aainstitutetext: Universite Claude Bernard Lyon 1, CNRS/IN2P3, IP2I UMR 5822,
4 rue Enrico Fermi, F-69100 Villeurbanne, Francebbinstitutetext: Quantum
Theory Center (QTC) & D-IAS, Southern Denmark Univ., Campusvej 55, 5230 Odense
M, Denmarkccinstitutetext: Department of Physics, University of Johannesburg,
PO Box 524, Auckland Park 2006, South Africaddinstitutetext: Institut für
Theoretische Physik und Astrophysik, Uni Würzburg, Emil-Hilb-Weg 22, D-97074
Würzburg, Germany

# Coloured spin-1 states in composite Higgs models

Giacomo Cacciapaglia<EMAIL_ADDRESS>a,c Aldo Deandrea
<EMAIL_ADDRESS>d Manuel Kunkel<EMAIL_ADDRESS>d
Werner Porod<EMAIL_ADDRESS>
###### Abstract

Strong dynamics for composite Higgs models predict spin-1 resonances which are
expected to be in the same mass range as the usually considered top-partners.
We study here QCD-coloured vector and axial-vector states stemming from
composite Higgs dynamics in several relevant models based on an underlying
gauge-fermion description. These states can come as triplet, sextet and octet
representation. All models considered have a colour octet vector state in
common which can be singly produced at hadron colliders as it mixes with the
gluon. We explore the rich and testable phenomenology of these coloured spin-1
states at the LHC and future colliders.

##  1 Introduction

The exploration of composite Higgs models has garnered significant attention
in the realm of theoretical particle physics, as these models offer a possible
explanation for the nature of the Higgs boson discovered at CERN and a
dynamical origin for the breaking of the electroweak symmetry in the Standard
Model (SM) Englert:1964et ; Higgs:1964pj ; Guralnik:1964eu . By positing the
Higgs boson as a composite state that originates from a new strongly
interacting sector, composite Higgs models provide a potential solution to the
problem of hierarchy between the electroweak scale and the Planck scale: like
in quantum chromodynamics (QCD), the breaking scale is dynamically generated
via confinement and condensation of a new interaction. This idea is as old as
the SM itself Weinberg:1975gm ; Dimopoulos:1979es , starting from the first
Higgsless (Technicolor) theories Dimopoulos:1979za  and their effective
Lagrangian counterparts Casalbuoni:1985kq , to models where the Higgs emerges
as a meson Kaplan:1983fs ; Kaplan:1983sm . Composite model building has
resumed in the early 2000’s thanks to the idea of holography Contino:2003ve ;
Agashe:2004rs ; Hosotani:2005nz , freely adapted from supersymmetric string
theory inspired duality conjectures Maldacena:1997re .

The varied and rich phenomenology of composite Higgs models has been
extensively studied, both from the point of view of holography-inspired
effective models Contino:2010rs ; Panico:2015jxa  and from models based on
underlying gauge-fermion theories Cacciapaglia:2020kgq , the latter close in
spirit to QCD. While we do not attempt to summarise the main features, which
have been described several times, we want to recall the essential ingredients
of such theories, which are directly related to the electroweak symmetry
breaking. The Higgs typically emerges as a pseudo-Nambu Goldstone boson (pNGB)
Contino:2003ve  from the spontaneous breaking of the global symmetry in the
strong sector. Its potential and mass are generated by explicit breaking
terms: the gauging of the electroweak symmetry, the couplings of the top quark
Agashe:2004rs  and (eventually) a mass term for the underlying fermions
Galloway:2010bp ; Cacciapaglia:2014uja . In this framework, composite spin-1
states matching the electroweak gauge bosons and top partners have been widely
considered. From the Higgs sector point of view, they are the minimal
components required from the strong sector.

Nevertheless, the strong dynamics of composite Higgs models is much richer
than this. Whether it consists of an unspecified conformal field theory in the
holographic approach, or of a well-defined gauge-fermion theory, a more
extended spectrum is a generic prediction. In particular, the fact that top
partners Kaplan:1991dc  need to be charged under QCD interactions implies that
other coloured resonances beyond the top partners must exist. This implies the
presence of coloured spin-0 and spin-1 mesons, as well as fermions carrying
unusual colour charges. In this work we will focus on coloured spin-1
resonances, which are expected to exist in all types of composite Higgs
models. In holographic models, they emerge as Kaluza-Klein resonances of the
gluon field Guchait:2007ux . As we will show, however, a richer set of
coloured spin-1 states is to be expected.

For definiteness, we will focus on theories based on an underlying gauge-
fermion description, where the properties and quantum numbers of the
resonances can be classified. A systematic list of models describing the
minimal resonances needed by the Higgs sector has been presented by Ferretti
and Karateev Ferretti:2013kya . Consistent models with a single species of
fermions can only be based on $\text{SU}(3)$ Vecchi:2015fma  – like in QCD –
or $G_{2}$ Ferretti:2013kya  with fermions in the fundamental. However, models
with two separate species in different irreducible representations (irreps) of
the gauge group offer the intriguing possibility of sequestering QCD
interactions from the sector responsible for the electroweak symmetry breaking
Barnard:2013zea ; Ferretti:2013kya . Theoretical and phenomenological
considerations lead to the definition of 12 minimal models, whose
characteristics are fully specified Ferretti:2016upr ; Belyaev:2016ftv  in
terms of the confining gauge group and the irreps and multiplicities of the
two species of fermions. Upon confinement, both fermion species condense, as
confirmed by Lattice results for $\text{SU}(4)$ and $\text{Sp}(4)$ gauge
symmetries Ayyar:2017uqh ; Bennett:2023wjw , hence generating two sets of
pNGBs Ferretti:2016upr  (plus one coming from a global anomaly-free U(1)
Belyaev:2016ftv ). The symmetry breaking patterns are uniquely determined by
the type of irrep the two species belong to Cacciapaglia:2014uja , leading to
the classification in Table 1. The top partners emerge as so called “chimera”
baryons formed of the two species, where two different patterns can be
realised: $\psi\psi\chi$ and $\psi\chi\chi$, where $\psi$ only carry
electroweak charges while $\chi$ carry QCD colour and hypercharge. In the
former case, the $\chi$’s QCD triplet carries hypercharge $2/3$, in the latter
case $-1/3$.

QCD EW | SU(4)/Sp(4) | SU(5)/SO(5) | SU(4)2/SU(4)
---|---|---|---
SU(6)/Sp(6) |  |  |
| M5 ($\psi\chi\chi$) |
SU(6)/SO(6) | M8-9 ($\psi\psi\chi$) | M3-4 ($\psi\psi\chi$) | M10-11 ($\psi\psi\chi$)
| M1-2 ($\psi\chi\chi$) |
SU(3)2/SU(3) |  |  | M12 ($\psi\psi\chi$)
| M6-7 ($\psi\chi\chi$) |
Table 1:  Classification of the 12 models based on the global symmetry
breaking patterns in the electroweak and QCD sectors. In parenthesis, we
indicate the template for the chimera baryons, representing the top partners.

The phenomenology of the resonances from these 12 models have been studied in
the literature, covering some of the resonance types. So far, studies have
focused on the pNGBs charged under electroweak quantum numbers
Ferretti:2016upr ; Agugliaro:2018vsu ; Cacciapaglia:2022bax , the singlets
stemming from the global $\text{U}(1)$’s Ferretti:2016upr ; Belyaev:2016ftv ;
Cacciapaglia:2019bqz ; BuarqueFranzosi:2021kky , QCD coloured pNGBs
Cacciapaglia:2015eqa ; Belyaev:2016ftv ; Cacciapaglia:2020vyf , top partners
with non-standard decays Bizot:2018tds ; Xie:2019gya ; Cacciapaglia:2019zmj
or colour assignment Cacciapaglia:2021uqh , and spin-1 resonances carrying
electroweak charges BuarqueFranzosi:2016ooy . We also note that the spectra
and couplings of such resonances can be computed on the Lattice, and some
results are available for models based on $\text{Sp}(4)$ Bennett:2017kga ;
Bennett:2019cxd ; Bennett:2019jzz ; Bennett:2020hqd ; Bennett:2020qtj ;
Bennett:2022yfa ; Bennett:2023gbe ; Bennett:2023mhh , like models M5 and M8,
and based on $\text{SU}(4)$ Ayyar:2017qdf ; Ayyar:2018glg ; Ayyar:2018ppa ;
Ayyar:2018zuk ; Ayyar:2019exp ; Golterman:2020pyx ; Hasenfratz:2023sqa , like
models M6 and M11. Computations based on holography are also available
Erdmenger:2020lvq ; Erdmenger:2020flu ; Erdmenger:2023hkl .

In this work, we will focus on the phenomenology of spin-1 resonances that
carry QCD charges and emerge as bound states of the $\chi$ species. Their
properties emerge from three types of cosets, $\text{SU}(6)/\text{SO}(6)$,
$\text{SU}(6)/\text{Sp}(6)$ and $\text{SU}(3)\times\text{SU}(3)/\text{SU}(3)$,
and the hypercharge assignment for the colour triplet $\chi$, which stems from
the types of chimera baryons. The spectrum contains both a set of vectors
$\mathcal{V}^{\mu}$ and of axial-vectors $\mathcal{A}^{\mu}$, which decay
respectively into two or three pNGBs. The latter property originates from the
symmetric nature of the cosets. Mixing of the ubiquitous octet with the QCD
gluons will also generate direct couplings to quarks, while the colour
triplets and sextets may or may not couple to a pair of quarks depending on
their baryon number. To properly characterise the phenomenology of these
states, we will employ the hidden symmetry approach Bando:1987br  to write an
effective Lagrangian, and use the results to study their collider
phenomenology.

The paper is structured as follows: in Section 2 we briefly review the hidden
symmetry approach and present results in the allowed cases. In Section 3 we
analyse the phenomenology at the LHC and future high energy proton colliders.
Finally, we offer our conclusions in Section 4.

##  2 Hidden gauge symmetry approach

The hidden local symmetry method is based on the idea that the nonlinear
$\sigma$ model on the manifold $G/H$ is gauge equivalent to the $\sigma$ model
based on $G\times H_{\mathrm{local}}$. The gauge bosons corresponding to the
local symmetry can be identified with composite spin-1 mesons. The general
procedure for building an effective Lagrangian including these new spin-1
resonances Bando:1984ej ; Casalbuoni:1985kq ; Casalbuoni:1988xm  consists,
therefore, in a generalised group structure that splits the unitary matrix
$U(x)$ describing the Goldstone bosons into factors transforming under an
extended symmetry $G^{\prime}$.

The generators of the group $G$ can be indicated with $T^{A}$ where
$A=1,\ldots,d_{G}$ and $d_{G}$ is the dimension of the group $G$. These
generators can be separated into two classes, $T^{A}=\\{S^{a},X^{I}\\}$: the
unbroken generators $S^{a}$ with $a=1,\ldots,d_{H}$ belonging to the unbroken
subgroup $H\subset G$, and the broken generators $X^{I}$ with
$I=1,\ldots,d_{G}-d_{H}$ belonging to the coset $G/H$. The elements of $G$ are
of the form $g=e^{i\alpha^{A}T^{A}}$ and those of $H$ of the form
$h=e^{i\beta^{a}S^{a}}$. The elements of $G$ can be parameterised by $g=Uh$
with $U$ in the coset $G/H$

$U=e^{i\pi^{I}X^{I}}\;.$ (1)

For cosets of the type $\text{SU}(N)/\text{SO}(N)$,
$\text{SU}(2N)/\text{Sp}(2N)$ and
$\text{SU}(N)_{L}\times\text{SU}(N)_{R}/\text{SU}(N)_{V}$, the two classes of
generators are determined by the following constraints:

$S^{a}\Sigma_{0}+\Sigma_{0}S^{aT}=0\;,\;\;\;X^{I}\Sigma_{0}-\Sigma_{0}X^{IT}=0,$
(2)

see Peskin:1980gc ; Preskill:1980mz ; BuarqueFranzosi:2023xux  for details.
The Lagrangian in the condensate (“chiral”) phase is built using the standard
chiral Lagrangian elements:

$\Omega_{\mu}=iU^{\dagger}D_{\mu}U\;,\;\;\;D_{\mu}U=(\partial_{\mu}-i\,j_{\mu})\,U\;,$
(3)

with $\Omega_{\mu}$ the Maurer-Cartan form and $j_{\mu}$ the current
$j_{\mu}=v^{a}_{\mu}S^{a}+a^{\hat{a}}_{\mu}X^{\hat{a}}$. The form
$\Omega_{\mu}$ can be further decomposed into projections along the unbroken
and broken parts:

$\displaystyle e_{\mu}$ $\displaystyle=2\Tr(S^{a}\Omega_{\mu})\ S^{a}\,,$ (4)
$\displaystyle d_{\mu}$ $\displaystyle=2\Tr(X^{I}\Omega_{\mu})\ X^{I}\,,$ (5)

which will be explicitly used in writing the Lagrangian. The notation for the
current $j_{\mu}$ indicates that vector resonances $v^{a}_{\mu}$ are
associated to the unbroken generators of $H$, while axial-vectors
$a^{I}_{\mu}$ to the broken ones. This is a formal definition, while a direct
correspondence to vector and axial currents of fermions is only recovered in
QCD-like cases based on $\text{SU}(N)_{L}\times\text{SU}(N)_{R}$ group
symmetries.

For concreteness, in the rest of the section we will provide some details on
the effective construction for one of the cosets, based on
$\text{SU}(6)/\text{SO}(6)$. We will show how to extend the results to the
other two cosets (c.f. Tab. 1) at the end of the section.

