SLAC-PUB-17572

SU(4) Flavorful Portal Matter

George N. Wojcik1,2 and Thomas G. Rizzo1<EMAIL_ADDRESS><EMAIL_ADDRESS>
1SLAC National Accelerator Laboratory, Menlo Park, CA, 94025 USA

2Department of Physics, University of Wisconsin-Madison, Madison, WI 53703 USA

###### Abstract

In this paper, we present a model which attempts to unify a new dark sector
force with a local $SU(3)$ flavor symmetry. Dark Matter (DM) and its potential
interactions with the Standard Model (SM) continue to present a rich framework
for model building. In the case of thermal DM of a mass between a few MeV and
a few GeV, a compelling and much-explored framework is that of a dark
photon/vector portal, which posits a new $U(1)$ “dark photon” which only
couples to the SM via small kinetic mixing (KM) with the SM hypercharge. This
mixing can be mediated at the one-loop level by portal matter (PM) fields
which are charged under both the dark $U(1)$ and the SM gauge group. In
earlier work, one of the authors has noted that models with appropriate portal
matter content to produce finite and calculable kinetic mixing can arise from
non-minimal dark sectors, in which the dark $U(1)$ is a subgroup of a larger
gauge symmetry under which SM particles might have non-trivial
representations. We expand on this idea here by constructing a model in which
in which the dark $U(1)$ is unified with another popular extension to the SM
gauge group, a local $SU(3)$ flavor symmetry. The full dark/flavor symmetry
group is $SU(4)_{F}\times U(1)_{F}$, incorporating the local $SU(3)$ flavor
symmetry with PM appearing as a vector-like “fourth generation” to supplement
the three generations of the SM. To ensure finite contributions to KM, the SM
gauge group is arranged into Pati-Salam multiplets. The new extended dark
gauge group presents a variety of interesting experimental signatures,
including non-trivial consequences of the flavor symmetry being unified with
the dark sector.

##  1 Introduction

In spite of composing roughly 80% of the matter in the universe, the precise
identity of dark matter (DM) remains undetermined. Models to produce the
appropriate abundance of DM, however, strongly suggest that it is subject to
interactions other than gravity, and null results (thus far) in searches for
historically favored DM candidates such as axions [1, 2] and WIMPs [3] have
led to a proliferation of new ideas on the identity, production mechanisms,
and experimental signatures of DM [5, 4]. In this work, we will be exploring
one of the more recently popular DM models, that of a kinetic mixing/vector
portal[6, 7]. In these setups, interaction between the Standard Model (SM)
fields and the DM is mediated by a massive vector dark photon $A_{D}$,
associated with a broken dark gauge group $U(1)_{D}$. The DM particle is
uncharged under the SM gauge group, but possesses non-zero $U(1)_{D}$ charge,
while the SM fields are just the opposite: Charged under the SM group, but not
under $U(1)_{D}$. Small couplings between the dark photon and the SM fields
occur due to kinetic mixing (KM) between $U(1)_{D}$ and $U(1)_{Y}$, which
after electroweak symmetry breaking ultimately give every SM particle a
coupling to $A_{D}$ of $e\epsilon Q$, where $e$ is the electromagnetic
coupling constant, $Q$ is the electric charge of a given particle, and
$\epsilon$ is a small value which parameterizes the strength of the kinetic
mixing. The simplest realizations of these models present a rich phenomenology
with a remarkably simple setup: The only parameters the DM and dark photon
masses (as well as potentially the mass of a scalar associated with $U(1)_{D}$
breaking), the $U(1)_{D}$ gauge coupling, and the KM parameter $\epsilon$.
However, these minimal setups leave little framework for addressing a number
of questions: What is the origin of the small KM parameter $\epsilon$? Is
$U(1)_{D}$ the only dark sector gauge group, or is it embedded in some larger
gauge symmetry? Could the dark sector gauge forces be connected in some way to
other ongoing questions in particle physics?

In the region of parameter space where DM and the dark photon have masses
ranging between $\sim$a few MeV to $\sim$a few GeV, the favored value of
$\epsilon$ generally lies in the range of $\epsilon\sim 10^{-(3-5)}$ [8, 9],
which can in turn suggests that the KM can arise from a simple one-loop vacuum
polarization-like diagram featuring “portal matter” fields charged under both
$U(1)_{Y}$ and $U(1)_{D}$ [7]. The kinetic mixing parameter then has the
dependence

$\displaystyle\epsilon\propto\sum_{i}Q_{Y_{i}}Q_{D_{i}}\log\bigg{(}\frac{m_{i}^{2}}{\mu^{2}}\bigg{)},$
(1)

where the sum over $i$ denotes a sum over all fermions (a nearly identical
expression arising from complex scalar loops also occurs, differing only in
the omitted proportionality constant in front of the sum), $Q_{Y_{i}}$ denotes
the hypercharge of particle $i$, $Q_{D_{i}}$ denotes its $U(1)_{D}$ charge,
and $\mu$ is a renormalization scale. The sum in Eq.(1) is finite and
calculable when $\sum_{i}Q_{D_{i}}Q_{Y_{i}}=0$. In two previous works by one
of our authors [11, 10], henceforth referred to as I and II, respectively, the
theory and phenomenology of these portal matter fields was explored. In I, it
was argued that fermionic portal matter fields must be vector-like in order to
avoid constraints due to gauge anomalies and precision electroweak
measurements. Additionally, in order to ensure that the fields are unstable
(and hence conform to cosmological constraints), they should only appear in
the same representations under the SM gauge group as SM particles. In short,
portal matter fermions should be vector-like copies of SM fields [12], a
possibility rarely explored in the literature [13]. The discussion in I
limited itself to the simplest possible constructions, in which a pair of such
vector-like fermions, with opposite dark charges, generate finite KM through a
small mass splitting between them. Although this setup provided for
interesting phenomenology, the inclusion of portal matter fields which
satisfied the condition $\sum_{i}Q_{D_{i}}Q_{Y_{i}}=0$ was ultimately _ad hoc_
: In spite of the critical importance that this cancellation had to the model,
it did not happen “naturally”. Furthermore, the $U(1)_{D}$ itself was still
minimal – the potentially new effects arising from a non-minimal set of dark
gauge symmetries in which $U(1)_{D}$ could be embedded, such as the
possibility that the SM fields were non-trivially charged under some part of
the extended gauge group orthogonal to $U(1)_{D}$, were not explored. II began
the process of addressing both of these questions. First, it was noted that
the required cancellation to render Eq.(1) finite would naturally occur when
the portal matter was placed in the vector-like representation
$\mathbf{5}+\mathbf{\overline{5}}$ of $SU(5)$, which would then be broken down
to the SM gauge group. Inspired by $E_{6}$ theories in which the $\mathbf{27}$
of $E_{6}$ breaks down to a
$(\mathbf{\overline{5}},\mathbf{2})+(\mathbf{1},\mathbf{2})+(\mathbf{5},\mathbf{1})+(\mathbf{10},\mathbf{1})$
of $SU(5)\times SU(2)$ (allowing for the complete SM to be contained in a
$\mathbf{\overline{5}}+\mathbf{10}$, with a vector-like set of portal fields
in $\mathbf{\overline{5}}+\mathbf{5}$), II then developed a model based on a
gauge group $SU(5)\times SU(2)_{I}\times U(1)_{Y_{I}}$. This setup exhibited a
number of intriguing properties. For example, the extended dark gauge sector
led to new heavy gauge bosons associated with the dark sector but coupled to
the SM, as well as non-standard dark photon couplings emerging from mass
mixing with SM gauge bosons (the latter a consequence of the dark group
$U(1)_{D}$ itself only emerging as a combination of $SU(2)_{I}$ and
$U(1)_{Y_{I}}$ generators). The model in II also exhibited potentially
interesting flavor physics behavior, including flavor-changing neutral
currents and flavor-dependent couplings of new gauge bosons to SM fermions.
However, the setup as written had comparatively little underlying theoretical
structure for the shape and magnitudes that these effects might take.

We wish now to expand on the work of II, and in particular the flavor physics
questions it raised, by considering the following: If the extended gauge group
containing $U(1)_{D}$ can include groups under which the SM particles are non-
trivial representations, might we create a model in which the dark photon
gauge symmetry $U(1)_{D}$ might be unified, either partially or completely,
with some sort of flavor symmetry? With that goal in mind, we develop a
specific model: First, we extend the SM to a Pati-Salam symmetry
$SU(4)_{c}\times SU(2)_{L}\times SU(2)_{R}$ [14], which, as a semisimple
group, we shall prove in Section 2.1 will guarantee that KM from contributions
of the form of Eq.(1) will remain finite and calculable. For our “dark
sector”, in which the dark photon $U(1)_{D}$ shall be embedded, we choose the
$SU(4)_{F}\times U(1)_{F}$ group. We shall see that this group is large enough
to contain both a $U(1)_{D}$ under which the SM fields can remain uncharged,
_and_ an embedded $SU(3)_{F}$ group describing quark flavor. Multiplets
containing the SM fields then are placed in fundamental (and antifundamental)
representations of the $SU(4)_{F}$ group, with the three SM generations
forming triplets (and antitriplets) under $SU(3)_{F}$ and portal matter fields
representing a “fourth generation” that are singlets under $SU(3)_{F}$. From
this outline, we create a phenomenologically realistic theory in which an
$SU(3)_{F}$ flavor symmetry is partially unified with a vector portal for DM.

Our paper is laid out as follows: In Section 2, we discuss the field content
of the model and the masses of any new exotic particles we require. In
addition to explicitly computing the masses and mixings of all
phenomenologically relevant new fermions and gauge bosons, we comment
qualitatively on several important aspects of the scalar sector, since a
rigorous treatment of this sector of the model is complex enough to lie beyond
the scope of the present work. In this Section, we also note that while the UV
model is enormously complicated, containing a variety of new heavy fermions,
the overwhelming majority of the new particles introduced will have a high
enough mass scale to have very little phenomenological relevance. As such,
while we make brief comments about the model at the high scale, we shall focus
the majority of the paper on the comparatively few new particles which are
light enough to have observable effects in the present generation of
experiments. In Section 3, we establish several important phenomenological
tools required for our later analysis: In particular, we explicitly determine
the couplings of the model’s fermions to both the new gauge bosons and to the
SM gauge bosons and the Higgs, and determine what a “natural” range for our
model’s parameters might be. In this Section, we also explicitly compute the
magnitude of the KM effect, i.e., $\epsilon$ in the model, and comment briefly
on how it constrains our parameter space. In Section 4, we compute the
phenomenologically significant flavor-changing neutral currents appearing in
the model, and compare them with existing experimental constraints. In Section
5, we offer a brief survey of the phenomenology of the new fields that may be
produced at collider experiments, in particular focusing on how our model here
differs from the results seen in I and II. Finally, Section 6 summarizes our
results and discusses possible avenues with which this work might be expounded
upon.

##  2 Model Setup

In this Section, we shall outline the components of our model and demonstrate
how realistic flavor structure, at least in the highly-constrained quark
sector, may be effected. It should be noted that in order to attain the
observed quark masses and CKM mixings, the model presented here is exceedingly
complicated and requires prodigious fine-tuning. However, this finely-tuned
sector of the model is associated with the new $SU(3)_{F}$ flavor symmetry,
which generally must be broken at multi-TeV scales. At energies which may be
probed experimentally, we shall find that the model is mostly agnostic to the
particulars of how the $SU(3)_{F}$ symmetry is broken. In the interest of
completeness, then, we present a construction of our model in which the
$SU(3)_{F}$ symmetry is broken in a manner consistent with the observed quark
masses and CKM parameters, but we note that the behavior of the model is in
most cases insensitive to the specifics of the model in the multi-TeV regime.
When we discuss the phenomenology of the model in later Sections, we shall
explicitly note when the results may be sensitive to modifications of the
multi-TeV sector of the theory.

###  2.1 Fermion Content

To begin our construction, it is useful to make several assumptions. First, to
ensure that our fermions will always produce a finite and calculable kinetic
mixing between SM hypercharge and any new $U(1)$’s appearing in our model, we
will restrict our efforts to theories where the gauge group can be written
$\mathcal{G}\times\mathcal{G}_{P}$, where $\mathcal{G}_{P}$ is an arbitrary
gauge group containing a dark $U(1)$ symmetry, and $\mathcal{G}$ is some
_semisimple_ group containing the SM. Our inclusion of the condition that
$\mathcal{G}$ is semisimple now guarantees that once $\mathcal{G}$ is broken
down to a group which includes $U(1)$ factors, any kinetic mixing between
these $U(1)$ factors and any $U(1)$ contained in $\mathcal{G}_{P}$ generated
by one-loop contributions of the form of Eq.(1) will be finite.

We can prove this claim of finiteness straightforwardly: To start, we consider
a field in the representation $\mathcal{R}_{1}\times\mathcal{R}_{2}$ of
$\mathcal{G}\times\mathcal{G}_{P}$. Then, consider an arbitrary $U(1)$
embedded in $\mathcal{G}$, the generator for which we’ll call $T_{Y}$, and an
arbitrary $U(1)$ embedded in $\mathcal{G}_{P}$, the generator for which we’ll
call $T_{P}$. We can demonstrate that for _any_ representation
$\mathcal{R}_{1}\times\mathcal{R}_{2}$ of $\mathcal{G}\times\mathcal{G}_{P}$,
the loop-level contribution to the kinetic mixing between the two $U(1)$
groups is finite and calculable. We know from Eq.(1)that this contribution to
the KM coefficient $\epsilon$ will be finite and calculable as long as the
trace $Tr[T_{Y}T_{P}]=0$, where the generators $T_{Y}$ and $T_{P}$ are given
for the representation $\mathcal{R}_{1}\times\mathcal{R}_{2}$. However, since
this representation is a direct product, this trace is simply $Tr[T_{Y}]\times
Tr[T_{P}]$. For a semisimple group, the trace $Tr[T_{Y}]$ is always zero for
any $U(1)$ generator and any representation. As as a result,
$Tr[T_{Y}T_{P}]=0\times Tr[T_{P}]=0$, and the contribution of any
representation under $\mathcal{G}\times\mathcal{G}_{P}$ to KM will always be
finite and calculable for semisimple $\mathcal{G}$.111There are two caveats
worth mentioning here: First, this works equally well assuming
$\mathcal{G}_{P}$, rather than $\mathcal{G}$, is semisimple, however, given
the large number of extensions of the SM gauge group to a semisimple one
(Pati-Salam, trinification, and all GUT models), we have elected here to
restrict our attention to the scenario where $\mathcal{G}$ is semisimple.
Second, we note that in the high-energy theory, the absence of an independent
$U(1)$ factor in $\mathcal{G}$ (since $U(1)$ is not simple) means that KM
arising from vacuum polarization diagrams as described in the Introduction are
forbidden by gauge invariance. However, we note that these terms still arise
via effective operators featuring the insertion of scalar vev’s when these
gauge symmetries are broken– we shall argue in Section 3.3 that this does not
vitiate our KM calculation there, at least for the purposes of evaluating the
magnitude of the mixing.

For our purposes, we shall select (as mentioned in the Introduction) the Pati-
Salam gauge group $SU(4)_{c}\times SU(2)_{L}\times SU(2)_{R}$ as our
semisimple group $\mathcal{G}$ [14]. To ensure that any new exotic fermions
occur in representations that will quickly decay into SM fields, we then
assume that all fermions in our model occur in the traditional Pati-Salam
multiplets $(\mathbf{4},\mathbf{2},\mathbf{1})$ and
$(\mathbf{\overline{4}},\mathbf{1},\mathbf{2})$, or their conjugate
representations.222This is not strictly necessary, since there exist other
Pati-Salam representations that contain only fields with SM-like quantum
numbers. However, we shall stick to the familiar canonical choices for
simplicity. As our work above for a general $\mathcal{G}$ would suggest, it is
straightforward to see that each of these multiplets has $Tr[Y_{SM}]=0$, and
hence, if we only introduce new matter in these representations, then
regardless of its transformation properties under $\mathcal{G}_{P}$, the new
fermions will invariably produce a finite and calculable kinetic mixing
between the SM hypercharge and any new $U(1)\subset\mathcal{G}_{P}$. Dark
matter, if we wish it to be fermionic, can then be introduced as a Pati-Salam
singlet (or collection of such singlets) or as the SM singlet component of a
$(\mathbf{4},\mathbf{1},\mathbf{2})$ multiplet, with some non-trivial charge
under $U(1)_{D}$. For the sake of brevity, we shall employ some further
simplifying notation: We shall denote the groups $SU(N)_{A}\times
SU(M)_{B}\times...\times U(1)_{C}$ by $N_{A}M_{B}...1_{C}$, so that, for
example, $\mathcal{G}_{PS}=4_{c}2_{L}2_{R}$, and the SM gauge group
$SU(3)_{c}\times SU(2)_{L}\times U(1)_{Y}$ is simply written
$3_{c}2_{L}1_{Y}$.

Our choice of $\mathcal{G}_{P}$ is then further restricted by our requirement
that the model incorporate three generations of the usual SM fermion fields,
all of which must be uncharged under $U(1)_{D}$. A simple method of
accomplishing this is to assume that $\mathcal{G}_{P}$ contains an $SU(3)$
flavor symmetry, $SU(3)_{F}$, the generators of which are orthogonal to the
generators of the $U(1)_{D}$ symmetry. That is, we can decompose
$\mathcal{G}_{P}\rightarrow SU(3)_{F}\times\mathcal{G^{\prime}}_{P}$, with
$\mathcal{G^{\prime}}_{P}\supset U(1)_{D}$. Then, if we can find a model which
gives the complete set of SM fermions in fundamental or antifundamental
representations of $SU(3)_{F}$ with zero charge under $U(1)_{D}$, we’ll have
recreated all three generations of the SM. The sole remaining requirement is
that all other additional fermions in the model be vector-like once
$\mathcal{G^{\prime}}_{P}$ is broken down to $U(1)_{D}$, leaving the SM as the
only ‘light’ chiral matter content.

Precisely this sort of construction can be arranged if we take, e.g.,
$\mathcal{G}_{P}=SU(4)_{F}\times U(1)_{F}=4_{F}1_{F}$, and write the matter
content of our theory in terms of left-handed multiplets (not to be confused
with the $SU(2)_{L}$ gauge group) of the form $4_{c}2_{L}2_{R}4_{F}1_{F}$ as

$\displaystyle(\mathbf{4},\mathbf{2},\mathbf{1},\mathbf{4},-1/4)+(\mathbf{\overline{4}},\mathbf{1},\mathbf{2},\mathbf{\overline{4}},+1/4)+(\mathbf{\overline{4}},\mathbf{2},\mathbf{1},\mathbf{1},+1)+(\mathbf{4},\mathbf{1},\mathbf{2},\mathbf{1},-1).$
(2)

By breaking $4_{F}1_{F}\rightarrow 3_{F}1_{F^{\prime}}1_{F}\rightarrow
3_{F}1_{D}$, we can then decompose this field content into

$\displaystyle(\mathbf{4},\mathbf{2},\mathbf{1},\mathbf{3},0)+(\mathbf{\overline{4}},\mathbf{1},\mathbf{2},\mathbf{\overline{3}},0)+(\mathbf{4},\mathbf{2},\mathbf{1},\mathbf{1},-1)+(\mathbf{\overline{4}},\mathbf{2},\mathbf{1},\mathbf{1},+1)+(\mathbf{4},\mathbf{1},\mathbf{2},\mathbf{1},-1)+(\mathbf{\overline{4}},\mathbf{1},\mathbf{2},\mathbf{1},+1),$
(3)

where multiplets are now labelled according to $4_{c}2_{L}2_{R}3_{F}1_{D}$. As
is readily apparent from Eq.(3), all of our criteria are satisfied: The SM is
contained in a chiral $SU(3)_{F}$ triplet and an antitriplet, which only gain
mass at the scale of electroweak symmetry breaking, while all additional
fields, which might serve as portal matter to contribute to kinetic mixing,
are vector-like and therefore able to obtain far larger masses. Inspection of
Eq.(2) also shows this model to be anomaly-free.

