aainstitutetext: Department of Physics, Durham University, Durham DH1 3LE, U.
K.bbinstitutetext: Institute for Particle Physics Phenomenology, Durham
University, Durham DH1 3LE, U.K.ccinstitutetext: Physik-Institut, Universität
Zürich, 8057 Zürich, Switzerlandddinstitutetext: INFN, Sezione di Trieste,
SISSA, Via Bonomea 265, 34136, Trieste, Italy

# Radiative effects in the scalar sector of vector leptoquark models

Rachel Houtz<EMAIL_ADDRESS>c Julie Pagès
<EMAIL_ADDRESS>d and Sokratis Trifinopoulos
<EMAIL_ADDRESS>
###### Abstract

Gauge models with massive vector leptoquarks at the TeV scale provide a
successful framework for addressing the $B$-physics anomalies. Among them, the
4321 model has been considered as the low-energy limit of some complete
theories of flavor. In this work, we study the renormalization group evolution
of this model, laying particular emphasis on the scalar sector. We find that,
despite the asymptotic freedom of the gauge couplings, Landau poles can arise
at relatively low scales due to the fast running of quartic couplings.
Moreover, we discuss the possibility of radiative electroweak symmetry
breaking and characterize the fine-tuning associated with the hierarchy
between the electroweak scale and the additional TeV-scale scalars. Finally,
the idea of scalar fields unification is explored, motivated by ultraviolet
embeddings of the 4321 model.

###### Keywords:

††preprint:  IPPP/22/26, ZU-TH-13/22

hep-ph/***

##  1 Introduction

Recently, there has been a resurgence of interest in the study of leptoquark
fields in light of discrepancies in semi-leptonic $B$-decays, collectively
known as $B$-physics anomalies LHCb:2014vgu ; LHCb:2017avl ; LHCb:2019hip ;
LHCb:2021trn ; BaBar:2012obs ; BaBar:2013mob ; Belle:2015qfa ; LHCb:2015gmp ;
LHCb:2017smo ; LHCb:2017rln , which challenge the assumed lepton flavor
universal nature of fundamental interactions. In particular, the vector
leptoquark $U_{1}$ with Standard Model (SM) quantum numbers $({\bf 3},{\bf
1})_{2/3}$, originally proposed in Ref. Alonso:2015sja ; Calibbi:2015kma ;
Barbieri:2015yvd , has been identified as the only single-mediator solution
Buttazzo:2017ixm ; Angelescu:2021lln . The search for an ultraviolet (UV)
completion of the $U_{1}$ simplified model naturally leads to variations on
the Pati-Salam (PS) gauge group, $\mathcal{G}_{\rm PS}\equiv SU(4)\times
SU(2)_{L}\times SU(2)_{R}$ Pati:1974yy , as it features quark-lepton
unification. The $U_{1}$ arises as a massive gauge boson after spontaneous
symmetry breaking of the PS group to the SM. However, the original PS model
predicts a leptoquark with flavor-universal couplings which has to be very
heavy in order to avoid the strict bounds from processes involving the light
generations.

The minimal variation containing the $U_{1}$ leptoquark at TeV scale while
being phenomenological viable is $\mathcal{G}_{4321}\equiv SU(4)\times
SU(3)^{\prime}\times SU(2)_{L}\times U(1)_{X}$ Diaz:2017lit ; DiLuzio:2017vat
, where $SU(2)_{L}$ is identified with SM weak isospin, and both color and
hypercharge are diagonal subgroups (for examples of other variations, see
Refs. Barbieri:2017tuq ; Calibbi:2017qbu ; Fornal:2018dqn ; Blanke:2018sro ).
Models based on $\mathcal{G}_{4321}$ are called 4321 models and generally fall
into two categories based on the charge assignments of the SM matter fields:
i) flavor universal 4321 DiLuzio:2017vat ; DiLuzio:2018zxy , where all would-
be SM fermions are singlets under $SU(4)$ and have the usual SM charges and
ii) flavor non-universal 4321 Greljo:2018tuh ; Bordone:2018nbg ;
Cornella:2019hct ; Fuentes-Martin:2020bnh ; Guadagnoli:2020tlx ; Fuentes-
Martin:2020hvc ; Baker:2021llj , where the would-be third generation quarks
and leptons are unified into $SU(4)$ multiplets. This non-universality in
charge assignments translates into non-universality of the couplings of the
new heavy vector boson, among them the $U_{1}$, to the different generations.
The flavor non-universal models can also be viewed as the low-energy limit of
more fundamental theories that envision larger symmetry groups broken at
various scales and aspiring to dynamically explain the SM Yukawa hierarchies
(see e.g. the three-site PS model Bordone:2017bld , PS variations in five
dimensions Fuentes-Martin:2020pww ; Fuentes-Martin:2022xnb  or the twin PS
model King:2021jeo ). In the following, we will refer to UV models reducing to
the 4321 model at low-energy as UV embeddings of 4321.

For both types of 4321 model, the spontaneous symmetry breaking to the SM
requires the presence of new scalars at the TeV scale. The purpose of this
work is to investigate the Renormalization Group (RG) evolution of the
parameters in the extended scalar sector of the 4321 model. The system of RG
equations together with boundary conditions is in general under-determined,
since the number of parameters is larger than the number of formal and
phenomenological bounds, even in the most simplified version of the model. To
study this system, we employ different benchmark scenarios for the boundary
conditions. We consider two types of scenarios: i) agnostic towards further
assumptions in the UV, and ii) inspired by UV embeddings. The former naturally
allows for more freedom in the choice of boundary conditions and is hence
suitable for examining the UV behavior in general terms, while the latter
introduces additional constraints that have to be satisfied at a certain UV
scale. Since we are interested in extrapolating the model as far as possible
in the UV, we will focus on the flavor non-universal 4321 model because it has
been shown that the universal model exhibits Landau poles at energy scales
$\mathcal{O}(100-1000)~{}\rm TeV$ DiLuzio:2018zxy .

An interesting feature of the setup is the possibility of triggering
Electroweak Symmetry Breaking (EWSB) via radiative effects given by the RG
flow. Additional scalars coupled to the Higgs sector have the potential to
radiatively trigger EWSB, well-established in supersymmetric theories
Ibanez:1982fr ; Inoue:1982pi ; Ibanez:1982ee ; Ellis:1982wr ; Ellis:1983bp ;
Alvarez-Gaume:1983drc  and examined in non-supersymmetric extensions of the SM
in Ref. Babu:2016gpg . In these models, the positive mass squared parameter of
the Higgs boson at high energy flows to negative values at low energies, in a
similar way as in the Coleman-Weinberg mechanism Coleman:1973jx . Here, we
show that the additional TeV scalar states responsible for the 4321 breaking
can also radiatively trigger EWSB. We will establish the conditions for
radiative EWSB, and in particular, consider the sensitivity of the resulting
Higgs mass to the 4321 parameters arranged in the UV.

The paper is structured as follows. In Section 2 we present the non-universal
4321 model with special emphasis on the scalar potential. In Section 3 the
system of RG equations for the parameters of the scalar sector (Appendix A) is
scrutinized. After identifying the most important contributions driving the
behavior of the solution in Section 3.1, we determine the energy scale at
which the first Landau pole appears. In Section 3.2 we identify the conditions
required to trigger radiative EWSB and characterize the sensitivity of the
physical Higgs mass to the UV values of the relevant parameters. Finally, in
Section 3.3 we consider benchmarks satisfying constraints from UV models
(Appendix C) which unify the $\Omega$ fields and the SM Higgs with a second
Higgs doublet (Appendix B.2). We conclude in Section 4.

##  2 The model

###  2.1 Gauge and fermion content

##### Gauge structure and heavy vectors.

The SM gauge group $\mathcal{G}_{\rm SM}=SU(3)_{c}\times SU(2)_{L}\times
U(1)_{Y}$, with respective couplings $g_{s},g_{2},g_{Y}$, is a subgroup of the
4321 gauge group $\mathcal{G}_{4321}=SU(4)\times SU(3)^{\prime}\times
SU(2)_{L}\times U(1)_{X}$, with respective couplings
$g_{4},g_{3},g_{2},g_{1}$. The $SU(4)$ contains $SU(3)_{4}\times U(1)_{4}$ as
a subgroup. $SU(3)_{c}$ is diagonally embedded in $SU(3)_{4}\times
SU(3)^{\prime}$, while $U(1)_{Y}$ is embedded in $U(1)_{4}\times U(1)_{X}$
with $Y=\sqrt{\frac{2}{3}}T^{15}+X$ where $T^{15}$ is the rank 4 diagonal
generator of $SU(4)$. The spontaneous symmetry breaking
$\mathcal{G}_{4321}\rightarrow\mathcal{G}_{\rm SM}$ gives rise to 15 heavy
gauge bosons: a leptoquark $U_{1}\sim(\bm{3},\bm{1})_{2/3}$, a coloron
$G^{\prime}\sim(\bm{8},\bm{1})_{0}$, and a
$Z^{\prime}\sim(\bm{1},\bm{1})_{0}$, with masses given by

$\displaystyle\begin{aligned}
M_{U}&=\frac{g_{4}}{2}\sqrt{\omega_{1}^{2}+\omega_{3}^{2}}\,,\qquad
M_{G^{\prime}}=\sqrt{\frac{g_{4}^{2}+g_{3}^{2}}{2}}\,\omega_{3}\,,\\\
M_{Z^{\prime}}&=\frac{1}{2\sqrt{6}}\sqrt{\left(3g_{4}^{2}+2g_{1}^{2}\right)\left(3\omega_{1}^{2}+\omega_{3}^{2}\right)}\,,\end{aligned}$
(1)

where $\omega_{1}$ and $\omega_{3}$ are the Vacuum Expectation Values (VEVs)
of the scalars responsible for the spontaneous symmetry breaking (see Section
2.2).

Two cases have previously been considered for the VEVs ratio:

  1. i)

$\omega_{1}=\omega_{3}$ and $g_{1,3}\ll g_{4}$ as in Refs. Fuentes-
Martin:2020luw ; Fuentes-Martin:2020hvc  in which case there is a residual
global symmetry: $SU(4)_{V}$ custodial from $SU(4)\times SU(4)^{\prime}$ with
$SU(4)^{\prime}$ the global symmetry containing as subgroup
$SU(3)^{\prime}\times U(1)^{\prime}$.111The gauged $U(1)_{X}$ is then a
combination of $U(1)^{\prime}$ and another $U(1)$, see Ref. Fuentes-
Martin:2020bnh . In this case, the heavy vectors have degenerate masses i.e.
$M_{U_{1}}=M_{G^{\prime}}=M_{Z^{\prime}}=g_{4}\,\omega_{3}/\sqrt{2}$.

  2. ii)

$\omega_{3}>\omega_{1}$ as in Ref. DiLuzio:2018zxy  which is a preferred
phenomenological limit to avoid significant contributions to $\Delta F=2$
observables. In this case and in the limit $g_{1,3}\ll g_{4}$, we obtain
$M_{U_{1}}=\sqrt{2}M_{Z^{\prime}}=M_{G^{\prime}}/\sqrt{2}$.

While the limit (i) is particularly motivated in composite scalar sector (see
for example Ref. Fuentes-Martin:2020bnh ), the limit (ii) offers better
compatibility between the case of fundamental scalars and phenomenology. For
completeness the couplings of the SM gauge bosons, as well as of the heavy
bosons $U_{1}$, $G^{\prime}$ and $Z^{\prime}$ are given by DiLuzio:2017vat

$\displaystyle\begin{aligned}
&g_{s}=\frac{g_{4}g_{3}}{\sqrt{g_{4}^{2}+g_{3}^{2}}}~{},&\qquad&g_{Y}=\frac{g_{1}g_{4}}{\sqrt{g_{4}^{2}+\frac{2}{3}g_{1}^{2}}}~{},&\\\
&g_{U}=g_{4}~{},&\quad&g_{G^{\prime}}=\sqrt{g_{4}^{2}-g_{s}^{2}}~{},&\qquad&g_{Z^{\prime}}=\frac{1}{2\sqrt{6}}\sqrt{g_{4}^{2}-\frac{2}{3}g_{Y}^{2}}\,.\end{aligned}$
(2)

##### Fermion content.

In the non-universal 4321 model, only the 3rd generation of quarks and leptons
are charged under $SU(4)$. The light generation fields,
$q_{L}^{i},u_{R}^{i},d_{R}^{i},\ell_{L}^{i},e_{R}^{i}$ with $i=1,2$, are
singlet under $SU(4)$ and have the same quantum numbers as in the SM, with
hypercharge as their $U(1)_{X}$ charge and $SU(3)_{c}$ representation as their
$SU(3)^{\prime}$ ones. The $SU(3)^{\prime}$ of 4321 can therefore be seen as
the color group for light generations. The model also contain one or more
vector-like fermions $\chi_{L,R}^{j}$, where $j$ runs over the number of
vector-like fermions. After being integrated out, they generate couplings
between the $3^{\rm rd}$ and the light generations of the SM fermions. The
$3^{\rm rd}$ generation fermions are unified into quadruplet with quarks
(partners) and lepton (partners) as follows

$\displaystyle\begin{aligned}
\Psi_{L}&\sim(\bm{4},\bm{1},\bm{2})_{0}\equiv\begin{pmatrix}q_{L}^{3}\\\
\ell_{L}^{3}\end{pmatrix}\,,\quad&\Psi_{R}^{+}&\sim(\bm{4},\bm{1},\bm{1})_{1/2}\equiv\begin{pmatrix}u_{R}^{3}\\\
\nu_{R}^{3}\end{pmatrix}\,,\quad\\\
\chi_{L,R}^{j}&\sim(\bm{4},\bm{1},\bm{2})_{0}\equiv\begin{pmatrix}Q_{L,R}^{j}\\\
L_{L,R}^{j}\end{pmatrix}\,,\quad&\Psi_{R}^{-}&\sim(\bm{4},\bm{1},\bm{1})_{-1/2}\equiv\begin{pmatrix}d_{R}^{3}\\\
e_{R}^{3}\end{pmatrix}\,.\end{aligned}$ (3)

The vector-like fermions $(Q_{L,R}^{j},L_{L,R}^{j})$ and the 3rd generation
fields $(q^{3}_{L},\ell_{L}^{3})$ in the broken phase of 4321 have the same SM
charges as the light generations fields. In the following, we will consider
only one vector-like field $\chi_{L,R}$ ($j=1$).

###  2.2 Scalar sector

####  2.2.1 Scalar content and Lagrangian

The scalar content of the 4321 model used in the analysis is presented in
Table 1. The scalar potential is

$V=V_{\Omega}+V_{H}+V_{\Omega H}\,,$ (4)

where $V_{\Omega},V_{H},V_{\Omega H}$ are defined below.

$\displaystyle\begin{array}[]{|c|c|c|c|c|}\hline\cr&SU(4)&SU(3)^{\prime}&SU(2)_{L}&U(1)_{X}\\\
\hline\cr H&1&1&2&1/2\\\ \Omega_{1}&\overline{4}&1&1&-1/2\\\
\Omega_{3}&\overline{4}&3&1&1/6\\\ \hline\cr\end{array}$ Table 1: Scalar
content of the 4321 model.

