# Composite Higgs Models with a Hidden Sector

Ann E. Nelson<EMAIL_ADDRESS>Department of Physics, University of
Washington, Seattle, WA  Michael Park<EMAIL_ADDRESS>Department of Physics,
University of Washington, Seattle, WA  Department of Physics, University of
Washington, Bothell, WA  Devin G. E. Walker<EMAIL_ADDRESS>Department of Physics and Astronomy, Dartmouth College, Hanover, NH

###### Abstract

We discuss the phenomenology of composite Higgs models that naturally produce
a Standard Model-like Higgs boson with a mass of $126$ GeV. The effective
theory below the compositeness scale is weakly coupled in these models, and
the goldstone sector acts as a portal between the third generation of quarks
and a hidden gauge sector. The addition of hidden-sector fermions gives rise
to a calculable effective scalar potential with a naturally light scalar
resonance. The generic prediction of these theories is the existence of
additional pseudo-Nambu Goldstone bosons with electroweak-scale cross sections
and masses. In this paper we analyze the collider signatures for some simple
concrete realizations of this framework. We find that despite the existence of
additional weakly and strongly coupled particles that are kinematically within
reach of current experiments, the generic signatures are difficult to resolve
at the LHC, and could remain well-hidden in the absence of an $e^{+}e^{-}$
Higgs factory such as the CEPC or a surface detector such as MATHUSLA.

###### Contents

  1. I Introduction
  2. II $SU(5)/SO(5)$ Intermediate Model
    1. II.1 Fermion Sector
      1. II.1.1 Color-charged Fermions
      2. II.1.2 Hidden-Sector Fermions
      3. II.1.3 Theory Space
    2. II.2 Electroweak Symmetry Breaking
    3. II.3 Phenomenology
      1. II.3.1 Resonant $t\bar{t}$ Final States
      2. II.3.2 Multi-top Final States
      3. II.3.3 Hidden-Sector Final States
  3. III $SU(4)/Sp(4)$ Intermediate Model
    1. III.1 Fermion Sector
      1. III.1.1 Color-charged Fermions
      2. III.1.2 Hidden-Sector Fermions
      3. III.1.3 Theory Space
    2. III.2 Electroweak Symmetry Breaking
    3. III.3 Phenomenology
      1. III.3.1 Rare Higgs Decays
  4. IV Conclusions and Outlook
  5. A Fermion Contribution to $SU(4)/Sp(4)$ Higgs Potential
  6. B Fermion Contribution to $SU(5)/SO(5)$ Higgs Potential
  7. C Multi-top Final States in $SU(5)/SO(5)$

##  I Introduction

Despite its success in discovering a Standard Model-like Higgs boson, the
Large Hadron Collider (LHC) has yet to provide a satisfying explanation for
the mechanism of electroweak symmetry breaking (EWSB). To date there is no
discovery leading the way to new physics, and many of the popular explanatory
frameworks are becoming constrained into finely tuned regions of their
parameter spaces. Theoretical development over the last several decades has
largely been motivated by criteria of naturalness and parsimony Craig (2013).
While there is a strong logical and historical motivation for this notion of
naturalness, there is also an arguably comparable motivation for cautious
skepticism in our conceptions about parsimony. In this paper we study a class
of effective theories that generically give rise to a composite Higgs boson
with Standard Model (SM)-like properties, and analyze their collider
signatures. A light Higgs with a large quartic is achieved via the
introduction of a hidden-sector that couples to the SM through the Higgs
portal, thus preserving naturalness at the expense of parsimony. This work
adds to a growing abundance of effective models for electroweak symmetry
breaking with highly subtle hadron-collider signatures, and motivates several
concrete analyses for further scrutiny.

The composite Higgs hypothesis Kaplan and Georgi (1984) offers an attractive
and radically conservative Wilczek and Devine (1988) possibility for the
origin of a naturally macroscopic electroweak scale. In this framework the
compositeness scale is naturally higher than the electroweak symmetry breaking
scale, as the Higgs arises as a pseudo-Nambu-Goldstone boson (pNGB) Kaplan
_et al._ (1984); Georgi and Kaplan (1984); Georgi  _et al._ (1984); Dugan  _et
al._ (1985); Arkani-Hamed  _et al._ (2001, 2002a, 2002b); Low  _et al._
(2002); Kaplan and Schmaltz (2003); Chang and Wacker (2004); Skiba and Terning
(2003); Chang (2003); Piai  _et al._ (2004) from the spontaneous breaking of
an approximate global symmetry by the compositeness dynamics. These models
retain a weakly coupled description of the EWSB sector above the chiral
symmetry breaking scale $f$, with a full UV completion appearing at a higher
scale associated with the new strong dynamics $\Lambda$ Weinberg (1976);
Susskind (1979). The most parsimonious constructions of composite Higgs
theories have historically found tension with measurements of precision
electro-weak (PEW) observables due to their generic assumption of additional
gauge structure above the scale $f$. The Intermediate Higgs (IH) or “natural
composite” scenario Katz  _et al._ (2005a, b); Contino  _et al._ (2007);
Gripaios  _et al._ (2009) relaxes this assumption, which results in a
quadratic sensitivity to the compositeness scale $\Lambda$ from loops of gauge
bosons. However this contribution is numerically small enough to avoid serious
fine tuning for a new physics scale as high as $\Lambda\sim 10$ TeV. The large
radiative corrections to the Higgs mass from the top sector are canceled by
new vector-like quarks with TeV-scale masses, utilizing the mechanism of
collective symmetry breaking, and resulting in partially composite third
generation Kaplan (1991).

The recently measured mass of the SM Higgs boson $m_{h}=126$ GeV Chatrchyan
_et al._ (2012); Aad  _et al._ (2012) provides a significant piece of
additional data for the composite Higgs hypothesis. In the IH scenario, the
scalar potential is highly constrained by the explicit symmetry breaking
pattern of the approximate global symmetries. Again the most parsimonious
constructions have historically found difficulty generating a sufficiently
light Higgs mass or an otherwise viable Higgs potential Low (2004); Hubisz and
Meade (2005); Bazzocchi  _et al._ (2005); Pappadopulo and Vichi (2011); Vecchi
(2013); Bellazzini  _et al._ (2014); Low  _et al._ (2015). In this paper we
consider the effects of extending the composite sector to include additional
hidden-sector interactions via Yukawa couplings between the pNGB matrix and a
new multiplet of hidden-sector fermions. We find that the radiative
corrections to the Higgs potential from these states at $f$ generically
produce a pattern of EWSB that is consistent with current experimental data.
The invariant prediction of these theories is the existence of new vector-like
quarks with masses of $\mathcal{O}(f)$ and new weakly interacting scalars with
masses of $\mathcal{O}(gf)$. All of the pNGB scalars remain uneaten in these
models and manifest as physical resonances, however their weak scale cross
sections are typically below threshold for discovery via direct production at
the LHC. Furthermore the extended Yukawa sector can generically lead to non-
standard decays of the new quarks that significantly weakens the limits from
LHC searches for simplified top-partner models Khachatryan  _et al._ (2016a);
Sirunyan  _et al._ (2017). The variety of possible coset spaces thus provides
a wide range of theoretically motivated collider signatures that should now be
considered more seriously.

In this paper we select two contrasting examples for detailed analysis.
Section II discusses a model based on the $SU(5)/SO(5)$ coset space Georgi and
Kaplan (1984), which generates a rich variety of new scalar resonances. Here
we focus on the low-energy phenomenology as well as the conditions for a
realistic pattern of EWSB. Large couplings between the pNGBs and the new
vector-like quarks in this model lead to an enhanced production of high-
multiplicity tops and bottoms, resulting in cascade decays with many leptons
and $b$-quarks in the final state. The hidden-sector interactions result in a
fairly generic hidden-valley phenomenology which we review in the context of
these models. Section III discusses the $SU(4)/Sp(4)$ coset space Georgi and
Kaplan (1984), which generates only one new pNGB resonance beyond the Higgs
multiplet. We briefly discuss the tension between the fermionic sector of this
model and electroweak precision constraints on the bottom quark interactions
and how to alleviate this. We then focus on the difficulties in resolving the
goldstone sector of this theory, which could remain well hidden until future
Higgs factories come online. In Section IV we conclude with a summary of these
results.

##  II $SU(5)/SO(5)$ Intermediate Model

The symmetry breaking pattern $SU(5)/SO(5)$ produces $24-10=14$ pseudo Nambu-
Goldstone Bosons (pNGBs). The sigma field transforms under $SU(5)$ as $V\Sigma
V^{T}$ where $V$ is an $SU(5)$ matrix. It is convenient to specify an $SO(5)$
symmetric background field $\Sigma_{0}$, under which the unbroken $SU(5)$
generators $T$ and the broken generators $X$ satisfy

$\begin{matrix}T\Sigma_{0}+\Sigma_{0}T^{T}=0\\\\[8.61108pt]
X\Sigma_{0}-\Sigma_{0}X^{T}=0\\\
\end{matrix}\qquad\qquad\qquad\Sigma_{0}=\begin{pmatrix}\quad&\quad&\mathbbm{1}\\\
\quad&1&\quad\\\ \mathbbm{1}&\quad&\quad\\\ \end{pmatrix}.$ (1)

The $SO(5)$ symmetry contains the global custodial $SO(4)_{c}=SU(2)_{L}\otimes
SU(2)_{R}$ subgroup, where $SU(2)_{R}$ is approximate and contains $U(1)_{Y}$.
The generators of this global symmetry $L^{a}$ and $R^{a}$ can be expressed as

$\displaystyle L^{a}$
$\displaystyle=\frac{1}{2}\begin{pmatrix}\sigma^{a}&\quad&\quad\\\
\quad&$0$&\quad\\\ \quad&\quad&-\sigma^{a*}\end{pmatrix}$ (2) $\displaystyle
R^{a}$
$\displaystyle=\frac{1}{2}\left\\{\begin{pmatrix}\quad&\quad&-i\sigma^{2}\\\
\quad&$0$&\quad\\\
i\sigma^{2}&\quad&\quad\end{pmatrix},\begin{pmatrix}\quad&\quad&-\sigma^{2}\\\
\quad&$0$&\quad\\\
-\sigma^{2}&\quad&\quad\end{pmatrix},\begin{pmatrix}\mathbbm{1}&\quad&\quad\\\
\quad&$0$&\quad\\\ \quad&\quad&-\mathbbm{1}\end{pmatrix}\right\\}$ (3)

To describe the quantum numbers, we embed the $SU(2)_{L}\otimes U(1)_{Y}$
gauge symmetry of the Standard Model into the global $SU(2)_{L}\otimes
SU(2)_{R}$ with $Y=R^{3}$. These generators remain unbroken in the reference
vacuum $\Sigma=\Sigma_{0}$ and the pNGB’s are fluctuations about this
background in the direction of the broken generators, $\Pi\equiv\pi^{a}X^{a}$.
Under under the gauged $SU(2)_{L}\otimes U(1)_{Y}$ they transform as
$1_{0}\oplus 2_{\pm 1/2}\oplus 3_{0}\oplus 3_{\pm 1}$ and may be parameterized
as

$\displaystyle\begin{matrix}\Sigma(x)=e^{2i\Pi/f}\Sigma_{0}\\\
\end{matrix}\qquad\qquad\Pi$ $\displaystyle=$
$\displaystyle\frac{1}{2}\begin{pmatrix}\frac{1}{\sqrt{10}}\eta+\Phi&H^{T}&\widetilde{\Phi}\\\
H^{*}&-\frac{4}{\sqrt{10}}\eta~{}\mathbbm{1}&H^{c}\\\
\widetilde{\Phi}^{\dagger}&H^{c\dagger}&-\frac{1}{\sqrt{10}}\eta+\Phi^{*}\\\
\end{pmatrix}$ (4)

All of the goldstone modes in this theory remain uneaten and appear as
physical fluctuations. These include the SM Higgs doublet $H=(H_{+}~{}H_{0})$,
a parity-odd electroweak singlet $\eta$, and a set of Georgi-Machacek scalars
$\Phi$ and $\widetilde{\Phi}$ Georgi and Machacek (1985); Hartling  _et al._
(2014) which may be written as

$\widetilde{\Phi}=\frac{1}{\sqrt{2}}\begin{pmatrix}\sqrt{2}\,\tilde{\phi}_{++}&~{}\tilde{\phi}_{+}\\\
\tilde{\phi}_{+}&~{}\sqrt{2}\,\tilde{\phi}_{0}\\\
\end{pmatrix}\qquad\qquad\Phi=\frac{1}{\sqrt{2}}\begin{pmatrix}\phi_{0}~{}&~{}\phi_{+}\\\
\phi_{+}^{*}\,&\phi_{0}\\\ \end{pmatrix}$ (5)

Under the $SU(2)_{L}\otimes SU(2)_{R}$ global symmetry the pNGBs transform as
$(2,2)\oplus(3,3)\oplus(1,1)$. The SM Higgs field transforms as a bi-doublet,
while the real and complex triplets $\Phi$ and $\widetilde{\Phi}$ transform
together as a bi-triplet. The fermionic sector of this theory gives a large
negative contribution to the mass of the Higgs, while the remaining goldstone
masses are dominated by a positive contribution from the one-loop gauge
interactions. The resulting vacuum breaks the gauge symmetry down to
$U(1)_{\rm EM}$ and can be parameterized by a single angle $\theta$.

$\displaystyle\theta^{2}\equiv\frac{1}{4}\frac{H^{\dagger}H}{f^{2}}\qquad\qquad\langle\Sigma\rangle=\begin{pmatrix}0&0&~{}~{}0~{}~{}&~{}~{}~{}1~{}~{}~{}&0\\\
0&-\sin^{2}\theta&~{}~{}\frac{i}{\sqrt{2}}\sin
2\theta~{}~{}&~{}~{}~{}0~{}~{}~{}&\cos^{2}\theta\\\ \frac{i}{\sqrt{2}}\sin
2\theta&0&~{}~{}\cos 2\theta~{}~{}&~{}~{}~{}0~{}~{}~{}&\frac{i}{\sqrt{2}}\sin
2\theta\\\ 1&0&~{}~{}0~{}~{}&~{}~{}~{}0~{}~{}~{}&0\\\
0&\cos^{2}\theta&~{}~{}\frac{i}{\sqrt{2}}\sin
2\theta~{}~{}&~{}~{}~{}0~{}~{}~{}&-\sin^{2}\theta\end{pmatrix}$ (6)

The $SU(5)$ global symmetry is explicitly broken by the $SU(2)_{L}\times
U(1)_{Y}$ gauge couplings. The covariant derivative can be expanded to second
order in the vacuum giving tree level masses to the weak gauge bosons in terms
of the $SU(2)_{L}\times U(1)_{Y}$ couplings $g,g^{\prime}$. The vacuum
expectation value of the Higgs bi-doublet thus gives mass to the $W$ and $Z$
bosons which preserves the remnant custodial $SU(2)_{c}$, thus guaranteeing
tree level relation $m_{W}/m_{Z}=\cos\theta_{w}$.

$D_{\mu}\Sigma=\partial_{\mu}\Sigma-\left[igW_{\mu}^{a}(L^{a}\Sigma+\Sigma
Q^{aT})+ig^{\prime}B_{\mu}(Y\Sigma+\Sigma Y^{T})\right]$ (7) $\displaystyle
m_{W}^{2}=\frac{g^{2}}{2}\,f^{2}\sin^{2}2\theta$ $\displaystyle
m_{Z}^{2}=\frac{g^{2}+g^{\prime 2}}{2}\,f^{2}\sin^{2}2\theta.$ (8)

###  II.1 Fermion Sector

If the $SU(5)$ symmetry arises accidentally from the dynamics of a strongly
coupled theory then the Yukawa sector could contain interactions between the
$\Sigma$ field and composite fermions in $SU(5)$ multiplets. Yukawa couplings
that softly break the global symmetry can result in potentially large
contributions to the scalar potential. However these interactions can
naturally be constructed with a collective symmetry breaking property that
guarantees the absence of quadratic divergences to the Higgs mass from fermion
loops. Despite the absence of such quadratic divergences, constructing soft-
symmetry breaking interactions that reproduce the measured properties of the
Higgs boson remains non-trivial. We find that this goal can be achieved in a
relatively simple way by extending the fermion sector of this theory to
include two vector-like multiplets of fermions in the fundamental
representation of $SU(5)$. One of the multiplets $(\psi,\,\bar{\psi})$ is
color-charged and mixes with the SM quarks resulting in a partially composite
third generation. The other multiplet $(\chi,\,\bar{\chi})$ is assumed to be
charged under a hidden-sector gauge group $G$ that confines at some scale
$\widetilde{\Lambda}<f$.

####  II.1.1 Color-charged Fermions

The most serious quadratic divergence to the Higgs potential from the top
quark can be eliminated by extending the fermion sector of the Standard Model
to include new vector-like quarks $(\psi,\,\bar{\psi})$ in the fundamental
representation of $SU(5)$. The states in this multiplet mix with the massless
chiral third generation $q$ and $(\bar{t},\bar{b})$ to produce partially
composite top and bottom mass eigenstates. The gauge quantum numbers of the
vector-like quarks are fixed by the requirement of partial compositeness and
the proposed embedding of $SU(2)_{L}\times U(1)_{Y}\subset SU(5)$. The
corresponding embedding of the gauge eigenstate components in
$(\psi,\,\bar{\psi})$ is given in Equation 9, and their transformation
properties are given in Table 1

$\psi=\left(Q~{}~{}T~{}~{}P\right)\qquad\qquad\bar{\psi}=\left(\overline{P}~{}~{}\overline{T}~{}~{}\overline{Q}\right)^{T}$
(9)

|  $\left(Q~{}\overline{Q}\right)$ |  $\left(T~{}\overline{T}\right)$ |  $\left(P~{}\overline{P}\right)$ |  $q$ |  $\bar{t}$ |  $\bar{b}$
---|---|---|---|---|---|---
$SU(3)_{c}$ | $(3~{}\overline{3})$ | $(3~{}\overline{3})$ | $(3~{}\overline{3})$ | $3$ | $\overline{3}$ | $\overline{3}$
$SU(2)_{L}$ | $(2~{}2)$ | $(\mathbbm{1}~{}\mathbbm{1})$ | $(2~{}2)$ | $2$ | $\mathbbm{1}$ | $\mathbbm{1}$
$U(1)_{Y}$ | $\left(\frac{1}{6}~{}\text{-}\frac{1}{6}\right)$ | $\left(\frac{2}{3}~{}\text{-}\frac{2}{3}\right)$ | $\left(\frac{7}{6}~{}\text{-}\frac{7}{6}\right)$ | $\frac{1}{6}$ | $\text{-}\frac{2}{3}$ | $\frac{1}{3}$
$G$ | $(\mathbbm{1}~{}\mathbbm{1})$ | $(\mathbbm{1}~{}\mathbbm{1})$ | $(\mathbbm{1}~{}\mathbbm{1})$ | $\mathbbm{1}$ | $\mathbbm{1}$ | $\mathbbm{1}$
Table 1: Gauge quantum numbers for the new vector-like quarks. These include
two new $SU(2)_{L}$ doublets $Q=(Q_{T},\,Q_{B})$ and $P=(P_{T},\,P_{B})$, in
addition to a vector-like singlet $T$. The new vector-like states mix with the
SM chiral doublet $q=(q_{T},\,q_{B})$ and singlets $(\bar{t},\,\bar{b})$.

