# Gravitational Wave Signatures of Lepton Universality Violation

Bartosz Fornal  Department of Physics and Astronomy, University of Utah, Salt
Lake City, UT 84112, USA

###### Abstract

We analyze the prospects for using gravitational waves produced in early
universe phase transitions as a complementary probe of the flavor anomalies in
$B$ meson decays. We focus on the Left-Right SU(4) Model, for which the
strength of the observed lepton universality violation and consistency with
other experiments impose a vast hierarchy between the symmetry breaking
scales. This leads to a multipeaked gravitational wave signature within the
reach of upcoming gravitational wave detectors.

##  I Introduction

Although the Standard Model (SM) does not provide all the answers to
fundamental questions in particle physics and needs to be augmented by new
physics, nearing half a century since its formulation Glashow (1961); Weinberg
(1967); Salam (1968); Fritzsch and Gell-Mann ; Fritzsch  _et al._ (1973) it
has certainly stood the test of time with respect to its predictive power. A
huge number of models beyond the SM have been constructed proposing solutions
to the outstanding problems, however, it is not certain which of them, if any,
is realized in nature. At this time, guidance from experiment is especially
important in order to achieve further progress on the theory side.

So far, among the strongest experimental hints of new physics are the
indications of lepton universality violation in $B$ meson decays, the so-
called $R_{K^{(*)}}$ and $R_{D^{(*)}}$ anomalies. Although the $R_{D^{(*)}}$
anomalies (reported by BaBar Lees  _et al._ (2013), Belle Sato  _et al._
(2016) and LHCb Aaij  _et al._ (2018a)) have not been confirmed in the most
recent set of Belle data Abdesselam  _et al._ (2019a), and the $R_{K^{*}}$
anomaly reported by LHCb Aaij  _et al._ (2017a) has become less significant
Abdesselam  _et al._ (2019b), the $R_{K}$ anomaly Aaij  _et al._ (2014) has
persisted with new LHCb data Aaij  _et al._ (2019).

From an effective theory point of view Kosnik (2012); Alonso  _et al._ (2014);
Bhattacharya  _et al._ (2015); Alonso  _et al._ (2015); Bhattacharya  _et al._
(2017); Geng  _et al._ (2017); Alok  _et al._ (2019); Buttazzo  _et al._
(2017); Kumar  _et al._ (2019) the observed signals of lepton universality
violation are best accounted for by either the vector leptoquark
$(3,1)_{\frac{2}{3}}$ or $(3,3)_{\frac{2}{3}}$. A natural origin of the former
in the context of flavor anomalies has been proposed in Assad  _et al._
(2018), where it was suggested that this leptoquark can be the gauge boson of
a Pati-Salam-type unified model. This has been followed by several efforts
aimed at explaining either both the $R_{K^{(*)}}$ and $R_{D^{(*)}}$ anomalies
Calibbi  _et al._ (2017); Di Luzio  _et al._ (2017); Bordone  _et al._ (2018);
Barbieri and Tesi (2018); Blanke and Crivellin (2018); Greljo and Stefanek
(2018); Di Luzio  _et al._ (2018); Fuentes-Martin  _et al._ (2019); Fuentes-
Martin and Stangl (2020); Guadagnoli  _et al._ (2020) or just the
$R_{K^{(*)}}$ Balaji  _et al._ (2019); Fornal  _et al._ (2019) through the
vector leptoquark $(3,1)_{\frac{2}{3}}$ in UV complete models. Other
explanations of the anomalies include $Z^{\prime}$ bosons Cline  _et al._
(2017); Marzocca (2018); Guadagnoli  _et al._ (2018); Li  _et al._ (2018);
Faber  _et al._ (2018); Allanach and Davighi (2018) and scalar leptoquarks Cai
_et al._ (2017); Heeck and Teresi (2018); Becirevic  _et al._ (2018); Bigaran
_et al._ (2019).

Since the latest results on $R_{D^{(*)}}$ from Belle Abdesselam  _et al._
(2019a) are consistent with the SM, lowering the overall significance of those
anomalies, in this paper we focus on the solution to just the $R_{K^{(*)}}$
anomalies offered by the _Left-Right SU(4) Model_ Fornal  _et al._ (2019).
This is the only model for the flavor anomalies proposed so far which does not
require any mixing between quarks and new vector-like fermions. Apart from the
existing experimental searches for lepton universality violation, the only
other conventional way of looking for signatures of this model is to produce
the vector leptoquark in particle colliders. However, given its large mass of
$\sim 10\ {\rm TeV}$, this would require using the $100\ {\rm TeV}$ Future
Circular Collider, whose construction has not yet been approved.

A new window of opportunities for probing particle physics models has recently
been opened by gravitational wave experiments. The gravitational wave
detectors LIGO Aasi  _et al._ (2015) and Virgo Acernese  _et al._ (2015), in
addition to observing signals from astrophysical phenomena such as black hole
and neutron star mergers, have unique capabilities of detecting the imprints
of cosmic events in the early universe, providing access to regions of
parameter space unexplored so far in various extensions of the SM. This will
be even more promising with future experiments such as the Laser
Interferometer Space Antenna (LISA) Amaro-Seoane  _et al._ (2017), Cosmic
Explorer (CE) Reitze  _et al._ (2019), Einstein Telescope (ET) Punturo  _et
al._ (2010), DECIGO Kawamura  _et al._ (2011) or Big Bang Observer (BBO)
Crowder and Cornish (2005).

One class of particle physics signals that gravitational wave detectors are
sensitive to arises from early universe phase transitions. If the scalar
potential has a nontrivial vacuum structure, the universe could have settled
in a state which, as the temperature dropped, became metastable. The universe
would then undergo a transition via thermal fluctuations from the false vacuum
to the true vacuum. During such a first order phase transition, bubbles of
true vacuum would form in different patches of the universe and start
expanding. Gravitational waves would be generated from bubble wall collisions,
magnetohydrodynamic turbulence and sound shock waves of the early universe
plasma generated by the bubble’s violent expansion. At the Lagrangian level of
a theory, a phase transition is triggered by spontaneous symmetry breaking. In
models with a rich gauge structure, multiple steps of symmetry breaking can
occur, resulting in a chain of phase transitions, each generating
gravitational waves.

First order phase transitions from symmetry breaking have been studied with
respect to their predictions regarding the gravitational wave signals in
various models of new physics (see, e.g., Apreda  _et al._ (2002); Grojean and
Servant (2007); Leitao  _et al._ (2012); Schwaller (2015); Huang and Zhang
(2019); Huang and Yu (2018); Demidov  _et al._ (2018); Hashino  _et al._
(2018); Madge and Schwaller (2019); Ahriche  _et al._ (2019); Brdar  _et al._
(2019a); Croon  _et al._ (2019); Angelescu and Huang (2019); Hasegawa  _et
al._ (2019); Dev  _et al._ (2019); Brdar  _et al._ (2019b); Wang  _et al._
(2020); Greljo  _et al._ (2020); Von Harling  _et al._ (2020); Hall  _et al._
(2020); Huang  _et al._ (2020)). Here we investigate the complementarity
between gravitational wave experiments and direct searches for lepton
universality violation. Such a connection has recently been made in Greljo
_et al._ (2020) in the context of the Pati-Salam Cubed Model Bordone  _et al._
(2018), which consists of three copies of the Pati-Salam gauge group, each for
a different family of particles. In the Left-Right SU(4) Model which we are
considering, the gauge group is common to all the families. The symmetry
breaking pattern consists of three steps, each leading to a distinct peak in
the gravitational wave spectrum. The position of the two lower-frequency peaks
in the three-peaked gravitational wave spectrum is determined by the magnitude
of the flavor anomalies, offering a way to discriminate the model.

##  II Left-right SU(4) model

In this section we provide a summary of the most important properties of the
model (for further details, see Fornal  _et al._ (2019)).