###  2.1 Setup for $\text{SU}(6)/\text{SO}(6)$

Following the hidden symmetry prescriptions, we consider a model based on the
symmetry $G^{\prime}=\text{SU}(6)_{0}\times\text{SU}(6)_{1}$, where
$\text{SU}(6)_{0}$ is partly gauged by the SM gauge bosons (gluons and
hypercharge) and $\text{SU}(6)_{1}$ is fully gauged by the heavy resonances.
The enlarged symmetry is broken to $\text{SO}(6)_{0}\times\text{SO}(6)_{1}$ by
two sets of pNGBs, $\pi_{0}$ and $\pi_{1}$, so that a linear combination of
them gives mass to the axial resonances. Furthermore,
$\text{SO}(6)_{0}\times\text{SO}(6)_{1}$ is broken to the diagonal subgroup
$\text{SO}(6)$ by a second set of pNGBS, $k$, which gives mass to the vector
resonances.

We parameterise the two sets of pNGBs as

$U_{0}=\exp[\frac{i\sqrt{2}}{f_{0}}\,\pi_{0}^{I}X^{I}]\quad\text{and{
\quad}}U_{1}=\exp[\frac{i\sqrt{2}}{f_{1}}\,\pi_{1}^{I}X^{I}]\,,$ (6)

transforming under $\text{SU}(6)_{i}$ as

$U_{i}\to g_{i}U_{i}h(g_{i},\pi_{i})^{\dagger}.$ (7)

We also define a Maurer-Cartan form for each sector:

$\Omega_{i,\mu}=iU_{i}^{\dagger}D_{\mu}U_{i}\,,$ (8)

with covariant derivatives

$\displaystyle D_{\mu}U_{0}$
$\displaystyle=(\partial_{\mu}-i\hat{g}_{s}\mathbf{G}_{\mu}-i\hat{g}^{\prime}\mathbf{B}_{\mu})\,U_{0}\,,$
(9) $\displaystyle D_{\mu}U_{1}$
$\displaystyle=(\partial_{\mu}-i\tilde{g}\boldsymbol{\mathcal{V}}_{\mu}-i\tilde{g}\boldsymbol{\mathcal{A}}_{\mu})\,U_{1}\,,$
(10)

where the gauge fields act via the commutator, $[\mathbf{G}_{\mu},U_{0}]$ etc,
and

$\displaystyle\mathbf{B}_{\mu}=B_{\mu}\,T_{X},\quad\mathbf{G}_{\mu}=G^{a}_{\mu}\,T_{G}^{a},\quad\boldsymbol{\mathcal{V}}_{\mu}=\mathcal{V}_{\mu}^{a}\,S^{a},\quad\boldsymbol{\mathcal{A}}_{\mu}=\mathcal{A}^{I}_{\mu}\,X^{I}\,,$
(11)

where $T_{X}$ and $T_{G}^{a}$ are the generators of $\text{SO}(6)_{0}$
corresponding to hypercharge and QCD colour, respectively. The colour
multiplets are embedded in the $\text{SO}(6)$ matrices as

where $\boldsymbol{\phi}_{8}=\frac{1}{2}\phi_{8}^{a}\lambda^{a}$ with the
Gell-Mann matrices $\lambda^{a}$,
$\boldsymbol{\phi}_{6}=\boldsymbol{\phi}_{6}^{T}$, and
$\boldsymbol{\phi}_{3}=-\boldsymbol{\phi}_{3}^{T}$. To employ the CCWZ
construction Coleman:1969sm ; Callan:1969sn , we define the components of the
Maurer-Cartan forms $d_{i,\mu}$ and $e_{i,\mu}$ parallel and orthogonal to
$\text{SO}(6)_{i}$ as in Eqs. 4 and 5. They transform under $\text{SU}(6)_{i}$
as

$\displaystyle d_{i,\mu}$ $\displaystyle\to
h(g_{i},\pi_{i})\,d_{i,\mu}\,h^{\dagger}(g_{i},\pi_{i})\,,$ (12)
$\displaystyle e_{i,\mu}$ $\displaystyle\to
h(g_{i},\pi_{i})(e_{i,\mu}+i\partial_{\mu})h^{\dagger}(g_{i},\pi_{i})\,.$ (13)

We refer the reader to sec. A.2 for the explicit calculation of the CCWZ
symbols.

For the $\text{SO}(6)_{0}\times\text{SO}(6)_{1}\to\text{SO}(6)$ breaking we
introduce a second set of pNGBs

$K=\exp[\frac{i}{f_{K}}k^{a}\,S^{a}]$ (14)

transforming as:

$K\to h(g_{0},\pi_{0})\,K\,h^{\dagger}(g_{1},\pi_{1})$ (15)

with covariant derivative:

$D_{\mu}K=\partial_{\mu}K-ie_{0,\mu}K+iKe_{1,\mu}\,.$ (16)

###  2.2 The Lagrangian

From the previous considerations, and in a similar way to what was obtained in
the corresponding $\text{SU}(4)$ case in BuarqueFranzosi:2016ooy , the most
general, leading-order Lagrangian reads:

$\displaystyle\mathcal{L}=$
$\displaystyle-\frac{1}{2}\Tr\mathbf{G}_{\mu\nu}\mathbf{G}^{\mu\nu}-\frac{1}{2}\Tr\mathbf{B}_{\mu\nu}\mathbf{B}^{\mu\nu}-\frac{1}{2}\Tr\boldsymbol{\mathcal{F}}_{\mu\nu}\boldsymbol{\mathcal{F}}^{\mu\nu}$
$\displaystyle+\frac{f_{0}^{2}}{2}\Tr
d_{0,\mu}d_{0}^{\mu}+\frac{f_{1}^{2}}{2}\Tr d_{1,\mu}d_{1}^{\mu}$
$\displaystyle+\frac{f^{2}_{K}}{2}\Tr
D^{\mu}K(D_{\mu}K)^{\dagger}+rf_{1}^{2}\,\Tr d_{0,\mu}Kd^{\mu}_{1}K^{\dagger}$
$\displaystyle+\mathcal{L}_{\mathrm{fermions}}$ (17)

where

$\boldsymbol{\mathcal{F}}_{\mu}=\boldsymbol{\mathcal{V}}_{\mu}+\boldsymbol{\mathcal{A}}_{\mu}$
(18)

contains all the massive resonances. We recall that for a generic gauge field
$\mathbf{V}_{\mu}$,

$\mathbf{V}_{\mu\nu}=\partial_{\mu}\mathbf{V}_{\nu}-\partial_{\nu}\mathbf{V}_{\mu}-ig[\mathbf{V}_{\mu},\mathbf{V}_{\nu}]\,,$
(19)

where $g$ is the appropriate coupling. In the unitary gauge, where
$K=\mathbb{1}$, the kinetic term for $K$ simplifies to

$\displaystyle\Tr D_{\mu}K(D^{\mu}K)^{\dagger}$ $\displaystyle=\Tr
e_{0,\mu}e_{0}^{\mu}+\Tr e_{1,\mu}e_{1}^{\mu}-2\Tr e_{0,\mu}e_{1}^{\mu}\,.$
(20)

Expanding the above Lagrangian will allow us to compute the mass eigenstates
(elementary vectors and resonances do mix) and their couplings.

###  2.3 Vector boson masses and mixing

The masses and mixing of the vector resonances stem from the pNGB matrix $K$.
The three terms in Eq. 20 read

$\displaystyle\Tr(e_{0,\mu}e_{0}^{\mu})$
$\displaystyle\supset\hat{g}_{s}^{2}\Tr(\mathbf{G}_{\mu}\mathbf{G}^{\mu})+\hat{g}^{\prime\,2}\Tr(\mathbf{B}_{\mu}\mathbf{B}^{\mu})=\frac{\hat{g}^{2}_{s}}{2}G^{a}_{\mu}G^{\mu,a}+\frac{\hat{g}^{\prime\,2}}{2}B_{\mu}B^{\mu}\,,$
(21) $\displaystyle\Tr(e_{1,\mu}e_{1}^{\mu})$
$\displaystyle\supset\tilde{g}^{2}\Tr(\boldsymbol{\mathcal{V}}_{\mu}\boldsymbol{\mathcal{V}}^{\mu})=\frac{1}{2}\tilde{g}^{2}\mathcal{V}_{\mu}^{a}\mathcal{V}^{a,\mu}\,,$
(22) $\displaystyle\Tr(e_{0,\mu}e_{1}^{\mu})$
$\displaystyle\supset\Tr((\hat{g}_{s}\mathbf{G}_{\mu}+\hat{g}^{\prime}\mathbf{B}_{\mu})\tilde{g}\boldsymbol{\mathcal{V}}^{\mu}))=\frac{1}{2}\hat{g}_{s}\tilde{g}\,G^{a}_{\mu}\mathcal{V}_{8}^{a,\mu}+\frac{1}{2}\hat{g}^{\prime}\tilde{g}\,B_{\mu}\mathcal{V}_{1}^{\mu}\,,$
(23)

where, from the last line, we see that the colour octet and singlet components
mix with gluons and the hypercharge gauge boson, respectively.

The Lagrangian contains a simple mass term for the colour-triplet state:

$M_{\mathcal{V}_{3}}=\frac{\tilde{g}f_{K}}{\sqrt{2}}.$ (24)

For the other states, a mixed mass term emerges. Starting with the colour
octets:

$\displaystyle\mathcal{L}$
$\displaystyle\supset\frac{f_{K}^{2}}{4}\tilde{g}^{2}\,\mathcal{V}^{a}_{8,\mu}\mathcal{V}_{8}^{a,\mu}+\frac{f_{K}^{2}}{4}\hat{g}_{s}^{2}\,G^{a}_{\mu}G^{a,\mu}-\frac{f_{K}^{2}}{2}\hat{g}_{s}\tilde{g}\,G^{a}_{\mu}\mathcal{V}_{8}^{a,\mu}=\frac{1}{2}V_{8,\mu}^{a,T}\
\mathcal{M}_{8}^{2}\ V^{a,\mu}_{8}\,,$ (25)

where

$V^{a}_{8,\mu}=\matrixquantity(G^{a}_{\mu}\\\
\mathcal{V}^{a}_{8,\mu}),\qquad\mathcal{M}_{8}^{2}=\frac{f_{K}^{2}}{2}\matrixquantity(\hat{g}_{s}^{2}&-\hat{g}_{s}\tilde{g}\\\
-\hat{g}_{s}\tilde{g}&\tilde{g}^{2})\,.$ (26)

Diagonalising the mass matrix, we find a massless eigenstate, which
corresponds to the physical gluon octet, and a massive state, corresponding to
the octet vector resonance. The latter have mass

$M_{\mathcal{V}_{8}}=\frac{f_{K}}{\sqrt{2}}\sqrt{\tilde{g}^{2}+\hat{g}_{s}^{2}}\,.$
(27)

With some abuse of notation, we can switch to the physical mass eigenstates by
replacing

$\matrixquantity(G^{a}_{\mu}\\\
\mathcal{V}^{a}_{8,\mu})\to\matrixquantity(\cos\beta_{8}&-\sin\beta_{8}\\\
\sin\beta_{8}&\cos\beta_{8})\matrixquantity(G^{a}_{\mu}\\\
\mathcal{V}^{a}_{8,\mu}),\qquad\tan\beta_{8}=\frac{\hat{g}_{s}}{\tilde{g}}\lesssim
1\,.$ (28)

Finally, the gauge coupling associated to the massless gluons reads

$g_{s}=\hat{g}_{s}\cos\beta_{8}=\tilde{g}\sin\beta_{8}=\frac{\hat{g}_{s}\tilde{g}}{\sqrt{\hat{g}_{s}^{2}+\tilde{g}^{2}}}\,,$
(29)

and this corresponds to the physical coupling of QCD interactions.

A similar mixing pattern emerges in the singlet, leading to

$M_{\mathcal{V}_{1}}=\frac{f_{K}}{\sqrt{2}}\sqrt{\tilde{g}^{2}+\hat{g}^{\prime\,2}},\qquad\tan\beta_{1}=\frac{\hat{g}^{\prime}}{\tilde{g}}\,,$
(30)

with the caveat that the hypercharge will also mix with a spin-1 resonance
stemming from the electroweak sector of the composite theory. Such a mixing
has been studied for the $\text{SU}(4)/\text{Sp}(4)$ coset in
BuarqueFranzosi:2016ooy . Combining the two sectors will, therefore, lead to a
more complicated mixing pattern. We will not further pursue the analysis of
the electroweak sector in this work, as we are interested in the phenomenology
of the coloured resonances, which are more abundantly produced at hadron
colliders.