The most general gauge invariant fermion interactions include terms that
softly break the $SU(5)$ global symmetry. For simplicity we assume that the SM
bottom quark mass term arises from the Yukawa interactions of an incomplete
$SU(5)$ multiplet. This introduces an explicit source of $SU(5)$ breaking and
a quadratic contribution to the Higgs potential from loops of bottom quarks.
However due to the relatively small value of the bottom quark Yukawa coupling,
these effects are numerically negligible compared to the leading logarithmic
contribution from the top sector. We thus express the Yukawa interactions for
the color-charged fermions as

${\cal
L}_{\psi}=\,y_{1}\,f\,\psi\,\Sigma\,\bar{\psi}+y_{2}f\,q\,\overline{Q}+y_{3}\,f\,T\,\bar{t}\,+y_{4}\,f\,s_{2\theta}\,q_{B}\,\bar{b}+{\rm
h.c.}$ (10)

| $~{}~{}\overline{T}~{}~{}$ | $~{}~{}\overline{Q}_{T}~{}~{}$ | $~{}~{}\overline{P}_{T}~{}~{}$ | $~{}~{}\bar{t}~{}~{}$
---|---|---|---|---
$T$ | $y_{1}fc_{2\theta}$ | $\frac{i}{\sqrt{2}}y_{1}fs_{2\theta}$ | $\frac{i}{\sqrt{2}}y_{1}fs_{2\theta}$ | $y_{3}f$
$Q_{T}$ | $\frac{i}{\sqrt{2}}y_{1}fs_{2\theta}$ | $y_{1}fc^{2}_{\theta}$ | $-y_{1}fs^{2}_{\theta}$ | $0$
$P_{T}$ | $\frac{i}{\sqrt{2}}y_{1}fs_{2\theta}$ | $-y_{1}fs^{2}_{\theta}$ | $y_{1}fc^{2}_{\theta}$ | $0$
$q_{T}$ | $0$ | $y_{2}f$ | $0$ | $0$

| $~{}~{}\overline{Q}_{B}~{}~{}$ | $~{}~{}\overline{P}_{B}~{}~{}$ | $~{}~{}\bar{b}~{}~{}$
---|---|---|---
$Q_{B}$ | $y_{1}fc^{2}_{\theta}$ | $-y_{1}fs^{2}_{\theta}$ | 0
$P_{B}$ | $-y_{1}fs^{2}_{\theta}$ | $y_{1}fc^{2}_{\theta}$ | $0$
$q_{B}$ | $y_{2}f$ | $0$ | $y_{4}s_{2\theta}$

Table 2: Mass matrices for the top-like quarks $M_{T}$ (left) and the bottom-
like quarks $M_{B}$ (right) in the gauge eigenbasis.

The top and bottom quark mass matrices $M_{T}$, $M_{B}$ can be expressed in
terms of the Higgs vacuum expectation value $\theta$ defined in Equation (6),
and the UV-insensitivity of the Yukawa interactions is guaranteed by the
following identity

$\displaystyle\frac{\partial}{\partial\theta}\operatorname{tr}M_{T}^{\dagger}M_{T}$
$\displaystyle=\frac{\partial}{\partial\theta}\operatorname{tr}(M_{T}^{\dagger}M_{T})^{2}=0.$
(11)

Diagonalizing $M_{T}$ and $M_{B}$ gives seven mass eigenstates. The up-type
sector contains four charge $\pm 2/3$ states which we label in order of
descending mass as $t^{\prime\prime\prime}$, $t^{\prime\prime}$, $t^{\prime}$,
$t$, and the lightest of these states corresponds to the SM top quark. The
down-type sector contains three mass eigenstates. Two of these $b^{\prime}$,
$b$ have charge $\mp 1/3$ and include the SM bottom quark, and there is
additionally a “peculiar” charge $\pm 5/3$ quark $p^{\prime}$, which can lead
to interesting phenomenological signatures. The heavy mass eigenstates are
independent of the Higgs VEV at leading order, and the numerically
diagonalized mass spectrum is shown in Figure (1). Their analytic expressions
have been computed in Appendix B to leading order in $\sin^{2}2\theta$ and are
approximated by the following scalings.

$\begin{aligned} m_{t}^{2}&\sim\frac{f^{2}}{2}y_{t}^{2}\sin^{2}2\theta\\\
m_{t^{\prime}}^{2}&\sim f^{2}y_{1}^{2}\\\ m_{t^{\prime\prime}}^{2}&\sim
f^{2}\left(y_{1}^{2}+y_{2}^{2}\right)\\\ m_{t^{\prime\prime\prime}}^{2}&\sim
f^{2}\left(y_{1}^{2}+y_{3}^{2}\right)\end{aligned}\qquad\qquad\qquad\begin{aligned}
m_{b}^{2}&\sim\frac{f^{2}}{2}y_{b}^{2}\sin^{2}2\theta\\\
m_{p^{\prime}}^{2}&\sim f^{2}y_{1}^{2}\\\ m_{b^{\prime}}^{2}&\sim
f^{2}\left(y_{1}^{2}+y_{2}^{2}\right)\\\ &\end{aligned}$ (12)

The top-sector mass matrix $M_{T}$ also has the property that $\det
M_{T}^{\dagger}M_{T}\propto\sin^{2}2\theta$, thus producing one massless state
when $\theta=n\pi/2$ for $n\in\mathbb{Z}$. The states corresponding to the SM
top and bottom quarks $t$ and $b$ remain massless at zeroth order in $\theta$,
and couple to the Higgs via the following Yukawa couplings

$y_{t}=\frac{y_{1}y_{2}y_{3}}{\sqrt{y_{1}^{2}+y_{2}^{2}}\sqrt{y_{1}^{2}+y_{3}^{2}}}\qquad\qquad
y_{b}=\frac{y_{1}y_{4}}{\sqrt{y_{1}^{2}+y_{2}^{2}}}$ (13)

####  II.1.2 Hidden-Sector Fermions

A realistic pattern of EWSB via radiative effects can be achieved throughout
this parameter space by introducing an additional multiplet of vector-like
fermions $(\chi,\,\bar{\chi})$. In this analysis we will assume that
$(\chi,\,\bar{\chi})$ are charged under a hidden-sector gauge group $G$,
transforming in the
$(\Box,5)\oplus(\overline{\mbox{$\Box$}\raisebox{7.11317pt}{}},5)$
representation of $G\times SU(5)$. If the hidden-sector gauge group $G$
confines at some scale $\widetilde{\Lambda}<f$, then the low energy spectrum
will be comprised of hidden-sector mesons, baryons, and glueballs. Due to a
conserved $U(1)_{B}$ in the hidden-sector, the baryons can be stable on
cosmological time scales and may thus be an attractive candidate for dark
matter if the lightest baryon is electrically neutral. The direct and indirect
detection signatures for these scenarios have been well studied in the context
of asymmetric dark matter Kaplan  _et al._ (2009); Laha and Braaten (2014);
Laha (2015); Mitridate  _et al._ (2017); Forestell  _et al._ (2018); Elor  _et
al._ (2018); Braaten  _et al._ (2018) and will not be discussed further. Here
we choose a hypercharge assignment for the hidden-sector fermions that can
naturally accomodate a number of low-lying neutral mass eigenstates. The
hidden-sector baryons in this framework thus become a plausible candidate for
dark matter, although in general this will depend on the discrete
$\mathbb{Z}_{N}$ subgroup at the center of $G$ and the mass splitting
interactions of the hidden baryon spectrum. The embedding of the gauge
eigenstate components in $(\chi,\,\bar{\chi})$ is given in Equation 14 and the
gauge quantum numbers of these new states are given in Table 3.

$\chi=\left(X~{}~{}N~{}~{}Y\right)\qquad\qquad\bar{\chi}=\left(\overline{Y}~{}~{}\overline{N}~{}~{}\overline{X}\right)^{T}$
(14)

|  $\left(X~{}\overline{X}\right)$ |  $\left(N~{}\overline{N}\right)$ |  $\left(Y~{}\overline{Y}\right)$ |  $\left(n~{}\bar{n}\right)$
---|---|---|---|---
$SU(3)_{c}$ | $(\mathbbm{1}~{}\mathbbm{1})$ | $(\mathbbm{1}~{}\mathbbm{1})$ | $(\mathbbm{1}~{}\mathbbm{1})$ | $(\mathbbm{1}~{}\mathbbm{1})$
$SU(2)_{L}$ | $(2~{}2)$ | $(\mathbbm{1}~{}\mathbbm{1})$ | $(2~{}2)$ | $(\mathbbm{1}~{}\mathbbm{1})$
$U(1)_{Y}$ | $\left(\frac{1}{2}~{}\text{-}\frac{1}{2}\right)$ | $\left(0~{}0\right)$ | $\left(\text{-}\frac{1}{2}~{}\frac{1}{2}\right)$ | $\left(0~{}0\right)$
$G$ | $(\Box~{}\overline{\mbox{$\Box$}\raisebox{6.40186pt}{}})$ | $(\Box~{}\overline{\mbox{$\Box$}\raisebox{6.40186pt}{}})$ | $(\Box~{}\overline{\mbox{$\Box$}\raisebox{6.40186pt}{}})$ | $(\Box~{}\overline{\mbox{$\Box$}\raisebox{6.40186pt}{}})$
Table 3: Gauge quantum numbers for the new vector-like hidden-sector fermions.
These include two $SU(2)_{L}$ doublets $X=(X_{+}~{}~{}X_{0})$ and
$Y=(Y_{-}~{}~{}Y_{0})$, as well as two singlets $N$ and $n$.

A realistic Higgs potential from one-loop fermion corrections can be obtained
by introducing symmetry breaking interactions between the hidden-sector
$SU(5)$ multiplet $(\chi,\,\bar{\chi})$ and an additional massless vector-like
pair of hidden-fermions $(n,\,\bar{n})$. The Yukawa interactions thus include
an $SU(5)$ symmetric coupling to the $\Sigma$-field in addition to terms that
softly break the global symmetry

${\cal
L}_{\chi}=\tilde{y}_{1}\,f\,\chi\,\Sigma\,\bar{\chi}+\tilde{y}_{2}fn\,\overline{N}+\tilde{y}_{3}fN\,\overline{n}+{\rm
h.c.}$ (15)

| $~{}~{}\overline{N}~{}~{}$ | $~{}~{}\overline{X}_{0}~{}~{}$ | $~{}~{}\overline{Y}_{0}~{}~{}$ | $~{}~{}\bar{n}~{}~{}$
---|---|---|---|---
$N$ | $\tilde{y}_{1}fc_{2\theta}$ | $\frac{i}{\sqrt{2}}\tilde{y}_{1}fs_{2\theta}$ | $\frac{i}{\sqrt{2}}\tilde{y}_{1}fs_{2\theta}$ | $\tilde{y}_{3}f$
$X_{0}$ | $\frac{i}{\sqrt{2}}\tilde{y}_{1}fs_{2\theta}$ | $\tilde{y}_{1}fc^{2}_{\theta}$ | $-\tilde{y}_{1}fs^{2}_{\theta}$ | 0
$Y_{0}$ | $\frac{i}{\sqrt{2}}\tilde{y}_{1}fs_{2\theta}$ | $-\tilde{y}_{1}fs^{2}_{\theta}$ | $\tilde{y}_{1}fc^{2}_{\theta}$ | $0$
$n$ | $\tilde{y}_{2}f$ | $0$ | $0$ | $0$

| $~{}~{}\overline{X}_{+}~{}~{}$ | $~{}~{}\overline{Y}_{-}~{}~{}$
---|---|---
$X_{+}$ | $\tilde{y}_{1}fc^{2}_{\theta}$ | $-\tilde{y}_{1}fs^{2}_{\theta}$
$Y_{-}$ | $-\tilde{y}_{1}fs^{2}_{\theta}$ | $\tilde{y}_{1}fc^{2}_{\theta}$

Table 4: Mass matrix for the neutral hidden-sector fermions $M_{N}$ (left) and
the charged hidden-sector fermions $M_{C}$ (right) in the gauge eigenbasis.

In this scenario the mass matrix for the neutral hidden-sector states $M_{N}$
has the property $\det M_{N}^{\dagger}M_{N}\propto\cos^{2}2\theta$, resulting
in one massless particle when $\theta=(n+1/2)\pi/2$ for $n\in\mathbb{Z}$. The
absence of a quadratic senstivity to the Higgs mass from loops of hidden-
sector fermions is guaranteed by the identity

$\displaystyle\frac{\partial}{\partial\theta}\operatorname{tr}M_{N}^{\dagger}M_{N}$
$\displaystyle=\frac{\partial}{\partial\theta}\operatorname{tr}(M_{N}^{\dagger}M_{N})^{2}=0$
(16)

The hidden-sector mass matrix for $M_{N}$ produces four electrically neutral
mass eigenstates which we label in in order of descending mass as
$n^{\prime\prime\prime}$, $n^{\prime\prime}$, $n^{\prime}$, $n$. The lightest
of these states is generically comparable in mass to the heaviest goldstone
modes, and thus stable against pair production via resonant scalar decay. The
down-type hidden-sector mass eigenstates include a charge $\pm 1$ vector-like
pair $(x_{+},\,\bar{x}_{+})$ and a charge $\mp 1$ vector-like pair
$(y_{-},\,\bar{y}_{-})$. Since these hidden-sector states are color-neutral,
their relatively low production cross sections make all but the lightest state
$n$ inconsequential for collider phenomenology. For the remainder of this
section we thus consider a simplified parameter space for this sector in which
$\tilde{y}_{3}=\tilde{y}_{2}=\tilde{y}_{1}$. Analytic expressions for general
couplings have been derived in Appendix B to leading order in
$\sin^{2}2\theta$. With this simplified parameter space the hidden-sector
fermion masses scale approximately as

$\begin{aligned}
m_{n}^{2}&\sim\frac{f^{2}}{4}\tilde{y}_{1}^{2}\cos^{2}2\theta\\\\[4.30554pt]
m_{n^{\prime}}^{2}&\sim f^{2}\tilde{y}_{1}^{2}\\\\[4.30554pt]
m_{n^{\prime\prime}}^{2}&\sim m_{n^{\prime\prime\prime}}^{2}\sim
2f^{2}\tilde{y}_{1}^{2}\\\ \end{aligned}\qquad\qquad\qquad\begin{aligned} &\\\
m_{y_{-}}^{2}&\sim m_{x_{+}}^{2}\sim f^{2}\tilde{y}_{1}^{2}\\\ &\end{aligned}$
(17)

####  II.1.3 Theory Space

The fermion interactions introduce a number of new parameters to the Standard
Model. The new color-charged interactions introduce four Yukawa couplings
$y_{i}$, two of which are fixed by the Standard Model top and bottom quark
masses as implied by Equation 13. This leaves two free parameters $y_{1}$ and
$y_{2}$ which are bound from below by the top-quark Yukawa coupling
$y_{t}<y_{1}y_{2}/\sqrt{y_{1}^{2}+y_{2}^{2}}$. The large observed top quark
mass in this framework thus descends from the $\mathcal{O}(1)$ values of the
composite sector Yukawa couplings. The new hidden-sector interactions
introduce three new Yukawa couplings which we have reduced to a single
parameter by assuming $\tilde{y}_{1}=\tilde{y}_{2}=\tilde{y}_{3}$. The fermion
content of this theory has been chosen to allow for a pattern of electroweak
symmetry breaking that can be tuned to agree with the measured properties of
the Higgs boson. In this simplified parameterization, for each point in the
two-dimensional parameter space of $(y_{1},\,y_{2})$ there exists a unique
value for $\tilde{y}_{1}$ which gives a Higgs mass and quartic that is
consistent with Standard Model predictions. The conditions of EWSB thus
generically imply a mass scale for the new hidden-fermions that is
$\mathcal{O}({\rm few})$ times lower than the masses of the new color-charged
states. The masses of the new fermions as a function of the chiral symmetry
breaking scale $f$ are illustrated for two benchmark points in Figure 1.

Figure 1: Mass spectrum for the new fermions at two benchmark points. One
corresponding to $y_{2}/y_{3}=2/2$ (left) and $y_{2}/y_{3}=3/\frac{3}{2}$ GeV
(right). The hidden-fermion masses are fixed relative to the color-charged
states by the Higgs mass.

These Yukawa interactions also introduce additional sources of $SU(2)_{c}$
violation which can generate large contributions to the $T$-parameter at loop-
level. These contributions will vanish as the chiral symmetry breaking scale
$f$ increases due to the decoupling of these vector-like states, but can
become unacceptably large at low values of $f$. The $T$-parameter may be
computed from the fermionic contributions to the $W$ and $Z$-boson self-
energies $\Pi_{WW}$ and $\Pi_{ZZ}$ as defined in Equation 18. We find that
these $SU(2)_{c}$ violating interactions provide unacceptable contributions to
the $T$-parameter when $f$ becomes on the order of $600$ GeV, as illustrated
in Figure 2.

$\displaystyle\alpha(m_{Z})\,T$
$\displaystyle=\frac{\Pi_{WW}(0)}{m_{W}^{2}}-\frac{\Pi_{ZZ}(0)}{m_{Z}^{2}}$
(18)

Figure 2: Constraints from the $T$-parameter at $f=600$ GeV (left), $800$ GeV
(middle), and $1000$ GeV (right). The light/medium/dark gray correspond to the
$1\sigma$/$2\sigma$/$3\sigma$ deviations from experimentally measured values.
The black corresponds to regions inconsistent with the top quark mass.

###  II.2 Electroweak Symmetry Breaking

The scalar effective potential in this theory is sculpted by the radiative
corrections from $SU(5)$ breaking interactions at $f$, which may be extracted
from the quadratically and logarithmically sensitive terms in the Coleman
Weinberg potential. The absence of additional gauge structure above the chiral
symmetry breaking scale implies that the contribution from $W$ and $Z$ loops
is quadratically sensitive to the compositeness scale $f$, which we take to be
$\Lambda\sim 4\pi f$. The form of this gauge contribution $\delta V_{G}$ can
be extracted from the covariant derivative and expressed in terms of the gauge
boson mass matrix $M_{V}$

$\displaystyle\delta V_{G}(c,\theta)$
$\displaystyle=f^{2}\operatorname{tr}M_{V}^{2}(\Sigma)$ (19)
$\displaystyle=f^{4}c\left(g^{2}\operatorname{tr}\left[L^{a}\Sigma(L^{a}\Sigma)^{*}\right]+{g^{\prime}}^{2}\operatorname{tr}\left[Y\Sigma(Y\Sigma)^{*}\right]\right)$
(20) $\displaystyle=-f^{2}f_{G}^{2}\,\cos 2\theta$ (21) $\displaystyle
f_{G}^{2}$ $\displaystyle=\frac{f^{2}c}{2}\left(3g^{2}+g^{\prime 2}\right)$
(22)

Here $c$ is a UV sensitive constant which parameterizes the leading gauge
contribution to the effective potential for the pNGBs. Dimensional analysis
gives $c$ of order 1, while a technicolor-like UV completion of the model
requires $c$ to be positive, that is, the gauge contribution to vacuum
alignment prefers vanishing gauge boson masses. The analogous term in the
effective low energy description of QCD gives the (positive) mass squared
splitting between the $\pi^{+}$ and $\pi^{0}$ Piai  _et al._ (2004).

