aainstitutetext: Department of Physics, University of Basel,
Klingelbergstrasse 82, CH-4056 Basel, Switzerlandbbinstitutetext: Albert
Einstein Center for Fundamental Physics, Institute for Theoretical Physics,
University of Bern, CH-3012 Bern, Switzerland

# Flavor Hierarchies From SU(2) Flavor and Quark–Lepton Unification

Admir<EMAIL_ADDRESS>b Anders Eller
<EMAIL_ADDRESS>a Hector<EMAIL_ADDRESS>
###### Abstract

In our recent attempt to explain flavor hierarchies Greljo:2023bix , a gauged
SU(2) flavor symmetry acting on left-handed fermions provides a ground to
introduce three independent rank-one contributions to the Yukawa matrices: a
renormalizable one for the third family, a mass-suppressed one for the second
family, and an additional loop-suppressed factor for the first family. Here,
we demonstrate how minimal quark-lepton unification à la Pati-Salam, relating
down-quarks to charged leptons, can significantly improve this mechanism. We
construct and thoroughly analyze a renormalizable model, performing a
comprehensive one-loop matching calculation that reveals how all flavor
hierarchies emerge from a single ratio of two scales. The first signatures may
appear in the upcoming charged lepton flavor violation experiments.

##  1 Introduction

The peculiar pattern of fermion masses and their mixing under electroweak (EW)
interactions stands as a riddle at the heart of the Standard Model (SM), known
as _the Flavor Puzzle_. All the flavor of the SM stems from Yukawa couplings
of the three generations of fermions with the single Higgs field. Contrary to
naive expectations, there are large hierarchies between the size of the
coupling parameters, although no power counting is suggested by the model.
Observations indicate a consistent mass hierarchy of approximately two orders
of magnitude between consecutive generations of all electrically charged
fermions: up quarks, down quarks, and charged leptons. Also, the misalignment
between the up- and down-quark Yukawa matrices, as captured by the
Cabibbo–Kobayashi–Maskawa (CKM) matrix Cabibbo:1963yz ; Kobayashi:1973fv , is
characterized by hierarchies between diagonal and off-diagonal elements
spanning three orders of magnitude in total.

The assumption of an underlying Beyond the Standard Model (BSM) explanation
for these hierarchies may provide a valuable hint as to the nature of new
physics, and there has been no lack of effort in this direction; Refs.
Froggatt:1978nt ; Weinberg:1972ws ; Barr:1990td ; Babu:1990hu ; Kaplan:1991dc
; Leurer:1992wg ; Leurer:1993gy ; Kaplan:1993ej ; Barbieri:1994cx ;
Barbieri:1995uv ; Barbieri:1996ae ; Barbieri:1996ww ; Barbieri:1999pe ;
Randall:1999ee ; Arkani-Hamed:1999ylh ; King:2003rf ; Grinstein:2010ve ;
Feruglio:2015jfa ; Calibbi:2016hwq ; Ema:2016ops ; Panico:2016ull ;
Bordone:2017bld ; Greljo:2018tuh ; Linster:2018avp ; Allanach:2018lvl ;
Alonso:2018bcg ; Greljo:2019xan ; Smolkovic:2019jow ; Baur:2019kwi ;
Fedele:2020fvh ; Nilles:2020nnc ; King:2020qaj ; Baker:2020vkh ; Babu:2020tnf
; Feruglio:2021dte ; Altmannshofer:2021qwx ; Davighi:2022fer ;
Cornella:2023zme ; Asadi:2023ucx ; Davighi:2023evx ; Davighi:2023iks ;
Barbieri:2023qpf ; Fuentes-Martin:2024fpx ; Altmannshofer:2022aml  constitute
a representative selection of early and recent examples. The goal of the
flavor model building is to explain the many hierarchies observed in the
fermion masses and mixings from a few (or no) ultraviolet (UV) hierarchies. We
would judge the success of a mechanism based on how simple its UV realizations
are and how well it produces the SM flavor pattern.

In physics, structures resulting from symmetries are widespread. Indeed, a
compelling idea put forward very early on is that flavor hierarchies are a
consequence of (accidental or gauged) flavor symmetries Froggatt:1978nt .
Relatively simple breaking patterns may be obtained from a horizontal
$\mathrm{U}(2)$ symmetry discriminating between the third and light
generations Babu:1990hu ; Barbieri:1995uv . It was recently observed
Greljo:2023bix ; Antusch:2023shi  that imposing a single $\mathrm{U}(2)$
symmetry, which charges only one chirality of leptons and quarks respectively,
still results in an accidental $\mathrm{U}(2)^{5}$ Feldmann:2008ja ;
Kagan:2009bn ; Barbieri:2011ci ; Barbieri:2012uh ; Blankenburg:2012nx ;
Fuentes-Martin:2019mun ; Faroughy:2020ina ; Greljo:2022cah  of the SM Yukawa
couplings in agreement with observations, allowing for a remarkably simple UV
realization of the underlying mechanism. The key premise is to charge only the
EW doublet on the quark side, which is involved in both the up and down Yukawa
interactions. This minimal assignment generates the desired mass hierarchies
and, at the same time, perturbative left-handed mixings entering the CKM
matrix.

The model of Greljo:2023bix  is based on an $\mathrm{SU}(2)_{q+\ell}$ gauge
symmetry under which the two light generations of left-handed quarks and
leptons each form a doublet. This provides a basis to introduce three
independent rank-one contributions to the Yukawa matrices, as shown in Fig. 1.
To begin with, the Higgs field can directly couple (and give mass) only to the
third-generation fermions at the renormalizable level. Suppressed Higgs
couplings with light generations are generated through the introduction of a
symmetry-breaking scalar doublet, whose vacuum expectation value (VEV) breaks
$\mathrm{SU}(2)_{q+\ell}$. In particular, the inclusion of heavy vector-like
quark and lepton fields generates at the tree level an effective dimension-5
operator between the flavor-breaking scalar, the Higgs, and light fermions;
see Fig. 1 (center). When a single copy of vector-like fermions (VLFs) is
present, the corresponding rank of the Yukawa matrices is only raised by one
unit, generating small non-zero Yukawa couplings for the second generation
only.

Figure 1: The mechanism for flavor hierarchies proposed in Greljo:2023bix .
Three independent rank-one contributions to the Yukawa matrices: a
renormalizable one for the third family (left), a mass-suppressed one for the
second family (center), and an additionally loop-suppressed one for the first
family (right). The fields shown in the representative (and leading) Feynman
diagrams are listed in Table 1.

In the final step, the effective Yukawa couplings for the first generation are
loop-generated, reusing the same vector-like fermions; see Fig. 1 (right). A
clever construction that does not generate any additional unsuppressed
dimension-5 operators (which might cause degeneracy between the light
generation masses) is to complete the loops with new scalar leptoquark fields,
such that a vector-like lepton (quark) enters that diagrams for the effective
quark (lepton) Yukawa couplings. Technically, the dominant effect comes from
the renormalization group mixing from a tree-level operator obtained after
integrating out vector-like fermions solely. Remarkably, as long as the
leptoquarks are lighter than the vector-like fermions, the hierarchy between
the first two families is approximately a loop factor with a slight
(logarithmic) dependence on the mass ratio.

The model proposed in Ref. Greljo:2023bix  manages to explain the six mass
hierarchies between consecutive generations of fermions in addition to the
three hierarchies between the CKM matrix elements. All of this is in terms of
two small input parameters $\epsilon_{q}$ and $\epsilon_{\ell}$, the ratios of
the flavor symmetry-breaking scale to the two vector-like fermion masses. The
similarity of the down quark and charged lepton mass hierarchies imply
$\epsilon_{q}\approx\epsilon_{\ell}$, which, in this model, requires the two
independent vector-like fermion masses to coincide. The model involves a
single new gauge symmetry, SM$\times\mathrm{SU}(2)_{q+\ell}$, and six new
matter fields. At first, this might seem unnecessarily convoluted compared to
some mechanisms to explain the flavor puzzle, but when judging realizations in
UV complete models, it compares very favorably. In fact, it admits further
simplification and unification, as we have detailed in this paper.

The field content of the $\mathrm{SU}(2)_{q+\ell}$ model Greljo:2023bix  is
very suggestive of a minimal quark–lepton unification in the manner of
Pati–Salam Pati:1974yy  embedding color, lepton number and baryon number into
an $\mathrm{SU}(4)$ gauge group, but without introducing
$\mathrm{SU}(2)_{\mathrm{R}}$ Smirnov:1995jq ; FileviezPerez:2013zmv . The
embedding of the gauge groups collects both vector-like fermions into a single
vector-like fourplet, thereby explaining the coincidence of scales between the
two, predicting $\epsilon_{q}=\epsilon_{\ell}$. Additionally, the four scalar
fields introduced in the original model are embedded into just two scalar
multiplets. Quark–lepton unification thus provides a simple extension to the
model, reducing the BSM matter content to four new fields and further unifying
the SM chiral fermions.

The goal of the present paper is to embed the model of Ref. Greljo:2023bix
into the $\mathrm{SU}(4)$ gauge group, unifying quarks and leptons. We will
demonstrate how the SM flavor structure emerges from a single hierarchy in the
UV. A full-fledged renormalizable model is defined in Section 2, while the
matching calculation of the effective SM Yukawa couplings is presented in
Section 3. In Section 4, we show how the SM masses are recovered from a
probability distribution by varying order-one parameters. Finally, in Section
5, we discuss $K_{L}\to\mu e$ and muon conversion on heavy nuclei as the main
phenomenological constraints on the model, pointing to a symmetry-breaking
scale above a few PeV. Future sensitivity in upgraded muon conversion
experiments provides an opportunity for discovery if the flavor-breaking scale
is low enough. Conclusions are left for Section 6 while the details of the
scalar potential are relegated to the Appendix.

