cancel in any phenomenological results we shall discuss, we have omitted them
above for the sake of brevity.

###  3.2 SM Gauge Bosons and the Higgs

Having discussed the couplings of the fermions to new gauge bosons in the
theory, we now address the coupling matrices for usual SM fields, namely the
$Z$ and $W$ gauge bosons and the light SM-like Higgs doublet embedded in the
bidoublet $H$. We begin our discussion with the $Z$ boson. Writing the
fermions as 9-component vectors in flavor space as outlined in Eq.(2.3), we
can write the coupling matrix of the $Z$ as

$\displaystyle\frac{g}{c_{w}}(\mathbf{C}^{u,d,e,\nu}_{Z})_{L}=\frac{g}{c_{w}}\mathcal{U}^{u,d,e,\nu\dagger}_{L}\begin{pmatrix}(T_{3L}-Qs_{w}^{2})\mathbf{1}_{3\times
3}&\mathbf{0}_{3\times 3}&\mathbf{0}_{3\times 1}&\mathbf{0}_{3\times
1}&\mathbf{0}_{3\times 1}\\\ \mathbf{0}_{3\times
3}&-Qs_{w}^{2}\mathbf{1}_{3\times 3}&\mathbf{0}_{3\times
1}&\mathbf{0}_{3\times 1}&\mathbf{0}_{3\times 1}\\\ \mathbf{0}_{1\times
3}&\mathbf{0}_{1\times 3}&(T_{3L}-Qs_{w}^{2})&0&0\\\ \mathbf{0}_{1\times
3}&\mathbf{0}_{1\times 3}&0&-Qs_{w}^{2}&0\\\ \mathbf{0}_{1\times
3}&\mathbf{0}_{1\times
3}&0&0&-Qs_{w}^{2}\end{pmatrix}\mathcal{U}^{u,d,e,\nu}_{L},$ (57)
$\displaystyle\frac{g}{c_{w}}(\mathbf{C}^{u,d,e,\nu}_{Z})_{R}=\frac{g}{c_{w}}\mathcal{U}^{u,d\dagger}_{R}\begin{pmatrix}-Qs_{w}^{2}\mathbf{1}_{3\times
3}&\mathbf{0}_{3\times 3}&\mathbf{0}_{3\times 1}&\mathbf{0}_{3\times
1}&\mathbf{0}_{3\times 1}\\\ \mathbf{0}_{3\times
3}&-Qs_{w}^{2}\mathbf{1}_{3\times 3}&\mathbf{0}_{3\times
1}&\mathbf{0}_{3\times 1}&\mathbf{0}_{3\times 1}\\\ \mathbf{0}_{1\times
3}&\mathbf{0}_{1\times 3}&(T_{3L}-Qs_{w}^{2})&0&0\\\ \mathbf{0}_{1\times
3}&\mathbf{0}_{1\times 3}&0&-Qs_{w}^{2}&0\\\ \mathbf{0}_{1\times
3}&\mathbf{0}_{1\times
3}&0&0&-Qs_{w}^{2}\end{pmatrix}\mathcal{U}^{u,d,e,\nu}_{R},$

up to $O(\epsilon)$ corrections due to kinetic mixing, which won’t be
phenomenologically significant here, since they’ll only represent a uniform
small correction to the couplings of the three portal matter states, arising
from mixing with the dark photon $A_{D}$. Here, $T_{3L}$ refers to the left-
handed isospin of the SM fermion species and $Q$ refers to its electric
charge. We can then find the coupling matrices for the phenomenologically
relevant mass eigenstates by simply truncating the above matrices to exclude
the extremely heavy states and rotating by the approximate diagonalization
matrices given in Eqs.(30), (31), (37), and (38). Up to numerically negligible
$O(\vec{\gamma}_{P}/v_{P})$ terms, we arrive at

$\displaystyle(\mathbf{C}^{u}_{Z})_{L}\approx\begin{pmatrix}\frac{1}{2}-\frac{2}{3}s_{w}^{2}&0&0&0&0&0\\\
0&\frac{1}{2}-\frac{2}{3}s_{w}^{2}&0&0&0&0\\\
0&0&\frac{1}{2}-\frac{2}{3}s_{w}^{2}&\frac{r}{2}\frac{m_{t}}{M_{t}}&0&0\\\
0&0&\frac{r}{2}\frac{m_{t}}{M_{t}}&-\frac{2}{3}s_{w}^{2}&0&0\\\
0&0&0&0&\frac{1}{2}-\frac{2}{3}s_{w}^{2}&0\\\
0&0&0&0&0&-\frac{2}{3}s_{w}^{2}\end{pmatrix},$ (58)
$\displaystyle(\mathbf{C}^{u}_{Z})_{R}\approx\begin{pmatrix}-\frac{2}{3}s_{w}^{2}&0&0&0&0&0\\\
0&-\frac{2}{3}&0&0&0&0\\\ 0&0&-\frac{2}{3}s_{w}^{2}&0&0&0\\\
0&0&0&-\frac{2}{3}s_{w}^{2}&0&0\\\
0&0&0&0&\frac{1}{2}-\frac{2}{3}s_{w}^{2}&0\\\
0&0&0&0&0&-\frac{2}{3}s_{w}^{2}\end{pmatrix},$

where we remind the reader that the variable $r$ is defined in Eq.(30). We see
that at this level of approximation, the sole new coupling for the $Z$ boson
(other than its diagonal couplings to the new fermions, which directly follow
their $SU(2)_{L}\times U(1)_{Y}$ quantum numbers) is between the top quark and
its vector-like partner, which can be quite large – the suppression ratio
$m_{t}/M_{t}$ for this coupling can be as high as $\sim 0.1$, for top partner
masses near 1 TeV. We note that there do exist $O(m_{t}^{2}/M_{t}^{2})$
corrections to the $Z\bar{t}t$ coupling, however, since these corrections
would be at most on the order of a few percent, they are well within current
constraints for modifications of the top-$Z$ coupling [28]. Of course, such
small variations in the top quark couplings to the $Z$ may be probed by
precision measurements made at future $e^{+}e^{-}$ colliders. In the down-like
and charged lepton sectors, we have an analogous result to Eq.(58), however,
assuming that the $b$ and $\tau$ partners are too massive to influence the
low-energy phenomenology of the theory, we find no significant departures from
the SM behavior of the $Z$ and diagonal coupling matrices.666As an aside, we
note that even in the event of a much lighter $b$ partner, brought about by a
percent-level tuning of the Higgs Yukawa coupling to the down quarks, which
might then have a mass comparable to the top partner mass $M_{t}\sim
1\;\textrm{TeV}$, we would not observe a measurable effect in the tightly-
constrained $Z\bar{b}b$ coupling, since this would still be suppressed by
$O(m_{b}^{2}/M_{b}^{2})\sim 10^{-6}$, assuming $M_{b}\sim M_{t}$, which is
several orders of magnitude below present constraints [29]

Next, we address the $W$ boson couplings, restricting our attentions to the
quark sector in order to avoid ambiguities arising in the neutrino sector in
this model. We may write our coupling matrices here as

$\displaystyle\frac{g}{\sqrt{2}}(\mathbf{C}_{W})_{L}=\frac{g}{\sqrt{2}}\mathcal{U}^{u\dagger}_{L}\begin{pmatrix}\mathbf{1}_{3\times
3}&\mathbf{0}_{3\times 3}&\mathbf{0}_{3\times 1}&\mathbf{0}_{3\times
1}&\mathbf{0}_{3\times 1}\\\ \mathbf{0}_{3\times 3}&\mathbf{0}_{3\times
3}&\mathbf{0}_{3\times 1}&\mathbf{0}_{3\times 1}&\mathbf{0}_{3\times 1}\\\
\mathbf{0}_{1\times 3}&\mathbf{0}_{1\times 3}&1&0&0\\\ \mathbf{0}_{1\times
3}&\mathbf{0}_{1\times 3}&0&0&0\\\ \mathbf{0}_{1\times 3}&\mathbf{0}_{1\times
3}&0&0&0\end{pmatrix}\mathcal{U}^{d}_{L},$ (59)
$\displaystyle\frac{g}{\sqrt{2}}(\mathbf{C}_{W})_{R}=\frac{g}{\sqrt{2}}\mathcal{U}^{u\dagger}_{R}\begin{pmatrix}\mathbf{0}_{3\times
3}&\mathbf{0}_{3\times 3}&\mathbf{0}_{3\times 1}&\mathbf{0}_{3\times
1}&\mathbf{0}_{3\times 1}\\\ \mathbf{0}_{3\times 3}&\mathbf{0}_{3\times
3}&\mathbf{0}_{3\times 1}&\mathbf{0}_{3\times 1}&\mathbf{0}_{3\times 1}\\\
\mathbf{0}_{1\times 3}&\mathbf{0}_{1\times 3}&1&0&0\\\ \mathbf{0}_{1\times
3}&\mathbf{0}_{1\times 3}&0&0&0\\\ \mathbf{0}_{1\times 3}&\mathbf{0}_{1\times
3}&0&0&0\end{pmatrix}\mathcal{U}^{d}_{R}.$

Up to numerically insignificant $O(\vec{\gamma}_{P}/v_{P})$ corrections, we
can derive the coupling matrix for the left-handed phenomenologically relevant
fermions using Eqs.(30) and (31) once again, yielding a coupling matrix (and
hence the CKM matrix) of

$\displaystyle(\mathbf{C}_{W})_{L}\approx\begin{pmatrix}(\mathbf{W}_{d})_{11}&(\mathbf{W}_{d})_{12}&(\mathbf{W}_{d})_{13}&0&0\\\
(\mathbf{W}_{d})_{21}&(\mathbf{W}_{d})_{22}&(\mathbf{W}_{d})_{23}&0&0\\\
(1-\frac{r^{2}}{2}\frac{m_{t}^{2}}{M_{t}^{2}})(\mathbf{W}_{d})_{31}&(1-\frac{r^{2}}{2}\frac{m_{t}^{2}}{M_{t}^{2}})(\mathbf{W}_{d})_{32}&(1-\frac{r^{2}}{2}\frac{m_{t}^{2}}{M_{t}^{2}})(\mathbf{W}_{d})_{33}&0&0\\\
r\frac{m_{t}}{M_{t}}(\mathbf{W}_{d})_{31}&r\frac{m_{t}}{M_{t}}(\mathbf{W}_{d})_{32}&r\frac{m_{t}}{M_{t}}(\mathbf{W}_{d})_{33}&0&0\\\
0&0&0&1&0\\\ 0&0&0&0&0\end{pmatrix},$ (60)

where $r$ remains as defined in Eq.(30). Note, at least to this order, that
the first row of this coupling matrix remains unitary when restricted to the
SM quarks. Here, we emphasize that in the truncated coupling matrix, only five
down-like quarks remain physically relevant (the three SM quarks and two
portal matter quarks), while six up-like quarks do (the three SM quarks, the
top partner, and two portal matter quarks), as can be seen from the original
definition of the truncated flavor-space vectors in Eq.(2.3). We also note
that, unlike elsewhere in this work, we have retained the
$O(m_{t}^{2}/M_{t}^{2})$ terms in the coupling matrix here; we shall see that
terms of this order represent the leading contribution of the top partner’s
loop-induced correction to neutral meson mixing, and as such, we must retain
these terms for numerical consistency. The right-handed coupling matrix
$(\mathbf{C}_{W})_{R}$ is substantially less phenomenologically interesting–
since the only right-handed fermions which couple to the $W$ boson are portal
matter fields, and they do so diagonally (at least in the limit, which holds
to $O(10^{-6})$ as discussed in Section 2.3, that mixing between the portal
matter states is numerically negligible), the $W$ does not exhibit numerically
significant couplings to the right-handed SM quarks, nor does it facilitate
decays of any of the new fermions in the model.

We conclude our exploration of the couplings in our model by considering the
SM Higgs field, or rather the combination of elements of the bidoublet scalar
$H$ that corresponds to such a field. As noted in Section 2.2, we can estimate
the scalar eigenstate corresponding to the SM Higgs field as in Eq.(7), which
in turn allows us, with reference to the Yukawa action of Eq.(8), to write the
Higgs coupling matrices to the quarks as

$\displaystyle
y_{u,d}\mathbf{C}^{u,d}_{H}=y_{u,d}\mathcal{U}^{u,d\dagger}_{R}\begin{pmatrix}\mathbf{0}_{3\times
3}&\mathbf{0}_{3\times 3}&\mathbf{0}_{3\times 1}&\mathbf{0}_{3\times
1}&\mathbf{0}_{3\times 1}\\\ \mathbf{1}_{3\times 3}&\mathbf{0}_{3\times
3}&\mathbf{0}_{3\times 1}&\mathbf{0}_{3\times 1}&\mathbf{0}_{3\times 1}\\\
\mathbf{0}_{1\times 3}&\mathbf{0}_{1\times 3}&0&0&0\\\ \mathbf{0}_{1\times
3}&\mathbf{0}_{1\times 3}&0&0&0\\\ \mathbf{0}_{1\times 3}&\mathbf{0}_{1\times
3}&1&0&0\end{pmatrix}\mathcal{U}^{u,d}_{L},$ (61)

where the constants $y_{u,d}$ are given in Eq.(2.3). An analogous matrix for
the charged leptons should be identical to that of the down-like quarks, up to
radiative corrections to the parameter $y_{d}$. Rather than relying on our
truncated rotation matrices here, which by removing the heavy partners of the
up and charm quarks, omits the seesaw mechanism by which these SM quarks
acquire mass, it is more instructive here to simply work in the limit where
$\vec{\gamma}_{P,d}\rightarrow 0$ using the rotation matrices of Eqs.(23) and
(33). Doing this, and then truncating the matrix to remove the extremely heavy
fermions as earlier yields the coupling matrix

$\displaystyle
y_{u}\mathbf{C}^{u}_{H}\approx\frac{\sqrt{2}}{v}\begin{pmatrix}m_{u}&0&0&0&0&0\\\
0&m_{c}&0&0&0&0\\\
0&0&m_{t}&m_{t}\frac{m_{t}}{M_{t}}\bigg{(}\frac{M_{t}^{2}-y_{S}^{2}v_{S}^{2}}{y_{S}^{2}v_{S}^{2}}\bigg{)}^{1/2}&0&0\\\
0&0&sgn(y_{u}u_{3})m_{t}\bigg{(}\frac{M_{t}^{2}-y_{S}^{2}v_{S}^{2}}{y_{S}^{2}v_{S}^{2}}\bigg{)}^{1/2}&m_{t}\frac{m_{t}}{M_{t}}\bigg{(}\frac{M_{t}^{2}-y_{S}^{2}v_{S}^{2}}{y_{S}^{2}v_{S}^{2}}\bigg{)}&0&0\\\
0&0&0&0&0&0\\\ 0&0&0&0&0&0\end{pmatrix},$ (62)

where the complete coupling is given by this matrix plus that generated by its
Hermitian conjugate. Notably, among the SM quarks the Higgs coupling matrix is
simply given by the normal SM Higgs coupling matrix – this conclusion holds up
to $O(m_{t}^{2}/M_{t}^{2})\sim O(10^{-2})$ corrections to the $H\bar{t}t$
coupling. There are, however, additional couplings between the top partner and
the top quark itself – the largest of these terms is, in fact $O(1)$. In
practice, these couplings will contribute significantly to the decay of the
top partner to the top quark, as is often the case in models with additional
vector-like quarks mixed with the third generation [30]. In the down-like
quark and charged lepton sectors, the results are analogous, however, given
the fact that the heavy $b$ and $\tau$ partners are likely too massive to be
observed, the approximate Higgs couplings in these sectors precisely matches
the SM result, up to insignificant numerical corrections.

