compat=1.0.0

# Unitarity constraints on large multiplets of arbitrary gauge groups

André Milagre111E-mail<EMAIL_ADDRESS>and Luís Lavoura222E-mail:
<EMAIL_ADDRESS>
Universidade de Lisboa, Instituto Superior Técnico, CFTP,
Av. Rovisco Pais 1, 1049-001 Lisboa, Portugal

###### Abstract

We impose partial-wave unitarity on $2\to 2$ tree-level scattering processes
to derive constraints on the dimensions of large scalar and fermionic
multiplets of arbitrary gauge groups. We apply our results to scalar and
fermionic extensions of the Standard Model, and also to the Grand Unified
Theories (GUTs) based on the groups $SU(5)$, $SO(10)$, and $E_{6}$. We find
scenarios within the latter two GUTs that violate the unitarity condition;
this may require a reevaluation of the validity of perturbation theory in
those scenarios.

##  1 Introduction

The conservation of probability in quantum-mechanical processes implies
unitarity of the $S$ matrix. When this condition is phrased in terms of the
partial waves of a scattering amplitude, one may derive constraints on the
magnitude of the zeroth partial wave, if one wants perturbation theory to be
trusted in the theory in question. Historically, a famous application of
partial-wave unitarity bounds was made by Lee, Quigg, and Thacker [1, 2] to
derive the upper bound $m_{H}\lesssim 1\,\textup{TeV}$ on the mass $m_{H}$ of
the Higgs boson. The same reasoning has been applied by Logan and her
collaborators [3] to constrain the quantum numbers of large scalar multiplets
in extensions of the Standard Model (SM); they have shown that one complex
scalar multiplet with more than eight components (or one real multiplet with
more than nine components) is enough to violate perturbation theory. They also
found constraints on the hypercharges of such multiplets. More recently, by
working with fields transforming in the trivial, fundamental, and adjoint
representations of $SU(N)\times U(1)$, upper bounds on Yukawa [4] and gauge
[5] couplings have been derived by using perturbative unitarity. One-loop
corrections to the Higgs–$WW$ and Higgs–$ZZ$ couplings have also been used to
constrain the dimension of large electroweak representations of vector-like
fermions [6].

The goal of this work is to extend Ref. [3] by using the unitarity bounds on
the zeroth partial wave of $2\to 2$ tree-level scattering amplitudes to
constrain the dimensions of large scalar _and fermionic_ multiplets that
transform under _arbitrary_ gauge groups. We compute both the fermion-pair and
the scalar-pair annihilation into gauge-boson pairs in the high-energy limit.
The calculations are performed in the unmixed basis of each gauge group and
before spontaneous symmetry breaking. Therefore the scalars, the fermions, and
the gauge bosons are taken to be massless, and the latter only have transverse
polarizations.

In Section 2 we provide a brief review of the unitarity constraints on the
zeroth partial wave of $2\to 2$ tree-level scattering amplitudes; we then
explicitly calculate the zeroth partial wave for a single fermion pair or
scalar pair annihilating into a gauge-boson pair of a single gauge group. In
Section 3 we introduce the technique of coupled-channel analysis to constrain
the dimension of a single fermion or scalar multiplet under a single gauge
group; we then generalize the reasoning to multiple multiplets transforming
under multiple gauge groups. We apply our results to derive constraints on
extensions of the SM and on the $SU(5)$, $SO(10)$, and $E_{6}$ Grand Unified
Theories (GUTs) in Section 4, and draw our conclusions in Section 5. We use
Appendix A to carefully derive the scattering amplitudes used in Section 3. In
Appendix B we substantiate an assumption made about the largest eigenvalue of
the coupled-channel matrix, by using a pedagogical example. In Appendix C we
provide useful properties for the relevant irreducible representations of the
various gauge groups used in this paper.

##  2 $2\to 2$ scatterings and unitarity constraints

###  2.1 Interactions of the scalars and fermions with the gauge bosons

We consider a quantum field theory symmetric under a Lie Group $\mathcal{G}$.
In this theory, a fermion multiplet is placed in a representation
$\textbf{{n}}_{\psi}$ of $\mathcal{G}$, while a scalar multiplet is placed in
a representation $\textbf{{n}}_{\varphi}$ of $\mathcal{G}$:

$\displaystyle\textbf{{n}}_{\psi}$ $\displaystyle=\begin{pmatrix}\psi_{1}\\\
\psi_{2}\\\ \vdots\\\ \psi_{n}\end{pmatrix},$
$\displaystyle\textbf{{n}}_{\varphi}=\begin{pmatrix}\varphi_{1}\\\
\varphi_{2}\\\ \vdots\\\ \varphi_{n}\end{pmatrix}.$ (1)

Without loss of generality, we let $n$ denote both the dimension of
$\textbf{{n}}_{\psi}$ and the one of $\textbf{{n}}_{\varphi}$.

The covariant derivative acting on either multiplet is defined as333A sum over
repeated indices is always implied in this paper. [7]

$\displaystyle D_{\mu}\textbf{{n}}_{\psi}$ $\displaystyle=$
$\displaystyle\partial_{\mu}\textbf{{n}}_{\psi}+igV^{a}_{\mu}\,T_{a}\textbf{{n}}_{\psi},$
(2a) $\displaystyle D_{\mu}\textbf{{n}}_{\varphi}$ $\displaystyle=$
$\displaystyle\partial_{\mu}\textbf{{n}}_{\varphi}+igV^{a}_{\mu}\,T_{a}\textbf{{n}}_{\varphi}.$
(2b)

The $T_{a}$ are the $n\times n$ matrices that represent the generators of
$\mathcal{G}$; the $V_{\mu}^{a}$ are $\tilde{n}$ gauge bosons which transform
in the $\tilde{n}$-dimensional adjoint representation of $\mathcal{G}$; $g$ is
the gauge coupling constant.

The interactions of the fermions with the gauge bosons derive from

$\displaystyle\mathcal{L}$ $\displaystyle\supset$ $\displaystyle
i\,\overline{\textbf{{n}}}_{\psi}\not{D}\textbf{{n}}_{\psi}$ (3a)
$\displaystyle=$ $\displaystyle
i\,\overline{\psi}_{i}\not{\partial}\psi_{i}-g\left(T_{a}\right)_{ji}\overline{\psi}_{j}\,\not{V}^{a}\psi_{i}.$
(3b)

The scalars interact with gauge bosons through

$\displaystyle\mathcal{L}$ $\displaystyle\supset$ $\displaystyle
D_{\mu}\textbf{{n}}_{\varphi}^{\dagger}\,D^{\mu}\textbf{{n}}_{\varphi}$ (4a)
$\displaystyle=$
$\displaystyle\left(\partial_{\mu}\varphi_{i}\right)^{\dagger}\left(\partial^{\mu}\varphi_{i}\right)+ig\left(T_{a}\right)_{ji}V_{\mu}^{a}\left(\varphi_{i}\,\partial^{\mu}\varphi_{j}^{\dagger}-\varphi_{j}^{\dagger}\,\partial^{\mu}\varphi_{i}\right)+g^{2}\left(T_{a}T_{b}\right)_{ji}V_{\mu}^{a}\,V^{\mu,b}\,\varphi_{j}^{\dagger}\,\varphi_{i}.$
(4b)

If $\textbf{{n}}_{\varphi}$ is in a real representation of $\mathcal{G}$, then
one must drop the $\dagger$’s from Eq. (4) and add a factor $1/2$ to its
right-hand side.

###  2.2 Unitarity bounds

We want to consider scattering processes of the type
$\psi_{i}\overline{\psi}_{j}\to V^{a}V^{b}$ or
$\varphi_{i}\varphi^{\dagger}_{j}\to V^{a}V^{b}$. The amplitude of each
process may be decomposed into partial-wave amplitudes according to the
formula [8]

$\mathcal{M}\left(\cos\theta\right)=16\pi\sum_{J=0}^{\infty}a_{J}\left(2J+1\right)P_{J}\left(\cos{\theta}\right),$
(5)

where $\theta$ is the scattering angle and $J$ (which is a non-negative
integer) is the total orbital angular momentum of the final state. The $a_{J}$
are numerical coefficients. The functions $P_{J}\left(\cos{\theta}\right)$ are
the Legendre polynomials; in particular,

$P_{0}\left(\cos{\theta}\right)=1.$ (6)

The Legendre polynomials satisfy the orthogonality relation

$\int_{-1}^{+1}\\!P_{J}\left(\cos{\theta}\right)P_{J^{\prime}}\left(\cos{\theta}\right)\,\mathrm{d}\cos{\theta}=\frac{2}{2J+1}\,\delta_{JJ^{\prime}}.$
(7)

Therefore, the zeroth partial wave is given by

$a_{0}\equiv\frac{1}{32\pi}\,\int_{-1}^{+1}\mathcal{M}\left(\cos{\theta}\right)\mathrm{d}\cos\theta.$
(8)

Tree-level zeroth partial-wave unitarity dictates that [8, 9]:

$\displaystyle\left|\mathrm{Re}\,a_{0}\right|$ $\displaystyle\leq$
$\displaystyle\frac{1}{2},$ (9a) $\displaystyle\left|a_{0}\right|$
$\displaystyle\leq$ $\displaystyle 1.$ (9b)

In this work, we want to constrain the dimension of large fermionic or scalar
multiplets by using condition (9a). In order to find the most stringent
constraint we employ the technique of coupled-channel analysis. The zeroth
partial waves will be calculated for each process
$\psi_{i}\overline{\psi}_{j}\to V^{a}V^{b}$ and/or
$\varphi_{i}\varphi^{\dagger}_{j}\to V^{a}V^{b}$, and will then be assembled
in a coupled-channel matrix for various pair of indices $\left(i,j\right)$ and
$\left(a,b\right)$. The unitarity bound arises from applying condition (9a) to
the largest (in modulus) eigenvalue of the coupled-channel matrix [9, 10, 11].
It should be pointed out that the partial wave for a process involving two
identical initial-state or final-state particles receives an extra factor
$1\left/\sqrt{2}\right.$; if the two initial-state particles are identical and
the two final-state particles are also identical, then the extra factor is
$1/2$.

###  2.3 Fermionic Scattering

We want to calculate the zeroth partial wave of the process

$\psi_{i}\,\overline{\psi}_{j}\to V^{a}V^{b}.$ (10)

Three channels contribute to the corresponding amplitude:

$i\mathcal{M}(ij\to ab)\ =\ \leavevmode\hbox to0pt{\vbox
to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } \feynman \vertex(a1) at (-1.7,1.)
{$\psi_{i}$}; \vertex(a2) at (0,0.8) ; \vertex(a3) at (1.7,1.)
{$V^{a}_{\mu}$}; \vertex(b1) at (-1.7,-1.) {$\overline{\psi}_{j}$};
\vertex(b2) at (0,-0.8) ; \vertex(b3) at (1.7,-1.) {$V^{b}_{\nu}$};
\par\par\diagram*{ (a1) -- [fermion, momentum=$p_{1}$](a2), (a2) -- [photon,
momentum=$p_{3}$](a3), (a2) -- [fermion, edge label=$\psi_{k}$](b2), (b1) --
[anti fermion, momentum'=$p_{2}$](b2), (b2) -- [photon,
momentum'=$p_{4}$](b3),}; \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\ \ +\ \
\leavevmode\hbox to0pt{\vbox to0pt{\pgfpicture\makeatletter\hbox{\hskip
0.0pt\lower 0.0pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } \feynman \vertex(a1) at (-1.7,1.)
{$\psi_{i}$}; \vertex(a2) at (0,0.8) ; \vertex(a3) at (1.7,1.)
{$V^{b}_{\nu}$}; \vertex(b1) at (-1.7,-1.) {$\overline{\psi}_{j}$};
\vertex(b2) at (0,-0.8) ; \vertex(b3) at (1.7,-1.) {$V^{a}_{\mu}$};
\par\par\diagram*{ (a1) -- [fermion, momentum=$p_{1}$](a2), (a2) -- [photon,
momentum=$p_{4}$](a3), (a2) -- [fermion, edge label=$\psi_{k}$](b2), (b1) --
[anti fermion, momentum'=$p_{2}$](b2), (b2) -- [photon,
momentum'=$p_{3}$](b3),}; \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\ \ +\ \
\leavevmode\hbox to0pt{\vbox to0pt{\pgfpicture\makeatletter\hbox{\hskip
0.0pt\lower 0.0pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } \feynman \vertex(a1) at (-1.7,1.1)
{$\psi_{i}$}; \vertex(a2) at (-0.9,0) ; \vertex(a3) at (1.7,1.1)
{$V_{\mu}^{a}$}; \vertex(b1) at (-1.7,-1.1) {$\overline{\psi}_{j}$};
\vertex(b2) at (0.9,0) ; \vertex(b3) at (1.7,-1.1) {$V_{\nu}^{b}$};
\par\par\diagram*{ (a1) -- [fermion, momentum=$p_{1}$](a2), (b1) -- [anti
fermion, momentum'=$p_{2}$](a2), (a2) -- [photon, edge label=$V^{c}$](b2),
(b2) -- [photon, momentum=$p_{3}$](a3), (b2) -- [photon,
momentum'=$p_{4}$](b3),}; \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys<EMAIL_ADDRESS>(11)

