# Non-uniform Lattices of Large Systole Containing a Fixed 3-Manifold Group

Paige Hillen  University of California, Santa Barbara

(Date: March 2024)

###### Abstract.

Let $d$ be a square free positive integer and $\mathbb{Q}(\sqrt{d})$ a totally
real quadratic field over $\mathbb{Q}$. We show there exists an arithmetic
lattice $\mathcal{L}$ in $SL(8,\mathbb{R})$ with entries in the ring of
integers of $\mathbb{Q}(\sqrt{d})$ and a sequence of lattices $\Gamma_{n}$
commensurable to $\mathcal{L}$ such that the systole of the locally symmetric
finite volume manifold

$\Gamma_{n}\diagdown SL(8,\mathbb{R})\diagup SO(8)$

goes to infinity as $n\rightarrow\infty$, yet every $\Gamma_{n}$ contains the
same hyperbolic 3-manifold group $\Pi$, a finite index subgroup of the
arithmetic hyperbolic 3-manifold vol3. Notably, such an example does not exist
in rank one, so this is a feature unique to higher rank lattices.

PAIGE HILLEN

##  1\. Introduction

The relationship between the the systolic genus, the minimal genus surface
subgroup, and the systole, the minimal length of a non-contractible closed
geodesic, is notably different in higher rank. In 2012, Belolipetsky [1] (see
also [2]) showed that in the hyperbolic setting, the systolic genus is bounded
from below in terms of the systole. In contrast, Long and Reid [13] found a
family of sequences of lattices in $SL(3,\mathbb{R})$ (commensurable to an
arbitrary non-uniform arithmetic lattice not commensurable to
$SL(3,\mathbb{Z})$) with systole going to infinity, yet each contains a genus
3 surface subgroup. They found a similar result in the uniform case. Hence in
higher rank, the systolic genus is not linked to the systole in the same way
as in the hyperbolic setting.

Our result continues in this line of research. We expand Long and Reid’s
result to the existence of a fixed 3-manifold group in a sequence of
commensurable lattices with arbitrarily large systole. More specifically, we
find an infinite family of non-uniform arithmetic lattices in
$SL(8,\mathbb{R})$ each with a sequence of abstractly commensurable lattices
whose systole $\rightarrow\infty$, however every lattice in the sequence
contains the same hyperbolic 3-manifold group. Our non-uniform arithmetic
lattices are indexed by square free numbers $d\geq 2$: for each such $d$, we
consider the integral special unitary group

$SU(I_{8},\mathcal{O}_{d},\tau):=\\{A\in
SL(8,\mathcal{O}_{d}):\tau(A)^{T}A=I_{8}\\}<SL(8,\mathbb{R})$

where $I_{8}=$ the identity matrix in $SL(8,\mathbb{R})$, $\mathcal{O}_{d}$ is
the ring of integers of $\mathbb{Q}(\sqrt{d})$ and
$\tau\in\text{Gal}(\mathbb{Q}(\sqrt{d})/\mathbb{Q})$ is the non-trivial
involution sending $\sqrt{d}$ to $-\sqrt{d}$.

######  Theorem 1.

Fix square free $d\in\mathbb{Z}_{>1}$ and let

$\mathcal{L}=SU(I_{8};\mathcal{O}_{d},\tau).$

Then there exists a sequence of irreducible non-uniform arithmetic lattices
$\Gamma_{n}<SL(8,\mathbb{R})$ abstractly commensurable to $\mathcal{L}$ such
that

$\text{sys}(\Gamma_{n})\rightarrow\infty$

as $n\rightarrow\infty$, yet every $\Gamma_{n}$ contains a fixed hyperbolic
3-manifold group $\Pi$, a finite index subgroup of vol3.

In this setting, rank and determinant up to $\tau$-Hermitian square classify
SU-equivalent $\tau$-Hermitian forms [10]. In turn, equivalent
$\tau$-Hermitian forms yield abstractly commensurable integral special unitary
groups. Hence $SU(I_{8},\mathcal{O}_{d},\tau)$ is commensurable to
$SU(J,\mathcal{O}_{d},\tau)$ for any non-degenerate $\tau$-Hermitian form $J$
over $\mathbb{Q}(\sqrt{d})^{8}$ such that det$(J)$ is a $\tau$-Hermitian
square.

Critical to the proof is an 8 dimensional version of the family of discrete
and faithful representations $\rho_{v}$ of the hyperbolic 3-manifold vol3
found by Cooper, Long, and Thistlethwaite [9] (see also [7]). From this
family, we obtain a sequence of representations of vol3 into lattices
$SU(J,\mathcal{O}_{d},\tau)$ for non-degenerate forms $J$. To ensure the
systole is going to infinity, we intersect with congruence subgroups
$\Lambda_{p}$ for primes $p$. After intersecting, we cannot guarantee vol3 is
contained in the lattice, but by carefully choosing the primes $p$, we can
guarantee the fundamental group of a certain fixed finite sheeted cover of
vol3 is still contained in each lattice.

It is worth remarking that our finite index subgroup of vol3, $\Pi$, contains
surface subgroups. Therefore our result provides an alternative proof to Long
and Reid’s [13] result that systolic genus can be bounded in a sequence of
commensurable higher rank lattices whose systole is diverging to infinity. The
image of surface subgroups of vol3 in $SL(4,\mathbb{R})$ under Cooper, Long,
and Thistlethwaite’s representation $\rho_{v}$ are interesting in their own
right. The representation restricted to a surface subgroup is not on the
Hitchin component, since at $v=2$ the representation lies in SO(3,1) and is
hence not Borel Anosov. However, for $v\geq 2$, $\rho_{v}$ is discrete and
faithful. If there is a value of $v$ such that $\rho_{v}$ is Borel Anosov on
some surface subgroup, this would provide an example of a Borel Anosov surface
representation outside the Hitchin component, answering a question by Canary
[4].

##  2\. Background

Consider the following measures of the geometry and topology of a space:

######  Definition 2.1.

The systole of a Riemannian manifold $M$, denoted sys$(M)$, is the minimal
length of a non-contractible closed geodesic in $M$. We will interchangeably
refer to the systole of $M$ as the systole of $\pi_{1}(M)$.

######  Definition 2.2.

Let $S_{g}$ denote the closed surface with genus g. The minimal $g$ such that
$\pi_{1}(S_{g})$ injects into $\pi_{1}(M)$ is called the systolic genus of
$M$, denoted sysg$(M)$ (See [1]).

For hyperbolic manifolds, the behavior of the systole puts some restrictions
on the behavior of the systolic genus. This is clear for hyperbolic surfaces,
since Besicovitch’s inequality

$\text{sys}(S_{g})^{2}\leq 2\text{area}(S_{g})$

combined with Gauss-Bonnet theorem (for a hyperbolic metric on $S_{g}$),

$\text{area}(S_{g},\omega)\leq 4\pi(g-1)$

together show that a sequence of closed hyperbolic surfaces with systole
$\rightarrow\infty$ requires the genera of the surfaces also tends toward
infinity. In fact, Belolipetsky showed this generalizes to higher dimensions
in the following sense (see Theorem 5.1 in [1]):

######  Theorem 2.

Let $M_{k}=\mathbb{H}^{n}/\Gamma_{k}$ be a sequence of finite volume closed
hyperbolic $n$-manifolds such that sys$(\Gamma_{k})\rightarrow\infty$ as
$k\rightarrow\infty$. Then the systolic genus sysg$(M_{k})\rightarrow\infty$
as well.

Intuitively, if the manifolds are getting complicated enough for the systole
to grow arbitrarily large, their topology must be getting complicated as well.
However, once we leave the hyperbolic setting, this is no longer true. In
particular, the presence of flats seems to allow enough space for systoles to
grow, without the systolic genus increasing.

######  Definition 2.3.

A discrete subgroup $\Gamma<SL(n,\mathbb{R})$ is a lattice if the quotient
orbifold

$M_{\Gamma}:=\Gamma\backslash SL(n,\mathbb{R})/SO(n)$

has finite volume. Note that $M_{\Gamma}$ is a manifold if and only if
$\Gamma$ is torsion free. A lattice $\Gamma$ is uniform (or cocompact) if
$M_{\Gamma}$ is compact. Otherwise, $\Gamma$ is non-uniform.

In the theory of lattices, passing to a finite index subgroup usually results
in only minor differences. Since we often like to ignore these minor
differences, we usually care about lattices up to commensurability.

######  Definition 2.4.

Lattices $\Gamma_{1},\Gamma_{2}<SL(n,\mathbb{R})$ are commensurable if

$[\Gamma_{1}:\Gamma_{1}\cap\Gamma_{2}]<\infty$ and
$[\Gamma_{2}:\Gamma_{1}\cap\Gamma_{2}]<\infty$

Equivalently, their corresponding manifolds $M_{\Gamma_{1}}$ and
$M_{\Gamma_{2}}$ have a common finite-sheeted cover (i.e.
$M_{\Gamma_{1}\cap\Gamma_{2}})$. Lattices
$\Gamma_{1},\Gamma_{2}<SL(n,\mathbb{R})$ are abstractly commensurable if a
finite index subgroup of $\Gamma_{1}$ is isomorphic to a finite index subgroup
of $\Gamma_{2}$.

We review a construction of a family of non-uniform arithmetic lattices in
$SL(n,\mathbb{R})$. See [15] Chapter 6.8 for more details.

######  Construction 1.

Fix a square-free $d\in\mathbb{Z}_{>1}$ and $n\geq 3$. Then

  * •

$F=\mathbb{Q}(\sqrt{d})$ is a totally real algebraic number field,

  * •

$\tau\in\text{Gal}(F/\mathbb{Q})$ is the non-trivial involution sending
$\sqrt{d}\mapsto-\sqrt{d}$, and

  * •

$\mathcal{O}_{d}=\begin{cases}\mathbb{Z}[\sqrt{d}]&d\equiv 2,3\text{ mod }4\\\
\mathbb{Z}[\frac{1+\sqrt{d}}{2}]&d\equiv 1\text{ mod }4\\\ \end{cases}$ is the
ring of integers of $F$.