The absence of quadratically divergent one-loop contributions from the Yukawa
interactions follows from the properties of the color-charged and hidden-
sector fermion mass matrices in Equation 11 and Equation 16. The most
significant contribution to the scalar potential from fermions $\delta V_{F}$
in this effective theory thus comes from the logarithmically sensitive term.
The above relations again guarantee that the $\Lambda$ dependent terms cancel
order by order in $\theta$. The result is thus sensitive only to scales of
$\mathcal{O}(f)$, and can be written in terms of ratios of mass eigenstates.

$\displaystyle\delta V_{F}(\theta)$
$\displaystyle=-\frac{3}{16\pi^{2}}\operatorname{tr}(M_{F}^{\dagger}M_{F})^{2}\log\frac{M_{F}^{\dagger}M_{F}}{\Lambda^{2}}$
(23)
$\displaystyle=-\frac{3}{16\pi^{2}}\displaystyle\sum_{F}|m_{F}^{2}(\theta)|^{2}\log|m_{F}^{2}(\theta)|$
(24)

The collective symmetry breaking property of these mass matrices guarantees
that these contributions to the effective potential are extremely well
approximated by their leading order terms in $\det M_{F}^{\dagger}M_{F}$. As
argued in Appendix B, higher order terms in $\det M_{F}^{\dagger}M_{F}$ are
negligible due to corresponding suppression by powers of
$(\operatorname{tr}M_{F}^{\dagger}M_{F})^{4}$. The most significant
logarithmic contributions from the color-charged top-like sector $\delta
V_{T}$ and the neutral hidden-sector $\delta V_{N}$ thus enter with opposite
sign, and the $\theta$ dependence is numerically well approximated by $\delta
V_{F}\propto\pm\cos 4\theta$. The dominant contribution to the Higgs potential
from fermions may thus be parameterized by two scales $f_{T}$ and $f_{N}$,
which are functions of the color-charged and hidden-sector Yukawa couplings
respectively.

$\displaystyle\delta V_{F}(\theta)$ $\displaystyle=\delta
V_{T}(y_{i},\theta)+\delta V_{N}(\tilde{y}_{i},\theta)$ (25)
$\displaystyle\approx f^{2}\left(f_{T}^{2}-f_{N}^{2}\right)\cos 4\theta$ (26)
$f_{T}^{2}=\frac{3f^{2}}{32\pi^{2}}\frac{y_{1}^{2}y_{2}^{2}y_{3}^{2}}{y_{2}^{2}-y_{3}^{2}}\log\frac{y_{1}^{2}+y_{2}^{2}}{y_{1}^{2}+y_{3}^{2}}\qquad\qquad
f_{N}^{2}=\frac{3f^{2}}{32\pi^{2}}\tilde{y}_{1}^{4}$ (27)

In this simplified parameter space, the hidden-sector fermion mass spectrum is
fixed by the masses of the color-charged fermions and the conditions of
electroweak symmetry breaking. The $\tilde{y}_{i}$ dependence of this
potential for arbitrary hidden-sector Yukawa couplings are derived in Appendix
B.

Since all of the Yukawa and gauge interactions are invariant under the axial
$U(1)_{a}\subset SU(5)$, the goldstone excitation $\eta$ in this direction is
left massless by the contributions considered thus far. In order to give a
mass to this gauge-singlet large enough to avoid constraints from hadronic
physics, we introduce an explicit source of $U(1)_{a}$ violation in the form
of a spurion term that proportional to the reference vacuum $M_{0}\equiv
m_{0}\Sigma_{0}$. Such a contribution would descend naturally from fermion
mass terms in the UV completion.

$\displaystyle\delta V_{0}(m_{0},\theta)$
$\displaystyle=-f^{3}\operatorname{tr}[M_{0}\Sigma]$ (28)
$\displaystyle=-f^{2}f_{0}^{2}\,\cos 2\theta$ (29) $\displaystyle f_{0}^{2}$
$\displaystyle=2\,fm_{0}$ (30)

In the absence of additional symmetry breaking terms, the scalar potential is
fully parameterized by the Yukawa coupings $y_{i}$ and $\tilde{y}_{i}$, the
1-loop gauge coefficient $c$, and spurion coefficient $m_{0}$. This potential
has a generic EWSB vacuum when $f_{T}^{2}-f_{N}^{2}\gg f_{0}^{2}+f_{G}^{2}>0$
and can be expressed as

$\displaystyle V_{\rm H}(\theta)$
$\displaystyle=-f^{2}\left(f_{0}^{2}+f_{G}^{2}\right)\cos
2\theta+f^{2}\left(f_{T}^{2}-f_{N}^{2}\right)\cos 4\theta$ (31)
$\displaystyle\langle\theta^{2}\rangle$ $\displaystyle=\frac{\langle
H^{\dagger}H\rangle}{4f^{2}}=\frac{3}{2}\left[\frac{4(f_{T}^{2}-f_{N}^{2})-\left(f_{0}^{2}+f_{G}^{2}\right)}{16(f_{T}^{2}-f_{N}^{2})-\left(f_{0}^{2}+f_{G}^{2}\right)}\right]$
(32)

The order parameter $\langle\theta^{2}\rangle$ is assumed to develop a vacuum
expectation value at $v^{2}=4f^{2}\langle\theta^{2}\rangle=(246\text{
GeV})^{2}$. The observed Higgs mass $m_{h}=126$ GeV thus fixes the relation
between the color-charged and hidden sector Yukawa couplings, as well as
between the gauge renormalization coefficient $c$ and the axial-breaking
spurion coefficient $m_{0}$.

$\displaystyle V_{\rm H}(\theta)\xrightarrow{H\rightarrow v+h}$
$\displaystyle\quad\frac{m_{h}^{2}}{2}h^{2}+\frac{\lambda}{4}h^{4}+\mathcal{O}(h^{6})$
(33) $\displaystyle f_{T}^{2}-f_{N}^{2}$
$\displaystyle=\frac{m_{h}^{2}}{24}\left(\frac{3}{2}\frac{1}{\langle\theta^{2}\rangle}-1\right)$
(34) $\displaystyle f_{0}^{2}+f_{G}^{2}$
$\displaystyle=\frac{2m_{h}^{2}}{3}\left(\frac{3}{8}\frac{1}{\langle\theta^{2}\rangle}-1\right)$
(35)

The mass spectrum of the non-SM scalars can also be extracted from these
higher dimensional operators. The fermionic contributions to the potential of
the scalars $\widetilde{\Phi}$ and $\Phi$ are suppressed by
$\mathcal{O}(\theta^{2})$. Their masses are thus roughly independent of the
Yukawa couplings $y_{i}$ and dominated by the positive contribution from gauge
interactions.

$m_{\widetilde{\Phi}}^{2}\sim 4f^{2}\left[c\left(g^{2}+\frac{g^{\prime
2}}{2}\right)+\frac{m_{0}}{f}\right]\qquad\qquad m_{\Phi}^{2}\sim
4f^{2}\left[c\,g^{2}+\frac{m_{0}}{f}\right]$ (36) $m_{\eta}^{2}\sim 4fm_{0}$
(37)

The mass of the gauge-singlet $\eta$ is simply proportional to the coefficient
of the spurion operator $m_{0}$, which is a function of the Higgs mass $m_{h}$
and the gauge renormalization coefficient $c$. The mass splitting between the
Georgi-Machacek scalars $\widetilde{\Phi}/\Phi$ and the gauge-singlet $\eta$
thus grows as a function of $c$, with $\eta$ developing a vacuum expectation
value at $c\sim 0.35$. These features of the scalar mass spectrum are shown in
Figure (3).

Figure 3: Mass spectrum for the goldstone scalars as a function of the
renormalization constant $c$ for $f=600$ GeV (left), $f=800$ GeV (middle), and
$f=1000$ GeV (right)

###  II.3 Phenomenology

In this $SU(5)/SO(5)$ model the spectrum of new states contains strongly
interacting quarks with masses that scale as $M_{T}\sim y_{i}f$, hidden-sector
fermions with masses that scale as $M_{N}\sim\tilde{y}_{i}f$, and weakly
interacting scalars with masses that scale as $M_{S}\sim gf$. At $\sqrt{s}=13$
TeV the neutral spin-0 states are singly produced via gluon fusion with
$\mathcal{O}({\rm pb})$ cross sections, and there is also zoo of new particle
states that are pair produced with $\mathcal{O}({\rm fb})$ cross sections.
These include Drell-Yan production of the charged goldstone scalars, strong
production of the lightest quark partners
$t^{\prime}\bar{t}^{\prime}/p^{\prime}\bar{p}^{\prime}$, as well as weak
production of the lightest hidden-fermions $n\bar{n}$. The lightest hidden-
sector states will be a spectrum of glueballs which could be long-lived, and
produced via $n\bar{n}$ annihilation or through Higgs decay. The direct
production cross section for these states at two benchmark points are shown in
Figure 4.

Figure 4: $\sqrt{s}=13$ TeV cross sections for new particle states in their
dominant production modes at $f=800$ GeV. Cross sections are shown as a
function of the gauge renormalization coefficient $c$ at two benchmark points
corresponding to $y_{2}=y_{3}=2$ (left) and $y_{2}=3$, $y_{3}=3/2$ (right).

Despite the abundance of new particle states in this theory, their production
cross sections and dominant decay paths would make them extremely difficult to
resolve in current LHC data sets. Drell-Yan production of the doubly charged
scalar $\tilde{\phi}_{++}$ is appreciable in large parts of the parameter
space due to the $Q_{++}^{2}$ enhancement to its cross section relative to the
other charged states. However, existing limits on this process typically
assume an $\mathcal{O}(1)$ branching fraction to pairs of same sign leptons
Aad  _et al._ (2015); Collaboration (2017) rendering them inapplicable to the
goldstone scalars in this model, which are leptophobic by assumption. Pairs of
the lightest top quark partner $t^{\prime}\bar{t}^{\prime}$ and peculiar quark
$p^{\prime}\bar{p}^{\prime}$ can also be strongly produced with cross sections
as high as $\sim 5$ fb in the viable regions of parameter space. The strongest
limits on vector-like quarks are set by the CMS search at $\sqrt{s}=$13 TeV in
the single lepton channel Khachatryan  _et al._ (2016a); Sirunyan  _et al._
(2017). These limits are driven by kinematically optimized searches for
$t\bar{t}+X$ and $b\bar{b}+X$ where $X=W,Z,h$ is highly boosted, and are
interpreted in the context of simplified models that assume ${\rm
BR}(t^{\prime}\shortrightarrow Z\,t)+{\rm BR}(t^{\prime}\shortrightarrow
h\,t)+{\rm BR}(t^{\prime}\shortrightarrow W\,b)=1$. In this model the lowest
lying vector-like partners are composed mostly of the hypercharge-$7/6$
doublet $(P_{T},P_{B})$ and their branching fractions are dominated by decays
to top and bottom quarks in association with the scalar states in
$\widetilde{\Phi}$. These generic experimental signatures are thus
qualitatively different from those of simplified models of top and bottom
partners, rendering existing limits on vector-like quarks largely
inapplicable.

####  II.3.1 Resonant $t\bar{t}$ Final States

Gluon fusion of neutral pseudo-scalars $\phi_{0}$, $\eta$ and the complex
scalar $\tilde{\phi}_{0}$ generically leads to a resonant $t\bar{t}$ final
state. The CP properties of the pNGB matrix forbids the two-body decays of the
goldstone scalars into pairs of gauge and Higgs bosons and their branching
fractions are thus dominated by decays to the third generation quarks whenever
these channels are kinematically open. In the bulk of this parameter space we
find ${\rm BR}(\tilde{\phi}_{0}/\phi_{0}/\eta\shortrightarrow
t\bar{t})\sim{\rm BR}(\tilde{\phi}_{+}/\phi_{+}\shortrightarrow t\bar{b})\sim
1$, although the gauge singlet $\eta$ can also have a significant branching
fraction to $gg$ and $b\bar{b}$ when its mass is below the top threshold as
computed in Section III.3. Currently the strongest limit on the process
$gg\rightarrow\tilde{\phi}_{0}/\phi_{0}\rightarrow t\bar{t}$ comes from the
ATLAS search at $\sqrt{s}=8$ TeV in the single lepton channel Aaboud  _et al._
(2017a). The gluon fusion process is mediated by loops of heavy vector-like
quarks. Their effects may thus be parameterized by a set of dimension-5
couplings $c^{S}_{gg}$, which can be expressed as a sum over the well known
gluon vertex function for the fermion triangle graph $V^{S}_{gg}$

Figure 5: Interfering diagrams for $t\bar{t}$ production via spin-0 and spin-1
mediators (left). Invariant mass distributions for the $t\bar{t}$ pairs
(right). The contribution from the spin-0 mediator (S) is shown in blue and
the combined contribution (S+I) is shown in red.

$\mathcal{L}\supset\displaystyle\sum_{S\in{\tilde{\phi},\phi,\eta}}-\frac{1}{4}c^{S}_{gg}\,S\,G^{\mu\nu}\widetilde{G}_{\mu\nu}$
(38)
$c^{S}_{gg}=\displaystyle\sum_{t^{\prime\prime\prime}t^{\prime\prime}t}V^{S}_{gg}(i)\equiv\frac{\alpha_{S}}{\sqrt{8}\pi}\displaystyle\sum_{t^{\prime\prime\prime}t^{\prime\prime}t}y^{S}_{ii}\,\frac{\tau_{i}f(\tau_{i})}{m_{i}}\\\
$ (39)

Searches for scalar resonances decaying to $t\bar{t}$ pairs is complicated by
destructive interference between the spin-0 and spin-1 mediated s-channel
diagrams in Figure 5. This results in a “peak-dip” structure for the invariant
mass distribution of the $t\bar{t}$ pairs rather than the usual Breit-Wigner
resonance. The analysis performed by ATLAS searches for this kinematic feature
in resolved $t\bar{t}$ pairs at invariant masses in the range of
$m_{t\bar{t}}=500-800$ GeV111Invariant masses below $500$ GeV are not
considered due to the importance of higher order corrections involving Higgs
decays to virtual top quarks in that regime.. The results are interpreted in
the context of a Two-Higgs-Doublet model (2HDM) and the cuts are kinematically
optimized for the case of a scalar, a pseudo-scalar, and the case of a
degenerate scalar-pseudoscalar pair. The limits are reported as a function of
the 2HDM parameter $\tan\beta$ in the alignment limit where the $W/Z$
couplings are equal to their SM values. In this limit, $\tan\beta$ can be
interpreted as the ratio of the top quark Yukawa coupling to the SM Higgs and
heavy Higgs bosons, $\tan\beta\rightarrow y^{H}_{tt}/y^{\widetilde{H}}_{tt}$.
Here $H$ and $\widetilde{H}$ refer to the $SU(2)_{L}$ doublets containing the
SM and heavy Higgses respectively.

The strength of this kinematic feature is a function of ratios of the
amplitudes for the two processes, which include factors involving the vertex
function $V^{H}_{gg}$. For each scalar mass eigenstate $S$ we can thus define
an analagous angle $\beta^{S}_{\rm eff}$ by replacing the vertex function in
the numerator with its analogous dimension-5 coupling as shown in Equation 41.
If the vector-like quark masses are heavy and decoupled, the contribution to
the gluon fusion vertex function is dominated by the top quark loop. In this
regime the $\mathcal{O}(1)$ top-quark Yukawa coupling implies the approximate
relation $\cot\beta^{S}_{\rm eff}\sim y_{tt}^{S}/y_{tt}^{H}$.

Figure 6: Expected and observed limits from CMS searches for $t\bar{t}$
resonances mediated by a pseudo-scalar (left) and a degenerate scalar/pseudo-
scalar (right), as a function of $\cot\beta_{\rm eff}$. Maximum values for
$\cot\beta_{\rm eff}$ are also shown.

$\displaystyle\cot^{2}\beta$
$\displaystyle=\bigg{(}\frac{y^{\widetilde{H}}_{tt}}{y^{H}_{tt}}\bigg{)}^{2}=\frac{y^{\widetilde{H}}_{tt}\,V^{\widetilde{H}}_{gg}(t)}{y^{H}_{tt}\,V^{H}_{gg}(t)}$
(40) $\displaystyle\cot\beta^{S}_{\rm eff}$
$\displaystyle\equiv\sqrt{\frac{y^{S}_{tt}}{y^{H}_{tt}}\frac{c^{S}_{gg}}{V^{H}_{gg}(t)}}\xrightarrow{\langle\theta\rangle\rightarrow
0}\frac{y^{S}_{tt}}{y^{H}_{tt}}$ (41)

The ATLAS limits are strongest in the mass range of $\mathcal{O}(500\,{\rm
GeV})$, but are insensitive to new scalars with Yukawa couplings to the top
quark that are below $\mathcal{O}(1)$. These limits have been reparameterized
as a function of $\cot\beta^{S}_{\rm eff}$ in Figure 6, along with the
translated $2\sigma$ ATLAS limits at $\sqrt{s}=8$ TeV. The masses of the
Georgi-Machacek scalars $\phi_{0}$ and $\tilde{\phi}_{0}$ and the gauge-
singlet $\eta$ can be varied as a function of the spurion coefficient $m_{0}$
throughout the mass range of interest, and are roughly independent of the
magnitude of their Yukawa couplings. Throughout the parameter space of this
model we find that $\cot\beta^{\phi_{0}}_{\rm
eff}<\cot\beta^{\tilde{\phi}_{0}}_{\rm eff}\ll 1$. The ability to resolve
these neutral scalars are thus well beyond the reach of near-term LHC limits
in this channel.

####  II.3.2 Multi-top Final States

Pair production of doubly-charged scalars and heavy vector-like quarks
democratically provide an $\mathcal{O}({\rm fb})$ contribution to
$\sigma(pp\shortrightarrow t\bar{t}t\bar{t})$ via final states that include
$(Wb)^{n}$ with $n\geq 4$. Despite the kinematic suppression of these
production modes, the abundance of new particle states can lead to a
measurable multi-top signature when combined over all processes. An anomalous
production of final states with a high multiplicity top and bottom quarks is
thus a general expectation for this class of theories. The most sensitive
searches for these final states are driven by multi-lepton analyses that look
for anomalously high b-jet multiplicities within samples containing three or
more leptons, or two leptons of the same sign Sirunyan  _et al._ (2018a). The
dominant contributions to the multi-top final state come from diagrams such as
those in Figure 7.