###  2.4 Axial masses and scalar mixing

From the $d_{1}^{2}$ term, we obtain a mass for the axial vectors:

$\displaystyle\frac{f_{1}^{2}}{2}\Tr
d_{1,\mu}d_{1}^{\mu}\supset\frac{f_{1}^{2}}{2}\Tr(\tilde{g}^{2}\boldsymbol{\mathcal{A}}_{\mu}\boldsymbol{\mathcal{A}}^{\mu}-2\tilde{g}\partial_{\mu}\boldsymbol{\pi}_{1}\boldsymbol{\mathcal{A}}^{\mu})\,,$
(31)

while a mixing with $\pi_{0}$ is generated by the $d_{0,\mu}d_{1}^{\mu}$ term.
The mixing terms can be removed with an appropriate choice of gauge fixing,
leaving a common mass term for all the axial vectors:

$\displaystyle M_{\mathcal{A}}=\frac{\tilde{g}f_{1}}{\sqrt{2}}\,.$ (32)

The mesons $\pi_{0}$ and $\pi_{1}$ undergo a non-trivial mixing, analogous to
the case studied in BuarqueFranzosi:2016ooy , hence we will simply recall the
basics here. As the $d_{i}$ forms give

$\displaystyle
d_{i,\mu}=-\frac{\sqrt{2}}{f_{i}}D_{\mu}\boldsymbol{\pi}_{i}+\cdots\,,$ (33)

at leading order in the expansion, the Lagrangian contains a kinetic mixing of
the form:

$\displaystyle\mathcal{L}\supset\Tr(D_{\mu}\boldsymbol{\pi}_{0}D^{\mu}\boldsymbol{\pi}_{0}+D_{\mu}\boldsymbol{\pi}_{1}D^{\mu}\boldsymbol{\pi}_{1}+2r\frac{f_{1}}{f_{0}}\,D_{\mu}\boldsymbol{\pi}_{0}D^{\mu}\boldsymbol{\pi}_{1})\,.$
(34)

Hence, one can define decoupled and canonically normalised fields $\pi_{A}$
and $\pi_{B}$ as

$\displaystyle\pi_{0}$
$\displaystyle=\frac{\pi_{A}}{\sqrt{2}\sqrt{1+r\,f_{1}/f_{0}}}-\frac{\pi_{B}}{\sqrt{2}\sqrt{1-r\,f_{1}/f_{0}}}\,,$
(35) $\displaystyle\pi_{1}$
$\displaystyle=\frac{\pi_{A}}{\sqrt{2}\sqrt{1+r\,f_{1}/f_{0}}}+\frac{\pi_{B}}{\sqrt{2}\sqrt{1-r\,f_{1}/f_{0}}}\,.$
(36)

A linear combination of these states is eaten by the $\mathcal{A}_{\mu}$. The
physical $\pi_{P}$ and unphysical $\pi_{U}$ states are given by

$\displaystyle\pi_{A}=\cos\alpha\,\pi_{P}-\sin\alpha\,\pi_{U},\qquad\pi_{B}=\sin\alpha\,\pi_{P}+\cos\alpha\,\pi_{U}$
(37)

where

$\tan\alpha=-\sqrt{\frac{1+r\,f_{1}/f_{0}}{1-r\,f_{1}/f_{0}}}.$ (38)

Combining the above redefinitions yields

$\pi_{0}=\pi_{P}\,\frac{1}{\sqrt{1-r^{2}f_{1}^{2}/f_{0}^{2}}},\qquad\pi_{1}=\pi_{U}-\pi_{P}\,\frac{rf_{1}/f_{0}}{\sqrt{1-r^{2}f_{1}^{2}/f_{0}^{2}}}.$
(39)

In the unitary gauge, only the $\pi_{P}$ remain in the spectrum, and they
correspond to the pNGBs from the coset $\text{SU}(6)/\text{SO}(6)$.

###  2.5 Decay channels

We are now ready to determine the main decay modes for the heavy spin-1
resonances. They are generated by three types of interactions:

  * •

Couplings to pNGBs from the chiral Lagrangian in the strong sector, Eq. (2.2);

  * •

Couplings to quarks via the mixing of the colour octet to gluons;

  * •

Partial compositeness couplings to top and bottom quarks.

The first type stems directly from the pNGB embedding in the effective
Lagrangian. We recall that, in the unitary gauge, the relevant terms simplify
to

$\displaystyle\mathcal{L}\supset\,$ $\displaystyle\frac{f_{0}^{2}}{2}\Tr
d_{0,\mu}d_{0}^{\mu}+\frac{f_{1}^{2}}{2}\Tr
d_{1,\mu}d_{1}^{\mu}+rf_{1}^{2}\,\Tr d_{0,\mu}d_{1}^{\mu}$
$\displaystyle+\frac{f^{2}_{K}}{2}\Tr
e_{0,\mu}e_{0}^{\mu}+\frac{f^{2}_{K}}{2}\Tr e_{1,\mu}e_{1}^{\mu}-f^{2}_{K}\Tr
e_{0,\mu}e_{1}^{\mu}\,.$ (40)

We are interested in terms linear in the vector fields
$\mathcal{V}_{\mu}/\mathcal{A}_{\mu}$ and with the smallest number of pNGBs.
It turns out that these only come as two independent traces:

$\displaystyle\mathcal{O}_{V}$
$\displaystyle=i\Tr([\boldsymbol{\pi},\partial_{\mu}\boldsymbol{\pi}]\boldsymbol{V}^{\mu}),$
(41) $\displaystyle\mathcal{O}_{\mathcal{A}}$
$\displaystyle=\Tr([\boldsymbol{\pi},[\boldsymbol{\pi},\partial_{\mu}\boldsymbol{\pi}]]\boldsymbol{\mathcal{A}}^{\mu}),$
(42)

where $V=\mathcal{V},G,B$ is a generic vector. We recover explicitly that
vectors couple to two pNGBs, while axial resonances can only couple to three
pNGBs. Both $\mathcal{O}_{V}$ and $\mathcal{O}_{\mathcal{A}}$ are hermitian.
After transforming the pions and vectors to the physical fields, we find that
these operators come with coefficients

$\displaystyle C_{\mathcal{V}_{3}}$
$\displaystyle=\frac{\tilde{g}(r^{2}-1)f_{K}^{2}}{f_{0}^{2}\ (1-R^{2})}\equiv
g_{\rho\pi\pi}\,,$ (43) $\displaystyle C_{\mathcal{V}_{8}}$
$\displaystyle=\frac{(r^{2}-1)f_{K}^{2}}{f_{0}^{2}\
\left(1-R^{2}\right)}(\tilde{g}\cos\beta_{8}+\hat{g}_{s}\sin\beta_{8})+2\frac{1+R^{2}}{1-R^{2}}\hat{g}_{s}\sin\beta_{8}$
$\displaystyle=\frac{g_{\rho\pi\pi}}{\cos\beta_{8}}+2\frac{1+R^{2}}{1-R^{2}}g_{s}\tan\beta_{8}\,,$
(44) $\displaystyle C_{\mathcal{A}}$
$\displaystyle=\frac{\sqrt{2}\tilde{g}r}{3}(1-r^{2})(f_{1}^{2}-3f_{K}^{2})\equiv
g_{a3\pi}\,;$ (45)

where $R=rf_{1}/f_{0}$. The details of this calculation are presented in sec.
A.4. The colour structure of the couplings among the various components are
determined uniquely by the above traces in the $\text{SU}(6)$ space.

The second type of couplings originates from the mixing of the gluon with
$\mathcal{V}_{8}$, hence yielding a universal couplings of the massive
resonance to quarks:

$\displaystyle\mathcal{L}_{\mathrm{fermions}}\supset
i\bar{q}\not{D}q\supset\hat{g}_{s}\,\bar{q}\,\not{G}^{a}t^{a}_{\mathbf{3}}\,q\to-
g_{s}\tan\beta_{8}\,\bar{q}\,\not{\mathcal{V}}_{8}^{a}t^{a}_{\mathbf{3}}\,q\equiv
C_{\mathcal{V}_{8}}^{qq}\mathcal{O}_{\mathcal{V}_{8}}^{qq}\,,$ (46)

where $t_{\mathbf{3}}^{a}=\lambda^{a}/2$ are the colour generators for the
fundamental irrep. Note that the massless octet inherits a coupling
$\hat{g}_{s}\cos\beta_{8}\equiv g_{s}$, hence consistent with QCD gauge
invariance. A coupling to two gluons, instead, is not generated, as shown in
sec. A.3.

Finally, the third type is generated by the coupling of the spin-1 resonances
to the baryons Erkol:2006sa ; Aliev:2009ei  that mix to top quarks via the
partial compositeness mechanism. While the couplings generated by the strong
dynamics are inherently vector-like, the chiral mixing of the physical states
generates chiral couplings to the mass eigenstates. Details of the origin of
these couplings are presented in sec. A.5. Such couplings always exist for the
colour octet states, and they can be parameterised as

$\displaystyle\mathcal{L}_{\text{PC}}\supset$
$\displaystyle\;\bar{t}\not{\mathcal{V}}_{8}^{a}t^{a}_{\mathbf{3}}\left(g_{\rho
tt,LL}P_{L}+g_{\rho
tt,RR}P_{R}\right)t+\bar{b}\not{\mathcal{V}}_{8}^{a}t^{a}_{\mathbf{3}}\left(g_{\rho
bb,LL}P_{L}\right)b+$
$\displaystyle\bar{t}\not{\mathcal{A}}_{8}^{a}t^{a}_{\mathbf{3}}\left(-g_{att,LL}P_{L}+g_{att,RR}P_{R}\right)t+\bar{b}\not{\mathcal{A}}_{8}^{a}t^{a}_{\mathbf{3}}\left(-g_{abb,LL}P_{L}\right)b\,,$
(47)

where $P_{L,R}$ are chiral projectors and we only consider the electric part
of the coupling. In the models under consideration, we have that
$g_{\rho/abb,LL}\simeq g_{\rho/att,LL}$ while the bottom coupling is only
left-handed at leading order. Note that all the above couplings are of order
$\tilde{g}$, while the chiralities are distinguished by the different mixing
angles from partial compositeness. The non-octet resonances, $\mathcal{V}_{3}$
and $\mathcal{A}_{6}$, couple to a pair of quarks via partial compositeness
only in models where the two resonances have baryon number $2/3$ and charge
$\pm 4/3$, hence leading to two top decay channels. The effective couplings
can be parameterised as

$\mathcal{L}_{\text{PC}}\supset g_{\rho tt,LR}\
\overline{t^{c}}\not{\mathcal{V}}_{3}t+g_{att,LR}\
\overline{t}\not{\mathcal{A}}_{6}t^{c}+\text{h.c.}$ (48)

where the superscript $c$ indicates charge conjugation. As the currents
contain effectively one left-handed and one right-handed top, the couplings
must be suppressed by the EW scale $v$ divided by the Higgs decay constant as
compared to the octet couplings.

Note finally that, while the first two types of couplings are completely
determined by the chiral Lagrangian in Eq. (2.2), the third one is more model
dependent. In fact, the value of the couplings depend on the quantum numbers
of baryons that mix with the elementary top fields, and on the value of the
mixing angles. 111Couplings to light quarks could also be generated by partial
compositeness, however their couplings will be generically suppressed by the
small mixing required by the lightness of the quark masses. Hence, such
contributions can be neglected compared to the mixing with the gluon. Hence,
they cannot be predicted in a model-independent way and we will leave them as
free parameters.

###  2.6 Independent parameters

The effective Lagrangian for the coloured spin-1 resonances contains six free
parameters: $f_{0}$, $f_{1}$, $f_{K}$, $r$, $\hat{g}_{s}$, and $\tilde{g}$
(the mixing angles depend only on $\tilde{g}$). As we have seen, $\hat{g}_{s}$
can be fixed by the physical coupling of the massless gluons, as in Eq. (29).
We can trade $f_{1}$ and $f_{K}$ for masses:

$\displaystyle f_{1}=\frac{\sqrt{2}M_{\mathcal{A}}}{\tilde{g}},\qquad
f_{K}=\frac{\sqrt{2}M_{\mathcal{V}_{3}}}{\tilde{g}}=\frac{\sqrt{2}M_{\mathcal{V}_{8}}}{\sqrt{\tilde{g}^{2}+\hat{g}_{s}^{2}}}=\frac{\sqrt{2}M_{\mathcal{V}_{1}}}{\sqrt{\tilde{g}^{2}+\hat{g}^{\prime}{}^{2}}}\,,$
(49)

hence we can choose as input parameters $M_{\mathcal{V}_{8}}$ and the ratio

$\displaystyle\xi=\frac{M_{\mathcal{A}}}{M_{\mathcal{V}_{8}}}\,.$ (50)

Note that the relation between the two vector masses only depends on the octet
mixing angle, i.e. on $\tilde{g}$, as
$M_{\mathcal{V}_{3}}=M_{\mathcal{V}_{8}}\cos\beta_{8}$. We can further use as
an input the physical decay constant of the pNGBs $f_{\chi}$, which enters the
couplings of the physical $\pi_{P}$ states and reads:

$\displaystyle f_{\chi}=\sqrt{f_{0}^{2}-r^{2}f_{1}^{2}}\,.$ (51)

Finally, another input parameter can be the coupling of the vectors to the
pion, $g_{\rho\pi\pi}$, which can be measured on the lattice, for instance. It
relates to the Lagrangian parameters as follows:

$\displaystyle g_{\rho\pi\pi}=\left.C_{\mathcal{V}_{8}}\right|_{\beta_{8}\to
0}=C_{\mathcal{V}_{3/6}}=\frac{\tilde{g}(r^{2}-1)f_{K}^{2}}{f_{0}^{2}\
(1-R^{2})}\,.$ (52)

Solving Eqs. 51 and 52 for $f_{0}$ and $r$ yields

$\displaystyle
f_{0}=\sqrt{\frac{f_{1}^{2}f_{\chi}^{2}}{f_{K}^{2}}\frac{g_{\rho\pi\pi}}{\tilde{g}}+f_{1}^{2}+f_{\chi}^{2}}\,,\qquad
r=\sqrt{1+\frac{f_{\chi}^{2}g_{\rho\pi\pi}}{f_{K}^{2}\tilde{g}}}\,.$ (53)

In summary, this leaves us with five independent input parameters:

$\displaystyle\tilde{g},\quad g_{\rho\pi\pi},\quad
M_{\mathcal{V}_{8}},\quad\xi,\quad f_{\chi}\,.$ (54)

As already mentioned, in addition we have the couplings to top and bottom
quarks generated by top partial compositeness.

###  2.7 Generalisation to $\text{SU}(6)/\text{Sp}(6)$ and
$\text{SU}(3)\times\text{SU}(3)/\text{SU}(3)$

For the $\text{SU}(6)/\text{Sp}(6)$ case, the computation of the effective
Lagrangian follows the same patterns as described above, with the only
difference in the broken and unbroken generators. Effectively, this implies
that colour charges of the non-octet states are interchanged:
$\mathcal{V}_{6}$ and $\mathcal{A}_{3}$ (as well as $\pi_{3}$). The
coefficients of the various couplings and mass values, however, follow the
same results as above.

For the case $\text{SU}(3)\times\text{SU}(3)/\text{SU}(3)$, the action of the
symmetries are slightly different in form. However, for this coset all vector
and axial resonances, as well as the pNGBs, transform as octets. Hence, the
effective interactions are the same as above, once the non-octet states are
removed.

