warn luatex=false aainstitutetext: School of Physics & Astronomy, University
of Glasgow, Glasgow G12 8QQ, UK bbinstitutetext: School of Physics &
Astronomy, University of Southampton, Southampton SO17 1BJ, UK
ccinstitutetext: Instituto de Física Corpuscular, CSIC-Universitat de
València, 46980 Paterna, Spain ddinstitutetext: Departament de Física Teòrica,
Universitat de València, 46100 Burjassot, Spain

# Minimal complete tri-hypercharge theories of flavour

Mario Fernández Navarro b Stephen F. King c,d and Avelino Vicente
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

The tri-hypercharge proposal introduces a separate gauged weak hypercharge
assigned to each fermion family as the origin of flavour. This is arguably one
of the simplest setups for building “gauge non-universal theories of flavour”
or “flavour deconstructed theories”. In this paper we propose and study two
minimal but ultraviolet complete and renormalisable tri-hypercharge models. We
show that both models, which differ only by the heavy messengers that complete
the effective theory, are able to explain the observed patterns of fermion
masses and mixings (including neutrinos) with all fundamental coefficients
being of $\mathcal{O}(1)$. In fact, both models translate the complicated
flavour structure of the Standard Model into three simple physical scales
above electroweak symmetry breaking, completely correlated with each other,
that carry meaningful phenomenology. In particular, the heavy messenger sector
determines the origin and size of fermion mixing, which controls the size and
nature of the flavour-violating currents mediated by the two heavy
$Z^{\prime}$ gauge bosons of the theory. The phenomenological implications of
the two minimal models are compared. In both models the lightest $Z^{\prime}$
remains discoverable in dilepton searches at the LHC Run 3.

##  1 Introduction

The Standard Model (SM) of particle physics, though highly successful, offers
no insight into the origin of the three fermion families nor the pattern of
fermion Yukawa couplings, which results in a hierarchical mass spectrum for
quarks and charged leptons that may be expressed at low energies as
PDG:2022ynf

$\displaystyle m_{t}\sim\frac{v_{\mathrm{SM}}}{\sqrt{2}}\,,\qquad$
$\displaystyle
m_{c}\sim\lambda^{3.3}\frac{v_{\mathrm{SM}}}{\sqrt{2}}\,,\qquad$
$\displaystyle m_{u}\sim\lambda^{7.5}\frac{v_{\mathrm{SM}}}{\sqrt{2}}\,,$ (1)
$\displaystyle
m_{b}\sim\lambda^{2.5}\frac{v_{\mathrm{SM}}}{\sqrt{2}}\,,\qquad$
$\displaystyle
m_{s}\sim\lambda^{5.0}\frac{v_{\mathrm{SM}}}{\sqrt{2}}\,,\qquad$
$\displaystyle m_{d}\sim\lambda^{7.0}\frac{v_{\mathrm{SM}}}{\sqrt{2}}\,,$ (2)
$\displaystyle
m_{\tau}\sim\lambda^{3.0}\frac{v_{\mathrm{SM}}}{\sqrt{2}}\,,\qquad$
$\displaystyle
m_{\mu}\sim\lambda^{4.9}\frac{v_{\mathrm{SM}}}{\sqrt{2}}\,,\qquad$
$\displaystyle m_{e}\sim\lambda^{8.4}\frac{v_{\mathrm{SM}}}{\sqrt{2}}\,,$ (3)

where $v_{\mathrm{SM}}\simeq 246\,\mathrm{GeV}$ is the SM vacuum expectation
value (VEV) and $\lambda\simeq 0.224$ is the Wolfenstein parameter which
parameterises the CKM matrix as

$\displaystyle V_{us}\sim\lambda\,,\qquad$ $\displaystyle
V_{cb}\sim\lambda^{2}\,,\qquad$ $\displaystyle V_{ub}\sim\lambda^{3}\,.$ (4)

In addition, the discovery of very small neutrino masses and large PMNS mixing
deSalas:2020pgw ; Gonzalez-Garcia:2021dve ,

$\displaystyle\tan\theta_{23}\sim 1\,,\qquad$
$\displaystyle\tan\theta_{12}\sim\frac{1}{\sqrt{2}}\,,\qquad$
$\displaystyle\theta_{13}\sim\frac{\lambda}{\sqrt{2}}\,,$ (5)

has made the flavour puzzle difficult to ignore.

A tentative approach to explain the flavour puzzle consists of embedding the
SM in a larger gauge symmetry that contains a separate gauge group for each
fermion family, with the light Higgs doublet(s) originating from the third
family group. This general idea is sometimes denoted in the literature as
either “gauge non-universal theories of flavour” or “flavour deconstructed
theories”. Different possibilities have been discussed since the early 80s
Salam:1979p ; Rajpoot:1980ib ; Georgi:1981gj ; Li:1981nk ; Ma:1987ds ;
Ma:1988dn ; Li:1992fi ; Hill:1994hp ; Barbieri:1994cx ; Carone:1995ge ;
Muller:1996dj ; Malkawi:1996fs ; Dvali:2000ha ; Asaka:2004ry ; Babu:2007mb ,
with new ideas appearing also in recent times Craig:2011yk ; Panico:2016ull ;
Bordone:2017bld ; Greljo:2018tuh ; Fuentes-Martin:2020bnh ; Allwicher:2020esa
; Fuentes-Martin:2020pww ; Fuentes-Martin:2022xnb ; Davighi:2022fer ;
Davighi:2022bqf ; Davighi:2023iks ; FernandezNavarro:2023rhv ; Davighi:2023evx
; FernandezNavarro:2023hrf ; Davighi:2023xqn ; Capdevila:2024gki ;
Barbieri:2023qpf ; Fuentes-Martin:2024fpx . Remarkably, these theories
distinguish the three fermion families (hence avoiding family replication) and
are able to explain the flavour structure of the SM with all fundamental
couplings being of $\mathcal{O}(1)$. Out of them, arguably the simplest one
involves just assigning one separate gauged weak hypercharge to each fermion
family FernandezNavarro:2023rhv

$SU(3)_{c}\times SU(2)_{L}\times U(1)_{Y_{1}}\times U(1)_{Y_{2}}\times
U(1)_{Y_{3}}\,.$ (6)

Under the extended gauge group in Eq. (6) the $i$-th fermion family only
carries $Y_{i}$ hypercharge, with the other hypercharges set equal to zero
(see Table 1), where $Y=Y_{1}+Y_{2}+Y_{3}$ is equal to the usual SM weak
gauged hypercharge. Anomalies cancel separately for each family, as in the SM,
but without family replication. Two Higgs doublets are introduced which carry
third family hypercharge, $H^{u,d}_{3}\sim(\mathbf{1,2})_{(0,0,\pm 1/2)}$,
leading to renormalisable Yukawa couplings only involving the third family.
Yukawa couplings involving the first and second families are generated only
after the gauge symmetry is broken to the SM gauge group, leading to a
hierarchical pattern of Yukawa couplings.

The so-called “tri-hypercharge” (TH) gauge group in Eq. (6) is broken down to
the SM gauge group by the VEVs of appropriate scalars that carry family
hypercharges which add up to zero. The breaking chain

$\displaystyle SU(3)_{c}\times SU(2)_{L}\times U(1)_{Y_{1}}\times
U(1)_{Y_{2}}\times U(1)_{Y_{3}}$ (7)
$\displaystyle\xrightarrow{v_{12}}SU(3)_{c}\times SU(2)_{L}\times
U(1)_{Y_{1}+Y_{2}}\times U(1)_{Y_{3}}$ (8)
$\displaystyle\xrightarrow{v_{23}}SU(3)_{c}\times SU(2)_{L}\times
U(1)_{Y_{1}+Y_{2}+Y_{3}}\,.$ (9)

leads to a successful generation of charged fermion mass hierarchies and quark
mixing. There generally exists a hierarchy between $v_{12}$ and $v_{23}$ that
is related to the origin of the fermion mass hierarchies, as we shall see.
Typically $v_{23}\sim\mathcal{O}(1-10\,\mathrm{TeV})$ and
$v_{12}\sim\mathcal{O}(100-10^{4}\,\mathrm{TeV})$ are allowed by current data.

In FernandezNavarro:2023rhv  two of us provided a general recipe for model
building in the tri-hypercharge framework based on a spurion formalism, we
proposed simple models in an effective field theory (EFT) framework (without
specifying the heavy messengers that complete the theory) and studied the main
phenomenological predictions of the setup, which are connected to heavy gauge
bosons that arise after the spontaneous breaking of the TH group. Then, in
FernandezNavarro:2023hrf  the three of us built an ultraviolet (UV) complete
tri-hypercharge theory that arises from a gauge unified framework based on
assigning a separate $SU(5)$ gauge group to each fermion family, with the
three $SU(5)$s related by a cyclic $\mathbb{Z}_{3}$ permutation symmetry that
enforces a single gauge coupling at the grand unification (GUT) scale.

In this paper we provide two examples of minimal but ultraviolet complete tri-
hypercharge models that contain the minimal amount of total degrees of freedom
and representations, being amongst the most minimal models in the literature
Salam:1979p ; Rajpoot:1980ib ; Georgi:1981gj ; Li:1981nk ; Ma:1987ds ;
Ma:1988dn ; Li:1992fi ; Hill:1994hp ; Barbieri:1994cx ; Carone:1995ge ;
Muller:1996dj ; Malkawi:1996fs ; Dvali:2000ha ; Asaka:2004ry ; Babu:2007mb ;
Craig:2011yk ; Panico:2016ull ; Bordone:2017bld ; Greljo:2018tuh ; Fuentes-
Martin:2020bnh ; Allwicher:2020esa ; Fuentes-Martin:2020pww ; Fuentes-
Martin:2022xnb ; Davighi:2022fer ; Davighi:2022bqf ; Davighi:2023iks ;
FernandezNavarro:2023rhv ; Davighi:2023evx ; FernandezNavarro:2023hrf ;
Davighi:2023xqn ; Capdevila:2024gki ; Barbieri:2023qpf ; Fuentes-
Martin:2024fpx . In particular, the rank of the tri-hypercharge symmetry is
the most minimal. We show that both models, which differ only by the heavy
messengers that complete the effective theory, are able to explain the
observed patterns of fermion masses and mixings (including neutrinos) with all
fundamental coefficients being of $\mathcal{O}(1)$.

In fact, we shall see that the two minimal models presented here translate the
complicated flavour structure of the Standard Model into three simple physical
scales above electroweak symmetry breaking, completely correlated with each
other and with the flavour hierarchies of the SM, that carry meaningful
phenomenology. In this regard, we find that the heavy messengers crucially
dictate the origin and size of fermion mixing, which controls the size and
nature of the flavour-violating currents mediated by the two heavy
$Z^{\prime}$ gauge bosons of the theory. The phenomenological implications of
the two minimal models are compared, where the lightest $Z^{\prime}$ is
discoverable in dilepton searches at the LHC Run 3 in both cases.

The layout of the paper is as follows. In Section 2 we first introduce the
tri-hypercharge theory from a bottom-up perspective and an EFT point of view,
and then we show two minimal UV-complete models. In the model of Section 2.1
all the heavy messengers that complete the effective theory are vector-like
fermions (hence having the simplest scalar sector and potential), whereas the
model of Section 2.2 offers a combination of heavy Higgs doublets and vector-
like fermions that contains the minimal amount of degrees of freedom and
representations. In Section 3 we discuss the neutrino sector that is common
for both complete models. Then in Section 4 we discuss the model-dependent
phenomenology associated to the two complete models, which is mostly dominated
by flavour-changing neutral currents (FCNCs). Finally, Section 5 concludes the
paper. Additional technical details are given in three separate appendices.

Field | $U(1)_{Y_{1}}$ | $U(1)_{Y_{2}}$ | $U(1)_{Y_{3}}$ | $SU(3)_{c}\times SU(2)_{L}$
---|---|---|---|---
$q_{1}$ | $1/6$ | 0 | 0 | $(\mathbf{3,2})$
$u_{1}^{c}$ | $-2/3$ | 0 | 0 | $(\mathbf{\overline{3},1})$
$d_{1}^{c}$ | $1/3$ | 0 | 0 | $(\mathbf{\overline{3},1})$
$\ell_{1}$ | $-1/2$ | 0 | 0 | $(\mathbf{1,2})$
$e_{1}^{c}$ | $1$ | 0 | 0 | $(\mathbf{1,1})$
$q_{2}$ | 0 | $1/6$ | 0 | $(\mathbf{3,2})$
$u_{2}^{c}$ | 0 | $-2/3$ | 0 | $(\mathbf{\overline{3},1})$
$d_{2}^{c}$ | 0 | $1/3$ | 0 | $(\mathbf{\overline{3},1})$
$\ell_{2}$ | 0 | $-1/2$ | 0 | $(\mathbf{1,2})$
$e_{2}^{c}$ | 0 | $1$ | 0 | $(\mathbf{1,1})$
$q_{3}$ | 0 | 0 | $1/6$ | $(\mathbf{3,2})$
$u_{3}^{c}$ | 0 | 0 | $-2/3$ | $(\mathbf{\overline{3},1})$
$d_{3}^{c}$ | 0 | 0 | $1/3$ | $(\mathbf{\overline{3},1})$
$\ell_{3}$ | 0 | 0 | $-1/2$ | $(\mathbf{1,2})$
$e_{3}^{c}$ | 0 | 0 | $1$ | $(\mathbf{1,1})$
$\nu_{1}^{c}$ | 0 | 0 | 0 | $(\mathbf{1,1})$
$\nu_{2}^{c}$ | 0 | 0 | 0 | $(\mathbf{1,1})$
$H_{3}^{u}$ | 0 | 0 | $1/2$ | $(\mathbf{1,2})$
$H_{3}^{d}$ | 0 | 0 | $-1/2$ | $(\mathbf{1,2})$
$\phi_{q12}$ | $-1/6$ | $1/6$ | 0 | $(\mathbf{1,1})$
$\phi_{\ell 12}$ | $1/2$ | $-1/2$ | 0 | $(\mathbf{1,1})$
$\phi_{q23}$ | 0 | $-1/6$ | $1/6$ | $(\mathbf{1,1})$
$\phi_{\ell 23}$ | 0 | $1/2$ | $-1/2$ | $(\mathbf{1,1})$
Table 1: Field content of the low-energy (EFT) tri-hypercharge theory. $q_{i}$
and $\ell_{i}$ (where $i=1,2,3$) are left-handed (LH) $SU(2)_{L}$ doublets of
chiral quarks and leptons, while $u_{i}^{c}$, $d_{i}^{c}$ and $e_{i}^{c}$ are
the CP conjugated right-handed (RH) quarks and leptons (so that they become
left-handed). Two singlet neutrinos $\nu_{1,2}^{c}$, also taken as left-handed
fields, will be relevant for the origin of neutrino masses. $H_{3}^{u,d}$ are
$SU(2)_{L}$ Higgs doublets, while $\phi_{q12,\ell 12,q23,\ell 23}$ are the so-
called scalar hyperons which spontaneously break the three gauge hypercharges
down to SM hypercharge (the diagonal subgroup $Y=Y_{1}+Y_{2}+Y_{3}$).

##  2 Minimal Models

We consider that at relatively high energies the gauge symmetry contains one
hypercharge abelian factor for each fermion family. These are spontaneously
broken via the VEVs of $\phi$ scalars (see Table 1) carrying non-zero family
hypercharges that add up to zero, so that they are gauge singlets under the SM
symmetry, and are denoted as hyperons,

$\displaystyle SU(3)_{c}\times SU(2)_{L}\times U(1)_{Y_{1}}\times
U(1)_{Y_{2}}\times U(1)_{Y_{3}}$ (10)
$\displaystyle\overset{\langle\phi_{q_{12}}\rangle,\left\langle\phi_{\ell
12}\right\rangle}{\longrightarrow}SU(3)_{c}\times SU(2)_{L}\times
U(1)_{Y_{1}+Y_{2}}\times U(1)_{Y_{3}}$ (11)
$\displaystyle\overset{\left\langle\phi_{q23}\right\rangle,\left\langle\phi_{\ell
23}\right\rangle}{\longrightarrow}SU(3)_{c}\times SU(2)_{L}\times
U(1)_{Y_{1}+Y_{2}+Y_{3}}\,.$ (12)

At renormalisable level, only third family Yukawa couplings are allowed. Two
(third family) Higgs doublets $H_{3}^{u,d}\sim(\mathbf{1,2})_{(0,0,\pm 1/2)}$
perform spontaneous breaking of the electroweak symmetry, and explain the mass
hierarchies $m_{b,\tau}/m_{t}$ if we assume a type-II two Higgs doublet model
(2HDM) with $\tan\beta=\langle H_{3}^{u}\rangle/\langle
H_{3}^{d}\rangle\sim\lambda^{-2}\approx 20$.

The masses of the light charged fermions arise from non-renormalisable
operators involving the hyperon scalars. With the four111For a discussion
about models with less than four hyperons, we refer the reader to Appendix B.
hyperons of Table 1, one can generate all the spurions222In particular, we
obtain the spurions $\eta\sim(-\frac{1}{6},-\frac{1}{3},\frac{1}{2})$ and
$\hat{\eta}\sim(-\frac{1}{6},0,\frac{1}{6})$, needed to generate quark mixing,
by combining the four hyperons of Table 1 as
$\eta\sim\phi_{q12}\tilde{\phi}_{\ell 23}$ and
$\hat{\eta}\sim\phi_{q12}\phi_{q23}$. introduced in Ref.
FernandezNavarro:2023rhv , which populate the Yukawa matrices. By working in
an EFT framework, we can write (ignoring $\mathcal{O}(1)$ dimensionless
coefficients),