Let $J\in M_{n\times n}(F)$ be sesqui-symmetric matrix with respect to $\tau$
(i.e. $J^{T}=\tau(J)$). We view $J$ as a $\tau$-Hermitian form on the
$F$-vector space $F^{n}$. The associated special unitary group is

$SU(J;F,\tau)=\\{M\in SL(n,F):M^{*}JM=J\\}\vspace{0.2cm}$

where $M^{*}:=\tau(M)^{T}$. The integer points of this group form the
associated integral special unitary group

$SU(J;\mathcal{O}_{d},\tau)=\\{M\in
SL(n,\mathcal{O}_{d}):M^{*}JM=J\\}\vspace{0.2cm}.$

######  Definition 2.5.

$\tau$-Hermitian forms $J$ and $J^{\prime}$ on $F^{n}$ are SU-equivalent if
$J^{\prime}=P^{*}JP$ for some change of basis matrix $P\in GL(n,F)$.

Observe that

$SU(P^{*}JP;F,\tau)=P^{-1}(SU(J;F,\tau))P.$

The corresponding integral groups may not be conjugate, but $P\in GL(n,F)$ is
in the commensurator of $SU(J;\mathcal{O},\tau)$. Hence
$SU(J;\mathcal{O},\tau)$ and $SU(P^{*}JP;\mathcal{O},\tau)$ are abstractly
commensurable. Our interest in commensurability classes of arithmetic lattices
with entries in $\mathcal{O}_{d}$ leads to the question: when are two
$\tau$-Hermitian forms over $F^{n}$ equivalent? By Landherr [10], (see [12]
section 3) we have the following classification:

Let $F=\mathbb{Q}[\sqrt{d}]$ and $\tau\in\text{Gal}(F/\mathbb{Q})$ the
involution sending $\sqrt{d}\mapsto-\sqrt{d}$. Suppose $n\geq 3$. Then an
equivalence class of $\tau$-Hermitian forms on $F^{n}$ is uniquely determined
by

  * •

the rank of $J$, and

  * •

the discriminant up to Hermitian square.111Since
$(\sqrt{d})\tau(\sqrt{d})=-d$, there is no signature in this setting.

######  Proposition 1.

Suppose $J$ is a full rank $\tau$-Hermitian form on $F^{n}$. Then the group
$SU(J;\mathcal{O}_{d},\tau)$ as constructed above is a non-uniform irreducible
arithmetic lattice. Moreover, the commensurability class is uniquely
determined by the determinant of $J$ up to Hermitian squares.

###### Proof.

That $SU(J;\mathcal{O}_{d},\tau)$ is an arithmetic lattice follows from
Proposition (6.8.14) in [15]. The same proposition tells us our lattice is
non-uniform if and only if there exists a nonzero $x\in F^{n}$ such that
$x^{*}Jx=0$.
By the classification above, $SU(J;\mathcal{O}_{d},\tau)$ is commensurable to
$SU(\text{diag}(1,-1,-\text{det}(J),1\dots,1);\mathcal{O}_{d},\tau)$. For
$x=[1,1,0,\dots,0]$, clearly

$x^{*}\text{diag}(1,-1,-\text{det}(J),1,\dots,1)x=0.$

Thus $SU(J;\mathcal{O}_{d},\tau)$ is non-uniform. ∎

Next, we find a way to control the systole of lattices such as those built by
construction 1. There is a 1-1 correspondence between closed geodesics in
$M_{\Gamma}$ and $\Gamma-$conjugacy classes of semi-simple elements in
$\Gamma$. After a nice choice of scaling, the length of the geodesic
corresponding to the conjugacy class of a semi simple $\gamma\in\Gamma$ is
equal to the translation length of $\gamma$ on the geodesic it leaves
invariant in $SL(n,\mathbb{R})/SO(n)$. Let $l(\gamma)$ denote this length.
Hence

$\text{sys}(M_{\Gamma})=\text{inf}\\{l(\gamma):\text{ semi-simple
}\gamma\in\Gamma\\}$

The translation lengths are then bounded from below in terms of the trace
([11], theorem 3.1):

######  Theorem 3.

Trace-Length Bounds. Let $\gamma\in SL(n,\mathbb{R})$ be semi-simple with
$|\text{tr}(\gamma)|\geq 1$. Then

$l(\gamma)\geq\sqrt{2}\text{arccosh}\Big{(}\text{max}\Big{\\{}1,\frac{\text{tr}(\gamma)}{n}\Big{\\}}\Big{)}.$

Since arccosh is a monotonically increasing function on $[1,\infty)$, we can
control the lower bound for the systole of $M_{\Gamma}$ by controlling the
lower bound for the traces of semisimple elements in $\Gamma$. Our tool for
controlling the lower bound of the trace is principal congruence subgroups.

######  Definition 2.6.

Let $\mathfrak{p}$ be a prime ideal in $\mathcal{O}$, the ring of integers of
some algebraic number field $F$. Then

$\Lambda_{\mathfrak{p}}=\text{Ker}(\pi:SL(n,\mathcal{O})\rightarrow
SL(n,\mathcal{O}/\mathfrak{p}))$

is the principal congruence subgroup of level $\mathfrak{p}$.

Since $\mathcal{O}/\mathfrak{p}$ is a finite field, $\Lambda_{\mathfrak{p}}$
is a normal subgroup of finite index in $SL(n,\mathcal{O})$. Rephrasing
corollary 5.2 in [11],

######  Theorem 4.

Let $p$ be a prime which does not ramify in $\mathcal{O}_{d}$ and let
$\Lambda_{p}=\Lambda_{\langle p\rangle}$ be as above. Suppose
$\gamma\in\Lambda_{p}$ is a nontrivial semisimple element, and let $l(\gamma)$
denote the length of the geodesic which $\gamma$ leaves invariant. Then

$l(\gamma)\geq\frac{2\sqrt{2}}{n}\text{arccosh}\Big{(}\frac{p-n}{n}\Big{)}.$

Since $\Lambda_{p}$ is a finite index subgroup of $SL(n,\mathcal{O})$, given
any lattice $\mathcal{L}<SL(n,\mathcal{O})$, we have
$\Lambda_{p}\cap\mathcal{L}$ is a finite index subgroup of $\mathcal{L}$.
Hence $\Lambda_{p}\cap\mathcal{L}$ is a lattice commensurable to $\mathcal{L}$
with systole bounded from below in terms of $p$.

##  3\. Result

Theorem 1. Fix square free $d\in\mathbb{Z}_{>1}$ and let

$\mathcal{L}=SU(I_{8};\mathcal{O}_{d},\tau).$

Then there exists a sequence of irreducible non-uniform arithmetic lattices
$\Gamma_{n}<SL(8,\mathbb{R})$ abstractly commensurable to $\mathcal{L}$ such
that

$\text{sys}(\Gamma_{n})\rightarrow\infty$

as $n\rightarrow\infty$, yet every $\Gamma_{n}$ contains a fixed hyperbolic
3-manifold group $\Pi$, a finite index subgroup of vol3.

Remark. The 3-manifold group $\Pi$ contains surface groups, so each
$\Gamma_{n}$ contains a fixed surface group. In particular, the systolic genus
of these lattices is bounded from above for all $n$.

The hyperbolic 3-manifold vol3 is an arithmetic manifold with the third lowest
volume in the census. The fundamental group of vol3, which we will also refer
to as vol3, has presentation

$\text{vol3}=\langle a,b\ |\ aabbABAbb;\ aBaBabaaa\rangle\vspace{0.15cm}$

where $A=a^{-1}$ and $B=b^{-1}$. The hyperbolic representation of vol3 into
$SO(3,1)$ admits discrete and faithful deformations in $SL(4,\mathbb{R})$. An
explicit one-parameter family of these deformations was found by Cooper, Long,
and Thistlethwaite [7]. An orbifold cover of vol3, denoted vol3$/\langle
u\rangle$, has a simpler representation.

$\rho_{v}:\text{vol3}/\langle u\rangle\rightarrow
SL(4,\mathbb{Q}(v,\sqrt{v^{2}-4}))\vspace{0.15cm}$

Two elements, denoted here $u$ and $c$, generate vol3$/\langle u\rangle$. To
recover the manifold group, one can use the relations $a=u^{2}c$ and
$b=(aua)^{-1}u$. In practice, we work with the orbifold cover when interacting
with the explicit matrix representation.

The hyperbolic representation is at $v=2$ and for real values $v\geq 2$ the
representation is the holonomy of a real projective structure on vol3, and is
thus discrete and faithful. It is not clear how to specialize $v$ to ensure
the entries of the image all lie in a ring of integers over some field, which
is a requirement for them to live in an arithmetic lattice. As luck would have
it, there are 16 specific elements $\\{1,g_{1},\dots,g_{15}\\}\in$
vol3$/\langle u\rangle$ such that

  * •

$\mathcal{B}_{v}=\\{\rho_{v}(1),\rho_{v}(g_{1}),\dots,\rho_{v}(g_{15})\\}$ is
a basis for the vector space $M_{4\times 4}(\mathbb{R})$.

  * •

The action of vol3$/\langle u\rangle$ on $\mathcal{B}_{v}$ yields a
representation $\eta_{v}:\text{vol3}/\langle u\rangle\rightarrow
SL(16,\mathbb{Z}[\frac{v}{2},\frac{\sqrt{v^{2}-4}}{2}])$

For the remainder of this paper, we make the substitution $t=\frac{v}{2}$.
Hence

$\eta_{t}:\text{vol3}/\langle u\rangle\rightarrow
SL(16,\mathbb{Z}[t,\sqrt{t^{2}-1}]))$

is discrete and faithful for $t\geq 1$.

In an attempt to find an integral representation of smaller dimension, we
study invariant subspaces of this 16 dimensional representation. By
considering eigenspaces of elements in the centralizer of
$\eta_{t}(\text{vol3}/\langle u\rangle)$, we were able to find an 8
dimensional invariant subspace, whose corresponding representation has entries
in $\mathbb{Z}[t,\sqrt{t^{2}-1}]$. Let

$\omega_{t}:\text{vol3}/\langle u\rangle\rightarrow
SL(8,\mathbb{Z}[t,\sqrt{t^{2}-1}])$

denote this representation. The matrices $\omega_{t}(u)$ and $\omega_{t}(c)$
are listed in the appendix along with the relations which recover $a$ and $b$
from $u$ and $c$. This representation is conjugate to two copies of the
original representation $\rho_{t}$, and hence faithful.