We work in the high-energy limit and we therefore neglect the masses of the
fermions and gauge bosons. In that limit, fermions become chiral, meaning
$\overline{\psi}_{j}$ has opposite chirality to $\psi_{i}$. Consequently, if
the two final-state gauge bosons have opposite polarizations (either
left–right or right–left), then the sum of the three diagrams in Eq. (11)
vanishes. On the other hand, if the two final-state gauge bosons have the same
polarization (either left–left or right–right), then the third diagram (the
$s$ channel) in Eq. (11) vanishes and only the first two are nonzero. The
amplitudes for the $t$\- and $u$-channels read

$\displaystyle\mathcal{M}_{t}\left(ij\to ab\right)$
$\displaystyle=-g^{2}\left(T_{b}T_{a}\right)_{ji}\,\frac{\overline{v}_{j}\,\not{\varepsilon}^{*}\left(p_{4}\right)\left(\not{p}_{1}-\not{p}_{3}\right)\not{\varepsilon}^{*}\left(p_{3}\right)u_{i}}{t},$
(12a) $\displaystyle\mathcal{M}_{u}(ij\to ab)$
$\displaystyle=-g^{2}\left(T_{a}T_{b}\right)_{ji}\,\frac{\overline{v}_{j}\,\not{\varepsilon}^{*}\left(p_{3}\right)\left(\not{p}_{1}-\not{p}_{4}\right)\not{\varepsilon}^{*}\left(p_{4}\right)u_{i}}{u},$
(12b)

respectively, where

$u_{i}$ and $\overline{v}_{j}$ are the the spinors of the initial-state
$\psi_{i}$ and $\overline{\psi}_{j}$, respectively,

$\varepsilon\left(p_{3}\right)$ and $\varepsilon\left(p_{4}\right)$ are the
polarization four-vectors of the final-state $V^{a}$ and $V^{b}$,
respectively,

$t\equiv\left(p_{1}-p_{3}\right)^{2}$ and
$u\equiv\left(p_{1}-p_{4}\right)^{2}$ are the Mandelstam variables.

Since the gauge bosons have the same polarization, either $V^{a}$ and $V^{b}$
are both right-handed (RR configuration), or they are both left-handed (LL
configuration). Using the results of Appendix A.1, the total amplitude for
each scenario is found to be

$\displaystyle\mathcal{M}_{\textup{RR}}(ij\to ab)$ $\displaystyle=$
$\displaystyle-g^{2}\sin\theta\left[\left(T_{a}T_{b}\right)_{ji}\tan^{2}\frac{\theta}{2}+\left(T_{b}T_{a}\right)_{ji}\right],$
(13a) $\displaystyle\mathcal{M}_{\textup{LL}}(ij\to ab)$ $\displaystyle=$
$\displaystyle
g^{2}\sin\theta\left[\left(T_{a}T_{b}\right)_{ji}+\left(T_{b}T_{a}\right)_{ji}\cot^{2}\frac{\theta}{2}\right].$
(13b)

We use Eq. (8) for $a_{0}$ and444Notice the factors $\pi$ in the integrals
(14). Those factors do not appear in the corresponding integrals for
$\varphi_{i}\varphi_{j}^{\dagger}\to V^{a}V^{b}$. They are responsible for a
factor $\pi/2$ in the comparison of the $a_{0}$’s for fermionic-pair and
scalar-pair scattering to gauge-boson pair.

$\displaystyle\int_{-1}^{+1}\sin{\theta}\,\tan^{2}{\frac{\theta}{2}}\
\mathrm{d}\cos{\theta}=\int_{-1}^{+1}\sin{\theta}\,\cot^{2}{\frac{\theta}{2}}\
\mathrm{d}\cos{\theta}$ $\displaystyle=$ $\displaystyle\frac{3\pi}{2},$ (14a)
$\displaystyle\int_{-1}^{+1}\sin{\theta}\ \mathrm{d}\cos{\theta}$
$\displaystyle=$ $\displaystyle\frac{\pi}{2},$ (14b)

to obtain

$\displaystyle a_{0}^{\textup{RR}}\left(ij\to ab\right)$ $\displaystyle=$
$\displaystyle-\frac{g^{2}}{64}\left(3T_{a}T_{b}+T_{b}T_{a}\right)_{ji},$
(15a) $\displaystyle a_{0}^{\textup{LL}}\left(ij\to ab\right)$
$\displaystyle=$
$\displaystyle\frac{g^{2}}{64}\left(T_{a}T_{b}+3T_{b}T_{a}\right)_{ji}.$ (15b)

Note that if $V^{a}=V^{b}$, then the results (15) must be divided by
$\sqrt{2}$.

###  2.4 Scalar scattering

In this section, we calculate the zeroth partial-wave of the process

$\varphi_{i}\,\varphi^{\dagger}_{j}\to V^{a}V^{b}.$ (16)

Four diagrams contribute to the corresponding amplitude:

$\begin{split}i\mathcal{M}(ij\to ab)\ =&\ \leavevmode\hbox to0pt{\vbox
to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } \feynman \vertex(a1) at (-1.7,1.)
{$\varphi_{i}$}; \vertex(a2) at (0,0.8) ; \vertex(a3) at (1.7,1.)
{$V^{a}_{\mu}$}; \vertex(b1) at (-1.7,-1.) {$\varphi_{j}^{\dagger}$};
\vertex(b2) at (0,-0.8) ; \vertex(b3) at (1.7,-1.) {$V^{b}_{\nu}$}; \diagram*{
(a1) -- [scalar, momentum=$p_{1}$](a2), (a2) -- [photon,
momentum=$p_{3}$](a3), (a2) -- [scalar, edge label=$\varphi_{k}$](b2), (b1) --
[scalar, momentum'=$p_{2}$](b2), (b2) -- [photon, momentum'=$p_{4}$](b3),};
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\ \ +\ \
\leavevmode\hbox to0pt{\vbox to0pt{\pgfpicture\makeatletter\hbox{\hskip
0.0pt\lower 0.0pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } \feynman \vertex(a1) at (-1.7,1.)
{$\varphi_{i}$}; \vertex(a2) at (0,0.8) ; \vertex(a3) at (1.7,1.)
{$V^{b}_{\nu}$}; \vertex(b1) at (-1.7,-1.) {$\varphi_{j}^{\dagger}$};
\vertex(b2) at (0,-0.8) ; \vertex(b3) at (1.7,-1.) {$V^{a}_{\mu}$}; \diagram*{
(a1) -- [scalar, momentum=$p_{1}$](a2), (a2) -- [photon,
momentum=$p_{4}$](a3), (a2) -- [scalar, edge label=$\varphi_{k}$](b2), (b1) --
[scalar, momentum'=$p_{2}$](b2), (b2) -- [photon, momentum'=$p_{3}$](b3),};
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\ \ +\ \ \\\ &\ \ +\ \
\leavevmode\hbox to0pt{\vbox to0pt{\pgfpicture\makeatletter\hbox{\hskip
0.0pt\lower 0.0pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } \feynman \vertex(a1) at (-1.7,1.1)
{$\varphi_{i}$}; \vertex(a2) at (-0.9,0) ; \vertex(a3) at (1.7,1.1)
{$V_{\mu}^{a}$}; \vertex(b1) at (-1.7,-1.1) {$\varphi^{\dagger}_{j}$};
\vertex(b2) at (0.9,0) ; \vertex(b3) at (1.7,-1.1) {$V_{\nu}^{b}$}; \diagram*{
(a1) -- [scalar, momentum=$p_{1}$](a2), (b1) -- [scalar,
momentum'=$p_{2}$](a2), (a2) -- [photon, edge label=$V^{c}$](b2), (b2) --
[photon, momentum=$p_{3}$](a3), (b2) -- [photon, momentum'=$p_{4}$](b3),};
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\ \ +\ \
\leavevmode\hbox to0pt{\vbox to0pt{\pgfpicture\makeatletter\hbox{\hskip
0.0pt\lower 0.0pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } \feynman \vertex(a1) at (-1.2,1.2)
{$\varphi_{i}$}; \vertex(a3) at (1.2,1.2) {$V_{\mu}^{a}$}; \vertex(c) at
(0,0); \vertex(b1) at (-1.2,-1.2) {$\varphi^{\dagger}_{j}$}; \vertex(b3) at
(1.2,-1.2) {$V_{\nu}^{b}$}; \diagram*{ (a1) -- [scalar, momentum=$p_{1}$](c),
(b1) -- [scalar, momentum=$p_{2}$](c), (c) -- [photon, momentum'=$p_{3}$](a3),
(c) -- [photon, momentum'=$p_{4}$](b3),}; \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys<EMAIL_ADDRESS>(17)

We work in the high-energy limit and we therefore neglect the masses of the
scalars and gauge bosons. In that limit, if the two final-state gauge bosons
have opposite polarizations, then the sum of the four diagrams in Eq. (17)
vanishes. On the other hand, if the two final-state gauge bosons have the same
polarization, then both the third and fourth diagrams in Eq. (11) vanish and
only the first two diagrams are nonzero. The amplitudes for those two diagrams
read

$\displaystyle\mathcal{M}_{t}\left(ij\to ab\right)$
$\displaystyle=g^{2}\left(T_{b}T_{a}\right)_{ji}\frac{\left[\left(2p_{2}-p_{4}\right)\cdot\varepsilon^{*}\left(p_{4}\right)\right]\left[\left(2p_{1}-p_{3}\right)\cdot\varepsilon^{*}\left(p_{3}\right)\right]}{t},$
(18a) $\displaystyle\mathcal{M}_{u}\left(ij\to ab\right)$
$\displaystyle=g^{2}\left(T_{a}T_{b}\right)_{ji}\frac{\left[\left(2p_{2}-p_{3}\right)\cdot\varepsilon^{*}\left(p_{3}\right)\right]\left[\left(2p_{1}-p_{4}\right)\cdot\varepsilon^{*}\left(p_{4}\right)\right]}{u},$
(18b)

respectively. Once again, either $V^{a}$ and $V^{b}$ are both right-handed (RR
configuration), or they are both (LL configuration). For each scenario, using
the results of Appendix A.2 we arrive at the following total amplitudes:

$\mathcal{M}_{\textup{RR}}(ij\to ab)\ =\ \mathcal{M}_{\textup{LL}}(ij\to ab)\
=\
-2g^{2}\left[\left(T_{a}T_{b}\right)_{ji}\sin^{2}\frac{\theta}{2}+\left(T_{b}T_{a}\right)_{ji}\cos^{2}\frac{\theta}{2}\right].$
(19)

Therefore, using Eq. (8) and

$\int_{-1}^{+1}\sin^{2}{\frac{\theta}{2}}\
\mathrm{d}\cos{\theta}=\int_{-1}^{+1}\cos^{2}{\frac{\theta}{2}}\
\mathrm{d}\cos{\theta}\ =\ 1,$ (20)

the corresponding zeroth partial waves read

$a_{0}^{\textup{RR}}\left(ij\to ab\right)=a_{0}^{\textup{LL}}\left(ij\to
ab\right)=-\frac{g^{2}}{16\pi}\left(T_{a}T_{b}+T_{b}T_{a}\right)_{ji}.$ (21)

These partial waves must be divided by $\sqrt{2}$ when $V^{a}=V^{b}$.
Furthermore, if $\textbf{{n}}_{\varphi}$ is a real representation and
$\varphi_{i}=\varphi_{j}$, then one must once again divide by $\sqrt{2}$.
Notice the presence of $\pi$ in the denominator of Eq. (21), while $\pi$ is
absent from Eqs. (15).