Fermion fields
---
$\mathcal{G}$ | SM rep. | masses
$\hat{\Psi}_{L}=(4,1,2,0)$ | $Q_{L}=(3,2)_{\frac{1}{6}}$ |
| $L_{L}=(1,2)_{-\frac{1}{2}}$ |
$\hat{\Psi}_{R}^{d}=(1,4,1,-\tfrac{1}{2})$ | $d_{R}=(3,1)_{-\frac{1}{3}}$ | SM masses for
| $e_{R}=(1,1)_{-1}$ | quarks and leptons
$\hat{\Psi}_{R}^{u}=(1,4,1,\tfrac{1}{2})$ | $u_{R}=(3,1)_{\frac{2}{3}}$ | $m_{\nu}\sim\frac{v^{2}}{v_{10}}$
| $\nu_{R}=(1,1)_{0}$ | $M_{\nu_{R}}\sim v_{10}$
$\hat{\chi}_{L}=(\bar{4},1,2,0)$ | $Q^{\prime}_{L}=(\bar{3},2)_{-\frac{1}{6}}$ |
| $L^{\prime}_{L}=(1,2)_{\frac{1}{2}}$ | $(M_{Q^{\prime}})_{ij}=\frac{Y_{ij}}{\sqrt{2}}v_{\Sigma}$
$\hat{\chi}_{R}=(1,\bar{4},2,0)$ | $Q^{\prime}_{R}=(\bar{3},2)_{-\frac{1}{6}}$ | $(M_{L^{\prime}})_{ij}=\frac{Y_{ij}}{\sqrt{2}}z\,v_{\Sigma}$
| $L^{\prime}_{R}=(1,2)_{\frac{1}{2}}$ |

Scalar fields
---
$\mathcal{G}$ | SM rep. | masses
$\hat{\Sigma}_{R}=(1,4,1,\tfrac{1}{2})$ | $(3,1)_{\frac{2}{3}}$ , $(1,1)_{0}$ | radial mode
|  | $\sqrt{2\lambda_{R}}\,v_{R}\\!$
$\hat{\Sigma}_{L}=(4,1,1,\tfrac{1}{2})$ | $(3,1)_{\frac{2}{3}}$ , $(1,1)_{0}$ | radial mode
|  | $\ \ \sqrt{2\lambda_{L}}\,v_{L}\\!\ \ $
|  $(8,1)_{0}$ , $(3,1)_{\frac{2}{3}}$ , $(1,1)_{0}$ , | radial modes
$\hat{\Sigma}=(\bar{4},4,1,0)$ | $(\bar{3},1)_{\text{--}\frac{2}{3}}$ , $(1,1)_{0}$ | $\sim\sqrt{\lambda^{(\prime)}_{\Sigma}}\,v_{\Sigma}$
|  $(8,2)_{\frac{1}{2}}$ , $(3,2)_{\frac{7}{6}}$ , $(1,2)_{\frac{1}{2}}$ , | $\sim M$
$\hat{H}_{d}=(4,\bar{4},2,\tfrac{1}{2})$ |  $(\bar{3},2)_{\text{--}\frac{1}{6}}$ , $(1,2)_{\frac{1}{2}}$ | $\\!m_{S^{\prime}_{1}}=m_{h}\\!$
|  $(8,2)_{\text{--}\frac{1}{2}}$, $(3,2)_{\frac{1}{6}}$, $(1,2)_{\text{--}\frac{1}{2}}$, |
$\\!\hat{H}_{u}=(4,\bar{4},2,\text{--}\tfrac{1}{2})$ |  $(\bar{3},2)_{\text{--}\frac{7}{6}}$ , $(1,2)_{\text{--}\frac{1}{2}}$ | $\sim M$
$\hat{\Phi}_{10}\\!=\\!(1,\overline{10},1,\text{--}1)$ | $(\bar{6},1)_{\text{--}\frac{4}{3}}$ , $(\bar{3},1)_{\text{--}\frac{2}{3}}$ , $(1,1)_{0}$ | $\sim\,v_{10}$

Vector fields
---
$\mathcal{G}$ | SM rep. | masses
| $g$, $W^{a}_{\mu}$, $Z$ | SM particles
$G^{A}_{R\mu}$ | $X_{R}=(3,1)_{\frac{2}{3}}$ |
| $X_{R}^{\dagger}=(\bar{3},1)_{-\frac{2}{3}}$ | $M_{X_{R}}=\frac{1}{2}g_{R}v_{R}$
$Z^{\prime}_{\mu}$ | $Z^{\prime}_{R}=(1,1)_{0}$ | $M_{Z^{\prime}_{R}}=\frac{1}{2}\sqrt{{g^{\prime}}_{\\!\\!1}^{2}+\frac{3}{2}g_{R}^{2}}\ v_{R}$
| $X_{L}=(3,1)_{\frac{2}{3}}$ |
$G^{A}_{L\mu}$ | $X_{L}^{\dagger}=(\bar{3},1)_{-\frac{2}{3}}$ | $M_{X_{L}}=\frac{1}{2}g_{L}v_{L}$
| $Z^{\prime}_{L}=(1,1)_{0}$ | $\\!M_{Z^{\prime}_{L}}=\sqrt{\frac{3{g^{\prime}_{1}}^{\\!2}(g_{L}^{2}+g_{R}^{2})+\frac{9}{2}\,g_{L}^{2}g_{R}^{2}}{8({g^{\prime}_{1}}^{\\!2}\\!+\frac{3}{2}g_{R}^{2})}}\ v_{L}$
$W^{a}_{\mu}$ | $G^{\prime}=(8,1)_{0}$ | $M_{G^{\prime}}=\frac{1}{\sqrt{2}}\sqrt{g_{L}^{2}+g_{R}^{2}}\ v_{\Sigma}$

Table 1: The fermion, scalar and vector particle content of the model. The
masses were calculated assuming the hierarchical vev structure $v_{10}\gg M\gg
v_{R}\gg v_{L}\gg v_{\Sigma}$.

The model is based on the gauge group

$\displaystyle\mathcal{G}={\rm SU}(4)_{L}\times{\rm SU}(4)_{R}\times{\rm
SU}(2)_{L}\times{\rm U}(1)^{\prime}\ .$ (1)

The fermion, scalar and vector particle contents are provided in Table 1. The
gauge group $\mathcal{G}$ is broken by the vacuum expectation values (vevs) of
the scalar fields $\hat{\Sigma}_{R}$, $\hat{\Sigma}_{L}$ and $\hat{\Sigma}$.
The parameters of the scalar potential can be chosen such that the following
vev structure is obtained,

$\displaystyle\langle\hat{\Sigma}_{R}^{i}\rangle=\frac{v_{R}}{\sqrt{2}}\,\delta^{i4},\
\ \ \ \langle\hat{\Sigma}_{L}^{i}\rangle=\frac{v_{L}}{\sqrt{2}}\,\delta^{i4},$
$\displaystyle\langle\hat{\Sigma}\rangle=\frac{v_{\Sigma}}{\sqrt{2}}\,{\rm
diag}(1,1,1,z)\ ,\hskip 19.91692pt$ (2)

where $z>0$. The ${\rm SU}(4)_{R}$ symmetry is broken at a high scale $v_{R}$
in order to suppress right-handed lepton flavor changing currents and comply
with the stringent experimental bounds, whereas the other scales, $v_{L}$ and
$v_{\Sigma}$, are constrained by the size of the $R_{K^{(*)}}$ anomalies to be
much lower than $v_{R}$. We make an additional assumption that there is also a
hierarchy between the scales $v_{L}$ and $v_{\Sigma}$, i.e.,

$\displaystyle v_{R}\gg v_{L}\gg v_{\Sigma}\ .$ (3)

This implies the following symmetry breaking pattern (with the numerical
choice for the vevs explained below):