Figure 7: Dominant contributions to the final state $(Wb)^{n}$ for $n\geq 4$.
These include Drell-Yan production of charged scalars (left) as well as strong
production of the lightest top partner (middle) and peculiar quark (right).

In this model Drell-Yan production of the charged scalars are mediated by
current interactions that can be extracted from the covariant derivative as
shown in Equation 158. The form of these couplings agree with those computed
in the low energy effective theory for Georgi-Machacek scalars Hartling  _et
al._ (2014) in the limit where only the Higgs gets a vacuum expectation value.

Figure 8: Branching fractions of the lightest top-quark partner (left) and the
peculiar bottom-quark partner (right) as a function of $y_{1}$ at a benchmark
points corresponding to $f=800$ GeV and $y_{2}=2$

$\displaystyle J_{\rm EM}^{\mu}$
$\displaystyle=gs_{w}\left(\phi_{+}\mathchoice{\vbox{\halign{#\cr\leftrightarrowfill@{\scriptstyle}\crcr\nointerlineskip\cr$\hfil\displaystyle\partial\hfil$\crcr}}}{\vbox{\halign{#\cr\leftrightarrowfill@{\scriptstyle}\crcr\nointerlineskip\cr$\hfil\textstyle\partial\hfil$\crcr}}}{\vbox{\halign{#\cr\leftrightarrowfill@{\scriptscriptstyle}\crcr\nointerlineskip\cr$\hfil\scriptstyle\partial\hfil$\crcr}}}{\vbox{\halign{#\cr\leftrightarrowfill@{\scriptscriptstyle}\crcr\nointerlineskip\cr$\hfil\scriptscriptstyle\partial\hfil$\crcr}}}^{\mu}\phi_{-}+\tilde{\phi}_{+}\mathchoice{\vbox{\halign{#\cr\leftrightarrowfill@{\scriptstyle}\crcr\nointerlineskip\cr$\hfil\displaystyle\partial\hfil$\crcr}}}{\vbox{\halign{#\cr\leftrightarrowfill@{\scriptstyle}\crcr\nointerlineskip\cr$\hfil\textstyle\partial\hfil$\crcr}}}{\vbox{\halign{#\cr\leftrightarrowfill@{\scriptscriptstyle}\crcr\nointerlineskip\cr$\hfil\scriptstyle\partial\hfil$\crcr}}}{\vbox{\halign{#\cr\leftrightarrowfill@{\scriptscriptstyle}\crcr\nointerlineskip\cr$\hfil\scriptscriptstyle\partial\hfil$\crcr}}}^{\mu}\tilde{\phi}_{-}+2\,\tilde{\phi}_{++}\mathchoice{\vbox{\halign{#\cr\leftrightarrowfill@{\scriptstyle}\crcr\nointerlineskip\cr$\hfil\displaystyle\partial\hfil$\crcr}}}{\vbox{\halign{#\cr\leftrightarrowfill@{\scriptstyle}\crcr\nointerlineskip\cr$\hfil\textstyle\partial\hfil$\crcr}}}{\vbox{\halign{#\cr\leftrightarrowfill@{\scriptscriptstyle}\crcr\nointerlineskip\cr$\hfil\scriptstyle\partial\hfil$\crcr}}}{\vbox{\halign{#\cr\leftrightarrowfill@{\scriptscriptstyle}\crcr\nointerlineskip\cr$\hfil\scriptscriptstyle\partial\hfil$\crcr}}}^{\mu}\tilde{\phi}_{--}+\tilde{\phi}_{0}^{\dagger}\mathchoice{\vbox{\halign{#\cr\leftrightarrowfill@{\scriptstyle}\crcr\nointerlineskip\cr$\hfil\displaystyle\partial\hfil$\crcr}}}{\vbox{\halign{#\cr\leftrightarrowfill@{\scriptstyle}\crcr\nointerlineskip\cr$\hfil\textstyle\partial\hfil$\crcr}}}{\vbox{\halign{#\cr\leftrightarrowfill@{\scriptscriptstyle}\crcr\nointerlineskip\cr$\hfil\scriptstyle\partial\hfil$\crcr}}}{\vbox{\halign{#\cr\leftrightarrowfill@{\scriptscriptstyle}\crcr\nointerlineskip\cr$\hfil\scriptscriptstyle\partial\hfil$\crcr}}}^{\mu}\tilde{\phi}_{0}\right)$
(74) $\displaystyle J_{\rm Z}^{\mu}$
$\displaystyle=\frac{g}{2c_{w}}\bigg{[}\left(1-2s_{w}^{2}+c_{\langle\theta\rangle}^{2}\right)\phi_{+}\mathchoice{\vbox{\halign{#\cr\leftrightarrowfill@{\scriptstyle}\crcr\nointerlineskip\cr$\hfil\displaystyle\partial\hfil$\crcr}}}{\vbox{\halign{#\cr\leftrightarrowfill@{\scriptstyle}\crcr\nointerlineskip\cr$\hfil\textstyle\partial\hfil$\crcr}}}{\vbox{\halign{#\cr\leftrightarrowfill@{\scriptscriptstyle}\crcr\nointerlineskip\cr$\hfil\scriptstyle\partial\hfil$\crcr}}}{\vbox{\halign{#\cr\leftrightarrowfill@{\scriptscriptstyle}\crcr\nointerlineskip\cr$\hfil\scriptscriptstyle\partial\hfil$\crcr}}}^{\mu}\phi_{-}+\left(1-2s_{w}^{2}-c_{\langle\theta\rangle}^{2}\right)\tilde{\phi}_{+}\mathchoice{\vbox{\halign{#\cr\leftrightarrowfill@{\scriptstyle}\crcr\nointerlineskip\cr$\hfil\displaystyle\partial\hfil$\crcr}}}{\vbox{\halign{#\cr\leftrightarrowfill@{\scriptstyle}\crcr\nointerlineskip\cr$\hfil\textstyle\partial\hfil$\crcr}}}{\vbox{\halign{#\cr\leftrightarrowfill@{\scriptscriptstyle}\crcr\nointerlineskip\cr$\hfil\scriptstyle\partial\hfil$\crcr}}}{\vbox{\halign{#\cr\leftrightarrowfill@{\scriptscriptstyle}\crcr\nointerlineskip\cr$\hfil\scriptscriptstyle\partial\hfil$\crcr}}}^{\mu}\tilde{\phi}_{-}$
(91) $\displaystyle\hskip
142.26378pt+2\,(1-2s_{w}^{2})\,\tilde{\phi}_{++}\mathchoice{\vbox{\halign{#\cr\leftrightarrowfill@{\scriptstyle}\crcr\nointerlineskip\cr$\hfil\displaystyle\partial\hfil$\crcr}}}{\vbox{\halign{#\cr\leftrightarrowfill@{\scriptstyle}\crcr\nointerlineskip\cr$\hfil\textstyle\partial\hfil$\crcr}}}{\vbox{\halign{#\cr\leftrightarrowfill@{\scriptscriptstyle}\crcr\nointerlineskip\cr$\hfil\scriptstyle\partial\hfil$\crcr}}}{\vbox{\halign{#\cr\leftrightarrowfill@{\scriptscriptstyle}\crcr\nointerlineskip\cr$\hfil\scriptscriptstyle\partial\hfil$\crcr}}}^{\mu}\tilde{\phi}_{--}+c_{\langle\theta\rangle}^{2}\,\tilde{\phi}_{0}^{\dagger}\mathchoice{\vbox{\halign{#\cr\leftrightarrowfill@{\scriptstyle}\crcr\nointerlineskip\cr$\hfil\displaystyle\partial\hfil$\crcr}}}{\vbox{\halign{#\cr\leftrightarrowfill@{\scriptstyle}\crcr\nointerlineskip\cr$\hfil\textstyle\partial\hfil$\crcr}}}{\vbox{\halign{#\cr\leftrightarrowfill@{\scriptscriptstyle}\crcr\nointerlineskip\cr$\hfil\scriptstyle\partial\hfil$\crcr}}}{\vbox{\halign{#\cr\leftrightarrowfill@{\scriptscriptstyle}\crcr\nointerlineskip\cr$\hfil\scriptscriptstyle\partial\hfil$\crcr}}}^{\mu}\tilde{\phi}_{0}\bigg{]}$
(108) $\displaystyle J_{\rm+}^{\mu}$
$\displaystyle=g\bigg{[}c_{\langle\theta\rangle}^{2}\,\left(\tilde{\phi}_{++}\mathchoice{\vbox{\halign{#\cr\leftrightarrowfill@{\scriptstyle}\crcr\nointerlineskip\cr$\hfil\displaystyle\partial\hfil$\crcr}}}{\vbox{\halign{#\cr\leftrightarrowfill@{\scriptstyle}\crcr\nointerlineskip\cr$\hfil\textstyle\partial\hfil$\crcr}}}{\vbox{\halign{#\cr\leftrightarrowfill@{\scriptscriptstyle}\crcr\nointerlineskip\cr$\hfil\scriptstyle\partial\hfil$\crcr}}}{\vbox{\halign{#\cr\leftrightarrowfill@{\scriptscriptstyle}\crcr\nointerlineskip\cr$\hfil\scriptscriptstyle\partial\hfil$\crcr}}}^{\mu}\tilde{\phi}_{-}+\tilde{\phi}_{+}\mathchoice{\vbox{\halign{#\cr\leftrightarrowfill@{\scriptstyle}\crcr\nointerlineskip\cr$\hfil\displaystyle\partial\hfil$\crcr}}}{\vbox{\halign{#\cr\leftrightarrowfill@{\scriptstyle}\crcr\nointerlineskip\cr$\hfil\textstyle\partial\hfil$\crcr}}}{\vbox{\halign{#\cr\leftrightarrowfill@{\scriptscriptstyle}\crcr\nointerlineskip\cr$\hfil\scriptstyle\partial\hfil$\crcr}}}{\vbox{\halign{#\cr\leftrightarrowfill@{\scriptscriptstyle}\crcr\nointerlineskip\cr$\hfil\scriptscriptstyle\partial\hfil$\crcr}}}^{\mu}\tilde{\phi}_{0}-\phi_{+}\mathchoice{\vbox{\halign{#\cr\leftrightarrowfill@{\scriptstyle}\crcr\nointerlineskip\cr$\hfil\displaystyle\partial\hfil$\crcr}}}{\vbox{\halign{#\cr\leftrightarrowfill@{\scriptstyle}\crcr\nointerlineskip\cr$\hfil\textstyle\partial\hfil$\crcr}}}{\vbox{\halign{#\cr\leftrightarrowfill@{\scriptscriptstyle}\crcr\nointerlineskip\cr$\hfil\scriptstyle\partial\hfil$\crcr}}}{\vbox{\halign{#\cr\leftrightarrowfill@{\scriptscriptstyle}\crcr\nointerlineskip\cr$\hfil\scriptscriptstyle\partial\hfil$\crcr}}}^{\mu}\phi_{0}\right)$
(133) $\displaystyle\hskip
128.0374pt+s_{\langle\theta\rangle}^{2}\,\left(\tilde{\phi}_{++}\mathchoice{\vbox{\halign{#\cr\leftrightarrowfill@{\scriptstyle}\crcr\nointerlineskip\cr$\hfil\displaystyle\partial\hfil$\crcr}}}{\vbox{\halign{#\cr\leftrightarrowfill@{\scriptstyle}\crcr\nointerlineskip\cr$\hfil\textstyle\partial\hfil$\crcr}}}{\vbox{\halign{#\cr\leftrightarrowfill@{\scriptscriptstyle}\crcr\nointerlineskip\cr$\hfil\scriptstyle\partial\hfil$\crcr}}}{\vbox{\halign{#\cr\leftrightarrowfill@{\scriptscriptstyle}\crcr\nointerlineskip\cr$\hfil\scriptscriptstyle\partial\hfil$\crcr}}}^{\mu}\phi_{-}-\tilde{\phi}_{+}\mathchoice{\vbox{\halign{#\cr\leftrightarrowfill@{\scriptstyle}\crcr\nointerlineskip\cr$\hfil\displaystyle\partial\hfil$\crcr}}}{\vbox{\halign{#\cr\leftrightarrowfill@{\scriptstyle}\crcr\nointerlineskip\cr$\hfil\textstyle\partial\hfil$\crcr}}}{\vbox{\halign{#\cr\leftrightarrowfill@{\scriptscriptstyle}\crcr\nointerlineskip\cr$\hfil\scriptstyle\partial\hfil$\crcr}}}{\vbox{\halign{#\cr\leftrightarrowfill@{\scriptscriptstyle}\crcr\nointerlineskip\cr$\hfil\scriptscriptstyle\partial\hfil$\crcr}}}^{\mu}\phi_{0}+\tilde{\phi}_{0}\mathchoice{\vbox{\halign{#\cr\leftrightarrowfill@{\scriptstyle}\crcr\nointerlineskip\cr$\hfil\displaystyle\partial\hfil$\crcr}}}{\vbox{\halign{#\cr\leftrightarrowfill@{\scriptstyle}\crcr\nointerlineskip\cr$\hfil\textstyle\partial\hfil$\crcr}}}{\vbox{\halign{#\cr\leftrightarrowfill@{\scriptscriptstyle}\crcr\nointerlineskip\cr$\hfil\scriptstyle\partial\hfil$\crcr}}}{\vbox{\halign{#\cr\leftrightarrowfill@{\scriptscriptstyle}\crcr\nointerlineskip\cr$\hfil\scriptscriptstyle\partial\hfil$\crcr}}}^{\mu}\phi_{+}\right)\bigg{]}$
(158)

These current interactions also mediate transitions between the different
scalar mass eigenstates via “weak-strahling” processes. However these
branching fractions are sub-leading relative to their decays to pairs of
fermions via $\mathcal{O}(1)$ Yukawa couplings. The mass, charge, and CP
properties of the $\tilde{\phi}_{++}$ scalar in this model forbid all two-body
decays and its branching fraction is thus dominated by the 3-body decay ${\rm
BR}(\tilde{\phi}_{++}\shortrightarrow W^{+}t\bar{b})\sim 1$. Pair production
of this doubly charged scalar will thus always result in the final state
$\tilde{\phi}_{++}\tilde{\phi}_{--}\rightarrow(Wb)^{4}$, effectively providing
an $\mathcal{O}({\rm fb})$ contribution to the four top-quark cross section
from the process shown in Figure 7. The other major contribution to an
anomalous multi-top cross section comes from strong production of pairs of
heavy vector-like quarks, which are the lightest top quark partner
$t^{\prime}\bar{t}^{\prime}$ and peculiar quark $p^{\prime}\bar{p}^{\prime}$.
The lightest top partner $t^{\prime}$ decays mostly via the neutral current
$t^{\prime}\rightarrow\tilde{\phi}_{0}t$ but there is also a sub-leading
branching fraction to the top quark via emission of $Z$ and $h$ bosons. These
two sub-leading contributions are approximately equal due to the goldstone
equivalence theorem, as shown in Figure 8. Flavor-changing charged-current
decays of $t^{\prime}$ are suppressed due to the fact that it mostly an
electroweak doublet implying that ${\rm BR}(t^{\prime}\shortrightarrow
W^{+}b)\sim 0$. The lightest bottom-type quark partner $p^{\prime}$ has a
peculiar charge of $5/3$ and therefore decays exclusively via emission of
charged bosons. Pair production of these heavy color-charged fermions thus
generically results in final states involving pairs of $\widetilde{\Phi}$
scalars and additional third generation quarks. At large values of the $SU(5)$
symmetric Yukawa coupling $y_{1}$, the mass splitting between the new fermion
and scalar states becomes large enough to attenuate the kinematic advantage of
two-body final states. In this regime we find an increasingly significant
contribution from 3-body decays, which can result in exotic signatures
containing a very high multiplicity of leptons and b-jets. In this model the
contribution to the multi-top cross section from any one production mode is
generally below threshold for detection by current experimental searches.
However we find that combining the effective contribution to this final state
across multiple channels makes these searches sensitive to large parts of the
parameter space. The various combinations of decay paths that contribute to
the effective $t\bar{t}t\bar{t}$ cross section are enumerated in Appendix C.
The aggregate contribution from all of these channels allows for a direct
translation of the limit on anomalous $4$-top production to the parameter
space of this theory, as illustrated in Figure 9.

Figure 9: Total cross section (fb) for production of final state $(Wb)^{n}$
with $n\geq 4$ at $f=600$ GeV (left) and $f=800$ GeV (right). Limits from
multi-top searches shown in red.

####  II.3.3 Hidden-Sector Final States

The lightest states in the hidden-sector are a spectrum of glueballs which may
be labeled by their $J^{PC}$ quantum numbers. The masses and widths of these
glueballs take a large contribution from the non-perturbative dynamics of the
hidden-sector gauge group $G$, thus preventing a detailed quantitative
analysis of their collider phenomenology. However naive dimensional analysis
implies that the glueball masses should be proportional to the hidden-sector
confinement scale $\widetilde{\Lambda}$, and their widths should be
proportional to some high power of $\widetilde{\Lambda}$. If the hidden-sector
gauge group is $G=SU(3)$, computations on the lattice indicate the existence
of at least a dozen stable glueball mass eigenstates Morningstar and Peardon
(1999); Chen  _et al._ (2006); Gockeler  _et al._ (2006). The lightest state
has $J^{PC}=0^{++}$ and a mass $m_{0^{++}}\sim 7\widetilde{\Lambda}$, while
the higher excited states all lie within $\mathcal{O}({\rm few})\times
m_{0^{++}}$. Despite the relative compression of this mass spectrum, if
$m_{0^{++}}\gg\widetilde{\Lambda}$ then the production of higher-mass
glueballs will be sub-dominant due to a Boltzmann suppression in the thermal
partition function. We will thus assume going forward that an $\mathcal{O}(1)$
fraction of all glueballs produced are of the type $0^{++}$. The lowest-lying
glueball $0^{++}$ can decay back to Standard Model particles through the Higgs
portal and its branching ratios are thus approximately equal to those of a
Higgs boson of the same mass. The widths of low-mass Higgs bosons are well
known Gunion  _et al._ (2000) and are illustrated in Figure 10. The parametric
form of the glueball width has been worked in out in Juknevich (2010), and is
given by the expression in Equation 159.

Figure 10: Branching ratio of the lightest hidden-sector glueball to Standard
Model particles as a function of the glueball mass $m_{0^{++}}$.