##  3 Coupled-channel analysis

###  3.1 Matrix of zeroth partial-waves

As stated before, the strongest constraints are achieved by enforcing the
unitarity condition (9a) on _the largest_ (_in modulus_) _eigenvalue_ of a
coupled-channel matrix of zeroth partial waves. In general, that largest
eigenvalue555When we say “largest eigenvalue” we really mean the eigenvalue
with the largest modulus. corresponds to an eigenvector that couples
symmetrically, i.e. in group-invariant fashion, both pairs of initial states
and pairs of final states. The concept is illustrated by the example in
Appendix B. We define the symmetric linear combinations of states as

$\displaystyle\left[\textbf{{n}}_{\psi}\textbf{{n}}_{\psi}\right]^{\textup{sym}}$
$\displaystyle\equiv\frac{1}{\sqrt{n}}\,\sum_{i=1}^{n}\psi_{i}\,\bar{\psi}_{i},$
$\displaystyle\left[VV\right]^{\textup{sym}}_{\textup{RR/LL}}$
$\displaystyle\equiv\frac{1}{\sqrt{2\tilde{n}}}\,\sum_{a=1}^{\tilde{n}}\left(V^{a}V^{a}\right)_{\textup{RR/LL}}.$
(22)

The factor $1\left/\sqrt{2}\right.$ in the second Eq. (22) accounts for the
identical gauge bosons in the final state. We _assume_ —based on our
experience in particular cases, like the one in Appendix B—that the process
with the largest zeroth partial wave is666If our assumption turns out not to
be true in some cases, then the unitarity bounds that we derive will not be
the strongest ones, yet they will remain _necessary_ conditions for unitarity.
$\left[\textbf{{n}}_{\psi}\textbf{{n}}_{\psi}\right]^{\textup{sym}}\to\left[VV\right]^{\textup{sym}}_{\textup{RR/LL}}$.
In order to find the corresponding eigenvalues, we move to the basis

$\left\\{\left[VV\right]^{\textup{sym}}_{\textup{RR}},\left[VV\right]^{\textup{sym}}_{\textup{LL}},\left[\textbf{{n}}_{\psi}\textbf{{n}}_{\psi}\right]^{\textup{sym}}\right\\},$
(23)

in which the coupled-channel matrix takes the form

$a_{0}^{\psi}=\begin{pmatrix}\textbf{0}_{2\times 2}&M\\\
{M}^{T}&\textbf{0}\end{pmatrix},\qquad
M=\begin{pmatrix}a_{0}\left(\left[\textbf{{n}}_{\psi}\textbf{{n}}_{\psi}\right]^{\textup{sym}}\to\left[VV\right]^{\textup{sym}}_{\textup{RR}}\right)\\\\[-8.61108pt]
\\\
a_{0}\left(\left[\textbf{{n}}_{\psi}\textbf{{n}}_{\psi}\right]^{\textup{sym}}\to\left[VV\right]^{\textup{sym}}_{\textup{LL}}\right)\end{pmatrix},$
(24)

where

$a_{0}\left(\left[\textbf{{n}}_{\psi}\textbf{{n}}_{\psi}\right]^{\textup{sym}}\to\left[VV\right]^{\textup{sym}}_{\textup{RR/LL}}\right)=\frac{1}{\sqrt{2n\tilde{n}}}\,\sum_{i=1}^{n}\sum_{a=1}^{\tilde{n}}a_{0}^{\textup{RR/LL}}\left(ii\to
aa\right).$ (25)

We derive explicit expressions for these matrix elements by resorting to the
group-theoretical relation [12]

$\sum_{a=1}^{\tilde{n}}T_{a}T_{a}=C\left(\textit{{n}}\right)\times\textbf{1}_{n\times
n},$ (26)

where $C\left(\textit{{n}}\right)$ is the quadratic Casimir invariant of the
$n$-dimensional representation n and $\textbf{1}_{n\times n}$ is the $n\times
n$ identity matrix. From Eqs. (15a) and (25) we derive

$\displaystyle
a_{0}\left(\left[\textbf{{n}}_{\psi}\textbf{{n}}_{\psi}\right]^{\textup{sym}}\to\left[VV\right]^{\textup{sym}}_{\textup{RR}}\right)$
$\displaystyle=$
$\displaystyle-\frac{1}{\sqrt{2n\tilde{n}}}\,\frac{g^{2}}{16}\,\sum_{i=1}^{n}\sum_{a=1}^{\tilde{n}}\left(T_{a}T_{a}\right)_{ii},$
(27a) $\displaystyle=$
$\displaystyle-\frac{1}{\sqrt{2n\tilde{n}}}\,\frac{g^{2}}{16}\,\sum_{a=1}^{\tilde{n}}\textup{Tr}\left(T_{a}T_{a}\right)$
(27b) $\displaystyle=$
$\displaystyle-\frac{1}{\sqrt{2n\tilde{n}}}\,\frac{g^{2}}{16}\,C\left(\textbf{{n}}_{\psi}\right)\textup{Tr}\left(\textbf{1}_{n\times
n}\right)$ (27c) $\displaystyle=$
$\displaystyle-\frac{g^{2}}{16}\,\sqrt{\frac{n}{2\tilde{n}}}\,C\left(\textbf{{n}}_{\psi}\right).$
(27d)

In a similar fashion, the other matrix elements are

$a_{0}\left(\left[\textbf{{n}}_{\psi}\textbf{{n}}_{\psi}\right]^{\textup{sym}}\to\left[VV\right]^{\textup{sym}}_{\textup{LL}}\right)=\frac{g^{2}}{16}\,\sqrt{\frac{n}{2\tilde{n}}}\,C\left(\textbf{{n}}_{\psi}\right)$
(28)

and

$a_{0}\left(\left[\textbf{{n}}_{\varphi}\textbf{{n}}_{\varphi}\right]^{\textup{sym}}\to\left[VV\right]^{\textup{sym}}_{\textup{RR}}\right)=a_{0}\left(\left[\textbf{{n}}_{\varphi}\textbf{{n}}_{\varphi}\right]^{\textup{sym}}\to\left[VV\right]^{\textup{sym}}_{\textup{LL}}\right)=-\frac{g^{2}}{8\pi}\,\sqrt{\frac{n}{2\tilde{n}}}\,C\left(\textbf{{n}}_{\varphi}\right).$
(29)

We plug Eqs. (27) and (28) into Eq. (24) and we find the matrix of zeroth
partial waves to be

$a_{0}^{\psi}=\frac{g^{2}}{16}\,\sqrt{\frac{n}{2\tilde{n}}}\,C\left(\textbf{{n}}_{\psi}\right)\begin{pmatrix}0&0&-1\\\
0&0&1\\\ -1&1&0\end{pmatrix}.$ (30)

Its largest eigenvalue of course is

$\left(a_{0}^{\psi}\right)^{\mathrm{max}}=\frac{g^{2}}{16}\,\sqrt{\frac{n}{\tilde{n}}}\,C\left(\textbf{{n}}_{\psi}\right).$
(31)

Note that considering simultaneously the final states
$\left[VV\right]^{\textup{sym}}_{\textup{RR}}$ and
$\left[VV\right]^{\textup{sym}}_{\textup{LL}}$ is effectively equivalent to
multiplying by a factor $\sqrt{2}$ the zeroth partial wave of either.
Henceforward, we use this trick to account for both polarization
configurations and we drop the subscripts RR and LL.

For a scalar multiplet, we have instead

$\left(a_{0}^{\varphi}\right)^{\mathrm{max}}=\frac{g^{2}}{8\pi}\,\sqrt{\frac{n}{\tilde{n}}}\,C\left(\textbf{{n}}_{\varphi}\right).$
(32)

The difference between the scalar and fermionic case is just a factor $\pi/2$:

$\left(a_{0}^{\psi}\right)^{\mathrm{max}}=\frac{\pi}{2}\left(a_{0}^{\varphi}\right)^{\mathrm{max}}.$
(33)

The factor $\pi$ arises from the integrations in Eqs. (14).

###  3.2 Multiple Particles and Multiple Symmetry Groups

The above analysis has assumed the existence of only one matter (either
fermionic or scalar) multiplet transforming non-trivially under only one
symmetry group $\mathcal{G}$. We extend this situation to a scenario where
there are $N_{F}$ fermionic multiplets $F_{1},F_{2},\ldots$ and $N_{S}$ scalar
multiplets $S_{1},S_{2},\ldots$, that transform under a single symmetry group
$\mathcal{G}$. In this scenario, the matrix of zeroth partial waves is defined
in the basis777Here we account for both gauge-boson polarizations by adding
the extra $\sqrt{2}$ to the zeroth partial wave.:

$\left\\{\left[VV\right]^{\textup{sym}},\left[F_{1}F_{1}\right]^{\textup{sym}},\left[F_{2}F_{2}\right]^{\textup{sym}},\ldots,\left[S_{1}S_{1}\right]^{\textup{sym}},\left[S_{2}S_{2}\right]^{\textup{sym}},\ldots\right\\},$
(34)

The largest eigenvalue of the corresponding matrix of zeroth partial-waves is
then simply given by

$\displaystyle\left(a_{0}\right)^{\mathrm{max}}$ $\displaystyle=$
$\displaystyle\sqrt{\sum_{i=1}^{N_{F}}\left[\left(a_{0}^{\psi}\right)^{\mathrm{max}}\right]^{2}+\sum_{i=1}^{N_{S}}\left[\left(a_{0}^{\varphi}\right)^{\mathrm{max}}\right]^{2}}$
(35a) $\displaystyle=$
$\displaystyle\frac{g^{2}}{8\pi\sqrt{\tilde{n}}}\,\sqrt{\frac{\pi^{2}}{4}\,\sum_{i=1}^{N_{F}}n_{F_{i}}\left[C\left(\textbf{{n}}_{F_{i}}\right)\right]^{2}+\sum_{i=1}^{N_{S}}n_{S_{i}}\left[C\left(\textbf{{n}}_{S_{i}}\right)\right]^{2}},$
(35b)

where we have used the results of Eqs. (31) and (32) for
$\left(a_{0}^{\psi}\right)^{\mathrm{max}}$ and
$\left(a_{0}^{\varphi}\right)^{\mathrm{max}}$, respectively. In Eq. (35b),
$n_{F_{i}}$ is the number of fermions in the irreducible representation
$\textbf{{n}}_{F_{i}}$ of the gauge group, and $n_{S_{i}}$ is the number of
fermions in the irreducible representation $\textbf{{n}}_{S_{i}}$ of the gauge
group.

This formalism may be further extended to a scenario where there are multiple
matter (either fermionic or scalar) multiplets $A,B,C,D,\ldots$ transforming
under the direct product of multiple symmetry groups
$\mathcal{G}\equiv\mathcal{G}_{1}\times\mathcal{G}_{2}\times\mathcal{G}_{3}\times\mathcal{G}_{4}\times\cdots$.
The representation of matter field $A$ under $\mathcal{G}$ is written

$A\sim\left(\textbf{{n}}_{A}^{\mathcal{G}_{1}},\textbf{{n}}_{A}^{\mathcal{G}_{2}},\textbf{{n}}_{A}^{\mathcal{G}_{3}},\textbf{{n}}_{A}^{\mathcal{G}_{4}},\cdots\right).$
(36)

We work in the basis

$\left\\{\left[VV\right]_{\mathcal{G}_{1}}^{\textup{sym}},\,\left[VV\right]_{\mathcal{G}_{2}}^{\textup{sym}},\,\left[VV\right]_{\mathcal{G}_{3}}^{\textup{sym}},\,\ldots,\,\left[AA\right]^{\textup{sym}},\,\left[BB\right]^{\textup{sym}},\,\left[CC\right]^{\textup{sym}},\,\ldots\right\\},$
(37)

where $\left[VV\right]_{\mathcal{G}_{i}}^{\textup{sym}}$ is the symmetric
linear combination of gauge bosons of the group $\mathcal{G}_{i}$, as defined
in Eq. (22).888We insert an extra factor $\sqrt{2}$ to account for both gauge-
boson polarizations. If the matter multiplet $A$ transforms non-trivially
under the gauge group $\mathcal{G}_{i}$, then there are

$\prod_{k\neq i}n_{A}^{\mathcal{G}_{k}}\equiv n_{A}^{\mathcal{G}_{1}}\times
n_{A}^{\mathcal{G}_{2}}\times\cdots\times n_{A}^{\mathcal{G}_{i-1}}\times
n_{A}^{\mathcal{G}_{i+1}}\times\cdots$ (38)

copies of that multiplet of the group $\mathcal{G}_{i}$ which interact with
$\left[VV\right]_{\mathcal{G}_{i}}^{\textup{sym}}$. We keep the original
definition of $\left[AA\right]^{\textup{sym}}$ provided in Eq. (22), but
multiply the corresponding zeroth partial wave by the square root of the
factor in Eq. (38). In the basis (37), the full matrix of zeroth partial waves
is written as

$a_{0}=\begin{pmatrix}\textbf{0}&M^{\prime}\\\
{M^{\prime}}^{T}&\textbf{0}\end{pmatrix},\qquad\mbox{with}\quad
M^{\prime}=\begin{pmatrix}\left(a_{0}^{A}\right)^{\mathrm{max}}_{\mathcal{G}_{1}}&\left(a_{0}^{B}\right)^{\mathrm{max}}_{\mathcal{G}_{1}}&\left(a_{0}^{C}\right)^{\mathrm{max}}_{\mathcal{G}_{1}}&\cdots\\\
\left(a_{0}^{A}\right)^{\mathrm{max}}_{\mathcal{G}_{2}}&\left(a_{0}^{B}\right)^{\mathrm{max}}_{\mathcal{G}_{2}}&\left(a_{0}^{C}\right)^{\mathrm{max}}_{\mathcal{G}_{2}}&\cdots\\\
\left(a_{0}^{A}\right)^{\mathrm{max}}_{\mathcal{G}_{3}}&\left(a_{0}^{B}\right)^{\mathrm{max}}_{\mathcal{G}_{3}}&\left(a_{0}^{C}\right)^{\mathrm{max}}_{\mathcal{G}_{3}}&\cdots\\\
\vdots&\vdots&\vdots&\ddots\\\ \end{pmatrix}.$ (39)

Each matrix element of $M^{\prime}$ is defined as

$\left(a_{0}^{A}\right)^{\mathrm{max}}_{\mathcal{G}_{i}}\equiv\frac{g_{\mathcal{G}_{i}}^{2}}{8\pi\sqrt{\tilde{n}_{\mathcal{G}_{i}}}}\,\sqrt{\prod_{k}n_{A}^{\mathcal{G}_{k}}}\
\times\left\\{\begin{array}[]{ll}\displaystyle{\frac{\pi}{2}\,C\left(\textbf{{n}}_{A}^{\mathcal{G}_{i}}\right)}&\quad\Leftarrow
A=\psi,\\\ C\left(\textbf{{n}}_{A}^{\mathcal{G}_{i}}\right)&\quad\Leftarrow
A=\varphi,\end{array}\right.$ (40)

where $g_{\mathcal{G}_{i}}$ is the gauge coupling constant of
$\mathcal{G}_{i}$, $\tilde{n}_{\mathcal{G}_{i}}$ the dimension of its adjoint
representation, and $C(\textbf{{n}}_{A}^{\mathcal{G}_{i}})$ the quadratic
Casimir invariant of the representation of the matter field $A$ under
$\mathcal{G}_{i}$. Notice that the product in Eq. (40) runs over _all_ $k$,
not just over $k\neq i$ as in Eq. (38); the extra factor
$n_{A}^{\mathcal{G}_{i}}$ comes from Eqs. (31) or (32).

