KIAS-A22001

Phenomenological aspects of composite Higgs scenarios:
exotic scalars and vector-like quarks

A. Banerjee1, D. Buarque Franzosi2, G. Cacciapaglia3, A. Deandrea3, G.
Ferretti1, T. Flacke4, B. Fuks5, M. Kunkel6, L. Panizzi7, W. Porod6, L.
Schwarze6

1 Department of Physics, Chalmers University of Technology, Fysikgården, 41296
Göteborg, Sweden

2 Stockholm University, Department of Physics, 106 91 Stockholm, Sweden

3 Univ. Lyon, Université Claude Bernard Lyon 1, CNRS/IN2P3, IP2I UMR5822,
F-69622, Villeurbanne, France

4 Center for AI and Natural Sciences, KIAS, Seoul 02455, Korea

5 Laboratoire de Physique Théorique et Hautes Energies (LPTHE), UMR 7589,
Sorbonne Université et CNRS, 4 place Jussieu, 75252 Paris Cedex 05, France

6 Institut für Theoretische Physik und Astrophysik, Uni Würzburg, Emil-Hilb-
Weg 22, D-97074 Würzburg, Germany

7 Department of Physics and Astronomy, Uppsala University, Box 516, SE-751 20
Uppsala, Sweden

Submitted to the Proceedings of the US Community Study

on the Future of Particle Physics (Snowmass 2021)

> ABSTRACT
>
>
>
>
> Composite Higgs models usually contain additional pseudo Nambu Goldstone
> bosons and vector-like quarks. We discuss various aspects related to their
> LHC phenomenology and provide summary plots of exclusion limits using
> currently available information. We also describe a general parametrisation
> implemented in a software for Monte Carlo simulations and study the
> $SU(5)/SO(5)$ scenario as a concrete example.

###### Contents

  1. 1 Executive summary
  2. 2 Composite Higgs and partial compositeness: an outline
    1. 2.1 A general parametrisation
  3. 3 LHC bounds for pair production of pNGBs and VLQs
    1. 3.1 Electroweak pNGBs
    2. 3.2 Vector-like quarks
  4. 4 Case study: $SU(5)/SO(5)$
  5. A The general Lagrangian for models with VLQs and pNGBs
    1. A.1 Couplings in $SU(5)/SO(5)$ coset

##  1 Executive summary

Models of electroweak (EW) symmetry breaking by a composite Higgs [1] with
partial compositeness [2] in the top-quark sector give promising solutions to
the hierarchy problem [3, 4, 5, 6, 7] of the Standard Model (SM). At the
effective field theory level they can be described by specifying the pattern
of symmetry breaking involved. Generically these models predict the existence
of light scalars in addition to the Higgs boson, all emerging as pseudo Nambu-
Goldstone bosons (pNGBs). Partial compositeness also implies the presence of
vector-like quarks (VLQs). This proliferation of new particles requires a
systematic approach to study this class of models and to facilitate a seamless
transition between theory, data, and simulation codes.

Current experimental analyses already allow to set limits on the masses of
such particles in specific scenarios depending on their interactions. For
pNGBs, we provide an overview plot of the limits for pair production cross-
sections times branching ratios in various final states (Fig. 2). We also work
out a concrete example based on the $SU(5)/SO(5)$ coset (Fig. 7). For VLQs, we
emphasize the relevance of their exotic decays by presenting a plot
summarizing current limits in the VLQ mass versus pNGB mass plane (Fig. 4).

Furthermore, we provide a general parametrisation describing the interactions
of the new particles arising in composite scenarios, which can be used for
phenomenological studies and is implemented in a numerical model for Monte
Carlo simulations[8]. A software toolbox of a new kind is also being
developed, FSMOG (FeynRules plugin for Simplified MOdel Generation) [9], to
automatize the construction of the most general interaction Lagrangian given a
specific particle content, which has a much wider range of applications beyond
the realm of composite Higgs models.

##  2 Composite Higgs and partial compositeness: an outline

There are many approaches to strongly-coupled EW symmetry breaking scenarios
[4, 5, 6, 7]. The most investigated ones include, e.g., 5D holographic
theories [10] and multi-site deconstructed models like Little Higgs models
[11]. Instead of using the holographic duality, we consider directly a class
of composite Higgs models based on 4D asymptotically free (hypercolor) gauge
theories. We assume that the hypercolor theory, after going through a near
conformal running [12, 13], confines at the multi-TeV scale. To include top
partial compositeness, its fundamental degrees of freedom contain fermionic
matter in two inequivalent irreducible representations (irreps) of hypercolor
[14, 15], chosen in order to sequester the EW coset from the composite states
carrying QCD color.

The strong sector is associated with a global symmetry $\mathcal{G}$ that
breaks spontaneously to a subgroup $\mathcal{H}$ below the confinement scale,
such that the SM gauge symmetry is contained in the unbroken subgroup
$\mathcal{H}$. The symmetry breaking pattern induced by the fermionic
condensates gives rise to an EW coset containing the Higgs boson as a pNGB.
The three minimal cosets arising from real, pseudoreal and complex irreps,
containing at least a pNGB Higgs doublet and preserving custodial symmetry,
are $SU(5)/SO(5)$ [16, 17] (real), $SU(4)/Sp(4)$ [18, 19] (pseudoreal), and
$SU(4)\times SU(4)^{\prime}/SU(4)_{D}$ [20, 21] (complex), respectively.
Holographic/little Higgs constructions, reviewed in [4] and [11], lead to
additional cosets, for instance the well-known minimal model $SO(5)/SO(4)$
(MCHM) [3] or $SO(6)/SO(4)\times SO(2)$ [22]. Most of the low energy
phenomenology discussed here is also applicable to this wider variety of
models. Below the condensation scale, the colored VLQs originate from the
trilinear fermionic operators combining the two irreps in a hypercolor
invariant way. They are assumed to mix linearly with the SM quarks of the
third generation through partial compositeness. We do not review these
constructions here but refer to the existing literature: see [14, 15] for the
initial constructions (a streamlined classification can be found in [23]).
Additional models of this type are presented in [17, 24, 25, 26, 27, 28, 29,
30, 31, 32].

Composite Higgs models have two major types of experimental signatures in the
low energy range. First, the pNGB nature of the Higgs boson implies
modifications of its couplings with the other SM particles [33, 34, 35, 36,
37]. The other kind of signature involves additional particles beyond the SM
(BSM), predicted by these models, for example, exotic EW pNGBs [38, 23, 39,
40], colored pNGBs [41, 39], VLQs [42, 43, 44, 45, 46, 47, 48] and other
colored fermionic states in non-triplet irreps [49], vector resonances [50,
51, 52, 53, 54], and axion-like particles [55, 56]. In this contribution we
focus on EW pNGBs and exotic decay modes of VLQs.

To construct the low energy theory [57, 58] involving the pNGBs of the EW
sector, the VLQs, and the SM quarks and gauge bosons one must choose a
symmetry breaking pattern $\mathcal{G}/\mathcal{H}$, VLQs in $\mathcal{H}$
irreps, and an embedding of the SM quarks into an irrep of $\mathcal{G}$, see
[59, 60, 6, 42, 61] for detailed constructions.