$\displaystyle{\cal L}$
$\displaystyle=\begin{pmatrix}q_{1}&q_{2}&q_{3}\end{pmatrix}\begin{pmatrix}{\displaystyle\frac{\phi_{\ell
12}}{\Lambda_{1}}\frac{{\phi}_{\ell
23}}{\Lambda_{2}}}&{\displaystyle\frac{{\phi}_{q12}}{\Lambda_{1}}\frac{{\phi}_{\ell
23}}{\Lambda_{2}}}&{\displaystyle\frac{{\phi}_{q12}}{\Lambda_{1}}\frac{{\phi}_{q23}}{\Lambda_{2}}}\\\
\vspace{-0.14cm}&\vspace{-0.14cm}&\vspace{-0.14cm}\\\
{\displaystyle\frac{\phi_{\ell
12}}{\Lambda_{1}}\frac{{\widetilde{\phi}}_{q12}}{\Lambda_{1}}\frac{{\phi}_{\ell
23}}{\Lambda_{2}}}&{\displaystyle\frac{{\phi}_{\ell
23}}{\Lambda_{2}}}&{\displaystyle\frac{{\phi}_{q23}}{\Lambda_{2}}}\\\
\vspace{-0.14cm}&\vspace{-0.14cm}&\vspace{-0.14cm}\\\
{\displaystyle\frac{\phi_{\ell 12}}{\Lambda_{1}}\frac{{\phi}_{\ell
23}}{\Lambda_{1}}}{\displaystyle\frac{{\widetilde{\phi}}_{q23}}{\Lambda_{2}}}&{\displaystyle\frac{{\phi}_{\ell
23}}{\Lambda_{2}}}{\displaystyle\frac{{\widetilde{\phi}}_{q23}}{\Lambda_{2}}}&1\end{pmatrix}\begin{pmatrix}u_{1}^{c}\\\
u_{2}^{c}\\\ u_{3}^{c}\end{pmatrix}H_{3}^{u}$ (13)
$\displaystyle+\begin{pmatrix}q_{1}&q_{2}&q_{3}\end{pmatrix}\begin{pmatrix}{\displaystyle\frac{\tilde{\phi}_{\ell
12}}{\Lambda_{1}}\frac{\tilde{\phi}_{\ell
23}}{\Lambda_{2}}}&{\displaystyle\frac{\phi_{q12}}{\Lambda_{1}}\frac{\tilde{\phi}_{\ell
23}}{\Lambda_{2}}}&{\displaystyle\frac{\phi_{q12}}{\Lambda_{1}}\frac{{\phi}_{q23}}{\Lambda_{2}}}\\\
\vspace{-0.14cm}&\vspace{-0.14cm}&\vspace{-0.14cm}\\\
{\displaystyle\frac{\tilde{\phi}_{\ell
12}}{\Lambda_{1}}\frac{{\widetilde{\phi}}_{q12}}{\Lambda_{1}}\frac{\tilde{\phi}_{\ell
23}}{\Lambda_{1}}}&{\displaystyle\frac{\tilde{\phi}_{\ell
23}}{\Lambda_{2}}}&{\displaystyle\frac{{\phi}_{q23}}{\Lambda_{2}}}\\\
\vspace{-0.14cm}&\vspace{-0.14cm}&\vspace{-0.14cm}\\\
{\displaystyle\frac{\phi_{q12}^{2}}{\Lambda_{1}^{2}}\frac{{\phi}_{q23}^{2}}{\Lambda_{2}^{2}}}&{\displaystyle\frac{{\phi}_{q23}^{2}}{\Lambda_{2}^{2}}}&1\end{pmatrix}\begin{pmatrix}d_{1}^{c}\\\
d_{2}^{c}\\\ d_{3}^{c}\end{pmatrix}H_{3}^{d}$ (14)
$\displaystyle+\begin{pmatrix}\ell_{1}&\ell_{2}&\ell_{3}\end{pmatrix}\begin{pmatrix}{\displaystyle\frac{\tilde{\phi}_{\ell
12}}{\Lambda_{1}}\frac{\tilde{\phi}_{\ell
23}}{\Lambda_{2}}}&{\displaystyle\frac{\phi_{\ell
12}}{\Lambda_{1}}\frac{\tilde{\phi}_{\ell
23}}{\Lambda_{2}}}&{\displaystyle\frac{\phi_{\ell
12}}{\Lambda_{1}}\frac{\phi_{\ell 23}}{\Lambda_{2}}}\\\
\vspace{-0.14cm}&\vspace{-0.14cm}&\vspace{-0.14cm}\\\
{\displaystyle\frac{\tilde{\phi}_{\ell
12}}{\Lambda_{1}}\frac{\tilde{\phi}_{\ell
12}}{\Lambda_{1}}\frac{\tilde{\phi}_{\ell
23}}{\Lambda_{2}}}&{\displaystyle\frac{\tilde{\phi}_{\ell
23}}{\Lambda_{2}}}&{\displaystyle\frac{\phi_{\ell 23}}{\Lambda_{2}}}\\\
\vspace{-0.14cm}&\vspace{-0.14cm}&\vspace{-0.14cm}\\\
{\displaystyle\frac{\tilde{\phi}_{\ell
12}^{2}}{\Lambda_{1}}\frac{\tilde{\phi}_{\ell
23}^{2}}{\Lambda_{2}}}&{\displaystyle\frac{\tilde{\phi}_{\ell
23}^{2}}{\Lambda_{2}^{2}}}&1\end{pmatrix}\begin{pmatrix}e_{1}^{c}\\\
e_{2}^{c}\\\ e_{3}^{c}\end{pmatrix}H_{3}^{d}\,+\mathrm{h.c.}$ (15)

Notice that the 23-hyperons generate the mass hierarchy $m_{2}/m_{3}$ along
with naturally suppressed $V_{cb}$ and $V_{ub}$ CKM elements. Then, the
12-hyperons discriminate the first family, generating the mass hierarchy
$m_{1}/m_{2}$, the Cabbibo angle $V_{us}$ and an extra suppression for
$V_{ub}$ with respect to $V_{cb}$, while the origin of small neutrino masses
and PMNS mixing is later discussed in the complete model, see Section 3.

In the following we UV-complete the above EFT by including heavy messengers
explicitly. Their masses provide the heavy scales $\Lambda_{1}$ and
$\Lambda_{2}$. Notice that we do not need to generate all the entries in the
Yukawa matrices of Eqs. (13-15) to explain the flavour structure of the SM,
and indeed some of them will not be generated at tree-level in our complete
models.

In the next two Sections we will show two minimal UV completions, one where
all the heavy messengers are vector-like (VL) fermions (hence having the
simplest scalar sector and potential), and another one with a combination of
heavy Higgs doublets and VL fermions that contains the minimal amount of
degrees of freedom and representations.

###  2.1 Model 1

Our first proposal for a minimal, complete and renormalisable model includes
three vector-like $SU(2)_{L}$ singlet fermions for each charged sector, as
shown in Table 2, that act as heavy messengers of the effective Yukawa
operators introduced before. The full set of renormalisable Yukawa couplings
and bare mass terms in the charged fermion sector of the model is given by

$\mathcal{L}_{Y}^{u}=\left(\begin{array}[]{c|cccccc}&u_{1}^{c}&u_{2}^{c}&u_{3}^{c}&U_{12}&U_{13}&U_{23}\\\
\hline\cr q_{1}&0&0&0&0&y_{15}^{u}H_{3}^{u}&0\\\
q_{2}&0&0&0&0&0&y_{26}^{u}H_{3}^{u}\\\ q_{3}&0&0&y_{3}^{u}H_{3}^{u}&0&0&0\\\
\overline{U}_{12}&y_{41}^{u}\phi_{\ell
12}&y_{42}^{u}\phi_{q12}&0&M_{U_{12}}&y_{45}^{u}\tilde{\phi}_{\ell 23}&0\\\
\overline{U}_{13}&0&0&0&y_{54}^{u}\phi_{\ell
23}&M_{U_{13}}&y_{56}^{u}\phi_{q12}\\\
\overline{U}_{23}&0&y_{62}^{u}\phi_{\ell
23}&y_{63}^{u}\phi_{q23}&0&y_{65}^{u}\tilde{\phi}_{q12}&M_{U_{23}}\end{array}\right)+\mathrm{h.c.}$
(16)
$\mathcal{L}_{Y}^{d}=\left(\begin{array}[]{c|cccccc}&d_{1}^{c}&d_{2}^{c}&d_{3}^{c}&D_{12}&D_{13}&D_{23}\\\
\hline\cr q_{1}&0&0&0&0&y_{15}^{d}H_{3}^{d}&0\\\
q_{2}&0&0&0&0&0&y_{26}^{d}H_{3}^{d}\\\ q_{3}&0&0&y_{3}^{d}H_{3}^{d}&0&0&0\\\
\overline{D}_{12}&y_{41}^{d}\tilde{\phi}_{\ell
12}&y_{42}^{d}\phi_{q12}&0&M_{D_{12}}&y_{45}^{d}\phi_{\ell 23}&0\\\
\overline{D}_{13}&0&0&0&y_{54}^{d}\tilde{\phi}_{\ell
23}&M_{D_{13}}&y_{56}^{d}\phi_{q12}\\\
\overline{D}_{23}&0&y_{62}^{d}\tilde{\phi}_{\ell
23}&y_{63}^{d}\phi_{q23}&0&y_{65}^{u}\tilde{\phi}_{q12}&M_{D_{23}}\end{array}\right)+\mathrm{h.c.}$
(17)
$\mathcal{L}_{Y}^{e}=\left(\begin{array}[]{c|cccccc}&e_{1}^{c}&e_{2}^{c}&e_{3}^{c}&E_{12}&E_{13}&E_{23}\\\
\hline\cr\ell_{1}&0&0&0&0&y_{15}^{e}H_{3}^{d}&0\\\
\ell_{2}&0&0&0&0&0&y_{26}^{e}H_{3}^{d}\\\
\ell_{3}&0&0&y_{3}^{e}H_{3}^{d}&0&0&0\\\
\overline{E}_{12}&y_{41}^{e}\tilde{\phi}_{\ell 12}&y_{42}^{e}\phi_{\ell
12}&0&M_{E_{12}}&y_{45}^{e}\phi_{\ell 23}&0\\\
\overline{E}_{13}&0&0&0&y_{54}^{e}\tilde{\phi}_{\ell
23}&M_{E_{13}}&y_{56}^{e}\phi_{\ell 12}\\\
\overline{E}_{23}&0&y_{62}^{e}\tilde{\phi}_{\ell 23}&y_{63}^{e}\phi_{\ell
23}&0&y_{65}^{e}\tilde{\phi}_{\ell
12}&M_{E_{23}}\end{array}\right)+\mathrm{h.c.}$ (18)

Field | $U(1)_{Y_{1}}$ | $U(1)_{Y_{2}}$ | $U(1)_{Y_{3}}$ | $SU(3)_{c}\times SU(2)_{L}$
---|---|---|---|---
$H_{3}^{u,d}$ | 0 | 0 | $\pm 1/2$ | $(\mathbf{1,2})$
$\phi_{q_{12}}$ | -1/6 | 1/6 | 0 | $(\mathbf{1,1})$
$\phi_{\ell_{12}}$ | 1/2 | -1/2 | 0 | $(\mathbf{1,1})$
$\phi_{q_{23}}$ | 0 | -1/6 | 1/6 | $(\mathbf{1,1})$
$\phi_{\ell_{23}}$ | 0 | 1/2 | -1/2 | $(\mathbf{1,1})$
$U_{12}$ | -1/6 | -1/2 | 0 | $(\mathbf{\overline{3},1})$
$U_{13}$ | -1/6 | 0 | -1/2 | $(\mathbf{\overline{3},1})$
$U_{23}$ | 0 | -1/6 | -1/2 | $(\mathbf{\overline{3},1})$
$D_{12}$ | -1/6 | 1/2 | 0 | $(\mathbf{\overline{3},1})$
$D_{13}$ | -1/6 | 0 | 1/2 | $(\mathbf{\overline{3},1})$
$D_{23}$ | 0 | -1/6 | 1/2 | $(\mathbf{\overline{3},1})$
$E_{12}$ | 1/2 | 1/2 | 0 | $(\mathbf{1,1})$
$E_{13}$ | 1/2 | 0 | 1/2 | $(\mathbf{1,1})$
$E_{23}$ | 0 | 1/2 | 1/2 | $(\mathbf{1,1})$
$N_{12}$ | 1/2 | -1/2 | 0 | $(\mathbf{1,1})$
$N_{13}$ | 1/2 | 0 | -1/2 | $(\mathbf{1,1})$
$N_{23}$ | 0 | 1/2 | -1/2 | $(\mathbf{1,1})$
Table 2: Scalar and vector-like fermion content of Model 1. Notice that for
vector-like fermions (highlighted in yellow), only the left-handed fields
$U,D,E,N$, with the same electric charges as the SM fermions
$u^{c},d^{c},e^{c},\nu^{c}$ in Table 1, are displayed, while their vector-like
partners with conjugate quantum numbers are implicitly assumed to be present
but not explicitly shown.

After integrating out the heavy VL charged fermions, the effective Yukawa
couplings for chiral charged fermions take the form

$\displaystyle{\cal L}$
$\displaystyle=\begin{pmatrix}q_{1}&q_{2}&q_{3}\end{pmatrix}\begin{pmatrix}c_{11}^{u}{\displaystyle\frac{\phi_{\ell
12}}{M_{U_{13}}}\frac{{\phi}_{\ell
23}}{M_{U_{12}}}}&c_{12}^{u}{\displaystyle\frac{{\phi}_{q12}}{M_{U_{13}}}\frac{{\phi}_{\ell
23}}{M_{U_{23}}}}&{\displaystyle
c_{13}^{u}\frac{{\phi}_{q12}}{M_{U_{13}}}\frac{{\phi}_{q23}}{M_{U_{23}}}}\\\
\vspace{-0.14cm}&\vspace{-0.14cm}&\vspace{-0.14cm}\\\
c_{21}^{u}{\displaystyle\frac{\phi_{\ell
12}}{M_{U_{12}}}}{\displaystyle\frac{\phi_{q12}}{M_{U_{13}}}}{\displaystyle\frac{\phi_{\ell
23}}{M_{U_{23}}}}&c_{22}^{u}{\displaystyle\frac{{\phi}_{\ell
23}}{M_{U_{23}}}}&c_{23}^{u}{\displaystyle\frac{{\phi}_{q23}}{M_{U_{23}}}}\\\
\vspace{-0.14cm}&\vspace{-0.14cm}&\vspace{-0.14cm}\\\
0&0&c_{33}^{u}\end{pmatrix}\begin{pmatrix}u_{1}^{c}\\\ u_{2}^{c}\\\
u_{3}^{c}\end{pmatrix}H_{3}^{u}$ (19)
$\displaystyle+\begin{pmatrix}q_{1}&q_{2}&q_{3}\end{pmatrix}\begin{pmatrix}c_{11}^{d}{\displaystyle\frac{\tilde{\phi}_{\ell
12}}{M_{D_{13}}}\frac{\tilde{\phi}_{\ell
23}}{M_{D_{12}}}}&c_{12}^{d}{\displaystyle\frac{\phi_{q12}}{M_{D_{13}}}\frac{\tilde{\phi}_{\ell
23}}{M_{D_{23}}}}&c_{13}^{d}{\displaystyle\frac{\phi_{q12}}{M_{D_{13}}}\frac{{\phi}_{q23}}{M_{D_{23}}}}\\\
\vspace{-0.14cm}&\vspace{-0.14cm}&\vspace{-0.14cm}\\\
c_{21}^{d}{\displaystyle\frac{\tilde{\phi}_{\ell
12}}{M_{D_{12}}}}{\displaystyle\frac{\tilde{\phi}_{q12}}{M_{D_{13}}}}{\displaystyle\frac{\tilde{\phi}_{\ell
23}}{M_{D_{23}}}}&c_{22}^{d}{\displaystyle\frac{\tilde{\phi}_{\ell
23}}{M_{D_{23}}}}&c_{23}^{d}{\displaystyle\frac{{\phi}_{q23}}{M_{D_{23}}}}\\\
\vspace{-0.14cm}&\vspace{-0.14cm}&\vspace{-0.14cm}\\\
0&0&c_{33}^{d}\end{pmatrix}\begin{pmatrix}d_{1}^{c}\\\ d_{2}^{c}\\\
d_{3}^{c}\end{pmatrix}H_{3}^{d}$ (20)
$\displaystyle+\begin{pmatrix}\ell_{1}&\ell_{2}&\ell_{3}\end{pmatrix}\begin{pmatrix}c_{11}^{e}{\displaystyle\frac{\tilde{\phi}_{\ell
12}}{M_{E_{13}}}\frac{\tilde{\phi}_{\ell
23}}{M_{E_{12}}}}&{\displaystyle{\displaystyle c_{12}^{e}\frac{\phi_{\ell
12}}{M_{E_{13}}}\frac{\tilde{\phi}_{\ell
23}}{M_{E_{23}}}}}&c_{13}^{e}{\displaystyle{\displaystyle\frac{\phi_{\ell
12}}{M_{E_{13}}}\frac{{\phi}_{\ell 23}}{M_{E_{23}}}}}\\\
\vspace{-0.14cm}&\vspace{-0.14cm}&\vspace{-0.14cm}\\\
c_{21}^{e}{\displaystyle\frac{\tilde{\phi}_{\ell
12}}{M_{E_{12}}}}{\displaystyle\frac{\tilde{\phi}_{\ell
12}}{M_{E_{13}}}}{\displaystyle\frac{\tilde{\phi}_{\ell
23}}{M_{E_{23}}}}&c_{22}^{e}{\displaystyle\frac{\tilde{\phi}_{\ell
23}}{M_{E_{23}}}}&{\displaystyle c_{23}^{e}\frac{\phi_{\ell
23}}{M_{E_{23}}}}\\\ \vspace{-0.14cm}&\vspace{-0.14cm}&\vspace{-0.14cm}\\\
0&0&c_{33}^{e}\end{pmatrix}\begin{pmatrix}e_{1}^{c}\\\ e_{2}^{c}\\\
e_{3}^{c}\end{pmatrix}H_{3}^{d}\,+\mathrm{h.c.}\,,$ (21)

where

$c^{u}=\begin{pmatrix}y_{15}^{u}y_{54}^{u}y_{41}^{u}&y_{15}^{u}y_{56}^{u}y_{62}^{u}&{\displaystyle
y_{15}^{u}y_{56}^{u}y_{63}^{u}}\\\
y_{26}^{u}y_{65}^{u}y_{54}^{u}y_{41}^{u}&y_{26}^{u}y_{62}^{u}&y_{26}^{u}y_{63}^{u}\\\
0&0&y_{3}^{u}\end{pmatrix}\,,$ (22)

$H^{d}_{3}$$\tilde{\phi}_{\ell
23}$$q_{2}$$D_{23}$$\overline{D}_{23}$$d_{2}^{c}$

(a)

$H^{d}_{3}$$\tilde{\phi}_{\ell 23}$$\tilde{\phi}_{\ell
12}$$q_{1}$$D_{13}$$\overline{D}_{13}$$D_{12}$$\overline{D}_{12}$$d_{1}^{c}$

(b)

$H^{d}_{3}$$\phi_{q23}$$q_{2}$$D_{23}$$\overline{D}_{23}$$d_{3}^{c}$

(c)

$H^{d}_{3}$$\phi_{q12}$$\tilde{\phi}_{\ell
23}$$q_{1}$$D_{13}$$\overline{D}_{13}$$D_{23}$$\overline{D}_{23}$$d_{2}^{c}$

(d)

Figure 1: Diagrams at the origin of light down-quark masses and mixing in
Model 1. Similar diagrams are obtained in the up-sector by replacing
$d_{i}^{c}\rightarrow u_{i}^{c}$, $D_{12,13,23}\rightarrow U_{12,13,23}$,
$\tilde{\phi}_{\ell 12,23}\rightarrow\phi_{\ell 12,23}$ and
$H_{3}^{d}\rightarrow H_{3}^{u}$, and also for the charged lepton sector by
replacing $q_{i}\rightarrow\ell_{i}$, $d_{i}^{c}\rightarrow e_{i}^{c}$,
$D_{12,13,23}\rightarrow E_{12,13,23}$ and $\phi_{q12,23}\rightarrow\phi_{\ell
12,23}$.

and similarly for the down and charged lepton sectors by replacing
$u\rightarrow d,e$. In terms of Feynman diagrams, the effective Yukawa
couplings are generated by mass insertions of the VL fermions as shown in Fig.
1, with an extra (welcomed) mass insertion in all the diagrams involving the
first family. Once the hyperons develop VEVs, the effective Yukawa matrices
above generate naturally small Yukawa couplings for light charged fermions.
Now we shall fix the ratios of hyperon VEVs over messenger masses in order to
explain the flavour structure of the SM.

Firstly, notice how the 23-messengers always appear together with the
23-hyperons in the Yukawa matrices, and similarly the 12,13-messengers which
distinguish the first family always appear together with 12-hyperons. This
suggests the presence of one scale at which all the 23-messengers live - that
we associate with $\Lambda_{2}$ from the EFT formalism - and another, heavier
scale where all the 12,13-messengers live - that we associate with
$\Lambda_{1}$. The exception is the ratio $\phi_{\ell 23}/M_{12}$ appearing in
the (1,1) entries, which provides a connection between both scales. This is a
highly non-generic feature in this class of models, which was not anticipated
by the EFT framework.