##  4\. Proof

For the remainder of this paper, fix square free $d\in\mathbb{Z}_{>1}$ and let

$\mathcal{L}=SU(I_{8};\mathcal{O}_{d},\tau).$

We start by finding a sequence of arithmetic lattices commensurable to
$\mathcal{L}$ which contain vol3.

######  Lemma 1.

There exists a family of Hermitian forms $J_{t}\in SL(8,\mathbb{Q}(t))$ such
that

  * •

For $t>>0$, $J_{t}$ is full rank with det$(J_{t})$ equal to a square in
$\mathbb{Q}(t)$, and

  * •

$\omega_{t}(\text{vol3})<SU(J_{t};\mathbb{Q}(t,\sqrt{t^{2}-1}),\tau)$.

Moreover, there exists a sequence $t_{n}\rightarrow\infty$ such that

$\omega_{t_{n}}(\text{vol3})<SU(J_{t_{n}};\mathcal{O}_{d},\tau)$

for all $n\in\mathbb{N}$. By Proposition 1, the
$SU(J_{t_{n}};\mathcal{O}_{d},\tau)$ are abstractly commensurable to
$\mathcal{L}$ for $n>>0$.

###### Proof.

The first part of the lemma follows from a computation: We solve for
$J=J_{t}\in GL(8,\mathbb{R})$ such that $\omega_{t}(u)^{*}J\omega_{t}(u)=J$
and $\omega_{t}(v)^{*}J\omega_{t}(v)=J$. By replacing $J$ with $J+J^{*}$, we
can ensure $J$ is sesqui-symmetric. There are 4 free variables in the solution
for $J$, and by making a choice of numerical value for each free
variable222Even leaving all four free variables in $J$ as unknowns, the
determinant of $J$ is a square in $\mathbb{Q}(t)$. Thus choosing values in
$\mathbb{Q}(t)$ for the free variables does not change the resulting
commensurability class of the lattice. we obtain a $\tau$-Hermitian form $J$
which is full rank (for all but finitely many choices of $t$) . Indeed,

$\text{det}(J_{t})=\frac{16(3-4t^{2})^{4}}{(1-4t^{2})^{2}}$

which is a square in $\mathbb{Q}(t)$. Moreover, for $t>>0$, this polynomial is
nonzero, so $J_{t}$ is full rank.

To guarantee $\omega_{t}($vol3$)$ lies in an integral special unitary group,
is it necessary for the entries of $\omega_{t}(a)$ and $\omega_{t}(b)$ to lie
in $\mathcal{O}_{d}$. It is sufficient to choose $t\in\mathbb{N}$ so that
$\sqrt{t^{2}-1}\in\mathcal{O}_{d}$. Equivalently, we need to find infinitely
many integral solutions $(t,y)$ to Pell’s equation:

$t^{2}-dy^{2}=1.$

It is well known that for any positive non-square $d\in\mathbb{Z}$, Pell’s
equation has a fundamental solution $(t_{1},y_{1})\in\mathbb{N}^{2}$ and the
other solutions are exactly the integers $(t_{n},y_{n})$ such that

$u^{n}=t_{n}+y_{n}\sqrt{d}$

for $u=t_{1}+y_{1}\sqrt{d}$. Therefore, for this sequence
$\\{t_{n}\\}_{n=1}^{\infty}$,

$\omega_{t_{n}}(vol3)<SU(J_{t_{n}};\mathcal{O}_{d},\tau)$

for all $n\in\mathbb{N}$. We can write $t_{n}=\frac{1}{2}(u^{n}+u^{-n})$, so
the sequence $t_{n}\rightarrow\infty$ as $n\rightarrow\infty$. ∎

Our next goal is to find finite index subgroups of our above arithmetic
lattices $SU(J_{t_{n}},\mathcal{O}_{d},\tau)$ whose systole goes to infinity
as $n\rightarrow\infty$. This is accomplished by intersecting with the
principal congruence subgroups

$\Lambda_{p}:=Ker(\pi_{p}:SL(8,\mathcal{O}_{d})\rightarrow
SL(8,\mathcal{O}_{d}/p\mathcal{O}_{d}))$

for carefully chosen primes $p$. By Theorem 4 , the systole of a lattice in
$\Lambda_{p}$ can be made arbitrarily large by making $p$ large [11].

A straightforward way to proceed from here, would be to argue
$\omega_{t_{n}}(\text{vol3})$ is also contained in our congruence subgroups,
$\Lambda_{p}$. It is unclear how to arrange this, but by carefully choosing
primes $p_{n}$, we can arrange $\omega_{t_{n}}(\Pi)<\Lambda_{p_{n}}$ for
$\Pi<$ vol3 a finite index subgroup. More specifically, set

$\Pi=\text{Ker}(\omega_{0}:\text{vol3}\rightarrow
SL(8,\mathbb{Z}[i]))\vspace{0.15cm}.$

At $t=0$, one can check that $|\omega_{0}(\text{vol3})|=320$ computationally,
so indeed $\Pi$ is finite index in vol3.

Consider the following diagram:

$\displaystyle
vol3$$\displaystyle\Pi$$\omega_{t}$$\omega_{0}$$SU(J_{t};\mathcal{O}_{d},\tau)$$SL(8,\mathcal{O}_{d})$$SL(8,\mathcal{O}_{d}/p\mathcal{O}_{d})$$Ker(\pi_{p})=\Lambda_{p}$$\pi_{p}$$SL(8,\mathbb{Z}[i])$$\pi_{p}$$SL(8,\mathbb{Z}[i]/p\mathbb{Z}[i])$$f_{*}$

The primes we choose will divide $t$. Since our choice of $t$ solves Pell’s
equation, we have $(pk)^{2}-1=dy^{2}$ for integers $k$ and $d$. Thus

$f:\mathbb{Z}[i]/p\mathbb{Z}[i]\rightarrow\mathcal{O}_{d}/p\mathcal{O}_{d}$

induced by sending $\overline{i}\mapsto\overline{y\sqrt{d}}$ is a ring
homomorphism. This induces a group homomorphism

$f_{*}:SL(8,\mathbb{Z}[i]/p\mathbb{Z}[i])\rightarrow
SL(8,\mathcal{O}_{d}/p\mathcal{O}_{d}).$

Our purpose in constructing this diagram is in the observation that, as long
as the diagram commutes, we can guarantee $\omega_{t}(\Pi)<\Lambda_{p}$. In
fact, by inspecting the representation (see Appendix), $p$ dividing $t$ is
sufficient to guarantee the diagram commutes.
Suppose we have primes $p_{n}\rightarrow\infty$ dividing $t_{n}$. We then set

$\Gamma_{n}=\Lambda_{p_{n}}\cap
SU(J_{t_{n}};\mathcal{O}_{d},\tau)\vspace{0.15cm}$

As a principal congruence subgroup, $\Lambda_{p_{n}}$ is a finite index
subgroup of $SL(8,\mathcal{O}_{d})$. Thus $\Gamma_{n}$ is a finite index
subgroup of the lattice $SU(J_{t_{n}};\mathcal{O}_{d},\tau)$, and whence also
a lattice commensurable to $\mathcal{L}$. In the next lemma, we find such a
sequence of primes $p_{n}$.

######  Lemma 2.

Let $\\{t_{n}\\}$ be as in Lemma 1. Possibly passing to a subsequence of
$t_{n}$, there exists a sequence of primes $p_{n}\rightarrow\infty$ with each
$p_{n}$ dividing $t_{n}$. Hence

$\omega_{t_{n}}(\Pi)<\Lambda_{p_{n}}$

for all $n\in\mathbb{N}$.

###### Proof.

For each $n$, we would like a prime $p_{n}$ dividing $t_{n}$, but not dividing
$t_{m}$ for $m<n$. The latter condition will ensure a subsequence of $p_{n}$
is strictly increasing. Some results from number theory will help us
accomplish this. We start with the following definition from [3]:

######  Definition 4.1.

A pair of algebraic integers $(\alpha,\beta)$ is called a Lucas pair if
$\alpha+\beta$ and $\alpha\beta$ are non-zero coprime rational integers and
$\frac{\alpha}{\beta}$ is not a root of unity.

By [5], (or see Theorem A in [3]) if $(\alpha,\beta)$ is a Lucas pair, then
for $n>>0$ the $n$-th term of the sequence

$S_{n}:=\alpha^{n}+\beta^{n}$

has a primitive prime divisor, i.e. a prime $p_{n}$ such that $p_{n}$ divides
$S_{n}$ but not $S_{m}$ for $m<n$.

Claim: Let $u=t_{1}+y_{1}\sqrt{d}$ be as in the proof of Lemma 1. Then
$(u,1/u)$ is a Lucas pair.

Observe that

$2t_{n}=u^{n}+\Big{(}\frac{1}{u}\Big{)}^{n}$

Therefore, pending the claim above, the sequence $2t_{n}$ has a corresponding
sequence of primitive prime divisors, $p_{n}$ for $n>>0$. Passing to a
subsequence, we may assume the $p_{n}$ are strictly increasing, each prime
$p_{n}\neq 2$ and does not ramify in $\mathcal{O}_{d}$. Hence each $p_{n}$
divides $t_{n}$ and $p_{n}\rightarrow\infty$.

Proof of Claim: Since $u\in\mathbb{Z}[\sqrt{d}]\subseteq\mathcal{O}_{d}$, we
know $u$ is an algebraic integer. Since $t_{1}^{2}-dy_{1}^{2}=1$,

$\displaystyle\frac{1}{u}$ $\displaystyle=t_{1}-y_{1}\sqrt{d}$

Hence $\frac{1}{u}$ is also an algebraic integer. Further,
$u+\frac{1}{u}=2t_{1}$ and $u(\frac{1}{u})=1$, so $u+\frac{1}{u}$ and
$u(\frac{1}{u})$ are non-zero coprime rational integers. Moreover
$\frac{u}{1/u}=u^{2}$ is not a root of unity. Hence $(u,\frac{1}{u})$ is a
Lucas pair.

