accidental global
symmetries of the Lagrangian of an unknown origin. Although one expects both
of those symmetries to be violated if grand unification is realized in Nature
Georgi and Glashow (1974); Fritzsch and Minkowski (1975), so far no sign of
related processes, such as proton decay, have been observed in experiments Abe
_et al._ (2017), which excludes the minimal non-supersymmetric version of
those theories. If grand unification does not happen, then global baryon and
lepton number may be a low-energy manifestation of some more fundamental gauge
symmetries unbroken at high energy scales. Indeed, this line of reasoning is
supported by the self-consistency of quantum theories of gravity, in which
only gauge symmetries can be properly accommodated, unless unnatural
conditions are introduced Kallosh  _et al._ (1995).

The first attempt to promote baryon and lepton number to the status of $\rm
U(1)$ gauge symmetries dates back to the 1970s Pais (1973), and was followed
by further theoretical efforts throughout the subsequent two decades Rajpoot
(1988); Foot  _et al._ (1989); Carone and Murayama (1995); Georgi and Glashow
(1996). However, the first phenomenologically viable model of this type was
constructed only recently in Fileviez Perez and Wise (2010), and later
modified to avoid all current experimental bounds Duerr  _et al._ (2013);
Fileviez Perez  _et al._ (2014). The idea of gauging baryon and lepton number
was later successfully incorporated into a supersymmetric framework Arnold
_et al._ (2013), theories unifying baryon number and color Fornal  _et al._
(2015); Fornal and Tait (2016), and generalized to a non-Abelian gauged lepton
number Fornal  _et al._ (2017). Models with gauged baryon and lepton numbers
have very attractive features: they explain the stability of the proton, have
a natural realization of the seesaw mechanism for neutrino masses, contain an
attractive baryonic dark matter candidate Duerr and Fileviez Perez (2015);
Ohmer and Patel (2015); Fileviez Perez  _et al._ (2019), and can accommodate
high scale leptogenesis Fileviez Perez  _et al._ (2021). Thus far, in all of
the existing ${\rm U}(1)$ formulations of theories with gauged baryon and
lepton number, each of the symmetries was broken by the vacuum expectation
value of a single scalar. However, there is no reason to expect that the
scalar sector is this minimal.

To this end, in this paper we investigate ways to probe the composition of
high-scale symmetry breaking sectors, i.e., when at least one of the two
sectors consists of more than one scalar breaking the symmetry. Although we
focus on the class of theories with gauged baryon and lepton number, most of
our analysis is general and can be applied to other theories with two broken
${\rm U}(1)$ gauge symmetries. Conventional particle physics experiments are
not able to differentiate between the two scenario, or even probe them at all
if the symmetry breaking scale is high. Nevertheless, as we demonstrate below,
gravitational wave detectors have opened up a completely new set of
opportunities to probe such models.

A renaissance period for gravitational wave physics was initiated by the first
direct detection of a gravitational wave signal coming from a black hole
merger by the Laser Interferometer Gravitational Wave Observatory (LIGO)
within the LIGO/Virgo collaboration Abbott  _et al._ (2016). By now, over one
hundred of such events, involving also neutron stars, have been recorded.
Those discoveries provide an ideal opportunity to test general relativity, but
they are not directly related to particle physics. The gravitational waves
which enable probing particle physics models, although not yet discovered, are
expected to come in the form of a stochastic gravitational wave background
produced in the early Universe by phenomena such as inflation Turner (1997),
first order phase transitions Kosowsky  _et al._ (1992), domain walls
Hiramatsu  _et al._ (2010) and cosmic strings Vachaspati and Vilenkin (1985a);
Sakellariadou (1990). Although for such signals to be detectable at LIGO the
underlying particle physics models require a large fine-tuning of parameters,
future gravitational wave experiments, such as the Laser Interferometer Space
Antenna (LISA) Amaro-Seoane  _et al._ (2017), Cosmic Explorer Reitze  _et al._
(2019), Einstein Telescope Punturo  _et al._ (2010), DECIGO Kawamura  _et al._
(2011), and Big Bang Observer Crowder and Cornish (2005), will be sensitive to
more generic scenarios.

The most model-independent stochastic gravitational wave background comes from
cosmic strings, which are topological defects formed via the Kibble mechanism
Kibble (1976) upon a spontaneous breaking of a ${\rm U}(1)$ symmetry. They
correspond to one-dimensional field configurations along the direction in
which the symmetry remains unbroken. The dynamics of the produced cosmic
string network provides a long-lasting source of gravitational radiation
resulting in a mostly flat stochastic gravitational wave background, with its
strength dependent only on the scale of the ${\rm U}(1)$ breaking. Cosmic
string signatures have been considered in the context of grand unified
theories Buchmuller  _et al._ (2019); King  _et al._ (2020), neutrino seesaw
models Blanco-Pillado and Olum (2017); Ringeval and Suyama (2017); Cui  _et
al._ (2018, 2019); Guedes  _et al._ (2018); Dror  _et al._ (2020); Zhou and
Bian (2020), new physics at the high scale Gouttenoire  _et al._ (2020a), as
well as baryon and lepton number violation Fornal and Shams Es Haghi (2020).
For a review of gravitational waves signatures of cosmic strings see
Gouttenoire  _et al._ (2020b), and for the constraints from LIGO/Virgo data
see Abbott  _et al._ (2021).

The other topological defects which can be produced in the early Universe are
domain walls, created when a $Z_{2}$ symmetry is spontaneously broken. They
are two-dimensional field configurations existing at the boundaries of regions
corresponding to different vacua. In order for domain walls not to overclose
the Universe, they need to annihilate away. This is possible when there exists
a small energy density difference between the two vacua (the so-called
potential bias). Domain wall annihilation leads to a stochastic gravitational
wave background which is peaked at some frequency, but its strength and the
peak frequency are, as in the case of cosmic strings, independent of the exact
particle physics details of the model – the spectrum is determined by only two
parameters: the scale of the symmetry breaking and the potential bias. Domain
wall signatures have been considered in many theoreies beyond the Standard
Model, including new electroweak scale physics Eto  _et al._ (2018a, b); Chen
_et al._ (2020); Battye  _et al._ (2020), supersymmetry Kadota  _et al._
(2015), axions Craig  _et al._ (2021); Blasi  _et al._ (2023), grand
unification Dunsky  _et al._ (2022), models with left-right symmetry Borah and
Dasgupta (2022), baryon/lepton number violation Fornal  _et al._ (2023), and
models of leptogenesis Barman  _et al._ (2022). The physics of domain walls
and the expected gravitational wave spectrum are reviewed in Saikawa (2017);
the bounds on domain walls from LIGO/Virgo data can be found in Jiang and
Huang (2022).

The most model-dependent gravitational wave signatures arise from cosmological
first order phase transitions. Those occur when the effective potential
develops a new minimum with a lower energy density than the high-temperature
one. If there exists a potential barrier between the two minima, the
transition is first order and bubbles of true vacuum are being nucleated in
various points in space. Such bubbles of true vacuum expand, eventually
filling up the entire Universe. Gravitational waves are emitted from bubble
collisions, turbulence, and sound shock waves in the primordial plasma
generated by the violent expansion of the bubbles. The position of the
gravitational wave peak is highly dependent on the temperature at which bubble
nucleation occurs. First order phase transitions have been analyzed in a
plethora of particle physics models, including, again, electroweak scale new
physics Grojean and Servant (2007); Vaskonen (2017); Dorsch  _et al._ (2017);
Bernon  _et al._ (2018); Chala  _et al._ (2018); Angelescu and Huang (2019);
Alves  _et al._ (2019); Han  _et al._ (2021); Benincasa  _et al._ (2022),
supersymmetry Craig  _et al._ (2020); Fornal  _et al._ (2021), axions Dev  _et
al._ (2019); Von Harling  _et al._ (2020); Delle Rose  _et al._ (2020), grand
unification Croon  _et al._ (2019); Huang  _et al._ (2020); Okada  _et al._
(2021), baryon/lepton number violation Hasegawa  _et al._ (2019); Fornal and
Shams Es Haghi (2020)), neutrino seesaw models Brdar  _et al._ (2019); Okada
and Seto (2018); Di Bari  _et al._ (2021); Zhou  _et al._ (2022), new flavor
physics Greljo  _et al._ (2020); Fornal (2021), dark gauge groups Schwaller
(2015); Breitbach  _et al._ (2019); Croon  _et al._ (2018); Hall  _et al._
(2020), models with conformal invariance Ellis  _et al._ (2020a); Kawana
(2022), and dark matter Baldes (2017); Azatov  _et al._ (2021); Costa  _et
al._ (2022a, b); Fornal and Pierre (2022); Kierkla  _et al._ (2023); Azatov
_et al._ (2022). A comprehensive review of gravitational waves from first
order phase transitions can be found in Caldwell  _et al._ (2022); Athron  _et
al._ (2023), while the most recent constraints from LIGO/Virgo data were
derived in Badger  _et al._ (2023). For recent progress on supercooled phase
transitions see Ellis  _et al._ (2019); Lewicki and Vaskonen (2020a, b); Ellis
_et al._ (2020a).