In fact, assuming for simplicity the two 23-VEVs to be degenerate and denoted
generically as $\langle\phi_{23}\rangle$, the two 12-VEVs to be degenerate and
denoted as $\langle\phi_{12}\rangle$, the 23-messenger masses to be degenerate
and denoted as $M_{23}$ and the 12,13-messenger masses to be degenerate and
denoted as $M_{12,13}$, then we find that the approximate scaling

$\langle\phi_{23}\rangle\ll M_{23}\sim\langle\phi_{12}\rangle\ll M_{12}\sim
M_{13}$ (23)

provides an excellent explanation of the SM flavour structure, which is
translated to three simple and physical scales of new physics (NP) above
electroweak symmetry breaking, as shown in Fig. 2. Indeed, we can explain the
whole flavour structure of the SM in terms of just three small parameters,

$\frac{\langle\phi_{23}\rangle}{M_{23}}\sim\lambda^{3}\,,\qquad\frac{\langle\phi_{12}\rangle}{M_{12,13}}\sim\lambda\,,\qquad\frac{\langle\phi_{23}\rangle}{M_{12,13}}\sim\lambda^{5}\,.$
(24)

The third ratio $\langle\phi_{23}\rangle/M_{12,13}\sim\lambda^{5}$, which only
appears in the (1,1) entry of the Yukawa matrices, provides an extra
(welcomed) suppression for first family fermion masses. Moreover, this
predicts a highly non-generic333Notice that this is a highly non-generic
feature in this class of models. Typically, alternative theories only contain
ratios of the form $\langle\phi_{23}\rangle/M_{23}$ and
$\langle\phi_{12}\rangle/M_{12}$ with no connection between the 23 and 12
VEVs. relation between the two VEV scales as follows (up to $\mathcal{O}(1)$
variation),

$\langle\phi_{23}\rangle/\langle\phi_{12}\rangle=v_{23}/v_{12}\sim\lambda^{4}\,,$
(25)

and also, together with the first ratio of Eq. (24), it suggests that
$M_{23}\sim v_{12}$. A typical benchmark compatible with experimental searches
then is, as depicted in Fig. 2,

$v_{23}\sim\mathcal{O}(10\,\mathrm{TeV}),\quad v_{12}\sim
M_{U_{23},D_{23},E_{23}}\sim\mathcal{O}(10^{3}\,\mathrm{TeV}),\quad
M_{U_{12,13},D_{12,13},E_{12,13}}\sim\mathcal{O}(10^{4}\,\mathrm{TeV})\,,$
(26)

where, thanks to the non-generic relation of Eq. (25), testing experimentally
any of these scales effectively tests all of them (and in particular, setting
bounds on any of them also sets bounds on the rest). Notice that our model
then translates the complicated flavour structure of the SM into just three
simple and physical scales of new physics above electroweak symmetry breaking.

However, we do not expect the masses of different messengers nor the VEVs of
different hyperons to be exactly degenerate - they may naturally vary by
$\mathcal{O}(1)$ factors. Therefore, we take advantage of these natural
$\mathcal{O}(1)$ variations with respect to Eq. (24) to explain the slightly
more accentuated mass hierarchies of the up sector and the tinier mass of the
electron with respect to first family quarks. More specifically, we assume the
following numerical values for the ratios appearing in the Yukawa matrices,
which are consistent with $\mathcal{O}(1)$ variations from the spectrum of
Fig. 2 (or similarly $\mathcal{O}(1)$ variations from the three fixed ratios
of Eq. (24)),

${\displaystyle\frac{\langle{\phi}_{\ell
23}\rangle}{M_{U_{23}}}}\sim\lambda^{4}\,,\qquad{\displaystyle\frac{\langle{\phi}_{q23}\rangle}{M_{U_{23}}}}\sim\lambda^{3}\,,\qquad\frac{\langle{\phi}_{\ell
12}\rangle}{M_{U_{12,13}}}\sim\frac{\langle{\phi}_{q12}\rangle}{M_{U_{12,13}}}\sim\lambda^{2}\,,$
(27) ${\displaystyle\frac{\langle{\phi}_{\ell
23}\rangle}{M_{D_{23}}}}\sim\lambda^{3}\,,\qquad{\displaystyle\frac{\langle{\phi}_{q23}\rangle}{M_{D_{23}}}}\sim\lambda^{2}\,,\qquad\frac{\langle{\phi}_{\ell
12}\rangle}{M_{D_{12,13}}}\sim\frac{\langle{\phi}_{q12}\rangle}{M_{D_{12,13}}}\sim\lambda\,,$
(28) ${\displaystyle\frac{\langle{\phi}_{\ell
23}\rangle}{M_{E_{23}}}}\sim\lambda^{3}\,,\qquad\frac{\langle{\phi}_{\ell
12}\rangle}{M_{E_{12,13}}}\sim\lambda^{2}\,,$ (29)

and the (1,1) entries of the effective Yukawa matrices are further suppressed
by fixed ratios that involve 23-VEVs in the numerator and 12-masses in the
denominator,

${\displaystyle\frac{\langle{\phi}_{\ell
23}\rangle}{M_{U_{12,13}}}}\sim\lambda^{5}\,,\qquad{\displaystyle\frac{\langle{\phi}_{\ell
23}\rangle}{M_{D_{12,13}}}}\sim\lambda^{4}\,,\qquad{\displaystyle\frac{\langle{\phi}_{\ell
23}\rangle}{M_{E_{12,13}}}}\sim\lambda^{5}\,.$ (30)

Figure 2: Illustrative spectrum of NP scales in Models 1 and 2. Notice that
the hyperons $\phi$, third family Higgs doublets $H^{u,d}_{3}$, messenger
neutrinos $N$ and messenger up-type quarks $U$ are common to both models.

We then predict the approximate numerical textures (up to $\mathcal{O}(1)$
coefficients) for the effective Yukawa couplings

$\displaystyle{\cal L}$
$\displaystyle=\begin{pmatrix}q_{1}&q_{2}&q_{3}\end{pmatrix}\begin{pmatrix}{\displaystyle\lambda^{7}}&{\displaystyle\lambda^{6}}&\lambda^{5}\\\
\lambda^{8}&\lambda^{4}&\lambda^{3}\\\
0&0&1\end{pmatrix}\begin{pmatrix}u_{1}^{c}\\\ u_{2}^{c}\\\
u_{3}^{c}\end{pmatrix}v_{\mathrm{SM}}$ (31)
$\displaystyle+\begin{pmatrix}q_{1}&q_{2}&q_{3}\end{pmatrix}\begin{pmatrix}{\displaystyle\lambda^{5}}&{\displaystyle\lambda^{4}}&\lambda^{3}\\\
\lambda^{5}&\lambda^{3}&\lambda^{2}\\\
0&0&1\end{pmatrix}\begin{pmatrix}d_{1}^{c}\\\ d_{2}^{c}\\\
d_{3}^{c}\end{pmatrix}\lambda^{2}v_{\mathrm{SM}}$ (32)
$\displaystyle+\begin{pmatrix}\ell_{1}&\ell_{2}&\ell_{3}\end{pmatrix}\begin{pmatrix}\lambda^{7}&\lambda^{5}&\lambda^{5}\\\
\lambda^{7}&{\displaystyle\lambda^{3}}&{\displaystyle\lambda^{3}}\\\
0&0&1\end{pmatrix}\begin{pmatrix}e_{1}^{c}\\\ e_{2}^{c}\\\
e_{3}^{c}\end{pmatrix}\lambda^{2}v_{\mathrm{SM}}+\mathrm{h.c.}\,,$ (33)

which can explain the flavour structure of the SM with all fundamental
couplings being $\mathcal{O}(1)$.

Notice that according to the above textures, CKM mixing in this model
originates mostly from the down sector. However, if one assumes for simplicity
degenerate VEVs and masses as in Eq. (24), then the Yukawa textures would be
similar (up to an overall normalisation factor) in both the up and down
sectors - then the alignment of the CKM depends on the off-diagonal
$c_{ij}^{u,d}$ coefficients, as in the model of Ref. FernandezNavarro:2023hrf
. In this case, all the coefficients $c_{ij}^{u,d,e}$ would still be of
$\mathcal{O}(1)$, with the exception of $c_{11}^{u}$ and $c_{11}^{d}$ which
would need to be of $\mathcal{O}(0.1)$ and of $\mathcal{O}(4)$, respectively.
This is completely acceptable as $c_{11}^{u,d}$ are given as a product of
three fundamental couplings in the complete theory.

On the other hand, the prediction of significant 1-2 RH quark mixing,
especially in the down sector, is of great importance for the phenomenology of
the heaviest $Z^{\prime}_{12}$ boson. Similarly, the presence of CKM-like
mixing in the charged lepton sector implies that the two $Z^{\prime}$ bosons
of the model potentially mediate charged lepton flavour-violating processes
(CLFV) at acceptable rates, as discussed later in Section 4. Most of the PMNS
mixing is however generated from the neutrino sector, as discussed later in
Section 3.

###  2.2 Model 2

Now we envisage an alternative complete and renormalisable model which
contains less VL fermion messengers by exchanging several of them by heavy
Higgs doublets. Here we exchange the VL fermions of $D$-type and $E$-type by
the pair of Higgs doublets $H_{1}^{d}\sim(\mathbf{1,2})_{(-\frac{1}{2},0,0)}$
and $H_{2}^{d}\sim(\mathbf{1,2})_{(0,-\frac{1}{2},0)}$, as shown in Table 3.
These messengers are motivated by the tri-hypercharge model of Ref.
FernandezNavarro:2023hrf  that originates from a gauge unified framework.
However, we still need to consider VL quarks to generate the effective Yukawa
operators that provide quark mixing. We could use the VL quark doublets
$Q_{i}\sim(\mathbf{3,2})_{\frac{1}{6}_{i}}$ as in FernandezNavarro:2023hrf ,
with $i=1,2,3$. Nevertheless, here we shall stick with the more minimal set of
up-type VL quark singlets $U_{12}$, $U_{13}$ and $U_{23}$ that were already
present in Model 1. On top of generating quark mixing, the $U_{ij}$ VL quarks
will also generate the small masses of light up-quarks, and will provide
unique phenomenological predictions as we shall see in Section 4.

Field | $U(1)_{Y_{1}}$ | $U(1)_{Y_{2}}$ | $U(1)_{Y_{3}}$ | $SU(3)_{c}\times SU(2)_{L}$
---|---|---|---|---
$\;\;\,H_{3}^{u,d}$ | 0 | 0 | $\pm 1/2$ | $(\mathbf{1,2})$
$H_{2}^{d}$ | 0 | $-1/2$ | 0 | $(\mathbf{1,2})$
$H_{1}^{d}$ | $-1/2$ | 0 | 0 | $(\mathbf{1,2})$
$\phi_{q_{12}}$ | -1/6 | 1/6 | 0 | $(\mathbf{1,1})$
$\phi_{\ell_{12}}$ | 1/2 | -1/2 | 0 | $(\mathbf{1,1})$
$\phi_{q_{23}}$ | 0 | -1/6 | 1/6 | $(\mathbf{1,1})$
$\phi_{\ell_{23}}$ | 0 | 1/2 | -1/2 | $(\mathbf{1,1})$
$U_{12}$ | -1/6 | -1/2 | 0 | $(\mathbf{\overline{3},1})$
$U_{13}$ | -1/6 | 0 | -1/2 | $(\mathbf{\overline{3},1})$
$U_{23}$ | 0 | -1/6 | -1/2 | $(\mathbf{\overline{3},1})$
$N_{12}$ | 1/2 | -1/2 | 0 | $(\mathbf{1,1})$
$N_{13}$ | 1/2 | 0 | -1/2 | $(\mathbf{1,1})$
$N_{23}$ | 0 | 1/2 | -1/2 | $(\mathbf{1,1})$
Table 3: Scalar and vector-like fermion content of Model 2. As in Table 2, for
vector-like fermions (highlighted in yellow), the conjugate partners are also
included but not explicitly shown.

With respect to Model 1, Model 2 contains a smaller number of total degrees of
freedom and representations, but also more scalars fields and hence more terms
in the scalar potential. We refer to Appendix A, where we discuss the scalar
potential of the two models and briefly comment as well on the stability of
the spectrum in Fig. 2 under radiative corrections.

We remark that the full set of renormalisable Yukawa couplings in the up
sector is also described in this model by Eq. (16). In contrast, the
renormalisable Yukawa couplings in the down and charged lepton sector are
simply given by

$\mathcal{L}_{Y}^{d,e}=\sum_{i=1,2,3}y_{i}^{d}q_{i}H_{i}^{d}d_{i}^{c}+y_{i}^{e}\ell_{i}H_{i}^{d}e_{i}^{c}+\mathrm{h.c.}$
(34)

Notice however that the Higgs doublets $H_{1}^{d}$ and $H_{2}^{d}$ are assumed
to not get a VEV at the electroweak scale, but instead we assume them to be
very heavy and act as heavy messengers of the diagonal entries in the
effective Yukawa couplings of down quarks and charged leptons, as shown in
Fig. 3. For this we use the $d=3$ couplings
$f_{12}^{dd}\widetilde{H}_{1}^{d}H_{2}^{d}\widetilde{\phi}_{\ell 12}$ and
$f_{23}^{dd}\widetilde{H}_{2}^{d}H_{3}^{d}\widetilde{\phi}_{\ell 23}$ that
appear in the scalar potential and couple the heavy Higgs doublets to the
hyperons, see Appendix A.

After integrating out the heavy messengers, the effective Yukawa couplings for
chiral fermions take the form

$\displaystyle{\cal L}_{d,e}$
$\displaystyle=\begin{pmatrix}q_{1}&q_{2}&q_{3}\end{pmatrix}\mathrm{diag}\left(c_{11}^{d}{\displaystyle\frac{\tilde{\phi}_{\ell
12}}{M_{H_{1}^{d}}}}{\displaystyle\frac{\tilde{\phi}_{\ell
23}}{M_{H_{2}^{d}}}},\,{\displaystyle c_{22}^{d}\frac{\tilde{\phi}_{\ell
23}}{M_{H_{2}^{d}}}},\,c_{33}^{d}\right)\begin{pmatrix}d_{1}^{c}\\\
d_{2}^{c}\\\ d_{3}^{c}\end{pmatrix}H_{3}^{d}$ (35)
$\displaystyle+\begin{pmatrix}\ell_{1}&\ell_{2}&\ell_{3}\end{pmatrix}\mathrm{diag}\left(c_{11}^{e}{\displaystyle\frac{\tilde{\phi}_{\ell
12}}{M_{H_{1}^{d}}}}{\displaystyle\frac{\tilde{\phi}_{\ell
23}}{M_{H_{2}^{d}}}},\,{\displaystyle c_{22}^{e}\frac{\tilde{\phi}_{\ell
23}}{M_{H_{2}^{d}}}},\,c_{33}^{e}\right)\begin{pmatrix}e_{1}^{c}\\\
e_{2}^{c}\\\ e_{3}^{c}\end{pmatrix}H_{3}^{d}\,+\mathrm{h.c.}\,,$ (36)

where

$c^{d,e}=\mathrm{diag}\left(y_{1}^{d,e}\frac{f_{12}^{dd}}{M_{H_{1}^{d}}}\frac{f_{23}^{dd}}{M_{H_{2}^{d}}},\,y_{2}^{d,e}\frac{f_{23}^{dd}}{M_{H_{2}^{d}}},\,y_{3}^{d,e}\right)$
(37)

and we assume for simplicity $f_{12}^{dd}\sim M_{H_{1}^{d}}$ and
$f_{23}^{dd}\sim M_{H_{2}^{d}}$. In contrast, the up-quark sector remains
described by Eq. (19).

$\tilde{\phi}_{\ell
23}$$H^{d}_{3}$$q_{2}{,}\,\ell_{2}$$d^{c}_{2}{,}\,e^{c}_{2}$$H_{2}^{d}$$\tilde{H}_{2}^{d}$

(a)

$\tilde{\phi}_{\ell 12}$$\tilde{\phi}_{\ell
23}$$H^{d}_{3}$$q_{1}{,}\,\ell_{1}$$d^{c}_{1}{,}\,e^{c}_{1}$$H_{1}^{d}$$\tilde{H}_{1}^{d}$$H_{2}^{d}$$\tilde{H}_{2}^{d}$

(b)

Figure 3: Diagrams at the origin of light down-quarks and charged lepton
masses in Model 2. The up-sector is described by diagrams such as those of
Fig. 1.

We can see that in this model the down-quark and charged lepton Yukawa
matrices are diagonal at tree-level, therefore CKM and PMNS mixing fully
originate from the up and neutrino sectors, respectively. The up-quark Yukawa
couplings take a similar form as in Model 1, and we can similarly describe all
the flavour structure of the SM in terms of three naturally small parameters
as in Eq. (24), obtaining three simple NP scales above electroweak symmetry
breaking as shown in Fig. 2. However, here we shall take advantage of the
$\mathcal{O}(1)$ variation of the VL fermion masses and VEVs of the hyperons
(as discussed in Section 2.1) to choose slightly different ratios as

${\displaystyle\frac{\langle{\phi}_{\ell
23}\rangle}{M_{U_{23}}}}\sim\lambda^{3}\,,\qquad{\displaystyle\frac{\langle{\phi}_{q23}\rangle}{M_{U_{23}}}}\sim\lambda^{2}\,,\qquad\frac{\langle{\phi}_{\ell
12}\rangle}{M_{U_{12,13}}}\sim\lambda^{2}\,,\qquad\frac{\langle{\phi}_{q12}\rangle}{M_{U_{12,13}}}\sim\lambda\,.$
(38)

and the (1,1) entry of the effective up-Yukawa matrix is further suppressed by
the fixed ratio,

${\displaystyle\frac{\langle{\phi}_{\ell
23}\rangle}{M_{U_{12,13}}}}\sim\lambda^{5}\,.$ (39)

This choice is consistent with the spectrum of Fig. 2 up to $\mathcal{O}(1)$
variations, now with the addition of the two heavy Higgs doublets that live
with the 23-messengers and with the 12,13-messengers, respectively. This
provides the following values for the ratios involving the heavy Higgs
doublets,

${\displaystyle\frac{\langle{\phi}_{\ell
23}\rangle}{M_{H_{2}^{d}}}}\sim\lambda^{3}\,,\qquad\frac{\langle{\phi}_{\ell
12}\rangle}{M_{H_{1}^{d}}}\sim\lambda^{3}\,.$ (40)

Notice that, as in Model 1, the presence of the ratio $\phi_{\ell
23}/M_{U_{12}}$ provides a highly non-generic and predictive connection
between both VEVs as per Eq. (25), namely

$\langle\phi_{\ell 23}\rangle/\langle\phi_{\ell
12}\rangle=v_{23}/v_{12}\sim\lambda^{3}\sim 0.01\,.$ (41)

Therefore, a typical benchmark for the scales of Model 2 compatible with
phenomenology is

$v_{23}\sim\mathcal{O}(10\,\mathrm{TeV}),\quad v_{12}\sim
M_{U_{23},H_{2}^{d}}\sim\mathcal{O}(10^{3}\,\mathrm{TeV})\,,\quad
M_{U_{12,13},H_{1}^{d}}\sim\mathcal{O}(10^{5}\,\mathrm{TeV})\,.$ (42)

We conclude this section with the approximate textures (up to $\mathcal{O}(1)$
coefficients) predicted for the effective Yukawa couplings by Model 2,

$\displaystyle{\cal L}$
$\displaystyle=\begin{pmatrix}q_{1}&q_{2}&q_{3}\end{pmatrix}\begin{pmatrix}{\displaystyle\lambda^{7}}&{\displaystyle\lambda^{4}}&\lambda^{3}\\\
\lambda^{6}&\lambda^{3}&\lambda^{2}\\\
0&0&1\end{pmatrix}\begin{pmatrix}u_{1}^{c}\\\ u_{2}^{c}\\\
u_{3}^{c}\end{pmatrix}v_{\mathrm{SM}}$ (43)
$\displaystyle+\begin{pmatrix}q_{1}&q_{2}&q_{3}\end{pmatrix}\begin{pmatrix}\lambda^{6}&0&0\\\
0&\lambda^{3}&0\\\ 0&0&1\end{pmatrix}\begin{pmatrix}d_{1}^{c}\\\ d_{2}^{c}\\\
d_{3}^{c}\end{pmatrix}\lambda^{2}v_{\mathrm{SM}}$ (44)
$\displaystyle+\begin{pmatrix}\ell_{1}&\ell_{2}&\ell_{3}\end{pmatrix}\begin{pmatrix}\lambda^{6}&0&0\\\
0&{\displaystyle\lambda^{3}}&{\displaystyle 0}\\\
0&0&1\end{pmatrix}\begin{pmatrix}e_{1}^{c}\\\ e_{2}^{c}\\\
e_{3}^{c}\end{pmatrix}\lambda^{2}v_{\mathrm{SM}}+\mathrm{h.c.}$ (45)

Since here CKM mixing originates from the up sector and the charged lepton
matrix is diagonal at tree-level, the leading flavour phenomenology involves
FCNCs in the up sector, unlike Model 1 where the leading phenomenology
involves FCNCs in the down sector along with CLFV processes.

