# Yamabe Solitons on Conformal Almost Contact Complex Riemannian Manifolds
with Vertical Torse-Forming Vector Field

Mancho Manev  University of Plovdiv Paisii Hilendarski, Faculty of Mathematics
and Informatics, Department of Algebra and Geometry, 24 Tzar Asen St., Plovdiv
4000, Bulgaria & Medical University – Plovdiv, Faculty of Pharmacy, Department
of Medical Physics and Biophysics, 15A Vasil Aprilov Blvd, Plovdiv 4002,
Bulgaria<EMAIL_ADDRESS>
###### Abstract.

A Yamabe soliton is considered on an almost contact complex Riemannian
manifold (also known as an almost contact B-metric manifold) which is obtained
by a contact conformal transformation of the Reeb vector field, its dual
contact 1-form, the B-metric, and its associated B-metric. The case when the
potential is a torse-forming vector field of constant length on the vertical
distribution determined by the Reeb vector field is studied. In this way,
manifolds from one of the main classes of the studied manifolds are obtained.
The same class contains the conformally equivalent manifolds of cosymplectic
manifolds by the usual conformal transformation of the given B-metric. An
explicit 5-dimensional example of a Lie group is given, which is characterized
in relation to the obtained results.

###### Key words and phrases:

Yamabe soliton; almost contact B-metric manifold; almost contact complex
Riemannian manifold; torse-forming vector field; contact conformal
transformations.

###### 2010 Mathematics Subject Classification:

Primary 53C25, 53D15, 53C50; Secondary 53E50 53D35, 70G45

##  1\. Introduction

The concept of Yamabe flow was first introduced by R. Hamilton in [10] to
construct Yamabe metrics on compact Riemannian manifolds. On a Riemannian or
pseudo-Riemannian manifold, a time-dependent metric $g(t)$ is said to evolve
by the Yamabe flow if $g$ satisfies the equation

$\frac{\partial}{\partial t}g=-\tau g,$

where $\tau(t)$ is the scalar curvature of $g(t)$. According to mathematical
physics, the Yamabe flow corresponds to the case of fast diffusion of the
porous medium equation [4].

A self-similar solution of the Yamabe flow is called a Yamabe soliton and is
determined by ([1])

$\frac{1}{2}\mathcal{L}_{v}g=(\tau-\sigma)g,$

where $\mathcal{L}_{v}g$ is the Lie derivative of $g$ along a vector field $v$
and $\sigma$ is a constant. Various authors have studied Yamabe solitons on
manifolds equipped with different types of structures (see e.g. [2, 3, 5, 9,
11, 21]).

In [16], we began the study of Yamabe solitons on _almost contact B-metric
manifolds_. These manifolds are also known as _almost contact complex
Riemannian manifolds_ , abbreviated as _accR manifolds_. The geometry of these
manifolds is largely determined by the presence of a pair of B-metrics that
are related each other by the almost contact structure.

Recall that the conformal class of the metric is preserved by the Yamabe flow.
With this reason in mind, we study Yamabe solitons and conformal
transformations together in the case under consideration.

The so-called contactly conformal transformations were studied in [15]. They
deformed not only the metric but also the Reeb vector field and its associated
contact 1-form using the pair of B-metrics. The partial case when this
transformation changes only the B-metrics is studied in [17].

The Ganchev–Mihova–Gribachev classification of the studied manifolds given in
[8] consists of eleven basic classes. We call four of them the main classes,
since the covariant derivatives of the structure tensors with respect to the
Levi-Civita connection of each of the B-metrics are expressed only by the pair
of B-metrics and the corresponding traces. Let us recall [17], where it is
proved that the direct sum of the four main classes is closed under the action
of contactly conformal transformations.

The main results in the present paper concern two of the four main classes,
namely $\mathcal{F}_{1}$ and $\mathcal{F}_{5}$, as well as their intersection
$\mathcal{F}_{0}$. The first of them is defined in [8] as an analogue of the
unique main class $\mathcal{W}_{1}$ of almost complex manifolds with Norden
metric according to the Ganchev–Borisov classification in [7]. Furthermore, it
is known from [17] that any $\mathcal{F}_{0}$-manifold can be transformed into
an $\mathcal{F}_{1}$-manifold by a usual conformal transformation of the
B-metric. An example of an $\mathcal{F}_{5}$-manifold as an isotropic
hypersurface with respect to the associated B-metric in an even-dimensional
real space is given in [8] and it is noted that the class $\mathcal{F}_{5}$ is
analogous in some sense to the class of $\alpha$-Sasakian manifolds (we can
add—to the class of $\beta$-Kenmotsu manifolds) in the theory of almost
contact metric manifolds. Results on the geometric properties of manifolds
from these main classes, as well as examples of them, can be found e.g. in
[13]–[19].

In [16] we introduced Yamabe solitons on accR manifolds and started their
study for two of the simplest types of these manifolds, namely cosymplectic
and Sasaki-like. We found that the resulting manifolds in both cases belong to
one of the main classes, the only one that contains the conformally equivalent
manifolds of the cosymplectic ones by the usual conformal transformations.

The aim of this paper is to study another possibility for the initial
manifold, which is determined by a natural condition, also intensively studied
in relation with solitons, namely the use of a torse-forming vector field.

The structure of the present paper is as follows. Section 1 is the present
introduction. Section 2 recalls the basic facts for the investigated
manifolds, the relevant transformations of the structure tensors on them, and
the notion of the Yamabe soliton on a transformed accR manifold. Section 3 is
devoted to the investigation of the case described in the title of the paper.
Section 4 supports the main theorem by providing an explicit example of a
hypersurface as a manifold equipped with the investigated structures and
having arbitrary dimension.

##  2\. Preliminaries

###  2.1. Almost contact complex Riemannian manifolds

Let
$(\mathcal{M},\allowbreak{}\varphi,\allowbreak{}\xi,\allowbreak{}\eta,\allowbreak{}{}g)$
be an almost contact manifold with B-metric, also known as an _almost contact
complex Riemannian manifold_ (abbreviated accR manifold) (see e.g. [8, 12]).
This means that $\mathcal{M}$ is a $(2n+1)$-dimensional differentiable
manifold with an almost contact structure $(\varphi,\xi,\eta)$ and a pseudo-
Riemannian metric $g$ of signature $(n+1,n)$ such that [8]

(1)
$\displaystyle\varphi\xi=0,\quad\varphi^{2}=-\iota+\eta\otimes\xi,\quad\eta\circ\varphi=0,\quad\eta(\xi)=1,$
(2) $\displaystyle g(X,Y)=-g(\varphi X,\varphi Y)+\eta(X)\eta(Y)$

for arbitrary $X$, $Y$ of the algebra $\mathfrak{X}(\mathcal{M})$ on the
smooth vector fields on $\mathcal{M}$, where $\iota$ denotes the identity on
$\mathfrak{X}(\mathcal{M})$.

From now on, we denote by $X$, $Y$, $Z$ arbitrary elements of
$\mathfrak{X}(\mathcal{M})$ or vectors in the tangent space $T_{p}\mathcal{M}$
of $\mathcal{M}$ at an arbitrary point $p$ in $\mathcal{M}$.

The given B-metric has its associated B-metric $\tilde{g}$ on
$(\mathcal{M},\allowbreak{}\varphi,\allowbreak{}\xi,\allowbreak{}\eta,\allowbreak{}{}g)$
defined by

$\tilde{g}(X,Y)=g(X,\varphi Y)\allowbreak+\eta(X)\eta(Y).$

A classification of the accR manifolds is given in [8] by Ganchev, Mihova and
Gribachev. This classification includes eleven basic classes
$\mathcal{F}_{1}$, $\mathcal{F}_{2}$, $\dots$, $\mathcal{F}_{11}$. Their
intersection is the special class $\mathcal{F}_{0}$ defined by condition for
the vanishing of $F$. Hence $\mathcal{F}_{0}$ is the class of cosymplectic
accR manifolds, where the structures $\varphi$, $\xi$, $\eta$, $g$,
$\tilde{g}$ are $\nabla$-parallel.

This classification is made in terms of the tensor $F$ of type (0,3) defined
on
$(\mathcal{M},\allowbreak{}\varphi,\allowbreak{}\xi,\allowbreak{}\eta,\allowbreak{}{}g)$
by

$F(X,Y,Z)=g\bigl{(}\left(\nabla_{X}\varphi\right)Y,Z\bigr{)},$

where $\nabla$ is the Levi-Civita connection of $g$. The following properties
of $F$ are consequences of (1) and (2):

$F(X,Y,Z)=F(X,Z,Y)=F(X,\varphi Y,\varphi
Z)+\eta(Y)F(X,\xi,Z)+\eta(Z)F(X,Y,\xi)$

and relations of $F$ with $\nabla\xi$ and $\nabla\eta$ are:

(3) $\left(\nabla_{X}\eta\right)Y=g\left(\nabla_{X}\xi,Y\right)=F(X,\varphi
Y,\xi).$

The following 1-forms are associated with $F$ and known as Lee forms of the
manifold:

(4) $\theta=g^{ij}F(E_{i},E_{j},\cdot),\qquad\theta^{*}=g^{ij}F(E_{i},\varphi
E_{j},\cdot),\qquad\omega=F(\xi,\xi,\cdot),$

where $g^{ij}$ are the contravariant components of $g$ with respect to a basis
$\left\\{E_{i};\xi\right\\}$ $(i=1,2,\dots,2n)$ of $T_{p}\mathcal{M}$. These
Lee forms satisfy the following general identities [17]

(5) $\theta^{*}\circ\varphi=-\theta\circ\varphi^{2},\qquad\omega(\xi)=0.$

###  2.2. Conformal transformations of the accR structure

The author and K. Gribachev introduced the so-called _contactly conformal
transformation of the B-metric_ in [17], using the pair of B-metrics $g$ and
$\tilde{g}$, as well as $\eta\otimes\eta$. This transformation deforms $g$
into a new B-metric $\bar{g}$ for
$(\mathcal{M},\allowbreak{}\varphi,\allowbreak{}\xi,\allowbreak{}\eta,\allowbreak{}{}g)$.
This transformation is later generalized in [15] to a _contact conformal
transformation of the accR structure_ , which yields a new almost contact
structure with B-metric $(\varphi,\bar{\xi},\bar{\eta},\bar{g})$ as follows

(6)
$\begin{array}[]{l}\bar{\xi}=e^{-w}\xi,\qquad\bar{\eta}=e^{w}\eta,\\\\[4.0pt]
\bar{g}=e^{2u}\cos{2v}\,g+e^{2u}\sin{2v}\,\tilde{g}+\left(e^{2w}-e^{2u}\cos{2v}-e^{2u}\sin{2v}\right)\eta\otimes\eta,\end{array}$

where $u,v,w$ are differentiable functions on $\mathcal{M}$. Hereafter, we
call a transformation of this type an _accR transformation_ for short.

Expression of the corresponding tensor $\bar{F}$ by $F$ under accP
transformation is given in [15], see also [19, (22)]. Moreover, the relations
between the Lee forms of the manifolds
$(\mathcal{M},\allowbreak{}\varphi,\allowbreak{}\xi,\allowbreak{}\eta,\allowbreak{}{}g)$
and $(\mathcal{M},\varphi,\bar{\xi},\bar{\eta},\bar{g})$ are the following

(7)
$\bar{\theta}=\theta+2n\,\alpha,\qquad\bar{\theta}^{*}=\theta^{*}+2n\,\beta,\qquad\bar{\omega}=\omega+\mathrm{d}w\circ\varphi,$

where we use the following notations for brevity

(8)
$\alpha=\mathrm{d}u\circ\varphi+\mathrm{d}v,\qquad\beta=\mathrm{d}u-\mathrm{d}v\circ\varphi.$

The following mutually equivalent properties follow immediately from (8)

(9)
$\alpha\circ\varphi^{2}+\beta\circ\varphi=\alpha\circ\varphi-\beta\circ\varphi^{2}=0.$

It is well known the following expression of the Lie derivative in terms of
the covariant derivative with respect to the Levi-Civita connection
$\bar{\nabla}$ of $\bar{g}$

(10)
$\left(\mathcal{L}_{\bar{\xi}}\bar{g}\right)(X,Y)=\bar{g}\left(\bar{\nabla}_{X}\bar{\xi},Y\right)+\bar{g}\left(X,\bar{\nabla}_{Y}\bar{\xi}\right).$

##  3\. Main results

In [16], it is said that the contact transformed B-metric $\bar{g}$ generates
a Yamabe soliton with potential the Reeb vector field $\bar{\xi}$ and soliton
constant $\sigma$ on a conformal accR manifold
$(\mathcal{M},\varphi,\bar{\xi},\bar{\eta},\bar{g})$, if the following
condition is satisfied

(11) $\frac{1}{2}\mathcal{L}_{\bar{\xi}}\bar{g}=(\bar{\tau}-\sigma)\bar{g},$

where $\bar{\tau}$ is the scalar curvature of $\bar{g}$.