$\Gamma(0^{++}\shortrightarrow h^{*}\shortrightarrow
XX)=\left(\frac{\kappa\,v\,m_{0^{++}}^{3}}{24\pi^{2}f^{2}(m_{h}^{2}-m_{0^{++}}^{2})}\right)^{2}\Gamma(h^{*}\shortrightarrow
XX)$ (159)

Here $\kappa$ is an $\mathcal{O}(1)$ constant that parameterizes the non-
perturbative contribution to the width, and lattice computations for $G=SU(3)$
give $\kappa\sim 3$ Chen  _et al._ (2006); Meyer (2009). For this analysis we
remain agnostic about the details of the hidden-sector gauge group by treating
$\kappa$ and $m_{0^{++}}$ as the input quantities that parameterize the
relevant glueball properties.

If $m_{0^{++}}>m_{h}/2$ then the primary production mode of the $0^{++}$
glueball will be through decays of a broad $n\bar{n}$-onium resonance, which
has an $\mathcal{O}({\rm fb})$ production cross section in the bulk of this
parameter space, often exceeding the pair production rate of the new color-
charged states $t^{\prime}\bar{t}^{\prime}$ and $p^{\prime}\bar{p}^{\prime}$.
Weak production of $n\bar{n}$ pairs is enhanced by an abundance of new
production modes via s-channel diagrams involving the new goldstone states,
and also by the conditions of EWSB which generically result in a hierarchy
between the color-charged and hidden-sector Yukawa couplings
$m_{n}<m_{t^{\prime}}$. However in this regime, the hidden-sector states would
be extremely difficult to resolve in any near-term LHC search due to the
overwhelming QCD backgrounds to its generic final states. For example if the
hidden-sector gauge group is QCD-like, then a relatively light glueball mass
$m_{0^{++}}\ll m_{n}\sim 0.5\,{\rm TeV}$ may result in a $n\bar{n}$-onium
state that decays to a high multiplicity of glueballs, the lightest of which
would decay promptly with an $\mathcal{O}(1)$ branching fraction to pairs of
bottom quarks. This $\mathcal{O}({\rm fb})$ contribution to high-multiplicity
bottom-quark final states would be extremely challenging to probe. However
given the impressive recent advances on object identification using deep-
learning algorithms, it is conceivable that future analysis techniques may
become sensitive to the resonance structure of such glueball decays between
correlated $b\bar{b}$ pairs. As the lightest glueball mass begins to saturate
the kinematic limit $m_{0^{++}}\rightarrow m_{n}$, the $n\bar{n}$-onium state
will decay promptly to pairs of the $0^{++}$ glueball, which has a prompt
$\mathcal{O}(1)$ branching fraction to $W^{+}W^{-}$ in this regime. The hidden
sector will thus provide an $\mathcal{O}({\rm fb})$ contribution to the cross
section $\sigma(pp\shortrightarrow n\bar{n}\shortrightarrow
W^{+}W^{-}W^{+}W^{-})$ that would be similarly challenging to resolve. The
non-resonant same-sign dilepton and multi-lepton final states are suppressed
by $\mathcal{O}(10^{-2})$ branching fractions and the rates are thus orders of
magnitude too low to be constrained by current non-resonant multi-lepton
searches Khachatryan  _et al._ (2016b); Aaboud  _et al._ (2017b). In the
regime $m_{0^{++}}>2m_{W}$ where all $W$-bosons are on-shell, then the fully
hadronic decay modes of the $W^{+}W^{-}W^{+}W^{-}$ final state would contain
interesting hierarchical resonance structures that could also conceivably be
exploited by future advances in analysis techniques.

Figure 11: The glueball decay length $c\tau$ (m) as a function of $m_{0^{++}}$
and $f$. Results are shown for various values of the constant $\kappa$
pameterizing non-perturbative effects with $\kappa=6$ (left), $\kappa=3$
(middle), and $1/2$ (right). Here we define “prompt” decays as those
corresponding to $c\tau<10\,\mu\rm{m}$ and “invisible” as those corresponding
to $c\tau>10\,\rm{m}$.

As a consequence of Equation 159, the lifetime of the $0^{++}$ glueball is a
strong function of the hidden-sector confinement scale $\widetilde{\Lambda}$
and its lifetime can either be prompt or extremely long-lived. The decay
lengths at a generic value of the Yukawa couplings are illustrated in Figure
11, with the middle plot corresponding to the QCD-like scenario with
$\kappa=3$. Here we define “prompt” decays as those corresponding to
$c\tau<10\,\mu\rm{m}$ and “invisible” as those corresponding to
$c\tau>10\,\rm{m}$. In the regime where $m_{0^{++}}<m_{h}/2$ then the $0^{++}$
glueball can be very long-lived Burdman  _et al._ (2008); Craig  _et al._
(2015); Curtin and Verhaaren (2015); Burdman and Lichtenstein (2018), and Big-
Bang Nucleosynthesis (BBN) places a limit on long-lived particles for which
$c\tau>10^{7}\,{\rm m}$. For a QCD-like hidden-sector this would place a lower
limit on the lightest glueball mass $m_{0^{++}}$ of about $\mathcal{O}({\rm
few})\,{\rm GeV}$. A simplified parameterization of the decay length relative
to the BBN limit is given by Equation 160.

$c\tau(0^{++})\sim 1.5\times 10^{7}\,{\rm m}\,\left(\frac{1.5\,{\rm
GeV}}{m_{0^{++}}}\right)^{7}\left(\frac{f}{800\,{\rm GeV}}\right)^{4}\\\ $
(160)

In this parameter space $c\tau(0^{++})\sim 10\,{\rm m}$ corresponds
approximately to $m_{0^{++}}\sim 10\,{\rm GeV}$, thus placing the glueball
decay vertex outside of the LHC detectors at lower masses. For simplicity we
thus restrict our analysis to glueball masses $m_{0^{++}}>2m_{b}$ where we
have approximately ${\rm BR}(0^{++}\shortrightarrow
b\bar{b})\sim\mathcal{O}(1)$, although similar limits hold at lower masses
when ${\rm BR}(0^{++}\shortrightarrow\tau^{+}\tau^{-})\sim\mathcal{O}(1)$. The
primary production mode of the $0^{++}$ glueball in this regime occurs via
decays of the Higgs mediated by loops of hidden-sector fermions. The branching
ratio of the Higgs boson to hidden-sector glueballs in this framework can be
estimated by the Higgs partial width to pairs of hidden-sector gluons, and is
parametrically given by the relation in Equation 161.

$\frac{{\rm BR}(h\rightarrow 0^{++}0^{++})}{{\rm BR}(h\rightarrow
gg)}\sim\left(\frac{\tilde{\alpha}(m_{h})}{\alpha_{S}}\frac{v^{2}}{f^{2}}\right)^{2}\sqrt{1-\frac{4m_{0^{++}}^{2}}{m_{h}^{2}}}\\\
$ (161)

Figure 12: Higgs branching ratio to hidden-glueballs as a function of
$m_{0^{++}}$ and $f$ for a fast and slow running hidden-sector gauge couplings
$\tilde{\alpha}=0.2$ (left) and $0.4$ GeV (right). Projected $95\%$ confidence
limits are shown for the proposed MATHUSLA surface detector (red) as well as
the proposed CEPC Higgs factory (yellow). The light/medium/dark shades
correspond to different values of the glueball width parameter
$\kappa=6/3/0.5$.

This branching ratio ${\rm BR}(h\rightarrow 0^{++}0^{++})$ depends on the
running of the hidden-sector coupling constant $\tilde{\alpha}$ from the
hidden confinement scale $\widetilde{\Lambda}$ to the Higgs mass $m_{h}$. In
this model we have assumed that the lightest states charged under the hidden-
sector gauge group $G$ have masses on the order of $f>\widetilde{\Lambda}$ and
the $\beta$-function for $\tilde{\alpha}$ thus depends purely on the quadratic
Casimir of $G$. Weak-scale perturbativity of the coupling constant
$\tilde{\alpha}(m_{h})<0.5$ generically implies a small branching fraction for
$h\rightarrow 0^{++}0^{++}$ that is below sensitivity projections for the HL-
LHC Dawson  _et al._ (2013); Collaboration (2013); CMS (2016); ATL (2013,
2014). However future Higgs factories such as the proposed Circular Electron-
Positron Collider (CEPC) or the International Linear Collider (ILC) CEP
(2015); Fujii  _et al._ (2015); Bicer  _et al._ (2014) will be able to exclude
a wide range of values for $\tilde{\alpha}(m_{h})$. The expected sensitivities
to a variety of rare Higgs decays have been projected for a long-term run at
the CEPC Liu  _et al._ (2017), which will be sensitive to the decay topology
${\rm BR}(h\shortrightarrow 0^{++}0^{++}\shortrightarrow b\bar{b}b\bar{b})\sim
6\times 10^{-4}$, as illustrated in Figure 12. Here we have restricted these
limits to regions in which the $0^{++}$ glueballs decay within the tracker
radius of the proposed CEPC detectors $\sim 1810\,{\rm mm}$, and assume that
object identification would be inefficient from calorimeter data alone. The
large region of parameter space with macroscopic glueball decay lengths also
provides additional motivation for a more careful exploration of the lifetime
frontier via surface detectors such as the proposed MATHUSLA experiment Curtin
_et al._ (2018). Preliminary studies indicate that such a surface detector
could exclude rare branching fractions of the Higgs boson as low as ${\rm
BR}(h\shortrightarrow XX)\sim 10^{-5}$ for detector-optimized values of
$c\tau(0^{++})$. The long-term projections for these rare Higgs decays
assuming $3\,{\rm ab}^{-1}$ of MATHUSLA data are also illustrated in Figure
12.

##  III $SU(4)/Sp(4)$ Intermediate Model

The symmetry breaking pattern $SU(4)/Sp(4)$ produces $15-10=5$ pseudo Nambu-
Goldstone Bosons (pNGBs). The nonlinear sigma model describing the low energy
effective theory may be expressed in terms of an antisymmetric unitary matrix
$\Sigma$, which transforms under $SU(4)$ as $V\Sigma V^{T}$ where $V$ is an
$SU(4)$ matrix. It is convenient to specify a background field $\Sigma_{0}$,
which is invariant under the $Sp(4)$ subgroup containing $SU(2)_{L}\otimes
SU(2)_{R}$. The generators for the $SU(2)_{L}\otimes U(1)_{Y}$ gauge symmetry
$L^{a}$ and $Y$ are embedded into an approximate global custodial symmetry via
$Y=R^{3}$, which remains unbroken in the reference vacuum $\Sigma=\Sigma_{0}$.

$\Sigma_{0}=\begin{pmatrix}i\sigma_{2}&\quad\\\ \quad&i\sigma_{2}\\\
\end{pmatrix}\qquad\qquad L^{a}=\frac{1}{2}\begin{pmatrix}\sigma^{a}&\quad\\\
\quad&~{}~{}0\end{pmatrix}\qquad
R^{a}=\frac{1}{2}\begin{pmatrix}0~{}~{}&\quad\\\
\quad&\sigma^{a}\end{pmatrix}$ (162)

The pNGB’s thus transform under the electroweak gauge group as $1_{0}\oplus
2_{\pm 1/2}$ giving one real electroweak singlet and a SM Higgs doublet. The
Nambu-Goldstone bosons are fluctuations about this background in the direction
of the broken generators, $\Pi\equiv\pi^{a}X^{a}$, and may be parameterized as

$\displaystyle\Pi$ $\displaystyle=$
$\displaystyle\frac{1}{2}\begin{pmatrix}\frac{1}{\sqrt{2}}\eta~{}\mathbbm{1}&~{}~{}(H^{c}~{}H)\\\
(H^{c}~{}H)^{\dagger}&~{}~{}-\frac{1}{\sqrt{2}}\eta~{}\mathbbm{1}\\\
\end{pmatrix}$ (163)

Here $H=(H_{+}~{}H_{0})$ and $H^{c}=i\sigma_{2}H^{*}$. Vacuum misalignment
with respect to the weak gauge interactions can be parameterized by an angle
$\theta$. This vacuum, which is invariant under the custodial $SU(2)_{c}$
generated by $R^{a}+L^{a}$, can be written as

$\theta^{2}\equiv\frac{1}{2}\frac{H^{\dagger}H}{f^{2}}\qquad\qquad\langle\Sigma\rangle=\begin{pmatrix}0&\cos\theta&0&i\sin\theta\\\
-\cos\theta&0&-i\sin\theta&0\\\ 0&i\sin\theta&0&\cos\theta\\\
-i\sin\theta&0&-\cos\theta&0\end{pmatrix},$ (164)

The gauge couplings explicitly break the $SU(4)$ global symmetry. The
covariant derivative can be expanded to second order in the vacuum giving tree
level masses to the weak gauge bosons in terms of the $SU(2)_{L}\times
U(1)_{Y}$ couplings $g,g^{\prime}$. The tree level relation
$m_{W}/m_{Z}=\cos\theta_{w}$ is guaranteed by the preservation of the remnant
custodial symmetry $SU(2)_{c}\subset Sp(4)$

$\displaystyle m_{W}^{2}=\frac{g^{2}}{2}\,f^{2}\sin^{2}\theta$ $\displaystyle
m_{Z}^{2}=\frac{g^{2}+g^{\prime 2}}{2}\,f^{2}\sin^{2}\theta.$ (165)

###  III.1 Fermion Sector

In this model the Yukawa sector of the Standard Model is extended to include
interactions between the $\Sigma$ field and composite fermions in complete
$SU(4)$ multiplets. Additional gauge invariant interactions softly break the
global symmetry, generating a significant negative contribution to the Higgs
potential, thus driving electroweak symmetry breaking. The mass and quartic
couplings of the Higgs can be tuned to their experimentally measured values by
extending the fermion sector of this theory to include two vector-like
multiplets of fermions in the fundamental representation of $SU(4)$. The first
multiplet $(\psi,\,\bar{\psi})$ is charged under $SU(3)_{c}$ and mixes with
the third generation of Standard Model quarks resulting in a partially
composite top and bottom. The second multiplet $(\chi,\,\bar{\chi})$ is
charged under a hidden-sector gauge group $G$ that confines at some scale
$\widetilde{\Lambda}<f$.

####  III.1.1 Color-charged Fermions

The new color-charged vector-like states $(\psi,\,\bar{\psi})$ mix with the
chiral third-generation of SM quarks, which are singlets under the global
$SU(4)$ symmetry. The gauge eigenstate components of these multiplets are
fixed by the embedding of $SU(2)_{L}\times U(1)_{Y}\subset SU(4)$, and are
shown in Equation 166. The gauge quantum numbers of the new color-charged
fermions are given in Table 5.

$\psi=\left(Q~{}~{}T~{}~{}B\right)\qquad\qquad\bar{\psi}=\left(\overline{Q}~{}~{}\overline{T}~{}~{}\overline{B}\right)^{T}$
(166)

|  $\left(Q~{}\overline{Q}\right)$ |  $\left(T~{}\overline{T}\right)$ |  $\left(B~{}\overline{B}\right)$ |  $q$ |  $\bar{t}$ |  $\bar{b}$
---|---|---|---|---|---|---
$SU(3)_{c}$ | $(3~{}\overline{3})$ | $(3~{}\overline{3})$ | $(3~{}\overline{3})$ | $3$ | $\overline{3}$ | $\overline{3}$
$SU(2)_{L}$ | $(2~{}2)$ | $(\mathbbm{1}~{}\mathbbm{1})$ | $(\mathbbm{1}~{}\mathbbm{1})$ | $2$ | $\mathbbm{1}$ | $\mathbbm{1}$
$U(1)_{Y}$ | $\left(\frac{1}{6}~{}\text{-}\frac{1}{6}\right)$ | $\left(\frac{2}{3}~{}\text{-}\frac{2}{3}\right)$ | $\left(\text{-}\frac{1}{3}~{}\frac{1}{3}\right)$ | $\frac{1}{6}$ | $\text{-}\frac{2}{3}$ | $\frac{1}{3}$
$G$ | $(\mathbbm{1}~{}\mathbbm{1})$ | $(\mathbbm{1}~{}\mathbbm{1})$ | $(\mathbbm{1}~{}\mathbbm{1})$ | $\mathbbm{1}$ | $\mathbbm{1}$ | $\mathbbm{1}$
Table 5: Gauge quantum numbers for the new vector-like quarks. These include a
new $SU(2)_{L}$ doublet $Q=(Q_{T},\,Q_{B})$ in addition to two vector-like
singlets $T$ and $B$. The new vector-like states mix with the SM chiral
doublet $q=(q_{T},\,q_{B})$ and singlets $(\bar{t},\,\bar{b})$.

The most general gauge invariant Yukawa couplings for the color-charged
fermions are given by Equation 167, and collective symmetry breaking
properties of these interactions guarantee the absence of quadratic
divergences to the scalar potential from fermion-loops. The resulting top-like
mass matrix $M_{T}$ has the property $\det
M_{T}^{\dagger}M_{T}\propto\sin^{2}\theta$, thus producing one massless state
when $\theta=n\pi$ for $n\in\mathbb{Z}$.

${\cal
L}_{\psi}=y_{1}\,f\,\psi\,\Sigma\,\bar{\psi}+y_{2}f\,q\,\overline{Q}+y_{3}\,f\,T\,\bar{t}\,+y_{4}\,f\,B\,\bar{b}+{\rm
h.c.}$ (167)

| $~{}~{}\overline{Q}_{T}~{}~{}$ | $~{}~{}\overline{T}~{}~{}$ | $~{}~{}\bar{t}~{}~{}$
---|---|---|---
$Q_{T}$ | $y_{1}f\cos\theta$ | $-iy_{1}f\sin\theta$ | $0$
$T$ | $-iy_{1}f\sin\theta$ | $y_{1}f\cos\theta$ | $y_{3}f$
$q_{T}$ | $y_{2}f$ | $0$ | $0$

| $~{}~{}\overline{Q}_{B}~{}~{}$ | $~{}~{}\overline{B}~{}~{}$ | $~{}~{}\bar{b}~{}~{}$
---|---|---|---
$Q_{B}$ | $y_{1}f\cos\theta$ | $-iy_{1}f\sin\theta$ | $0$
$B$ | $-iy_{1}f\sin\theta$ | $y_{1}f\cos\theta$ | $y_{3}f$
$q_{B}$ | $0$ | $y_{2}f$ | $0$

Table 6: Mass matrices for the top-like quarks $M_{T}$ (left) and bottom-like
quarks $M_{B}$ (right) in the gauge eigenbasis.