∎

Now we put these pieces together to prove the main theorem:

Theorem 1. Fix square free $d\in\mathbb{Z}_{>1}$ and let

$\mathcal{L}=SU(I_{8};\mathcal{O}_{d},\tau).$

Then there exists a sequence of irreducible non-uniform arithmetic lattices
$\Gamma_{n}<SL(8,\mathbb{R})$ abstractly commensurable to $\mathcal{L}$ such
that

$\text{sys}(\Gamma_{n})\rightarrow\infty$

as $n\rightarrow\infty$, yet every $\Gamma_{n}$ contains a fixed hyperbolic
3-manifold group $\Pi$, a finite index subgroup of vol3.

###### Proof.

For each $n\in\mathbb{N}$, let

$\Lambda_{p_{n}}=\text{Ker}(\pi_{p_{n}}:SL(8,\mathcal{O}_{d})\rightarrow
SL(8,\mathcal{O}_{d}/p_{n}\mathcal{O}_{d}))$

be the principal congruence subgroup for $p_{n}$ as in Lemma 2. Now define

$\Gamma_{n}=\Lambda_{p_{n}}\cap SU(J_{t_{n}};\mathcal{O}_{d},\tau)$

for $t_{n}$ from Lemma 1. Since $\Lambda_{p_{n}}$ is a finite index subgroup
of $SL(8,\mathcal{O}_{d})$, $\Gamma_{n}$ is a finite index subgroup of
$SU(J_{t_{n}};\mathcal{O}_{d},\tau)$. Therefore the $\Gamma_{n}$ are lattices
abstractly commensurable to $\mathcal{L}$. Since $\omega_{t_{n}}$ is
injective, the diagram commutes by Lemma 2, and

$\Pi<\text{Ker}(\pi_{p_{n}}\circ\omega_{0}(\text{vol3})),$

we must have $\Pi\cong\omega_{t_{n}}(\Pi)<\Gamma_{n}$. By Theorem 4,
sys$(\Gamma_{n})\rightarrow\infty$ as $n\rightarrow\infty$.

∎

##  5\. Examples

######  Example 1.

We describe how to find the first few terms of the sequence $(t_{n},p_{n})$ in
the case that $d=3$. Since $3\not\equiv 1$(mod $4$), we have
$\mathcal{O}_{3}=\mathbb{Z}[\sqrt{3}]$.

To guarantee the entries of $\omega_{t}(u)$ and $\omega_{t}(v)$ lie in
$\mathcal{O}_{3}$, we solve for $t$ in Pell’s equation:

$t^{2}-3y^{2}=1.$

The fundamental solution is $(t_{1},y_{1})=(2,1)$. Let $u=2+\sqrt{3}$. Powers
of $u$ allow us to find al the other solutions. In order to ensure the diagram
commutes and $\omega_{t}(\Pi)<\Lambda_{p}$ we need to find corresponding
primes $p_{n}$ such that $t_{n}=0\text{ mod }p_{n}$.

$n$ | $u^{n}$ | $t_{n}$ | $p_{n}$
---|---|---|---
$1$ | $2+\sqrt{3}$ | $2$ | $2$
$2$ | $7+4\sqrt{3}$ | $7$ | $7$
$3$ | $26+15\sqrt{3}$ | $26$ | $13$
$4$ | $97+56\sqrt{3}$ | $97$ | $97$
$5$ | $362+209\sqrt{5}$ | $362$ | $181$

Observe each prime does not divide any of the previous values of $t_{n}$.

######  Example 2.

Now let $d=5$. Since $5\equiv 1$ (mod $4$),
$\mathcal{O}_{5}=\mathbb{Z}[\frac{1+\sqrt{5}}{2}]$. We first solve Pell’s
equation:

$t^{2}-5y^{2}=1.$

The fundamental solution is $(t_{1},y_{1})=(9,4)$. Following the same process
as above, we find the first 5 terms of the sequences $t_{n}$ and $p_{n}$.

$n$ | $u^{n}$ | $t_{n}$ | $p_{n}$
---|---|---|---
$1$ | $9+4\sqrt{5}$ | $9$ | $3$
$2$ | $161+72\sqrt{5}$ | $161$ | $7$
$3$ | $2889+1292\sqrt{5}$ | $2889$ | $107$
$4$ | $51841+23184\sqrt{5}$ | $51841$ | $1103$
$5$ | $930249+416020\sqrt{5}$ | $930249$ | $2521$

##  6\. Further Questions and Observations

  1. (1)

Thin 3-manifold Group: Our 8 dimensional representation $\omega$ is conjugate
to two copies of the original representation $\rho$. By the proof of Theorem
2.1 in [8], $\rho_{t}(\text{vol3})$ is Zariski dense in $SL(4,\mathbb{R})$.
Therefore the Zariski closure of $\omega_{t}($vol3$)$ in $SL(8,\mathbb{R})$ is
a subgroup isomorphic to $SL(4,\mathbb{R})$. We also observe that for any of
our chosen values of $t$, $\omega_{t}(\Pi)$ is infinite index in
$SU(J_{t};\mathcal{O}_{d},\tau)$. To see this, observe that
$SU(J_{t};\mathcal{O}_{d},\tau)$ is an irreducible arithmetic lattice of
$\mathbb{R}-$rank $\geq 2$. Thus, by Margulis’s Normal Subgroup Theorem, if
$\omega_{t}($vol3$)$ were finite index in $SU(J_{t};\mathcal{O}_{d},\tau)$,
then the abelianization of $\omega_{t}($vol3$)$ would be finite. However, vol3
is an arithmetic hyperbolic 3-manifold with positive first Betti number. Thus
vol3 must have infinite virtual first Betti number, so the abelianization
cannot be finite (see [6]). If would be interesting to know whether
$\omega_{t}($vol3$)$ is infinite index in the intersection of
$SU(J_{t};\mathcal{O}_{d},\tau)$ and Zcl$(\omega_{t}($vol3$))\cong
SL(4,\mathbb{R})$. If so, then $\omega_{t}($vol3$)$ is a thin 3-manifold
group.

  2. (2)

Non-Hitchin Surface Subgroups: At $v=2$ the image $\rho_{2}(\text{vol3})$ lies
in $SO(3,1)$, so every non-trivial semi-simple element has a pair of
eigenvalues with the same modulus. Thus $\rho_{2}$ is not Borel Anosov (for
example, by the characterization of Anosov in theorem 4.3 of [14]). Moreover,
the restriction to any subgroup is also not Borel Anosov. Hence for any value
of $v\geq 2$ and any surface subgroup $H$, $\rho_{v}(H)$ is a discrete and
faithful representation in $SL(4,\mathbb{R})$ which is not on the Hitchin
component. This is notable, since many examples of interesting representations
of surface subgroups in the literature are in the Hitchin component. In
particular, if one can find a surface subgroup of $\rho_{v}($vol3$)$ which is
Borel Anosov for some value of $v>2$, then it would be an example of a non-
Hitchin Borel Anosov representation, answering a question posed by Canary [4].

##  7\. Appendix

###  7.1. The original 4 dimensional representation of vol3

The hyperbolic 3-manifold vol3 has presentation

$\text{vol3}=\langle a,b\ |\ aabbABAbb;\ aBaBabaaa\rangle\vspace{0.1cm}$

where $A=a^{-1}$ and $B=b^{-1}$. The orbifold cover is generated by $u$ and
$c$, of order 4 and 2 respectively. To recover vol3, we have $a=u^{2}c$ and
$b=(aua)^{-1}u$. The image of $u$ and $c$ under the original 4D representation
from section 2.4 of [9] with the substitution $t=\frac{v}{2}$ are:

$\rho_{t}(u)=\left(\begin{array}[]{cccc}1&0&0&0\\\ 0&1&0&0\\\
0&0&\sqrt{\frac{(t^{2}-1)}{(2+t^{2})}}&1\\\
0&0&-\frac{(1+2t^{2})}{(2+t^{2})}&-\sqrt{\frac{(t^{2}-1)}{(2+t^{2})}}\end{array}\right)$

$\rho_{t}(c)=\left(\begin{array}[]{cccc}\frac{1}{2}(t+\sqrt{2+t^{2}})&0&\frac{1}{2}(1-t^{2}-t\sqrt{2+t^{2}})&0\\\
0&\frac{1}{2}(t-\sqrt{2+t^{2}})&0&\frac{1}{2}(-1+t^{2}-t\sqrt{2+t^{2}})\\\
1&0&\frac{1}{2}(-t-\sqrt{2+t^{2}})&0\\\
0&-1&0&\frac{1}{2}(-t+\sqrt{2+t^{2}})\end{array}\right)$

There is one $\tau-$Hermitian form (up to scalar multiples) which both
generators preserve:

$M_{t}=\left(\begin{array}[]{cccc}-\frac{(2\sqrt{2+t^{2}})}{(2t+t^{3}-\sqrt{2+t^{2}}+t^{2}\sqrt{2+t^{2}})}&0&0&0\\\
0&\frac{2(2+t^{2})}{((1+2t^{2})(1-t^{2}+t\sqrt{2+t^{2}}))}&0&0\\\ 0&0&1&0\\\
0&0&0&\frac{(2+t^{2})}{(1+2t^{2})}\end{array}\right)$ Note that for $t=1$, we
have $\sqrt{t^{2}-1}=0$, so the involution $\tau$ is trivial. Since $M_{1}$ is
a diagonal matrix with signature $(3,1)$, it is clear that $\rho_{1}$ lies in
$SO(3,1)$.