In this work, we examine how gravitational wave signals from domain walls,
cosmic strings, and phase transitions interplay with each other, producing
novel features in the expected spectrum. The two new gravitational wave
signatures which have not been considered in the literature so far are:$(1)$
Two coexisting signals from domain wall annihilation, forming a characteristic
sharp double-peak in the spectrum; $(2)$ Domain wall signal over a flat cosmic
string contribution, leading to an unusually-shaped peak. Although we focus on
a specific model, our results involving cosmic strings and domain walls, in
particular signatures $(1)$ and $(2)$, are general, since they do not depend
on the details of the model.

##  II The model

The gravitational wave signatures we propose to search for, as will be
discussed in Section VII, are anticipated in a large class of models with a
two-step symmetry breaking pattern. In this paper, to provide a concrete
realization of such scenarios, we focus on a model with gauged baryon and
lepton number, based on the gauge group

$\displaystyle{\rm SU}(3)_{c}\times{\rm SU}(2)_{L}\times{\rm
U}(1)_{Y}\times{\rm U}(1)_{B}\times{\rm U}(1)_{L}\ .$ (1)

Below we describe the possible symmetry breaking patterns in the model and the
new particles along with their masses.

### Scalar sector

As mentioned in Section I, we consider extending the usual single-scalar
symmetry breaking sector to possibly include two scalars per each symmetry
breaking. Each of the fields breaking the ${\rm U}(1)_{L}$ symmetry comes in
the representation

$\displaystyle\Phi_{Li}=(1,1,0,0,2)\ ,$ (2)

while each of the scalars breaking ${\rm U}(1)_{B}$ is

$\displaystyle\Phi_{Bi}=(1,1,0,3,3)\ .$ (3)

Within this framework, there are four possible cases:

  * $(a)$

$\Phi_{B}$ breaks ${\rm U(1)}_{B}$ and $\Phi_{L}$ breaks ${\rm U(1)}_{L}$,

  * $(b)$

$\Phi_{B1}$, $\Phi_{B2}$ break ${\rm U(1)}_{B}$ and $\Phi_{L}$ breaks ${\rm
U(1)}_{L}$,

  * $(c)$

$\Phi_{B}$ breaks ${\rm U(1)}_{B}$ and $\Phi_{L1}$, $\Phi_{L2}$ break ${\rm
U(1)}_{L}$,

  * $(d)$

$\Phi_{B1}$, $\Phi_{B2}$ break ${\rm U(1)}_{B}$ and $\Phi_{L1}$, $\Phi_{L2}$
break ${\rm U(1)}_{L}$.

We assume that the mixed terms involving scalars breaking different ${\rm
U}(1)$ gauge groups have negligible coefficients. This implies that for a
given ${\rm U}(1)$, if only one scalar breaks the symmetry, the scalar
potential is

$\displaystyle V(\Phi)$ $\displaystyle=$
$\displaystyle-m^{2}|\Phi|^{2}+\lambda|\Phi|^{4}\ ,$ (4)

whereas if two scalars participate in symmetry breaking,

$\displaystyle V(\Phi_{1},\Phi_{2})$ $\displaystyle=$
$\displaystyle-\,m_{1}^{2}|\Phi_{1}|^{2}+\lambda_{1}|\Phi_{1}|^{4}-m_{2}^{2}|\Phi_{2}|^{2}+\lambda_{2}|\Phi_{2}|^{4}$
(5) $\displaystyle+$
$\displaystyle\left[(\lambda_{4}|\Phi_{1}|^{2}\\!+\\!\lambda_{5}|\Phi_{2}|^{2}\\!+\\!\lambda_{6}\Phi_{1}^{*}\Phi_{2})\Phi_{1}^{*}\Phi_{2}+{\rm
h.c.}\right]$ $\displaystyle-$
$\displaystyle(m_{12}^{2}\Phi_{1}^{*}\Phi_{2}+{\rm
h.c.})+\lambda_{3}|\Phi_{1}|^{2}|\Phi_{2}|^{2}\ .$

The scalars develop the following vacuum expectation values,

$\displaystyle\langle\Phi_{i}\rangle=\frac{v_{i}}{\sqrt{2}}\ .$ (6)

Whenever two scalars take part in the symmetry breaking, we define
$v\equiv\sqrt{v_{1}^{2}+v_{2}^{2}}\ .$ This way one can collectively describe
the ${\rm U}(1)_{B}$ breaking scale as $v=v_{B}$, and the ${\rm U}(1)_{L}$
breaking scale as $v=v_{L}$, independent of whether the vacuum expectation
value comes from a single scalar or two scalars.

If lepton number is broken at a higher scale than baryon number, the symmetry
breaking pattern is:

$\begin{array}[]{c}{\rm SU}(3)_{c}\times{\rm SU}(2)_{L}\times{\rm
U}(1)_{Y}\times{\rm U}(1)_{B}\times{\rm U}(1)_{L}\\\\[3.0pt] \hskip
31.86707pt\bigg{\downarrow}\hskip
8.53581pt{\scriptstyle\langle\Phi_{Li}\rangle\ \neq\ 0}\\\\[12.0pt] {\rm
SU}(3)_{c}\times{\rm SU}(2)_{L}\times{\rm U}(1)_{Y}\times{\rm
U}(1)_{B}\\\\[3.0pt] \hskip 31.86707pt\bigg{\downarrow}\hskip
8.53581pt{\scriptstyle\langle\Phi_{Bi}\rangle\ \neq\ 0}\\\\[12.0pt] {\rm
SU}(3)_{c}\times{\rm SU}(2)_{L}\times{\rm U}(1)_{Y}\ ,\\\\[7.0pt] \end{array}$

followed by the usual electroweak symmetry breaking by the Standard Model
Higgs. We note that the order of ${\rm U}(1)_{B}$ and ${\rm U}(1)_{L}$
breaking may be reversed.

### Fermion sector

To provide a concrete quantitative example, we consider the model with gauged
${\rm U}(1)_{B}$ and ${\rm U}(1)_{L}$ proposed in Fileviez Perez  _et al._
(2014), which involves the minimal fermionic particle content for a theory
with such a gauge group. It is straightforward to check that all gauge
anomalies are cancelled if the Standard Model quark fields $Q_{L}^{j}$,
$u_{R}^{j}$, $d_{R}^{j}$ and lepton fields $l_{L}^{j}$, $e_{R}^{j}$ are
augmented by

$\displaystyle\nu_{R}^{j}$ $\displaystyle=$ $\displaystyle(1,1,0,0,1)\ ,$
$\displaystyle\Psi_{L}$ $\displaystyle=$
$\displaystyle\begin{pmatrix}\psi_{L}^{+}\\\\[2.0pt]
\psi_{L}^{0}\end{pmatrix}=\left(1,2,\tfrac{1}{2},\tfrac{3}{2},\tfrac{3}{2}\right)\
,$ $\displaystyle\Psi_{R}$ $\displaystyle=$
$\displaystyle\begin{pmatrix}\psi_{R}^{0}\\\
\psi_{R}^{-}\end{pmatrix}=\left(1,2,\tfrac{1}{2},-\tfrac{3}{2},-\tfrac{3}{2}\right)\
,$ $\displaystyle\Sigma_{L}$ $\displaystyle=$
$\displaystyle\tfrac{1}{2}\begin{pmatrix}\sigma^{0}&\sqrt{2}\,\sigma^{+}\\\\[2.0pt]
\sqrt{2}\,\sigma^{-}&-\sigma^{0}\end{pmatrix}=\left(1,3,0,-\tfrac{3}{2},-\tfrac{3}{2}\right)\
,$ $\displaystyle\chi_{L}$ $\displaystyle=$
$\displaystyle\left(1,1,0,-\tfrac{3}{2},-\tfrac{3}{2}\right)\ ,$ (7)

where $j$ is the family index. Among the fields above, $\nu_{R}^{j}$ are the
right-handed neutrinos, whereas $\chi_{L}$ is a Majorana dark matter candidate
discussed in Section III.

### Particle masses

The scalars $\Phi_{Bi}$ and $\Phi_{Li}$ generate masses for the new fermions
through the following Lagrangian terms,

$\displaystyle-\mathcal{L}$ $\displaystyle\supset$
$\displaystyle\sum_{i}\\!\left(Y_{\Psi}^{i}\overline{\Psi}_{R}\Psi_{L}\Phi_{Bi}^{*}\\!+\\!Y_{\Sigma}^{i}{\rm
Tr}(\Sigma_{L}^{2})\Phi_{Bi}\\!+\\!Y_{\chi}^{i}\chi_{L}\chi_{L}\Phi_{Bi}\right)$
(8) $\displaystyle+$ $\displaystyle
y_{\nu}^{j}\,\bar{l}_{L}^{j}H\nu_{R}^{j}+\sum_{i}Y_{\nu}^{ij}\nu_{R}^{j}\nu_{R}^{j}\Phi_{Li}+{\rm
h.c.}\ ,\ \ \ \ $

which provide vector-like masses to the new fermions, as well as introduce a
type I seesaw mechanism for the neutrinos. For example, assuming that the
Yukawa couplings are $y_{\nu}^{j}\sim 1$ and $Y_{\nu}^{j}\sim 10^{-2}$, the
measured neutrino mass splittings are reproduced if $v_{L}\sim 10^{5}\ {\rm
PeV}$. The mass matrices for the new fermions are provided in Fileviez Perez
_et al._ (2014).