###  2.1 A general parametrisation

Here and in the appendix we present the generic Lagrangian, focusing on the
interactions of the pNGBs and the VLQs. At this stage we only use invariance
under the electromagnetic $U(1)_{\textrm{em}}$ and the QCD color $SU(3)_{c}$
to construct the interactions. This introduces a large number of arbitrary
couplings that will have to be fixed in a specific model implementation. We
take this approach since it can be easily implemented in software tools.

We consider the particle content shown in Tab. 1. As mentioned above, even
this choice reflects some restrictions on the type of degrees of freedom
present. However, this field content is sufficient for the discussion below.

Fields | Spin | $SU(3)_{c}$ | $U(1)_{\textrm{em}}$
---|---|---|---
$S^{0}_{i}$ | $0$ | $\bf 1$ | 0
$S^{\pm}_{i}$ | $0$ | $\bf 1$ | $\pm 1$
$S^{\pm\pm}_{i}$ | $0$ | $\bf 1$ | $\pm 2$
$\pi^{q}_{r}$ | $0$ | $\mathbf{r}$ | $q$
$T$ | $1/2$ | $\bf 3$ | $2/3$
$B$ | $1/2$ | $\bf 3$ | $-1/3$
$X$ | $1/2$ | $\bf 3$ | $5/3$
Table 1: BSM particle content considered in the Lagrangian presented in this
section. For the colored scalars $\pi^{q}_{r}$, the subscript
$r=\mathbf{3},\mathbf{6},\mathbf{8}$ denotes the $SU(3)_{c}$ representation.
We allow for multiple colorless scalars $S^{0,\pm,\pm\pm}_{i}$ $(i=1,2\dots)$
since they generically arise in composite Higgs models.

The interaction Lagrangian can be expressed as follows

$\displaystyle\mathcal{L}_{\rm int}$
$\displaystyle=\mathcal{L}_{SSV}+\mathcal{L}_{SSVV}+\mathcal{L}_{SVV}+\mathcal{L}_{SV\tilde{V}}+\mathcal{L}_{\Psi\Psi
V}+\mathcal{L}_{\Psi fV}+\mathcal{L}_{\Psi fS}+\mathcal{L}_{ffS}$
$\displaystyle+\mathcal{L}_{\pi\pi V+\pi\pi VV}+\mathcal{L}_{\pi
V\tilde{V}}+\mathcal{L}_{\Psi f\pi}+\mathcal{L}_{ff\pi}\,,$ (1)

where $\Psi\equiv\\{T,B,X\\}$ collectively denotes the VLQs. In addition to
the field content in Tab. 1, $V$ and $f$ denote the SM gauge bosons and third
generation quarks. The explicit expressions for each term of the above
Lagrangian are presented in the appendix.

For the time being, we use a FeynRules implementation of the above Lagrangian,
limited to the EW pNGBs, and VLQs [8], through which a UFO model [62] has been
generated. The UFO implementation allows the simulation of processes up to
one-loop in QCD. A more automatized tool, FSMOG (FeynRules plugin for
Simplified MOdel Generation), is being developed to facilitate the writing of
the most general Lagrangian. The program will take as input fields of given
spin, charge, and color and output the FeynRules file containing the most
general interactions between these fields and the SM ones, up to a certain
mass dimension. At present this is limited to dimension-4 operators for
simplicity, however future versions of FSMOG will include commonly used higher
dimension operators. FSMOG is still under development, but the current version
can be obtained from the authors [9]. In this implementation the reduction of
number of independent couplings for a specific model is achieved by using a
FeynRules-style restriction card.

##  3 LHC bounds for pair production of pNGBs and VLQs

###  3.1 Electroweak pNGBs

In this section we confine ourselves to the case where the only accessible
degrees of freedom are the pNGBs from the EW coset. We present the bounds on
their production cross sections times branching ratios as function of their
mass in various channels. In this way, the bounds are applicable to any BSM
scenario with scalars.

Figure 1: Examples of pNGB pair production via Drell-Yan processes with
subsequent decays into SM particles.

The direct production of EW pNGBs can occur either via Drell-Yan processes or
via vector boson fusion. In case of pair production the Drell-Yan processes

$pp\to
S_{i}^{\pm\pm}S_{j}^{\mp}\,,\,S_{i}^{\pm}S_{j}^{0}\,,\,S_{i}^{++}S_{j}^{--}\,,\,S_{i}^{+}S_{j}^{-}\,,\,S_{i}^{0}S_{j}^{0}$
(2)

clearly dominate. The vector boson fusion processes also allow single
production via the anomaly induced couplings, see Eq. 11 in the appendix.
However, the corresponding cross sections are rather small as these couplings
are loop induced. Nevertheless, these couplings become important for the pNGB
decays if couplings to SM fermions are either forbidden or strongly
suppressed. In this case the dominant decays are

$\displaystyle S^{++}_{i}$ $\displaystyle\to W^{+}W^{+}$ (3a) $\displaystyle
S^{+}_{i}$ $\displaystyle\to W^{+}\gamma,\,W^{+}Z$ (3b) $\displaystyle
S^{0}_{i}$ $\displaystyle\to W^{+}W^{-},\,\gamma\gamma,\,\gamma Z,\,ZZ.$ (3c)

Combining these with the various production channels leads to a plethora of
final states containing 4 gauge bosons, see for example left diagram in Fig.
1. A few of these processes have been searched for directly: Ref. [63] sets
bounds on the cross section times branching ratio of $S^{++}S^{--}\to WWWW$
and $S^{\pm\pm}S^{\mp}\to WWWZ$, which are included in Fig. 2 .

Given a coset, the anomaly couplings of pNGBs to vector bosons are fixed up to
an overall constant, while the couplings to fermions are more model-dependent.
It is typically possible to construct both fermiophobic as well as
fermiophilic models. If present, the couplings of pNGBs to quarks in Eq. 17 of
the appendix are expected to scale with $m_{q}/f$, i.e.

$\kappa_{t}^{S_{i}^{0}}\sim\frac{m_{t}}{f},\qquad\kappa_{b}^{S_{i}^{0}}\sim\frac{m_{b}}{f},\qquad\kappa_{tb,L/R}^{S_{i}^{+}}\sim\frac{m_{t}}{f},$
(4)

up to factors of order 1, where $f$ is the pNGB decay constant. In this case
the branching ratios into third-generation quarks dominate over the loop-
induced anomaly decays into vector bosons. The corresponding decay channels
are

$\displaystyle S^{++}$ $\displaystyle\to W^{+}t\overline{b},$ (5a)
$\displaystyle S^{+}$ $\displaystyle\to t\overline{b},$ (5b) $\displaystyle
S^{0}$ $\displaystyle\to t\overline{t},\,b\overline{b}.$ (5c)

Note that the three-body decay $S^{++}\to W^{+}t\overline{b}$ proceeds via an
off-shell $S^{+}$, see right diagram in Fig. 1. To our knowledge, this channel
has not been studied in the literature before. The corresponding branching
ratio is usually close to 100% as the decay $S^{++}\to W^{+}W^{+}$ is loop-
induced. In this section we assume that all pNGBs have the same mass. Effects
induced due to mass differences are more model-dependent, hence we defer the
corresponding discussion to Sec. 4 for a specific model.