A vector field $\vartheta$ on a (pseudo-)Riemannian manifold $(\mathcal{M},g)$
is called _torse-forming vector field_ if it satisfies the following condition
for arbitrary vector field $x\in\mathfrak{X}(\mathcal{M})$

(12) $\nabla_{x}\vartheta=f\,x+\gamma(x)\vartheta,$

where $f$ is a differentiable function and $\gamma$ is a 1-form [22, 23]. The
1-form $\gamma$ is called the _generating form_ and the function $f$ is called
the _conformal scalar_ of $\vartheta$ [20].

Further, we consider a torse-forming vector field $\vartheta$ on an accR
manifold
$(\mathcal{M},\allowbreak{}\varphi,\allowbreak{}\xi,\allowbreak{}\eta,\allowbreak{}{}g)$,
i. e. (12) is valid.

In addition, those vector fields that have a special arrangement regarding the
structure under consideration are naturally distinguished. The almost contact
structure on $\mathcal{M}$ gives rise to two mutually orthogonal distributions
with respect to the B-metric $g$, namely the contact (or horizontal)
distribution $\mathcal{H}=\ker(\eta)=\operatorname{im}(\varphi)$ and the
vertical distribution
$\mathcal{H}^{\bot}=\operatorname{span}(\xi)=\ker(\varphi)$.

For these reasons, we study the case where the torse-forming vector field
$\vartheta$ is vertical, i. e. $\vartheta\in
H^{\bot}=\operatorname{span}(\xi)$. Therefore, $\vartheta$ is collinear to
$\xi$, i. e. the following equality holds

(13) $\vartheta=k\,\xi,$

where $k$ is a nonzero function on $\mathcal{M}$ and obviously
$k=\eta(\vartheta)$ holds true.

By virtue of (12) and (13), we obtain

(14) $\mathrm{d}k(x)\,\xi+k\nabla_{x}\xi=fx+k\gamma(x)\,\xi,$

which after applying $\eta$ gives

(15) $\mathrm{d}k(x)=f\eta(x)+k\gamma(x).$

Thus, (1), (14) and (15) imply

$\nabla_{x}\xi=-\frac{f}{k}\,\varphi^{2}x.$

The last formula due to (3) gives the following

$\left(\nabla_{x}\eta\right)(y)=-\frac{f}{k}\,g(\varphi x,\varphi y),$ (16)
$F(x,y,\xi)=-\frac{f}{k}\,g(x,\varphi y).$

Equality (16) shows that the following basic properties for the operation of
the structure on the considered manifold are fulfilled

(17) $F(x,y,\xi)=F(y,x,\xi)=-F(\varphi x,\varphi
y,\xi)=F(\varphi^{2}x,\varphi^{2}y,\xi).$

Using (4), we obtain the following results for the Lee forms

(18) $\theta(\xi)=0,\qquad\theta^{*}(\xi)=2n\,\frac{f}{k},\qquad\omega=0.$

Then, by virtue of (1) and (18), the Lee forms of the accR manifold with a
vertical torse-forming vector field $\vartheta$ satisfy the following

(19)
$\theta=-\theta\circ\varphi^{2},\qquad\theta^{*}=-\theta^{*}\circ\varphi^{2}+2n\,\frac{f}{k}\eta,\qquad\omega=0.$

In [13], it is proved that
$(\mathcal{M},\allowbreak{}\varphi,\allowbreak{}\xi,\allowbreak{}\eta,\allowbreak{}{}g)$
belongs to a certain class $\mathcal{F}_{i}$ $(i=1,\dots,11)$ from the
Ganchev–Mihova–Gribachev classification if and only if $F$ satisfies the
condition $F=F^{i}$, where the components $F^{i}$ of $F$. The components of
$F$ for the classes $\mathcal{F}_{1}$ and $\mathcal{F}_{5}$ mentioned in the
main result of this paper are as follows

(20) $\begin{split}&F^{1}(x,y,z)=\frac{1}{2n}\left\\{g(\varphi x,\varphi
y)\theta(\varphi^{2}z)+g(x,\varphi y)\theta(\varphi z)\right.\\\
&\phantom{F^{1}(x,y,z)=\frac{1}{2n}\left\\{\right.}\left.+g(\varphi x,\varphi
z)\theta(\varphi^{2}y)+g(x,\varphi z)\theta(\varphi y)\right\\},\\\\[4.0pt]
&F^{5}(x,y,z)=-\frac{\theta^{*}(\xi)}{2n}\Bigl{\\{}g(x,\varphi
y)\eta(z)+g(x,\varphi z)\eta(y)\Bigr{\\}}.\end{split}$

In addition, it is said that an accR manifold belongs to a direct sum of two
or more basic classes, i. e.
$(\mathcal{M},\allowbreak{}\varphi,\allowbreak{}\xi,\allowbreak{}\eta,\allowbreak{}{}g)\in\mathcal{F}_{i}\oplus\mathcal{F}_{j}\oplus\cdots$,
if and only if the fundamental tensor $F$ on
$(\mathcal{M},\allowbreak{}\varphi,\allowbreak{}\xi,\allowbreak{}\eta,\allowbreak{}{}g)$
is the sum of the corresponding components $F^{i}$, $F^{j}$, $\ldots$ of $F$,
i. e. the following condition is satisfied $F=F^{i}+F^{j}+\cdots$ for
$i,j\in\\{1,2,\dots,11\\}$, $i\neq j$.

Bearing in mind (16), (17) and the expressions for the components $F^{i}$ from
[13], we establish the vanishing of the components for
$i\in\\{4,6,7,8,9,11\\}$, which means that the common class of the studied
accR manifolds is
$\mathcal{F}_{1}\oplus\mathcal{F}_{2}\oplus\mathcal{F}_{3}\oplus\mathcal{F}_{5}\oplus\mathcal{F}_{10}$.
Moreover, among the basic classes, only $\mathcal{F}_{5}$ can contain such
manifolds. Note that $\mathcal{F}_{5}$-manifolds are counterparts of
$\beta$-Kenmotsu manifolds in the case of almost contact metric manifolds.

Let us consider
$(\mathcal{M},\allowbreak{}\varphi,\allowbreak{}\xi,\allowbreak{}\eta,\allowbreak{}{}g)$
with a vertical torse-forming vector field $\vartheta$ and an accR structure
such that it is an $\mathcal{F}_{5}$-manifold, i. e. $F=F^{5}$ from (20) is
valid. In this case, due to $\theta=\omega=0$,
$\theta^{*}=\theta^{*}(\xi)\eta$ and (19), we have

$\theta=0,\qquad\theta^{*}=2n\,\frac{f}{k}\eta,\qquad\omega=0.$

Furthermore, the expression of $\bar{F}$ through $F$ under accP transformation
given in [15] takes the form

(21) $\displaystyle\bar{F}(X,Y,Z)={}-e^{2u}$
$\displaystyle\Bigl{\\{}\lambda(Z)g(\varphi X,\varphi Y)+\lambda(Y)g(\varphi
X,\varphi Z)$ $\displaystyle+\mu(Z)g(X,\varphi Y)+\mu(Y)g(X,\varphi
Z)\Bigr{\\}}$ $\displaystyle{}+e^{2w}$
$\displaystyle\eta(X)\Bigl{\\{}\eta(Y)\mathrm{d}w(\varphi
Z)+\eta(Z)\mathrm{d}w(\varphi Y)\Bigr{\\}},$

where the 1-forms $\lambda$ and $\mu$ are expressed by $\alpha$ and $\beta$
introduced in (8) as follows

$\begin{split}\lambda(Z)&=\cos{2v}\,\alpha(Z)+\sin{2v}\,\left\\{\beta(Z)+\frac{f}{k}\eta(Z)\right\\},\\\
\mu(Z)&=\cos{2v}\,\left\\{\beta(Z)+\frac{f}{k}\eta(Z)\right\\}-\sin{2v}\,\alpha(Z).\end{split}$

From (21), bearing in mind (3) and the first two equalities of (6), we obtain

(22) $\displaystyle\bar{g}\left(\bar{\nabla}_{X}\bar{\xi},Y\right)=$
$\displaystyle\ e^{2u-w}\Bigl{\\{}\lambda(\xi)g(X,\varphi Y)-\mu(\xi)g(\varphi
X,\varphi Y)\Bigr{\\}}$ $\displaystyle+e^{w}\eta(X)\mathrm{d}w(\varphi^{2}Y).$

Using the last equality of (6), we have the following system of equalities

$\begin{array}[]{l}\bar{g}(\varphi\cdot,\varphi\cdot)=e^{2u}\cos{2v}\,g(\varphi\cdot,\varphi\cdot)-e^{2u}\sin{2v}\,g(\cdot,\varphi\cdot),\\\\[4.0pt]
\bar{g}(\cdot,\varphi\cdot)=e^{2u}\cos{2v}\,g(\cdot,\varphi\cdot)+e^{2u}\sin{2v}\,g(\varphi\cdot,\varphi\cdot),\end{array}$

which can be expressed vice versa in the form

(23)
$\begin{array}[]{l}g(\varphi\cdot,\varphi\cdot)=e^{-2u}\cos{2v}\,\bar{g}(\varphi\cdot,\varphi\cdot)+e^{-2u}\sin{2v}\,\bar{g}(\cdot,\varphi\cdot),\\\\[4.0pt]
g(\cdot,\varphi\cdot)=e^{-2u}\cos{2v}\,\bar{g}(\cdot,\varphi\cdot)-e^{-2u}\sin{2v}\,\bar{g}(\varphi\cdot,\varphi\cdot).\end{array}$

Applying (23) in (22), we obtain

(24) $\displaystyle\bar{g}\left(\bar{\nabla}_{X}\bar{\xi},Y\right)=$
$\displaystyle\ \alpha(\bar{\xi})\,\bar{g}(X,\varphi
Y)-\left\\{\beta(\bar{\xi})+\frac{f}{k}e^{-w}\right\\}\bar{g}(\varphi
X,\varphi Y)$ $\displaystyle+\bar{\eta}(X)\mathrm{d}w(\varphi^{2}Y).$

Note that according to (8), the following equalities are satisfied

(25)
$\alpha(\bar{\xi})=\mathrm{d}v(\bar{\xi}),\qquad\beta(\bar{\xi})=\mathrm{d}u(\bar{\xi}).$

Combining the expression in (24) with (8), (10) and (25) gives the following
formula

(26) $\displaystyle\left(\mathcal{L}_{\bar{\xi}}\bar{g}\right)(X,Y)=$
$\displaystyle\ 2\left\\{\mathrm{d}v(\bar{\xi})\bar{g}(X,\varphi
Y)-\left[\mathrm{d}u(\bar{\xi})+\frac{f}{k}e^{-w}\right]\bar{g}(\varphi
X,\varphi Y)\right\\}$
$\displaystyle+\bar{\eta}(X)\mathrm{d}w(\varphi^{2}Y)+\bar{\eta}(Y)\mathrm{d}w(\varphi^{2}X).$

######  Theorem 3.1.

An accR manifold of the main class $\mathcal{F}_{5}$ with a vertical torse-
forming vector field $\vartheta$ can be transformed by an accR transformation
so that the transformed B-metric is a Yamabe soliton with potential the
transformed Reeb vector field and a soliton constant $\sigma$ if and only if
the functions $(u,v,w)$ of the used transformation satisfy the conditions

(27) $\displaystyle\mathrm{d}u(\xi)=-\frac{f}{k},\qquad\mathrm{d}v(\xi)=0,$
(28) $\displaystyle\mathrm{d}w=\mathrm{d}w(\xi)\eta.$

Moreover, the obtained Yamabe soliton has a constant scalar curvature with
value $\bar{\tau}=\sigma$ and the obtained accR manifold belongs to the
subclass of the main class $\mathcal{F}_{1}$ determined by the conditions:

(29)
$\bar{\theta}=2n\left\\{\mathrm{d}u\circ\varphi+\mathrm{d}v\right\\},\qquad\bar{\theta}^{*}=-2n\left\\{\mathrm{d}u\circ\varphi^{2}+\mathrm{d}v\circ\varphi\right\\},\qquad\bar{\omega}=0.$

As a corollary, if $(u,v)$ are a pair of $\varphi$-holomorphic functions then
the transformed manifold belongs to the special class $\mathcal{F}_{0}$ of
cosymplectic accR manifolds.