Below the scale of electroweak symmetry breaking the fermion interactions can
be expressed in terms of their low energy gauge eigenstates. The top-like and
bottom-like mass matrices $M_{T}$ and $M_{B}$ are a function of Higgs’ vacuum
expectation value $\theta=\langle h\rangle/(\sqrt{2}f)$, and produce six mass
eigenstates upon diagonalization. The up-type sector contains three charge
$\pm 2/3$ states which we label in order of descending mass as
$t^{\prime\prime}$, $t^{\prime}$, $t$, and the lightest of these states
corresponds to the SM top quark. The down-type sector contains three charge
$\mp 1/3$ mass eigenstates which we label in order of descending mass as
$b^{\prime\prime}$, $b^{\prime}$, $b$, and the lightest of these states
corresponds to the SM bottom quark. The effective Yukawa couplings for the SM
top and bottom quarks may be expressed as

$y_{t}=\frac{y_{1}y_{2}y_{3}}{\sqrt{y_{1}^{2}+y_{2}^{2}}\sqrt{y_{1}^{2}+y_{3}^{2}}}\qquad\qquad\qquad
y_{b}=\frac{y_{1}y_{2}y_{4}}{\sqrt{y_{1}^{2}+y_{2}^{2}}\sqrt{y_{1}^{2}+y_{3}^{2}}}$
(168)

Note that a non-zero top quark mass requires $y_{1},y_{2},y_{3}>0$
simultaneously, revealing the collective symmetry breaking properties of the
top sector mass matrix. The masses of the color-charged fermions are computed
in Appendix A to second order in $\sin^{2}\theta$, and the numerically
diagonalized mass spectrum for the heavy fermions are shown in Figure (13). To
leading order in $\sin^{2}\theta$ these masses scale approximately as

$\begin{aligned} m_{t}^{2}&\sim f^{2}y_{t}^{2}\sin^{2}\theta\\\
m_{t^{\prime}}^{2}&\sim f^{2}\left(y_{1}^{2}+y_{2}^{2}\right)\\\
m_{t^{\prime\prime}}^{2}&\sim
f^{2}\left(y_{1}^{2}+y_{3}^{2}\right)\end{aligned}\qquad\qquad\qquad\begin{aligned}
m_{b}^{2}&\sim f^{2}y_{b}^{2}\sin^{2}\theta\\\ m_{b^{\prime}}^{2}&\sim
f^{2}\left(y_{1}^{2}+y_{4}^{2}\right)\\\ m_{b^{\prime\prime}}^{2}&\sim
f^{2}\left(y_{1}^{2}+y_{2}^{2}\right)\end{aligned}$ (169)

####  III.1.2 Hidden-Sector Fermions

The new hidden-sector vector-like states $(\chi,\,\bar{\chi})$ transform in
the $(\Box,4)\oplus(\overline{\mbox{$\Box$}\raisebox{7.11317pt}{}},4)$
representation of $G\times SU(4)$. The hidden-sector gauge group $G$ confines
at some scale $\widetilde{\Lambda}<f$ resulting in a low energy spectrum
comprised of unstable mesons, stable baryons, and potentially long-lived
glueballs. The lightest baryons in this sector are a natural candidate for
dark matter if they are electrically neutral, however a detailed analysis of
hidden-baryon spectrum is beyond the scope of this analysis. The gauge quantum
numbers of the hidden-fermions are given in Table 7.

$\chi=\left(X~{}~{}N~{}~{}C\right)\qquad\qquad\bar{\chi}=\left(\overline{X}~{}~{}\overline{N}~{}~{}\overline{C}\right)^{T}$
(170)

|  $\left(X~{}\overline{X}\right)$ |  $\left(N~{}\overline{N}\right)$ |  $\left(C~{}\overline{C}\right)$ |  $\left(n~{}\bar{n}\right)$
---|---|---|---|---
$SU(3)_{c}$ | $(\mathbbm{1}~{}\mathbbm{1})$ | $(\mathbbm{1}~{}\mathbbm{1})$ | $(\mathbbm{1}~{}\mathbbm{1})$ | $(\mathbbm{1}~{}\mathbbm{1})$
$SU(2)_{L}$ | $(2~{}2)$ | $(\mathbbm{1}~{}\mathbbm{1})$ | $(\mathbbm{1}~{}\mathbbm{1})$ | $(\mathbbm{1}~{}\mathbbm{1})$
$U(1)_{Y}$ | $\left(\frac{1}{2}~{}\text{-}\frac{1}{2}\right)$ | $\left(0~{}0\right)$ | $\left(\text{-}1~{}1\right)$ | $\left(0~{}0\right)$
$G$ | $(\Box~{}\overline{\mbox{$\Box$}\raisebox{6.40186pt}{}})$ | $(\Box~{}\overline{\mbox{$\Box$}\raisebox{6.40186pt}{}})$ | $(\Box~{}\overline{\mbox{$\Box$}\raisebox{6.40186pt}{}})$ | $(\Box~{}\overline{\mbox{$\Box$}\raisebox{6.40186pt}{}})$
Table 7: Gauge quantum numbers for the new vector-like quarks. These include
one $SU(2)_{L}$ doublet $X=(X_{+}~{}~{}X_{0})$, one charged singlet $C$, and
two neutral singlets $N$ and $n$.

A Standard Model-like Higgs boson can be obtained from fermionic loop-
corrections by introducing symmetry breaking interactions between the
multiplet $(\chi,\,\bar{\chi})$ and an additional massless vector-like pair of
hidden-sector fermions $(n,\,\bar{n})$. The Yukawa interactions thus include
an $SU(4)$ symmetric coupling to the $\Sigma$-field in addition to terms that
softly break the global symmetry.

${\cal
L}_{\chi}=\tilde{y}_{1}\,f\,\chi\,\Sigma\,\bar{\chi}+\tilde{y}_{2}f\,n\,\overline{N}+\tilde{y}_{3}\,f\,N\,\bar{n}+{\rm
h.c.}$ (171)

| $~{}~{}\overline{X}_{0}~{}~{}$ | $~{}~{}\overline{N}~{}~{}$ | $~{}~{}\bar{n}~{}~{}$
---|---|---|---
$X_{0}$ | $\tilde{y}_{1}f\cos\theta$ | $-i\tilde{y}_{1}f\sin\theta$ | $0$
$N$ | $-i\tilde{y}_{1}f\sin\theta$ | $\tilde{y}_{1}f\cos\theta$ | $\tilde{y}_{3}f$
$n$ | $0$ | $\tilde{y}_{2}f$ | $0$

| $~{}~{}\overline{X}_{+}~{}~{}$ | $~{}~{}\overline{C}~{}~{}$
---|---|---
$X_{+}$ | $\tilde{y}_{1}f\cos\theta$ | $-i\tilde{y}_{1}f\sin\theta$
$C$ | $-i\tilde{y}_{1}f\sin\theta$ | $\tilde{y}_{1}f\cos\theta$

Table 8: Mass matrices for the neutral hidden sector fermions $M_{N}$ (left)
and charged hidden-sector fermions $M_{C}$ (right) in the gauge eigenbasis.

The hidden-sector interactions are constructed in such a way that the neutral
mass matrix $M_{N}$ obeys the condition $\det
M_{N}^{\dagger}M_{N}\propto\cos^{2}\theta$, thus producing one massless state
when $\theta=(n+1/2)\pi$. Diagonalizing $M_{N}$ produces three electrically
neutral mass eigenstates which we label in in order of descending mass as
$n^{\prime\prime}$, $n^{\prime}$, $n$. Diagonalizing the charged hidden-sector
mass matrix $M_{C}$ results in a charge $\pm 1$ vector-like pair
$(x_{+},\,\bar{x}_{+})$ and a charge $\mp 1$ vector-like pair
$(c_{-},\,\bar{c}_{-})$. The precision electroweak constraints described in
the following section push the fermion masses of this theory to very high
scales, and the details of their mass spectrum are thus irrelevant for hadron-
collider phenemenology. We thus consider a simplified parameter space in which
$\tilde{y}_{3}=\tilde{y}_{2}=\tilde{y}_{1}$, though the masses are computed
for general Yukawa couplings in Appendix A to second order in
$\sin^{2}\theta$. In this simplified parameter space the eigenvalues of the
mass matrix scale approximately as

$\begin{aligned}
m_{n}^{2}&\sim\frac{f^{2}}{4}\tilde{y}_{1}^{2}\cos^{2}\theta\\\\[10.0pt]
m_{n^{\prime}}^{2}&\sim m_{n^{\prime\prime}}^{2}\sim
2f^{2}\tilde{y}_{1}^{2}\end{aligned}\qquad\qquad\begin{aligned} &\\\\[10.0pt]
m_{x_{+}}^{2}&\sim m_{c_{-}}^{2}\sim f^{2}\tilde{y}_{1}^{2}\end{aligned}$
(172)

####  III.1.3 Theory Space

The new color-charged fermion interactions introduce four Yukawa couplings
$y_{i}$, two of which are fixed by the Standard Model top and bottom quark
masses as implied by Equation 167. This leaves two free parameters $y_{1}$ and
$y_{2}$ which are bound from below by the top-quark Yukawa coupling
$y_{t}<y_{1}y_{2}/\sqrt{y_{1}^{2}+y_{2}^{2}}$. The new hidden-sector
interactions introduce three new Yukawa couplings, which we reduce to a single
parameter by assuming $\tilde{y}_{1}=\tilde{y}_{2}=\tilde{y}_{3}$. The
conditions of EWSB described in the following section thus fixes a unique
value for $\tilde{y}_{1}$ at each point in the two-dimensional parameter space
of $(y_{1},\,y_{2})$. In this framework, the experimentally measured Higgs
mass implies a mass scale for the new hidden-sector fermions that is
$\mathcal{O}({\rm few})$ times lower than their color-charged counterparts.
The masses of the new fermionic states as a function of the chiral symmetry
breaking scale $f$ are illustrated for two benchmark points in Figure 13

Figure 13: Mass spectrum for the new quark partners at two benchmark points.
One corresponding to $y_{1}/y_{2}=7/2$ (left) and $y_{1}/y_{2}=10/2$ GeV
(right). The hidden-fermion masses are fixed relative to the color-charged
states by the Higgs mass.

The most constraining precision electroweak observable for the $SU(4)/Sp(4)$
model comes from the branching fraction of $Z\rightarrow b\bar{b}$. The
doublet-singlet mixing between the SM bottom quark and its vector-like
partners will generically induce large corrections to the bottom quark neutral
currents if $y_{2}$ is significantly larger than $y_{1}$. The extreme
precision of experimental measurements on the parameter
$R_{b}=\Gamma(Z\rightarrow b\bar{b})/\Gamma(Z\rightarrow{\rm had})=0.21629\pm
0.00066$ thus excludes all regions except those in which $y_{1}\gg y_{2}$, as
shown in Figure 14. The precision electroweak constraints from $R_{b}$ thus
push the masses of the vector-like fermions to very high scales that are
likely beyond the reach of a $14\,{\rm TeV}$ machine via direct production.
These constraints may be relaxed by introducing additional states with bottom-
like gauge quantum numbers that reduce the doublet-singlet mixing. However,
for the remainder of this paper we will assume the minimal matter content for
this model in which the fermions are heavy and decoupled. The contributions to
the $T$-parameter from fermion loops are less constraining in this model and
are also shown in Figure 14.

Figure 14: Constraints from precision electroweak measurements at $f=1200$ GeV
(left), $f=1500$ GeV (middle), and $f=2000$ GeV (right). The light/medium/dark
shades correspond to the $1\sigma/2\sigma/3\sigma$ deviations from
experimentally measured values. The black corresponds to regions inconsistent
with the top quark mass.

###  III.2 Electroweak Symmetry Breaking

The one-loop corrections to the scalar effective potential can be extracted
from the quadratically and logarithmically sensitive terms in the Coleman
Weinberg potential. The Higgs mass receives a large positive contribution from
gauge loops that is quadratically sensitive to the compositeness scale
$\Lambda$, and proportional to the gauge boson mass matrix squared. Taking
$\Lambda\sim 4\pi f$ we find

$\displaystyle\delta V_{G}(c,\theta)$
$\displaystyle=f^{2}\operatorname{tr}M_{V}^{2}(\Sigma)$ (173)
$\displaystyle=-f^{2}f_{G}^{2}\cos 2\theta$ (174) $\displaystyle f_{G}^{2}$
$\displaystyle=\frac{f^{2}}{4}c\,(3g^{2}+g^{\prime 2})$ (175)

Collective symmetry breaking ensures the absence of quadratically divergent
contributions from the Yukawa sector, and a logarithmic contribution that is
well approximated by the leading order terms in $\det M_{T}^{\dagger}M_{T}$
and $\det M_{N}^{\dagger}M_{N}$. As argued in Appendix A the higher order
contributions are suppressed by corresponding powers of
$(\operatorname{tr}M_{T}^{\dagger}M_{T})^{3}$, and are thus proportional to a
function of the Yukawa couplings that has a numerically negligible global
maximum. The leading contribution to the Higgs effective potential from the
color-charged and hidden-sector fermions are thus numerically well
approximated by a simple $\delta V_{F}\propto\pm\cos 2\theta$ dependence, and
contribute with opposite sign. Analytical expressions for the coefficients of
this potential are computed for general values of the Yukawa couplings $y_{i}$
and $\tilde{y}_{i}$ in Appendix A, as well as higher order corrections. In the
simplified parameter space, the fermionic contribution to the scalar effective
potential may be expressed to leading order as

$\displaystyle\delta V_{F}(\theta)$ $\displaystyle\approx
f^{2}\left(f_{T}^{2}-f_{N}^{2}\right)\cos 2\theta$ (176)
$f_{T}^{2}=\frac{3f^{2}}{16\pi^{2}}\frac{y_{1}^{2}y_{2}^{2}y_{3}^{2}}{y_{2}^{2}-y_{3}^{2}}\log\frac{y_{1}^{2}+y_{2}^{2}}{y_{1}^{2}+y_{3}^{2}}\qquad\qquad
f_{N}^{2}=\frac{3f^{2}}{16\pi^{2}}\tilde{y}_{1}^{4}$ (177)

A realistic pattern of EWSB in this model requires the introduction of a
$U(1)_{a}$ violating spurion analogous to the operator of Equation 30. This
spurion descends naturally from fermion mass terms in the UV complete theory,
and gives a contribution to the Higgs potential that is periodic in
$\theta\mod 2\pi$

$\displaystyle\delta V_{0}(m_{0},\theta)$
$\displaystyle=-f^{3}\operatorname{tr}[M_{0}\Sigma]+{\rm h.c.}$ (178)
$\displaystyle=-f^{2}f_{0}^{2}\cos\theta$ (179) $\displaystyle f_{0}^{2}$
$\displaystyle=4fm_{0}$ (180)

In the absence of additional matter content at the scale $f$, the leading
contributions to the scalar effective potential are fixed by the Yukawa
coupings $y_{i}$ and $\tilde{y}_{i}$, the 1-loop gauge coefficient $c$, and
spurion coefficient $m_{0}$. This potential has a generic EWSB vacuum when
$f_{T}^{2}-f_{N}^{2}-f_{G}^{2}\gg f_{0}^{2}>0$ and the vacuum expectation
value of the order parameter $\theta$ can be expressed as a function of these
scales.

$V_{\rm
H}(\theta)=-f^{2}f_{0}^{2}\cos\theta+f^{2}\left(f_{T}^{2}-f_{N}^{2}-f_{G}^{2}\right)\cos
2\theta$ (181) $\langle\theta^{2}\rangle=\frac{\langle
H^{\dagger}H\rangle}{2f^{2}}=6\left[\frac{4\left(f_{T}^{2}-f_{N}^{2}-f_{G}^{2}\right)-f_{0}^{2}}{16\left(f_{T}^{2}-f_{N}^{2}-f_{G}^{2}\right)-f_{0}^{2}}\right]$
(182)

Fixing $v^{2}=2f^{2}\langle\theta^{2}\rangle=(246\text{ GeV})^{2}$ and
expanding about this symmetry breaking vacuum, we find that the coefficients
of the one-loop corrections are fixed by the measured value of the physical
Higgs boson $m_{h}=126$ GeV.

$\displaystyle f_{T}^{2}-f_{N}^{2}-f_{G}^{2}$
$\displaystyle=\frac{m_{h}^{2}}{12}\left(\frac{1}{2}\frac{1}{\langle\theta^{2}\rangle}-1\right)$
(183) $\displaystyle f_{0}^{2}$
$\displaystyle=\frac{4m_{h}^{2}}{3}\left(\frac{3}{2}\frac{1}{\langle\theta^{2}\rangle}-1\right)$
(184)

In this framework the mass of the gauge-singlet pseudoscalar $\eta$ takes a
large contribution from the spurion coefficient $m_{0}$ via Equation 184. At
leading order in $\theta$, the size of this coefficient is fixed by the Higgs
mass to be $m_{0}\sim fm_{h}^{2}/v^{2}$ resulting in a heavy pseudoscalar
$m_{\eta}\sim\mathcal{O}({\rm TeV})$. The pseudoscalar mass can be made into a
free parameter by introducing an additional symmetry breaking spurion that is
proportional to an alternative $Sp(4)$ preserving background field
$\widetilde{M}_{0}\equiv\tilde{m}_{0}\widetilde{\Sigma}_{0}$

$\displaystyle\delta\widetilde{V}_{0}(\tilde{m}_{0},\alpha)$
$\displaystyle=-f^{2}\operatorname{tr}[\widetilde{M}_{0}\Sigma]^{2}+{\rm
h.c.}$ (185)
$\displaystyle=8f^{2}\tilde{m}_{0}^{2}\,\eta^{2}\,\frac{\sin^{2}\alpha}{\alpha^{2}}$
(186) $\widetilde{\Sigma}_{0}=\begin{pmatrix}i\sigma_{2}&\quad\\\
\quad&-i\sigma_{2}\\\
\end{pmatrix}\qquad\alpha=\sqrt{\frac{\eta^{2}+H^{2}}{2f^{2}}}$ (187)

The spurion given by Equation 186 would descend naturally from a four-fermion
interaction in the UV completion at the compositeness scale $\Lambda$. This
term provides a contribution to the potential for $\eta$ only. In the absence
of additional sources of symmetry breaking terms, the pseudoscalar mass is
given as a function of the spurion coefficients $m_{0}$ and $\tilde{m}_{0}$

$m_{\eta}^{2}\sim
4f^{2}\left(\frac{m_{0}}{f}-\frac{8\tilde{m}_{0}^{2}}{f^{2}}\right)$ (188)

###  III.3 Phenomenology

In the $SU(4)/Sp(4)$ model, precision electroweak constraints push the masses
of the BSM fermions to very high values, putting them firmly beyond the reach
of near-term LHC limits. These constraints can be relaxed by introducing
additional vector-like partners in a way that suppresses mixing between the SM
bottom quark and the heavy singlet. However the branching fractions of the
lightest BSM states to the singlet $\eta$ are generically small. The lightest
color-charged states in such models would thus be virtually indistinguishable
from the top and bottom partners of simplified models Khachatryan  _et al._
(2016a); Sirunyan  _et al._ (2017) and their phenomenology will not be
discussed further here. The phenomenological signatures of of the hidden-
sector in this model are similar to those described in Section II.3.3 and will
also not be reiterated here. In the absence of kinematically accessible BSM
fermions, the production of new particles is limited to weak production of the
goldstone mode $\eta$, which can be singly produced via gluon fusion, or pair
produced via Higgs decay $h\rightarrow\eta\eta$ if it is sufficiently light.
The large tree-level Yukawa couplings in this model result in ${\rm
BR}(\eta\shortrightarrow t\bar{t})\sim 1$ when $m_{\eta}>2m_{t}$ and ${\rm
BR}(\eta\shortrightarrow b\bar{b})\sim 1$ when $m_{\eta}<2m_{t}$.