The potential for the $\Omega_{1,3}$ scalar fields reads DiLuzio:2018zxy

$\displaystyle V_{\Omega}$ $\displaystyle=m_{\Omega_{3}}^{2}{\rm
Tr}[\Omega_{3}^{\dagger}\Omega_{3}]+m_{\Omega_{1}}^{2}\Omega_{1}^{\dagger}\Omega_{1}+\frac{\rho_{1}}{2}(\Omega_{1}^{\dagger}\Omega_{1})^{2}+\frac{\rho_{2}}{2}{\rm
Tr}[\Omega_{3}^{\dagger}\Omega_{3}]^{2}+\frac{\rho^{\prime}_{2}}{2}{\rm
Tr}[\Omega_{3}^{\dagger}\Omega_{3}\Omega_{3}^{\dagger}\Omega_{3}]$
$\displaystyle\ \ \ \ \ +\rho_{3}{\rm
Tr}[\Omega_{3}^{\dagger}\Omega_{3}]\Omega_{1}^{\dagger}\Omega_{1}+\rho_{4}\,\Omega_{1}^{\dagger}\Omega_{3}\Omega_{3}^{\dagger}\Omega_{1}+\frac{\rho_{5}}{3!}\left([\Omega_{3}\Omega_{3}\Omega_{3}\Omega_{1}]+\rm
h.c.\right)\,,$ (5)

where
$[\Omega_{3}\Omega_{3}\Omega_{3}\Omega_{1}]=\epsilon_{\alpha\beta\gamma\delta}\epsilon^{abc}(\Omega_{3})^{\alpha}_{a}(\Omega_{3})^{\beta}_{b}(\Omega_{3})^{\gamma}_{c}(\Omega_{1})^{\delta}$.
Notice that the $\rho_{5}$ term in Eq. (5) breaks explicitly a $U(1)$ global
symmetry of the scalar potential, thus preventing the appearance of a massless
Goldstone boson after spontaneous symmetry breaking.

The Higgs potential is the same as in the SM

$V_{H}=\mu_{H}^{2}H^{\dagger}H+\frac{\lambda}{2}(H^{\dagger}H)^{2}\,,$ (6)

and the mixing terms between the Higgs sector and the $\Omega$ sector are

$V_{\Omega
H}=\eta_{1}\,H^{\dagger}H\,\Omega_{1}^{\dagger}\Omega_{1}+\eta_{3}\,H^{\dagger}H\,{\rm
Tr}[\Omega_{3}^{\dagger}\Omega_{3}]\,.$ (7)

The roles of each scalar are as follows (the explicit Yukawa terms are written
in Appendix B.1):

  * •

$\Omega_{3}$ breaks 4321. After getting a vev, since this scalar is charged
under both $SU(3)$ light and $SU(4)$ heavy, it will mix the light generation
quarks with the vector-like quarks. Therefore, it is responsible for the mass
mixing between light quarks and the $3^{\rm rd}$ family quarks, and also for
the small coupling between the $U_{1}$ leptoquark, the colorons and the
$Z^{\prime}$ with the light quarks.

  * •

$\Omega_{1}$ breaks 4321. After getting a vev, since this scalar is charged
under $SU(4)$ heavy and has the right $U(1)_{X}$ charge, it will mix the light
generation leptons with the vector-like leptons. It has the same role as
$\Omega_{3}$ but for leptons instead of quarks (mass mixing between light and
$3^{\rm rd}$ family and small coupling of light leptons to heavy gauge boson
from 4321 breaking).

  * •

$H$ breaks the electroweak symmetry. This scalar is the same as the SM Higgs.
It gives mass to all quarks and charged leptons.

The two scalar fields $\Omega_{1,3}$ break spontaneously
$\mathcal{G}_{4321}\to\mathcal{G}_{\rm SM}$ by acquiring the VEVs

$\langle\Omega_{3}\rangle=\frac{1}{\sqrt{2}}\begin{pmatrix}\omega_{3}&0&0\\\
0&\omega_{3}&0\\\ 0&0&\omega_{3}\\\
0&0&0\end{pmatrix}\quad\text{and}\quad\langle\Omega_{1}\rangle=\frac{1}{\sqrt{2}}\begin{pmatrix}0\\\
0\\\ 0\\\ \omega_{1}\end{pmatrix}\,.$ (8)

The $\beta$-functions for the couplings of the potential in Eq. (4) are given
in Appendix A. The scalar content presented in Table 1 is the minimal content
needed for the analysis. More scalars might be needed for the phenomenology of
the 4321 model Cornella:2019hct , however they are not expected to alter
significantly the conclusions of this analysis (see Appendix B.3).

####  2.2.2 Radial and Goldstone modes

The $\Omega_{1,3}$ scalars can be decomposed as

$\Omega_{1}^{\dagger}=\begin{pmatrix}\tilde{T}_{1}\\\
\frac{\omega_{1}}{\sqrt{2}}+\tilde{S}_{1}^{*}\end{pmatrix}\,,\qquad\Omega_{3}^{\dagger}=\begin{pmatrix}\left(\frac{\omega_{3}}{\sqrt{2}}+\frac{\tilde{S}_{3}^{*}}{\sqrt{3}}\right)\mathbb{1}_{3}+\,{\tilde{O}^{a*}_{3}}\,t^{a}\\\
\tilde{T}_{3}^{\dagger}\end{pmatrix}\,,$ (9)

where $\mathbb{1}_{3}$ is the $3\times 3$ identity matrix and $t^{a}$ are the
$SU(3)$ generators. The complex scalars have the following representations
under the SM gauge group: $\tilde{S}_{3}\sim(\bm{1},\bm{1})_{0}$,
$\tilde{T}_{3}\sim(\bm{3},\bm{1})_{2/3}$,
$\tilde{O}_{3}\sim(\bm{8},\bm{1})_{0}$ and
$\tilde{S}_{1}\sim(\bm{1},\bm{1})_{0}$,
$\tilde{T}_{1}\sim(\bm{3},\bm{1})_{2/3}$.

The scalar spectrum contains the following mass eigenstates:

  * •

two real octets: the massless $O_{\rm
GB}^{a}=\sqrt{2}\,\text{Im}\,\tilde{O}_{3}^{a}$ and its massive orthogonal
combination $O_{R}^{a}$;

  * •

two complex triplet: the massless $T_{\rm
GB}=s_{\beta}\tilde{T}_{1}-c_{\beta}\tilde{T}_{3}$, with
$\tan\beta=\omega_{1}/\omega_{3}$, and its massive orthogonal combination
$T_{R}$;

  * •

four real singlets, all combination of $\tilde{S}_{1}$ and $\tilde{S}_{3}$:
the massless $S_{\rm GB}$ and three massive radials $S_{0},S_{1},S_{2}$.

The Goldstone boson modes $O_{\rm GB}^{a}$, $S_{\rm GB}$, and $T_{\rm GB}$
fields are eaten by the coloron $G^{\prime}$, the $Z^{\prime}$, and the
leptoquark $U_{1}$, respectively and the radial modes acquire the following
masses:

$\displaystyle\bullet\quad
M_{\,O_{R}}^{2}=\omega_{3}(\rho^{\prime}_{2}\omega_{3}-\rho_{5}\omega_{1})\qquad\qquad\qquad\text{with}~{}\rho^{\prime}_{2}\omega_{3}>\rho_{5}\omega_{1}~{},$
(10) $\displaystyle\bullet\quad
M_{T_{R}}^{2}=\frac{1}{2}\left(\rho_{4}-\rho_{5}\frac{\omega_{3}}{\omega_{1}}\right)\left(\omega_{1}^{2}+\omega_{3}^{2}\right)\qquad\text{with}~{}\rho_{4}\omega_{1}>\rho_{5}\omega_{3}~{},$
(11) $\displaystyle\bullet\quad
M_{S_{0}}^{2}=\frac{\rho_{5}}{2}\frac{\omega_{3}}{\omega_{1}}\left(3\omega_{1}^{2}+\omega_{3}^{2}\right)\qquad\qquad\qquad\text{with}~{}\rho_{5}>0~{},$
(12) $\displaystyle\bullet\quad
M_{S_{1}}^{2}=\frac{1}{2}\left(\rho_{1}\omega_{1}^{2}+(3\rho_{2}+\rho^{\prime}_{2})\omega_{3}^{2}+\frac{\rho_{5}}{2}\frac{\omega_{3}}{\omega_{1}}(\omega_{1}^{2}-\omega_{3}^{2})\right)-\frac{u}{2\omega_{1}}~{},$
(13) $\displaystyle\bullet\quad
M_{S_{2}}^{2}=\frac{1}{2}\left(\rho_{1}\omega_{1}^{2}+(3\rho_{2}+\rho^{\prime}_{2})\omega_{3}^{2}+\frac{\rho_{5}}{2}\frac{\omega_{3}}{\omega_{1}}(\omega_{1}^{2}-\omega_{3}^{2})\right)+\frac{u}{2\omega_{1}}\,,$
(14)

with $u$ defined as

$\displaystyle u^{2}$
$\displaystyle=\left[\rho_{1}\omega_{1}^{3}+(3\rho_{2}+\rho^{\prime}_{2})\omega_{1}\omega_{3}^{2}+\frac{\rho_{5}}{2}\omega_{3}(\omega_{1}^{2}-\omega_{3}^{2})\right]^{2}-4\omega_{1}\omega_{3}\bigg{[}\rho_{1}\omega_{1}^{3}\left(\frac{\rho_{5}}{2}\omega_{1}\
+(3\rho_{2}+\rho^{\prime}_{2})\omega_{3}\right)$
$\displaystyle-3\omega_{3}\left(\rho_{3}^{2}\omega_{1}^{3}+\rho_{3}\rho_{5}\omega_{3}\omega_{1}^{2}+\frac{\rho_{5}^{2}}{3}\omega_{3}^{2}\omega_{1}+\frac{\rho_{5}}{6}(3\rho_{2}+\rho^{\prime}_{2})\omega_{3}^{3}\right)\bigg{]}\,.$
(15)

####  2.2.3 Electroweak symmetry breaking

In this section, we identify the conditions for the spontaneous symmetry
breaking of the electroweak sector. Writing the potential Eq. (4) in terms of
the radial degrees of freedom aligned along the vacua, we obtain

$\displaystyle\begin{aligned}
V(s_{1},s_{3},h)=~{}&m_{\Omega_{1}}^{2}s_{1}^{2}+m_{\Omega_{3}}^{2}s_{3}^{2}+\frac{\rho_{1}}{2}s_{1}^{4}+\frac{1}{6}(3\rho_{2}+\rho_{2}^{\prime})s_{3}^{4}+\rho_{3}s_{1}^{2}s_{3}^{2}+\frac{2\rho_{5}}{3\sqrt{3}}s_{1}s_{3}^{3}\\\
&+\frac{\mu_{H}^{2}}{2}h^{2}+\frac{\lambda}{8}h^{4}+\frac{\eta_{1}}{2}h^{2}s_{1}^{2}+\frac{\eta_{3}}{2}h^{2}s_{3}^{2}~{},\end{aligned}$
(16)

where $s_{1}=\frac{\omega_{1}}{\sqrt{2}}\,+$ Re$(\tilde{S}_{1})$ and
$s_{3}=\sqrt{\frac{3}{2}}\,\omega_{3}\,+$ Re$(\tilde{S}_{3})$.

A necessary requirement for our potential to have a minimum at $\langle
h\rangle=v$, $\langle s_{1}\rangle=\frac{\omega_{1}}{\sqrt{2}}$ and $\langle
s_{3}\rangle=\sqrt{\frac{3}{2}}\omega_{3}$ is that the first derivative of the
potential evaluated at this point vanishes, i.e.

$\left.\frac{\partial V(s_{1},s_{3},h)}{\partial h_{i}}\right|_{\langle
h\rangle,\langle s_{1}\rangle,\langle s_{3}\rangle}=0\quad\text{for
}h_{i}=h,s_{1},s_{3}\,.$ (17)

These three equations give the set of conditions:

$\displaystyle\begin{aligned}
\mu_{H}^{2}+\frac{\eta_{1}}{2}\omega_{1}^{2}+\frac{3}{2}\eta_{3}\omega_{3}^{2}+\frac{\lambda}{2}v^{2}&=0~{},\\\
m_{\Omega_{1}}^{2}+\frac{\eta_{1}}{2}v^{2}+\frac{\rho_{1}}{2}\omega_{1}^{2}+\frac{3}{2}\rho_{3}\omega_{3}^{2}+\frac{\rho_{5}}{2}\frac{\omega_{3}^{3}}{\omega_{1}}&=0~{},\\\
m_{\Omega_{3}}^{2}+\frac{\eta_{3}}{2}v^{2}+\frac{1}{2}(3\rho_{2}+\rho_{2}^{\prime})\omega_{3}^{2}+\frac{\rho_{3}}{2}\omega_{1}^{2}+\frac{\rho_{5}}{2}\omega_{3}\omega_{1}&=0~{},\end{aligned}$
(18)

where for each the vanishing VEV solution has been ignored. From this point
forth, since the smallness of $\rho_{5}$ is technically natural, we neglect
its contribution to the effective mass and quartic coupling of the Higgs. In
the limit where $v\ll\omega_{1},\omega_{3}$, the term proportional to $v^{2}$
can be omitted in the second and third equations in Eq. (18). The Hessian
matrix $[M^{2}_{s_{3},s_{1},h}]_{ij}=~{}\dfrac{\partial^{2}\,V}{\partial
h_{i}\partial h_{j}}$ with $h_{i}\in(s_{3},s_{1},h)$ evaluated around the
stationary points, solutions of Eq. (18), describes the curvature around these
points and give us the conditions for them to be local minima. It is given by

$M^{2}_{s_{3},s_{1},h}=\begin{pmatrix}2(3\rho_{2}+\rho_{2}^{\prime})\omega_{3}^{2}&2\sqrt{3}\rho_{3}\omega_{1}\omega_{3}&\sqrt{6}\eta_{3}v\omega_{3}~{},\\\
2\sqrt{3}\rho_{3}\omega_{1}\omega_{3}&2\rho_{1}\omega_{1}^{2}&\sqrt{2}\eta_{1}v\omega_{1}~{},\\\
\sqrt{6}\eta_{3}v\omega_{3}&\sqrt{2}\eta_{1}v\omega_{1}&\lambda
v^{2}\end{pmatrix}\,.$ (19)

The squared mass matrix has to be positive definite, which, for a symmetric
matrix, is equivalent to all its leading principal minors being positive

$\displaystyle\begin{aligned} D_{1}=\,&3\rho_{2}+\rho_{2}^{\prime}>0\\\
D_{12}=\,&\rho_{1}D_{1}-3\rho_{3}^{2}>0\\\ D_{123}=\,&\lambda
D_{12}-(3\eta_{3}^{2}\rho_{1}-6\eta_{1}\eta_{3}\rho_{3}+\eta_{1}^{2}D_{1})>0\,.\end{aligned}$
(20)

The other minors are guaranteed to be positive from the inequalities above,
but it is useful to write them explicitly as they put clear constraints on the
mixing parameters $\eta_{1}$ and $\eta_{3}$, namely

$|\eta_{1}|<\sqrt{\lambda\rho_{1}}\qquad\text{and}\qquad|\eta_{3}|<\sqrt{\frac{\lambda}{3}(3\rho_{2}+\rho_{2}^{\prime})}\,.$
(21)

In the limit where $v\ll\omega_{1},\omega_{3}$, the eigenvalue of the square
mass matrix Eq. (19) corresponding to the physical Higgs mass is

$m_{h_{\rm phys}}^{2}=\lambda_{\rm
eff}\,v^{2}=\frac{D_{123}}{D_{12}}\,v^{2}+O\left(\frac{v^{2}}{\omega_{1,3}^{2}}\right)\,.$
(22)

Given the positive definite conditions Eq. (20), it is clear from its
definition that $\lambda_{\rm eff}>0$. Moreover, we can define an effective
Higgs mass parameter as

$\mu_{\rm eff}^{2}=-\frac{\lambda_{\rm
eff}}{2}v^{2}=\mu_{H}^{2}-\frac{\eta_{1}D_{1}-3\eta_{3}\rho_{3}}{D_{12}}m_{\Omega_{1}}^{2}-3\frac{\eta_{3}\rho_{1}-\eta_{1}\rho_{3}}{D_{12}}m_{\Omega_{3}}^{2}\,.$
(23)

The tree-level threshold corrections from integrating out $s_{1}$ and $s_{3}$
in the potential Eq. (16) yields an effective potential

$V_{\rm eff}=\mu_{\rm eff}^{2}H^{\dagger}H+\frac{\lambda_{\rm
eff}}{2}(H^{\dagger}H)^{2}~{},$ (24)

with the same expressions of $\lambda_{\rm eff}$ and $\mu_{\rm eff}^{2}$ as in
Eq. (22) and Eq. (23), respectively. Notice the equivalence between the
squared mass matrix diagonalization Eq. (19) with a VEV hierarchy
approximation and the tree-level threshold corrections by integrating out the
heavy states.222In Ref. Elias-Miro:2012eoi , the threshold effects in the
presence of an additional singlet scalar has been exploited as a mechanism to
avoid instabilities of the electroweak vacuum.  The EWSB happens when
$\mu_{\rm eff}^{2}$ flips sign, from positive to negative.