###  3.3 Kinetic Mixing

Having addressed the fermion and gauge boson spectra here, it is useful at
this point to comment on the magnitude and effects of the kinetic mixing
between $U(1)_{F}$ and $U(1)_{Y}$ that will arise from the one-loop
contributions of the model’s fermion fields. For simplicity, we shall assume
that kinetic mixing vanishes until the scale at which the Pati-Salam group is
broken down to the SM and the dark/flavor group remains $SU(4)_{F}\times
U(1)_{F}$. At this scale, the only Abelian groups which may enjoy kinetic
mixing are the $U(1)_{F}$ and $U(1)_{Y}$ groups, so we shall compute this
mixing here.777As was noted in Section 2.1, in the UV theory as written (with
all SM gauge symmetries are contained in the non-Abelian Pati-Salam group
factors), kinetic mixing mediated by a vacuum polarization-like diagram is
forbidden. However, higher-order operators stemming from insertions of scalar
vev’s will still generate kinetic mixing here, and a truly concerned reader
can assume that the $U(1)_{F}$ is unified with either the $SU(4)_{c}$ or
$SU(2)_{R}$, the two Pati-Salam groups which contain the $U(1)_{Y}$ symmetry,
at some higher scale, and some form of symmetry breaking at this scale breaks
the resulting theory down to the SM gauge group by $U(1)_{F}$. As in II, we
note that both SM and portal matter fields will contribute to the kinetic
mixing via vacuum polarization-like diagrams at one loop. In the original
basis, the SM hypercharge boson $\hat{B}$ will mix with the $U(1)_{F}$ boson
$\hat{B}_{F}$ via a term of the form,

$\displaystyle\mathcal{L}_{KM}=\frac{\epsilon}{2c_{w}c_{P}}\hat{B}_{\mu\nu}\hat{B}_{F}^{\mu\nu},$
(63)

where $c_{w}$ is the familiar Weinberg angle and $c_{P}\equiv\cos\theta_{P}$
is the cosine of the angle $\theta_{P}$ described directly above Eq.(46), and
$\hat{B}_{\mu\nu}$ and $\hat{B}_{F}^{\mu\nu}$ are the field strength tensors
of the $U(1)_{Y}$ and $U(1)_{F}$ fields, respectively. Given this
normalization convention, the kinetic mixing term $\epsilon$ here becomes

$\displaystyle\epsilon=\sqrt{\frac{2}{3}}\frac{(gs_{w})(g_{4}s_{P})}{24\pi^{2}}\sum_{i}\bigg{(}\frac{Y_{i}}{2}\bigg{)}Q^{F}_{i}\log\frac{m_{i}^{2}}{\mu^{2}},$
(64)

where $s_{w}$ and $s_{P}$ are simply the sines of the same angles referenced
in $c_{w}$ and $c_{P}$, $g$ is the $SU(2)_{L}$ coupling constant, and $g_{4}$
is the $SU(4)_{F}$ coupling constant. The sum over $i$ is performed over all
the fermions in the theory, $Y_{i}/2$ is the SM hypercharge of fermion $i$,
$Q^{F}_{i}$ is its charge under $U(1)_{F}$, $m_{i}$ is its mass, and $\mu$ is
an arbitrary mass scale which will cancel out of the final calculation.
Ignoring the mixing between various states of different representations under
$U(1)_{Y}$ or $U(1)_{F}$, the effects of which are numerically negligible
anyway, gives

$\displaystyle\epsilon\approx(4.2\times
10^{-4})\bigg{(}\frac{g_{4}s_{P}}{gs_{w}}\bigg{)}\bigg{[}$
$\displaystyle\log\frac{m_{P1}^{e}}{m_{P1}^{u}}+\log\frac{m_{P1}^{\nu}}{m_{P1}^{d}}+2\log\frac{m_{P2}^{d}}{m_{P2}^{u}}+2\log\frac{m_{P2}^{e}}{m_{P2}^{u}}+\frac{4}{5}\log\frac{m_{P3}^{d}}{m_{P3}^{u}}+\frac{4}{5}\log\frac{m_{P3}^{e}}{m_{P3}^{u}}$
(65)
$\displaystyle+\sum_{i}\bigg{(}\frac{1}{5}\log\frac{m_{di}}{m_{ui}}+\frac{3}{5}\log\frac{m_{ei}}{m_{ui}}+\frac{1}{5}\log\frac{m_{\nu
i}}{m_{ui}}+\frac{4}{5}\log\frac{M_{di}}{M_{ui}}+\frac{4}{5}\log\frac{M_{ei}}{M_{ui}}\bigg{)}\bigg{]},$

where here we have used the same mass labelling convention as Eq.(12), with
the sub/superscripts $u$ referring to the up-like quark sector, $d$ to the
down-like quark sector, $e$ to the charged lepton sector, and $\nu$ to the
neutrino sector (of course, since the model’s neutrino sector remains
incomplete, the contributions from it to this mixing are somewhat suspect; we
include them here for the sake of completeness). We note that some of
logarithms of ratios of the exotic particles’ masses may have either sign. In
the absence of significant hierarchies, we would anticipate that $\epsilon$
would be of $O(10^{-4})$ if $g_{4}s_{P}$ were approximately equal to $gs_{W}$,
however, we note that there exist several mass ratios in Eq.(65) that are
necessarily quite hierarchical. For example, if we assume that each SM
neutrino possesses a mass of approximately $0.1\;\textrm{eV}$ and that the
masses of the the portal matter fields $m_{P1,P2,P3}^{u,d,e,\nu}$ are close to
degenerate (that is, $m_{Pi}^{u}\sim m_{Pi}^{d}\sim m_{Pi}^{e}\sim
m_{Pi}^{\nu}$ for $i=1,2,3$, at least for the purposes of computing the logs
of their ratios), then we can estimate the magnitude of Eq.(65) by noting that
$\log(m_{di,ei}/m_{ui})\sim-\log(M_{di,ei}/M_{ui})$, which holds as long as
there are no significant hierarchies between the couplings to the scalar $H$
among the charged leptons, up-like quarks, and down-like quarks. We then
arrive at a rough numerical estimate of

$\displaystyle\epsilon\sim(3\times
10^{-3})\bigg{(}\frac{g_{4}s_{P}}{gs_{w}}\bigg{)},$ (66)

which is in fact an order of magnitude larger than the numerical coefficient
in front of Eq.(65) might suggest. While this level of kinetic mixing is still
not unfeasible if $g_{4}s_{P}\simeq gs_{w}=e$, it does suggest that this
coupling is unlikely to be much greater than this, and that to produce smaller
$O(10^{-4})$ values for the kinetic mixing parameter, the coupling combination
$g_{4}s_{P}$ should likely be somewhat smaller, perhaps closer to $\sim 0.1e$.

With $\epsilon$ computed, the treatment of kinetic mixing is ultimately
entirely analogous to that of I and II (albeit with no additional mixing due
to scalars charged under both the dark and SM gauge groups, which occur in II
but not here), with dark photon couplings to SM fields of $\epsilon eQ$, where
$e$ is the proton charge and $Q$ is the electromagnetic charge of a given
fermion. As these results are well-known, we do not reproduce them here.

###  3.4 Low-Energy Parameters From the High-Energy Model

With the field content, mass spectra, and coupling terms for the model now
determined, we have only one remaining task before being able to explore this
setup’s phenomenological implications: Properly identifying the parameters
with which we might conduct a probe of the model space. Over the course of our
development of the model in Section 2, we have noted that in spite of the
large number of new particles present in the model at high energy, only a
handful of these can be expected to have any significant effect at scales that
can be experimentally probed now or in the near future. It stands to reason,
then, that the large number of parameters in our model at high energy can in
fact be reduced to a more manageable quantity at low energy. For the sake of
clarity, we shall distinguish now between these two pictures: The “high-
energy” model shall refer to the complete model with the field content
outlined in Section 2. The “low-energy” model shall refer to the model in
which only the fields of mass $\mathrel{\mathchoice{\lower
3.01385pt\vbox{\halign{$\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$\cr<\crcr\sim\crcr}}}}O(10\;\textrm{TeV})$,
namely those which might have an observable effect on current and upcoming
experiments, are retained.

In Section 2.3, we found that the quark sector of the model is uniquely
specified by the SM Higgs Yukawa-coupling parameters $y_{H}\sim O(1)$ and
$\alpha\in[-\pi,\pi]$, the two portal matter Yukawa couplings $y_{P1}\sim
O(1)$ and $y_{P2}\sim O(1)$, the vector-like mass term $M$, the scalar vev
parameters $\vec{\gamma}_{P}$, $\vec{\gamma}_{d}$, $v_{P}$, and
$y_{S}v_{S}$888For the purposes of this analysis, we have combined the
coupling $y_{S}$ with the vev parameter $v_{S}$ of the singlet scalar
$\Phi_{S}$. Because the only instances of these parameters occurring
separately happen when considering interactions of the physical scalar arising
from $\Phi_{S}$, a detailed analysis of which we have omitted here, this
simplification is sufficient for our purposes., and finally the signs of the
eigenvalues of the matrices $\mathbf{u}$ (defined in Eq.(13)) and $\mathbf{d}$
(its analogous quantity in the down-like sector). In the limit where particles
which we estimate to be too heavy to be experimentally observable decouple
from the theory, our work simplifies substantially. Referencing our
expressions for the masses and eigenvectors of the fermion fields in Section
2.3, we see that up to signs of various quantities (which we shall see do not
affect any physical results up to numerically small corrections), we note that
we can completely specify the physics of the accessible quark sector, namely
the masses in Eqs.(2.3) and (22), their analogous values in the down-like
sector, and the mass eigenvectors given in Eqs.(30), (31), (37), and (38),
simply with the vev parameters $v_{P}$, $y_{S}v_{S}$, and $\vec{\gamma}_{P}$,
the mass of the top quark partner $M_{t}$, and the masses of the accessible
portal matter fields, $m^{u}_{P1}=m^{d}_{P1}$ and $m^{u}_{P2}=m^{d}_{P2}$.

Moving on to the gauge boson sector, we see that can enjoy a similar drastic
reduction in independent parameters. Consulting Section 3.1, we see that the
coupling matrices of $Z_{F}$, $Z_{P}$, and $A_{D}$, the three new gauge bosons
that are light enough to potentially have experimentally observable effects,
depend only on the CKM matrix, the mass eigenvectors of the fermions, the
$SU(4)_{F}$ coupling constant $g_{4}$, and the angle $\theta_{P}$ which
functions as a Weinberg-like angle for the group $SU(4)_{F}\times U(1)_{F}$.
Turning to Section 2.4, we note that while the mass of $Z_{P}$, given in
Eq.(48), is entirely specified by the parameters $g_{4}$, $v_{P}$, and
$\theta_{P}$, the masses of the gauge bosons $Z_{F}$ and $A_{D}$ arise as
complicated functions of parameters which aren’t otherwise relevant in the
low-energy theory, namely the parameters $u_{1,2}$ (defined in Eq.(13)) and
$d_{1,2,3}$ (their counterparts in the down-like quark sector), the Yukawa
couplings $y_{A}$ and $y_{B}$, and, in the case of $A_{D}$, the
$U(1)_{D}$-breaking vev components $\vec{\gamma}_{d}$. Because the mass of
$A_{D}$ depends on one set of parameters that the mass of $Z_{F}$ doesn’t,
meanwhile, there is no obvious rigid relationship between these masses. It is
therefore easier to simply specify the masses of $Z_{F}$ and $A_{D}$ as
independent low-energy parameters in their own right, for the purposes of
probing the phenomenology of the model.

Finally, we note that the above treatment has neglected to include any
discussion of the leptons in the model. While we have noted earlier that the
neutrino sector will lie largely unaddressed in this work, we cannot afford
the same luxury with the charged leptons, so the fact that we are only
introducing phenomenological parameters which cover the emergence of new
physics in the quark sector bears some discussion. In the charged lepton
sector, we note that we would require specifying some additional parameters,
since we’ve posited that the charged lepton spectrum will be generated by the
same mechanism as that of the down-like quarks, up to radiative corrections in
the couplings to the scalars $\Phi_{A}$, $\Phi_{B}$, the parameters $y_{H}$
and $\alpha$, and the mass term $M$. Among the accessible elements of the
theory, these radiative corrections would result in a modification of the
rotation matrix $\mathbf{W}_{l}$ (defined analogously to $\mathbf{W}_{u}$ is
defined in the up-like sector in Eq.(13)), rather than it being precisely
equal to $\mathbf{W}_{d}$, its counterpart in the down-like quark sector which
must be approximately given by the CKM matrix. Referencing the couplings of
various interactions in Section 3.1, we see that the matrix $\mathbf{W}_{l}$
plays a role in the lepton-flavor-changing currents facilitated by the dark
photon $A_{D}$, and the couplings of the accessible leptonic portal matter
states to SM leptons (which in turn will govern the lifetimes and branching
fractions of these new particles). However, we note that constraints on
lepton-flavor-violating two-body decays of the $\tau$ lepton are much less
restrictive than those in the quark sector [31], while the most restricted
leptonic flavor-changing processes mediated by $A_{D}$, $\mu^{-}\rightarrow
e^{-}A_{D}$, is essentially always kinematically disallowed in the range of
dark photon masses that we consider ($m_{A_{D}}>100\;\textrm{MeV}$).999It
should be noted that for lighter dark photon masses, the two-body decay of a
muon to a dark photon and an electron would have a distinctive experimental
signal, with a sharp peak in the electron energy spectrum. Limits on flavor
violating decays in the quark sector remain more stringent than these
constraints for now, but null results in searches for $\mu^{+}\rightarrow
e^{+}X$ decays, where $X$ is some undetected boson, from the upcoming Mu3e
experiment can be expected to constrain this ratio to $\sim 10^{-8}$, which
may allow it to begin to compete with current limits on the model from flavor-
violating $K$ meson decays [32]. Meanwhile, discovery limits on the color-
singlet leptonic portal matter will likely be far less constraining than those
of the colored quark portal matter particles. As a result, we note that the
new physics arising from additional parameters we must include to describe the
charged lepton sector is likely beneath any notice, and we therefore have no
need to expand our parameter space beyond what is necessary to specify the
kinematically accessible fermions in the quark sector.

Our full parameterization of the lower energy observable sector of the model
then simply consists of two vev parameter $v_{P}$ and $y_{S}v_{S}$, a three-
component complex vector of vev’s $\vec{\gamma}_{P}$, two gauge coupling
parameters $g_{4}$ and $\theta_{P}$, and five particle masses: the mass
$M_{t}$ of the heavy top partner, the masses $m^{u}_{P1}=m^{d}_{P1}$ and
$m^{u}_{P2}=m^{d}_{P2}$ of the accessible portal matter fields, the mass
$m_{Z_{F}}$ of the “flavor $Z$” boson $Z_{F}$, and the mass $m_{A_{D}}$ of the
dark photon. For the reader’s convenience, we have listed the parameters that
are relevant in the high- and low-energy theories, as well as their
approximate ranges, in several tables. Table 4 contains the parameters which
are given in the underlying high-energy theory, and retained unchanged as
independent parameters in our probe of the sector of the theory which is
experimentally accessible, as well as the ranges that we have assumed here.