In order to effect realistic flavor mixing within this setup, we finally
introduce additional vector-like multiplets to act as seesaw partners to the
SM fermions, analogous to the treatment of the charged leptons and quarks in
the model construction of [15]. Specifically, we find that adding a vector-
like multiplet in the
$(\mathbf{4},\mathbf{1},\mathbf{2},\mathbf{4},-1/4)+(\mathbf{\overline{4}},\mathbf{1},\mathbf{2},\mathbf{\overline{4}},+1/4)$
representation to the matter content of Eq.(2) allows us, with appropriate
selections for the model’s scalar content, to exhibit the observed mass
hierarchy for quarks and charged leptons, as well as the SM CKM matrix. We
also note that in order to ensure the existence of light, SM-like fermions,
certain mass terms will have to be forbidden by imposing a discrete $Z_{2}$
symmetry on our model, discussed in detail in Section 2.3. Our final matter
content can then be written as

$\displaystyle(\mathbf{4},\mathbf{2},\mathbf{1},\mathbf{4},-1/4)+(\mathbf{\overline{4}},\mathbf{1},\mathbf{2},\mathbf{\overline{4}},+1/4)$
$\displaystyle+(\mathbf{\overline{4}},\mathbf{2},\mathbf{1},\mathbf{1},+1)+(\mathbf{4},\mathbf{1},\mathbf{2},\mathbf{1},-1)$
(4)
$\displaystyle+(\mathbf{4},\mathbf{1},\mathbf{2},\mathbf{4},-1/4)+(\mathbf{\overline{4}},\mathbf{1},\mathbf{2},\mathbf{\overline{4}},+1/4).$

For convenience, we have listed this matter content, labelled by Pati-Salam
and SM multiplets, in Table 1. We have also labelled the various multiplets in
Table 1 based on their SM quantum numbers and their role in the theory:
$q_{L}$, $d^{c}_{R}$, and $u^{c}_{R}$ represent (up to small mass mixing with
vector-like fermions) $SU(2)_{L}$ doublet, down-like $SU(2)_{L}$ singlet, and
up-like $SU(2)_{L}$ singlet quarks, respectively, while $l_{L}$, $e^{c}_{R}$,
and $\nu^{c}_{R}$ represent the same for $SU(2)_{L}$ doublet leptons, charged
$SU(2)_{L}$ singlet leptons, and sterile neutrinos respectively. The
multiplets labelled by capitalized letters represent (again up to small mass
mixing) the new vector-like particles in the theory, which act as portal
matter and serve as seesaw partners to the SM states.

The addition of the new vector-like states in Eq.(4), however, raises two
issues that warrant some discussion. First, we note that these extra states
vitiate any discrete left-right symmetry that previously existed in the model,
and which is frequently assumed in Pati-Salam constructions. Our selection
here is ultimately one of convenience: The additional vector-like fields
represent the _minimal_ additions we need to effect the observed SM flavor
structure, at least via the method outlined in Section 2.3. In principle,
nothing prevents us from adding a corresponding vector-like $SU(2)_{L}$
multiplet to restore the left-right discrete symmetry. In practice, however,
the addition of these states significantly complicates the numerical structure
of the model, so for our purposes here we simply include the extra $SU(2)_{R}$
vector-like multiplet. If we wish to maintain a left-right discrete symmetry,
we can assume that the particles of the corresponding $SU(2)_{L}$ vector-like
multiplet are significantly more massive than those of the $SU(2)_{R}$
multiplet, and therefore do not have a significant numerical effect on the
theory at low energy. We also note that even if we were to include an
$SU(2)_{L}$ multiplet with similar particle masses to those of its $SU(2)_{R}$
counterpart, the effect on low-energy phenomenology would be minimal; we shall
see in Section 2.3 that the natural masses of the particles in the extra
vector-like multiplets are so high that only one of these vector-like
particles, acting as a seesaw partner to the top quark, will be accessible at
the LHC or any likely future planned colliders. We therefore can determine
that the effect of adding an extra vector-like multiplet to the theory to
restore left-right symmetry would, even if the masses of the particles in the
multiplet were comparable, likely only result in the introduction of a single
additional particle, a vector-like $SU(2)_{L}$ doublet quark, at accessible
energies. Given the substantial increase in numerical complexity coupled with
relatively limited phenomenological impact, we leave an exploration of the
strict enforcement of discrete left-right symmetry in this model to future
work.

We also note that the fermion content of Eq.(4) pushes us over the constraint
for asymptotically free QCD: If the Pati-Salam $SU(4)_{c}$ symmetry is broken
down to the SM QCD at a higher scale than any of these fermions acquire mass,
then the theory has 18 flavors, while QCD only remains asymptotically free for
16 or fewer flavors. While we wish to focus our efforts on the low-energy
implications of this model, rather than the complexities associated with its
full UV completion, the abandonment of asymptotic freedom does merit some
justification. We might note that as long as we don’t introduce any further
colored fermions (as we would need to in order to, for example, enforce a
discrete left-right symmetry), the matter content of Eq.4 should result in a
confined $SU(4)_{c}$ theory (even accounting for the existence of an
$SU(4)_{c}$ adjoint scalar to break this symmetry). In fact, using the rough
order-of-magnitude estimates for the various fermionic masses in the theory
given in Section 2.3, we find that $\alpha_{s}$ remains comfortably
perturbative even if $SU(4)_{c}$ is broken as high as the Planck scale, due to
the fact that a number of fermion fields must have masses in the high multi-
TeV range in order to recreate the observed fermion mass hierarchy. Given that
the Pati-Salam symmetry is generally broken at significantly sub-Planckian
energies [16, 17, 18], we can safely assume that for this construction, our
abandonment of asymptotic freedom does not present a severe difficulty. For
more general cases, we also note that it has been conjectured that it’s not
unreasonable to abandon asymptotic freedom at some intermediate scales, such
as the TeV-scale, in exchange for asymptotic _safety_ , where a UV interacting
fixed point is reinstated by physics at some high energy before the Planck
scale [19]. As our discussion in this paper is limited to physics well below
the Planck scale, we won’t conjecture as to how this asymptotic freedom is
restored, and only note that such a restoration may be possible, even in
setups with an even greater number of excess fermions than the minimal
construction we employ here.

Fermion |  $4_{c}2_{L}2_{R}4_{F}1_{F}$ Multiplet | SM Label | $SU(3)_{c}$ | $Y/2$ | $SU(3)_{F}$ | $U(1)_{D}$
---|---|---|---|---|---|---
$\Psi_{a}$ | $(\mathbf{4},\mathbf{2},\mathbf{1},\mathbf{4},-1/4)$ | $q_{L}$ | $\mathbf{3}$ | $+1/6$ | $\mathbf{3}$ | 0
$l_{L}$ | $\mathbf{1}$ | $-1/2$ | $\mathbf{3}$ | 0
$Q_{L}$ | $\mathbf{3}$ | $+1/6$ | $\mathbf{1}$ | -1
$L_{L}$ | $\mathbf{1}$ | $-1/2$ | $\mathbf{1}$ | -1
$\Psi_{b}$ | $(\mathbf{\overline{4}},\mathbf{2},\mathbf{1},\mathbf{1},+1)$ | $Q_{R}^{c}$ | $\mathbf{\overline{3}}$ | $-1/6$ | $\mathbf{1}$ | +1
$L_{R}^{c}$ | $\mathbf{1}$ | $+1/2$ | $\mathbf{1}$ | +1
$\Psi_{c}$ | $(\mathbf{\overline{4}},\mathbf{1},\mathbf{2},\mathbf{\overline{4}},+1/4)$ | $d_{R}^{c}$ | $\mathbf{\overline{3}}$ | +1/3 | $\mathbf{\overline{3}}$ | 0
$u_{R}^{c}$ | $\mathbf{\overline{3}}$ | -2/3 | $\mathbf{\overline{3}}$ | 0
$e_{R}^{c}$ | $\mathbf{1}$ | +1 | $\mathbf{\overline{3}}$ | 0
$\nu_{R}^{c}$ | $\mathbf{1}$ | 0 | $\mathbf{\overline{3}}$ | 0
$(D_{1})_{R}^{c}$ | $\mathbf{\overline{3}}$ | +1/3 | $\mathbf{1}$ | +1
$(U_{1})_{R}^{c}$ | $\mathbf{\overline{3}}$ | -2/3 | $\mathbf{1}$ | +1
$(E_{1})_{R}^{c}$ | $\mathbf{1}$ | +1 | $\mathbf{1}$ | +1
$(N_{1})_{R}^{c}$ | $\mathbf{1}$ | 0 | $\mathbf{1}$ | +1
$\Psi_{d}$ | $(\mathbf{4},\mathbf{1},\mathbf{2},\mathbf{1},-1)$ | $(D_{1})_{L}$ | $\mathbf{3}$ | -1/3 | $\mathbf{1}$ | -1
$(U_{1})_{L}$ | $\mathbf{3}$ | +2/3 | $\mathbf{1}$ | -1
$(E_{1})_{L}$ | $\mathbf{1}$ | -1 | $\mathbf{1}$ | -1
$(N_{1})_{L}$ | $\mathbf{1}$ | 0 | $\mathbf{1}$ | -1
$\Psi_{e}$ | $(\mathbf{4},\mathbf{1},\mathbf{2},\mathbf{4},-1/4)$ | $(D_{2})_{L}$ | $\mathbf{3}$ | -1/3 | $\mathbf{3}$ | 0
$(U_{2})_{L}$ | $\mathbf{3}$ | +2/3 | $\mathbf{3}$ | 0
$(E_{2})_{L}$ | $\mathbf{1}$ | -1 | $\mathbf{3}$ | 0
$(N_{2})_{L}$ | $\mathbf{1}$ | 0 | $\mathbf{3}$ | 0
$(D_{3})_{L}$ | $\mathbf{3}$ | -1/3 | $\mathbf{1}$ | -1
$(U_{3})_{L}$ | $\mathbf{3}$ | +2/3 | $\mathbf{1}$ | -1
$(E_{3})_{L}$ | $\mathbf{1}$ | -1 | $\mathbf{1}$ | -1
$(N_{3})_{L}$ | $\mathbf{1}$ | 0 | $\mathbf{1}$ | -1
$\Psi_{f}$ | $(\mathbf{\overline{4}},\mathbf{1},\mathbf{2},\mathbf{\overline{4}},+1/4)$ | $(D_{2})_{R}^{c}$ | $\mathbf{\overline{3}}$ | +1/3 | $\mathbf{\overline{3}}$ | 0
$(U_{2})_{R}^{c}$ | $\mathbf{\overline{3}}$ | -2/3 | $\mathbf{\overline{3}}$ | 0
$(E_{2})_{R}^{c}$ | $\mathbf{1}$ | +1 | $\mathbf{\overline{3}}$ | 0
$(N_{2})_{R}^{c}$ | $\mathbf{1}$ | 0 | $\mathbf{\overline{3}}$ | 0
$(D_{3})_{R}^{c}$ | $\mathbf{\overline{3}}$ | +1/3 | $\mathbf{1}$ | +1
$(U_{3})_{R}^{c}$ | $\mathbf{\overline{3}}$ | -2/3 | $\mathbf{1}$ | +1
$(E_{3})_{R}^{c}$ | $\mathbf{1}$ | +1 | $\mathbf{1}$ | +1
$(N_{3})_{R}^{c}$ | $\mathbf{1}$ | 0 | $\mathbf{1}$ | +1
Table 1: The fermion content of the model, grouped by their representation
under $4_{c}2_{L}2_{R}4_{F}1_{F}$

###  2.2 Scalar Content

In order to break the gauge symmetry in our model from its original GUT-scale
$4_{c}2_{L}2_{R}4_{F}1_{F}$ to the SM gauge group, and then down to
$3_{c}1_{\textrm{em}}$, as well as generate the appropriate spectrum for SM
fermions, we must posit the existence of a significant number of scalar
fields. For simplicity, we shall assume that the Pati-Salam symmetries
$SU(4)_{c}$ and $SU(2)_{R}$ are broken at some exceedingly high scale (or
scales). We shall see that this is not in fact an unreasonable assumption – if
we were to assume a discrete left-right symmetry, for example, renormalization
group running of the $SU(2)_{L}$ and $SU(2)_{R}$ in a minimal Pati-Salam model
suggests symmetry breaking around $\sim 10^{13-14}\;\textrm{GeV}$ [18, 16,
17], while the highest scales we shall require for breaking the
$SU(4)_{F}\times U(1)_{F}$ symmetry and reproducing the fermion spectra and
mixings will not exceed $O(10^{8-9}\;\textrm{GeV})$. Assuming the Pati-Salam
symmetry is broken, then, we are now left with the need for scalars which
break the $SU(4)_{F}\times U(1)_{F}$ symmetry down to a dark $U(1)$, which we
shall call $U(1)_{D}$, which is then broken entirely at some low scale at
roughly $O(0.1-1\;\textrm{GeV})$, as well as some scalar content to perform
the role of the SM Higgs, breaking $2_{L}1_{Y}\rightarrow 1_{\textrm{em}}$. To
accomplish these tasks, we posit 5 Higgs scalars, given in the
$4_{c}2_{L}2_{R}4_{F}1_{F}$ representations as

$\displaystyle\Phi_{A}\sim(\mathbf{1},\mathbf{1},\mathbf{1},\mathbf{15},0),$
$\displaystyle\Phi_{B}\sim(\mathbf{1},\mathbf{1},\mathbf{3},\mathbf{15},0),$
$\displaystyle\Phi_{S}\sim(\mathbf{1},\mathbf{1},\mathbf{1},\mathbf{1},0),$
(5)
$\displaystyle\Phi_{P}\sim(\mathbf{1},\mathbf{1},\mathbf{1},\mathbf{4},+3/4),$
$\displaystyle H\sim(\mathbf{1},\mathbf{2},\mathbf{2},\mathbf{1},0).$

For convenience, we have listed these scalars, along with rough orders of
magnitude for their vacuum expectation values necessary to achieve the
observed SM quark mass spectrum (which we shall derive in Section 2.3), in
Table 2. The scalar content that we have selected here can be easily deduced
from phenomenological considerations: The scalars $\Phi_{A}$, $\Phi_{B}$,
$\Phi_{S}$, and $H$ are all immediately required for a treatment of the quark
masses analogous to that of [15], generalized to treat our larger model. In
[15], the masses and mixings of the quarks and leptons are reproduced with the
aid of a traditional SM Higgs doublet, two $SU(3)_{F}$ adjoint scalars, and a
singlet. To adapt the construction of that work to the present scenario, we
require that the $SU(3)_{F}$ adjoints in that work are promoted to $SU(4)_{F}$
adjoints, that one of the two $SU(4)_{F}$ adjoints also be a triplet of
$SU(2)_{R}$ in order to effect the large discrepancy between up-like and down-
like quark masses (and the non-trivial CKM matrix) without relying on
renormalization group evolution, and that the scalar $H$ containing the SM
Higgs field is promoted to a bidoublet under $SU(2)_{L}\times SU(2)_{R}$. It
is useful to note that the scalar content of Eq.(2.2) does not contain any
scalars that are charged under both $U(1)_{Y}$ and $U(1)_{F}$. Loop-level
kinetic mixing between these two $U(1)$ symmetries, then, such as that
mediated by the fermion content in our model, does NOT occur in the scalar
sector. In the scalar content of Table 2, the addition of the singlet
$\Phi_{S}$ may seem superfluous, and so merits some further clarification. As
we shall see in Section 2.3, the most general set of Yukawa couplings that can
be written with the contents of Tables 1 and 2 does _not_ reproduce the
fermion spectrum accurately. In order to forbid undesirable Yukawa terms, the
authors of [15] impose a $Z_{2}$ parity under which certain fermion fields are
odd and others are even. However, in both this work and [15], this $Z_{2}$
parity forbids one mass term that is still necessary for the recreation of
realistic fermion masses, and so this term is reintroduced with the
$Z_{2}$-odd scalar $\Phi_{S}$.

Scalar | $SU(2)_{L}$ | $SU(2)_{R}$ | $SU(4)_{F}$ | $U(1)_{F}$ | $\langle\Phi\rangle$
---|---|---|---|---|---
$\Phi_{A}$ | $\mathbf{1}$ | $\mathbf{1}$ | $\mathbf{15}$ | $0$ | $\begin{pmatrix}\langle A\rangle&\vec{\gamma}_{A}\\\ \vec{\gamma}_{A}^{\dagger}&-Tr[\langle A\rangle]\end{pmatrix}\sim\begin{pmatrix}10^{3}-10^{9}\;\textrm{GeV}&0.1-1\;\textrm{GeV}\\\ 0.1-1\;\textrm{GeV}&10^{9}\;\textrm{GeV}\end{pmatrix}$
$\Phi_{B}$ | $\mathbf{1}$ | $\mathbf{3}$ | $\mathbf{15}$ | 0 | $\frac{\sigma_{3}}{2}\otimes\begin{pmatrix}\langle B\rangle&\vec{\gamma}_{B}\\\ \vec{\gamma}_{B}^{\dagger}&-Tr[\langle B\rangle]\end{pmatrix}\sim\begin{pmatrix}10^{3}-10^{9}\;\textrm{GeV}&0.1-1\;\textrm{GeV}\\\ 0.1-1\;\textrm{GeV}&10^{9}\;\textrm{GeV}\end{pmatrix}$
$\Phi_{P}$ | $\mathbf{1}$ | $\mathbf{1}$ | $\mathbf{4}$ | +3/4 | $(\vec{\gamma}_{P},v_{P})\sim(0.1-1\;\textrm{GeV},\;10^{3}-10^{4}\;\textrm{GeV})$
$\Phi_{S}$ | $\mathbf{1}$ | $\mathbf{1}$ | $\mathbf{1}$ | 0 | $v_{S}\sim 10^{3}-10^{4}\;\textrm{GeV}$
$H$ | $\mathbf{2}$ | $\mathbf{2}$ | $\mathbf{1}$ | 0 | $\frac{v}{\sqrt{2}}\begin{pmatrix}\cos\beta&0\\\ 0&\sin\beta\end{pmatrix}$
Table 2: The scalars introduced to break the $4_{F}1_{F}$ flavor symmetry and
provide masses to the fermions. Here, $\langle A\rangle$ and $\langle
B\rangle$ are $3\times 3$ Hermitian matrices, $\vec{\gamma}_{A,B,P}$ are
3-component complex vectors, and $v_{P}$, $v_{S}$, and $v_{EW}$ are simply
real vevs, with $v\approx 246\;\textrm{GeV}$ being the SM Higgs vev. In the
definition of $\Phi_{B}$’s vev, $\sigma_{3}$ denotes the third Pauli matrix.

It is helpful here to use the labelling of Table 2 to illustrate the rough
pattern of symmetry breaking from the scale where the Pati-Salam group is
broken down to the SM group $3_{c}2_{L}1_{Y}$ down to the scale where the dark
photon gauge symmetry $U(1)_{D}$ is broken – or in other words, how the
$SU(4)_{F}\times U(1)_{F}$ symmetry is broken.

$\displaystyle SU(4)_{F}\times U(1)_{F}\xrightarrow{\langle
A,B\rangle}U(1)_{F}^{\prime}\times
U(1)_{F}\xrightarrow{v_{P}}U(1)_{D}\xrightarrow{\vec{\gamma}_{A,B,P}}\textrm{Nothing}.$
(6)

In Eq.(6), we have simply listed how the various vev terms from Table 2 break
the $SU(4)_{F}\times U(1)_{F}$ symmetry down, arranged by the rough scale at
which each breaking occurs. Consulting Table 2, we can see that $SU(4)_{F}$ is
broken by $\langle A\rangle$ and $\langle B\rangle$ down to
$U(1)_{F}^{\prime}$ at a series of scales spanning between $\sim
10^{9}\;\textrm{GeV}$ and $10^{3}\;\textrm{GeV}$ (in practice, we shall that
this symmetry breaking usually completes at a substantially higher scale than
$10^{3}\;\textrm{GeV}$), while $v_{P}$ then breaks $U(1)_{F}^{\prime}\times
U(1)_{F}$ down to $U(1)_{D}$ at a scale of roughly $10^{3-4}$ GeV, and finally
the small vev’s $\vec{\gamma_{A,B,P}}$ break $U(1)_{D}$ entirely at a scale of
$\sim 0.1-1\;\textrm{GeV}$. The contents of Table 2 highlights two notable
characteristics of our vev arrangements. First, there exist substantial
hierarchies between the various vev scales in the model, the values of which
span nine orders of magnitude. As we’ll see in Section 2.3, these hierarchies
are necessary to recreate the observed hierarchy in SM quark masses, as well
as ensure that the dark photon remains at roughly the scale of $\sim
1\;\textrm{GeV}$. Due to the significant complexity of the scalar sector here,
with numerous possible potential terms, a detailed exploration on the
nauralness of these hierarchies is beyond the scope of this paper, however it
is clear that any potential which might produce these vev’s is likely quite
finely tuned.

The second notable characteristic of our vev’s evinced in Table 2 is that the
vacuum expectation values of certain components of the scalar fields are
exceedingly large (we remind the reader that even the largest scales here,
$\sim 10^{9}\;\textrm{GeV}$, are still well below what we can anticipate for
the breaking of the symmetries $SU(4)_{c}$ and $SU(2)_{R}$, which as noted
before we assume to be $\sim 10^{13-14}\;\textrm{GeV}$). As a result, we can
anticipate (and, in Sections 2.3 and 2.4, prove) that much of the new physics
which shall arise in this model appears at scales far in excess of what can be
directly probed in the foreseeable future. The task left to us is therefore
identifying the elements of our model construction that yield new physics at
scales that we do expect to be probed in the near future. As with the problem
of quantitatively evaluating the vev arrangement’s tuning, performing this
task in detail is far beyond the scope of the present work, given the large
number of multiplets and the resulting high degree of complexity for the
scalar potential.