###  7.2. The explicit 8 dimensional representation of vol3:

Our 8D representation which is integral for values of $t$ solving Pell’s
equation is generated by:

$\omega_{t}(u)=\left(\begin{array}[]{cccccccc}1&0&0&2t&-2t&2t&0&0\\\
0&1&0&t-\sqrt{t^{2}-1}&-t+\sqrt{t^{2}-1}&t-\sqrt{t^{2}-1}&1&0\\\
0&0&1&-1&1&-1&0&0\\\ 0&0&1&0&0&0&0&0\\\
0&0&1&0&0&1&-t+\sqrt{t^{2}-1}&-t-\sqrt{t^{2}-1}\\\
0&0&1&-1&0&1&-t+\sqrt{t^{2}-1}&-t-\sqrt{t^{2}-1}\\\ 0&0&0&0&0&0&0&-1\\\
0&0&0&0&0&0&1&0\end{array}\right)$

$\omega_{t}(c)=\left(\begin{array}[]{cccccccc}0&0&1&0&0&0&0&0\\\
0&0&0&1&0&0&0&0\\\ 1&0&0&0&0&0&0&0\\\ 0&1&0&0&0&0&0&0\\\ 0&0&0&0&0&0&-1&1\\\
0&0&0&0&0&0&-1&0\\\ 0&0&0&0&0&-1&0&0\\\
0&0&0&0&1&-1&0&0\par\end{array}\right)$

## References

  * [1] M. Belolipetsky, On $2$-systoles of hyperbolic $3$-manifolds, Geom. Funct. Anal. 23 (2013), 813–827.
  * [2] M. Belolipetsky and C. Dória, Free subgroups of 3-manifold groups, Groups Geom. Dyn. 14 (2020), 243–254
  * [3] Y. Bilu, G. Hanrot, and P. M. Voutier, Existence of primitive divisors of Lucas and Lehmer numbers, J. Reine Angew. Math. 539 (2001), 75–122.
  * [4] R. Canary, Anosov Representations: Informal Lecture Notes, (2021).
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  * [6] D. Cooper, D. D. Long and A.W. Reid, On the betti numbers of arithmetic hyperbolic 3-manifolds, Geometry and Topology. 11 (2007), 226–2276.
  * [7] D. Cooper, D. D. Long and M. Thistlethwaite, Computing varieties of representations of hyperbolic 3-manifolds into $SL(4,\mathbb{R})$, Experimental Math. 15 (2006), 291–305.
  * [8] D. Cooper, D. D. Long, and M. Thistlethwaite, Constructing noncongruence subgroups of flexible hyperbolic 3-manifold groups, Proceedings of the American Mathematical Society. 137 (2009), 3943–3949.
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Department of Mathematics,
University of California,
Santa Barbara, CA 93106.
Email<EMAIL_ADDRESS>


|

# NLP-CIC at SemEval-2020 Task 9$\mathpunct{:}$ Analysing sentiment in code-
switching language using a simple deep-learning classifier

Jason Angel
CIC, Instituto Politécnico Nacional
Mexico City, Mexico
<EMAIL_ADDRESS>Segun Taofeek Aroyehun
CIC, Instituto Politécnico Nacional
Mexico City, Mexico
<EMAIL_ADDRESS>Antonio Tamayo
CIC, Instituto Politécnico Nacional
Mexico City, Mexico
<EMAIL_ADDRESS>Alexander Gelbukh
CIC, Instituto Politécnico Nacional
Mexico City, Mexico
www.gelbukh.com

###### Abstract

Code-switching is a phenomenon in which two or more languages are used in the
same message. Nowadays, it is quite common to find messages with languages
mixed in social media. This phenomenon presents a challenge for sentiment
analysis. In this paper, we use a standard convolutional neural network model
to predict the sentiment of tweets in a blend of Spanish and English
languages. Our simple approach achieved a F1-score of $0.71$ on test set on
the competition. We analyze our best model capabilities and perform error
analysis to expose important difficulties for classifying sentiment in a code-
switching setting.

##  1 Introduction

00footnotetext:  This work is licensed under a Creative Commons Attribution
4.0 International Licence. Licence details:
http://creativecommons.org/licenses/by/4.0/.

The phenomenon of combining two or more languages in the same message is known
as code-switching or code-mixing [Gumperz, 1982, Myers-Scotton, 1993]. Code-
switching is an indicator of bilingual competence [Hamers and Blanc, 1999],
and it is also motivated by social and cultural factors such as social status,
race, age, etc. [Kim, 2006]. Although this phenomenon has been studied
extensively in linguistics [Pfaff, 1979, Poplack, 1980, Gumperz, 1982, Myers-
Scotton, 1993, Milroy and Muysken, 1995, Lipski, 2005, Martínez, 2010, Auer,
2013], it is still challenging for machines to process mixed natural
languages. Code-switching is notoriously present on social media posts and
chats such as Twitter, Facebook or WhatsApp; consequently making it more
difficult to process the sentiment expressed in such contents.

In this work, we present a Convolutional Neural Network (CNN) system to
predict the sentiment of a given code-mixed tweet. The sentiment labels are
either positive, negative, or neutral, and the languages involved are English
and Spanish. Our best model utilizes only Spanish word embeddings from tweets
[Deriu et al., 2017] and does not require manual feature engineering.

##  2 Related work

Sentiment analysis is a widely studied task in monolingual, multilingual and
cross-lingual settings. For instance, monolingual opinion mining in a
multilingual context [Boiy and Moens, 2009], multilingual sentiment analysis
[Balahur and Turchi, 2014], and cross-lingual polarity detection [Demirtas and
Pechenizkiy, 2013]. Language-independent approaches to sentiment analysis
include the use of emoticons [Davies and Ghahramani, 2011] or emoticons and
noisy labels [Narr et al., 2012]. Sentiment analysis has not been extensively
studied on code-switched content. A possible reason is the paucity of large
annotated data covering several language pairs or combinations. The Sentimix
shared task is an effort to address this problem on the Spanish-English and
Hindi-English language pairs for sentiment analysis.

Analysing opinions in tweets that blend Spanish and English is a difficult
task [Vilares et al., 2017]. Vilares et al. [Vilares et al., 2017] obtained an
accuracy of 59.34% using lexical, syntactic, and N-gram features. They
concluded that the task is challenging because of the presence of noise,
difficulties with language identification and POS tagging, and the lack of
annotated code-mixed lexicons and a large dataset.

Another line of work related to code-switching is the development of
contextual and static multilingual or cross-lingual text representations which
cover multiple languages in the same vector space. Examples include LASER
[Artetxe and Schwenk, 2019], MUSE [Lample et al., 2017], and multilingual
BERT111https://github.com/google-
research/bert/blob/a9ba4b8d7704c1ae18d1b28c56c0430d41407eb1/multilingual.md.
These representations can be used to encode inputs for deep learning models.
The effectiveness of these text representation approaches on code-switched
texts remains an open question.

##  3 Methodology

We describe the Spanglish dataset and our submitted models.

###  3.1 The Sentimix Spanglish dataset

The Sentimix Spanglish dataset [Patwa et al., 2020] consists of a list of
tweets blending English and Spanish text. The dataset is divided into train,
development, and test sets holding $12002$, $2998$, and $3789$ samples for
each set respectively. This dataset provides detailed annotations per word,
which besides the English and Spanish tags includes named entities, in-word
mixes, ambiguous and foreign words. A major part of the dataset is Spanish,
being the mode language for $65.1\%$ of all tweets, whereas English is only
$20.5\%$. We noticed that language statistics are proportionally distributed
across the three partitions of the dataset. With regards to sentiment labels,
Table 1 presents the label distribution for training and development sets.

Set | Sentiment | Samples | Proportion
---|---|---|---
| Negative (N) | 2023 | 16.86%
train | Neutral (T) | 3974 | 33.11%
| Positive (P) | 6005 | 50.03%
| Negative (N) | 506 | 16.88%
development | Neutral (T) | 994 | 33.16%
| Positive (P) | 1498 | 49.97%
Table 1: Distribution of sentiment labels in the dataset

###  3.2 Text normalization as pre-processing

We normalize the input text using Ekphrasis
222https://github.com/cbaziotis/ekphrasis to address the noise in social media
text. Specifically, the normalization process consists of:

  * •

Mapping URL, email, percent, money, phone, user, time, date, and numbers to a
unique descriptive token.

  * •

Labeling stylistic patterns such as uppercase, elongated, repeated,
emphasized, and censored words. Examples of these can be seen in Table 3.

  * •

Word transformation suitable for social media content, such as word segmenter,
spellchecker, and tokenizer.

We apply the normalization to the entire dataset and noticed it had a marginal
influence over the Spanish text. We did not apply normalization to the Spanish
part of the text.

###  3.3 Neural Network architectures

#### CNN model.

We used a standard architecture [Kim, 2014] with standard values as
hyperparameters, consisting of a single convolution layer with multiple filter
sizes of 2, 3 and 4, each of them with 100 filters, followed by max-pooling
and a dropout layer, to finally stack a fully connected layer which outputs
the results. We used ReLU as activation function, Adam as optimizer and cross-
entropy loss function as the optimization objective.The hyperparameters are:
vocabulary size of 15000, batch size of 64, and dropout probability of 0.5. We
predict using the best epoch result obtained out of 5 epochs.

Our embedding vectors were initialized using only 200 dimension Spanish word
embeddings 333https://www.spinningbytes.com/resources/wordembeddings/ which
were trained on a collection of tweets [Deriu et al., 2017]. We did not use
English embeddings because in previous experiments we noticed it does not
contribute enough to the performance gain. This is probably due to the fact
that Spanish is the majority language in the dataset, as we mentioned before.

#### GRU model.

This submission used a bidirectional Gated Recurrent Unit (GRU)444We chose GRU
over LSTM because the former has less parameters to learn. over English-
Spanish aligned word embeddings of dimension $300$ [Conneau et al., 2017]. The
hidden state of the GRU is of dimension 512. We used dropout with probability
of $0.1$ on the outputs of the embedding and GRU layers. Also, layer
normalization is applied on the hidden representation generated by the GRU
layer. A fixed representation for a given tweet is derived by taking the
average of the backward and forward hidden representation of the GRU layer.
This serves as input to a dense layer with softmax activation function which
output a probability distribution over the three labels. We used AdamW as
optimizer to minimize the negative log-likelihood loss function for $10$
epochs. We use a batch size of $256$ and learning rate of $0.001$.

##  4 Results and discussion

We obtained a precision score of $0.807$ and a recall score of $0.647$ on the
competition, thus a $0.71$ F1 score was calculated for the competition
rank555https://competitions.codalab.org/competitions/20789#learn_the_details-
result (Codalab user name (our team): ajason08). Moreover, according to the
competition guidelines, we also report class-wise F1 scores for test set, in
which we achieved a lower score 666The Codalab scorer is using a less strict
metric, probably a weighted scheme.. Under this metric, our aforementioned
architecture achieved results shown at Table 2. These results are similar to
those obtained on our experiments on the development set, where we achieved a
performance of 0.45 using macro-F1.

| CNN model | GRU model
---|---|---
| Precision | Recall | F1-score | Precision | Recall | F1-score
Positive | 0.901 | 0.71 | 0.794 | 0.854 | 0.404 | 0.548
Negative | 0.547 | 0.375 | 0.445 | 0.155 | 0.341 | 0.213
Neutral | 0.082 | 0.403 | 0.136 | 0.065 | 0.374 | 0.110
Table 2: Class-wise performance scores

###  4.1 Error analysis

Here an additional analysis is done to examine the results of our best model
(the CNN model) via removing the text normalization stage and running cross-
validation. Moreover, we perform error analysis to find common situations in
which our system fails to make the correct prediction. These analysis were
done using development set labels.