##  4 Results

In the following sections, we will calculate the largest eigenvalue of the
matrix of zeroth partial waves for the gauge groups $SU(3)\times SU(2)\times
U(1)$ (i.e. the SM), $SU(5)$, $SO(10)$, and $E_{6}$. We shall impose the
unitarity bound (9a) to constrain the dimension of additional fermionic or
scalar multiplets in various scenarios. We follow the group-theory conventions
of Mathematica’s package GroupMath [13]. The Dynkin indices and the quadratic
Casimir invariants for the relevant irreducible representations of the various
gauge groups are listed in Appendix C. A more comprehensive compilation of
group-theory data may be found in Ref. [14].

###  4.1 $SU(3)\times SU(2)\times U(1)$

The SM is invariant under the gauge group $\mathcal{G}_{\textup{SM}}\equiv
SU(3)_{\mathrm{colour}}\times SU(2)_{L}\times U(1)_{Y}$. Using the convention
in Eq. (36), the matter fields of the SM transform as

$\displaystyle L_{L,f}$ $\displaystyle\sim$
$\displaystyle\left(\textbf{1},\,\textbf{2},\,-\frac{1}{2}\right),$ (41a)
$\displaystyle Q_{L,f}$ $\displaystyle\sim$
$\displaystyle\left(\textbf{3},\,\textbf{2},\,\frac{1}{6}\right),$ (41b)
$\displaystyle\ell_{R,f}$ $\displaystyle\sim$
$\displaystyle\left(\textbf{1},\,\textbf{1},\,-1\right),$ (41c) $\displaystyle
u_{R,f}$ $\displaystyle\sim$
$\displaystyle\left(\textbf{3},\,\textbf{1},\,\frac{2}{3}\right),$ (41d)
$\displaystyle d_{R,f}$ $\displaystyle\sim$
$\displaystyle\left(\textbf{3},\,\textbf{1},\,-\frac{1}{3}\right),$ (41e)
$\displaystyle H$ $\displaystyle\sim$
$\displaystyle\left(\textbf{1},\,\textbf{2},\,\frac{1}{2}\right).$ (41f)

There are three copies of each of the multiplets (41a)–(41e), which we label
through subscripts $f=1,2,3$. An appropriate basis for the scattering states
is

$\displaystyle\left\\{\left[VV\right]_{SU(3)}^{\textup{sym}},\
\left[VV\right]_{SU(2)}^{\textup{sym}},\
\left[VV\right]_{U(1)}^{\textup{sym}},\right.$
$\displaystyle\left[L_{L,1}L_{L,1}\right]^{\mathrm{sym}},\
\left[Q_{L,1}Q_{L,1}\right]^{\mathrm{sym}},\
\left[\ell_{R,1}\ell_{R,1}\right]^{\mathrm{sym}},\
\left[u_{R,1}u_{R,1}\right]^{\mathrm{sym}},\
\left[d_{R,1}d_{R,1}\right]^{\mathrm{sym}},$
$\displaystyle\left[L_{L,2}L_{L,2}\right]^{\mathrm{sym}},\
\left[Q_{L,2}Q_{L,2}\right]^{\mathrm{sym}},\
\left[\ell_{R,2}\ell_{R,2}\right]^{\mathrm{sym}},\
\left[u_{R,2}u_{R,2}\right]^{\mathrm{sym}},\
\left[d_{R,2}d_{R,2}\right]^{\mathrm{sym}},$
$\displaystyle\left.\left[L_{L,3}L_{L,3}\right]^{\mathrm{sym}},\
\left[Q_{L,3}Q_{L,3}\right]^{\mathrm{sym}},\
\left[\ell_{R,3}\ell_{R,3}\right]^{\mathrm{sym}},\
\left[u_{R,3}u_{R,3}\right]^{\mathrm{sym}},\
\left[d_{R,3}d_{R,3}\right]^{\mathrm{sym}},\
\left[HH\right]^{\mathrm{sym}}\vphantom{\left[VV\right]_{SU(3)}^{\textup{sym}},}\right\\}.$
(42)

The corresponding matrix $M^{\prime}$, in the notation of Eq. (39), is
$3\times 16$ and may be written in terms of a $3\times 5$ sub-matrix
$M^{\prime}_{F}$ and a $3\times 1$ sub-matrix $M^{\prime}_{H}$ as

$M^{\prime}=\left(M^{\prime}_{F},\ \ M^{\prime}_{F},\ \ M^{\prime}_{F},\ \
M^{\prime}_{H}\right),$ (43)

with

$\displaystyle M_{F}^{\prime}$ $\displaystyle=$
$\displaystyle\frac{1}{576}\begin{pmatrix}0&24\sqrt{3}\,g_{SU(3)}^{2}&0&12\sqrt{6}\,g_{SU(3)}^{2}&12\sqrt{6}\,g_{SU(3)}^{2}\\\\[4.0pt]
9\sqrt{6}\,g_{SU(2)}^{2}&27\sqrt{2}\,g_{SU(2)}^{2}&0&0&0\\\\[4.0pt]
9\sqrt{2}\,g_{U(1)}^{2}&\sqrt{6}\,g_{U(1)}^{2}&36\,g_{U(1)}^{2}&16\sqrt{3}\,g_{U(1)}^{2}&4\sqrt{3}\,g_{U(1)}^{2},\end{pmatrix},$
(44a) $\displaystyle M^{\prime}_{H}$ $\displaystyle=$
$\displaystyle\frac{1}{32\pi}\begin{pmatrix}0\\\ \sqrt{6}\,g_{SU(2)}^{2}\\\
\sqrt{2}\,g_{U(1)}^{2}\end{pmatrix}.$ (44b)

Each matrix element of $M^{\prime}_{F}$ and $M^{\prime}_{H}$ has been computed
by using Eq. (40).

We use the gauge coupling constants measured at the Fermi scale
$\sqrt{s}\approx 10^{2}$ GeV [15]:

$g_{SU(3)}\approx 1.23,\quad\quad\quad g_{SU(2)}\approx 0.652,\quad\quad\quad
g_{U(1)}\approx 0.357.$ (45)

We then numerically999There is an analytic formula for the largest eigenvalue
in terms of $g_{SU(3)}$, $g_{SU(2)}$, and $g_{U(1)}$. However, its size is
large enough that it is not meaningful to write it down here. calculate the
largest eigenvalue of the matrix $a_{0}$ of Eq. (39) to be

$\left(a_{0}\right)^{\mathrm{max}}\approx 0.269.$ (46)

We thus reassuringly conclude that the SM is perturbative, since $a_{0}=0.269$
is well compatible with the unitarity condition
$\left|\textup{Re}\,a_{0}\right|\leq 1/2$ of Eq. (9a).

If we suppose the existence of an additional fermionic multiplet

$\psi\sim\left(\textbf{{n}}^{SU(3)}_{\psi},\,\textbf{{n}}^{SU(2)}_{\psi},\,Y_{\psi}\right),$
(47)

then we must add the extra state $\left[\psi\psi\right]^{\textup{sym}}$ to the
basis vector (42). For each $\textbf{{n}}^{SU(3)}_{\psi}$,
$\textbf{{n}}^{SU(2)}_{\psi}$, and $Y_{\psi}$ we use Eqs. (39) and (40) to
construct the matrix $a_{0}$ and we then compute its largest eigenvalue. In
order to fulfill the unitarity condition
$\left|\textup{Re}\,\left(a_{0}\right)^{\textup{max}}\right|\leq\frac{1}{2}$,
when the dimensions of the representations of $SU(3)_{\mathrm{colour}}$ and
$SU(2)_{L}$ that $\psi$ sits in are fixed, the hypercharge $Y_{\psi}$ cannot
be arbitrarily large. We thus constrain the absolute value of $Y_{\psi}$; the
results are shown in the left panel of Table 1. Obviously, smaller
$n^{SU(3)}_{\psi}$ and smaller $n^{SU(2)}_{\psi}$ allow for a larger
$\left|Y_{\psi}\right|$. In general, any fermionic multiplet added to the
Standard Model must satisfy $n^{SU(3)}_{\psi}\leq 8$ and $n^{SU(2)}_{\psi}\leq
7$.

One may instead introduce an additional (complex) scalar multiplet $\varphi$
which transforms under $\mathcal{G}_{\textup{SM}}$ as

$\varphi\sim\left(\textbf{{n}}^{SU(3)}_{\varphi},\,\textbf{{n}}^{SU(2)}_{\varphi},\,Y_{\varphi}\right)$
(48)

In a similar fashion, $\left|Y_{\varphi}\right|$ may be constrained by
imposing partial-wave unitarity. For each combination of
$\textbf{{n}}^{SU(3)}_{\varphi}$ and $\textbf{{n}}^{SU(2)}_{\varphi}$, the
upper bound on $\left|Y_{\varphi}\right|$ is presented in the right panel of
Table 1. We conclude that scalar multiplets are less constrained; they must
satisfy $n^{SU(3)}_{\varphi}\leq 10$ and $n^{SU(2)}_{\varphi}\leq 8$. The
first column of the right panel of Table 1 corresponds to the scenario studied
in Ref. [3]; our constraints are, however, more stringent and accurate,
because we consider not only the contributions from the extra scalar multiplet
$\varphi$ but also the contributions from the SM matter fields of Eqs. (41).

Table 1: Upper bounds on $\left|Y_{\psi}\right|$ (left panel) and on
$\left|Y_{\varphi}\right|$ (right panel) imposed by the tree-level unitarity
condition $\left|\textup{Re}\,a_{0}\right|\leq\frac{1}{2}$, in an extension of
the Standard Model through either a fermionic multiplet $\psi$ (left panel) or
a scalar multiplet $\varphi$ (right panel). The symbol (—) stands for a
negative, hence un-physical number.

| $\textbf{{n}}^{SU(3)}_{\psi}$
---|---
| $\left|Y_{\psi}\right|^{\mathrm{max}}$ | 1 | 3 | 6 | 8
$\textbf{{n}}^{SU(2)}_{\psi}$ | 1 | 7.91 | 5.96 | 4.41 | 4.04
2 | 6.65 | 4.96 | 2.65 | 2.13
| 3 | 6.00 | 4.39 | — | —
| 4 | 5.52 | 3.87 | — | —
| 5 | 5.05 | 2.95 | — | —
| 6 | 4.37 | — | — | —
| 7 | 2.13 | — | — | —

| $\textbf{{n}}^{SU(3)}_{\varphi}$
---|---
| $|Y_{\varphi}|^{\mathrm{max}}$ | 1 | 3 | 6 | 8 | 10
$\textbf{{n}}^{SU(2)}_{\varphi}$ | 1 | 9.92 | 7.51 | 6.04 | 5.60 | 3.03
2 | 8.34 | 6.29 | 4.78 | 4.40 | —
| 3 | 7.53 | 5.64 | 3.94 | 3.56 | —
| 4 | 6.97 | 5.15 | 2.86 | 2.10 | —
| 5 | 6.51 | 4.62 | — | — | —
| 6 | 6.03 | 3.74 | — | — | —
| 7 | 5.38 | — | — | — | —
| 8 | 4.08 | — | — | — | —

###  4.2 Grand Unified Theory $SU(5)$

Another interesting case study is the $SU(5)$ Grand Unified Theory (GUT). In
this model, the three gauge coupling constants of the SM unify at an energy
scale $\sqrt{s}\approx 10^{16}\,\textup{GeV}$. We use the following estimate
for the $SU(5)$ gauge coupling constant at the unification scale101010This
estimate is conservative—some authors obtain larger values for $g_{SU(5)}$,
and then the unitarity bounds will be stronger. The result for $g_{SU(5)}$
depends on the particle content of the GUT, hence on the details of the
Renormalization Group evolution of the three SM gauge coupling constants that
merge into $g_{SU(5)}$. [16, 17]:

$g_{SU(5)}\approx 0.60.$ (49)

The adjoint representation of $SU(5)$ is 24-dimensional; the $SU(5)$ GUT has

$\tilde{n}_{SU(5)}=24$ (50)

gauge bosons. In addition, there are three generations of two $SU(5)$
fermionic multiplets [18]:

$\displaystyle\psi_{5,f}$ $\displaystyle\sim\overline{\textbf{5}},$
$\displaystyle\psi_{10,f}$ $\displaystyle\sim\textbf{10}.$ (51)

with the generation index $f\in\left\\{1,2,3\right\\}$.