##  2 The model: Quark–lepton unification with gauged $\mathrm{SU}(2)$ flavor

As a starting point for our model, we embed $\mathrm{SU}(3)_{c}$ and
$\mathrm{U}(1)_{\rm B-L}$ into an $\mathrm{SU}(4)$ gauge group in the manner
of Pati and Salam Pati:1974yy , identifying leptons with a “fourth color”. The
SM hypercharge emerges from the product of $\mathrm{SU}(4)$ with a new Abelian
group $\mathrm{U}(1)_{\mathrm{R}}$ acting on the right-handed SM fermions. At
low scales, we expect to find the SM symmetry embedded as
$\mathrm{SU}(3)_{c}\times\mathrm{U}(1)_{Y}\subseteq\mathrm{SU}(4)\times\mathrm{U}(1)_{\mathrm{R}}$.
This construction corresponds to the minimal quark–lepton unification
Smirnov:1995jq ; FileviezPerez:2013zmv ; FileviezPerez:2023rxn  and can
manifest at energies as low as the PeV-scale. The
$\mathrm{SU}(4)\times\mathrm{U}(1)_{\mathrm{R}}$ gauge group is further
augmented with the $\mathrm{SU}(2)_{\mathrm{L}}$ of the SM in addition to the
$\mathrm{SU}(2)_{q+\ell}$ that acts on the two light generations of the left-
handed SM fermions Babu:1990hu ; Greljo:2023bix . The latter group is the
origin of the flavor structure of the model and is responsible for producing
the SM mass hierarchies. Thus, the UV gauge group is given by

$G_{{\scriptscriptstyle\mathrm{UV}}}=\mathrm{SU}(4)\times\mathrm{SU}(2)_{\mathrm{L}}\times\mathrm{U}(1)_{\mathrm{R}}\times\mathrm{SU}(2)_{q+\ell}.$
(1)

The full matter content of our theory is shown in Table 1. The SM fermions fit
into the four irreducible representations under
$G_{{\scriptscriptstyle\mathrm{UV}}}$:
$\psi_{\mathrm{L}}=(q_{\mathrm{L}},\,\ell_{\mathrm{L}})$ for the two light
generation left-handed fermions which form an $\mathrm{SU}(2)_{q+\ell}$
doublet; $\psi^{3}_{\mathrm{L}}=(q^{3}_{\mathrm{L}},\,\ell^{3}_{\mathrm{L}})$
for the third generation left-handed fermions; three copies of
$\psi^{p}_{d,\mathrm{R}}=(d^{p}_{\mathrm{R}},\,e^{p}_{\mathrm{R}})$ make up
all generations for the down-type quarks and charged leptons; and three copies
of $\psi^{p}_{u,\mathrm{R}}=(u^{p}_{\mathrm{R}},\,\nu^{p}_{\mathrm{R}})$,
which contain the three right-handed up-type quarks. To complete
$\psi^{p}_{u,\mathrm{R}}$, one must also include three right-handed neutrinos
in the SM field content. We postpone the discussion of neutrino masses and
mixings to Section 4.3. In order to generate the correct flavor structure, we
additionally include a single vector-like fermion
$\Psi_{\mathrm{L},\mathrm{R}}$, which transforms as the third-generation left-
handed fermions.

[] Field | $\mathrm{SU}(4)$ | $\mathrm{SU}(2)_{\mathrm{L}}$ | $\mathrm{U}(1)_{\mathrm{R}}$ | $\mathrm{SU}(2)_{q+\ell}$
---|---|---|---|---
$\psi_{\mathrm{L}}$ | $\mathbf{4}$ | $\mathbf{2}$ | $\phantom{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}}0\phantom{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}}$ | $\mathbf{2}$
$\psi_{\mathrm{L}}^{3}$ | $\mathbf{4}$ | $\mathbf{2}$ | $\phantom{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}}0\phantom{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}}$ | $\mathbf{1}$
$\psi_{u,\mathrm{R}}^{p}$ | $\mathbf{4}$ | $\mathbf{1}$ | $\phantom{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}}1/2\phantom{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}}$ | $\mathbf{1}$
$\psi_{d,\mathrm{R}}^{p}$ | $\mathbf{4}$ | $\mathbf{1}$ | $\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}1/2\phantom{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}}$ | $\mathbf{1}$
$\Psi_{\mathrm{L,R}}$ | $\mathbf{4}$ | $\mathbf{2}$ | $\phantom{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}}0\phantom{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}}$ | $\mathbf{1}$
$\chi$ | $\mathbf{4}$ | $\mathbf{1}$ | $\phantom{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}}1/2\phantom{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}}$ | $\mathbf{1}$
$H_{1}$ | $\mathbf{1}$ | $\mathbf{2}$ | $\phantom{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}}1/2\phantom{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}}$ | $\mathbf{1}$
$\Sigma_{\mathrm{H}}$ | $\mathbf{15}$ | $\mathbf{2}$ | $\phantom{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}}1/2\phantom{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}}$ | $\mathbf{1}$
$\Sigma_{\mathrm{\Phi}}$ | $\mathbf{15}$ | $\mathbf{1}$ | $\phantom{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}}0\phantom{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}}$ | $\mathbf{2}$
Table 1: The matter field content of the model and their representations under
the gauge group. The first four rows are chiral fermions, $\Psi$ is a vector-
like fermion, while the remaining fields are scalars. There are three copies
of the right-handed chiral fermions ($p=1,2,3$), but other fields do not
replicate.

The flavor hierarchies of Greljo:2023bix  were produced through the use of
five scalar fields. These fields can be embedded into just two irreducible
representations of $G_{{\scriptscriptstyle\mathrm{UV}}}$: $\Sigma_{H}$ and
$\Sigma_{\Phi}$. Two additional scalars are required, so our model comprises
four scalar fields: $\chi$, which is used to break
$\mathrm{SU}(4)\times\mathrm{U}(1)_{\mathrm{R}}$ to
$\mathrm{SU}(3)_{c}\times\mathrm{U}(1)_{Y}$; $H_{1}$, which is used to break
the EW symmetry and provide fermion masses; $\Sigma_{H}$, which contains both
the leptoquarks that run in the loops as well as an additional Higgs that can
split the masses between the quark and lepton sectors; and, lastly,
$\Sigma_{\Phi}$, which contains additional necessary leptoquarks and is used
to break the flavor symmetry $\mathrm{SU}(2)_{q+\ell}$. The scalar field
$\Sigma_{\Phi}$ contains an SM singlet component $\Phi$, the VEV of which will
break the flavor symmetry. At first glance, it seems like the inclusion of
$\Sigma_{\Phi}$ is not a minimal configuration since it is sufficient for
$\chi$ and a $\mathbf{2}_{q+\ell}$ doublet $\Phi$ to develop VEVs to break
$G_{{\scriptscriptstyle\mathrm{UV}}}$ into the SM gauge group. However, to
provide masses to the first generation as shown in Greljo:2023bix , we need
the leptoquark $S$, which is a triplet under the color symmetry and a doublet
under the flavor symmetry. $\Sigma_{\Phi}$ contains both $S$ and $\Phi$ and
is, therefore, the smallest representation that simultaneously ensures the
breaking of the flavor symmetry and the masses for the first family. For the
decomposition of the fields under
$G_{\scriptscriptstyle\mathrm{SM}}\times\mathrm{SU}(2)_{q+\ell}$, see Table 2.

As per usual in non-supersymmetric BSM models with new scalar states, the
scalar potential in our model is rather complicated and contains in excess of
60 free parameters; we refer the reader to Appendix A for the full potential.
Of these there are only four parameters with the mass dimension, which control
the scales on the model (see Fig. 2). The rest are marginal couplings and are
assumed to be of $\mathcal{O}(1)$, apart from complying with the vacuum
stability and perturbative unitarity bounds. With the full potential in hand,
we solved the tadpole equations and verified that our VEV configuration does
indeed minimize the potential. Scanning through the parameter space, we easily
find suitable points that produce condensates for $\chi$, $\Phi$, and one of
the Higgses while still ensuring all other scalar fields obtain positive
masses.

[] UV Field | Component | $\mathrm{SU}(3)$ | $\mathrm{SU}(2)_{\mathrm{L}}$ | $\mathrm{U}(1)_{Y}$ | $\mathrm{SU}(2)_{q+\ell}$
---|---|---|---|---|---
$\Sigma_{\mathrm{H}}$ | $\Theta_{H}$ | $\mathbf{8}$ | $\mathbf{2}$ | $\phantom{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}}0\phantom{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}}$ | $\mathbf{1}$
$R_{u}$ | $\mathbf{3}$ | $\mathbf{2}$ | $\phantom{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}}7/6\phantom{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}}$ | $\mathbf{1}$
$R_{d}$ | $\mathbf{3}$ | $\mathbf{2}$ | $\phantom{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}}1/6\phantom{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}}$ | $\mathbf{1}$
$H_{2}$ | $\mathbf{1}$ | $\mathbf{2}$ | $\phantom{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}}1/2\phantom{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}}$ | $\mathbf{1}$
$\Sigma_{\mathrm{\Phi}}$ | $\Theta_{\Phi}$ | $\mathbf{8}$ | $\mathbf{1}$ | $\phantom{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}}0\phantom{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}}$ | $\mathbf{2}$
$S_{1}$ | $\mathbf{3}$ | $\mathbf{1}$ | $\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}1/3\phantom{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}}$ | $\mathbf{2}$
$S_{2}$ | $\mathbf{3}$ | $\mathbf{1}$ | $\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}1/3\phantom{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}}$ | $\mathbf{2}$
$\Phi$ | $\mathbf{1}$ | $\mathbf{1}$ | $\phantom{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}}0\phantom{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}}$ | $\mathbf{2}$
Table 2: Decomposition of the scalar fields under
$G_{\scriptscriptstyle\mathrm{SM}}\times\mathrm{SU}(2)_{q+\ell}$.