The spontaneous breaking of ${\rm U}(1)_{L}$ and ${\rm U}(1)_{B}$ leads to the
appearance of vector gauge bosons $Z_{L}$ and $Z_{B}$. Given the charges of
the scalars breaking the two symmetries, the corresponding masses are

$\displaystyle m_{Z_{L}}=2g_{L}v_{L}\ ,\ \ \ \ \ \ \ \ m_{Z_{B}}=3g_{B}v_{B}\
,$ (9)

where $g_{L}$ and $g_{B}$ are the ${\rm U}(1)_{L}$ and ${\rm U}(1)_{B}$ gauge
couplings, respectively, whose values are free parameters.

##  III Dark matter and matter-antimatter asymmetry

Apart from providing a natural framework accommodating a type I seesaw
mechanism generating small neutrino masses via ${\rm U}(1)_{L}$ breaking, the
model also contains a phenomenologically viable dark matter candidate
$\chi_{L}$ Fileviez Perez  _et al._ (2014); Ohmer and Patel (2015) and can
account for the matter-antimatter asymmetry through leptogenesis Fileviez
Perez  _et al._ (2021). We discuss the most relevant aspects of those
highlights of the model below.

### Dark matter

After ${\rm U}(1)_{B}$ breaking, there remains a residual discrete ${Z}_{2}$
symmetry under which the new fermions transform as

$\displaystyle\Psi_{L}$ $\displaystyle\to$ $\displaystyle-\Psi_{L}\ ,\ \ \ \
\overline{\Psi}_{R}\to-\overline{\Psi}_{R}\ ,$ $\displaystyle\Sigma$
$\displaystyle\to$ $\displaystyle-\Sigma\ ,\ \ \ \ \ \ \ \chi_{L}\to-\chi_{L}\
.$ (10)

If $\chi_{L}$ is the lightest of the new fermions, there is no decay channel
available for it, thus it becomes a good candidate for particle dark matter.

It was argued in Duerr and Fileviez Perez (2015); Fileviez Perez  _et al._
(2019) that in models with gauged baryon and lepton number consistency with
the dark matter relic abundance of $h^{2}\Omega_{\rm DM}\approx 0.12$ Aghanim
_et al._ (2020) imposes an upper bound on the ${\rm U(1)}_{B}$ breaking scale.
In particular, if the dark matter annihilation happens via the resonant
$s$-channel process

$\displaystyle\chi_{L}\,\chi_{L}\to Z_{B}^{*}\to\bar{q}\,q\ ,$ (11)

a dependence between the parameters $v_{B}$, $Y_{\chi}$, and $g_{B}$ arises,
and the perturbativity requirement leads to $g_{B}v_{B}\lesssim 20\ {\rm
TeV}$. This was the reason why in Fornal and Shams Es Haghi (2020), where the
gravitational wave signal from a model with gauged baryon and lepton number
was considered, a low scale of ${\rm U}(1)_{B}$ was imposed.

However, as was demonstrated in Ohmer and Patel (2015), in the model we are
considering other dark matter annihilation channels remain unsuppressed,
including the nonresonant $t$-channel process

$\displaystyle\chi_{L}\,\chi_{L}\to\Phi_{B}\,\Phi_{B}\ ,$ (12)

whose cross section can be sufficiently large to explain the dark matter relic
density. Therefore, the arguments in Duerr and Fileviez Perez (2015); Fileviez
Perez  _et al._ (2019) do not apply in our case, and the scale of ${\rm
U}(1)_{B}$ breaking can be high. Alternatively, the aforementioned bound on
the ${\rm U}(1)_{B}$ breaking scale can always be avoided by assuming
nonthermal dark matter production.

### Leptogenesis

There cannot exist any primordial baryon or lepton number asymmetry above the
scales of ${\rm U}(1)_{B}$ and ${\rm U}(1)_{L}$ breaking. An excess of matter
over antimatter can only arise once one of those two symmetries is broken. A
natural setting to achieve this below the scale of ${\rm U}(1)_{L}$ breaking
is offered by high-scale leptogenesis (see Davidson  _et al._ (2008) and
references therein), in which a lepton number asymmetry is generated through
the out-of-equilibrium decays of the lightest right-handed neutrino,

$\displaystyle N_{1}\to H\,l_{L}\ .$ (13)

The $CP$ asymmetry is introduced through the standard interference between the
tree-level diagram for the process in Eq. (13) and the one-loop diagrams
involving $H$, $l_{L}$, and the two heavier right-handed neutrinos $N_{2}$,
$N_{3}$ in the loop.

The generated lepton asymmetry is calculated by solving the Boltzmann
equations for the evolution of the lightest right-handed neutrino abundance
$Y_{N_{1}}=n_{N_{1}}/s$ (where $n_{N_{1}}$ is the $N_{1}$ particle density and
$s$ is the co-moving entropy density) and the $B\\!-\\!L$ asymmetry $Y_{B-L}$
Buchmuller  _et al._ (2005),

$\displaystyle\frac{dY_{N_{1}}}{dz}$ $\displaystyle=$
$\displaystyle-(D+S)(Y_{N_{1}}-Y_{N_{1}}^{\rm eq})\ ,$
$\displaystyle\frac{dY_{B-L}}{dz}$ $\displaystyle=$
$\displaystyle-\epsilon_{1}D(Y_{N_{1}}-Y_{N_{1}}^{\rm eq})-WY_{B-L}\ ,$ (14)

where $z=m_{N_{1}}/T$, the term $D$ accounts for decays and inverse decays,
$S$ represents $\Delta L=1$ scatterings, $W$ describes the washout effects,
and $\epsilon_{1}$ is the $CP$ asymmetry parameter. In our case, the Boltzmann
equations are slightly different than in the standard leptogenesis scenario,
since the right-handed neutrinos have an extra interaction with the $Z_{L}$
gauge boson. Those equations were solved in Fileviez Perez  _et al._ (2021),
and the amount of the generated lepton asymmetry $\Delta L$ was determined for
various values of model parameters.

The produced lepton asymmetry is then partially converted into a baryon
asymmetry through the electroweak sphalerons. Above the scale of ${\rm
U}(1)_{B}$ breaking the sphaleron-induced interactions have the form

$\displaystyle(QQQL)^{3}\overline{\Psi}_{R}\Psi_{L}\Sigma_{L}^{4}\ ,$ (15)

and it was shown in Duerr and Fileviez Perez (2015) that if the breaking of
${\rm U}(1)_{B}$ occurs close to the electroweak scale, then the final baryon
asymmetry predicted by the model is given by

$\displaystyle|\Delta B|=\frac{32}{99}|\Delta L|\ .$ (16)

This can explain the observed baryon-to-photon ratio Workman  _et al._ (2022)

$\displaystyle\eta\approx 6\times 10^{-10}$ (17)

if the scale of ${\rm U}(1)_{L}$ breaking satisfies the relation

$\displaystyle v_{L}\gtrsim 4000\ {\rm PeV}\ .$ (18)

The requirement for such a high ${\rm U}(1)_{L}$ symmetry breaking scale
provides the desired setting accommodating the type I seesaw mechanism for the
neutrinos.

##  IV Cosmic string spectrum

Spontaneous breaking of a ${\rm U}(1)$ gauge symmetry leads to the production
of topological defects in the form of cosmic strings Kibble (1976), which
correspond to one-dimensional field configurations along the direction of the
unbroken symmetry. The network of produced cosmic strings is described
collectively by the string tension $\mu$, equal to the energy stored in a
string per unit length, and depends solely on the scale at which the ${\rm
U}(1)$ gauge symmetry is broken Vilenkin and Shellard (2000); Gouttenoire  _et
al._ (2020b),

$\displaystyle G\mu=2\pi\left(\frac{v}{M_{Pl}}\right)^{2},$ (19)

where $M_{Pl}=1.22\times 10^{13}\ {\rm PeV}$ is the Planck mass, the
gravitational constant $G=6.7\times 10^{-39}\ \rm GeV^{-2}$, and the winding
number was taken to be one. The constraints from the cosmic microwave
background measurements set an upper limit on the string tension of
$G\mu\lesssim 10^{-7}$ Ade  _et al._ (2014), which corresponds to the
following bound on the scale of symmetry breaking,

$\displaystyle v\lesssim 1.5\times 10^{9}\ {\rm PeV}\ .$ (20)

Through its dynamics, the cosmic string network provides a long-lasting source
of gravitational radiation and leads to a stochastic gravitational wave
background roughly constant across a wide range of frequencies.