We found that removing the normalization step reduces the model performance in
some points. Specifically, the macro-F1 score without the presence of
Ekphrasis normalization is 0.42, hence a performance reduction of 3%.

We present five categories that highlight some of the errors that our best
model made. The analysis was made over a stratified sample of $300$ examples
on the development set. Due to space limitations, we show a few examples of
these categories along with the gold label (L) and our prediction (P) in Table
3. We consider the following categories:

See pages - of table3_aspdf.pdf

  * •

Difficult: examples our best model classified wrongly and we consider hard
because of informal style which leads to out-of-vocabulary issues, and the
high level of common sense or pragmatics understanding probably involved.
Also, we noticed that often the effect of code-switching is sarcastic.

  * •

Negative tendency: the use of highly negative words (e.g. vulgar expressions)
seems to bias the model towards predicting a negative sentiment, however, this
only occurs in the English portion of the text. We notice it because some
English words appears in the embedding vocabulary, probably from people doing
Spanish-English code-switching.

  * •

Advertising: These are tweets promoting some product or service. Surprisingly,
most of them have neutral or negative label. Conversely, our best model rates
them mostly as conveying positive or neutral sentiment.

  * •

Ambiguous labels: we hypothesize that the model gets confused when it finds
samples with a very similar narrative but different labels. For example, the
tweet “mami esta llorando con la voz kids"777”My mom is crying while watching
’La Voz Kids’” is referring to a hilarious situation, and it receives a
“neutral" label, however a second tweet “llego a casa a encontrarme a mami
llorando viendo inside out”888”I arrive at home and I found my mom crying
while watching ’Inside Out’” although is referring to a very similar
situation, it receives a “positive" label, we noticed that for both tweets the
source of the sentiment comes from the relation between the adult and emotive
show. We think these instances are a source of confusion for the model.

  * •

Doubtful labels: some tweet labels can be related to subjectivity in the
annotation process. We consider that these examples are incorrectly labeled.

##  5 Conclusions

Code-switching is an interesting problem holding an important presence in
social media, which combined with informal writing style increases the
challenges for social media processing such as sentiment analysis. We
experimented with the Sentimix Spanglish dataset using CNN model and only
Spanish embeddings. We achieve a precsion, recall, and F1 score of 0.80, 0.64,
and 0.71 respectively. Our analyses suggest that a deep learning model can be
easily biased by the presence of cue words such as vulgar expressions for
sentiment analysis. We found that this occurs mostly when the cue word is in
English. This observation requires a deeper analysis. We also highlight the
need to address complex language usage such as informality and sarcasm.

Furthermore, we also pointed out that subjectivity in the annotation of
sentiment labels is a problem that deserves to be addressed. We plan to test
contextual multilingual embeddings (e.g. Bert) and leverage the language tags
and other non-linguistic constructs such as hashtags and emojis.

## References

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# Search for the anomalous $ZZZ$ and $ZZ\gamma$ gauge couplings through the
process $e^{+}e^{-}\to ZZ$ with unpolarized and polarized beams

V<EMAIL_ADDRESS>Department of Physics, Kutahya
Dumlupinar University, 43100, Turkiye.
S<EMAIL_ADDRESS>Department of Medical Imaging Techniques,
Zonguldak Bülent Ecevit University, 67100, Zonguldak, Turkiye.
E<EMAIL_ADDRESS>Department of Physics, Sinop University,
Turkiye.
M<EMAIL_ADDRESS>Department of Physics, Sivas Cumhuriyet
University, 58140, Sivas, Turkiye.

###### Abstract

This work offers the constraints on the anomalous neutral triple gauge
couplings (aNTGC) for the process $e^{+}e^{-}\to ZZ$ at the CLIC with
$\sqrt{s}=3$ TeV. The realistic CLIC detector environments and their effects
are considered in our analysis. The study is planned for the decays of
producted $Z$ bosons to a pair of charged leptons (electrons or muons) and
neutrino pairs. The bounds on the aNTGCs defining $CP$-conserving
$C_{\widetilde{B}W}/{\Lambda^{4}}$ coupling and three $CP$-violating
$C_{WW}/{\Lambda^{4}}$, $C_{BW}/{\Lambda^{4}}$, and $C_{BB}/{\Lambda^{4}}$
couplings are obtained. Also, the effects and advantages of polarization for
incoming electron beams in these calculations are investigated.

###### pacs:

12.60.-i, 14.70.Hp, 14.70.Bh
Keywords: Electroweak interaction, Models beyond the Standard Model, Anomalous
couplings.

##  I Introduction

Gauge boson pair productions in the examined processes has a significant role
in the tests of the non-Abelian structure of the Standard Model (SM) and
exploring physics beyond the SM. Thus, $Z\gamma$ and $ZZ$ diboson productions
can be studied with the anomalous neutral triple gauge couplings (aNTGCs)
originating from high-dimensional operators defined by the effective
Lagrangian method in a model-independent approach. Howeever, $ZZ$ production
from diboson productions has the smallest cross-section among all diboson
processes but is highly competitive for measurements and searches due to the
generally high signal-to-background ratio with leptonic or neutrino decay
channels Aaboud:2019bza .

On the other hand, CLIC is a potential future collider that stands out due to
its ability to test new physics scenarios through high-energy $e^{-}e^{+}$
collisions in a clean experimental environment with low background compared to
$pp$ colliders. It also provides the opportunity for detailed study of the
various processes and production mechanisms that occur in $e^{-}e^{+}$
colliders. The CLIC will be operated at three energy stages, center-of-mass
energies of $\sqrt{s}=$380 GeV, 1.5 TeV and 3 TeV Boland:2016yqz , with an
integrated luminosity of 5 ab-1 at $\sqrt{s}=$3 TeV stage. CLIC is planned
with $\pm 80\%$ polarization for the electron beam Roloff:2018tvb . The weight
of each event is calculated by dividing the integrated luminosity between
negative and positive polarizations of the electron beam in a 4:1 ratio,
resulting in integrated luminosities of $\mathcal{P}(e^{-})=-80\%$ for ${\cal
L}_{\text{int}}=4$ ab-1 and $\mathcal{P}(e^{-})=+80\%$ for ${\cal
L}_{\text{int}}=1$ ab-1 Roloff:2018tvb . Beam polarizations offer advantages
in future $e^{+}e^{-}$ colliders, including enhanced signals, reduced
backgrounds, improved control of systematic uncertainties, and the ability to
study the chiral properties of SM and BSM processes through additional
observables.

While the Cabibbo-Kobayashi-Maskawa (CKM) mixing matrix can explain
$CP$-violation, the SM does not account for sufficient $CP$-violation effects
to explain the current matter-antimatter asymmetry in the universe, resulting
in the baryogenesis problem Shaposhnikovab:1987vbn ; Nelson:1992ded .
Therefore, there is a need for new sources of $CP$-violation, one of
Sakharov’s three conditions, to explain the baryon asymmetry of the universe
Sakharov:1967hla . Investigating the $CP$-conserving and $CP$-violation
couplings in the anomalous $Z\gamma\gamma$, $ZZ\gamma$ and $ZZZ$ vertex has
the potential to reveal a new source of $CP$-violation and thus increases
motivation to study the neutral trilinear gauge couplings.

New physics effects beyond the SM can be characterized in terms of the SM
effective field theory (SMEFT), which is used to investigate aNGC within the
SM gauge group. This involves adding high-dimensional operators that are
invariant under $SU(2)_{L}\times U(1)_{Y}$ to the SM Lagrangian, and
determining anomalous couplings from the effective Lagrangian obtained after
electroweak symmetry breaking. Processes involving neutral gauge boson
couplings can be sensitive to dimension-8 operators, which are not contributed
by dimension-6 operators Degrande:2014ydn . Thus, the effective Lagrangian
that accounts for SM interactions and the effects of physics beyond the SM is
given by Degrande:2014ydn

$\displaystyle{\cal L}^{\text{NTGC}}={\cal
L}_{\text{SM}}+\sum_{i}\frac{C_{i}}{\Lambda^{4}}({\cal O}_{i}+{\cal
O}_{i}^{\dagger})$ (1)

where $\Lambda$ represents the new physics scale. On the other hand, $C_{i}$
coefficients are dimensionless parameters and ${\cal O}_{i}$ dimension-8
operators are defined by

$\displaystyle{\cal
O}_{\widetilde{B}W}=iH^{\dagger}\widetilde{B}_{\sigma\rho}W^{\sigma\nu}\\{D_{\nu},D^{\rho}\\}H,$
(2) $\displaystyle{\cal
O}_{BW}=iH^{\dagger}B_{\sigma\rho}W^{\sigma\nu}\\{D_{\nu},D^{\rho}\\}H,$ (3)
$\displaystyle{\cal
O}_{WW}=iH^{\dagger}W_{\sigma\rho}W^{\sigma\nu}\\{D_{\nu},D^{\rho}\\}H,$ (4)
$\displaystyle{\cal
O}_{BB}=iH^{\dagger}B_{\sigma\rho}B^{\sigma\nu}\\{D_{\nu},D^{\rho}\\}H$ (5)

where

$\displaystyle
B_{\sigma\rho}=\left(\partial_{\sigma}B_{\rho}-\partial_{\rho}B_{\sigma}\right),$
(6) $\displaystyle
W_{\sigma\rho}=\sigma^{i}\left(\partial_{\sigma}W_{\rho}^{i}-\partial_{\rho}W_{\sigma}^{i}+g\epsilon_{ijk}W_{\sigma}^{j}W_{\rho}^{k}\right),$
(7)

with $\langle\sigma^{i}\sigma^{j}\rangle=\delta^{ij}/2$ and

$\displaystyle
D_{\mu}\equiv\partial_{\mu}-i\frac{g^{\prime}}{2}B_{\mu}Y-ig_{W}W_{\mu}^{i}\sigma^{i}.$
(8)

Here, $H$ is the Higgs field, $B_{\sigma\rho}$ and $W_{\sigma\rho}$ are the
field strength tensors and $D_{\mu}$ shows the covariant derivative.