Table 5 contains the parameters which must be specified to generate the
complete high-energy theory, but can be substituted for other parameters in
the lower-energy theory. Finally, in Table 6, we present the parameters which
may be used in lieu of those of Table 5 when probing the model parameter space
where only phenomenologically accessible new particles are included. Notably,
while there are simple expressions for $m_{P1}^{u,d}$ and $m_{P2}^{u,d}$ in
terms of the parameters of Table 5 in Eq.(22), allowing for an easy estimate
of the range these parameters might take on in our model, the natural ranges
for the other three parameters in Table 6, namely the masses $M_{t}$,
$m_{A_{D}}$, and $m_{Z_{F}}$, are not immediately obvious: There are no such
compact expressions for these masses, at least in terms readily corresponding
to those in Table 5. Since $m_{A_{D}}$ is the only parameter which directly
depends on the magnitude of $\vec{\gamma}_{P}$, rather than the ratio
$\vec{\gamma}_{P}/v_{P}$, we find that by specifying $v_{P}$,
$\vec{\gamma}_{P}$, and $\vec{\gamma}_{d}$ properly we can reproduce virtually
any $m_{A_{D}}$ between 0.1 and 1 GeV, so it is not unreasonable to treat this
as a free parameter in this range, but the masses $M_{t}$ and $m_{Z_{F}}$ are
more restricted. We can, however, produce estimates for the ranges of these
parameters via a simple numerical probe of the high-energy parameter space.

Before beginning this exercise, however, we may simplify our task with a
handful of observations. For $m_{Z_{F}}$, we see from Eqs.(40), (41), and (42)
that the mass matrix which produces $Z_{F}$ (namely, the first 8 rows and
columns of the matrix given in Eq.(40), up to tiny $O(\vec{\gamma}_{P,d})$
corrections) should only have terms which depend on the matrices $\mathbf{u}$
(as defined in Eq.(13)) and $\mathbf{d}$, its down-like quark counterpart –
even dependence on the parameter $M$ cancels out in this portion of the gauge
boson mass squared matrix. Furthermore, inspection of Eq.(2.3), in particular
the SM fermion mass expression which may be rewritten (in the up-like sector,
with a corresponding expression applying in the down-like sector) as
$u_{i}^{2}\approx y_{S}^{2}v_{S}^{2}(y_{u}^{2}v^{2}/(2m_{ui}^{2})-1)$, that
the eigenvalues $u_{1,2,3}$ and $d_{1,2,3}$ of $\mathbf{u}$ and $\mathbf{d}$
are all directly proportional to $y_{S}v_{S}$. That is, for all other model
parameters held constant, the matrices $\mathbf{u}$ and $\mathbf{d}$ scale
directly with $y_{S}v_{S}$. This in turn allows us to say that the mass matrix
which governs $m_{Z_{F}}$, which in turn consists solely of combinations of
$\mathbf{u}$ and $\mathbf{d}$, must be directly proportional to $y_{S}v_{S}$,
and therefore the mass $m_{Z_{F}}$ is directly proportional to $y_{S}v_{S}$ as
well. Meanwhile, a similar argument can be made regarding the mass of the top
partner $M_{t}$, noting in Eq.(2.3) that we may rewrite the equation for the
mass of this fermion as
$M_{t}^{2}=y_{S}^{2}v_{S}^{2}y_{u}^{2}v^{2}/(2m_{t}^{2})$, and hence this mass
is also directly proportional to $y_{S}v_{S}$. Therefore, when probing the
masses $m_{Z_{F}}$ and $m_{A_{D}}$, we only need to consider the range of
these parameters at one specified value for $y_{S}v_{S}$ and extrapolate from
there, and not probe the entire range we list in Table 4. Similarly, we need
not probe the entire range of the $SU(4)_{F}\times U(1)_{F}$ gauge coupling
parameters $g_{4}$ and $\theta_{P}$, rather noting that $m_{Z_{F}}$,
consisting up to tiny corrections entirely of $SU(4)_{F}$ bosons, is
proportional to $g_{4}$.

To perform our numerical examination, we next generate a sample of $10^{5}$
points in parameter space (we note that larger samples grant us the same
results for our parameter ranges, indicating that this sample size is likely
large enough to fully probe this setup), specifying
$y_{S}v_{S}=1\;\textrm{TeV}$, and selecting $y_{H}$, $\alpha$, $\beta$, $M$,
$y_{A}$, and $y_{B}$ randomly (i.e., with flat priors) in the ranges described
in Table 5. We also impose several additional conditions on $y_{H}$, $\alpha$,
and $\beta$, the three parameters which govern the SM-like Higgs bidoublet
sector. Specifically, we require that $y_{u}$ and $y_{d}$, as defined in
Eq.(2.3), are such that $y_{u}^{2}>(2m_{t}^{2})/v^{2}$ and $y_{d}^{2}>1/100$.
The condition on $y_{u}$ ensures that the SM quark spectrum can be reproduced
using the relations of Eq.(2.3). The condition on $y_{d}$ ensures that this
Yukawa coupling is not hierarchically small, which would in turn yield a heavy
$b$ partner with a mass roughly comparable to that of the heavy top partner, a
region of parameter space that we have already established lies beyond the
scope of this paper. To achieve a full sample of $10^{5}$ points which meet
the the aforementioned constraints on $y_{u}$ and $y_{d}$, we generate
$y_{H}$, $\alpha$, and $\beta$, then re-generate these points (again with flat
priors) if the original set fails them, until a point is found that meets the
constraints; this in turn ensures that we uniformly sample the parameter space
of $y_{H}$, $\alpha$, and $\beta$ that meet the $y_{u}$ and $y_{d}$
constraints without bias. Finally, we note that outside of numerically
negligible corrections, the particular values of the parameters
$\vec{\gamma}_{P}$, $\vec{\gamma}_{d}$, $v_{P}$, $y_{P1}$, and $y_{P2}$ have
no effect on the masses we wish to probe here, so there is no need to specify
them. Given these conditions, we estimate the expected “natural” ranges of
$M_{t}$ and $m_{Z_{F}}$, as between the $5\%$ and $95\%$ quantile of their
values in this scan – because we engage in a sampling of all of the parameters
in our high energy theory with flat priors, we expect the results of our
numerical probe to at least qualitatively reflect the necessary fine-tuning to
effect certain values of these parameters. The results are then given in the
ranges quoted in Table 6. We should note that although our sampling of the
high-energy parameters is uniform over their ranges, the same cannot be said
for the ranges of the parameters $M_{t}$ and $m_{Z_{F}}$. To get a sense of
the shapes of the $M_{t}$ and $m_{Z_{F}}$ distributions, we depict probability
histograms of them in Figure 1. Consulting this Figure, we see that while
$M_{t}$ has a straightforward (if nonuniform) distribution in which smaller
values of $M_{t}/(y_{S}v_{S})$ are slightly favored over larger ones, the
distribution of $m_{Z_{F}}$ is somewhat more complex, with a large peak
appearing near $m_{Z_{F}}\sim 30g_{4}y_{S}v_{S}$ followed by a steady drop-
off. As a result of this long tail in the distribution, the range between the
5% and 95% quantiles of our sample given in Table 6 is unusually large. As we
shall see in Section 4.2.2, however, the phenomenologically interesting region
of parameter space lies in the region where
$m_{Z_{F}}\mathrel{\mathchoice{\lower
3.01385pt\vbox{\halign{$\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$\cr<\crcr\sim\crcr}}}}60g_{4}y_{S}v_{S}$,
so we shall restrict our attentions there, merely noting that ample parameter
space exists for significantly larger $m_{Z_{F}}$ without significant fine-
tuning.

$y_{S}v_{S}$ | $v_{P}$ | $\vec{\gamma}_{P}$ | $g_{4}$ | $\theta_{P}$
---|---|---|---|---
1-10 TeV | 1-10 TeV | 0.1-1 GeV | (0.1-0.6) | 0-$\frac{\pi}{2}$
Table 4: The parameters which must be set to specify a unique point in parameter space in both the complete high-energy model and the low-energy model, as described in the text. Here, $g$ denotes the electroweak coupling constant. The philosophy behind the chosen ranges of $y_{S}v_{S}$, $v_{P}$, and $\vec{\gamma}_{P}$ are discussed in Section 2.3, while we assume that $g_{4}\lesssim g$ to keep the magnitude of the kinetic mixing $\lesssim O(10^{-3})$, as discussed in Section 3.3 $y_{H}$ | $\alpha$ | $\beta$ | $y_{A}$ | $y_{B}$ | $y_{P1}$ | $y_{P2}$ | $M/(y_{S}v_{S})$ | $\vec{\gamma}_{d}$
---|---|---|---|---|---|---|---|---
$\pm[1/3,3]$ | $[-\pi,\pi]$ | $[0,\arctan(0.8)]$ | $\pm[1/3,3]$ | $\pm[1/3,3]$ | $\pm[1/3,3]$ | $\pm[1/3,3]$ | $[-10^{6},10^{6}]$ | 0.1-1 GeV
Table 5: The parameters which must be set to specify a unique point in parameter space in the complete high-energy model, but which can be eliminated in favor of a simpler set of parameters in the low-energy model. The ranges quoted here assume that the Yukawa couplings are all of $O(1)$ and $\beta$ is restricted based on the requirement for perturbativity in the left-right-symmetric model Higgs sector. The magnitudes of $M$ and $\vec{\gamma}_{d}$ are discussed in Section 2.3. $M_{t}$ | $m_{P1}^{u}=m_{P1}^{d}$ | $m_{P2}^{u}=m_{P2}^{d}$ | $m_{A_{D}}$ | $m_{Z_{F}}$
---|---|---|---|---
$(1.1-3.2)y_{S}v_{S}$ | $(1/3-3)v_{P}$ | $(1/3-3)v_{P}$ | 0.1-1 GeV | $g_{4}y_{S}v_{S}(17-210)$
Table 6: The parameters which which may be used to specify a unique point in
the parameter space of the low-energy model in lieu of the high-energy model
parameters in Table 5. The large range of $m_{Z_{F}}$, and the shape of the
$m_{Z_{F}}$ distribution in the numerical probe, is discussed in the text and
Figure 1.

Figure 1: (Top) A probability histogram of the values of $m_{Z_{F}}$ in units
of $g_{4}y_{S}v_{S}$ obtained in our numerical sample, described in the text.
The green region denotes the values of $m_{Z_{F}}$ between the 5% and 95%
quantiles. (Bottom) Same as above, but the probability histogram now depicts
values of $M_{t}/(y_{S}v_{S})$.

To better illustrate the transition from the enormously complicated high-
energy model to the comparatively compact low-energy model, and to get an idea
of the sensitivity (or lack thereof) of low-energy parameters to the specific
realization of the complicated pattern of $SU(3)_{F}$ breaking in the high-
energy model, it is useful to explicitly summarize the scales of symmetry
breaking effected in the high-energy model and compare it to the scale of the
new physics fields which survive in the low-energy model. Specifically, we
note that in the high-energy setup, the gauge symmetry $SU(4)_{F}\times
U(1)_{F}$ undergoes the symmetry breaking pattern

$\displaystyle\mathbf{4}_{F}\mathbf{1}_{F}\xrightarrow{\gg\textrm{TeV}}\mathbf{1}_{F}^{\prime}\mathbf{1}_{F}\xrightarrow{O(\textrm{TeV})}\mathbf{1}_{D}\xrightarrow{O(\textrm{GeV})}\textrm{Nothing}.$
(67)

In particular, we see that the $SU(3)_{F}$ symmetry breaking, which required
prodigious fine-tuning in Section 2, generally takes place at the multi-TeV
scale. In fact, the _only_ fields in the low-energy model which remain
sensitive to the $SU(3)_{F}$ breaking are the “flavor $Z$” field $Z_{F}$ and
the vector-like top quark partner. Given that the emergence of a vector-like
top partner of mass $M_{t}\sim O(\textrm{TeV})$ is well-motivated as long as a
seesaw mechanism is used to generate the quark mass hierarchy, we can
comfortably assert that, with the exception of any observable effects of the
$Z_{F}$ boson, _the phenomenology of the low-energy model is applicable to a
wide range of high-energy constructions and agnostic to the complicated
specifics of $SU(3)_{F}$ breaking_.

##  4 Phenomenology: Flavor-Changing Neutral Currents

Having set up the model and determined the relevant parameters we can begin to
analyze the sector of the model within experimental reach, the sole remaining
task left to us is to actually do the necessary exploration. To begin, we note
that our significant expansion of the model’s flavor sector has resulted in
substantial additional sources of flavor-changing couplings, in particular
flavor-changing neutral currents (FCNC’s), which have the potential to
contribute to highly constrained processes. To examine these flavor-changing
effects, then, we explore the impact of the model’s FCNC’s in two sectors:
rare meson decays and neutral meson oscillation.

###  4.1 Flavor-Changing $A_{D}$ Interactions

We begin our discussion of potential phenomenological effects with a brief
foray into the most constrained couplings which emerge here, namely the light
flavor-changing neutral currents mediated by the dark photon. In particular,
the tiny flavor-changing couplings among the SM quarks in Eq.(52) allow for
$b\rightarrow s+A_{D}$, $b\rightarrow d+A_{D}$, and $s\rightarrow d+A_{D}$
transitions. As in II, we shall assume that the on-shell long-lived light dark
photon will escape our detector and/or decay to dark matter,101010the dark
photon decay to dark matter will be the dominant decay if it is kinematically
accessible, since its coupling isn’t suppressed by a small kinetic mixing
factor, so this assumption is both completely reasonable and trivially
realizable with the addition of a dark matter particle of mass less than
$m_{A_{D}}/2$, for example in the form of a scalar singlet under the SM gauge
group. and therefore these interactions can facilitate meson decays which
mimic to some extent the highly-constrained rare decay channels $B\rightarrow
K\nu\overline{\nu}$, $B\rightarrow\pi\nu\overline{\nu}$, and
$K\rightarrow\pi\nu\overline{\nu}$.

It should be noted that the naive bounds from the three-body $B\rightarrow
K\nu\overline{\nu}$, $B\rightarrow\pi\nu\overline{\nu}$, and
$K\rightarrow\pi\nu\overline{\nu}$ are not precisely applicable to the two-
body decay of a $B$ and $K$ mesons to a lighter meson and a long-lived dark
photon: The latter will result in a sharply peaked momentum distribution for
the visible light meson in the final state. This difference has a significant
effect on our analysis of $K\rightarrow\pi\,A_{D}$ constraints in particular:
Searches in the “golden channel” $K^{+}\rightarrow\pi^{+}\nu\overline{\nu}$,
which are normally highly constraining, are substantially weakened in cases
where $A_{D}\sim m_{\pi}$ due to high backgrounds from the
$K^{+}\rightarrow\pi^{+}\pi_{0}$ channel [33, 34, 35]. Instead, it is more
instructive to consider the constraints from $K_{L}\rightarrow\pi_{0}X$
searches such as [35], where $X$ is simply some light long-lived invisible
particle. Because the background $K_{L}\rightarrow 2\pi_{0}$ is CP-violating
(and therefore suppressed), unlike the decay $K^{+}\rightarrow\pi^{+}\pi_{0}$,
searches for $K_{L}\rightarrow\pi_{0}X$ are not subject to the same ruinous
kinematic cuts when the mass of the $X$ particle is close to the mass of the
pion. So, to derive for our model’s constraints from
$K_{L}\rightarrow\pi_{0}X$ branching fractions, we compute the branching
fraction for this decay arising from $A_{D}$ emission and compare it to the
90% CL upper limits from [35] obtained for various mass values. In the case of
the other flavor-changing decays we consider, a more careful analysis of
$B^{+}\rightarrow\pi^{+},K^{+}\nu\overline{\nu}$ searches, which lack the same
troubling kinematic cuts as appear in
$K^{+}\rightarrow\pi^{+}\nu\overline{\nu}$ searches, doesn’t result in
significant differences from the constraints derived by simply quoting the 90%
CL upper bounds of $\mathcal{B}(B^{+}\rightarrow\pi^{+}\nu\overline{\nu})\leq
0.8\times 10^{-5}$ [36] and $\mathcal{B}(B^{+}\rightarrow
K^{+}\nu\overline{\nu})\leq 1.6\times 10^{-5}$ [37] as direct limits on the
corresponding flavor-changing decays mediated by $A_{D}$ (although it should
be noted that we should treat these constraints as order-of-magnitude, not
precise, limits barring a more detailed analysis of the final-state light
meson spectrum). Since we shall see that these constraints are much less
rigorous than those which are derived from the
$K_{L}\rightarrow\pi_{0}\,A_{D}$ system, we also note that the changes in
restrictions on the parameter space from $O(1)$ changes in these constraints
are not qualitatively significant.