### Dynamics of cosmic strings

There are two main processes governing the behavior of a cosmic string
network: formation of string loops and stretching due to the expansion of the
Universe. The first of those contributions, creation of string loops, happens
when long strings intersect and intercommute. The newly created string loops
oscillate and emit gravitational radiation, mainly from cusps and kinks
propagating through the string loop, and from kink-kink collisions Olum and
Blanco-Pillado (2000); Moore  _et al._ (2002).

A competition between these two effects leads to the so-called scaling regime,
in which there is a large number of string loops and a small number of Hubble-
size strings Kibble (1985); Bennett and Bouchet (1988, 1989); Albrecht and
Turok (1989); Allen and Shellard (1990). There is a continuous flow of energy
from long strings to string loops, and then to gravitational radiation through
their decays. This gravitational radiation makes up a fixed fraction of the
energy density of the Universe Hindmarsh and Kibble (1995).

To describe this process quantitatively, we follow the steps outlined in Cui
_et al._ (2019); Gouttenoire  _et al._ (2020b). We consider a string loop
created at time $t_{0}$ with initial length $l(t_{0})=\alpha\,t_{0}$, where
$\alpha$ is a constant loop size parameter. The loop oscillates emitting
gravitational waves with frequencies

$\displaystyle\tilde{f}=\frac{2k}{l}\ ,$ (21)

where $k$ is a positive integer. Rescaling this result by the scale factor
$a(t)$, one obtains the currently observed frequency,

$\displaystyle f=\frac{a(t_{e})}{a(T)}\,\tilde{f}\ ,$ (22)

where $t_{e}$ is the time of the emission and $T$ is the time today. The
spectrum of the emitted gravitational waves from a single oscillating string
loop is given by Blanco-Pillado  _et al._ (2014); Blanco-Pillado and Olum
(2017)

$\displaystyle P_{(k,n)}=\frac{\Gamma
G\mu^{2}}{k^{n}}\bigg{(}\sum_{p=1}^{\infty}\frac{1}{p^{n}}\\!\bigg{)}^{-1},$
(23)

where for the contribution from cusps $n=4/3$, from kinks $n=5/3$, and from
kink-kink collisions $n=2$, while the overall factor $\Gamma\simeq 50$
Vachaspati and Vilenkin (1985b). Due to the constant emission of gravitational
radiation, the string loop shrinks and its length at the time of the
gravitational wave emission is

$\displaystyle l(t_{e})=\alpha\,t_{0}-\Gamma G\mu\,(t_{e}-t_{0})\ ,$ (24)

causing the loop to vanish after the time ${\alpha\,t_{0}}/{(\Gamma G\mu)}$.

The only model-dependent quantity describing the cosmic string network is the
loop distribution function $F(l,t_{0})$ for the created loops. Adopting the
well-established model developed in Martins and Shellard (1996a, b, 2002),
describing the string network just by the mean string velocity and the
correlation length, leads in the scaling regime to the following formula,

$\displaystyle F(l,t_{0})=\frac{\sqrt{2}\,{C}_{\rm
eff}}{\alpha\,t_{0}^{4}}\,\delta(l-\alpha\,t_{0})\ ,$ (25)

where the constant ${C}_{\rm eff}$ depends on the era in the evolution of the
Universe (for radiation $C_{\rm eff}=5.4$, whereas for the matter dominated
era $C_{\rm eff}=0.39$ Cui  _et al._ (2019)).

### Gravitational wave spectrum

The stochastic gravitational wave background generated by the dynamics of the
cosmic string network is Cui  _et al._ (2019); Gouttenoire  _et al._ (2020b)

$\displaystyle h^{2}\Omega_{\rm CS}(f)$ $\displaystyle=$
$\displaystyle\frac{2h^{2}\mathcal{F}_{\alpha}}{\rho_{c}\alpha^{2}f}\,\sum_{k,n}\,{k\,P_{(k,n)}}\int_{t_{F}}^{T}\\!dt_{e}\
\frac{C_{\text{eff}}(t_{0k})}{t_{0k}^{\,4}}\ \ \ \ $ (26)
$\displaystyle\times$
$\displaystyle\left(\frac{a(t_{e})}{a(T)}\right)^{\\!5}\left(\frac{a(t_{0k})}{a(t_{e})}\right)^{\\!3}\theta(t_{0k}-t_{F})\
,$

where

$\displaystyle
t_{0k}=\frac{1}{\alpha}\left(\frac{2k}{f}\frac{a(t_{e})}{a(T)}+\Gamma
G\mu\,t_{e}\right)\ .$ (27)

In Eqs. (26) and (27) the parameters are the following: $\mathcal{F}_{\alpha}$
is the fraction of the loops contributing to the gravitational wave signal,
estimated to be $\mathcal{F}_{\alpha}\approx 0.1$ Blanco-Pillado  _et al._
(2014) since the majority of the energy is lost by long strings going into
highly boosted smaller loops which provide only a subdominant contribution;
$\rho_{c}$ is the critical density of the Universe; the loop size parameter
$\alpha=0.1$ provides an accurate estimate of the loop size distribution
Blanco-Pillado  _et al._ (2014); Blanco-Pillado and Olum (2017); $t_{F}$ is
the time at which the cosmic string network was formed, related to the density
of the Universe at that time via $\sqrt{\rho(t_{F})}=\mu$ Gouttenoire  _et
al._ (2020b), $t_{0k}$ is the instance when the loop was produced, and
$\theta(x)$ is the Heaviside step function. We also note that, as argued in
Cui  _et al._ (2019); Gouttenoire  _et al._ (2020b), the largest contribution
to the gravitational wave signal comes from the cusps.

Figure 1: Stochastic gravitational wave background from cosmic strings for
four different symmetry breaking scales. Shaded regions correspond to the
sensitivity of future gravitational wave detectors: LISA (green), DECIGO
(blue), Big Bang Observer (purple), Einstein Telescope (red), and Cosmic
Explorer (gray). Figure 2: Reach of future detectors in probing the scale of a
${\rm U}(1)$ symmetry breaking leading to the production of cosmic strings.
The colors for each experiment correspond to those adopted in Fig. 1.

The resulting stochastic gravitational wave background is presented in Fig. 1
for four values of the symmetry breaking scale, in the range of frequencies
relevant for the upcoming gravitational wave experiments, whose sensitivities
are denoted by the colored regions. If $v\gtrsim 10^{4}\ {\rm PeV}$, the
signal can be seen by all the detectors: LISA Amaro-Seoane  _et al._ (2017),
DECIGO Kawamura  _et al._ (2011), Big Bang Observer Crowder and Cornish
(2005), Einstein Telescope Punturo  _et al._ (2010), and Cosmic Explorer
Reitze  _et al._ (2019). This is illustrated in more detail in Fig. 2, which
shows the reach of each experiment in terms of the symmetry breaking scale
leading to a cosmic string signal. The lower bound is detector-specific,
whereas the upper bound reflects the cosmic microwave background constraint in
Eq. (20).

The cosmic string network will be produced if either ${\rm U}(1)_{B}$ or ${\rm
U}(1)_{L}$ is spontaneously broken by a single scalar. Therefore, among the
cases enumerated in Section II, the gravitational wave signals discussed here
are relevant in case $(a)$ for both baryon and lepton number breaking, in case
$(b)$ only for lepton number breaking, and in case $(c)$ only for baryon
number breaking.

##  V Domain wall spectrum

Another kind of topological defects, appearing when a ${\rm U}(1)$ symmetry is
broken by two scalars, are domain walls. As in the case of ${\rm SU}(2)$
breaking discussed in Fornal and Pierre (2022), the effective potential
$V(\phi_{1},\phi_{2},T)$, which at high temperature has just one vacuum at
$(\phi_{1},\phi_{2})=(0,0)$, at lower temperature develops four vacua. They
come in two pairs related via a gauge transformation, $\Phi_{i}\to
e^{i\theta}\Phi_{i}$, and only two of them, say $\vec{\phi}_{\rm vac1}$ and
$\vec{\phi}_{\rm vac2}$, correspond to disconnected manifolds, and thus are
physically distinct Ginzburg and Krawczyk (2005); Battye  _et al._ (2011).
When a transition takes place, patches of the Universe end up in either one of
those vacua, leading to the creation of domain walls, i.e., two-dimensional
field configurations on the boundaries of $\vec{\phi}_{\rm vac1}$ and
$\vec{\phi}_{\rm vac2}$.

If the two vacua have identical energy densities, domain walls remain stable
and considerably affect the evolution of the Universe, introducing
unacceptably large density fluctuations Saikawa (2017). Therefore, for a
phenomenologically viable scenario, the ${Z}_{2}$ symmetry between the two
vacua needs to be softly broken. It cannot be strongly broken, since then
patches of the Universe would transition preferentially to the lower energy
density vacuum and domain walls would not form. In our case, the soft breaking
of the ${Z}_{2}$ symmetry removing the degeneracy between the vacua is
provided by the terms involving $m_{12}^{2}$, $\lambda_{4}$ and $\lambda_{5}$
in the Lagrangian in Eq. (5).