It is rather simple to organize the symmetry-breaking pattern: A non-zero VEV
of $\chi$ is responsible for the breaking
$G_{{\scriptscriptstyle\mathrm{UV}}}$ to
$G_{\scriptscriptstyle\mathrm{SM}}\times\mathrm{SU}(2)_{q+\ell}$. With an
$\mathrm{SU}(4)$ rotation, the VEV can always be arranged such that

$\expectationvalue{\chi}=\matrixquantity(0\\\ 0\\\ 0\\\ v_{\chi}).$ (2)

In the broken phase, the adjoint scalar fields then decompose as in Table 2.
In matrix form

$\Sigma_{H}=\matrixquantity(\Theta_{H}&R_{u}\\\
\widetilde{R}_{d}&0)+T_{15}H_{2},\quad\Sigma_{\Phi}=\matrixquantity(\Theta_{\Phi}&S_{1}\\\
\widetilde{S}_{2}&0)+T_{15}\Phi,$ (3)

where for any $\mathrm{SU}(2)$ doublet we use the notation
$\widetilde{F}^{i}=\varepsilon^{ij}F_{j}^{*}$, and $T_{15}$ is the diagonal
generator of $\mathrm{SU}(4)$ in the fundamental representation,

$T_{15}=\frac{1}{2\sqrt{6}}\matrixquantity(1&\phantom{N}&\phantom{N}&\phantom{N}\\\
\phantom{N}&1&\phantom{N}&\phantom{N}\\\
\phantom{N}&\phantom{N}&1&\phantom{N}\\\
\phantom{N}&\phantom{N}&\phantom{N}&-3).$ (4)

In Greljo:2023bix , the masses for the first family are generated through the
use of three leptoquarks: $R_{u},R_{d},S$. These leptoquarks are embedded into
the adjoint fields $\Sigma_{H},\Sigma_{\Phi}$, with the only difference being
that the latter contains two leptoquarks $S_{1},S_{2}$ with the quantum
numbers of $S$. To break the flavor symmetry in the same way, $\Phi$ develops
a VEV according to

$\expectationvalue{\Phi}=\matrixquantity(0\\\
v_{\Phi}),\quad\rm{where}\quad\langle\widetilde{\Phi}\rangle=\matrixquantity(v_{\Phi}\\\
0).$ (5)

The placement of the VEV picks out light flavors.

The flavor hierarchies are controlled by $\varepsilon=v_{\Phi}/M_{\Psi}\sim
10^{-2}$, where $M_{\Psi}$ is the VLF mass. There is freedom in choosing
$v_{\chi}$. However, simple naturalness considerations of the scalar potential
suggest $v_{\chi}\sim v_{\Phi}$ since the portal operators generically
transmit VEVs into dimensionful couplings among the remaining
scalars.444Scalar leptoquarks entering the loop diagrams for the first family
masses (Fig. 1 right) must have a mass no larger than $\mathcal{O}(M_{\Psi})$.
Otherwise, the hierarchy between the first two families becomes excessively
large. For $\mathcal{O}(1)$ scalar quartics, this condition implies
$v_{\chi}\lesssim M_{\Psi}$. Thus, without tuning, the $\mathrm{SU}(4)$
breaking can not be decoupled from the flavor breaking when assuming all
dimensionless couplings are $\mathcal{O}(1)$. Thus, for simplicity, we will
assume that $\chi$ and $\Phi$ develop VEVs at the same scale, as expected in
the absence of tuning. As we will see in Section 5, current experimental data
indicates that the symmetry-breaking should occur at the PeV scale or higher.

Indeed, it is an experimental necessity that the EW symmetry breaking occurs
at a much lower scale than the breaking of $G_{\scriptscriptstyle\mathrm{UV}}$
to $G_{\scriptscriptstyle\mathrm{SM}}$ (see Section 5). Admittedly, a tuning
of the scalar potential is required in order to get such a separation of
scales of more than three orders of magnitude. This is an irreducible problem
known to all BSM models with fundamental scalars in the absence of a
protection mechanism such as supersymmetry, and we will not resolve this issue
here. After the gauge symmetry breaks to $G_{\scriptscriptstyle\mathrm{SM}}$,
the two Higgs fields $(H_{1},H_{2})$ get an effective mass matrix

$M^{2}_{H}=\begin{pmatrix}m_{H_{1}}^{\prime
2}&-\frac{\left(\lambda_{51}+\lambda_{52}\right)v_{\Phi}^{2}+3\lambda_{5}v_{\chi}^{2}}{2\sqrt{6}}\\\
-\frac{\left(\lambda_{51}+\lambda_{52}\right)v_{\Phi}^{2}+3\lambda_{5}v_{\chi}^{2}}{2\sqrt{6}}&m_{\Sigma_{H}}^{\prime
2}\end{pmatrix},$ (6)

mixing the two components. The diagonal components receive contributions from
various quartic interactions with $\langle\Sigma_{\Phi}\rangle$ and
$\langle\chi\rangle$ insertions, which, for our purposes, are simply absorbed
into the mass terms. A realistic limit with an SM-like Higgs boson is obtained
when $m_{H_{1}}^{\prime
2}\ll\frac{\left(\lambda_{51}+\lambda_{52}\right)v_{\Phi}^{2}+3\lambda_{5}v_{\chi}^{2}}{2\sqrt{6}}\lesssim
m_{\Sigma_{H}}^{\prime 2}$. This ensures a small but non-zero mixing between
the two states, where we consider a single tuning of $m_{H_{1}}^{2}$ to ensure
that the state consisting mostly of $H_{1}$ gets a small, negative mass,
thereby developing an EW symmetry-breaking VEV. In this scenario, the ratio
between VEVs of $H_{1}$ and $H_{2}$ is given by

$\tan\beta\equiv\frac{v_{2}}{v_{1}}=-\frac{\left(\lambda_{51}+\lambda_{52}\right)v_{\Phi}^{2}+3\lambda_{5}v_{\chi}^{2}}{2\sqrt{6}\,m_{\Sigma_{H}}^{\prime
2}}\,,$ (7)

where
$v_{1}^{2}+v_{2}^{2}=v_{\scriptscriptstyle\mathrm{EW}}^{2}=($174\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$)^{2}$.
A modest order of magnitude difference in the scales $v_{\Phi,\chi}$ and
$m_{\Sigma_{H}}$, easily gives $\tan\beta\lesssim 0.01$ without any additional
tuning. A small $\tan\beta$ is particularly interesting for phenomenology, as
will be discussed in Section 4, suggesting
$v_{\Phi,\chi}<m^{\prime}_{\Sigma_{H}}<M_{\Psi}$. A rough sketch of a
realistic benchmark for the various scales is shown in Fig. 2.

Figure 2: A benchmark scenario for the mass thresholds of the model on a
logarithmic scale.

##  3 Matching to the SM Yukawas

The fermion masses are generated through a combination of renormalizable and
dimension-5 effective operators produced at both tree and one-loop levels,
starting from the UV Lagrangian and integrating out heavy fields. We will
examine these contributions individually and establish their role in
determining the rank of the mass matrices.

###  3.1 Rank 1

The symmetries of the model only allow the third generation to acquire masses
at the renormalizable level. The relevant Yukawa interactions are

$\mathcal{L}_{\scriptscriptstyle\mathrm{UV}}\supset-
x_{\mathrm{u}}^{p}\overline{\psi}_{\mathrm{L}}^{3}\widetilde{H}_{1}\psi_{u,\mathrm{R}}^{p}-X_{\mathrm{u}}^{p}\overline{\psi}_{\mathrm{L}}^{3}\widetilde{\Sigma}_{H}\psi_{u,\mathrm{R}}^{p}-x_{\mathrm{d}}^{p}\overline{\psi}_{\mathrm{L}}^{3}H_{1}\psi_{d,\mathrm{R}}^{p}-X_{\mathrm{d}}^{p}\overline{\psi}_{\mathrm{L}}^{3}\Sigma_{H}\psi_{d,\mathrm{R}}^{p}+\text{h.c.}.$
(8)

We denote the couplings to the singlet Higgs field $H_{1}$ with lowercase
letters and the couplings to the adjoint Higgs field $\Sigma_{H}$ with
uppercase letters. When the EW symmetry is broken by the Higgs fields
developing VEVs, the resulting SM Yukawa matrices $Y_{f}^{pr}$ ($p,r=1,2,3$)
will depend on a fixed linear combination of the vectors $x_{u(d)}^{p}$ and
$X_{u(d)}^{p}$. Importantly, these will contribute only to the third rows,
resulting in the Yukawa matrices having rank 1.

###  3.2 Rank 2

To raise the rank of the Yukawa matrices and provide the second generation
with masses, we make use of the renormalizable interactions between the VLF
$\Psi$ and the scalar fields. After integrating out $\Psi$, we will be left
with dimension-5 operators that generate masses for the second generation. The
relevant part of the Lagrangian for the matching calculation is

$\displaystyle\mathcal{L}_{\scriptscriptstyle\mathrm{UV}}\supset-$
$\displaystyle
y_{\mathrm{u}}^{p}\overline{\Psi}_{\mathrm{L}}\widetilde{H}_{1}\psi_{u,\mathrm{R}}^{p}$
$\displaystyle-$ $\displaystyle
Y_{\mathrm{u}}^{p}\overline{\Psi}_{\mathrm{L}}\widetilde{\Sigma}_{H}\psi_{u,\mathrm{R}}^{p}$
$\displaystyle-$ $\displaystyle
y_{\mathrm{d}}^{p}\overline{\Psi}_{\mathrm{L}}H_{1}\psi_{d,\mathrm{R}}^{p}$
$\displaystyle-$ $\displaystyle
Y_{\mathrm{d}}^{p}\overline{\Psi}_{\mathrm{L}}\Sigma_{H}\psi_{d,\mathrm{R}}^{p}$
$\displaystyle-$ $\displaystyle
Y_{\mathrm{\Phi}}\overline{\psi}_{\mathrm{L}}\Sigma_{\Phi}\Psi_{\mathrm{R}}$
$\displaystyle-$
$\displaystyle\widetilde{Y}_{\Phi}\overline{\psi}_{\mathrm{L}}\widetilde{\Sigma}_{\Phi}\Psi_{\mathrm{R}}+\text{h.c.}\,.$
(9)