##  3 Neutrinos

As first discussed in FernandezNavarro:2023rhv , the tri-hypercharge symmetry
imposes selection rules on the Weinberg operator that potentially complicate
the explanation of large neutrino mixing. Nevertheless, this problem can be
solved via minimal extensions of the traditional seesaw mechanism. With
respect to FernandezNavarro:2023rhv , here we propose a different, more
minimal model for the neutrino sector which does not require the addition of
any extra scalar beyond those already present in Models 1 and 2.

In order to describe neutrino masses and mixing, we use the two complete
singlet neutrinos $\nu_{1,2}^{c}$ and the two hyperons $\phi_{\ell 23}$ and
$\phi_{\ell 12}$ already included in Table 1, along with thee vector-like
neutrinos $N_{12}$, $N_{13}$ and $N_{23}$ common to both Models 1 and 2 as
shown in Tables 2 and 3. The rationale is that the vector-like neutrinos,
thanks to the presence of the hyperons $\phi_{\ell 23}$ and $\phi_{\ell 12}$,
can mediate couplings between the light lepton doublets and the (very heavy)
singlet neutrinos $\nu_{1,2}^{c}$, as shown in Fig. 4, which would otherwise
be forbidden by the selection rules.

However, the diagrammatic approach of Fig. 4 requires to employ the mass
insertion approximation where $M_{N_{23}}\gg\langle\phi_{\ell 23}\rangle$,
hence typically predicting suppressed neutrino mixing in conflict with
measured values deSalas:2020pgw ; Gonzalez-Garcia:2021dve . Nevertheless, the
diagrams in Fig. 4 give the intuition that if $M_{N_{23}}\sim\langle\phi_{\ell
23}\rangle$ (and similarly $M_{N_{12,13}}\sim\langle\phi_{\ell 12}\rangle$),
then large neutrino mixing can be obtained. In order to verify this, we need
to disregard the mass insertion approximation, because in the regime
$M_{N_{23}}\sim\langle\phi_{23}\rangle$ the EFT suggested by the diagrams in
Fig. 4 is not consistent.

The way to go is to introduce formally the three messenger neutrinos $N_{12}$,
$N_{13}$ and $N_{23}$ in the theory and employ the seesaw formula. We can
construct the Dirac and Majorana mass matrices in the usual way by writing all
gauge invariant couplings as

$m_{D}=\left(\begin{array}[]{c|cccccccc}&\overline{N}_{12}&\overline{N}_{13}&\overline{N}_{23}&N_{12}&N_{13}&N_{23}&\nu_{1}^{c}&\nu_{2}^{c}\\\
\hline\cr\ell_{1}&0&0&0&0&y_{13}H_{3}^{u}&0&0&0\\\
\ell_{2}&0&0&0&0&0&y_{24}H_{3}^{u}&0&0\\\
\ell_{3}&0&0&0&0&0&0&y_{35}H_{3}^{u}&y_{36}H_{3}^{u}\end{array}\right)\,,$
(46)
$M_{N}=\left(\begin{array}[]{c|cccccccc}&\overline{N}_{12}&\overline{N}_{13}&\overline{N}_{23}&N_{12}&N_{13}&N_{23}&\nu_{1}^{c}&\nu_{2}^{c}\\\
\hline\cr\overline{N}_{12}&0&0&0&m_{N_{12}}&x_{15}\widetilde{\phi}_{\ell
23}&0&z_{17}\phi_{\ell 12}&z_{18}\phi_{\ell 12}\\\
\overline{N}_{13}&0&0&0&x_{24}\phi_{\ell
23}&m_{N_{13}}&x_{26}\widetilde{\phi}_{\ell 12}&0&0\\\
\overline{N}_{23}&0&0&0&0&x_{35}\phi_{\ell 12}&m_{N_{23}}&z_{37}\phi_{\ell
23}&z_{38}\phi_{\ell 23}\\\ N_{12}&m_{N_{12}}&x_{24}\phi_{\ell
23}&0&0&0&0&z_{47}\widetilde{\phi}_{\ell 12}&z_{48}\widetilde{\phi}_{\ell
12}\\\ N_{13}&x_{15}\widetilde{\phi}_{\ell
23}&m_{N_{13}}&x_{26}\widetilde{\phi}_{\ell 12}&0&0&0&0&0\\\
N_{23}&0&x_{35}\phi_{\ell 12}&m_{N_{23}}&0&0&0&z_{67}\widetilde{\phi}_{\ell
23}&z_{68}\widetilde{\phi}_{\ell 23}\\\ \nu_{1}^{c}&z_{17}\phi_{\ell
12}&0&z_{37}\phi_{\ell 23}&z_{47}\widetilde{\phi}_{\ell
12}&0&z_{67}\widetilde{\phi}_{\ell 23}&M_{11}&M_{12}\\\
\nu_{2}^{c}&z_{18}\phi_{\ell 12}&0&z_{38}\phi_{\ell
23}&z_{48}\widetilde{\phi}_{\ell 12}&0&z_{68}\widetilde{\phi}_{\ell
23}&M_{12}&M_{22}\end{array}\right)\,,$ (47)

where we can take advantage of a global $U(2)$ symmetry relating $\nu_{1}^{c}$
and $\nu_{2}^{c}$ to set $M_{12}=0$ without loss of generality. The full
neutrino mass matrix is given as

$M_{\nu}=\left(\begin{array}[]{cc}0&m_{D}\\\
m_{D}^{\mathrm{T}}&M_{N}\end{array}\right)\,.$ (48)

Since $m_{D}$ only contains couplings to the Higgs doublet $H^{u}_{3}$ that
breaks electroweak symmetry, while $M_{N}$ in all cases contains heavy scales
in the multi-TeV region or above, we are in the regime $m_{D}\ll M_{N}$ where
we can safely apply the seesaw formula.

$H^{u}_{3}$$\phi_{\ell
23}$$H^{u}_{3}$$\ell_{2}$$N_{23}$$\overline{N}_{23}$$\nu^{c}_{i}$$\nu^{c}_{i}$$\ell_{3}$

(a)

$H^{u}_{3}$$\phi_{\ell 23}$$H^{u}_{3}$$\phi_{\ell
23}$$\ell_{2}$$N_{23}$$\overline{N}_{23}$$\nu^{c}_{i}$$\nu^{c}_{i}$$N_{23}$$\overline{N}_{23}$$\ell_{2}$

(b)

Figure 4: Illustrative diagrams for the generation of effective Majorana
masses for active neutrinos, where $i=1,2$.

But first we note that after exchanging the hyperons by their VEVs
$\langle\phi_{\ell 12}\rangle\sim v_{12}$ and $\langle\phi_{\ell
23}\rangle\sim v_{23}$, the submatrix in $M_{N}$ that corresponds to the VL
messengers contains two eigenvalues at $v_{12}$ and one at $v_{23}$ (as we
expect from the spectrum in Fig. 2) only for the choice of bare masses
$m_{N_{12}}\sim v_{23}$, $m_{N_{13}}\sim v_{23}$ and $m_{N_{23}}\sim v_{12}$,
namely

$\left(\begin{array}[]{c|ccc}&N_{12}&N_{13}&N_{23}\\\
\hline\cr\overline{N}_{12}&v_{23}&x_{15}v_{23}&0\\\
\overline{N}_{13}&x_{24}v_{23}&v_{23}&x_{26}v_{12}\\\
\overline{N}_{23}&0&x_{35}v_{12}&v_{12}\end{array}\right)\overset{\mathrm{diag}}{\longrightarrow}\left(\begin{array}[]{c|ccc}&\hat{N}_{23}&\hat{N}_{13}&\hat{N}_{12}\\\
\hline\cr\hat{\overline{N}}_{23}&v_{23}&0&0\\\
\hat{\overline{N}}_{13}&0&v_{12}&0\\\
\hat{\overline{N}}_{12}&0&0&v_{12}\end{array}\right)\,,$ (49)

where we have assumed $\mathcal{O}(1)$ coefficients $x_{ij}$ and $v_{23}\ll
v_{12}$ as predicted from Eq. (25). The choice of bare masses $m_{N_{ij}}$
above may seem contrary to the prescription $M_{N_{23}}\sim v_{23}$ (and
similarly $M_{N_{12,13}}\sim v_{12}$) that we anticipated from the
diagrammatic approach, but we need to note that $m_{N_{ij}}$ are not
necessarily the physical masses of the messenger neutrinos. Indeed, in the
regime where the messenger neutrinos and the VEVs of the hyperons are of the
same order, the mixing terms among $N_{ij}$ are very relevant. After taking
them into account, our choice of bare masses delivers $M_{N_{23}}\sim v_{23}$
and $M_{N_{12,13}}\sim v_{12}$, as the physical masses of the messenger
neutrinos. After applying the seesaw formula to extract the light neutrino
masses, we verify that messenger neutrinos at these scales mediate
$\mathcal{O}(1)$ couplings between the light lepton doublets and the singlet
neutrinos. Indeed, we obtain that to leading order all the $v_{12}$ and
$v_{23}$ scales cancel between the numerator and the denominator. Namely, we
find

$\displaystyle m_{\nu}\simeq m_{D}M_{N}^{-1}m_{D}^{\mathrm{T}}$
$\displaystyle=\left(\begin{array}[]{ccc}{\displaystyle
y_{13}^{2}x_{24}^{2}\frac{M_{11}z_{18}^{2}+M_{22}z_{17}^{2}}{M_{11}M_{22}x_{26}^{2}x_{35}^{2}}}&{\displaystyle-
y_{13}y_{24}x_{24}^{2}\frac{M_{11}z_{18}^{2}+M_{22}z_{17}^{2}}{M_{11}M_{22}x_{26}^{2}x_{35}}}&{\displaystyle-
y_{13}\frac{M_{11}z_{18}y_{36}+M_{22}z_{17}y_{35}}{M_{11}M_{22}x_{26}x_{35}}}\\\
{\displaystyle-
y_{13}y_{24}x_{24}^{2}\frac{M_{11}z_{18}^{2}+M_{22}z_{17}^{2}}{M_{11}M_{22}x_{26}^{2}x_{35}}}&{\displaystyle
y_{24}^{2}x_{24}^{2}\frac{M_{11}z_{18}^{2}+M_{22}z_{17}^{2}}{M_{11}M_{22}x_{26}^{2}}}&{\displaystyle
y_{24}x_{24}\frac{M_{11}z_{18}y_{36}+M_{22}z_{17}y_{35}}{M_{11}M_{22}x_{26}}}\\\
{\displaystyle-
y_{13}\frac{M_{11}z_{18}y_{36}+M_{22}z_{17}y_{35}}{M_{11}M_{22}x_{26}x_{35}}}&{\displaystyle
y_{24}x_{24}\frac{M_{11}z_{18}y_{36}+M_{22}z_{17}y_{35}}{M_{11}M_{22}x_{26}}}&{\displaystyle\frac{M_{11}y_{36}^{2}+M_{22}y_{35}^{2}}{M_{11}M_{22}}}\end{array}\right)v_{\mathrm{SM}}^{2}$
(53)
$\displaystyle+\mathcal{O}\left(v_{\mathrm{SM}}^{2}\frac{M_{11,22}}{M_{11}M_{22}}\frac{v_{23}}{v_{12}}\right)\,.$
(54)

If we assume $M_{11}\ll M_{22}$ as in sequential dominance King:1998jw ;
King:1999mb ; King:2002nf , and $x_{ij}=z_{ij}=1$ for simplicity, then we have

$m_{\nu}\simeq
m_{D}M_{N}^{-1}m_{D}^{\mathrm{T}}\approx\left(\begin{array}[]{ccc}y_{13}^{2}&-y_{13}y_{24}&-y_{13}y_{35}\\\
-y_{13}y_{24}&y_{24}^{2}&y_{24}y_{35}\\\
-y_{13}y_{35}&y_{24}y_{35}&y_{35}^{2}\end{array}\right)\frac{v_{\mathrm{SM}}^{2}}{M_{11}}$
(55)

The above result is similar to what one would obtain in a minimal type-I
seesaw extension of the SM. Indeed the texture above predicts two very light
active neutrinos (and one massless) if $M_{11}\approx 10^{15}$ GeV, and is
capable to explain large neutrino mixing by fitting the $\mathcal{O}(1)$
coefficients to existing data. Notably, this implementation of the seesaw
mechanism requires no addition of extra scalars beyond those already
introduced to explain the flavour structure of the charged fermion sector.
Moreover, there is no need to impose small couplings nor assuming that the
scales $v_{12}$ or $v_{23}$ are very heavy in order to achieve small neutrino
masses (meaning that such scales can be in the multi-TeV region), as is
commonly assumed in the literature.

We remark that the three neutrino messengers $N_{12}$, $N_{13}$ and $N_{23}$
obtain masses at the same scale of the $Z^{\prime}_{12}$ and $Z^{\prime}_{23}$
bosons to which they couple and which could live in the multi-TeV region,
meaning that they could carry meaningful phenomenology if such $Z^{\prime}$
bosons are discovered.

##  4 Phenomenology

###  4.1 Gauge boson couplings and complete fermion mixing

The phenomenology of the tri-hypercharge framework is mostly driven by the
$Z^{\prime}_{12}$ and $Z^{\prime}_{23}$ gauge bosons that arise from the 12
and 23 breaking, respectively, and to some extent also to the $Z$ boson, as
firstly discussed in FernandezNavarro:2023rhv  where the couplings of such
gauge bosons to chiral fermions can be found.

The only difference with respect to the study in FernandezNavarro:2023rhv  is
the inclusion of the heavy messengers that mediate the effective Yukawa
couplings in the complete models presented here. Beyond dictating the origin
and size of chiral fermion mixing, which controls the size and nature of
potential flavour-changing neutral currents (FCNCs) mediated by the gauge
bosons, the vector-like fermion messengers also couple to the gauge bosons and
mix with chiral fermions, hence potentially providing new sources of flavour-
violation beyond those that originate from chiral fermion mixing. The fermion
couplings of the gauge bosons may be represented as $6\times 6$ matrices in
flavour space, with the vector-like fermion families $F_{12}$, $F_{13}$ and
$F_{23}$ (with $F=U,D,E,N$ for Model 1, and $F=U,N$ only in Model 2) taken as
fourth, fifth and sixth family respectively. For simplicity, we start by
treating separately the couplings of chiral and vector-like fermions with the
gauge bosons, in order to write in the interaction basis444Here we depart from
our purely left-handed and two-component convention to work in a four-
component, left-right notation more familiar for phenomenological studies.,

$\displaystyle\mathcal{L}_{\mathrm{gauge}}$
$\displaystyle\supset(\kappa_{ij}^{f_{L,R}}\overline{f}_{L,R}^{i}\gamma_{\mu}f_{L,R}^{j}+\kappa_{ab}^{F}\overline{F}_{L,R}^{a}\gamma_{\mu}F_{L,R}^{b})Z^{\prime\mu}_{12}+(\xi_{ij}^{f_{L,R}}\overline{f}_{L,R}^{i}\gamma_{\mu}f_{L,R}^{j}+\xi_{ab}^{F}\overline{F}_{L,R}^{a}\gamma_{\mu}F_{L,R}^{b})Z^{\prime\mu}_{23}$
(56)
$\displaystyle+(\omega_{ij}^{f_{L,R}}\overline{f}_{L,R}^{i}\gamma_{\mu}f_{L,R}^{j}+\omega_{ab}^{F}\overline{F}_{L,R}^{a}\gamma_{\mu}F_{L,R}^{b})Z^{\mu}\,,$

where $f=u,d,e,\nu$ with $i,j=1,2,3$ and $a,b=4,5,6$. The coupling matrices
above are obtained from the covariant derivatives in FernandezNavarro:2023rhv

$\displaystyle\kappa_{ij}^{f_{L,R}}=\mathrm{diag}(-Y_{1}^{f_{L,R}^{1}}g_{1}\sin\theta_{12}^{W},\;Y_{2}^{f_{L,R}^{2}}g_{2}\cos\theta_{12}^{W},\;0)\,,$
(57)
$\displaystyle\kappa_{ab}^{F}=\mathrm{diag}(Y_{1}^{F_{12}}g_{1}\sin\theta_{12}^{W}-Y_{2}^{F_{12}}g_{2}\cos\theta_{12}^{W},\;Y_{1}^{F_{13}}g_{1}\sin\theta_{12}^{W},\;-Y_{2}^{F_{23}}g_{2}\cos\theta_{12}^{W})\,,$
$\displaystyle\xi_{ij}^{f_{L,R}}=\mathrm{diag}(-Y_{12}^{f_{L,R}^{1}}g_{12}\sin\theta_{23}^{W},\;-Y_{12}^{f_{L,R}^{2}}g_{12}\sin\theta_{23}^{W},\;Y_{3}^{f_{L,R}^{3}}g_{3}\cos\theta_{23}^{W})\,,$
$\displaystyle\xi_{ab}^{F}=\mathrm{diag}(Y_{12}^{F_{12}}g_{12}\sin\theta_{23}^{W},\;Y_{12}^{F_{13}}g_{12}\sin\theta_{23}^{W}-Y_{3}^{F_{13}}g_{3}\cos\theta_{23}^{W},\;Y_{12}^{F_{23}}g_{12}\sin\theta_{23}^{W}-Y_{3}^{F_{23}}g_{3}\cos\theta_{23}^{W})\,,$
$\displaystyle\omega_{ij}^{f_{L,R}}=(T_{f_{L,R}}^{3}g_{L}\cos\theta^{W}-Y^{f_{L,R}}g_{Y}\sin\theta^{W})\mathbb{I}_{\mathrm{3\times
3}}\,,\qquad\omega_{ab}^{F}=Y^{F}\,g_{Y}\sin\theta^{W}\mathbb{I}_{\mathrm{3\times
3}}\,,$

where $Y_{i}^{f_{L,R}^{i}}$ refers to the $i$-th hypercharge of the
$f_{L,R}^{i}$ fermion (and similarly for the vector-like fermions $F$),
$g_{Y}$ and $g_{L}$ are the conventional gauge couplings of SM hypercharge and
$SU(2)_{L}$ respectively, $T_{f_{L,R}}^{3}$ is the third component of weak
isospin, $Y=Y_{1}+Y_{2}+Y_{3}$ is SM hypercharge, $\theta^{W}$ is the
conventional weak mixing angle and the weak-like mixing angles
$\theta_{12,23}^{W}$ are defined as

$\mathrm{sin}\theta_{12}^{W}=\frac{g_{1}}{\sqrt{g_{1}^{2}+g_{2}^{2}}},\qquad\mathrm{sin}\theta_{23}^{W}=\frac{g_{12}}{\sqrt{g_{12}^{2}+g_{3}^{2}}}\,,$
(58)

where the gauge couplings are related by the following matching conditions

$g_{12}=\frac{g_{1}g_{2}}{\sqrt{g_{1}^{2}+g_{2}^{2}}},\qquad
g_{Y}=\frac{g_{12}g_{3}}{\sqrt{g_{12}^{2}+g_{3}^{2}}}\,.$ (59)