### Dynamics of domain walls

The profile of the domain wall configuration $\vec{\phi}_{dw}(z)$ is the
solution to the equation Chen  _et al._ (2020)

$\displaystyle\frac{d^{2}\vec{\phi}_{dw}(z)}{dz^{2}}-\vec{\nabla}_{\\!\phi}V_{\rm
eff}\big{[}\vec{\phi}_{dw}(z)\big{]}=0\ ,$ (28)

where the $z$-axis was chosen to be perpendicular to the domain wall, and the
boundary conditions are,

$\displaystyle\vec{\phi}_{dw}(-\infty)=\vec{\phi}_{\rm vac1}\ ,\ \ \ \ \
\vec{\phi}_{dw}(\infty)=\vec{\phi}_{\rm vac2}.$ (29)

As mentioned earlier, there are only two parameters which describe the
gravitational wave spectrum from domain walls. The first of them is the domain
wall tension $\sigma$ given by

$\displaystyle\sigma=\int_{-\infty}^{\infty}dz\\!\left[\frac{1}{2}\bigg{(}\frac{d\vec{\phi}_{dw}(z)}{dz}\bigg{)}^{2}+V_{\rm
eff}\big{[}\vec{\phi}_{dw}(z)\big{]}\right].$ (30)

In the model we are considering, to a good approximation

$\displaystyle\sigma\,\sim\,v^{3}\ .$ (31)

The other parameter is the potential bias $\Delta\rho$, i.e., the energy
density difference between the vacua, in our case equal to

$\displaystyle\Delta\rho=\big{(}m_{12}^{2}+\tfrac{1}{2}\lambda_{4}v_{1}^{2}+\tfrac{1}{2}\lambda_{5}v_{2}^{2}\big{)}\,v_{1}v_{2}\
.$ (32)

The created domain walls are unstable and undergo annihilation, emitting
gravitational radiation, provided that Saikawa (2017)

$\displaystyle\Delta\rho\gtrsim\left(\frac{\sigma}{M_{Pl}}\right)^{2}\ .$ (33)

If in Eq. (32) the term involving $m_{12}^{2}$ is the dominant one, then Eq.
(33) takes the form $m_{12}\gtrsim{v^{2}}/{M_{Pl}}$. For example, if the
symmetry breaking scale is $v\sim 10^{3}\ \rm PeV$, this implies
$m_{12}\gtrsim 100\ {\rm MeV}$. An independent constraint arises from the
necessity of domain wall annihilation happening before Big Bang
nucleosynthesis, so the ratios of the produced elements are not altered, but
the bound in Eq. (33) remains stronger.

### Gravitational wave spectrum

Domain wall annihilation leads to a stochastic gravitational wave background
given by Kadota  _et al._ (2015); Saikawa (2017),

$\displaystyle h^{2}\Omega_{\rm DW}(f)$ $\displaystyle\approx$ $\displaystyle
7.1\times 10^{-33}\left(\frac{\sigma}{\rm PeV^{3}}\right)^{4}\left(\frac{\rm
TeV^{4}}{\Delta\rho}\right)^{2}\left(\frac{100}{g_{*}}\right)^{\frac{1}{3}}$
(34) $\displaystyle\times$
$\displaystyle\\!\\!\bigg{[}\\!\left(\frac{f}{f_{d}}\right)^{3}\\!\theta(f_{d}-f)+\left(\frac{f_{d}}{f}\right)\theta(f-f_{d})\bigg{]},\
\ \ \ \ \ \ $

where for the area parameter we used $\mathcal{A}=0.8$ and for the efficiency
parameter we adopted the value $\tilde{\epsilon}_{\rm gw}=0.7$ Hiramatsu  _et
al._ (2014),$\theta$ denotes the step function, and the peak frequency $f_{d}$
is

$\displaystyle f_{d}\approx(0.14\ {\rm Hz})\,\sqrt{\frac{\rm
PeV^{3}}{\sigma}\frac{\Delta\rho}{\rm TeV^{4}}}\ .$ (35)

The slope of the signal falls $\sim f^{3}$ to the left of the peak when moving
toward lower frequencies, and falls like $f^{-1}$ to the right of the peak
when moving toward higher frequencies. The cosmic microwave background
constraint on the strength of the signal at the peak is
$h^{2}\Omega(f)<2.9\times 10^{-7}$ Clarke  _et al._ (2020), which translates
to the following condition on the parameters,

$\displaystyle\frac{\sigma}{\sqrt{\Delta\rho}}\lesssim 2.5\times 10^{12}\ \rm
PeV\ ,$ (36)

stronger than the bound imposed by Eq. (33).

Figure 3: Stochastic gravitational wave background from domain walls for
various symmetry breaking scales. Shaded regions correspond to the sensitivity
of future gravitational wave detectors, as in Fig. 1. Figure 4: Regions of
parameter space $(v,\Delta\rho)$ for which the signal-to-noise ratio of the
gravitational wave signal generated by domain wall annihilation is greater
than five upon one year of data taking by various experiments. The choice of
colors matches that in Fig. 3, including the color of the dots which
correspond to the four curves.

Several examples of gravitational wave spectra from domain wall annihilation,
plotted using Eq. (34), are shown in Fig. 3 for representative values of the
parameters $v$ and $\Delta\rho$. The reach of the upcoming gravitational wave
detectors is also shown, including LISA, Big Bang Observer, DECIGO, Einstein
Telescope, and Cosmic Explorer. A more detailed look at their sensitivity is
provided by Fig. 4, which shows the full parameter space that can be probed by
those experiments. The lower bound on the domain wall parameter $\Delta\rho$
is a reflection of the cosmic microwave background constraint from Eq. (36).
The parameter $\Delta\rho$ depends in general on all three fundamental
Lagrangian parameters $m_{12}$, $\lambda_{4}$, and $\lambda_{5}$ through the
relation in Eq. (32). Under the assumption that the term involving $m_{12}$ is
dominant, the experimental sensitivity plot in the plane $(v,m_{12})$ would be
the same as in figure 4 of Fornal  _et al._ (2023).

##  VI First order phase transition spectrum

Perhaps the most anticipated stochastic gravitational wave signal to be
discovered is the one generated by a first order phase transition in the early
Universe, predicted in a large class of theories beyond the Standard Model.
Such a signature in the case of ${\rm U}(1)_{B}$ breaking has been considered
in Fornal and Shams Es Haghi (2020), but in this work we adopt a different
assumption for the Yukawa couplings and keep our analysis more general, so
that it can be applied to both gauged ${\rm U}(1)_{B}$ and gauged ${\rm
U}(1)_{L}$. A first order phase transition can occur either when the symmetry
breaking sector consists of a single scalar, or contains multiple scalars.
Since the generalization is straightforward, we concentrate on the case with a
single scalar.

### Effective potential

The effective potential for the background field $\phi$ consists of the tree-
level part, the one-loop Coleman-Weinberg zero temperature correction, and the
finite temperature contribution. Upon imposing the condition that the minimum
of the zero temperature potential and the mass of $\phi$ remain at their tree-
level values (i.e., the cutoff regularization scheme), the effective potential
is given by

$\displaystyle V_{\rm eff}(\phi,T)=-\frac{1}{2}\lambda
v^{2}\phi^{2}+\frac{1}{4}\lambda\phi^{4}$
$\displaystyle+\sum_{i}\frac{n_{i}m_{i}^{2}(\phi)}{64\pi^{2}}\bigg{\\{}\\!m_{i}^{2}(\phi)\\!\left[\log\\!\left(\frac{m_{i}^{2}(\phi)}{m_{i}^{2}(v)}\right)\\!-\\!\frac{3}{2}\right]\\!+\\!2m_{i}^{2}(v)\\!\bigg{\\}}$
$\displaystyle+\
\frac{T^{4}}{2\pi^{2}}\sum_{i}n_{i}\int_{0}^{\infty}dx\,x^{2}\log\left(1\mp
e^{-\sqrt{{m_{i}^{2}(\phi)}/{T^{2}}+x^{2}}}\right)$ $\displaystyle+\
\frac{T}{12\pi}\sum_{j}n^{\prime}_{j}\left\\{m_{j}^{3}(\phi)-\left[m_{j}^{2}(\phi)+\Pi_{j}(T)\right]^{\frac{3}{2}}\right\\}\
.$ (37)

In the expression above the sums are over all particles charged under the
${\rm U}(1)$ including the Goldstone bosons $\chi_{\rm GB}$, $m_{i}(\phi)$ are
the field-dependent masses, $n_{i}$ is the number of degrees of freedom for a
given particle ($n_{Z^{\prime}}=3$, $n_{\phi}=1$, $n_{\chi_{\rm GB}}=1$),
$n_{j}^{\prime}$ is similar but includes only scalars and longitudinal
components of vector bosons ($n^{\prime}_{Z^{\prime}}=1$,
$n^{\prime}_{\phi}=1$, $n^{\prime}_{\chi_{\rm GB}}=1$), and for the Goldstones
one needs to replace $m_{\chi_{\rm GB}}(v)\to m_{\phi}(v)$. We will assume
that all new Yukawa couplings are small, so that the only relevant field-
dependent masses are

$\displaystyle m_{Z^{\prime}}(\phi)=xg\phi\ ,\ \ \ \
m_{\phi}(\phi)=\sqrt{\lambda(3\phi^{2}-v^{2})}\ ,$ $\displaystyle m_{\chi_{\rm
GB}}(\phi)=\sqrt{\lambda(\phi^{2}-v^{2})}\ .$ (38)

In the limit $\lambda\ll g$, the thermal masses are

$\displaystyle\Pi_{Z^{\prime}}(T)$ $\displaystyle=$
$\displaystyle\frac{1}{3}\left(x^{2}+\frac{9}{2}\right)g^{2}T^{2}\ ,$
$\displaystyle\Pi_{\phi}(T)$ $\displaystyle=$ $\displaystyle\Pi_{\chi_{\rm
GB}}(T)=\frac{1}{4}x^{2}g^{2}T^{2}\ ,$ (39)

where for gauged baryon number $g=g_{B}$ and $x=3$, while for gauged lepton
number $g=g_{L}$ and $x=2$.