$\begin{array}[]{c}{\rm SU}(4)_{L}\times{\rm SU}(4)_{R}\times{\rm
SU}(2)_{L}\times{\rm U}(1)^{\prime}\\\\[3.0pt] \hskip
39.83385pt\bigg{\downarrow}\hskip 8.53581pt{\scriptstyle v_{R}\ \sim\ 5000\
{\rm TeV}}\\\\[12.0pt] {\rm SU}(4)_{L}\times{\rm SU}(3)_{R}\times{\rm
SU}(2)_{L}\times{\rm U}(1)^{\prime\prime}\\\\[3.0pt] \hskip
31.86707pt\bigg{\downarrow}\hskip 8.53581pt{\scriptstyle v_{L}\ \sim\ 40\ {\rm
TeV}}\\\\[12.0pt] {\rm SU}(3)_{L}\times{\rm SU}(3)_{R}\times{\rm
SU}(2)_{L}\times{\rm U}(1)_{Y}\\\\[3.0pt] \hskip
28.45274pt\bigg{\downarrow}\hskip 8.53581pt{\scriptstyle v_{\Sigma}\ \sim\ 7\
{\rm TeV}}\\\\[12.0pt] {\rm SU}(3)_{c}\times{\rm SU}(2)_{L}\times{\rm
U}(1)_{Y}\\\\[7.0pt] \end{array}$

The ${\rm U}(1)^{\prime}$ charge $Y^{\prime}$, the ${\rm U}(1)^{\prime\prime}$
charge $Y^{\prime\prime}$ and the SM hypercharge $Y$ are related via

$\displaystyle Y^{\prime\prime}=Y^{\prime}+\tfrac{1}{6}\,{\rm diag}(1,1,1,-3)\
,$ $\displaystyle Y=\,Y^{\prime\prime}+\tfrac{1}{6}\,{\rm diag}(1,1,1,-3)\ .$
(4)

The covariant derivative can be written as

$\displaystyle\begin{aligned} D_{\mu}=\partial_{\mu}&+ig_{L}\hskip
1.13809ptG_{L\mu}^{A}T_{L}^{A}+ig_{R}\hskip 1.13809ptG_{R\mu}^{A}T_{R}^{A}\\\
&+ig_{2}\,W_{\mu}^{a}\,T^{a}+ig_{1}^{\prime}\,Y^{\prime}_{\mu}\,Y^{\prime}\
,\end{aligned}$ (5)

where the index $A=1,...,15$, the index $a=1,2,3$, and$T_{L}^{A}$,
$T_{R}^{A}$, $T^{a}$, $Y^{\prime}$ are the ${\rm SU}(4)_{L}$, ${\rm
SU}(4)_{R}$, ${\rm SU}(2)_{L}$, ${\rm U}(1)^{\prime}$ generators,
respectively. At the low scale, the gauge couplings $g_{L}$, $g_{R}$,
$g^{\prime}_{1}$ are related to the SM gauge couplings $g_{s}$, $g_{1}$ via

$\displaystyle g_{s}=\\!\frac{g_{L}g_{R}}{\sqrt{g_{L}^{2}+g_{R}^{2}}}\ ,\ \ \
g_{1}\\!=\\!\frac{g^{\prime}_{1}g_{L}g_{R}}{\sqrt{\tfrac{2}{3}{g^{\prime}_{1}}^{\\!2}(g_{L}^{2}+g_{R}^{2})+g_{L}^{2}g_{R}^{2}}}\
.\ \ \ \ \ \ \ $ (6)

The Lagrangian terms describing the fermion masses are

$\displaystyle\mathcal{L}_{f}$ $\displaystyle=$
$\displaystyle\big{[}\,y_{ij}^{d}\,\overline{\hat{\Psi}^{i}_{L}}\hat{H}_{d}\hat{\Psi}_{R}^{d\hskip
0.56905ptj}+y_{ij}^{u}\,\overline{\hat{\Psi}^{i}_{L}}\hat{H}_{u}\hat{\Psi}_{R}^{uj}+Y_{ij}\,\overline{\hat{\chi}^{i}_{L}}\,\hat{\Sigma}\,\hat{\chi}_{R}^{j}\,\big{]}$
(7) $\displaystyle+$ $\displaystyle{\rm h.c.}+y_{ij}^{u\hskip
0.56905pt\prime}\,\overline{(\hat{\Psi}_{R}^{ui})^{c}}\,\hat{\Phi}_{10}\hat{\Psi}_{R}^{uj}\
,$

where the scalar field $\hat{\Phi}_{10}=(1,\overline{10},1,-1)$ develops a
high-scale vev $v_{10}\sim 10^{13}\ {\rm GeV}$ and provides a seesaw mechanism
for the neutrino masses, $m_{\nu}\sim v^{2}/v_{10}$ with $v$ being the SM
Higgs vev. After symmetry breaking down to the SM gauge group, the fermion
mass terms become

$\displaystyle\mathcal{L}_{f}$ $\displaystyle\supset$
$\displaystyle\Big{[}\,y_{ij}^{d}\,\overline{L^{i}_{L}}{S}_{1}e_{R}^{j}+y_{ij}^{d}\,\overline{Q^{i}_{L}}{S}_{2}d_{R}^{j}+y_{ij}^{u}\,\overline{L^{i}_{L}}{S}_{3}^{*}\nu_{R}^{j}$
(8) $\displaystyle+$ $\displaystyle
y_{ij}^{u}\,\overline{Q^{i}_{L}}{S}_{4}^{*}u_{R}^{j}+\tfrac{1}{\sqrt{2}}Y_{ij}v_{\Sigma}\big{(}\overline{{Q^{\prime}}^{i}_{\\!\\!L}}\,{Q^{\prime}}_{\\!\\!R}^{j}+z\,\overline{{L^{\prime}}^{i}_{\\!\\!L}}\,{L^{\prime}}_{\\!\\!R}^{j}\big{)}\,\Big{]}$
$\displaystyle+$ $\displaystyle{\rm h.c.}+y_{ij}^{u\hskip
0.56905pt\prime}\,v_{10}\,\overline{(\nu_{R}^{i})^{c}}\,\nu_{R}^{j}\ .$

The scalar sector is described by the Lagrangian

$\displaystyle\mathcal{L}_{s}$ $\displaystyle=$
$\displaystyle|D_{\mu}\hat{\Sigma}_{R}|^{2}\\!+\\!|D_{\mu}\hat{\Sigma}_{L}|^{2}\\!+\\!|D_{\mu}\hat{\Sigma}|^{2}\\!+\\!|D_{\mu}\hat{H}_{d}|^{2}\\!+\\!|D_{\mu}\hat{H}_{u}|^{2}$
(9) $\displaystyle+$
$\displaystyle|D_{\mu}\hat{\Phi}_{10}|^{2}+V(\hat{\Sigma}_{R},\hat{\Sigma}_{L},\hat{\Sigma},\hat{H}_{d},\hat{H}_{u},\hat{\Phi}_{10})\
,$

where the scalar potential contains the following terms(traces are implicit),

$\displaystyle V\supset$ $\displaystyle-$
$\displaystyle\mu_{R}^{2}\,|\hat{\Sigma}_{R}|^{2}+\lambda_{R}|\hat{\Sigma}_{R}|^{4}-\mu_{L}^{2}\,|\hat{\Sigma}_{L}|^{2}+\lambda_{L}|\hat{\Sigma}_{L}|^{4}$
(10) $\displaystyle-$
$\displaystyle\mu_{\Sigma}^{2}\,|\hat{\Sigma}|^{2}+\lambda_{\Sigma}(\hat{\Sigma}\hat{\Sigma}^{\dagger})^{2}+\lambda_{\Sigma}^{\prime}|\hat{\Sigma}\hat{\Sigma}^{\dagger}|^{2}$
$\displaystyle+$
$\displaystyle\lambda_{12}|\hat{\Sigma}_{L}|^{2}|\hat{\Sigma}_{R}|^{2}+\lambda_{13}|\hat{\Sigma}_{L}|^{2}|\hat{\Sigma}|^{2}+\lambda_{23}|\hat{\Sigma}_{R}|^{2}|\hat{\Sigma}|^{2}$
$\displaystyle+$
$\displaystyle\lambda^{\prime}_{13}|\hat{\Sigma}_{L}\hat{\Sigma}|^{2}+\lambda^{\prime}_{23}|\hat{\Sigma}_{R}^{\dagger}\hat{\Sigma}|^{2}+\big{[}\kappa\,\hat{\Sigma}_{L}\hat{\Sigma}\,\hat{\Sigma}_{R}^{\dagger}+{\rm
h.c.}\big{]}$ $\displaystyle+$ $\displaystyle
M_{u}^{2}|\hat{H}_{u}|^{2}+M_{d}^{2}|\hat{H}_{d}|^{2}\ .$

In Eq. (10) we left out the cross terms between $\hat{\Sigma}_{R}$,
$\hat{\Sigma}_{L}$, $\hat{\Sigma}$ and $\hat{H}_{u}$, $\hat{H}_{d}$; the full
form of the scalar potential is given in Fornal  _et al._ (2019).