###### Proof.

The condition that $\bar{g}$ generates a Yamabe soliton with potential
$\bar{\xi}$ on a conformal accR manifold
$(\mathcal{M},\varphi,\bar{\xi},\bar{\eta},\bar{g})$ means that (11) is
satisfied. In this case, (26) takes the form

(30) $\displaystyle 2\left(\bar{\tau}-\sigma\right)\left\\{-\bar{g}(\varphi
X,\varphi Y)+\bar{\eta}(X)\bar{\eta}(Y)\right\\}$
$\displaystyle=2\left\\{\mathrm{d}v(\bar{\xi})\bar{g}(X,\varphi
Y)-\left[\mathrm{d}u(\bar{\xi})+\frac{f}{k}e^{-w}\right]\bar{g}(\varphi
X,\varphi Y)\right\\}$
$\displaystyle\phantom{=}+\bar{\eta}(X)\mathrm{d}w(\varphi^{2}Y)+\bar{\eta}(Y)\mathrm{d}w(\varphi^{2}X).$

The substitution $\bar{\xi}$ for $Y$ in (30) gives

(31) $2\left(\bar{\tau}-\sigma\right)\bar{\eta}(X)=\mathrm{d}w(\varphi^{2}X).$

Then we replace $X$ with $\bar{\xi}$ and get an expression of the scalar
curvature for $\bar{g}$ as follows

(32) $\bar{\tau}=\sigma.$

Hence, (31) implies the vanishing of $\mathrm{d}w\circ\varphi^{2}$, which is
equivalent to the condition in (28).

After applying (32) and (28) to (30), we obtain (27).

Note that $\mathcal{L}_{\bar{\xi}}\bar{g}$ vanishes due to (32) and (11), i.
e. $\bar{\xi}$ is a Killing vector field in the considered case. For more
results concerning Killing vector fields on Riemannian manifolds, see [6].

As a next step, we substitute (23) in (21) and get

(33) $\displaystyle\bar{F}(X,Y,Z)={}$ $\displaystyle-\alpha(Z)\bar{g}(\varphi
X,\varphi Y)-\left\\{\beta(Z)+\frac{f}{k}\eta(Z)\right\\}\bar{g}(X,\varphi Y)$
$\displaystyle-\alpha(Y)\bar{g}(\varphi X,\varphi
Z)-\left\\{\beta(Y)+\frac{f}{k}\eta(Y)\right\\}\bar{g}(X,\varphi Z)$
$\displaystyle+\bar{\eta}(X)\left\\{\bar{\eta}(Y)\mathrm{d}{w}(\varphi
Z)+\bar{\eta}(Z)\mathrm{d}{w}(\varphi Y)\right\\}.$

Then, computing the Lee forms for $\bar{F}$ in (33) by (4), (8) and (9), we
obtain the following expressions

(34)
$\alpha=\frac{1}{2n}\bar{\theta},\qquad\beta=\frac{1}{2n}\bar{\theta}^{*}-\frac{f}{k}\eta,\qquad\mathrm{d}w\circ\varphi=\bar{\omega}.$

By virtue of (25), (27), (28) and (34), we obtain

$\bar{\theta}(\bar{\xi})=0,\qquad\bar{\theta}^{*}(\bar{\xi})=0,\qquad\bar{\omega}=0,$

where the first two equalities are equivalent to the following properties,
respectively

(35)
$\bar{\theta}=-\bar{\theta}\circ\varphi^{2},\qquad\bar{\theta}^{*}=-\bar{\theta}^{*}\circ\varphi^{2}.$

Thus, (8), (27), (28), (34) and (35) imply (29).

Substituting (28), (34) and (35) in (33), we get

(36) $\displaystyle\bar{F}(X,Y,Z)=\frac{1}{2n}\Bigl{\\{}$
$\displaystyle\bar{g}(\varphi X,\varphi
Y)\,\bar{\theta}(\varphi^{2}Z)+\bar{g}(X,\varphi
Y)\,\bar{\theta}^{*}(\varphi^{2}Z)$ $\displaystyle\\!+\bar{g}(\varphi
X,\varphi Z)\,\bar{\theta}(\varphi^{2}Y)+\bar{g}(X,\varphi
Z)\,\bar{\theta}^{*}(\varphi^{2}Y)\Bigr{\\}}.$

The resulting expression of $\bar{F}$ in (36) can be written in the form

(37) $\bar{F}=\bar{F}^{1},$

using the corresponding component of $\bar{F}$ in (20) and the relation in the
first equality of (5). The expression of $\bar{F}$ in (37) means that the
transformed manifold $(\mathcal{M},\varphi,\bar{\xi},\bar{\eta},\bar{g})$
belongs to the main class $\mathcal{F}_{1}$, according to the classification
of Ganchev–Mihova–Gribachev in [8].

The assumption that $(u,v)$ is a $\varphi$-holomorphic pair, i. e. $(u,v)$
satisfy the conditions

$\mathrm{d}u\circ\varphi=\mathrm{d}v\circ\varphi^{2},\qquad\mathrm{d}u\circ\varphi^{2}=-\mathrm{d}v\circ\varphi,$

and (29) yield the vanishing of all Lee forms of the obtained
$\mathcal{F}_{1}$-manifold, which means that it is in $\mathcal{F}_{0}$. ∎

As a result of (27) and (28), we derive the following conclusion. The
situation of Theorem 3.1 occurs when the functions $(u,v,w)$ of the used accR
transformation satisfy the conditions:

  * •

$v$ is a vertical constant, i. e. constant on $\mathcal{H}^{\bot}$;

  * •

$w$ is a horizontal constant, i. e. constant on $\mathcal{H}$.

##  4\. Example

We recall a known example of an $\mathcal{F}_{5}$-manifold given in [8] as
Example 3. The space
$\mathbb{R}^{2n+2}=\left\\{(x^{1},\dots,x^{n+1};x^{n+2},\dots,x^{2n+2})|\
x^{i}\in\mathbb{R}\right\\}$ is considered as a complex Riemannian manifold
with the canonical complex structure $J$ and the metric $G$ defined for any
$X=\lambda^{i}\frac{\partial}{\partial x^{i}}+\mu^{i}\frac{\partial}{\partial
x^{n+i+1}}$ by

$J\frac{\partial}{\partial x^{i}}=\frac{\partial}{\partial x^{n+i+1}},\quad
J\frac{\partial}{\partial x^{n+i+1}}=-\frac{\partial}{\partial x^{i}},\quad
G(X,X)=-\delta_{ij}\lambda^{i}\lambda^{j}+\delta_{ij}\mu^{i}\mu^{j},$

where $\delta_{ij}$ is the Kronecker delta for $i,j\in\\{1,\dots,n+1\\}$.

The manifold
$(\mathcal{M},\allowbreak{}\varphi,\allowbreak{}\xi,\allowbreak{}\eta,\allowbreak{}{}g)$
is constructed as a hypersurface of
$(\mathbb{R}^{2n+2},\allowbreak{}{}J,\allowbreak{}{}G)$ determined by

$G(Z,JZ)=0,\qquad G(Z,Z)=\cosh^{2}t,\qquad t>0,$

where $Z$ is the position vector of a point in $\mathbb{R}^{2n+2}$.

The Reeb vector field is defined by $\xi=-JN$, where $N=\frac{1}{\cosh t}JZ$
is the unit normal of $\mathcal{M}$ and $N$ is time-like, i. e. $G(N,N)=-1$.

The structure tensors $\varphi$ and $\eta$ are determined by the following
condition for any tangent vector field $X$ on $\mathcal{M}$

$JX=\varphi X+\eta(X)J\xi.$

The B-metric $g$ is the restriction of $G$ on $\mathfrak{X}(\mathcal{M})$.

In [8], it is shown that the constructed manifold
$(\mathcal{M},\allowbreak{}\varphi,\allowbreak{}\xi,\allowbreak{}\eta,\allowbreak{}{}g)$
belongs to $\mathcal{F}_{5}$ because $F=F^{5}$, where

(38) $\theta=0,\qquad\theta^{*}=\frac{2n}{\cosh t}\eta,\qquad\omega=0.$

Now, we define the following functions on
$\mathbb{R}^{2n+1}=\left\\{\left(x^{1},\dots,x^{n};\allowbreak{}x^{n+1},\allowbreak{}\dots,\right.\right.$
$\left.\left.\allowbreak{}x^{2n};\allowbreak{}{}t\right)\right\\}$

(39)
$\begin{split}u&=\frac{1}{2}\sum_{i=1}^{n}\ln\left\\{\left(x^{i}\right)^{2}+\left(x^{n+i}\right)^{2}\right\\}+\ell(t),\\\\[4.0pt]
v&=\sum_{i=1}^{n}\arctan\frac{x^{i}}{x^{n+i}},\\\\[4.0pt] w&=h(t),\end{split}$

where $\ell(t)$ is an arbitrary twice differentiable function on $\mathbb{R}$
such that $\ell^{\prime}\neq 0$ and $h(t)$ is an arbitrary differentiable
function on $\mathbb{R}$. Then, we apply an accR transformation determined by
these functions $(u,v,w)$.

We obtain the expressions of their partial derivatives for
$i\in\\{1,2,\dots,n\\}$ as follows:

$\begin{array}[]{ll}\dfrac{\partial u}{\partial x^{i}}=-\dfrac{\partial
v}{\partial x^{n+i}}=\dfrac{x^{i}}{(x^{i})^{2}+(x^{n+i})^{2}},&\dfrac{\partial
u}{\partial t}=\ell^{\prime},\\\\[9.0pt] \dfrac{\partial u}{\partial
x^{n+i}}=\dfrac{\partial v}{\partial
x^{i}}=\dfrac{x^{n+i}}{(x^{i})^{2}+(x^{n+i})^{2}},&\dfrac{\partial w}{\partial
t}=h^{\prime}.\end{array}$

Then we find that the functions from (39) have the properties

(40)
$\mathrm{d}u\circ\varphi=\mathrm{d}v,\qquad\mathrm{d}u(\xi)=\ell^{\prime},\qquad\mathrm{d}v(\xi)=0,\qquad\mathrm{d}w=h^{\prime}\eta.$

As a next step, we apply an accR transformation with the functions $(u,v,w)$
defined by (39).

Taking into account (8), (7), (38) and (40), we obtain the corresponding Lee
forms, which coincide with the results in (29) if and only if the following
condition is valid

$\ell^{\prime}=-\frac{1}{\cosh t}.$

A solution of this differential equation is

$\ell=-\arctan(\sinh t).$

Then, the Lee forms of the obtained manifold are determined by

$\bar{\theta}=4n\,\mathrm{d}u\circ\varphi,\qquad\bar{\theta}^{*}=-4n\,\mathrm{d}v\circ\varphi,\qquad\bar{\omega}=0.$

Thus, the transformed manifold
$(\mathcal{M},\varphi,\bar{\xi},\bar{\eta},\bar{g})$ is an
$\mathcal{F}_{1}$-manifold with a Yamabe soliton with potential $\bar{\xi}$
and a constant scalar curvature $\bar{\tau}=\sigma$, according ot Theorem 3.1.

## Conclusions

Steady-state solutions of geometric flows, in particular Yamabe solitons and
the like are still a very current topic in differential geometry. The paper
reveals new properties of these types of solitons in a contact conformally
transformed manifold of an understudied type having a vertical torse-forming
potential by considering the interrelation between the original and the
deformed structure. Since Yamabe solitons are not yet sufficiently studied,
any contribution in this direction may bring new perspectives on the geometry
of the manifold.

