# Towards the precise description of Composite Higgs models at colliders

Diogo Buarque Franzosi  Stockholm University, Department of Physics, 106 91
Stockholm, Sweden<EMAIL_ADDRESS>
###### Abstract

We present a framework to study the interactions among Nambu-Goldstone bosons
(NGB), pseudo-NGB (pNGBs) and gauge bosons in Composite Higgs (CH) models at
high energies, including operators of order $\mathcal{O}(p^{4})$ and
$\mathcal{O}(p^{2}g^{2})$ in the chiral expansion and topological terms. The
set of (p)NGBs comprises the longitudinal modes of electroweak bosons, the
Higgs boson, and possibly other scalar states from the dynamical spontaneous
electroweak symmetry breaking. The framework is implemented in a collider
simulation tool especially suited for the study of Goldstone Boson Scattering
(GBS), which includes vector boson scattering (VBS), di-Higgs production via
vector boson fusion (VBF) and the pair production of other pNGBs via VBF.

###### Contents

  1. I Introduction
  2. II CCWZ formalism for CH models to lowest order
  3. III The next-to-leading order Lagrangian
  4. IV Towards predictions: GBS at the LHC
    1. IV.1 Di-Higgs via VBF and VBS at $\mathcal{O}(p^{4})$ and tree-level
    2. IV.2 pNGB production and decay in non-minimal models
  5. V Conclusion and prospects
  6. A Explicit matrices for “minimal cosets”
  7. B Explicit breaking terms and spurions

##  I Introduction

Composite Higgs (CH) models are promising alternatives to the Standard Model
(SM), describing the electroweak (EW) symmetry breaking (SB) dynamically via a
fermionic condensate, solving the hierarchy problem Weinberg (1976); Susskind
(1979); Dimopoulos and Susskind (1979), as well as the little hierarchy
between the compositeness scale and the Higgs mass via the vacuum misalignment
mechanism Dugan et al. (1985) with a SM-like light Higgs boson appearing as a
pseudo-Nambu-Goldstone boson (pNGB) Kaplan and Georgi (1984); Georgi and
Kaplan (1984). It also opens several paths towards explaining dark matter with
composite states, unification and SM fermion masses hierarchy via partial
compositeness (PC) Kaplan (1991) with large anomalous dimension of fermionic
operators in a near-conformal phase Holdom (1981).

The description of CH interactions below the condensation scale is provided by
the chiral expansion of the Coleman-Callan-Wess-Zumino (CCWZ) formalism
Coleman et al. (1969); Callan et al. (1969), usual in the CH literature (see
Contino (2011); Bellazzini et al. (2014); Panico and Wulzer (2016) for
reviews). In CH models an approximate global symmetry $G$ is spontaneously
broken to the stability group $H$ giving origin to a set of (p)NGBs. Besides
the 3 exact NGB _eaten_ by the weak bosons, CH models contain the Higgs boson
as well as other pNGBs in non-minimal scenarios. Models with an explicit
matter content of hyperfermions with PC mechanism via a four-dimensional gauge
theory have typically non-minimal cosets Barnard et al. (2014); Ferretti and
Karateev (2014); Vecchi (2017); Ferretti (2016); Csáki et al. (2018); Guan et
al. (2020).

The interactions among the (p)NGBs, including the Higgs and longitudinal modes
of weak bosons, are fixed by the non-linear symmetry of the CCWZ construction.
These interactions can be observed via precise measurement of SM processes or
the production of other pNGBs. In this paper we present the CCWZ Lagrangian
for a generic CH coset, with a minimal set of operators that allow the
description of the gauge interactions and the misalignment of the vacuum. We
include operators at orders $\mathcal{O}(p^{2})$, $\mathcal{O}(p^{4})$ and
$\mathcal{O}(p^{2}g^{2})$ in the chiral expansion, as well as the topological
Wess-Zumino-Witten terms (WZW) Wess and Zumino (1971); Witten (1979). The
presented $\mathcal{O}(p^{4})$ and $\mathcal{O}(p^{2}g^{2})$ terms are
intended as a minimum self-consistent set of operators that can be used to
renormalize one-loop amplitudes in the chiral perturbation framework.

A framework for the automatic simulation of CH processes in collider
experiments based on the FeynRules package Christensen and Duhr (2009); Alloul
et al. (2014), Universal Feynrules Output (UFO) Degrande et al. (2012) and
MG5_aMC@NLO Alwall et al. (2011) is presented. Predictions for
(pseudo-)Goldstone boson scattering (GBS) are used as examples. GBS processes
give the ultimate test of the dynamical origin of EWSB. It includes Vector
Boson Scattering (VBS), di-Higgs production via Vector Boson Fusion (VBF) and
the pair production of extra pNGBs via VBF. For the SM processes VBS and di-
Higgs via VBF distributions are computed up to $\mathcal{O}(p^{4})$ at tree
level. Predictions for pNGB pair production via VBF for $SU(4)/Sp(4)$ and
$SU(5)/SO(5)$ are also produced at $\mathcal{O}(p^{2})$ at tree level.

The paper is organized as follows. In sec. (II) we provide the general CCWZ
framework and the contruction of the lowest order Lagrangian, including the
WZW terms. In sec. (III) we extend the formalism to include operators of
$\mathcal{O}(p^{4})$ and $\mathcal{O}(p^{2}g^{2})$. In sec. (IV) we show some
predictions as example. We conclude in sec. (V) with prospects for the future.

##  II CCWZ formalism for CH models to lowest order

In this section we repeat the CCWZ formalism for CH models in order to fix the
notation. The basic idea of the CCWZ construction is to define objects which
are invariant under the global symmetry $G$, that are spontaneously broken to
the subgroup $H$ via a “gauge” choice.

The generators of $G$, $T^{A}$, $A=1,\cdots,d_{G}$, are normalized with the
convention $\langle T^{A}T^{B}\rangle=\frac{1}{2}\delta^{AB}$.
$\langle\rangle$ is the trace. They are split in the unbroken generators
belonging to $H$, $T^{a}\equiv\tilde{T}^{a}\equiv S^{a}$, $a=1,\cdots,d_{H}$
and the broken ones that belong to the coset $G/H$,
$T^{d_{H}+\hat{a}}\equiv\hat{T}^{\hat{a}}\equiv X^{\hat{a}}$,
$\hat{a}=1,\cdots,d_{G}-d_{H}$. For the cosets which can be realized with
fermion condensates $SU(2N)/SO(2N)$, $SU(2N)/Sp(2N)$ and $SU(N)_{L}\times
SU(N)_{R}/SU(N)_{V}$ Peskin (1980); Preskill (1981), we can get the generators
from the equations

$S^{a}\Sigma_{0}+\Sigma_{0}S^{aT}=0,\quad
X^{\hat{a}}\Sigma_{0}-\Sigma_{0}X^{\hat{a}T}=0,$ (1)

with the $\Sigma_{0}$ a (anti-)symmetric matrix in the (pseudo-)real rep.
$SU(N)/SO(N)$ ($SU(N)/Sp(N)$) case. We choose explicitly

$\Sigma_{0}=\begin{cases}1_{2N},\quad\text{for }SU(2N)/SO(2N)\\\
\left(\begin{array}[]{ccc}&-\epsilon_{N}&\\\ \epsilon_{N}&&\\\
&&1\end{array}\right),\quad\text{for }SU(2N+1)/SO(2N+1)\\\
\left(\begin{array}[]{cc}&-1_{N}\\\ 1_{N}&\end{array}\right),\quad\text{for
}SU(2N)/Sp(2N)\\\ \left(\begin{array}[]{cc}&1_{N}\\\
1_{N}&\end{array}\right),\quad\text{for }SU(N)_{L}\times
SU(N)_{R}/SU(N)_{V}\end{cases}$ (2)

$1_{N}$ is the $N$-dimension identity matrix and $\epsilon_{N}$ is the fully
anti-symmetric $N$-dimension matrix with entries given by the Levi-Civita
symbol with $\epsilon_{12\dots N}=1$.