We can, however, briefly comment on some of the scalars that will likely be
relevant at accessible energies. First, we note that if the left-right
symmetry breaking scale is quite high (as we have already assumed), the SM
Higgs field is well-approximated as the sole element of the bidoublet $H$ that
achieves a non-zero vev, so that

$\displaystyle H=\begin{pmatrix}\phi_{1}^{0}&\phi_{2}^{+}\\\
\phi_{1}^{-}&-\phi_{2}^{0*}\end{pmatrix},\;\;\langle
H\rangle=\frac{v}{\sqrt{2}}\begin{pmatrix}\cos\beta&0\\\
0&\sin\beta\end{pmatrix},\;\;h_{SM}\approx\phi_{1}^{0}\cos\beta+\phi_{2}^{0}\sin\beta,$
(7)

where $h_{SM}$ denotes the real scalar identified with the SM Higgs boson, and
$v\approx 246\;\textrm{GeV}$ is the SM Higgs vev. In immediate analogy with
conventional left-right symmetric models, additional physical scalars
associated with the bidoublet $H$ will all have masses at approximately the
scale of $SU(2)_{R}$ breaking [20, 21, 22], and hence given our assumptions be
unobservable at present or immediately foreseeable experiments. Finally, we
note that to ensure perturbativity for the Yukawa couplings of the heavy
scalars arising from the bidoublet, again in analogy with well-known left-
right symmetric model building principles, we must also require that
$\tan\beta\mathrel{\mathchoice{\lower
3.01385pt\vbox{\halign{$\matth\displaystyle\hfil#\hfil$\cr<\crcr\sim\crcr}}}{\lower
3.01385pt\vbox{\halign{$\matth\textstyle\hfil#\hfil$\cr<\crcr\sim\crcr}}}{\lower
3.01385pt\vbox{\halign{$\matth\scriptstyle\hfil#\hfil$\cr<\crcr\sim\crcr}}}{\lower
3.01385pt\vbox{\halign{$\matth\scriptscriptstyle\hfil#\hfil$\cr<\crcr\sim\crcr}}}}0.8$
[21].333This specifically stems from the fact that the heavy doublet’s Yukawa
coupling to SM quarks is enhanced by $\sec(2\beta)$, so having $\tan\beta$
overly close to 1 will result in an unacceptably large magnitude of heavy
scalar couplings to the third-generation quarks. In principle, one might
accomodate this bound by having $\tan\beta\mathrel{\mathchoice{\lower
3.01385pt\vbox{\halign{$\matth\displaystyle\hfil#\hfil$\cr>\crcr\sim\crcr}}}{\lower
3.01385pt\vbox{\halign{$\matth\textstyle\hfil#\hfil$\cr>\crcr\sim\crcr}}}{\lower
3.01385pt\vbox{\halign{$\matth\scriptstyle\hfil#\hfil$\cr>\crcr\sim\crcr}}}{\lower
3.01385pt\vbox{\halign{$\matth\scriptscriptstyle\hfil#\hfil$\cr>\crcr\sim\crcr}}}}1.2$,
but this is identical to simply choosing an appropriate
$\tan\beta\mathrel{\mathchoice{\lower
3.01385pt\vbox{\halign{$\matth\displaystyle\hfil#\hfil$\cr<\crcr\sim\crcr}}}{\lower
3.01385pt\vbox{\halign{$\matth\textstyle\hfil#\hfil$\cr<\crcr\sim\crcr}}}{\lower
3.01385pt\vbox{\halign{$\matth\scriptstyle\hfil#\hfil$\cr<\crcr\sim\crcr}}}{\lower
3.01385pt\vbox{\halign{$\matth\scriptscriptstyle\hfil#\hfil$\cr<\crcr\sim\crcr}}}}0.8$
and changing the Yukawa coupling parameters, in particular the angle $\alpha$,
in the Yukawa action given in Eq.(8).

Apart from the SM Higgs, there is only one other scalar that we shall consider
here. Specifically, given the exceptionally low scale at which the symmetry
$U(1)_{D}$ is broken in comparison to the other scales of the theory, we
anticipate that some combination of the scalars described in this section will
emerge as a physical “dark Higgs” boson of mass $\sim
O(|\vec{\gamma}_{P,A,B}|)$, as the physical counterpart to the combination of
Goldstone bosons which become the longitudinal component of $A_{D}$– such a
scalar appears in, for example, the discussions of I and II. Because the
precise structure of the scalar sector is too complex to be able to probe in
detail in this work, we instead shall simply deduce the contribution of this
scalar to relevant processes (in particular, in Sections 5.1 and 5.2 when
discussing the decay processes of heavy exotic fermions) via the Equivalence
Theorem [23] where applicable, and comment briefly upon the potential effect
of these scalars where such a quantitative treatment fails.

###  2.3 Fermion Spectrum

In order to determine the structure of the model’s scalar vevs, we turn to the
CKM mixing matrix and the spectrum of SM quark masses for guidance. In order
to recreate the observed flavor structure of the SM, we present a construction
based heavily on that of [15]. In the unbroken theory, we shall write the
Yukawa sector of the model as

$\displaystyle\mathcal{L}_{Y}$
$\displaystyle=-y_{H}\cos\alpha\Psi_{f}^{T}i\sigma_{2}H\Psi_{a}-y_{H}\sin\alpha\Psi_{f}^{T}i\sigma_{2}\tilde{H}\Psi_{a}-y_{P1}\Psi_{b}^{T}i\sigma_{2}\Phi_{P}^{\dagger}\Psi_{a}-y_{P2}\Psi_{c}^{T}i\sigma_{2}\Phi_{P}\Psi_{d}$
(8) $\displaystyle-
y_{A}\Psi_{f}^{T}i\sigma_{2}\Phi_{A}\Psi_{e}-y_{B}\Psi_{f}^{T}i\sigma_{2}\Phi_{B}\Psi_{e}-M\Psi_{f}^{T}i\sigma_{2}\Psi_{e}-y_{S}\Psi_{c}^{T}i\sigma_{2}\Phi_{S}\Psi_{e}\;+h.c.$

Here, $\tilde{H}\equiv\sigma_{2}H^{*}\sigma_{2}$, $\sigma_{2}$ is the second
Pauli matrix, $\alpha$ is an arbitrary real angle between $-\pi$ and $\pi$,
and $M$ is an arbitrary mass term not forbidden by any symmetries. For
simplicity, we assume that all Yukawa coupling parameters here are real, as
would be the case in a model with no explicit CP violation. The expression in
Eq.(8) is not, a priori, the most general set of Yukawa couplings that can be
written with the scalar and fermion content of Tables 1 and 2. However as
mentioned in Section 2.2, any additional terms can be easily eliminated by
imposing a discrete $Z_{2}$ symmetry. Specifically, we outline our parity
assignments for the fermions and scalars in Table 3.

$Z_{2}$ Parity | Fermions | Scalars
---|---|---
+ | $\Psi_{a},\,\Psi_{b},\,\Psi_{e},\,\Psi_{f}$ | $H,\,\Phi_{P},\,\Phi_{A},\,\Phi_{B}$
- | $\Psi_{c},\,\Psi_{d}$ | $\Phi_{S}$
Table 3: The parity of the model’s fermion and scalar fields under $Z_{2}$, in
order to ensure that Eq.(8) is the most general set of Yukawa couplings that
we can write.

Following the parity assignments of Table 3, Eq.(8) becomes the most general
set of Yukawa couplings that we can construct. We can now write the up- and
down-like quark mass matrices explicitly by referencing Tables 1 and 2. The
mass terms for the up- and down-like quarks take the form

$\displaystyle\mathbf{\overline{U}}_{R}\mathcal{M}_{u}\mathbf{U}_{L},$ (9)
$\displaystyle\mathbf{\overline{D}}_{R}\mathcal{M}_{d}\mathbf{D}_{L},$

where

$\displaystyle\mathbf{\overline{U}}_{R}\equiv(\overline{u}_{R},(\overline{U}_{2})_{R},\overline{Q}_{R}^{u},(\overline{U}_{1})_{R},(\overline{U}_{3})_{R}),$
$\displaystyle\mathbf{U}_{L}\equiv(u_{L},(U_{2})_{L},Q_{L}^{u},(U_{1})_{L},(U_{3})_{L}),$
(10)
$\displaystyle\mathbf{\overline{D}_{R}}\equiv(\overline{d}_{R},(\overline{D}_{2})_{R},\overline{Q}_{R}^{d},(\overline{D}_{1})_{R},(\overline{D}_{3})_{R}),$
$\displaystyle\mathbf{D}_{L}\equiv(d_{L},(D_{2})_{L},Q_{L}^{d},(D_{1})_{L},(D_{3})_{L}).$

Here, $Q_{L,R}^{u(d)}$ denotes the up(down)-like component of the $SU(2)_{L}$
doublet $Q_{L,R}$. We also remind the reader that $u_{L,R}$ and
$(U_{2})_{L,R}$ are triplets of $SU(3)_{F}$, and therefore have three
components in flavor space, in contrast to $Q_{L,R}^{u,d}$, $(U_{1})_{L,R}$,
and $(U_{3})_{L,R}$, which each have only one. So, $\mathbf{U}_{L,R}$ and
$\mathbf{D}_{L,R}$ are both 9-component vectors in flavor space, where we have
grouped the $SU(3)_{F}$ triplets $u_{L,R}$ and $(U_{2})_{L,R}$ together in the
first 6 components, leaving the last 3 components for the $SU(3)_{F}$
singlets. Notably, the $SU(3)_{F}$ triplets are all uncharged under
$U(1)_{D}$, while the $SU(3)_{F}$ singlets all possess a common non-zero
charge under this symmetry. The mass matrices $\mathcal{M}_{u,d}$ are then
given by

$\displaystyle\mathcal{M}_{u}=\begin{pmatrix}\mathbf{0}_{3\times
3}&y_{S}v_{S}\mathbf{1}_{3\times 3}&\mathbf{0}_{3\times
1}&y_{P2}\vec{\gamma}_{P}&\mathbf{0}_{3\times 1}\\\
\frac{y_{u}v}{\sqrt{2}}\mathbf{1}_{3\times
3}&\mathbf{M}_{u}+M\mathbf{1}_{3\times 3}&\mathbf{0}_{3\times
1}&\mathbf{0}_{3\times 1}&\vec{\gamma}_{u}\\\
y_{P1}\vec{\gamma}_{P}^{\dagger}&\mathbf{0}_{1\times 3}&y_{P1}v_{P}&0&0\\\
\mathbf{0}_{1\times 3}&\mathbf{0}_{1\times 3}&0&y_{P2}v_{P}&y_{S}v_{S}\\\
\mathbf{0}_{1\times
3}&\vec{\gamma}_{u}^{\dagger}&\frac{y_{u}v}{\sqrt{2}}&0&-Tr[\mathbf{M}_{u}]+M\end{pmatrix},$
$\displaystyle\mathcal{M}_{d}=\begin{pmatrix}\mathbf{0}_{3\times
3}&y_{S}v_{S}\mathbf{1}_{3\times 3}&\mathbf{0}_{3\times
1}&y_{P2}\vec{\gamma}_{P}&\mathbf{0}_{3\times 1}\\\
\frac{y_{d}v}{\sqrt{2}}\mathbf{1}_{3\times
3}&\mathbf{M}_{d}+M\mathbf{1}_{3\times 3}&\mathbf{0}_{3\times
1}&\mathbf{0}_{3\times 1}&\vec{\gamma}_{d}\\\
y_{P1}\vec{\gamma}_{P}^{\dagger}&\mathbf{0}_{1\times 3}&y_{P1}v_{P}&0&0\\\
\mathbf{0}_{1\times 3}&\mathbf{0}_{1\times 3}&0&y_{P2}v_{P}&y_{S}v_{S}\\\
\mathbf{0}_{1\times
3}&\vec{\gamma}_{d}^{\dagger}&\frac{y_{d}v}{\sqrt{2}}&0&-Tr[\mathbf{M}_{d}]+M\end{pmatrix},$
$\displaystyle\mathbf{M}_{u}\equiv y_{A}\langle A\rangle+y_{B}\langle
B\rangle,\;\;\mathbf{M}_{d}\equiv y_{A}\langle A\rangle-y_{B}\langle
B\rangle,$ $\displaystyle\vec{\gamma}_{u}\equiv
y_{A}\vec{\gamma}_{A}+y_{B}\vec{\gamma}_{B},\;\;\vec{\gamma}_{d}\equiv
y_{A}\vec{\gamma}_{A}-y_{B}\vec{\gamma}_{B}.$ $\displaystyle y_{u}\equiv
y_{H}\cos(\alpha-\beta),\;\;\;y_{d}\equiv y_{H}\sin(\alpha+\beta),$

where we remind the reader that $v$, $\beta$, $v_{S}$, $v_{P}$,
$\vec{\gamma}_{P,A,B}$, $\langle A\rangle$, and $\langle B\rangle$ are defined
in Table 2, while all remaining parameters here appear in Eq.(8). In general,
we can exploit $SU(4)_{F}$ gauge freedom to render $\mathbf{M}_{u}$ or
$\mathbf{M}_{d}$ diagonal, and eliminate either $\vec{\gamma}_{u}$ (if we
diagonalize $\mathbf{M}_{u}$) or $\vec{\gamma}_{d}$ (if we diagonalize
$\mathbf{M}_{d}$). This corresponds to moving to a basis in which either the
combination of $\Phi_{A}$ and $\Phi_{B}$ vev’s which couple to the up-like or
down-like quarks are diagonal. Although we shall find it most convenient to
work in a basis in which $\mathbf{M}_{u}$ is diagonal and
$\vec{\gamma}_{u}=0$, for now we will make no such choice, so that our
analytical results for the up-like quarks will apply straightforwardly to the
down-like sector as well. Continuing our analysis, the matrices
$\mathcal{M}_{u}$ and $\mathcal{M}_{d}$ can be bi-diagonalized as usual to
produce mass eigenstates. That is, there exist unitary matrices
$\mathcal{U}_{L,R}^{u,d}$ such that

$\displaystyle(\mathcal{U}_{L}^{u})^{\dagger}\mathcal{M}_{u}^{\dagger}\mathcal{M}_{u}\mathcal{U}_{L}^{u}=(\mathcal{U}_{R}^{u})^{\dagger}\mathcal{M}_{u}\mathcal{M}_{u}^{\dagger}\mathcal{U}_{R}^{u}=diag(m_{u1}^{2},m_{u2}^{2},m_{u3}^{2},M_{u1}^{2},M_{u2}^{2},M_{u3}^{2},(m_{P1}^{u})^{2},(m_{P2}^{u})^{2},(m_{P3}^{u})^{2}),$
(12)

where the $m_{ui}$ denotes the mass of the $i^{th}$-generation up-like SM
quark, $M_{ui}$ is the mass of the (also uncharged under $U(1)_{D}$) heavy
vector-like partner to this quark, and $m_{P1,2,3}^{u}$ denote the masses of
the three portal matter fields which are charged under $U(1)_{D}$. An
analogous expression holds for $\mathcal{M}_{d}$. As an example, we shall now
determine the matrices $\mathcal{U}_{L,R}^{u}$, with a completely analogous
treatment holding for the determination of $\mathcal{U}_{L,R}^{d}$.

We start by determining $\mathcal{U}_{L}^{u}$. To begin, we find it useful to
define

$\displaystyle\mathbf{u}\equiv\mathbf{M}_{u}+M\mathbf{1}_{3\times
3},\;\;\;\mathbf{u}_{D}=diag(u_{1},u_{2},u_{3})=\mathbf{W}_{u}^{\dagger}\mathbf{u}\mathbf{W}_{u},\;\;\;X_{u}\equiv-
Tr[\mathbf{M}_{u}]+M,$ (13)

where $\mathbf{u}_{D}$ is the diagonalized form of $\mathbf{u}$ and
$\mathbf{W}_{u}$ is a $3\times 3$ unitary matrix. Then, we may write (dropping
terms of $O(\vec{\gamma}_{P,u}^{2})$, which shall be significantly smaller
than the other mass scales occurring in the matrix)

$\displaystyle\mathcal{M}_{u}^{\dagger}\mathcal{M}_{u}=\begin{pmatrix}\frac{y_{u}^{2}v_{u}^{2}}{2}\mathbf{1}_{3\times
3}&\frac{y_{u}v}{\sqrt{2}}\mathbf{u}&y_{P1}^{2}v_{P}\vec{\gamma}_{P}&\mathbf{0}_{3\times
1}&\frac{y_{u}v}{\sqrt{2}}\vec{\gamma}_{u}\\\
\frac{y_{u}v}{\sqrt{2}}\mathbf{u}&\mathbf{u}^{2}+y_{S}^{2}v_{S}^{2}\mathbf{1}_{3\times
3}&\frac{y_{u}v}{\sqrt{2}}\vec{\gamma_{u}}&y_{P2}y_{S}v_{S}\vec{\gamma}_{P}&(\mathbf{u}+X_{u}\mathbf{1}_{3\times
3})\cdot\vec{\gamma}_{u}\\\
y_{P1}^{2}v_{P}\vec{\gamma}_{P}^{\dagger}&\frac{y_{u}v}{\sqrt{2}}\vec{\gamma}_{u}^{\dagger}&y_{P1}^{2}v_{P}^{2}+\frac{y_{u}^{2}v^{2}}{2}&0&\frac{y_{u}v}{\sqrt{2}}X_{u}\\\
\mathbf{0}_{1\times
3}&y_{P2}y_{S}v_{S}\vec{\gamma}_{P}^{\dagger}&0&y_{P2}^{2}v_{P}^{2}&y_{P2}v_{P}y_{S}v_{S}\\\
\frac{y_{u}v}{\sqrt{2}}\vec{\gamma}_{u}^{\dagger}&\vec{\gamma}_{u}^{\dagger}\cdot(\mathbf{u}+X_{u}\mathbf{1}_{3\times
3})&\frac{y_{u}v}{\sqrt{2}}X_{u}&y_{P2}v_{P}y_{S}v_{S}&y_{S}^{2}v_{S}^{2}+X_{u}^{2}\end{pmatrix}.$
(14)

To derive expressions for the mass eigenvectors here, it is easiest to split
the problem into two basic parts. First, we shall determine the unitary matrix
$(\mathcal{U}_{L}^{u})_{0}$ which diagonalizes the matrix of Eq.(14) in the
limit where all components of $\vec{\gamma}_{P,u}$ are zero, that is, in the
absence of all mixing between the fields with no charge under $U(1)_{D}$
(namely, $u_{L}$, and $(U_{2})_{L}$, the first six rows/columns of Eq.(14))
and those with non-vanishing $U(1)_{D}$ charge ($Q_{L}^{u}$,$(U_{1})_{L}$, and
$(U_{3})_{L}$, the last three rows/columns). With this diagonalization done,
we can then determine the effects of $\vec{\gamma}_{P,u}$ as small
perturbations, in a rotation matrix $(\mathcal{U}_{L}^{u})_{\gamma}$. The
matrix which diagonalizes Eq.(14) is then simply given as

$\displaystyle\mathcal{U}_{L}^{u}\approx(\mathcal{U}_{L}^{u})_{0}(\mathcal{U}_{L}^{u})_{\gamma}.$
(15)

We now begin with our determination of $(\mathcal{U}_{L}^{u})_{0}$. Since
there’s no mixing between the $U(1)_{D}$-charged and $U(1)_{D}$-uncharged
fermions in this limit, we shall treat the two separately. We start with the
mass matrix for the fermions with no $U(1)_{D}$ charge, which takes the form,

$\displaystyle\begin{pmatrix}\frac{y_{u}^{2}v^{2}}{2}\mathbf{1}_{3\times
3}&\frac{y_{u}v}{\sqrt{2}}\mathbf{u}\\\
\frac{y_{u}v}{\sqrt{2}}\mathbf{u}&\mathbf{u}^{2}+y_{S}^{2}v_{S}^{2}\end{pmatrix}.$
(16)

We can break this matrix down into three $2\times 2$ matrices via rotation by
the unitary block matrix

$\displaystyle\begin{pmatrix}\mathbf{1}_{3\times 3}&\mathbf{0}_{3\times 3}\\\
\mathbf{0}_{3\times 3}&\mathbf{W}_{u}\end{pmatrix},$ (17)

where we recall that
$\mathbf{W}_{u}^{\dagger}\mathbf{u}\mathbf{W}_{u}=diag(u_{1},u_{2},u_{3})$.
Then, each $2\times 2$ matrix can be easily diagonalized, yielding (assuming
that $v\ll v_{S},u_{i}$)