As discussed in Fornal  _et al._ (2019), it is possible to tune the parameters
of the scalar potential such that only one linear combination of the fields
$S_{1}$, $S_{2}$, $S_{3}$, $S_{4}$ is light, reproducing the SM scalar sector
at low energies ($S^{\prime}_{1}\equiv H$). The remaining fields
$S^{\prime}_{2,3,4}$ and all other components of $\hat{H}_{u}$ and
$\hat{H}_{d}$ have masses set by the hard mass parameters $M_{u}$, $M_{d}$,
which we take to be $M_{u}=M_{d}\equiv M\gg v_{R}$. The relative mass
hierarchies between the SM down-type quarks and charged leptons are reproduced
reasonably well within this minimal setup. One can also introduce into the
model the scalar representation $\hat{\Phi}_{15}=(15,1,1,0)$ that develops a
vev at a high scale and leads to terms
$\overline{\hat{\Psi}^{i}_{L}}\hat{H}_{d}\hat{\Psi}_{R}^{d\,j}\hat{\Phi}_{15}/\Lambda$
providing distinct contributions to the quark and lepton masses.

The Lagrangian terms involving the fermion and vector fields are given by

$\displaystyle\mathcal{L}_{v}$ $\displaystyle=$
$\displaystyle\overline{\hat{\Psi}}_{L}i\not{D}\,{\hat{\Psi}_{L}}+\overline{\hat{\Psi}}{}^{u}_{R}\,i\not{D}\,{\hat{\Psi}_{R}^{u}}+\overline{\hat{\Psi}}{}_{R}^{d}\,i\not{D}\,{\hat{\Psi}_{R}^{d}}$
(11) $\displaystyle+$
$\displaystyle\overline{\hat{\chi}}{}_{R}\,i\not{D}\,{\hat{\chi}_{R}}+\overline{\hat{\chi}}{}_{L}\,i\not{D}\,{\hat{\chi}_{L}}\
,$

which, at the low scale, result in the following interactions between quarks,
leptons and gauge leptoquarks,

$\displaystyle\mathcal{L}_{v}$ $\displaystyle\supset$
$\displaystyle\frac{g_{L}}{\sqrt{2}}\,X_{L\mu}\Big{[}L^{u}_{ij}\,(\bar{u}^{i}\gamma^{\mu}P_{L}\,\nu^{j})+L^{d}_{ij}\,(\bar{d}^{i}\gamma^{\mu}P_{L}\,e^{j})\Big{]}$
$\displaystyle+$
$\displaystyle\frac{g_{R}}{\sqrt{2}}\,X_{R\mu}\Big{[}R^{u}_{ij}\,(\bar{u}^{i}\gamma^{\mu}P_{R}\,\nu^{j})+R^{d}_{ij}\,(\bar{d}^{i}\gamma^{\mu}P_{R}\,e^{j})\Big{]}\\!+{\rm
h.c.},$

where $L^{u}$, $L^{d}$, $R^{u}$ and $R^{d}$ are mixing matrices. They are all
unitary and related to the Cabibbo-Kobayashi-Maskawa matrix and the
Pontecorvo-Maki-Nakagawa-Sakata matrix via $L^{u}=V_{\rm CKM}L^{d}U_{\rm
PMNS}\,$ and $\,R^{u}=V_{\rm CKM}R^{d}U_{\rm PMNS}$.

To circumvent the stringent experimental constraints on lepton universality
violation Britton  _et al._ (1992, 1994); Czapek  _et al._ (1993); Ambrose
_et al._ (1998a, b, 2000); Appel  _et al._ (2000); Sher  _et al._ (2005);
Ambrosino  _et al._ (2009); Aubert  _et al._ (2006, 2007a, 2007b, 2008);
Aaltonen  _et al._ (2009); Aaij  _et al._ (2018b, 2017b, 2017c); Aubert  _et
al._ (2007c); Miyazaki  _et al._ (2007, 2010); Bertl  _et al._ (2006), the
scale of ${\rm SU}(4)_{R}$ breaking needs to be $v_{R}\gtrsim 5000\ {\rm TeV}$
for a generic unitary matrix $R^{d}$ Fornal  _et al._ (2019). At the same
time, in order for the vector leptoquark $X_{L}$ to explain the $R_{K^{(*)}}$
anomalies, one requires Fornal  _et al._ (2019)

$\displaystyle\frac{M_{X_{L}}}{g_{L}\sqrt{{\rm
Re}\left(L_{22}^{d}L_{32}^{d*}-L_{21}^{d}L_{31}^{d*}\right)}}\approx 23\
\rm{TeV}\ .$ (13)

Because of the unitarity of the matrix $L^{d}$, this relation can be fulfilled
only if $M_{X_{L}}\lesssim(23\ {\rm TeV})\,g_{L}$. The experimental
constraints then force $L^{d}$ to be of the form

$\displaystyle L^{d}\,\approx{e^{i\phi}}\left(\
\begin{matrix}\vspace{0.5mm}\\!\delta_{1}&\delta_{2}&\ 1\\\
\vspace{1mm}\\!e^{i\phi_{1}}\cos\theta&e^{i\phi_{2}}\sin\theta&\ \delta_{3}\\\
\\!-e^{-i\phi_{2}}\sin\theta&e^{-i\phi_{1}}\cos\theta&\
\delta_{4}\end{matrix}\ \ \right),$ (14)

where $\delta_{i}\lesssim 0.02$. The allowed leptoquark mass in Eq. (13) is
maximized for $\theta=\pi/4$ and $\phi_{1}+\phi_{2}=0$, which implies

$\displaystyle v_{L}\lesssim 46\ {\rm TeV}\ .$ (15)

Given the assumption $v_{L}\gg v_{\Sigma}$, the lower bound on $v_{\Sigma}$ is
of relevance. It arises from LHC dijet searches for colorons Sirunyan  _et
al._ (2020) and translates to $v_{\Sigma}>6.6\ {\rm TeV}$. We take
$v_{\Sigma}=7\ {\rm TeV}$. The only particles other than $G^{\prime}$ with
masses governed by $v_{\Sigma}$ are the radial modes of $\hat{\Sigma}$ and the
vector-like fermions $Q^{\prime}$ and $L^{\prime}$. The former do not couple
to SM quarks, and our choice $v_{\Sigma}=7\ {\rm TeV}$ is consistent with
experimental bounds, even for $\lambda^{(\prime)}_{\Sigma}$ as small as $\sim
3\times 10^{-3}$. The latter do not mix with SM quarks, so for $Y_{ij}\sim 1$
this choice of $v_{\Sigma}$ is also consistent with collider searches, even
for a relatively small $z$.

In the subsequent analysis, we consider the hierarchical symmetry breaking
pattern with the following vevs,

$\displaystyle v_{R}\approx 5000\ {\rm TeV}\ ,\ \ \ \ v_{L}\approx 40\ {\rm
TeV}\ ,\ \ \ \ v_{\Sigma}\approx 7\ {\rm TeV}\ .\ \ \ \ \ \ \ \ $ (16)

The vev structure in Eq. (II) can be realized if the parameters of the scalar
potential satisfy the
conditions:$\lambda^{\prime}_{13}>4\lambda_{\Sigma}(v_{\Sigma}/v_{L})^{2}$,
$\lambda^{\prime}_{23}>4\lambda_{\Sigma}(v_{\Sigma}/v_{R})^{2}$,
$\lambda^{(\prime)}_{\Sigma}>0$ and $\kappa<0$. We also note that the
hierarchy between the vevs in Eq. (16) is not protected against radiative
corrections andrequires a tuning of the parameters $\lambda_{12}$,
$\lambda_{13}$ and $\lambda_{23}$.