Figure 5: The effective potential of the model, $V_{\rm eff}(\phi_{B},T)$,
plotted for $v_{B}=100\ \rm TeV$, $g_{B}=0.25$, $\lambda_{B}=0.006$, and
several temperatures.

For a range of $\lambda$ and $g$ values the effective potential develops a
vacuum at $\phi\neq 0$ (true vacuum) with a lower energy density than the high
temperature vacuum at $\phi=0$ (false vacuum), separated by a potential bump,
which are precisely the conditions needed for a first order phase transition
to take place. The changing shape of the effective potential is shown in Fig.
5 for a particular choice of parameters, in the case of gauged baryon number.

### Dynamics of the phase transition

A first order phase transition from the false vacuum to the true vacuum of a
given patch of the Universe corresponds to the nucleation of a bubble which
then starts expanding. This process is initiated at the nucleation temperature
$T_{*}$, which is determined from the condition that the bubble nucleation
rate Linde (1983) becomes comparable with the Hubble expansion,

$\displaystyle\bigg{(}\frac{S(T_{*})}{2\pi
T_{*}}\bigg{)}^{3/2}T_{*}^{4}\,e^{-{S(T_{*})}/{T_{*}}}\approx H(T_{*})^{4}\ ,$
(40)

where $S(T)$ is the Euclidean action given by

$\displaystyle S(T)=\int
d^{3}r\left[\frac{1}{2}\left(\frac{d\phi_{b}}{dr}\right)^{2}+V_{\rm
eff}(\phi_{b},T)\right],$ (41)

with $\phi_{b}$ being the solution of the bubble equation,

$\displaystyle\frac{d^{2}\phi}{dr^{2}}+\frac{2}{r}\frac{d\phi}{dr}-\frac{dV_{\rm
eff}(\phi,T)}{d\phi}=0\ ,$ (42)

subject to the boundary conditions

$\displaystyle\frac{d\phi}{dr}\bigg{|}_{r=0}=0\ ,\ \ \ \ \
\phi(\infty)=\phi_{\rm false}\ .$ (43)

Since $H(T)\approx(T^{2}/M_{Pl})\sqrt{4\pi^{3}g_{*}/45}$, Eq. (40) becomes

$\displaystyle\frac{S(T_{*})}{T_{*}}\approx
4\log\\!\left(\frac{M_{Pl}}{T_{*}}\right)\\!-\\!\log\left[\left(\frac{4\pi^{3}g_{*}}{45}\right)^{\\!\\!2}\\!\left(\frac{2\pi\,T_{*}}{S(T_{*})}\right)^{\\!\\!\frac{3}{2}}\right]\\!.\
\ $ (44)

The phase transition parameters relevant for determining the gravitational
wave signal are: the bubble wall velocity $v_{w}$, the nucleation temperature
$T_{*}$, the phase transition strength $\alpha$, and its duration
$1/\tilde{\beta}$. In our analysis we assume $v_{w}=c$, but other choices are
also possible Espinosa  _et al._ (2010); Caprini  _et al._ (2016). The other
three parameters, $T_{*}$, $\alpha$, and $\tilde{\beta}$, are determined from
the behavior of the effective potential with changing temperature. As such,
those parameters encode information about the details of the particle physics
model considered.

The phase transition strength is calculated as the ratio of the energy density
difference between the false and true vacuum, and that of radiation, both
taken at nucleation temperature,

$\displaystyle\alpha=\frac{\rho_{\rm vac}(T_{*})}{\rho_{\rm rad}(T_{*})}\ .$
(45)

Those two quantities are obtained from the relations

$\displaystyle\rho_{\rm vac}(T)$ $\displaystyle=$ $\displaystyle V_{\rm
eff}(\phi_{\rm false},T)-V_{\rm eff}(\phi_{\rm true},T)$ $\displaystyle-$
$\displaystyle T\frac{\partial}{\partial T}{\left[V_{\rm eff}(\phi_{\rm
false},T)-V_{\rm eff}(\phi_{\rm true},T)\right]}\ ,$ $\displaystyle\rho_{\rm
rad}(T)$ $\displaystyle=$ $\displaystyle\frac{\pi^{2}}{30}g_{*}T^{4}\ ,$ (46)

where $g_{*}$ is the number of degrees of freedom active when the bubbles are
nucleated. The parameter $\tilde{\beta}$, related to the time scale of the
phase transition, is determined via

$\displaystyle\tilde{\beta}=T_{*}\frac{d}{dT}\bigg{(}\frac{S(T)}{T}\bigg{)}\bigg{|}_{T=T_{*}}\
.$ (47)

### Gravitational wave spectrum

The dynamics of the nucleated bubbles generates gravitational waves through
sound shock waves in the early Universe plasma, bubble collisions, and
magnetohydrodynamic turbulence. The expected contribution of each of those
processes to the stochastic gravitational wave background was determined
through numerical simulations, and the corresponding empirical formulas were
derived. The resulting gravitational wave signal is the sum of the three
contributions,

$\displaystyle h^{2}\Omega_{\rm
PT}(f)=h^{2}\Omega_{s}(f)+h^{2}\Omega_{c}(f)+h^{2}\Omega_{t}(f)\ .$ (48)

The expected signal from sound waves is Hindmarsh  _et al._ (2014); Caprini
_et al._ (2016)

$\displaystyle h^{2}\Omega_{s}(f)\,$ $\displaystyle\approx$
$\displaystyle\,\frac{1.9\times
10^{-5}}{\tilde{\beta}}\left(\frac{\kappa_{s}\,\alpha}{\alpha+1}\right)^{2}\left(\frac{100}{g_{*}}\right)^{\frac{1}{3}}\Upsilon$
(49) $\displaystyle\times$
$\displaystyle\frac{(f/f_{s})^{3}}{\big{[}1+0.75(f/f_{s})^{2}\big{]}^{7/2}}\
,$

where $f_{s}$ is the peak frequency, $\kappa_{s}$ is the fraction of the
latent heat transformed into the bulk motion of the plasma Espinosa  _et al._
(2010), and $\Upsilon$ is the suppression factor Ellis  _et al._ (2020b); Guo
_et al._ (2021),

$\displaystyle f_{s}$ $\displaystyle=$ $\displaystyle(0.19\ {\rm
Hz})\left(\frac{T_{*}}{1\ {\rm
PeV}}\right)\left(\frac{g_{*}}{100}\right)^{\frac{1}{6}}\tilde{\beta}\ ,$
$\displaystyle\kappa_{s}$ $\displaystyle=$
$\displaystyle\frac{\alpha}{0.73+0.083\sqrt{\alpha}+\alpha}\ ,$
$\displaystyle\Upsilon\,$ $\displaystyle=$ $\displaystyle
1-\frac{1}{\Big{[}1+\frac{8\pi^{\frac{1}{3}}}{\tilde{\beta}}\big{(}\frac{{\alpha+1}}{3\kappa_{s}\alpha}\big{)}^{\frac{1}{2}}\Big{]}^{\frac{1}{2}}}\
.$ (50)

The signal from bubble wall collisions is Kosowsky  _et al._ (1992); Huber and
Konstandin (2008); Caprini  _et al._ (2016) (see Lewicki and Vaskonen (2021)
for recent updates)

$\displaystyle h^{2}\Omega_{c}(f)\,$ $\displaystyle\approx$
$\displaystyle\,\frac{4.9\times
10^{-6}}{\tilde{\beta}^{2}}\left(\frac{\kappa_{c}\,\alpha}{\alpha+1}\right)^{2}\left(\frac{100}{g_{*}}\right)^{\frac{1}{3}}$
(51) $\displaystyle\times$
$\displaystyle\frac{(f/f_{c})^{2.8}}{1+2.8(f/f_{c})^{3.8}}\ ,\ \ \ \ \ \ $

where now $f_{c}$ is the peak frequency and $\kappa_{c}$ is the fraction of
the latent heat deposited into the bubble front Kamionkowski  _et al._ (1994),