Contur pool | Description | final state(s)
---|---|---
ATLAS-13-LL-GAMMA | dilepton and $\geq 1$ photon [64] |  $W\gamma W\gamma$,
ATLAS-13-GAMMA | inclusive multiphotons [65] |  $W\gamma W\gamma$, $W\gamma WZ$
|  | $W\gamma\gamma\gamma$
ATLAS-13-GAMMA-MET | photon and MET [66] | $W\gamma WZ$
ATLAS-13-4L | four leptons [67] | $WZWZ$
ATLAS-13-L1L2METJET | unlike dilepton, MET and jets [68] |  $W\gamma W\gamma$, $WZWZ$
ATLAS-13-MMJET |  $\mu^{+}\mu^{-}$ at the $Z$ pole, plus optional jets [69] | $WZWZ$
CMS-13-EEJET |  $e^{+}e^{-}$ at the $Z$ pole, plus optional jets [70] | $WZWZ$
CMS-13-MMJET |  $\mu^{+}\mu^{-}$ at the $Z$ pole, plus optional jets [70] | $WZWZ$
Table 2: Contur pools that are most sensitive to the final states listed in
the last column. Figure 2: Upper limits on the cross section times branching
ratio of the Drell-Yan production of electroweak pNGBs. The bounds are
obtained from recasts implemented in Contur and MadAnalysis5 except for the
ones presented in [63]. The reference cross sections $\sigma$ for pair
production are calculated for a custodial quintuplet from a benchmark model
that is discussed in detail in Sec. 4.

We determine the extent to which the currently recasted analyses can constrain
processes for which no direct searches exist. To this end, we have implemented
simplified models in FeynRules [71] at leading order. We have generated
$10^{5}$ events using MadGraph5_aMC@NLO [72] version 3.3.1 with the NNPDF 2.3
set of parton densities [73, 74] and hadronized them with Pythia8 [75]. To
obtain constraints on the parameter space, the events are analysed with
MadAnalysis5, [76, 77, 78, 79] version 1.9.60. MadAnalysis5 performs an event
reconstruction using Delphes 3 [80] and the anti-$k_{T}$ algorithm [81]
implemented in FastJet [82], and calculates an exclusion using the CLs
prescription [83]. We also tested the generated events against the SM
measurements implemented in Rivet [84] version 3.1.5 and have calculated
exclusions from the respective YODA files with Contur [85, 86]. Here, we have
used all available analyses in Contur version 2.1.1, as well as Refs. [65, 66,
87] from the upcoming release 2.2.1.

For final states with quarks (in particular top quarks) we found BSM searches
implemented in MadAnalysis5 to be the most constraining.***An analysis using
BSM searches implemented in CheckMATE [88, 89] is in preparation. In contrast,
in case of vector boson final states, we obtain the bounds in general from
measurements implemented in Rivet, on which Contur is based. In Fig. 2 we
present the bounds on the pair-production cross section times branching ratio
for various final states. Beyond our recast results, we also included the
results of the ATLAS search for $pp\to S^{++}S^{--}\to WWWW$ and $pp\to
S^{\pm\pm}S^{\mp}\to WWWZ$ [63] for completeness. In Tab. 2 we give the most
sensitive Contur pools for the various vector boson final states. In the case
of quark final states the strongest bounds stem from the four-top search CMS-
TOP-18-003 [90], which is implemented in MadAnalysis5 [91].

For reference, in Fig. 2 we also show the production cross section for members
of a custodial quintuplet, which is part of the benchmark scenario based on
the $SU(5)/SO(5)$ coset discussed in Sec. 4. Note that the references are
production cross sections, while the bounds shown are constraints on
$\sigma\times\mathrm{Br}$. To obtain bounds on $m_{S}$, these reference cross
sections must be multiplied by the (model-dependent) branching ratios into the
respective final states. Comparing the theoretical cross sections to the
obtained constraints, it is clear that the currently implemented analyses can
give bounds on $m_{S}$ of at most 400 GeV (in the $S^{\mp}S^{0}$-channel, if
the respective branching ratios to $W\gamma$ and $\gamma\gamma$ are close to
1). We want to stress that other models give cross section of the same size.
For example, the cross sections shown for $S^{++}S^{--}$ and
$S^{\pm\pm}S^{\mp}$ are the same as for the $SU(2)_{L}$ triplet used in the
ATLAS analysis [63]. The main model dependence enters not in the pair
production cross section but in the branching ratios of the pNGB decays.

###  3.2 Vector-like quarks

Figure 3: Illustration of VLQ pair production with subsequent decays into
pNGBs and third generation quarks.

Coloured particles have a privileged role in the determination of model-
independent constraints from searches at the LHC, since the cross-section of
QCD pair production depends exclusively on their masses. VLQs have been
extensively searched at the LHC by both ATLAS and CMS, under the general
assumption that the SM is minimally extended with one VLQ, directly decaying
into SM final states. Strong bounds on their masses (above the TeV) have been
obtained regardless of their decay channels [92, 93, 94, 95, 96, 97, 98, 99,
100, 101, 102, 103].

We have seen, however, that in the scenarios at hand VLQs can also interact
with further lighter exotic states, such as new pNGBs, changing their decay
patterns as indicated in Fig. 3. Exclusion bounds obtained under the
aforementioned hypotheses thus require a reinterpretation to account for the
reduced branching ratios into SM final states. Moreover, the new decay
channels can be tested against experimental data to obtain combined limits on
the masses of VLQs and the new states they can decay into. Experimental
searches targeting exotic decays of VLQs are not available yet, however
phenomenological analyses providing recasts of different searches have already
appeared [104, 105, 44, 106, 107, 108, 109, 110, 111, 112, 113].

Figure 4: 95% CL limits in the $\\{m_{\Psi},m_{S}\\}$ plane for VLQs decaying
to new pNGBs. For reference, current experimental bounds on the masses of VLQs
decaying to SM particles only and the masses of spin-0 states produced in
pairs are provided as vertical and horizontal lines respectively.

A summary of results from some of these analyses, corresponding to exclusive
decays (100% Br) of the VLQs into different exotic states, is presented in
Fig. 4. When additional decay channels are open, bounds on VLQ masses can be
significantly lower than those reported in current experimental searches. In
this case, results from Fig. 4 should be accompanied by maps of experimental
efficiencies or upper limits on the cross-sections in the same plane for each
relevant decay channel of the VLQ.†††An explicit example of this kind of
analysis has been proposed in [46] for pair production of $X_{5/3}$ decaying
to light leptons, not motivated by composite scenarios. The representation
proposed in Fig. 4 has thus the purpose to provide an overview and a general
reference for where realistic bounds can be placed.

To allow an easy comparison with experimental results, current bounds on the
masses of VLQs decaying to SM particles and bounds on the masses of pair-
produced pNGBs have also been included as vertical and horizontal lines
respectively. Notice that except for the colour sextet $\pi_{6}$, a residual
model dependence affects the exclusion limits: the pair production cross-
section of an EW pNGB depends on its coupling to the $Z$ boson, which is
associated with its representation under the EW gauge group. Experimental
bounds on charged scalars are obtained under specific assumptions, for which
we refer to the corresponding literature [114, 115, 63].

##  4 Case study: $SU(5)/SO(5)$

As a concrete example, we now consider a model based on the $SU(5)/SO(5)$
coset after having integrated out the VLQs. The UV theory can be constructed
as in [17] and it’s being studied on the lattice [116, 117]. This coset
delivers a total of 14 pNGBs, which include the Higgs bidoublet, a singlet
$\eta$, and a bitriplet under $SU(2)_{L}\times SU(2)_{R}$. Here we focus on
the bitriplet, as direct $\eta$ production at the LHC is strongly suppressed.
The bitriplet decomposes as a quintuplet
$\eta_{5}\equiv(\eta^{0}_{5},\eta^{\pm}_{5},\eta^{\pm\pm}_{5})$, a triplet
$\eta_{3}\equiv(\eta^{0}_{3},\eta^{\pm}_{3})$ and a singlet $\eta_{1}^{0}$ of
custodial $SU(2)_{C}$.