##  III Effective potential

Because of the vast hierarchy of scales in the model and small cross terms in
the scalar potential, the three steps of symmetry breaking can be considered
independently from one another. Denoting the background fields as

$\displaystyle\phi_{R}\equiv{\rm Re}(\hat{\Sigma}_{R})_{4}\sqrt{2}\ ,\ \ \ \
\phi_{L}\equiv{\rm Re}(\hat{\Sigma}_{L})_{4}\sqrt{2}\ ,$ $\displaystyle\hskip
42.67912pt\phi_{\Sigma}\equiv{\rm Re}(\hat{\Sigma})^{1}_{1}\sqrt{2}\ ,$ (17)

the effective potential splits into three pieces,

$\displaystyle V_{\rm eff}=V_{\rm eff}^{(R)}(\phi_{R})+V_{\rm
eff}^{(L)}(\phi_{L})+V_{\rm eff}^{(\Sigma)}(\phi_{\Sigma})\ .$ (18)

Before analyzing the phase transitions, we first discuss the effective
potential in the general case. We adopt the collective notation for the
background fields $\phi=\phi_{R},\phi_{L},\phi_{\Sigma}$, the vevs
$v=v_{R},v_{L},v_{\Sigma}$ and the quartic couplings
$\lambda=\lambda_{R},\lambda_{L},\lambda_{\Sigma},\lambda^{\prime}_{\Sigma}$.
Each piece of the effective potential consists of a tree-level part, a one-
loop Coleman-Weinberg correction and a finite temperature contribution,

$\displaystyle V_{\rm eff}(\phi,T)=V_{\rm
tree}(\phi)+V_{\rm{loop}}(\phi)+V_{\rm temp}(\phi,T)\ .$ (19)

Using the fact that the minimum of the tree-level potential for
$\phi=\phi_{R},\phi_{L}$ is at $v=\mu/\sqrt{\lambda}$, one can write

$\displaystyle V_{\rm
tree}(\phi)=-\frac{1}{2}\lambda\,v^{2}\phi^{2}+\frac{1}{4}\lambda\,\phi^{4}\
.$ (20)

The tree-level potential for $\phi=\phi_{\Sigma}$ contains terms involving
$\lambda_{\Sigma}$ and $\lambda^{\prime}_{\Sigma}$ with different
$z$-dependence.

To obtain the Coleman-Weinberg term, we implement the cutoff regularization
scheme and assume that the minimum of the one-loop potential and the mass of
$\phi$ are the same as their tree-level values Anderson and Hall (1992). In
this scheme, the one-loop zero temperature correction is

$\displaystyle V_{\rm loop}(\phi)$ $\displaystyle=$ $\displaystyle\sum_{\rm
particles}\frac{n_{i}}{64\pi^{2}}\bigg{\\{}m_{i}^{4}(\phi)\left[\log\left(\frac{m_{i}^{2}(\phi)}{m_{i}^{2}(v)}\right)-\frac{3}{2}\right]$
(21) $\displaystyle\hskip 5.69054pt+\
2\,m_{i}^{2}(\phi)\,m_{i}^{2}(v)\bigg{\\}}\ ,\ \ \ \ \ $

where the sum is over all particles charged under the gauge group that
undergoes symmetry breaking, including the Goldstone bosons $\chi_{GB}$,
$n_{i}$ is the number of degrees of freedom with an extra minus sign for
fermions, and $m_{i}(\phi)$ are the background field-dependent masses. For the
contribution of the Goldstone bosons one needs to replace $m_{\chi_{GB}}(v)\to
m_{\Phi}(v)$, where $\Phi$ is the radial mode.

The temperature-dependent part of the potential consists of the one-loop
finite temperature contribution $V_{\rm temp}^{(1)}(\phi,T)$ and, in case of
bosonic degrees of freedom, the Daisy diagrams contribution $V_{\rm
temp}^{(2)}(\phi,T)$. The corresponding formulae are given by Quiros (2007)

$\displaystyle V_{\rm temp}^{(1)}(\phi,T)$ $\displaystyle=$
$\displaystyle\frac{T^{4}}{2\pi^{2}}\sum_{\rm
particles}n_{i}\int_{0}^{\infty}\\!dy\,y^{2}$ (22) $\displaystyle\times$
$\displaystyle\log\left(1\mp
e^{-\sqrt{{m_{i}^{2}(\phi)}/{T^{2}}+y^{2}}}\right),\ \ \ \ \ \ \ \ $

where the minus sign is for bosons and the plus sign is for fermions, and

$\displaystyle V_{\rm temp}^{(2)}(\phi,T)=\\!\frac{T}{12\pi}\sum_{\rm
bosons}\\!n^{\prime}_{i}\left\\{m_{i}^{3}(\phi)\\!-\\!\left[m_{i}^{2}(\phi)+\Pi_{i}(T)\right]^{\frac{3}{2}}\right\\}\\!.$

The thermal masses $\Pi_{i}(T)$ can be calculated following the prescription
provided in Comelli and Espinosa (1997).

Figure 1: The effective potential $V_{\rm eff}(\phi_{R},T)$ for the
temperatures: $T=0$, $T_{c}=1.1\ {\rm PeV}$ and $T=1.4\ {\rm PeV}$, assuming
$v_{R}=5\ {\rm PeV}$, $\ \lambda_{R}=0.011$ and after subtracting off the term
$V_{\rm eff}(0,T)$.

##  IV Phase transitions

A strong first order phase transition is required to produce a gravitational
wave signal. This occurs when the effective potential develops a barrier
separating the false vacuum from the true vacuum. We perform a scan over the
parameters of the model and identify the regions of parameter space which
yield gravitational wave signals most promising for detection. As a benchmark
scenario, we adopt the values $\lambda_{R}(v_{R})=0.011$,
$\lambda_{L}(v_{L})=0.029$ and
$\lambda^{(\prime)}_{\Sigma}(v_{\Sigma})=0.036$, as this choice of parameters
leads to a strong gravitational wave signal. For such small quartics the only
relevant contributions to the field-dependent masses and thermal masses are
those involving the gauge couplings $g_{R/L}$.

To properly estimate those contributions, we first analyze the running of the
gauge couplings. We match $g_{R}$ and $g_{L}$ to the SM strong coupling
$g_{s}$ at the scale $v_{\Sigma}=7\ {\rm TeV}$ via Eq. (6) and choose
$g_{R}(v_{\Sigma})=g_{L}(v_{\Sigma})$. This implies that
$g_{R}(v_{\Sigma})=g_{L}(v_{\Sigma})\simeq 1.44$. We then perform the running
using the renormalization group equations

$\displaystyle\frac{\partial
g_{R/L}(\mu)}{\partial\log\mu}=-\left(11-\frac{n_{s}}{6}-\frac{2n_{f}}{3}\right)\frac{g^{3}_{R/L}(\mu)}{16\pi^{2}}\
,$ (24)

where $n_{s}$ is the number of complex scalars and $n_{f}$ is the number of
Dirac fermions in the fundamental representation of the gauge group ${\rm
SU}(4)_{R}/{\rm SU}(4)_{L}$ with masses below the scale $\mu$. We find that
$g_{R}(v_{R})\simeq 1.01$ and $g_{L}(v_{L})\simeq 1.32$.