####  2.2.4 Bounded from below conditions

In order for the potential to be Bounded From Below (BFB) and thus for the
vacuum to be stable, there must be restrictions on the parameters in the
quartic part of the potential $V_{4}$. For the potential $V_{\Omega}$, the BFB
conditions have not previously been derived in the literature, and a full
stability analysis of the potential is beyond the scope of this paper. A
rigorous treatment of vacuum stability for similar models using the
copositivity criteria can be found in Refs. Kannike:2012pe ; Kannike:2016fmd .
Below we list some conditions, obtained by studying the behavior of $V_{4}$
along specific directions of the fields $\Omega_{1}$ and $\Omega_{3}$ and
minimizing the potential against extra parameters such as $\theta$ and
$\alpha$ as in Ref. Chauhan:2019fji . In what follows, we consider strong
stability requirements, i.e. $V_{4}>0$ for arbitrary large values of fields
components. Stability in the marginal sense $V_{4}\geq 0$ (if $V_{4}=0$ then
one must also require the quadratic part $V_{2}\geq 0$) is obtained by making
the following inequalities non-strict. For the $\Omega$ potential, we obtained

$\displaystyle\begin{aligned}
&|\Omega_{3}|=0~{},&\quad&|\Omega_{1}|\to\infty&\Rightarrow\quad&\rho_{1}>0~{},\\\
&|\Omega_{3}|\to\infty~{},&\quad&|\Omega_{1}|=0&\Rightarrow\quad&\rho_{2}+\rho_{2}^{\prime}>0~{},\quad
3\rho_{2}+\rho_{2}^{\prime}>0~{},\\\
&|\Omega_{3}|^{2}=r\cos\theta~{},&\quad&|\Omega_{1}|^{2}=r\sin\theta&\Rightarrow\quad&\rho_{3}>-\sqrt{\rho_{1}(\rho_{2}+\rho_{2}^{\prime})}~{},\quad\rho_{3}+\rho_{4}>-\sqrt{\rho_{1}(\rho_{2}+\rho_{2}^{\prime})}~{},\\\
&\Omega_{3}=re^{i\alpha}~{},&\quad&\Omega_{1}=re^{i\left(\alpha-\frac{\pi}{2}\right)}&\Rightarrow\quad&|\rho_{5}|<\frac{1}{4}(\rho_{1}+3(3\rho_{2}+\rho_{2}^{\prime})+6\rho_{3})~{},\end{aligned}$
(25)

with $r\to\infty$. For the Higgs potential, the well-known condition is
$\lambda>0$. For the mixed part of the potential $V_{\Omega H}$, using the
parametrization $|H|^{2}=r\sin\theta$, we obtained

$\displaystyle|\Omega_{3}|=0~{},$ $\displaystyle|\Omega_{1}|^{2}=r\cos\theta$
$\displaystyle\Rightarrow~{}$
$\displaystyle\eta_{1}>-\sqrt{\lambda\rho_{1}}~{},$ (26)
$\displaystyle|\Omega_{3}|^{2}=r\cos\theta~{},$ $\displaystyle|\Omega_{1}|=0$
$\displaystyle\Rightarrow~{}$
$\displaystyle\eta_{3}>-\min\left[\sqrt{\lambda(\rho_{2}+\rho_{2}^{\prime})},\sqrt{\lambda(3\rho_{2}+\rho_{2}^{\prime})}\right]~{}.$

These relations are necessary, but not sufficient, conditions to ensure the
vacuum stability.

##  3 Analysis of the RG equations system

In this section, we study the RG equations system of the model. In what
follows, we explain the procedure for solving this system and list the
constraints at the IR boundary.

Methodology. We solve numerically the system of RG equations using the
$\beta$-functions for the 4321 model presented in Appendix A (neglecting the
terms related to the second Higgs doublet) from the scale $\Lambda_{\rm UV}$
to the matching scale $\Lambda_{4}$, and the SM $\beta$-functions from
$\Lambda_{4}$ down to $m_{Z}$. Starting with the full particle content, we
first integrate out the heavy gauge bosons of 4321 together with the radials
of the scalars $\Omega_{1,3}$ and the vector-like fermion(s) at $\Lambda_{4}$,
then the top quark and eventually the Higgs boson at their respective masses.
Performing the matching between 4321 and the SM at a common scale is a
simplification, because the spectrum of the heavy modes is non-degenerate.
However, thanks to the relations imposed by the $SU(4)$ algebra Eq. (1), the
mass differences between the gauge bosons is still expected to be relatively
small and no large logarithms are generated. We choose this scale to be the
mass of the leptoquark $\Lambda_{4}\approx M_{U}$. Concerning the threshold
effects at $\Lambda_{4}$, we take into account the tree-level corrections due
to the decoupling of the heavy scalars $\Omega_{1,3}$ and neglect any
subleading loop-level corrections (see Eqs. (22) and (23)).

Infrared boundary conditions. We require that all studied benchmarks respect
the following phenomenological and formal constraints:

  * •

We initialize the running of the SM gauge couplings, the parameters of the
Higgs potential and the top Yukawa coupling as follows
ParticleDataGroup:2020ssz :

$\displaystyle\alpha_{Y}(m_{Z})$
$\displaystyle=0.01017~{},\quad\alpha_{2}(m_{Z})=0.03380~{},\quad\alpha_{s}(m_{Z})=0.1179~{},$
$\displaystyle\mu_{H}^{2}(m_{h_{\rm phys}})$ $\displaystyle=-(m_{h_{\rm
phys}})^{2}/2~{},\quad\lambda(m_{h_{\rm phys}})=0.25923~{},\quad
y_{t}(m_{t})=0.93419~{}.$

where $\alpha_{i}=g_{i}^{2}/4\pi$, $m_{h_{\rm phys}}=125.25~{}\rm GeV$ and
$m_{t}=y_{t}\,v/\sqrt{2}$ with $v=246$ GeV.

  * •

In order to address the $B$-physics anomalies within this framework, assuming
both left-handed and right-handed couplings of the leptoquark, the key
condition is Cornella:2021sby 333The fit of Ref. Cornella:2021sby  includes
the charged-current anomalies, encoded by the ratios $R_{D}^{(*)}$
BaBar:2012obs ; BaBar:2013mob ; Belle:2015qfa ; LHCb:2015gmp ; LHCb:2017smo ;
LHCb:2017rln  (see Table 2.1), the neutral-current anomalies $R_{K}^{(*)}$
LHCb:2014vgu ; LHCb:2017avl ; LHCb:2019hip ; LHCb:2021trn  as well as
$B_{s}\to\mu^{+}\mu^{-}$ (see Table 2.2) and a plethora of low-energy
constraints (see Table 3.1).

$g_{U}=(1.1\pm 0.2)\times\left(\frac{M_{U}}{2~{}{\rm TeV}}\right)~{}.$ (27)

Using Eq. (1) and Eq. (LABEL:eq:4321_gauge_couplings), the above relation is
equivalent to $\sqrt{\omega_{1}^{2}+\omega_{3}^{2}}\in[3,4.5]~{}\rm TeV$.

  * •

The lower bounds for the masses of the 4321 gauge bosons are set by LHC
searches at high $p_{T}$ Baker:2019sli ; Cornella:2021sby . Since the masses
of the gauge bosons are correlated, the exclusion limits extracted from
searches of resonant coloron production with $t\bar{t}$ and $b\bar{b}$ final
states impose the leading constraints on the whole mass spectrum:
$M_{G^{\prime}}\gtrsim 4~{}\rm TeV$ and $M_{U},M_{Z^{\prime}}\gtrsim 3~{}\rm
TeV$ .

  * •

The colored radial modes $O_{R}$ and $T_{R}$ can be produced via QCD
interactions at colliders. In order to avoid direct detection bounds at the
LHC, their masses should be above $2~{}\rm TeV$ (see Ref. Diaz:2017lit  for
the color triplet and Ref. Bai:2018jsr  for the octet). For the singlet radial
modes $S_{0,1,2}$, the collider bounds are much milder since the only
available ones originate from direct Higgs searches Ilnicka:2018def . Taking
the minimum lower mass bound to be $500~{}\rm GeV$ and the mixing couplings
$\eta_{1,3}$ to be $\mathcal{O}(0.1)$, they can be safely avoided. Note that
the above mass bounds translate into constraints for the coupling values
$\rho_{i}(\Lambda_{4})$ and the VEVs $\omega_{1,3}$ (see Eqs. (10-14)).

  * •

Flavor constraints involving the radial $T_{R}$, e.g. one-loop FCNC via the
Yukawa interaction, impose an important lower bound on the ratio
$\tan\beta_{T}=\omega_{3}/\omega_{1}$. As discussed in Ref. DiLuzio:2018zxy ,
in case of the universal 4321, the radial $T_{R}$ together with the vector-
like leptons enter box diagrams that generate contributions to the amplitudes
of $B_{s}-\bar{B_{s}}$ and $D-\bar{D}$ mixing. These contributions are
proportional to $\tan\beta_{T}^{-4}$ and for $\tan\beta_{T}\gtrsim 2$ they
become negligible. In the present case of the non-universal 4321 only the
$D-\bar{D}$ mixing is affected, and still requires a similar suppression
realized by assuming a hierarchy of VEVs i.e. $\omega_{3}>\omega_{1}$.

  * •

We require that the BFB conditions Eq. (25) and Eq. (26) are satisfied at all
scales.

  * •

The mass parameters $m_{\Omega_{1,3}}^{2}$ at scale $\Lambda_{4}$ are set to
the minimum of the potential by satisfying Eq. (18).

###  3.1 General dependencies and perturbativity range

In Table 2 we give a benchmark that satisfies the IR boundary conditions and
in Figure 1 we show the resulting RG evolution of the couplings. The fit to
the $B$-physics anomalies and the collider bounds on the 4321 gauge bosons
prefer a large $g_{4}$ coupling i.e. $g_{4}(\Lambda_{4})\gtrsim 2$. On the
other hand, the experimental bounds on the scalar masses are marginally
surpassed by taking the couplings of the scalar sector within the range
$\mathcal{O}(0.1-1)$. The only exception is the coupling $\rho_{5}$ which
mainly fixes the mass of $S_{0}$ and therefore can be kept one order of
magnitude smaller. In fact, as we argued in Section 2.2, the smallness of this
parameter is technically natural. In fact, larger values for $\rho_{5}$ would
imply also larger values for the rest of the $\rho_{i}$ parameters in order to
keep the $u$ parameter of Eq. (15) real i.e. $u^{2}>0$, and as we will see
next, this severely restricts the extrapolation the 4321 model in the UV.

$\Omega$ VEVs | values | scalar couplings | values
---|---|---|---
$\omega_{1}$ | $1.5~{}\rm TeV$ | $\lambda$ | 0.2
$\omega_{3}$ | $4~{}\rm TeV$ | $\rho_{1}$ | 0.5
mass parameters |  | $\rho_{2}$ | 0.1
$\mu_{H}^{2}$ | $(1.6~{}\rm{TeV})^{2}$ | $\rho_{2}^{\prime}$ | 0.5
$m_{\Omega_{1}}^{2}$ | $-(1.8~{}\rm{TeV})^{2}$ | $\rho_{3}$ | 0.1
$m_{\Omega_{3}}^{2}$ | $-(2.5~{}\rm{TeV})^{2}$ | $\rho_{4}$ | 1
gauge coupling |  | $\rho_{5}$ | 0.01
$g_{4}$ | $3$ | $\eta_{1}$ | -0.1
$\eta_{3}$ | -0.1
Table 2: Benchmark for the boundary conditions of the model parameters at
scale $\Lambda_{4}$.

Even though $g_{4}$ is an asymptotically free gauge coupling, the running is
slow and its value remains $\mathcal{O}(1)$ even at high energies.
Consequently, since the $\beta$-functions of the various parameters are
explicitly or indirectly dependent on $g_{4}$, the whole RG evolution is
effectively driven by it. In particular, $\beta(\rho_{1})$ contains a term,
which solely depends on $g_{4}^{4}$ and has the largest numerical prefactor in
comparison to other similar terms appearing in the rest of the
$\beta$-functions (see Eq. (A)). The reason is that the purely gauge
correction to the $\Omega_{1}$ quartic term (box diagram) includes all
possible contractions of $SU(4)$, contrary for example to the $\Omega_{3}$
quartic for which the different contractions are split between the two terms
with couplings $\rho_{2}$ and $\rho_{2}^{\prime}$. As a result $\rho_{1}$ is a
monotonically increasing function of the energy scale $\mu$ and exhibits the
fastest running among the couplings leading to a Landau pole.

Depending on the values of $g_{4}$ and $\rho_{1}$ at $\Lambda_{4}$ we can
estimate the energy scale where the scalar couplings grow larger than $\sim
4\pi$ and where the theory becomes non-perturbative. In Figure 2 we plot this
scale as a function of the mass of the radial $S_{1}$, which is the scalar
mass that is most sensitive to $\rho_{1}(\Lambda_{4})$. For the benchmark
values in Table 2 and the minimum requirement of radials heavier than
$500~{}\rm GeV$, we find that the perturbativity range of the theory for
$g_{4}=2$ ($M_{U}\approx 4.5$ TeV) reaches almost $\mathcal{O}(10^{8}~{}\rm
TeV)$, while for $g_{4}=3$ ($M_{U}\approx 6.5$ TeV) it only reaches
$\mathcal{O}(10^{5}~{}\rm TeV)$. Furthermore, we see that if we require
heavier radials this range drastically drops. Notice that the $g_{2}=3$ curve
falls slower than the $g_{4}=2$ curve for higher $\rho_{1}$ values, because
the $-\frac{45}{2}g_{4}^{2}\rho_{1}$ term in Eq. (A) starts to dominate.
Characteristically, the Landau poles appear at even lower energies when one
further requires all the radial scalar states heavier than $2~{}\rm{TeV}$. In
this case, couplings blow up at $\mathcal{O}(100~{}\text{TeV})$. Considering
the above, we argue that the limit of degenerate radials or radials with
masses equal or heavier than the leptoquark, which has been used in the
literature for simplicity (see for example Refs. Fuentes-Martin:2019ign ;
Fuentes-Martin:2020luw ; Fuentes-Martin:2020hvc ; Cornella:2021sby ), is not
favored. Notice also that our results are relevant to all current versions of
4321 models and consequently impose generic upper bounds for the cut-off scale
of 4321 model. In Section 3.3 we will see that the indicated scale for new
physics is even lower if one wishes to unify in the UV.

In this context of fundamental scalars, the question can be posed whether new
degrees of freedom can appear at intermediate energies to tame this effect by
generating appropriate (possibly fine-tuned) cancellations in the
$\beta$-functions. This, however, seems difficult to arrange in our new
physics setup (see Appendix B). The reason is that in order to cancel the
$\frac{99}{16}g_{4}^{4}$ term in Eq. (A), besides having the opposite sign, a
cancellation term must either have a coupling as large as $g_{4}$ or for a
coupling less than 1 an unrealistically large prefactor of $\sim 500(100)$ for
$g_{4}=3(2)$. In the former case, couplings such as Yukawa or scalar
couplings, if large, would quickly develop Landau poles themselves. Thus, it
is desirable that the large couplings in the IR be asymptotically free.
However in the simplest case where they are e.g. gauge couplings, the breaking
of the additional gauge symmetry would require the presence of even more
scalars and scalar couplings with similar issues in their RG running. As a
result, a solution within this context, if at all feasible, would probably be
highly fine-tuned.