##  3 Phenomenology

| Models |  $\chi$ ($R,Y,B$) | $\pi$ | $\mathcal{V}^{\mu}$ | $\mathcal{A}^{\mu}$ | $\Psi$ | di-quark
---|---|---|---|---|---|---|---
C1 | M1-2 |  (R,$-\frac{1}{3}$,$\frac{1}{6}$) | $8_{0},\,{\color[rgb]{1,0,0}6_{-2/3}}$ | $8_{0},\,1_{0},\,{\color[rgb]{1,0,0}3_{2/3}}$ | $8_{0},\,{\color[rgb]{1,0,0}6_{-2/3}}$ | $8,\,1,\,{\color[rgb]{1,0,0}3},\,{\color[rgb]{1,0,0}6}$ | none
C2 | M3-4, M8-11 |  (R,$\frac{2}{3}$,$\frac{1}{3}$) | $8_{0},\,{\color[rgb]{0,0,1}6_{4/3}}$ | $8_{0},\,1_{0},\,{\color[rgb]{0,0,1}3_{-4/3}}$ | $8_{0},\,{\color[rgb]{0,0,1}6_{4/3}}$ | ${\color[rgb]{1,0,0}3}$ | $\pi_{6},\mathcal{V}^{\mu}_{3},\mathcal{A}^{\mu}_{6}$
C3 | M5 |  (Pr,$-\frac{1}{3}$,$\frac{1}{6}$) | $8_{0},\,{\color[rgb]{1,0,0}3_{2/3}}$ | $8_{0},\,1_{0},\,{\color[rgb]{1,0,0}6_{-2/3}}$ | $8_{0},\,{\color[rgb]{1,0,0}3_{2/3}}$ | $8,\,1,\,{\color[rgb]{1,0,0}3},\,{\color[rgb]{1,0,0}6}$ | none
C4 | M6-7 |  (C,$-\frac{1}{3}$,$\frac{1}{6}$) | $8_{0}$ | $8_{0},\,1_{0}$ | $8_{0}$ | $8,\,1,\,{\color[rgb]{1,0,0}3},\,{\color[rgb]{1,0,0}6}$ | none
C5 | M12 |  (C,$\frac{2}{3}$,$\frac{1}{3}$) | $8_{0}$ | $8_{0},\,1_{0}$ | $8_{0}$ | ${\color[rgb]{1,0,0}3}$ | none
Table 2:  Properties of the spin-0 ($\pi$), spin-1 ($\mathcal{V}^{\mu}$,
$\mathcal{A}^{\mu}$) and spin-1/2 ($\Psi$) lightest resonances in the 12
models, grouped in 5 classes. Each class is determined by the properties of
the $\chi$ species, listed in the second column by irrep type (R for real, Pr
for pseudo-real and C for complex). For the resonances, the colours indicate
the baryon numbers, with black for $B=0$, red for $B=\pm 1/3$ and blue for
$B=\pm 2/3$. In the last column we indicate the bosons that can decay into a
di-quark state ($tt$).

The twelve models under consideration allow us to predict the quantum numbers
of the lightest coloured resonances. Following the properties of the fermion
species $\chi$, they can be grouped into five classes, as shown in Tab. 2. For
the fermionic states, the electroweak charges depend on the configuration of
the $\psi$ fermions inside the chimera baryons, and a full classification is
possible, but beyond our purposes. In fact, we will assume here a lattice and
QCD inspired mass hierarchy, where the baryon-like states are heavier than the
spin-1 states, which are heavier than the pNGBs. Henceforth, the heavy baryons
do not have a direct relevance for the phenomenology of the spin-1 states,
except for the fact that their couplings can generate a direct coupling of the
spin-1 resonances to a pair of tops via the top partial compositeness mixing,
as discussed in the previous section.

The coloured spin-1 resonances, therefore, can be produced via their QCD
interactions at hadron colliders. This leads to pair production for all types
of states. The only one that also features single-production is the vector
colour octet, as it inherits a universal coupling to all quarks via its mixing
to the gluons. As the masses of the spin-1 resonances are expected to be of
the same order, we will first study the LHC limits on the vector colour octet
to determine the smallest allowed mass. Before doing that, however, it is
important to recall the properties of the coloured pNGBs, which appear in the
decays of all spin-1 resonances. Finally we will present first results for
future high energy hadron colliders, which could access pair production of all
the resonances.

###  3.1 Coloured pNGB decays

The phenomenology of the coloured pNGBs have been studied in several works and
contexts Cacciapaglia:2015eqa ; Belyaev:2016ftv ; Cacciapaglia:2020vyf , hence
we will here only remind their main features.

A colour octet pNGB is ubiquitous to all models. It always features two types
of couplings: a coupling to gauge bosons generated by a topological anomaly
and one to tops generated by partial compositeness Belyaev:2016ftv ;
Cacciapaglia:2020vyf . Which one dominates, however, depends crucially on the
details of the model, as their origin is rather different in nature. Note that
the anomaly dominantly consists of couplings to two gluons, however it also
generates suppressed couplings to $g\gamma$ and $gZ$, which provide
interesting and clean final states Belyaev:1999xe ; Cacciapaglia:2020vyf .
Nevertheless, to simplify the analysis and focus on existing searches, we will
neglect the single-gluon decay channels in the following.

The decays of the non-octets depend crucially on the scenario at hand.
Following the classification in Table 2, we distinguish four cases:

$\displaystyle\text{C1}:$ $\displaystyle\qquad\pi_{8}\to
t\bar{t},\,gg;\,\,\pi_{6}\to bb\,,$ (61) $\displaystyle\text{C2}:$
$\displaystyle\qquad\pi_{8}\to t\bar{t},\,gg;\,\,\pi_{6}\to tt\,,$ (62)
$\displaystyle\text{C3}:$ $\displaystyle\qquad\pi_{8}\to
t\bar{t},\,gg;\,\,\pi_{3}\to\bar{b}\bar{s}\text{ or
}t\bar{\nu},\,b\tau^{+}\,,$ (63) $\displaystyle\text{C4-5}:$
$\displaystyle\qquad\pi_{8}\to t\bar{t},\,gg\,.$ (64)

In C2, the sextet has baryon number $2/3$ and charge $4/3$, hence partial
compositeness will generate an unsuppressed coupling to two right-handed tops
Cacciapaglia:2015eqa . In C1 and C3, the sextet and triplet have baryon number
$\mp 1/3$ and charges $\mp 2/3$, respectively, hence they are not allowed to
decay into standard model fermions by partial compositeness alone. Their
decays must, therefore, be generated by specific operators that need to
violate either baryon or lepton number. Considering the standard model gauge
quantum numbers Carpenter:2021rkl , the allowed final states are listed above.
The di-quark final state violates baryon number by one unit, $\Delta B=1$, and
we consider preferential couplings to heavier flavours (while this is not
strictly required). For the triplet, decays to a quark and a lepton can be
envisioned, violating lepton number by one unit, $\Delta L=1$. They can be
generated in some models by partial compositeness extended to leptons
Cacciapaglia:2021uqh , hence naturally involving the third (heavier) family.
In such case, the triplet effectively behaves like a composite leptoquark
Gripaios:2009dq . We remark that the $B$ or $L$ violating couplings can be
rather small, however they provide the only decay channel for the sextet and
triplet states. Depending on the value of these couplings, therefore, they
decay promptly, as we consider in the following, or lead to displaced vertices
and anomalously massive hadronic tracks Kraan:2004tz .

The ubiquitous colour octet can be searched at the LHC via QCD pair
production: in the following, we will assume dominant top couplings, hence
leading to a four tops final state. A recent reinterpretation Darme:2021gtt
of a CMS search CMS:2019rvj  leads to a conservative lower bound of $1.25$ TeV
for the colour octet mass. Note that in C2 models, the contribution of the
sextet can further push up this limit. Dedicated searches also exist for the
leptoquark decays of the triplet in C3 models, where both $b\tau$ and $t\nu$
final states have been searched for by ATLAS and CMS ATLAS:2021jyv ;
CMS:2020wzx ; CMS:2023qdw ; ATLAS:2023uox ; ATLAS:2024huc  yielding bounds
between $1.25$ and $1.46$ TeV, depending on the branching ratios in the two
channels. The bounds on this mass are significantly lower of about $770$ GeV
if $\pi_{3}$ decays dominantly into light quarks ATLAS:2017jnp ; CMS:2022usq .

###  3.2 Colour octet single production at the LHC

Figure 1: Feynman diagrams of $\mathcal{V}_{8}$ single production and decay
into quarks or pNGBs. Figure 2: Sample branching ratios for $g_{\rho\pi\pi}=1$
and $M_{\mathcal{V}_{8}}=4.5$ TeV. We also fix $m_{\pi}=1.4$ TeV, $f_{\chi}=1$
TeV and the coupling to top quarks to 1. Note that here $q=u,d,s,c,b$.

Via mixing to the gluons, the vector octet $\mathcal{V}_{8}$ inherits a
coupling to all quarks, see Eq. 46, allowing it to be singly produced. This
coupling is suppressed by a mixing angle that depends only on $\tilde{g}$,
hence it is reduced for large $\tilde{g}$, see also Eq. (29). Nevertheless,
even for moderate values, the single production cross section dominates over
pair production. Henceforth, the colour octet is the main resonance to be
hunted at hadron colliders, and it has been considered in the literature in
various composite contexts Lillie:2007yh . Typically, decays into two quarks
are considered, while we will also include decays into two coloured pNGBs as
shown in Fig. 1.

In the models under consideration, the possible decay modes of the vector
octet can be classified as follows:

$\displaystyle\text{C1-2}:\qquad$ $\displaystyle\mathcal{V}_{8}\to
q\bar{q},\;b\bar{b},\;t\bar{t},\;\pi_{8}\pi_{8},\;\pi_{6}\pi_{6}^{c},$ (65)
$\displaystyle\text{C3}:$ $\displaystyle\mathcal{V}_{8}\to
q\bar{q},\;b\bar{b},\;t\bar{t},\;\pi_{8}\pi_{8},\;\pi_{3}\pi_{3}^{c},$ (66)
$\displaystyle\text{C4-5}:$ $\displaystyle\mathcal{V}_{8}\to
q\bar{q},\;b\bar{b},\;t\bar{t},\;\pi_{8}\pi_{8},$ (67)

where C1 and C2 are distinguished by the decays of the sextet pNGB. The decays
into light quarks $q=u,d,c,s$ feature flavour-independent branching ratios,
while bottom and top quark channels receive additional contributions from
partial compositeness, see sec. 2.5, leading to different branching ratios.
Finally, the relative strength of the pNGB channels is determined purely by
colour factors, assuming their masses are equal, and we find

$\frac{\text{BR}(\pi_{6}\pi_{6}^{c})}{\text{BR}(\pi_{8}\pi_{8})}=\frac{10}{3}\,,\qquad\frac{\text{BR}(\pi_{3}\pi_{3}^{c})}{\text{BR}(\pi_{8}\pi_{8})}=\frac{2}{3}\,.$
(68)

The importance of each channel depends on the parameter space, and we provide
some benchmarks in Fig. 2. The relevance of the decays into light quarks and
pNGBs depends mainly on the $\tilde{g}$ and $g_{\rho\pi\pi}$ couplings. On the
one hand, the partial width to light quarks is controlled by the mixing angle
to gluons and it decreases for increasing $\tilde{g}$. On the other hand, the
partial width to pNGBs receives a dominant contribution proportional to
$g_{\rho\pi\pi}$: the dependence on $\tilde{g}$ is such that this partial
width also decreases for increasing $\tilde{g}$. For very small $\tilde{g}$,
instead, the second term in Eq. (44) starts becoming relevant, thus explaining
the drop in the $qq$ branching ratio observed in Fig. 2. The scaling in
$\tilde{g}$ also explains why the total width of $\mathcal{V}_{8}$ increases
for small values and for large octet masses. Finally, the branching ratio to
top (and bottom) receives a dominant contribution from partial compositeness,
which do not scale with $\tilde{g}$ and hence dominates for large values.

Figure 3: Isocurves of width/mass for $\xi=1.0$ and $f_{\chi}=1$ TeV for
different values of $\tilde{g}$ for classes C1-2. The coupling to top quarks
is fixed to 1.

One caveat is that the coupling $g_{\rho\pi\pi}$ and the coupling to baryons
are expected to be large in the strong theory, hence the colour octet will
tend to have large width as compared to its mass. To quantify this important
effect, we show in Fig. 3 curves of fixed width over mass ratios for different
values of $\tilde{g}$ as a function of $g_{\rho\pi\pi}$ and the octet mass.
The plots highlight the fact that the width can be larger than 50% of the
mass, especially for small values of $\tilde{g}$, hence invalidating the
treatment of the octet in a standard narrow width approximation.

Finally, the current bounds on the colour octet mass crucially depend on the
branching ratios in the three channels: light quarks, pNGBs and tops, as they
are controlled by different couplings. As a model-independent estimate of the
bounds we, therefore, decided to show limits assuming 100% branching ratios in
the three channels, as shown in Fig. 4 for the four nonequivalent classes C1,
C2, C3 and C4-5. The lines are extracted from the following searches:

  * •

di-jet: Search for high mass di-quark resonances CMS:2019gwf ;

  * •

di-top: Search for $t\bar{t}$ resonances ATLAS:2020lks ;

  * •

pNGBs: Recasts of SUSY searches ATLAS:2021twp ; ATLAS:2019fag ; ATLAS:2022ihe
implemented in CheckMATE222 For the simulation of signal events, we
implemented the relevant interactions as a FeynRules Alloul:2013bka  model at
leading order. For each mass point, we generate $10^{4}$ events using
MadGraph5_aMC@NLO Alwall:2014hca  version 3.5.3, in association with the
parton densities in the NNPDF 2.3 set Ball:2012cx ; Buckley:2014ana . We then
interfaced the events with Pythia8 Sjostrand:2014zea  for showering and
hadronisation. The resulting showered signal events are analysed with
CheckMATE Drees:2013wra ; Dercks:2016npn  (commit number 1cb3f7). To this end,
events are reconstructed using Delphes 3 deFavereau:2013fsa  and the
anti-$k_{T}$ algorithm Cacciari:2008gp  implemented in FastJet Cacciari:2011ma
. We also ran the events against the searches and SM measurements implemented
in MadAnalysis5 Conte:2012fm ; Conte:2014zja ; Dumont:2014tja ; Conte:2018vmg
version 1.10.9beta and Rivet Bierlich:2019rhm  version 3.1.8 in combination
with Contur Butterworth:2019wnt ; Buckley:2021neu  version 2.2.4, but both
yielded subdominant bounds compared to CheckMATE..