The next step is to rotate the full $6\times 6$ matrices containing the gauge
boson couplings to the mass basis (denoted with hat notation) for each charged
sector555Although our general formalism for the gauge couplings also applies
to the neutrino sector, we do not discuss neutrino phenomenology here since
the most competitive FCNCs generated in our model involve charged fermions
only.,

$d_{L}$ sector |  $d_{R}$ sector |  $e_{L}$ sector |  $e_{R}$ sector
---|---|---|---
$s_{12}^{d_{L}}\simeq\frac{c_{12}^{d}}{c_{22}^{d}}\frac{\langle\phi_{q12}\rangle}{M_{D_{13}}}\sim\lambda$ | $s_{12}^{d_{R}}\simeq\frac{c_{21}^{d}}{c_{22}^{d}}\frac{\langle\phi_{\ell 12}\rangle}{M_{D_{12}}}\frac{\langle\phi_{q12}\rangle}{M_{D_{13}}}\sim\lambda^{2}$ | $s_{12}^{e_{L}}\simeq\frac{c_{12}^{e}}{c_{22}^{e}}\frac{\langle\phi_{\ell 12}\rangle}{M_{E_{13}}}\sim\lambda^{2}$ | $s_{12}^{e_{R}}\simeq\frac{c_{21}^{e}}{c_{22}^{e}}\frac{\langle\phi_{\ell 12}\rangle}{M_{E_{12}}}\frac{\langle\phi_{\ell 12}\rangle}{M_{E_{13}}}\sim\lambda^{4}$
$s_{13}^{d_{L}}\simeq\frac{c_{13}^{d}}{c_{33}^{d}}\frac{\langle\phi_{q12}\rangle}{M_{D_{13}}}\frac{\langle\phi_{q23}\rangle}{M_{D_{23}}}\sim\lambda^{3}$ | $s_{26}^{d_{R}}\simeq y_{62}^{d}\frac{\langle\phi_{\ell 23}\rangle}{M_{D_{23}}}\sim\lambda^{3}$ | $s_{13}^{e_{L}}\simeq\frac{c_{13}^{e}}{c_{33}^{e}}\frac{\langle\phi_{\ell 12}\rangle}{M_{E_{13}}}\frac{\langle\phi_{\ell 23}\rangle}{M_{E_{23}}}\sim\lambda^{5}$ | $s_{26}^{e_{R}}\simeq y_{62}^{e}\frac{\langle\phi_{\ell 23}\rangle}{M_{E_{23}}}\sim\lambda^{3}$
$s_{23}^{d_{L}}\simeq\frac{c_{23}^{d}}{c_{33}^{d}}\frac{\langle\phi_{q23}\rangle}{M_{D_{23}}}\sim\lambda^{2}$ | $s_{36}^{d_{R}}\simeq y_{63}^{d}\frac{\langle\phi_{q23}\rangle}{M_{D_{23}}}\sim\lambda^{2}$ | $s_{23}^{e_{L}}\simeq\frac{c_{23}^{e}}{c_{33}^{e}}\frac{\langle\phi_{\ell 23}\rangle}{M_{E_{23}}}\sim\lambda^{3}$ | $s_{36}^{e_{R}}\simeq y_{63}^{d}\frac{\langle\phi_{\ell 23}\rangle}{M_{E_{23}}}\sim\lambda^{3}$
$s_{26}^{d_{L}}\simeq y_{26}^{d}\frac{\langle H_{3}^{d}\rangle}{M_{D_{13}}}\sim 10^{-6}$ | $s_{14}^{d_{R}}\simeq y_{41}^{d}\frac{\langle\phi_{\ell 12}\rangle}{M_{D_{12}}}\sim\lambda$ | $s_{26}^{e_{L}}\simeq y_{26}^{e}\frac{\langle H_{3}^{d}\rangle}{M_{E_{13}}}\sim 10^{-6}$ | $s_{14}^{e_{R}}\simeq y_{41}^{e}\frac{\langle\phi_{\ell 12}\rangle}{M_{E_{12}}}\sim\lambda^{2}$
$s_{15}^{d_{L}}\simeq y_{15}^{d}\frac{\langle H_{3}^{d}\rangle}{M_{D_{13}}}\sim 10^{-7}$ | $s_{24}^{d_{R}}\simeq y_{42}^{d}\frac{\langle\phi_{q12}\rangle}{M_{D_{12}}}\sim\lambda$ | $s_{15}^{e_{L}}\simeq y_{15}^{e}\frac{\langle H_{3}^{d}\rangle}{M_{E_{13}}}\sim 10^{-7}$ | $s_{24}^{e_{R}}\simeq y_{42}^{e}\frac{\langle\phi_{\ell 12}\rangle}{M_{E_{12}}}\sim\lambda^{2}$
$s_{56}^{d_{L}}\simeq y_{56}^{d}\frac{\langle\phi_{q12}\rangle}{M_{D_{13}}}\sim\lambda$ | $s_{56}^{d_{R}}\simeq y_{65}^{d}\frac{\langle\phi_{q12}\rangle}{M_{D_{13}}}\sim\lambda$ | $s_{56}^{e_{L}}\simeq y_{56}^{e}\frac{\langle\phi_{\ell 12}\rangle}{M_{E_{13}}}\sim\lambda^{2}$ | $s_{56}^{e_{R}}\simeq y_{65}^{e}\frac{\langle\phi_{\ell 12}\rangle}{M_{E_{13}}}\sim\lambda$
$s_{45}^{d_{L}}\simeq y_{45}^{d}\frac{\langle\phi_{\ell 23}\rangle}{M_{D_{12,13}}}\sim\lambda^{4}$ | $s_{45}^{d_{R}}\simeq y_{54}^{d}\frac{\langle\phi_{\ell 23}\rangle}{M_{D_{12,13}}}\sim\lambda^{4}$ | $s_{45}^{e_{L}}\simeq y_{45}^{e}\frac{\langle\phi_{\ell 23}\rangle}{M_{E_{12,13}}}\sim\lambda^{5}$ | $s_{45}^{e_{R}}\simeq y_{54}^{e}\frac{\langle\phi_{\ell 23}\rangle}{M_{E_{12,13}}}\sim\lambda^{5}$

Table 4: Approximate charged fermion mixing in Model 1. The angles not shown
are zero at tree-level. We also neglect up-quark mixing, assuming that the CKM
mostly originates from the down sector as the textures in Eqs. (13) and (14)
suggest.

$\hat{\kappa}^{\psi_{L,R}}=V_{\psi_{L,R}}^{\dagger}\left(\begin{array}[]{cc}\kappa_{ij}^{f_{L,R}}&0_{3\times
3}\\\ 0_{3\times
3}&\kappa_{ab}^{F}\end{array}\right)V_{\psi_{L,R}}\,,\quad\hat{\xi}^{\psi_{L,R}}=V_{\psi_{L,R}}^{\dagger}\left(\begin{array}[]{cc}\xi_{ij}^{f_{L,R}}&0_{3\times
3}\\\ 0_{3\times 3}&\xi_{ab}^{F}\end{array}\right)V_{\psi_{L,R}}\,,$ (60)
$\hat{\omega}^{\psi_{L,R}}=V_{\psi_{L,R}}^{\dagger}\left(\begin{array}[]{cc}\omega_{ij}^{f_{L,R}}&0_{3\times
3}\\\ 0_{3\times 3}&\omega_{ab}^{F}\end{array}\right)V_{\psi_{L,R}}\,,$ (61)

where $\psi=(f,F)$ generically denotes fermions with the same electric charge,
without distinguishing between chiral and vector-like. The mixing matrices
$V_{\psi_{L,R}}$, obtained by perturbatively diagonalising the full mass
matrices of Eqs. (16-18), are $6\times 6$ unitary matrices that connect the
interaction basis and mass basis as

$\psi_{L,R}=V_{\psi_{L,R}}\hat{\psi}_{L,R}\,.$ (62)

If we consider only the mixing angles that are non-zero at tree-level, then we
can write

$V_{\psi_{L}}=V_{45}^{\psi_{L}}V_{56}^{\psi_{L}}V_{25}^{\psi_{L}}V_{36}^{\psi_{L}}V_{23}^{\psi_{L}}V_{13}^{\psi_{L}}V_{12}^{\psi_{L}}\,,$
(63)
$V_{\psi_{R}}=V_{45}^{\psi_{R}}V_{56}^{\psi_{R}}V_{24}^{\psi_{R}}V_{14}^{\psi_{R}}V_{36}^{\psi_{R}}V_{26}^{\psi_{R}}V_{12}^{\psi_{R}}\,,$
(64)

where each $V_{\alpha\beta}^{\psi_{L,R}}$ contains a mixing angle that
quantifies the mixing between the $\alpha$ and $\beta$ families (see
King:2017anf ; King:2021jeo ; FernandezNavarro:2022gst  for more details on
the notation). Their approximate numerical values and their scaling with
parameters of the model are given in Tables 4 and 5.

From the complete gauge coupling matrices in Eqs. (60) and (61) we notice that
particular GIM-like mechanisms arise which protect from certain FCNCs. For
example $(f_{L1},F_{L13})$ and $(f_{L2},F_{L23})$ couple universally to
$Z^{\prime}_{12}$, and similarly $(f_{L1},f_{L2})$ and
$(f_{R1},f_{R2},F_{R12})$ couple universally to $Z^{\prime}_{23}$.
Nevertheless, significant VL-chiral mixing in the RH sector (see Tables 4 and
5) can induce large FCNCs among $F_{13,23}$ and light chiral fermions mediated
by $Z^{\prime}_{23}$, while in the LH sector all VL-chiral mixing is
suppressed by very small angles of $\mathcal{O}(\langle
H_{3}^{u,d}\rangle/M_{13,23})$.

$u_{L}$ sector |  $u_{R}$ sector
---|---
$s_{12}^{u_{L}}\simeq\frac{c_{12}^{u}}{c_{22}^{u}}\frac{\langle\phi_{q12}\rangle}{M_{U_{13}}}\sim\lambda$ | $s_{12}^{u_{R}}\simeq\frac{c_{21}^{u}}{c_{22}^{u}}\frac{\langle\phi_{\ell 12}\rangle}{M_{U_{12}}}\frac{\langle\phi_{q12}\rangle}{M_{U_{13}}}\sim\lambda^{3}$
$s_{13}^{u_{L}}\simeq\frac{c_{13}^{u}}{c_{33}^{u}}\frac{\langle\phi_{q12}\rangle}{M_{U_{13}}}\frac{\langle\phi_{q23}\rangle}{M_{U_{23}}}\sim\lambda^{3}$ | $s_{26}^{u_{R}}\simeq y_{62}^{u}\frac{\langle\phi_{\ell 23}\rangle}{M_{U_{23}}}\sim\lambda^{3}$
$s_{23}^{u_{L}}\simeq\frac{c_{23}^{u}}{c_{33}^{u}}\frac{\langle\phi_{q23}\rangle}{M_{U_{23}}}\sim\lambda^{2}$ | $s_{36}^{u_{R}}\simeq y_{63}^{u}\frac{\langle\phi_{q23}\rangle}{M_{U_{23}}}\sim\lambda^{2}$
$s_{26}^{u_{L}}\simeq y_{26}^{u}\frac{\langle H_{3}^{u}\rangle}{M_{U_{13}}}\sim 10^{-4}$ | $s_{14}^{u_{R}}\simeq y_{41}^{u}\frac{\langle\phi_{\ell 12}\rangle}{M_{U_{12}}}\sim\lambda^{2}$
$s_{15}^{u_{L}}\simeq y_{15}^{u}\frac{\langle H_{3}^{u}\rangle}{M_{U_{13}}}\sim 10^{-5}$ | $s_{24}^{u_{R}}\simeq y_{42}^{u}\frac{\langle\phi_{q12}\rangle}{M_{U_{12}}}\sim\lambda$
$s_{56}^{u_{L}}\simeq y_{56}^{u}\frac{\langle\phi_{q12}\rangle}{M_{U_{13}}}\sim\lambda$ | $s_{56}^{u_{R}}\simeq y_{65}^{u}\frac{\langle\phi_{q12}\rangle}{M_{U_{13}}}\sim\lambda$
$s_{45}^{d_{L}}\simeq y_{45}^{d}\frac{\langle\phi_{\ell 23}\rangle}{M_{U_{12,13}}}\sim\lambda^{4}$ | $s_{45}^{d_{R}}\simeq y_{54}^{d}\frac{\langle\phi_{\ell 23}\rangle}{M_{U_{12,13}}}\sim\lambda^{5}$
Table 5: Approximate charged fermion mixing in Model 2. The angles not shown
are zero at tree-level.

On the other hand, we find that all right-handed fermions of each charged
sector couple universally to $Z$, i.e. $\omega_{ij}^{f_{R}}=\omega_{ab}^{F}$,
therefore protecting from FCNCs proportional to the large RH mixing angles. In
contrast, left-handed chiral fermions and the left-handed components of the
vector-like fermions couple differently to $Z$, i.e.
$\omega_{ij}^{f_{L}}\neq\omega_{ab}^{F}$. Giving the mixing structure of the
model, this induces 1-2 left-handed FCNCs mediated by $Z$, which are
nevertheless suppressed by small mixing angles of $\mathcal{O}(\langle
H_{3}^{u,d}\rangle/M_{13,23})$.

###  4.2 Bounds from FCNCs

The messengers of the complete models presented here are too heavy to be
directly probed at existing collider facilities. Nevertheless, they crucially
dictate the origin and size of fermion mixing which controls the size and
nature of the FCNCs mediated by the gauge bosons.

For example, the heavy messengers (vector-like fermions) of Model 1 predict
significant left-handed mixing and also significant 1-2 right-handed mixing in
the three charged sectors, while the heavy messengers of Model 2 only predict
significant mixing among up-quarks (and neutrinos). On top of this, the
vector-like fermions mix with RH chiral fermions with mildly suppressed mixing
angles of $\mathcal{O}(\left\langle\phi\right\rangle/M)$, and with LH chiral
fermions with further suppressed mixing angles of $\mathcal{O}(\langle
H_{3}^{u,d}\rangle/M)$, as shown in Tables 4 and 5. In this manner, the VL-
chiral fermion mixing can induce extra flavour-violating couplings beyond
those obtained from chiral-chiral fermion mixing.

In the following we explore the leading flavour bounds over the heavy gauge
bosons and vector-like fermions of our two models, but we note that performing
a complete and comprehensive study of all possible flavour-violating
observables is beyond the scope of this work.

Model | Observable | Mediator |  Bound (TeV)
---|---|---|---
1 |  $K-\bar{K}$ (Re) | $Z^{\prime}_{12}$ | $M_{Z^{\prime}_{12}}/g_{1}>340\times|\mathrm{Re}\left[\frac{c_{12}^{d}}{c_{22}^{d}}\frac{c_{21}^{d}}{c_{22}^{d}}\right]|$
$K-\bar{K}$ (Im) | $Z^{\prime}_{12}$ | $M_{Z^{\prime}_{12}}/g_{1}>3\cdot 10^{3}\times|\mathrm{Im}\left[\frac{c_{12}^{d}}{c_{22}^{d}}\frac{c_{21}^{d}}{c_{22}^{d}}\right]|$
$\mu\rightarrow e\gamma$ | $Z^{\prime}_{12}$ | $M_{Z^{\prime}_{12}}/g_{1}>30\times|c_{12}^{e}/c_{22}^{e}|$
$Z^{\prime}_{23}$ | $M_{Z^{\prime}_{23}}/g_{3}>8\times|y_{62}^{e}(y_{65}^{e}y_{15}^{e})^{*}|$
$\mu\rightarrow 3\,e$ | $Z^{\prime}_{12}$ | $M_{Z^{\prime}_{12}}/g_{1}>30\times|c_{12}^{e}/c_{22}^{e}|$
2 |  $D-\bar{D}$ (Re) | $Z^{\prime}_{12}$ | $M_{Z^{\prime}_{12}}/g_{1}>150\times|\mathrm{Re}\left[\frac{c_{12}^{u}}{c_{22}^{u}}\frac{c_{21}^{u}}{c_{22}^{u}}\right]|$
$D-\bar{D}$ (Im) | $Z^{\prime}_{12}$ | $M_{Z^{\prime}_{12}}/g_{1}>500\times|\mathrm{Im}\left[\frac{c_{12}^{u}}{c_{22}^{u}}\frac{c_{21}^{u}}{c_{22}^{u}}\right]|$
Table 6: Bounds in TeV on the mass of the heavy gauge bosons
$M_{Z^{\prime}_{12}}\sim v_{12}$ and $M_{Z^{\prime}_{23}}\sim v_{23}$,
assuming $g_{1}\approx g_{2}$ for the gauge couplings. The model dependence is
due to the different fermion mixing predicted in Model 1 and 2. Re and Im
refer to the real and imaginary parts of the effective operator responsible
for a given contribution, respectively.

#### Bounds on the $\boldsymbol{Z^{\prime}_{12}}$ mass

We start by discussing the phenomenology of the $Z^{\prime}_{12}$ boson which
potentially mediates dangerous FCNCs, being very sensitive to the fermion
mixing dictated by the heavy messengers in both models. We show in more detail
the EFT formalism used to extract the bounds of this section in Appendix C.

In Model 1 the leading signals are expected in the down sector, and to a
lesser extent in the charged lepton sector. In particular, we find two leading
contributions to $K-\bar{K}$ mixing given by the operators

$Q_{1}^{sd}=(\bar{s}_{L}\gamma_{\mu}d_{L})^{2}\,,\qquad
Q_{5}^{sd}=(\bar{s}_{L}^{\alpha}d_{R}^{\beta})(\bar{s}_{R}^{\beta}d_{L}^{\alpha})\,,$
(65)

generated from tree-level $Z^{\prime}_{12}$ exchange (with $\alpha$ and
$\beta$ being colour indices), whose sizes are connected to the 1-2 LH and RH
mixing angles $s_{12}^{d_{L}}\sim\lambda c_{12}^{d}$ and
$s_{12}^{d_{R}}\sim\lambda^{2}c_{21}^{d}$. However, the experimental bounds
over the $Q_{5}^{sd}$ operator are much stronger than those over $Q_{1}^{sd}$.

As shown in Table 6, by assuming $g_{1}\approx g_{2}$ for the gauge couplings
in the multi-TeV region (as expected by the GUT origin
FernandezNavarro:2023hrf ) and assuming all the dimensionless coefficients to
be $\mathcal{O}(1)$ as expected, we find typical bounds
$M_{Z^{\prime}_{12}}\gtrsim\mathcal{O}(100\,\mathrm{TeV})$ from the real part
of $Q_{5}^{sd}$ and
$M_{Z^{\prime}_{12}}\gtrsim\mathcal{O}(10^{3}\,\mathrm{TeV})$ from the
imaginary part UTfit:2007eik ; Bona:2024bue ; UTfit2023 , although the latter
is completely dependent on the size of complex phases in the dimensionless
couplings. Then from $\mu\rightarrow e\gamma$ and $\mu\rightarrow 3\,e$
PDG:2022ynf ; MEGII:2023ltw  we find the less stringent bound
$M_{Z^{\prime}_{12}}\gtrsim\mathcal{O}(30\,\mathrm{TeV})$ with respect to
$K-\bar{K}$, due to the fact that both LH and RH charged lepton mixing are
further suppressed with respect to down-quark mixing by extra $\lambda$
factors. In any case, the bounds are consistent with our expectation of
$M_{Z^{\prime}_{12}}\sim v_{12}\sim\mathcal{O}(10^{3}\,\mathrm{TeV})$, but
current precision in $K-\bar{K}$ mixing is starting to test the complex phases
of some dimensionless coefficients in the down-quark Yukawa matrix.