For $ZZ$ production at high-energy scale, the most significant contribution of
physics beyond the SM is provided by the interference between the SM and the
dimension-8 operators. Unless interference between the SM and dimension-8 and
dimension-10 operators are both influentially suppressed, the ${\cal
O}({\Lambda^{-8}})$ square of dimension-8 operators cannot be expected to
contain a contribution from physics beyond the SM when the energy scale is
high. At the tree-level, while dimension-6 operators have no effect on the
anomalous neutral gauge boson couplings, the contributions of dimension-8
operators are of the order ${\upsilon^{2}\hat{s}}/{\Lambda^{4}}$.
Nevertheless, dimension-6 operators can have
${\upsilon^{2}\hat{s}}/{\Lambda^{4}}$ order effects on the aNGC at one-loop.
Therefore, the contribution of dim-8 operators appears to be more important
than the one-loop contribution of dimension-6 operators with
$\Lambda\lesssim\sqrt{4\pi\hat{s}/\alpha}$ (Here, $\upsilon$ and $\alpha$ are
vacuum expectation values and fine-structure constant) Degrande:2014ydn .

In the literature, effective Lagrangian for the aNTGCs with dim-6 and dim-8
operators is given by Gounaris:2000wlp

$\displaystyle\begin{split}{\cal
L}=&\frac{e}{m_{Z}^{2}}\Bigg{[}-[f_{4}^{\gamma}(\partial_{\mu}F^{\mu\beta})+f_{4}^{Z}(\partial_{\mu}Z^{\mu\beta})]Z_{\alpha}(\partial^{\alpha}Z_{\beta})+[f_{5}^{\gamma}(\partial^{\sigma}F_{\sigma\mu})+f_{5}^{Z}(\partial^{\sigma}Z_{\sigma\mu})]\widetilde{Z}^{\mu\beta}Z_{\beta}\\\
&-[h_{1}^{\gamma}(\partial^{\sigma}F_{\sigma\mu})+h_{1}^{Z}(\partial^{\sigma}Z_{\sigma\mu})]Z_{\beta}F^{\mu\beta}-[h_{3}^{\gamma}(\partial_{\sigma}F^{\sigma\rho})+h_{3}^{Z}(\partial_{\sigma}Z^{\sigma\rho})]Z^{\alpha}\widetilde{F}_{\rho\alpha}\\\
&-\bigg{\\{}\frac{h_{2}^{\gamma}}{m_{Z}^{2}}[\partial_{\alpha}\partial_{\beta}\partial^{\rho}F_{\rho\mu}]+\frac{h_{2}^{Z}}{m_{Z}^{2}}[\partial_{\alpha}\partial_{\beta}(\square+m_{Z}^{2})Z_{\mu}]\bigg{\\}}Z^{\alpha}F^{\mu\beta}\\\
&+\bigg{\\{}\frac{h_{4}^{\gamma}}{2m_{Z}^{2}}[\square\partial^{\sigma}F^{\rho\alpha}]+\frac{h_{4}^{Z}}{2m_{Z}^{2}}[(\square+m_{Z}^{2})\partial^{\sigma}Z^{\rho\alpha}]\bigg{\\}}Z_{\sigma}\widetilde{F}_{\rho\alpha}\Bigg{]}\end{split}$
(9)

where $\widetilde{Z}_{\mu\nu}=1/2\epsilon_{\mu\nu\rho\sigma}Z^{\rho\sigma}$
$(\epsilon^{0123}=+1)$ is defined by the field strength tensor
$Z_{\mu\nu}=\partial_{\mu}Z_{\nu}-\partial_{\nu}Z_{\mu}$ and similarly the
electromagnetic field tensor $F_{\mu\nu}$ is also defined. Nevertheless,
$f_{4}^{V}$, $h_{1}^{V}$, $h_{2}^{V}$ are the $CP$-violating couplings and
$f_{5}^{V}$, $h_{3}^{V}$, $h_{4}^{V}$ are the $CP$-conserving couplings
$(V=\gamma$, $Z)$. In the SM, these couplings are zero at tree-level. In Eq.
(9), $h_{2}^{V}$ and $h_{4}^{V}$ couplings are concerned with dim-8 operators
while the other couplings to dim-6 operators.

In $SU(2)_{L}\times U(1)_{Y}$ gauge invariance, the couplings in Eq. (9) are
related to the couplings of the operators given in Eqs. (2-5) Rahaman:2020 .
The $CP$-conserving anomalous couplings for the $ZZV^{*}$ vertex with the two
on-shell $Z$ bosons and one off-shell $V^{*}=\gamma,Z$ boson are written by
Degrande:2014ydn

$\displaystyle f_{5}^{Z}=0,$ (10) $\displaystyle
f_{5}^{\gamma}=\frac{\upsilon^{2}m_{Z}^{2}}{4c_{\omega}s_{\omega}}\frac{C_{\widetilde{B}W}}{\Lambda^{4}}.$
(11)

Nevertheless, the $CP$-violating anomalous couplings for the $ZZV^{*}$ vertex
are given as follows

$\displaystyle
f_{4}^{Z}=\frac{m_{Z}^{2}\upsilon^{2}\left(c_{\omega}^{2}\frac{C_{WW}}{\Lambda^{4}}+2c_{\omega}s_{\omega}\frac{C_{BW}}{\Lambda^{4}}+4s_{\omega}^{2}\frac{C_{BB}}{\Lambda^{4}}\right)}{2c_{\omega}s_{\omega}},$
(12) $\displaystyle
f_{4}^{\gamma}=-\frac{m_{Z}^{2}\upsilon^{2}\left(-c_{\omega}s_{\omega}\frac{C_{WW}}{\Lambda^{4}}+\frac{C_{BW}}{\Lambda^{4}}(c_{\omega}^{2}-s_{\omega}^{2})+4c_{\omega}s_{\omega}\frac{C_{BB}}{\Lambda^{4}}\right)}{4c_{\omega}s_{\omega}},$
(13)

where $s_{\omega}$ and $c_{\omega}$ are the sine and cosine of weak mixing
angles $\theta_{w}$.

On the theoretical side, contributions to NTGC in the context of the minimal
supersymmetric standard model (MSSM) Gounaris:2000jsx ; Choudhury:2001yvz ;
Gounaris:2006edf , little Higgs model Dutta:2009tvq , and two-Higgs doublet
model (THDM) Chang:1995gwx ; Grzadkowski:2016evb ; Belusca:2018hhd  have been
discussed previously. There has recently been extensive research on the
potential effects of NTGC in collider phenomenology, for example, $e^{-}e^{+}$
collider Rodriguez:2009rnw ; Ananthanarayan:2014cal ; Rahaman:2016nzs ;
Rahaman:2017qed ; Ellis:2020ekm ; Fu:2021jec ; Ellis:2021rop ; Spor:2022ssp ;
Yang:2022tgw , $pp$ collider Senol:2018gvg ; Rahaman:2019tnp ; Senol:2019ybv ;
Senol:2020hbh ; Yilmaz:2020ser ; Yilmaz:2021dbm ; Hernandez:2021wsz ;
Lombardi:2022tgv  and $\mu^{-}\mu^{+}$ collider Senol:2022psb ; Spor:2022ywc .

The current experimental limits on $C_{BB}/{\Lambda^{4}}$,
$C_{BW}/{\Lambda^{4}}$, $C_{\widetilde{B}W}/{\Lambda^{4}}$ and
$C_{WW}/{\Lambda^{4}}$ couplings at the $95\%$ Confidence Level (C.L.) by
ATLAS and CMS Collaborations at the LHC with center-of-mass energy of 13 TeV
are given in Table 1.

Table 1: The experimental limits at 95$\%$ C.L. on anomalous $C_{BB}/{\Lambda^{4}}$, $C_{BW}/{\Lambda^{4}}$, $C_{\widetilde{B}W}/{\Lambda^{4}}$ and $C_{WW}/{\Lambda^{4}}$ couplings. | Couplings (TeV-4)
---|---
${C_{BB}}/{\Lambda^{4}}$ | ${C_{BW}}/{\Lambda^{4}}$ | ${C_{\widetilde{B}W}}/{\Lambda^{4}}$ | ${C_{WW}}/{\Lambda^{4}}$
ATLAS Aaboud:2018ybz  $Z\gamma\rightarrow\nu\bar{\nu}\gamma$ | [-0.24; 0.24] | [-0.65; 0.64] | [-1.10; 1.10] | [-2.30; 2.30]
($\sqrt{s}=13$ TeV, ${\cal L}_{\text{int}}=36.1$ fb-1)
ATLAS Aaboud:2018onm  $ZZ\rightarrow\ell^{+}\ell^{-}\ell^{\prime+}\ell^{\prime-}$ | [-2.70; 2.80] | [-3.30; 3.30] | [-5.90; 5.90] | [-3.00; 3.00]
($\sqrt{s}=13$ TeV, ${\cal L}_{\text{int}}=36.1$ fb-1)
CMS Sirunyan:2021edk  $ZZ\rightarrow\ell^{+}\ell^{-}\ell^{\prime+}\ell^{\prime-}$ | [-1.20; 1.20] | [-1.40; 1.30] | [-2.30; 2.50] | [-1.40; 1.20]
($\sqrt{s}=13$ TeV, ${\cal L}_{\text{int}}=137$ fb-1)

##  II Generation of events

Our aim in the SMEFT framework is to determine the constraints on the aNTGCs
in $ZZ$ diboson production at the CLIC. We present the tree-level Feynman
diagrams for the process $e^{+}e^{-}\to ZZ$ in Fig. 1, where black dots
indicate the anomalous $ZZ\gamma$ and $ZZZ$ couplings representing new
physics, while the two other diagrams show the SM contributions.

Figure 1: Schematic diagrams for the process $e^{-}e^{+}\to ZZ$ including the
anomalous contribution of $ZZZ$ and $ZZ\gamma$ vertices and the SM.