A general feature of GUTs is the over-abundance of particles in their scalar
sectors. But, if perturbativity is to be preserved, the largest eigenvalue of
the matrix of zeroth partial-waves—given in Eq. (35)—must satisfy the
unitarity condition (9a). As a result, the number of fermions $N_{F}$, the
number of scalars $N_{S}$, and the dimensions of their corresponding $SU(5)$
irreducible representations (‘irreps’) are constrained by

$\displaystyle\left|\mathrm{Re}\,\left(a_{0}\right)^{\textup{max}}\right|^{2}\leq\frac{1}{4}$
$\displaystyle\Leftrightarrow$
$\displaystyle\sum_{i}^{N_{S}}n_{S_{i}}\,C\left(\textbf{{n}}_{S_{i}}\right)^{2}+\frac{\pi^{2}}{4}\,\sum_{i}^{N_{F}}n_{F_{i}}\,C\left(\textbf{{n}}_{F_{i}}\right)^{2}\leq\frac{16\pi^{2}\tilde{n}_{SU(5)}}{g_{SU(5)}^{4}}\approx
29,243.$ (52)

This implies an upper bound on the size of the scalar sector. Note that, for
the most usual set of fermionic multiplets in (51), the second term in the
left-hand side of Eq. (52) takes the value

$\frac{\pi^{2}}{4}\,\sum_{i}^{N_{F}}n_{F_{i}}\,C\left(\textbf{{n}}_{F_{i}}\right)^{2}=\frac{3\pi^{2}}{4}\,\left(5\,C\left(\psi_{5}\right)^{2}+10\,C\left(\psi_{10}\right)^{2}\right)\
\approx\ 1,173.$ (53)

We firstly consider four minimal scenarios found in the literature [18, 19,
20], all of which employ a 24-dimensional scalar multiplet to spontaneously
break $SU(5)$ into $\mathcal{G}_{\textup{SM}}$, but differ in the set of
scalars used to further break $\mathcal{G}_{\textup{SM}}$ into the gauge group
$\mathcal{G}_{31}\equiv SU(3)_{\mathrm{colour}}\times
U(1)_{\mathrm{electromagnetism}}$ and to generate fermion masses. For each
scenario below, we use Eq. (35) and the results of Appendix C.4 to compute the
largest eigenvalue of the matrix of zeroth partial waves:

$\displaystyle SU(5)\xrightarrow[]{\ \ \mathbf{24}\ \
}\mathcal{G}_{\textup{SM}}\xrightarrow[]{\ \ \mathbf{5}\ \ }\mathcal{G}_{31}\
\ \Rightarrow\ \ \left(a_{0}\right)^{\mathrm{max}}$ $\displaystyle\approx
0.12,$ (54a) $\displaystyle SU(5)\xrightarrow[]{\ \ \mathbf{24}\ \
}\mathcal{G}_{\textup{SM}}\xrightarrow[]{\ \ \mathbf{5}\,\oplus\,\mathbf{45}\
\ }\mathcal{G}_{31}\ \ \Rightarrow\ \ \left(a_{0}\right)^{\mathrm{max}}$
$\displaystyle\approx 0.18,$ (54b) $\displaystyle SU(5)\xrightarrow[]{\ \
\mathbf{24}\ \
}\mathcal{G}_{\textup{SM}}\xrightarrow[]{\mathbf{5}\,\oplus\,\mathbf{5}\,\oplus\,\mathbf{5}\,\oplus\,\mathbf{45}}\mathcal{G}_{31}\
\ \Rightarrow\ \ \left(a_{0}\right)^{\mathrm{max}}$ $\displaystyle\approx
0.18,$ (54c) $\displaystyle SU(5)\xrightarrow[]{\ \ \mathbf{24}\ \
}\mathcal{G}_{\textup{SM}}\xrightarrow[]{\mathbf{5}\,\oplus\,\mathbf{40}\,\oplus\,\mathbf{45}}\mathcal{G}_{31}\
\ \Rightarrow\ \ \left(a_{0}\right)^{\mathrm{max}}$ $\displaystyle\approx
0.22.$ (54d)

We find that every scenario satisfies the unitarity condition (9a).

A natural extension of these four scenarios is the addition of either another
fermionic multiplet $\psi\sim\textbf{{n}}_{\psi}$ or another scalar multiplet
$\varphi\sim\textbf{{n}}_{\varphi}$. Constraints on the dimensionality of such
multiplets are attained through Eq. (52). Regarding the former possibility,
one can immediately see that unitarity is violated for the irrep 70’ and for
any fermionic multiplet with dimension $n_{\psi}\geq 105$. For scalars the
bounds are less strict: in the first three scenarios, unitarity is violated
for the irreps 126’, 175” and for any multiplet with dimension
$n_{\varphi}\geq 200$; in the fourth scenario, unitarity is furthermore
violated for the irreps 160 and 175’.

The five-dimensional representation of $SU(5)$ contributes very little to
$\left(a_{0}\right)^{\mathrm{max}}$. We checked that, in order to break
perturbative unitarity, one must add to the minimal $SU(5)$ GUT close to one
thousand 5-plets of scalars.

We summarize the results of this subsection in Table 2.

Table 2: Summary of the $SU(5)$ extensions considered. Fermion Content | Scalar Content |  $\left|\textup{Re}\,\left(a_{0}\right)^{\mathrm{max}}\right|\leq\frac{1}{2}$ if
---|---|---
$3\,(\psi_{5}\oplus\psi_{10})\oplus\textbf{{n}}_{\psi}$ | $\textbf{5}\oplus\textbf{24}$ | $n_{\psi}\leq 75\,\backslash\,\\{\textbf{70'}\\}$
$\textbf{5}\oplus\textbf{24}\oplus\textbf{45}$
$3(\textbf{5})\oplus\textbf{24}\oplus\textbf{45}$
$\textbf{5}\oplus\textbf{24}\oplus\textbf{40}\oplus\textbf{45}$
$3\,(\psi_{5}\oplus\psi_{10})$ | $\textbf{5}\oplus\textbf{24}\oplus\textbf{{n}}_{\varphi}$ | $n_{\varphi}\leq 175\,\backslash\,\\{\textbf{126'},\textbf{175''}\\}$
$\textbf{5}\oplus\textbf{24}\oplus\textbf{45}\oplus\textbf{{n}}_{\varphi}$
$3(\textbf{5})\oplus\textbf{24}\oplus\textbf{45}\oplus\textbf{{n}}_{\varphi}$
$3\,(\psi_{5}\oplus\psi_{10})$ | $\textbf{5}\oplus\textbf{24}\oplus\textbf{40}\oplus\textbf{45}\oplus\textbf{{n}}_{\varphi}$ | $n_{\varphi}\leq 126\vee n_{\varphi}=\textbf{175}$
$3\,(\psi_{5}\oplus\psi_{10})$ | $\textbf{24}\oplus N_{5}(\textbf{5})$ | $N_{5}\leq 953$
$\textbf{24}\oplus\textbf{45}\oplus N_{5}(\textbf{5})$ | $N_{5}\leq 889$
$\textbf{24}\oplus\textbf{40}\oplus\textbf{45}\oplus N_{5}(\textbf{5})$ | $N_{5}\leq 829$

###  4.3 Grand Unified Theory $SO(10)$

In the $SO(10)$ GUT, the three SM coupling constants unify at the energy scale
$\sqrt{s}\approx 10^{16}\,\textup{GeV}$ at the value [21]

$g_{SO(10)}\approx 0.60.$ (55)

The adjoint representation of $SO(10)$ is 45-dimensional, thus

$\tilde{n}_{SO(10)}=45.$ (56)

Each generation of 15 SM fermions is contained in a single 16-dimensional
multiplet $\psi_{16,f}\sim\textbf{16}$ ($f\in\left\\{1,2,3\right\\}$), which
also accommodates a right-handed neutrino. The scalar sector of the $SO(10)$
GUT may be divided in two parts: the scalars required to break $SO(10)$ into
$\mathcal{G}_{\textup{SM}}$, and the scalars needed to generate the fermion
masses and/or to break $\mathcal{G}_{\textup{SM}}$ into $\mathcal{G}_{31}$.
There are many possible ways to break $SO(10)$ into
$\mathcal{G}_{\textup{SM}}$. The most common minimal models have one of the
following symmetry-breaking chains [22, 23]:

$\displaystyle SO(10)\xrightarrow[]{\mathbf{144}}\mathcal{G}_{\textup{SM}},$
(57a) $\displaystyle
SO(10)\xrightarrow[]{\mathbf{45}}\mathcal{G}_{421}\xrightarrow[]{\mathbf{16}/\mathbf{126}}\mathcal{G}_{\textup{SM}},$
(57b) $\displaystyle
SO(10)\xrightarrow[]{\mathbf{45}}\mathcal{G}_{421}\xrightarrow[]{\mathbf{45}/\mathbf{210}}\mathcal{G}_{3211}\xrightarrow[]{\mathbf{16}/\mathbf{126}}\mathcal{G}_{\textup{SM}},$
(57c) $\displaystyle
SO(10)\xrightarrow[]{\mathbf{16}/\mathbf{126}}\mathcal{G}_{5}\xrightarrow[]{\mathbf{45}/\mathbf{54}/\mathbf{210}}\mathcal{G}_{\textup{SM}},$
(57d) $\displaystyle
SO(10)\xrightarrow[]{\mathbf{45}/\mathbf{210}}\mathcal{G}_{51}\xrightarrow[]{\mathbf{16}/\mathbf{126}}\mathcal{G}_{\textup{SM}},$
(57e) $\displaystyle
SO(10)\xrightarrow[]{\mathbf{45}/\mathbf{210}}\mathcal{G}_{51}\xrightarrow[]{\mathbf{16}/\mathbf{126}}\mathcal{G}_{5}\xrightarrow[]{\mathbf{45}/\mathbf{54}/\mathbf{210}}\mathcal{G}_{\textup{SM}},$
(57f) $\displaystyle
SO(10)\xrightarrow[]{\mathbf{54}/\mathbf{210}}\mathcal{G}_{422}\xrightarrow[]{\mathbf{16}/\mathbf{126}}\mathcal{G}_{\textup{SM}},$
(57g) $\displaystyle
SO(10)\xrightarrow[]{\mathbf{54}/\mathbf{210}}\mathcal{G}_{422}\xrightarrow[]{\mathbf{45}}\mathcal{G}_{421}\xrightarrow[]{\mathbf{16}/\mathbf{126}}\mathcal{G}_{\textup{SM}},$
(57h) $\displaystyle
SO(10)\xrightarrow[]{\mathbf{54}/\mathbf{210}}\mathcal{G}_{422}\xrightarrow[]{\mathbf{45}/\mathbf{210}}\mathcal{G}_{3211}\xrightarrow[]{\mathbf{16}/\mathbf{126}}\mathcal{G}_{\textup{SM}},$
(57i) $\displaystyle
SO(10)\xrightarrow[]{\mathbf{54}/\mathbf{210}}\mathcal{G}_{422}\xrightarrow[]{\mathbf{45}/\mathbf{210}}\mathcal{G}_{3221}\xrightarrow[]{\mathbf{16}/\mathbf{126}}\mathcal{G}_{\textup{SM}},$
(57j) $\displaystyle
SO(10)\xrightarrow[]{\mathbf{45}/\mathbf{210}}\mathcal{G}_{3221}\xrightarrow[]{\mathbf{16}/\mathbf{126}}\mathcal{G}_{\textup{SM}},$
(57k) $\displaystyle
SO(10)\xrightarrow[]{\mathbf{45}/\mathbf{210}}\mathcal{G}_{3211}\xrightarrow[]{\mathbf{16}/\mathbf{126}}\mathcal{G}_{\textup{SM}},$
(57l) $\displaystyle
SO(10)\xrightarrow[]{\mathbf{45}/\mathbf{210}}\mathcal{G}_{3221}\xrightarrow[]{\mathbf{45}/\mathbf{210}}\mathcal{G}_{3211}\xrightarrow[]{\mathbf{16}/\mathbf{126}}\mathcal{G}_{\textup{SM}},$
(57m)

where

$\displaystyle\mathcal{G}_{5}$ $\displaystyle\equiv$ $\displaystyle SU(5),$
(58a) $\displaystyle\mathcal{G}_{51}$ $\displaystyle\equiv$ $\displaystyle
SU(5)\times U(1),$ (58b) $\displaystyle\mathcal{G}_{421}$
$\displaystyle\equiv$ $\displaystyle SU(4)\times SU(2)\times U(1),$ (58c)
$\displaystyle\mathcal{G}_{422}$ $\displaystyle\equiv$ $\displaystyle
SU(4)\times SU(2)\times SU(2),$ (58d) $\displaystyle\mathcal{G}_{3211}$
$\displaystyle\equiv$ $\displaystyle SU(3)\times SU(2)\times U(1)\times U(1),$
(58e) $\displaystyle\mathcal{G}_{3221}$ $\displaystyle\equiv$ $\displaystyle
SU(3)\times SU(2)\times SU(2)\times U(1).$ (58f)

Except for the direct route (57a), all other paths employ either a 16 or a 126
together with a combination of 45’s, 54’s, and 210’s.