## References

  * [1] Barbosa, E.; Ribeiro Jr., E. On conformal solutions of the Yamabe flow. Arch. Math. 101 (2013), 79–89.
  * [2] Cao, H.D.; Sun, X.; Zhang, Y. On the structure of gradient Yamabe solitons. Math. Res. Lett. 19 (4) (2012), 767–774.
  * [3] Chen, B.-Y.; Deshmukh, S. A note on Yamabe solitons. Balk. J. Geom. Appl. 23 (1) (2018), 37–43.
  * [4] Chow, B.; Lu, P.; Ni L. Hamilton’s Ricci Flow, In Graduate Studies in Mathematics 77; American Mathematical Society: Providence, RI; Science Press: Beijing, New York, 2006.
  * [5] Daskalopoulos, P.; Sesum, N. The classification of locally conformally flat Yamabe solitons. Adv. Math. 240 (2013), 346–369.
  * [6] Deshmukh, S.; Belova, O. On Killing vector fields on Riemannian manifolds. Math. 9 (2021), 259.
  * [7] Ganchev, G.; Borisov, A. Note on almost contact manifolds with Norden metric. C. R. Acad. Bulg. Sci. 39 (1986), 31–34.
  * [8] Ganchev, G.; Mihova, V.; Gribachev, K. Almost contact manifolds with B-metric. Math. Balkanica (N.S.) 7 (1993), 261–276.
  * [9] Ghosh, A. Yamabe soliton and Quasi Yamabe soliton on Kenmotsu manifold. Math. Slovaca 70 (1) (2020), 151–160.
  * [10] Hamilton, R.S. The Ricci flow on surfaces, Mathematics and General Relativity, Contemp. Math. Santa Cruz, CA, 1986, vol. 71, Amer. Math. Soc. Providence, RI, 1988, 237–262.
  * [11] Hui, S.K.; Mandal, Y.C. Yamabe solitons on Kenmotsu manifolds, Commun. Korean Math. Soc. 34 (2019), 321–331.
  * [12] Ivanov, S.; Manev, H.; Manev, M. Sasaki-like almost contact complex Riemannian manifolds. J. Geom. Phys. 105 (2016), 136–148.
  * [13] Manev, H. On the structure tensors of almost contact B-metric manifolds. Filomat 29 (3) (2015), 427–436.
  * [14] Manev, H.; Mekerov, D. Lie groups as 3-dimensional almost contact B-metric manifolds, J. Geom. 106 (2015), 229–242.
  * [15] Manev, M. Contactly conformal transformations of general type of almost contact manifolds with B-metric. Applications. Math. Balkanica (N.S.) 11 (1997), 347–357.
  * [16] Manev, M. Yamabe solitons on conformal Sasaki-like almost contact B-metric manifolds. Math. 10 (2022), 658.
  * [17] Manev, M.; Gribachev, K. Contactly conformal transformations of almost contact manifolds with B-metric. Serdica Math. J. 19 (1993), 287–299.
  * [18] Manev, M.; Gribachev, K. Conformally invariant tensors on almost contact manifolds with B-metric. Serdica Math. J. 20 (1994), 133–147.
  * [19] Manev, M.; Ivanova, M. Canonical type connections on almost contact manifold with B-matric. Ann. Global Anal. Geom. 43 (2013), no. 4, 397–408.
  * [20] Mihai, A.; Mihai, I. Torse forming vector fields and exterior concurrent vector fields on Riemannian manifolds and applications. J. Geom. Phys. 73 (2013), 200–208.
  * [21] Roy, S.; Dey, S.; Bhattacharyya, A. Yamabe Solitons on (LCS)n-manifolds. J. Dyn. Syst. Geom. Theor. 18 (2) (2020), 261–279.
  * [22] Schouten, J.A. Ricci Calculus. An Introduction to Tensor Analysis and Its Geometrical Applications. Springer-Verlag: Berlin, Heidelberg, 1954.
  * [23] Yano, K. On torse forming direction in a Riemannian space. Proc. Imp. Acad. Tokyo 20 (1944), 340–345.


# On long-lived electroweak-singlet colored scalars

Christian T Preuss  and German Valencia

###### Abstract

There has been much recent interest in long-lived massive particles at the
LHC, understood as those with lifetimes between tens of micrometers and
several meters. In this context we consider the possibility of long-lived
electroweak singlet scalars charged under color $\mathrm{SU}(3)$ with masses
near a TeV. The shortest lifetime of interest is already longer than typical
hadronization scales. These exotic new particles would therefore appear as
color singlet bound states of the new scalars with quarks and gluons and it is
their color charge that prevents them from decaying. In particular we consider
color representations consistent with maintaining asymptotic freedom, those
with dimensionality $d_{R}\leq 15$. We find that only the octets can decay,
and they do so into multi-jet final states through the two-gluon channel. The
other representations are stable and form fractionally charged color singlets,
with the decuplet being the only one that can form electrically neutral color
singlets.

###### Contents

  1. 1 Introduction
  2. 2 Couplings to gluons
    1. 2.1 Asymptotic freedom
    2. 2.2 Production via gluon fusion at the LHC
  3. 3 The scalar potential
  4. 4 Scalar decay modes
    1. 4.1 Tree-level decays
    2. 4.2 One-loop decays
    3. 4.3 Mixed decays to quarks and gluons
  5. 5 Constraints from $H\to gg$
  6. 6 Exotic showers and hadrons
    1. 6.1 Phenomenological modeling
    2. 6.2 A first estimate of LHC limits
  7. 7 Summary and conclusions
  8. A Some $\mathrm{SU}(3)$ relations and notation
  9. B Scalar loop functions

##  1 Introduction

Recently there has been renewed interest in the study of long-lived particles
at the LHC, what is dubbed the lifetime frontier. In the context of LHC
studies the interesting lifetime range lies between tens of micrometers and
several meters. Recent reviews that highlight different theoretical scenarios
that may give rise to such long-lived particles and the experimental
strategies to search for them are, for example, Refs. [1, 2, 3].

One intriguing possibility consists of long-lived scalars charged under the
color group. Due to color confinement, such particles would hadronize before
decaying (if they decay at all) and show up in the detectors as exotic
hadrons. Examples of these kinds of particles which have been studied at
length in the literature are the so called $R$-hadrons [4]. In this case,
gluinos or squarks are pair produced and are either long-lived or stable due
to $R$-parity. Other scenarios have also been considered for the case of color
triplets [5, 6]. For the electroweak singlet scalars we consider here, it is
their color charge that prevents them from decaying, as will be outlined
below.

We present a bottom-up simplified model in which we extend the standard model
(SM) with a set of electroweak singlet scalars in a representation $R$ of the
color $\mathrm{SU}(3)$. These objects would then be copiously pair produced by
gluon fusion at the LHC if their masses are below the order of a TeV. Being
electroweak singlets, their exotic color structure can prevent them from
decaying and color thus plays the role of $R$-parity in making these particles
stable. Carrying a color charge, these hypothetical scalars would also combine
with quarks and gluons to form color-singlets on typical hadronization time
scales. Experimental searches would thus mimic the strategies used for
$R$-hadrons.

We find that the requirement of maintaining asymptotic freedom restricts the
possible $\mathrm{SU}(3)$ representations to
$R=\boldsymbol{3},~{}\boldsymbol{6},~{}\boldsymbol{8},~{}\boldsymbol{10},~{}\boldsymbol{15},~{}\boldsymbol{15^{\prime}}$.
With masses below $\sim 1$ TeV, their production cross-sections would be in
the $\sim 0.1-1000$ pb range and they can potentially introduce significant
corrections to the Higgs decay which then serves to constrain them. We find
that the octet is the only one of these examples that could decay into two
gluons, and for most of parameter space would be short-lived. The others are
stable and would hadronize into fractionally charged color singlets with the
exception of the color decuplet which can form an electrically neutral exotic
hadron.

This manuscript is structured as follows. In section 2, we outline the
extension of the SM towards including the new colored scalars based on
arguments of conserving asymptotic freedom. Section 3 focusses on the scalar
potential of the new particles. Section 4 then discusses all possible decay
modes at leading-order (LO) and next-to-leading order (NLO). Before concluding
in section 7, we briefly discuss implications on hadrons created by such new
colored scalars in section 6.

##  2 Couplings to gluons

We begin by considering those model-independent couplings that depend only on
the color representation of the scalars and that determine their tree-level
couplings to gluons. For a complex electroweak-singlet scalar field $S$
transforming with respect to an irreducible representation $R$ of
$\mathrm{SU}(3)$ its interaction with the gluon field $A^{\mu}$ is dictated by
the covariant derivative in the quadratic part of the Lagrangian,

$\mathcal{L}=\left[\left(\partial_{\mu}+\text{i}g_{\text{s}}A_{\mu}^{A}T_{R}^{A}\right)S\right]^{\dagger}\left(\partial^{\mu}+\text{i}g_{\text{s}}A^{A,\mu}T_{R}^{A}\right)S-m_{S}^{2}S^{\dagger}S.$
(1)

The coupling strength is the usual strong coupling $g_{\text{s}}$ and in
writing eq. 1 we have omitted explicit color indices. For real representations
of $\mathrm{SU}(3)$, the scalar fields will be real and an additional factor
of $1/2$ is required.

###  2.1 Asymptotic freedom

The $\mathrm{SU}(3)$ representations that we consider are first restricted by
requiring the model to maintain asymptotic freedom at scales above the scalar
masses. To this end, we recall the well known QCD one-loop beta function, to
which vector, fermion, and scalar fields charged under the $\mathrm{SU}(3)$
contribute as,

$\beta_{g}=-\frac{g_{\text{s}}^{3}}{16\pi^{2}}\left(\frac{11}{3}t_{2}(V)-\frac{4}{3}n_{\text{F}}t_{2}(F)-\frac{1}{3}t_{2}(S)\right).$
(2)

Here, $t_{2}(V)$, $t_{2}(F)$, and $t_{2}(S)$ denote the Dynkin index of the
representation in which the different fields transform respectively. For the
SM augmented by complex scalars in representations with dimension up to 15,
eq. 2 takes the form

$\beta_{g}=-\frac{g_{\text{s}}^{3}}{16\pi^{2}}\left(11-\frac{2}{3}n_{\text{F}}-\frac{1}{6}n_{3}-\frac{5}{6}n_{6}-n_{8}-\frac{5}{2}n_{10}-\frac{10}{3}n_{15}-\frac{35}{6}n_{15^{\prime}}\right).$
(3)

In what follows we will only allow one scalar multiplet at a time, and this
implies that asymptotic freedom holds as long as111Similar considerations have
been used for fermions in higher color representations [7, 8].

$t_{2}(S)<21.$ (4)

All possible multiplets satisfying eq. 4 are collected in table 1 and their
corresponding pair-production cross-section at the LHC is plotted in fig. 2.
The lowest-dimensional representation to fail eq. 4 is $(3,1)$ for which
$t_{2}(\mathbf{24})=25$.

Table 1: Dynkin indices of all representations of colored scalars allowed by asymptotic freedom. Label | Representation | Dynkin Index $t_{2}$
---|---|---
$(0,0)$ | $\boldsymbol{1}$ | $0$
$(1,0)$ | $\boldsymbol{3}$ | $1/2$
$(2,0)$ | $\boldsymbol{6}$ | $5/2$
$(1,1)$ | $\boldsymbol{8}$ | $3$
$(3,0)$ | $\boldsymbol{10}$ | $15/2$
$(2,1)$ | $\boldsymbol{15}$ | $10$
$(4,0)$ | $\boldsymbol{15}^{\prime}$ | $35/2$

###  2.2 Production via gluon fusion at the LHC

Equation 1 determines the scalar pair-production cross-section from gluon
fusion which proceeds at LO through the diagrams in fig. 1. For complex
scalars it is given in terms of the quadratic Casimir of the representation,
$c_{2}(R)$, by [9]

$\displaystyle\frac{\,\text{d}\sigma}{\,\text{d}t}$
$\displaystyle=\frac{2\pi\alpha_{\text{s}}^{2}}{s^{2}}\frac{c_{2}(R){\dim}~{}R}{({\dim}~{}A)^{2}}\left[\left(c_{2}(R)-\frac{1}{4}c_{2}(A)\right)+\frac{c_{2}(A)}{4}\frac{(u-t)^{2}}{s^{2}}\right]$
$\displaystyle\qquad\times\left[1+\frac{2m_{S}^{2}t}{(m_{S}^{2}-t)^{2}}+\frac{2m_{S}^{2}u}{(m_{S}^{2}-u)^{2}}+\frac{4m_{S}^{4}}{(m_{S}^{2}-t)(m_{S}^{2}-u)}\right].$
(5)

The terms depending on the Casimir of the adjoint representation, $c_{2}(A)$,
originate from the s-channel diagram and $s,t,u$ are the usual Mandelstam
variables. For the case of real scalar fields, eq. 5 must be multiplied by an
additional factor of $1/2$. Many detailed phenomenological studies for octets,
particularly electroweak doublets, exist in the literature [10, 9, 11, 12, 13,
14, 15] and we have checked that the corresponding results agree with ours. In
particular, eq. 5 is half as large as the corresponding cross-section for
squark pair production from gluon fusion for which there are two complex
scalar doublets [16].

Figure 1: Leading-order diagrams for scalar pair production from gluon fusion.

In fig. 2, we present the leading-order $pp\to SS$ cross-section for all
representations satisfying eq. 4 at the LHC with
$\sqrt{s}=13~{}\mathrm{T}e\mathrm{V}$ using the PDF4LHC15_nlo_mc parton
distribution function set [17] with factorization scale
$\mu_{\mathrm{F}}=2m_{S}$, accessed via the ManeParse package [18] and LHAPDF6
[19]. Calculations of these cross-sections beyond leading order exist for the
triplets (squarks) [20] where NLO enhancements over the LO are found to be
described approximately by modest $K$-factors near 1.3 at
$\sqrt{s}=14~{}\mathrm{T}e\mathrm{V}$ with weak dependence on the squark mass.
Next-to-leading order calculations for scalar octets (sgluons) exist in the
literature [21] as well, and a $K$-factor in the range (1.5-1.8) is found at
$\sqrt{s}=14~{}\mathrm{T}e\mathrm{V}$, with the variation stemming from the
sgluon mass. Leading-order color sextet (diquark) production at the LHC has
been studied previously in [22].