By redefining fields, we have chosen a basis with no mass mixing between
$\Psi_{\mathrm{L}}$ and $\psi^{3}_{\mathrm{L}}$. In addition, note that
$\Sigma_{\Phi}$ and $\widetilde{\Sigma}_{\Phi}$ share quantum numbers. Thus,
the kinetic term for $\Sigma_{\Phi}$ exhibits an accidental $\mathrm{SU}(2)$
symmetry. We exploit this symmetry by choosing a basis where
$\widetilde{Y}_{\Phi}=0$, ensuring that the first generation of fermions
remains massless at the tree level. Integrating out $\Psi$ at the tree level,
we are left with the dimension-5 operators

$\displaystyle\mathcal{L}_{\scriptscriptstyle\mathrm{EFT}}\supset\frac{Y_{\Phi}}{M_{\Psi}}\Big{(}$
$\displaystyle
y_{\mathrm{u}}^{p}\overline{\psi}_{\mathrm{L}}\Sigma_{\Phi}\widetilde{H}_{1}\psi_{u,\mathrm{R}}^{p}$
$\displaystyle+$ $\displaystyle
Y_{\mathrm{u}}^{p}\overline{\psi}_{\mathrm{L}}\Sigma_{\Phi}\widetilde{\Sigma}_{H}\psi_{u,\mathrm{R}}^{p}$
$\displaystyle+$ $\displaystyle
y_{\mathrm{d}}^{p}\overline{\psi}_{\mathrm{L}}\Sigma_{\Phi}H_{1}\psi_{d,\mathrm{R}}^{p}$
$\displaystyle+$ $\displaystyle
Y_{\mathrm{d}}^{p}\overline{\psi}_{\mathrm{L}}\Sigma_{\Phi}\Sigma_{H}\psi_{d,\mathrm{R}}^{p}+\text{h.c.}\Big{)}.$
(10)

###  3.3 Rank 3

Already at the one-loop level, the rank of the mass matrices is increased to
3. As a result, the first family becomes massive, and the loop factor nicely
explains the hierarchy between the two light families. Even though the same
couplings from Eq. (9) are involved in the loop diagrams (see Fig. 1 right),
the leptoquark contributions, together with the independent scalar quartics,
provide enough freedom to raise the rank and correctly fit the first-family
parameters. We use Matchete Fuentes-Martin:2022jrf  to compute the effective
dimension-5 Yukawa operators at the one loop. Due to their length and
complexity, we only list one of the representative terms which increases the
rank:

$\displaystyle\mathcal{L}_{\scriptscriptstyle\mathrm{EFT}}\supset\frac{15}{32}\frac{1}{16\pi^{2}}\frac{1}{M_{\Psi}}\quantity[\log\quantity(\frac{M_{\Psi}^{2}}{\mu^{2}})-1]Y_{u}^{p}Y_{\Phi}\lambda_{45}^{*}\Tr[\widetilde{\Sigma}_{H}\widetilde{\Sigma}_{\Phi}]\overline{\psi}_{\mathrm{L}}\psi_{u,\mathrm{R}}^{p}.$
(11)

When the scalar fields develop VEVs, the leading expressions for the resulting
mass matrices are rather simple and will be presented in full below.

###  3.4 The mass matrices

The fermion masses are generated after the Higgs doublets develop VEVs
according to

$\expectationvalue{H_{i}}=\matrixquantity(0\\\ v_{i}).$ (12)

$H_{1}$ will contribute equally to the quark and lepton sectors since it is a
singlet under $\mathrm{SU}(4)$, while $H_{2}$ will split the down-quark and
charged-lepton masses since it is embedded in the adjoint field $\Sigma_{H}$.

Utilizing the remaining freedom to rotate the right-handed fields,
$\mathrm{U}(3)_{\psi_{u,\mathrm{R}}}\times\mathrm{U}(3)_{\psi_{d,\mathrm{R}}}$,
we choose a basis where the tree-level quark Yukawa matrices are upper-
triangular,555The conditions are $2\sqrt{6}v_{1}y_{f}^{1}+v_{2}Y_{f}^{1}=0$
and $v_{1}x_{f}^{r}+\frac{v_{2}X_{f}^{r}}{2\sqrt{6}}=0$ where $r=1,2$.

$M_{f}^{\scriptscriptstyle\mathrm{Tree}}=\matrixquantity(0&0&0\\\
0&-\frac{v_{\Phi}Y_{\Phi}\quantity(2\sqrt{6}v_{1}y_{f}^{2}+v_{2}Y_{f}^{2})}{24M_{\Psi}}&-\frac{v_{\Phi}Y_{\Phi}\quantity(2\sqrt{6}v_{1}y_{f}^{3}+v_{2}Y_{f}^{3})}{24M_{\Psi}}\\\
0&0&v_{1}x_{f}^{3}+\frac{v_{2}X_{f}^{3}}{2\sqrt{6}}),\quad
f\in\quantity{u,d}.$ (13)

Including the one-loop results only for the first generation, where it makes
the difference, we finally get

$M_{u}=\matrixquantity(-b\quantity(2\sqrt{6}v_{1}\lambda_{50}^{*}+v_{2}\lambda_{u}^{*})Y_{u}\\\
-a\quantity(2\sqrt{6}v_{1}y_{u}+v_{2}Y_{u})\\\
v_{1}x_{u}+\frac{v_{2}X_{u}}{2\sqrt{6}}),\quad
M_{d}=\matrixquantity(b\quantity(2\sqrt{6}v_{1}\lambda_{49}^{*}+v_{2}\lambda_{d}^{*})Y_{d}\\\
-a\quantity(2\sqrt{6}v_{1}y_{d}+v_{2}Y_{d})\\\
v_{1}x_{d}+\frac{v_{2}X_{d}}{2\sqrt{6}}).$ (14)

Here

$a=\frac{Y_{\Phi}}{24}\frac{v_{\Phi}}{M_{\Psi}},\quad
b=\frac{1}{16\pi^{2}}\frac{Y_{\Phi}}{24}\frac{v_{\Phi}}{M_{\Psi}}\quantity[\log\quantity(\frac{M_{\Psi}^{2}}{\mu^{2}})-1],$
(15)

are the suppression factors associated with the tree-level and one-loop
operators, respectively, and

$\lambda_{u}=-11\lambda_{45}-\lambda_{46}+3\lambda_{47}+2\lambda_{48},\quad\lambda_{d}=-11\lambda_{45}-\frac{391}{128}\lambda_{46}+\lambda_{47}+2\lambda_{48}.$
(16)

Note that the quark mass matrices are upper-triangular with the hierarchical
rows, which ensures small left-handed rotation matrices in agreement with the
CKM matrix.

In our chosen basis, the charged lepton mass matrix $M_{e}$ is not upper-
triangular due to its dependence on the same Yukawa couplings as the down-type
quark mass matrix, i.e., $\mathrm{U}(3)_{\psi_{d,\mathrm{R}}}$ is already used
up to set the latter into a desired form. Therefore,

$M_{e}=\leavevmode\resizebox{389.89749pt}{}{$\matrixquantity(-3b\quantity(2\sqrt{6}v_{1}\lambda_{49}^{*}+v_{2}\lambda_{e})Y_{d}^{1}&-3b\quantity(2\sqrt{6}v_{1}\lambda_{49}^{*}+v_{2}\lambda_{e})Y_{d}^{2}&-3b\quantity(2\sqrt{6}v_{1}\lambda_{49}^{*}+v_{2}\lambda_{e})Y_{d}^{3}\\\
-12av_{2}Y_{d}^{1}&3a\quantity(2\sqrt{6}v_{1}y_{d}^{2}-3v_{2}Y_{d}^{2})&3a\quantity(2\sqrt{6}v_{1}y_{d}^{3}-3v_{2}Y_{d}^{3})\\\
-\sqrt{\frac{2}{3}}v_{2}X_{d}^{1}&-\sqrt{\frac{2}{3}}v_{2}X_{d}^{2}&v_{1}x_{d}^{3}-\frac{1}{2}\sqrt{\frac{3}{2}}v_{2}X_{d}^{3})$},$
(17)

where

$\lambda_{e}=3\lambda_{45}+\frac{149}{128}\lambda_{46}-3\lambda_{47}+2\lambda_{48}.$
(18)

##  4 Producing the flavor hierarchies

Starting from the mass matrices in Eqs. (14) and (17), in Section 4.1, we will
derive quark and charged-lepton masses and the CKM elements in terms of the UV
parameters. Subsequently, in Section 4.2, we will conduct a detailed numerical
scan of the UV parameter space to predict the SM flavor hierarchies. Finally,
we will discuss the neutrino sector in Section 4.3.

###  4.1 Perturbative diagonalization

Since the quark mass matrices are upper-triangular with hierarchical rows, we
perturbatively diagonalize them to obtain the quark masses and the CKM matrix.
The perturbative diagonalization is performed as described in the Appendix of
Ref. Greljo:2023bix , and we report only the leading terms in $b\ll a\ll 1$
expansion. The singular value decomposition gives us the rotation matrices
through

$M_{u(d)}=L_{u(d)}\widehat{M}_{u(d)}R_{u(d)}^{\dagger},$ (19)

where $\widehat{M}_{u(d)}$ is diagonal. The upper-triangular structure of the
quark mass matrices ensures that the off-diagonal elements of $R_{u(d)}$ are
negligible, while the hierarchy between the rows ensures that the off-diagonal
elements of $L_{u(d)}$ are small and hierarchical, satisfying

$\quantity[L_{u(d)}]_{ij}\propto\frac{\quantity[\widehat{M}_{u(d)}]_{ii}}{\quantity[\widehat{M}_{u(d)}]_{jj}},\quad
i\leq j.$ (20)

The CKM matrix is then determined by $V_{\mathrm{CKM}}=L_{u}^{\dagger}L_{d}$.
From the diagonal $\widehat{M}_{u(d)}$, we get the quark masses

$\displaystyle m_{u}$
$\displaystyle=-b\quantity(2\sqrt{6}v_{1}\lambda_{50}^{*}+v_{2}\lambda_{u}^{*})Y_{u}^{1},\quad$
$\displaystyle m_{d}$
$\displaystyle=b\quantity(2\sqrt{6}v_{1}\lambda_{49}^{*}+v_{2}\lambda_{d}^{*})Y_{d}^{1},$
(21) $\displaystyle m_{c}$
$\displaystyle=-a\quantity(2\sqrt{6}v_{1}y_{u}^{2}+v_{2}Y_{u}^{2}),\quad$
$\displaystyle m_{s}$
$\displaystyle=-a\quantity(2\sqrt{6}v_{1}y_{d}^{2}+v_{2}Y_{d}^{2}),$ (22)
$\displaystyle m_{t}$
$\displaystyle=v_{1}x_{u}^{3}+\frac{v_{2}X_{u}^{3}}{2\sqrt{6}},\quad$
$\displaystyle m_{b}$
$\displaystyle=v_{1}x_{d}^{3}+\frac{v_{2}X_{d}^{3}}{2\sqrt{6}}.$ (23)

While using the full expression in the numerical scan, we omit writing it here
due to its complexity and only note that it satisfies
$\quantity[V_{\mathrm{CKM}}]_{ii}=1$ at the leading order.