A straightforward calculation employing the coupling constants in Eq.(52)
gives us the decay width of the process $K_{L}\rightarrow\pi_{0}A_{D}$

$\displaystyle\Gamma_{K_{L}\rightarrow\pi_{0}\,A_{D}}=g_{4}^{2}s^{2}_{P}\frac{|\xi_{1}|^{2}|\xi_{2}|^{2}}{24\pi}\frac{m_{K}^{3}}{m_{A_{D}}^{2}}\bigg{(}1+\frac{(m_{A_{D}}^{2}-m_{\pi}^{2})^{2}}{m_{K}^{4}}-\frac{2(m_{A_{D}}^{2}+m_{\pi}^{2})}{m_{K}^{2}}\bigg{)}^{3/2}|f_{+}^{K^{0}\pi^{0}}|^{2},$
(68)
$\displaystyle\xi_{i}\equiv\frac{(\mathbf{W}_{d}\vec{\gamma}_{P})_{i}}{v_{P}}$

where $m_{K}$ and $m_{\pi}$ are the masses of the neutral $K$ and $\pi$
mesons, respectively, $f_{+}^{K^{0}\pi^{0}}$ is a hadronic form factor we can
extract from [38] (up to percent level corrections, this factor is equal to 1
for the process $K_{L}\rightarrow\pi_{0}A_{D}$), and we remind the reader
that, also up to percent level corrections, $\mathbf{W}_{d}$ is equal to the
CKM matrix. Expressions for $B^{+}\rightarrow K^{+}A_{D}$ and
$B^{+}\rightarrow\pi^{+}A_{D}$ decays can be easily determined by replacing
the appropriate indices of $\xi$ and meson masses in Eq.(68), and extracting
the appropriate hadronic form factors from [39]. Consulting the definition of
$\xi_{i}$ in Eq.(68), we see that these decay processes should naturally
undergo substantial suppression, since $\vec{\gamma}_{P}\sim
0.1-1\;\textrm{GeV}$ and $v_{P}\mathrel{\mathchoice{\lower
3.01385pt\vbox{\halign{$\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$\cr>\crcr\sim\crcr}}}}1\;\textrm{TeV}$.
We might expect, therefore, that a natural value of $\xi_{i}$ would be in the
realm or $10^{-4}$, leading to a $\sim 10^{-16}$ suppression of these decay
processes, as was noted in II. In spite of this suppression, we shall find, as
in II, that constraints on these decays are severe enough to provide
meaningful limits on the values of the $\xi_{i}$’s beyond even their naturally
small magnitudes.

We can straightforwardly derive expressions for the maximum allowed magnitudes
for products of $\xi_{i}$ values, upon which the constrained meson decays
depend, for various representative choices of the parameters $m_{A_{D}}$ and
$g_{4}s_{P}$ by comparing our results to upper experimental limits on the
branching ratios of these mesons to the corresponding two-neutrino final
states (or in the case of $K_{L}\rightarrow\pi_{0}\,A_{D}$, the mass-dependent
limits on the branching ratios $\mathcal{B}(K_{L}\rightarrow\pi_{0}\,X)$). The
maximum values of the relevant components of
$(\mathbf{W}_{d}^{\dagger}\vec{\gamma}_{P})/v_{P}$ as functions of $m_{A}$ at
various values of $g_{4}s_{P}$ are depicted in Figure 2.

Figure 2: Maximum values of the products $\xi_{1}\xi_{2}$ (top left),
$\xi_{1}\xi_{2}$ (top right), and $\xi_{2}\xi_{3}$ (bottom) based on the 90%
CL measured limits on $\mathcal{B}(K_{L}\rightarrow\pi_{0}X)$ (where $X$ is
some invisible long-lived particle) [35],
$\mathcal{B}(B^{+}\rightarrow\pi^{+}\nu\overline{\nu})$ [36], and
$\mathcal{B}(B^{+}\rightarrow K^{+}\nu\overline{\nu})$ [37], respectively.
Benchmark values of $g_{4}s_{P}$ are taken as 0.1 (blue), 0.2 (red), and 0.3
(magenta). The $\xi$ parameters are described in the text.

From Figure 2, we can derive some approximate constraints on the parameters
$\xi_{i}$, which suggest that some of their magnitudes must be well below the
natural value of $\sim O(10^{-4})$ that we might expect. Up to $O(1)$
variations as depicted in this Figure, we arrive at the rough constraints

$\displaystyle\xi_{1}\xi_{2}\mathrel{\mathchoice{\lower
3.01385pt\vbox{\halign{$\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$\cr<\crcr\sim\crcr}}}}10^{-11},\;\;\;\xi_{1}\xi_{3}\mathrel{\mathchoice{\lower
3.01385pt\vbox{\halign{$\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$\cr<\crcr\sim\crcr}}}}10^{-8},\;\;\;\xi_{2}\xi_{3}\mathrel{\mathchoice{\lower
3.01385pt\vbox{\halign{$\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$\cr<\crcr\sim\crcr}}}}10^{-8},$
(93)

where the first (and most stringent) constraint, coming from
$K_{L}\rightarrow\pi_{0}A_{D}$, only applies when
$m_{A_{D}}\mathrel{\mathchoice{\lower
3.01385pt\vbox{\halign{$\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$\cr<\crcr\sim\crcr}}}}260\;\textrm{MeV}$,
past which the study of [35] doesn’t offer constraints due to overly large
$K_{L}\rightarrow\pi^{+}\pi^{-}\pi_{0}$ backgrounds. The much less severe
constraints from the other decay processes have no such similar restriction on
the dark photon mass for which they apply; these processes remain
kinematically accessible over the entire range of $m_{A_{D}}$ we consider.
Because the constraints in Eq.(93) only apply to the products of $\xi_{i}$
terms, there is a fair amount of flexibility in how these constraints might be
satisfied. For example, we might meet the harsh constraint on the product of
$\xi_{1}\xi_{2}$ by allowing one of either $\xi_{1}$ or $\xi_{2}$ to be $\sim
10^{-7}$, several orders of magnitude below its expected magnitude of $\sim
10^{-4}$, while the other stays at the natural higher magnitude, or we can
require that $\xi_{1}\sim\xi_{2}\sim 3\times 10^{-6}$. Given that the other
constraints here are or comparatively little significance, we can outline
several phenomenologically allowed benchmark values of the parameters
$\xi_{i}$, listed in Table 7. An astute reader may be concerned that, in
suppressing various $\xi_{i}$ (and hence $\vec{\gamma}_{P}$) values to such
low values, we may anticipate that contributions to fermion mixing from
$\vec{\gamma}_{d}$, which we have neglected in our treatment of the low-energy
phenomenology, may be significant. However, in practice we find that even
assuming all components of $\xi_{i}$ are as small as $O(10^{-10})$, the
$\vec{\gamma}_{P}$ effects still dominate all phenomenologically significant
couplings in the model.

Benchmark | $\xi_{1}$ | $\xi_{2}$ | $\xi_{3}$
---|---|---|---
A1 | $\mathrel{\mathchoice{\lower 3.01385pt\vbox{\halign{$\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$\cr<\crcr\sim\crcr}}}}10^{-4}$ | 0 | 0
A2 | 0 | $\mathrel{\mathchoice{\lower 3.01385pt\vbox{\halign{$\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$\cr<\crcr\sim\crcr}}}}10^{-4}$ | 0
A3 | 0 | 0 | $\mathrel{\mathchoice{\lower 3.01385pt\vbox{\halign{$\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$\cr<\crcr\sim\crcr}}}}10^{-4}$
B1 | $\mathrel{\mathchoice{\lower 3.01385pt\vbox{\halign{$\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$\cr<\crcr\sim\crcr}}}}10^{-4}$ | 0 | $\mathrel{\mathchoice{\lower 3.01385pt\vbox{\halign{$\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$\cr<\crcr\sim\crcr}}}}10^{-4}$
B2 | 0 | $\mathrel{\mathchoice{\lower 3.01385pt\vbox{\halign{$\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$\cr<\crcr\sim\crcr}}}}10^{-4}$ | $\mathrel{\mathchoice{\lower 3.01385pt\vbox{\halign{$\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$\cr<\crcr\sim\crcr}}}}10^{-4}$
C1 | $\mathrel{\mathchoice{\lower 3.01385pt\vbox{\halign{$\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$\cr<\crcr\sim\crcr}}}}10^{-4}$ | $\mathrel{\mathchoice{\lower 3.01385pt\vbox{\halign{$\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$\cr<\crcr\sim\crcr}}}}10^{-4}$ | $\mathrel{\mathchoice{\lower 3.01385pt\vbox{\halign{$\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$\cr<\crcr\sim\crcr}}}}10^{-4}$
C2 | $\mathrel{\mathchoice{\lower 3.01385pt\vbox{\halign{$\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$\cr<\crcr\sim\crcr}}}}10^{-6}$ | $\mathrel{\mathchoice{\lower 3.01385pt\vbox{\halign{$\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$\cr<\crcr\sim\crcr}}}}10^{-6}$ | $\mathrel{\mathchoice{\lower 3.01385pt\vbox{\halign{$\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$\cr<\crcr\sim\crcr}}}}10^{-6}$
Table 7: The different benchmarks for $\xi_{i}$ magnitudes which can satisfy
the rough constraints of Eq.(93). Scenarios A1, A2, and A3 correspond to cases
where one $\xi_{i}$ is hierarchically larger than the others, with the number
denoting which generation, first, second, or third, has the dominant
$\xi_{i}$. Scenario B1(B2) assumes that either $\xi_{1}(\xi_{2})$ is roughly
equal in magnitude to $\xi_{3}$, while $\xi_{2}(\xi_{1})$ is hierarchically
smaller. Scenario C1, in which all $\xi_{i}$’s are approximately equal, is
only viable if $m_{A_{D}}\gtrsim 250\;\textrm{MeV}$, and hence the decay
$K_{L}\rightarrow\pi_{0}A_{D}$ is either hidden by
$K_{L}\rightarrow\pi^{+}\pi^{-}\pi_{0}$ backgrounds or is kinematically
inaccessible; scenario C2 is the equivalent of C1 in the event that
$m_{A_{D}}\lesssim 250\;\textrm{MeV}$.

Before moving on, we note that, as mentioned in Section 2.2, we expect the
model should contain a physical scalar of mass $\sim m_{A_{D}}$, i.e., the
‘dark Higgs’, which should generically have similar flavor-changing couplings
as the dark photon. If we likewise assume that the light scalar is long-lived
or can primarily only decay to dark photons as we would expect, we might
anticipate that it will make additional contributions to the branching
fractions we have discussed in this section, of comparable magnitude to those
facilitated by the dark photon. However, given the complexity of the scalar
sector within this model, computing the exact magnitude of these contributions
is highly non-trivial, and will likely depend on a number of additional
parameters in the scalar potential. For the sake of simplicity, then, we shall
not compute the light scalar contributions to these flavor-changing decays. We
can assume that at worst, these likely manifest $O(1)$ corrections to the
constraints appearing in Figure 2, and given the highly non-trivial nature of
the scalar sector, it is just as plausible that large regions of parameter
space in the scalar potential exist such that these decays are either
kinematically forbidden or highly suppressed by a small phase space factor.

Finally, it should be noted that similar transitions exist in the up-like
quark and charged lepton sectors (e.g., $t\rightarrow cA_{D}$, $c\rightarrow
uA_{D}$, $\mu\rightarrow eA_{D}$). However, in the case of up-like quarks the
experimental constraints on decays mediated by these processes are much
weaker, while the case against including restrictions in the leptonic sector
was discussed in Section 3.4. In either case, these processes provide no
significant limit on our model parameters at present.

###  4.2 Neutral Meson Oscillation

While the dark photon’s flavor-changing couplings are capable of facilitating
distinctive and highly-constrained meson decays, they are hardly the only
potentially dangerous flavor-changing couplings in the theory. In particular,
we find that $\overline{B}_{d}-B_{d}$ and $\overline{B}_{s}-B_{s}$
oscillations can suffer significant contributions to their dispersive
amplitudes, given by
$(M_{12})_{q}=\langle\overline{B}^{0}_{q}|\mathcal{H}^{\Delta
B=2}_{eff}|B^{0}_{q}\rangle/(2m_{B_{q}})$ (where $\mathcal{H}^{\Delta
B=2}_{eff}$ is simply the effective Hamiltonian for the relevant flavor-
changing interactions) from two additional sources: The flavor-changing
$Z_{F}$ couplings, identified in Eq.(56), and the one-loop contributions of
the heavy top partner to the box diagram which generates meson mixing in the
SM. We note that effects in $\overline{K}^{0}-K^{0}$ mixing, by contrast, are
generally quite muted: The contribution to this process due to the top partner
loops is suppressed by CKM factors, while the corresponding FCNC’s mediated by
$Z_{F}$ vanish at $O(\lambda^{4})$. To obtain our results, we shall compute
the new physics contributions to $(M_{12})_{q}$ and parameterize them as [40,
27, 41]

$\displaystyle(M_{12})_{q}=(M_{12})_{q}^{SM}(1+h_{q}e^{2i\sigma_{q}}),$ (94)

where $(M_{12})_{q}^{SM}$ is the SM contribution to $B_{q}$ meson mixing,
while $h_{q}$ and $\sigma_{q}$ are real parameters for the contribution of new
physics to these processes. $(M_{12})_{q}^{SM}$ is in turn given by [42]

$\displaystyle(M_{12})_{q}^{SM}\approx\frac{G_{F}^{2}m_{W}^{2}}{12\pi^{2}}f_{B_{q}}^{2}m_{B_{q}}B_{B_{q}}(\lambda^{(q)}_{t})^{2}S_{0}(x_{t}),$
$\displaystyle
S_{0}(x_{t})\equiv\bigg{(}\frac{x_{t}(4-11x_{t}+x_{t}^{2})}{4(1-x_{t})^{2}}-\frac{3x_{t}^{3}\log(x_{t})}{2(1-x_{t})^{3}}\bigg{)},$
(95) $\displaystyle\lambda^{(q)}_{i}\equiv
V_{ib}^{*}V_{iq},\;\;\;x_{i}\equiv\frac{(m_{i}^{\textrm{pole}})^{2}}{m_{W}^{2}},$

We can then compare the new physics contributions to the limits in a fit of
CKM observables to new physics in the meson mixing sector in [27] in order to
extract approximate constraints on our model parameters. In particular, this
fit gives

$\displaystyle h_{d}\leq 0.26,\;\;h_{s}\leq 0.12$ (96)

at the 95% CL level as of Summer 2019. We note that simply referring to these
ranges represents only a very approximate assessment of the constraints
afforded by flavor-changing neutral currents in this model; for example, the
above ranges assume uncorrelated $h_{d}$ and $h_{s}$ values, which we shall
see is not the case here, and ignore non-trivial constraints on the phase
$\sigma_{d}$, which we shall see may substantially tighten constraints on
$h_{d}$. In spite of these limitations, however, we find that the limits in
Eq.(96) will afford us an approximate picture of the effect our model has on
meson oscillation parameters.