$\displaystyle f_{c}$ $\displaystyle=$ $\displaystyle(0.037\ {\rm
Hz})\left(\frac{T_{*}}{1\ {\rm
PeV}}\right)\left(\frac{g_{*}}{100}\right)^{\frac{1}{6}}\tilde{\beta}\ ,$
$\displaystyle\kappa_{c}$ $\displaystyle=$
$\displaystyle\frac{\frac{4}{27}\sqrt{\frac{3}{2}\alpha}+0.72\,\alpha}{1+0.72\,\alpha}\
.$ (52)

The final contribution is provided by turbulence Caprini and Durrer (2006);
Caprini  _et al._ (2009),

$\displaystyle h^{2}\Omega_{t}(f)\,$ $\displaystyle\approx$
$\displaystyle\,\frac{3.4\times
10^{-4}}{\tilde{\beta}}\left(\frac{\epsilon\,\kappa_{s}\,\alpha}{\alpha+1}\right)^{\frac{3}{2}}\left(\frac{100}{g_{*}}\right)^{\frac{1}{3}}$
(53) $\displaystyle\times$
$\displaystyle\frac{({f}/{f_{t}})^{3}}{\big{(}1+{8\pi
f}/{f_{*}}\big{)}\big{(}1+{f}/{f_{t}}\big{)}^{{11}/{3}}}\ ,$

where $\epsilon=0.05$ Caprini  _et al._ (2016), while the peak frequency
$f_{t}$ and the parameter $f_{*}$ are

$\displaystyle f_{t}$ $\displaystyle=$ $\displaystyle(0.27\ {\rm
Hz})\left(\frac{g_{*}}{100}\right)^{\frac{1}{6}}\left(\frac{T_{*}}{1\ {\rm
PeV}}\right)\tilde{\beta}\ ,$ $\displaystyle f_{*}$ $\displaystyle=$
$\displaystyle(0.17\ {\rm
Hz})\left(\frac{g_{*}}{100}\right)^{\frac{1}{6}}\left(\frac{T_{*}}{1\ {\rm
PeV}}\right)\ .$ (54)

Figure 6: Stochastic gravitational wave background from first order phase
transitions triggered by ${\rm U}(1)_{B}$ breaking assuming the model
parameters $g_{B}=0.25$ and $\lambda_{B}=0.006$, for several choices of the
symmetry breaking scale. The shaded regions correspond to the sensitivity of
future gravitational wave detectors, as in Fig. 1.

The resulting gravitational wave signals from the first order phase transition
triggered by ${\rm U}(1)_{B}$ breaking are shown in Fig. 6 for several
symmetry breaking scales: $10\ {\rm TeV}$ (light brown curve), $100\ {\rm
TeV}$ (brown curve), $1\ {\rm PeV}$ (purple curve), and $10\ {\rm PeV}$ (black
curve). The gauge coupling was assumed to be $g_{B}=0.25$, and the quartic
coupling was chosen to be $\lambda_{B}=0.006$. Spectra with peaks at larger
frequencies correspond to higher symmetry breaking scales.

The effect of the suppression factor $\Upsilon$ reducing the sound wave
contribution in Eq. (49) is that the bubble collision and turbulence
components become distinguishable in the spectrum. Although the main peak is
still due to sound waves, the slope at lower frequencies is dominated by
bubble collisions, whereas for higher frequencies the turbulence contribution
visibly changes the slope of the curve. Without the suppression factor, the
spectrum would be determined, to a good approximation, just by the sound wave
component in the region relevant for future detectors.

Depending on the Lagrangian parameters, the signal may be detectable in
upcoming gravitational wave experiments: LISA, DECIGO, Big Bang Observer,
Einstein Telescope, and Cosmic Explorer. To study this more quantitatively, in
Fig. 7 we show part of the $(g_{B},\lambda_{B})$ parameter space for which the
signal can be detected in each experiment when $v_{B}=1\ {\rm PeV}$.
Specifically, the upper boundary for each detector corresponds to a signal-to-
noise ratio of five upon a single year of data taking, while the lower
boundary arises when either $S(T)/T$ is too large to satisfy Eq. (44) or the
new vacuum has an energy density larger than that of the high temperature
vacuum.

Figure 7: Regions of parameter space $(g_{B},\lambda_{B})$ assuming $v_{B}=1\
{\rm PeV}$ where the gravitational wave signal from a first order phase
transition has a signal-to-noise ratio greater than five upon one year of data
taking in various experiments. The choice of colors matches that in Fig. 6,
and the dot corresponds to the purple curve.

As mentioned earlier, our analysis of the first order phase transition signal
from ${\rm U}(1)_{B}$ breaking differs from the one in Fornal and Shams Es
Haghi (2020) in several aspects. The fermionic particle content in Eq. (II) is
different, and we chose the corresponding Yukawa couplings to be small, which
is a more minimal scenario than $Y=0.6$ in Fornal and Shams Es Haghi (2020).
Our analysis is also applicable to ${\rm U}(1)_{L}$ breaking, since we
calculate the thermal masses in the general case – this result will be used in
Section VII. Finally, when determining the gravitational wave signal we
included the effect of bubble collisions, which was not considered in Fornal
and Shams Es Haghi (2020), but which increases the reach of upcoming detectors
due to the enhancement of the signal in the lower frequency region.

##  VII Gravitational wave signatures

In this section we demonstrate the diversity of gravitational wave signatures
expected within the framework of the model, searchable in near-future
experiments. The cases enumerated in Section II, corresponding to the possible
structures of the scalar sectors, give rise to the coexistence in the spectrum
of the following gravitational wave signals from first order phase transitions
(PT), cosmic strings (CS), and domain walls (DW):

  * $(a)$

${\rm(PT+PT),(PT+CS),(CS+CS);}$

  * $(b,c)$

${\rm(PT+PT),(PT+CS),(PT+DW),(CS+DW);}$

  * $(d)$

${\rm(PT+PT),(PT+DW),(DW+DW).}$

We discuss below explicit examples of how those signatures are realized in our
model. Two of them, ${\rm(DW+DW)}$ and ${\rm(CS+DW)}$ have not been considered
in the literature before, whereas ${\rm(PT+CS)}$, ${\rm(PT+PT)}$ and
${\rm(PT+DW)}$ have been already proposed. The case ${\rm(CS+CS)}$ does not
give rise to any new features, since the signal is dominated by the cosmic
string contribution from the higher symmetry breaking due to the flatness of
the cosmic string spectrum.

### Domain walls + domain walls

Figure 8: First novel gravitational wave signature of the model consisting of
a double domain wall peak, realized when each of the ${\rm U}(1)$ symmetries
is broken by two scalars – case $(d)$.
Figure 9: Second novel gravitational wave signature consisting of a domain
wall peak over a cosmic string background, realized when one ${\rm U}(1)$
symmetry is broken by one scalar and the other ${\rm U(1)}$ is broken by two
scalars – cases $(b)$ and $(c)$.

A new gravitational wave signature arises when each of the two ${\rm U}(1)$
symmetries is broken by two scalars, leading to the production of domain walls
at two different energy scales during the evolution of the Universe. The
signal consists of two sharp domain wall peaks. The slope on the left side of
each peak depends on the frequency like $\sim f^{3}$, whereas the slope on the
right side of the peak falls like $\sim 1/f$. There is a nontrivial structure
created between the two peaks, which can be used to distinguish this type of
signal from others. If the two symmetry breaking scales are high, this
signature can be searched for in all the upcoming gravitational wave
experiments we considered: LISA, DECIGO, Big Bang Observer, Einstein
Telescope, and Cosmic Explorer. A realization of this scenario in our model is
shown in Fig. 8, where the parameters for the ${\rm U(1)}_{B}$ breaking were
chosen to be $v_{B}=10^{3}\ {\rm PeV}$ and $\Delta\rho=10^{-5}\ {\rm
PeV^{4}}$, whereas for the ${\rm U(1)}_{L}$ breaking they are $v_{L}=5\times
10^{4}\ {\rm PeV}$ and $\Delta\rho=1.6\times 10^{5}\ {\rm PeV^{4}}$.

### Cosmic strings + domain walls

Another gravitational wave signature, not considered in the literature before,
is realized when one of the ${\rm U}(1)$ symmetries is broken by one scalar,
leading to cosmic string production, whereas the other ${\rm U}(1)$ is broken
by two scalars, resulting in domain wall creation. If the two symmetry
breaking scales are high, their contributions may overlap and produce a very
unusual domain wall peak over the cosmic string background. An example is
shown in Fig. 9, where the symmetry breaking scale for ${\rm U}(1)_{L}$ was
chosen to be $v_{L}=10^{6}\ {\rm PeV}$, whereas the parameters for ${\rm
U}(1)_{B}$ breaking are $v_{B}=8\times 10^{3}\ {\rm PeV}$ and $\Delta\rho=2\
{\rm PeV^{4}}$. For this particular selection of parameters, Big Bang Observer
and DECIGO can probe the peak area, but for lower ${\rm U}(1)_{B}$ breaking
scales this structure is accessible to LISA, whereas higher symmetry breaking
scales would make it detectable by Einstein Telescope and Cosmic Explorer.

Figure 10: Gravitational wave signature with a first order phase transition
peak over a cosmic string background, first proposed in Fornal and Shams Es
Haghi (2020), realized when one ${\rm U}(1)$ is broken by only one scalar and
the other ${\rm U}(1)$ is broken either by one or two scalars – cases $(a)$,
$(b)$, $(c)$.