$\displaystyle
diag(m_{ui}^{2},M_{ui}^{2})=\begin{pmatrix}\cos(\rho^{u}_{i})&\sin(\rho^{u}_{i})\\\
-\sin(\rho^{u}_{i})&\cos(\rho^{u}_{i})\end{pmatrix}\begin{pmatrix}\frac{y_{u}^{2}v^{2}}{2}&\frac{y_{u}v}{\sqrt{2}}u_{i}\\\
\frac{y_{u}v}{\sqrt{2}}u_{i}&u_{i}^{2}+y_{S}^{2}v_{S}^{2}\end{pmatrix}\begin{pmatrix}\cos(\rho^{u}_{i})&-\sin(\rho^{u}_{i})\\\
\sin(\rho^{u}_{i})&\cos(\rho^{u}_{i})\end{pmatrix}$
$\displaystyle\cos(\rho^{u}_{i})\approx
1-\frac{1}{2}\frac{m_{ui}^{2}}{M_{ui}^{2}}\bigg{(}\frac{M_{ui}^{2}-y_{S}^{2}v_{S}^{2}}{y_{S}^{2}v_{S}^{2}}\bigg{)},\;\;\sin(\rho^{u}_{i})\approx-
sgn(y_{u}u_{i})\frac{m_{ui}}{M_{ui}}\bigg{(}\frac{M_{ui}^{2}-y_{S}^{2}v_{S}^{2}}{y_{S}^{2}v_{S}^{2}}\bigg{)}^{1/2},$
(18) $\displaystyle
m_{ui}^{2}\approx\frac{y_{u}^{2}v^{2}}{2}\frac{y_{S}^{2}v_{S}^{2}}{(u_{i}^{2}+y_{S}^{2}v_{S}^{2})},\;\;M_{ui}^{2}\approx
u_{i}^{2}+y_{S}^{2}v_{S}^{2}$

Having diagonalized the mass matrix for the left-handed quarks with no charge
under $U(1)_{D}$, we next turn our attention to the left-handed fields which
are charged under $U(1)_{D}$. In the case of the up-like quark sector, these
are $Q_{L}^{u}$, $(U_{1})_{L}$, and $(U_{3})_{L}$. From Eq.(14), we see that
the mass matrix for these fields is

$\displaystyle\begin{pmatrix}y_{P1}^{2}v_{P}^{2}+\frac{y_{u}^{2}v^{2}}{2}&0&\frac{y_{u}v}{\sqrt{2}}X_{u}\\\
0&y_{P2}^{2}v_{P}^{2}&(y_{S}v_{S})(y_{P2}v_{P})\\\
\frac{y_{u}v}{\sqrt{2}}X_{u}&(y_{S}v_{S})(y_{P2}v_{P})&y_{S}^{2}v_{S}^{2}+X_{u}^{2}\end{pmatrix}.$
(19)

In the absence of extreme fine-tuning, we shall see that natural Yukawa
couplings yield $X_{u}$ at roughly the scale $X_{u}\sim 10^{5}y_{S}v_{S}$, so
we can assume that $v_{S},v_{P}\ll X_{u}$. Any mixing between the three fields
of Eq.(19), then, is exceedingly small and may be treated perturbatively.
Dropping terms of
$O(v_{P}^{2}/X_{u}^{2},v_{S}^{2}/X_{u}^{2},v^{2}/X_{u}^{2})$, the mass matrix
of Eq.(19) can be diagonalized by a rotation by the matrix

$\displaystyle\mathbf{W}^{P}_{L}\equiv\begin{pmatrix}1&\alpha_{12}^{L}&\alpha_{13}^{L}\\\
-\alpha_{12}^{L}&1&\alpha_{23}^{L}\\\
-\alpha_{13}^{L}&-\alpha_{23}^{L}&1\end{pmatrix},$ (20)

where

$\displaystyle\alpha_{12}^{L}\equiv\frac{(y_{u}v)(y_{S}v_{S})(y_{P2}v_{P})}{\sqrt{2}X_{u}(y_{P1}^{2}-y_{P2}^{2})v_{P}^{2}},\;\;\;\alpha_{13}^{L}\equiv\frac{y_{u}v}{\sqrt{2}X_{u}},\;\;\;\alpha_{23}^{L}\equiv\frac{(y_{S}v_{S})(y_{P2}v_{P})}{X_{u}^{2}}.$
(21)

In the end, we arrive at

$\displaystyle
diag((m_{P1}^{u})^{2},(m_{P2}^{u})^{2},(m_{P3}^{u})^{2})=\mathbf{W}^{P\dagger}_{L}\begin{pmatrix}y_{P1}^{2}v_{P}^{2}+\frac{y_{u}^{2}v^{2}}{2}&0&\frac{y_{u}v}{\sqrt{2}}X_{u}\\\
0&y_{P2}^{2}v_{P}^{2}&(y_{S}v_{S})(y_{P2}v_{P})\\\
\frac{y_{u}v}{\sqrt{2}}X_{u}&(y_{S}v_{S})(y_{P2}v_{P})&y_{S}^{2}v_{S}^{2}+X_{u}^{2}\end{pmatrix}\mathbf{W}^{P}_{L},$
(22) $\displaystyle(m_{P1}^{u})^{2}\equiv
y_{P1}^{2}v_{P}^{2},\;\;\;(m_{P2}^{u})^{2}\equiv
y_{P2}^{2}v_{P}^{2},\;\;\;(m_{P3}^{u})^{2}\equiv X_{u}^{2}.$

We can now apply the combined rotations of Eqs.(17), (2.3), and (20) in order
to arrive at the full rotation matrix $(\mathcal{U}^{u}_{L})_{0}$. The
combined rotation matrix is

$\displaystyle(\mathcal{U}_{L}^{u})_{0}\equiv\begin{pmatrix}\mathbf{W}_{u}\mathbf{c}_{\rho}^{u}&-\mathbf{W}_{u}\mathbf{s}_{\rho}^{u}&\mathbf{0}_{3\times
1}&\mathbf{0}_{3\times 1}&\mathbf{0}_{3\times 1}\\\
\mathbf{W}_{u}\mathbf{s}_{\rho}^{u}&\mathbf{W}_{u}\mathbf{c}_{\rho}^{u}&\mathbf{0}_{3\times
1}&\mathbf{0}_{3\times 1}&\mathbf{0}_{3\times 1}\\\ \mathbf{0}_{1\times
3}&\mathbf{0}_{1\times 3}&1&\alpha_{12}^{L}&\alpha_{13}^{L}\\\
\mathbf{0}_{1\times 3}&\mathbf{0}_{1\times
3}&-\alpha_{12}^{L}&1&\alpha_{23}^{L}\\\ \mathbf{0}_{1\times
3}&\mathbf{0}_{1\times 3}&-\alpha_{13}^{L}&-\alpha_{23}^{L}&1\end{pmatrix},$
(23) $\displaystyle\mathbf{c}_{\rho}^{u}\equiv
diag(\cos(\rho_{1}^{u}),\cos(\rho_{2}^{u}),\cos(\rho_{3}^{u})),\;\;\;\mathbf{s}_{\rho}^{u}\equiv
diag(\sin(\rho_{1}^{u}),\sin(\rho_{2}^{u}),\sin(\rho_{3}^{u})),$

where we remind the reader that $\mathbf{W}_{u}$ is defined in Eq.(13),
$\rho^{u}_{1}$, $\rho^{u}_{2}$, and $\rho^{u}_{3}$ are given in Eq.(2.3), and
the variables $\alpha_{ij}^{L}$ are defined in Eq.(21). The mass matrix of
Eq.(14) then takes the form

$\displaystyle(\mathcal{U}^{u}_{L})_{0}^{\dagger}\mathcal{M}_{u}^{\dagger}\mathcal{M}_{u}(\mathcal{U}^{u}_{L})_{0}=diag(m_{u}^{2},m_{c}^{2},m_{t}^{2},M_{u}^{2},M_{c}^{2},M_{t}^{2},(m_{P1}^{u})^{2},(m_{P2}^{u})^{2},(m_{P3}^{u})^{2})+O(\vec{\gamma}_{u},\vec{\gamma}_{P}).$
(24)

It is now straightforward to determine the perturbed eigenvectors of the
matrix in Eq.(24) up to $O(\vec{\gamma}_{u},\vec{\gamma}_{P})$. Once the dust
settles, we have

$\displaystyle(\mathcal{U}^{u}_{L})_{\gamma}\approx\begin{pmatrix}\mathbf{1}_{3\times
3}&\mathbf{0}_{3\times
3}&\vec{\Delta}_{P1L}^{u}&\vec{\Delta}_{P2L}^{u}&\vec{\Delta}_{P3L}^{u}\\\
\mathbf{0}_{3\times 3}&\mathbf{1}_{3\times
3}&\vec{\Delta}_{P1L}^{U}&\vec{\Delta}_{P2L}^{U}&\vec{\Delta}_{P3L}^{U}\\\
-\vec{\Delta}_{P1L}^{u\dagger}&-\vec{\Delta}_{P1L}^{U\dagger}&1&0&0\\\
-\vec{\Delta}_{P2L}^{u\dagger}&-\vec{\Delta}_{P2L}^{U\dagger}&0&1&0\\\
-\vec{\Delta}_{P3L}^{u\dagger}&-\vec{\Delta}_{P3L}^{U\dagger}&0&0&1\end{pmatrix},$
(25)

where

$\displaystyle(\vec{\Delta}_{P1L}^{u})_{i}=\frac{((y_{P1}^{2}v_{P}\mathbf{c}_{\rho}^{u}-\alpha_{12}^{L}y_{P2}y_{S}v_{S}\mathbf{s}_{\rho}^{u})\mathbf{W}_{u}^{\dagger}\vec{\gamma}_{P}+(\frac{y_{u}v}{\sqrt{2}}\mathbf{s}_{\rho}^{u}-\alpha_{13}^{L}(\frac{y_{u}v}{\sqrt{2}}\mathbf{c}_{\rho}^{u}+\mathbf{u}_{D}\mathbf{s}_{\rho}^{u}+X_{u}\mathbf{s}_{\rho}^{u}))\mathbf{W}_{u}^{\dagger}\vec{\gamma}_{u})_{i}}{m_{P1}^{2}-m_{ui}^{2}},$
$\displaystyle(\vec{\Delta}_{P2L}^{u})_{i}=\frac{((\alpha_{12}^{L}y_{P1}^{2}v_{P}\mathbf{c}_{\rho}^{u}+y_{P2}y_{S}v_{S}\mathbf{s}_{\rho}^{u})\mathbf{W}_{u}^{\dagger}\vec{\gamma}_{P}+(\alpha_{12}^{L}\frac{y_{u}v}{\sqrt{2}}\mathbf{s}_{\rho}^{u}-\alpha_{23}^{L}(\frac{y_{u}v}{\sqrt{2}}\mathbf{c}_{\rho}^{u}+\mathbf{u}_{D}\mathbf{s}_{\rho}^{u}+X_{u}\mathbf{s}_{\rho}^{u}))\mathbf{W}_{u}^{\dagger}\vec{\gamma}_{u})_{i}}{m_{P2}^{2}-m_{ui}^{2}}$
$\displaystyle(\vec{\Delta}_{P3L}^{u})_{i}=\frac{((\alpha_{13}^{L}y_{P1}^{2}v_{P}\mathbf{c}_{\rho}^{u}+\alpha_{23}^{L}y_{P2}y_{S}v_{S}\mathbf{s}_{\rho}^{u})\mathbf{W}_{u}^{\dagger}\vec{\gamma}_{P}+(\alpha_{13}^{L}\frac{y_{u}v}{\sqrt{2}}\mathbf{s}_{\rho}^{u}+\frac{y_{u}v}{\sqrt{2}}\mathbf{c}_{\rho}^{u}+\mathbf{u}_{D}\mathbf{s}_{\rho}^{u}+X_{u}\mathbf{s}_{\rho}^{u})\mathbf{W}^{\dagger}_{u}\vec{\gamma}_{u})_{i}}{m_{P3}^{2}-m_{ui}^{2}}$
(26)
$\displaystyle(\vec{\Delta}_{P1L}^{U})_{i}=\frac{((-y_{P1}^{2}v_{P}\mathbf{s}_{\rho}^{u}-\alpha_{12}^{L}y_{P2}y_{S}v_{S}\mathbf{c}_{\rho}^{u})\mathbf{W}_{u}^{\dagger}\vec{\gamma}_{P}+(\frac{y_{u}v}{\sqrt{2}}\mathbf{c}_{\rho}^{u}-\alpha_{13}^{L}(\mathbf{u}_{D}\mathbf{c}_{\rho}^{u}+X_{u}\mathbf{c}_{\rho}^{u}-\frac{y_{u}v}{\sqrt{2}}\mathbf{s}_{\rho}^{u}))\mathbf{W}_{u}^{\dagger}\vec{\gamma}_{u})_{i}}{(m_{P1}^{u})^{2}-M_{ui}^{2}}$
$\displaystyle(\vec{\Delta}_{P2L}^{U})_{i}=\frac{((-\alpha_{12}^{L}y_{P1}^{2}v_{P}\mathbf{s}_{\rho}^{u}+y_{P2}y_{S}v_{S}\mathbf{c}_{\rho}^{u})\mathbf{W}_{u}^{\dagger}\vec{\gamma}_{P}+(\alpha_{12}^{L}\frac{y_{u}v}{\sqrt{2}}\mathbf{c}_{\rho}^{u}-\alpha_{23}^{L}(\mathbf{u}_{D}\mathbf{c}_{\rho}^{u}+X_{u}\mathbf{c}_{\rho}^{u}-\frac{y_{u}v}{\sqrt{2}}\mathbf{s}_{\rho}^{u}))\mathbf{W}_{u}^{\dagger}\vec{\gamma}_{u})_{i}}{(m_{P2}^{u})^{2}-M_{ui}^{2}},$
$\displaystyle(\vec{\Delta}_{P3L}^{u})_{i}=\frac{((-\alpha_{13}^{L}y_{P1}^{2}v_{P}\mathbf{s}_{\rho}^{u}+\alpha_{23}^{L}y_{P2}y_{S}v_{S}\mathbf{c}_{\rho}^{u})\mathbf{W}_{u}^{\dagger}\vec{\gamma}_{P}+(\alpha_{13}^{L}\frac{y_{u}v}{\sqrt{2}}\mathbf{c}_{\rho}^{u}+\mathbf{u}_{D}\mathbf{c}_{\rho}^{u}+X_{u}\mathbf{c}_{\rho}^{u}-\frac{y_{u}v}{\sqrt{2}}\mathbf{s}_{\rho}^{u})\mathbf{W}_{u}^{\dagger}\vec{\gamma}_{u})_{i}}{(m_{P3}^{u})^{2}-M_{ui}^{2}},$

where $\mathbf{u}_{D}$ is the diagonalized form of $\mathbf{u}$ discussed in
Eq.(13).

The lengthy expressions of Eq.(2.3), while extremely cumbersome in their full
form, can be dramatically simplified by examining the numerical hierarchies
between various parameters, and dropping all but the numerically dominant
contributions to each expression. To get a better sense of the numerics of the
system, we begin by exploring the $SU(3)_{F}$ triplet sector, containing the
SM quarks and their heavy partners. Examining Eq.(2.3), we note that the
expressions for $m_{ui}$ and $M_{ui}$ only hold here up to
$O(v^{2}/v_{S}^{2})$. To maintain numerical accuracy, we then assume that
$y_{S}v_{S}\mathrel{\mathchoice{\lower
3.01385pt\vbox{\halign{$\matth\displaystyle\hfil#\hfil$\cr>\crcr\sim\crcr}}}{\lower
3.01385pt\vbox{\halign{$\matth\textstyle\hfil#\hfil$\cr>\crcr\sim\crcr}}}{\lower
3.01385pt\vbox{\halign{$\matth\scriptstyle\hfil#\hfil$\cr>\crcr\sim\crcr}}}{\lower
3.01385pt\vbox{\halign{$\matth\scriptscriptstyle\hfil#\hfil$\cr>\crcr\sim\crcr}}}}1\;\textrm{TeV}$–
given that $v\sim 246\;\textrm{GeV}$, this will ensure that the expressions
given here are accurate to within a few percent. This also ensures that there
is a significant hierarchy between $m_{ui}$ and $M_{ui}$, as can be seen from
the relation
$m_{ui}^{2}/M_{u_{i}}^{2}=2m_{ui}^{4}/(y_{S}^{2}v_{S}^{2}y_{u}^{2}v^{2})$
(which holds exactly in the limit that $\vec{\gamma}_{u,P}\rightarrow 0$). If
$y_{S}v_{S}$ were much smaller than 1 TeV, the heavy counterpart of the top
quark would have roughly the same mass as the SM top, and be subject to
extremely severe constraints from direct production searches at the LHC and
from modifications to top quark couplings; the former limit the mass of such a
vector-like quark to $\mathrel{\mathchoice{\lower
3.01385pt\vbox{\halign{$\matth\displaystyle\hfil#\hfil$\cr>\crcr\sim\crcr}}}{\lower
3.01385pt\vbox{\halign{$\matth\textstyle\hfil#\hfil$\cr>\crcr\sim\crcr}}}{\lower
3.01385pt\vbox{\halign{$\matth\scriptstyle\hfil#\hfil$\cr>\crcr\sim\crcr}}}{\lower
3.01385pt\vbox{\halign{$\matth\scriptscriptstyle\hfil#\hfil$\cr>\crcr\sim\crcr}}}}1.3-1.5\;\textrm{TeV}$,
due to null results in searches for pair production [24]. We can also use the
relation $M_{ui}^{2}=(y_{S}v_{S})^{2}(y_{u}v)^{2}/(2m_{ui}^{2})$, and its
trivially analogous expression for down-like quarks, to derive approximate
estimates of the masses of the various heavy quark partners.

Assuming that we want to reproduce the SM quark masses at the scale of $\sim
1\;\textrm{TeV}$, roughly the scale at which the lightest heavy partner quark,
the top partner, will be integrated out444A technically more correct procedure
(albeit one that still ignores the running of the CKM matrix) would be for
each SM quark’s mass to be determined near the scale of the mass of its
specific partner, so the model would reproduce the mass $m_{u}$ run up to
$\sim 10^{9}\;\textrm{GeV}$, $m_{c}$ near $\sim 10^{6}\;\textrm{GeV}$, $m_{t}$
near $1\;\textrm{TeV}$, etc. However, due to the additional quarks in the
model, this computation would be substantially more complicated than our
treatment here, and the numerical changes to the results would be minimal,
especially given the fact that the most pronounced discrepancies with our
calculation would arise in the least certain quark masses, $m_{u}$ and
$m_{d}$., we arrive at

$\displaystyle m_{u}(1\;\textrm{TeV})=1.07\;\textrm{MeV},$ $\displaystyle
M_{u}\sim(2\times 10^{5})y_{S}v_{S},$ $\displaystyle
m_{c}(1\;\textrm{TeV})=532\;\textrm{MeV},$ $\displaystyle M_{c}\sim
300y_{S}v_{S},$ $\displaystyle m_{t}(1\;\textrm{TeV})=144\;\textrm{GeV},$
$\displaystyle M_{t}\sim y_{S}v_{S},$ (27) $\displaystyle
m_{d}(1\;\textrm{TeV})=2.28\;\textrm{MeV},$ $\displaystyle M_{d}\sim(8\times
10^{4})y_{S}v_{S},$ $\displaystyle m_{s}(1\;\textrm{TeV})=46.5\;\textrm{MeV},$
$\displaystyle M_{s}\sim(4\times 10^{4})y_{S}v_{S},$ $\displaystyle
m_{b}(1\;\textrm{TeV})=2.40\;\textrm{GeV},$ $\displaystyle M_{b}\sim
70y_{S}v_{S}.$

Given these estimates, and assuming $y_{S}v_{S}\mathrel{\mathchoice{\lower
3.01385pt\vbox{\halign{$\matth\displaystyle\hfil#\hfil$\cr>\crcr\sim\crcr}}}{\lower
3.01385pt\vbox{\halign{$\matth\textstyle\hfil#\hfil$\cr>\crcr\sim\crcr}}}{\lower
3.01385pt\vbox{\halign{$\matth\scriptstyle\hfil#\hfil$\cr>\crcr\sim\crcr}}}{\lower
3.01385pt\vbox{\halign{$\matth\scriptscriptstyle\hfil#\hfil$\cr>\crcr\sim\crcr}}}}1\;\textrm{TeV}$,
only the heavy partner to the top quark, with a mass $M_{t}$, can reasonably
be expected to be experimentally directly observable in the foreseeable
future. We do, however, note that Eq.(2.3) assumes that the up-like and down-
like Yukawa couplings to the SM Higgs, that is, $y_{H}\cos(\alpha-\beta)$ and
$y_{H}\sin(\alpha+\beta)$ are both roughly of $O(1)$. While this may be
expected from naturalness, it is far from the only feasible scenario. If, for
example, $y_{H}\sin(\alpha+\beta)\sim O(10^{-2})$, equivalent to a value
arising in a number of multi-Higgs doublet scenarios which seek to explain the
mass hierarchy between the top and $b$ quark, we might expect that $M_{b}\sim
M_{t}$, and hence the heavy partner of the $b$ may also play a significant
phenomenological role. In this case, we would expect significant additional
constraints arising from both direct production of the $b$ partner at the LHC
and possibly significant modifications to the SM $Z\overline{b}b$ coupling.
For the sake of simplicity, however, we leave a detailed exploration of the
effects of a lighter $b$ partner within this model to future work.