The coloured heatmap indicates the Drell-Yan cross section, which only depends
on $\tilde{g}$ and the octet mass, while the region below and to the left of
the lines is excluded. The results show that the mass limits are roughly the
same for all cases, and comparable for the three decay modes. Hence, the mass
limits are in the range of $4$ to $5$ TeV. Note that the region for small
$\tilde{g}\lesssim 3$ cannot be trusted as it corresponds to widths above 50%
of the mass, c.f. Fig. 3.

The High-Luminosity run at the LHC will certainly allow to further improve the
mass limits on the vector octet, with the caveat that dedicated searches or
reinterpretations will be needed to take into account the large width.
However, even with the current bounds in Fig. 4, we can infer that pair
production of all spin-1 resonances will be very small at the LHC, hence
making their detection unlikely. In the next section, therefore, we will
discuss pair production at a future high energy hadron collider.

|
---|---
|
Figure 4:  Bounds on vector octet single production cross section in the four
cases C1, C2, C3 and C4-5. For each bound we assume 100% branching ratio into
that channel.

###  3.3 Pair production at future high-energy hadron colliders

Future hadron collider projects are expected to reach energies well above the
LHC, with expectations up to $100$ TeV FCC:2018vvp ; Tang:2022fzs . At such
energies, pair production of the vector and axial resonances will be
accessible. In principle, single production of the vector octet remains the
leading channel, with the caveat that at large masses the width will also
increase and hence affect the search strategy. In this section, we will focus
on pair production. The cross sections only depend on the QCD quantum numbers
of the spin-1 states, and they are the same for vector and axial-vector
states. In Fig. 5 we show them as a function of the masses for $pp$ collisions
at $100$ TeV centre of mass, using the NNPDF 2.3 PDF set Ball:2012cx ;
Buckley:2014ana . Hence, the most abundantly produced states will be the
sextets, followed by octets, while triplet production is about one order of
magnitude below.

Figure 5:  Pair production cross section at $\sqrt{s}=100$ TeV for vectors in
the sextet, octet and triplet representations. The same values apply for axial
vectors.

In the following, we describe the main features to be expected from pair
production, leaving a detailed analysis for future work. Firstly, we should
stress that single production of the vector octet remains the leading
discovery channel, including non-resonant effects that are relevant in the
case of very large width. Hence, at a future $100$ TeV collider, one would
expect to discover the octet before pair production becomes relevant.

The sextets, which feature the largest pair production cross sections, are
present in the classes C1, C2 and C3. The decays can be classified as follows:

$\displaystyle\mathcal{A}_{6}$ $\displaystyle\to$
$\displaystyle\pi_{8}\pi_{8}\pi_{6}\
(t\bar{t}t\bar{t}bb)\;\text{and}\;\pi_{6}\pi_{6}^{c}\pi_{6}\
(\bar{b}\bar{b}bbbb)\quad\text{in C1}\,,$ (71) $\displaystyle\mathcal{A}_{6}$
$\displaystyle\to$ $\displaystyle tt\;\text{or}\;\pi_{8}\pi_{8}\pi_{6}\
(t\bar{t}t\bar{t}tt)\;\text{and}\;\pi_{6}\pi_{6}^{c}\pi_{6}\
(\bar{t}\bar{t}tttt)\quad\text{in C2}\,,$ (72) $\displaystyle\mathcal{V}_{6}$
$\displaystyle\to$
$\displaystyle\pi_{8}\pi_{3}^{c}\to(t\bar{t})(\bar{b}\bar{s}\;\text{or}\;ql)\quad\text{in
C3}\,.$ (73)

The two-body decay of $\mathcal{V}_{6}$ in C3 is driven by $g_{\rho\pi\pi}$,
hence is will likely produce a large decay width. Instead, thanks to the three
body final state for $\mathcal{A}_{6}$ in pNGBs, we expect the axial widths to
remain small compared to the mass. In the cases C2, a competitive decay into
two tops is also present: the two-body top decay width is, in fact, suppressed
by $v^{2}/f^{2}\leq 0.04$ (for $f\geq 1$ TeV), while the three-pNGB channel is
suppressed by a phase space factor as compared to the two-body channel. Hence,
we expect the two to lead to competitive branching ratios. In both cases C1
and C2, the pair production of the sextet will lead to final states with many
top and bottom jets.

The axial colour octet $\mathcal{A}_{8}$ also has a sizeable pair cross
section, and it also has a leading decay channel into two tops in all cases.
Hence, it will generate 4-top final states with potentially large width
effects.

Finally, the colour triplets, present in C1, C2 and C3, lead to the following
decays:

$\displaystyle\mathcal{V}_{3}$ $\displaystyle\to$
$\displaystyle\pi_{8}\pi_{6}^{c}\ (t\bar{t}\bar{b}\bar{b})\;\quad\text{in
C1}\,,$ (74) $\displaystyle\mathcal{V}_{3}$ $\displaystyle\to$
$\displaystyle\bar{t}\bar{t}\;\text{or}\;\pi_{8}\pi_{6}^{c}\
(t\bar{t}\bar{t}\bar{t})\quad\text{in C2}\,,$ (75)
$\displaystyle\mathcal{A}_{3}$ $\displaystyle\to$
$\displaystyle\pi_{8}\pi_{8}\pi_{3}\;(t\bar{t}t\bar{t}+\bar{b}\bar{s}\;\text{or}\;ql)\;\text{or}\;\pi_{3}\pi_{3}^{c}\pi_{3}\;(\bar{b}\bar{s}bs\bar{b}\bar{s}\;\text{or}\;ql\bar{q}lql)\quad\text{in
C3}\,.$ (76)

For the vectors, the decays will be largely dominated by the
$\pi_{8}\pi_{6}^{c}$ channels, as the di-top coupling in C2 is suppressed by
$v/f$. The caveat remains that the vector widths may be large in most of the
allowed parameter space. Instead, the axial in C3 remains narrow, leading to
interesting final states rich in tops and possibly leptons, if the $\pi_{3}$
decays violate lepton number.

##  4 Conclusions

We have investigated the phenomenology of spin-1 resonances in Composite Higgs
Models carrying QCD charges, with particular attention on production and decay
modes, LHC bounds, and future hadron collider prospects. We have in particular
focused on models which allow for fermionic UV completions Ferretti:2016upr ;
Belyaev:2016ftv  as they provide detailed information on the quantum numbers
and properties of the bound states. We have worked out their properties for
three types of cosets, $\text{SU}({6})/\text{SO}({6})$,
$\text{SU}({6})/\text{Sp}({6})$ and
$\text{SU}({3})\times\text{SU}({3})/\text{SU}({3})$ and the most relevant
production and decay channels at present and future pp-colliders. The
considered cosets are symmetric and they, therefore, contain two sets of
spin-1 resonances: vector states that couple to two pNGBs and axial-vector
states that couple to three pNGBs.

In all scenarios, the vector $\mathcal{V}_{8}$ in the adjoint representations
of colour $\text{SU}({3})$ is present, and it mixes with the QCD gluon octet.
Thanks to the mixing, this state can be singly produced at hadron colliders
via Drell-Yan, whereas all other states can only be pair-produced. The
$\mathcal{V}_{8}$ can either decay into a quark pair or into two pNGBs leading
in all cases to the final states $q\bar{q}$ ($q\neq t$), $t\bar{t}$ and $4t$.
In a coset with triplet or sextet pNGBs, in addition one has a subset of the
following final states: $4b$, $2b2s$ or $2t2\nu$, $2b2\tau$ and $t\nu b\tau$.
We have investigated in all cases bounds on the mass ranging from $3.5$ TeV to
$6$ TeV from existing LHC data. We have focused on scenarios where
$\mathcal{V}_{8}$ has a sufficiently small decay width so that the narrow
width approximation holds. Hence, pair production is only relevant for a
future high energy hadron collider, where pairs of the sextets and octets will
be abundantly produced. We classified all permitted final states, which are
typically rich in top quarks and leptons. Further studies, including the large
width case and specific prospects for future 100 TeV pp-collider, as well as
more model dependent signatures will be analysed in a separate publication.

## Acknowledgements

We thank Alan Cornell for discussions and collaboration during the initial
stages of this project. We are grateful to the Mainz Institute for Theoretical
Physics (MITP) of the DFG Cluster of Excellence PRISMA+ (Project ID 39083149)
for its hospitality and support during the final stages of this work. This
work has been supported by the “DAAD, Frankreich” and “Partenariat Hubert
Curien (PHC)” PROCOPE 2021-2023, project number 57561441. M.K. and W.P. are
supported by DFG, project no. PO-1337/12-1. M.K. is supported by the
“Studienstiftung des deutschen Volkes”.

##  Appendix A Details on the calculation

###  A.1 Conventions

It is convenient to embed a field $\phi_{\mathbf{r}}$ in the $\mathbf{r}$
irrep of QCD within $3\times 3$ matrices in the notation of Carpenter:2021rkl
:

$\displaystyle\phi_{3}=\phi_{3,i}\ L^{i},\quad\phi_{6}=\phi_{6,s}\
K^{s},\quad\phi_{8}=\phi_{8}^{a}\ t^{a}_{\mathbf{3}},\quad\phi_{1}=\phi_{1}\
\mathbb{1}\,,$ (77)

where

$\displaystyle[L^{i}]^{jk}=L^{ijk}=\frac{1}{\sqrt{2}}\epsilon^{ijk},\quad
t^{a}_{\mathbf{3}}=\frac{1}{2}\lambda^{a}$ (78)

and

The matrices are normalised as follows:

$\displaystyle\Tr(L^{i}L_{j})=\frac{1}{2}\epsilon^{ikl}\epsilon_{jkl}=\delta^{i}_{\phantom{i}j},\quad\Tr(K^{s}K_{t})=\delta^{s}_{\phantom{s}t},\quad\Tr(t^{a}_{\mathbf{3}}t^{b}_{\mathbf{3}})=\frac{1}{2}\delta^{ab}.$
(79)

For the $\text{SU}(6)/\text{SO}(6)$ coset, the fields can be embedded within
the symmetric and anti-symmetric two-index irrep as follows:

(80) (81)

###  A.2 CCWZ symbols

In the first sector of the hidden-symmetry extended coset, the Maurer-Cartan
form reads:

$\displaystyle\Omega_{0,\mu}$
$\displaystyle=iU_{0}^{\dagger}(\partial_{\mu}-i\hat{g}_{s}\mathbf{G}_{\mu}-i\hat{g}^{\prime}\mathbf{B}_{\mu})U_{0}$
(82)
$\displaystyle=-\frac{\sqrt{2}}{f_{0}}\partial_{\mu}\boldsymbol{\pi}_{0}+\frac{i}{f_{0}^{2}}[\boldsymbol{\pi}_{0},\partial_{\mu}\boldsymbol{\pi}_{0}]+\frac{\sqrt{2}}{3f_{0}^{3}}[\boldsymbol{\pi}_{0},[\boldsymbol{\pi}_{0},\partial_{\mu}\boldsymbol{\pi}_{0}]]+\cdots$
$\displaystyle\quad+\hat{g}_{s}\left(\mathbf{G}_{\mu}-\frac{\sqrt{2}i}{f_{0}}[\boldsymbol{\pi}_{0},\mathbf{G}_{\mu}]-\frac{1}{f_{0}^{2}}[\boldsymbol{\pi}_{0},[\boldsymbol{\pi}_{0},\mathbf{G}_{\mu}]]+\cdots\right)$
$\displaystyle\quad+\hat{g}^{\prime}\left(\mathbf{B}_{\mu}-\frac{\sqrt{2}i}{f_{0}}[\boldsymbol{\pi}_{0},\mathbf{B}_{\mu}]-\frac{1}{f_{0}^{2}}[\boldsymbol{\pi}_{0},[\boldsymbol{\pi}_{0},\mathbf{B}_{\mu}]]+\cdots\right),$
(83)

where the dots indicate higher orders in the pNGB fields. Reading off the
components for $d_{0,\mu}$ and $e_{0,\mu}$, we find:

$\displaystyle d_{0,\mu}$
$\displaystyle=-\frac{\sqrt{2}}{f_{0}}\left(\partial_{\mu}\boldsymbol{\pi}_{0}-i\hat{g}_{s}[\boldsymbol{\pi}_{0},\mathbf{G}_{\mu}]-i\hat{g}^{\prime}[\boldsymbol{\pi}_{0},\mathbf{B}_{\mu}]\right)+\frac{\sqrt{2}}{3f_{0}^{3}}[\boldsymbol{\pi}_{0},[\boldsymbol{\pi}_{0},\partial_{\mu}\boldsymbol{\pi}_{0}]]+\cdots$
(84)
$\displaystyle=-\frac{\sqrt{2}}{f_{0}}D_{\mu}\boldsymbol{\pi}_{0}+\frac{\sqrt{2}}{3f_{0}^{3}}[\boldsymbol{\pi}_{0},[\boldsymbol{\pi}_{0},\partial_{\mu}\boldsymbol{\pi}_{0}]]+\cdots$
(85) $\displaystyle e_{0,\mu}$
$\displaystyle=\hat{g}_{s}\mathbf{G}_{\mu}+\hat{g}^{\prime}\mathbf{B}_{\mu}+\frac{i}{f_{0}^{2}}[\boldsymbol{\pi}_{0},\partial_{\mu}\boldsymbol{\pi}_{0}]-\frac{\hat{g}_{s}}{f_{0}^{2}}[\boldsymbol{\pi}_{0},[\boldsymbol{\pi}_{0},\mathbf{G}_{\mu}]]-\frac{\hat{g}^{\prime}}{f_{0}^{2}}[\boldsymbol{\pi}_{0},[\boldsymbol{\pi}_{0},\mathbf{B}_{\mu}]]+\cdots$
(86)
$\displaystyle=\hat{g}_{s}\mathbf{G}_{\mu}+\hat{g}^{\prime}\mathbf{B}_{\mu}+\frac{i}{f_{0}^{2}}[\boldsymbol{\pi}_{0},D_{\mu}\boldsymbol{\pi}_{0}]+\cdots$
(87)