In the $SU(N)^{2}/SU(N)$ case, we use

$T^{A}=\left(\begin{array}[]{cc}T_{L}^{A}&\\\
&-T_{R}^{AT}\end{array}\right),\quad
T_{L(R)}^{A}=\begin{cases}\lambda^{A}(1_{N}),\,A=1,\cdots,\text{dim}(SU(N))\\\
1_{N}(\lambda^{A}),\,A=\text{dim}(SU(N))+1,\cdots,2\text{dim}(SU(N))\end{cases}$
(3)

With this choice for $T^{A}$ and $\Sigma_{0}$ we get the unbroken
$T_{L}=T_{R}$ and the broken $T_{L}=-T_{R}$ generators as block diagonal
matrices:

$S^{a}=\text{diag}(\lambda^{a},-\lambda^{Ta}),\quad
X^{a}=\text{diag}(\lambda^{a},\lambda^{Ta})$ (4)

where $\lambda^{a}$ are the Gell-Mann matrices generators of $SU(N)$. The
advantage of this notation is that all subsequent discussion can be made
general. For the minimal CH coset $SO(5)/SO(4)$ we use Contino et al. (2011)

$\displaystyle X^{\hat{a}}$
$\displaystyle=-\frac{i}{\sqrt{2}}(\delta^{\hat{a}i}\delta^{5j}-\delta^{\hat{a}j}\delta^{5i}),$
(5) $\displaystyle S^{a}_{ij}$
$\displaystyle=-\frac{i}{2}(\frac{i}{2}\epsilon^{abc}(\delta^{bi}\delta^{cj}-\delta^{bj}\delta^{ci})+(\delta^{ai}\delta^{4j}-\delta^{aj}\delta^{4i})),$
(6) $\displaystyle S^{a+3}_{ij}$
$\displaystyle=-\frac{i}{2}(\frac{i}{2}\epsilon^{abc}(\delta^{bi}\delta^{cj}-\delta^{bj}\delta^{ci})-(\delta^{ai}\delta^{4j}-\delta^{aj}\delta^{4i})),$
(7)

with $a=1,2,3$.

An element of $G$ and of $H$ are $g=e^{i\alpha^{A}T^{A}}$ and
$h=e^{i\sigma^{a}S^{a}}$ respectively. Every element of $G$ can be
parameterized by $g=\xi(\pi)h$ with $\xi$ an element of the coset $G/H$

$\xi(\pi)=e^{i\pi^{A}X^{A}}$ (8)

The action of $g\in G$ in $\xi(\pi)$ is still an element $g^{\prime}\in G$ and
can be parametrized in the same form $g^{\prime}\equiv
g\xi(\pi)=\xi(\pi^{\prime})h(\pi,g)$ defining the non-linear character of
$\xi(\pi)$ transformation

$\xi(\pi)\to\xi(\pi^{\prime})=g\xi(\pi)h(\pi,g)^{-1}\,.$ (9)

It is also useful to define

$\Sigma\equiv\xi\Sigma_{0}\xi^{T}\to g\Sigma g^{T}\,.$ (10)

which follows from
$\Sigma_{0}h=\Sigma_{0}e^{\mathrm{i}\alpha^{a}S^{a}}=e^{-\mathrm{i}\alpha^{a}S^{aT}}\Sigma_{0}=h^{*}\Sigma_{0}$.

We can now embed the left $T_{L}^{i}$ and right $T_{R}^{i}$ generators of the
custodial subgroup $SU(2)_{L}\times SU(2)_{R}\subset H$ as unbroken
generators. This can be defined as any 3+3 generators that obey the SU(2) Lie
algebra

$[T_{L,R}^{a},T_{L,R}^{b}]=\mathrm{i}\epsilon^{abc}T_{L,R}^{c},\quad[T_{L}^{a},T_{R}^{b}]=0\,.$
(11)

We will eventually _gauge_ $SU(2)_{L}$ and the hypercharge as the third
component of $SU(2)_{R}$. We will not consider cases where $H$ has no
custodial group as subgroup. We keep the structure constant $\epsilon^{abc}$
as in the SM, in order to conserve the usual values of the gauge couplings
$g,g^{\prime}$. This enforces us to have in general a different normalization
for the generators

$\mbox{Tr}\;[T_{L,R}^{a}T_{L,R}^{b}]=t\delta^{ab}\,.$ (12)

In this manuscript we consider only condensates in the EW sector and will not
discuss QCD charged condensates, even though they are important in CH models
and in top quark partial compositeness contruction in particular Ferretti and
Karateev (2014); Ferretti (2016).

We can now identify the Higgs doublet $\Phi$ as a NGB transforming as a bi-
doublet of $\mbox{SU}(2)_{L}\times\mbox{SU}(2)_{R}$ just defined. The neutral
component $\tilde{h}$ (with associated generator $X_{h}$) acquires a vacuum
expectation value (vev) $\langle\tilde{h}\rangle$ to break EW symmetry.
Similarly, other NGB can acquire a vev. The $\xi$ object can thus be written
as

$\xi=\Omega e^{i\Pi/f}\,,\quad\text{with }\Pi=N\pi^{a}X^{a}\,.$ (13)

The fields $\pi^{a}$ are free of tadpoles, since the form of $\Omega$ is
provided by the minimization of the potential. $N$ is a normalization factor.
We note that the new fields are not simply related by
$\tilde{h}=\langle\tilde{h}\rangle+h$. $\Omega$ is given by the exponentiation
of the vev

$\Omega=e^{i\langle\tilde{\Pi}\rangle/f}\Rightarrow
e^{\mathrm{i}NX_{h}\langle\tilde{h}\rangle/f}=e^{\mathrm{i}NX_{h}\theta^{\prime}}$
(14)

where the arrow stands for the case where only the Higgs gets a vev and
$X_{h}$ is the respective generator.

To identify the Higgs and construct the $\Omega$ matrix we notice that it is
the real part of a bi-doublet of $\mbox{SU}(2)_{L}\times\mbox{SU}(2)_{R}$.
Under $h\in H$ transformation, we have

$\xi\to g\xi\hat{h}^{-1}(g,\pi)\Rightarrow\xi\xrightarrow{g=h}h\xi h^{-1}$
(15)

and so $\Pi\to h\Pi h^{-1}$ and by expanding we can get the eigensystem of
specific generators $[T,\Pi]=\lambda(T)\Pi$. The neutral part of the bidoublet
$H^{0}=h\pm\mathrm{i}\pi^{3}$ has eigenvalues
$\lambda(T_{L}^{3})=-\lambda(T_{R}^{3})=\pm 1/2$ and from that we can identify
the Higgs with the real component 111Alternatively, we can use the fact that
the Higgs is a singlet of $\mbox{SU}(2)_{V}$ while $\pi^{3}$ is part of a
triplet, and get rid of the triplet component.. After identifying the Higgs we
can exponentiate it to get the $\Omega$ matrix. We provide the explicit form
of $X_{h}$ and $\Omega$ for the “minimal cosets” $SU(4)/Sp(4)$, $SU(5)/SO(5)$
and $SU(4)\times SU(4)/SU(4)$ in appendix. (A).

In general other pNGB might acquire a vev, and in this case the whole vacuum
has to be exponetiated to get the $\Omega$ matrix. We will not explore this
possibility in this paper, although it can have interesting phenomenological
implications.