Figure 2: Leading-order scalar pair production cross-section $pp\to SS$ at the
LHC with $\sqrt{s}=13~{}\mathrm{T}e\mathrm{V}$ for complex scalars in
$\mathrm{SU}(3)$ representations $\mathbf{3}$, $\mathbf{6}$, $\mathbf{10}$,
$\mathbf{15}$, $\mathbf{15^{\prime}}$ and real scalars in the adjoint
representation $\mathbf{8}$.

##  3 The scalar potential

The renormalizable interactions of colored scalars are governed by the most
general scalar potential up to dimension four. For electroweak singlets, the
terms

$V_{R}=\kappa_{R}H^{\dagger}HS^{\dagger}S+\mu_{R}(S^{\dagger}S)^{2}\,,$ (6)

occur for all $\mathrm{SU}(3)$ representations. There are also further
representation-dependent terms, which will be detailed below for specific
cases.

Triple-scalar couplings are allowed when $R\otimes R\otimes R\
\supset\boldsymbol{1}$, and this is possible for all the representations we
consider, as can be seen from table 3. However, some of these couplings are
completely antisymmetric under scalar exchanges and therefore vanish when the
scalars carry no other quantum numbers beyond color. Completely symmetric
triple-scalar couplings occur except for the $\boldsymbol{3}$ and
$\boldsymbol{10}$ as shown below. We denote symmetric (anti-symmetric) triple
couplings by $\lambda_{R}$ ($\tilde{\lambda}_{R}$). The triple-scalar
couplings allowed under these conditions have dimensions of mass but they are
not related to the usual Higgs vacuum expectation value. These presumably
originate from the mechanism that gives mass to the scalars.

Model-dependent quartic-scalar couplings occur if $R\otimes R\otimes R\otimes
R\ \supset\boldsymbol{1}$ and this occurs for $R=\boldsymbol{8}$,
$\boldsymbol{10}$. In addition, for $R=\boldsymbol{10}$ there is a quartic
coupling involving three scalar particles and one anti-particle.

In the following, fundamental indices of the color group are denoted by small
Latin letters $i,j,k,\ldots$, while adjoint indices are denoted by capital
Latin letters $A,B,C,\ldots$. The potentials for
$R=\boldsymbol{3},~{}\boldsymbol{6}$, motivated by minimal flavor violation,
have been discussed in [23] albeit with non-zero $U(1)_{Y}$ charge.

#### Triplets

It is possible to construct a singlet from three triplets but not from four so
the most general potential is

$V_{\boldsymbol{3}}=m_{S}^{2}S^{\dagger}_{i}S^{i}+\kappa_{\boldsymbol{3}}H^{\dagger}HS^{\dagger}_{i}S^{i}+\mu_{\boldsymbol{3}}(S^{\dagger}_{i}S^{i})^{2}+(\tilde{\lambda}_{\boldsymbol{3}}\varepsilon_{ijk}S^{i}S^{j}S^{k}+\mathrm{h.c.}\,).$
(7)

Note that the $\tilde{\lambda}_{\boldsymbol{3}}$ term is completely
antisymmetric and therefore vanishes for identical scalars.

#### Sextets

For scalar sextets it is once again possible to construct a singlet with three
fields but not four, so the most general potential is given by:

$V_{\boldsymbol{6}}=m_{S}^{2}S^{\dagger}_{ij}S^{ij}+\kappa_{\boldsymbol{6}}H^{\dagger}HS^{\dagger}_{ij}S^{ij}+\mu_{\boldsymbol{6}a}(S^{\dagger}_{ij}S^{ij})^{2}+\mu_{\boldsymbol{6}b}S^{\dagger}_{ik}S^{\dagger}_{j\ell}S^{ij}S^{k\ell}+(\lambda_{\boldsymbol{6}}\varepsilon_{ikm}\varepsilon_{j\ell
n}S^{ij}S^{k\ell}S^{mn}+\mathrm{h.c.}\,).$ (8)

In this case the triple-scalar coupling is symmetric under the exchange of any
two sextets.

#### Octets

The octet is a real representation of $\mathrm{SU}(3)$ and, in the absence of
additional quantum numbers, we will use a real scalar field. As the octet
plays a special role as the adjoint representation, we write the potential for
this case using adjoint indices,222A real scalar in the model of [9] is
$S_{R}$ and the triple-scalar coupling would correspond to
$\lambda_{\boldsymbol{8}}=v(\lambda_{4}+\lambda_{5})/12$ in that model.

$V_{\boldsymbol{8}}=m_{S}^{2}S^{A}S^{A}+\kappa_{\boldsymbol{8}}H^{\dagger}HS^{A}S^{A}+\mu_{\boldsymbol{8}}(S^{A}S^{A})^{2}+\lambda_{\boldsymbol{8}}d_{ABC}S^{A}S^{B}S^{C}$
(9)

Another quartic term found in the literature, $\sim{\rm
Tr}(T^{A}T^{B}T^{C}T^{D})S^{A}S^{B}S^{C}S^{D}$ [10, 24], reduces to $\mu_{8}$
when the scalars do not carry other quantum numbers.

#### Decuplets

For scalar decuplets, the most general potential is given by:

$\displaystyle V_{\boldsymbol{10}}$
$\displaystyle=m_{S}^{2}S^{\dagger}_{ijk}S^{ijk}+\kappa_{\boldsymbol{10}}H^{\dagger}HS^{\dagger}_{ijk}S^{ijk}+\mu_{\boldsymbol{10}a}(S^{\dagger}_{ijk}S^{ijk})^{2}+\mu_{\boldsymbol{10}b}S^{\dagger}_{ijk}S^{\dagger}_{\ell
mn}S^{ijm}S^{\ell nk}$
$\displaystyle+\left(\tilde{\lambda}_{\boldsymbol{10}}\varepsilon_{i\ell
o}\varepsilon_{jmp}\varepsilon_{knq}S^{ijk}S^{\ell
mn}S^{opq}+\omega_{\boldsymbol{10}}\varepsilon_{i\ell
o}\varepsilon_{jmr}\varepsilon_{kps}\varepsilon_{nqt}S^{ijk}S^{\ell
mn}S^{opq}S^{rst}\right.$
$\displaystyle+\left.\rho_{\boldsymbol{10}}S^{\dagger}_{knq}~{}\varepsilon_{i\ell
o}\varepsilon_{jmp}S^{ijk}S^{\ell mn}S^{opq}+\mathrm{h.c.}\,\right)$ (10)

This color structure allows an anti-symmetric triple-scalar vertex, a second
quartic vertex with two particles and two anti-particles, a quartic vertex
with four particles and a quartic vertex with three particles and one anti-
particle.

####  $(2,1)$-Quindecuplets

For scalars transforming in the $\mathbf{15}$, we find the potential

$\displaystyle V_{\boldsymbol{15}}$ $\displaystyle=m_{S}^{2}S^{\dagger
k}_{ij}S^{ij}_{k}+\kappa_{\boldsymbol{15}}H^{\dagger}HS^{\dagger
k}_{ij}S^{ij}_{k}+\mu_{\boldsymbol{15}a}(S^{\dagger
k}_{ij}S^{ij}_{k})^{2}+\mu_{\boldsymbol{15}b}(S^{\dagger
k}_{ij}S^{\dagger\ell}_{mn}S^{im}_{k}S^{jn}_{\ell})+\mu_{\boldsymbol{15}c}(S^{\dagger
k}_{ij}S^{\dagger\ell}_{mn}S^{im}_{\ell}S^{jn}_{k})$
$\displaystyle+\left(\lambda_{\boldsymbol{15}}\varepsilon_{\ell
mn}S^{i\ell}_{j}S^{jm}_{k}S^{kn}_{i}+\tilde{\lambda}_{\boldsymbol{15}}\varepsilon_{ijk}\varepsilon_{\ell
mn}\varepsilon^{opq}S^{i\ell}_{o}S^{jm}_{p}S^{kn}_{q}+\mathrm{h.c.}\,\right)$
(11)

The $(2,1)$ representation allows a symmetric triple-scalar vertex, the term
with coupling $\lambda_{\boldsymbol{15}}$; as well as an antisymmetric one
with coupling $\tilde{\lambda}_{\boldsymbol{15}}$. It does not allow quartic
terms with three particles and one antiparticle nor ones with four particles.

####  $(4,0)$-Quindecuplets

For scalars transforming in the $\mathbf{15}^{\prime}$, we find the potential

$\displaystyle V_{\boldsymbol{15^{\prime}}}$
$\displaystyle=m_{S}^{2}S^{\dagger}_{ijk\ell}S^{ijk\ell}+\kappa_{\boldsymbol{15^{\prime}}}H^{\dagger}HS^{\dagger}_{ijk\ell}S^{ijk\ell}+\mu_{\boldsymbol{15^{\prime}}a}(S^{\dagger}_{ijk\ell}S^{ijk\ell})^{2}+\mu_{\boldsymbol{15^{\prime}}b}(S^{\dagger}_{ijk\ell}S^{\dagger}_{mnop}S^{ijmn}S^{k\ell
op})$
$\displaystyle+\mu_{\boldsymbol{15^{\prime}}c}(S^{\dagger}_{ijk\ell}S^{\dagger}_{mnop}S^{ijkm}S^{\ell
nop})+\left(\lambda_{\boldsymbol{15^{\prime}}}\varepsilon_{i\ell
o}\varepsilon_{jmp}\varepsilon_{knq}\varepsilon_{rst}S^{ijkr}S^{\ell
mns}S^{opqt}+\mathrm{h.c.}\,\right)$ (12)

The $(4,0)$ representation permits a symmetric triple-scalar coupling but no
quartic couplings.

We want to close this section by elaborating upon possible Yukawa couplings of
the colored scalars. Since we do not consider any Standard-Model extensions
beyond the colored scalars, we only have the SM fermion content at our
disposal, meaning that all fermions must either be color singlets or triplets.
In effect, only scalar singlets, triplets, sextets, and octets can have Yukawa
couplings. Moreover, if the scalars are $\mathrm{SU}(2)_{L}$ singlets and do
not carry electric charge, the only possible Yukawa coupling is to right
handed neutrinos. Insisting on $\mathrm{SU}(2)_{L}$ singlets but allowing for
$Y=Q\neq 0$, the only possibilities with colored scalars are for triplets and
sextets such as [25]

$\displaystyle V_{Y,\mathbf{3}}$ $\displaystyle\supset
g_{\text{Y},\boldsymbol{3}}^{e1}S^{i}\mathrm{bar}{d}_{iR}e^{c}_{R}+g_{\text{Y},\boldsymbol{3}}^{e2}S^{i}\mathrm{bar}{Q}_{iL}L^{c}_{L}+g_{\text{Y},\boldsymbol{3}}^{e3}S^{i}\mathrm{bar}{u}_{iR}e^{c}_{R}$
$\displaystyle\qquad+g_{\text{Y},\boldsymbol{3}}^{q1}\varepsilon_{ijk}S^{i}\mathrm{bar}{Q}_{L}^{c}Q_{L}+g_{\text{Y},\boldsymbol{3}}^{q2}\varepsilon_{ijk}S^{i}\mathrm{bar}{u}_{R}^{c}d_{R}+g_{\text{Y},\boldsymbol{3}}^{q3}\varepsilon_{ijk}S^{i}\mathrm{bar}{u}_{R}^{c}u_{R}+g_{\text{Y},\boldsymbol{3}}^{q4}\varepsilon_{ijk}S^{i}\mathrm{bar}{d}_{R}^{c}d_{R}+\mathrm{h.c.}\,$
(13) $\displaystyle V_{Y,\mathbf{6}}$ $\displaystyle\supset
g_{\text{Y},\boldsymbol{6}}^{q1}S^{ij}\mathrm{bar}{Q}_{iL}Q_{jL}^{c}+g_{\text{Y},\boldsymbol{6}}^{q2}S^{ij}\mathrm{bar}{d}_{iR}u_{jR}^{c}+g_{\text{Y},\boldsymbol{6}}^{q3}S^{ij}\mathrm{bar}{u}_{iR}u_{jR}^{c}+g_{\text{Y},\boldsymbol{6}}^{q4}S^{ij}\mathrm{bar}{d}_{iR}d_{jR}^{c}+\mathrm{h.c.}\,$
(14)

where we have omitted flavor indices. All of these couplings require the
scalars to carry electric charge, so are not of interest here.