### Phase transition + cosmic strings

If one of the symmetries is broken by a single scalar at the high scale, and
the other symmetry is broken by either one or two scalars at the low scale,
this can lead to a gravitational wave signature consisting of a phase
transition bump over a cosmic string background. This signature was first
proposed in Fornal and Shams Es Haghi (2020) in the context of a different
gauged baryon and lepton number model with a well-motivated large hierarchy
between symmetry breaking scales, and recently considered in another scenario
Ferrer  _et al._ (2023). In Fig. 10 we show an example of such a signal, where
${\rm U}(1)_{L}$ is broken at the scale $v_{L}=10^{6}\ {\rm PeV}$, whereas the
scale of ${\rm U}(1)_{B}$ breaking is $v_{B}=200\ \rm TeV$. The other
parameter values for this particular plot are $g_{B}=0.25$ and
$\lambda_{B}=0.006$. As in the previous case, by changing the scale of ${\rm
U}(1)_{B}$ breaking the bump can shift and become searchable not only by Big
Bang Observer and DECIGO, but also by Cosmic Explorer, Einstein Telescope, or
LISA.

### Phase transition + phase transition

Independently of the scalar sector structure, the breaking of two ${\rm U}(1)$
gauge symmetries can always result in a gravitational wave signal with two
first order phase transition peaks. Such a signature is generically expected
in theories with a multistep symmetry breaking pattern, and has been proposed
for various models of new physics Angelescu and Huang (2019); Greljo  _et al._
(2020); Fornal (2021). In Fig. 11 a realization of this signature is shown in
the case of our model, assuming that the ${\rm U}(1)_{B}$ symmetry is broken
by one scalar at the scale $v_{B}=20\ {\rm TeV}$ (the other parameters are
$g_{B}=0.25$ and $\lambda_{B}=0.006$), and the ${\rm U}(1)_{L}$ symmetry is
broken also by one scalar at the scale $v_{L}=5\ {\rm PeV}$ (with $g_{L}=0.20$
and $\lambda_{L}=0.0025$). We note that for the two contributions appropriate
formulas for the thermal masses were adopted, according to Eq. (VI). Such a
signal can be searched for in all the future gravitational wave detectors we
considered.

Figure 11: Gravitational wave signature consisting of two first order phase
transition peaks, similar to the ones proposed in Angelescu and Huang (2019);
Greljo  _et al._ (2020); Fornal (2021), arising when the two ${\rm U}(1)$
symmetries are broken by any number of scalars – realized in cases $(a)$,
$(b)$, $(c)$, $(d)$.

### Phase transition + domain wall

The final qualitatively different signature consists also of two peaks, but
this time one coming from a first order phase transition and the second one
arising from domain wall annihilation. Such a signal has very recently been
proposed in Fornal  _et al._ (2023). In our model it can be realized if there
is a large hierarchy between the ${\rm U}(1)_{B}$ and ${\rm U}(1)_{L}$
symmetry breaking scales. Figure 12 shows a realization of this scenario when
${\rm U}(1)_{L}$ is broken by two scalars at the scale $v_{L}=3\times 10^{4}\
{\rm PeV}$ (with a potential bias $\Delta\rho=6.3\times 10^{3}\ {\rm
PeV^{4}}$), whereas ${\rm U}(1)_{B}$ is broken by one scalar at the scale
$v_{B}=20\ {\rm TeV}$ (with the other parameters being $g_{B}=0.25$ and
$\lambda_{B}=0.006$). As pointed out in Fornal  _et al._ (2023), the two peaks
may appear in a different order, which would happen for a ${\rm U}(1)_{L}$
breaking scale of $v_{L}\sim 10^{3}\ {\rm PeV}$ and a ${\rm U}(1)_{B}$
breaking scale of $v_{B}\sim 10\ {\rm PeV}$. In both scenarios, the signature
can be searched for in the upcoming gravitational wave detectors we focused
on.

Although the signatures discussed above can be realized for any pattern of
symmetry breaking, the phenomenologically more attractive scenarios involve
${\rm U}(1)_{L}$ broken at the high scale, so that the bound in Eq. (18) is
satisfied and the theory can successfully accommodate leptogenesis, as
discussed in Section III. Additionally, with no motivation for a low ${\rm
U}(1)_{B}$ breaking scale in the model we are considering, the new signatures
shown in Figs. 8 and 9 can be naturally realized, and are quite appealing
given the reach of the upcoming gravitational wave experiments.

Figure 12: Gravitational wave signature containing a phase transition peak and
a domain wall peak, proposed in Fornal  _et al._ (2023), realized when at
least one ${\rm U}(1)$ symmetry is broken by two scalars – cases $(b)$, $(c)$,
$(d)$.

##  VIII Conclusions and Outlook

It is truly extraordinary that gravitational wave astronomy can join forces
with elementary particle physics to search for answers to fundamental
questions about the structure of the Universe and its earliest stages of
evolution. Indeed, processes happening at energies too high to be probed by
conventional particle physics detectors (such as high scale leptogenesis and
seesaw mechanism) can leave a remarkable imprint through the primordial
gravitational wave background emitted soon after the Big Bang. Detecting such
a signal would bring us closer to discovering which, if any, of the proposed
Standard Model extensions addressing the outstanding questions about dark
matter, matter-antimatter asymmetry, or neutrino masses, is realized in
Nature.

A stochastic gravitational wave background is expected to originate in the
early Universe within the framework of many particle physics models through
first order phase transitions, cosmic string dynamics and domain wall
annihilation. In particular, explaining the matter-antimatter asymmetry puzzle
requires a first order phase transition to happen, indicating the huge
importance of stochastic gravitational wave searches. The literature referred
to in Section I contains analyzes of such signatures in theories beyond the
Standard Model, however, the majority of the works focus on one single
component at a time, generally not looking at the possible interplay between
the contributions from different sources.

In this paper we highlighted the importance of searches for novel
gravitational wave signatures arising when multiple components are present in
the spectrum and add up producing new features in the signal. Such unique
signatures are expected in theories with more than one symmetry breaking, and
result from the interplay between the contributions from first order phase
transitions, cosmic strings, and/or domain walls. The new gravitational wave
signals we propose to look for are: $(1)$ Double-sharp-peak structure from
domain walls produced when two gauge symmetries are broken by multiple
scalars; $(2)$ Domain wall peak over a cosmic string plateau when one symmetry
is broken by a single scalar and the other symmetry is broken by multiple
scalars.

Although we demonstrate how those signatures arise in a specific model with
gauged baryon and lepton number, our results are applicable to a much wider
class of theories with two ${\rm U}(1)$ gauge symmetries broken at different
energy scales. Indeed, the new signals consist of the cosmic string and domain
wall contributions, thus they are fairly model-independent, since the cosmic
string component depends only on the symmetry breaking scale, whereas the
domain wall contribution depends on the symmetry breaking scale and the
potential bias. Our results can also be extended to models with non-Abelian
gauge groups. As already suggested in Fornal  _et al._ (2023), it would be
interesting to investigate the case when one of the symmetries is ${\rm
SU}(2)$ broken by two scalar triplets, as this can result in the production of
cosmic strings Hindmarsh  _et al._ (2016), and could perhaps lead to new
signals involving contributions from all three processes: first order phase
transitions, cosmic string dynamics, and domain wall annihilation.

The gravitational wave signatures discussed here can be searched for in
upcoming experiments, including LISA, Big Bang Observer, DECIGO, Cosmic
Explorer, and Einstein Telescope, enabling those detectors to probe the
structure of high-scale symmetry breaking sectors. This is especially relevant
for theories of leptogenesis such as the model we considered, in which,
contrary to Duerr  _et al._ (2013); Fornal and Shams Es Haghi (2020), the
scale of ${\rm U(1)_{B}}$ symmetry breaking is not bounded from above and can
also be high, allowing for signals $(1)$ and $(2)$ to be generated.

Finally, it is worth mentioning that a spontaneous breaking of a single gauge
symmetry can by itself lead to gravitational wave signatures combining signals
from a phase transition and cosmic strings, or a phase transition and domain
walls. Given the sensitivity of the experiments considered, the symmetry
breaking scale would have to be $\sim 100-1000\ {\rm PeV}$ for the combined
signal to be discoverable. The cosmic string contribution would then be
detectable by Big Bang Observer and DECIGO, the phase transition peak could be
seen by Cosmic Explorer and Einstein Telescope, and the domain wall peak would
be visible in LISA. Investigating this in more detail is an interesting
follow-up project, and could be tied to gravitational wave experiments
sensitive to lower frequencies, such as the pulsar timing arrays: NANOGrav
Arzoumanian  _et al._ (2018), PPTA Manchester  _et al._ (2013), EPTA Ferdman
_et al._ (2010), IPTA Hobbs  _et al._ (2010), or SKA Weltman  _et al._ (2020).

### Acknowledgments

This research was supported by the National Science Foundation under Grant No.
PHY-2213144.

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