For the second sector, containing the heavy spin-1 resonances, we find:

$\displaystyle\Omega_{1,\mu}$
$\displaystyle=iU_{1}^{\dagger}(\partial_{\mu}-i\tilde{g}\boldsymbol{\mathcal{V}}_{\mu}-i\tilde{g}\boldsymbol{\mathcal{A}}_{\mu})U_{1}$
(88)
$\displaystyle=-\frac{\sqrt{2}}{f_{1}}\partial_{\mu}\boldsymbol{\pi}_{1}+\frac{i}{f_{1}^{2}}[\boldsymbol{\pi}_{1},\partial_{\mu}\boldsymbol{\pi}_{1}]+\frac{\sqrt{2}}{3f_{1}^{3}}[\boldsymbol{\pi}_{1},[\boldsymbol{\pi}_{1},\partial_{\mu}\boldsymbol{\pi}_{1}]]+\cdots$
$\displaystyle\quad+\tilde{g}\left(\boldsymbol{\mathcal{V}}_{\mu}-\frac{\sqrt{2}i}{f_{1}}[\boldsymbol{\pi}_{1},\boldsymbol{\mathcal{V}}_{\mu}]-\frac{1}{f_{1}^{2}}[\boldsymbol{\pi}_{1},[\boldsymbol{\pi}_{1},\boldsymbol{\mathcal{V}}_{\mu}]]+\cdots\right)$
$\displaystyle\quad+\tilde{g}\left(\boldsymbol{\mathcal{A}}_{\mu}-\frac{\sqrt{2}i}{f_{1}}[\boldsymbol{\pi}_{1},\boldsymbol{\mathcal{A}}_{\mu}]-\frac{1}{f_{1}^{2}}[\boldsymbol{\pi}_{1},[\boldsymbol{\pi}_{1},\boldsymbol{\mathcal{A}}_{\mu}]]+\cdots\right).$
(89)

Reading off the two components:

$\displaystyle d_{1,\mu}$
$\displaystyle=\tilde{g}\boldsymbol{\mathcal{A}}_{\mu}-\frac{\sqrt{2}}{f_{1}}\partial_{\mu}\boldsymbol{\pi}_{1}-\frac{\sqrt{2}i\tilde{g}}{f_{1}}[\boldsymbol{\pi}_{1},\boldsymbol{\mathcal{V}}_{\mu}]-\frac{\tilde{g}}{f_{1}^{2}}[\boldsymbol{\pi}_{1},[\boldsymbol{\pi}_{1},\boldsymbol{\mathcal{A}}_{\mu}]]+\frac{\sqrt{2}}{3f_{1}^{3}}[\boldsymbol{\pi}_{1},[\boldsymbol{\pi}_{1},\partial_{\mu}\boldsymbol{\pi}_{1}]]+\cdots$
(90) $\displaystyle e_{1,\mu}$
$\displaystyle=\tilde{g}\boldsymbol{\mathcal{V}}_{\mu}-\frac{\sqrt{2}i\tilde{g}}{f_{1}}[\boldsymbol{\pi}_{1},\boldsymbol{\mathcal{A}}_{\mu}]+\frac{i}{f_{1}^{2}}[\boldsymbol{\pi}_{1},\partial_{\mu}\boldsymbol{\pi}_{1}]-\frac{\tilde{g}}{f_{1}^{2}}[\boldsymbol{\pi}_{1},[\boldsymbol{\pi}_{1},\boldsymbol{\mathcal{V}}_{\mu}]]+\cdots$
(91)

The elements above are used as building blocks for the effective Lagrangian we
used in the main text.

###  A.3 Couplings to gluons

The couplings of the coloured resonances to QCD gluons stem from the kinetic
terms of the elementary and composite states. For simplicity, we consider here
only the couplings to vectors $\mathcal{V}_{r}$. In the hidden symmetry
approach, the gauge kinetic terms read

$\displaystyle\mathcal{L}_{\mathrm{gauge}}=-\frac{1}{2}\Tr\mathbf{G}_{\mu\nu}\mathbf{G}^{\mu\nu}-\frac{1}{2}\Tr\boldsymbol{\mathcal{F}}_{\mu\nu}\boldsymbol{\mathcal{F}}^{\mu\nu}$
(92)

before the mixing of the elementary gluon with $\mathcal{V}_{8}$. At zeroth
order in all couplings, $\mathcal{O}(g^{0})$, we have

$\displaystyle\mathcal{L}_{g^{0}}\supset-\frac{1}{4}(\partial_{\mu}G_{\nu}^{a}-\partial_{\nu}G_{\mu}^{a})^{2}-\frac{1}{4}(\partial_{\mu}\mathcal{V}_{8\nu}^{a}-\partial_{\nu}\mathcal{V}_{8\mu}^{a})^{2}-\frac{1}{2}\big{|}\partial_{\mu}\mathcal{V}_{3\nu,i}-\partial_{\nu}\mathcal{V}_{3\mu,i}\big{|}^{2}.$
(93)

At linear order, $\mathcal{O}(g^{1})$, there are derivative terms, which can
be written as

$\displaystyle\mathcal{L}_{g}=-\frac{1}{2}\hat{g}_{s}\,$ $\displaystyle
f^{abc}\,(\partial_{\mu}G^{a}_{\nu}-\partial_{\nu}G^{a}_{\mu})G^{b}_{\mu}G^{c}_{\nu}$
$\displaystyle-4i\tilde{g}\Big{[}$
$\displaystyle(\partial_{\mu}\mathcal{V}_{8\nu}^{a}-\partial_{\nu}\mathcal{V}_{8\mu}^{a})\mathcal{V}_{8}^{b,\mu}\mathcal{V}_{8}^{c\nu}\Tr(t^{a}_{\mathbf{3}}t^{b}_{\mathbf{3}}t^{c}_{\mathbf{3}})$
$\displaystyle-(\partial_{\mu}\mathcal{V}_{8\nu}^{a}-\partial_{\nu}\mathcal{V}_{8\mu}^{a})\mathcal{V}_{3,i}^{c\mu}\mathcal{V}_{3,j}^{\nu}\Tr(t_{\mathbf{3}}^{a}L^{i}L^{j})$
$\displaystyle-(\partial_{\mu}\mathcal{V}_{3\nu,i}^{c}-\partial_{\nu}\mathcal{V}_{3\mu,i}^{c})\mathcal{V}_{3,j}^{\mu}\mathcal{V}_{8}^{a,\nu}\Tr(L^{i}L^{j}t^{a}_{\mathbf{3}})+\text{h.c.}\Big{]}$
(94)

with traces

$\displaystyle\Tr(t^{a}_{\mathbf{3}}t^{b}_{\mathbf{3}}t^{c}_{\mathbf{3}})=\frac{1}{4}(d^{abc}+if^{abc}),\qquad\Tr(L^{i}L^{j}t^{a}_{\mathbf{3}})=\frac{1}{2}[t^{a}_{\mathbf{3}}]^{ij}.$
(95)

The fully symmetric term with $d^{abc}$ falls out due to symmetry, hence
leaving

$\displaystyle\mathcal{L}_{g}$
$\displaystyle=-\frac{1}{2}\hat{g}_{s}\,f^{abc}\,(\partial_{\mu}G^{a}_{\nu}-\partial_{\nu}G^{a}_{\mu})G^{b}_{\mu}G^{c}_{\nu}+i\tilde{g}\Bigg{[}\frac{1}{2}(\partial_{\mu}\mathcal{V}_{8\nu}^{a}-\partial_{\nu}\mathcal{V}_{8\mu}^{a})\mathcal{V}_{8}^{b,\mu}\mathcal{V}_{8}^{c,\nu}\,if^{abc}$
$\displaystyle\quad-(\partial_{\mu}\mathcal{V}_{8\nu}^{a}-\partial_{\nu}\mathcal{V}_{8\mu}^{a})\mathcal{V}_{3,i}^{c\mu}\mathcal{V}_{3,j}^{\nu}\,[t^{a}_{\mathbf{3}}]_{ij}-\big{(}(\partial_{\mu}\mathcal{V}_{3\nu,i}^{c}-\partial_{\nu}\mathcal{V}_{3\mu,i}^{c})\mathcal{V}_{3,j}^{\mu}\mathcal{V}_{8}^{a,\nu}[t^{a}_{\mathbf{3}}]_{ij}+\text{h.c.}\big{)}\Bigg{]}.$
(96)

Finally, the $\mathcal{O}(g^{2})$ terms read

$\displaystyle\mathcal{L}_{g^{2}}$
$\displaystyle=-\frac{\hat{g}_{s}^{2}}{4}\,f^{abc}f^{ade}\,G^{b}_{\mu}G^{c}_{\nu}G^{d,\mu}G^{e,\nu}-\tilde{g}^{2}\Bigg{[}\frac{1}{4}\mathcal{V}_{8\mu}^{a}\mathcal{V}_{8\nu}^{b}\mathcal{V}_{8}^{c,\mu}\mathcal{V}_{8}^{d,\nu}\,f^{abe}f^{cde}$
$\displaystyle\quad+i\,\mathcal{V}_{8\mu}^{a}\mathcal{V}_{8\nu}^{b}\mathcal{V}_{3,i}^{c\mu}\mathcal{V}_{3,j}^{\nu}\,f^{abc}\,[t^{c}_{\mathbf{3}}]_{ij}+(\mathcal{V}_{3\mu,j}\mathcal{V}_{8\nu}^{a}-\mathcal{V}_{3\nu,j}\mathcal{V}_{8\mu}^{a})\mathcal{V}_{3,i}^{c\mu}\mathcal{V}_{8}^{b,\nu}\,[t^{b}_{\mathbf{3}}t^{a}_{\mathbf{3}}]_{ij}\Bigg{]}.$
(97)

We now take into account the octet mixing, redefining the fields to mass
eigenstates (c.f. main text) as follows

$\displaystyle
G_{\mu}\stackrel{{\scriptstyle\mathrm{phys}}}{{\longrightarrow}}\cos\beta_{8}\,G_{\mu}-\sin\beta_{8}\,\mathcal{V}_{8,\mu},\qquad\mathcal{V}_{8,\mu}\stackrel{{\scriptstyle\mathrm{phys}}}{{\longrightarrow}}\sin\beta_{8}\,G_{\mu}+\cos\beta_{8}\,\mathcal{V}_{8,\mu},$
(98)

and we only show the couplings involving two heavy vectors, which are
phenomenologically relevant for pair production via QCD interactions. We hence
neglect terms of $\mathcal{O}(\mathcal{V}^{3})$ and
$\mathcal{O}(\mathcal{V}^{4})$. The kinetic terms in Eq. 93 remain unaffected
by the field redefinition. Instead, the $\mathcal{O}(g^{1})$ terms lead to the
following couplings (up to two $\mathcal{V}$ fields):

$\displaystyle\mathcal{L}_{g}=$
$\displaystyle-\frac{1}{2}g_{s}\,f^{abc}\,(\partial_{\mu}G^{a}_{\nu}-\partial_{\nu}G^{a}_{\mu})G^{b\mu}G^{c\nu}$
$\displaystyle-\frac{1}{2}g_{s}\,f^{abc}\,(\partial_{\mu}G^{a}_{\nu}-\partial_{\nu}G^{a}_{\mu})\mathcal{V}^{b\mu}_{8}\mathcal{V}^{c\nu}_{8}-g_{s}f^{abc}(\partial_{\mu}\mathcal{V}^{a}_{8\nu}-\partial_{\nu}\mathcal{V}^{a}_{8\mu})G^{b\mu}\mathcal{V}^{c\nu}_{8}$
$\displaystyle-
ig_{s}(\partial_{\mu}G_{\nu}^{a}-\partial_{\nu}G_{\mu}^{a})\mathcal{V}_{3,i}^{c\mu}\mathcal{V}_{3,j}^{\nu}\,[t^{a}_{\mathbf{3}}]_{ij}-g_{s}\big{(}i(\partial_{\mu}\mathcal{V}_{3\nu,i}^{c}-\partial_{\nu}\mathcal{V}_{3\mu,i}^{c})\mathcal{V}_{3,j}^{\mu}G^{a,\nu}[t^{a}_{\mathbf{3}}]_{ij}+\text{h.c.}\big{)}.$
(99)

Note that there is no $\mathcal{V}_{8}$-$G$-$G$ coupling. Finally we turn to
the $\mathcal{O}(g^{2})$ terms:

$\displaystyle\mathcal{L}_{g^{2}}=$
$\displaystyle-\frac{g_{s}^{2}}{4}\,f^{abe}f^{cde}\,G^{a}_{\mu}G^{b}_{\nu}G^{c,\mu}G^{d,\nu}$
$\displaystyle-\frac{g_{s}^{2}}{2}\,f^{abe}f^{cde}\,\big{(}G^{a}_{\mu}G^{b}_{\nu}\mathcal{V}^{c\mu}_{8}\mathcal{V}_{8}^{d\nu}+G^{a}_{\mu}\mathcal{V}^{b}_{8\nu}G^{c\mu}\mathcal{V}_{8}^{d\nu}+G^{a}_{\mu}\mathcal{V}^{b}_{8\nu}\mathcal{V}^{c\mu}_{8}G^{d\nu}\big{)}$
$\displaystyle-
ig_{s}^{2}\,G_{\mu}^{a}G_{\nu}^{b}\mathcal{V}_{3,i}^{c\mu}\mathcal{V}_{3,j}^{\nu}\,f^{abc}\,[t^{c}_{\mathbf{3}}]_{ij}-g_{s}^{2}(\mathcal{V}_{3\mu,j}G_{\nu}^{a}-\mathcal{V}_{3\nu,j}G_{\mu}^{a})\mathcal{V}_{3,i}^{c\mu}G^{b,\nu}\,[t^{b}_{\mathbf{3}}t^{a}_{\mathbf{3}}]_{ij}.$
(100)

We remark that couplings with three $\mathcal{V}$ and one gluon are also
present, but they are only relevant for triple production:

$\mathcal{L}_{g^{2}}\supset-\frac{1}{2}g_{s}(\tilde{g}c_{8}^{3}-\hat{g}_{s}s_{8}^{3})\,f^{abe}f^{cde}\,\big{(}G^{a}_{\mu}V^{b}_{8\nu}V^{c\mu}_{8}V_{8}^{d\nu}+V^{a}_{8\mu}G^{b}_{\nu}V^{c\mu}_{8}V_{8}^{d\nu}\big{)}$
(101)

and the coupling is not fully fixed by gauge invariance Zerwekh:2012bf .