The construction of the Lagrangian in the condensate phase follows the usual
chiral Lagrangian prescription, with the definition of some basic objects,

$\omega_{\mu}=\xi^{\dagger}\nabla_{\mu}\xi\,,\quad\nabla_{\mu}\xi=(\partial_{\mu}-ij_{\mu})\xi,\quad
j_{\mu}=v_{\mu}^{a}S^{a}+a_{\mu}^{\hat{a}}X^{\hat{a}}$ (16)

which transform under $G$ as

$\omega_{\mu}\to h\omega_{\mu}h^{\dagger}+h\partial_{\mu}h^{\dagger}\,,\quad
j_{\mu}\to gj_{\mu}g^{\dagger}+ig\partial_{\mu}g^{\dagger}.$ (17)

$\omega_{\mu}$ can be further decomposed into projections to the unbroken and
broken directions as

$\displaystyle x_{\mu}$ $\displaystyle=2\langle X^{\hat{a}}\omega_{\mu}\rangle
X^{\hat{a}},\quad x_{\mu}\to hx_{\mu}h^{\dagger},$ (18) $\displaystyle
s_{\mu}$ $\displaystyle=2\langle S^{a}\omega_{\mu}\rangle S^{a},\quad
s_{\mu}\to hs_{\mu}h^{\dagger}+h\partial_{\mu}h^{\dagger}\,.$ (19)

We can also define a field-strength that transforms homogeneously

$s_{\mu\nu}=\partial_{\mu}s_{\nu}-\partial_{\nu}s_{\mu}+[s_{\mu},s_{\nu}]\to
hs_{\mu\nu}h^{\dagger}\,.$ (20)

The incorporation of propagating EW bosons is similar to the incorporation of
virtual photons in the QCD chiral theory Urech (1995). The vector current is
simply the gauge interactions. We can thus turn off the fictitious gauge
bosons and keep only the real EW bosons

$\displaystyle j_{\mu}$ $\displaystyle=$ $\displaystyle
g{\bf\widetilde{W}}_{\mu}+g^{\prime}{\bf B_{\mu}}$ (21) $\displaystyle{\bf
B_{\mu}}$ $\displaystyle=$ $\displaystyle B_{\mu}\
T_{R}^{3},\quad{\bf\widetilde{W}_{\mu}}=\sum_{a=1}^{3}\widetilde{W}_{\mu}^{a}\
T_{L}^{a},$ (22)

where $\widetilde{W}^{k}_{\mu}$ ($k=1,\,2,\,3$) and $B_{\mu}$ are the
elementary electroweak gauge bosons associated with the $SU(2)_{L}$ and $U(1)$
hypercharge groups.

Hyperfermion current masses can be parameterized via the following spurion
transforming homogeneously

$\chi=f^{2}\xi^{\dagger}\mathcal{M}\xi^{*}\Sigma_{0}\to h\chi h^{\dagger}$
(23)

where $\mathcal{M}$ is the hyperfermion mass matrix with dimensionless
coefficients of order $m/f$ where $m$ are hyperquark current masses. We can
also define a hermitian and an anti-hermitian combination

$\chi_{\pm}=\frac{1}{2}(\chi\pm\chi^{\dagger})\,.$ (24)

The fact that the condensate are EW charged also require the definition of
gauge spurions

$\Gamma^{g,i}=\xi^{\dagger}T_{L}^{i}\xi\to
h\Gamma^{g,i}h^{\dagger},\quad\Gamma^{g^{\prime}}=\xi^{\dagger}T_{R}^{3}\xi\to
h\Gamma^{g^{\prime}}h^{\dagger}.$ (25)

With the basic objects defined above and a power counting scheme Manohar and
Georgi (1984); Gavela et al. (2016), we can construct the lowest order
Lagrangian.

$\displaystyle\mathcal{L}_{LO}$ $\displaystyle=\frac{f^{2}}{N^{2}}\langle
x^{\mu}x_{\mu}+\chi_{+}\rangle-\frac{1}{4}W_{\mu\nu}^{a}W^{a,\mu\nu}-\frac{1}{4}B_{\mu\nu}B^{\mu\nu}$
$\displaystyle+C_{g}f^{4}g^{2}\langle\Gamma^{g,i}\Sigma_{0}\Gamma^{g,iT}\Sigma_{0}\rangle+C_{g}^{\prime}f^{4}g^{\prime
2}\langle\Gamma^{g^{\prime}}\Sigma_{0}\Gamma^{g^{\prime}T}\Sigma_{0}\rangle.$
(26)

The coefficients $C_{g,g^{\prime}}$ are order 1. The corresponding $C$ in QCD
with the addition of the photon, that gives the mass difference between the
charged and neutral pions is $~{}0.8$ Urech (1995).

The Lagrangian sets the LO relation between the NGB decay constant and the EW
scale $v=246\mbox{ ${\mathrm{GeV}}$}$,

$v=f\sin\theta\,.$ (27)

The SM fermion kinetic term is provided by the usual SM covariant term, while
their Yukawa couplings are model dependent. The usual approach in dynamical EW
symmetry breaking is the existence of another type of interaction at the
flavor scale $\Lambda_{F}\sim 10^{4}\mbox{ ${\mathrm{TeV}}$}$, which induces
4-fermion interactions containing both the hyper fermions and SM fermions,
which at low energy after condensation give origin to the SM fermion masses
and their Yukawa couplings with the pNGBs. The form of these Yukawa terms
depends on the physics at the flavor scale but can be described by spurionic
fields transforming under $G$. For sake of simplicity we follow Alanne et al.
(2018) and concentrate here on spurions transforming in the fundamental (F),
two-index symmetric (S) or anti-symmetric (A) and the adjoint (Adj)
representations. Each one transforms as

$\displaystyle\Xi_{F}\to g\Xi_{F},\quad\Xi_{S/A}\to
g\Xi_{S/A}g^{T},\quad\Xi_{Adj}\to g\Xi_{Adj}g^{\dagger}.$ (28)

The SM fermions can be embedded in these representations after identifying the
object that transforms according to their SM quantum numbers. We can then
define the fermion fields in 2-index irreps and fundamental transforming as

$\displaystyle\psi$
$\displaystyle=\begin{cases}\xi^{\dagger}\Xi_{A/S}\xi^{*}\Sigma_{0}\\\
\xi^{\dagger}\Xi_{Adj}\xi\end{cases}\to h\psi h^{\dagger},$ (29)
$\displaystyle\psi^{F}$ $\displaystyle=\xi^{\dagger}\Xi_{F}\to h\psi$ (30)

The Yukawa terms follow from the bilinear terms like $\bar{\psi}\psi$. The
Lagrangian describing SM fermions at lowest order is thus

$\displaystyle\mathcal{L}_{\psi}$
$\displaystyle=\bar{Q}i\not{D}Q+\bar{L}i\not{D}L+\bar{u}i\not{D}u+\bar{d}i\not{D}d+\bar{e}i\not{D}e$
(31)
$\displaystyle-\frac{f}{4\pi}\left(Y^{1}_{u}\langle\bar{\psi}_{Q}\psi_{u}\rangle+Y^{1}_{d}\langle\bar{\psi}_{Q}\psi_{d}\rangle+Y^{1}_{e}\langle\bar{\psi}_{L}\psi_{e}\rangle+Y^{1}_{\nu}\langle\bar{\psi}_{L}\psi_{\nu}\rangle+\text{h.c.}\right)$
(32)
$\displaystyle-\frac{f}{4\pi}\left(Y^{2}_{u}\langle\bar{\psi}_{Q}\rangle\langle\psi_{u}\rangle+Y^{2}_{d}\langle\bar{\psi}_{Q}\rangle\langle\psi_{d}\rangle+Y^{2}_{e}\langle\bar{\psi}_{L}\rangle\langle\psi_{e}\rangle+Y^{2}_{\nu}\langle\bar{\psi}_{L}\rangle\langle\psi_{\nu}\rangle+\text{h.c.}\right)$
(33)
$\displaystyle-\frac{f}{4\pi}\left(Y^{F}_{u}\bar{\psi}^{F}_{Q}\psi^{F}_{u}+Y^{F}_{d}\bar{\psi}^{F}_{Q}\psi^{F}_{d}+Y^{F}_{e}\langle\bar{\psi}^{F}_{L}\psi^{F}_{e}+Y^{F}_{\nu}\bar{\psi}^{F}_{L}\psi^{F}_{\nu}+\text{h.c.}\right)$
(34)

where $Q$ are left-handed quarks, $L$ are the left handed, $u,d$ the up and
down right handed quarks and $e$ the right-handed electron. All should be
understood as sum over flavour space. $Y_{u,d,e,\nu}$ are matrices in flavour
space with coefficients related to the pre-Yukawas from the flavor scale, and
$\psi_{f}$ are the spurion embedding the corresponding fermion $f$ and defined
as 30. Explicit embeddings can be found in Golterman and Shamir (2018); Alanne
et al. (2018).

The set of terms in the effective potential that needs to be added to
renormalize SM fermion loops can be derived from the interactions above. We
leave this for future work and will be concerned with only pNGBs and EW bosons
in loops.