##  4 Scalar decay modes

We now examine possible decay modes at tree and one-loop level, and show that
only the octet can decay. To conserve color, any scalar decay must include
quarks, gluons or more scalars.

###  4.1 Tree-level decays

Decays into standard-model fermion pairs, $S\to f\mathrm{bar}f$, could proceed
at tree-level through Yukawa interactions. As the fermions in the SM transform
under the color group as singlets or triplets, the $f\mathrm{bar}f$ state can
only be an $\mathrm{SU}(3)$ $\boldsymbol{1}$, $\boldsymbol{3}$,
$\boldsymbol{6}$, or $\boldsymbol{8}$. Requiring that the new scalars be
electroweak singlets forbids any couplings of the form listed in eq. 14 so
decays to fermion pairs are not possible. If instead, we allow scalars with
non-zero hyper-charge, these would decay into two or more jets for
$\boldsymbol{3}$, $\boldsymbol{6}$, or $\boldsymbol{8}$ and small Yukawa
couplings would be required to ensure longevity.

Since we consider only one (degenerate) multiplet at a time, the scalars
cannot decay into each other either.

###  4.2 One-loop decays

The decay $S\to gg$ is in principle allowed as a loop-induced process. Without
Yukawa couplings, however, it can only proceed via intermediate colored
scalars as shown in fig. 3. This process then depends on the existence of
appropriate triple-scalar couplings and can proceed only for scalars in the
octet, $\mathbf{8}$, as we will show below.

Figure 3: Lowest-order loop-induced diagrams for $S\to gg$.

Color conservation indicates that only those representations that appear in
the Clebsch-Gordan decomposition of the direct product of two adjoint
representations can decay into two gluons,

$\mathbf{8}\otimes\mathbf{8}=\mathbf{1}\oplus_{2}\mathbf{8}\oplus\mathbf{10}\oplus\overline{\mathbf{10}}\oplus\mathbf{27}$
(15)

Of these, we only need to consider the adjoint $\mathbf{8}$ and
(anti-)decuplet $\mathbf{10}$ ($\overline{\mathbf{10}}$) representations as
per eq. 4. It is straightforward to confirm that these are the only
representations to consider even if a few additional gluons are added in the
final state.

We begin by examining the case of the $\mathbf{10}$ (or
$\overline{\mathbf{10}}$). The color flow (indicated by the arrows) shows that
for the left-hand diagram in fig. 3 to be non-vanishing, either an $SSg$ or an
$S^{\dagger}S^{\dagger}g$ vertex is required, while for the seagull diagram in
fig. 3 an $SSgg$ vertex is needed. However, gluons only appear in the
quadratic part of the Lagrangian which couples a $\mathbf{10}$ with a
$\overline{\mathbf{10}}$ due to the constraint of the Lagrangian being a color
singlet. Hence, neither of the one-loop diagrams appears for scalars
transforming as (anti-)decuplets. This argument indicates that the diagrams in
fig. 3 only occur for real representations. We note that the decuplet also
cannot decay into two gluons at the two-loop level when it carries no quantum
numbers other than color. This is true, because the required triple-scalar
vertex, cf. fig. 4 (the gluons can be attached to any scalar line), is
antisymmetric and therefore vanishes for identical scalars.

Figure 4: Generic two-loop diagrams for the $S_{\mathbf{10}}\to gg$ decay
combining vertices with $\kappa_{\mathbf{10}}$ and
$\mu^{\prime}_{\mathbf{10}}$. Arrows indicate particles/antiparticles (or
color/anticolor representations). The gluons can be emitted from any of the
scalar lines as shown or both at a time from a seagull vertex.

Following the above reasoning, the only possibility left for $S\to gg$ to
occur is for color octets where the answer at one-loop is already known. In
adjoint indices it is given by [11, 14]

${\cal
L}_{Sgg}=\frac{9\alpha_{\text{s}}}{8\pi}~{}\frac{\lambda_{\mathbf{8}}}{m_{S}^{2}}~{}I_{s}(1)~{}d_{ABC}G_{\mu\nu}^{A}G^{B\mu\nu}S^{C},$
(16)

where the loop factor is $I_{s}(1)=\left(\frac{\pi^{2}}{9}-1\right)$ for the
case where all intervening scalars are degenerate in mass, cf. appendix B. The
decay width is then

$\Gamma(S\to
gg)=\frac{9}{8\pi}\left(\frac{\alpha_{\text{s}}}{8\pi}\right)^{2}~{}\frac{\lambda_{\mathbf{8}}^{2}}{m_{S}}~{}30~{}I_{s}(1)^{2},$
(17)

with 30 being a color factor.

###  4.3 Mixed decays to quarks and gluons

Color conservation implies that the $S_{\mathbf{15}}$ can decay into states
like $qg$ or $qgg$ and the $S_{\mathbf{15}^{\prime}}$ into $qgg$. To proceed
at dimension six (eight) these processes would require operators of the form
($\ell,q$ denote a generic lepton and quark respectively)

$\displaystyle{\cal
O}\sim\mathrm{bar}{\ell}\sigma_{\mu\nu}qG^{\mu\nu}S_{\mathbf{15}},$
$\displaystyle{\cal
O}\sim\mathrm{bar}{\ell}qG^{\mu\nu}G_{\mu\nu}S_{\mathbf{15}^{\prime}},$ (18)

where the gluons have to appear explicitly to satisfy the color requirement.
The necessary fermion bilinears, however, cannot couple to scalars with zero
hyper-charge as we saw in our discussion of Yukawa couplings.

##  5 Constraints from $H\to gg$

For all representations we have discussed here, there is an additional new
contribution to the Higgs decay $H\to gg$, and correspondingly to the Higgs
boson production cross-section from gluon fusion, proceeding through the
diagrams in fig. 5 [26, 24]. This contribution will be proportional to the
$HSS$ coupling, which is contained in the term in the scalar potential
proportional to $\kappa_{R}$ in eq. 6.

Figure 5: Lowest-order loop-induced diagrams for the decay $H\to gg$,
including the standard-model contribution mediated by heavy quarks (left) and
new contributions mediated by colored scalars (middle and right).

After adding the SM top-quark loop contribution we find333In this notation,
for example, the complex $S^{\pm}$ scalar field in [9] would contribute with
$\kappa_{R}=\frac{\lambda_{1}}{2}$ and $t_{2}(\mathbf{8})=3$.

$\Gamma(H\to gg)=\frac{m_{H}^{3}}{8\pi
v^{2}}\left(\frac{\alpha_{\text{s}}}{\pi}\right)^{2}\left|I_{q}\left(\frac{m_{t}^{2}}{m_{H}^{2}}\right)+\kappa_{R}~{}\frac{v^{2}}{4m_{H}^{2}}t_{2}(R)~{}I_{s}\left(\frac{m_{S}^{2}}{m_{H}^{2}}\right)\right|^{2}$
(19)

with the known loop functions listed in appendix B. The corresponding
corrections to the partial decay width of the Higgs $\Gamma(H\to gg)$ are
illustrated in fig. 6. From the figure we see that for $|\kappa_{R}|\lesssim
1$, and for all the representations satisfying eq. 4 scalars with masses
$m_{S}\gtrsim 1$ TeV have modest effects. The parameter space for lighter
scalars, on the other hand, can be significantly constrained by Higgs decays.

##  6 Exotic showers and hadrons

As shown above, electroweak singlet scalars in color representations
$R=\boldsymbol{3},~{}\boldsymbol{6},~{}\boldsymbol{10},~{}\boldsymbol{15},~{}\boldsymbol{15^{\prime}}$
cannot decay and are therefore long-lived. The radiation of additional gluons
in form of bremsstrahlung is, however, possible for all the scalars we have
considered here. After production, these scalars will therefore lose energy
via a QCD showering process and, once a scale of the order of the
hadronization scale $\Lambda_{\mathrm{QCD}}\approx 1~{}\mathrm{G}e\mathrm{V}$
is reached, hadronize by picking up quarks and gluons to form exotic color
singlets. This situation is comparable to the one for so-called $R$-hadrons
[4, 27], where exotic hadrons are formed from long-lived super-symmetric
particles such as the stop or the gluino. Irrespective of their origin, we
will call such exotic hadrons $R$-hadrons in the following.

Figure 6: Corrections to the decay $H\to gg$ for different scalar masses and representations. Table 2: List of possible $R$-hadrons for each long-lived representation. Here, $u$ and $d$ denote generic up- and down-type quarks, respectively. Representation |  $R$-hadrons
---|---
$\boldsymbol{3}$ |  $R_{S\mathrm{bar}{u}}^{-2/3}$, $R_{S\mathrm{bar}{d}}^{+1/3}$, $R_{Suu}^{+4/3}$, $R_{Sdd}^{-2/3}$
$\boldsymbol{6}$ |  $R_{S\mathrm{bar}{u}\mathrm{bar}{u}}^{-4/3}$, $R_{S\mathrm{bar}{d}\mathrm{bar}{d}}^{+2/3}$, $R_{Sug}^{+2/3}$, $R_{Sdg}^{-1/3}$
$\boldsymbol{10}$ | $R_{Sgg}^{0}$
$\boldsymbol{15}$ |  $R_{S\mathrm{bar}{u}g}^{-2/3}$, $R_{S\mathrm{bar}{d}g}^{+1/3}$
$\boldsymbol{15}^{\prime}$ |  $R_{S\mathrm{bar}{u}gg}^{-2/3}$, $R_{S\mathrm{bar}{d}gg}^{+1/3}$

The simplest $R$-hadrons that can be formed from the long-lived scalars are
collected in table 2 with the corresponding Clebsch-Gordan decompositions
shown in table 4. Except for the decuplet, all scalars would form fractionally
charged hadrons. In addition, color singlets containing the
$(4,0)$-quindecuplet can only be formed with a minimum of three additional
partons.

Another intriguing possibility is the formation of quarkonium-like bound
states following the production of scalar anti-scalar pairs. This was
considered in [28] for color octets, and we will not pursue it further.

###  6.1 Phenomenological modeling

Multi-purpose event generators such as Sherpa [29], Herwig [30], and Pythia
[31] provide general tools for phenomenological studies of Standard Model
phenomena as well as extensions to it. It would therefore be beneficial to
implement our models considered above in such a framework to give access to a
detailed modeling of not only the hard production process (as described by the
leading-order cross-section in eq. 5), but also the modeling of
logarithmically enhanced QCD bremsstrahlung and non-perturbative effects such
as hadron fragmentation. In addition, Monte Carlo event generators provide the
input for dedicated detector simulation programs such as Geant4 [32] or
Delphes [33, 34], giving access to realistic particle-level analysis
environments.

Monte Carlo event generators, however, routinely use the Les Houches standard
to store color information [35], which only allows for two color tags per
particle. (Strictly speaking it only allows for a color-anticolor combination,
but this can be tweaked to enable color-color or anticolor-anticolor
combinations, cf. e.g. [36].) Implementations of models with colored particles
in representations other than the (anti-)triplet or octet have therefore been
limited mainly to sextets. An explicit “color flow” representation of the
Standard-Model vertices has been presented in [37] in the context of the
O’Mega [38] matrix element generator in Whizard [39, 40]. Sextet production
including subsequent parton showering has been studied in [22], while a
general shower framework for heavy colored particles has been developed in
[41] and extended to showers off stops $\tilde{t}$ in [42]. As far as the non-
perturbative modeling is concerned, both the Herwig and Pythia event
generation frameworks provide dedicated modules for the hadronization of long-
lived colored particles, in the form of $R$-hadrons, cf. [36, 43] for a
description of the model implemented in Pythia 8\. A comparison of Pythia’s
string and Herwig’s cluster fragmentation models for the case of $R$-hadrons
was considered in [44].

A Monte Carlo implementation of the decuplet and quindecuplet models
considered in this work necessitates the extension of the format in which
color information is stored and interpreted in event generators. As the bare
minimum, a third color tag has to be introduced and leading-order matrix
elements projected onto a “leading-color”, i.e. planar, decomposition in terms
of these. It must be emphasized that while this may enable the parton-
showering of such colored particles, it does not automatically include non-
perturbative modeling of showered events. To this end, hadronization models
require a more fundamental extension to utilize the additional color
representations in the formation of color-singlet hadrons. Such an
implementation is therefore beyond the scope of this work. We will instead
obtain a first estimate of LHC bounds in the next section.

###  6.2 A first estimate of LHC limits

To estimate the limits that could be placed at LHC we can readily adapt some
of the existing results for $R$-hadrons found in the literature, cf. e.g. [45,
46, 47].