Meanwhile, in the mass matrix of Eq.(19), we note that
$X_{u}=-Tr[\mathbf{M}_{u}]+M$ should, in the absence of extreme fine tuning,
be roughly the same order of magnitude as the largest eigenvalue of the matrix
$\mathbf{u}$. We have established, however, that this eigenvalue is
approximately $\sim 10^{5}\times y_{S}v_{S}$. The mass
$m_{P3}^{u}\approx|X_{u}|$ will naturally be extremely large, on the order of
$\sim 10^{8}\;\textrm{GeV}$.

Our expressions in Eq.(2.3) can then be dramatically simplified by merely
assuming that the extremely heavy particles (the fermion with mass
$\approx|X_{u}|$ and the heavy partners of the up and charm quarks) are
virtually entirely decoupled from the lower mass fields (the SM quarks, the
fermions with masses $\sim v_{P}$, and the heavy partner to the top quark) –
in practice, this involves taking the limits where
$|u_{1,2}|,|X_{u}|,M_{u1,u2}\rightarrow\infty$ in the expressions of Eq.(2.3).
This permits us to write

$\displaystyle\vec{\Delta}_{P1L}^{u}\approx\frac{\mathbf{W}_{u}^{\dagger}\vec{\gamma}_{P}}{v_{P}},$
$\displaystyle(\vec{\Delta}_{P2L}^{u})_{3}\approx-
sgn(y_{u}y_{s}y_{P2}u_{3})\frac{m_{t}}{M_{t}}\frac{(M_{t}^{2}-y_{S}^{2}v_{S}^{2})^{1/2}}{m_{P2}^{u}}\frac{(\mathbf{W}_{u}^{\dagger}\vec{\gamma}_{P})_{3}}{v_{P}}$
(28) $\displaystyle(\vec{\Delta}_{P1L}^{U})_{3}\approx
sgn(y_{u}u_{3})\frac{m_{t}}{M_{t}}\bigg{(}\frac{M_{t}^{2}-y_{S}^{2}v_{S}^{2}}{y_{S}^{2}v_{S}^{2}}\bigg{)}^{1/2}\frac{(m_{P1}^{u})^{2}}{((m_{P1})^{2}-M_{t}^{2})}\frac{(\mathbf{W}_{u}^{\dagger}\vec{\gamma}_{P})_{3}}{v_{P}}$
$\displaystyle(\vec{\Delta}_{P2L}^{U})_{3}\approx
sgn(y_{P2})\frac{y_{S}v_{S}m_{P2}^{u}}{(m_{P2}^{u})^{2}-M_{t}^{2}}\frac{(\mathbf{W}_{u}^{\dagger}\vec{\gamma}_{P})_{3}}{v_{P}},$

where the rest of the $\vec{\Delta}$ terms are either negligibly small or
parameterize mixing between the extremely heavy states. The expressions in
Eq.(2.3) already provide some useful information: In particular, in the limit
where the super-heavy states decouple the mixing of the phenomenologically
accessible states is independent of $\vec{\gamma}_{u}$, the parameter
corresponding to the $U(1)_{D}$-breaking components of the scalar vev’s
$\langle\Phi_{A}\rangle$ and $\langle\Phi_{B}\rangle$. Additionally, there
exists a suppression factor of $m_{t}/M_{t}$ in front of
$(\vec{\Delta}_{P2L}^{u})_{3}$ and $(\vec{\Delta}_{P1L}^{U})_{3}$,
corresponding to the mixing between the top quark and the $m_{P2}^{u}$ state,
and the mixing between the heavy top partner and the $m_{P1}^{u}$ state,
respectively – these couplings between $SU(3)_{F}$ triplet and singlet states
only arise due to mixing between the top quark and its heavy partner, and
therefore vanish in the limit where the top partner decouples from the SM.

Armed with Eq.(2.3), it is not difficult to extract a complete expression for
$\mathcal{U}_{L}^{u}$, or at least the components of this rotation matrix that
are relevant for the phenomenologically accessible particles. Taking the same
decoupling limit in the expression for the diagonalization matrix
$(\mathcal{U}_{L}^{u})_{0}$ in Eq.(23) (which here corresponds to letting
$\rho_{1,2}^{u},\alpha_{ij}^{L}\rightarrow 0$), we can use Eqs.(15), (23),
(25), and (2.3) in order to extract the approximate mass eigenvectors for each
left-handed up-like (and, through trivial generalization, down-like) quark
with a phenomenologically accessible mass. Truncating our 9-dimensional
flavor-space quark vectors to omit the super-heavy particles, we can write
these vectors as

$\displaystyle(\overline{\mathbf{U}}_{R})_{\textrm{dec}}$
$\displaystyle\equiv(\overline{u}_{R},\overline{T}_{R},\overline{Q}^{u}_{R},(\overline{U}_{1})_{R}),$
$\displaystyle(\mathbf{U}_{L})_{\textrm{dec}}$
$\displaystyle\equiv(u_{L},T_{L},Q^{u}_{L},(U_{1})_{L}),$ (29)
$\displaystyle(\overline{\mathbf{D}}_{R})_{\textrm{dec}}$
$\displaystyle\equiv(\overline{d}_{R},\overline{Q}^{d}_{R},(\overline{D}_{1})_{R}),$
$\displaystyle(\mathbf{D}_{L})_{\textrm{dec}}$
$\displaystyle\equiv(d_{L},Q^{d}_{L},(D_{1})_{L}).$

Notably, the above truncation in the up-like sector, in which the top quark’s
heavy partner is retained, is much easier in a basis in which
$\mathbf{W}_{u}=\mathbf{1}_{3\times 3}$. Otherwise, as we can see from
Eq.(2.3), the top partner will be represented by a non-trivial combination of
the elements of the $SU(3)_{F}$ triplets $(U_{2})_{L}$ and $(U_{2})_{R}$,
which would require retaining rows and columns of $\mathbf{U}^{u}_{L,R}$ that
correspond to heavy states until a rotation into a basis in which
$\mathbf{W}_{u}=\mathbf{1}_{3\times 3}$ could be performed. Therefore, for the
following expressions, we have assumed that
$\mathbf{W}_{u}=\mathbf{1}_{3\times 3}$; as we mentioned at the beginning of
this section, it is always possible to use $SU(4)_{F}$ freedom to work in such
a basis from the beginning of our analysis. Using Eqs.(23) and (25), we can
write the truncated up-like quark rotation matrix as

$\displaystyle\mathcal{U}^{u}_{L}\approx\begin{pmatrix}1&0&0&0&\frac{(\vec{\gamma}_{P})_{1}}{v_{P}}&0\\\
0&1&0&0&\frac{(\vec{\gamma}_{P})_{2}}{v_{P}}&0\\\
0&0&1&r\frac{m_{t}}{M_{t}}&\frac{(\vec{\gamma}_{P})_{3}}{v_{P}}&rq\bigg{(}\frac{z_{P2}^{3}}{1-z_{P2}^{2}}\bigg{)}\frac{m_{t}}{M_{t}}\frac{(\vec{\gamma}_{P})_{3}}{v_{P}}\\\
0&0&-r\frac{m_{t}}{M_{t}}&1&r\bigg{(}\frac{z_{P1}^{2}}{1-z_{P1}^{2}}\bigg{)}\frac{m_{t}}{M_{t}}\frac{(\vec{\gamma}_{P})_{3}}{v_{P}}&q\bigg{(}\frac{z_{P2}}{1-z_{P2}^{2}}\bigg{)}\frac{(\vec{\gamma}_{P})_{3}}{v_{P}}\\\
-\frac{(\vec{\gamma}_{P})_{1}^{*}}{v_{P}}&-\frac{(\vec{\gamma}_{P})_{2}^{*}}{v_{P}}&-\frac{(\vec{\gamma}_{P})_{3}^{*}}{v_{P}}&-r\bigg{(}\frac{1}{1-z_{P1}^{2}}\bigg{)}\frac{m_{t}}{M_{t}}\frac{(\vec{\gamma}_{P})^{*}_{3}}{v_{P}}&1&0\\\
0&0&rqz_{P2}\frac{m_{t}}{M_{t}}\frac{(\vec{\gamma}_{P})_{3}^{*}}{v_{P}}&-\frac{qz_{P2}}{1-z_{P2}^{2}}\frac{(\vec{\gamma}_{P})_{3}^{*}}{v_{P}}&0&1\end{pmatrix}$
(30) $\displaystyle r\equiv
sgn(y_{u}u_{3})\bigg{(}\frac{M_{t}^{2}-y_{S}^{2}v_{S}^{2}}{y_{S}^{2}v_{S}^{2}}\bigg{)}^{1/2},\;\;q\equiv\frac{y_{S}v_{S}}{M_{t}},\;\;z_{P1}\equiv
sgn(y_{P2})\frac{M_{t}}{m_{P1}^{u}},\;\;z_{P2}\equiv
sgn(y_{P2})\frac{M_{t}}{m_{P2}^{u}},$

where we have dropped terms of $O(m_{t}^{2}/M_{t}^{2})$ and higher. Note that
each of the variables $r$, $q$, $z_{P1}$, and $z_{P2}$ are all $O(1)$
parameters, if we assume that the portal scale $v_{P}$ is close to the heavy
top partner mass scale $v_{S}$. The result for the down-like quarks is
dramatically simpler, since we assume that the heavy $b$ quark partner, unlike
the top partner, is too massive to be phenomenologically relevant. In this
case, we simply have

$\displaystyle\mathcal{U}^{d}_{L}\approx\begin{pmatrix}\mathbf{W}_{d}&\frac{\vec{\gamma}_{P}}{v_{P}}&\mathbf{0}_{3\times
1}\\\ -\frac{\vec{\gamma}_{P}^{\dagger}\mathbf{W}_{d}}{v_{P}}&1&0\\\
\mathbf{0}_{1\times 3}&0&1\end{pmatrix}.$ (31)

Having determined the relevant components of the left-handed rotation matrix
$\mathcal{U}_{L}^{u}$, we can follow an analogous procedure for the right-
handed rotation matrix $\mathcal{U}_{R}^{u}$ identified in Eq.(12). For the
sake of brevity, we shall simply summarize the results here. The right-handed
fermion mass matrix is given (up to $O(\vec{\gamma}_{P,u})$) by

$\displaystyle\mathcal{M}_{u}\mathcal{M}_{u}^{\dagger}=\begin{pmatrix}y_{S}^{2}v_{S}^{2}\mathbf{1}_{3\times
3}&y_{S}v_{S}\mathbf{u}&\mathbf{0}_{3\times
1}&y_{P2}^{2}v_{P}\vec{\gamma}_{P}&y_{S}v_{S}\vec{\gamma}_{u}\\\
y_{S}v_{S}\mathbf{u}&\frac{y_{u}^{2}v^{2}}{2}+\mathbf{u}^{2}&\frac{y_{u}v}{\sqrt{2}}y_{P1}\vec{\gamma}_{P}&y_{S}v_{S}\vec{\gamma}_{u}&(\mathbf{u}+X_{u}\mathbf{1}_{3\times
3})\vec{\gamma}_{u}\\\ \mathbf{0}_{1\times
3}&\frac{y_{u}v}{\sqrt{2}}y_{P1}\vec{\gamma}_{P}^{\dagger}&y_{P1}^{2}v_{P}^{2}&0&\frac{y_{u}v}{\sqrt{2}}y_{P1}v_{P}\\\
y_{P2}^{2}v_{P}\vec{\gamma}_{P}^{\dagger}&y_{S}v_{S}\vec{\gamma}_{u}^{\dagger}&0&y_{P2}^{2}v_{P}^{2}+y_{S}^{2}v_{S}^{2}&X_{u}y_{S}v_{S}\\\
y_{S}v_{S}\vec{\gamma}_{u}^{\dagger}&\vec{\gamma}_{u}^{\dagger}(\mathbf{u}+X_{u}\mathbf{1}_{3\times
3})&\frac{y_{u}v}{\sqrt{2}}y_{P1}v_{P}&X_{u}y_{S}v_{S}&\frac{y_{u}^{2}v^{2}}{2}+X_{u}^{2}\end{pmatrix}.$
(32)

In the same framework as our treatment of $\mathcal{U}_{L}^{u}$, we shall
split the result into a matrix $(\mathcal{U}_{R}^{u})_{0}$ which diagonalizes
$\mathcal{M}_{u}\mathcal{M}_{u}^{\dagger}$ in the limit where
$\vec{\gamma}_{P,u}\rightarrow 0$, and a matrix
$(\mathcal{U}_{R}^{u})_{\gamma}$ which provides the leading-order corrections
due to the $\vec{\gamma}_{P,u}$ terms. For $(\mathcal{U}_{R}^{u})_{0}$, we
arrive at

$\displaystyle(\mathcal{U}_{R}^{u})_{0}=\begin{pmatrix}-\mathbf{W}_{u}\mathbf{c}^{u}_{\eta}sgn(y_{S}y_{u}\mathbf{u}_{D})&-\mathbf{W}_{u}\mathbf{s}^{u}_{\eta}sgn(\mathbf{u}_{D})&\mathbf{0}_{3\times
1}&\mathbf{0}_{3\times 1}&\mathbf{0}_{3\times 1}\\\
-\mathbf{W}_{u}\mathbf{s}^{u}_{\eta}sgn(y_{S}y_{u}\mathbf{u}_{D})&\mathbf{W}_{u}\mathbf{c}^{u}_{\eta}sgn(\mathbf{u}_{D})&\mathbf{0}_{3\times
1}&\mathbf{0}_{3\times 1}&\mathbf{0}_{3\times 1}\\\ \mathbf{0}_{1\times
3}&\mathbf{0}_{1\times
3}&sgn(y_{P1})&\alpha_{12}^{R}sgn(y_{P2})&\alpha_{13}^{R}sgn(X_{u})\\\
\mathbf{0}_{1\times 3}&\mathbf{0}_{1\times
3}&-\alpha_{12}^{R}sgn(y_{P1})&sgn(y_{P2})&\alpha_{23}^{R}sgn(X_{u})\\\
\mathbf{0}_{1\times 3}&\mathbf{0}_{1\times
3}&-\alpha_{13}^{R}sgn(y_{P1})&-\alpha_{23}^{R}sgn(y_{P2})&sgn(X_{u})\end{pmatrix},$
(33) $\displaystyle\mathbf{c}_{\eta}^{u}\equiv
diag(\cos(\eta_{1}^{u}),\cos(\eta_{2}^{u}),\cos(\eta_{3}^{u})),\;\;\;\mathbf{s}_{\eta}^{u}\equiv
diag(\sin(\eta_{1}^{u}),\sin(\eta_{2}^{u}),\sin(\eta_{3}^{u})),$
$\displaystyle\cos(\eta_{i}^{u})=\bigg{(}\frac{M_{ui}^{2}-y_{S}^{2}v_{S}^{2}}{M_{ui}^{2}}\bigg{)}^{1/2},\;\;\;\sin(\eta_{i}^{u})=-\frac{y_{S}v_{S}}{M_{ui}}sgn(u_{i}),$
$\displaystyle\alpha_{12}^{R}\equiv\frac{y_{u}v}{\sqrt{2}X_{u}}\frac{(y_{S}v_{S})(y_{P1}v_{P})}{(y_{P1}^{2}-y_{P2}^{2})v_{P}^{2}},\;\;\;\alpha_{13}^{R}\equiv\frac{y_{u}v}{\sqrt{2}X_{u}}\frac{y_{P1}v_{P}}{X_{u}}\bigg{(}1-\frac{y_{S}^{2}v_{S}^{2}}{(y_{P1}^{2}-y_{P2}^{2})v_{P}^{2}}\bigg{)},\;\;\;\alpha_{2}3^{R}\equiv\frac{y_{S}v_{S}}{X_{u}}.$

The sole non-trivial difference in the result of Eq.(33) and that of Eq.(23)
is the introduction of sign flips (or rather, phase rotations) of the right-
handed quarks in order to ensure that the mass terms in the final action
correspond to real positive fermion masses– these rotations could have just as
well been done in the left-handed sector, without altering the physical
results of the model. The leading perturbations to $(\mathcal{U}_{R}^{u})_{0}$
arising from the $\vec{\gamma}_{P,u}$ terms of the mass matrix of Eq.(32) are
then given by

$\displaystyle(\mathcal{U}_{R}^{u})_{\gamma}=\begin{pmatrix}\mathbf{1}_{3\times
3}&\mathbf{0}_{3\times
3}&\vec{\Delta}^{u}_{P1R}&\vec{\Delta}^{u}_{P2R}&\vec{\Delta}^{u}_{P3R}\\\
\mathbf{0}_{3\times 3}&\mathbf{1}_{3\times
3}&\vec{\Delta}^{U}_{P1R}&\vec{\Delta}^{U}_{P2R}&\vec{\Delta}^{U}_{P3R}\\\
-\vec{\Delta}^{u\dagger}_{P1R}&-\vec{\Delta}^{U\dagger}_{P1R}&1&0&0\\\
-\vec{\Delta}^{u\dagger}_{P2R}&-\vec{\Delta}^{U\dagger}_{P2R}&0&1&0\\\
-\vec{\Delta}^{u\dagger}_{P3R}&-\vec{\Delta}^{U\dagger}_{P3R}&0&0&1\end{pmatrix},$
(34)

where

$\displaystyle(\vec{\Delta}^{u}_{P1R})_{i}=\frac{-\bigg{(}(\frac{y_{u}v}{\sqrt{2}}y_{P1}\mathbf{s}^{u}_{\eta}-\alpha_{12}^{R}y_{P2}^{2}v_{P}\mathbf{c}^{u}_{\eta})\mathbf{W}_{u}^{\dagger}\vec{\gamma}_{P}-(\alpha_{12}^{R}y_{S}v_{S}\mathbf{s}^{u}_{\eta}+\alpha_{13}^{R}(y_{S}v_{S}\mathbf{c}^{u}_{\eta}+\mathbf{u}_{D}\mathbf{s}^{u}_{\eta}+X_{u}\mathbf{s}^{u}_{\eta}))\mathbf{W}_{u}^{\dagger}\vec{\gamma}_{u}\bigg{)}_{i}}{((m_{P1}^{u})^{2}-m_{ui}^{2})sgn(y_{P1}y_{S}y_{u}u_{i})}$
$\displaystyle(\vec{\Delta}^{u}_{P2R})_{i}=\frac{-\bigg{(}(\alpha_{12}^{R}\frac{y_{u}v}{\sqrt{2}}y_{P1}\mathbf{s}^{u}_{\eta}+y_{P2}^{2}v_{P}\mathbf{c}^{u}_{\eta})\mathbf{W}_{u}^{\dagger}\vec{\gamma}_{P}+(y_{S}v_{S}\mathbf{s}^{u}_{\eta}-\alpha_{23}^{R}(y_{S}v_{S}\mathbf{c}^{u}_{\eta}+\mathbf{u}_{D}\mathbf{s}^{u}_{\eta}+X_{u}\mathbf{s}^{u}_{\eta}))\mathbf{W}_{u}^{\dagger}\vec{\gamma}_{u}\bigg{)}_{i}}{((m_{P2}^{u})^{2}-(m_{ui}^{2}))sgn(y_{P2}y_{S}y_{u}u_{i})}$
$\displaystyle(\vec{\Delta}_{P3R}^{u})_{i}=\frac{-\bigg{(}(\alpha_{13}^{R}\frac{y_{u}v}{\sqrt{2}}y_{P1}\mathbf{s}^{u}_{\eta}+\alpha_{23}^{R}y_{P2}^{2}v_{P}\mathbf{c}^{u}_{\eta})\mathbf{W}_{u}^{\dagger}\vec{\gamma}_{u}+(\alpha_{23}^{R}y_{S}v_{S}\mathbf{s}^{u}_{\eta}+y_{S}v_{S}\mathbf{c}^{u}_{\eta}+\mathbf{u}_{D}\mathbf{s}^{u}_{\eta}+X_{u}\mathbf{s}^{u}_{\eta})\mathbf{W}_{u}^{\dagger}\vec{\gamma}_{u}\bigg{)}_{i}}{((m_{P2}^{u})^{2}-m_{ui}^{2})sgn(X_{u}y_{S}y_{u}u_{i})}$
(35)
$\displaystyle(\vec{\Delta}^{U}_{P1R})_{i}=\frac{\bigg{(}(\frac{y_{u}v}{\sqrt{2}}y_{P1}\mathbf{c}^{u}_{\eta}+\alpha_{12}^{R}y_{P2}^{2}v_{P}\mathbf{s}^{u}_{\eta})\mathbf{W}_{u}^{\dagger}\vec{\gamma}_{P}-(\alpha_{12}^{R}y_{S}v_{S}\mathbf{c}^{u}_{\eta}+\alpha_{13}^{R}(-y_{S}v_{S}\mathbf{s}^{u}_{\eta}+\mathbf{u}_{D}\mathbf{c}^{u}_{\eta}+X_{u}\mathbf{c}^{u}_{\eta}))\mathbf{W}_{u}^{\dagger}\vec{\gamma}_{u}\bigg{)}_{i}}{((m_{P1}^{u})^{2}-M_{ui}^{2})sgn(y_{P1}u_{i})},$
$\displaystyle(\vec{\Delta}^{U}_{P2R})_{i}=\frac{\bigg{(}(\alpha_{12}^{R}\frac{y_{u}v}{\sqrt{2}}y_{P1}\mathbf{c}^{u}_{\eta}-y_{P2}^{2}v_{P}\mathbf{s}^{u}_{\eta})\mathbf{W}_{u}^{\dagger}\vec{\gamma}_{P}+(y_{S}v_{S}\mathbf{c}^{u}_{\eta}-\alpha_{23}^{R}(-y_{S}v_{S}\mathbf{s}^{u}_{\eta}+\mathbf{u}_{D}\mathbf{c}^{u}_{\eta}+X_{u}\mathbf{c}^{u}_{\eta}))\mathbf{W}_{u}^{\dagger}\vec{\gamma}_{u}\bigg{)}_{i}}{((m_{P2}^{u})^{2}-M_{ui}^{2})sgn(y_{P2}u_{i})},$
$\displaystyle(\vec{\Delta}^{U}_{P3R})_{i}=\frac{\bigg{(}(\alpha_{13}^{R}\frac{y_{u}v}{\sqrt{2}}y_{P1}\mathbf{c}^{u}_{\eta}-\alpha_{23}^{R}y_{P2}^{2}v_{P}\mathbf{s}^{u}_{\eta})\mathbf{W}_{u}^{\dagger}\vec{\gamma}_{P}+(\alpha_{23}^{R}y_{S}v_{S}\mathbf{c}^{u}_{\eta}-y_{S}v_{S}\mathbf{s}^{u}_{\eta}+\mathbf{u}_{D}\mathbf{c}^{u}_{\eta}+X_{u}\mathbf{c}^{u}_{\eta})\mathbf{W}_{u}^{\dagger}\vec{\gamma}_{u}\bigg{)}_{i}}{((m_{P3}^{u})^{2}-M_{ui}^{2})sgn(X_{u}u_{i})}.$