The topological Wess-Zumino-Witten terms Wess and Zumino (1971); Witten (1983)
are universal and model independent, besides being phenomenologically relevant
in particular for the description of bosonic decays of pNGBs. They are given
in differential form by Kaymakcalan et al. (1984); Ferretti (2016); Brauner
and Kolešová (2019)

$\displaystyle S_{\mathrm{W}ZW}\supset\frac{i\rm{dim}(\psi)}{48\pi^{2}}\int$
$\displaystyle\langle\bigg{(}dA_{L}A_{L}dUU^{\dagger}+A_{L}dA_{L}dUU^{\dagger}+dA_{R}A_{R}U^{\dagger}dU+A_{R}dA_{R}U^{\dagger}dU$
$\displaystyle-
dA_{L}dUA_{R}U^{\dagger}+dA_{R}dU^{\dagger}A_{L}U\bigg{)}\rangle.$ (35)

For $\rm SU(4)/Sp(4)$ and $\rm SU(5)/SO(5)$: $A_{L}=A$,
$A_{R}=-A^{T}=-\epsilon A\epsilon$, while for $\rm SU(4)\times
SU(4)^{\prime}/SU(4)_{D}$: $A_{L}=A_{R}=A$. Expanding in the first order in
the pNGB and integrating by parts yields

$\mathcal{L}_{WZW}=\frac{\rm{dim}(\psi)}{48\pi^{2}f}\langle
2F_{\mu\nu}\widetilde{F}^{\mu\nu}(\Omega\Pi\Omega^{\dagger}+\Omega^{\dagger}\Pi\Omega)+\Omega^{\dagger}F_{\mu\nu}\Omega\Pi\Omega\widetilde{F}^{\mu\nu}\Omega^{\dagger}+\Omega
F_{\mu\nu}\Omega^{\dagger}\Pi\Omega^{\dagger}\widetilde{F}^{\mu\nu}\Omega\rangle\,.$
(36)

$\widetilde{F}^{\mu\nu}\equiv\frac{1}{2}\epsilon^{\mu\nu\rho\sigma}{F}_{\rho\sigma}$.

##  III The next-to-leading order Lagrangian

To construct the $\mathcal{O}(p^{4})$ Lagrangian we need to define some other
objects. The vector field strength

$j_{\mu\nu}=\partial_{\mu}j_{\nu}-\partial_{\nu}j_{\mu}+[j_{\mu},j_{\nu}]\to
gj_{\mu\nu}g^{\dagger},\quad f_{\mu\nu}=\xi^{\dagger}j_{\mu\nu}\xi\to
hf_{\mu\nu}h^{\dagger}$ (37)

and its projections $f_{\mu\nu}=\tilde{f}_{\mu\nu}+\hat{f}_{\mu\nu}$ as in
eqs. (18)–(19).

The Leutwyler-Gasser terms Gasser and Leutwyler (1984, 1985) at NLO, worked
out for a generic coset in Bijnens and Lu (2009), are given by

$\displaystyle\mathcal{L}_{p^{4}}$
$\displaystyle=\frac{1}{16\pi^{2}}\big{\\{}L_{0}\langle
x^{\mu}x^{\nu}x_{\mu}x_{\nu}\rangle+L_{1}\langle x^{\mu}x_{\mu}\rangle\langle
x^{\nu}x_{\nu}\rangle+L_{2}\langle x^{\mu}x^{\nu}\rangle\langle
x_{\mu}x_{\nu}\rangle+L_{3}\langle x^{\mu}x_{\mu}x^{\nu}x_{\nu}\rangle$
$\displaystyle+L_{4}\langle
x^{\mu}x_{\mu}\rangle\langle\chi_{+}\rangle+L_{5}\langle
x^{\mu}x_{\mu}\chi_{+}\rangle+L_{6}\langle\chi_{+}\rangle^{2}+L_{7}\langle\chi_{-}\rangle^{2}+\frac{1}{2}L_{8}\langle\chi_{+}^{2}+\chi_{-}^{2}\rangle$
$\displaystyle-\mathrm{i}L_{9}\langle\tilde{f}_{\mu\nu}x^{\mu}x^{\nu}\rangle+\frac{1}{4}L_{10}\langle\tilde{f}_{\mu\nu}^{2}-\hat{f}_{\mu\nu}^{2}\rangle\big{\\}}$
(38)

We add a $1/(4\pi)^{2}$ from naive dimension analysis (NDA) power counting
Manohar and Georgi (1984) and so the coefficients are expected to be
$\mathcal{O}(1)$.

Once gauge interactions are turned on, we need to add the gauge spurions
$\Gamma$ (eq. (25)) and to define their covariant derivative

$D_{\mu}\Gamma=\partial_{\mu}\Gamma+[s_{\mu},\Gamma]\to hD_{\mu}\Gamma
h^{\dagger}\,.$ (39)

Following Urech (1995) we obtain the corresponding terms for the Lagrangian,

$\displaystyle\mathcal{L}_{g^{2}p^{2}}$
$\displaystyle=\frac{g^{2}f^{2}}{16\pi^{2}}\big{\\{}K_{1}\langle
x_{\mu}x^{\mu}\rangle\langle\Gamma^{2}\rangle+K_{2}\langle
x_{\mu}x^{\mu}\rangle\langle\Gamma^{g,i}\Sigma_{0}\Gamma^{g,iT}\Sigma_{0}\rangle+K_{3}\langle
x_{\mu}\Gamma\rangle\langle x^{\mu}\Gamma\rangle$ $\displaystyle+K_{5}\langle
x_{\mu}x^{\mu}\Gamma^{2}\rangle+K_{6}(\langle
x_{\mu}x^{\mu}\Gamma^{g,i}\Sigma_{0}\Gamma^{g,iT}\Sigma_{0}\rangle+\text{h.c.})+K_{7}\langle\chi_{+}\rangle\langle\Gamma^{2}\rangle$
$\displaystyle+K_{8}\langle\chi_{+}\rangle\langle\Gamma^{g,i}\Sigma_{0}\Gamma^{g,iT}\Sigma_{0}\rangle+K_{9}\langle\chi_{+}\Gamma^{2}\rangle+K_{10}(\langle\chi_{+}\Gamma^{g,i}\Sigma_{0}\Gamma^{g,iT}\Sigma_{0}\rangle+\text{h.c.})$
$\displaystyle+K_{11}(\langle\chi_{-}\Gamma^{g,i}\Sigma_{0}\Gamma^{g,iT}\Sigma_{0}\rangle+\text{h.c.})+K_{12}(\langle
x^{\mu}D_{\mu}\Gamma^{g,i}\Sigma_{0}\Gamma^{g,iT}\Sigma_{0}\rangle+\text{h.c.})$
$\displaystyle+K_{13}D_{\mu}\Gamma^{g,i}\Sigma_{0}(D^{\mu}\Gamma^{g,i})^{T}\Sigma_{0}+K_{14}D_{\mu}\Gamma^{g,i}D^{\mu}\Gamma^{g,iT}\big{\\}}$
(40)

Similar terms replacing $g$ with $g^{\prime}$ should also be added.

##  IV Towards predictions: GBS at the LHC

The framework described in sections II and III is implemented for some cases
of interest in the the FeynRules package Alloul et al. (2014); Christensen and
Duhr (2009). The available models are222The FeynRules and UFO models can be
retrieved from the High Energy Model Database
https://hepmdb.soton.ac.uk/hepmdb:0223.0338.:

  * •

the minimal CH (MCH) model based on the coset $SO(5)/SO(4)$, including terms
of $\mathcal{O}(p^{4})$, $\mathcal{L}_{p^{4}}$ (topological terms in $L_{WZW}$
vanish in this coset).

  * •

the $SU(4)/Sp(4)$ model including topological terms $L_{WZW}$, suited for the
phenomenological study of the extra singlet pNGB and its bosonic decays.

  * •

the $SU(5)/SO(5)$ model including topological terms $L_{WZW}$, suited for the
study of the 14 (p)NGBs and their bosonic decays.