Figure 5: Feynman diagrams corresponding to three-body decays of
$\eta_{3}^{0}$.

A detailed description of how all the couplings can be derived, including the
VLQ sector, has been given in [61]. For this section we consider the exotic
pNGBs to be fermiophobic. In Appendix A.1, we collect the relevant couplings,
which depend on the pNGB decay constant $f$, which we set to 1 TeV. The
remaining parameters to be chosen are the masses for the custodial multiplets
$m_{1,3,5}$ ‡‡‡In principle members of different custodial multiplets can mix
as discussed in [40].. Note that the exotic pNGBs do not acquire a vacuum
expectation value in this model.

Figure 6: Three-body decays of $\eta_{3}^{0}$ for the scenario $m_{1,3}\ll
m_{5}$. When kinematically accessible, the dominant channel is
$\eta_{3}^{0}\to W^{\pm}\eta_{3}^{\mp(*)}$.

Figure 7: Exclusion bounds on the pNGB masses in $m_{1}-m_{3}$ plane (left
panel) and $m_{5}-m_{3}$ plane (right panel). In both cases we assume the
pNGBs are fermiophobic and produced in pair through Drell-Yan processes. The
black (grey) lines represent the 2-$\sigma$ (1-$\sigma$) exclusion. Red points
are excluded with CL above 2-$\sigma$ and yellow ones between 1 and
2-$\sigma$.

In this scenario, these pNGBs are mainly produced through Drell-Yan processes
(see Eq. 2). In the presence of a mass splitting between the custodial
multiplets the decay pattern changes with respect to the discussion in Sec. 3.
In such case the lowest lying multiplet will still decay only via the anomaly.
However, the heavier pNGBs will decay dominantly into a possibly off-shell
vector boson and a member of the lower lying multiplet, as soon as the mass
splitting is larger than about 1 GeV. As an example we take $m_{3}\geq
m_{5,1}$, in which case the $\eta_{3}$ members decay dominantly as

$\displaystyle\eta_{3}^{+}$
$\displaystyle\to\eta_{5}^{++}W^{-(*)}\,,\,\,\eta_{5}^{+}Z^{(*)}\,,\,\eta^{0}_{5}W^{+(*)}\,,\,\eta^{0}_{1}W^{+(*)}\,;$
(6) $\displaystyle\eta_{3}^{0}$
$\displaystyle\to\eta_{5}^{\pm}W^{\mp(*)}\,,\,\,\eta_{5}^{0}Z^{(*)}\,,\,\,\eta_{1}^{0}Z^{(*)}\,.$

Since $\Gamma_{\eta_{5,1}}\ll\Gamma_{W,Z}$, the decays into off-shell
$\eta_{5,1}$ and on-shell vector bosons are strongly suppressed. Note that in
scenarios where $\eta_{3}$ is the lightest multiplet, $\eta^{0}_{3}$ is forced
to have three body decays since it does not couple to the anomaly, see Fig. 5.
We give examples of branching ratios of $\eta^{0}_{3}$ for different scenarios
in Fig. 6. We use the same set-up as in Sec. 3 to determine bounds at the LHC.
We find again that Contur gives the strongest bounds which are displayed in
Fig. 7 for two limiting scenarios: $m_{1,3}\ll m_{5}$ (left plot) and
$m_{3,5}\ll m_{1}$ (right plot). The bounds on the right side are much
stronger due to the presence of the additional singly and doubly charged
states in the quintuplet.

## Acknowledgements

We thank Jon Butterworth for helpful discussions on Contur. A.B., D.B.F., G.F.
and L.P. acknowledge support from the Knut and Alice Wallenberg foundation
under the grant KAW 2017.0100 (SHIFT project). G.C., A.D., M.K, W.P. and L.S.
have been supported by the “DAAD, Frankreich” and “Partenariat Hubert Curien
(PHC)” PROCOPE 2021-2023, project number 57561441. T.F, M.K., W.P and L.S were
supported under the framework of the international cooperation program “GenKo”
managed by the National Research Foundation of Korea (No. 2022K2A9A2A15000153,
FY2022) and DAAD, P33 - projekt-id 57608518. G.C., A.D., T.F. and B.F.
acknowledge support from the Campus-France STAR project n. 43566UG, “Higgs and
Dark Matter connections”. T.F. is supported by KIAS Individual Grant
(AP083701) via the Center for AI and Natural Sciences at Korea Institute for
Advanced Study. G.C. and A.D. are grateful to the LABEX Lyon Institute of
Origins (ANR-10-LABX-0066) Lyon for its financial support within the program
“Investissements d’Avenir” of the French government operated by the National
Research Agency (ANR).

##  Appendix A The general Lagrangian for models with VLQs and pNGBs

In this appendix we provide the explicit expressions of each of the terms
shown in Eq. 1 which is also implemented in the software tools, albeit with a
limited particle content. The terms $\mathcal{L}_{SSV}$, $\mathcal{L}_{SSVV}$,
and $\mathcal{L}_{\pi\pi V+\pi\pi VV}$ are simply the couplings of the pNGBs
with the gauge bosons, arising from the covariant derivative.