###  A.4 Couplings to pNGBs

We recall that the Lagrangian in unitary gauge reads:

$\displaystyle\mathcal{L}=$
$\displaystyle-\frac{1}{2g_{s}^{2}}\Tr\mathbf{G}_{\mu\nu}\mathbf{G}^{\mu\nu}-\frac{1}{2g^{\prime\,2}}\Tr\mathbf{B}_{\mu\nu}\mathbf{B}^{\mu\nu}-\frac{1}{2\tilde{g}^{2}}\Tr\boldsymbol{\mathcal{F}}_{\mu\nu}\boldsymbol{\mathcal{F}}^{\mu\nu}$
$\displaystyle+\frac{f_{0}^{2}}{2}\Tr
d_{0,\mu}d_{0}^{\mu}+\frac{f_{1}^{2}}{2}\Tr
d_{1,\mu}d_{1}^{\mu}+rf_{1}^{2}\,\Tr d_{0,\mu}d_{1}^{\mu}$
$\displaystyle+\frac{f^{2}_{K}}{2}\Tr
e_{0,\mu}e_{0}^{\mu}+\frac{f^{2}_{K}}{2}\Tr e_{1,\mu}e_{1}^{\mu}-f^{2}_{K}\Tr
e_{0,\mu}e_{1}^{\mu}$ $\displaystyle+\mathcal{L}_{\mathrm{fermions}}\,,$ (102)

where it is the $d^{2}$\- and $e^{2}$-terms that contain couplings of the
spin-1 resonances to the pNGBs. It turns out that these interactions only come
as two independent
traces333$\Tr(\partial_{\mu}\boldsymbol{\pi}[\boldsymbol{\pi},\boldsymbol{V}^{\mu}])=-\Tr([\boldsymbol{\pi},\partial_{\mu}\boldsymbol{\pi}]\boldsymbol{V}^{\mu})$
and
$\Tr([\boldsymbol{\pi},[\boldsymbol{\pi},\partial\boldsymbol{\pi}]]\boldsymbol{\mathcal{A}})=-\Tr([\boldsymbol{\pi},\partial\boldsymbol{\pi}][\boldsymbol{\pi},\boldsymbol{\mathcal{A}}])=\Tr(\partial\boldsymbol{\pi}[\boldsymbol{\pi},[\boldsymbol{\pi},\boldsymbol{\mathcal{A}}]])$.:

$\displaystyle\mathcal{O}_{V}$
$\displaystyle=i\Tr([\boldsymbol{\pi},\partial_{\mu}\boldsymbol{\pi}]\boldsymbol{V}^{\mu}),$
(103) $\displaystyle\mathcal{O}_{\mathcal{A}}$
$\displaystyle=\Tr([\boldsymbol{\pi},[\boldsymbol{\pi},\partial_{\mu}\boldsymbol{\pi}]]\boldsymbol{\mathcal{A}}^{\mu}),$
(104)

where $V=\mathcal{V},G,B$ is a generic vector. Both $\mathcal{O}_{V}$ and
$\mathcal{O}_{\mathcal{A}}$ are hermitian. Traces with two pNGBs and one
axial-vector vanish as they contain three broken generators of the coset.

When rotating the pNGBs to the physical eigenstates with Eq. 39, $\pi_{0}$ and
$\pi_{1}$ only differ in the prefactor:

$\pi_{0}=\pi\,\frac{1}{\sqrt{1-R^{2}}},\qquad\pi_{1}=-\pi\,\frac{R}{\sqrt{1-R^{2}}},\qquad\mathrm{with}\quad
R=r\frac{f_{1}}{f_{0}},$ (105)

where we switched to unitary gauge, $\pi_{U}\to 0$ and $\pi_{P}\to\pi$. In the
operators $\mathcal{O}_{V}$ and $\mathcal{O}_{\mathcal{A}}$, it is therefore
sufficient to keep track of the number of $\pi_{0}$ and $\pi_{1}$ fields:

$\displaystyle\mathcal{O}_{V}^{k,l}$
$\displaystyle=\left(\frac{1}{\sqrt{1-R^{2}}}\right)^{k}\left(-\frac{R}{\sqrt{1-R^{2}}}\right)^{l}i\Tr([\boldsymbol{\pi},\partial_{\mu}\boldsymbol{\pi}]\boldsymbol{V}^{\mu}),$
(106) $\displaystyle\mathcal{O}_{\mathcal{A}}^{k,l}$
$\displaystyle=\left(\frac{1}{\sqrt{1-R^{2}}}\right)^{k}\left(-\frac{R}{\sqrt{1-R^{2}}}\right)^{l}\Tr([\boldsymbol{\pi},[\boldsymbol{\pi},\partial_{\mu}\boldsymbol{\pi}]]\boldsymbol{\mathcal{A}}^{\mu}).$
(107)

We can now collect the terms that facilitate the vector and axial vector
decays, starting with the $d^{2}$-terms. In the first sector we have

$\displaystyle\frac{f_{0}^{2}}{2}\Tr(d_{0,\mu}d_{0}^{\mu})$
$\displaystyle\supset\Tr(D_{\mu}\boldsymbol{\pi}_{0}D^{\mu}\boldsymbol{\pi}_{0})$
(108) $\displaystyle\supset
2ig_{s}\Tr(\partial_{\mu}\boldsymbol{\pi}_{0}[\boldsymbol{\pi}_{0},\mathbf{G}^{\mu}])+2ig^{\prime}\Tr(\partial_{\mu}\boldsymbol{\pi}_{0}[\boldsymbol{\pi}_{0},\mathbf{B}^{\mu}])$
(109)
$\displaystyle=-2g_{s}\,\mathcal{O}_{G}^{2,0}-2g^{\prime}\,\mathcal{O}_{B}^{2,0},$
(110)

Analogously, in the second sector we get

$\displaystyle\frac{f_{1}^{2}}{2}\Tr(d_{1,\mu}d_{1}^{\mu})$
$\displaystyle\supset
2i\tilde{g}\Tr(\partial_{\mu}\boldsymbol{\pi}_{1}[\boldsymbol{\pi}_{1},\boldsymbol{\mathcal{V}}^{\mu}])+\frac{\sqrt{2}\tilde{g}}{3f_{1}}\Tr([\boldsymbol{\pi}_{1},[\boldsymbol{\pi}_{1},\partial_{\mu}\boldsymbol{\pi}_{1}]]\boldsymbol{\mathcal{A}}^{\mu})$
$\displaystyle\quad+\frac{\sqrt{2}\tilde{g}}{f_{1}}\Tr(\partial_{\mu}\boldsymbol{\pi}_{1}[\boldsymbol{\pi}_{1},[\boldsymbol{\pi}_{1},\boldsymbol{\mathcal{A}}^{\mu}]])$
(111)
$\displaystyle=2i\tilde{g}\Tr(\partial_{\mu}\boldsymbol{\pi}_{1}[\boldsymbol{\pi}_{1},\boldsymbol{\mathcal{V}}^{\mu}])+\frac{4\sqrt{2}\tilde{g}}{3f_{1}}\Tr([\boldsymbol{\pi}_{1},[\boldsymbol{\pi}_{1},\partial_{\mu}\boldsymbol{\pi}_{1}]]\boldsymbol{\mathcal{A}}^{\mu})$
(112)
$\displaystyle=-2\tilde{g}\,\mathcal{O}_{\mathcal{V}}^{0,2}+\frac{4\sqrt{2}\tilde{g}}{3f_{1}}\,\mathcal{O}_{\mathcal{A}}.$
(113)

Next we have the mixed $d_{0}d_{1}$-term, which contributes

$\displaystyle rf_{1}^{2}\,\Tr d_{0,\mu}d_{1}^{\mu}$ (114)
$\displaystyle\supset
rf_{1}^{2}\,\Tr\\!\Bigg{(}\frac{2i\tilde{g}}{f_{0}f_{1}}\partial_{\mu}\boldsymbol{\pi}_{0}[\boldsymbol{\pi}_{1},\boldsymbol{\mathcal{V}}^{\mu}]+\frac{\sqrt{2}\tilde{g}}{f_{0}f_{1}^{2}}\partial_{\mu}\boldsymbol{\pi}_{0}[\boldsymbol{\pi}_{1},[\boldsymbol{\pi}_{1},\boldsymbol{\mathcal{A}}^{\mu}]]$
$\displaystyle\qquad\qquad\qquad-\frac{2i}{f_{0}f_{1}}[\boldsymbol{\pi}_{0},(g_{s}\mathbf{G}_{\mu}+g^{\prime}\mathbf{B}_{\mu})]\partial^{\mu}\boldsymbol{\pi}_{1}+\frac{\sqrt{2}\tilde{g}}{3f_{0}^{3}}[\boldsymbol{\pi}_{0},[\boldsymbol{\pi}_{0},\partial_{\mu}\boldsymbol{\pi}_{0}]]\boldsymbol{\mathcal{A}}^{\mu}\Bigg{)}$
(115)
$\displaystyle=rf_{1}^{2}\left(-\frac{2\tilde{g}}{f_{0}f_{1}}\,\mathcal{O}_{\mathcal{V}}^{1,1}+\frac{\sqrt{2}\tilde{g}}{f_{0}f_{1}^{2}}\,\mathcal{O}_{\mathcal{A}}^{1,2}+\frac{2g_{s}}{f_{0}f_{1}}\,\mathcal{O}_{G}^{1,1}+\frac{2g^{\prime}}{f_{0}f_{1}}\,\mathcal{O}_{B}^{1,1}+\frac{\sqrt{2}\tilde{g}}{3f_{0}^{3}}\,\mathcal{O}_{\mathcal{A}}^{3,0}\right).$
(116)

Now on to the $e^{2}$-terms:

$\displaystyle\frac{f_{K}^{2}}{2}\Tr e_{0,\mu}e_{0}^{\mu}$
$\displaystyle\supset\frac{f_{K}^{2}}{2f_{0}^{2}}\Tr(2i(g_{s}\mathbf{G}_{\mu}+g^{\prime}\mathbf{B}_{\mu})[\boldsymbol{\pi}_{0},\partial^{\mu}\boldsymbol{\pi}_{0}])=\frac{f_{K}^{2}}{f_{0}^{2}}\left(g_{s}\,\mathcal{O}_{G}^{2,0}+g^{\prime}\,\mathcal{O}_{B}^{2,0}\right),$
(117) $\displaystyle\frac{f_{K}^{2}}{2}\Tr e_{1,\mu}e^{\mu}_{1}$
$\displaystyle\supset\frac{f_{K}^{2}}{2f_{1}^{2}}\Tr(2i\tilde{g}\boldsymbol{\mathcal{V}}_{\mu}[\boldsymbol{\pi}_{1},\partial^{\mu}\boldsymbol{\pi}_{1}]+\frac{2\sqrt{2}\tilde{g}}{f_{1}}[\boldsymbol{\pi}_{1},\partial_{\mu}\boldsymbol{\pi}_{1}][\boldsymbol{\pi}_{1},\boldsymbol{\mathcal{A}}^{\mu}])$
(118)
$\displaystyle=\frac{f_{K}^{2}}{f_{1}^{2}}\left(\tilde{g}\,\mathcal{O}_{\mathcal{V}}^{0,2}-\frac{\sqrt{2}\tilde{g}}{f_{1}}\,\mathcal{O}_{\mathcal{A}}^{0,3}\right).$
(119)

And finally $e_{0}e_{1}$:

$\displaystyle-f_{K}^{2}\Tr e_{0,\mu}e_{1}^{\mu}$ $\displaystyle\supset-
f_{K}^{2}\Tr(\frac{i}{f_{1}^{2}}\mathbf{V}_{\mu}[\boldsymbol{\pi}_{1},\partial_{\mu}\boldsymbol{\pi}_{1}]+\frac{i\tilde{g}}{f_{0}^{2}}[\boldsymbol{\pi}_{0},\partial_{\mu}\boldsymbol{\pi}_{0}]\boldsymbol{\mathcal{V}}^{\mu}+\frac{\sqrt{2}\tilde{g}}{f_{0}^{2}f_{1}}[\boldsymbol{\pi}_{0},\partial_{\mu}\boldsymbol{\pi}_{0}][\boldsymbol{\pi}_{1},\boldsymbol{\mathcal{A}}^{\mu}])$
(120)
$\displaystyle=-f_{K}^{2}\left(\frac{g_{s}}{f_{1}^{2}}\,\mathcal{O}_{G}^{0,2}+\frac{g^{\prime}}{f_{1}^{2}}\,\mathcal{O}_{B}^{0,2}+\frac{\tilde{g}}{f_{0}^{2}}\,\mathcal{O}_{\mathcal{V}}^{2,0}-\frac{\sqrt{2}\tilde{g}}{f_{0}^{2}f_{1}}\,\mathcal{O}_{\mathcal{A}}^{2,1}\right).$
(121)