Figure 15: Cross sections for single pseudo-scalar production at 13 TeV (left)
and branching fractions as a function of the pseudo-scalar mass $m_{\eta}$
(right)

In the high-mass regime the discovery of such a state would be extremely
challenging due to the low production cross section and the interference
effects of its two-body decays, as described in Section II.3. The loop-induced
interactions of the $\eta$ boson are roughly independent of the Yukawa-
couplings due to the dominant invariant contribution from the top-quark loop.
The cross sections and branching fractions of the $\eta$ boson are shown in
Figure 15 for a generic value of the Yukawa couplings. The pseudo-scalar
$\eta$ only has tree level couplings to the third generation quarks and their
BSM partners. Its branching fractions are thus dominated by decays to top and
bottom quark pairs whenever these channels are kinematically open. However
there will also be effective couplings to pairs of gauge bosons from
dimension-5 operators, which arise via 1-loop diagrams analogous to gluon
fusion. The relevant dimension-5 interactions are given in Equation 189. The
couplings $\tilde{c}^{\eta}_{V_{1}V_{2}}$ have been computed in FeynCalc
Shtabovenko  _et al._ (2016), and the partial widths to various pairs of gauge
bosons are given by the expression in Equation 190.

$\displaystyle\mathcal{L}$ $\displaystyle\supset$
$\displaystyle\frac{1}{4}\eta\left(\tilde{c}^{\eta}_{ww}\,W_{\mu\nu}\tilde{W}^{\mu\nu}+\tilde{c}^{\eta}_{zz}\,Z_{\mu\nu}\tilde{Z}^{\mu\nu}+\tilde{c}^{\eta}_{z\gamma}\,Z_{\mu\nu}\tilde{F}^{\mu\nu}+\tilde{c}^{\eta}_{\gamma\gamma}\,F_{\mu\nu}\tilde{F}^{\mu\nu}+\tilde{c}^{\eta}_{gg}\,G_{\mu\nu}\tilde{G}^{\mu\nu}\right)$
(189) $\Gamma(\eta\rightarrow V_{1}V_{2})=\frac{1}{16\pi
m_{\eta}}\left|\tilde{c}^{\eta}_{V_{1}V_{2}}\right|^{2}\left[\lambda(m_{\eta}^{2},m_{V_{1}}^{2},m_{V_{2}}^{2})+6m_{V_{1}}^{2}m_{V_{2}}^{2}\right]$
(190)

####  III.3.1 Rare Higgs Decays

If the pseudo-scalar mass is comparable to that of the Higgs boson then its
dominant branching fraction to bottom quarks would make it vulnerable to
sophisticated searches for boosted $b\bar{b}$ pairs such as those recently
performed by CMS Sirunyan  _et al._ (2018b); Vernieri (2017). However these
techniques are unable to resolve boosted objects with low invariant masses
$m_{\eta}<50$ GeV due to overwhelming multijet backgrounds Collaboration
(2018). In this regime, the most viable kinematic pathway for resolving the
pseudoscalar $\eta$ would be via its interactions with the Higgs boson, which
can decay to pairs of $\eta$ bosons via loop induced couplings as shown in
Figure 16. The branching ratio ${\rm BR}(h\rightarrow\eta\eta)$ is thus loop-
suppressed and such decays would likely be impossible to resolve at a future
hadron collider such as the HL-LHC, which will constrain the decay modes of
the Higgs at a relative precision not exceeding $\mathcal{O}(10\%)$ Dawson
_et al._ (2013); Collaboration (2013); CMS (2016); ATL (2013, 2014).

Figure 16: Feynman diagram for $h\rightarrow(b\bar{b})(b\bar{b})$ via the rare
Higgs decay mode $h\rightarrow\eta\eta$

The presence of such a low-mass goldstone mode provides additional motivation
for precision measurements at a future Higgs factory. One existing proposal
for such a machine is the Circular Electron-Positron Collider (CEPC), which is
expected to measure various couplings of the Higgs boson at a relative
precision of $\mathcal{O}(0.1-1\%)$ CEP (2015); Fujii  _et al._ (2015); Bicer
_et al._ (2014). The partial width for the rare Higgs decay
$h\rightarrow\eta\eta$ is an increasing function of the Yukawa couplings
$y_{i}$ and is saturated at the kinematic limit, as shown in Figure 17. The
branching fraction for this process lies in the range of ${\rm
BR}(h\shortrightarrow\eta\eta)\sim 10^{-5}-10^{-3}$ throughout the parameter
space of this model. The dominant branching fraction of the pseudoscalar to
$b\bar{b}$ pairs would thus lead to a rare $h\rightarrow(b\bar{b})(b\bar{b})$
signal that could be effectively probed at an $e^{+}e^{-}$ collider. An
estimate of the expected reach for this process at the CEPC has been computed
at $5\text{ ab}^{-1}$ Liu  _et al._ (2017), and we find that translating these
limits puts a large part of this parameter space within reach of a long-term
run at the CEPC, as shown in Figure 17.

Figure 17: Branching ratios for rare decays of the Higgs boson to pairs of
pseudoscalars. The expected reach at CEPC from rare Higgs decays
$h\rightarrow(b\bar{b})(b\bar{b})$ are also shown at $5\text{ ab}^{-1}$.

##  IV Conclusions and Outlook

A pseudo Nambu-Goldstone Higgs with a top coupling arising from mixing with
top partners offers the theoretically compelling possibility of a natural and
calculable model for electroweak symmetry breaking. We have considered two of
the most economical scenarios, expanded to include a Higgs coupling to a
hidden fermion sector. The addition of a hidden sector coupled to the pseudo
Nambu-Goldstone bosons allows for a relatively simple way to obtain a Higgs
potential that is consistent with the experimentally measured masses of Higgs
and weak gauge bosons, as well as a possible dark matter candidate. We have
performed analyses on the most relevant collider signatures of the new
fermions and new spinless particles, and find that for most of the viable
parameter space a new particle discovery will be challenging at the LHC. Given
low energy precision electroweak constraints, the lack of an LHC discovery of
any non-standard model particles to date would thus be a general expectation
for both of these composite Higgs models, though for different reasons.
Nonetheless our analysis suggests several options for probing these
nonstandard phenomena at the LHC and future colliders.

In the $SU(5)/SO(5)$ model these challenges arise due to the serendipitous
subtlety of the final state signatures which arise from electroweak
interactions that are preferential to the third generation of quarks. The
backgrounds from QCD processes generically overwhelm or interfere with the
most obvious processes that could be used to resolve new particle states
beyond the Standard Model. However the high multiplicity of new BSM states
will generally aggregate to produce a measurable deviation in the production
of final states with a high multiplicity of third generation quarks in the
high-luminosity limit. The presence of a confining hidden-sector gauge group
could also lead to a potentially measurable branching fraction of the Higgs
boson to displaced pairs of bottom quarks, thus providing additional
motivation for experiments focused on the long-lived particle frontier, such
as the proposed MATHUSLA experiment. In the $SU(4)/Sp(4)$ model, the
experimental challenges arise primarily due to the stark absence of new
particle states involved with the EWSB dynamics. The goldstone sector of this
theory consists a single spin-0 gauge-singlet state, and precision electroweak
constraints imply a mass range for new strongly interacting particles that is
likely well beyond the reach of a $14\,{\rm TeV}$ hadron collider. However the
Higgs boson generically has a decay rate to the gauge-singlet goldstone mode
that could be resolved at a future Higgs factory such as the CEPC.

## Acknowledgments

The work of A.E.N. is partially supported by the DOE under grant DE-SC0011637
and by the Kenneth K. Young Memorial Endowed Chair. The work of D.G.E.W. was
partially supported by Burke faculty fellowship. We would like to thank
Calibourne D. Smith and Scott Thomas for helpful conversations. A.E.N.
acknowledges the hospitality of the Aspen Center for Physics, which is
supported by National Science Foundation grant PHY-1607611. We thank the
Galileo Galilei Institute for Theoretical Physics for the hospitality and the
INFN for partial support during the completion of this work. This work was
partially supported by a grant from the Simons Foundation (341344, LA).

##  Appendix A Fermion Contribution to $SU(4)/Sp(4)$ Higgs Potential

In the $SU(4)/Sp(4)$ model, the top-sector mass matrix for the color-charged
quarks $M_{T}$ takes the form given in Table 9. The mass matrices can be
expressed in terms of a single angle $\theta\equiv v/\sqrt{2}f$, representing
the magnitude of the Higgs VEV relative to the chiral symmery breaking scale.

| $~{}~{}\overline{X}_{0}~{}~{}$ | $~{}~{}\overline{N}~{}~{}$ | $~{}~{}\bar{n}~{}~{}$
---|---|---|---
$X_{0}$ | $\tilde{y}_{1}f\cos\theta$ | $-i\tilde{y}_{1}f\sin\theta$ | $0$
$N$ | $-i\tilde{y}_{1}f\sin\theta$ | $\tilde{y}_{1}f\cos\theta$ | $\tilde{y}_{3}f$
$n$ | $0$ | $\tilde{y}_{2}f$ | $0$
Table 9: Mass matrices for the color-charged (left) and hidden-sector (right)
top-like fermions in the gauge eigenbasis.

The UV insensitivity of the Yukawa sector provides powerful constraints on the
form of the scalar effective potential, and is guaranteed by the following
properties of the mass matrix

$\partial_{\theta}\operatorname{tr}M_{T}^{\dagger}M_{T}=\partial_{\theta}\operatorname{tr}(M_{T}^{\dagger}M_{T})^{2}=0$
(191)

The fermion masses are periodic with respect to $\theta\mod 2\pi$. The color-
charged mass matrix $M_{T}$ produces one massless state when $\theta=n\pi$ for
$n\in\mathbb{Z}$. For notational simplicity we define the following
combinations of Yukawa couplings

$y_{\pm}^{2}=(y_{1}^{2}+y_{2}^{2})\pm(y_{1}^{2}+y_{3}^{2})\qquad\qquad\qquad
y_{123}^{2}\equiv y_{1}^{2}y_{2}^{2}y_{3}^{2}$ (192)

In terms of these Yukawa parameters, the traces and determinants of the
fermion mass matrices take a simple form

$\operatorname{tr}M_{T}^{\dagger}M_{T}=f^{2}y_{+}^{2}\qquad\qquad\operatorname{tr}(M_{T}^{\dagger}M_{T})^{2}=\frac{f^{2}}{2}(y_{+}^{4}+y_{-}^{4})$
(193) $\det M_{T}^{\dagger}M_{T}=f^{6}y_{123}^{2}\sin^{2}\theta$ (194)

These expressions allow for a direct computation of the one-loop fermion
corrections order-by-order in the Higgs VEV. We thus expand the fermion masses
as a power series in $\sin^{2}\theta$

$\displaystyle m_{t}^{2}$
$\displaystyle=\displaystyle\sum_{n=0}^{\infty}m_{n}^{2}\sin^{2n}\theta\qquad\qquad
m_{0}^{2}=0$ (195) $\displaystyle m_{t^{\prime}}^{2}$
$\displaystyle=\displaystyle\sum_{n=0}^{\infty}m^{\prime\,2}_{n}\sin^{2n}\theta\qquad\qquad
m^{\prime 2}_{0}=\frac{f^{2}}{2}(y_{+}^{2}+y_{-}^{2})$ (196) $\displaystyle
m_{t^{\prime\prime}}^{2}$
$\displaystyle=\displaystyle\sum_{n=0}^{\infty}m^{\prime\prime\,2}_{n}\sin^{2n}\theta\qquad\qquad
m^{\prime\prime 2}_{0}=\frac{f^{2}}{2}(y_{+}^{2}-y_{-}^{2})$ (197)

Computing to fourth order in $\sin\theta$, we find that the color-charged
fermion mass eigenstates take the following form

$\displaystyle\frac{m_{t}^{2}}{f^{2}}$
$\displaystyle=\frac{4y_{123}^{2}}{y_{+}^{4}-y_{-}^{4}}\sin^{2}\theta-\frac{64y_{123}^{4}y_{+}^{2}}{(y_{+}^{4}-y_{-}^{4})^{3}}\sin^{4}\theta$
(198) $\displaystyle\frac{m_{t^{\prime}}^{2}}{f^{2}}$
$\displaystyle=\frac{y_{+}^{2}-y_{-}^{2}}{2}-\frac{2y_{123}^{2}}{y_{-}^{2}(y_{+}^{2}-y_{-}^{2})}\sin^{2}\theta+\frac{4y_{123}^{4}(y_{+}^{2}-3y_{-}^{2})}{y_{-}^{6}(y_{+}^{2}-y_{-}^{2})^{3}}\sin^{4}\theta$
(199) $\displaystyle\frac{m_{t^{\prime\prime}}^{2}}{f^{2}}$
$\displaystyle=\frac{y_{+}^{2}+y_{-}^{2}}{2}+\frac{2y_{123}^{2}}{y_{-}^{2}(y_{+}^{2}+y_{-}^{2})}\sin^{2}\theta-\frac{4y_{123}^{4}(y_{+}^{2}+3y_{-}^{2})}{y_{-}^{6}(y_{+}^{2}+y_{-}^{2})^{3}}\sin^{4}\theta$
(200)

The UV insensitivity of the Yukawa sector additionally guarantees that their
contributions to the scalar effective potential can be expressed in terms of
ratios of mass eigenstates. Computing to fourth order in $\sin\theta$, we find
that the top-sector contribution to the Higgs effective potential takes the
following form

$\displaystyle\delta V_{T}(y_{i},\theta)$
$\displaystyle=-\frac{3}{16\pi^{2}}\operatorname{tr}(M_{T}^{\dagger}\\!M_{T})^{2}\log\frac{M_{T}^{\dagger}\\!M_{T}}{\Lambda^{2}}$
(201)
$\displaystyle=-\frac{3}{16\pi^{2}}\displaystyle\sum_{t^{\prime\prime}t^{\prime}t}|m_{q}^{2}(y_{i},\theta)|^{2}\log|m_{q}^{2}(y_{i},\theta)|$
(202) $\displaystyle\approx C_{0}+C_{2}\sin^{2}\theta+C_{4}\sin^{4}\theta$
(203)

The coefficients of the $\theta$ dependent terms may be expressed as

$\displaystyle C_{2}$
$\displaystyle=-\frac{3f^{4}}{16\pi^{2}}\frac{2y_{123}^{2}}{y_{-}^{2}}\log\frac{y_{+}^{2}-y_{-}^{2}}{y_{+}^{2}+y_{-}^{2}}$
(204) $\displaystyle C_{4}$
$\displaystyle=-\frac{3f^{4}}{16\pi^{2}}\frac{4y_{123}^{4}}{(y_{+}^{2}-y_{-}^{2})^{2}}\Bigg{(}\frac{2y_{+}^{2}}{y_{-}^{2}}-\frac{y_{+}^{6}-3y_{-}^{4}y_{+}^{2}-6y_{-}^{6}}{3y_{-}^{6}}\log\frac{8y_{123}^{2}}{(y_{+}^{2}-y_{-}^{2})^{3}}$
(205) $\displaystyle\hskip
142.26378pt+\frac{y_{+}^{6}-3y_{-}^{4}y_{+}^{2}+6y_{-}^{6}}{3y_{-}^{6}}\log\frac{8y_{123}^{2}}{(y_{+}^{2}+y_{-}^{2})^{3}}\Bigg{)}$
(206)

The collective symmetry breaking properties of $M_{T}$ guarantees that the
potential is proportional to $\det M_{T}^{\dagger}M_{T}$. Higher order
contributions are thus suppressed by powers of $y_{123}^{2}/y_{+}^{6}$, which
is a function of the Yukawa couplings that has a global maximum at $1/54$

$\frac{C_{4}}{C_{2}}\sim\frac{y_{123}^{2}}{y_{+}^{6}}\left(1+\mathcal{O}\left(\frac{y_{-}^{4}}{y_{+}^{4}}\right)\right)\qquad\qquad\qquad\left|\frac{y_{123}^{2}}{y_{+}^{6}}\right|\leq\frac{1}{54}$
(207)

For the color-charged contribution we thus find that the potential is
numerically well approximated by the second order approximation in
$\sin\theta$, which has been plotted against the numerical result in Figure
18. The fermionic contribution to the Higgs potential is thus extremely well
approximated by a $\cos 2\theta$ dependence.

$\displaystyle\delta V_{T}(y_{i},\theta)$
$\displaystyle=-2f^{2}f_{T}^{2}\sin^{2}\theta$ (208) $\displaystyle\sim
f^{2}f_{T}^{2}\cos 2\theta$ (209) $\displaystyle f_{T}^{2}$
$\displaystyle=\frac{3f^{2}}{16\pi^{2}}\frac{y_{123}^{2}}{y_{-}^{2}}\log\frac{y_{+}^{2}-y_{-}^{2}}{y_{+}^{2}+y_{-}^{2}}$
(210)

Figure 18: Fermionic contribution to the Higgs potential from the color-charge
top-like sector. The numerical result is shown in black dots while the
$\mathcal{O}(\sin^{2}\theta)$ approximation is shown in red.

The contribution to the Higgs effective potential from hidden-sector fermions
follows a similar story to the color-charged case. The neutral hidden-sector
fermion mass matrix $M_{N}$ is given in Table 10, and the insensitivity of the
Higgs potential to loops of hidden-sector fermions is guaranteed by Equation
211

| $~{}~{}\overline{X}_{0}~{}~{}$ | $~{}~{}\overline{N}~{}~{}$ | $~{}~{}\bar{n}~{}~{}$
---|---|---|---
$X_{0}$ | $\tilde{y}_{1}f\cos\theta$ | $-i\tilde{y}_{1}f\sin\theta$ | $0$
$N$ | $-i\tilde{y}_{1}f\sin\theta$ | $\tilde{y}_{1}f\cos\theta$ | $\tilde{y}_{3}f$
$n$ | $0$ | $\tilde{y}_{2}f$ | $0$
Table 10: Mass matrices for the color-charged (left) and hidden-sector (right)
top-like fermions in the gauge eigenbasis.

$\partial_{\theta}\operatorname{tr}M^{\dagger}_{N}M_{N}=\partial_{\theta}\operatorname{tr}(M^{\dagger}_{N}M_{N})^{2}=0$
(211)

The mass matrix for the neutral states $M_{N}$ produces one massless state
when $\theta=(n+1/2)\pi$, for $n\in\mathbb{Z}$ which can be seen from the form
of the mass determinant.