####  4.2.1 Neutral Meson Oscillation: Top Partner Loop Contribution

We now move on from the flavor effects of the dark photon $A_{D}$ to the
effects of heavier new physics on flavor observables, in particular the highly
constrained measurements related to neutral meson oscillation. To start, we
consider the effect of the heavy top partner fermion. In spite of being a
loop-level interaction, the contribution of the heavy top quark partner to
neutral meson mixing processes, in particular those of the $B_{s}$ and $B_{d}$
mesons, is significant enough to warrant discussion. We note that the new
physics contributions to meson mixing from the top partner take two forms:
Additional loop diagrams featuring the heavy top quark, and, as can be seen in
Eq.(60), tree-level modifications of the SM CKM matrix elements $V_{td}$,
$V_{ts}$, and $V_{tb}$. As discussed in [27], the fit performed in that work
extrapolates these CKM matrix elements from unitarity, rather than directly,
and as such the tree-level modifications to these parameters must be included
as “new physics” that contributes to $h_{d,s}$ and $\sigma_{d,s}$. We can now
determine the contribution of the top partner here in direct analogy to the SM
calculation done in, for example, [42]. Inserting Eq.(60) for the CKM
couplings of the top partner, we find that up to $O(10^{-2})$ corrections due
to subdominant loops featuring up and charm quarks, the top partner’s
contribution to $(M_{12})_{q}$ is

$\displaystyle\frac{(M_{12})_{q}^{\textrm{T}}}{(M_{12})_{q}^{SM}}=h_{q}^{T}e^{2i\sigma^{T}_{q}}\approx\frac{r^{2}}{S_{0}(x_{t})}\frac{m_{t}^{2}}{M_{t}^{2}}(-2\tilde{S}(x_{t},x_{t})+2\tilde{S}(0,x_{t})+2\tilde{S}(x_{t},x_{T})+r^{2}\frac{m_{t}^{2}}{M_{t}^{2}}\tilde{S}(x_{T},x_{T})),$
(97)
$\displaystyle\tilde{S}(x_{i},x_{j})=\frac{4-7x_{i}x_{j}}{4(1-x_{i}-x_{j}+x_{i}x_{j})}+\frac{(4-(8-x_{i})x_{j})x_{i}^{2}\log(x_{i})}{4(x_{i}-1)^{2}(x_{i}-x_{j})}+\frac{(4-(8-x_{j})x_{i})x_{j}^{2}\log(x_{j})}{4(x_{j}-1)^{2}(x_{j}-x_{i})},$
$\displaystyle
x_{T}\equiv\frac{M_{t}^{2}}{m_{W}^{2}},\;\;\;x_{t}\equiv\frac{(m_{t}^{\textrm{pole}})^{2}}{m_{W}^{2}},$

where we have assumed that the QCD corrections for the heavy top partner loops
are approximately equal to those of the top quark itself (as done in, for
example, [43]), $S_{0}(x_{t})$ is the Inami-Lim function given here in
Eq.(4.2), and we remind the reader that $r$ is the variable first defined in
Eq.(30). It is useful to point out that here, as elsewhere in this work, when
we refer to an SM quark mass such as $m_{t}$, we are explicitly referring to
its value at a scale of $\sim 1\;\rm TeV$ (as we noted in Section 2.3), the
scale at which we assumed the SM fermion masses were generated via a seesaw
mechanism. Hence, $m_{t}$ in Eq.(97) refers to the top quark mass _at this
scale_ , or in this case $m_{t}(\sim 1~{}\rm TeV)\approx 144\;\textrm{GeV}$,
and not its pole mass, which is explicitly denoted by
$m_{t}^{\textrm{pole}}\approx 173\;\textrm{GeV}$. As both the pole mass
$m_{t}^{\textrm{pole}}$ and the RG-evolved mass $m_{t}$ appear in Eq.(97),
some care must be taken in order to use the formula correctly. Since the scale
$1\;\textrm{TeV}$ is dramatically closer to the top partner pole mass than it
is to that of the top quark, we neglect any effects of RG running in $M_{t}$.
In the limit where $x_{T}\gg x_{t}$, which applies here, we can combine
Eqs.(4.2) and (97) as

$\displaystyle
h_{d,s}^{T}\approx\frac{r^{2}x_{t}}{2S_{0}(x_{t})}\frac{m_{t}^{2}}{M_{t}^{2}}\bigg{[}\frac{6x_{t}}{(-1+x_{t})^{2}}-\frac{3(-1+3x_{t})}{(-1+x_{t})^{3}}\log(x_{t})-1+\frac{r^{2}}{2}\bigg{(}\frac{m_{t}}{m_{t}^{\textrm{pole}}}\bigg{)}^{2}-2\log\bigg{(}\frac{m_{t}^{\textrm{pole}}}{M_{t}}\bigg{)}\bigg{]},$
(98) $\displaystyle\sigma_{d,s}^{T}=0.$

Inserting $m_{t}^{\textrm{pole}}\approx 173\;\textrm{GeV}$, $m_{t}\approx
144\;\textrm{GeV}$, and $x_{t}\approx 4.63$, we arrive at

$\displaystyle h_{d,s}^{T}\approx
0.064\bigg{(}\frac{1\;\textrm{TeV}}{y_{S}v_{S}}\bigg{)}^{2}\bigg{(}1-\frac{y_{S}^{2}v_{S}^{2}}{M_{t}^{2}}\bigg{)}\bigg{[}1+0.102\bigg{(}\frac{M_{t}^{2}}{y_{S}^{2}v_{S}^{2}}-1\bigg{)}-0.592\log\bigg{(}\frac{1\;\textrm{TeV}}{M_{t}}\bigg{)}\bigg{]},$
(99)

where for clarity we have explicitly inserted the equation
$r^{2}=M_{t}^{2}/(y_{S}^{2}v_{S}^{2})-1$ in the above expression. Notably, we
see that the heavy top partner contributes the _same_ correction to both
$B_{d}$ and $B_{s}$ oscillation, and also that this correction can be quite
significant, even of $O(10\%)$. Obtaining a realistic range for
$M_{t}/(y_{S}v_{S})$ from Table 6, we can then consider the correction
$h_{d,s}^{T}$ as a function of the parameters $y_{S}v_{S}$ and
$M_{t}/(y_{S}V_{S})$. In the case of $h_{d}$, the top partner loops represent
the dominant new physics contribution to this parameter, so our computation of
this variable can be completed here. We depict the results for $h_{d}$ for
various values of $y_{S}v_{S}$ and $M_{t}/(y_{S}v_{S})$ in Figure 3 (we remind
the reader that $M_{t}>y_{S}v_{S}$, which can be straightforwardly seen from
the equation for $M_{t}^{2}=M_{u3}^{2}$ in the last line of Eq.(2.3)).

Figure 3: (Top) Value of $h_{d}$ as a function of $y_{S}v_{S}$ for, from top
to bottom, $M_{t}/(y_{S}v_{S})=$ 3, 2.5, 2, 1.5, and 1.1, respectively.
(Bottom) Value of $h_{d}$ as a function of $M_{t}/(y_{S}v_{S})$ for, from top
to bottom, $y_{S}v_{S}=$ 1, 1.5, 2, 2.5, and 3 TeV, respectively.

From Figure 3, we can see several interesting characteristics of this new
physics contribution. First, comparing the magnitude of the effects in this
Figure to the constraints in Eq.(96), we see that for the region of parameter
space we are considering, the effects of the top partner loops on meson mixing
parameters should lie within current experimental constraints. We do, however,
note that for regions with the largest effect, which can reach $h_{d}\sim
0.14$, the results exceed the expected sensitivity of LHCb and Belle-II data
in the coming years, which may be sensitive to $h_{d}\sim 0.07$ within the
next decade [27]. Furthermore, we note that a quick reference to Eq.(97)
allows us to see that $\sigma_{d}=0$ from this contribution (indicating a
_positive_ proportional contribution to $(M_{12})_{d}$), while the best-fit
value of $\sigma_{d}$ in the fit of [27] is $\sigma_{d}=-1.40^{+0.97}_{-0.23}$
(or roughly corresponding to a _negative_ proportional contribution to the
mixing matrix– see [40] for a full chart of allowed values in the
$h_{d}-\sigma_{d}$ plane, albeit with slightly outdated data but exhibiting
similar favored regions of $h_{d}-\sigma_{d}$ as the present constraints). We
might therefore expect that constraining our model more rigorously, with a CKM
fit of our own, might result in non-trivial constraints on the scenarios with
low $y_{S}v_{S}$ values from measurements of $B_{d}$ oscillation. However, the
sharp dropoff of these contributions as $y_{S}v_{S}$ increases likely removes
this sensitivity as $y_{S}v_{S}\mathrel{\mathchoice{\lower
3.01385pt\vbox{\halign{$\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$\cr>\crcr\sim\crcr}}}}1.75\;\textrm{TeV}$,
unless these constraints are an order of magnitude more sensitive than the
generic boundaries found in [27]. In practice, therefore, we find that the
loop-level contributions of the top partner to neutral meson mixing are
presently not subject to any substantial constraints; however additional data
from LHCb and Belle-II in the coming years will likely constrain the parameter
space near the lower end of the range $y_{S}v_{S}\sim 1\;\textrm{TeV}$ from
effects in this sector.

####  4.2.2 Neutral Meson Oscillation: $Z_{F}$ Contribution

While the top partner loops represent by far the dominant effect in the
$B_{d}$-meson mixing system, the $B_{s}$ mixing system suffers additional
contributions from another source, i.e., the “flavor $Z$” boson, $Z_{F}$. In
particular, we can see from the coupling in Eq.(56) that the dominant
contribution of the $Z_{F}$ boson will be to $B_{s}$ mixing alone. We can
compute this contribution straightforwardly by following the treatment of,
e.g., [44]. In this case, we can write the effective flavor-changing
Hamiltonian as

$\displaystyle\mathcal{H}^{\Delta
B=2}_{Z_{F}}=\mathcal{C}_{1}(m_{Z_{F}})\mathcal{O}_{1}(m_{Z_{F}})+\tilde{\mathcal{C}}_{1}(m_{Z_{F}})\tilde{\mathcal{O}}_{1}(m_{Z_{F}})+\mathcal{C}_{4}(m_{Z_{F}})\mathcal{O}_{4}(m_{Z_{F}})+\mathcal{C}_{5}(m_{Z_{F}})\mathcal{O}_{5}(m_{Z_{F}}),$
(100)

with

$\displaystyle\mathcal{O}_{1}=(\overline{b}_{L}^{\alpha}\gamma_{\mu}q_{L}^{\alpha})(\overline{b}_{L}^{\beta}\gamma_{\mu}q_{L}^{\beta}),\;\;\tilde{\mathcal{O}}_{1}=(\overline{b}_{R}^{\alpha}\gamma_{\mu}q_{R}^{\alpha})(\overline{b}_{R}^{\beta}\gamma_{\mu}q_{R}^{\beta}),\;\;\mathcal{O}_{4}=(\overline{b}_{R}^{\alpha}s_{L}^{\alpha})(\overline{b}_{L}^{\beta}s_{R}^{\beta}),\;\;\mathcal{O}_{5}=(\overline{b}_{R}^{\alpha}s_{L}^{\beta})(\overline{b}_{L}^{\beta}s_{R}^{\alpha}),$
(101)

using operators defined identically to those of [44]. Our computation of
$(M_{12})_{s}$ stemming from these effects then simply requires identifying
the Wilson coefficients $\mathcal{C}_{i}$ and extracting the expectation
values of the operators $\mathcal{O}_{i}$ from [44]. Using Eq.(56), we can
straightforwardly see that at some high scale $\mu_{\textrm{high}}$, we have

$\displaystyle\mathcal{C}_{1}(\mu_{\textrm{high}})=\tilde{\mathcal{C}}_{1}(\mu_{\textrm{high}})=\frac{3g_{4}^{2}}{4m_{Z_{F}}^{2}}A^{2}\lambda^{4}(\rho+i\eta)^{2},$
$\displaystyle~{}~{}\mathcal{C}_{4}(\mu_{\textrm{high}})=0,$
$\displaystyle\mathcal{C}_{5}(\mu_{\textrm{high}})=-\frac{3g_{4}^{2}}{m_{Z_{F}}^{2}}A^{2}\lambda^{4}(\rho+i\eta)^{2}.$
(102)

We can then run these coefficients down to hadronic scales using anomalous
dimension matrices which can be extracted from [45]. For the sake of
simplicity, we shall assume that $\mu_{\textrm{high}}=10\;\textrm{TeV}$, which
should be within a factor of $\sim O(\textrm{a few})$ from $m_{Z_{F}}$,
consulting the range for this parameter given in Table 6 and assuming
$g_{4}\sim 0.3$, $y_{S}v_{S}\sim 1\;\textrm{TeV}$. We also assume, consulting
the likely ranges for these masses in Table 6, that at the scale $\sim
10\;\textrm{TeV}$, only five non-SM quark flavors are dynamical: The top
partner, two up-like portal matter quarks, and two down-like portal matter
quarks, and that for the purposes of our RGE calculation all five of these
extra flavors can be integrated out at the same scale, which we take to be 1
TeV. Running the coupling constants of Eq.(102) down to the $b$ quark mass,
extracting the expectation value of the operators $\mathcal{O}_{1,4,5}$ (and
the bag parameter as well as the decay constant from the SM contribution to
$B_{s}$ oscillation given in Eq.(4.2)) from [44], we arrive at

$\displaystyle\frac{(M_{12})_{s}^{Z_{F}}}{(M_{12})_{s}^{SM}}=h_{s}^{Z_{F}}e^{2i\sigma_{s}^{Z_{F}}}\approx(0.024)e^{2i(-0.099)\pi}\bigg{(}\frac{10\;\textrm{TeV}}{m_{Z_{F}}}\bigg{)}^{2}\bigg{(}\frac{g_{4}}{0.3}\bigg{)}^{2}.$
(103)

We note that while this correction has a magnitude which varies depending on
the $SU(4)_{F}$ coupling $g_{4}$ and the mass $m_{Z_{F}}$, its phase remains
fixed. This is the result of the coupling constants in Eq.(102) all having the
same complex phase, that of $\rho+i\eta$, which is of course fixed by the
Wolfenstein parameters. The result of Eq.(103) suggests, as in the case of the
loop-level corrections due to the top partner, that we may anticipate roughly
$O(\textrm{a few}\%)$ corrections from $Z_{F}$ exchange: That is, these two
processes have roughly comparable effects. We should, then, estimate total
contribution to $h_{s}$ as

$\displaystyle
h_{s}=|h_{s}^{Z_{F}}e^{2i\sigma_{s}^{Z_{F}}}+h_{s}^{T}e^{2i\sigma_{s}^{T}}|,$
(104)

with $h_{s}^{Z_{F}}$ and $\sigma_{s}^{Z_{F}}$ given by Eq.(103), $h_{s}^{T}$
given by Eq.(99), and $\sigma_{s}^{T}=0$, as noted in Section 4.2.1. Taking
numerical results of Eq.(104) and comparing the results to the constraints on
$h_{s}$ in Eq.(96), we can also extract some rough constraints on the
parameters $y_{S}v_{S}$ for various choices of the proportional mass
parameters $M_{t}/(y_{S}v_{S})$ and $m_{Z_{F}}/(g_{4}y_{S}v_{S})$. In Figure
4, we depict the combined effect of the $Z_{F}$ and top partner contributions
to the variable $h_{s}$.

Figure 4: The full value of $h_{s}$, including tree-level $Z_{F}$ and top
partner loop contributions, for $m_{Z_{F}}/(g_{4}y_{S}v_{S})=20$ (blue), $40$
(red), and $60$ (magenta). These plots assume that $M_{t}/(y_{S}v_{S})=3$ (top
left), $2.5$ (top right), $2$ (bottom left), and $1.5$ (bottom right). For
comparison, the 95% CL limit on $h_{s}$ is depicted in each plot as a dashed
orange line.