To illustrate the power of the developed tools we discuss the effect of
$\mathcal{O}(p^{4})$ terms at tree-level in di-Higgs production and VBS and
the pair production of extra pNGBs in the two implemented non-minimal models.
All simulation is performed with the export of the FeynRules model to the
Universal FeynRules Output (UFO) Degrande et al. (2012), that can be imported
in the MG5_aMC@NLO program Alwall et al. (2011) for simulations of proton-
proton scattering. The simulations are performed at 14 TeV in the proton-
proton center-of-mass energy with the NNPDF 3.1 NLO LUXQED parton distribution
function (PDF) set with $\alpha_{s}(\mu)=0.118$ Ball et al. (2013).

###  IV.1 Di-Higgs via VBF and VBS at $\mathcal{O}(p^{4})$ and tree-level

To study the Higgs and weak boson physics at high energy we neglect possible
extra pNGBs, having in mind that the Higgs and EW bosons obey universal
relations. We also neglect the the spurion contributions, $\chi\to 0$ in
$\mathcal{L}_{4}$ and the $\mathcal{L}_{g^{2}p^{2}}$ (eq. (40)) as a first
approximation. As pointed out in Liu et al. (2019), the modification in these
interactions can be well approximated by the MCH SO(5)/SO(4) description. In
SO(5)/SO(4) the $L_{0}$ and $L_{3}$ terms are redundant and can be rewritten
in terms of $L_{1}$, $L_{2}$. An interesting particularity in the
$SO(5)/SO(4)$ coset is that the stability group $H$ is not simple, $SO(4)\sim
SU(2)_{L}\times SU(2)_{R}$. This fact allow us to further break the
projections into the unbroken generators into two independent terms. For
example $\tilde{f}_{\mu\nu}$ can be broken in

$\displaystyle O_{L}$ $\displaystyle\equiv O_{L}^{a}S^{a}=2\langle
S^{a}\tilde{O}\rangle S^{a},\,a=1,2,3\quad$ (41) $\displaystyle O_{R}$
$\displaystyle\equiv O_{R}^{a}S^{a}=2\langle S^{a}\tilde{O}\rangle
S^{a},\,a=4,5,6\,.$ (42)

Therefore, we use a different basis for the operators

$\displaystyle\mathcal{L}^{MCH}_{4}$
$\displaystyle=\frac{1}{16\pi^{2}}\big{\\{}L_{1}\langle
x^{\mu}x_{\mu}\rangle\langle x^{\nu}x_{\nu}\rangle+L_{2}\langle
x^{\mu}x^{\nu}\rangle\langle x_{\mu}x_{\nu}\rangle+L_{3}^{-}\langle
s^{\mu\nu}_{L}s_{\mu\nu,L}-s^{\mu\nu}_{R}s_{\mu\nu,R}\rangle$
$\displaystyle-\mathrm{i}L_{9}^{+}\langle(f^{\mu\nu}_{L}+f^{\mu\nu}_{R})x_{\mu}x_{\nu}\rangle-\mathrm{i}L_{9}^{-}\langle(f^{\mu\nu}_{L}-f^{\mu\nu}_{R})x_{\mu}x_{\nu}\rangle$
$\displaystyle+\frac{1}{4}L_{10}^{+}\langle\hat{f}_{\mu\nu}\hat{f}^{\mu\nu}\rangle+\frac{1}{4}L_{10}^{-}\langle
f_{L}^{\mu\nu}f_{\mu\nu,L}-f_{R}^{\mu\nu}f_{\mu\nu,R}\rangle\big{\\}}$ (43)

The coefficients carrying a minus sign break the $Z_{2}$ symmetry
$R\leftrightarrow L$. Notice that in several models the $Z_{2}$ symmetry is
present in the strong theory and is only softly broken. For instance $O(4)$ is
a subgroup of $Sp(4)$. However, the breaking of $Z_{2}$ can be achieved in
other cosets by gauge spurions $\Gamma^{g,g^{\prime}}$ and the appropriate
choice of the hyperfermion spurion $\chi$. The Lagrangian is equivalent to
Contino et al. (2011); Liu et al. (2019), using the NDA power counting
$m_{\rho}=4\pi f$ and the identification,

$c_{1}=L_{1},\,c_{2}=L_{2},\,c_{3}=L_{3}^{-},\,c_{4}^{\pm}=L_{9}^{\pm},\,c_{5}^{\pm}=L_{10}^{\pm}\,.$
(44)

The implementation in FeynRules requires the fields to have canonically
normalized kinetic terms and diagonal masses. The higher dimension operators
induce corrections to the kinetic terms, as

$\mathcal{L}_{4}\supset-\frac{1}{4}B_{\mu\nu}B^{\mu\nu}[1+X_{B}]-\frac{1}{4}W^{i}_{\mu\nu}W^{i\mu\nu}[1+X_{W}]-\frac{1}{4}W^{3}_{\mu\nu}B^{\mu\nu}X_{BW}$
(45)

and one must thus redefine the fields to fulfill these requirements. This is
done by the following replacements:

$\displaystyle W^{i}_{\mu}$ $\displaystyle\to W^{i}_{\mu}(1+X_{W})$ (46)
$\displaystyle A_{\mu}$ $\displaystyle\to
A_{\mu}(1+X_{W}s_{W,0}^{2}+X_{B}c_{W,0}^{2}+2s_{W,0}c_{W,0}X_{BW})+$
$\displaystyle
Z_{\mu}(s_{W,0}c_{W,0}(X_{W}-X_{B})+(c_{W,0}^{2}-s_{W,0}^{2})X_{BW}+\delta
s_{W}/c_{W,0}),$ (47) $\displaystyle Z_{\mu}$ $\displaystyle\to
Z_{\mu}(1+X_{W}c_{W,0}^{2}+X_{B}s_{W,0}^{2}-2s_{W,0}c_{W,0}X_{BW})+$
$\displaystyle
A_{\mu}(s_{W,0}c_{W,0}(X_{W}-X_{B})+(c_{W,0}^{2}-s_{W,0}^{2})X_{BW}-\delta
s_{W}/cw).$ (48)

Also the gauge couplings have to be modified,

$\displaystyle g_{2}$ $\displaystyle\to g_{2}/(1+X_{W})\sim g_{2}(1-X_{W})$
(49) $\displaystyle g_{1}$ $\displaystyle\to g_{1}/(1+X_{B})\sim
g_{1}(1-X_{B})$ (50) $\displaystyle s_{W,0}^{2}$
$\displaystyle=(1-M_{W}^{2}/M_{Z}^{2})$ (51) $\displaystyle s_{W}$
$\displaystyle=s_{W,0}(1+\delta s_{W}^{2})$ (52) $\displaystyle\delta s_{W}$
$\displaystyle=X_{BW}c_{W,0}(1-2s_{W,0}^{2})$ (53)

The exact form of the mixing coefficients in terms of the Lagrangian
coefficients are

$\displaystyle X_{W}$
$\displaystyle=\frac{e^{2}}{8s_{W,0}^{2}}[6L_{9}^{+}+L_{10}^{+}+4(4L_{3}^{+}+2L_{9}^{-}+L_{10}^{-})c_{\theta}+(2L_{9}^{+}-L_{10}^{+})c_{2\theta}],$
(54) $\displaystyle X_{B}$
$\displaystyle=\frac{e^{2}}{8c_{W,0}^{2}}[6L_{9}^{+}+L_{10}^{+}-4(4L_{3}^{-}+2L_{9}^{-}+L_{10}^{-})c_{\theta}+(2L_{9}^{+}-L_{10}^{+})c_{2\theta}],$
(55) $\displaystyle X_{WB}$
$\displaystyle=\frac{e^{2}}{4c_{W,0}s_{W,0}}(2L_{9}^{+}-L_{10}^{+})s_{\theta}^{2}$
(56)

The coefficients $L_{3}^{-}$, $L_{9}^{\pm}$ and $L_{10}^{+}$ contribute to
anomalous trilinear gauge couplings (aTGC)

$\displaystyle\delta g_{1,z}=\left(\frac{m_{Z}^{2}}{(4\pi
f)^{2}}\right)(-2L_{3}^{-}+L_{9}^{-})\cos\theta+L_{9}^{+},\quad\delta\kappa_{\gamma}=\left(\frac{m_{W}^{2}}{(4\pi
f)^{2}}\right)(-4(L_{9}^{+}-2L_{10}^{+}))$

and $\delta\kappa_{z}=\delta
g_{1,z}-\frac{s_{W}^{2}}{c_{W}^{2}}\delta\kappa_{\gamma}$. Using the fit done
in Falkowski et al. (2016) we can put bounds on those coefficients. But only
with these modifications it is not possible to lift the degeneracy since there
can be cancellations. On the other hand, if we assume $Z_{2}$ symmetry
($L_{9}^{-}=L_{3}=0$), a bound can be set on $L_{9}^{+}$ and $L_{10}^{+}$, as
shown in fig. (1). We use $\theta=0.3$ ($f=832\mbox{ ${\mathrm{GeV}}$}$). The
bounds are typically weak, especially considering order 1 coefficients (in QCD
for instance $L_{i}\lesssim 1$ Ecker (1995)).