In Model 2, quark mixing is mostly generated from the up sector and both the
down-quark and charged lepton mass matrices are diagonal at tree-level.
Therefore, in this model the phenomenology of $Z^{\prime}_{12}$ is dominated
by the two operators

$Q_{1}^{cu}=(\bar{c}_{L}\gamma_{\mu}u_{L})^{2}\,,\qquad
Q_{5}^{cu}=(\bar{c}_{L}^{\alpha}u_{R}^{\beta})(\bar{c}_{R}^{\beta}u_{L}^{\alpha})\,,$
(66)

that contribute to $D-\bar{D}$ mixing and are controlled by the size of
$s_{12}^{u_{L}}\sim\lambda c_{12}^{u}$ and
$s_{12}^{u_{R}}\sim\lambda^{3}c_{21}^{u}$. We find the leading bound
$M_{Z^{\prime}_{12}}\gtrsim\mathcal{O}(100\,\mathrm{TeV})$ from the real part
of $Q_{5}^{cu}$ UTfit:2007eik ; Bona:2024bue ; UTfit2023  and
$M_{Z^{\prime}_{12}}\gtrsim\mathcal{O}(500\,\mathrm{TeV})$ from the imaginary
part, respectively. In both Models 1 and 2 we find that contributions to FCNCs
via VL-chiral fermion mixing are subleading for the phenomenology of the
$Z^{\prime}_{12}$ boson, which is mostly driven by the size of the 1-2 mixing
angles.

#### Bounds on the $\boldsymbol{Z^{\prime}_{23}}$ mass

$\mu_{R}$$e_{L}$$Z^{\prime}_{12}$$\mu_{R}$$\mu_{L}$

(a)

$\mu_{R}$$e_{L}$$Z^{\prime}_{23}$$E_{23R}$$E_{23L}$

(b)

Figure 5: Leading contributions to $\mu\rightarrow e\gamma$ mediated by
$Z^{\prime}_{12}$ (left) and by $Z^{\prime}_{23}$ (right). The crosses denote
mass insertions, with the photon lines not shown.

In contrast, the $Z^{\prime}_{23}$ boson is mostly protected from the most
dangerous FCNCs by an approximate $U(2)^{5}$ flavour symmetry. The exception
to this appears in Model 1 only, where we find that $Z^{\prime}_{23}$ mediates
a 1-loop contribution to $\mu\rightarrow e\gamma$ with $E_{23}$ running in the
loop, as shown in Fig. 5 (right), that appears due to the mixing of the
$E_{23}$ VL fermion666This is a consequence of non-minimal breaking of the
approximate $U(2)^{5}$ flavour symmetry by the messengers $F_{23}$ in Model
1\. This is avoided in the (less minimal) model of FernandezNavarro:2023hrf ,
where the leading breaking of $U(2)^{5}$ is done minimally, SM-like via the VL
fermion $Q_{3}\sim(\mathbf{3,2})_{(0,0,\frac{1}{6})}$ and heavy Higgs doublets
$H_{2}^{u,d}\sim(\mathbf{1,2})_{(0,\frac{1}{2},0)}$. Alternatively, in Model 2
the $U(2)^{5}$ symmetry is broken minimally in the down and charged lepton
sectors, and non-minimally in the up sector where experimental bounds are much
weaker. with $\mu$ and $e$. This contribution to $\mu\rightarrow e\gamma$ is
chirally enhanced by a factor $M_{E_{23}}/m_{\mu}\sim 10^{7}$ but also
suppressed by small mixing angles
$s_{26}^{e_{R}}\sim\mathcal{O}(\langle\phi_{\ell
23}\rangle/M_{E_{23}}\sim\lambda^{3})$ and
$s_{16}^{e_{L}}\sim\mathcal{O}(s_{12}^{e_{L}}\langle
H_{3}^{d}\rangle/M_{E_{23}}\sim 10^{-7})$. As shown in Table 6 and Fig. 6(a),
this contribution leads to a bound in the few TeV region, stronger than the
bounds obtained from $3\rightarrow 2$ flavour transitions such as $B_{s}$
meson mixing or $B_{s}\rightarrow\mu\mu$. In Fig. 6(a) we also depict in green
the region of parameter space that could explain the anomaly in $b\rightarrow
s\nu\nu$ transitions recently observed by Belle II Belle-II:2023esi , although
this region is excluded by electroweak precision and by the leading flavour-
violating observables. Similarly, we expect significant enhancements of
$b\rightarrow s\mu\mu$ or $b\rightarrow s\tau\tau$ (see e.g. Capdevila:2017iqn
; Alguero:2023jeh ; Bordone:2023ybl ) to be in conflict with electroweak
precision and with the leading flavour observables.

(a)

(b)

Figure 6: Parameter space of the 23-symmetry breaking, where
$M_{Z^{\prime}_{23}}$ is the mass of the $Z^{\prime}_{23}$ gauge boson and
$g_{3}$ is the gauge coupling of the $U(1)_{Y_{3}}$ gauge group. We consider
respectively the non-generic fermion mixing predicted by Model 1 (left) and by
Model 2 (right) where no significant FCNCs are mediated by $Z^{\prime}_{23}$,
hence only the model independent observables appear in the right plot. The
benchmark of equal couplings is motivated by a possible GUT origin
FernandezNavarro:2023rhv , where the three gauge couplings are of the same
order in the multi-TeV.

This bound is well compatible with the model-independent phenomenology of
$Z^{\prime}_{23}$, which is dominated by the production of dilepton tails at
the LHC and the modification of electroweak precision observables via
mixing777As shown in FernandezNavarro:2023rhv , this mass mixing is present
even if kinetic mixing among the three gauge $U(1)$ factors is negligible, as
it happens when they originate from a semi-simple embedding like in
FernandezNavarro:2023hrf . with the SM $Z$ boson FernandezNavarro:2023rhv ;
Davighi:2023evx . In particular, assuming all Yukawa couplings of the theory
to be of $\mathcal{O}(1)$, and assuming $g_{3}(10\,\mathrm{TeV})\simeq 0.6$ as
suggested by a GUT origin FernandezNavarro:2023hrf , we obtain the same bound
$M_{Z^{\prime}_{23}}>5\,\mathrm{TeV}$ from both $\mu\rightarrow e\gamma$ and
from dilepton searches at the LHC. Therefore, to be conservative we set
$v_{23}\sim\mathcal{O}(10\,\mathrm{TeV})$ and, thanks to the predictive
relation in Eq. (25), this fixes the the whole spectrum of both models as in
Fig. 2, and in particular the mass of $Z^{\prime}_{12}$ as
$M_{Z^{\prime}_{12}}\sim v_{12}\sim\mathcal{O}(10^{3}\,\mathrm{TeV})$, well
compatible with all the bounds shown in Table 6.

In contrast with Model 1, in Model 2 CKM mixing originates from the up sector,
with the down-quark and charged lepton Yukawa couplings being diagonal at
tree-level. Therefore, $Z^{\prime}_{23}$ can only mediate FCNCs involving up-
quarks. We find that the approximate $U(2)^{5}$ symmetry protects from
contributions to $D-\overline{D}$ mixing, even in the presence of the VL
quarks $U_{ij}$. Indeed only $t\rightarrow c,u$ FCNCs can be sizable, but
these are poorly bounded from the experiment, leading to no significant bounds
over the parameter space in Fig. 6(b) beyond the model-independent ones that
correspond to the production of dilepton tails at the LHC and the modification
of electroweak precision observables.

#### Bounds on vector-like fermion masses

Finally, we discuss the bounds over the masses of the heavy messenger fermions
obtained from FCNCs. These arise from contributions that are sensitive to the
VL fermion masses directly, rather than sensitive to ratios of the form
$\langle\phi\rangle/M_{F}$ which we consider fixed by the flavour model.

For starters, as shown in Section 4.1, mixing between vector-like and chiral
fermions induces FCNCs mediated by the SM $Z$ boson and involving left-handed
fermions only. In particular, the $Z$ boson mediates a purely left-handed
contribution to $K-\bar{K}$ mixing via the effective operator $Q_{1}^{sd}$,
and similarly for $D-\bar{D}$ mixing via $Q_{1}^{cu}$. It also mediates purely
left-handed contributions to $\mu\rightarrow 3\,e$ at tree-level. These FCNCs
are suppressed by small mixing angles of $\mathcal{O}(\langle
H_{3}^{u}\rangle/M_{23,13}\sim 10^{-5})$ in the up-sector and by
$\mathcal{O}(\langle H_{3}^{d}\rangle/M_{23,13}\sim 10^{-7})$ in the down and
charged lepton sectors. Thanks to the relation
$M_{F_{23}}/M_{F_{12,13}}\sim\lambda$ fixed by the flavour model, we can
express all contributions in terms of the mass of the lightest vector-like
fermions, $M_{F_{23}}$, and obtain bounds over this scale.

Order | Observable |  Bound (TeV)
---|---|---
|  $D-\bar{D}$ (Re) | $M_{U_{23}}>2.5\times|\mathrm{Re}\left[y_{26}^{u}(y_{65}^{u}y_{15}^{u})^{*}\right]|$
|  $D-\bar{D}$ (Im) | $M_{U_{23}}>4\times\left|\mathrm{Im}\left[y_{26}^{u}(y_{65}^{u}y_{15}^{u})^{*}\right]\right|$
Tree-level |  $K-\bar{K}$ (Re) | $M_{D_{23}}>0.1\times|\mathrm{Re}\left[y_{26}^{d}(y_{65}^{d}y_{15}^{d})^{*}\right]|$
($Z$-mediated) |  $K-\bar{K}$ (Im) | $M_{D_{23}}>0.3\times|\mathrm{Im}\left[y_{26}^{d}(y_{65}^{d}y_{15}^{d})^{*}\right]|$
| $\mu\rightarrow 3\,e$ | $M_{E_{23}}>3\times|y_{26}^{e}(y_{65}^{e}y_{15}^{e})^{*}|$
|  $D-\bar{D}$ (Re) | $M_{U_{23}}>15\times|\mathrm{Re}\left[y_{26}^{u}(y_{65}^{u}y_{15}^{u})^{*}\right]|$
1-loop |  $D-\bar{D}$ (Im) | $M_{U_{23}}>130\times\left|\mathrm{Im}\left[y_{26}^{u}(y_{65}^{u}y_{15}^{u})^{*}\right]\right|$
(box with $H_{3}^{u,d}$) |  $K-\bar{K}$ (Re) | $M_{D_{23}}>14\times|\mathrm{Re}\left[y_{26}^{d}(y_{65}^{d}y_{15}^{d})^{*}\right]|$
|  $K-\bar{K}$ (Im) | $M_{D_{23}}>170\times|\mathrm{Im}\left[y_{26}^{d}(y_{65}^{d}y_{15}^{d})^{*}\right]|$
Table 7: Bounds in TeV over the mass of the heavy vector-like fermions,
extracted from FCNCs mediated by the SM $Z$ boson and from 1-loop box diagrams
involving $F_{23}$ and $H_{3}^{u,d}$. Re and Im refer to the real and
imaginary parts of the effective operator responsible for a given
contribution, respectively. Notice that $D_{23}$ and $E_{23}$ are only present
in Model 1, while $U_{23}$ is present in both Models 1 and 2.

The $Z$-mediated FCNCs are then directly sensitive to the masses of
$M_{F_{23}}$, rather than to any ratio of NP scales, providing direct bounds
over the masses of heavy VL messengers in the theory. As shown in Table 7, we
find bounds in the few TeV range over $M_{F_{23}}$, well below their expected
values of $\mathcal{O}(10^{3}\,\mathrm{TeV})$. Particularly weak are the
bounds over $M_{D_{23}}$ with respect to those over $M_{U_{23}}$, due to the
fact that the bounds over $Q_{1}^{sd}$ and $Q_{1}^{cu}$ are of similar size
UTfit:2007eik ; Bona:2024bue ; UTfit2023  but the former is suppressed by the
smaller VEV $\langle H_{3}^{d}\rangle$. This is due to the fact that
$\tan\beta=\langle H_{3}^{u}\rangle/\langle
H_{3}^{d}\rangle\simeq\lambda^{-2}\approx 20$ is imposed to explain the
$m_{b,\tau}/m_{t}$ mass hierarchies, as discussed at the beginning of Section
2.

Going beyond tree-level, we find 1-loop contributions to four-fermion
operators that arise from box diagrams with scalars and VL fermions running in
the loop. We find the leading contributions888Diagrams involving hyperons are
further suppressed by factors of $\lambda$ or involve the heavier fermions
$F_{12,13}$, hence resulting in weaker bounds. to arise from box diagrams
involving the $H_{3}^{u,d}$ Higgs doublets and VL fermions $F_{23}$ running in
the loop, as shown in Fig. 7. These diagrams are not affected by any CKM or
mass suppression but are directly sensitive to the heavy masses of the
$F_{23}$ VL fermions, and to fundamental Yukawa couplings of the model
expected to be of $\mathcal{O}(1)$. Box diagrams of this kind have been
computed previously in the literature Ishiwata:2015cga ; Bobeth:2016llm ;
Alves:2023ufm . Adapting these results to our model, assuming $M_{F_{23}}\gg
M_{H_{3}^{u,d}}$, we obtain effective contributions to $Q_{1}^{sd,cu}$ as

$\left.C_{1}^{sd}\right|_{\mathrm{1-loop}}\simeq\frac{1}{8}\frac{\left(y_{26}^{d}(y_{65}^{d}y_{15}^{d})^{*}\lambda\right)^{2}}{16\pi^{2}M_{D_{23}}^{2}}\,,\qquad\left.C_{1}^{uc}\right|_{\mathrm{1-loop}}\simeq\frac{1}{8}\frac{\left(y_{26}^{u}(y_{65}^{u}y_{15}^{u})^{*}\lambda\right)^{2}}{16\pi^{2}M_{U_{23}}^{2}}\,.$
(67)

As shown in Table 7, from the experimental bounds over the $Q_{1}^{sd,cu}$
operators we obtain bounds over $M_{D_{23}}$ and $M_{U_{23}}$ of
$\mathcal{O}(100\,\mathrm{TeV})$, which are compatible with our expectations
of $M_{U_{23},D_{23}}\gtrsim\mathcal{O}(10^{3}\,\mathrm{TeV})$ according to
the spectrum of Fig. 2.

$c_{L}$$u_{L}$$u_{L}$$c_{L}$$H^{u}_{3}$$U_{23R}$$H^{u}_{3}$$U_{23R}$ (a)

$s_{L}$$d_{L}$$d_{L}$$s_{L}$$H^{d}_{3}$$D_{23R}$$H^{d}_{3}$$D_{23R}$ (b)

Figure 7: 1-loop box diagrams contributing to $D-\bar{D}$ mixing (left) and to
$K-\bar{K}$ mixing (right), with VL fermions $D_{23}$, $U_{23}$ and Higgs
doublets $H_{3}^{u,d}$ running in the loop.

Finally, we comment that a box diagram similar to those of Fig. 7 but
involving charged leptons also arises in Model 1, contributing to
$\mu\rightarrow 3\,e$. However, the associated bounds turn out to be much
weaker than those from meson mixing, roughly
$M_{E_{23}}\apprge\mathcal{O}(1\,\mathrm{TeV})$ and hence of the same order as
the bounds from tree-level $Z$-mediated processes.

##  5 Conclusions

Tri-hypercharge is based on assigning a separate gauge hypercharge to each
fermion family. This simple framework has been shown to explain successfully
the origin of hierarchies in the flavour structure of the SM
FernandezNavarro:2023rhv , and may arise from a gauge unified framework
FernandezNavarro:2023hrf .

In this paper we have proposed and studied two minimal, ultraviolet complete,
renormalisable tri-hypercharge theories of flavour. These models are the most
minimal such models, in the sense of number of total degrees of freedom and
representations, which can account for the quark and lepton (including
neutrino) masses and mixing parameters in a natural way, namely where all
observed mass and mixing hierarchies are explained in terms of ratios of high
energy mass scales. These two models are similar from the low-energy point of
view but differ in the heavy messenger sector that completes the theory, with
consequently different flavour-changing phenomenology.

In the first model all the heavy messengers are vector-like, $SU(2)$ singlet
fermions, hence containing the simplest scalar potential, and in the second
model several of the vector-like fermions are replaced by just two heavy Higgs
doublets, hence being more minimal in terms of total degrees of freedoms and
representations but having more terms in the scalar potential.

Both models provide a very successful explanation of the hierarchical flavour
structure of the SM, which is explained in terms of just three naturally small
parameters defined as ratios of new physics scales (see Eq. (24)), with all
dimensionless couplings of the complete theory being of $O(1)$. Indeed, both
models translate the complicated flavour structure of the SM into three simple
and physical (not so) high energy scales of new physics above the electroweak
scale, as shown in Fig. 2, which have meaningful, non-generic phenomenological
consequences which can potentially be tested experimentally. In contrast to
other alternative proposals in the literature, the three NP scales of our
model are completely correlated in order to explain the flavour structure of
the SM, highlighting the predictivity and minimality of the framework. This
means that by testing any scale of the model, one is also essentially testing
the other two.

On top of this, both models are able to explain small neutrino masses and
large neutrino mixing via the implementation of a simple seesaw mechanism
which requires no extra scalars beyond those present in the theory to explain
the charged fermion sector, being then more minimal than the mechanism
suggested in FernandezNavarro:2023rhv ; FernandezNavarro:2023hrf . Remarkably,
the implementation of the seesaw mechanism in our framework does not require
the presence of very small couplings nor assumes that the scales of symmetry
breaking of the theory are very high. The smallness of neutrino masses is
explained via very heavy singlet neutrinos, which couple to the active
neutrinos thanks to messenger neutrinos that are expected to be not very
heavy, perhaps in the multi-TeV to PeV region, that could be tested via their
couplings to the heavy gauge bosons of the theory.

Indeed, two heavy $Z^{\prime}$ bosons are predicted to live at different,
hierarchical scales which are connected to the origin of fermion mass
hierarchies. These heavy gauge bosons, along with the SM $Z$ boson, couple to
the heavy messengers of the complete theories and predict different
phenomenology for both models.

The lighter $Z^{\prime}_{23}$ boson is protected from the most dangerous FCNCs
by an accidental $U(2)^{5}$ flavour symmetry, although a contribution to
$\mu\rightarrow e\gamma$ arises only in Model 1 due to non-minimal breaking of
the $U(2)^{5}$ symmetry by the heavy messenger fermions. However, this
contribution turns out to be compatible with the model-independent
phenomenology of the $Z^{\prime}_{23}$ boson, which is dominated by dilepton
production at the LHC and by electroweak precision observables, and allows for
$M_{Z^{\prime}_{23}}\sim\mathcal{O}(10\,\mathrm{TeV})$. Taking this as our
benchmark mass for $Z^{\prime}_{23}$ fixes all the scales of our model, and in
particular the mass of the heavier gauge boson $Z^{\prime}_{12}$ as
$M_{Z^{\prime}_{12}}\sim\mathcal{O}(10^{3}\,\mathrm{TeV})$.

In Model 1, the heavier $Z^{\prime}_{12}$ boson mediates contributions to
neutral kaon mixing, $\mu\rightarrow e\gamma$ and $\mu\rightarrow 3\,e$ at
acceptable rates for the expected mass
$M_{Z^{\prime}_{12}}\gtrsim\mathcal{O}(10^{3}\,\mathrm{TeV})$. In contrast, in
Model 2, $Z^{\prime}_{12}$ only mediates contributions to neutral $D$-meson
mixing which suggests
$M_{Z^{\prime}_{12}}\gtrsim\mathcal{O}(100\,\mathrm{TeV})$. Finally, the
vector-like messenger fermions also generate FCNCs via box diagrams involving
the light Higgs doublets, and induce tree-level FCNCs mediated by the SM $Z$
boson. In either case, the resulting bounds are compatible with the heavy
masses expected for the messenger fermions.