Comparing the results of Figure 4 to those of Figure 2, which should be
identical to $h_{s}^{T}$, or the new physics parameter in the absence of any
$Z_{F}$ contributions, we see that for lower masses of $Z_{F}$, in particular
$m_{Z_{F}}=20g_{4}y_{S}v_{S}$ (the lower end of the range we consider), the
effect of the $Z_{F}$ boson on $h_{s}$ is significant, on the order of $0.06$
for $y_{S}v_{S}=1\;\textrm{TeV}$. For lower values of $M_{t}/(y_{S}v_{S})$,
the $Z_{F}$ contribution at these low masses is even the same magnitude as the
contribution due to vector-like quarks. However, some caveats are required
here. First, a $Z_{F}$ mass as low as $20g_{4}y_{S}v_{S}\sim 7y_{S}v_{S}$ is
the result of somewhat finely-tuned parameters in the high-energy theory: Our
numerical probe of the high-energy parameter space in Section 3.4 places such
a mass only just above the 5% quantile. Larger values of $m_{Z_{F}}$, such as
$60g_{4}y_{S}v_{S}\sim 20y_{S}v_{S}$, which is close to the median $m_{Z_{F}}$
of our sample, generally offer only an $O(10\%)$ correction to the mixing
result purely from considering top partner loops. For larger values of
$m_{Z_{F}}$, which can reach as high as $200g_{4}y_{S}v_{S}\sim 70y_{S}v_{S}$
in our sample, the effects of the $Z_{F}$ boson quickly become negligible. We
also note that, as discussed in Section 2.4, the $Z_{F}$ boson’s mass may be
sensitive to modifications in the model to accomodate the SM neutrino sector,
or generally to a broad array of changes to the specifics of the high energy
model. In particular, the mass of any gauge bosons corresponding to the
$SU(3)_{F}$ flavor group may increase dramatically. As a result, we might see
any observable new physics contribution from $Z_{F}$ vanish.

These concerns aside, however, we do see in Figure 4 that any constraint from
$B_{s}$ meson mixing is quite mild – while there do exist points in parameter
space which can exceed the 95% CL upper bound for $h_{s}\simeq 0.12$, we see
that they tend to only occur in the “optimal” scenario for these corrections:
The only scenarios we depict in which the present $h_{s}$ cannot be evaded by
simply requiring $y_{S}v_{S}\mathrel{\mathchoice{\lower
3.01385pt\vbox{\halign{$\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$\cr>\crcr\sim\crcr}}}}1.1\;\textrm{TeV}$,
a mere 10% increase from the lowest value of $y_{S}v_{S}$ we consider, occur
when we have assumed $m_{Z_{F}}=20g_{4}y_{S}v_{S}$, near its 5% quantile value
in our numerical sample, and $M_{t}\geq 2.5y_{S}v_{S}$. As in the case of
$B_{d}$ mixing, however, we can note that future improvements to $h_{s}$
observations from Belle-II and LHCb measurements (and improvements in lattice
computations of hadronic matrix elements), which might be expected to reduce
the 95% CL bound on $h_{s}$ to $\sim 0.06$ [27], may begin to place more
meaningful constraints on our model parameter space for the $Z_{F}$ and top
partner masses.

##  5 Phenomenology: Portal Bosons and Exotic Matter

Having discussed the low energy flavor-violating implications of the model, we
can turn to some other simple processes in the model, in particular focusing
on where our results differ from those of I and II. We shall organize this
discussion by considering each of the likely kinematically accessible heavier
exotic particles, the $Z_{P}$, top partner, and the portal matter separately.

###  5.1 Phenomenology: Portal Matter Fermions

We shall begin our discussion of the phenomenology of our new exotic particles
with the portal matter: The fermion fields which possess non-trivial
$U(1)_{D}$ charge. From Section 2.3, we note that our theory anticipates two
such fields for each sector of the theory (up-like quarks, down-like quarks,
charged leptons, and neutrinos) that we can expect to have masses potentially
accessible to experiment in the foreseeable future. For convenience, we shall
label the two portal matter states based on their masses: $P^{u}_{1}$, for
example, will be the up-like portal matter quark with mass $m_{P1}^{u}$ (as
given in Eq.(22)), while $P^{u}_{2}$ will be the up-like portal matter quark
with mass $m_{P2}^{u}$ (again given in Eq.(22)). Analogous definitions exist
for the portal matter in the down-like quark and leptonic sectors. For the
sake of simplicity, and because the constraints on color triplet new particles
are dramatically stronger than those for color singlets, we shall limit our
discussion here to the portal matter in the quark sector, $P_{1,2}^{u,d}$.

Much of the behavior of the portal matter phenomenology in the present setup
is analogous to that of I and II, and as such does not bear particular
repeating. However, the additional structure present in this model does
introduce some non-trivial complications to the existing frameworks discussed
there. In particular, the couplings of the portal matter states to various
generations of SM matter are entirely controlled by the orientation of the
vector $\vec{\gamma}_{P}$ in three-dimensional flavor space: Couplings of
$i^{th}$-generation up-like quarks are proportional to
$(\vec{\gamma}_{P})_{i}/v_{P}$, while couplings of the $i^{th}$-generation
down-like quarks are proportional to
$\xi_{i}=(\mathbf{W}_{d}^{\dagger}\vec{\gamma}_{P})_{i}/v_{P}$, where we
remind the reader that $\mathbf{W}_{d}$ is well-approximated by the CKM
matrix. Since the CKM matrix is nearly diagonal, this in turn limits the
freedom with which we may couple different quark portal matter states to
various generations: If the benchmark A1 is taken from the $\xi_{i}$
benchmarks in Table 7, for example, the branching fractions of the up-like
portal matter to the second- and third-generation up-like quarks will be
suppressed by CKM factors of $\mathrel{\mathchoice{\lower
3.01385pt\vbox{\halign{$\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$\cr<\crcr\sim\crcr}}}}O(10^{-2})$.
Finally, we note that the universal dependence on $\vec{\gamma}_{P}$ to
generate couplings between portal matter and SM states limits the freedom with
which discrepancies may appear between the mixing of $P_{1}$ states with the
SM and that of $P_{2}$ states: With the exception of the third generation of
the up-like sector, where corrections occur due to mixing between the SM top
quark and its non-portal partner, the magnitude of $P_{1}$ and $P_{2}$
couplings to SM states are the same, with $P_{1}$ states coupled to left-
handed ($SU(2)_{L}$ doublet) SM states and $P_{2}$ states coupled to right-
handed ($SU(2)_{L}$ singlet) ones. So, for example, the branching fractions of
$P_{1}^{d}$ and $P_{2}^{d}$ portal matter particles to each of $b$, $s$, and
$d$ quarks are always _equal_. Meanwhile, the fact that the $P_{1}$ states
overwhelmingly couple to $SU(2)_{L}$ doublet SM fields and the $P_{2}$
overwhelmingly couple to $SU(2)_{L}$ singlet fields is unsurprising: Given
that the $P_{1}$ states are $SU(2)_{L}$ doublets, mixing between $P_{1}$
states and right-handed ($SU(2)_{L}$ singlet) fields violates $SU(2)_{L}$, and
so only occurs due to mass terms in Eqs.(14) and (32) proportional to the
$SU(2)_{L}$-breaking vev, $v/\sqrt{2}\sim 170\;\textrm{GeV}$. Mixing between
$P_{1}$ states and left-handed fermions, however, has no such restriction, and
can be proportional to the significantly larger vev terms $v_{S}$ and $v_{P}$.
An analogous argument applies for $P_{2}$ states preferring to couple to
$SU(2)_{L}$ singlet fields over doublets.

Regarding the production of portal matter states, the QCD pair production
cross section of these fermions will be the same as the results already given
I and II for 13 TeV and 14 TeV center-of-mass energy at the LHC, respectively.
Additionally, as in II, there exist potentially significant non-QCD
contributions to the $q\overline{q}\rightarrow\bar{P}^{u}_{i}P^{u}_{i}$
process from $t-$channel dark photon exchange. We note that the dark photon
exchange contribution has one piece of additional structure compared to that
of II: Because all three generations may mix with the portal matter states,
depending on the orientation of the vector $\vec{\gamma}_{P}$, the
contribution of the $t$-channel $A_{D}$ exchange to portal matter pair
production may stem from portal matter couplings to multiple different SM
quarks. This is in contrast to II, in which each portal matter field was only
coupled to a single SM quark, and therefore portal matter pair production
would only suffer a $t$-channel dark photon contribution from that specific
quark flavor. However, we also note that the effect of this exchange varies
radically depending on the generation in question, with portal matter coupling
to lighter quarks having a dramatically stronger effect than coupling with
heavier quarks, due to the greater content of the lighter quarks in proton
parton distribution functions. We can then see, consulting the results of II
for the down-like sector of our model, that restricting considerations of
mixing in this contribution to just that of the lightest generation for which
the mixing is non-trivial (in the language of Section 4.1, the $\xi_{i}$ for
the lowest index $i$ that isn’t hierarchically smaller than any other
$\xi_{i}$) will likely be accurate to within $O(10\%)$ corrections, at worst.
Given the even larger hierarchies among quark masses in the up-like sector, we
anticipate this approximation to be even more accurate for up-like portal
matter.

Finally, we can discuss the decay branching fractions of portal matter fields
to SM states. As in I and II, the dominant decay process is the emission of a
dark photon to decay into an SM state with the same SM quantum numbers as the
portal matter field. Consulting the couplings in Eqs.(50), (51), and (52), we
find that the decay widths for these processes are given by

$\displaystyle\Gamma_{P_{1,2}^{d}\rightarrow
d_{i}A_{D}}=\frac{g_{4}^{2}s_{P}^{2}(m_{P1,P2}^{d})^{3}}{48\pi
m_{A_{D}}^{2}}\frac{|(\mathbf{W}_{d}^{\dagger}\vec{\gamma}_{P})_{i}|^{2}}{v_{P}^{2}},\;i=1,2,3,$
$\displaystyle\Gamma_{P_{1}^{u}\rightarrow
u_{i}A_{D}}=\frac{g_{4}^{2}s_{P}^{2}(m_{P1}^{u})^{3}}{48\pi
m_{A_{D}}^{2}}\frac{|(\vec{\gamma}_{P})_{i}|^{2}}{v_{P}^{2}},\;i=1,2,3,$ (105)
$\displaystyle\Gamma_{P_{2}^{u}\rightarrow
u_{i}A_{D}}=\frac{g_{4}^{2}s_{P}^{2}(m_{P2}^{u})^{3}}{48\pi
m_{A_{D}}^{2}}\frac{|(\vec{\gamma}_{P})_{i}|^{2}}{v_{P}^{2}},\;i=1,2,$
$\displaystyle\Gamma_{P_{2}^{u}\rightarrow
tA_{D}}=\frac{g_{4}^{2}s_{P}^{2}(m_{P2}^{u})^{3}}{48\pi
m_{A_{D}}^{2}}\frac{|(\vec{\gamma}_{P})_{3}|^{2}}{v_{P}^{2}}\bigg{(}\frac{M_{t}^{2}-y_{S}^{2}v_{S}^{2}}{M_{t}^{2}}\bigg{)},$

where $d_{1}\equiv d$, $d_{2}\equiv s$, $d_{3}\equiv b$, analogous definitions
hold for the up-like quarks, and we have neglected the effects of the top
quark mass (which should only introduce $O(m_{t}^{2}/(m^{u}_{P1,2})^{2})\sim
O(10^{-2})$ corrections). For clarity, we note that the enormous ratios
$(m^{u,d}_{P1,P2})^{2}/(m_{A_{D}})^{2}$ in these expressions are cancelled by
the comparably tiny ratios $|(\vec{\gamma}_{P})_{i}|^{2}/v_{P}^{2}$, leading
to results with reasonable magnitudes – this is in fact analogous to the
discussion in Section 3.3 of II in which the dark photon coupling between SM
and portal matter states is strongly suppressed, but the suppression is
counteracted by the dark photon’s small mass to give an $O(1)$ overall
strength for the interaction. In addition to these decay widths, we would
expect equal (at least in the limit where $m_{A_{D}}\rightarrow 0$)
contributions from a light scalar, corresponding to a physical scalar mode
associated with $U(1)_{D}$ breaking, from Goldstone boson equivalence. As the
scalar sector is not well-explored here, and such changes would not affect the
branching fractions of the portal matter anyway, we shall ignore this scalar
from here on out, with one exception discussed in Section 5.2. As discussed in
I, assuming that the dark photon is long-lived or only decays to DM (which we
will assume here), the signal for a portal matter pair production event here
should consist of two (possibly fat) jets, which may be from $t$, $b$, or
lighter quarks depending on $\vec{\gamma}_{P}$’s orientation in flavor space,
plus a significant amount of missing energy stemming from the dark photon
decays. Perhaps more interesting in this model are the possibilities afforded
us by the existence of the heavy top partner: In addition to decays to SM
particles, decays with comparable rates should exist for the portal matter
fields into the top partner, provided the latter is lighter enough that these
channels are kinematically accessible. We should note that, because the masses
$M_{t}$, $m_{P1}^{u}$, and $m_{P2}^{u}$ are simply free parameters in our
theory with masses of at least $O(\textrm{TeV})$, we have little a priori
guidance about whether the top partner is heavier than the portal fields or
vice versa. It therefore behooves us to consider both the possibility that a
portal matter field can decay into a top partner, and, in Section 5.2, whether
the reverse process might occur in a different region of parameter space.
Turning our attention back to the decay of portal matter to a top partner, up
to $O(m_{t}^{2}/M_{t}^{2})$ corrections and assuming that $m_{A_{D}}$ and
$\vec{\gamma}_{P}$ are much smaller than any other mass scale in the
expression, we find that these decay widths are given by

$\displaystyle\Gamma_{P^{u}_{1}\rightarrow
TA_{D}}\approx\frac{g_{4}^{2}s_{P}^{2}(m^{u}_{P1})^{3}}{48\pi
m_{A}^{2}}\bigg{(}1-\frac{M_{t}^{4}}{(m_{P1}^{u})^{4}}\bigg{)}\frac{m_{t}^{2}}{y_{S}^{2}v_{S}^{2}}\frac{|(\vec{\gamma}_{P})_{3}|^{2}}{v_{P}^{2}},$
(106) $\displaystyle\Gamma_{P^{u}_{2}\rightarrow
TA_{D}}\approx\frac{g_{4}^{2}s_{P}^{2}(m^{u}_{P2})^{3}}{48\pi
m_{A}^{2}}\bigg{(}1-\frac{M_{t}^{4}}{(m^{u}_{P2})^{4}}\bigg{)}\frac{y_{S}^{2}v_{S}^{2}}{M_{t}^{2}}\frac{|(\vec{\gamma}_{P})_{3}|^{2}}{v_{P}^{2}}.$

The somewhat strange dependence of these decay widths on the fourth power of
the ratio $M_{t}/m_{P1,2}^{u}$ bears a brief discussion here: It occurs
because the phase space factor in the decay (which approaches
$1-M_{t}^{2}/(m^{u}_{P1,P2})^{2}$ in the limit where $m_{A_{D}}\rightarrow 0$)
multiplies the combination of coupling constants appearing in the squared
amplitude, which are proportional to $1+M_{t}^{2}/(m^{u}_{P1,P2})^{2}$.
Meanwhile, we also note that the decay of $P_{1}^{u}$ to a top partner quark
is suppressed by a factor of
$m_{t}^{2}/(y_{S}^{2}v_{S}^{2})\mathrel{\mathchoice{\lower
3.01385pt\vbox{\halign{$\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$\cr<\crcr\sim\crcr}}}}10^{-2}$,
and so is unlikely to have much phenomenological impact. However, the decay of
$P_{2}^{u}$ to the heavy top partner can be quite significant. The branching
fraction for this decay should be given by

$\displaystyle\mathcal{B}(P_{2}^{u}\rightarrow
T\,A_{D})=\frac{\bigg{(}1-\frac{M_{t}^{4}}{(m_{P2}^{u})^{4}}\bigg{)}\frac{y_{S}^{2}v_{S}^{2}}{M_{t}^{2}}|(\vec{\gamma}_{P})_{3}|^{2}}{(\vec{\gamma}_{P}^{*}\cdot\vec{\gamma}_{P})-\frac{M_{t}^{4}}{(m_{P2}^{u})^{4}}\frac{y_{S}^{2}v_{S}^{2}}{M_{t}^{2}}|(\vec{\gamma}_{P})_{3}|^{2}}.$
(107)

Note that because $M_{t}<(m^{u}_{P2})$, $y_{S}v_{S}<M_{t}$, and
$|(\vec{\gamma}_{P})_{3}|^{2}$ will always be less than
$\vec{\gamma}_{P}^{*}\cdot\vec{\gamma}_{P}$ here, the above branching fraction
will always be positive. To get a feel for the magnitude of this effect, we
show this branching fraction as a function of the ratio $M_{t}/m_{P1}^{u}$ in
Figure 5 for some of the benchmarks of $\vec{\gamma}_{P}$ orientation outlined
in Table 7. We note that the benchmarks in Table 7 list values of
$\xi_{i}\equiv(\mathbf{W}_{d}^{\dagger}\vec{\gamma}_{P})_{i}/v_{P}$, not
$\vec{\gamma}_{P}$, however, the corrections due to letting
$\xi_{i}\approx(\vec{\gamma}_{P})_{i}/v_{P}$ to Eq.(107) will be suppressed by
$O(\lambda^{2})\sim 10^{-2}$ CKM factors and so are negligible for
demonstrative purposes.