The left-handed rotations in the lepton sector are also small and hierarchical
due to the hierarchies between the rows in Eq. 17. By contrast, the right-
handed rotations are can be sizeable depending on the region of parameter
space. Consider now the limit $\tan\beta\to 0$, in which $M_{e}$ becomes
upper-triangular, causing the right-handed rotations to be negligible. In this
limit it follows from Eqs. 14 and 17 that the third rows of $M_{d}$ and
$M_{e}$ become identical while the first two rows differ only by a
multiplicative factor. In this limit, $m_{b}=m_{\tau}=v_{1}x_{d}^{3}$.
Empirically, the bottom-tau unification is remarkably successful at high
energies, which is the primary motivation for considering this limit. However,
obtaining the correct size necessitates restricting
$x_{d}^{3}\sim\mathcal{O}(10^{-2})$. As we will demonstrate in the numerical
scan below, this is the only dimensionless parameter that must be somewhat
small compared to $\mathcal{O}(1)$ expectations. In fact, as long as
$\tan\beta\lesssim y_{b,\tau}\sim 10^{-2}$, only a single parameter needs to
be tuned. For $\tan\beta\sim 1$, four parameters must be restricted to
magnitudes of $10^{-2}$, see Eq. 17. Finally, the $\tan\beta\to\infty$ limit
predicts a wrong mass ratio $m_{b}/m_{\tau}=1/3$ and is ruled out. In the
numerical studies, we will focus on $\tan\beta\lesssim 10^{-2}$.

###  4.2 Numerical scan

Figure 3: Histogram showing the probability of obtaining the correct order of
magnitude for the SM flavor parameters when the UV parameters take on random
numbers drawn from a flat distribution with the magnitude $\leq 1$. The black
lines display the running SM values at the renormalization scale
$1\text{\,}\mathrm{P}\mathrm{e}\mathrm{V}$. See Section 4.2 for details.

We randomly generate one million parameter points from a flat distribution,
applying the following constraints on the dimensionless couplings in the UV
Lagrangian: $-1\leq Y_{\Phi}\leq 1$, $\absolutevalue{x_{d}^{3}}\leq
10^{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}2}$, and all other contributing
Yukawa and scalar couplings are allowed to be complex, with their magnitudes
constrained to $\leq 1$. We take the symmetry breaking and the renormalization
scales equal, $v_{\Phi}=\mu=$1\text{\,}\mathrm{P}\mathrm{e}\mathrm{V}$$, and
fix the ratio between the VLF mass and the symmetry breaking scale to
$\varepsilon=v_{\Phi}/M_{\Psi}=10^{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}2}$.
Additionally, we set $\tan\beta=0.001$, but we have verified that the results
remain qualitatively unchanged as long as $\tan\beta\lesssim 10^{-2}$.

In Fig. 3, we plot the predicted histograms for the SM flavor parameters. The
first three rows display the values of the Yukawa couplings,
$y_{i}=m_{i}/v_{\scriptscriptstyle\mathrm{EW}}$, while the fourth row shows
the magnitudes of the CKM elements. The black solid lines represent the
singular values of the running SM Yukawa couplings at the renormalization
scale $\mu$ as given in Greljo:2023bix , and the CKM matrix elements, which do
not run significantly, as provided by the PDG Workman:2022ynf . These
distributions illustrate the overall satisfactory agreement between the
predicted and observed parameters. Somewhat ad-hoc restriction of a single
parameter $\absolutevalue{x_{d}^{3}}\leq
10^{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}2}$ is critical; otherwise, for
$\absolutevalue{x_{d}^{3}}\leq 1$, the bottom-quark and tau distributions
would look like the top-quark distribution, shown in the first row, third
column.

By repeating a similar exercise with other parameters, we confirmed that the
structure is primarily controlled by $\varepsilon$, with only a mild
dependence on other parameters. The varying broadness of the distributions
between $y_{i}$ for different generations is due to the fact that the first,
second, and third family Yukawas are proportional to a product of three, two,
and to a single UV coupling, respectively (see Fig. 1). Remember that their
magnitudes are drawn from a uniform distribution in the $[0,1]$ range.

Another interesting aspect is revealed when studying the correlations between
the observed parameters. To visualize the effect that $\beta$ has on the
bottom-tau splitting, we generated one million parameter points for two
different values of $\beta$: $0.001$ and $0.01$ while other configurations of
the scan remain unchanged. In Fig. 4, we present histograms showing the
correlation between the masses of the down-type quarks and charged leptons.
The black dot marks the running masses at the appropriate scale. In the
$\tan\beta\to 0$ limit, there is a strong correlation across all three
generations, which is in remarkable agreement with observations! Increasing
$\beta$ decreases the correlation, and this effect is most noticeable for the
third generation.

Figure 4: Two-dimensional histograms showing the correlation between the
masses of the down quarks and charged leptons for two choices of $\tan\beta$.
The black points mark the SM values at the chosen renormalization scale. See
Section 4.2 for details.

###  4.3 Neutrino sector

Up to this point, we have focused on the flavor structure in the charged
fermion sector, neglecting the neutrino masses and the
Pontecorvo–Maki–Nakagawa–Sakata (PMNS) Maki:1962mu  mixing matrix. Unlike the
hierarchies observed in the charged sector, the neutrino flavor appears to be
anarchic. Indeed, the PMNS matrix stands in stark contrast to the CKM matrix,
exhibiting large mixing angles. Arguably, the only relevant hierarchy in the
neutrino sector is related to the overall neutrino mass scale, which is many
orders of magnitude below the electron mass. In this section, we will briefly
discuss how to correctly reproduce neutrino parameters within this model.

At face value, the model predicts a Dirac mass matrix

$M_{\nu}=\leavevmode\resizebox{389.89749pt}{}{$\matrixquantity(3b\quantity(2\sqrt{6}v_{1}\lambda_{50}^{*}+v_{2}\lambda_{\nu})Y_{u}^{1}&3b\quantity(2\sqrt{6}v_{1}\lambda_{50}^{*}+v_{2}\lambda_{\nu})Y_{u}^{2}&3b\quantity(2\sqrt{6}v_{1}\lambda_{50}^{*}+v_{2}\lambda_{\nu})Y_{u}^{3}\\\
-12av_{2}Y_{u}^{1}&3a\quantity(2\sqrt{6}v_{1}y_{u}^{2}-3v_{2}Y_{u}^{2})&3a\quantity(2\sqrt{6}v_{1}y_{u}^{3}-3v_{2}Y_{u}^{3})\\\
-\sqrt{\frac{2}{3}}v_{2}X_{u}^{1}&-\sqrt{\frac{2}{3}}v_{2}X_{u}^{2}&v_{1}x_{u}^{3}-\frac{1}{2}\sqrt{\frac{3}{2}}v_{2}X_{u}^{3})$}$
(24)

for the neutrinos, where

$\lambda_{\nu}=3\lambda_{45}+3\lambda_{46}-\lambda_{47}+2\lambda_{48}.$ (25)

This alone incorrectly implies that neutrino masses are similar in size to the
up-quark masses, and the PMNS matrix is close to a unit matrix.

An elegant explanation of the smallness of neutrino masses in the context of
minimal quark-lepton unification is provided by the inverse seesaw mechanism
FileviezPerez:2013zmv . Our model is minimally expanded by introducing three
left-handed gauge-singlet fermions $S_{i}$ ($i=1,2,3$). As a result, the
Lagrangian gets supplemented by the following gauge-invariant and
renormalizable operators:

$-\mathcal{L}\supset\frac{1}{2}\mu_{S}^{ij}\bar{S}^{c}_{i}S_{j}+Y^{ij}_{R}\bar{S}_{i}\chi^{\dagger}\psi_{u,\mathrm{R}}^{j}+{\rm
h.c.}\,,$ (26)

where $M^{ij}_{R}\equiv Y^{ij}_{R}\langle\chi\rangle$. We define
$n_{i}=(\nu_{Li},\nu_{Ri}^{c},S_{i})$, where $c$ denotes charge conjugation.
The neutral lepton mass matrix, which is $9\times 9$, is given by

$\mathcal{L}_{M_{n}}=-\frac{1}{2}\bar{n}_{i}M^{ij}_{n}n_{j}^{c}+{\rm
h.c.}\,,\quad M_{n}=\begin{pmatrix}0&M_{\nu}&0\\\ M_{\nu}^{T}&0&M_{R}^{T}\\\
0&M_{R}&\mu_{S}\end{pmatrix},$ (27)

The global lepton number is an approximate symmetry softly broken by
$\mu^{ij}_{S}$. For $\mu_{S}\ll M_{\nu}<M_{R}$, the mass matrix for active
neutrinos approximates to

$m_{\nu}\approx M_{\nu}M_{R}^{-1}\mu_{S}(M_{R}^{-1})^{T}M^{T}_{\nu}~{}.$ (28)

Thus, the overall smallness of the neutrino masses is explained by the small
breaking of the lepton number via $\mu_{S}$. It is evident that the observed
neutrino mass splittings and the PMNS mixing angles can be accurately fitted
by introducing hierarchies into $\mu_{S}$, given that $M_{\nu}$ is
hierarchical. The origin of these hierarchies is beyond the present work.