In the same manner as the left-handed eigenvectors, the expressions in
Eq.(2.3) simplify dramatically in the limit where the $m_{P3}^{u}$ quark and
the heavy up and charm partners decouple from the theory. In this limit, we
arrive at

$\displaystyle(\vec{\Delta}_{P1R}^{u})_{3}\approx\frac{m_{t}}{m_{P1}^{u}}\frac{(\mathbf{W}_{u}^{\dagger}\vec{\gamma}_{P})_{3}}{v_{P}},$
$\displaystyle(\vec{\Delta}_{P2R}^{u})_{i}\approx-
sgn(y_{u}y_{S}u_{i}y_{P2})\bigg{(}\frac{M_{ui}^{2}-y_{S}^{2}v_{S}^{2}}{M_{ui}^{2}}\bigg{)}^{1/2}\frac{(\mathbf{W}_{u}^{\dagger}\vec{\gamma}_{P})_{i}}{v_{P}},$
(36) $\displaystyle(\vec{\Delta}_{P1R}^{U})_{3}\approx
sgn(y_{u}u_{3})\frac{m_{P1}^{u}m_{t}}{(m_{P1}^{u})^{2}-M_{t}^{2}}\bigg{(}\frac{M_{t}^{2}-y_{S}^{2}v_{S}^{2}}{y_{S}^{2}v_{S}^{2}}\bigg{)}^{1/2}\frac{(\mathbf{W}_{u}^{\dagger}\vec{\gamma}_{P})_{3}}{v_{P}},$
$\displaystyle(\vec{\Delta}_{P2R}^{U})_{3}\approx
sgn(y_{P2})\frac{(m_{P2}^{u})^{2}}{(m_{P2}^{u})^{2}-M_{t}^{2}}\bigg{(}\frac{y_{S}v_{S}}{M_{t}}\bigg{)}\frac{(\mathbf{W}_{u}^{\dagger}\vec{\gamma}_{P})_{3}}{v_{P}},$

with all other $\vec{\Delta}$ terms for the right-handed quarks being either
numerically negligible or parameterizing mixing between the extremely heavy
fermions. We can then insert the results of Eq.(2.3) into Eq.(34) in order to
derive the rotation matrix $(\mathcal{U}^{u}_{R})_{\gamma}$, which when
combined with $(\mathcal{U}^{u}_{R})_{0}$ given in Eq.(33) will give us the
mass eigenstates for the right-handed up-like quarks in the model. As in the
case of the left-handed diagonalization matrices, we shall restrict our
attentions to the limit in which only the SM quarks, the heavy top partner,
and the two lighter portal matter fields mix among one another, with the other
much more massive additional fermions decoupled from the low-energy theory.
Using the truncated flavor-space vectors of Eq.(2.3), we can write

$\displaystyle\mathcal{U}^{u}_{R}\approx\mathbf{N}_{1}\begin{pmatrix}1&0&0&0&0&\frac{(\vec{\gamma}_{P})_{1}}{v_{P}}\\\
0&1&0&0&0&\frac{(\vec{\gamma}_{P})_{2}}{v_{P}}\\\
0&0&qr&q&rq\bigg{(}\frac{z_{P1}^{3}}{1-z_{P1}^{2}}\bigg{)}\frac{m_{t}}{M_{t}}\frac{(\vec{\gamma}_{P})_{3}}{v_{P}}&\bigg{(}\frac{1-q^{2}r^{2}z_{P2}^{2}}{1-z_{P2}^{2}}\bigg{)}\frac{(\vec{\gamma}_{P})_{3}}{v_{P}}\\\
0&0&-q&qr&\frac{z_{P1}}{q}\bigg{(}\frac{1-q^{2}z_{P1}^{2}}{1-z_{P1}^{2}}\bigg{)}\frac{m_{t}}{M_{t}}\frac{(\vec{\gamma}_{P})_{3}}{v_{P}}&r\frac{q^{2}z_{P2}^{2}}{1-z_{P2}^{2}}\frac{(\vec{\gamma}_{P})_{3}}{v_{P}}\\\
0&0&z_{P1}\frac{m_{t}}{M_{t}}\frac{(\vec{\gamma}_{P})_{3}^{*}}{v_{P}}&-r\bigg{(}\frac{z_{P1}}{1-z_{P1}^{2}}\bigg{)}\frac{m_{t}}{M_{t}}\frac{(\vec{\gamma}_{P})_{3}^{*}}{v_{P}}&1&0\\\
-\frac{(\vec{\gamma_{P}})_{1}^{*}}{v_{P}}&-\frac{(\vec{\gamma_{P}})_{2}^{*}}{v_{P}}&-qr\frac{(\vec{\gamma_{P}})_{3}^{*}}{v_{P}}&-\frac{q}{1-z_{P2}^{2}}\frac{(\vec{\gamma}_{P})_{3}^{*}}{v_{P}}&0&1\end{pmatrix}\mathbf{N}_{2},$
(37) $\displaystyle\mathbf{N_{1}}\equiv
diag(1,1,1,sgn(y_{u}y_{S}),1,1),\;\;\mathbf{N}_{2}\equiv
diag(-sgn(u_{1}y_{S}y_{u}),-sgn(u_{2}y_{S}y_{u}),-1,1,sgn(y_{P1}),sgn(y_{P2})),$

up to $O(m_{t}^{2}/M_{t}^{2})$ corrections, where $r$, $q$, $z_{P1}$, and
$z_{P2}$ are all defined as in Eq.(30). Considering the case of the down-like
quarks, where just as we did for $\mathcal{U}^{d}_{L}$ we have assumed that
the heavy $b$ partner has decoupled from the theory, we arrive at

$\displaystyle\mathcal{U}^{d}_{R}\approx\begin{pmatrix}\mathbf{W}_{d}&\mathbf{0}_{3\times
1}&\frac{\vec{\gamma}_{P}}{v_{P}}\\\ \mathbf{0}_{1\times 3}&1&0\\\
-\frac{\vec{\gamma}_{P}^{\dagger}\mathbf{W}_{d}}{v_{P}}&0&1\end{pmatrix}\begin{pmatrix}-sgn(y_{S}y_{d}\mathbf{d}_{D})&\mathbf{0}_{3\times
1}&\mathbf{0}_{3\times 1}\\\ \mathbf{0}_{1\times 3}&sgn(y_{P1})&0\\\
\mathbf{0}_{1\times 3}&0&sgn(y_{P2})\end{pmatrix}.$ (38)

At this point, we have determined the rotation matrices $\mathcal{U}^{u}_{L}$
and $\mathcal{U}^{u}_{R}$ necessary to diagonalize the up-like quark mass
matrix, and by trivial generalization, these results will also give us the
rotation matrices $\mathcal{U}^{d}_{L,R}$ needed to diagonalize the down-like
quark mass matrices. With this knowledge, we can now comment briefly on how to
ensure that this construction reproduces the observed CKM matrix. Considering
Eq.(23) and its down-like equivalent, we can see that in the limit where
$\vec{\gamma}_{P,u,d}\rightarrow 0$, the CKM matrix (as defined by the
coupling matrix to the SM $W$ boson, which couples to the left-handed SM
quarks but not their heavy partners) is approximately given by

$\displaystyle
V_{CKM}=\mathbf{c}^{u}_{\rho}\mathbf{W}_{u}^{\dagger}\mathbf{W}_{d}\mathbf{c}^{d}_{\rho},$
(39)

where we remind the reader that $\mathbf{W}_{u}$ and $\mathbf{W}_{d}$ are the
unitary matrices which diagonalize the Hermitian matrices $\mathbf{u}$ and
$\mathbf{d}$ (as defined in Eq.(13) for $\mathbf{u}$, with an analogous
expression for $\mathbf{d}$) respectively. Since
$\mathbf{c}^{u}_{\rho}\approx\mathbf{c}^{d}_{\rho}\approx 1$ up to
$O(m_{q}^{2}/M_{q}^{2})$ corrections (where $q=u,d,s,c,b,t$), the CKM matrix
can be well-approximated as $\mathbf{W}_{u}^{\dagger}\mathbf{W}_{d}$, or in
other words, by specifying $\mathbf{u}$ and $\mathbf{d}$ such that the clash
of their diagonalization matrices matches the observed CKM.

Our recipe for reconstructing the quark sector in this construction is
therefore compete, and it is useful at this point to take stock of the
parameters which we may freely specify while still producing SM quark masses
and mixings consistent with observation. First, we recall that there are two
$SU(4)_{F}$ adjoint scalars in the model, $\Phi_{A}$ and $\Phi_{B}$, with
vev’s that we can write $\langle\Phi_{A}\rangle$ and
$\sigma_{2}/2\otimes\langle\Phi_{B}\rangle$. Then, we can use $SU(4)$ freedom
to work in a basis in which the combination
$y_{A}\langle\Phi_{A}\rangle+y_{B}\langle\Phi_{B}\rangle$ is diagonal, which
we can see from the action in Eq.(8) results in $\gamma_{u}=0$ and
$\mathbf{u}=\mathbf{u}_{D}=diag(u_{1},u_{2},u_{3})$. Then, we have
$V_{CKM}\approx\mathbf{W}_{d}$, that is, the rotation matrix to diagonalize
$\mathbf{d}$ is uniquely determined by the need to recreate the CKM matrix
($\mathbf{W}_{u}$ is, of course, simply the identity matrix in this basis).
Turning to quark masses, we note that if we specify $y_{u}\equiv
y_{H}\cos(\alpha-\beta)\sim O(1)$, $y_{d}\equiv y_{H}\sin(\alpha+\beta)\sim
O(1)$, and $y_{S}v_{S}\sim 1\;\textrm{TeV}$, Eqs.(2.3) and (33) uniquely
determine the eigenvalues of $\mathbf{u}$ and $\mathbf{d}$ up to a sign, and
therefore also give us the masses of the heavy partner quarks and the mixing
angles $\rho^{u,d}_{1,2,3}$ and $\eta^{u,d}_{1,2,3}$. If we additionally
specify $y_{P1}\sim 1$, $y_{P2}\sim 1$, $v_{P}\mathrel{\mathchoice{\lower
3.01385pt\vbox{\halign{$\matth\displaystyle\hfil#\hfil$\cr>\crcr\sim\crcr}}}{\lower
3.01385pt\vbox{\halign{$\matth\textstyle\hfil#\hfil$\cr>\crcr\sim\crcr}}}{\lower
3.01385pt\vbox{\halign{$\matth\scriptstyle\hfil#\hfil$\cr>\crcr\sim\crcr}}}{\lower
3.01385pt\vbox{\halign{$\matth\scriptscriptstyle\hfil#\hfil$\cr>\crcr\sim\crcr}}}}1\;\textrm{TeV}$,
and the mass term $M$ (as seen in Eqs.(8), (2.3), and (13)), we can then
specify the masses and mixings of the portal matter fermions (that is, those
that consist primarily of fields with non-zero charge under $U(1)_{D}$).
Notably, in our present construction, the masses of the some of the portal
matter fermions are degenerate between the up- and down-quark sectors
(specifically, $m_{P1}^{u}=m_{P1}^{d}$ and $m_{P2}^{u}=m_{P2}^{d}$) up to
radiative corrections. In practice, we might expect such a relation to not
hold precisely, due to differing renormalization group evolution of the Yukawa
couplings $y_{P1,2}$ to the up- and down-like sectors, but for the purposes of
our simple numerical study we shall take this relation at face value. The sole
remaining parameters we may specify are the $U(1)_{D}$-breaking terms, in form
of the complex vectors $\vec{\gamma}_{P,d}\mathrel{\mathchoice{\lower
3.01385pt\vbox{\halign{$\matth\displaystyle\hfil#\hfil$\cr<\crcr\sim\crcr}}}{\lower
3.01385pt\vbox{\halign{$\matth\textstyle\hfil#\hfil$\cr<\crcr\sim\crcr}}}{\lower
3.01385pt\vbox{\halign{$\matth\scriptstyle\hfil#\hfil$\cr<\crcr\sim\crcr}}}{\lower
3.01385pt\vbox{\halign{$\matth\scriptscriptstyle\hfil#\hfil$\cr<\crcr\sim\crcr}}}}1\;\textrm{GeV}$
(recalling that in our basis, $\vec{\gamma}_{u}=0$), which in turn give us the
effects of mixing between states of different $U(1)_{D}$ charge, as
parameterized in Eqs.(2.3) and (2.3).

In summary, then, we can generate a point in the model parameter space that
produces the observed quark masses and mixings by specifying the $O(1)$ real
parameters $y_{H}$, $y_{P1}$, $y_{P2}$, as well as $y_{S}v_{S}$, the angles
$\alpha$ and $\beta$, the real vev’s $v_{P},v_{S}\mathrel{\mathchoice{\lower
3.01385pt\vbox{\halign{$\matth\displaystyle\hfil#\hfil$\cr>\crcr\sim\crcr}}}{\lower
3.01385pt\vbox{\halign{$\matth\textstyle\hfil#\hfil$\cr>\crcr\sim\crcr}}}{\lower
3.01385pt\vbox{\halign{$\matth\scriptstyle\hfil#\hfil$\cr>\crcr\sim\crcr}}}{\lower
3.01385pt\vbox{\halign{$\matth\scriptscriptstyle\hfil#\hfil$\cr>\crcr\sim\crcr}}}}1\;\textrm{TeV}$,
and the complex $U(1)_{D}$-breaking vev’s $\vec{\gamma}_{P}$ and
$\vec{\gamma}_{d}$. Altogether, these selections generate a unique point in
parameter space up to the signs of the eigenvalues of the matrices
$\mathbf{u}$ and $\mathbf{d}$.

Having addressed the quark sector, we move on to discussing the lepton sector
of the theory. Notably, because none of the scalars introduced thus far break
the group $SU(4)_{c}$, the Yukawa couplings in Eq.(8) do not account for any
discrepancy between down-like quark and charged lepton masses at tree level.
However, because we’ve already assumed that the scale for $SU(4)_{c}$ breaking
is incredibly high (and indeed must be, due to constraints on processes
mediated by new $SU(4)_{c}$ gauge bosons), we can assume that discrepancies
between the charged lepton and down-like quark masses are due to significant
renormalization group effects. Specifically, we can posit differing running of
the couplings $y_{A}$, $y_{B}$, and $y_{H}$, as well as the mass term $M$, for
color triplets versus singlets. In practice, the task of generating a specific
numerical realization of the leptonic Yukawa couplings necessary to reproduce
the observed charged lepton spectrum (consistent with the vev structure
necessary to reproduce the SM quark masses and CKM matrix) is analytically
onerous and not terribly enlightening. However, we can expect that the effect
of the specific realization of the charged lepton sector is largely
phenomenologically irrelevant: Because the spectrum of charged leptons
resembles that of the down-like quarks (at least in order of magnitude), it
suffices here to point out that such a realization is almost certainly
achievable with choices of leptonic $y_{A}$, $y_{B}$, $y_{H}$, and $M$ of
roughly the same order of magnitude as those for the down-like quark sector,
as would be consistent with RG evolution. Phenomenologically, then, we would
expect that the $SU(3)_{F}$ heavy partner leptons here are too heavy to be
realistically producible in existing and currently planned collider
experiments, like their counterparts in the down-like quark sector. Therefore,
the only potentially accessible new fermions arising from the charged leptons
would be the $SU(3)_{F}$ singlet portal matter fields, specifically those
which should have masses of $O(\textrm{TeV})$. However, due to their lack of
color charge, experimental constraints on their production at the LHC are
somewhat weaker than those of the quark portal matter fields with comparable
masses, so we shall not address their phenomenology in detail here.

Meanwhile, the neutrino sector within this model presents substantial
additional challenges. In particular, the tiny neutrino masses $\sim
0.1\;\textrm{eV}$ and near-maximal neutrino flavor mixings are in general
inconsistent with any sort of high-energy degeneracy with the up-like quark
sector broken by renormalization group running, as the Yukawa couplings in
Eq.(8) suggest. However, there do exist extensions of the model that can
account for this discrepancy. For example, including scalars in the
representations $(\mathbf{15},\mathbf{1},\mathbf{1},\mathbf{15},0)$,
$(\mathbf{15},\mathbf{1},\mathbf{3},\mathbf{15},0)$,
$(\mathbf{15},\mathbf{1},\mathbf{1},\mathbf{1},0)$ and/or
$(\mathbf{15},\mathbf{1},\mathbf{3},\mathbf{1},0)$ could allow for
dramatically different Yukawa couplings and mixings between the quark and
lepton sectors, at the expense of a yet larger scalar sector and extreme fine-
tuning to reproduce the tiny neutrino masses. Alternatively, the introduction
of a scalar in the representation
$(\mathbf{10},\mathbf{1},\mathbf{3},\mathbf{10},-1/2)$ with the appropriate
vev would permit the inclusion of a large Majorana mass for the right-handed
$SU(3)_{F}$ triplet neutrinos, which would in turn suppress the SM neutrino
masses via a seesaw mechanism. In both of these scenarios, the extension of
the scalar sector would also result in substantial additional contributions to
the masses of the new $SU(3)_{F}$ gauge bosons– given the fact that the SM
neutrino masses are so small, it’s even probable that given these setups the
vev’s of some of these scalars would dominate the mass terms of the gauge
bosons. Given the model-building ambiguity here and the substantial complexity
that such considerations would introduce, however, we determine that a full
exploration of potential ways to realize the neutrino mass spectrum and mixing
within this framework is beyond the scope of our present work. In our
discussion of the gauge boson masses in Section 2.4, however, we shall note
that even in the absence of any additional mass terms which might arise from
added scalar content to the model, the new gauge bosons associated with the
$SU(3)_{F}$ symmetry will be extremely heavy, with only one of these bosons
approaching a low enough mass scale to have a potentially observable effect
even in highly constrained measurements of flavor-changing neutral currents.
As a result, we can expect that any modifications to the model setup that must
be made to accommodate the neutrino masses and mixings should have a minimal
effect on the experimentally observable new physics in the model considered
here.