Figure 1: RG running of the gauge and top Yukawa couplings (left) and quartic
couplings of the scalar potential (right) for the benchmark presented in Table
2. Figure 2: Dependence of the cut-off scale of the 4321 model as a function
of the mass of the $S_{1}$ radial. The mass of $S_{1}$ is mostly governed by
the value of $\rho_{1}$ at $\Lambda_{4}$, shown in the upper horizontal axis.
The bands corresponds to a $50\%$ variation around the matching scale, set at
central value $\Lambda_{4}=M_{U}$.

###  3.2 Radiative electroweak symmetry breaking

Another feature emerges when studying the RG behavior of the 4321 model above
$\Lambda_{4}$: EWSB triggered by radiative effects. Spontaneous EWSB requires
$\mu_{\text{eff}}^{2}<0$, where $\mu^{2}_{\text{eff}}$ is defined in Eq. (23).
For the 4321 model benchmark given in Table 2, $\mu_{\text{eff}}^{2}$ flows
from positive in the UV to negative in the IR, radiatively triggering EWSB as
shown in Figure 5.

Figure 3: The RG evolution of the scalar sector mass squared parameters
$\mu_{H}^{2}$, $m_{\Omega_{1}}^{2},$ $m_{\Omega_{3}}^{2}$, and
$\mu_{\text{eff}}^{2}$. For each, the quantity plotted is defined as
$X\equiv\text{sign}(X^{2})|X|$
($X=\mu_{H},m_{\Omega_{1}},m_{\Omega_{3}},\mu_{\text{eff}}$). The inset plot
focuses on where $\mu_{\text{eff}}$ passes through $0$, radiatively triggering
EWSB near $\Lambda_{4}=4.5$ TeV. We also show the effect of individually
varying each quartic coupling as $|\delta(\lambda,\eta,\rho)_{i}|\leq 5\%$ at
$\Lambda_{\text{UV}}$. This variation propagates along the RG flow of the mass
squared parameters, represented by a band enclosed by dashed lines. We also
overlay the effect on $\mu_{\text{eff}}$ from the smaller variations
$\delta(\lambda,\eta,\rho)_{i}$ in the $\pm 1\%$ range with a dark gray
band.555We thank Ulrich Haisch for pointing out that the perturbative
determination of $\mu_{\rm eff}$ in Eq. (23) breaks down at energies much
higher than the matching scale. Therefore, the black curve in Fig. 5 cannot be
trusted in the utmost right part of the figure. Nevertheless, we expect it to
be reliable in the region where radiative EWSB happens, i.e. in the window of
the inset plot.

To see how radiative EWSB occurs, consider the limit $\eta_{1,3}\to 0$, so
that the Higgs is decoupled from $\Omega_{1,3}$. In that limit,
$\mu_{\text{eff}}^{2}=\mu_{H}^{2}$, and the RG flow of $\mu_{H}^{2}$ is
controlled by its $\beta$-function given in Eq. (35). In the absence of the
$\Omega_{1,3}$ parameters, the one-loop $\beta$-function is proportional to
$\mu_{H}^{2}$ and thus a $\mu_{H}^{2}$ sign flip from radiative effects alone
is not possible. Instead, as $\mu_{H}$ approaches $0$, it also approaches an
approximate fixed point, and $\beta_{\mu_{H}^{2}}$ vanishes. Consequently, we
see that the coupling of the Higgs to the additional scalars is primarily
responsible for triggering EWSB radiatively, as has been pointed out in
Babu:2016gpg . In Babu:2016gpg , it was argued that the new scalar states must
couple to the Higgs with positive mixing couplings in order to ensure
$\mu_{H}$ flows to negative values in the IR. Here since our $\Omega_{1,3}$
masses are negative (for 4321 spontaneous symmetry breaking), we choose
$\eta_{1,3}<0$ to reproduce the same behavior.

Note that once $\eta_{1,3}\neq 0$, however, the relevant parameter to track
over the RG flow is $\mu_{\text{eff}}^{2}$ rather than $\mu_{H}$, and its
running is a complicated combination of radiative contributions to
$\mu_{H}^{2},m_{\Omega_{1}}^{2},$ and $m_{\Omega_{3}}^{2}$, as one can infer
from Eq. (23). In this case, it is not clear that $\eta_{1,3}$ are the
dominant contributions to radiative EWSB. Instead, radiative EWSB will be
controlled not just by the mixing couplings $\eta_{1,3}$, but also by the
$\rho_{i}$ self-couplings in the $\Omega_{1,3}$ system, see Eqs. (5, 37-38,
A-50, 52). For the benchmark used in Fig. 5, the
$\eta_{3},\rho_{1,2,3,4},\rho_{2}^{\prime}$ all contribute significantly to
EWSB breaking in the IR. The coupling $\eta_{1}$ has a smaller impact due to
the fact that $\Omega_{1}$ has fewer degrees of freedom than $\Omega_{3}$.

Figure 5 shows that for this benchmark, there is a close coincidence of scales
between $\Lambda_{4}$ and the scale at which $\mu_{\text{eff}}^{2}$ changes
signs. This is due to the scalar mass mixing that defines
$\mu_{\text{eff}}^{2}$, equivalent to accounting for the tree-level threshold
effects near the $\mathcal{G}_{4321}\to\mathcal{G}_{\text{SM}}$ breaking scale
Elias-Miro:2012eoi . As explained in Section 2.2.3, in the presence of
$\Omega_{1,3}$ scalar fields, EWSB is a collective effect of the full set of
scalar fields, and so as the RG evolution scale $\mu$ approaches
$\Lambda_{4}$, the onset of $\langle\Omega_{1,3}\rangle\neq 0$ is contributing
to the emergence of EWSB.

The steep decrease of $\mu_{\text{eff}}^{2}$ from positive to negative values
near $\Lambda_{4}$ signals a careful arrangement of parameters required in the
UV to reproduce the measured Higgs mass in the IR. The Higgs mass parameter
$\mu_{\text{eff}}$ in the IR is indeed sensitive to small deviations of
parameters in the UV, as demonstrated in Figure 4. We present the response of
$\mu_{\text{eff}}^{2}$ in the IR to varying the quartic couplings at
$\Lambda_{\text{UV}}$ according to

$\displaystyle\delta\eta_{i}$
$\displaystyle=\frac{\eta_{i}-\eta_{i}^{0}}{\eta_{i}^{0}}\
,\quad\delta\rho_{i}=\frac{\rho_{i}-\rho_{i}^{0}}{\rho_{i}^{0}}\ ,$ (28)

where $\eta_{i}^{0},\rho_{i}^{0}$ are the benchmark UV couplings values given
in Table 2. One can employ one of the traditional measures Barbieri:1987fn  of
fine-tuning $\frac{100}{\Delta_{\rm FT}}\%$, defined as

$\Delta_{\rm
FT}=\underset{y_{i}=\\{\lambda,\eta,\rho\\}_{i}}{\max}\left|\frac{\delta\mu_{\text{eff}}^{2}}{\delta
y_{i}}\right|~{},\quad\delta\mu_{\text{eff}}^{2}=\frac{\mu_{\text{eff}}^{2}(y_{i})-\mu_{\text{eff}}^{2}(y^{0}_{i})}{\mu_{\text{eff}}^{2}(y^{0}_{i})}~{},$
(29)

where $\mu_{\text{eff}}^{2}(y_{i})$ is evaluated at the IR boundary
$\mu=m_{h_{\rm phys}}$. The maximum fine-tuning is at the sub-percent level
$\Delta_{\rm FT}=\mathcal{O}(10^{3})$. In every coupling except
$\rho_{5}$,666The coupling $\rho_{5}$ has been chosen to be much smaller than
the other couplings for this benchmark, removing $\mu_{\text{eff}}$’s
sensitivity to $\rho_{5}$. variations of $\mathcal{O}(1\%)$ can yield positive
$\mu_{\text{eff}}^{2}$ values in the IR, spoiling EWSB altogether, as shown in
the shaded regions of Figure 4.

Figure 4: Here we show the sensitivity of $\mu_{\text{eff}}^{2}$ (evaluated at
the IR scale $\mu=m_{h_{\rm phys}}$) to small deviations in the
Higgs-$\Omega_{1,3}$ mixing couplings (left) and the $\Omega_{1,3}$ quartic
couplings (right) at $\Lambda_{\text{UV}}=10^{3}$ TeV. For reference, the
value of $\mu_{\text{eff}}^{2}$ that reproduces the measured Higgs mass is
shown by a dashed, black line. The region where $\mu_{\text{eff}}^{2}$ fails
to radiatively trigger EWSB is shaded in gray.

The variations at $\Lambda_{\text{UV}}$ propagate over the RG evolution of the
theory, leading to deviations in $\mu_{H}^{2}$ and $m_{\Omega_{1,3}}^{2}$ at
$\Lambda_{4}$, denoted by the dashed bands in Figure 5. After the spontaneous
symmetry breaking at $\Lambda_{4}$, deviations in the mass parameters lead to
a spread in possible $\mu_{\text{eff}}^{2}$ values in the IR, represented by a
gray band in Figure 5.

Returning to the limit of $\eta_{1,3}\to 0$ where the scalars $\Omega_{1,3}$
are decoupled, we find this tuning problem is postponed to the UV. As
mentioned above, in this limit we cannot rely on radiative effects to trigger
EWSB, and instead $\mu_{\text{eff}}^{2}\approx-\left(88\text{ GeV}\right)^{2}$
must be arranged directly at $\Lambda_{\text{UV}}$. A mass hierarchy between
$\mu_{\text{eff}}^{2}$ and $m_{\Omega_{1,3}}^{2}\sim\text{ TeV}^{2}$ will
endure throughout the flow from the UV to IR. This imposed hierachy of scales
at $\Lambda_{\text{UV}}$ is the trade-off for alleviating $\mu_{\text{eff}}$’s
sensitivity to the initial conditions $\eta_{i}(\Lambda_{\text{UV}}),\
\rho_{i}(\Lambda_{\text{UV}})$. The tuning apparent in the sensitive
dependence of both the Higgs mass and the radiative EWSB mechanism on
couplings set at $\Lambda_{\text{UV}}$ is thus rooted in the little hierarchy
problem, as discussed in Allwicher:2020esa .

###  3.3 Unification in the ultraviolet

In this section, we examine the implications of UV constraints on the RG
evolution. The constraints we employ are inspired by the UV constructions
presented in Refs. Bordone:2017bld ; King:2021jeo , in which the 4321 model is
seen as an effective theory whose gauge group is embedded in multiple copies
of the PS group. The scale $\Lambda_{\rm UV}$ is then the scale where the
symmetry group of the UV theory breaks spontaneously to the 4321 gauge group.
We also choose this scale as the matching scale between the two theories (for
more details see Appendix C). In those UV embeddings the scalar sectors are
considerably more complex. However, they feature two common properties:

  1. i)

The fields $\Omega_{1,3}$ unify in the same multiplet. As derived in Appendix
C.2, this implies that the following relations should hold at the matching
scale $\Lambda_{\rm UV}$:

$m_{\Omega_{1}}=m_{\Omega_{3}}~{},\qquad\rho_{1}=\rho_{2}+\rho_{2}^{\prime}~{},\quad\rho_{2}=\rho_{3}~{},\quad\rho_{2}^{\prime}=\rho_{4}~{}.$
(30)

  2. ii)

A second Higgs doublet with 4321 quantum numbers
$H_{2}\sim(\mathbf{1},\mathbf{1},\mathbf{2})_{-1/2}$ and mass parameter
$\mu_{2}$ is present. The Higgs potential is that of a Two Higgs Doublet Model
(2HDM) as written in Eq. (57) and the mixing potential is given by Eq. (58).
The UV constraints derived in Appendix C.2 give:

$\mu_{H}=\mu_{2}~{},\qquad\lambda=\lambda_{2}=\lambda_{3}~{},\qquad\eta_{1}=\eta_{2}=\eta_{3}=\eta_{4}~{}.$
(31)

The 2HDM slightly alters the procedure outlined in the beginning of the
section. First of all, we assume a simplified 2HDM potential (57) in the Higgs
basis i.e. $\mu_{2}^{2}$ remains positive at all energies and the $H_{2}$
field does not take a VEV. However, in this case, Eq. (31) implies that
radiative EWSB must happen in order to generate a negative $\mu_{\rm
eff}^{2}$. Then collider bounds can be simply avoided by requiring that the
masses of the heavy Higgs radials are above $500~{}\rm GeV$.777Notice that
this bound is sufficient also for other types of 2HDMs, e.g. where flavor
constraints are relevant Haller:2018nnx . Regardless of the choice of
$\lambda_{i}$, this constraint is readily satisfied for $\mu_{2}^{2}(500~{}\rm
GeV)\gtrsim(500~{}\rm GeV)^{2}$ (see Eq. (60)). Moreover, the $\lambda_{i}$
must satisfy the BFB conditions for the 2HDM (see Eq. (61)).

Here, we solve the RG equations by using the full $\beta$-functions listed in
Appendix A. From the scale $\Lambda_{4}$ down to the scale $\mu_{2}$, we use
the $\beta$-functions of the 2HDM (see Appendix B.2) and at $\mu_{2}$ the
degenerate heavy Higgs radials are collectively integrated out. Because
$\mu_{2}$ is much closer to the electroweak scale, the threshold corrections
in this case are much smaller and can be omitted. On the other hand, threshold
corrections at $\Lambda_{\rm UV}$ could originate from different scales and
induce important deviations to the expectations Eq. (30) and Eq. (31). We
quantify this uncertainty by allowing deviations up to 10 % from the exact
equalities. Despite that, due to the fast running of certain parameters, it is
not a priori clear whether the RG evolution can be reconciled with even
approximate conditions.

Our goal becomes then to search for sets of initial parameter values that
satisfies the IR boundary conditions and via their RG evolution yield sets of
values that approximately satisfy the UV boundary conditions. To obtain such
sets, we perform a numerical scan over the parameter space by solving the RG
equations for $\mathcal{O}(10^{5})$ random sets for benchmarks $g_{4}=2,3$ and
$\Lambda_{\rm UV}=10,10^{2},10^{3},10^{4},10^{5}~{}\rm TeV$. In order to
increase the efficiency of the scan in this high-dimensional space, we employ
a Markov Chain Monte Carlo algorithm (Hastings-Metropolis) for the generation
of trial sets. Additionally, we require that the UV values of the couplings
are still perturbative. It is worth mentioning that the non-trivial part of
the search concerns only the $\rho$ and $m_{\Omega_{1,3}}$ parameters. This is
because in the absence of concrete IR conditions for $\eta$, $\lambda_{2,3}$,
$\mu_{2}$ and a loose dependence on $g_{4}$, one can always adjust the IR
values in order to find appropriate UV ones.888We note that in multi-site UV
constructions, like the $\rm PS^{3}$ model, the $\Omega$ and the Higgs fields
may unify into their respective multiplets at different scales. Because, as
mentioned, the problem of matching to the conditions Eq. (31) is much less
constrained, one could also achieve this without significant deviations from
our discussion.

In Figure 5 we present the distribution of the obtained initial values of a
representative pair of parameters, namely $\rho_{3}$ and $\rho_{4}$. The first
observation is that we find no acceptable sets for unification scales as low
as $10~{}\rm TeV$ and very few sets for scales as high as $10^{5}~{}\rm TeV$
(and only in the case of $g_{4}=2$). This is explained by the fact that the
different $\rho$ parameters exhibit different running behaviors and thus
convergence to the UV conditions is possible only in the intermediate scales
$10^{3}-10^{4}~{}\rm TeV$. For instance, the relation
$\rho_{1}=\rho_{2}+\rho_{2}^{\prime}$ cannot be satisfied at higher energies
due to the faster running of $\rho_{1}$ and the fact that $\rho_{2}^{\prime}$
is monotonically decreasing as opposed to the other two. In total, we found
more sets for smaller $g_{4}$ because, as already discussed, the overall
running in this case is slower and thus more suitable for achieving
unification.