To further break $\mathcal{G}_{\textup{SM}}$ into $\mathcal{G}_{31}$, we
consider models with a 10 or a 120 (which generate the masses of the Standard
Model quarks and leptons) and a $\overline{\textbf{126}}$ (which furthermore
generates neutrino masses via a seesaw mechanism). Schematically:

$\displaystyle\mathcal{G}_{\textup{SM}}\xrightarrow[]{\
\mathbf{10}\,\oplus\,\overline{\mathbf{126}}\ }\mathcal{G}_{31},$ (59a)
$\displaystyle\mathcal{G}_{\textup{SM}}\xrightarrow[]{\
\mathbf{120}\,\oplus\,\overline{\mathbf{126}}\ }\mathcal{G}_{31}.$ (59b)

We use

$\sum_{i}^{N_{S}}n_{S_{i}}\,C\left(n_{S_{i}}\right)^{2}+\frac{\pi^{2}}{4}\,\sum_{i}^{N_{F}}n_{F_{i}}\,C\left(n_{F_{i}}\right)^{2}\
\leq\ \frac{16\pi^{2}\tilde{n}_{SO(10)}}{g_{SO(10)}^{4}}\ \approx\ 54,831,$
(60)

where $N_{S}$ is the total number of scalars in representations
$\textbf{{n}}_{S_{i}}$ and $N_{F}$ is the total number of fermions in
representations $\textbf{{n}}_{F_{i}}$; thus, if there are three fermionic
$\mathbf{16}$ representations,

$\sum_{i}^{N_{S}}n_{S_{i}}\,C\left(n_{S_{i}}\right)^{2}\ \leq\
\frac{16\pi^{2}\tilde{n}_{SO(10)}}{g_{SO(10)}^{4}}-\frac{\pi^{2}}{4}\times
48\times\left(\frac{45}{8}\right)^{2}\ \approx\ 51,083.$ (61)

The condition (61) may be used to evaluate the validity of perturbation theory
in each scenario presented in Eqs. (57) and (59). The results are shown in
Table 3, wherein the scalar contents of the grey-shaded scenarios violate the
partial-wave unitarity condition
$\left|\textup{Re}\,\left(a_{0}\right)^{\textup{max}}\right|\leq 1/2$. Since
$\left(a_{0}\right)^{\textup{max}}$ is an increasing function of the number of
scalar multiplets, we omit all other scenarios for which a subset of scalars
already violates unitarity. Thus, we advocate a reexamination of the validity
of perturbation theory in $SO(10)$ models that have, at least, one of the
following sets of scalar irreps:

$\displaystyle\left(\mathbf{10}\ \mbox{or
higher}\right)\oplus\mathbf{210}\oplus\mathbf{210};$ (62a)
$\displaystyle\left(\mathbf{45}\ \mbox{or
higher}\right)\oplus\mathbf{\overline{126}}\oplus\mathbf{210};$ (62b)
$\displaystyle\left(\mathbf{10}\ \mbox{or
higher}\right)\oplus\mathbf{120}\oplus\mathbf{126}\oplus\mathbf{\overline{126}}.$
(62c)

Table 3: Largest zeroth partial waves of minimal $SO(10)$ GUTs. Every scenario assumes a fermionic sector composed of three families of 16 multiplets.  $\ \ \ SO(10)\to\mathcal{G}_{\textup{SM}}\ \ \ $ | $\ \ \ \mathcal{G}_{\textup{SM}}\to\mathcal{G}_{31}\ \ \ $ | $\left(a_{0}\right)^{\mathrm{max}}$
---|---|---
144 | $\textbf{10}\oplus\overline{\textbf{126}}$ | 0.43
144 | $\textbf{120}\oplus\overline{\textbf{126}}$ | 0.49
$\textbf{16}\oplus\textbf{45}$ | $\textbf{10}\oplus\overline{\textbf{126}}$ | 0.35
$\textbf{16}\oplus\textbf{45}$ | $\textbf{120}\oplus\overline{\textbf{126}}$ | 0.43
$\textbf{16}\oplus\textbf{54}$ | $\textbf{10}\oplus\overline{\textbf{126}}$ | 0.37
$\textbf{16}\oplus\textbf{54}$ | $\textbf{120}\oplus\overline{\textbf{126}}$ | 0.44
$\textbf{16}\oplus\textbf{210}$ | $\textbf{10}\oplus\overline{\textbf{126}}$ | 0.50
$\textbf{16}\oplus\textbf{210}$ | $\textbf{120}\oplus\overline{\textbf{126}}$ | 0.55
$\textbf{45}\oplus\textbf{126}$ | $\textbf{10}\oplus\overline{\textbf{126}}$ | 0.46
$\textbf{45}\oplus\textbf{126}$ | $\textbf{120}\oplus\overline{\textbf{126}}$ | 0.52
$\textbf{54}\oplus\textbf{126}$ | $\textbf{10}\oplus\overline{\textbf{126}}$ | 0.47
$\textbf{54}\oplus\textbf{126}$ | $\textbf{120}\oplus\overline{\textbf{126}}$ | 0.53
$\textbf{126}\oplus\textbf{210}$ | $\textbf{10}\oplus\overline{\textbf{126}}$ | 0.58
$\textbf{126}\oplus\textbf{210}$ | $\textbf{120}\oplus\overline{\textbf{126}}$ | 0.63
$\textbf{16}\oplus\textbf{45}\oplus\textbf{45}$ | $\textbf{10}\oplus\overline{\textbf{126}}$ | 0.37
$\textbf{16}\oplus\textbf{45}\oplus\textbf{45}$ | $\textbf{120}\oplus\overline{\textbf{126}}$ | 0.44
$\textbf{16}\oplus\textbf{45}\oplus\textbf{54}$ | $\textbf{10}\oplus\overline{\textbf{126}}$ | 0.38
$\textbf{16}\oplus\textbf{45}\oplus\textbf{54}$ | $\textbf{120}\oplus\overline{\textbf{126}}$ | 0.46
$\textbf{16}\oplus\textbf{45}\oplus\textbf{210}$ | $\textbf{10}\oplus\overline{\textbf{126}}$ | 0.51
$\textbf{16}\oplus\textbf{45}\oplus\textbf{210}$ | $\textbf{120}\oplus\overline{\textbf{126}}$ | 0.57
$\textbf{16}\oplus\textbf{54}\oplus\textbf{54}$ | $\textbf{10}\oplus\overline{\textbf{126}}$ | 0.40
$\textbf{16}\oplus\textbf{54}\oplus\textbf{54}$ | $\textbf{120}\oplus\overline{\textbf{126}}$ | 0.47
$\textbf{16}\oplus\textbf{54}\oplus\textbf{210}$ | $\textbf{10}\oplus\overline{\textbf{126}}$ | 0.52
$\textbf{45}\oplus\textbf{45}\oplus\textbf{126}$ | $\textbf{10}\oplus\overline{\textbf{126}}$ | 0.47

One may want to expand the scalar sector of the remaining $SO(10)$ models by
incorporating low-dimensional irreps. Specifically, we consider adding either
$N_{10}$ copies of ten-dimensional scalar multiplets 10, or (but not
simultaneously) $N_{16}$ copies of sixteen-dimensional scalar multiplets 16.
According to Eq. (61), $N_{10}$ and $N_{16}$ must be bounded from above. In
Table 4 we present upper bounds on $N_{10}$ and $N_{16}$ for some $SO(10)$
models that do not violate tree-level partial-wave unitarity, yet have rather
low (smaller than 10) maximal $N_{10}$ and/or maximal $N_{16}$.

Table 4: Upper bounds on $N_{10}$ and $N_{16}$ for some allowed $SO(10)$ models. We always assume the fermionic content to be three $\mathbf{16}$ of $SO(10)$. $SO(10)\to\mathcal{G}_{\textup{SM}}\to\mathcal{G}_{31}$ | $N_{10}^{\mathrm{max}}$ | $N_{16}^{\mathrm{max}}$
---|---|---
$\mathbf{120}\oplus\mathbf{126}\oplus\mathbf{144}$ | 9 | 0
$\mathbf{120}\oplus\mathbf{126}\oplus\mathbf{144}$ | 0 | 3
$\mathbf{10}\oplus\mathbf{126}\oplus\mathbf{210}$ | 2 | 0
$\mathbf{10}\oplus\mathbf{126}\oplus\mathbf{210}$ | 0 | 1
$\mathbf{10}\oplus\mathbf{54}\oplus\mathbf{54}\oplus\mathbf{126}\oplus\mathbf{126}$ | 3 | 0
$\mathbf{10}\oplus\mathbf{54}\oplus\mathbf{54}\oplus\mathbf{126}\oplus\mathbf{126}$ | 0 | 1

###  4.4 Grand Unified Theory $E_{6}$

We now study the $E_{6}$ Grand Unified Theory. At the GUT energy scale
$\sqrt{s}\approx 10^{16}\,\textup{GeV}$, we adopt the conservative estimate
for the unified coupling constant [24]

$g_{E_{6}}\approx 0.55.$ (63)

The adjoint representation of $E_{6}$ is 78-dimensional, thus

$\tilde{n}_{E_{6}}=78.$ (64)

The fermionic sector is composed of three generations of a 27-dimensional
multiplet $\psi_{27}^{f}\sim\textbf{27}$, $f\in\left\\{1,2,3\right\\}$. Each
$\mathbf{27}$ accommodates the 15 SM fermions of Eqs. (41a)–(41e) along with
12 new fermions, out of which one is a right-handed neutrino [25].

The tensor product of two 27’s decomposes as

$\textbf{27}\otimes\textbf{27}=\textbf{27}\oplus\textbf{351}\oplus\textbf{351'}$
(65)

under $E_{6}$. Therefore, fermion masses can be generated through a Higgs-like
mechanism by introducing any linear combination of 27, 351, and 351’ scalar
multiplets. However, we expect the masses of the 12 fermions beyond the SM to
be orders of magnitude larger than the masses of the SM fermions. This
splitting of mass scales can only be achieved by including at least one scalar
multiplet 351’ [26, 27, 28]. With this in mind, before specifying the scalar
multiplets needed to break $E_{6}$ into $\mathcal{G}_{\textup{SM}}$, we
compute the largest eigenvalue of the matrix of zeroth partial-waves for a
model composed of three 27 fermionic multiplets and one 351’ scalar multiplet.
We define the matrix of zeroth partial-waves in the basis:

$\left\\{\left[VV\right]^{\textup{sym}}_{E_{6}},\,\left[\psi_{27}^{1}\,\psi_{27}^{1}\right]_{\textup{sym}},\,\left[\psi_{27}^{2}\,\psi_{27}^{2}\right]_{\textup{sym}},\,\left[\psi_{27}^{3}\,\psi_{27}^{3}\right]_{\textup{sym}},\,\left[\textbf{351'}\,\textbf{351'}\right]_{\textup{sym}}\right\\}.$
(66)

By using Eq. (35) we immediately find that the largest eigenvalue violates the
unitarity condition of Eq. (9a):

$\left(a_{0}\right)^{\mathrm{max}}\ =\
\frac{g_{E_{6}}^{2}}{8\pi\sqrt{78}}\sqrt{\frac{3\pi^{2}}{4}\left[\,\sqrt{27}\,C\left(\textbf{27}\right)\right]^{2}+\left[\sqrt{351}\,C\left(\textbf{351'}\right)\right]^{2}}\
\approx\ 0.51.$ (67)

The situation can only worsen with the introduction of further scalar or
fermionic multiplets. To give one out of many examples found in the
literature, Ref. [29] considers a model with three 27 fermionic multiplets and
a scalar sector composed of $\textbf{27}\oplus\textbf{78}\oplus\textbf{351'}$.
By running the three SM couplings up to the GUT energy scale, where they
unify, Ref. [29] obtains $g_{E_{6}}\approx 0.70$. The corresponding largest
eigenvalue in this scenario equals

$\left(a_{0}\right)^{\mathrm{max}}\approx 0.86,$ (68)

which obviously violates the unitarity condition (9a). For this reason, we
raise questions about the viability of _any_ perturbative Grande Unified
Theory with $E_{6}$ gauge symmetry, capable of generating the correct mass
hierarchy in the fermion sector.

##  5 Conclusion

We have computed the largest (in modulus) eigenvalue of the matrix of zeroth
partial waves in the scattering of a fermion–antifermion pair and of a
scalar–scalar pair into two gauge bosons. We have made the educated guess that
the largest eigenvalue is always obtained when the fermion–antifermion,
scalar–scalar, and gauge boson–gauge boson pairs are in symmetric combinations
under the gauge group; if this assumption fails in some cases—which we do not
expect to happen—then the bounds that we have obtained will still be valid,
even if they are not the strongest possible ones. We have found that no
problem of violation of the unitarity requirement occurs in most extensions of
the SM and/or of the $SU(5)$ Grand Unified Theory. On the other hand, many
schemes of $SO(10)$ symmetry breaking display unitarity violation, and this is
also true of _all_ the $E_{6}$-based GUTs. In those GUTs, perturbative
calculations seem to be unwarranted.

#### Acknowledgements:

We were supported by the Portuguese Foundation for Science and Technology
through projects UIDB/00777/2020 and UIDP/00777/2020. The work of L.L. was
furthermore supported by projects CERN/FIS-PAR/0002/2021 and CERN/FIS-
PAR/0019/2021.