###  2.4 Gauge Boson Spectrum

Having selected our scalar vacuum expectation values such that the SM quark
masses and mixings can be faithfully recreated, we now turn to how these
selections will generate masses for the new gauge bosons in the theory. As
noted in Section 2.2, we shall assume for the sake of simplicity that the
Pati-Salam symmetry group, $SU(4)_{c}\times SU(2)_{L}\times SU(2)_{R}$, is
broken down to the SM gauge group $SU(3)_{c}\times SU(2)_{L}\times U(1)_{Y}$
at a significantly higher energy scale than any of the symmetry breakings of
the group $SU(4)_{F}\times U(1)_{F}$. While this is certainly feasible, we do
also note that even if that assumption is dropped, the $SU(2)_{R}$ symmetry
must still be broken at a scale much higher than is presently observable, due
to the vev of the field
$\Phi_{B}\sim(\mathbf{1},\mathbf{1},\mathbf{3},\mathbf{15},0)$, which we can
estimate from our results in Section 2.3 will break this symmetry at a scale
of roughly $\sim 10^{8}\;\textrm{GeV}$. Assuming that the new gauge bosons
associated with the Pati-Salam extension to the SM are too heavy to be
relevant, we turn to the mass matrices of the gauge bosons corresponding to
the $SU(4)_{F}\times U(1)_{F}$ generators. In this case, the squared mass
matrix of the 16 gauge bosons here takes the form,

$\displaystyle\mathcal{M}_{G}^{2}=\begin{pmatrix}g_{4}^{2}\mathbf{M}_{SU(4)}&\frac{3}{2}g_{4}g_{1}\vec{M}_{SU(4)\times
U(1)}\\\ \frac{3}{2}g_{4}g_{1}\vec{M}^{T}_{SU(4)\times
U(1)}&\frac{9}{8}g_{1}^{2}|\langle\Phi_{P}\rangle|^{2}\end{pmatrix},$ (40)
$\displaystyle(\mathbf{M}_{SU(4)})_{ij}\equiv-2(Tr[[t^{i},\langle\Phi_{A}\rangle][t^{j},\langle\Phi_{A}\rangle]]+Tr[[t^{i},\langle\Phi_{B}\rangle][t^{j},\langle\Phi_{B}\rangle]])+\langle\Phi_{P}\rangle^{\dagger}\\{t^{i},t^{j}\\}\langle\Phi_{P}\rangle,$
$\displaystyle(\vec{M}_{SU(4)\times
U(1)})_{i}\equiv\langle\Phi_{P}\rangle^{\dagger}t^{i}\langle\Phi_{P}\rangle,$

where $g_{4}$ and $g_{1}$ are the $SU(4)_{F}$ and $U(1)_{F}$ coupling
constants respectively, $[A,B]$ denotes the commutator of $A$ and $B$,
$\\{A,B\\}$ denotes the anticommutator, and $t^{i}$ denotes the $i^{th}$
generator matrix for the fundamental representation of the group $SU(4)$, in
the basis described in Appendix A. Note that here, $\mathbf{M}_{SU(4)}$ is a
$15\times 15$ real symmetric matrix, while $\vec{M}_{SU(4)\times U(1)}$ is a
15-component real vector, and both of these terms have dimensions of mass
squared.

Referring to Eq.(40), our first task to determine the mass eigenstates of
$SU(4)\times U(1)$ gauge bosons is to determine what the vev’s
$\langle\Phi_{A}\rangle$ and $\langle\Phi_{B}\rangle$ are, given our
requirements that the SM quark masses and mixing matrices are properly
reproduced. To that end, we consider the combinations of $\Phi_{A}$ and
$\Phi_{B}$ that are coupled to the up- and down-like quarks, referring to
Eq.(2.3) and several useful variable definitions in Eq.(13). Comparing these
to the vev’s of $\Phi_{A}$ and $\Phi_{B}$ as given in Table 2, we arrive at

$\displaystyle\begin{pmatrix}\mathbf{u}_{D}&\mathbf{0}_{3\times 1}\\\
\mathbf{0}_{1\times 3}&X_{u}\end{pmatrix}=y_{A}\begin{pmatrix}\langle
A\rangle&\vec{\gamma}_{A}\\\ \vec{\gamma}_{A}^{\dagger}&-Tr[\langle
A\rangle]\end{pmatrix}+y_{B}\begin{pmatrix}\langle
B\rangle&\vec{\gamma}_{B}\\\ \vec{\gamma}_{B}^{\dagger}&-Tr[\langle
B\rangle]\end{pmatrix}+M\mathbf{1}_{4\times 4},$ (41)
$\displaystyle\begin{pmatrix}V_{CKM}^{\dagger}\mathbf{d}_{D}V_{CKM}&\vec{\gamma}_{d}\\\
\vec{\gamma}_{d}^{\dagger}&X_{d}\end{pmatrix}=y_{A}\begin{pmatrix}\langle
A\rangle&\vec{\gamma}_{A}\\\ \vec{\gamma}_{A}^{\dagger}&-Tr[\langle
A\rangle]\end{pmatrix}-y_{B}\begin{pmatrix}\langle
B\rangle&\vec{\gamma}_{B}\\\ \vec{\gamma}_{B}^{\dagger}&-Tr[\langle
B\rangle]\end{pmatrix}+M\mathbf{1}_{4\times 4},$

where we remind the reader that we can use $SU(4)$ gauge freedom to work in a
basis where $\mathbf{u}$ is diagonal and $\vec{\gamma}_{u}=0$, as we have done
in Section 2.3, and that $V_{CKM}$ is the CKM matrix. We recall that in that
Section, the procedure we developed to produce a point in parameter space that
recreates the SM quark masses and mixings already leaves us with
$\mathbf{u}_{D}$, $\mathbf{d}_{D}$, and $M$ specified. Then, simply selecting
the Yukawa couplings $y_{A}$ and $y_{B}$ here will allow us to uniquely
determine the matrices $\langle A\rangle$ and $\langle B\rangle$, as well as
the vectors $\vec{\gamma}_{A}$ and $\vec{\gamma}_{B}$. Specifically, we have

$\displaystyle\langle
A\rangle=\frac{1}{2y_{A}}(\mathbf{u}_{D}+V_{CKM}^{\dagger}\mathbf{d}_{D}V_{CKM}-2M),\;\;\langle
B\rangle=\frac{1}{2y_{B}}(\mathbf{u}_{D}-V_{CKM}^{\dagger}\mathbf{d}_{D}V_{CKM}),$
(42)
$\displaystyle\vec{\gamma}_{A}=\frac{1}{2y_{A}}\vec{\gamma}_{d},\;\;\vec{\gamma}_{B}=-\frac{1}{2y_{B}}\vec{\gamma}_{d}.$

Hence, given a construction which generates the SM quark masses and mixings,
we now can readily determine the necessary vev’s $\langle\Phi_{A}\rangle$ and
$\langle\Phi_{B}\rangle$ by only specifying two additional parameters, the
Yukawa couplings $y_{A}$ and $y_{B}$. With this information, we can now
discuss the spectrum of the gauge bosons which results. In general, the
eigenvalues of the mass matrix in Eq.(40) are highly complicated expressions
and difficult to present in a compact closed form. However, we can avail
ourselves of the hierarchies present between eigenvalues of $\mathbf{u}_{D}$
and $\mathbf{d}$, as well as the nearly-diagonal form of the CKM matrix, in
order to dramatically simplify matters. Specifically, we will employ the
Wolfenstein parameterization of the CKM matrix,

$\displaystyle\begin{pmatrix}1-\frac{\lambda^{2}}{2}-\frac{\lambda^{4}}{8}&\lambda&A\lambda^{3}(\rho-i\eta)\\\
-\lambda&1-\frac{\lambda^{2}}{2}-(1+4A^{2})\frac{\lambda^{4}}{8}&A\lambda^{2}\\\
-A(\rho+i\eta-1)\lambda^{3}&-A(\lambda^{2}+\frac{\lambda^{4}}{2}(2(\rho+i\eta)-1))&1-\frac{A^{2}\lambda^{4}}{2}\end{pmatrix},$
(43) $\displaystyle\lambda=0.22453,\;\;A=0.836,\;\;\rho=0.122,\;\;\eta=0.355,$

and note that, numerically, the eigenvalues of $\mathbf{u}_{D}$ and
$\mathbf{d}$ ($u_{1-3}$ and $d_{1-3}$, respectively) satisfy
$u_{2}\sim\lambda^{4}u_{1}$, $u_{3}\sim\lambda^{8}u_{1}$,
$d_{2}\sim\lambda^{2}d_{1}$, and $d_{3}\sim\lambda^{4}d_{1}$ for values of
these parameters that reproduce the SM quark masses. By rewriting Eq.(40)
using the identities in Eq.(42), and making the substitutions
$u_{2}\rightarrow\lambda^{4}\upsilon_{2}$,
$u_{3}\rightarrow\lambda^{8}\upsilon_{3}$,
$d_{2}\rightarrow\lambda^{2}\delta_{2}$, and
$d_{3}\rightarrow\lambda^{4}\delta_{3}$, where $u_{1}\sim
d_{1}\sim\upsilon_{2}\sim\delta_{2}\sim\upsilon_{3}\sim\delta_{3}$, we can
determine an approximate hierarchy between gauge boson mass eigenvalues by
expanding in the parameter $\lambda$. To start, we note that 13 of the gauge
bosons acquire masses that are generally in excess of any currently observable
energy scale. In particular, we find that 10 gauge bosons possess masses that
are of $O(u_{1},d_{1})\sim 10^{8}\;\textrm{GeV}$, while three more have masses
of $O(\lambda^{2}u_{1},\lambda^{2}d_{1})\sim 10^{7}\;\textrm{GeV}$. Even more
conveniently, the gauge bosons which acquire masses at these scales correspond
to the operators facilitating the most constrained flavor-changing effects,
those involving light quarks: These gauge bosons in fact must develop high
masses, because the scales of the scalar vev’s which contribute to these
operators are ultimately set by the masses of the light quarks’ seesaw
partners, which dictate how we constructed our scalar vev’s. As we have seen
in Section 2.3, these masses are enormous, and therefore in our construction
the masses of the gauge bosons which couple to the light quarks must be
similarly large.

For particles at the scale of these heavy bosons (around
$10^{7-8}\;\textrm{GeV})$, the only even remotely observable effects stem from
the imaginary part of effective 4-quark operators that facilitate
$K^{0}-\overline{K}^{0}$ mixing [25, 26]. However, from the $\lambda$
expansions outlined above, it can be seen that effects which give a non-
trivial phase to the coefficients of these 4-quark operators only appear at
the $O(\lambda^{3})$ level or higher. Since the masses of the potentially
offending gauge bosons are already near the limit of the sensitivity of this
tree-level probe, this essentially rules out any observable effect from the
heavy bosons. Given that any observable signatures of these bosons are likely
well outside the sensitivity of any experiment, we can shift our attention to
the three remaining gauge bosons that acquire mass at a far lower scale.

The first of these, which we shall refer to as $Z_{F}$ (the “flavor $Z$”),
consists almost entirely of a combination of the generators of the $SU(3)_{F}$
group embedded in $SU(4)_{F}$ (corresponding to the first eight $t^{i}$
generators, and the first eight rows/columns of the mass matrix in Eq.(40)),
up to numerically negligible corrections due to $\vec{\gamma}_{d,P}$ terms.
There does not exist a compact numerically accurate approximation for the mass
of $Z_{F}$. However, expanding in the Wolfenstein parameter $\lambda$ as above
yields the result that $m_{Z_{F}}\sim O(\lambda^{4}d_{1},\lambda^{4}u_{1})\sim
10^{5}\;\textrm{GeV}$. While this initially seems quite large, we note that a
boson of this mass may still mediate observable tree-level flavor-changing
neutral currents [27, 15, 26]. As such, it behooves us to determine how
$Z_{F}$ couples to the quarks, and crucially what flavor-changing interactions
it can mediate. Expanding Eq.(40) in the Wolfenstein parameter $\lambda$
offers an appealing approximate expression for the relevant combinations of
generators corresponding to this state. Specifically, up to $O(\lambda^{2})$,
the $SU(4)_{F}$ generator combination corresponding to $Z_{F}$ becomes

$\displaystyle
O(\lambda^{7})\;t^{1}+O(\lambda^{7})\;t^{2}+O(\lambda^{4})\;t^{3}+O(\lambda^{5})\;t^{4}+O(\lambda^{5})\;t^{5}+\sqrt{3}A\lambda^{2}(\rho-1)\;t^{6}+\sqrt{3}A\lambda^{2}\eta\;t^{7}+t^{8}.$
(44)

Consulting the explicit form of the generator matrices $t^{i}$ in Appendix A,
we see that $Z_{F}$ only mediates significant flavor-changing neutral currents
between the second and third generations – in both the up- and down-like
sectors, flavor-changing neutral currents featuring the first generation of
quarks only appear at the $O(\lambda^{5})\sim 10^{-4}$ level, at most. Our
chief phenomenological concern regarding $Z_{F}$, then, is the constraint
arising from tree-level flavor-changing neutral currents that it mediates
between the second and third generation of quarks. We shall discuss
phenomenological constraints arising from this type of interaction in Section
4.2.2.

The next two light gauge bosons we consider will arise from the generators
that are left unbroken by $\langle\Phi_{A}\rangle$ and
$\langle\Phi_{B}\rangle$, at least in the approximation where
$\vec{\gamma}_{d}\rightarrow 0$. In particular, we see that in the limit where
$\vec{\gamma}_{d,P}\rightarrow 0$, $\langle\Phi_{A}\rangle$ and
$\langle\Phi_{B}\rangle$ break $SU(4)_{F}\times U(1)_{F}$ down to
$U(1)_{F}^{\prime}\times U(1)_{F}$, where the generator for
$U(1)_{F}^{\prime}$ is given by the matrix $t^{15}$. In the limit where
$\vec{\gamma}_{P}\rightarrow 0$, the scalar
$\langle\Phi_{P}\rangle=(\vec{\gamma}_{P},v_{P})$ breaks
$U(1)_{F}^{\prime}\times U(1)_{F}$ down to $U(1)_{D}$ at the scale $v_{P}\sim
1-10\;\textrm{TeV}$. The $\vec{\gamma}_{d}$ and $\vec{\gamma}_{P}$ terms then
finally break the $U(1)_{D}$ symmetry at $\sim 0.1-1\;\textrm{GeV}$. The two
mass eigenstates we consider here, then, will be well-approximated as
combinations of the $U(1)_{F}$ and $U(1)_{F}^{\prime}$ bosons, with one
achieving a mass of $O(v_{P})\sim 1-10\;\textrm{TeV}$, the scale at which
$U(1)_{F}^{\prime}\times U(1)_{F}$ breaks down to $U(1)_{D}$, and the other
achieving mass only at the scale where $U(1)_{D}$ is broken, at approximately
$0.1-1\;\textrm{GeV}$. Referring to Eq.(40), we see that in the limit where
$\vec{\gamma}_{d,P}\rightarrow 0$, the squared mass matrix for the eventual
mass eigenstates, $Z_{P}$ and $A_{D}$ (i.e., the dark photon), becomes

$\displaystyle\begin{pmatrix}B^{\prime}&B\end{pmatrix}\begin{pmatrix}\frac{3g_{4}^{2}v_{P}^{2}}{4}&-\sqrt{\frac{3}{2}}\frac{3g_{4}g_{1}v_{P}^{2}}{4}\\\
-\sqrt{\frac{3}{2}}\frac{3g_{4}g_{1}v_{P}^{2}}{4}&\frac{9g_{1}^{2}v_{P}^{2}}{8}\end{pmatrix}\begin{pmatrix}B^{\prime}\\\
B\end{pmatrix},$ (45)

where $B$ and $B^{\prime}$ refer to the $U(1)_{F}$ and $U(1)_{F}^{\prime}$
gauge bosons, respectively. It is convenient to define an angle $\theta_{P}$
such that $g_{1}=\sqrt{2/3}g_{4}\tan\theta_{P}$. Then, we may write the mass
squared matrix as

$\displaystyle\frac{3g_{4}^{2}v_{P}^{2}}{4}\begin{pmatrix}B^{\prime}&B\end{pmatrix}\begin{pmatrix}1&-\tan\theta_{P}\\\
-\tan\theta_{P}&\tan^{2}\theta_{P}\end{pmatrix}\begin{pmatrix}B^{\prime}\\\
B\end{pmatrix}.$ (46)

The mass eigenstates for these gauge bosons can then be easily determined, up
to $O(\vec{\gamma}_{P,d}^{2}/v_{P}^{2})\sim 10^{-4}$ corrections. Given the
definition of $\theta_{P}$ we’ve provided, we see that the eigenstates are
simply

$\displaystyle A_{D}\approx B^{\prime}s_{P}+Bc_{P},\;\;Z_{P}\approx
B^{\prime}c_{P}-Bs_{P},$ (47)

where $A_{D}$ refers to the dark photon, with a mass of $\sim
0.1-1\;\textrm{GeV}$, while $Z_{P}$ refers to the heavier gauge boson, a dark
“portal Z”, with $O(v_{P})$ mass. Here we have also used the abbreviations
$c_{P}\equiv\cos\theta_{P}$ and $s_{P}\equiv\sin\theta_{P}$, which for
simplicity we shall also employ going forward.

The masses of $A_{D}$ and $Z_{P}$ can also be straightforwardly determined. In
the case of $Z_{P}$, we can directly extract the mass from the mass matrix of
Eq.(46), arriving at

$\displaystyle m_{Z_{P}}^{2}\approx\frac{3g_{4}^{2}v_{P}^{2}}{4c^{2}_{P}},$
(48)

which is accurate up to numerically negligible
$O(\vec{\gamma}_{P,d}^{2}/v_{P}^{2})$ corrections. Meanwhile, the mass of the
dark photon can be given up to $O(\lambda^{2})\sim O(10^{-2})$ corrections by

$\displaystyle m_{A_{D}}^{2}$
$\displaystyle\approx\frac{2g_{4}^{2}s^{2}_{P}}{3}\bigg{(}\frac{7}{8}(\vec{\gamma}_{P}^{*}\cdot\vec{\gamma_{P}})+\mu_{1}^{A_{D}}(\vec{\gamma}_{d}^{*}\cdot\vec{\gamma}_{d})+\mu_{2}^{A_{D}}|(\vec{\gamma}_{d})_{1}|^{2}\bigg{)},$
(49) $\displaystyle\mu_{1}^{A_{D}}$
$\displaystyle\equiv\frac{4(4M-u_{1})^{2}}{y_{A}^{2}(d_{1}-u_{1})^{2}+y_{B}^{2}(d_{1}+u_{1}-8M)^{2}},$
$\displaystyle\mu_{2}^{A_{D}}$
$\displaystyle\equiv\frac{16M(d_{1}-u_{1})(y_{A}^{2}(d_{1}-u_{1})(-3M+u_{1})+y_{B}^{2}((-4M+u_{1})^{2}-Mu_{1}+d_{1}(-3M+u_{1})))}{(y_{A}^{2}(d_{1}-u_{1})^{2}+y_{B}^{2}(d_{1}+u_{1}-8M)(d_{1}+u_{1}-4M))^{2}+16y_{A}^{2}y_{B}^{2}M^{2}(d_{1}-u_{1})^{2}},$

where to arrive at this expression, we have exploited the fact that
$d_{1},u_{1}\gg d_{2,3},u_{2,3},v_{P}$. At this point, we have obtained the
masses and eigenvectors for the three new gauge bosons that may be
phenomenologically relevant – namely, the $SU(3)_{F}$ boson $Z_{F}$ with a
mass of $O(10^{5}\;\textrm{GeV})$, the “portal $Z$” field $Z_{P}$ with a mass
roughly of $O(1\;\textrm{TeV})$, and the dark photon field $A_{D}$, with a
mass of $O(0.1-1\;\textrm{GeV})$.

##  3 Couplings, Kinetic Mixing, and Low-Energy Parameters

Before computing the observable experimental signatures of this setup, it is
useful to explicitly determine the couplings between various
phenomenologically relevant particles in the model. In particular, in light of
our discussion of the fermion spectrum in Section 2.3, we shall focus on the
SM and new physics gauge boson and scalar couplings of the SM fermions, the
portal matter fermions, and the heavy top partner.

###  3.1 $SU(4)_{F}\times U(1)_{F}$ Gauge Bosons

We begin our discussion by deriving the couplings of the new gauge bosons that
arise in our setup due to our $SU(4)_{F}\times U(1)_{F}$ extension of the SM
gauge group. In Section 2.4, we have determined that only three of the new
gauge bosons are sufficiently light so as to have observable phenomenological
effects: A (comparatively) light $SU(3)_{F}$ boson that we have dubbed
$Z_{F}$, with a mass of roughly $O(10^{5}\;\textrm{GeV})$, a “portal $Z$”
field with a mass of $O(1-10\;\textrm{TeV})$, which we denote as $Z_{P}$, and
a dark photon, $A_{D}$, with a mass of approximately $O(0.1-1\;\textrm{GeV})$.