On the other hand, for lower unification scales the RG evolution spans a
shorter energy range and the running has to be accelerated in order to meet
the UV conditions. This is the reason why in Figure 5 we see that the initial
values of the parameters increase as $\Lambda_{\rm UV}$ decreases so that the
terms in the $\beta$-functions proportional to the couplings themselves are
enhanced. Altogether, the attempt to unify the couplings requires
$\Lambda_{\rm UV}$ approximately three orders of magnitude lower than the cut-
off scales discussed in Section 3.1 for the generic case. We conclude the
discussion by illustrating in Figure 6 the unification of the masses and
couplings using as an example the RG evolution generated by one of the
obtained sets.

Figure 5: Benchmark points in the $\rho_{3}(\Lambda_{4})$ –
$\rho_{4}(\Lambda_{4})$ plane satisfying the UV conditions Eqs. (30) and (31)
for different $g_{4}$ couplings and unification scales $\Lambda_{\text{UV}}$.

Figure 6: RG running of the gauge and top Yukawa couplings (left) and
couplings of the scalar potential (right) with a benchmark reaching the UV
conditions Eqs. (30) and (31). The y-axis, scalar masses, is defined as in
Figure 5.

##  4 Conclusions

In this work, we performed an analysis of the RG evolution of the 4321 model,
which revealed several interesting features inherent to the scalar sector of
the model. We observed that the large value of $g_{4}$ plays an important role
in the overall behavior of the running and it can lead to the appearance of
Landau poles even though the gauge coupling itself is asymptotically free.
Phenomenological constraints on the $\Omega_{1,3}$ radial modes prevent their
quartic couplings from assuming arbitrarily small IR values, leading these
quartic couplings to blow up, and compromising the validity of the 4321 model,
at energies far below the GUT scale. Additionally, we found that it is
possible to satisfy relations from motivated UV embeddings within the energy
range $\mathcal{O}(10^{2}-10^{4})~{}\rm TeV$.

Intriguingly, the emergence of Landau poles seems to be intrinsic to a large
class of the models proposed as combined explanations for both charged- and
neutral-current $B$-physics anomalies. Depending on the model, the problem may
appear in different sectors of the theory but typically involves the presence
of fundamental scalars (see Refs. DiLuzio:2018zxy ; Azatov:2018kzb ;
Marzocca:2021azj  for examples of Landau poles in the Yukawa couplings). As
discussed, it is highly non-trivial to address this problem at the level of
$\beta$-functions by introducing additional fields. This might indicate that
the 4321 gauge symmetry is broken not by fundamental scalars, but rather à la
technicolor by the condensate of a strongly coupled sector as in Ref. Fuentes-
Martin:2020bnh ; Fuentes-Martin:2022xnb . Another interesting possibility is
that the leptoquark arises directly as a composite resonance of a new strongly
coupled sector as in Refs. Barbieri:2016las ; Marzocca:2018wcf . For example,
the model in Ref. Marzocca:2018wcf  with composite scalar leptoquarks and
Higgs is free of low-energy Landau poles.

Upon studying the RG flow of the mass squared parameters in the scalar sector,
we found that the 4321 model can accommodate radiative EWSB due to the
additional scalar fields coupled to the Higgs. While radiative EWSB requires
positive contributions to the $\beta$-function generated by the
$H$-$\Omega_{1,3}$ mixing, the IR value of the parameter that controls EWSB,
$\mu_{\text{eff}}^{2}$, relies on the whole system of scalar quartic
couplings, mixing couplings, and $\Omega_{1,3}$ mass parameters. We quantified
the sensitivity of the Higgs mass to the UV values of the parameters. The
resulting fine-tuning reflected the little hierarchy problem associated with
the new TeV scalars.

Ultimately, if the B-physics anomalies are confirmed as clear signals of new
physics, leptoquarks arise as the most plausible explanation. Given that all
leptoquark models contain additional scalar particles, we stress the
importance of carefully examining their sectors, and in particular, the RG
evolution of the scalar couplings. As we have seen in the case of 4321, new
features can emerge upon analysis of radiative effects, constraining the
possibilities for UV constructions.

###### Acknowledgements.

We thank Gino Isidori for discussion and comments on this work, Ben Stefanek
for useful discussion and Sandro Mächler for crosschecks. S.T. acknowledges
support by MIUR grant PRIN 2017L5W2PT. R.H. is supported by the STFC under
grant ST/P001246/. J.P. acknowledges the European Research Council (ERC) under
the European Union’s Horizon 2020 research and innovation programme under
grant agreement 833280 (FLAY).

##  Appendix A $\beta$-functions of the 4321 model

The $\beta$-functions for the gauge couplings at one-loop level are given by
the master formula:

$\mu\frac{d\alpha_{i}}{d\mu}\equiv\frac{1}{16\pi^{2}}\,\beta_{\alpha_{i}}=-\frac{\alpha_{i}^{2}}{2\pi}b_{i}~{},$
(32)

where

$b_{i}=\frac{11}{3}C_{2}(G)-\frac{4}{3}\sum_{f}\kappa_{f}S_{2}(R_{f})-\sum_{s}\frac{1}{6}S_{2}(R_{s})~{}.$
(33)

The first term contains the quadratic Casimir of the gauge boson
representation $C_{2}(G)$, the second and the third the sums over the Dynkin
indices for the representations of all the fermions $S_{2}(R_{f})$
($\kappa_{f}=1/2$ or $\kappa_{f}=1$ for 2- or 4-component fermions) and the
scalars $S_{2}(R_{s})$, respectively. Taking into account the full particle
content of the 4321 model the resulting values for $b_{i}$ are

$\displaystyle\begin{aligned} U(1)^{\prime}:b_{1}&=-\frac{115}{18}~{},&\quad
SU(2)_{L}:b_{2}&=\frac{19}{6}-\frac{8}{3}n_{\rm VL}~{},~{}\\\
SU(3)_{1+2}:b_{3}&=\frac{23}{3}~{},&\quad
SU(4)_{3}:b_{4}&=\frac{38}{3}-\frac{4}{3}n_{\rm VL}~{}.\end{aligned}$ (34)

where $n_{\rm VL}$ is the number of vector-like fermions.

The $\beta$-functions for the couplings of the potential in Eq. (4) are given
at one-loop order:

$\displaystyle\beta_{\mu_{H}^{2}}$
$\displaystyle=\left(6\lambda+8y_{t}^{2}-\frac{3}{2}g_{1}^{2}-\frac{9}{2}g_{2}^{2}\right)\mu_{H}^{2}+8\eta_{1}m_{\Omega_{1}}^{2}+24\eta_{3}m_{\Omega_{3}}^{2}{\color[rgb]{.5,.5,.5}\,+\,(4\lambda_{3}+2\lambda_{4})\mu_{2}^{2}}~{},$
(35) $\displaystyle{\color[rgb]{.5,.5,.5}\beta_{\mu_{2}^{2}}}$
$\displaystyle{\color[rgb]{.5,.5,.5}\,=\left(6\lambda_{2}+8y_{t}^{2}-\frac{3}{2}g_{1}^{2}-\frac{9}{2}g_{2}^{2}\right)\mu_{2}^{2}+(4\lambda_{3}+2\lambda_{4})\mu_{H}^{2}+8\eta_{2}m_{\Omega_{1}}^{2}+24\eta_{4}m_{\Omega_{3}}^{2}~{},}$
(36) $\displaystyle\beta_{m_{\Omega_{1}}^{2}}$
$\displaystyle=\left(10\rho_{1}-\frac{3}{2}g_{1}^{2}-\frac{45}{4}g_{4}^{2}\right)m_{\Omega_{1}}^{2}+(24\rho_{3}+6\rho_{4})m_{\Omega_{3}}^{2}+4\eta_{1}\mu_{H}^{2}{\color[rgb]{.5,.5,.5}\,+\,4\eta_{2}\mu_{2}^{2}~{}},$
(37) $\displaystyle\beta_{m_{\Omega_{3}}^{2}}$
$\displaystyle=\left(26\rho_{2}+14\rho_{2}^{\prime}-\frac{1}{6}g_{1}^{2}-8g_{3}^{2}-\frac{45}{4}g_{4}^{2}\right)m_{\Omega_{3}}^{2}+4\eta_{3}\mu_{H}^{2}+(8\rho_{3}+2\rho_{4})m_{\Omega_{1}}^{2}{\color[rgb]{.5,.5,.5}\,+\,4\eta_{4}\mu_{2}^{2}}~{},$
(38) $\displaystyle\beta_{\lambda}$
$\displaystyle=\left(12\lambda+16y_{t}^{2}-3g_{1}^{2}-9g_{2}^{2}\right)\lambda+8\eta_{1}^{2}+24\eta_{3}^{2}-16y_{t}^{4}{\color[rgb]{.5,.5,.5}\,+\,4\lambda_{3}^{2}+4\lambda_{3}\lambda_{4}+2\lambda_{4}^{2}+2\lambda_{5}^{2}}$
$\displaystyle\ \ \ \ \
+\frac{3}{4}g_{1}^{4}+\frac{3}{2}g_{1}^{2}g_{2}^{2}+\frac{9}{4}g_{2}^{4}~{},$
(39) $\displaystyle{\color[rgb]{.5,.5,.5}\beta_{\lambda_{2}}}$
$\displaystyle{\color[rgb]{.5,.5,.5}\,=\left(12\lambda_{2}+16y_{t}^{2}-3g_{1}^{2}-9g_{2}^{2}\right)\lambda_{2}+8\eta_{2}^{2}+24\eta_{4}^{2}-16y_{t}^{4}+4\lambda_{3}^{2}+4\lambda_{3}\lambda_{4}+2\lambda_{4}^{2}+2\lambda_{5}^{2}}$
$\displaystyle\ \ \ \ \
{\color[rgb]{.5,.5,.5}+\,\frac{3}{4}g_{1}^{4}+\frac{3}{2}g_{1}^{2}g_{2}^{2}+\frac{9}{4}g_{2}^{4}~{},}$
(40) $\displaystyle{\color[rgb]{.5,.5,.5}\beta_{\lambda_{3}}}$
$\displaystyle{\color[rgb]{.5,.5,.5}\,=\left(4\lambda_{3}+6\lambda+6\lambda_{2}-3g_{1}^{2}-9g_{2}^{2}+16y_{t}^{2}\right)\lambda_{3}+2\lambda_{4}(\lambda+\lambda_{2})+2\lambda_{4}^{2}+2\lambda_{5}^{2}-16y_{t}^{4}}$
$\displaystyle\ \ \ \ \
{\color[rgb]{.5,.5,.5}+\,8\eta_{1}\eta_{2}+24\eta_{3}\eta_{4}+\frac{3}{4}g_{1}^{4}-\frac{3}{2}g_{1}^{2}g_{2}^{2}+\frac{9}{4}g_{2}^{4}~{},}$
(41) $\displaystyle{\color[rgb]{.5,.5,.5}\beta_{\lambda_{4}}}$
$\displaystyle{\color[rgb]{.5,.5,.5}\,=\left(4\lambda_{4}+2\lambda+2\lambda_{2}+8\lambda_{3}-3g_{1}^{2}-9g_{2}^{2}+16y_{t}^{2}\right)\lambda_{4}+8\lambda_{5}^{2}+3g_{1}^{2}g_{2}^{2}+16y_{t}^{4}~{},}$
(42) $\displaystyle{\color[rgb]{.5,.5,.5}\beta_{\lambda_{5}}}$
$\displaystyle{\color[rgb]{.5,.5,.5}\,=\left(2\lambda+2\lambda_{2}+8\lambda_{3}+12\lambda_{4}+16y_{t}^{2}-9g_{2}^{2}-3g_{1}^{2}\right)\lambda_{5}~{},}$
(43) $\displaystyle\beta_{\rho_{1}}$
$\displaystyle=\left(16\rho_{1}-3g_{1}^{2}-\frac{45}{2}g_{4}^{2}\right)\rho_{1}+24\rho_{3}^{2}+12\rho_{3}\rho_{4}+6\rho_{4}^{2}+4\eta_{1}^{2}{\color[rgb]{.5,.5,.5}\,+\,4\eta_{2}^{2}}$
$\displaystyle\ \ \ \ \
+\frac{3}{4}g_{1}^{4}+\frac{9}{4}g_{1}^{2}g_{4}^{2}+\frac{99}{16}g_{4}^{4}~{},$
(44) $\displaystyle\beta_{\rho_{2}}$
$\displaystyle=\left(32\rho_{2}+28\rho_{2}^{\prime}-\frac{1}{3}g_{1}^{2}-16g_{3}^{2}-\frac{45}{2}g_{4}^{2}\right)\rho_{2}+6{\rho_{2}^{\prime}}^{2}+8\rho_{3}^{2}+4\rho_{3}\rho_{4}+4\rho_{5}^{2}$
$\displaystyle\ \ \ \ \
+4\eta_{3}^{2}{\color[rgb]{.5,.5,.5}\,+\,4\eta_{4}^{2}}+\frac{1}{108}g_{1}^{4}+\frac{11}{6}g_{3}^{4}+\frac{27}{16}g_{4}^{4}-\frac{1}{9}g_{1}^{2}g_{3}^{2}-\frac{1}{12}g_{1}^{2}g_{4}^{4}+\frac{13}{2}g_{3}^{2}g_{4}^{2}~{},$
(45) $\displaystyle\beta_{\rho_{2}^{\prime}}$
$\displaystyle=\left(14{\rho_{2}^{\prime}}+12\rho_{2}-\frac{1}{3}g_{1}^{2}-16g_{3}^{2}-\frac{45}{2}g_{4}^{2}\right)\rho_{2}^{\prime}+2\rho_{4}^{2}-4\rho_{5}^{2}$
$\displaystyle\ \ \ \ \
+\frac{5}{2}g_{3}^{4}+\frac{9}{2}g_{4}^{4}+\frac{1}{3}g_{1}^{2}g_{3}^{2}+\frac{1}{3}g_{1}^{2}g_{4}^{2}-\frac{7}{2}g_{3}^{2}g_{4}^{2}~{},$
(46) $\displaystyle\beta_{\rho_{3}}$
$\displaystyle=\left(4\rho_{3}+10\rho_{1}+26\rho_{2}+14\rho_{2}^{\prime}-\frac{5}{3}g_{1}^{2}-8g_{3}^{2}-\frac{45}{2}g_{4}^{2}\right)\rho_{3}+2\rho_{4}^{2}+4\rho_{5}^{2}$
$\displaystyle\ \ \ \ \
+2(\rho_{1}+3\rho_{2}+\rho_{2}^{\prime})\rho_{4}+4\eta_{1}\eta_{3}{\color[rgb]{.5,.5,.5}\,+\,4\eta_{2}\eta_{4}}+\frac{1}{12}g_{1}^{4}+\frac{1}{4}g_{1}^{2}g_{4}^{2}+\frac{27}{16}g_{4}^{4}~{},$
(47) $\displaystyle\beta_{\rho_{4}}$
$\displaystyle=\left(8\rho_{4}+2\rho_{1}+2\rho_{2}+6\rho_{2}^{\prime}+8\rho_{3}-\frac{5}{3}g_{1}^{2}-8g_{3}^{2}-\frac{45}{2}g_{4}^{2}\right)\rho_{4}-4\rho_{5}^{2}$
$\displaystyle\ \ \ \ \ -g_{1}^{2}g_{4}^{2}+\frac{9}{2}g_{4}^{4}~{},$ (48)
$\displaystyle\beta_{\rho_{5}}$
$\displaystyle=\left[6(\rho_{2}-\rho_{2}^{\prime}+\rho_{3}-\rho_{4})-g_{1}^{2}-12g_{3}^{2}-\frac{45}{2}g_{4}^{2}\right]\rho_{5}~{},$
(49) $\displaystyle\beta_{\eta_{1}}$
$\displaystyle=\left(4\eta_{1}+6\lambda+10\rho_{1}-3g_{1}^{2}-\frac{9}{2}g_{2}^{2}-\frac{45}{4}g_{4}^{2}+8y_{t}^{2}\right)\eta_{1}$
$\displaystyle\ \ \ \ \
{\color[rgb]{.5,.5,.5}+\,4\lambda_{3}\eta_{2}+2\lambda_{4}\eta_{2}}+24\rho_{3}\eta_{3}+6\rho_{4}\eta_{3}+\frac{3}{4}g_{1}^{4}~{},$
(50) $\displaystyle{\color[rgb]{.5,.5,.5}\beta_{\eta_{2}}}$
$\displaystyle{\color[rgb]{.5,.5,.5}\,=\left(4\eta_{2}+6\lambda_{2}+10\rho_{1}-3g_{1}^{2}-\frac{9}{2}g_{2}^{2}-\frac{45}{4}g_{4}^{2}+8y_{t}^{2}\right)\eta_{2}}$
$\displaystyle\ \ \ \ \
{\color[rgb]{.5,.5,.5}+\,4\lambda_{3}\eta_{1}+2\lambda_{4}\eta_{1}+24\rho_{3}\eta_{4}+6\rho_{4}\eta_{4}+\frac{3}{4}g_{1}^{4}~{},}$
(51) $\displaystyle\beta_{\eta_{3}}$
$\displaystyle=\left(6\lambda+26\rho_{2}+14\rho_{2}^{\prime}-\frac{5}{3}g_{1}^{2}-\frac{9}{2}g_{2}^{2}-8g_{3}^{2}-\frac{45}{4}g_{4}^{2}+8y_{t}^{2}\right)\eta_{3}$
$\displaystyle\ \ \ \ \
{\color[rgb]{.5,.5,.5}+\,(4\lambda_{3}+2\lambda_{4})\eta_{4}}+(8\rho_{3}+2\rho_{4})\eta_{1}+4\eta_{3}^{2}+\frac{1}{12}g_{1}^{4}~{},$
(52) $\displaystyle{\color[rgb]{.5,.5,.5}\beta_{\eta_{4}}}$
$\displaystyle{\color[rgb]{.5,.5,.5}\,=6\lambda_{2}\eta_{4}+(4\lambda_{3}+2\lambda_{4})\eta_{3}+2(13\rho_{2}+7\rho_{2}^{\prime})\eta_{4}+(8\rho_{3}+2\rho_{4})\eta_{2}+4\eta_{4}^{2}}$
$\displaystyle\ \ \ \ \
{\color[rgb]{.5,.5,.5}-\left(\frac{5}{3}g_{1}^{2}+\frac{9}{2}g_{2}^{2}+8g_{3}^{2}+\frac{45}{4}g_{4}^{2}\right)\eta_{4}+\frac{1}{12}g_{1}^{4}+8\eta_{4}Y_{12}~{},}$
(53) $\displaystyle\beta_{y_{t}}$
$\displaystyle=\left(\frac{11}{2}y_{t}^{2}-\frac{9}{4}g_{2}^{2}-\frac{45}{4}g_{4}^{2}-\frac{3}{4}g_{1}^{2}\right)y_{t}~{}.$
(54)