${(1)}$ _First phase transition_ : ${\rm SU}(4)_{R}\to{\rm SU}(3)_{R}$
This transition is triggered when the field $\hat{\Sigma}_{R}$ develops the
vev as in Eq. (II) with $v_{R}\approx 5000\ {\rm TeV}$. The relevant
background field-dependent masses are

$\displaystyle m_{X_{R}}(\phi_{R})=\tfrac{1}{2}g_{R}\phi_{R}\ ,\ \ \ \ \
m_{Z^{\prime}_{R}}(\phi_{R})=\tfrac{M_{Z^{\prime}_{R}}}{v_{R}}\,{\phi_{R}}\ .\
\ \ \ \ $ (25)

The numbers of degrees of freedom corresponding to the gauge bosons $X_{R}$
and $Z^{\prime}_{R}$ are: $n_{X_{R}}\\!=\\!18$ and $n_{Z^{\prime}_{R}}\\!=3$.
Thethermal masses are given by

$\displaystyle\Pi_{X_{R}}^{L}(T)=\Pi_{Z^{\prime}_{R}}^{L}(T)=\tfrac{8}{3}g_{R}^{2}T^{2}\
,$
$\displaystyle\Pi_{\Phi}(T)=\Pi_{\chi_{GB}}(T)\approx\tfrac{1}{8}\,\Big{(}3g_{R}^{2}+2\tfrac{M_{Z^{\prime}_{R}}^{2}}{v_{R}^{2}}\Big{)}\,T^{2}\
,\ \ \ \ $ (26)

where we dropped terms involving the small quartic coupling. The superscript
$L$ for the gauge boson thermal masses denotes longitudinal components, $\Phi$
is the radial mode and $\chi_{GB}$ are the Goldstone bosons. The corresponding
numbers of degrees of freedom are: $n^{L}_{X_{R}}=6$,
$n^{L}_{Z^{\prime}_{R}}=1$, $n_{\Phi}=1$ and $n_{\chi_{GB}}=7$.

Figure 1 shows the full $\phi_{R}$-dependent part of the effective potential,
$V_{\rm eff}(\phi_{R},T)-V_{\rm eff}(0,T)$, for the parameter values discussed
above and for three different temperatures: $T=0$, $T_{c}=1.1\ {\rm PeV}$ and
$T=1.4\ {\rm PeV}$. At the critical temperature $T_{c}$ the two vacua become
degenerate. The order parameter is equal to
$\xi^{(R)}\equiv{(\phi_{R})_{c}}/{T_{c}}\approx 4$, indicating a strong first
order phase transition.

${(2)}$ _Second phase transition_ : ${\rm SU}(4)_{L}\to{\rm SU}(3)_{L}$
This transition happens when the field $\hat{\Sigma}_{L}$ develops the vev
$v_{L}\approx 40\ {\rm TeV}$. The corresponding background field-dependent
masses and thermal masses are obtained from Eqs. (25) and (IV) upon
substituting $R\to L$. The critical temperature is $T_{c}\approx 15\ {\rm
TeV}$ and the order parameter $\xi^{(L)}\approx 3$.

${(3)}$ _Third phase transition_ : ${\rm SU}(3)_{R}\times{\rm
SU}(3)_{L}\to{\rm SU}(3)_{c}$
This symmetry breaking is triggered when $\hat{\Sigma}$ develops the vev as in
Eq. (II) with $v_{\Sigma}\approx 7\ {\rm TeV}$. For a small $z$ the
contribution of the cross terms to the effective potential is small, as are
those of the vector-like fermions $Q^{\prime}$ and $L^{\prime}$, even with
Yukawas $Y_{ij}\sim 1$ (see, e.g., Angelescu and Huang (2019); Davoudiasl  _et
al._ (2013) for the corresponding formulae). Therefore, the only relevant
background field-dependent mass is that of $G^{\prime}$. For transverse
components

$\displaystyle
m_{G^{\prime}}(\phi_{\Sigma})=\tfrac{1}{\sqrt{2}}\sqrt{g_{R}^{2}+g_{L}^{2}}\,\phi_{\Sigma}\
,$ (27)

with the number of degrees of freedom $n_{G^{\prime}}^{T}=16$. For
thelongitudinal modes of $G^{\prime}$ and the SM gluon, the masses
$m_{i}^{2}(\phi_{\Sigma})+\Pi_{i}^{L}(T)$ are given by the eigenvalues of the
matrix

$\displaystyle\mathcal{M}^{2}_{i}(\phi_{\Sigma},T)=\frac{1}{2}\begin{pmatrix}g_{R}^{2}(\phi_{\Sigma}^{2}+4T^{2})&-g_{R}g_{L}\phi_{\Sigma}^{2}\\\
-g_{R}g_{L}\phi_{\Sigma}^{2}&g_{L}^{2}(\phi_{\Sigma}^{2}+4T^{2})\end{pmatrix}.\
\ \ \ \ \ $ (28)

The numbers of degrees of freedom are $n^{L}_{G^{\prime}}=n^{L}_{g}=8$. The
thermal masses for the radial modes and Goldstone bosons are

$\displaystyle\Pi_{\Sigma}(T)\approx(g_{R}^{2}+g_{L}^{2})\,T^{2}$ (29)

with $n_{\Sigma}=32$. A strong first order phase transition occurs since
$\xi^{(\Sigma)}\approx 2$ for the critical temperature $T_{c}\approx 1.7\ {\rm
TeV}$.

##  V Gravitational wave signals

As a result of a first order phase transition, bubbles of true vacuum are
nucleated, they expand (with velocity $v_{w}$) and eventually fill up the
entire universe. The bubble nucleation rate per unit volume is given by the
expression Linde (1983)

$\displaystyle\Gamma(T)\approx\left(\frac{S_{3}(T)}{2\pi
T}\right)^{\frac{3}{2}}T^{4}\,e^{-\frac{S_{3}(T)}{T}}\ ,$ (30)

where $S_{3}(T)$ is the Euclidean action

$\displaystyle S_{3}(T)=4\pi\int
dr\,r^{2}\left[\frac{1}{2}\left(\frac{d\phi_{b}}{dr}\right)^{2}+V_{\rm
eff}(\phi_{b},T)\right].\ \ \ \ \ $ (31)

Here $\phi_{b}(r)$ is the ${\rm SO}(3)$ symmetric bounce solution describing
the profile of the expanding bubble, i.e., the solution of the equation

$\displaystyle\frac{d^{2}\phi_{b}}{dr^{2}}+\frac{2}{r}\frac{d\phi_{b}}{dr}-\frac{dV_{\rm
eff}(\phi,T)}{d\phi}\bigg{|}_{\phi=\phi_{b}}=0$ (32)

with the boundary conditions

$\displaystyle\frac{d\phi_{b}}{dr}\bigg{|}_{r=0}=0\ ,\ \ \ \ \
\phi_{b}(\infty)=\phi_{\rm false}\ ,$ (33)

where $\phi_{\rm false}=0$ is the field value of the false vacuum.

The phase transition begins at the temperature $T_{*}$, called the nucleation
temperature, at which $\Gamma(T_{*})\approx H^{4}$, where $H$ is the Hubble
value at that time. This is equivalent to

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

where $M_{P}=1.22\times 10^{19}\ {\rm GeV}$ is the Planck mass.