To start, we consider the couplings of the dark photon $A_{D}$ to the fermions
of the theory before KM takes place. In the limit where
$\vec{\gamma}_{P,d}\rightarrow 0$, the dark photon is given as a simple
combination of two $U(1)$ bosons, described in Eq.(47). In fact, the results
of Eq.(47) are exceedingly numerically accurate, even once contributions from
$\vec{\gamma}_{P,d}$ terms are included: Mixing between $A_{D}$, the gauge
boson $Z_{P}$, and the gauge bosons corresponding to the $SU(3)_{F}$ embedded
in $SU(4)_{F}$ only occurs at second order in the quantities
$\vec{\gamma}_{P,d}$, and is hence numerically negligible, while mixing
between $A_{D}$ and the gauge bosons corresponding to the other $SU(4)_{F}$
generators is suppressed by the latter’s enormous masses – in Section 2.4, we
found these to be of $O(10^{7-8})\;\textrm{GeV}$. Thus, we can use the
combinations given in Eq.(47) to derive the couplings of $A_{D}$ and $Z_{P}$
without concerning ourselves with any additional complicating effects. Noting
that the $B^{\prime}$ boson corresponds to the $15^{\textrm{th}}$ generator of
$SU(4)_{F}$ ($t^{15}$, as listed in Appendix A), and $B$ is simply the
$U(1)_{F}^{\prime}$ boson, we can straightforwardly find a coupling matrix by
writing the fermions as 9-dimensional vectors in flavor space, as in Eq.(2.3).
We find that for up-like and down-like quarks, charged leptons, and neutrinos
(at least ignoring any extra structure the model may need to accommodate
neutrino flavor phenomenology), the dark photon couplings are given by

$\displaystyle
g_{4}\sqrt{\frac{2}{3}}s_{P}(\mathbf{C}^{u,d,e,\nu}_{A_{D}})_{L,R}=-g_{4}\sqrt{\frac{2}{3}}s_{P}\mathcal{U}^{u,d,e,\nu\dagger}_{L,R}\begin{pmatrix}\mathbf{0}_{3\times
3}&\mathbf{0}_{3\times 3}&\mathbf{0}_{3\times 3}\\\ \mathbf{0}_{3\times
3}&\mathbf{0}_{3\times 3}&\mathbf{0}_{3\times 3}\\\ \mathbf{0}_{3\times
3}&\mathbf{0}_{3\times 3}&\mathbf{1}_{3\times
3}\end{pmatrix}\mathcal{U}^{u,d,e,\nu}_{L,R},$ (50)

where we have used the fact that the $U(1)_{F}$ coupling constant, $g_{1}$, is
given in terms of the $SU(4)_{F}$ coupling constant by the relation
$g_{1}=\sqrt{2/3}g_{4}\tan\theta_{P}$. The overall negative sign in this
expression is simply an artifact of the sign conventions we have selected for
the definitions of $U(1)_{F}$ and $U(1)_{D}$. At this point, we have omitted
the additional couplings arising from kinetic mixing with the SM hypercharge
field as noted earlier – these will introduce shifts of $e\epsilon
Q\mathbf{1}_{9\times 9}$ to the above expression, where $e$ is the proton
charge, $\epsilon$ is the kinetic mixing parameter and $Q$ is the electric
charge of the given field. While these shifts are obviously significant in
dark matter phenomenology, since they encapsulate the means by which, for
example, a dark matter field might annihilate to form SM particles, they are
of less importance to us when considering other constraints here: Because the
$e\epsilon Q$ shift is always proportional to the identity matrix for a given
fermion electric charge, it does not facilitate flavor-changing neutral
currents or portal matter decay to SM particles. Restricting our attention to
only fermions which are light enough to be phenomenologically relevant, namely
the SM quarks, the heavy top partner, and the lightest two portal matter
fields in each sector, we can rewrite the coupling matrices using the
truncated flavor vectors of Eq.(2.3). In the up-like quark sector, we arrive
at the truncated coupling matrices

$\displaystyle(\mathbf{C}_{A_{D}}^{u})_{L}\approx\begin{pmatrix}&&&0&-\frac{(\vec{\gamma}_{P})_{1}}{v_{P}}&0\\\
&(\mathbf{C}_{A_{D}}^{u})_{L}^{SM}&&0&-\frac{(\vec{\gamma}_{P})_{2}}{v_{P}}&0\\\
&&&0&-\frac{(\vec{\gamma}_{P})_{3}}{v_{P}}&qrz_{P2}\frac{m_{t}}{M_{t}}\frac{(\vec{\gamma}_{P})_{3}}{v_{P}}\\\
0&0&0&0&-\frac{r}{(1-z_{P1}^{2})}\frac{m_{t}}{M_{t}}\frac{(\vec{\gamma}_{P})_{3}}{v_{P}}&-\frac{qz_{P2}}{1-z_{P2}^{2}}\frac{(\vec{\gamma}_{P})_{3}}{v_{P}}\\\
-\frac{(\vec{\gamma}_{P})_{1}^{*}}{v_{P}}&-\frac{(\vec{\gamma}_{P})_{2}^{*}}{v_{P}}&-\frac{(\vec{\gamma}_{P})_{3}^{*}}{v_{P}}&-\frac{r}{(1-z_{P1}^{2})}\frac{m_{t}}{M_{t}}\frac{(\vec{\gamma}_{P})_{3}^{*}}{v_{P}}&1&0\\\
0&0&qrz_{P2}\frac{m_{t}}{M_{t}}\frac{(\vec{\gamma}_{P})_{3}^{*}}{v_{P}}&-\frac{qz_{P2}}{1-z_{P2}^{2}}\frac{(\vec{\gamma}_{P})_{3}^{*}}{v_{P}}&0&1\end{pmatrix},$
(51)
$\displaystyle(\mathbf{C}_{A_{D}}^{u})_{R}\approx\mathbf{N}_{2}\begin{pmatrix}&&&0&0&\frac{(\vec{\gamma}_{P})_{1}}{v_{P}}\\\
&(\mathbf{C}_{A_{D}}^{u})_{R}^{SM}&&0&0&\frac{(\vec{\gamma}_{P})_{2}}{v_{P}}\\\
&&&0&-z_{P1}\frac{m_{t}}{M_{t}}\frac{(\vec{\gamma}_{P})_{3}}{v_{P}}&qr\frac{(\vec{\gamma}_{P})_{3}}{v_{P}}\\\
0&0&0&0&-\frac{rz_{P1}}{(1-z_{P1}^{2})}\frac{m_{t}}{M_{t}}\frac{(\vec{\gamma}_{P})_{3}}{v_{P}}&-\frac{q}{1-z_{P2}^{2}}\frac{(\vec{\gamma}_{P})_{3}}{v_{P}}\\\
0&0&-z_{P1}\frac{m_{t}}{M_{t}}\frac{(\vec{\gamma}_{P})_{3}^{*}}{v_{P}}&-\frac{rz_{P1}}{(1-z_{P1}^{2})}\frac{m_{t}}{M_{t}}\frac{(\vec{\gamma}_{P})_{3}^{*}}{v_{P}}&1&0\\\
\frac{(\vec{\gamma}_{P})_{1}^{*}}{v_{P}}&\frac{(\vec{\gamma}_{P})_{2}^{*}}{v_{P}}&qr\frac{(\vec{\gamma}_{P})_{3}^{*}}{v_{P}}&-\frac{q}{1-z_{P2}^{2}}\frac{(\vec{\gamma}_{P})_{3}^{*}}{v_{P}}&0&1\end{pmatrix}\mathbf{N}_{2},$
$\displaystyle(\mathbf{C}_{A_{D}}^{u})_{L}^{SM}\equiv\frac{(\vec{\gamma}_{P}\otimes\vec{\gamma}_{P}^{\dagger})}{v_{P}^{2}},\;\;\;(\mathbf{C}_{A_{D}}^{u})_{R}^{SM}\equiv\begin{pmatrix}\frac{|(\vec{\gamma}_{P})_{1}|^{2}}{v_{P}^{2}}&\frac{(\vec{\gamma}_{P})_{1}(\vec{\gamma}_{P})_{2}^{*}}{v_{P}^{2}}&qr\frac{(\vec{\gamma}_{P})_{1}(\vec{\gamma}_{P})_{3}^{*}}{v_{P}^{2}}\\\
\frac{(\vec{\gamma}_{P})_{2}(\vec{\gamma}_{P})_{1}^{*}}{v_{P}^{2}}&\frac{|(\vec{\gamma}_{P})_{2}|^{2}}{v_{P}^{2}}&qr\frac{(\vec{\gamma}_{P})_{2}(\vec{\gamma}_{P})_{3}^{*}}{v_{P}^{2}}\\\
qr\frac{(\vec{\gamma}_{P})_{3}(\vec{\gamma}_{P})_{1}^{*}}{v_{P}^{2}}&qr\frac{(\vec{\gamma}_{P})_{3}(\vec{\gamma}_{P})_{2}^{*}}{v_{P}^{2}}&q^{2}r^{2}\frac{|(\vec{\gamma}_{P})_{3}|^{2}}{v_{P}^{2}}\end{pmatrix},$

where $q$, $r$, $z_{P1}$, and $z_{P2}$ are defined as in Eq.(30), and
$\mathbf{N}_{2}$ is defined as in Eq.(37). The down-like quark sector,
meanwhile, has its truncated coupling matrices given by

$\displaystyle(\mathbf{C}_{A_{D}}^{d})_{L}\approx\begin{pmatrix}\frac{(\mathbf{W}_{d}^{\dagger}\vec{\gamma_{P}})\otimes(\vec{\gamma}_{P}^{\dagger}\mathbf{W}_{d})}{v_{P}^{2}}&-\frac{\mathbf{W}_{d}^{\dagger}\vec{\gamma}_{P}}{v_{P}}&\mathbf{0}_{3\times
1}\\\ -\frac{\vec{\gamma}_{P}^{\dagger}\mathbf{W}_{d}}{v_{P}}&1&0\\\
\mathbf{0}_{1\times 3}&0&1\end{pmatrix},$ (52)
$\displaystyle(\mathbf{C}_{A_{D}}^{d})_{R}\approx\begin{pmatrix}\frac{(\mathbf{W}_{d}^{\dagger}\vec{\gamma_{P}})\otimes(\vec{\gamma}_{P}^{\dagger}\mathbf{W}_{d})}{v_{P}^{2}}&\mathbf{0}_{3\times
1}&-\frac{\mathbf{W}_{d}^{\dagger}\vec{\gamma}_{P}}{v_{P}}\\\
\mathbf{0}_{1\times 3}&1&0\\\
-\frac{\vec{\gamma}^{\dagger}_{P}\mathbf{W}_{d}}{v_{P}}&0&1\end{pmatrix},$

where we remind the reader that $\mathbf{W}_{d}$ is the unitary matrix first
described in Eq.(13), and that in the $SU(4)_{F}$ basis we have chosen,
$\mathbf{W}_{d}$ is equal to the CKM matrix. The corresponding coupling matrix
for the charged leptons can be given in complete analogy to Eq.(52), with the
sole exception that the matrix $\mathbf{W}_{d}$ must be replaced by the
appropriate rotation matrix given the lepton couplings to the scalars
$\Phi_{A}$ and $\Phi_{B}$.

In I and II, it was seen that in the event that the portal matter mixes with
electrons, there exists a parity-violating interaction of the right-handed
electrons with the dark photon field. Consulting Eq.(52), we observe that the
analogous terms emerge at the $O(\vec{\gamma}_{P}^{2}/v_{P}^{2})$ level,
however, they occur _identically_ for the right- and left-handed fields (at
least up to corrections due to the super-heavy fermions that we have omitted
from our truncated coupling matrices). As a result, no such parity-violating
interaction occurs in this model, in contrast to those of I and II.
Generically, this can be expected to be the case due to the more left-right
symmetric form of the model: The left-handed electrons receive a chiral
correction from their coupling from the $SU(2)_{L}$ doublet portal matter,
while the right-handed ones receive the _same_ correction from $SU(2)_{L}$
singlet portal matter. The couplings in the neutrino sector may be
qualitatively similar to those that we have already explored, however, we note
that the recreation of light neutrino masses and the observed mixing matrix
will likely require significant modifications to the neutrino mass matrix,
which are beyond the scope of this paper. Notably, we have retained
$O(\vec{\gamma}_{P}^{2}/v_{P}^{2})$ terms in the components of Eqs.(51) and
(52) which correspond to mixing among the SM quarks.555In spite of the fact
that our expressions for $\mathcal{U}^{u}_{L,R}$ and for $A_{D}$’s mass
eigenvector are only valid up to $O(\vec{\gamma}_{P}/v_{P})$, these
expressions (at least for flavor-changing interactions, which are of
phenomenological interest here) are numerically valid, because the
$O(\vec{\gamma}_{P}^{2}/v_{P}^{2})$ corrections to the matrices
$\mathcal{U}^{u}_{L,R}$ don’t contribute to the
$O(\vec{\gamma}_{P}^{2}/v_{P}^{2})$ flavor-changing couplings in the SM, and
we shall see that the gauge boson $Z_{P}$, which may mix with $A_{D}$ at the
$O(\vec{\gamma}_{P}^{2}/v_{P}^{2})$ level, has SM flavor-universal couplings
up to $O(\vec{\gamma}_{P}/v_{P})$ corrections. In spite of their minute
magnitude, with $\vec{\gamma}_{P}/v_{P}\sim 10^{-(3-4)}$, we have kept them
here because they facilitate highly-constrained flavor-changing neutral
currents mediated by the extremely light $A_{D}$ boson. In Section 4.1, we
shall in fact derive significant non-trivial constraints on the model
parameter space from these interactions. We do note in passing that similar
$O(\vec{\gamma}_{P}^{2}/v_{P}^{2})$ terms exist elsewhere in Eqs.(51) and
(52), however, we have omitted them here since they have a negligible effect
on any observable physics.

Apart from the flavor-changing interactions we observe in the SM, the dark
photon also facilitates $O(\vec{\gamma}_{P}/v_{P})$ interactions between the
portal matter fermion fields and those uncharged under $U(1)_{D}$, namely the
SM quarks and the heavy top partner. In practice, due to $A_{D}$’s tiny mass
compared to other gauge bosons in the theory, these interactions make
meaningful contributions to a number of interesting processes; most notably,
they dominate the decay of portal matter to an SM quark, or the decay of a top
partner to portal matter (or vice versa, depending on which process is
kinematically allowed). The presence of these couplings for the extremely
light dark photon field can substantially simplify our later discussions of
the couplings of heavier gauge bosons: From the Equivalence Theorem [23], and
the fact that $|\vec{\gamma}_{P}|\sim m_{A_{D}}\sim 0.1-1\;\textrm{GeV}$, we
would anticipate that the overall strength of the $A_{D}$-facilitated
interaction between portal and non-portal matter would undergo a substantial
enhancement over what the $\vec{\gamma}_{P}/v_{P}$ suppression in its coupling
would suggest, since this suppression would be cancelled by $m_{A_{D}}$ (at
least for the longitudinal mode of $A_{D}$). Since obviously such an
enhancement doesn’t exist for $\vec{\gamma}_{P}/v_{P}$-suppressed couplings
for heavier gauge bosons, which in turn mediate the same $U(1)_{D}$-breaking
couplings as we see emerging from $A_{D}$, we therefore can omit a detailed
evaluation of $O(\vec{\gamma}_{P}/v_{P})$-suppressed effects in the couplings
of $Z_{F}$, $Z_{P}$, the SM electroweak gauge bosons and the SM-like Higgs; in
all cases we consider, these effects are overwhelmed by those arising from
$A_{D}$.

We next turn to the couplings for the “portal $Z$” boson, $Z_{P}$. Again
referencing Eq.(47), this time to get the approximate mass eigenvector for
$Z_{P}$, we arrive at the coupling matrices

$\displaystyle-\frac{g_{4}}{2\sqrt{6}c_{P}}(\mathbf{C}^{u,d,e,\nu}_{Z_{P}})_{L}=-\frac{g_{4}}{2\sqrt{6}c_{P}}\mathcal{U}^{u,d,e,\nu\dagger}_{L}\begin{pmatrix}\mathbf{1}_{6\times
6}&\mathbf{0}_{6\times 1}&\mathbf{0}_{6\times 1}&\mathbf{0}_{6\times 1}\\\
\mathbf{0}_{1\times 6}&1-4c^{2}_{P}&0&0\\\ \mathbf{0}_{1\times
6}&0&4s^{2}_{P}&0\\\ \mathbf{0}_{1\times
6}&0&0&1-4c^{2}_{P}\end{pmatrix}\mathcal{U}^{u,d,e,\nu}_{L},$ (53)
$\displaystyle-\frac{g_{4}}{2\sqrt{6}c_{P}}(\mathbf{C}^{u,d,e,\nu}_{Z_{P}})_{R}=-\frac{g_{4}}{2\sqrt{6}c_{P}}\mathcal{U}^{u,d,e,\nu\dagger}_{R}\begin{pmatrix}\mathbf{1}_{6\times
6}&\mathbf{0}_{6\times 1}&\mathbf{0}_{6\times 1}&\mathbf{0}_{6\times 1}\\\
\mathbf{0}_{1\times 6}&4s^{2}_{P}&0&0\\\ \mathbf{0}_{1\times
6}&0&1-4c^{2}_{P}&0\\\ \mathbf{0}_{1\times
6}&0&0&1-4c^{2}_{P}\end{pmatrix}\mathcal{U}^{u,d,e,\nu}_{R}.$

We can then determine what the couplings are for the quarks that are light
enough to remain phenomenologically relevant, as we’ve already done for
$A_{D}$, by inserting our results for $\mathcal{U}^{u,d}_{L,R}$ from Eqs.(30),
(31), (37), and (38) into Eq.(53). In fact, up to $O(\vec{\gamma}_{P}/v_{P})$
corrections, which we note are negligible compared to similar interactions
arising from $A_{D}$, we find that we can write the truncated coupling
matrices as

$\displaystyle(\mathbf{C}^{u}_{Z_{P}})_{L}\approx
diag(1,1,1,1,1-4c^{2}_{P},4s^{2}_{P}),$
$\displaystyle(\mathbf{C}^{u}_{Z_{P}})_{R}\approx
diag(1,1,1,1,4s^{2}_{P},1-4c^{2}_{P}),$ (54)
$\displaystyle(\mathbf{C}^{d,e}_{Z_{P}})_{L}\approx
diag(1,1,1,1-4c^{2}_{P},4s^{2}_{P}),$
$\displaystyle(\mathbf{C}^{d,e}_{Z_{P}})_{R}\approx
diag(1,1,1,4s^{2}_{P},1-4c^{2}_{P}),$

where we have noted that in the absence of corrections due to mixing with
heavy partner states and $\vec{\gamma}_{P}$ terms, the coupling matrix for
charged leptons here is the same as that for down-like quarks. We shall not
explicitly determine the coupling matrices in the neutrino sector in this
work. As noted at the end of Section 2.3, a realistic neutrino mass matrix in
this model would likely involve vacuum expectation values of still more
additional $SU(4)_{F}\times U(1)_{F}$ scalars, which would complicate the
already highly intricate structure of the model while remaining unlikely to
influence most of the new physics at an experimentally observable scale.

We finally address couplings which arise from the gauge boson $Z_{F}$, the
“flavor $Z$” that represents the only gauge boson from the $SU(3)_{F}$ flavor
symmetry which possesses a low enough mass to have some phenomenological
impact. Consulting Eq.(44), we see that the coupling matrix for this gauge
boson may be written,

$\displaystyle\frac{g_{4}}{2\sqrt{3}}(\mathbf{C}^{u,d,e,\nu}_{Z_{F}})_{L,R}=\frac{g_{4}}{2\sqrt{3}}\mathcal{U}^{u,d,e,\nu\dagger}_{L,R}\begin{pmatrix}\bm{\Lambda}_{3\times
3}&\mathbf{0}_{3\times 3}&\mathbf{0}_{3\times 3}\\\ \mathbf{0}_{3\times
3}&\bm{\Lambda}_{3\times 3}&\mathbf{0}_{3\times 3}\\\ \mathbf{0}_{3\times
3}&\mathbf{0}_{3\times 3}&\mathbf{0}_{3\times
3}\end{pmatrix}\mathcal{U}^{u,d,e,\nu}_{L,R},$ (55)
$\displaystyle\bm{\Lambda}\equiv\begin{pmatrix}1&0&0\\\
0&1&3A\lambda^{2}(\rho-i\eta-1)\\\
0&3A\lambda^{2}(\rho+i\eta-1)&-2\end{pmatrix},$

where we remind the reader that $A$, $\lambda$, $\rho$, and $\eta$ are the
Wolfenstein parameters. Notably, $Z_{F}$ couples to SM fields and their heavy
partners equivalently, and as such doesn’t facilitate any couplings between
them. However, we see that this gauge boson can produce flavor-changing
neutral currents in the SM quark sector. Focusing on this possibility, we
consider what the coupling matrices for the SM quarks look like in our setup,
arriving at

$\displaystyle(\mathbf{C}^{u}_{Z_{F}})_{L,R}^{SM}=\bm{\Lambda},\;\;\;(\mathbf{C}^{d}_{Z_{F}})_{L,R}^{SM}=\mathbf{W}_{d}^{\dagger}\bm{\Lambda}\mathbf{W}_{d}\approx\begin{pmatrix}1&0&0\\\
0&1&3A\lambda^{2}(\rho-i\eta)\\\ 0&3A\lambda^{2}(\rho+i\eta)&-2\end{pmatrix},$
(56)

where to derive the expression for $(\mathbf{C}^{d}_{Z_{F}})_{L,R}^{SM}$, we
have used the fact that $\mathbf{W}_{d}$ is simply given by the CKM matrix,
and used the Wolfenstein paramterization of the CKM given in Eq.(43), keeping
terms up to $O(\lambda^{2})$. Note that in spite of its appearance (and
explicit dependence on Wolfenstein parameters), the coupling of Eq.(56) is
_not_ simply an artifact of our choice of $SU(4)_{F}$ gauge: Effecting an
$SU(4)_{F}$ transformation here to a frame which, for example,
$\mathbf{W}_{d}$ is equal to the identity matrix and $\mathbf{W}_{u}$ is non-
trivial should correspondingly alter the gauge boson mass matrix so that the
resulting coupling is preserved. We do, however, note that the right-handed
coupling expressions in Eq.(56) ignore the sign flips (phase rotations) of
various right-handed quark fields observed in Eqs.(37) and (38). Since they