$\tilde{y}_{\pm}^{2}=(\tilde{y}_{1}^{2}+\tilde{y}_{2}^{2})\pm(\tilde{y}_{1}^{2}+\tilde{y}_{3}^{2})\qquad\qquad\qquad\tilde{y}_{123}^{2}\equiv\tilde{y}_{1}^{2}\tilde{y}_{2}^{2}\tilde{y}_{3}^{2}$
(212) $\det(M_{N}^{\dagger}M_{N})=f^{6}\tilde{y}_{123}^{2}\cos^{2}\theta$
(213)

The hidden-sector fermion mass eigenstates thus take a simple form when
expressed as an expansion in $\cos^{2}\theta$

$\displaystyle\frac{m_{n}^{2}}{f^{2}}$
$\displaystyle=\frac{4\tilde{y}_{123}^{2}}{\tilde{y}_{+}^{4}-\tilde{y}_{-}^{4}}\cos^{2}\theta-\frac{64\tilde{y}_{123}^{4}\tilde{y}_{+}^{2}}{(\tilde{y}_{+}^{4}-\tilde{y}_{-}^{4})^{3}}\cos^{4}\theta$
(214) $\displaystyle\frac{m_{n^{\prime}}^{2}}{f^{2}}$
$\displaystyle=\frac{\tilde{y}_{+}^{2}-\tilde{y}_{-}^{2}}{2}-\frac{2\tilde{y}_{123}^{2}}{\tilde{y}_{-}^{2}(\tilde{y}_{+}^{2}-\tilde{y}_{-}^{2})}\cos^{2}\theta+\frac{4\tilde{y}_{123}^{4}(\tilde{y}_{+}^{2}-3\tilde{y}_{-}^{2})}{\tilde{y}_{-}^{6}(\tilde{y}_{+}^{2}-\tilde{y}_{-}^{2})^{3}}\cos^{4}\theta$
(215) $\displaystyle\frac{m_{n^{\prime\prime}}^{2}}{f^{2}}$
$\displaystyle=\frac{\tilde{y}_{+}^{2}+\tilde{y}_{-}^{2}}{2}+\frac{2\tilde{y}_{123}^{2}}{\tilde{y}_{-}^{2}(\tilde{y}_{+}^{2}+\tilde{y}_{-}^{2})}\cos^{2}\theta-\frac{4\tilde{y}_{123}^{4}(\tilde{y}_{+}^{2}+3\tilde{y}_{-}^{2})}{\tilde{y}_{-}^{6}(\tilde{y}_{+}^{2}+\tilde{y}_{-}^{2})^{3}}\cos^{4}\theta$
(216)

Similarly for the hidden-sector fermions, we find that the contribution is
well approximated by a $\cos 2\theta$ dependence, but with an opposite sign
that descends from the form of the determinants in Equation 213

$\displaystyle\delta V_{N}(\tilde{y}_{i},\theta)$
$\displaystyle=-2f^{2}f_{N}^{2}\cos^{2}\theta$ (217)
$\displaystyle\sim-f^{2}f_{N}^{2}\cos 2\theta$ (218) $\displaystyle f^{2}_{N}$
$\displaystyle=\frac{3f^{2}}{16\pi^{2}}\frac{\tilde{y}_{123}^{2}}{\tilde{y}_{-}^{2}}\log\frac{\tilde{y}_{+}^{2}-\tilde{y}_{-}^{2}}{\tilde{y}_{+}^{2}+\tilde{y}_{-}^{2}}$
(219)

##  Appendix B Fermion Contribution to $SU(5)/SO(5)$ Higgs Potential

In the $SU(5)/SO(5)$ model, the top-sector mass matrix for the color-charged
quarks $M_{T}$ and the hidden sector fermions $\widetilde{M}_{T}$ takes the
form given in Table 11. The mass matrices can be expressed in terms of a
single angle $\theta\equiv v/2f$, representing the magnitude of the Higgs VEV
relative to the chiral symmery breaking scale.

| $~{}~{}\overline{T}~{}~{}$ | $~{}~{}\overline{Q}_{T}~{}~{}$ | $~{}~{}\overline{P}_{T}~{}~{}$ | $~{}~{}\bar{t}~{}~{}$
---|---|---|---|---
$T$ | $y_{1}fc_{2\theta}$ | $\frac{i}{\sqrt{2}}y_{1}fs_{2\theta}$ | $\frac{i}{\sqrt{2}}y_{1}fs_{2\theta}$ | $y_{3}f$
$Q_{T}$ | $\frac{i}{\sqrt{2}}y_{1}fs_{2\theta}$ | $y_{1}fc^{2}_{\theta}$ | $-y_{1}fs^{2}_{\theta}$ | $0$
$P_{T}$ | $\frac{i}{\sqrt{2}}y_{1}fs_{2\theta}$ | $-y_{1}fs^{2}_{\theta}$ | $y_{1}fc^{2}_{\theta}$ | $0$
$q_{T}$ | $0$ | $y_{2}f$ | $0$ | $0$

| $~{}~{}\overline{N}~{}~{}$ | $~{}~{}\overline{X}_{0}~{}~{}$ | $~{}~{}\overline{Y}_{0}~{}~{}$ | $~{}~{}\bar{n}~{}~{}$
---|---|---|---|---
$N$ | $\tilde{y}_{1}fc_{2\theta}$ | $\frac{i}{\sqrt{2}}\tilde{y}_{1}fs_{2\theta}$ | $\frac{i}{\sqrt{2}}\tilde{y}_{1}fs_{2\theta}$ | $\tilde{y}_{3}f$
$X_{0}$ | $\frac{i}{\sqrt{2}}\tilde{y}_{1}fs_{2\theta}$ | $\tilde{y}_{1}fc^{2}_{\theta}$ | $-\tilde{y}_{1}fs^{2}_{\theta}$ | 0
$Y_{0}$ | $\frac{i}{\sqrt{2}}\tilde{y}_{1}fs_{2\theta}$ | $-\tilde{y}_{1}fs^{2}_{\theta}$ | $\tilde{y}_{1}fc^{2}_{\theta}$ | $0$
$n$ | $\tilde{y}_{2}f$ | $0$ | $0$ | $0$

Table 11: Mass matrix for the top-sector fermions $M_{T}$ in the gauge
eigenbasis. Mass matrices for the color-charged (left) and hidden-sector
(right) top-like fermions.

The fermion masses are periodic with respect to $\theta\mod\pi$. The
determinant of the color-charged mass matrix guarantees the existence of a
massless fermion whenever $\theta=n\pi/2$ for $n\in\mathbb{Z}$, which we may
associated with the Standard Model top quark. Similarly the determinant of the
hidden-sector mass matrix squared produces one massless state when
$\theta=(n+1/2)\pi/2$

$\det
M_{T}^{\dagger}M_{T}=f^{8}y_{1}^{2}y_{123}^{2}\sin^{2}2\theta\qquad\qquad\det
M^{\dagger}_{N}M_{N}=f^{8}\tilde{y}_{1}^{2}\tilde{y}_{123}^{2}\cos^{2}2\theta$
(220)

Diagonalizing these mass matrices produces four charge $\pm 2/3$ mass
eigenstates and four neutral mass eigenstates, which can be computed order by
order in $\sin 2\theta$ using the constraints described in Appendix A. In
terms of the Yukawa couplings that were defined in Equation 192, the
eigenvalues of the top-sector mass matrices take the following form to
$\mathcal{O}(\sin^{2}2\theta)$.

$\begin{aligned}
\frac{m_{t}^{2}}{f^{2}}&=\frac{2y_{123}^{2}}{y_{+}^{4}-y_{-}^{4}}\sin^{2}2\theta\\\
\frac{m_{t^{\prime}}^{2}}{f^{2}}&=y_{1}^{2}\\\
\frac{m_{t^{\prime\prime}}^{2}}{f^{2}}&=\frac{y_{+}^{2}-y_{-}^{2}}{2}-\frac{y_{123}^{2}}{y_{-}^{2}(y_{+}^{2}-y_{-}^{2})}\sin^{2}2\theta\\\
\frac{m_{t^{\prime\prime\prime}}^{2}}{f^{2}}&=\frac{y_{+}^{2}+y_{-}^{2}}{2}+\frac{y_{123}^{2}}{y_{-}^{2}(y_{+}^{2}+y_{-}^{2})}\sin^{2}2\theta\end{aligned}\qquad\begin{aligned}
\frac{m_{n}^{2}}{f^{2}}&=\frac{2\tilde{y}_{123}^{2}}{\tilde{y}_{+}^{4}-\tilde{y}_{-}^{4}}\cos^{2}2\theta\\\
\frac{m_{n^{\prime}}^{2}}{f^{2}}&=\tilde{y}_{1}^{2}\\\
\frac{m_{n^{\prime\prime}}^{2}}{f^{2}}&=\frac{\tilde{y}_{+}^{2}-\tilde{y}_{-}^{2}}{2}-\frac{\tilde{y}_{123}^{2}}{\tilde{y}_{-}^{2}(\tilde{y}_{+}^{2}-\tilde{y}_{-}^{2})}\cos^{2}2\theta\\\
\frac{m_{n^{\prime\prime\prime}}^{2}}{f^{2}}&=\frac{\tilde{y}_{+}^{2}+\tilde{y}_{-}^{2}}{2}+\frac{\tilde{y}_{123}^{2}}{\tilde{y}_{-}^{2}(\tilde{y}_{+}^{2}+\tilde{y}_{-}^{2})}\cos^{2}2\theta\end{aligned}$
(221)

The insensitivity of these mass matrices to UV scales is again guaranteed by
the relations in Equation 191. The contributions from these states to the
scalar potential may thus be expressed in terms of ratios of mass eigenstates,
and are extremely well approximated by their leading order terms in $\sin
2\theta$, as shown in Figure 19. This contribution to the Higgs potential is
thus approximately given by the $\cos 4\theta$ dependence in Equation 223.

$\displaystyle\delta V_{T}(y_{i},\theta)+\delta V_{N}(\tilde{y}_{i},\theta)$
$\displaystyle=-2f^{2}\left(f_{T}^{2}\sin^{2}2\theta+f_{N}^{2}\cos^{2}2\theta\right)$
(222) $\displaystyle\sim f^{2}(f_{T}^{2}-f_{N}^{2})\cos 4\theta$ (223)
$f_{T}^{2}=\frac{3f^{2}}{32\pi^{2}}\frac{y_{123}^{2}}{y_{-}^{2}}\log\frac{y_{+}^{2}-y_{-}^{2}}{y_{+}^{2}+y_{-}^{2}}\qquad\qquad
f^{2}_{N}=\frac{3f^{2}}{32\pi^{2}}\frac{\tilde{y}_{123}^{2}}{\tilde{y}_{-}^{2}}\log\frac{\tilde{y}_{+}^{2}-\tilde{y}_{-}^{2}}{\tilde{y}_{+}^{2}+\tilde{y}_{-}^{2}}$
(224)

The fermion contribution to the Higgs potential is thus a function of two
independent scales $f_{T}^{2}$ and $f^{2}_{N}$ for the color-charged and
hidden Yukawa sectors respectively

Figure 19: Fermionic contribution to the Higgs potential from the color-charge
top-like sector. The numerical result is shown in black dots while the
$\mathcal{O}(\sin^{2}2\theta)$ approximation is shown in red.

##  Appendix C Multi-top Final States in $SU(5)/SO(5)$

The $SU(5)/SO(5)$ theory generically predicts an enhancement of the total
$t\bar{t}t\bar{t}$ cross section due to the Drell-Yan production of doubley
charged scalars and the strong production of pairs of vector-like quarks. Here
we enumerate all of the possible production and decay modes that can
contribute to the final state $(Wb)^{n}$ with $n\geq 4$. These decay channels
can result in complex correlations between the final state particles that
could be resolved with targeted kinematic techniques in the high-luminosity
limit.

Production Mode | Decay | Final State
---|---|---
$gg\rightarrow p^{\prime}\bar{p}^{\prime}$ | $p^{\prime}\bar{p}^{\prime}\rightarrow(\tilde{\phi}_{++}b)(\tilde{\phi}_{--}W^{+}\bar{t})\rightarrow(W^{+}t\bar{b}b)(W^{-}\bar{t}bW^{+}\bar{t})$ | $(Wb)^{6}$
| $p^{\prime}\bar{p}^{\prime}\rightarrow(\tilde{\phi}_{++}b)(\tilde{\phi}_{0}W^{-}\bar{t})\rightarrow(W^{+}t\bar{b}b)(t\bar{t}W^{-}\bar{t})$ |
| $p^{\prime}\bar{p}^{\prime}\rightarrow(\tilde{\phi}_{+}t)(\tilde{\phi}_{--}W^{+}\bar{t})\rightarrow(t\bar{b}t)(W^{-}\bar{t}bW^{+}\bar{t})$ |
| $p^{\prime}\bar{p}^{\prime}\rightarrow(\tilde{\phi}_{+}t)(\tilde{\phi}_{0}W^{-}\bar{t})\rightarrow(t\bar{b}t)(t\bar{t}W^{-}\bar{t})$ |
| $p^{\prime}\bar{p}^{\prime}\rightarrow(\tilde{\phi}_{++}W^{-}t)(\tilde{\phi}_{-}W^{-}\bar{b})\rightarrow(W^{+}t\bar{b}W^{-}t)(\bar{t}bW^{-}\bar{b})$ |
| $p^{\prime}\bar{p}^{\prime}\rightarrow(\tilde{\phi}_{+}W^{+}b)(\tilde{\phi}_{0}W^{-}\bar{t})\rightarrow(t\bar{b}W^{+}b)(t\bar{t}W^{-}\bar{t})$ |
| $p^{\prime}\bar{p}^{\prime}\rightarrow(\tilde{\phi}_{++}b)(W^{-}\bar{t})\rightarrow(W^{+}t\bar{b}b)(W^{-}\bar{t})$ | $(Wb)^{4}$
| $p^{\prime}\bar{p}^{\prime}\rightarrow(\tilde{\phi}_{+}t)(W^{-}\bar{t})\rightarrow(t\bar{b}t)(W^{-}\bar{t})$ |
| $p^{\prime}\bar{p}^{\prime}\rightarrow(W^{+}t)(\tilde{\phi}_{-}W^{-}\bar{b})\rightarrow(W^{+}t)(\bar{t}bW^{-}\bar{b})$ |
| $p^{\prime}\bar{p}^{\prime}\rightarrow(\tilde{\phi}_{++}b)(\tilde{\phi}_{--}\bar{b})\rightarrow(W^{+}t\bar{b}b)(W^{-}\bar{t}b\bar{b})$ | $(Wb)^{4}bb$
| $p^{\prime}\bar{p}^{\prime}\rightarrow(\tilde{\phi}_{+}t)(\tilde{\phi}_{-}\bar{t})\rightarrow(t\bar{b}t)(\bar{t}b\bar{t})$ |
| $p^{\prime}\bar{p}^{\prime}\rightarrow(\tilde{\phi}_{+}W^{+}b)(\tilde{\phi}_{-}W^{-}\bar{b})\rightarrow(t\bar{b}W^{+}b)(\bar{t}bW^{-}\bar{b})$ |
| $p^{\prime}\bar{p}^{\prime}\rightarrow(\tilde{\phi}_{++}b)(\tilde{\phi}_{-}\bar{t})\rightarrow(W^{+}t\bar{b}b)(\bar{t}b\bar{t})$ |
| $p^{\prime}\bar{p}^{\prime}\rightarrow(\tilde{\phi}_{++}b)(\tilde{\phi}_{-}W^{-}\bar{b})\rightarrow(W^{+}t\bar{b}b)(\bar{t}bW^{-}\bar{b})$ |
| $p^{\prime}\bar{p}^{\prime}\rightarrow(\tilde{\phi}_{+}t)(\tilde{\phi}_{-}W^{-}\bar{b})\rightarrow(t\bar{b}t)(\bar{t}bW^{-}\bar{b})$ |
| $p^{\prime}\bar{p}^{\prime}\rightarrow(W^{+}t)(\tilde{\phi}_{--}W^{+}\bar{t})\rightarrow(W^{+}t)(W^{-}\bar{t}bW^{+}\bar{t})$ | $(Wb)^{4}WW$
| $p^{\prime}\bar{p}^{\prime}\rightarrow(W^{+}t)(\tilde{\phi}_{0}W^{-}\bar{t})\rightarrow(W^{+}t)(t\bar{t}W^{-}\bar{t})$ |
| $p^{\prime}\bar{p}^{\prime}\rightarrow(\tilde{\phi}_{0}W^{+}t)(\tilde{\phi}_{0}W^{-}\bar{t})\rightarrow(t\bar{t}W^{+}t)(t\bar{t}W^{-}\bar{t})$ | $(Wb)^{6}WW$
| $p^{\prime}\bar{p}^{\prime}\rightarrow(\tilde{\phi}_{++}W^{-}t)(\tilde{\phi}_{--}W^{+}\bar{t})\rightarrow(W^{+}t\bar{b}W^{-}t)(W^{-}\bar{t}bW^{+}\bar{t})$ |
| $p^{\prime}\bar{p}^{\prime}\rightarrow(\tilde{\phi}_{++}W^{-}t)(\tilde{\phi}_{0}W^{-}\bar{t})\rightarrow(W^{+}t\bar{b}W^{-}t)(t\bar{t}W^{-}\bar{t})$ |

Production Mode | Decay | Final State
---|---|---
$gg\rightarrow t^{\prime}\bar{t}^{\prime}$ | $t^{\prime}\bar{t}^{\prime}\rightarrow(\tilde{\phi}_{0}t)(\tilde{\phi}_{0}\bar{t})\rightarrow(t\bar{t}t)(t\bar{t}\bar{t})$ | $(Wb)^{6}$
| $t^{\prime}\bar{t}^{\prime}\rightarrow(W^{+}W^{-}t)(h\bar{t})\rightarrow(W^{+}W^{-}t)(b\bar{b}\bar{t})$ | $(Wb)^{4}$
| $t^{\prime}\bar{t}^{\prime}\rightarrow(\tilde{\phi}_{0}t)(h\bar{t})\rightarrow(t\bar{t}t)(b\bar{b}\bar{t})$ | $(Wb)^{4}bb$
| $t^{\prime}\bar{t}^{\prime}\rightarrow(\tilde{\phi}_{0}t)(Z\bar{t})\rightarrow(t\bar{t}t)(Z\bar{t})$ | $(Wb)^{4}Z$
| $t^{\prime}\bar{t}^{\prime}\rightarrow(\tilde{\phi}_{0}t)(W^{+}W^{-}\bar{t})\rightarrow(t\bar{t}t)(W^{+}W^{-}\bar{t})$ | $(Wb)^{4}WW$
$q\bar{q}\rightarrow\tilde{\phi}_{++}\tilde{\phi}_{--}$ | $\tilde{\phi}_{++}\tilde{\phi}_{--}\rightarrow(W^{+}t\bar{b})(W^{-}\bar{t}b)$ | $(Wb)^{4}$
Table 12: Dominant production and decay modes for the multi-top final state.

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