Each phase transition continues until most of the universe is filled with
bubbles of true vacuum. The inverse of the duration of this process, the so-
called $\tilde{\beta}$ parameter, is given by

$\displaystyle\tilde{\beta}\equiv
T_{*}\frac{d}{dT}\\!\left(\frac{S_{3}(T)}{T}\right)\bigg{|}_{T=T_{*}}\ .$ (35)

The strength of a phase transition, denoted by $\alpha$, is defined as

$\displaystyle\alpha\equiv\frac{\rho_{\rm vac}(T_{*})}{\rho_{\rm rad}(T_{*})}\
,$ (36)

where $\rho_{\rm vac}(T_{*})$ is the energy density of the false vacuum (i.e.,
the latent heat released during the phase transition) Ellis  _et al._ (2019),

$\displaystyle\rho_{\rm vac}(T_{*})=\Delta V_{\rm
eff}(T_{*})-T_{*}\frac{\partial\Delta V_{\rm eff}(T)}{\partial
T}\bigg{|}_{T=T_{*}}$ (37)

with $\Delta V_{\rm eff}(T)=V_{\rm eff}(\phi_{\rm false},T)-V_{\rm
eff}(\phi_{\rm true},T)\ ,$
and ${\rho_{\rm rad}(T_{*})}$ is the energy density of radiation at nucleation
temperature,

$\displaystyle\rho_{\rm rad}(T_{*})=\frac{\pi^{2}}{30}g_{*}T^{4}_{*}\ .$ (38)

In the expressions above $\phi_{\rm true}$ is the field value of the true
vacuum, whereas $g_{*}$ is the number of relativistic degrees of freedom at
the time of the transition. In our benchmarkscenario: $g_{*}^{(1)}\simeq 274$,
$g_{*}^{(2)}\simeq 252$ and $g_{*}^{(3)}\simeq 228$. The four parameters:
$\alpha$, $\tilde{\beta}$, $v_{w}$ and $T_{*}$ determine the size and peak
frequency of the stochastic gravitational wave signal. In our analysis we set
the bubble wall velocity to $v_{w}=0.6\,c$ (for an extensive discussion of
bubble expansion, see Espinosa  _et al._ (2010); Caprini  _et al._ (2016)).

Figure 2: Contributions to the gravitational wave signal of the phase
transition ${\rm SU}(4)_{R}\to{\rm SU}(3)_{R}$ arising from sound waves,
bubble collisions and magnetohydrodynamic turbulence.

There are three sources of gravitational waves generated from phase
transitions: sound waves, bubble collisions and magnetohydrodynamic
turbulence. Those three contributions combine linearly to give the total
gravitational wave signal

$\displaystyle h^{2}\Omega_{\rm GW}\approx h^{2}\Omega_{\rm
sound}+h^{2}\Omega_{\rm collision}+h^{2}\Omega_{\rm turbulence}\ .\ \ \ \ \ $
(39)

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

$\displaystyle h^{2}\Omega_{\rm sound}(\nu)$ $\displaystyle\ \approx\ $
$\displaystyle(1.86\times 10^{-5})\
\frac{v_{w}}{\tilde{\beta}}\left(\frac{\kappa_{s}\,\alpha}{1+\alpha}\right)^{2}\left(\frac{100}{g_{*}}\right)^{\frac{1}{3}}$
(40) $\displaystyle\times$
$\displaystyle\frac{\big{(}\frac{\nu}{\nu_{s}}\big{)}^{3}}{\Big{[}1+0.75\,\big{(}\frac{\nu}{\nu_{s}}\big{)}^{2}\,\Big{]}^{\frac{7}{2}}}\
,$

where the model-dependent parameter $\kappa_{s}$ is the fraction of the latent
heat that is transformed into the bulk motion of the plasma, approximated by
Espinosa  _et al._ (2010)

$\displaystyle\kappa_{s}\approx\frac{\alpha}{0.73+0.083\sqrt{\alpha}+\alpha}\
,$ (41)

and $\nu_{s}$ is the peak frequency given by

$\displaystyle\nu_{s}$ $\displaystyle=$ $\displaystyle(0.019\ {\rm
Hz})\,\frac{\tilde{\beta}}{v_{w}}\,\left(\frac{g_{*}}{100}\right)^{\frac{1}{6}}\left(\frac{T_{*}}{100\
{\rm TeV}}\right)\ .\ \ \ \ \ \ \ $ (42)

The contribution from bubble collisions is Kosowsky  _et al._ (1992); Huber
and Konstandin (2008); Caprini  _et al._ (2016)

$\displaystyle h^{2}\Omega_{\rm collision}(\nu)$ $\displaystyle\ \approx\ $
$\displaystyle(1.66\times
10^{-5})\,\frac{1}{\tilde{\beta}^{2}}\left(\frac{\kappa_{c}\,\alpha}{1+\alpha}\right)^{2}\left(\frac{100}{g_{*}}\right)^{\frac{1}{3}}$
(43) $\displaystyle\times$
$\displaystyle\left(\frac{v_{w}^{3}}{1+2.4\,v_{w}^{2}}\right)\frac{\big{(}\frac{\nu}{\nu_{c}}\big{)}^{2.8}}{1+2.8\big{(}\frac{\nu}{\nu_{c}}\big{)}^{3.8}}\
,$

where $\kappa_{c}$ is the fraction of the latent heat that is deposited into a
thin shell close to the bubble front Kamionkowski  _et al._ (1994),

$\displaystyle\kappa_{c}\approx\frac{0.715\,\alpha+\frac{4}{27}\sqrt{{\frac{3\alpha}{2}}}}{1+0.715\,\alpha}\
,$ (44)

and the peak frequency $\nu_{c}$ is

$\displaystyle\nu_{c}$ $\displaystyle=$ $\displaystyle(0.010\ {\rm
Hz})\,\tilde{\beta}\,\left(\frac{g_{*}}{100}\right)^{\frac{1}{6}}\left(\frac{T_{*}}{100\
{\rm TeV}}\right)$ (45) $\displaystyle\times$
$\displaystyle\left(\frac{1}{1.8-0.1v_{w}+v_{w}^{2}}\right)\ .$

Figure 3: Regions of parameter space for which a phase transition ${\rm
SU}(4)_{R}\to{\rm SU}(3)_{R}$ with $v_{R}=5\ {\rm PeV}$ gives rise to a
gravitational wave signal detectable at the Einstein Telescope (blue) and
Cosmic Explorer (blue and purple) with a signal-to-noise ratio greater than
five after one year of collecting data (see text for details). The red dot
denotes the benchmark parameters adopted in Fig. 4.

The contribution from turbulence is Caprini and Durrer (2006); Caprini  _et
al._ (2009)

$\displaystyle h^{2}\Omega_{\rm turbulence}(\nu)$ $\displaystyle\ \approx\ $
$\displaystyle(3.35\times
10^{-4})\,\frac{v_{w}}{\tilde{\beta}}\\!\left(\frac{\kappa_{t}\,\alpha}{1+\alpha}\right)^{\\!\frac{3}{2}}\\!\left(\frac{100}{g_{*}}\right)^{\\!\frac{1}{3}}$
(46) $\displaystyle\times$
$\displaystyle\frac{\big{(}\frac{\nu}{\nu_{t}}\big{)}^{3}}{\big{(}1+\frac{8\pi\nu}{h_{*}}\big{)}\big{(}1+\frac{\nu}{\nu_{t}}\big{)}^{\frac{11}{3}}}\
,$

where $\kappa_{t}=\epsilon\,\kappa_{s}$ denotes the fraction of the latent
heat transformed into magnetohydrodynamic turbulence (following Caprini  _et
al._ (2016), we take $\epsilon=0.05$), the peak frequency $\nu_{t}$ is

$\displaystyle\nu_{t}$ $\displaystyle=$ $\displaystyle(0.027\ {\rm
Hz})\,\frac{\tilde{\beta}}{v_{w}}\,\left(\frac{g_{*}}{100}\right)^{\frac{1}{6}}\left(\frac{T_{*}}{100\
{\rm TeV}}\right)\ \ \ \ \ $ (47)

and the parameter $h_{*}$ Caprini  _et al._ (2016)

$\displaystyle h_{*}=(0.0165\ {\rm
Hz})\left(\frac{g_{*}}{100}\right)^{\frac{1}{6}}\left(\frac{T_{*}}{100\ {\rm
TeV}}\right)\ .$ (48)

Figure 4: Gravitational wave signature of the Left-Right SU(4) Model (black
line) for the benchmark scenario described in Eq. (16). Overplotted are the
sensitivities of the future gravitational wave experiments: LISA in the C1
configuration Caprini  _et al._ (2016) (red), Big Bang Observer Yagi and Seto
(2011) (light green), DECIGO Yagi and Seto (2011) (dark green), Einstein
Telescope Sathyaprakash  _et al._ (2012) (blue) and Cosmic Explorer Reitze
_et al._ (2019) (purple). The three peaks correspond to the phase transitions:
$(1)\,$ ${\rm SU}(4)_{R}\to{\rm SU}(3)_{R}$, $(2)\,$ ${\rm SU}(4)_{L}\to{\rm
SU}(3)_{L}$ and $(3)\,$ ${\rm SU}(3)_{R}\times{\rm SU}(3)_{L}\to{\rm
SU}(3)_{c}$ discussed in Sec. IV.