##  Appendix A Explicit calculation of the scattering amplitudes

As we will be considering scattering amplitudes of processes of the type
$A_{i}(p_{1})A^{\dagger}_{j}(p_{2})\to V^{a}(p_{3})V^{b}(p_{4})$ (with
$A=\psi,\varphi$), it is important to define the four-momenta of each
particle. In the $x-z$ plane these can be written as:

$\displaystyle p_{1}^{\mu}$
$\displaystyle\approx\frac{\sqrt{s}}{2}(1,\sin{\theta},0,\cos{\theta}),$ (69a)
$\displaystyle p_{2}^{\mu}$
$\displaystyle\approx\frac{\sqrt{s}}{2}(1,-\sin{\theta},0,-\cos{\theta}),$
(69b) $\displaystyle p_{3}^{\mu}$
$\displaystyle\approx\frac{\sqrt{s}}{2}(1,0,0,1),$ (69c) $\displaystyle
p_{4}^{\mu}$ $\displaystyle\approx\frac{\sqrt{s}}{2}(1,0,0,-1),$ (69d)

where the center-of-mass frame of reference was adopted and the high-energy
limit ($\sqrt{s}\gg m$) was taken. In this limit the corresponding chiral
spinors associated with $\psi_{i}$ and $\overline{\psi}_{j}$ (i.e. $u_{i}$,
$\overline{v}_{j}$) and gauge boson transverse polarization vectors
($\varepsilon^{\mu}_{L/R}$) can be written in the helicity basis as [30, 31]:

$\displaystyle u_{i}(p_{1})$
$\displaystyle\approx\frac{s^{1/4}}{\sqrt{2}}\begin{pmatrix}-\sin{\frac{\theta}{2}}\\\
\cos{\frac{\theta}{2}}\\\ \sin{\frac{\theta}{2}}\\\
-\cos{\frac{\theta}{2}}\end{pmatrix},$ $\displaystyle\overline{v}_{j}(p_{2})$
$\displaystyle\approx\frac{s^{1/4}}{\sqrt{2}}\begin{pmatrix}\cos{\frac{\theta}{2}}\\\
\sin{\frac{\theta}{2}}\\\ \cos{\frac{\theta}{2}}\\\
\sin{\frac{\theta}{2}}\end{pmatrix},$ (70a)
$\displaystyle\varepsilon^{\mu}_{L}(p_{3})$
$\displaystyle=\frac{1}{\sqrt{2}}(0,1,-i,0),$
$\displaystyle\varepsilon^{\mu}_{R}(p_{3})$
$\displaystyle=\frac{1}{\sqrt{2}}(0,1,i,0),$ (70b)
$\displaystyle\varepsilon^{\mu}_{L}(p_{4})$
$\displaystyle=\frac{1}{\sqrt{2}}(0,-1,i,0),$
$\displaystyle\varepsilon^{\mu}_{R}(p_{4})$
$\displaystyle=\frac{1}{\sqrt{2}}(0,-1,-i,0).$ (70c)

###  A.1 Fermions

To simplify Eqs. (12a) and (12b), we first contract the four vectors of Eqs.
(70) with the Dirac gamma matrices. Working in the Pauli-Dirac representation,
one obtains:

$\displaystyle\not{p}_{1}$
$\displaystyle=\frac{\sqrt{s}}{2}\begin{pmatrix}1&0&-c_{\theta}&-s_{\theta}\\\
0&1&-s_{\theta}&c_{\theta}\\\ c_{\theta}&s_{\theta}&-1&0\\\
s_{\theta}&-c_{\theta}&0&-1\end{pmatrix},$ $\displaystyle\not{p}_{2}$
$\displaystyle=\frac{\sqrt{s}}{2}\begin{pmatrix}1&0&c_{\theta}&s_{\theta}\\\
0&1&s_{\theta}&-c_{\theta}\\\ -c_{\theta}&-s_{\theta}&-1&0\\\
-s_{\theta}&c_{\theta}&0&-1\end{pmatrix},$ (71a)

$\displaystyle\not{p}_{3}$
$\displaystyle=\frac{\sqrt{s}}{2}\begin{pmatrix}1&0&-1&0\\\ 0&1&0&1\\\
1&0&-1&0\\\ 0&-1&0&-1\end{pmatrix},$ $\displaystyle\not{p}_{4}$
$\displaystyle=\frac{\sqrt{s}}{2}\begin{pmatrix}1&0&1&0\\\ 0&1&0&-1\\\
-1&0&-1&0\\\ 0&1&0&-1\end{pmatrix},$ (71b)

$\displaystyle\not{\varepsilon}_{R}^{*}(p_{3})$
$\displaystyle=\sqrt{2}\begin{pmatrix}0&0&0&0\\\ 0&0&-1&0\\\ 0&0&0&0\\\
1&0&0&0\end{pmatrix},$ $\displaystyle\not{\varepsilon}_{L}^{*}(p_{3})$
$\displaystyle=\sqrt{2}\begin{pmatrix}0&0&0&-1\\\ 0&0&0&0\\\ 0&1&0&0\\\
0&0&0&0\end{pmatrix},$ (71c)

$\displaystyle\not{\varepsilon}_{R}^{*}(p_{4})$
$\displaystyle=\sqrt{2}\begin{pmatrix}0&0&0&1\\\ 0&0&0&0\\\ 0&-1&0&0\\\
0&0&0&0\end{pmatrix},$ $\displaystyle\not{\varepsilon}_{L}^{*}(p_{4})$
$\displaystyle=\sqrt{2}\begin{pmatrix}0&0&0&0\\\ 0&0&1&0\\\ 0&0&0&0\\\
-1&0&0&0\end{pmatrix},$ (71d)

where $c_{\theta}$ and $s_{\theta}$ are the cosine and sine of the scattering
angle, respectively. The numerator of Eqs. (12a) and (12b) can be explicitly
written for the RR and LL polarization configurations as:

$\displaystyle\left[\overline{v}_{j}\cdot\not{\varepsilon}_{R}^{*}(p_{4})\cdot(\not{p}_{1}-\not{p}_{3})\cdot\not{\varepsilon}_{R}^{*}(p_{3})\cdot
u_{i}\right]$ $\displaystyle=-s\sin{\theta}\sin^{2}{\frac{\theta}{2}},$ (72a)
$\displaystyle\left[\overline{v}_{j}\cdot\not{\varepsilon}_{R}^{*}(p_{3})\cdot(\not{p}_{1}-\not{p}_{4})\cdot\not{\varepsilon}_{R}^{*}(p_{4})\cdot
u_{i}\right]$ $\displaystyle=-s\sin{\theta}\sin^{2}{\frac{\theta}{2}},$ (72b)
$\displaystyle\left[\overline{v}_{j}\cdot\not{\varepsilon}_{L}^{*}(p_{4})\cdot(\not{p}_{1}-\not{p}_{3})\cdot\not{\varepsilon}_{L}^{*}(p_{3})\cdot
u_{i}\right]$ $\displaystyle=s\sin{\theta}\cos^{2}{\frac{\theta}{2}},$ (72c)
$\displaystyle\left[\overline{v}_{j}\cdot\not{\varepsilon}_{L}^{*}(p_{3})\cdot(\not{p}_{1}-\not{p}_{4})\cdot\not{\varepsilon}_{L}^{*}(p_{4})\cdot
u_{i}\right]$ $\displaystyle=s\sin{\theta}\cos^{2}{\frac{\theta}{2}},$ (72d)

while, in the high-energy limit, the Mandelstam variables $t$ and $u$ take the
form:

$\displaystyle t\equiv(p_{1}-p_{3})^{2}$
$\displaystyle\approx-s\sin^{2}{\frac{\theta}{2}},$ $\displaystyle
u\equiv(p_{1}-p_{4})^{2}$ $\displaystyle\approx-s\cos^{2}{\frac{\theta}{2}}.$
(73)

Summing Eqs. (12a) and (12b) and using Eqs. (72) and (73) we arrive at the
following formula for the total amplitude of the RR and LL processes:

$\displaystyle\mathcal{M}_{\textup{RR}}(ij\to ab)$ $\displaystyle=$
$\displaystyle-g^{2}\sin\theta\left[\left(T_{a}T_{b}\right)_{ji}\tan^{2}\frac{\theta}{2}+\left(T_{b}T_{a}\right)_{ji}\right],$
(74a) $\displaystyle\mathcal{M}_{\textup{LL}}(ij\to ab)$ $\displaystyle=$
$\displaystyle
g^{2}\sin\theta\left[\left(T_{a}T_{b}\right)_{ji}+\left(T_{b}T_{a}\right)_{ji}\cot^{2}\frac{\theta}{2}\right].$
(74b)

where the group generators should be evaluated at the appropriate initial
state $\psi_{i}\overline{\psi}_{j}$, as the notation implies.

###  A.2 Scalars

The first step in simplifying Eqs. (18a) and (18b) is to calculate the momenta
four-vectors coupling to the polarization vectors explicitly:

$\displaystyle(2p_{2}-p_{4})^{\mu}$
$\displaystyle=\frac{\sqrt{s}}{2}(1,-2\sin\theta,0,1-2\cos\theta),$ (75a)
$\displaystyle(2p_{1}-p_{3})^{\mu}$
$\displaystyle=\frac{\sqrt{s}}{2}(1,2\sin\theta,0,-1+2\cos\theta),$ (75b)
$\displaystyle(2p_{2}-p_{3})^{\mu}$
$\displaystyle=\frac{\sqrt{s}}{2}(1,-2\sin\theta,0,-1-2\cos\theta),$ (75c)
$\displaystyle(2p_{1}-p_{4})^{\mu}$
$\displaystyle=\frac{\sqrt{s}}{2}(1,2\sin\theta,0,1+2\cos\theta).$ (75d)

To perform the contractions with the polarization vectors, we need to consider
the RR and LL scenarios separately:

$\displaystyle(2p_{2}-p_{4})\cdot\varepsilon_{R}^{*}(p_{4})$
$\displaystyle=-\sqrt{\frac{s}{2}}\sin\theta,$
$\displaystyle(2p_{1}-p_{3})\cdot\varepsilon_{R}^{*}(p_{3})$
$\displaystyle=-\sqrt{\frac{s}{2}}\sin\theta,$ (76a)
$\displaystyle(2p_{2}-p_{4})\cdot\varepsilon_{L}^{*}(p_{4})$
$\displaystyle=-\sqrt{\frac{s}{2}}\sin\theta,$
$\displaystyle(2p_{1}-p_{3})\cdot\varepsilon_{L}^{*}(p_{3})$
$\displaystyle=-\sqrt{\frac{s}{2}}\sin\theta,$ (76b)
$\displaystyle(2p_{2}-p_{3})\cdot\varepsilon_{R}^{*}(p_{3})$
$\displaystyle=\sqrt{\frac{s}{2}}\sin\theta,$
$\displaystyle(2p_{1}-p_{4})\cdot\varepsilon_{R}^{*}(p_{4})$
$\displaystyle=\sqrt{\frac{s}{2}}\sin\theta,$ (76c)
$\displaystyle(2p_{2}-p_{3})\cdot\varepsilon_{L}^{*}(p_{3})$
$\displaystyle=\sqrt{\frac{s}{2}}\sin\theta,$
$\displaystyle(2p_{1}-p_{4})\cdot\varepsilon_{L}^{*}(p_{4})$
$\displaystyle=\sqrt{\frac{s}{2}}\sin\theta.$ (76d)

Summing Eqs. (18a) and (18b) and using Eqs. (73) and (76) we arrive at the
following formula for the total scattering amplitude of the RR and LL
processes:

$\displaystyle\mathcal{M}_{\textup{RR}}(ij\to ab)$
$\displaystyle=-2g^{2}\left[\left(T_{a}T_{b}\right)_{ji}\sin^{2}\frac{\theta}{2}+\left(T_{b}T_{a}\right)_{ji}\cos^{2}\frac{\theta}{2}\right],$
(77a) $\displaystyle\mathcal{M}_{\textup{LL}}(ij\to ab)$
$\displaystyle=-2g^{2}\left[\left(T_{a}T_{b}\right)_{ji}\sin^{2}\frac{\theta}{2}+\left(T_{b}T_{a}\right)_{ji}\cos^{2}\frac{\theta}{2}\right].$
(77b)

##  Appendix B Instructive example of the largest eigenvalue of $a_{0}$

We consider a fermionic $\mathbf{2}_{\psi}$ transforming in the doublet
representation of $SU(2)$. One then has $T_{a}=\tau_{a}/2$, where the
$\tau_{a}$ are the Pauli matrices. Hence,

$\displaystyle 4T_{1}T_{1}=4T_{2}T_{2}=4T_{3}T_{3}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}1&0\\\ 0&1\end{array}\right),$ (78c)
$\displaystyle 3T_{1}T_{2}+T_{2}T_{1}=-\left(3T_{2}T_{1}+T_{1}T_{2}\right)$
$\displaystyle=$ $\displaystyle\frac{1}{2}\left(\begin{array}[]{cc}i&0\\\
0&-i\end{array}\right),$ (78f) $\displaystyle
3T_{1}T_{3}+T_{3}T_{1}=-\left(3T_{3}T_{1}+T_{1}T_{3}\right)$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(\begin{array}[]{cc}0&-1\\\
1&0\end{array}\right),$ (78i) $\displaystyle
3T_{2}T_{3}+T_{3}T_{2}=-\left(3T_{3}T_{2}+T_{2}T_{3}\right)$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(\begin{array}[]{cc}0&i\\\
i&0\end{array}\right).$ (78l)