$\displaystyle\mathcal{L}_{SSV}$
$\displaystyle=\frac{ie}{s_{W}}W^{-\mu}\sum_{i,j}\left[K^{S^{0}_{i}S^{+}_{j}}_{W}S^{0}_{i}\overleftrightarrow{\partial_{\mu}}S^{+}_{j}+K^{S^{-}_{i}S^{++}_{j}}_{W}S^{-}_{i}\overleftrightarrow{\partial_{\mu}}S^{++}_{j}\right]+{\mathrm{h.c.}}$
$\displaystyle+\frac{ie}{s_{W}c_{W}}Z^{\mu}\left[\sum_{i<j}K^{S^{0}_{i}S^{0}_{j}}_{Z}S^{0}_{i}\overleftrightarrow{\partial_{\mu}}S^{0}_{j}+\sum_{i,j}\left(K^{S^{+}_{i}S^{-}_{j}}_{Z}S^{+}_{i}\overleftrightarrow{\partial_{\mu}}S^{-}_{j}+K^{S^{++}_{i}S^{--}_{j}}_{Z}S^{++}_{i}\overleftrightarrow{\partial_{\mu}}S^{--}_{j}\right)\right]$
$\displaystyle-
ieA^{\mu}\sum_{i}\left[S^{+}_{i}\overleftrightarrow{\partial_{\mu}}S^{-}_{i}+2S^{++}_{i}\overleftrightarrow{\partial_{\mu}}S^{--}_{i}\right],$
(7) $\displaystyle\mathcal{L}_{SSVV}$
$\displaystyle=e^{2}A_{\mu}A^{\mu}\sum_{i}\left[S^{+}_{i}S^{-}_{i}+4S^{++}_{i}S^{--}_{i}\right]$
$\displaystyle+\frac{e^{2}}{s_{W}c_{W}}A^{\mu}Z_{\mu}\sum_{i,j}\left[K^{S^{+}_{i}S^{-}_{j}}_{AZ}S^{+}_{i}S^{-}_{j}+K^{S^{++}_{i}S^{--}_{j}}_{AZ}S^{++}_{i}S^{--}_{j}\right]$
$\displaystyle+\frac{e^{2}}{s^{2}_{W}c^{2}_{W}}Z_{\mu}Z^{\mu}\left[\sum_{i\leq
j}K^{S^{0}_{i}S^{0}_{j}}_{ZZ}S^{0}_{i}S^{0}_{j}+\sum_{i,j}\left(K^{S^{+}_{i}S^{-}_{j}}_{ZZ}S^{+}_{i}S^{-}_{j}+K^{S^{++}_{i}S^{--}_{j}}_{ZZ}S^{++}_{i}S^{--}_{j}\right)\right]$
$\displaystyle+\frac{e^{2}}{s^{2}_{W}}W^{+}_{\mu}W^{-\mu}\left[\sum_{i\leq
j}K^{S^{0}_{i}S^{0}_{j}}_{WW}S^{0}_{i}S^{0}_{j}+\sum_{i,j}\left(K^{S^{+}_{i}S^{-}_{j}}_{WW}S^{+}_{i}S^{-}_{j}+K^{S^{++}_{i}S^{--}_{j}}_{WW}S^{++}_{i}S^{--}_{j}\right)\right]$
$\displaystyle+\frac{e^{2}}{s_{W}}A^{\mu}W^{-}_{\mu}\sum_{i,j}\left[K^{S^{0}_{i}S^{+}_{j}}_{AW^{-}}S^{0}_{i}S^{+}_{j}+K^{S^{-}_{i}S^{++}_{j}}_{AW^{-}}S^{-}_{i}S^{++}_{j}\right]+{\mathrm{h.c.}}$
$\displaystyle+\frac{e^{2}}{s^{2}_{W}c_{W}}Z^{\mu}W^{-}_{\mu}\sum_{i,j}\left[K^{S^{0}_{i}S^{+}_{j}}_{ZW^{-}}S^{0}_{i}S^{+}_{j}+K^{S^{-}_{i}S^{++}_{j}}_{ZW^{-}}S^{-}_{i}S^{++}_{j}\right]+{\mathrm{h.c.}}$
$\displaystyle+\frac{e^{2}}{s^{2}_{W}}W^{-}_{\mu}W^{-\mu}\left[\sum_{i\leq
j}K^{S^{+}_{i}S^{+}_{j}}_{W^{-}W^{-}}S^{+}_{i}S^{+}_{j}+\sum_{i,j}\left(K^{S^{0}_{i}S^{++}_{j}}_{W^{-}W^{-}}S^{0}_{i}S^{++}_{j}\right)\right]+{\mathrm{h.c.}},$
(8) $\displaystyle\mathcal{L}_{\pi\pi V+\pi\pi VV}$
$\displaystyle=\sum_{r=3,6,8}(D_{\mu}\pi^{q}_{r})^{\dagger}D^{\mu}\pi^{q}_{r},$
(9)

with
$D_{\mu}=\partial_{\mu}-ig_{s}G^{a\mu}T^{a}_{r}-ieq(A_{\mu}-\tan\theta_{W}Z_{\mu})$
and $s_{W}(c_{W})\equiv\sin\theta_{W}(\cos\theta_{W})$. Note that the colored
pNGBs do not carry any $SU(2)_{L}$ index in the models considered here and,
thus, the couplings to the $Z$ arise purely from hypercharge. The
$\mathcal{L}_{SVV}$ consists of the coupling of the Higgs boson to the EW
gauge bosons.

$\displaystyle\mathcal{L}_{SVV}=2M_{W}^{2}\frac{h}{v}\left[K^{h}_{WW}W_{\mu}^{+}W^{-\mu}+K^{h}_{ZZ}\frac{1}{2c_{W}^{2}}Z_{\mu}Z^{\mu}\right]\,.$
(10)

Any additional pNGB acquiring a vacuum expectation value contributes to the
masses of the gauge bosons and acquires a coupling of this type. We consider
here the case where only the Higgs doublet receives a vacuum expectation value
($v$). In this case, the neutral scalars easily evade all LEP collider bounds
and no corrections to the tree-level $\rho$-parameter arise. This choice is
well motivated by a study of the potential for models given in [23, 40].

The Lagrangian $\mathcal{L}_{SV\tilde{V}}$ and $\mathcal{L}_{\pi V\tilde{V}}$
contain the dimension five anomalous couplings of the pNGBs with the vector
boson field strengths. If one proceeds to integrate out the third family or
the heavy vector bosons, both $\mathcal{L}_{SVV}$ and
$\mathcal{L}_{SV\tilde{V}}$ include additional dimension five couplings such
as the coupling of the Higgs boson to the gluons.

$\displaystyle\mathcal{L}_{SV\tilde{V}}$
$\displaystyle=\frac{e^{2}}{16\pi^{2}v}\bigg{[}\sum_{i}S^{0}_{i}\left(\tilde{K}^{S^{0}_{i}}_{\gamma\gamma}F_{\mu\nu}\tilde{F}^{\mu\nu}+\frac{2}{s_{W}c_{W}}\tilde{K}^{S^{0}_{i}}_{\gamma
Z}F_{\mu\nu}\tilde{Z}^{\mu\nu}+\frac{1}{s_{W}^{2}c_{W}^{2}}\tilde{K}^{S^{0}_{i}}_{ZZ}Z_{\mu\nu}\tilde{Z}^{\mu\nu}\right.$
$\displaystyle\left.+\frac{2}{s_{W}^{2}}\tilde{K}^{S^{0}_{i}}_{WW}W^{+}_{\mu\nu}\tilde{W}^{-\mu\nu}\right)+\sum_{i}S^{+}_{i}\left(\frac{2}{s_{W}}\tilde{K}^{S^{+}_{i}}_{\gamma
W}F_{\mu\nu}\tilde{W}^{-\mu\nu}+\frac{2}{s_{W}^{2}c_{W}}\tilde{K}^{S^{+}_{i}}_{ZW}Z_{\mu\nu}\tilde{W}^{-\mu\nu}\right)+{\rm
h.c.}$
$\displaystyle+\frac{1}{s_{W}^{2}}\sum_{i}S^{++}_{i}\tilde{K}^{S^{++}_{i}}_{W^{-}W^{-}}W^{-}_{\mu\nu}\tilde{W}^{-\mu\nu}+{\rm
h.c.}\bigg{]},$ (11) $\displaystyle\mathcal{L}_{\pi V\tilde{V}}$
$\displaystyle=\frac{1}{16\pi^{2}f}\left(g_{s}^{2}\tilde{K}^{\pi^{0}_{8}}_{gg}\pi^{0}_{8}G_{\mu\nu}\tilde{G}^{\mu\nu}+2g_{s}e\tilde{K}^{\pi^{0}_{8}}_{g\gamma}\pi^{0}_{8}G_{\mu\nu}\tilde{F}^{\mu\nu}+\frac{2g_{s}e}{s_{W}c_{W}}\tilde{K}^{\pi^{0}_{8}}_{gZ}\pi^{0}_{8}G_{\mu\nu}\tilde{Z}^{\mu\nu}\right).$
(12)

The only terms in $\mathcal{L}_{\Psi\Psi V}$ relevant for VLQ pair production
are their couplings to the gluons

$\displaystyle\mathcal{L}_{\Psi\Psi V}$
$\displaystyle=\frac{g_{s}}{2}\left[\overline{T}\not{G}^{a}\lambda^{a}T+\overline{B}\not{G}^{a}\lambda^{a}B+\overline{X}\not{G}^{a}\lambda^{a}X\right],$
(13)

with $\lambda^{a}$ being the usual Gell-Mann matrcies. The terms
$\mathcal{L}_{\Psi fV}$, $\mathcal{L}_{\Psi fS}$, and $\mathcal{L}_{\Psi
f\pi}$ arise from partial compositeness by linearly coupling the VLQs ($\Psi$)
to the third quark family ($f$).