Finally we take into account the $\mathcal{V}_{8}^{\mu}$-$G^{\mu}$ and the
$\mathcal{V}_{1}^{\mu}$-$B^{\mu}$ mixings:

$\displaystyle
G_{\mu}\stackrel{{\scriptstyle\mathrm{phys}}}{{\longrightarrow}}\cos\beta_{8}\,G_{\mu}-\sin\beta_{8}\,\mathcal{V}_{8,\mu},\qquad\mathcal{V}_{8,\mu}\stackrel{{\scriptstyle\mathrm{phys}}}{{\longrightarrow}}\sin\beta_{8}\,G_{\mu}+\cos\beta_{8}\,\mathcal{V}_{8,\mu}$
(122)

and analogous for the singlet. For the full vector multiplet, this means

$\displaystyle\boldsymbol{\mathcal{V}}^{\mu}\stackrel{{\scriptstyle\mathrm{phys}}}{{\longrightarrow}}\boldsymbol{\mathcal{V}}_{3/6}^{\mu}+\cos\beta_{8}\,\boldsymbol{\mathcal{V}}_{8}^{\mu}+\cos\beta_{1}\,\boldsymbol{\mathcal{V}}_{1}^{\mu}+\sin\beta_{8}\,\mathbf{G}^{\mu}+\sin\beta_{1}\,\mathbf{B}^{\mu}$
(123)

where $\mathcal{V}_{3}$ ($\mathcal{V}_{6}$) contains both $\mathbf{3}$ and
$\mathbf{\bar{3}}$ ($\mathbf{6}$ and $\mathbf{\bar{6}}$). All in all, the
decays into pNGBs are described by

$\displaystyle\mathcal{L}_{\mathrm{decays}}=C_{\mathcal{A}}\mathcal{O}_{\mathcal{A}}+C_{\mathcal{V}_{1}}\mathcal{O}_{\mathcal{V}_{1}}+C_{\mathcal{V}_{8}}\mathcal{O}_{\mathcal{V}_{8}}+C_{\mathcal{V}_{3/6}}\mathcal{O}_{\mathcal{V}_{3/6}}$
(124)

with coefficients

$\displaystyle C_{\mathcal{V}_{1}}$
$\displaystyle=\frac{(r^{2}-1)f_{K}^{2}}{f_{0}^{2}\cdot\left(1-R^{2}\right)}(\tilde{g}c_{8}+g_{s}s_{8})+2\frac{1+R^{2}}{1-R^{2}}g_{s}s_{8},$
(125) $\displaystyle C_{\mathcal{V}_{8}}$
$\displaystyle=\frac{(r^{2}-1)f_{K}^{2}}{f_{0}^{2}\cdot\left(1-R^{2}\right)}(\tilde{g}c_{8}+g_{s}s_{8})+2\frac{1+R^{2}}{1-R^{2}}g_{s}s_{8},$
(126) $\displaystyle C_{\mathcal{V}_{3/6}}$
$\displaystyle=\frac{\tilde{g}(r^{2}-1)f_{K}^{2}}{f_{0}^{2}\cdot(1-R^{2})},$
(127) $\displaystyle C_{\mathcal{A}}$
$\displaystyle=\frac{\sqrt{2}\tilde{g}r}{3}(1-r^{2})(f_{1}^{2}-3f_{K}^{2}).$
(128)

We recall that the singlet will have additional mixing in the electroweak
sector of the theory, which we do not include here.

Finally we have to calculate the operators
$\mathcal{O}_{\mathcal{V}/\mathcal{A}}$. In the main text, we focus on the
phenomenology of the $\mathcal{V}_{8}$, so we calculate
$\mathcal{O}_{\mathcal{V}_{8}}$. In the $\text{SU}(6)/\text{SO}(6)$ coset,

$\displaystyle\mathcal{O}_{\mathcal{V}_{8}}=\frac{i}{4\sqrt{2}}\,\pi_{8}^{a}\overset{\leftrightarrow}{\partial}{}^{\mu}\pi_{8}^{b}\,\mathcal{V}^{c}_{8,\mu}\,f^{abc}+\frac{1}{2\sqrt{2}}\pi_{6,s}\overset{\leftrightarrow}{\partial}{}^{\mu}\pi_{6}^{c,t}\,\mathcal{V}_{8,\mu}^{a}\,[t^{a}_{\mathbf{6}}]_{t}^{\phantom{t}s},$
(129)

while in the $\text{SU}(6)/\text{Sp}(6)$ coset we have a triplet pNGB,

$\displaystyle\mathcal{O}_{\mathcal{V}_{8}}=\frac{i}{4\sqrt{2}}\,\pi_{8}^{a}\overset{\leftrightarrow}{\partial}{}^{\mu}\pi_{8}^{b}\,\mathcal{V}^{c}_{8,\mu}\,f^{abc}+\frac{1}{2\sqrt{2}}\pi_{3,i}\overset{\leftrightarrow}{\partial}{}^{\mu}\pi_{3}^{c,j}\,\mathcal{V}_{8,\mu}^{a}\,[t^{a}_{\mathbf{3}}]_{i}^{\phantom{i}j}.$
(130)

The operator

$\displaystyle\mathcal{L}=\lambda\,\mathcal{V}_{8,\mu}^{a}\,\pi^{x}\overset{\leftrightarrow}{\partial}{}^{\mu}\pi^{y}\,c^{axy}$
(131)

with $\text{SU}(3)$ tensor $c^{axy}$ yields

$\displaystyle\Gamma(\mathcal{V}_{8}\to\pi_{r}\pi_{r})=|\lambda_{8rr}|^{2}\mathcal{C}_{\mathbf{r}}\,\frac{M_{\mathcal{V}_{8}}}{3\times
2^{7}\pi}\left(1-\frac{4M_{\pi_{r}}^{2}}{M_{\mathcal{V}_{8}}^{2}}\right)^{3/2}$
(132)

with colour factor $\mathcal{C}_{\mathbf{r}}=c^{axy}c^{ayx}$. We have

$\displaystyle\lambda_{888}=\frac{iC_{\mathcal{V}_{8}}}{4\sqrt{2}},\qquad\lambda_{866}=\frac{C_{\mathcal{V}_{8}}}{2\sqrt{2}}=\lambda_{833}$
(133)

and

$\displaystyle\mathcal{C}_{\mathbf{r}}=C_{2}(\mathbf{r})\dim(\mathbf{r})\,\,\Rightarrow\,\,\mathcal{C}_{\mathbf{8}}=3\times
8=24,\quad\mathcal{C}_{\mathbf{6}}=\frac{10}{3}\times
6=20,\quad\mathcal{C}_{\mathbf{3}}=\frac{4}{3}\times 3=4,$ (134)

and therefore

$\displaystyle\mathrm{Br}(\mathcal{V}_{8}\to\pi_{8}\pi_{8}):\mathrm{Br}(\mathcal{V}_{8}\to\pi_{6}\pi_{6}^{c}):\mathrm{Br}(\mathcal{V}_{8}\to\pi_{3}\pi_{3}^{c})=3:10:2,$
(135)

assuming all scalars have the same mass.

###  A.5 Couplings to top and bottom quarks via partial compositeness

In the models under consideration Ferretti:2013kya , the top mass is generated
via partial compositeness, i.e. a linear mixing of the elementary top fields
to composite baryons. In the hidden symmetry framework, baryons can be
included as spin-1/2 resonances transforming under irreducible representations
$R_{\Psi}$ of the hidden symmetry $\mathcal{G}^{\prime}$, hence they couple to
vector and axial resonances via their gauging. However, note that as
$\mathcal{G}^{\prime}$ is broken down to $\mathcal{H}^{\prime}$, different
components of $R_{\Psi}$ will have a different mass as generated by the strong
dynamics.

In general, to provide a successful top mass generation, all models must
contain baryons with the same quantum numbers as the left-handed and right-
handed fields, $q_{L}$ and $t_{R}$. Hence all models contain at least two
fields, $\Psi_{Q}$ and $\Psi_{T}$, both being vector-like. The mixing pattern
with the elementary top fields Kaplan:1991dc  is such that the left-handed
components of $\Psi_{Q}$ and the right-handed component of $\Psi_{T}$ have
large mixing angles with the mass eigenstates, while the other two chiralities
have mixing angles suppressed by the ratio of the electroweak scale over the
Higgs decay constant, $v/f$. Hence, the couplings to the physical top and
bottom fields can be obtained from the baryon couplings with the
substitutions:

$\displaystyle\Psi_{T,L}\to s_{R}\frac{v}{f}\ t_{L}\,,\quad\Psi_{Q_{u},L}\to
s_{L}\ t_{L}\,,\quad\Psi_{Q_{d},L}\to s_{L}\ b_{L}\,,$
$\displaystyle\Psi_{T,R}\to s_{R}\ t_{R}\,,\quad\Psi_{Q_{u},R}\to
s_{L}\frac{v}{f}\ t_{R}\,;$ (136)

where $s_{L/R}=\sin\theta_{L/R}$, with $\theta_{L/R}$ being two independent
mixing angles. With this recipe, we can convert the couplings of vector and
axial resonances to baryons into couplings to physical top and bottom.

Independently on $R_{\Psi}$, the octets always couple to the baryons in all
models, with vector-like couplings given by

$\mathcal{L}_{\text{PC}}\supset\tilde{g}\mathcal{V}_{8,\mu}^{a}\left[\bar{\Psi}_{Q}\gamma^{\mu}t_{3}^{a}\Psi_{Q}+\bar{\Psi}_{T}\gamma^{\mu}t_{3}^{a}\Psi_{T}\right]+i\tilde{g}\mathcal{A}_{8,\mu}^{a}\left[\bar{\Psi}_{Q}\gamma^{\mu}\gamma^{5}t_{3}^{a}\Psi_{Q}+\bar{\Psi}_{T}\gamma^{\mu}\gamma^{5}t_{3}^{a}\Psi_{T}\right]\,.$
(137)

As the mixing to the physical top fields are chiral, the effective couplings
of the mass eigenstates can be parameterised as:

$\displaystyle\mathcal{L}_{\text{PC}}\supset$
$\displaystyle\;\bar{t}\not{\mathcal{V}}_{8}^{a}t^{a}_{\mathbf{3}}\left(g_{\rho
tt,LL}P_{L}+g_{\rho
tt,RR}P_{R}\right)t+\bar{b}\not{\mathcal{V}}_{8}^{a}t^{a}_{\mathbf{3}}\left(g_{\rho
bb,LL}P_{L}+g_{\rho bb,RR}P_{R}\right)b+$
$\displaystyle\bar{t}\not{\mathcal{A}}_{8}^{a}t^{a}_{\mathbf{3}}\left(-g_{att,LL}P_{L}+g_{att,RR}P_{R}\right)t+\bar{b}\not{\mathcal{A}}_{8}^{a}t^{a}_{\mathbf{3}}\left(-g_{abb,LL}P_{L}+g_{abb,RR}P_{R}\right)b\,,$
(138)

where $P_{L/R}$ are the usual chiral projectors and, from the above
substitutions, we have at leading order in $v/f$:

$\displaystyle g_{\rho tt,LL}=g_{\rho bb,LL}=\tilde{g}\cos\beta_{8}\
s_{L}^{2}\,,\quad g_{att,LL}=g_{abb,LL}=\tilde{g}\ s_{L}^{2}$ $\displaystyle
g_{\rho tt,RR}=\tilde{g}\cos\beta_{8}\ s_{R}^{2}\,,\quad g_{att,RR}=\tilde{g}\
s_{R}^{2}\,,\quad g_{\rho bb,RR}=g_{abb,RR}=0\,,$ (139)

where the $\cos\beta_{8}$ factor comes from the mixing of $\mathcal{V}_{8}$ to
gluons.

Regarding the non-octet resonances, $\mathcal{V}_{3}$ and $\mathcal{A}_{6}$,
they can couple to a ditop state only when the underlying fermion $\chi$
carries baryon number $1/3$, i.e. in the case $\psi\psi\chi$. Hence, all
baryons transform as the fundamental of $\text{SU}(6)$. In this case, the
baryons that mix with the elementary top and bottom can be embedded in the $6$
of $\text{SU}(6)$ as:

$\Psi_{6}=\begin{pmatrix}\Psi\\\ \Psi^{C}\end{pmatrix}\,.$ (140)

Hence, the triplet and sextet coupling to baryons will have the generic form:

$\mathcal{L}_{\text{PC}}\supset\tilde{g}\mathcal{V}_{3,\mu}\overline{\Psi^{c}}\gamma^{\mu}\Psi+i\tilde{g}\mathcal{A}_{6,\mu}\overline{\Psi}\gamma^{\mu}\Psi^{c}+\text{h.c.}$
(141)

with appropriate colour contractions. Taking into account the electroweak
charges, only the singlet $\Psi_{T}$ is allowed such couplings, hence the
effective couplings of $\mathcal{V}_{3}$ and $\mathcal{A}_{6}$ only involve
tops and can be parameterised as

$\mathcal{L}_{\text{PC}}\supset g_{\rho tt,LR}\
\overline{t^{c}}\not{\mathcal{V}}_{3}t+g_{att,LR}\
\overline{t}\not{\mathcal{A}}_{6}t^{c}+\text{h.c.}$ (142)

where

$g_{\rho tt,LR}=g_{att,LR}=\tilde{g}s_{R}^{2}\frac{v}{f}\,.$ (143)

The $v/f$ suppression compared to the octet couplings stems from the fact that
the couplings always involve one left-handed and one right-handed baryon,
hence at least one (the left-handed) will have a suppressed mixing angle to
the physical tops.

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