##  5 Phenomenology

The symmetry-breaking scale of our model is constrained by rare processes
mediated by the massive vector bosons resulting from this breaking. In
Pati–Salam-type models, the vector leptoquarks are known to give contributions
to the lepton flavor–violating decay $K_{L}\to\mu e$, which is forbidden in
the SM. This typically constrains the breaking scale to be of the order of a
few PeV Valencia:1994cj ; Smirnov:2007hv , and we will see that this general
picture is reproduced here. The $\mathrm{SU}(2)_{q+\ell}$ vector bosons are
also known to give unsuppressed contributions to these SM-forbidden decays
through a mechanism known as flavor transfer Darme:2023nsy ; Greljo:2023bix ,
but the resulting bound on $v_{\Phi}$ is weaker as we will show below. Typical
rotations into the fermion mass basis will also result in contributions to
muon conversion on heavy nuclei, a stringently constrained process with great
potential for improvements in planned experiments.

Integrating out the vector leptoquark and the flavored $Z^{\prime}$ at tree
level results in effective operators

$\mathcal{L}_{\scriptscriptstyle\mathrm{SMEFT}}\supset-\dfrac{1}{v_{\chi}^{2}}\Big{|}\overline{q}^{3}\gamma_{\mu}\ell^{3}+\overline{q}_{\alpha}\gamma_{\mu}\ell^{\alpha}+\overline{d}^{p}\gamma_{\mu}e^{p}\Big{|}^{2}-\dfrac{1}{v_{\Phi}^{2}}\Big{|}\overline{q}_{\alpha}\gamma_{\mu}t^{\alpha}_{a\beta}q^{\beta}+\overline{\ell}_{\alpha}\gamma_{\mu}t^{\alpha}_{a\beta}\ell^{\beta}\Big{|}^{2}$
(29)

at the high scale, where $t_{a}=\tfrac{1}{2}\sigma_{a}$ is the generator of
$\mathrm{SU}(2)_{q+\ell}$ rotations in the fundamental representation. We
assume here that the right-handed neutrinos are too heavy to play a role in
low-energy flavor physics and have ignored their contributions accordingly.
Next, we transition to the low-energy effective Lagrangian (LEFT) below the
scale of EW symmetry breaking to get to the mass basis fermions at the scale
$1\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$ relevant to the lepton
flavor–violating processes discussed above. Only the semi-leptonic operators
are relevant and shown here:666We have omitted a contribution to the left-
handed vector operator proportional to
$[L^{\dagger}_{e}\mathcal{I}L_{e}]_{pr}$, as lepton flavor–violating
contributions from this term are severely suppressed on account of a GIM-like
mechanism.

$\displaystyle\mathcal{L}_{\scriptscriptstyle\mathrm{LEFT}}$
$\displaystyle\supset\,-\left(\dfrac{[L^{\dagger}_{e}L_{d}]_{pt}[L^{\dagger}_{d}L_{e}]_{sr}}{v_{\chi}^{2}}+\dfrac{[L^{\dagger}_{e}\mathcal{I}L_{d}]_{pt}[L^{\dagger}_{d}\mathcal{I}L_{e}]_{sr}}{v_{\Phi}^{2}}\right)(\overline{e}^{p}_{\mathrm{L}}\gamma_{\mu}e^{r}_{\mathrm{L}})(\overline{d}^{s}_{\mathrm{L}}\gamma^{\mu}d^{t}_{\mathrm{L}})$
$\displaystyle-\dfrac{[L^{\dagger}_{e}\mathcal{I}L_{u}]_{pt}[L^{\dagger}_{u}\mathcal{I}L_{e}]_{sr}}{v_{\Phi}^{2}}(\overline{e}^{p}_{\mathrm{L}}\gamma_{\mu}e^{r}_{\mathrm{L}})(\overline{u}^{s}_{\mathrm{L}}\gamma^{\mu}u^{t}_{\mathrm{L}})$
(30)
$\displaystyle+\bigg{[}6.0\dfrac{[L^{\dagger}_{e}L_{d}]_{pt}[R^{\dagger}_{d}R_{e}]_{sr}}{v_{\chi}^{2}}(\overline{e}^{p}_{\mathrm{L}}e_{\mathrm{R}}^{r})(\overline{d}^{s}_{\mathrm{R}}d_{\mathrm{L}}^{t})\,+\,\mathrm{H.c.}\bigg{]}-\dfrac{[R^{\dagger}_{e}R_{d}]_{pt}[R^{\dagger}_{d}R_{e}]_{sr}}{v_{\chi}^{2}}(\overline{e}^{p}_{\mathrm{R}}\gamma_{\mu}e^{r}_{\mathrm{R}})(\overline{d}^{s}_{\mathrm{R}}\gamma^{\mu}d^{t}_{\mathrm{R}})$

where $\mathcal{I}=\mathop{\mathrm{diag}}(1,\,1,\,0)$. The numerical factor of
$6.0$ on the second line is a combination of a factor of $-2$ from Fierzing
the four-fermion operator and a factor of $\sim 3.0$ from running the operator
down from $\mu=$3\text{\,}\mathrm{P}\mathrm{e}\mathrm{V}$$ to
$\mu=$1\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$$ (as determined with DsixTools
Celis:2017hod ; Fuentes-Martin:2020zaz ). By contrast, the vector current
operators barely run at all (shifting only the operators by a few percent),
and we disregard that effect.

###  5.1 Lepton flavor–violating kaon decay

The non-observation of the $K_{L}\to\mu e$ decay poses a strong constraint on
the scale of symmetry breaking. The current experimental bound on the
branching rate is $\mathrm{Br}(K_{L}\to\mu e)<$4.7\text{\cdot}{10}^{-12}$$ @
90% CL BNL:1998apv . We calculated the NP contribution to this decay rate from
the LEFT Lagrangian in Section 5 using Ref. Marzocca:2021miv  and obtained

$\displaystyle\dfrac{\mathrm{Br}\big{(}K_{L}\to\mu
e\big{)}}{$4.7\text{\cdot}{10}^{-12}$}=\,$
$\displaystyle\Big{|}14\,\hat{v}^{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}2}_{\chi}\big{(}[L^{\dagger}_{e}L_{d}]_{ed}[R^{\dagger}_{d}R_{e}]_{s\mu}+[L^{\dagger}_{e}L_{d}]_{es}[R^{\dagger}_{d}R_{e}]_{d\mu}\big{)}^{\ast}$
$\displaystyle\qquad-0.10\,\hat{v}^{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}2}_{\chi}\big{(}[L^{\dagger}_{e}L_{d}]_{\mu
s}[L^{\dagger}_{d}L_{e}]_{de}+[L^{\dagger}_{e}L_{d}]_{\mu
d}[L^{\dagger}_{d}L_{e}]_{se}\big{)}$
$\displaystyle\qquad-0.10\,\hat{v}^{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}2}_{\Phi}\big{(}[L^{\dagger}_{e}\mathcal{I}L_{d}]_{\mu
s}[L^{\dagger}_{d}\mathcal{I}L_{e}]_{de}+[L^{\dagger}_{e}\mathcal{I}L_{d}]_{\mu
d}[L^{\dagger}_{d}\mathcal{I}L_{e}]_{se}\big{)}\Big{|}^{2}$ (31)
$\displaystyle+\Big{|}14\,\hat{v}^{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}2}_{\chi}\big{(}[L^{\dagger}_{e}L_{d}]_{\mu
d}[R^{\dagger}_{d}R_{e}]_{se}+[L^{\dagger}_{e}L_{d}]_{\mu
s}[R^{\dagger}_{d}R_{e}]_{de}\big{)}$
$\displaystyle\qquad+0.10\,\hat{v}^{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}2}_{\chi}\big{(}[R^{\dagger}_{e}R_{d}]_{\mu
s}[R^{\dagger}_{d}R_{e}]_{de}+[R^{\dagger}_{e}R_{d}]_{\mu
d}[R^{\dagger}_{d}R_{e}]_{se}\big{)}\Big{|}^{2},$

having normalized the VEVs by $\hat{v}_{f}\equiv
v_{f}/$1\text{\,}\mathrm{P}\mathrm{e}\mathrm{V}$$. The pseudo-scalar quark
current interpolates $K_{L}$ better than the axial vector, resulting in a
significant enhancement to the contributions stemming from mixed left–right
currents due to the vector leptoquark, which are also enhanced by the RG
running.777These are the contributions multiplying
$14\hat{v}^{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}2}_{\chi}$.

With no suppression of the NP contribution from the fermion rotation matrices,
the branching ratio in Section 5.1 bounds
$v_{\chi}\gtrsim$4\text{\,}\mathrm{P}\mathrm{e}\mathrm{V}$$. However, it may
be possible to evade this constraint. In a simple approximation, where
$L_{e}^{\dagger}L_{d}$ and $R_{d}^{\dagger}R_{e}$ are taken to be two-
dimensional rotation matrices (dominated by the 1–2 rotation angle), the
scalar contribution will vanish altogether if the rotation angles are
orthogonal. While the left-handed rotations are perturbative and, therefore,
close to the identity matrix, the right-handed rotation might be close to
maximal, leading to a vanishing scalar contribution. In this case, the bound
will come from the vector currents constraining
$v_{\chi},v_{\Phi}\gtrsim$300\text{\,}\mathrm{T}\mathrm{e}\mathrm{V}$$. That
being said, we consider the realizations with the symmetry-breaking scale of
at least a PeV to be the more realistic scenario.