Figure 5: (Top) The branching fraction of the $P^{u}_{2}$ quark to the top
partner, assuming that $M_{t}/(y_{S}v_{S})=1.5$ and that $\vec{\gamma}_{P}$ is
oriented according to the benchmarks $A_{3}$ (blue), $B_{1}$ (red), and
$C_{1}$ (magenta), described in Table 7. (Bottom) Same as above, but assuming
that $M_{t}/(y_{S}v_{S})=2$.

In Figure 5, we can see that when kinematically allowed, the branching
fraction of $P_{2}^{u}$ to the top partner can be quite significant. In cases
where the portal matter mixes predominantly with the top quark, it can even
account for nearly half of possible decays. As we shall see in Section 5.2, in
this scenario the top partner will then decay via the emission of an SM $Z$,
$W$, or Higgs into an SM top quark, reflecting the more conventional decay
channels for vector-like quarks. This behavior in turn can yield interesting
collider signals: Rather than simply producing a pair of jets plus missing
energy, as will overwhelmingly occur for the other portal fields in this
model, one of the $P^{u}_{2}$ fields decaying via the top partner might result
in, for example, missing energy, a pair of fat top quark jets, and a pair of
leptons from a $Z$ boson. This is an atypical signature.

###  5.2 Phenomenology: Top Quark Partner

Other than the portal fields, the top partner $T$ is the sole other exotic
fermion field that may be accessible at current or planned experiments in this
model. Anticipating that the top partner here will behave similarly to
conventional up-like vector-like quarks, we have quoted the constraint of
[24], which places a 95% CL limit on the mass of a generic electroweak singlet
vector-like quark with this electric charge as $>1.31\;\textrm{TeV}$. However,
we note that this constraint assumes that the top partner will always decay
via the standard processes $T\rightarrow h\,t$, $T\rightarrow Z\,t$, or
$T\rightarrow W^{+}\,b$. In our present construction, additional decay
channels may also be present, so it is useful to determine the extent to which
this assumption is valid.

The interactions of this top partner with the SM closely follow the familiar
behavior of vector-like quarks in other models; most notably, as we can see
from the couplings in Section 3.2, the SM $Z$, $W$, and Higgs all mediate
interactions between the top partner and the SM top, facilitating the
$T\rightarrow Z\,t$, $T\rightarrow W^{+}\,b$, and $T\rightarrow h\,t$ decay
processes which are assumed in [24], and which are by contrast highly
suppressed for the portal matter fields. Specifically, for these processes we
have (up to $O(m_{t}^{2}/M_{t}^{2})\sim 10^{-2}$ corrections)

$\displaystyle\Gamma_{T\rightarrow Wb}=2\Gamma_{T\rightarrow
Zt}=2\Gamma_{T\rightarrow
ht}=\frac{G_{F}m_{t}^{2}}{\sqrt{2}~{}8\pi}\bigg{(}\frac{M_{t}^{2}-y_{S}^{2}v_{S}^{2}}{y_{S}^{2}v_{S}^{2}}\bigg{)}M_{t},$
(108)

where $G_{F}=1.1663787\times 10^{-5}\;\textrm{GeV}^{-2}$ is the Fermi
constant. Notably, while there are _no_ decays of this quark into other SM up-
like quarks (at least up to highly-suppressed flavor-changing neutral currents
from, for example, the dark photon), there are CKM-suppressed decays of the
form $T\rightarrow W^{+}\,d$ and $T\rightarrow W^{+}\,s$; these are, however,
suppressed by at least $\lambda^{4}\sim O(10^{-3})$ compared to the
$T\rightarrow W^{+}\,b$ decay, and therefore have no observable experimental
effect at present.

In the event that all portal matter fields in the model are more massive than
$T$, the above exhausts the possible decay channels for the top partner,
indicating that it strongly resembles a conventional vector-like quark. The
scenario can become more complicated, however, when the up-like portal matter
is light enough to allow for the top partner to decay into it. In this case,
the widths for decays of $T$ to $P^{u}_{1}$ and $P^{u}_{2}$ are

$\displaystyle\Gamma_{T\rightarrow
P^{u}_{1}\,A_{D}}=\frac{g_{4}^{2}s_{P}^{2}(m_{P1}^{u})^{2}}{48\pi
m_{A_{D}}^{2}}\bigg{(}1-\frac{(m_{P1}^{u})^{4}}{M_{t}^{4}}\bigg{)}\bigg{(}1-\frac{y_{S}^{2}v_{S}^{2}}{M_{t}^{2}}\bigg{)}\frac{m_{t}^{2}}{y_{S}^{2}v_{S}^{2}}M_{t}\frac{|(\vec{\gamma}_{P})_{3}|^{2}}{v_{P}^{2}},$
(109) $\displaystyle\Gamma_{T\rightarrow
P^{u}_{2}\,A_{D}}=\frac{g_{4}^{2}s_{P}^{2}(m_{P2}^{u})^{2}}{48\pi
m_{A_{D}}^{2}}\bigg{(}1-\frac{(m_{P2}^{u})^{4}}{M_{t}^{4}}\bigg{)}\frac{y_{S}^{2}v_{S}^{2}}{M_{t}^{2}}M_{t}\frac{|(\vec{\gamma}_{P})_{3}|^{2}}{v_{P}^{2}}.$

We see that, analogous to the case for the reversed decay pattern discussed in
Section 5.1, the decay width from $T$ to $P_{1}^{u}$ is suppressed by a factor
of $m_{t}^{2}/(y_{S}^{2}v_{S}^{2})\mathrel{\mathchoice{\lower
3.01385pt\vbox{\halign{$\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$\cr<\crcr\sim\crcr}}}}10^{-2}$,
and so is likely not numerically significant, while the decay width from $T$
to $P_{2}^{u}$ has no such suppression and may be large. Furthermore, we note
(as discussed in Section 5.1), we would anticipate from Goldstone boson
equivalence that a physical scalar, the ‘dark’ Higgs, which we shall call
$S_{D}$, associated with the breaking of $U(1)_{D}$ will likely also exist,
with a decay width such that $\Gamma_{T\rightarrow
P^{u}_{2}\,S_{D}}=\Gamma_{T\rightarrow P^{u}_{2}\,A_{D}}$, at least in the
limit where $m_{A_{D}}\rightarrow 0$. Unlike in the case of portal matter
decays, for which each numerically significant decay channel has such an
identical channel with $S_{D}$, and hence branching fractions are unaffected
by ignoring the scalar, the top partner’s dominant SM decays $T\rightarrow
Z\,t$, $T\rightarrow h\,t$, and $T\rightarrow W^{+}\,b$ do not have an
equivalent $S_{D}$ channel, while the decay to portal matter does. Therefore,
when discussing the decay of the top partner to portal matter, we must include
the decay width of the process $T\rightarrow P^{u}_{2}\,S_{D}$ explicitly,
arriving at the combined branching fraction from Eqs.(108) and (109) of

$\displaystyle\mathcal{B}(T\rightarrow
P^{u}_{2}\,A_{D},S_{D})\approx\frac{2\Gamma_{T\rightarrow
P^{u}_{2}\,A_{D}}}{4\Gamma_{T\rightarrow ht}+2\Gamma_{T\rightarrow
P^{u}_{2}\,A_{D}}}.$ (110)

To get a sense of the possible magnitude of the branching fraction given in
Eq.(110), we depict it for various choices of $M_{t}$ and $M_{t}/(y_{S}v_{S})$
in Figure 6, assuming that $|(\vec{\gamma}_{P})_{3}|/v_{P}=10^{-4}$, the
“natural” magnitude we discussed in Section 4.1.

Figure 6: (Top) The branching fraction of the top partner quark to $P^{u}_{2}$
as a function of the ratio of their masses, assuming that $g_{4}s_{P}=0.3$,
$m_{A_{D}}=0.1\;\textrm{GeV}$, $|(\vec{\gamma}_{P})_{3}|/v_{P}=10^{-4}$, and
$M_{t}/(y_{S}v_{S})=1.5$, for $M_{t}=2$ TeV (blue), 4 TeV (red), and 6 TeV
(magenta). (Bottom) Same as above, but assuming that $M_{t}/(y_{S}v_{S})=2$.

In Figure 6, we have selected values of $g_{4}s_{P}$ and $m_{A_{D}}$ so as to
maximize the branching fraction to the portal matter fields, selecting
$m_{A_{D}}$ at the bottom of the range we consider, while selecting
$g_{4}s_{P}$ close to the upper end of the range that still permits
$O(10^{-3})$ kinetic mixing (as discussed in Section 3.3). Given that the
decay width of the top partner to $m_{P1}$ is proportional to the square of
$g_{4}s_{P}$, and the inverse square of $m_{A_{D}}$, we can anticipate that
larger values of $m_{A_{D}}$ and smaller couplings $g_{4}s_{P}$ will result in
dramatically smaller branching fractions – letting
$m_{A_{D}}=0.3\;\textrm{GeV}$, for example, will reduce the branching fraction
by a factor of 9. Even in the maximal case we depict in Figure 6, we generally
only attain limited branching fractions of $T$ to $P^{2}_{u}$ here, however –
we achieve a $\sim 30\%$ branching fraction at the largest when
$M_{t}=6\;\textrm{TeV}$, half that when $M_{t}=4\;\textrm{TeV}$, and in all
other cases we depict, the branching fraction is less than $10\%$. This
branching fraction can be enhanced if we assume a larger value of
$|(\vec{\gamma}_{P})_{3}|/v_{P}$ (for example, enhancing this parameter by a
factor of 3 would increase the decay width of $T$ to portal matter by a factor
of 9), however, considering a value of $|(\vec{\gamma}_{P})_{3}|/v_{P}$ much
larger than $10^{-4}$, as we have taken it to be in Figure 6, begins to be in
tension with constraints on flavor-changing $B$ meson decays discussed in
Section 4.1. In short, in contrast to the scenario with the possible decay of
portal matter to the top partner the reverse scenario is likely to have a
minimal effect on the signal of top partner production, except in certain
special corners of parameter space. In these corners, however, we might
anticipate some interesting potential top partner events – for example, top
partner pair production might give us one ‘conventional’ top partner which
decays via the channels $T\rightarrow Z\,t$, $T\rightarrow h\,t$, or
$T\rightarrow W\,b$, and the other that decays to a top (or, depending on the
orientation of $\vec{\gamma}_{P}$ in flavor space, a lighter up-like quark)
through sequentially emitting two dark photons, which would appear as missing
energy. Determining the degree to which the latter decay path might be
kinematically distinguishable from, for example, $T\rightarrow Z\,t$ with an
invisibly decaying $Z\rightarrow\bar{\nu}\nu$ may be of interest, but lies
beyond the scope of this work.

###  5.3 Phenomenology: $Z_{P}$ Production

To start, we shall consider the production of the “portal $Z$” boson $Z_{P}$,
representing the heavy counterpart to the lighter dark photon, with a mass of
$O(1\;\textrm{TeV})$. Notably, since the other gauge bosons with which $Z_{P}$
might couple (specifically, those associated with $SU(4)_{F}$ generators other
than $t^{15}$, so in principle any gauge boson except $A_{D}$) are all much
heavier than $Z_{P}$ itself, there is no region of parameter space in which
$Z_{P}$ might decay into any other gauge bosons at tree-level. Our sole
concern at this point, then, would be the decay of the $Z_{P}$ into fermion
pairs. Consulting the coupling matrix for $Z_{P}$ given in Eqs.(53) and (3.1),
we can then straightforwardly determine the decay widths of the $Z_{P}$ boson.
In particular, for decay into any particular SM fermion, we have

$\displaystyle\Gamma_{Z_{P}\rightarrow\overline{f}f}\approx
C_{f}\frac{g_{4}^{2}/c^{2}_{P}}{288\pi}m_{Z_{P}}\bigg{[}1+O\bigg{(}\frac{m_{f}^{2}}{m_{Z_{P}}^{2}}\bigg{)}\bigg{]},$
(111)

where $C_{f}$ is simply a color factor (3 for quarks, 1 for charged leptons
and neutrinos). Depending on the relative masses of the top partner and the
portal matter fields to $m_{Z_{P}}$, however, $Z_{P}$ may also decay into
these states. For the top partner, the decay width is

$\displaystyle\Gamma_{Z_{P}\rightarrow\overline{T}T}=\frac{g_{4}^{2}/c^{2}_{P}}{96\pi}m_{Z_{P}}\bigg{(}1+\frac{2M_{t}^{2}}{m_{Z_{P}}^{2}}\bigg{)}\sqrt{1-\frac{4M_{t}^{2}}{m_{Z_{P}}^{2}}}.$
(112)

The phenomenologically relevant portal matter fields, meanwhile, yield decay
processes with widths

$\displaystyle\Gamma_{Z_{P}\rightarrow\overline{P}_{i}^{u}P_{i}^{u}}=3~{}\frac{g_{4}^{2}/c^{2}_{P}}{64\pi}m_{Z_{P}}\bigg{[}\bigg{(}1+\frac{8}{9}x_{P}\bigg{)}\bigg{(}1-\frac{(m_{Pi}^{u})^{2}}{m_{Z_{P}}^{2}}\bigg{)}+\frac{8}{3}x_{P}\frac{(m_{Pi}^{u})^{2}}{m_{Z_{P}}^{2}}\bigg{]}\sqrt{1-\frac{4(m_{Pi}^{u})^{2}}{m_{Z_{P}}^{2}}},$
(113) $\displaystyle x_{P}\equiv s^{2}_{P}(1-4c^{2}_{P}),$

where $P_{i}^{u}$ refers to the portal matter field of mass $m_{Pi}^{u}$, and
analogous expressions exist for the portal matter in the down-like quark,
charged lepton, and neutrino sectors (note that the factor of 3 in front of
Eq.(113) is a color factor, and will hence be equal to one for the
corresponding case of neutrino and charged lepton portal matter). Because the
couplings of the $Z_{P}$ to SM states are flavor-universal, this model makes
easy contact with existing searches for high-mass spin-1 resonances. To
estimate constraints on our model, we employ the limits on leptonic cross
sections from [46], a dilepton (dielectron and dimuon)-channel null search
using 139 $\textrm{fb}^{-1}$ of 13 TeV data, and the analogous expected limits
from a null result with 3000 $\textrm{fb}^{-1}$ of LHC data at 14 TeV [47]. In
Figure 7, we depict the maximum value of $g_{4}/c_{P}$ allowed as a function
of $m_{Z_{P}}$ and the corresponding minimum allowed value of $v_{P}$, both
for current experimental limits and those expected from a null result at the
High-Luminosity LHC, assuming that the $Z_{P}$ boson only decays into SM final
states (in which case the branching ratios for the dielectron or dimuon
channels are simply 1/24).