$\displaystyle\mathcal{L}_{\Psi fV}$
$\displaystyle=\frac{e}{\sqrt{2}s_{W}}\kappa^{W}_{T,L}\overline{T}\not{W}^{+}P_{L}b+\frac{e}{2c_{W}s_{W}}\kappa^{Z}_{T,L}\overline{T}\not{Z}P_{L}t+\frac{e}{\sqrt{2}s_{W}}\kappa^{W}_{B,L}\overline{B}\not{W}^{-}P_{L}t$
$\displaystyle+\frac{e}{2c_{W}s_{W}}\kappa^{Z}_{B,L}\overline{B}\not{Z}P_{L}b+\frac{e}{\sqrt{2}s_{W}}\kappa^{W}_{X,L}\overline{X}\not{W}^{+}P_{L}t+L\leftrightarrow
R+{\mathrm{h.c.}}$ (14) $\displaystyle\mathcal{L}_{\Psi fS}$
$\displaystyle=\sum_{i}S^{+}_{i}\left[\kappa^{S^{+}_{i}}_{T,L}\overline{T}P_{L}b+\kappa^{S^{+}_{i}}_{X,L}\overline{X}P_{L}t+L\leftrightarrow
R\right]+{\rm
h.c.}+\sum_{i}S^{-}_{i}\left[\kappa^{S^{-}_{i}}_{B,L}\overline{B}P_{L}t+L\leftrightarrow
R\right]+{\rm h.c.}$
$\displaystyle+\sum_{i}S^{0}_{i}\left[\kappa^{S^{0}_{i}}_{T,L}\overline{T}P_{L}t+\kappa^{S^{0}_{i}}_{B,L}\overline{B}P_{L}b+L\leftrightarrow
R\right]+{\rm h.c.}$
$\displaystyle+\sum_{i}S^{++}_{i}\left[\kappa^{S^{++}_{i}}_{X,L}\overline{X}P_{L}b+L\leftrightarrow
R\right]+{\rm h.c.}$ (15) $\displaystyle\mathcal{L}_{\Psi f\pi}$
$\displaystyle=\pi^{0}_{8}\left[\kappa^{\pi^{0}_{8}}_{T,L}\overline{T}P_{L}t+\kappa^{\pi^{0}_{8}}_{B,L}\overline{B}P_{L}b+L\leftrightarrow
R\right]+{\rm h.c.}$
$\displaystyle+\left[\pi^{4/3}_{6}\left(\kappa^{\pi^{4/3}_{6}}_{T,L}\overline{T}CP_{R}\overline{t}^{T}+\kappa^{\pi^{4/3}_{6}}_{X,L}\overline{X}CP_{R}\overline{b}^{T}\right)+\pi^{2/3}_{6}\kappa^{\pi^{2/3}_{6}}_{B,L}B^{T}CP_{L}b+L\leftrightarrow
R\right]+{\rm h.c.}$ (16)

with $C$ being the charge conjugation operator. The terms $\mathcal{L}_{ffS}$
and $\mathcal{L}_{ff\pi}$ involving two SM third generation quarks are given
below with $y_{f}=\sqrt{2}m_{f}/v$.

$\displaystyle\mathcal{L}_{ffS}$
$\displaystyle=-\frac{h}{\sqrt{2}}\left[y_{t}\kappa^{h}_{t}\overline{t}t+y_{b}\kappa^{h}_{b}\overline{b}b\right]+\sum_{i}S^{0}_{i}\bigg{[}\overline{t}\left(\kappa^{S^{0}_{i}}_{t}+i\tilde{\kappa}^{S^{0}_{i}}_{t}\gamma_{5}\right)t+\overline{b}\left(\kappa^{S^{0}_{i}}_{b}+i\tilde{\kappa}^{S^{0}_{i}}_{b}\gamma_{5}\right)b\bigg{]}$
$\displaystyle+\sum_{i}S^{+}_{i}\left[\kappa^{S^{+}_{i}}_{tb,L}\overline{t}P_{L}b+L\leftrightarrow
R\right]+{\rm h.c.}$ (17) $\displaystyle\mathcal{L}_{ff\pi}$
$\displaystyle=\pi^{0}_{8}\left[\overline{t}(\kappa^{\pi^{0}_{8}}_{t}+i\tilde{\kappa}^{\pi^{0}_{8}}_{t}\gamma_{5})t+\overline{b}(\kappa^{\pi^{0}_{8}}_{b}+i\tilde{\kappa}^{\pi^{0}_{8}}_{b}\gamma_{5})b\right]$
$\displaystyle+\pi^{4/3}_{6}\overline{t}(\kappa^{\pi^{4/3}_{6}}_{t}+i\tilde{\kappa}^{\pi^{4/3}_{6}}_{t}\gamma_{5})C\overline{t}^{T}+\pi^{2/3}_{6}b^{T}(\kappa^{\pi^{2/3}_{6}}_{t}+i\tilde{\kappa}^{\pi^{2/3}_{6}}_{t}\gamma_{5})Cb+{\rm
h.c.}$ (18)

###  A.1 Couplings in $SU(5)/SO(5)$ coset

Now we present the explicit coupling strengths between the weak gauge bosons
and EW pNGBs in the custodial basis for $SU(5)/SO(5)$ coset, in terms of
$\sin\theta\equiv s_{\theta}=v/f$. The couplings presented below can be
directly used to construct the restriction cards for the Feynrules-based tools
we discussed earlier. In Tab. 3 we map the generic symbols for the scalar
fields used in the Eq. 1 to the specific symbols used for the $SU(5)/SO(5)$
coset. The coefficient $K^{h}_{VV}$ ($V=W,Z$), the universal modification of
the Higgs coupling with the weak gauge bosons appearing in $\mathcal{L}_{SVV}$
is given by $c_{\theta}$.