The ATLAS experiment, using $36.1~{}\mathrm{f}\mathrm{b}^{-1}$ of data at
$\sqrt{s}=13~{}\mathrm{T}e\mathrm{V}$, rules out at the 95% c.l. long-lived
sbottom and stop $R$-hadrons with masses below 1250 GeV and 1340 GeV
respectively. The production cross-section for sbottom or stop pairs from
gluon fusion444The leading-order gluon-fusion partonic cross-section for
complex triplet scalars is the same as for squarks, but the one for octets is
smaller than the one for gluinos, see for example [16, 20]. is twice as large
as the one shown in yellow for color triplets in fig. 2. Differences in the
lifetime are not important for these constraints, as the scalars considered
here are stable and those in [46] are assumed to live long enough to reach the
hadronic calorimeter (decay lengths of a few meters). For comparison, searches
(with significantly lower integrated luminosity) assuming the particles to be
stable find an upper bound for stop $R$-hadrons near
$1~{}\mathrm{T}e\mathrm{V}$, cf. the respective CMS [48] and ATLAS [49]
publications. The main difference would be in the constituents of the color-
singlet exotic hadron. Whereas for sbottom and stop we could have neutral
objects, for the color triplet considered here the exotic hadron would have a
fractional electric charge, cf. table 2. CMS searches looking for long-lived
fractionally charged objects find similar constraints as the ones above, cf.
[50, 48]. We thus expect bounds for exotic hadrons from color triplets to be
around a TeV, slightly weaker than those for squarks, with stronger limits for
the higher representations.

To obtain a first estimate for the scalar-mass constraints, we compare our
scalar pair-production cross-sections to the average cross-section
corresponding to the ATLAS sbottom and stop $R$-hadron mass limits. In
addition, we include a $K$-factor of $1.3$ for all representations. The
results are shown in fig. 7, where we have superimposed the first-estimate
exclusion cross-section as a horizontal line. From this figure, the expected
constraints based only on a scaling of the cross-section can be read off to
range from $m_{S}=1.1~{}\mathrm{T}e\mathrm{V}$ for color triplets to
$m_{S}\approx 2.1~{}\mathrm{T}e\mathrm{V}$ for scalars in the
$\mathbf{15^{\prime}}$.

Figure 7: Pair-production cross-sections for all long-lived scalars including
a $K$-factor of $1.3$, with the dashed line marking a first-estimate LHC
bound.

As LHC limits for long-lived charged particles rely on measurements of
ionization and as such do not directly apply to our neutral decuplet
$R^{0}_{Sgg}$, some remarks are in order:

  * •

First, the scalar decuplet can also hadronize with three quarks leading to a
charged $R^{+}_{S\mathrm{bar}{u}\mathrm{bar}{u}\mathrm{bar}{d}}$ for example.
The hadronization ratios into the different possible $R$-hadrons are not
known, but are expected to be comparable. In addition, collisions of an
$R$-hadron with the detector material can change the electric charge through
strong interaction reactions such as $R^{0}_{Sgg}+p\to
R^{+}_{S\mathrm{bar}{u}\mathrm{bar}{u}\mathrm{bar}{d}}+{\rm pions}$.

  * •

The neutral $R$-hadrons would also leave a hadronic shower in the hadronic
calorimeters providing another handle for their detection.

It is interesting to note that one can also obtain an upper limit on the mass
of neutral $R$-hadrons by requiring their annihilation cross-section be large
enough to avoid over-production of relic dark matter density [51, 52]. We can
estimate the annihilation cross-section in a spectator model by adjusting the
kinematic and color-average factors in eq. 5 to find

$\langle\sigma(R^{0}_{Sgg}R^{0}_{Sgg}\to
gg)v\rangle\approx\frac{\pi\alpha_{s}^{2}c_{2}(R)}{8({\dim}~{}R)m_{S}^{2}}\left(4c_{2}(R)-c_{2}(A)\right)\,.$
(20)

This result, in combination with $\langle\sigma v\rangle\gtrsim 4\times
10^{-9}~{}\mathrm{G}e\mathrm{V}^{-2}$ [51], suggests that
$m_{R^{0}_{Sgg}}\lesssim 3.2~{}\mathrm{T}e\mathrm{V}$. There is no lower bound
in this case since there could be many other contributions to the relic
density.

##  7 Summary and conclusions

In this manuscript, we have discussed scalar extensions to the Standard Model,
transforming non-trivially under the color group and as singlets under the
electroweak gauge group. Based on the requirement to leave asymptotic freedom
intact, the maximal dimension of the color representation is 15. Requiring the
new scalars to have zero hypercharge, the new particles are stable, with only
octets decaying to gluons at the one-loop level.

We have discussed how long-lived colored scalars will appear as exotic hadrons
in detector signals and obtained a first estimate of mass bounds at the LHC.
More detailed analyses can be obtained by implementing models of higher color
representations in multi-purpose event generators and we have outlined such
implementations, including a discussion of current technical restrictions. The
implementation in multi-purpose event generators is beyond the scope of the
current work, but will be important to obtain mass constraints from collider
experiments, as it facilitates the modeling of QCD as well as EW/QED
bremsstrahlung, hadronization as well as other non-perturbative corrections,
and subsequent detector simulation.

## Acknowledgements

We would like to thank Ulrik Egede, Xiao-Gang He, Peter Skands and Ray Volkas
for helpful discussions. CTP is supported by the Monash Graduate Scholarship,
the Monash International Postgraduate Research Scholarship, and the J. L.
William Scholarship.

##  Appendix A Some $\mathrm{SU}(3)$ relations and notation

For a general representation $R$, the Dynkin index is given by the trace of
the generators $T_{R}$ in the representation,

$\mathrm{Tr}(T_{R}^{A}T_{R}^{B})=t_{2}(R)\delta^{AB},$ (21)

and is related to the eigenvalue $c_{2}(R)$ of the quadratic Casimir operator,

$c_{2}(R)=\delta^{AB}T_{R}^{A}T_{R}^{B},$ (22)

in the representation $R$ by

$t_{2}(R)=\frac{\dim R}{\dim\mathfrak{su}(3)}c_{2}(R),$ (23)

where $\dim R$ and $\dim\mathfrak{su}(3)=8$ denote the dimensions of the
representation and the Lie algebra $\mathfrak{su}(3)$ of the gauge group
$\text{SU}(3)$, respectively. Labelling irreducible representations $R(p,q)$
of $\mathfrak{su}(3)$ by the Dynkin label $(p,q)$, the eigenvalue of the
quadratic Casimir is

$c_{2}(R(p,q))\equiv c_{2}(p,q)=\frac{p^{2}+q^{2}+3p+3q+pq}{3}\,.$ (24)

Moreover, the dimension of the representation $R(p,q)$ can be calculated by

$\dim R(p,q)=\frac{(p+1)(q+1)(p+q+2)}{2}\,.$ (25)

In the Standard Model, gluons transform with respect to the adjoint
representation $\boldsymbol{8}$ and fermions with respect to the fundamental
representation $\boldsymbol{3}$. Therefore, with $n_{\text{F}}=6$,
$N_{\text{C}}=3$, and $T_{\text{R}}=1/2$, in the Standard Model we have

$t_{2}(V)=t_{2}(\boldsymbol{8})=3,\quad\text{and}\quad
t_{2}(F)=t_{2}(\boldsymbol{3})=\frac{1}{2}.$ (26)

Some Clebsch-Gordan decompositions useful to determine the terms entering the
scalar potential are tabulated below. All decompositions are calculated with
LieART [53] and cross-checked with an independent implementation of the
algorithm in [54].

Table 3: Selected three- and four-particle Clebsch-Gordan decompositions. Direct Product | Decomposition
---|---
$\mathbf{3}\otimes\mathbf{3}\otimes\mathbf{3}$ | $\mathbf{1}\oplus_{2}\mathbf{8}\oplus\mathbf{10}$
$\mathbf{6}\otimes\mathbf{6}\otimes\mathbf{6}$ | $\mathbf{1}\oplus_{2}\mathbf{8}\oplus\mathbf{10}\oplus\overline{\mathbf{10}}\oplus_{3}\mathbf{27}\oplus\mathbf{28}\oplus_{2}\mathbf{35}$
$\mathbf{8}\otimes\mathbf{8}\otimes\mathbf{8}$ | $\oplus_{2}\mathbf{1}\oplus_{8}\mathbf{8}\oplus_{4}\mathbf{10}\oplus_{4}\overline{\mathbf{10}}\oplus_{6}\mathbf{27}\oplus_{2}\mathbf{35}\oplus_{2}\overline{\mathbf{35}}\oplus\mathbf{64}$
$\mathbf{10}\otimes\mathbf{10}\otimes\mathbf{10}$ | $\mathbf{1}\oplus_{2}\mathbf{8}\oplus\mathbf{10}\oplus\overline{\mathbf{10}}\oplus_{3}\mathbf{27}\oplus\mathbf{28}\oplus_{2}\mathbf{35}\oplus_{2}\overline{\mathbf{35}}\oplus\cdots$
$\mathbf{15}\otimes\mathbf{15}\otimes\mathbf{15}$ | $\oplus_{2}\mathbf{1}\oplus_{10}\mathbf{8}\oplus_{8}\mathbf{10}\oplus_{7}\overline{\mathbf{10}}\oplus_{15}\mathbf{27}\oplus_{4}\mathbf{28}\oplus\overline{\mathbf{28}}\oplus_{11}\mathbf{35}\oplus_{8}\overline{\mathbf{35}}\oplus\cdots$
$\mathbf{15^{\prime}}\otimes\mathbf{15^{\prime}}\otimes\mathbf{15^{\prime}}$ | $\mathbf{1}\oplus_{2}\mathbf{8}\oplus\mathbf{10}\oplus\overline{\mathbf{10}}\oplus_{3}\mathbf{27}\oplus\mathbf{28}\oplus\overline{\mathbf{28}}\oplus_{2}\mathbf{35}\oplus_{2}\overline{\mathbf{35}}\oplus\cdots$
$\mathbf{8}\otimes\mathbf{8}\otimes\mathbf{8}\otimes\mathbf{8}$ | $\oplus_{8}\mathbf{1}\oplus_{32}\mathbf{8}\oplus_{20}\mathbf{10}\oplus_{20}\overline{\mathbf{10}}\oplus_{33}\mathbf{27}\oplus_{2}\mathbf{28}\oplus_{2}\overline{\mathbf{28}}\oplus_{15}\mathbf{35}\oplus_{15}\overline{\mathbf{35}}\oplus\cdots$
$\mathbf{10}\otimes\mathbf{10}\otimes\mathbf{10}\otimes\mathbf{10}$ | $\mathbf{1}\oplus_{8}\mathbf{8}\oplus_{10}\mathbf{10}\oplus_{6}\overline{\mathbf{10}}\oplus_{15}\mathbf{27}\oplus_{6}\mathbf{28}\oplus_{4}\overline{\mathbf{28}}\oplus\cdots$
$\mathbf{10}\otimes\mathbf{10}\otimes\mathbf{10}\otimes\overline{\mathbf{10}}$ | $\mathbf{1}\oplus_{8}\mathbf{8}\oplus_{7}\mathbf{10}\oplus_{10}\overline{\mathbf{10}}\oplus_{18}\mathbf{27}\oplus_{10}\mathbf{28}\oplus_{3}\overline{\mathbf{28}}\oplus\cdots$
Table 4: Relevant Clebsch-Gordan decompositions needed to form a color-singlet bound state with one colored scalar. Direct Product | Decomposition
---|---
$\mathbf{3}\otimes\overline{\mathbf{3}}$ | $\mathbf{1}\oplus\mathbf{8}$
$\mathbf{8}\otimes\mathbf{8}$ | $\mathbf{1}\oplus_{2}\mathbf{8}\oplus\mathbf{10}\oplus\overline{\mathbf{10}}\oplus\mathbf{27}$
$\mathbf{3}\otimes{\mathbf{3}}\otimes{\mathbf{3}}$ | $\mathbf{1}\oplus_{2}\mathbf{8}\oplus\mathbf{10}$
$\mathbf{3}\otimes{\mathbf{3}}\otimes{\mathbf{8}}$ | $\mathbf{1}\oplus_{3}\mathbf{8}\oplus\mathbf{10}\oplus\overline{\mathbf{10}}\oplus\mathbf{27}$
${\mathbf{3}}\otimes{\mathbf{6}}\otimes\mathbf{8}$ | $\mathbf{1}\oplus_{3}\mathbf{8}\oplus_{2}\mathbf{10}\oplus\overline{\mathbf{10}}\oplus_{2}\mathbf{27}\oplus\mathbf{35}$
$\overline{\mathbf{3}}\otimes\overline{\mathbf{3}}\otimes\mathbf{6}$ | $\mathbf{1}\oplus_{2}\mathbf{8}\oplus\mathbf{10}\oplus\mathbf{27}$
$\overline{\mathbf{3}}\otimes{\mathbf{8}}\otimes\mathbf{15}$ | $\mathbf{1}\oplus_{4}\mathbf{8}\oplus_{3}\mathbf{10}\oplus_{2}\overline{\mathbf{10}}\oplus_{4}\mathbf{27}\oplus\cdots$
${\mathbf{8}}\otimes\mathbf{8}\otimes{\mathbf{10}}$ | $\mathbf{1}\oplus_{4}\mathbf{8}\oplus_{4}\mathbf{10}\oplus_{2}\overline{\mathbf{10}}\oplus_{5}\mathbf{27}\oplus\cdots$
$\overline{\mathbf{3}}\otimes{\mathbf{8}}\otimes{\mathbf{8}}\otimes\mathbf{15}^{\prime}$ | $\mathbf{1}\oplus_{6}\mathbf{8}\oplus_{8}\mathbf{10}\oplus_{3}\overline{\mathbf{10}}\oplus_{11}\mathbf{27}\oplus\cdots$

##  Appendix B Scalar loop functions

In this appendix, we collect the well-known scalar one-loop functions
appearing in the loop-induced diagrams $H\to gg$ and $S\to gg$.