Figure 7: (Top) The maximum allowed value of $g_{4}/c_{P}$ (left) and the
minimum allowed $v_{P}$ (right) for different values of $m_{Z_{P}}$, based on
limits from dilepton-channel searches with 139 $\textrm{fb}^{-1}$ of 13 TeV
LHC data [46], assuming $Z_{P}$ only decays into SM final states. For
comparison, the electroweak coupling $g/c_{w}$ (left) and the value of $v_{P}$
assuming $g_{4}/c_{P}=g/c_{w}$ for a given $m_{Z_{P}}$ value (right) have been
depicted as dashed orange lines. (Bottom) Same as above, but using the
expected constraints from a null result with 3 $\textrm{ab}^{-1}$ of 14 TeV
LHC data [47]

Examining Figure 7, we note that for realistic $O(1)$ values of $g_{4}/c_{P}$
(or, equivalently, $O(10\;\textrm{TeV})$ values of $v_{P}$), the present
constraint roughly requires $m_{Z_{P}}\mathrel{\mathchoice{\lower
3.01385pt\vbox{\halign{$\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$\cr>\crcr\sim\crcr}}}}5\;\textrm{TeV}$;
if we assume that $g_{4}/c_{P}=g/c_{w}$, where $g$ is the $SU(2)_{L}$ coupling
constant and $c_{w}$ is the cosine of the Weinberg angle, the limit is, for
example, $m_{Z_{P}}\geq 5.15\;\textrm{TeV}$. A null result at the HL-LHC
meanwhile suggests that these limits could be increased by approximately 1
TeV: If $g_{4}/c_{P}=g/c_{w}$, the expected limit from a null result of HL-LHC
data would be $m_{Z_{P}}\geq 6.2\;\textrm{TeV}$. However, we note that similar
to the setup in II, this circumstance can be somewhat modified in regions of
parameter space in which decays to exotic fermion species are kinematically
allowed. In particular, constraints may be somewhat relaxed by assuming that
the portal matter fields, $P_{1,2}^{u,d,e,\nu}$ are sufficiently light to
allow for $Z_{P}$ to decay into them (which, assuming $g_{4}/c_{P}\sim O(1)$,
would require at worst a mild $O(10^{-1})$ tuning of the Yukawa coupling
parameters $y_{P1}$ and $y_{P2}$). While the decay into the top partner can
reduce the branching fraction of the $Z_{P}$ to dilepton channels by
approximately $10\%$, the large number of portal matter states with roughly
degenerate masses (if we assume that decays to all of these portal matter
states are kinematically accessible, for example, $Z_{P}$ would have 8
different new particles, some color triplets, to which it can decay) can
provide as much as an order of magnitude reduction in the cross section
observable in dilepton final state searches. We depict the proportional
suppression factor of the branching ratio $\mathcal{B}(Z_{P}\rightarrow
l^{+}l^{-})$ assuming that some or all portal matter states are kinematically
accessible in Figure 8, where we note that, as Eq.(113) suggests, the
suppression factor is a function of $c_{P}$.

Figure 8: (Top) The proportional suppression of the branching ratio of $Z_{P}$
to leptons, compared to the scenario where $Z_{P}$ only decays to SM states,
as a function of $c_{P}$, assuming portal matter states have the same mass
$m_{P}$, assuming $m_{P}/m_{Z_{P}}=0.2$ (blue), 0.3 (red), and 0.4 (magenta).
(Bottom) Same as above, but assuming that only one of the two lighter portal
matter states in each sector, rather than both, are kinematically accessible
for this decay.

In Figure 8, we have specifically considered two sets of simple benchmarks for
the possibility that the portal matter states are kinematically accessible in
$Z_{P}$ decays – either that all portal matter states have the same mass
(which, we note, would not be an unreasonable approximation if $y_{P1}\approx
y_{P2}$, in which case all portal matter states would have degenerate masses
up to radiative corrections), or that only half of the portal matter states
are light enough for $Z_{P}$ to decay into them (as we might expect if, for
example, $y_{P1}<y_{P2}$). The suppression factors in Figure 8, in turn,
correspond to as much as $O(1)$ increases in the maximum allowed $g_{4}/c_{P}$
values (or equivalently, $O(1)$ decreases in the minimum $v_{P}$ values)
depicted in Figure 7. We note, however, that even under the best of
circumstances (for example, if the suppression factor is $\sim 0.2$, very near
the lowest we can achieve), the $g_{4}/c_{P}$ values are only increased by a
factor of $\sim 2$. A cursory inspection of Figure 7 indicates that this
amelioration is enough to perhaps allow $m_{Z_{P}}\sim 4\;\textrm{TeV}$ while
maintaining a reasonable $O(1)$ value of $g_{4}/c_{P}$ (at least based on 13
TeV data), but certainly cannot enable much smaller $m_{Z_{P}}$ values. Of
course, the potential ameliorating effects of these exotic decay channels on
dilepton searches may be counteracted by new distinctive signals due to the
exotic matter production. In particular, the production of particle-
antiparticle pairs of portal matter or top partner quarks via a resonant
$Z_{P}$ might have a significant effect on the overall production cross
section of these states, especially in the case of leptonic portal matter,
which otherwise may only be produced via electroweak and dark photon
interactions. Null results for portal matter and top partner searches, then,
will likely apply a non-trivial constraint on the possible mass of the $Z_{P}$
as well, however treating this situation quantitatively is beyond the scope of
this paper.

##  6 Discussion and Conclusions

The distinctive requirements for fermionic portal matter fields in vector
portal DM models provide an intriguing framework for UV extensions of these
models. In particular, constraints from SM anomalies and lifetimes of new SM-
charged particles suggest that these fields are vector-like, albeit with
collider signatures that are quite different from those seen with more
“conventional” vector-like fermions. Here, we have seen that highly complex
additional physics, not a priori related to DM, can in fact be incorporated
into a portal matter model by following simple rules: Extending the SM gauge
group to a semisimple one guarantees that KM with a $U(1)_{D}$ embedded in a
separate dark group will be finite and calculable, at which point the only
requirements are to ensure that the complete theory is anomaly-free and
contains only the SM and heavy vector-like particles.

To demonstrate the potential of following this recipe, we have developed a
model with Pati-Salam symmetry extended by $SU(4)_{F}\times
U(1)_{F}=4_{F}1_{F}$, in which the $4_{F}1_{F}$ symmetry contains both a dark
photon gauge group $U(1)_{D}$ and an $SU(3)$ flavor symmetry. The Pati-Salam
symmetry is assumed to be broken at a high scale $\sim 10^{13}\;\textrm{GeV}$,
at which point KM between $U(1)_{F}$ and $U(1)_{Y}$ occurs. Scalars in the
adjoint representation of $SU(4)_{F}$ then break the $4_{F}1_{F}$ down to
$1_{F^{\prime}}1_{F}$, completely breaking the embedded $SU(3)_{F}$ flavor
symmetry, at multi-TeV scales to avoid flavor constraints and generate the
fermion mass hierarchy observed in the SM. The $1_{F^{\prime}}1_{F}$ symmetry
is then broken down to $U(1)_{D}$ at a scale of $\sim 1-10\;\textrm{TeV}$, and
$U(1)_{D}$, the gauge symmetry corresponding to the DM vector portal, is
finally broken by small vev terms of $O(0.1-1\;\textrm{GeV})$. The presence of
diverse scales in the symmetry breaking in turn translates to diverse scales
of the new exotic particles proposed. However, we find that among the new
fermions, only certain portal matter fields, which we have labelled
$P_{1,2}^{u,d,e,\nu}$, and a vector-like new partner to the top quark, which
we call $T$, can be expected to be light enough to be observed at present
collider experiments, having masses at the scale of $\sim$ a few TeV. Among
the new gauge bosons we have anticipated, the only ones which we find to have
potentially observable effects are the dark photon of mass
$O(1\;\textrm{GeV})$ $A_{D}$, a TeV-scale boson with flavor-universal
couplings $Z_{P}$, and a TeV-scale flavor-changing neutral boson $Z_{F}$
associated with a combination of generators of the flavor group $SU(3)_{F}$.

With the model set up, we proceeded to explore the phenomenological
implications of the experimentally observable (that is, TeV-scale or less)
sector of the theory, in particular focusing on the phenomenology of quarks
and their portal matter partners, which we expect to have more
phenomenologically visible signals than the corresponding physics in the
lepton sector. We found that present constraints from flavor-changing neutral
currents arising from the dark photon, in particular from $K\rightarrow\pi
A_{D}$ transitions, placed harsh limits on the form of the vev parameters
which might break $U(1)_{D}$, and in turn can substantially influence the
branching fractions of portal matter fields to SM particles. We next noted
that the vector-like top partner $T$ and the new vector boson $Z_{P}$
contributed non-trivially to $\Delta B=2$ flavor-changing processes, the
$Z_{P}$ at tree-level and the $T$ at the one-loop level. These effects were
comparable in magnitude and heavily dependent on the value of the model
parameter $y_{S}v_{S}$, a dimensionful quantity which set the scale of both
$T$ and $Z_{F}$’s mass. However, even when combined the sum of these
contributions still provided only a limited constraint on our parameter space
based on present measurements, although Belle II and LHCb data may constrain
these more stringently in the near future [27].

Beyond the rich collider phenomenology already explored in I and II, much of
which is replicated in our model, we find additional model-building
flexibility arising from the potential for the portal matter fields
$P_{2}^{u}$ to decay into $T$ (or vice versa, depending on the particles’
relative masses), which can provide distinctive collider signatures appearing
in neither the treatments discussed in I and II nor more conventional models
of vector-like quarks [12, 43, 30].

Looking forward, we note that the present model represents a single specific
realization of a much broader recipe for developing models with non-minimal
portal matter sectors. There are a number of different avenues through which
this particular effort can be further explored, for example by incorporating
an explanation for the small masses and near-maximal mixing observed in the
neutrino sector, addressing the scalar sector rigorously, considering
scenarios in which the Pati-Salam symmetry (or some components of it) are
broken at a similar or lower scale to the one at which the $SU(4)_{F}\times
U(1)_{F}$ symmetries are broken, or explicitly incorporating either scalar or
fermionic DM in the construction. Perhaps a broader conclusion to be drawn
from this work, however, is the potential that portal matter model building
possesses to elaborate on other extensions of the SM. In particular, we have
seen that incorporating a local $SU(3)$ flavor symmetry with a model of portal
matter leads to rich phenomenological signatures, some of which do not appear
in more conventional portal matter models or models with local flavor
symmetries individually, such as a potential correlation between portal matter
lifetime, branching fractions, and the allowed dark photon masses, or decays
of portal matter to more conventional vector-like quarks. Presumably, similar
paths can be taken to explore potential links between portal matter models and
other popular extensions of the SM, predicting entirely different unique
signatures that may depart radically from those anticipated by conventional
new physics searches.

## Acknowledgements

GNW would like to thank T.D. Reuter for discussions relating to this work.
This work was supported by the Department of Energy, Contract DE-
AC02-76SF00515.

##  Appendix A $SU(4)$ Generator Matrices

Here we list the generators of the $SU(4)$ algebra, $t^{i}$. The first eight
generators correspond to the embedded group $SU(3)_{F}$ in $SU(4)_{F}$, and
are given by

$\displaystyle t^{1}=\frac{1}{2}\begin{pmatrix}0&1&0&0\\\ 1&0&0&0\\\
0&0&0&0\\\ 0&0&0&0\end{pmatrix},\;\;$ $\displaystyle
t^{2}=\frac{1}{2}\begin{pmatrix}0&-i&0&0\\\ i&0&0&0\\\ 0&0&0&0\\\
0&0&0&0\end{pmatrix},$ $\displaystyle
t^{4}=\frac{1}{2}\begin{pmatrix}0&0&1&0\\\ 0&0&0&0\\\ 1&0&0&0\\\
0&0&0&0\end{pmatrix},\;\;$ $\displaystyle
t^{5}=\frac{1}{2}\begin{pmatrix}0&0&-i&0\\\ 0&0&0&0\\\ i&0&0&0\\\
0&0&0&0\end{pmatrix},$ (114) $\displaystyle
t^{6}=\frac{1}{2}\begin{pmatrix}0&0&0&0\\\ 0&0&1&0\\\ 0&1&0&0\\\
0&0&0&0\end{pmatrix},\;\;$ $\displaystyle
t^{7}=\frac{1}{2}\begin{pmatrix}0&0&0&0\\\ 0&0&-i&0\\\ 0&i&0&0\\\
0&0&0&0\end{pmatrix},$ $\displaystyle
t^{3}=\frac{1}{2}\begin{pmatrix}1&0&0&0\\\ 0&-1&0&0\\\ 0&0&0&0\\\
0&0&0&0\end{pmatrix},\;\;$ $\displaystyle
t^{8}=\frac{1}{2\sqrt{3}}\begin{pmatrix}1&0&0&0\\\ 0&1&0&0\\\ 0&0&-2&0\\\
0&0&0&0\end{pmatrix}.$

The next 6 generators are then

$\displaystyle t^{9}=\frac{1}{2}\begin{pmatrix}0&0&0&1\\\ 0&0&0&0\\\
0&0&0&0\\\ 1&0&0&0\end{pmatrix},\;\;$ $\displaystyle
t^{10}=\frac{1}{2}\begin{pmatrix}0&0&0&-i\\\ 0&0&0&0\\\ 0&0&0&0\\\
i&0&0&0\end{pmatrix},$ $\displaystyle
t^{11}=\frac{1}{2}\begin{pmatrix}0&0&0&0\\\ 0&0&0&1\\\ 0&0&0&0\\\
0&1&0&0\end{pmatrix},\;\;$ $\displaystyle
t^{12}=\frac{1}{2}\begin{pmatrix}0&0&0&0\\\ 0&0&0&-i\\\ 0&0&0&0\\\
0&i&0&0\end{pmatrix},$ (115) $\displaystyle
t^{13}=\frac{1}{2}\begin{pmatrix}0&0&0&0\\\ 0&0&0&0\\\ 0&0&0&1\\\
0&0&1&0\end{pmatrix},\;\;$ $\displaystyle
t^{14}=\frac{1}{2}\begin{pmatrix}0&0&0&0\\\ 0&0&0&0\\\ 0&0&0&-i\\\
0&0&i&0\end{pmatrix}.$ (116)

Finally, $t^{15}$ is the generator corresponding to the embedded group
$U(1)_{F}^{\prime}$ in $SU(4)_{F}$, which mixes with $U(1)_{F}$ to form the
dark charge group. It is given by

$\displaystyle t^{15}=\frac{1}{2\sqrt{6}}\begin{pmatrix}1&0&0&0\\\ 0&1&0&0\\\
0&0&1&0\\\ 0&0&0&-3\end{pmatrix}.$ (117)

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