Fields in Eq. (1) | Fields in section 4
---|---
$S^{0}_{i}$ | $h,\eta,\eta^{0}_{1},\eta^{0}_{3},\eta^{0}_{5}$
$S^{\pm}_{i}$ | $\eta^{\pm}_{3},\eta^{\pm}_{5}$
$S^{\pm\pm}_{i}$ | $\eta^{\pm\pm}_{5}$
Table 3: Mapping between the generic symbols for the spin-0 fields to the
specific EW pNGBs in the custodial basis for $SU(5)/SO(5)$ coset

#### Coefficients appearing in $\mathcal{L}_{SSV}$

$\begin{array}[]{cccccccccc}\hline\cr\hline\cr&\lx@intercol\hfil
K^{S^{0}_{i}S^{+}_{j}}_{W}\hfil\lx@intercol&K^{S^{-}_{i}S^{++}_{j}}_{W}\\\
\hline\cr&\eta^{+}_{3}&\eta^{+}_{5}&\eta_{5}^{++}\\\ \hline\cr
h&0&0&\hbox{\multirowsetup$-$}\\\
\eta^{0}_{3}&-\frac{i}{2}&\frac{c_{\theta}}{2}&\\\
\eta^{0}_{5}&-\frac{c_{\theta}}{2\sqrt{3}}&\frac{i\sqrt{3}}{2}&\\\
\eta^{0}_{1}&\sqrt{\frac{2}{3}}c_{\theta}&0&\\\ \eta&0&0&\\\
\eta^{-}_{3}&\lx@intercol\hfil\hbox{\multirowsetup$-$}\hfil\lx@intercol&\frac{c_{\theta}}{\sqrt{2}}\\\
\eta^{-}_{5}&&&-\frac{i}{\sqrt{2}}\\\ \vphantom{--}\\\
\hline\cr\hline\cr\end{array}\quad\begin{array}[]{ccccccccccccc}\hline\cr\hline\cr&\lx@intercol\hfil
K^{S^{0}_{i}S^{0}_{j}}_{Z}\hfil\lx@intercol&\lx@intercol\hfil
K^{S^{+}_{i}S^{-}_{j}}_{Z}\hfil\lx@intercol&K^{S^{++}_{i}S^{--}_{j}}_{Z}\\\
\hline\cr&h&\eta^{0}_{3}&\eta^{0}_{5}&\eta^{0}_{1}&\eta&\eta^{-}_{3}&\eta^{-}_{5}&\eta_{5}^{--}\\\
\hline\cr
h&0&0&0&0&0&\lx@intercol\hfil\hbox{\multirowsetup$-$}\hfil\lx@intercol&\hbox{\multirowsetup$-$}\\\
\eta^{0}_{3}&&0&\frac{ic_{\theta}}{\sqrt{3}}&i\sqrt{\frac{2}{3}}c_{\theta}&0\\\
\eta^{0}_{5}&&&0&0&0\\\ \eta^{0}_{1}&&&&0&0\\\ \eta&&&&&0\\\
\eta^{+}_{3}&\lx@intercol\hfil\hbox{\multirowsetup$-$}\hfil\lx@intercol&-\frac{c_{2W}}{2}&-\frac{ic_{\theta}}{2}\\\
\eta^{+}_{5}&&&&&&&-\frac{c_{2W}}{2}\\\
\eta^{++}_{5}&\lx@intercol\hfil-\hfil\lx@intercol&-c_{2W}\\\
\hline\cr\hline\cr\end{array}$

#### Coefficients appearing in $\mathcal{L}_{SSVV}$

$\begin{array}[]{ccccccccccccc}\hline\cr\hline\cr&\lx@intercol\hfil
K^{S^{0}_{i}S^{0}_{j}}_{ZZ}\hfil\lx@intercol&\lx@intercol\hfil
K^{S^{+}_{i}S^{-}_{j}}_{ZZ}\hfil\lx@intercol&K^{S^{++}_{i}S^{--}_{j}}_{ZZ}\\\
\hline\cr&h&\eta^{0}_{3}&\eta^{0}_{5}&\eta^{0}_{1}&\eta&\eta^{-}_{3}&\eta^{-}_{5}&\eta_{5}^{--}\\\
\hline\cr
h&\frac{c_{2\theta}}{8}&0&0&0&0&\lx@intercol\hfil\hbox{\multirowsetup$-$}\hfil\lx@intercol&\hbox{\multirowsetup$-$}\\\
\eta^{0}_{3}&&\frac{3+5c_{2\theta}}{16}&0&0&0\\\
\eta^{0}_{5}&&&\frac{c_{2\theta}}{6}&\frac{3+5c_{2\theta}}{12\sqrt{2}}&-\sqrt{\frac{5}{6}}\frac{s^{2}_{\theta}}{2}\\\
\eta^{0}_{1}&&&&\frac{15+17c_{2\theta}}{96}&-\sqrt{\frac{5}{3}}\frac{s^{2}_{\theta}}{8}\\\
\eta&&&&&-\frac{5s^{2}_{\theta}}{16}\\\
\eta^{+}_{3}&\lx@intercol\hfil\hbox{\multirowsetup$-$}\hfil\lx@intercol&\frac{3-2s^{2}_{2W}+c_{2\theta}}{8}&\frac{ic_{2W}c_{\theta}}{2}\\\
\eta^{+}_{5}&&&&&&&\frac{c^{2}_{2W}+c_{2\theta}}{4}\\\
\eta^{++}_{5}&\lx@intercol\hfil-\hfil\lx@intercol&c^{2}_{2W}\\\
\hline\cr\hline\cr\end{array}$

$\begin{array}[]{ccccccccccccc}\hline\cr\hline\cr&\lx@intercol\hfil
K^{S^{0}_{i}S^{0}_{j}}_{WW}\hfil\lx@intercol&\lx@intercol\hfil
K^{S^{+}_{i}S^{-}_{j}}_{WW}\hfil\lx@intercol&K^{S^{++}_{i}S^{--}_{j}}_{WW}\\\
\hline\cr&h&\eta^{0}_{3}&\eta^{0}_{5}&\eta^{0}_{1}&\eta&\eta^{-}_{3}&\eta^{-}_{5}&\eta_{5}^{--}\\\
\hline\cr
h&\frac{c_{2\theta}}{4}&0&0&0&0&\lx@intercol\hfil\hbox{\multirowsetup$-$}\hfil\lx@intercol&\hbox{\multirowsetup$-$}\\\
\eta^{0}_{3}&&\frac{3+c_{2\theta}}{8}&0&0&0\\\
\eta^{0}_{5}&&&\frac{9+c_{2\theta}}{12}&-\frac{3+5c_{2\theta}}{12\sqrt{2}}&\sqrt{\frac{5}{6}}\frac{s^{2}_{\theta}}{2}\\\
\eta^{0}_{1}&&&&\frac{15+17c_{2\theta}}{48}&-\sqrt{\frac{5}{3}}\frac{s^{2}_{\theta}}{4}\\\
\eta&&&&&-\frac{5s^{2}_{\theta}}{8}\\\
\eta^{+}_{3}&\lx@intercol\hfil\hbox{\multirowsetup$-$}\hfil\lx@intercol&\frac{3c^{2}_{\theta}}{2}&-\frac{ic_{\theta}}{2}\\\
\eta^{+}_{5}&&&&&&&\frac{5+c_{2\theta}}{4}\\\
\eta^{++}_{5}&\lx@intercol\hfil-\hfil\lx@intercol&c_{\theta}^{2}\\\
\hline\cr\hline\cr\end{array}$