$\displaystyle I_{q}(x)$ $\displaystyle=$ $\displaystyle 2x-x\
(1-4x)f(x),~{}~{}~{}I_{s}(x)=-(1+2xf(x)),$ $\displaystyle f(x)$
$\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{ccc}\dfrac{1}{2}\left(\ln\left(\dfrac{1+\sqrt{1-4x}}{1-\sqrt{1-4x}}\right)-i\pi\right)^{2}&&\text{for
}x<\dfrac{1}{4}\\\ &&\\\
-2\left(\arcsin\left(\dfrac{1}{2\sqrt{x}}\right)\right)^{2}&&\text{for
}x>\dfrac{1}{4}\end{array}\right.$ (30)

## References

  * [1] O. Eberhardt, V. Miralles and A. Pich, _Constraints on coloured scalars from global fits_ , 2106.12235.
  * [2] J. Alimena et al., _Searching for long-lived particles beyond the Standard Model at the Large Hadron Collider_ , _J. Phys. G_ 47 (2020) 090501 [1903.04497].
  * [3] D. Curtin et al., _Long-Lived Particles at the Energy Frontier: The MATHUSLA Physics Case_ , _Rept. Prog. Phys._ 82 (2019) 116201 [1806.07396].
  * [4] G. R. Farrar and P. Fayet, _Phenomenology of the production, decay, and detection of new hadronic states associated with supersymmetry_ , _Physics Letters B_ 76 (1978) 575.
  * [5] J. Barnard, P. Cox, T. Gherghetta and A. Spray, _Long-Lived, Colour-Triplet Scalars from Unnaturalness_ , _JHEP_ 03 (2016) 003 [1510.06405].
  * [6] A. de la Puente and A. Szynkman, _Long-lived Colored Scalars at the LHC_ , _Eur. Phys. J. C_ 76 (2016) 124 [1504.07293].
  * [7] W. J. Marciano, _Exotic New Quarks and Dynamical Symmetry Breaking_ , _Phys. Rev. D_ 21 (1980) 2425.
  * [8] R. Chivukula, M. Golden and E. H. Simmons, _Six jet signals of highly colored fermions_ , _Phys. Lett. B_ 257 (1991) 403.
  * [9] A. V. Manohar and M. B. Wise, _Flavor changing neutral currents, an extended scalar sector, and the Higgs production rate at the CERN LHC_ , _Phys. Rev._ D74 (2006) 035009 [hep-ph/0606172].
  * [10] N. V. Krasnikov, _The Phenomenology of scalar color octets_ , _JETP Lett._ 62 (1995) 7 [hep-ph/9506431].
  * [11] M. I. Gresham and M. B. Wise, _Color octet scalar production at the LHC_ , _Phys. Rev. D_ 76 (2007) 075003 [0706.0909].
  * [12] T. Han, I. Lewis and Z. Liu, _Colored Resonant Signals at the LHC: Largest Rate and Simplest Topology_ , _JHEP_ 12 (2010) 085 [1010.4309].
  * [13] A. Hayreter and G. Valencia, _LHC constraints on color octet scalars_ , _Phys. Rev. D_ 96 (2017) 035004 [1703.04164].
  * [14] A. Hayreter and G. Valencia, _Color-octet scalar decays to a gluon and an electroweak gauge boson in the Manohar-Wise model_ , _Phys. Rev. D_ 102 (2020) 115033 [1810.04048].
  * [15] V. Miralles and A. Pich, _LHC bounds on colored scalars_ , _Phys. Rev. D_ 100 (2019) 115042 [1910.07947].
  * [16] S. Dawson, E. Eichten and C. Quigg, _Search for supersymmetric particles in hadron-hadron collisions_ , _Phys. Rev. D_ 31 (1985) 1581.
  * [17] J. Butterworth et al., _PDF4LHC recommendations for LHC Run II_ , _J. Phys. G_ 43 (2016) 023001 [1510.03865].
  * [18] D. B. Clark, E. Godat and F. I. Olness, _ManeParse : A Mathematica reader for Parton Distribution Functions_ , _Comput. Phys. Commun._ 216 (2017) 126 [1605.08012].
  * [19] A. Buckley, J. Ferrando, S. Lloyd, K. Nordström, B. Page, M. Rüfenacht et al., _LHAPDF6: parton density access in the LHC precision era_ , _Eur. Phys. J. C_ 75 (2015) 132 [1412.7420].
  * [20] W. Beenakker, R. Hopker, M. Spira and P. M. Zerwas, _Squark and gluino production at hadron colliders_ , _Nucl. Phys. B_ 492 (1997) 51 [hep-ph/9610490].
  * [21] D. Goncalves-Netto, D. Lopez-Val, K. Mawatari, T. Plehn and I. Wigmore, _Sgluon Pair Production to Next-to-Leading Order_ , _Phys. Rev. D_ 85 (2012) 114024 [1203.6358].
  * [22] P. Richardson and D. Winn, _Simulation of Sextet Diquark Production_ , _Eur. Phys. J. C_ 72 (2012) 1862 [1108.6154].
  * [23] J. M. Arnold, M. Pospelov, M. Trott and M. B. Wise, _Scalar Representations and Minimal Flavor Violation_ , _JHEP_ 01 (2010) 073 [0911.2225].
  * [24] B. A. Dobrescu, G. D. Kribs and A. Martin, _Higgs Underproduction at the LHC_ , _Phys. Rev. D_ 85 (2012) 074031 [1112.2208].
  * [25] A. Davies and X.-G. He, _Tree Level Scalar Fermion Interactions Consistent With the Symmetries of the Standard Model_ , _Phys. Rev. D_ 43 (1991) 225.
  * [26] X.-G. He and G. Valencia, _An extended scalar sector to address the tension between a fourth generation and Higgs searches at the LHC_ , _Phys. Lett. B_ 707 (2012) 381 [1108.0222].
  * [27] G. R. Farrar, R. Mackeprang, D. Milstead and J. P. Roberts, _Limit on the mass of a long-lived or stable gluino_ , _JHEP_ 02 (2011) 018 [1011.2964].
  * [28] A. Idilbi, C. Kim and T. Mehen, _Pair Production of Color-Octet Scalars at the LHC_ , _Phys. Rev. D_ 82 (2010) 075017 [1007.0865].
  * [29] Sherpa collaboration, _Event Generation with Sherpa 2.2_ , _SciPost Phys._ 7 (2019) 034 [1905.09127].
  * [30] J. Bellm et al., _Herwig 7.2 release note_ , _Eur. Phys. J. C_ 80 (2020) 452 [1912.06509].
  * [31] T. Sjöstrand, S. Ask, J. R. Christiansen, R. Corke, N. Desai, P. Ilten et al., _An introduction to PYTHIA 8.2_ , _Comput. Phys. Commun._ 191 (2015) 159 [1410.3012].
  * [32] GEANT4 collaboration, _GEANT4–a simulation toolkit_ , _Nucl. Instrum. Meth. A_ 506 (2003) 250.
  * [33] S. Ovyn, X. Rouby and V. Lemaitre, _DELPHES, a framework for fast simulation of a generic collider experiment_ , 0903.2225.
  * [34] DELPHES 3 collaboration, _DELPHES 3, A modular framework for fast simulation of a generic collider experiment_ , _JHEP_ 02 (2014) 057 [1307.6346].
  * [35] J. Alwall et al., _A Standard format for Les Houches event files_ , _Comput. Phys. Commun._ 176 (2007) 300 [hep-ph/0609017].
  * [36] N. Desai and P. Z. Skands, _Supersymmetry and Generic BSM Models in PYTHIA 8_ , _Eur. Phys. J. C_ 72 (2012) 2238 [1109.5852].
  * [37] W. Kilian, T. Ohl, J. Reuter and C. Speckner, _QCD in the Color-Flow Representation_ , _JHEP_ 10 (2012) 022 [1206.3700].
  * [38] T. Ohl, _O’mega: An Optimizing matrix element generator_ , _AIP Conf. Proc._ 583 (2002) 173 [hep-ph/0011243].
  * [39] T. Ohl, _O’Mega \ & WHIZARD: Monte Carlo event generator generation for future colliders_, _AIP Conf. Proc._ 578 (2001) 638 [hep-ph/0011287].
  * [40] P. Stienemeier, S. Braß, P. Bredt, W. Kilian, N. Kreher, T. Ohl et al., _WHIZARD 3.0: Status and News_ , in _International Workshop on Future Linear Colliders_ , 4, 2021, 2104.11141.
  * [41] H. Brooks and P. Skands, _Coherent showers in decays of colored resonances_ , _Phys. Rev. D_ 100 (2019) 076006 [1907.08980].
  * [42] I. Begic, _A coherent treatment of qcd radiation from supersymmetric quarks in the antenna shower formalism a coherent treatment of qcd radiation from supersymmetric quarks in the antenna shower formalism_ , Master’s thesis, Monash University, 2019.
  * [43] N. Desai, F. Domingo, J. S. Kim, R. R. d. A. Bazan, K. Rolbiecki, M. Sonawane et al., _Constraining electroweak and strongly charged long-lived particles with CheckMATE_ , 2104.04542.
  * [44] M. Fairbairn, A. C. Kraan, D. A. Milstead, T. Sjostrand, P. Z. Skands and T. Sloan, _Stable Massive Particles at Colliders_ , _Phys. Rept._ 438 (2007) 1 [hep-ph/0611040].
  * [45] CMS collaboration, _Search for decays of stopped exotic long-lived particles produced in proton-proton collisions at $\sqrt{s}=$ 13 TeV_, _JHEP_ 05 (2018) 127 [1801.00359].
  * [46] ATLAS collaboration, _Search for heavy charged long-lived particles in the ATLAS detector in 36.1 fb -1 of proton-proton collision data at $\sqrt{s}=13$ TeV_, _Phys. Rev. D_ 99 (2019) 092007 [1902.01636].
  * [47] ATLAS collaboration, _A search for the decays of stopped long-lived particles at $\sqrt{s}=13$ TeV with the ATLAS detector_, 2104.03050.
  * [48] CMS collaboration, _Search for long-lived charged particles in proton-proton collisions at $\sqrt{s}=$ 13 TeV_, _Phys. Rev. D_ 94 (2016) 112004 [1609.08382].
  * [49] ATLAS collaboration, _Search for heavy long-lived charged $R$-hadrons with the ATLAS detector in 3.2 fb-1 of proton–proton collision data at $\sqrt{s}=13$ TeV_, _Phys. Lett. B_ 760 (2016) 647 [1606.05129].
  * [50] CMS collaboration, _Search for Fractionally Charged Particles in $pp$ Collisions at $\sqrt{s}=7$ TeV_, _Phys. Rev. D_ 87 (2013) 092008 [1210.2311].
  * [51] G. Busoni, A. De Simone, T. Jacques, E. Morgante and A. Riotto, _Making the Most of the Relic Density for Dark Matter Searches at the LHC 14 TeV Run_ , _JCAP_ 03 (2015) 022 [1410.7409].
  * [52] A. Albert et al., _Recommendations of the LHC Dark Matter Working Group: Comparing LHC searches for dark matter mediators in visible and invisible decay channels and calculations of the thermal relic density_ , _Phys. Dark Univ._ 26 (2019) 100377 [1703.05703].
  * [53] R. Feger, T. W. Kephart and R. J. Saskowski, _LieART 2.0 – A Mathematica application for Lie Algebras and Representation Theory_ , _Comput. Phys. Commun._ 257 (2020) 107490 [1912.10969].
  * [54] S. Coleman, _The clebsch‐gordan series for su(3)_ , _Journal of Mathematical Physics_ 5 (1964) 1343 [https://doi.org/10.1063/1.1704245].