where only the top Yukawa coupling was considered from Eq. (55). The terms
written in gray describe the running/contribution from a second Higgs doublet
in the model (see Appendix B.2). All the $\beta$-functions were crosschecked
using the Mathematica package RGBeta Thomsen:2021ncy .

##  Appendix B Subleading contributions to the $\beta$-functions

In Section 2.2.1, we considered the minimal scalar content for the 4321 model
and neglected all Yukawa couplings except for the top. Below, we discuss the
influence of additional contributions to the $\beta$-functions, arising from
the Yukawa couplings of the vector-like fermions and from the addition of a
second Higgs doublet, or other extra scalars.

###  B.1 Yukawa sector

Neglecting the Yukawa couplings of the light generations, the Yukawa sector of
the 4321 model reads

$\displaystyle\begin{aligned} -\mathcal{L}_{\rm
Yuk}&=y_{t}\,\bar{\Psi}_{L}\tilde{H}\Psi_{R}^{+}+y_{b}\,\bar{\Psi}_{L}H\Psi_{R}^{-}+y_{+}\,\bar{\chi}_{L}\tilde{H}\Psi_{R}^{+}+y_{-}\,\bar{\chi}_{L}H\Psi_{R}^{-}\\\
&+\lambda_{q}\,\bar{q}_{L}\Omega_{3}\chi_{R}+\lambda_{\ell}\,\bar{\ell}_{L}\Omega_{1}\chi_{R}+M_{\chi}\,\bar{\chi}_{L}\chi_{R}\,+\text{h.c.}\,~{},\end{aligned}$
(55)

with $\tilde{H}=i\sigma_{2}H^{*}$ and where we used the freedom of rotating
the multiplet $\begin{pmatrix}\Psi_{L},\chi_{L}\end{pmatrix}$ to remove the
mixing term $\bar{\Psi}_{L}\,\chi_{R}$. The biggest couplings are $y_{t}\sim
1$ and $y_{+}\sim 1/4$, while the other couplings are expected to be smaller,
i.e. $\lambda_{q,\ell}\sim 10^{-1}$, $y_{b}\sim 10^{-2}$ and $y_{-}\sim
10^{-3}$ from phenomenological considerations Cornella:2021sby . The influence
of these terms on the scalar quartics $\beta$-functions were computed and did
not yield any relevant contribution. For example, the contribution

$\beta_{\rho_{1}}\supset-8|\lambda_{\ell}|^{4}+8\rho_{1}|\lambda_{\ell}|^{2}\,~{},$
(56)

should be compared to the $g_{4}^{4}$ term in Eq. (A).

###  B.2 Two higgs doublet model

A second Higgs doublet $H_{2}\sim(\bm{1},\bm{1},\bm{2})_{1/2}$ is
theoretically motivated by UV embeddings of 4321. For instance, in the $\rm
PS^{3}$ model Bordone:2017bld  $H$ and $H_{2}$ form a bi-doublet field
(doublet under $SU(2)_{L}$ and $SU(2)_{R}$ of the third site). Although a bi-
doublet can be formed by a doublet and its own dual, the presence of a
different doublet $H_{2}$ allows the splitting between the top and bottom
masses.

##### Potential.

Assuming a simplified 2HDM, the Higgs potential in Eq. (6) is augmented by

$\displaystyle V_{H}$
$\displaystyle\supset\mu_{2}^{2}H_{2}^{\dagger}H_{2}+\frac{\lambda_{2}}{2}(H_{2}^{\dagger}H_{2})^{2}+\lambda_{3}(H^{\dagger}H)(H_{2}^{\dagger}H_{2})+\lambda_{4}(H^{\dagger}H_{2})(H_{2}^{\dagger}H)+\frac{\lambda_{5}}{2}\left((H^{\dagger}H_{2})^{2}+\rm
h.c\right)$ (57)

and the mixing potential in Eq. (7) by

$V_{\Omega
H}\supset\eta_{2}\,H_{2}^{\dagger}H_{2}\,\Omega_{1}^{\dagger}\Omega_{1}+\eta_{4}\,H_{2}^{\dagger}H_{2}\,{\rm
Tr}[\Omega_{3}^{\dagger}\Omega_{3}]~{}.$ (58)

For the Yukawa couplings, the same terms with $H$ in Eq. (55) can be written
with $H_{2}$.

##### Radial modes.

In the Higgs basis where only $H$ takes a VEV, the two doublets can be
parametrized as

$H=\begin{pmatrix}\eta^{+}_{W}\\\
\dfrac{v+h}{\sqrt{2}}+i\,\eta_{Z}\end{pmatrix}~{},\qquad
H_{2}=\begin{pmatrix}h^{+}\\\
\dfrac{1}{\sqrt{2}}(h_{R}+i\,h_{I})\end{pmatrix}~{},$ (59)

where $\eta_{W}^{\pm},\eta_{Z}$ become the $W^{\pm}$ and $Z$ radial modes,
respectively, and $h$ is the physical Higgs boson. The scalar radial degrees
of freedom $h$, $h^{\pm}$, $h_{R}$ and $h_{I}$ aquire the following masses
Branco:2011iw ; Belyaev:2016lok

$\displaystyle\begin{aligned} &m_{h}^{2}=\lambda v^{2}~{},\\\
&m_{\pm}^{2}=\mu_{2}^{2}+\frac{\lambda_{3}}{2}v^{2}~{},\\\
&m_{R}^{2}=\mu_{2}^{2}+\frac{\lambda_{3}+\lambda_{4}+\lambda_{5}}{2}v^{2}~{},\\\
&m_{I}^{2}=\mu_{2}^{2}+\frac{\lambda_{3}+\lambda_{4}-\lambda_{5}}{2}v^{2}~{}.\end{aligned}$
(60)

##### BFB conditions.

For the potential (57), necessary and sufficient conditions to ensure
boundedness from below were found to be Klimenko:1984qx

$\lambda>0~{},\qquad\lambda_{2}>0~{},\qquad\lambda_{3}>-\sqrt{\lambda\,\lambda_{2}}~{},\qquad\lambda_{3}+\lambda_{4}-|\lambda_{5}|>-\sqrt{\lambda\,\lambda_{2}}~{}.$
(61)

For the mixed part of the potential $V_{\Omega H}$, similarly to Eq. (26), the
derived conditions are

$\displaystyle\begin{aligned}
&\eta_{2}>-\sqrt{\lambda_{2}\rho_{1}}~{},\qquad\eta_{4}>-\min\left[\sqrt{\lambda_{2}(\rho_{2}+\rho_{2}^{\prime})},\sqrt{\lambda_{2}(3\rho_{2}+\rho_{2}^{\prime})}\right]~{}.\end{aligned}$
(62)

#####  $\beta$-functions.

The $\beta$-functions above the 4321 breaking scale are presented in Appendix
A with the gray terms to be included. Another small difference is that the
coefficient in front of $y_{t}^{3}$ in Eq. (54) becomes $6$ instead of $11/2$.

Below 4321 breaking scale, we can run in the 2HDM until the mass threshold of
the heavier Higgs. The most general Yukawa sector is

$\displaystyle-\mathcal{L}_{\rm Yuk}^{\rm(2HDM)}$
$\displaystyle=y_{t}\,\bar{q}_{L}^{(3)}\tilde{H}u_{R}^{(3)}+y_{b}\,\bar{q}_{L}^{(3)}Hd_{R}^{(3)}+y_{\tau}\,\bar{\ell}_{L}^{(3)}He_{R}^{(3)}$
$\displaystyle+y_{t2}\,\bar{q}_{L}^{(3)}\tilde{H}_{2}u_{R}^{(3)}+y_{b2}\,\bar{q}_{L}^{(3)}H_{2}d_{R}^{(3)}+y_{\tau
2}\,\bar{\ell}_{L}^{(3)}H_{2}e_{R}^{(3)}~{}.$ (63)

In the case $\langle H\rangle=v$ and $\langle H_{2}\rangle=0$, we expect from
UV matching $y_{t}=y_{b2}=y_{\tau 2}$ and $y_{t2}=y_{b}=y_{\tau}$ (see
Appendix C.1).

The gauge and scalar contributions to the 2HDM $\beta$-functions are the same
as in Eqs.(35-36) and (39-43) taking $\eta_{1,2,3,4}\rightarrow 0$. The Yukawa
contributions are changed as follow

$\displaystyle\begin{aligned} \beta_{\mu_{H}^{2}}&\supset
6y_{t}^{2}\mu_{H}^{2}~{},\qquad&\beta_{\mu_{2}^{2}}&\supset{\color[rgb]{.5,.5,.5}8y_{t}^{2}\mu_{2}^{2}}~{},\\\
\beta_{\lambda}&\supset
12y_{t}^{2}\lambda-12y_{t}^{4}~{},\qquad&\beta_{\lambda_{2}}&\supset{\color[rgb]{.5,.5,.5}16y_{t}^{2}\lambda_{2}-16y_{t}^{4}}~{},\\\
\beta_{\lambda_{3}}&\supset
6y_{t}^{2}\lambda_{3}{\color[rgb]{.5,.5,.5}\,+\,8y_{t}^{2}\lambda_{3}-12y_{t}^{4}}~{},\qquad&\beta_{\lambda_{4}}&\supset
6y_{t}^{2}\lambda_{4}{\color[rgb]{.5,.5,.5}\,+\,8y_{t}^{2}\lambda_{4}+12y_{t}^{4}}~{},\\\
\beta_{\lambda_{5}}&\supset
6y_{t}^{2}\lambda_{5}{\color[rgb]{.5,.5,.5}\,+\,8y_{t}^{2}\lambda_{5}}~{}.\end{aligned}$
(64)

where the $y_{b}$ contributions were neglected. The difference in
multiplicative factors with respect to Eqs. (35-36) and (39-43) is due to the
fact that we neglected the neutrino Yukawa in equation Eq. (63), whereas it is
automatically included in the 4321 beta functions since both chiralities are
part of $SU(4)$ multiplets.

The top Yukawa couplings in equation Eq. (63) run as

$\displaystyle\beta_{y_{t}}$
$\displaystyle=y_{t}\left(-\frac{17}{12}g_{1}^{2}-\frac{9}{4}g_{2}^{2}-8g_{3}^{2}+\frac{9}{2}y_{t}^{2}{\color[rgb]{.5,.5,.5}\,+\,\frac{1}{2}y_{t}^{2}}\right)~{}.$
(65)

Without UV input, we could also consider a simpler 2HDM where only one of the
Higgs couple to fermions, i.e. $y_{t2}=y_{b2}=y_{\tau 2}=0$, in which case the
gray terms in all equations above Eq. (64) and Eq. (65) would be absent.

###  B.3 Extended scalar sector

For our analysis of the UV behavior, we only considered $\Omega_{1}$ and
$\Omega_{3}$. However, we mentioned that other scalars might be required for
phenomenological reasons. Below, we briefly review which additional scalars
have been considered in the literature, their role and their contribution to
the Lagrangian.

##### $H_{2}\sim(\bm{1},\bm{1},\bm{2})_{1/2}$

was already discussed in Appendix B.2.

##### $H_{15}\sim(\bm{15},\bm{1},\bm{2})_{1/2}$

from Refs. Bordone:2018nbg ; Cornella:2019hct  breaks the electroweak symmetry
with a VEV aligned along $T^{15}\propto\text{diag}(1,1,1,-3)$. This field
generates a splitting between the third generation quark and lepton in the
Yukawa couplings allowing for $m_{b}\neq m_{\tau}$.999The $m_{t}\neq m_{\nu}$
splitting can in principle also be realized this way but would require an
unnatural fine-tuning of $10^{-12}$. For a realistic neutrino mechanism see
Ref. Fuentes-Martin:2020pww . Its Lagrangian extension is similar to the one
of the second Higgs $H_{2}$. Three noticeable differences with the previous
case are:

  1. i)

$H$ and $H_{15}$ cannot come from the same bi-doublet in the UV.

  2. ii)

Since it is an adjoint of $SU(4)$ it can only couple to the third generation
fermions in the Yukawa, therefore exhibiting a $U(2)^{5}$ symmetry on the
light generations. On the contrary $H$ and $H_{2}$ can couple to the light
generations and their Yukawa couplings were just assumed to be small.

  3. iii)

It contains more fields in the broken phase of 4321 namely
$H_{15}\rightarrow(\bm{8},\bm{2})_{1/2}\oplus(\bm{3},\bm{2})_{7/6}\oplus(\bm{\overline{3}},\bm{2})_{-1/6}\oplus(\bm{1},\bm{2})_{1/2}$.
One can assume that the non-trivial $SU(3)_{c}$ representations decouple,
retrieving the standard 2HDM again. Otherwise, the theory below 4321 breaking
should also contain these modes.