We define the matrix of zeroth partial-waves in the basis

$\left\\{\left[VV\right]_{\textup{RR}},\,\left[VV\right]_{\textup{LL}},\,\left[\mathbf{2}_{\psi}\mathbf{2}_{\psi}\right]\right\\},$
(79)

where

$\displaystyle\left[VV\right]_{\textup{RR}}$ $\displaystyle=$
$\displaystyle\frac{\left(V^{1}V^{1}\right)_{\textup{RR}}}{\sqrt{2}},\,\left(V^{1}V^{2}\right)_{\textup{RR}},\,\left(V^{1}V^{3}\right)_{\textup{RR}},\,\frac{\left(V^{2}V^{2}\right)_{\textup{RR}}}{\sqrt{2}},\,\left(V^{2}V^{3}\right)_{\textup{RR}},\,\frac{\left(V^{3}V^{3}\right)_{\textup{RR}}}{\sqrt{2}},\quad\quad$
(80a) $\displaystyle\left[VV\right]_{\textup{LL}}$ $\displaystyle=$
$\displaystyle\frac{\left(V^{1}V^{1}\right)_{\textup{LL}}}{\sqrt{2}},\,\left(V^{1}V^{2}\right)_{\textup{LL}},\,\left(V^{1}V^{3}\right)_{\textup{LL}},\,\frac{\left(V^{2}V^{2}\right)_{\textup{LL}}}{\sqrt{2}},\,\left(V^{2}V^{3}\right)_{\textup{LL}},\,\frac{\left(V^{3}V^{3}\right)_{\textup{LL}}}{\sqrt{2}},\quad\quad$
(80b) $\displaystyle\left[\mathbf{2}_{\psi}\mathbf{2}_{\psi}\right]$
$\displaystyle=$
$\displaystyle\psi_{1}\overline{\psi}_{1},\,\psi_{2}\overline{\psi}_{2},\,\psi_{1}\overline{\psi}_{2},\,\psi_{2}\overline{\psi}_{1}.$
(80c)

The coupled-channel matrix of zeroth partial-waves then reads

$a_{0}^{\psi}=\begin{pmatrix}\textbf{0}_{12\times 12}&M\\\
M^{T}&\textbf{0}_{4\times 4}\end{pmatrix},$ (81)

where $M$ is the following $12\times 4$ matrix:

$\displaystyle M$
$\displaystyle=\frac{g^{2}}{128}\begin{pmatrix}-\sqrt{2}&-\sqrt{2}&0&0\\\\[-4.0pt]
-i&i&0&0\\\\[-4.0pt] 0&0&-1&1\\\\[-4.0pt] -\sqrt{2}&-\sqrt{2}&0&0\\\\[-4.0pt]
0&0&-i&-i\\\\[-4.0pt] -\sqrt{2}&-\sqrt{2}&0&0\\\\[-4.0pt]
\sqrt{2}&\sqrt{2}&0&0\\\\[-4.0pt] -i&i&0&0\\\\[-4.0pt] 0&0&-1&1\\\\[-4.0pt]
\sqrt{2}&\sqrt{2}&0&0\\\\[-4.0pt] 0&0&-i&-i\\\\[-4.0pt]
\sqrt{2}&\sqrt{2}&0&0\\\\[-4.0pt] \end{pmatrix}.$ (82)

The $16\times 16$ matrix $a_{0}^{\psi}$ of Eq. (81) has eigenvalues

$\pm\frac{g^{2}\sqrt{3}}{32\sqrt{2}},\ \pm\frac{g^{2}}{64},\
\pm\frac{ig^{2}}{64},\ 0.$ (83)

The eigenvalue with the largest modulus is

$\left(a_{0}^{\psi}\right)^{\mathrm{max}}=\frac{g^{2}\sqrt{3}}{32\sqrt{2}},$
(84)

and the corresponding (transposed) eigenvector is

$\left(-1,\,0,\,0,\,-1,\,0,\,-1,\,1,\,0,\,0,\,1,\,0,\,1,\,\sqrt{3},\,\sqrt{3},\,0,\,0\right)^{T}.$
(85)

This means that the largest zeroth partial-waves arise from the processes

$\displaystyle\left[\left(V^{1}V^{1}\right)_{\textup{RR}}+\left(V^{2}V^{2}\right)_{\textup{RR}}+\left(V^{3}V^{3}\right)_{\textup{RR}}\right]$
$\displaystyle\to$
$\displaystyle\left(\psi_{1}\overline{\psi}_{1}+\psi_{2}\overline{\psi}_{2}\right),$
(86a)
$\displaystyle\left[\left(V^{1}V^{1}\right)_{\textup{LL}}+\left(V^{2}V^{2}\right)_{\textup{LL}}+\left(V^{3}V^{3}\right)_{\textup{LL}}\right]$
$\displaystyle\to$
$\displaystyle\left(\psi_{1}\overline{\psi}_{1}+\psi_{2}\overline{\psi}_{2}\right).$
(86b)

We have found this feature to be a general property of any representation of
any symmetry group.

##  Appendix C Group Representations and Casimir Invariants

###  C.1 $U(1)$ Gauge Group

The dimension of the $U(1)$ adjoint representation is $\tilde{n}_{U(1)}=1$.
All $U(1)$ representations are one-dimensional, and given by a charge $Y$.
Table 5 shows these properties alongside the corresponding quadratic Casimir
invariant ($C(Y)$).

Charge | Dimension | Casimir ($C(Y)$)
---|---|---
$Y\in\mathbb{R}$ | 1 | $Y^{2}$

Table 5: Irreducible representations and their Casimir invariants of $U(1)$.

irrep | Dynkin | Dimension | Casimir
---|---|---|---
(n) | Index | ($n$) | ($C(\textbf{{n}})$)
$\textbf{{n}}\in\mathbb{N}$ | $\\{n-1\\}$ | $n$ | $\frac{1}{4}(n^{2}-1)$

Table 6: Irreducible representations and their Casimir invariants of $SU(2)$.

###  C.2 $SU(2)$ Gauge Group

The dimension of the $SU(2)$ adjoint representation is $\tilde{n}_{SU(2)}=3$.
Table 6 shows the $SU(2)$ irreducible representations (n), alongside their
Dynkin indices, dimensions ($n$), and quadratic Casimir invariants
($C(\textbf{{n}})$).

###  C.3 $SU(3)$ Gauge Group

The dimension of the adjoint representation is $\tilde{n}_{SU(3)}=8$. Table 7
shows the lowest-dimensional $SU(3)$ irreducible representations (n),
alongside their Dynkin indices, dimensions ($n$), and quadratic Casimir
invariants ($C(\textbf{{n}})$). For a more intuitive understanding, in Figure
1 we plot the quadratic Casimir invariant as a function of the dimension of
the irreducible representations found in Table 7. The primed marker
$\bullet\textbf{'}$ represents a primed representation n’.

irrep | Dynkin | Dimension | Casimir
---|---|---|---
(n) | Index | ($n$) | ($C(\textbf{{n}})$)
1 | $\\{0,0\\}$ | 1 | 0
3 | $\\{1,0\\}$ | 3 | 4/3
6 | $\\{0,2\\}$ | 6 | 10/3
8 | $\\{1,1\\}$ | 8 | 3
10 | $\\{3,0\\}$ | 10 | 6
15 | $\\{2,1\\}$ | 15 | 16/3
15’ | $\\{4,0\\}$ | 15 | 28/3
21 | $\\{0,5\\}$ | 21 | 40/3
24 | $\\{1,3\\}$ | 24 | 25/3

Table 7: Lowest-dimensional $SU(3)$ irreducible representations and their
Casimir invariants.

Figure 1: $SU(3)$ Casimir invariant as a function of the dimension of the
irreducible representation.

###  C.4 $SU(5)$ Gauge Group

The dimension of the adjoint representation is $\tilde{n}_{SU(5)}=24$. Table 8
shows the lowest-dimensional $SU(5)$ irreducible representations (n),
alongside their Dynkin indices, dimensions ($n$), and quadratic Casimir
invariants ($C(\textbf{{n}})$). In Figure 2 we plot the quadratic Casimir
invariant as a function of the dimension of irreducible representations found
in Table 8. The primed (double primed) marker $\bullet\textbf{'}$
($\bullet\textbf{''}$) represents a primed (double primed) representation n’
(n”).

irrep | Dynkin | Dimension | Casimir
---|---|---|---
(n) | Index | ($n$) | ($C(\textbf{{n}})$)
1 | $\\{0,0,0,0\\}$ | 1 | 0
5 | $\\{1,0,0,0\\}$ | 5 | 12/5
10 | $\\{0,1,0,0\\}$ | 10 | 18/5
15 | $\\{2,0,0,0\\}$ | 15 | 28/5
24 | $\\{1,0,0,1\\}$ | 24 | 5
35 | $\\{0,0,0,3\\}$ | 35 | 48/5
40 | $\\{0,0,1,1\\}$ | 40 | 33/5
45 | $\\{0,1,0,1\\}$ | 45 | 32/5
50 | $\\{0,0,2,0\\}$ | 50 | 42/5
70 | $\\{2,0,0,1\\}$ | 70 | 42/5
70’ | $\\{0,0,0,4\\}$ | 70 | 72/5
75 | $\\{0,1,1,0\\}$ | 75 | 8
105 | $\\{0,0,1,2\\}$ | 105 | 52/5
126 | $\\{2,0,1,0\\}$ | 126 | 10
126’ | $\\{5,0,0,0\\}$ | 126 | 20
160 | $\\{3,0,0,1\\}$ | 160 | 63/5
175 | $\\{1,1,0,1\\}$ | 175 | 48/5
175’ | $\\{1,2,0,0\\}$ | 175 | 12
175” | $\\{0,3,0,0\\}$ | 175 | 72/5

Table 8: Lowest-dimensional $SU(5)$ irreducible representations and their
Casimir invariants.

Figure 2: $SU(5)$ Casimir invariant as a function of the dimension of the
irreducible representation.

###  C.5 $SO(10)$ Gauge Group

The dimension of the adjoint representation is $\tilde{n}_{SO(10)}=45$. Table
9 shows the lowest-dimensional $SO(10)$ irreducible representations (n),
alongside their Dynkin indices, dimensions ($n$), and quadratic Casimir
invariants ($C(\textbf{{n}})$). In Figure 3 we plot the quadratic Casimir
invariant as a function of the dimension of irreducible representations found
in Table 9. The primed marker $\bullet\textbf{'}$ represents a primed
representation n’.

irrep | Dynkin | Dimension | Casimir
---|---|---|---
(n) | Index | ($n$) | ($C(\textbf{{n}})$)
1 | $\\{0,0,0,0,0\\}$ | 1 | 0
10 | $\\{1,0,0,0,0\\}$ | 10 | 9/2
16 | $\\{0,0,0,0,1\\}$ | 16 | 45/8
45 | $\\{0,1,0,0,0\\}$ | 45 | 8
54 | $\\{2,0,0,0,0\\}$ | 54 | 10
120 | $\\{0,0,1,0,0\\}$ | 120 | 21/2
126 | $\\{0,0,0,2,0\\}$ | 126 | 25/2
144 | $\\{1,0,0,1,0\\}$ | 144 | 85/8
210 | $\\{0,0,0,1,1\\}$ | 210 | 12
210’ | $\\{3,0,0,0,0\\}$ | 210 | 33/2

Table 9: Lowest-dimensional $SO(10)$ irreducible representations and their
Casimir invariants.

Figure 3: $SO(10)$ Casimir invariant as a function of the dimension of the
irreducible representation.

###  C.6 $E_{6}$ Gauge Group

The dimension of the adjoint representation is $\tilde{n}_{E_{6}}=78$. Table
10 shows the lowest-dimensional $E_{6}$ irreducible representations (n),
alongside their Dynkin indices, dimensions ($n$), and quadratic Casimir
invariants ($C(\textbf{{n}})$). In Figure 4 we plot the quadratic Casimir
invariant as a function of the dimension of irreducible representations found
in Table 10. The primed marker $\bullet\textbf{'}$ represents a primed
representation n’.

irrep | Dynkin | Dimension | Casimir
---|---|---|---
(n) | Index | ($n$) | ($C(\textbf{{n}})$)
1 | $\\{0,0,0,0,0,0\\}$ | 1 | 0
27 | $\\{1,0,0,0,0,0\\}$ | 27 | 26/3
78 | $\\{0,0,0,0,0,1\\}$ | 78 | 12
351 | $\\{0,0,0,1,0,0\\}$ | 351 | 50/3
351’ | $\\{0,0,0,0,2,0\\}$ | 351 | 56/3
650 | $\\{1,0,0,0,1,0\\}$ | 650 | 18

Table 10: Lowest-dimensional $E_{6}$ irreducible representations and their
Casimir invariants.

Figure 4: $E_{6}$ Casimir invariant as a function of the dimension of the
irreducible representation.

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