To find the gravitational wave signal from the three phase transitions, we
analyzed separately the $\phi_{R}$, $\phi_{L}$, $\phi_{\Sigma}$-dependent
pieces of the potential in Eq. (18). In each case we determined numerically
the temperature at which the shape of the potential yields the Euclidean
action $S(T_{*})$ satisfying Eq. (V). For this nucleation temperature, we
calculated the values of the parameters $\alpha$, $\tilde{\beta}$, and used
them to derive the gravitational wave spectrum via Eqs. (39)-(48).

Our calculation revealed that in the Left-Right SU(4) Model the sound wave
contribution dominates over the contributions from bubble collisions and
magnetohydrodynamic turbulence in most of the peak region, thus the shape of
the signal is well approximated by Eq. (40) and the peak frequency by Eq.
(42). This is illustrated in Fig. 2 for the phase transition associated with
${\rm SU}(4)_{R}\to{\rm SU}(3)_{R}$.

In order to assess how generic are the phase transitions leading to detectable
gravitational wave signals in the model, we performed a scan over the relevant
parameters. Figure 3 presents the regions of parameter space for the gauge
coupling $g_{R}$ versus the quartic coupling $\lambda_{R}$ which yield a first
order phase transition ${\rm SU}(4)_{R}\to{\rm SU}(3)_{R}$ (for $v_{R}=5\ {\rm
PeV}$) giving rise to signals detectable at the Einstein Telescope and Cosmic
Explorer, with a signal-to-noise ratio greater than five after one year of
collecting data. In particular, the lower boundaries of those regions
correspond to the signal-to-noise ratio of five for each experiment. Above the
upper boundary, either the value of $S_{3}(T)/T$ is too large to satisfy the
condition in Eq. (34) or the zero temperature vacuum at $v_{R}\neq 0$ has a
higher energy density than the vacuum at $v_{R}=0$, which is unphysical, since
it would lead to a second transition back to the vacuum with $v_{R}=0$.
Similarly sized regions of parameter space relevant for the phase transitions
${\rm SU}(4)_{L}\to{\rm SU}(3)_{L}$ (for $v_{L}=40\ {\rm TeV}$) and ${\rm
SU}(3)_{R}\times{\rm SU}(3)_{L}\to{\rm SU}(3)_{c}$ (for $v_{\Sigma}=7\ {\rm
TeV}$) yield signals detectable in other gravitational wave experiments.

Figure 4 shows the combined spectrum of gravitational waves from all three
phase transitions in our benchmarkscenario, with the corresponding parameters
summarized in Table 2. As expected, each of the phase transitions produces a
distinct peak in the spectrum, characterized by a large signal-to-noise ratio.
As seen from Eq. (42), the position of individual peaks depends linearly on
the nucleation temperature $T_{*}$, thus signals from phase transitions
corresponding to higher symmetry breaking scales appear at higher frequencies.
The peak frequency depends also linearly on $\tilde{\beta}$. The height of the
peak is governed by $\alpha$ and $\tilde{\beta}$ ; it increases with bigger
$\alpha$ and decreases with larger $\tilde{\beta}$. Of course all those
parameters depend on the values of the vevs, quartic couplings and gauge
couplings in the model.

Phase transition | $\alpha$ | $\tilde{\beta}$ | $T_{*}$
---|---|---|---
$(1)\ \ \ {\rm SU}(4)_{R}\to{\rm SU}(3)_{R}$ | 0.35 | 270 | 430 TeV
$(2)\ \ \ {\rm SU}(4)_{L}\to\,{\rm SU}(3)_{L}$ | 0.09 | 120 | 6.4 TeV
$(3)\ \ \ {\rm SU}(3)_{R}\times{\rm SU}(3)_{L}\to{\rm SU}(3)_{c}$ | 0.02 | 60 | 0.9 TeV
Table 2: Values of the parameters $\alpha$, $\tilde{\beta}$ and $T_{*}$ for
the three phase transitions giving rise to the gravitational wave signal in
Fig. 4.

Within the benchmark scenario, the gravitational wave signal generated by the
symmetry breaking ${\rm SU}(4)_{R}\to{\rm SU}(3)_{R}$ at the scale
$v_{R}\approx 5\ {\rm PeV}$ (peak $(1)$) falls within the sensitivity of the
future gravitational wave detectors Cosmic Explorer and Einstein Telescope.
The signal resulting from the second phase transition ${\rm SU}(4)_{L}\to{\rm
SU}(3)_{L}$ at the scale $v_{L}\approx 40\ {\rm TeV}$ (peak $(2)$) is well
within the reach of the Big Bang Observer and DECIGO. The third phase
transition ${\rm SU}(3)_{R}\times{\rm SU}(3)_{L}\to{\rm SU}(3)_{c}$ occurring
at the scale $v_{\Sigma}\approx 7\ {\rm TeV}$ (peak $(3)$) can also be probed
by the Big Bang Observer and DECIGO. Inaddition, it can be searched for by
LISA, but only if the C1 configuration Caprini  _et al._ (2016) is
implemented.

A unique property of the Left-Right SU(4) Model is that the range of peak
frequencies for the phase transitions ${\rm SU}(4)_{L}\to{\rm SU}(3)_{L}$ and
${\rm SU}(3)_{R}\times{\rm SU}(3)_{L}\to{\rm SU}(3)_{c}$ is constrained by the
size of the $R_{K^{(*)}}$ anomalies, as described by Eq. (15). In our
benchmark scenario we assumed that there is a maximal hierarchy between the
scales $v_{L}$ and $v_{\Sigma}$, which leads to two well-separated peaks in
the spectrum. However, if the two scales are comparable, then the size of the
flavor anomalies sets the symmetry breaking scale at $v_{L}\approx
v_{\Sigma}\lesssim 25\ {\rm TeV}$, resulting in a single peak shifted towards
lower frequencies compared to peak $(2)$ in Fig. 4. This is still within the
reach of the Big Bang Observer and DECIGO.

Finally, we point out that the scale of the symmetry breaking ${\rm
SU}(4)_{R}\to{\rm SU}(3)_{R}$ is not bounded from above. In particular, it can
be larger than $\sim 100\ {\rm PeV}$, shifting peak $(1)$ to higher
frequencies and escaping the detection at the Cosmic Explorer and Einstein
Telescope. A gravitational wave experiment sensitive to such high frequencies
would be necessary to probe this scenario.

##  VI Conclusions

Gravitational wave experiments have recently emerged as a powerful tool for
testing particle physics models. One class of signatures which those
experiments are sensitive to arises from first order phase transitions in the
early universe, making them valuable probes of the scalar sector in models
with spontaneous symmetry breaking.

In this paper we demonstrated, in the context of the Left-Right SU(4) Model,
that gravitational wave detectors can be used to look for signatures specific
to the flavor anomalies recently observed at the LHCb, BaBar and Belle
experiments. The measured magnitude of lepton universality violation implies
that there can be two peaks in the gravitational wave spectrum within the
sensitivity of the upcoming LISA, Big Bang Observer and DECIGO experiments.
There is also a possibility of a third peak which could be observed by the
Cosmic Explorer and Einstein Telescope.

If the hints of lepton universality violation are confirmed and a
gravitational wave signal with features similar to those of the Left-Right
SU(4) Model is discovered, this would be a strong motivation for building the
100 TeV collider, which could provide a complementary direct detection method
of testing the model.

### Acknowledgments

The author is grateful to Peter Stangl and Yue Zhao forinspiring discussions
and helpful comments. This research was supported in part by the U.S.
Department of Energyunder Award No. ${\rm DE}$-${\rm SC0009959}$.

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