Figure 1: Bounds on $L_{9}^{+}$ and $L_{10}^{+}$ assuming $Z_{2}$ symmetry
$L_{9}^{-}=L_{3}=0$ for $\theta=0.3$ ($f=832\mbox{ ${\mathrm{GeV}}$}$). The
bounds scale with $f_{NEW}^{2}/f^{2}$ becoming less constraining for large
$f_{NEW}$.

The Lagrangian 43 contains also anomalous quartic gauge couplings (aQGC). We
use the conventions of Éboli and Gonzalez-Garcia (2016) to define the
effective couplings,

$\displaystyle\mathcal{O}_{WW,1}$
$\displaystyle=W^{+\mu}W^{-}_{\mu}W^{+\nu}W^{-}_{\nu},$
$\displaystyle\mathcal{O}_{WW,2}$
$\displaystyle=W^{+\mu}W^{-\nu}W^{+}_{\mu}W^{-}_{\nu},$ (57)
$\displaystyle\mathcal{O}_{WZ,1}$
$\displaystyle=W^{+\mu}W^{-}_{\mu}Z^{\nu}Z_{\nu},$
$\displaystyle\mathcal{O}_{WZ,2}$
$\displaystyle=W^{+\mu}W^{-\nu}Z_{\mu}Z_{\nu},$ (58)
$\displaystyle\mathcal{O}_{ZZ,1}$
$\displaystyle=Z^{\mu}Z^{\mu}Z^{\nu}Z_{\nu},$ (59)
$\displaystyle\mathcal{O}_{WA,1}$
$\displaystyle=W^{+\mu}W^{-}_{\mu}A^{\nu}A_{\nu},$
$\displaystyle\mathcal{O}_{WA,2}$
$\displaystyle=W^{+\mu}W^{-\nu}A_{\mu}A_{\nu},$ (60)
$\displaystyle\mathcal{O}_{AZ,1}$
$\displaystyle=W^{+\mu}W^{-}_{\mu}A^{\nu}Z_{\nu},$
$\displaystyle\mathcal{O}_{AZ,2}$
$\displaystyle=W^{+\mu}W^{-\nu}A_{\mu}Z_{\nu}+\text{h.c.}$ (61)

Only zero derivative terms are generated at this order. The operators that
contribute to aTGC ($L_{3}^{-}$, $L_{9}^{\pm}$ and $L_{10}^{+}$) contribute
exclusively to a modification to quartic coupling with the same Lorentz
structure of the SM, while the so-called genuine quartic coupling operators
$L_{1}$ and $L_{2}$ contribute with a different Lorentz structure (and
$L_{10}^{-}$ do not contribute at all). The full QGC Lagrangian is

$\displaystyle\mathcal{L}_{QGC}$
$\displaystyle=-g^{2}(\mathcal{O}_{WW,1}-\mathcal{O}_{WW_{2}})\left[1+\frac{m_{W}^{2}}{16\pi^{2}f^{2}}\left(-8\cos\theta
L_{3}^{-}+L_{4}^{+}+\cos\theta L_{4}^{-}\right)\right]$
$\displaystyle+\frac{m_{W}^{4}}{\pi^{2}f^{4}}\left[2L_{1}\mathcal{O}_{WW,1}+L_{2}(\mathcal{O}_{WW,1}+\mathcal{O}_{WW,2})\right]$
$\displaystyle+g^{2}c_{W}^{2}(\mathcal{O}_{WZ,1}-\mathcal{O}_{WZ_{2}})\left[1+\frac{m_{Z}^{2}}{16\pi^{2}f^{2}}\left(8\cos\theta
L_{3}^{-}-(1-4s_{W}^{4})L_{4}^{+}+\cos\theta
L_{4}^{-}+s_{W}^{4}L_{5}^{+}\right)\right]$
$\displaystyle+\frac{m_{W}^{2}m_{Z}^{2}}{\pi^{2}f^{4}}\left(2L_{1}\mathcal{O}_{WZ,1}+L_{2}\mathcal{O}_{WZ,2}\right)$
$\displaystyle-e^{2}\frac{c_{W}}{s_{W}}(\mathcal{O}_{AZ,1}-\mathcal{O}_{AZ_{2}})\left[1+\frac{m_{Z}^{2}}{16\pi^{2}f^{2}}\left(-16\cos\theta
L_{3}^{-}+2(4s_{W}^{4}-4s_{W}^{2}-1)L_{4}^{+}-\cos\theta
L_{4}^{-}+s_{W}^{4}L_{5}^{+}\right)\right]$
$\displaystyle+\frac{m_{W}^{2}m_{Z}^{2}}{\pi^{2}f^{4}}\left(2L_{1}\mathcal{O}_{WZ,1}+L_{2}\mathcal{O}_{WZ,2}\right)$
$\displaystyle+\frac{m_{Z}^{4}}{\pi^{2}f^{4}}2\left(L_{1}\mathcal{O}_{ZZ,1}+L_{2}\mathcal{O}_{ZZ,2}\right)$
$\displaystyle+e^{2}(\mathcal{O}_{WA,1}-\mathcal{O}_{WA_{2}})\left[1+\frac{e^{2}v^{2}}{16\pi^{2}f^{2}}\left(2L_{4}^{+}-L_{5}^{+}\right)\right]$
(62)

With the model implementation and a better understanding of its modifications
to the SM interactions we perform the simulation of di-Higgs production via
VBF and VBS at the LHC.

A diagram depicting di-Higgs production via VBF $pp\to jjhh$ is shown in fig.
(2) (left). We adopt the selection cuts listed in tab. (1).

Figure 2: Diagrams depicting di-Higgs via VBF (left) and $W^{+}W^{-}$ VBS (right). $pp\to jjhh$ | $p_{T}(j)>20\mbox{ ${\mathrm{GeV}}$}$ | $|\eta(j)|<5$ | $m(jj)>200\mbox{ ${\mathrm{GeV}}$}$ | $\Delta R(jj)>0.4$
---|---|---|---|---
| $p_{T}(h)>30\mbox{ ${\mathrm{GeV}}$}$ | $|\eta(h)|<3.5$ |  |
$pp\to jjW^{+}W^{-}$ | $p_{T}(j)>20\mbox{ ${\mathrm{GeV}}$}$ | $|\eta(j)|<5$ | $m(jj)>250\mbox{ ${\mathrm{GeV}}$}$ | $\Delta\eta(jj)>2.5$
| $p_{T}(w)>30\mbox{ ${\mathrm{GeV}}$}$ | $|\eta(W^{\pm})|<3.5$ | $m(W^{+}W^{-})>500\mbox{ ${\mathrm{GeV}}$}$ |
$pp\to jj\eta\eta$ | $p_{T}(j)>20\mbox{ ${\mathrm{GeV}}$}$ | $|\eta(j)|<5$ | $m(jj)>200\mbox{ ${\mathrm{GeV}}$}$ | $\Delta R(jj)>0.4$
| $p_{T}(\eta)>30\mbox{ ${\mathrm{GeV}}$}$ | $|\eta(\eta)|<3.5$ |  |
$pp\to jj\eta_{5}^{++}\eta_{5}^{--}$ | $p_{T}(j)>20\mbox{ ${\mathrm{GeV}}$}$ | $|\eta(j)|<5$ | $m(jj)>200\mbox{ ${\mathrm{GeV}}$}$ | $\Delta R(jj)>0.4$
Table 1: Selection cuts.