In conclusion, the minimal models presented here highlight the simplicity and
effectiveness of the tri-hypercharge framework to explain the origin of
fermion masses and mixings, and they provide complete but simple setups to
investigate the potential experimental signals associated to this class of
models. On top of the model-independent phenomenology associated to direct
collider searches and electroweak precision observables, the models proposed
here predict complementary signals in flavour-violating observables that allow
to test the higher scales of the theory, and highlight the potential of the
flavour precision program to test the origin of flavour in the future.

## Acknowledgements

MFN is grateful to Xavier Ponce Díaz and Admir Greljo for interesting
discussions about flavour model-building during the La Thuile 2024 conference.
SFK would like to thank CERN for hospitality and acknowledges the STFC
Consolidated Grant ST/L000296/1 and the European Union’s Horizon 2020 Research
and Innovation programme under Marie Sklodowska-Curie grant agreement HIDDeN
European ITN project (H2020-MSCA-ITN-2019//860881-HIDDeN). MFN is supported by
the STFC under grant ST/X000605/1. AV acknowledges financial support from the
Spanish grants PID2020-113775GB-I00 (AEI/10.13039/ 501100011033) and
CIPROM/2021/054 (Generalitat Valenciana), as well as from MINECO through the
Ramón y Cajal contract RYC2018-025795-I.

##  Appendix A Complete scalar potential

We introduce a $\mathbb{Z}_{2}$ symmetry which enforces that the Higgs
doublets only couple respectively to up quarks (neutrinos) and down quarks
(charged leptons), as the notation of this paper suggests. This is achieved by
assuming that the following fields are odd under $\mathbb{Z}_{2}$,

$H_{3}^{u}\rightarrow-H_{3}^{u}\,,$ (68) $u_{i}^{c}\rightarrow-
u_{i}^{c}\,,\qquad U_{ij}\,(\overline{U}_{ij})\rightarrow-
U_{ij}\,(\overline{U}_{ij})\,,\qquad\nu_{i}^{c}\rightarrow-\nu_{i}^{c}\,,\qquad
N_{ij}\,(\overline{N}_{ij})\rightarrow-N_{ij}\,(\overline{N}_{ij})\,,$ (69)

while all the remaining fields of both Model 1 and 2 remain even. Notice that
such $\mathbb{Z}_{2}$ should be softly broken in order to avoid the formation
of domain walls in the early Universe, hence we allow dimensionful terms
breaking $\mathbb{Z}_{2}$ explicitly in the scalar potential, which is given
in Model 1 by

$\displaystyle V_{1}$
$\displaystyle=m_{q_{12}}^{2}\left|\phi_{q_{12}}\right|^{2}+m_{q_{23}}^{2}\left|\phi_{q_{23}}\right|^{2}+m_{\ell
12}^{2}\left|\phi_{\ell_{12}}\right|^{2}+m_{\ell
23}^{2}\left|\phi_{\ell_{23}}\right|^{2}$
$\displaystyle+\lambda_{q_{12}}\left|\phi_{q_{12}}\right|^{4}+\lambda_{q_{23}}\left|\phi_{q_{23}}\right|^{4}+\lambda_{\ell_{12}}\left|\phi_{\ell_{12}}\right|^{4}+\lambda_{\ell_{23}}\left|\phi_{\ell_{23}}\right|^{4}$
$\displaystyle+\left(\lambda_{q_{12}^{3}\ell_{12}}\phi_{q_{12}}^{3}\phi_{\ell_{12}}+\lambda_{q_{23}^{3}\ell_{23}}\phi_{q_{23}}^{3}\phi_{\ell_{23}}+\mathrm{h.c.}\right)$
$\displaystyle+\lambda_{q_{12}q_{23}}\left|\phi_{q_{12}}\right|^{2}\left|\phi_{q_{23}}\right|^{2}+\lambda_{q_{12}\ell_{12}}\left|\phi_{q_{12}}\right|^{2}\left|\phi_{\ell_{12}}\right|^{2}+\lambda_{q_{12}\ell_{23}}\left|\phi_{q_{12}}\right|^{2}\left|\phi_{\ell_{23}}\right|^{2}$
$\displaystyle+\lambda_{q_{23}\ell_{12}}\left|\phi_{q_{23}}\right|^{2}\left|\phi_{\ell_{12}}\right|^{2}+\lambda_{q_{23}\ell_{23}}\left|\phi_{q_{23}}\right|^{2}\left|\phi_{\ell_{23}}\right|^{2}+\lambda_{\ell_{12}\ell_{23}}\left|\phi_{q_{\ell
12}}\right|^{2}\left|\phi_{\ell_{23}}\right|^{2}$
$\displaystyle+M_{H_{3}^{u}}^{2}|H_{3}^{u}|^{2}+M_{H_{3}^{d}}^{2}|H_{3}^{d}|^{2}+\left(M_{H_{3}^{u,d}}^{2}H_{3}^{u}H_{3}^{d}+\mathrm{h.c.}\right)+\lambda_{H_{3}^{u}}|H_{3}^{u}|^{4}+\lambda_{H_{3}^{d}}|H_{3}^{d}|^{4}$
$\displaystyle\lambda_{H_{3}^{u}H_{3}^{d}}|H_{3}^{u}|^{2}|H_{3}^{d}|^{2}+\lambda_{\widetilde{H}_{3}^{u}\widetilde{H}_{3}^{d}}(H_{3}^{u}H_{3}^{d})(\widetilde{H}_{3}^{d}\widetilde{H}_{3}^{u})+\left(\lambda_{H_{3}^{u}H_{3}^{u}H_{3}^{d}H_{3}^{d}}H_{3}^{u}H_{3}^{u}H_{3}^{d}H_{3}^{d}+\mathrm{h.c.}\right)$
$\displaystyle+\lambda_{H_{3}^{u}q_{12}}|H_{3}^{u}|^{2}\left|\phi_{q_{12}}\right|^{2}+\lambda_{H_{3}^{u}q_{23}}|H_{3}^{u}|^{2}\left|\phi_{q_{23}}\right|^{2}+\lambda_{H_{3}^{u}\ell_{12}}|H_{3}^{u}|^{2}\left|\phi_{\ell_{12}}\right|^{2}+\lambda_{H_{3}^{u}\ell_{23}}|H_{3}^{u}|^{2}\left|\phi_{\ell_{23}}\right|^{2}$
$\displaystyle+\lambda_{H_{3}^{d}q_{12}}|H_{3}^{d}|^{2}\left|\phi_{q_{12}}\right|^{2}+\lambda_{H_{3}^{d}q_{23}}|H_{3}^{d}|^{2}\left|\phi_{q_{23}}\right|^{2}+\lambda_{H_{3}^{d}\ell_{12}}|H_{3}^{d}|^{2}\left|\phi_{\ell_{12}}\right|^{2}+\lambda_{H_{3}^{d}\ell_{23}}|H_{3}^{d}|^{2}\left|\phi_{\ell_{23}}\right|^{2}\,.$
(70)

The scalar potential of Model 2 contains all the terms of Model 1 plus a few
more involving the Higgs doublets $H_{1}^{d}$ and $H_{2}^{d}$,

$\displaystyle V_{2}$
$\displaystyle=V_{1}+\sum_{i=1,2}\left(M_{H_{i}^{d}}^{2}|H_{i}^{d}|^{2}+\lambda_{H_{i}^{d}}|H_{i}^{d}|^{4}+\lambda_{H_{3}^{u}H_{i}^{d}}|H_{3}^{u}|^{2}|H_{i}^{d}|^{2}+\lambda_{H_{3}^{d}H_{i}^{d}}|H_{3}^{d}|^{2}|H_{i}^{d}|^{2}\right.$
$\displaystyle+\lambda_{\widetilde{H}_{3}^{u}\widetilde{H}_{i}^{d}}(H_{3}^{u}H_{i}^{d})(\widetilde{H}_{i}^{d}\widetilde{H}_{3}^{u})+\lambda_{\widetilde{H}_{3}^{d}\widetilde{H}_{i}^{d}}(\widetilde{H}_{3}^{d}H_{i}^{d})(\widetilde{H}_{i}^{d}H_{3}^{d})$
$\displaystyle\left.+\lambda_{H_{i}^{d}q_{12}}|H_{i}^{d}|^{2}\left|\phi_{q_{12}}\right|^{2}+\lambda_{H_{i}^{d}q_{23}}|H_{i}^{d}|^{2}\left|\phi_{q_{23}}\right|^{2}+\lambda_{H_{i}^{d}\ell_{12}}|H_{i}^{d}|^{2}\left|\phi_{\ell_{12}}\right|^{2}+\lambda_{H_{i}^{d}\ell_{23}}|H_{i}^{d}|^{2}\left|\phi_{\ell_{23}}\right|^{2}\right)$
$\displaystyle+\lambda_{H_{1}^{d}H_{2}^{d}}|H_{1}^{d}|^{2}|H_{2}^{d}|^{2}+\lambda_{\widetilde{H}_{1}^{d}\widetilde{H}_{2}^{d}}(\widetilde{H}_{1}^{d}H_{2}^{d})(\widetilde{H}_{2}^{d}H_{1}^{d})$
$\displaystyle+\left(f_{12}^{dd}\widetilde{H}_{1}^{d}H_{2}^{d}\widetilde{\phi}_{\ell
12}+f_{23}^{dd}\widetilde{H}_{2}^{d}H_{3}^{d}\widetilde{\phi}_{\ell
23}+f_{23}^{du}H_{2}^{d}H_{3}^{u}\phi_{\ell
23}+\lambda_{H_{1}^{d}H_{2}^{d}\phi_{12}\phi_{23}}\widetilde{H}_{1}^{d}H_{3}^{d}\widetilde{\phi}_{\ell
12}\widetilde{\phi}_{\ell 23}+\mathrm{h.c.}\right)\ $ (71)

Notice that the $d=4$ couplings $\phi_{q_{12}}^{3}\phi_{\ell_{12}}$ and
$\phi_{q_{23}}^{3}\phi_{\ell_{23}}$ break the global phases associated to the
four hyperons, while the $d=4$ coupling $H_{3}^{u}H_{3}^{u}H_{3}^{d}H_{3}^{d}$
breaks the global phases associated to the third family Higgs doublets (along
with the $d=2$ coupling $H_{3}^{u}H_{3}^{d}$). For Model 2, the same
discussion applies but the extra $d=4$ coupling
$\widetilde{H}_{1}^{d}H_{3}^{d}\widetilde{\phi}_{\ell
12}\widetilde{\phi}_{\ell 23}$ and $d=3$ couplings $f_{12,23}^{dd}$,
$f_{23}^{du}$ break the global phases of $H_{1}^{d}$ and $H_{2}^{d}$, leaving
the three gauge hypercharges as the only remaining unbroken $U(1)$s. In this
manner, both Model 1 and Model 2 are free from the appearance of physical
Nambu-Goldstone bosons.

Having discussed the scalar sector of our two models, we comment that we
expect radiative corrections to disturb the spectrum of scales depicted in
Fig. 2 unless some of the couplings of the scalar potential are fine-tuned to
some degree. Indeed the light Higgs doublets $H^{u,d}_{3}$ that perform
electroweak symmetry breaking couple to heavy degrees of freedom at tree-
level, especially in Model 2 where we find that $H^{d}_{3}$ couples to the
heavy Higgs doublets and the hyperons. This is a consequence of the well-known
hierarchy problem. Then the radiative stability of the scales in Fig. 2
requires fine-tuning of the couplings of the scalar potential to some degree,
as discussed e.g. in Allwicher:2020esa  for a related framework. In order to
avoid such fine-tuning one would need to impose extra dynamics like
Supersymmetry or a strongly coupled sector in order to protect the masses of
fundamental scalars from radiative corrections, which is beyond the scope of
this paper.

##  Appendix B Models with less than four hyperons

In both minimal models presented in the main text, we have considered only the
four hyperons $\phi_{q12}^{(-\frac{1}{6},\frac{1}{6},0)}$, $\phi_{\ell
12}^{(\frac{1}{2},-\frac{1}{2},0)}$,
$\phi_{q23}^{(0,-\frac{1}{6},\frac{1}{6})}$ and $\phi_{\ell
23}^{(0,\frac{1}{2},-\frac{1}{2})}$. However, as first shown in
FernandezNavarro:2023rhv , a simple spurion analysis reveals that one can
exchange the $\ell$-type hyperons by cubic powers of the $q$-type hyperons,
i.e.

$\phi_{\ell 12}\sim\tilde{\phi}_{q12}^{3}\,,$ (72) $\phi_{\ell
23}\sim\tilde{\phi}_{q23}^{3}\,.$ (73)

Therefore, in principle it is possible to remove $\phi_{\ell 12}$ and
$\phi_{\ell 23}$ since $\phi_{q12}$ and $\phi_{q23}$ suffice to spontaneously
break the tri-hypercharge symmetry in the desired way of Eq. (12), and also
can generate all the spurions needed to populate the effective Yukawa
matrices. Assuming UV completions where all the heavy messengers are vector-
like fermions (like in Model 1 of the main text), the spurionic relations in
Eqs. (72) and (73) lead to models that contain only the two hyperons
$\phi_{q12}$ and $\phi_{q23}$ and the two Higgs doublets $H^{u,d}_{3}$ as the
only scalars of the theory, hence having the most economical scalar sector and
the simplest scalar potential (and being automatically free from the
appearance of physical Goldstone bosons).

Indeed by using only $\phi_{q12}$ and $\phi_{q23}$ we can write the effective
Yukawa matrices in an EFT framework as

$\displaystyle{\cal L}$
$\displaystyle=\begin{pmatrix}Q_{1}&Q_{2}&Q_{3}\end{pmatrix}\begin{pmatrix}\tilde{\phi}_{q12}^{3}\tilde{\phi}_{q23}^{3}&{\displaystyle\phi_{q12}\tilde{\phi}_{q23}^{3}}&{\phi}_{q12}{\phi}_{q23}\\\
\tilde{\phi}_{q12}^{4}\tilde{\phi}_{q23}^{3}&\tilde{\phi}_{q23}^{3}&{\phi}_{q23}\\\
\tilde{\phi}_{q12}^{4}\tilde{\phi}_{q23}^{4}&\tilde{\phi}_{q23}^{3}\tilde{\phi}_{q23}&1\end{pmatrix}\begin{pmatrix}u_{1}^{c}\\\
u_{2}^{c}\\\ u_{3}^{c}\end{pmatrix}H_{3}^{u}$ (74)
$\displaystyle+\begin{pmatrix}Q_{1}&Q_{2}&Q_{3}\end{pmatrix}\begin{pmatrix}\phi_{q12}^{3}\phi_{q23}^{3}&{\displaystyle{\phi}_{q12}\phi_{q23}^{3}}&{\displaystyle{\phi}_{q12}{\phi}_{q23}}\\\
{\phi}_{q12}^{2}\phi_{q23}^{3}&\phi_{q23}^{3}&{\phi}_{q23}\\\
{\phi}_{q12}^{2}{\phi}_{q23}^{2}&{\phi}_{q23}^{2}&1\end{pmatrix}\begin{pmatrix}d_{1}^{c}\\\
d_{2}^{c}\\\ d_{3}^{c}\end{pmatrix}H_{3}^{d}$ (75)
$\displaystyle+\begin{pmatrix}L_{1}&L_{2}&L_{3}\end{pmatrix}\begin{pmatrix}\phi_{q12}^{3}\phi_{q23}^{3}&\tilde{\phi}_{q12}^{3}\phi_{q23}^{3}&\tilde{\phi}_{q12}^{3}\tilde{\phi}_{q23}^{3}\\\
\phi_{q12}^{6}\phi_{q23}^{3}&\phi_{q23}^{3}&\tilde{\phi}_{q23}^{3}\\\
\phi_{q12}^{6}\phi_{q23}^{6}&\phi_{q23}^{6}&1\end{pmatrix}\begin{pmatrix}e_{1}^{c}\\\
e_{2}^{c}\\\ e_{3}^{c}\end{pmatrix}H_{3}^{d}\,+\mathrm{h.c.}\,,$ (76)

where the cut-off of the EFTs are not shown, we assume that a power of
$\Lambda_{1}^{-1}$ accompanies every insertion of $\phi_{q12}$ and a power of
$\Lambda_{2}^{-1}$ accompanies every insertion of $\phi_{q23}$. From the
inspection of the matrices above, we already notice that the 23-mass hierarchy
$m_{2}/m_{3}\sim\phi_{q23}^{3}/\Lambda_{2}^{3}$ is generated at dimension-7 in
the EFT while $V_{cb}\sim\phi_{q23}/\Lambda_{2}$, which is numerically close,
is generated at only dimension-5. Nevertheless, after exchanging the hyperons
by their VEVs, we can assume the following numerical values,

$\frac{\langle{\phi}_{q12}\rangle}{\Lambda_{1}}\sim{\displaystyle\frac{\langle{\phi}_{q23}\rangle}{\Lambda_{2}}}\sim\lambda\,,$
(77)

to obtain the approximate textures (up to $\mathcal{O}(1)$ coefficients)

$\displaystyle{\cal L}$
$\displaystyle=\begin{pmatrix}u_{1}&u_{2}&u_{3}\end{pmatrix}\begin{pmatrix}\lambda^{6}&\lambda^{4}&\lambda^{2}\\\
\lambda^{7}&\lambda^{3}&\lambda\\\
\lambda^{9}&\lambda^{5}&1\end{pmatrix}\begin{pmatrix}u_{1}^{c}\\\ u_{2}^{c}\\\
u_{3}^{c}\end{pmatrix}\frac{v_{\mathrm{SM}}}{\sqrt{2}}$ (78)
$\displaystyle+\begin{pmatrix}d_{1}&d_{2}&d_{3}\end{pmatrix}\begin{pmatrix}\lambda^{6}&\lambda^{4}&\lambda^{2}\\\
\lambda^{5}&\lambda^{3}&\lambda\\\
\lambda^{6}&\lambda^{4}&1\end{pmatrix}\begin{pmatrix}d_{1}^{c}\\\ d_{2}^{c}\\\
d_{3}^{c}\end{pmatrix}\lambda^{2}\frac{v_{\mathrm{SM}}}{\sqrt{2}}$ (79)
$\displaystyle+\begin{pmatrix}e_{1}&e_{2}&e_{3}\end{pmatrix}\begin{pmatrix}\lambda^{6}&\lambda^{6}&\lambda^{6}\\\
\lambda^{9}&\lambda^{3}&\lambda^{3}\\\
\lambda^{12}&\lambda^{6}&1\end{pmatrix}\begin{pmatrix}e_{1}^{c}\\\
e_{2}^{c}\\\
e_{3}^{c}\end{pmatrix}\lambda^{2}\frac{v_{\mathrm{SM}}}{\sqrt{2}}\,+\mathrm{h.c.}\,.$
(80)

The textures above are successful to explain the flavour structure of the SM,
albeit requiring some dimensionless coefficients of $\mathcal{O}(\lambda)$ to
generate the right $V_{cb}$ and $V_{ub}$. We consider this acceptable, and
perhaps could be avoided if the mass of a VL quark in the UV theory provides
an extra $\lambda$ factor beyond the simplified assumption of Eq. (77).

| $U(1)_{Y_{1}}$ | $U(1)_{Y_{2}}$ | $U(1)_{Y_{3}}$ | $SU(3)_{c}\times SU(2)_{L}$
---|---|---|---|---
$H_{3}^{u,d}$ | 0 | 0 | $\pm 1/2$ | $(\mathbf{1,2})$
$\phi_{q_{12}}$ | -1/6 | 1/6 | 0 | $(\mathbf{1,1})$
$\phi_{q_{23}}$ | 0 | -1/6 | 1/6 | $(\mathbf{1,1})$
$U_{23}$ | 0 | -1/6 | -1/2 | $(\mathbf{\overline{3},1})$
$\widetilde{U}_{23}$ | 0 | -1/3 | –1/3 | $(\mathbf{\overline{3},1})$
$\hat{U}_{23}$ | 0 | -1/2 | -1/6 | $(\mathbf{\overline{3},1})$
$U_{13}$ | -1/6 | 0 | -1/2 | $(\mathbf{\overline{3},1})$
$U_{2}$ | 0 | 2/3 | 0 | $(\mathbf{\overline{3},1})$
$U_{12}$ | -1/6 | 1/2 | 0 | $(\mathbf{\overline{3},1})$
$\widetilde{U}_{12}$ | -1/3 | -1/3 | 0 | $(\mathbf{\overline{3},1})$
$\hat{U}_{12}$ | -1/2 | 1/6 | 0 | $(\mathbf{\overline{3},1})$
$D_{23}$ | 0 | -1/6 | 1/2 | $(\mathbf{\overline{3},1})$
$D_{3}$ | 0 | 0 | 1/3 | $(\mathbf{\overline{3},1})$
$\tilde{D}_{23}$ | 0 | 1/6 | 1/6 | $(\mathbf{\overline{3},1})$
$D_{13}$ | -1/6 | 0 | 1/2 | $(\mathbf{\overline{3},1})$
$D_{2}$ | 0 | 1/3 | 0 | $(\mathbf{\overline{3},1})$
$\tilde{D}_{12}$ | 1/6 | 1/6 | 0 | $(\mathbf{\overline{3},1})$
$E_{23}$ | 0 | 1/2 | 1/2 | $(\mathbf{1,1})$
$\widetilde{E}_{23}$ | 0 | 2/3 | 1/3 | $(\mathbf{1,1})$
$\hat{E}_{23}$ | 0 | 5/6 | 1/6 | $(\mathbf{1,1})$
$E_{12}$ | 1/2 | 1/2 | 0 | $(\mathbf{1,1})$
$\widetilde{E}_{12}$ | 2/3 | 1/3 | 0 | $(\mathbf{1,1})$
$\hat{E}_{12}$ | 5/6 | 1/6 | 0 | $(\mathbf{1,1})$
$E_{13}$ | 1/2 | 0 | 1/2 | $(\mathbf{1,1})$
$E_{123}$ | 1/2 | 1/3 | 1/6 | $(\mathbf{1,1})$
$\widetilde{E}_{123}$ | 1/2 | 1/6 | 1/3 | $(\mathbf{1,1})$
Table 8: Scalar and vector-like content of the renormalisable model with two
hyperons (charged fermion sector only). As in Tables 2 and 3, for vector-like
fermions (highlighted in yellow), the conjugate partners are also included but
not explicitly shown.