##### $\Omega_{15}\sim(\bm{15},\bm{1},\bm{1})_{0}$

from Refs. DiLuzio:2018zxy ; Cornella:2019hct  breaks 4321 with the VEV
$\langle\Omega_{15}\rangle=\omega_{15}\,T^{15}$. Similarly to $H_{15}$, this
field splits quarks and leptons. This time, the splitting happens in the
coupling of the third generation quarks and leptons to their vector-like
partners, and after integrating out the vector-like fermions, to the $U_{1}$
leptoquark. The masses of the vector-like fermions are also split, allowing
the vector-like leptons to be lighter than the vector-like quarks. The Yukawa
terms can be written as

$-\mathcal{L}\supset\lambda_{15}\bar{\Psi}_{L}\Omega_{15}\chi_{R}+\lambda^{\prime}_{15}\bar{\chi}_{L}\Omega_{15}\chi_{R}+\text{h.c.}$
(66)

This field could also be used to split the leptons and quarks masses directly,
as with $H_{15}$, by introducing higher dimensional operators, e.g.
$\frac{1}{\Lambda}\bar{\Psi}_{L}\,\Omega_{15}\,\tilde{H}\,\Psi_{R}^{+}$. In
the broken phase of 4321, this field decomposes into
$\Omega_{15}\to(\bm{1},\bm{1})_{0}\oplus(\bm{3},\bm{1})_{2/3}\oplus(\bm{\overline{3}},\bm{1})_{-2/3}\oplus(\bm{8},\bm{1})_{0}$.

Both $H_{15}$ and $\Omega_{15}$ have a more involved scalar potential which
could give rise to the appearance of new Landau poles similarly to the
$\Omega_{1,3}$ potential. Moreover their mixing coupling with the
$\Omega_{1,3}$ sector will give contribution to the quartic couplings studied
in the main analysis with the same sign as the $g_{4}^{4}$ contribution in
Eqs. (A-A). Their effect will therefore only strengthen the conclusions about
the appearance of Landau poles.

Another important effect of extending the scalar content of the theory is its
influence on the running of the gauge couplings as can be seen in Eq. (33). At
one loop, the condition for asymptotic freedom is $b_{i}>0$. The
$\beta$-functions coefficients $b_{i}$ for $i=1,2,3,4$ with the minimal scalar
content are presented in Eq. (34). Since the IR value of $g_{4}$ is large and
drives most scalar quartic beta-functions, the faster it decreases in the UV,
the later the Landau poles arise. Adding more scalars is equivalent to
decreasing $b_{4}$ and slowing down its approach to asymptotic freedom. The
influence of each scalar on the $b_{i}$ coefficients is presented in Table 3.
We can conclude that adding all these scalars to the theory is not enough to
directly ruin the one-loop asymptotic freedom condition for gauge couplings,
but they would accelerate the appearance of the Landau poles in the scalar
sector as discussed in Section 3.1.

$\displaystyle\begin{array}[]{|c|c|c|c|c|c|c|c|}\hline\cr
S_{2}(R_{s})&H&H_{2}&H_{15}&\Omega_{15}&\Omega_{3}&\Omega_{1}\\\ \hline\cr
U(1)&1&1&15&0&2/3&2\\\ SU(2)_{L}&1&1&15&0&0&0\\\ SU(3)&0&0&0&0&4&0\\\
SU(4)&0&0&16&8&3&1\\\ \hline\cr\end{array}$ Table 3: Dynkin representation
multiplicity factor of each scalar.

##  Appendix C Constraints from UV embedding

Assuming a PS3 origin of the 4321 model as in Bordone:2017bld  yields specific
relations between the coefficient of the scalar potential and the Yukawa
couplings. By considering the representations of the fields in this UV setup,
their decompositions under the subgroup

${\rm PS}_{1+2}\times{\rm PS}_{3}\to SU(4)_{3}\times SU(3)_{1+2}\times
SU(2)_{L}\times U(1)^{\prime}~{},$ (67)

where ${\rm PS}_{i}=SU(4)_{i}\times SU(2)_{L,i}\times SU(2)_{R,i}$, read as

$\displaystyle\begin{aligned} \Phi\sim(1,2,\bar{2})_{3}&\to
H\sim(1,1,2)_{\frac{1}{2}}+\tilde{H_{2}}\sim(1,1,2)_{-\frac{1}{2}}~{},\\\
\Omega\sim(4,2,1)_{1+2}\times(\bar{4},\bar{2},1)_{3}&\to\Omega_{3}\sim(\bar{4},3,1)_{\frac{1}{6}}+\Omega_{3}^{\prime}\sim(\bar{4},3,3)_{\frac{1}{6}}\\\
&\quad+\Omega_{1}\sim(\bar{4},1,1)_{-\frac{1}{2}}+\Omega_{1}^{\prime}\sim(\bar{4},1,3)_{-\frac{1}{2}}~{},\\\
\Psi_{R}^{(3)}\sim(4,1,2)_{3}&\to\Psi_{R}^{+}\sim(4,1,1)_{\frac{1}{2}}+\Psi_{R}^{-}\sim(4,1,1)_{-\frac{1}{2}}~{}\,,\\\
\Psi_{L}^{(3)}\sim(4,2,1)_{3}&\to\Psi_{L}\sim(4,2,1)_{0}~{}.\end{aligned}$
(68)

###  C.1 Yukawa sector

We consider a Higgs bi-doublet field that couples mostly to the third
generation of fermions. Neglecting light generations coupling, two Yukawas
terms can be written with $\Phi^{c}=\sigma_{2}\Phi^{*}\sigma_{2}$. The complex
bi-doublet $\Phi$ can be written as

$\Phi=\begin{pmatrix}\tilde{H}_{2}&H\end{pmatrix}=\begin{pmatrix}{H_{2}^{0}}^{*}&H^{+}\\\
-H_{2}^{-}&H^{0}\end{pmatrix}~{},\qquad\Phi^{c}=\begin{pmatrix}\tilde{H}&H_{2}\end{pmatrix}=\begin{pmatrix}{H^{0}}^{*}&H_{2}^{+}\\\
-H^{-}&H_{2}^{0}\end{pmatrix}~{}.$ (69)

where $H$ and $H_{2}$ are $SU(2)_{L}$ doublets and
$\tilde{H}_{(2)}=i\sigma^{2}H_{(2)}^{*}$. Since
$\Psi_{R}^{(3)}=\begin{pmatrix}\Psi_{R}^{+}\\\ \Psi_{R}^{-}\end{pmatrix}$ the
Yukawa Lagrangian expands to

$\displaystyle\begin{aligned} -\mathcal{L}_{\rm
Yuk}&=\tilde{y}^{\prime}\,\bar{\Psi}_{L}^{(3)}\Phi\Psi_{R}^{(3)}+\tilde{y}\,\bar{\Psi}_{L}^{(3)}\Phi^{c}\Psi_{R}^{(3)}\\\
&=\tilde{y}^{\prime}\,\bar{\Psi}_{L}\tilde{H}_{2}\Psi_{R}^{+}+\tilde{y}^{\prime}\,\bar{\Psi}_{L}H\Psi_{R}^{-}+\tilde{y}\,\bar{\Psi}_{L}\tilde{H}\Psi_{R}^{+}+\tilde{y}\,\bar{\Psi}_{L}H_{2}\Psi_{R}^{-}~{}.\end{aligned}$
(70)

Generally, the Higgses will get the aligned VEVs

$\langle H\rangle=\begin{pmatrix}0\\\
\frac{v_{1}}{\sqrt{2}}\end{pmatrix}~{},\qquad\langle
H_{2}\rangle=\begin{pmatrix}0\\\ \frac{v_{2}}{\sqrt{2}}\end{pmatrix}~{}.$ (71)

and give rise to the masses

$\displaystyle\begin{aligned}
&m_{t}=m_{\nu}=\frac{1}{\sqrt{2}}(\tilde{y}^{\prime}\,v_{2}+\tilde{y}\,v_{1})=\frac{v}{\sqrt{2}}\underbrace{(\tilde{y}^{\prime}\,c_{\alpha}+\tilde{y}\,s_{\alpha})}_{\displaystyle
y_{t}}~{},\\\
&m_{b}=m_{\tau}=\frac{1}{\sqrt{2}}(\tilde{y}^{\prime}\,v_{1}+\tilde{y}\,v_{2})=\frac{v}{\sqrt{2}}\underbrace{(\tilde{y}^{\prime}\,s_{\alpha}+\tilde{y}\,c_{\alpha})}_{\displaystyle
y_{b}}~{}.\end{aligned}$ (72)

where we defined $v^{2}=v_{1}^{2}+v_{2}^{2}$ and
$\tan\alpha=\dfrac{v_{1}}{v_{2}}$. The choice of $\tan\alpha$ is arbitrary as
one is free to choose the values of the Yukawa couplings $\tilde{y}$ and
$\tilde{y}^{\prime}$ in order to reproduce $m_{t}$ and $m_{b}$. In the case
where $v_{1}=v,~{}v_{2}=0$, we have

$\tilde{y}=y_{t}\quad\text{and}\quad\tilde{y}^{\prime}=y_{b}~{}.$ (73)

###  C.2 Scalar sector

##### Higgs potential (2HDM).

The most general potential for a bi-doublet Higgs is Chauhan:2019fji

$\displaystyle\begin{aligned} V_{\rm\Phi}^{\rm
UV}=&~{}-\tilde{\mu}_{\Phi}^{2}{\rm
Tr}[\Phi^{\dagger}\Phi]-\left(\tilde{\mu}_{\Phi 12}^{2}{\rm
Tr}[\Phi^{\dagger}\Phi^{c}]+\rm h.c.\right)\\\ &+\tilde{\lambda}{\rm
Tr}[\Phi^{\dagger}\Phi]^{2}+\left(\tilde{\lambda}_{2}{\rm
Tr}[\Phi^{\dagger}\Phi^{c}]^{2}+\rm h.c.\right)\\\ &+\tilde{\lambda}_{3}{\rm
Tr}[\Phi^{\dagger}\Phi^{c}]{\rm
Tr}[{\Phi^{c}}^{\dagger}\Phi]+\left(\tilde{\lambda}_{4}{\rm
Tr}[\Phi^{\dagger}\Phi]{\rm Tr}[\Phi^{\dagger}\Phi^{c}]+\rm
h.c.\right)~{}.\end{aligned}$ (74)

Using the decomposition of the invariants

$\displaystyle\begin{aligned} {\rm
Tr}[\Phi^{\dagger}\Phi]&=H_{2}^{\dagger}H_{2}+H^{\dagger}H~{},\\\ {\rm
Tr}[\Phi^{\dagger}\Phi^{c}]&=\tilde{H}_{2}^{\dagger}\tilde{H}+H^{\dagger}H_{2}=2H^{\dagger}H_{2}~{}.\end{aligned}$
(75)

the potential reduces to the well-known 2HDM Branco:2011iw

$\displaystyle\begin{aligned} V_{\rm
H}=&-\tilde{\mu}_{\Phi}^{2}\left(H^{\dagger}H+H_{2}^{\dagger}H_{2}\right)-2\left(\tilde{\mu}_{\Phi
12}^{2}H^{\dagger}H_{2}+\rm h.c.\right)\\\
&+\tilde{\lambda}\left((H^{\dagger}H)^{2}+(H_{2}^{\dagger}H_{2})^{2}+2(H^{\dagger}H)(H_{2}^{\dagger}H_{2})\right)\\\
&+4\left(\tilde{\lambda}_{2}(H^{\dagger}H_{2})^{2}+\rm
h.c.\right)+4\tilde{\lambda}_{3}(H^{\dagger}H_{2})(H_{2}^{\dagger}H)\\\
&+2\left(\tilde{\lambda}_{4}(H_{2}^{\dagger}H_{2})(H_{2}^{\dagger}H)+\tilde{\lambda}_{4}(H^{\dagger}H)(H^{\dagger}H_{2})+\rm
h.c.\right)~{}.\end{aligned}$ (76)

Comparing with the simplified 2HDM potential Eq. (57), we find
$\tilde{\mu}_{\Phi 12}=\tilde{\lambda}_{4}=0$ and the relations between the
coefficients coming from the UV unification is

$\mu_{H}=\mu_{2}$ (77)

and

$\lambda=\lambda_{2}=\lambda_{3}~{}.$ (78)

#####  $\Omega$ potential.

The UV relations of the $\Omega$ potential coefficients would require
considering all degrees of freedom from the decomposition in Eq. (68). The
next step would be to decouple triplets of $SU(2)_{L}$, so that $\Omega$ only
takes a VEV along the $SU(2)_{L}$-preserving direction. Neglecting the
contribution from integrating out $\Omega_{1,3}^{\prime}$, we write a
multiplet where these degrees of freedom are omitted from the start, as well
as the $SU(2)_{\ell L}$ and $SU(2)_{3L}$ indices,

$\Omega=\begin{pmatrix}\Omega_{3}\\\
\Omega_{1}\end{pmatrix}\mathbb{1}_{2L}+\begin{pmatrix}\Omega^{\prime}_{3}\\\
\Omega^{\prime}_{1}\end{pmatrix}_{I}\tau^{I}_{2L}\approx\begin{pmatrix}\Omega_{3}\\\
\Omega_{1}\end{pmatrix}~{},$ (79)

where $\Omega_{3}$ and $\Omega_{1}$ are both anti-fundamental of $SU(4)_{3}$
and the vector is an $SU(4)_{\ell}$ multiplet. Keeping only the two $SU(4)$
indices, the different invariants composing the potential read

$\displaystyle\begin{aligned} V_{\Omega}^{\rm
UV}=~{}&-\tilde{\mu}_{\Omega}{\rm
Tr}[\Omega^{\dagger}\Omega]+\frac{\tilde{\rho}_{1}}{2}{\rm
Tr}[\Omega^{\dagger}\Omega]^{2}+\frac{\tilde{\rho}_{2}}{2}{\rm
Tr}[\Omega^{\dagger}\Omega\Omega^{\dagger}\Omega]~{},\\\
&+\frac{\tilde{\rho}_{3}}{4!}\,\epsilon_{\alpha\beta\gamma\sigma}\epsilon^{abcd}(\Omega)^{\alpha}_{a}(\Omega)^{\beta}_{b}(\Omega)^{\gamma}_{c}(\Omega)^{\sigma}_{d}~{},\end{aligned}$
(80)

which yields

$\displaystyle\begin{aligned} V_{\Omega}=&-\tilde{\mu}_{\Omega}\left({\rm
Tr}[\Omega_{3}^{\dagger}\Omega_{3}]+\Omega_{1}^{\dagger}\Omega_{1}\right)\\\
&+\frac{\tilde{\rho}_{1}}{2}\left({\rm
Tr}[\Omega_{3}^{\dagger}\Omega_{3}]^{2}+2{\rm
Tr}[\Omega_{3}^{\dagger}\Omega_{3}]\Omega_{1}^{\dagger}\Omega_{1}+(\Omega_{1}^{\dagger}\Omega_{1})^{2}\right)\\\
&+\frac{\tilde{\rho}_{2}}{2}\left({\rm
Tr}[\Omega_{3}^{\dagger}\Omega_{3}\Omega_{3}^{\dagger}\Omega_{3}]+2\Omega_{1}^{\dagger}\Omega_{3}\Omega_{3}^{\dagger}\Omega_{1}+(\Omega_{1}^{\dagger}\Omega_{1})^{2}\right)\\\
&+\frac{\tilde{\rho}_{3}}{3!}\,\epsilon_{\alpha\beta\gamma\sigma}\epsilon^{abc}(\Omega_{3})^{\alpha}_{a}(\Omega_{3})^{\beta}_{b}(\Omega_{3})^{\gamma}_{c}(\Omega_{1})^{\sigma}~{}.\end{aligned}$
(81)