Predictions for the distributions of the invariant mass of the di-Higgs system
$m(hh)$, the transverse momentum $p_{T}(h)$, and the pseudo-rapidity $\eta(h)$
of the hardest Higgs boson are shown in fig. (3), on the top, center and
bottom rows respectively. The left column compare the predictions of the LO
Lagrangian for different values of $\theta=0.2,\,0.3$ and the SM. The usual
growing behavior with energy is observed in the $m(hh)$ distribution for the
CH scenarios. In the middle column the contribution of each of the NLO
operators to each observable are shown separately for each $L_{i}$ coefficient
set to 1. These should be summed linearly to the distributions. It is
interesting to notice that the genuinely quartic couplings give the largest
contributions at high energy, but they are suppressed near threshold
production. In the right column the full NLO prediction with all $L_{i}=1$ are
compared to the LO predictions.

Figure 3: Distributions of di-Higgs via VBF.

VBS $W^{+}W^{-}$ production $pp\to jjW^{+}W^{-}$ is depicted in diagram fig.
(2) (right). We adopt the selection cuts shown in tab. (1). Predictions for
the distributions of the invariant mass of the $W^{+}W^{-}$ system
$m(W^{+}W^{-})$, the transverse momentum $p_{T}(W^{+})$, and the pseudo-
rapidity $\eta(W^{+})$ of $W^{+}$ are shown in fig. (4). The same format of
fig. (3) is used, with also similar observations about each NLO operator. We
also show predictions for the case where both final state $W^{+}$ and $W^{-}$
are polarized longitudinally. We used the polarized scattering implementation
in MG5_aMC@NLO Buarque Franzosi et al. (2020a). The corresponding
distributions are shown in fig. (5). It can indeed be noticed a larger new
physics effect in the longitudinal components compared to the unpolarized
scattering. This is expected since only the longitudinal component of
$W^{\pm}$ can be identified as the NGB at high energies, according to the
Equivalence Theorem.

Figure 4: Unpolarized VBS.

Figure 5: Longitudinal VBS.

###  IV.2 pNGB production and decay in non-minimal models

We consider here two symmetry breaking cosets: $SU(4)/Sp(4)$ and
$SU(5)/SO(5)$. We include only the LO kinetic term and the WZW terms and
neglect the SM fermion interactions.

The $SU(4)/Sp(4)$ model contains 5 (p)NGBs. The Higgs bi-doublet (2,2) and a
singlet $\eta$ (1,1) under custodial $SU(2)_{L}\times SU(2)_{R}$ group. The
pair production of $\eta$ via VBF and associate production has been considered
in several other publications (see e.g. Arbey et al. (2017)). The branching
ratios of $\eta$ are shown in fig. (6). We show only the 2-body decays
computed with the analytic expressions provided in the UFO model. For
$50\mbox{ ${\mathrm{GeV}}$}\lesssim m_{\eta}\lesssim 90\mbox{
${\mathrm{GeV}}$}$ the offshell $Z^{*}\gamma$ decay channel dominates, and for
$m_{\eta}\lesssim 50\mbox{ ${\mathrm{GeV}}$}$ loop induced $b\bar{b}$ channel
dominates. See Buarque Franzosi et al. (2020b) for more detail on the low
$\eta$ mass case.

Distributions for process $pp\to jj\eta\eta$ are shown in fig. (7). They are
the invariant mass of the $\eta\eta$ system, the transverse momentum and
pseudo-rapidity of the hardest $\eta$. Event generation has been performed
with the selection cuts shown in tab. (1).

Figure 6: $\eta$ branching ratios in $SU(4)/Sp(4)$ model computed at tree
level and including only 2-body decays via the WZW anomaly.

Figure 7: $pp\to\eta\eta jj$ in $SU(4)/Sp(4)$. Left: the invariant mass of the
$\eta\eta$ system. Middle: the transverse momentum. Right: pseudo-rapidity of
the hardest $\eta$.

The $SU(5)/SO(5)$ model contains ${\bf 14}\subset SO(5)$ (p)NGBs Ferretti
(2016); Agugliaro et al. (2019). ${\bf 14}$ decomposes into the Higgs bi-
doublet, a bi-triplet and a singlet under the custodial subgroup
$SU(2)_{L}\times SU(2)_{R}$, ${\bf(2,2)+(3,3)+(1,1)}$. We work in the
custodial basis, further decomposing these multiplets into $SU(2)_{V}$,
${\bf(3,3)\sim 5+3+1}$, which we name $\eta_{5}$, $\eta_{3}$, $\eta_{1}$
respectively and the singlet ${\bf(1,1)}$ which we call $\eta$. Typically
there is mixing between these states, and the different charges of each
$SU(2)_{V}$ multiplet can have different masses in general.

We further take as benchmark for the masses the following values:

$m_{\eta_{5}^{\pm\pm,\pm,0}}=600\mbox{ ${\mathrm{GeV}}$},\quad
m_{\eta_{3}^{\pm,0}}=400\mbox{ ${\mathrm{GeV}}$},\quad m_{\eta_{1}}=300\mbox{
${\mathrm{GeV}}$},\quad m_{\eta}=200\mbox{ ${\mathrm{GeV}}$}\,.$ (63)

The branching ratios into two body computed at tree level are shown in fig.
(8) for the neutral pNGBs and fig. (9) for the singly charged ones. They are
calculated for 2-body at LO only with the analytical expressions provided in
the UFO model. The singlet of $SO(5)$ $\eta$ does not couple to other pNGBs
and decay fully through the anomaly. The other states usually decay into di-
boson till a kinematic threshold into other pNGB opens up, and then tend to
decay into them. Near kinematic threshold 3-body decays via off-shell
propagating pNGBs can be important and are not shown in the figures.

The $\eta_{3}^{\pm,0}$ state has no couplings via anomaly and decays only via
the couplings to other pNGBs. If it is the lightest state, it decays to 3 body
mediated by an offshell pNGBs. As a example we take $m_{\eta^{0}}=300\mbox{
${\mathrm{GeV}}$}$ (keeping the other masses as in eq. (63)) and get total
width $\Gamma_{3}=0.32$ eV and the branching ratios $\Gamma_{VVV}$ in tab.
(2).

$VVV$ | $Z\gamma\gamma$ | $ZZ\gamma$ | $\gamma W^{+}W^{-}$ | $ZW^{+}W^{-}$ | $ZZZ$
---|---|---|---|---|---
$\Gamma_{VVV}$ (%) | 74.17 | 14.25 | 10.64 | 0.73 | 0.21
Table 2: Branching ratios of $\eta_{3}^{0}$ for $m_{\eta^{0}}=300\mbox{
${\mathrm{GeV}}$}$, $\theta=0.3$ and the other masses as eq. (63).

Figure 8: Two-body tree level branching ratios of neutral pNGBs in
$SU(5)/SO(5)$. $\theta=0.3$, apart from one pNGB with scanned mass, the other
pNGB masses are fixed to $m_{\eta_{5}^{\pm\pm,\pm,0}}=600\mbox{
${\mathrm{GeV}}$}$, $m_{\eta_{3}^{\pm,0}}=500\mbox{ ${\mathrm{GeV}}$}$,
$m_{\eta_{1}}=400\mbox{ ${\mathrm{GeV}}$}$, $m_{\eta}=300\mbox{
${\mathrm{GeV}}$}$.

Figure 9: Two-body tree level branching ratios of charged pNGBs in
$SU(5)/SO(5)$. $\theta=0.3$, apart from one pNGB with scanned mass, the other
pNGB masses are fixed to $m_{\eta_{5}^{\pm\pm,\pm,0}}=600\mbox{
${\mathrm{GeV}}$}$, $m_{\eta_{3}^{\pm,0}}=500\mbox{ ${\mathrm{GeV}}$}$,
$m_{\eta_{1}}=400\mbox{ ${\mathrm{GeV}}$}$, $m_{\eta}=300\mbox{
${\mathrm{GeV}}$}$.