###  5.2 Muon conversion on heavy nuclei

Muon conversion on heavy nuclei is closely correlated to the lepton
flavor–violating kaon decay, which is also caused by the Lagrangian in Section
5, but without any quark flavor violation. The SINDRUM-II experiment set the
best present bound on this process, obtaining the limit
$\mathrm{Cr}(\mu\,\mathrm{Au}\to e\,\mathrm{Au})<$7\text{\cdot}{10}^{-13}$$
for the conversion rate in muonic gold SINDRUMII:2006dvw . Scalar and vector
four-fermion operators contribute to this process at roughly the same rate,
and, using Ref. Kitano:2002mt , we determine the contribution from the heavy
vector bosons:

$\displaystyle\dfrac{\mathrm{Cr}(\mu\,\mathrm{Au}\to
e\,\mathrm{Au})}{$7\text{\cdot}{10}^{-13}$}=\,$
$\displaystyle\Big{|}0.5\,\hat{v}^{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}2}_{\Phi}[L^{\dagger}_{e}\mathcal{I}L_{u}]_{eu}[L^{\dagger}_{u}\mathcal{I}L_{e}]_{u\mu}+0.6\,\hat{v}^{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}2}_{\chi}[L^{\dagger}_{e}L_{d}]_{ed}[L^{\dagger}_{d}L_{e}]_{d\mu}$
(32)
$\displaystyle+0.6\,\hat{v}^{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}2}_{\Phi}[L^{\dagger}_{e}\mathcal{I}L_{d}]_{ed}[L^{\dagger}_{d}\mathcal{I}L_{e}]_{d\mu}+2.7\hat{v}_{\chi}^{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}2}[R^{\dagger}_{e}R_{d}]_{ed}[L^{\dagger}_{d}L_{e}]_{d\mu}+1.4\hat{v}_{\chi}^{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}2}[R^{\dagger}_{e}R_{d}]_{es}[L^{\dagger}_{d}L_{e}]_{s\mu}\Big{|}^{2}$
$\displaystyle+\Big{|}0.6\,\hat{v}^{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}2}_{\chi}[R^{\dagger}_{e}R_{d}]_{ed}[R^{\dagger}_{d}R_{e}]_{d\mu}+2.7\hat{v}_{\chi}^{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}2}[L^{\dagger}_{e}L_{d}]_{ed}[R^{\dagger}_{d}R_{e}]_{d\mu}+1.4\hat{v}_{\chi}^{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}2}[L^{\dagger}_{e}L_{d}]_{es}[R^{\dagger}_{d}R_{e}]_{s\mu}\Big{|}^{2}$

The numerical enhancement of the scalar operators is due to the RG-running
already present in the Wilson coefficients in Section 5.

Muon conversion violates lepton flavor by one unit while quark flavor is
conserved. Thus, in the interaction basis, the process would be forbidden, as
no flavor transfer Darme:2023nsy  is possible. We should, however, expect a
mixing of at least the left-handed down-type in order to recover the sizable
Cabbibo angle. In the limit, where the $[L_{d}]_{12}=\theta_{c}=0.22$ is the
only non-trivial mixing, the NP contribution in Eq. 32 gives the bounds
$v_{\chi}\gtrsim$0.8\text{\,}\mathrm{P}\mathrm{e}\mathrm{V}$$ and
$v_{\Phi}\gtrsim$0.4\text{\,}\mathrm{P}\mathrm{e}\mathrm{V}$$. While specific
realizations of the mixing matrices may either enhance or weaken the bounds,
they are a good estimate for realistic benchmarks in the absence of tuning.
Interestingly, the future Mu2e Bernstein:2019fyh  and COMET Moritsu:2022lem
experiments projects an improvement on the experimental sensitivity to the
level of ${10}^{-17}$.888Both Mu2e and COMET will be using aluminum as the
target nuclei, changing the numerical factors in Eq. (32). Hence, the
experimental sensitivity to the symmetry-breaking scales is expected to
improve by an order of magnitude. This provides an exciting avenue for
experimentally detecting the imprints of the model behind the flavor structure
posited in this paper.

##  6 Conclusion

The field of flavor physics has a bright future due to a comprehensive
experimental program anticipated within this decade. The interest is driven by
the potential to detect indirect effects of heavy BSM physics from scales well
beyond the reach of direct searches. Indirect signs of new physics might
manifest in phenomena such as rare flavor-changing neutral currents,
violations of charged lepton flavor, and electric dipole moments, each of
which is expected to see substantial experimental advancements LHCb:2018roe ;
LHCb:2021glh ; Belle-II:2018jsg ; Forti:2022mti ; Belle-II:2022cgf ;
NA62KLEVER:2022nea ; MEGII:2018kmf ; Bernstein:2019fyh ; Moritsu:2022lem ;
n2EDM:2021yah ; Wu:2019jxj ; Blondel:2013ia ; ACME:2018yjb ;
EuropeanStrategyforParticlePhysicsPreparatoryGroup:2019qin . Furthermore,
flavor physics is closely tied to some of the most significant structural
challenges of the SM. The origin of flavor remains an enduring enigma: one of
the most puzzling aspects is the presence of hierarchies in the masses (and
mixings) of charged fermions between successive generations, despite all of
them arising from the Yukawa interactions with a single Higgs field. This
longstanding puzzle has attracted a renewed interest in light of recent
experimental activities.

Perhaps the resolution of the flavor puzzle may lie already at the next hidden
layer of nature. It is worthwhile to explore novel possibilities for low-scale
flavor models, which could lead to deviations detectable in forthcoming
experiments. In this paper, we construct and thoroughly investigate one
particularly elegant example. Building upon our recent work Greljo:2023bix ,
we combine a gauged $\mathrm{SU}(2)$ flavor symmetry with a minimal quark-
lepton unification paradigm based on the $\mathrm{SU}(4)$ gauge group. The
flavor symmetry acts on left-chiral fermions, distinguishing the light-family
doublet from the third-family singlet. The model, detailed in Section 2,
predicts hierarchically distinct contributions to the rank of the Yukawa
matrices, which then translates into the hierarchies between families.

As illustrated in Fig. 1, the third row in the Yukawa matrices results from
unsuppressed dimension-4 Yukawa interactions. The second row emerges from
integrating out a single heavy vector-like fermion, yielding a tree-level
dimension-5 operator. The first row is populated by loop diagrams involving
the same fields and generating a loop-level dimension-5 operator. The matter
field content required to realize this mechanism is remarkably minimal, as
shown in Table 1. This is due to the elegant unification of multiple elements
from Ref. Greljo:2023bix  collected in Table 2.

The entire SM flavor structure is controlled by a single parameter
$\varepsilon=v_{\Phi}/M_{\Psi}$, a ratio of the flavor-breaking scale to the
VLF mass. After performing a detailed matching calculation to the SM Yukawas
in Section 3, we conduct a comprehensive numerical scan in Section 4. The key
result is presented in Fig. 3, showing a histogram of predictions for SM
flavor parameters, for $\varepsilon=10^{-2}$, obtained from randomly selecting
dimensionless UV couplings from a flat distribution in the range $|y_{x}|\leq
1$. Remarkably, reasonable order-of-magnitude predictions emerge for all
measured parameters, except for the $b/\tau$ Yukawa, which requires a single
UV parameter to be somewhat smaller than expected, $x_{d}^{3}\sim 10^{-2}$.
Even more remarkable is the predicted correlation between down-quark and
charged lepton masses shown in Fig. 4. For a small ratio of Higgs VEVs,
$\tan\beta\to 0$, this correlation is very precise and shows an extraordinary
agreement with the measured values. This exemplifies how quark-lepton
unification, a well-motivated concept on its own, aids in explaining the
flavor puzzle.

As demonstrated in Section 5, the leading signatures of the model are lepton
flavor–violating kaon decays and muon conversion on heavy nuclei, where the
experimental prospects are excellent. In fact, current limits already require
the scale of the model to be at least of the order of PeV, which introduces
another hierarchy problem related to the smallness of the electroweak scale.
Additionally, generating small neutrino masses can be straightforwardly
achieved using the inverse see-saw mechanism, as described in Section 4.3.
However, predicting large PMNS mixing angles requires additional structure.
These issues are not addressed in this work and are left for future research.

### Acknowledgments

This work has received funding from the Swiss National Science Foundation
(SNF) through the Eccellenza Professorial Fellowship “Flavor Physics at the
High Energy Frontier,” project number 186866, and the Ambizione grant
“Matching and Running: Improved Precision in the Hunt for New Physics,”
project number 209042.

##  Appendix A Scalar potential

To find the full scalar potential and ensure that no superfluous terms were
included or any possible terms omitted, we made use of the Mathematica package
Sym2Int Fonseca:2017lem . We have used Roman letters for
$\mathrm{SU}(2)_{\mathrm{L}}$ indices: $i,j$, and Greek letters for
$\mathrm{SU}(2)_{q+\ell}$ indices: $\alpha,\beta,\gamma,\delta$, and
surpressed all other indices. The full scalar potential is given below:

$\displaystyle V$
$\displaystyle=m_{\chi}^{2}\absolutevalue{\chi}^{2}+m_{H_{1}}^{2}\absolutevalue{H_{1}}^{2}+2m_{\Sigma_{H}}^{2}\Tr\\!\big{(}\absolutevalue{\Sigma_{H}}^{2})+2m_{\Sigma_{\Phi}}^{2}\Tr\\!\big{(}\absolutevalue{\Sigma_{\Phi}}^{2})$
$\displaystyle+\lambda_{1}\absolutevalue{\chi}^{4}+\lambda_{2}\absolutevalue{H_{1}}^{2}\absolutevalue{\chi}^{2}+\lambda_{3}\absolutevalue{H_{1}}^{2}\Tr\\!\big{(}\absolutevalue{\Sigma_{H}}^{2})+\lambda_{4}\absolutevalue{\chi}^{2}\Tr\\!\big{(}\absolutevalue{\Sigma_{H}}^{2})$
$\displaystyle+(\lambda_{5}H_{1i}^{\dagger}\chi^{\dagger}\Sigma_{H}^{i}\chi+\text{h.c.})+\lambda_{6}H_{1i}^{\dagger}\Tr\\!\big{(}\Sigma_{Hj}^{\dagger}\Sigma_{H}^{i})H^{j}+\lambda_{7}\chi^{\dagger}\Sigma_{H}^{i}\Sigma_{Hi}^{\dagger}\chi+\lambda_{8}\chi^{\dagger}\absolutevalue{\Sigma_{H}}^{2}\chi$
$\displaystyle+\lambda_{9}\absolutevalue{H_{1}}^{4}+\lambda_{10}\Tr\\!\big{(}\absolutevalue{\Sigma_{H}}^{4})+\lambda_{11}\quantity[\Tr\\!\big{(}\absolutevalue{\Sigma_{H}}^{2})]^{2}+\quantity(\lambda_{12}H_{1i}^{\dagger}\Tr\\!\big{(}\Sigma_{H}^{i}\Sigma_{H}^{j})H_{1j}^{\dagger}+\text{h.c.})$
$\displaystyle+\quantity(\lambda_{13}H_{1i}^{\dagger}\Tr\\!\big{(}\Sigma_{H}^{i}\absolutevalue{\Sigma_{H}}^{2})+\lambda_{14}H_{1i}^{\dagger}\Tr\\!\big{(}\Sigma_{H}^{i}\Sigma_{H}^{j}\Sigma_{Hj}^{\dagger})+\text{h.c.})+\lambda_{15}\Tr\\!\big{(}\Sigma_{Hi}^{\dagger}\Sigma_{H}^{j}\Sigma_{Hj}^{\dagger}\Sigma_{H}^{i})$
$\displaystyle+\lambda_{16}\Tr\\!\big{(}\Sigma_{Hi}^{\dagger}\Sigma_{H}^{j})\Tr\\!\big{(}\Sigma_{Hj}^{\dagger}\Sigma_{H}^{i})+\lambda_{17}\Tr\\!\big{(}\Sigma_{Hi}^{\dagger}\Sigma_{Hj}^{\dagger})\Tr\\!\big{(}\Sigma_{H}^{i}\Sigma_{H}^{j})$
$\displaystyle+\lambda_{18}\Tr\\!\big{(}\Sigma_{Hi}^{\dagger}\Sigma_{Hj}^{\dagger}\Sigma_{H}^{i}\Sigma_{H}^{j})+\lambda_{19}\Tr\\!\big{(}\Sigma_{Hi}^{\dagger}\absolutevalue{\Sigma_{H}}^{2}\Sigma_{H}^{i})+\lambda_{20}\absolutevalue{H_{1}}^{2}\Tr\\!\big{(}\absolutevalue{\Sigma_{\Phi}}^{2})$
$\displaystyle+\lambda_{21}\absolutevalue{\chi}^{2}\Tr\\!\big{(}\absolutevalue{\Sigma_{\Phi}}^{2})+\lambda_{22}\chi^{\dagger}\absolutevalue{\Sigma_{\Phi}}^{2}\chi+\lambda_{23}\chi^{\dagger}\Sigma_{\Phi}^{\alpha}\Sigma_{\Phi\alpha}^{\dagger}\chi+\lambda_{24}\epsilon_{\alpha\beta}\chi^{\dagger}\Sigma_{\Phi}^{\alpha}\Sigma_{\Phi}^{\beta}\chi$
$\displaystyle+\lambda_{25}\Tr\\!\big{(}\absolutevalue{\Sigma_{\Phi}}^{4})+\lambda_{26}\quantity[\Tr\\!\big{(}\absolutevalue{\Sigma_{\Phi}}^{2})]^{2}+\lambda_{27}\Tr\\!\big{(}\Sigma_{\Phi\alpha}^{\dagger}\Sigma_{\Phi}^{\beta})\Tr\\!\big{(}\Sigma_{\Phi\beta}^{\dagger}\Sigma_{\Phi}^{\alpha})$
$\displaystyle+\lambda_{28}\Tr\\!\big{(}\Sigma_{\Phi\alpha}^{\dagger}\Sigma_{\Phi\beta}^{\dagger})\Tr\\!\big{(}\Sigma_{\Phi}^{\alpha}\Sigma_{\Phi}^{\beta})+\lambda_{29}\Tr\\!\big{(}\Sigma_{\Phi\alpha}^{\dagger}\Sigma_{\Phi}^{\beta}\Sigma_{\Phi\beta}^{\dagger}\Sigma_{\Phi}^{\alpha})$
$\displaystyle+\lambda_{30}\Tr\\!\big{(}\Sigma_{\Phi\alpha}^{\dagger}\Sigma_{\Phi\beta}^{\dagger}\Sigma_{\Phi}^{\alpha}\Sigma_{\Phi}^{\beta})+\lambda_{31}\Tr\\!\big{(}\Sigma_{\Phi\alpha}^{\dagger}\absolutevalue{\Sigma_{\Phi}}^{2}\Sigma_{\Phi}^{\alpha})$
$\displaystyle+\quantity(\lambda_{32}\epsilon_{\alpha\beta}\epsilon_{\gamma\delta}\Tr\\!\big{(}\Sigma_{\Phi}^{\alpha}\Sigma_{\Phi}^{\gamma})\Tr\\!\big{(}\Sigma_{\Phi}^{\beta}\Sigma_{\Phi}^{\delta})+\lambda_{33}\epsilon_{\alpha\beta}\epsilon_{\gamma\delta}\Tr\\!\big{(}\Sigma_{\Phi}^{\alpha}\Sigma_{\Phi}^{\gamma}\Sigma_{\Phi}^{\beta}\Sigma_{\Phi}^{\delta})+\text{h.c.})$
$\displaystyle+\quantity(\lambda_{34}\epsilon_{\alpha\beta}\Tr\\!\big{(}\Sigma_{\Phi}^{\alpha}\Sigma_{\Phi\gamma}^{\dagger})\Tr\\!\big{(}\Sigma_{\Phi}^{\beta}\Sigma_{\Phi}^{\gamma})+\lambda_{35}\epsilon_{\alpha\beta}\Tr\\!\big{(}\Sigma_{\Phi}^{\alpha}\absolutevalue{\Sigma_{\Phi}}^{2}\Sigma_{\Phi}^{\beta})+\text{h.c.})$
$\displaystyle+\quantity(\lambda_{36}\epsilon_{\alpha\beta}\Tr\\!\big{(}\Sigma_{\Phi}^{\alpha}\Sigma_{\Phi}^{\gamma}\Sigma_{\Phi\gamma}^{\dagger}\Sigma_{\Phi}^{\beta})+\text{h.c.})+\lambda_{37}\Tr\\!\big{(}\absolutevalue{\Sigma_{H}}^{2})\Tr\\!\big{(}\absolutevalue{\Sigma_{\Phi}}^{2})$
$\displaystyle+\lambda_{38}\Tr\\!\big{(}\Sigma_{Hi}^{\dagger}\Sigma_{\Phi}^{\alpha})\Tr\\!\big{(}\Sigma_{\Phi\alpha}^{\dagger}\Sigma_{H}^{i})+\lambda_{39}\Tr\\!\big{(}\Sigma_{Hi}^{\dagger}\Sigma_{\Phi\alpha}^{\dagger})\Tr\\!\big{(}\Sigma_{H}^{i}\Sigma_{\Phi}^{\alpha})+\lambda_{40}\Tr\\!\big{(}\absolutevalue{\Sigma_{H}}^{2}\absolutevalue{\Sigma_{\Phi}}^{2})$
$\displaystyle+\lambda_{41}\Tr\\!\big{(}\Sigma_{Hi}^{\dagger}\Sigma_{\Phi\alpha}^{\dagger}\Sigma_{H}^{i}\Sigma_{\Phi}^{\alpha})+\lambda_{42}\Tr\\!\big{(}\Sigma_{Hi}^{\dagger}\absolutevalue{\Sigma_{\Phi}}^{2}\Sigma_{H}^{i})+\lambda_{43}\Tr\\!\big{(}\Sigma_{\Phi\alpha}^{\dagger}\absolutevalue{\Sigma_{H}}^{2}\Sigma_{\Phi}^{\alpha})$
$\displaystyle+\lambda_{44}\Tr\\!\big{(}\Sigma_{Hi}^{\dagger}\Sigma_{\Phi}^{\alpha}\Sigma_{H}^{i}\Sigma_{\Phi\alpha}^{\dagger})+\lambda_{45}\Tr\\!\big{(}\Sigma_{H}^{i}\Sigma_{Hi}^{\dagger}\Sigma_{\Phi}^{\alpha}\Sigma_{\Phi\alpha}^{\dagger})$
$\displaystyle+\quantity(\lambda_{46}\epsilon_{\alpha\beta}\Tr\\!\big{(}\Sigma_{Hi}^{\dagger}\Sigma_{\Phi}^{\alpha})\Tr\\!\big{(}\Sigma_{H}^{i}\Sigma_{\Phi}^{\beta})+\lambda_{47}\epsilon_{\alpha\beta}\Tr\\!\big{(}\Sigma_{\Phi}^{\alpha}\absolutevalue{\Sigma_{H}}^{2}\Sigma_{\Phi}^{\beta})+\text{h.c.})$
$\displaystyle+\quantity(\lambda_{48}\epsilon_{\alpha\beta}\Tr\\!\big{(}\Sigma_{\Phi}^{\alpha}\Sigma_{H}^{i}\Sigma_{Hi}^{\dagger}\Sigma_{\Phi}^{\beta})+\lambda_{49}\epsilon_{\alpha\beta}\Tr\\!\big{(}\Sigma_{\Phi}^{\alpha}\Sigma_{Hi}^{\dagger}\Sigma_{\Phi}^{\beta}\Sigma_{H}^{i})+\text{h.c.})$
$\displaystyle+\quantity(\lambda_{50}\epsilon_{\alpha\beta}H_{1i}^{\dagger}\Tr\\!\big{(}\Sigma_{H}^{i}\Sigma_{\Phi}^{\alpha}\Sigma_{\Phi}^{\beta})+\lambda_{51}\epsilon_{\alpha\beta}H_{1}^{i}\Tr\\!\big{(}\Sigma_{Hi}^{\dagger}\Sigma_{\Phi}^{\alpha}\Sigma_{\Phi}^{\beta})+\text{h.c.})$
$\displaystyle+\quantity(\lambda_{52}H_{1i}^{\dagger}\Tr\\!\big{(}\Sigma_{H}^{i}\absolutevalue{\Sigma_{\Phi}}^{2})+\lambda_{53}H_{1i}^{\dagger}\Tr\\!\big{(}\Sigma_{H}^{i}\Sigma_{\Phi}^{\alpha}\Sigma_{\Phi\alpha}^{\dagger})+\text{h.c.})~{}.$
(33)

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