The coefficients in the potential Eq. (5) are connected because of their UV
origin through the relations

$\rho_{1}=\rho_{2}+\rho_{2}^{\prime}~{},\qquad\rho_{2}=\rho_{3}~{},\qquad\rho_{2}^{\prime}=\rho_{4}$
(82)

and

$m_{\Omega_{1}}=m_{\Omega_{3}}~{}.$ (83)

##### Mixed $\Omega$-Higgses potential.

As for the mixed potential Eqs. (7) and (58), noticing that all mixed term
come from a single UV term (where again the $SU(2)_{L}$-triplet contributions
were neglected):

$V_{\Omega\Phi}^{\rm UV}=\tilde{\eta}~{}{\rm Tr}[\Omega^{\dagger}\Omega]{\rm
Tr}[\Phi^{\dagger}\Phi]~{},$ (84)

we find the UV relation between the coefficients in $V_{\Omega H}$

$\eta_{1}=\eta_{2}=\eta_{3}=\eta_{4}~{}.$ (85)

## References

  * (1) R. Aaij et al., “Test of lepton universality using $B^{+}\rightarrow K^{+}\ell^{+}\ell^{-}$ decays,” Phys. Rev. Lett., vol. 113, p. 151601, 2014.
  * (2) R. Aaij et al., “Test of lepton universality with $B^{0}\rightarrow K^{*0}\ell^{+}\ell^{-}$ decays,” JHEP, vol. 08, p. 055, 2017.
  * (3) R. Aaij et al., “Search for lepton-universality violation in $B^{+}\to K^{+}\ell^{+}\ell^{-}$ decays,” Phys. Rev. Lett., vol. 122, no. 19, p. 191801, 2019.
  * (4) R. Aaij et al., “Test of lepton universality in beauty-quark decays,” Nature Phys., vol. 18, no. 3, pp. 277–282, 2022.
  * (5) J. P. Lees et al., “Evidence for an excess of $\bar{B}\to D^{(*)}\tau^{-}\bar{\nu}_{\tau}$ decays,” Phys. Rev. Lett., vol. 109, p. 101802, 2012\.
  * (6) J. P. Lees et al., “Measurement of an Excess of $\bar{B}\to D^{(*)}\tau^{-}\bar{\nu}_{\tau}$ Decays and Implications for Charged Higgs Bosons,” Phys. Rev. D, vol. 88, no. 7, p. 072012, 2013.
  * (7) M. Huschle et al., “Measurement of the branching ratio of $\bar{B}\to D^{(\ast)}\tau^{-}\bar{\nu}_{\tau}$ relative to $\bar{B}\to D^{(\ast)}\ell^{-}\bar{\nu}_{\ell}$ decays with hadronic tagging at Belle,” Phys. Rev. D, vol. 92, no. 7, p. 072014, 2015.
  * (8) R. Aaij et al., “Measurement of the ratio of branching fractions $\mathcal{B}(\bar{B}^{0}\to D^{*+}\tau^{-}\bar{\nu}_{\tau})/\mathcal{B}(\bar{B}^{0}\to D^{*+}\mu^{-}\bar{\nu}_{\mu})$,” Phys. Rev. Lett., vol. 115, no. 11, p. 111803, 2015.  [Erratum: Phys.Rev.Lett. 115, 159901 (2015)].
  * (9) R. Aaij et al., “Measurement of the ratio of the $B^{0}\to D^{*-}\tau^{+}\nu_{\tau}$ and $B^{0}\to D^{*-}\mu^{+}\nu_{\mu}$ branching fractions using three-prong $\tau$-lepton decays,” Phys. Rev. Lett., vol. 120, no. 17, p. 171802, 2018.
  * (10) R. Aaij et al., “Test of Lepton Flavor Universality by the measurement of the $B^{0}\to D^{*-}\tau^{+}\nu_{\tau}$ branching fraction using three-prong $\tau$ decays,” Phys. Rev. D, vol. 97, no. 7, p. 072013, 2018\.
  * (11) R. Alonso, B. Grinstein, and J. Martin Camalich, “Lepton universality violation and lepton flavor conservation in $B$-meson decays,” JHEP, vol. 10, p. 184, 2015.
  * (12) L. Calibbi, A. Crivellin, and T. Ota, “Effective Field Theory Approach to $b\to s\ell\ell^{(^{\prime})}$, $B\to K^{(*)}\nu\overline{\nu}$ and $B\to D^{(*)}\tau\nu$ with Third Generation Couplings,” Phys. Rev. Lett., vol. 115, p. 181801, 2015.
  * (13) R. Barbieri, G. Isidori, A. Pattori, and F. Senia, “Anomalies in $B$-decays and $U(2)$ flavour symmetry,” Eur. Phys. J. C, vol. 76, no. 2, p. 67, 2016\.
  * (14) D. Buttazzo, A. Greljo, G. Isidori, and D. Marzocca, “B-physics anomalies: a guide to combined explanations,” JHEP, vol. 11, p. 044, 2017.
  * (15) A. Angelescu, D. Bečirević, D. A. Faroughy, F. Jaffredo, and O. Sumensari, “Single leptoquark solutions to the B-physics anomalies,” Phys. Rev. D, vol. 104, no. 5, p. 055017, 2021.
  * (16) J. C. Pati and A. Salam, “Lepton Number as the Fourth Color,” Phys. Rev. D, vol. 10, pp. 275–289, 1974.  [Erratum: Phys.Rev.D 11, 703–703 (1975)].
  * (17) B. Diaz, M. Schmaltz, and Y.-M. Zhong, “The leptoquark Hunter’s guide: Pair production,” JHEP, vol. 10, p. 097, 2017.
  * (18) L. Di Luzio, A. Greljo, and M. Nardecchia, “Gauge leptoquark as the origin of B-physics anomalies,” Phys. Rev. D, vol. 96, no. 11, p. 115011, 2017.
  * (19) R. Barbieri and A. Tesi, “$B$-decay anomalies in Pati-Salam SU(4),” Eur. Phys. J. C, vol. 78, no. 3, p. 193, 2018.
  * (20) L. Calibbi, A. Crivellin, and T. Li, “Model of vector leptoquarks in view of the $B$-physics anomalies,” Phys. Rev. D, vol. 98, no. 11, p. 115002, 2018\.
  * (21) B. Fornal, S. A. Gadam, and B. Grinstein, “Left-Right SU(4) Vector Leptoquark Model for Flavor Anomalies,” Phys. Rev. D, vol. 99, no. 5, p. 055025, 2019\.
  * (22) M. Blanke and A. Crivellin, “$B$ Meson Anomalies in a Pati-Salam Model within the Randall-Sundrum Background,” Phys. Rev. Lett., vol. 121, no. 1, p. 011801, 2018.
  * (23) L. Di Luzio, J. Fuentes-Martin, A. Greljo, M. Nardecchia, and S. Renner, “Maximal Flavour Violation: a Cabibbo mechanism for leptoquarks,” JHEP, vol. 11, p. 081, 2018.
  * (24) A. Greljo and B. A. Stefanek, “Third family quark–lepton unification at the TeV scale,” Phys. Lett. B, vol. 782, pp. 131–138, 2018\.
  * (25) M. Bordone, C. Cornella, J. Fuentes-Martín, and G. Isidori, “Low-energy signatures of the $\mathrm{PS}^{3}$ model: from $B$-physics anomalies to LFV,” JHEP, vol. 10, p. 148, 2018.
  * (26) C. Cornella, J. Fuentes-Martin, and G. Isidori, “Revisiting the vector leptoquark explanation of the B-physics anomalies,” JHEP, vol. 07, p. 168, 2019.
  * (27) J. Fuentes-Martín and P. Stangl, “Third-family quark-lepton unification with a fundamental composite Higgs,” Phys. Lett. B, vol. 811, p. 135953, 2020.
  * (28) D. Guadagnoli, M. Reboud, and P. Stangl, “The Dark Side of 4321,” JHEP, vol. 10, p. 084, 2020.
  * (29) J. Fuentes-Martín, G. Isidori, M. König, and N. Selimović, “Vector Leptoquarks Beyond Tree Level III: Vector-like Fermions and Flavor-Changing Transitions,” Phys. Rev. D, vol. 102, p. 115015, 2020.
  * (30) M. J. Baker, D. A. Faroughy, and S. Trifinopoulos, “Collider signatures of coannihilating dark matter in light of the B-physics anomalies,” JHEP, vol. 11, p. 084, 2021.
  * (31) M. Bordone, C. Cornella, J. Fuentes-Martin, and G. Isidori, “A three-site gauge model for flavor hierarchies and flavor anomalies,” Phys. Lett. B, vol. 779, pp. 317–323, 2018.
  * (32) J. Fuentes-Martin, G. Isidori, J. Pagès, and B. A. Stefanek, “Flavor non-universal Pati-Salam unification and neutrino masses,” Phys. Lett. B, vol. 820, p. 136484, 2021.
  * (33) J. Fuentes-Martin, G. Isidori, J. M. Lizana, N. Selimovic, and B. A. Stefanek, “Flavor hierarchies, flavor anomalies, and Higgs mass from a warped extra dimension,” 3 2022.
  * (34) S. F. King, “Twin Pati-Salam theory of flavour with a TeV scale vector leptoquark,” JHEP, vol. 11, p. 161, 2021.
  * (35) L. E. Ibanez and G. G. Ross, “SU(2)-L x U(1) Symmetry Breaking as a Radiative Effect of Supersymmetry Breaking in Guts,” Phys. Lett. B, vol. 110, pp. 215–220, 1982.
  * (36) K. Inoue, A. Kakuto, H. Komatsu, and S. Takeshita, “Aspects of Grand Unified Models with Softly Broken Supersymmetry,” Prog. Theor. Phys., vol. 68, p. 927, 1982.  [Erratum: Prog.Theor.Phys. 70, 330 (1983)].
  * (37) L. E. Ibanez, “Locally Supersymmetric SU(5) Grand Unification,” Phys. Lett. B, vol. 118, pp. 73–78, 1982.
  * (38) J. R. Ellis, D. V. Nanopoulos, and K. Tamvakis, “Grand Unification in Simple Supergravity,” Phys. Lett. B, vol. 121, pp. 123–129, 1983.
  * (39) J. R. Ellis, J. S. Hagelin, D. V. Nanopoulos, and K. Tamvakis, “Weak Symmetry Breaking by Radiative Corrections in Broken Supergravity,” Phys. Lett. B, vol. 125, p. 275, 1983.
  * (40) L. Alvarez-Gaume, J. Polchinski, and M. B. Wise, “Minimal Low-Energy Supergravity,” Nucl. Phys. B, vol. 221, p. 495, 1983.
  * (41) K. S. Babu, I. Gogoladze, and S. Khan, “Radiative Electroweak Symmetry Breaking in Standard Model Extensions,” Phys. Rev. D, vol. 95, no. 9, p. 095013, 2017.
  * (42) S. R. Coleman and E. J. Weinberg, “Radiative Corrections as the Origin of Spontaneous Symmetry Breaking,” Phys. Rev. D, vol. 7, pp. 1888–1910, 1973\.
  * (43) J. Fuentes-Martín, G. Isidori, M. König, and N. Selimović, “Vector leptoquarks beyond tree level. II. $\mathcal{O}(\alpha_{s})$ corrections and radial modes,” Phys. Rev. D, vol. 102, no. 3, p. 035021, 2020.
  * (44) J. Elias-Miro, J. R. Espinosa, G. F. Giudice, H. M. Lee, and A. Strumia, “Stabilization of the Electroweak Vacuum by a Scalar Threshold Effect,” JHEP, vol. 06, p. 031, 2012.
  * (45) K. Kannike, “Vacuum Stability Conditions From Copositivity Criteria,” Eur. Phys. J. C, vol. 72, p. 2093, 2012.
  * (46) K. Kannike, “Vacuum Stability of a General Scalar Potential of a Few Fields,” Eur. Phys. J. C, vol. 76, no. 6, p. 324, 2016.  [Erratum: Eur.Phys.J.C 78, 355 (2018)].
  * (47) G. Chauhan, “Vacuum Stability and Symmetry Breaking in Left-Right Symmetric Model,” JHEP, vol. 12, p. 137, 2019.
  * (48) P. A. Zyla et al., “Review of Particle Physics,” PTEP, vol. 2020, no. 8, p. 083C01, 2020.
  * (49) C. Cornella, D. A. Faroughy, J. Fuentes-Martin, G. Isidori, and M. Neubert, “Reading the footprints of the B-meson flavor anomalies,” JHEP, vol. 08, p. 050, 2021.
  * (50) M. J. Baker, J. Fuentes-Martín, G. Isidori, and M. König, “High- $p_{T}$ signatures in vector–leptoquark models,” Eur. Phys. J. C, vol. 79, no. 4, p. 334, 2019.
  * (51) Y. Bai and B. A. Dobrescu, “Collider Tests of the Renormalizable Coloron Model,” JHEP, vol. 04, p. 114, 2018.
  * (52) A. Ilnicka, T. Robens, and T. Stefaniak, “Constraining Extended Scalar Sectors at the LHC and beyond,” Mod. Phys. Lett. A, vol. 33, no. 10n11, p. 1830007, 2018.
  * (53) J. Fuentes-Martín, G. Isidori, M. König, and N. Selimović, “Vector Leptoquarks Beyond Tree Level,” Phys. Rev. D, vol. 101, no. 3, p. 035024, 2020.
  * (54) R. Barbieri and G. F. Giudice, “Upper Bounds on Supersymmetric Particle Masses,” Nucl. Phys. B, vol. 306, pp. 63–76, 1988.
  * (55) L. Allwicher, G. Isidori, and A. E. Thomsen, “Stability of the Higgs Sector in a Flavor-Inspired Multi-Scale Model,” JHEP, vol. 01, p. 191, 2021.
  * (56) J. Haller, A. Hoecker, R. Kogler, K. Mönig, T. Peiffer, and J. Stelzer, “Update of the global electroweak fit and constraints on two-Higgs-doublet models,” Eur. Phys. J. C, vol. 78, no. 8, p. 675, 2018.
  * (57) A. Azatov, D. Barducci, D. Ghosh, D. Marzocca, and L. Ubaldi, “Combined explanations of B-physics anomalies: the sterile neutrino solution,” JHEP, vol. 10, p. 092, 2018.
  * (58) D. Marzocca and S. Trifinopoulos, “Minimal Explanation of Flavor Anomalies: B-Meson Decays, Muon Magnetic Moment, and the Cabibbo Angle,” Phys. Rev. Lett., vol. 127, no. 6, p. 061803, 2021.
  * (59) R. Barbieri, C. W. Murphy, and F. Senia, “B-decay Anomalies in a Composite Leptoquark Model,” Eur. Phys. J. C, vol. 77, no. 1, p. 8, 2017.
  * (60) D. Marzocca, “Addressing the B-physics anomalies in a fundamental Composite Higgs Model,” JHEP, vol. 07, p. 121, 2018.
  * (61) A. E. Thomsen, “Introducing RGBeta: a Mathematica package for the evaluation of renormalization group $\beta$-functions,” Eur. Phys. J. C, vol. 81, no. 5, p. 408, 2021.
  * (62) G. C. Branco, P. M. Ferreira, L. Lavoura, M. N. Rebelo, M. Sher, and J. P. Silva, “Theory and phenomenology of two-Higgs-doublet models,” Phys. Rept., vol. 516, pp. 1–102, 2012.
  * (63) A. Belyaev, G. Cacciapaglia, I. P. Ivanov, F. Rojas-Abatte, and M. Thomas, “Anatomy of the Inert Two Higgs Doublet Model in the light of the LHC and non-LHC Dark Matter Searches,” Phys. Rev. D, vol. 97, no. 3, p. 035011, 2018.
  * (64) K. G. Klimenko, “On Necessary and Sufficient Conditions for Some Higgs Potentials to Be Bounded From Below,” Theor. Math. Phys., vol. 62, pp. 58–65, 1985.