As a example of GBS in this model we consider the doubly-charged scalar pair
production via VBF $pp\to jj\eta^{++}\eta^{--}$. The pair production of
$\eta_{5}^{\pm\pm}$ is bounded by ATLAS searches, giving a lower bound on its
mass $m_{5}\gtrsim 400\mbox{ ${\mathrm{GeV}}$}$ Aad et al. (2021). The Drell-
Yan (DY) production and interpretation has been studied in Banerjee et al.
(2022). For $m_{5}=400\mbox{ ${\mathrm{GeV}}$}$ and $\theta=0.3$ the pair
production production via VBF has total cross section $\sigma_{VBS}=0.135$ fb,
to be compared with $\sigma_{DY}=5.38$ fb for Drell-Yan (DY) production. Event
generation has been performed with the selection cuts shown in tab. (1).

In fig. (10) we show the invariant mass of the $\eta^{++}\eta^{--}$ system.

Figure 10: $pp\to\eta^{++}\eta^{--}jj$ in $SU(5)/SO(5)$. Invariant mass of the
$\eta^{++}\eta^{--}$ system.

##  V Conclusion and prospects

We have presented a framework for the study of (p)NGBs in CH models, including
VBS, di-Higgs via VBF, GBS in non-minimal models. We have implemented the
model in FeynRules and output it in the UFO format. We then showed examples of
simulations that can be further explored in more detailed phenomenological
analysis. We looked at the effect of $\mathcal{O}(p^{4})$ operators in VBS, in
quartic couplings modifications and in di-Higgs production via VBF. We also
computed the branching ratios and pair-production distributions for extra
pNGBs in non-minimal models of CH for some particular benchmarks.

Besides the suitability for phenomenological studies, the tool here presented
is intended as a first step towards the incorporation of loop corrections of
EW and chiral perturbation theory origin.

##  Appendix A Explicit matrices for “minimal cosets”

For $\mbox{SU}(4)/Sp(4)$ we have $N=\sqrt{2}$ and $t=1/2$. We use the matrices
in Ref.Cacciapaglia and Sannino (2014). $\hat{T}_{L}^{1,2,3}=S^{1,2,3}$
$\hat{T}_{R}^{1,2,3}=S^{4,5,6}$. $X_{h}=X_{4}$.

$\Omega=\exp(\mathrm{i}NX_{h}\theta)=\cos\frac{\theta}{2}+\mathrm{i}2\sqrt{2}X_{h}\sin\frac{\theta}{2}\,.$
(64)

For $SU(4)\times SU(4)/SU(4)$ we get $N=2$, $t=1$. Working with the $8\times
8$ matrices, we choose

$\hat{T}_{L,R}^{i}=\left(\begin{array}[]{cc}\tau_{L,R}^{i}&\\\
&\tau_{L,R}^{i}\end{array}\right)$ (65)

with

$\tau_{L}^{i}=\frac{1}{2\sqrt{2}}\left(\begin{array}[]{cc}\sigma_{i}&\\\
&0\end{array}\right)\quad\text{and}\quad\tau_{R}^{i}=\frac{1}{2\sqrt{2}}\left(\begin{array}[]{cc}0&\\\
&\sigma_{i}\end{array}\right)$ (66)

The Higgs generator can be chosen in one of 2 possible directions. We choose

$X_{h}=\left(\begin{array}[]{cc}\tau_{H}&\\\ &-\tau_{H}\end{array}\right)$
(67)

with

$\tau_{H}=\frac{i}{4}\left(\begin{array}[]{cc}0&\mbox{ ${\bf 1}_{2\times 2}$
}\\\ -\mbox{ ${\bf 1}_{2\times 2}$ }&\end{array}\right)$ (68)

So

$\Omega=\exp\left[\mathrm{i}2\theta
X_{h}\right]=\cos\frac{\theta}{2}+\mathrm{i}4X_{h}\sin\frac{\theta}{2}$ (69)

For $SU(5)/SO(5)$ we have $N=2$, $t=1$. The EW generators can be chosen in the
following way

$\displaystyle
T_{L}^{i}=\frac{1}{2\sqrt{2}}\left(\begin{array}[]{ccc}\sigma_{i}&&\\\
&\sigma_{i}&\\\ &&\end{array}\right)\quad\text{and}\quad
T_{R}^{i}=-\frac{1}{2\sqrt{2}}\left(\begin{array}[]{cc}\sigma_{i}^{T}\left(\begin{array}[]{cc}\mbox{
${\bf 1}_{2\times 2}$ }&\\\ &\mbox{ ${\bf 1}_{2\times 2}$
}\end{array}\right)&\\\ &\end{array}\right)$ (77)

The Higgs is identified by the generator

$\displaystyle
X_{h}=\frac{i}{2\sqrt{2}}\left(\begin{array}[]{ccc}&&\binom{0}{1}\\\
&&\binom{-1}{0}\\\ (0,-1)&(1,0)&\end{array}\right)$ (81)

So

$\Omega=\exp\left[\mathrm{i}2\theta
X_{h}\right]=1+4X_{h}^{2}(\cos\theta-1)+\mathrm{i}2X_{h}\sin\theta\,.$ (82)

##  Appendix B Explicit breaking terms and spurions

The gauging of the EW subgroup will explicitly break the global symmetry $G$
and the $j_{\mu}$ can be regarded as a spurionic field. Other interactions
besides the gauge of $H^{\prime}$ may also break $G$ and can be included in
the low energy chiral lagrangian via spurions. The only requirement is that
the local SM gauge group be unbroken, and preferably that the whole custodial
symmetry be respected. Following Alanne et al. (2018) we concentrate here on
spurions transforming in the fundamental (F), two-index symmetric (S) or anti-
symmetric (A) and the adjoint (Adj) representations. The best example of such
terms is given by hyperfermion masses $\bar{\psi}\mathcal{M}\psi$, which
transforms in general as $\Sigma$ (),

$\mathcal{M}\to
g\mathcal{M}g^{T},\quad\chi=\xi^{\dagger}\mathcal{M}\xi^{*}\Sigma_{0}\to h\chi
h^{\dagger}$ (83)

Notice that with our way to write the $SU(N)^{2}/SU(N)$ case, we can keep the
same structure of a simple group, without having to specify $g_{L}$ or
$g_{R}$. The other spurions representations transform as

$\displaystyle\Xi_{F}\to g\Xi_{F},\quad\Xi_{S/A}\to
g\Xi_{S/A}g^{T},\quad\Xi_{Adj}\to g\Xi_{Adj}g^{\dagger}.$ (84)

The A/S and Adj representation can be put together in the object

$\chi=\begin{cases}\xi^{\dagger}\Xi_{A/S}\xi^{*}\Sigma_{0}\\\
\xi^{\dagger}\Xi_{Adj}\xi\end{cases}\to h\chi h^{\dagger}$ (85)

For the symmetric and anti-symmetric cases, since we always want to add the
hermitian conjugate of the terms we can define a hermitian and an anti-
hermitian combination

$\chi_{\pm}=\frac{1}{2}(\chi\pm\chi^{\dagger})\,.$ (86)

For the adjoint $\chi_{+}=\chi$ and $\chi_{-}=0$. With those objects we can
construct different objects transforming homogeneously under $H$ ($O\to
hOh^{\dagger}$) that can be used to construct invariants for the Lagrangian,
such as

$\displaystyle\langle\chi_{+}\rangle,\,\langle\chi_{\pm}^{2}\rangle,\,\langle\chi_{\pm}\rangle^{2},\,\langle
x_{\mu}\chi_{+}\rangle,$ (87)
$\displaystyle\langle\chi_{+}\epsilon_{0}\chi_{+}^{T}\epsilon_{0}\rangle,\,\langle\chi_{-}\epsilon_{0}\chi_{-}^{T}\epsilon_{0}\rangle,$
(88)

## Acknowledgements

We thank Gabriele Ferretti and Avik Banerjee for reading of the manuscript and
for useful comments. We thank Ilaria Brivio for relevant discussions about the
kinetic normalization of the gauge bosons. Computational resources have been
provided by the supercomputing facilities of the Université catholique de
Louvain (CISM/UCL) and the Consortium des Équipements de Calcul Intensif en
Fédération Wallonie Bruxelles (CÉCI) funded by the Fond de la Recherche
Scientifique de Belgique (F.R.S.-FNRS) under convention 2.5020.11 and by the
Walloon Region.

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