$\begin{array}[]{cccccccccc}\hline\cr\hline\cr&\lx@intercol\hfil
K^{S^{+}_{i}S^{-}_{j}}_{AZ}\hfil\lx@intercol&K^{S^{++}_{i}S^{--}_{j}}_{AZ}\\\
\hline\cr&\eta^{-}_{3}&\eta^{-}_{5}&\eta_{5}^{--}\\\
\hline\cr\eta^{+}_{3}&c_{2W}&ic_{\theta}&\hbox{\multirowsetup$-$}\\\
\eta^{+}_{5}&&c_{2W}\\\
\eta^{++}_{5}&\lx@intercol\hfil-\hfil\lx@intercol&4c_{2W}\\\
\hline\cr\hline\cr\vphantom{--}\\\ \vphantom{--}\\\ \vphantom{--}\\\
\vphantom{--}\\\
\end{array}\quad\begin{array}[]{cccccccccc}\hline\cr\hline\cr&\lx@intercol\hfil
K^{S^{0}_{i}S^{+}_{j}}_{AW}\hfil\lx@intercol&K^{S^{-}_{i}S^{++}_{j}}_{AW}\\\
\hline\cr&\eta^{+}_{3}&\eta^{+}_{5}&\eta_{5}^{++}\\\ \hline\cr
h&0&0&\hbox{\multirowsetup$-$}\\\
\eta^{0}_{3}&-\frac{i}{2}&\frac{c_{\theta}}{2}\\\
\eta^{0}_{5}&-\frac{c_{\theta}}{2\sqrt{3}}&\frac{i\sqrt{3}}{2}\\\
\eta^{0}_{1}&\sqrt{\frac{2}{3}}c_{\theta}&0\\\ \eta&0&0\\\
\eta^{-}_{3}&\lx@intercol\hfil\hbox{\multirowsetup$-$}\hfil\lx@intercol&\frac{3c_{\theta}}{\sqrt{2}}\\\
\eta^{-}_{5}&&&-\frac{3i}{\sqrt{2}}\\\ \hline\cr\hline\cr\end{array}$

$\begin{array}[]{cccccccccc}\hline\cr\hline\cr&\lx@intercol\hfil
K^{S^{0}_{i}S^{+}_{j}}_{ZW}\hfil\lx@intercol&K^{S^{-}_{i}S^{++}_{j}}_{ZW}\\\
\hline\cr&\eta^{+}_{3}&\eta^{+}_{5}&\eta_{5}^{++}\\\ \hline\cr
h&0&0&\hbox{\multirowsetup$-$}\\\
\eta^{0}_{3}&\frac{i(s_{W}^{2}+c_{2\theta})}{2}&-\frac{(1+s_{W}^{2})c_{\theta}}{2}\\\
\eta^{0}_{5}&-\frac{(3-s^{2}_{W})c_{\theta}}{2\sqrt{3}}&\frac{i(3c_{2W}-c_{2\theta})}{4\sqrt{3}}\\\
\eta^{0}_{1}&-\sqrt{\frac{2}{3}}s^{2}_{W}c_{\theta}&-\frac{i(3+5c_{2\theta})}{4\sqrt{6}}\\\
\eta&0&\sqrt{\frac{5}{2}}\frac{is^{2}_{\theta}}{2}\\\
\eta^{-}_{3}&\lx@intercol\hfil\hbox{\multirowsetup$-$}\hfil\lx@intercol&\frac{(1-3s_{W}^{2})c_{\theta}}{\sqrt{2}}\\\
\eta^{-}_{5}&&&-\frac{i(3c_{2W}-c_{2\theta})}{2\sqrt{2}}\\\
\hline\cr\hline\cr\end{array}\quad\begin{array}[]{cccccccccc}\hline\cr\hline\cr&\lx@intercol\hfil
K^{S^{+}_{i}S^{+}_{j}}_{W^{-}W^{-}}\hfil\lx@intercol&K^{S^{0}_{i}S^{++}_{j}}_{W^{-}W^{-}}\\\
\hline\cr&\eta^{+}_{3}&\eta^{+}_{5}&\eta_{5}^{++}\\\
\hline\cr\eta^{+}_{3}&-\frac{c_{2\theta}}{4}&\frac{ic_{\theta}}{2}&\hbox{\multirowsetup$-$}\\\
\eta^{+}_{5}&&\frac{3-c_{2\theta}}{8}\\\
h&\lx@intercol\hfil\hbox{\multirowsetup$-$}\hfil\lx@intercol&0\\\
\eta^{0}_{3}&&&-\frac{ic_{\theta}}{2}\\\
\eta^{0}_{5}&&&-\frac{3-c_{2\theta}}{2\sqrt{6}}\\\
\eta^{0}_{1}&&&\frac{3+5c_{2\theta}}{8\sqrt{3}}\\\
\eta&&&-\frac{\sqrt{5}s^{2}_{\theta}}{4}\\\ \hline\cr\hline\cr\end{array}$

#### Coefficients appearing in $\mathcal{L}_{SV\tilde{V}}$

The coefficients for the anomaly terms are specified up to an overall factor
coming from the dimension of the hyperfermion irrep.

$\begin{array}[]{cccccccccc}\hline\cr\hline\cr&\tilde{K}^{S^{0}_{i}}_{\gamma\gamma}&\tilde{K}^{S^{0}_{i}}_{\gamma
Z}&\tilde{K}^{S^{0}_{i}}_{ZZ}&\tilde{K}^{S^{0}_{i}}_{W^{+}W^{-}}&\tilde{K}^{S^{+}_{i}}_{\gamma
W^{-}}&\tilde{K}^{S^{+}_{i}}_{ZW^{-}}&\tilde{K}^{S^{++}_{i}}_{W^{-}W^{-}}\\\
\hline\cr
h&0&\\!\\!\\!0&\\!\\!\\!0&\\!\\!\\!0&\lx@intercol\hfil\hbox{\multirowsetup$-$}\hfil\lx@intercol&\hbox{\multirowsetup$-$}\\\
\eta^{0}_{3}&0&\\!\\!\\!0&\\!\\!\\!0&\\!\\!\\!0\\\
\eta^{0}_{5}&-\frac{2s_{\theta}}{\sqrt{3}}&\\!\\!\\!-\frac{c_{2W}s_{\theta}}{\sqrt{3}}&\\!\\!\\!\frac{(1-3c_{4W}+2c_{2\theta})s_{\theta}}{12\sqrt{3}}&\\!\\!\\!\frac{s^{3}_{\theta}}{6\sqrt{3}}\\\
\eta^{0}_{1}&\sqrt{\frac{2}{3}}s_{\theta}&\\!\\!\\!\frac{c_{2W}s_{\theta}}{\sqrt{6}}&\\!\\!\\!\frac{(1+6c_{4W}-7c_{2\theta})s_{\theta}}{24\sqrt{6}}&\\!\\!\\!\frac{7s^{3}_{\theta}}{12\sqrt{6}}\\\
\eta&\sqrt{\frac{2}{5}}s_{\theta}&\\!\\!\\!\frac{c_{2W}s_{\theta}}{\sqrt{10}}&\\!\\!\\!\frac{(3c_{\theta}^{2}+c_{4W})s_{\theta}}{4\sqrt{10}}&\\!\\!\\!\frac{(3c_{2\theta}+5)s_{\theta}}{8\sqrt{10}}\\\
\eta^{+}_{3}&\lx@intercol\hfil\hbox{\multirowsetup$-$}\hfil\lx@intercol&-\frac{s_{2\theta}}{4}&\\!\\!\\!\frac{s_{W}^{2}s_{2\theta}}{4}\\\
\eta^{+}_{5}&&&&&\frac{is_{\theta}}{2}&\\!\\!\\!\frac{i(c_{2W}-c_{2\theta}-2)s_{\theta}}{12}\\\
\eta^{++}_{5}&\lx@intercol\hfil-\hfil\lx@intercol&-\frac{s_{\theta}^{3}}{3\sqrt{2}}\\\
\hline\cr\hline\cr\end{array}$

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