However, we find that in order to UV-complete the effective theory via
$SU(2)_{L}$ singlet VL fermions, a total of 23 VL fermion representations are
needed for the charged fermion sector only, as shown in Table 8, in contrast
with only 9 needed in Model 1 of the main text. Moreover, some of the VL
fermions need to be charged under the three family hypercharges. Hence we find
that in models with only two hyperons $\phi_{q12}$ and $\phi_{q23}$, despite
containing the simplest scalar sector and the most economical low-energy
theory, the UV-complete theories seem to be far from minimal.

In comparison with Model 1 of the main text, the model sketched here contains
two less hyperons but at least 14 more VL fermions. This counting is obtained
without exploring the neutrino sector, which will require the addition of more
neutrino messengers and the full neutrino mass matrix will potentially contain
various powers of the VEVs $\langle\phi_{q12}\rangle$ and
$\langle\phi_{q23}\rangle$, perhaps making it more difficult to explain large
neutrino mixing. Instead, if we seek for UV completions as in Model 2 of the
main text, we will need to introduce 9 heavy Higgs doublets for the down and
charged lepton sector (plus 7 VL fermions for the up sector), hence being less
minimal and also complicating the scalar potential.

Perhaps an alternative would be to introduce $\phi_{\ell
23}^{(0,\frac{1}{2},-\frac{1}{2})}$ only, without introducing $\phi_{\ell
12}^{(\frac{1}{2},-\frac{1}{2},0)}$. This is exactly the model with 3 hyperons
first discussed in Section 3.3 of FernandezNavarro:2023rhv . In this case, the
effective Yukawa matrix (showing only the up sector for simplicity) reads

${\cal
L}_{u}=\begin{pmatrix}Q_{1}&Q_{2}&Q_{3}\end{pmatrix}\begin{pmatrix}\tilde{\phi}_{q12}^{3}{\phi}_{\ell
23}&{\phi}_{q12}{\phi}_{\ell 23}&{\phi}_{q12}{\phi}_{q23}\\\
\tilde{\phi}_{q12}^{4}{\phi}_{\ell 23}&{\phi}_{\ell 23}&{\phi}_{q23}\\\
\tilde{\phi}_{q12}^{4}{\phi}_{\ell 23}\tilde{\phi}_{q23}&{\phi}_{\ell
23}\tilde{\phi}_{q23}&1\end{pmatrix}\begin{pmatrix}u_{1}^{c}\\\ u_{2}^{c}\\\
u_{3}^{c}\end{pmatrix}H^{u}_{3}+\mathrm{h.c.}$ (81)

hence now both $m_{2}/m_{3}$ and $V_{cb}$ are generated at dimension-5 in the
EFT expansion, but in contrast with the models of the main text now we
generate first family masses at dimension-8 rather than dimension-6, due to
the fact that $\phi_{\ell 12}^{(\frac{1}{2},-\frac{1}{2},0)}$ is not present
but generated as $\phi_{\ell 12}\sim\phi_{q12}^{3}$. As first noted in
FernandezNavarro:2023rhv , from the pure EFT point of view, this provides a
nice description of the SM flavour hierarchies thanks to further hyperon
insertions for the smallest (effective) Yukawa couplings.

However, we find that in order to UV-complete this effective theory via
$SU(2)_{L}$ singlet VL fermions, a total of 15 VL fermion representations are
needed for the charged fermion sector only, in contrast with only 9 needed in
Model 1 of the main text. Therefore, in comparison with Model 1 of the main
text, this model would contain only one less hyperon but 6 more VL fermions.
On top of this, in this model the hyperons $\phi_{q23}$ and $\phi_{\ell 23}$
have a quartic coupling in the scalar potential of the form
$\phi_{q23}^{3}\,\phi_{\ell 23}$. This breaks the global phases of both
hyperons. In contrast, $\phi_{q12}$ cannot couple to other scalars beyond
quadratic self-conjugate terms, hence a physical Goldstone boson potentially
arises associated to the unbroken phase of $\phi_{q12}$. This problem persists
in UV completions based on the addition of heavy Higgs doublets, as in Model 2
of the main text, where we find that at least 5 heavy Higgs doublets are
needed for the down and charged lepton sectors.

##  Appendix C EFT formalism for FCNCs

The leading FCNCs in the quark sector are connected to the following operators
contributing to $K-\bar{K}$ and $D-\overline{D}$ mixing

$\mathcal{L}_{\mathrm{EFT}}\supset-\left[C_{LL}^{sd}(\bar{s}_{L}\gamma_{\mu}d_{L})^{2}+C_{LR}^{sd}(\bar{s}_{L}\gamma_{\mu}d_{L})(\bar{s}_{R}\gamma_{\mu}d_{R})+C_{LL}^{cu}(\bar{c}_{L}\gamma_{\mu}u_{L})^{2}+C_{LR}^{cu}(\bar{c}_{L}\gamma_{\mu}u_{L})(\bar{c}_{R}\gamma_{\mu}u_{R})\right]\,,$
(82)

where

$\displaystyle C_{LL}^{sd,uc}$
$\displaystyle=\frac{\hat{\kappa}_{12}^{d_{L},u_{L}}\hat{\kappa}_{12}^{d_{L},u_{L}}}{2M_{Z^{\prime}_{12}}^{2}}+\frac{\hat{\xi}_{12}^{d_{L},u_{L}}\hat{\xi}_{12}^{d_{L},u_{L}}}{2M_{Z^{\prime}_{23}}^{2}}+\frac{\hat{\omega}_{12}^{d_{L},u_{L}}\hat{\omega}_{12}^{d_{L},u_{L}}}{2M_{Z}^{2}}\,,$
(83) $\displaystyle C_{LR}^{sd,uc}$
$\displaystyle=\frac{\hat{\kappa}_{12}^{d_{L},u_{L}}\hat{\kappa}_{12}^{d_{R},u_{R}}}{M_{Z^{\prime}_{12}}^{2}}+\frac{\hat{\xi}_{12}^{d_{L},u_{L}}\hat{\xi}_{12}^{d_{R},u_{R}}}{M_{Z^{\prime}_{23}}^{2}}\,.$
(84)

Now we apply a Fierz transformation to the LR operators in order to match to
the basis of Refs. UTfit:2007eik ; Bona:2024bue ; UTfit2023 , from which we
take the bounds, obtaining

$\mathcal{L}_{\mathrm{EFT}}\supset-\left[C_{1}^{sd}(\bar{s}_{L}\gamma_{\mu}d_{L})^{2}-2C_{LR}^{sd}(\bar{s}_{L}^{\alpha}d_{R}^{\beta})(\bar{s}_{R}^{\beta}d_{L}^{\alpha})+C_{1}^{cu}(\bar{c}_{L}\gamma_{\mu}u_{L})^{2}-2C_{LR}^{cu}(\bar{c}_{L}^{\alpha}u_{R}^{\beta})(\bar{c}_{R}^{\beta}u_{L}^{\alpha})\right]\,.$
(85)

After integrating out the gauge bosons, we also obtain the following effective
operators contributing to charged lepton flavour-violating processes,

$\displaystyle\mathcal{L}_{\mathrm{EFT}}\supset$
$\displaystyle-\frac{4G_{F}}{\sqrt{2}}\left[m_{\beta}\left[C_{e\gamma}\right]^{\alpha\beta}\bar{e}_{L}^{\alpha}\sigma^{\mu\nu}e_{R}^{\beta}F_{\mu\nu}+m_{\beta}\left[C_{e\gamma}\right]^{\beta\alpha}\bar{e}_{L}^{\beta}\sigma^{\mu\nu}e_{R}^{\alpha}F_{\mu\nu}\right.$
(86)
$\displaystyle+\left[C_{ee}^{V,LL}\right]^{\alpha\beta\alpha\alpha}(\bar{e}_{L}^{\alpha}\gamma_{\mu}e_{L}^{\beta})(\bar{e}_{L}^{\alpha}\gamma^{\mu}e_{L}^{\alpha})+\left[C_{ee}^{V,LR}\right]^{\alpha\beta\alpha\alpha}(\bar{e}_{L}^{\alpha}\gamma_{\mu}e_{L}^{\beta})(\bar{e}_{R}^{\alpha}\gamma^{\mu}e_{R}^{\alpha})$
$\displaystyle+\left[C_{ee}^{V,LR}\right]^{\alpha\alpha\alpha\beta}(\bar{e}_{L}^{\alpha}\gamma_{\mu}e_{L}^{\alpha})(\bar{e}_{R}^{\alpha}\gamma^{\mu}e_{R}^{\beta})+\left[C_{ee}^{V,RR}\right]^{\alpha\beta\alpha\alpha}(\bar{e}_{R}^{\alpha}\gamma_{\mu}e_{R}^{\beta})(\bar{e}_{R}^{\alpha}\gamma^{\mu}e_{R}^{\alpha})+\mathrm{h.c.}\,,$

where all Wilson coefficients are dimensionless, including those of the dipole
operators, and where $\alpha,\beta=e,\mu,\tau$ and $\alpha\neq\beta$. Our
models generates the four-lepton structures at tree-level with the following
Wilson coefficients

$\displaystyle\left[C_{ee}^{V,LL}\right]^{\alpha\beta\alpha\alpha}$
$\displaystyle=\eta_{\mathrm{RGE}}\left(\frac{\sqrt{2}}{4G_{F}}\right)\left[\frac{\hat{\kappa}_{\alpha\beta}^{e_{L}}\hat{\kappa}_{\alpha\alpha}^{e_{L}}}{M_{Z^{\prime}_{12}}^{2}}+\frac{\hat{\xi}_{\alpha\beta}^{e_{L}}\hat{\xi}_{\alpha\alpha}^{e_{L}}}{M_{Z^{\prime}_{23}}^{2}}+\frac{\hat{\omega}_{\alpha\beta}^{e_{L}}\hat{\omega}_{\alpha\alpha}^{e_{L}}}{M_{Z}^{2}}\right]\,,$
(87) $\displaystyle\left[C_{ee}^{V,RR}\right]^{\alpha\beta\alpha\alpha}$
$\displaystyle=\eta_{\mathrm{RGE}}\left(\frac{\sqrt{2}}{4G_{F}}\right)\left[\frac{\hat{\kappa}_{\alpha\beta}^{e_{R}}\hat{\kappa}_{\alpha\alpha}^{e_{R}}}{M_{Z^{\prime}_{12}}^{2}}+\frac{\hat{\xi}_{\alpha\beta}^{e_{R}}\hat{\xi}_{\alpha\alpha}^{e_{R}}}{M_{Z^{\prime}_{23}}^{2}}\right]\,,$
(88) $\displaystyle\left[C_{ee}^{V,LR}\right]^{\alpha\beta\alpha\alpha}$
$\displaystyle=\eta_{\mathrm{RGE}}\left(\frac{\sqrt{2}}{4G_{F}}\right)\left[\frac{\hat{\kappa}_{\alpha\beta}^{e_{L}}\hat{\kappa}_{\alpha\alpha}^{e_{R}}}{M_{Z^{\prime}_{12}}^{2}}+\frac{\hat{\xi}_{\alpha\beta}^{e_{L}}\hat{\xi}_{\alpha\alpha}^{e_{R}}}{M_{Z^{\prime}_{23}}^{2}}\right]\,,$
(89) $\displaystyle\left[C_{ee}^{V,LR}\right]^{\alpha\alpha\alpha\beta}$
$\displaystyle=\eta_{\mathrm{RGE}}\left(\frac{\sqrt{2}}{4G_{F}}\right)\left[\frac{\hat{\kappa}_{\alpha\alpha}^{e_{L}}\hat{\kappa}_{\alpha\beta}^{e_{R}}}{M_{Z^{\prime}_{12}}^{2}}+\frac{\hat{\xi}_{\alpha\alpha}^{e_{L}}\hat{\xi}_{\alpha\beta}^{e_{R}}}{M_{Z^{\prime}_{23}}^{2}}\right]\,,$
(90)

where the $\eta_{\mathrm{RGE}}$ coefficients encode RGE effects from running
from the high scale $M_{Z^{\prime}_{12}}\sim 10^{3}$ TeV or
$M_{Z^{\prime}_{23}}\sim 10$ TeV down to 1 GeV. By using DsixTools
Celis:2017hod ; Fuentes-Martin:2020zaz  we typically find
$\eta_{\mathrm{RGE}}\sim 0.8-0.9$ for the leptonic operators studied here. In
contrast, the dipole operators are generated at 1-loop, with the heavy vector-
like fermions also running in the loop, see Fig. 5. For example, in case of
$\mu\to e\gamma$ contributions, from $Z^{\prime}_{12}$ exchange we obtain

$\left[C_{e\gamma}\right]^{\alpha\beta}=\left(\frac{\sqrt{2}}{4G_{F}}\right)\frac{\eta_{\mathrm{RGE}}}{32\pi^{2}M_{Z_{12}^{\prime}}^{2}}\sum_{a=1}^{6}\left[\hat{\kappa}_{\beta
a}^{e_{L}}\hat{\kappa}_{\alpha
a}^{e_{L}}F\left(\frac{M^{2}_{a}}{M^{2}_{Z^{\prime}_{12}}}\right)+\frac{M_{a}}{m_{\beta}}\hat{\kappa}_{\beta
a}^{e_{R}}\hat{\kappa}_{\alpha
a}^{e_{L}}G\left(\frac{M^{2}_{a}}{M^{2}_{Z^{\prime}_{12}}}\right)\right]\,,$
(91)
$\left[C_{e\gamma}\right]^{\beta\alpha}=\left(\frac{\sqrt{2}}{4G_{F}}\right)\frac{\eta_{\mathrm{RGE}}}{32\pi^{2}M_{Z_{12}^{\prime}}^{2}}\sum_{a=1}^{6}\left[\hat{\kappa}_{\beta
a}^{e_{R}}\hat{\kappa}_{\alpha
a}^{e_{R}}F\left(\frac{M^{2}_{a}}{M^{2}_{Z^{\prime}_{12}}}\right)+\frac{M_{a}}{m_{\beta}}\hat{\kappa}_{\beta
a}^{e_{L}}\hat{\kappa}_{\alpha
a}^{e_{R}}G\left(\frac{M^{2}_{a}}{M^{2}_{Z^{\prime}_{12}}}\right)\right]\,,$
(92)

and similarly for $Z^{\prime}_{23}$ by replacing
$M_{Z^{\prime}_{12}}\rightarrow M_{Z^{\prime}_{23}}$ and
$\kappa\rightarrow\xi$. The loop functions $F(x)$ and $G(x)$ are

$F(x)=\frac{5x^{4}-14x^{3}+39x^{2}-38x-18x^{2}\log x+8}{12(1-x)^{4}}\,,\quad
G(x)=\frac{x^{3}+3x-6x\log x-4}{2(1-x)^{3}}\,.$ (93)

Then finally one can compute the branching fractions of $e_{\beta}\rightarrow
e_{\alpha}\gamma$ and $e_{\beta}\rightarrow 3\,e_{\alpha}$ as Kuno:1999jp

$\mathcal{B}(e_{\beta}\rightarrow
e_{\alpha}\gamma)=\left(\frac{4G_{F}}{\sqrt{2}}\right)^{2}\alpha_{\mathrm{QED}}\,m_{\beta}^{5}\tau_{\beta}\left(\left|\left[C_{e\gamma}\right]^{\alpha\beta}\right|^{2}+\left|\left[C_{e\gamma}\right]^{\beta\alpha}\right|^{2}\right)\,,$
(94) $\displaystyle\mathcal{B}(e_{\beta}\rightarrow 3\,e_{\alpha})=$
$\displaystyle
2\left(\left|\left[C_{ee}^{V,LL}\right]^{\alpha\beta\alpha\alpha}\right|^{2}+\left|\left[C_{ee}^{V,RR}\right]^{\alpha\beta\alpha\alpha}\right|^{2}\right)+\left(\left|\left[C_{ee}^{V,LR}\right]^{\alpha\beta\alpha\alpha}\right|^{2}+\left|\left[C_{ee}^{V,LR}\right]^{\alpha\alpha\alpha\beta}\right|^{2}\right)$
$\displaystyle+32\left[\mathrm{log}\left(\frac{m_{\beta}^{2}}{m_{\alpha}^{2}}\right)-\frac{11}{4}\right]\left(\left|e\left[C_{e\gamma}\right]^{\alpha\beta}\right|^{2}+\left|e\left[C_{e\gamma}\right]^{\beta\alpha}\right|^{2}\right)$
$\displaystyle+\mathrm{Re}\left[e\left[C_{e\gamma}\right]^{\alpha\beta}\left(\left[C_{ee}^{V,LL}\right]^{\alpha\beta\alpha\alpha}\right)^{*}\right]+\mathrm{Re}\left[e\left[C_{e\gamma}\right]^{\beta\alpha}\left(\left[C_{ee}^{V,RR}\right]^{\alpha\beta\alpha\alpha}\right)^{*}\right]\,,$
(95)

where $e=\sqrt{4\pi\alpha_{\mathrm{QED}}}$ is the QED gauge coupling.

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