We have conducted the production of signal events through the process
$e^{+}e^{-}\rightarrow ZZ\rightarrow\ell\ell\nu\nu$ and background events at
the CLIC using the Universal FeynRules Output (UFO) model file
Degrande:2014ydn  with MadGraph5$\\_$aMC@NLO Alwall:2014cvc . For each
coupling, 500k signal and background events were generated. We utilized the
CLIC detector card in Delphes v3.5.0 Favereau:2014qaz  to include realistic
detector effects, and applied PYTHIA v8.306 Bierlich:2022tge  for parton
showering and hadronization. The generated events were analyzed with the
ExRootAnalysis ExRootAnalysis  package and ROOT v6.26 Brun:1997cvb .

The total cross-sections of the $e^{-}e^{+}\rightarrow ZZ$ process at
$\sqrt{s}=3$ TeV at the CLIC are given in Figs. 2-4 for $-80\%$, unpolarized,
and $+80\%$ electron beam polarizations, respectively. The cross-sections are
plotted as a function of the anomalous couplings $C_{BB}/{\Lambda^{4}}$,
$C_{BW}/{\Lambda^{4}}$, $C_{\widetilde{B}W}/{\Lambda^{4}}$, and
$C_{WW}/{\Lambda^{4}}$. Here, only one coupling is run at a time while the
other three couplings were set to zero.

Figure 2: Cross-section of $e^{-}e^{+}\to ZZ$ process in terms of the
anomalous $C_{BB}/{\Lambda^{4}}$, $C_{BW}/{\Lambda^{4}}$,
$C_{\widetilde{B}W}/{\Lambda^{4}}$, $C_{WW}/{\Lambda^{4}}$ couplings at the
$\sqrt{s}=3$ TeV and $P_{e^{-}}=-80\%$. Here, only one coupling is set to 1
TeV-4 at a time while the other three couplings are set to zero.

Figure 3: Same as Fig. 2 but for unpolarized beams $P_{e^{-}}=0\%$.

Figure 4: Same as Fig. 2 but for polarized beams $P_{e^{-}}=80\%$.

The final state $\ell\ell\nu\nu$ of the process $e^{+}e^{-}\rightarrow ZZ$
consists of a pair of charged leptons and neutrinos, coming from the decay of
$Z$ bosons. This process with non-zero effective couplings is assumed as
signal including SM contribution and interference term that is coming from
crossing the diagrams of effective couplings and SM. The two main sources of
SM background processes considered in this paper are: $e^{+}e^{-}\rightarrow
ZZ\rightarrow\ell\ell\nu\nu$, which is the same final state as signal process,
and $e^{+}e^{-}\rightarrow WW\rightarrow\ell\nu\ell\nu$, which includes the
leptonic decay of both $W$ bosons.

The preselection criteria for the analysis include the presence of a dilepton
of the same flavor with opposite charge to form the $Z$ boson. At least two
leptons of the same flavor with opposite charge ($N_{\ell\,(e,\,\mu)}>=2$)
must be present in the event. The transverse momentum of the leading lepton
$\ell^{1}$ must be greater than 30 GeV and the sub-leading lepton $\ell^{2}$
greater than 20 GeV. A cut on the pseudorapidity between the two leptons is
also imposed with a limit of $|\eta|<2.5$. As can be seen from Fig. 5, the
event selection is optimized by imposing
$|E_{T}^{miss}-p_{T}^{\ell\ell}|/p_{T}^{\ell\ell}<0.5$, as the distribution of
transverse momentum balance ratio for the signal deviates significantly from
the SM background for values less than 0.5.

Figure 5: Distribution of transverse momentum balance ratio for signal with
$C_{\widetilde{B}W}/{\Lambda^{4}}=3$ TeV-4, $C_{BB}/{\Lambda^{4}}=3$ TeV-4,
$C_{BW}/{\Lambda^{4}}=3$ TeV-4, $C_{WW}/{\Lambda^{4}}=3$ TeV-4 and relevant
background processes

In the plane transverse to beam direction, the dilepton thrust axis is
indicated with
$\hat{t}=\vec{p}_{T}^{\,\,\ell^{1}}-\vec{p}_{T}^{\,\,\ell^{2}}/|\vec{p}_{T}^{\,\,\ell^{1}}-\vec{p}_{T}^{\,\,\ell^{2}}|$,
where $\vec{p}_{T}^{\,\,\ell^{i}}$ is the transverse momentum vector of lepton
$i$. The transverse momentum vector of the dilepton is
$\vec{p}_{T}^{\,\,\ell\ell}=\vec{p}_{T}^{\,\,\ell^{1}}+\vec{p}_{T}^{\,\,\ell^{2}}$.
The variable $a_{T}^{\ell\ell}$, which is equal to
$a_{T}^{\ell\ell}=|\vec{p}_{T}^{\,\,\ell\ell}\times\hat{t}|$, correspond to
one of two orthogonal components of $p_{T}^{\ell\ell}$ relative to the
dilepton thrust axis $\hat{t}$ Vesterinen:2009ymz ; Abazov:2015wcz . Fig. 6
shows that by applying $a_{T}^{\ell\ell}>40$ GeV, the signal can be separated
from the effect of the $WW$ background.

Figure 6: Transverse component of $p_{T}^{ll}$ according to the thrust axis
for signal with $C_{\widetilde{B}W}/{\Lambda^{4}}=3$ TeV-4,
$C_{BB}/{\Lambda^{4}}=3$ TeV-4, $C_{BW}/{\Lambda^{4}}=3$ TeV-4,
$C_{WW}/{\Lambda^{4}}=3$ TeV-4 and relevant background processes

As seen in Fig.7, choosing $E_{T}^{miss}>200$ provides a well seperation
region between the signals and backgrounds. On the other hand, by limiting the
distance between two charged leptons in the $\eta-\phi$ plane to $\Delta
R({\ell_{1},\ell_{2}})<3.0$, the $WW$ background can be suppressed as seen in
Fig. 8. Additionally, the events must not have any extra leptons with a
transverse momentum greater than 10 GeV.

Figure 7: Distribution of missing energy transverse for signal with
$C_{\widetilde{B}W}/{\Lambda^{4}}=3$ TeV-4, $C_{BB}/{\Lambda^{4}}=3$ TeV-4,
$C_{BW}/{\Lambda^{4}}=3$ TeV-4, $C_{WW}/{\Lambda^{4}}=3$ TeV-4 and relevant
background processes

Figure 8: Distance $\Delta R({\ell_{1},\ell_{2}})$ between leading and sub-
leading charged leptons for signal with $C_{\widetilde{B}W}/{\Lambda^{4}}=3$
TeV-4, $C_{BB}/{\Lambda^{4}}=3$ TeV-4, $C_{BW}/{\Lambda^{4}}=3$ TeV-4,
$C_{WW}/{\Lambda^{4}}=3$ TeV-4 and relevant background processes

If the mass of the $Z$ boson is $m_{Z}$, we determine the invariant mass to be
in the range of $|m_{\ell\ell}-m_{Z}|<20$ GeV to focus on events from the
decay of the two charged leptons seen in Fig. 9. The kinematic distributions
utilized to distinguish the signal from the background in this study are shown
in Figs. 5-9. Each plot is created by applying generator level cuts to
determine the optimum cut set to separate the signals and backgrounds events.
All selected cuts are given in a cut-flow chart for the analysis in Table 2.
Finally, distributions of the transverse momentum of dilepton system
$p^{\ell\ell}_{T}$ after applied cuts in Table 2 for every single coupling is
given in Fig. 10. While obtaining the sensitivities of the anomalous
$C_{\widetilde{B}W}/{\Lambda^{4}}$, $C_{WW}/{\Lambda^{4}}$,
$C_{BW}/{\Lambda^{4}}$, and $C_{BB}/{\Lambda^{4}}$ couplings, we have focused
on the variable transverse momentum of the dilepton system.

Figure 9: Invariant mass distribution of $Z$ $(\ell\ell)$ boson for signal
with $C_{\widetilde{B}W}/{\Lambda^{4}}=3$ TeV-4, $C_{BB}/{\Lambda^{4}}=3$
TeV-4, $C_{BW}/{\Lambda^{4}}=3$ TeV-4, $C_{WW}/{\Lambda^{4}}=3$ TeV-4 and
relevant background processes

Figure 10: Distribution of the transverse momentum of dilepton $p^{\ell\ell}_{T}$ for signal with $C_{\widetilde{B}W}/{\Lambda^{4}}=3$ TeV-4, $C_{BB}/{\Lambda^{4}}=3$ TeV-4, $C_{BW}/{\Lambda^{4}}=3$ TeV-4, $C_{WW}/{\Lambda^{4}}=3$ TeV-4 and relevant background processes Table 2: List of selected cuts for the analysis. Cuts |  Definitions
---|---
Cut-0 |  Preselection: $N_{\ell\,(e,\,\mu)}>=2$ and a pair of leptons
Cut-1 |  Transverse momentum and pseudo-rapidity of the leading and sub-leading charged leptons: $p_{T}^{\ell^{1}}>30$ GeV, $p_{T}^{\ell^{2}}>20$ GeV and $|\eta^{\ell^{1}}|<2.5$, $|\eta^{\ell^{2}}|<2.5$
Cut-2 |  $p_{T}$ balance: $|E_{T}^{miss}-p_{T}^{\ell\ell}|/p_{T}^{\ell\ell}<0.5$
Cut-3 |  Transverse component of $p_{T}^{\ell\ell}$ with respect to the thrust axis: $a_{T}^{\ell\ell}>40$
Cut-4 |  Missing energy transverse: $E_{T}^{miss}>200$ GeV
Cut-5 |  Minimum distance between leptons: $\Delta R({\ell_{1},\ell_{2}})<3.0$
Cut-6 |  Invariant mass: $|m_{\ell\ell}-m_{Z}|<20$ GeV

##  III Sensitivities on aNTGC

The sensitivities obtained at 95$\%$ C. L. on the anomalous
$C_{BB}/{\Lambda^{4}}$, $C_{BW}/{\Lambda^{4}}$,
$C_{\widetilde{B}W}/{\Lambda^{4}}$, and $C_{WW}/{\Lambda^{4}}$ couplings are
calculated with the process of $e^{-}e^{+}\,\rightarrow\,ZZ$ for the decay for
lepton pair and neutrino pair of $Z$ bosons using $\chi^{2}$ method for every
single bin given in the following equation:

$\displaystyle\chi^{2}=\sum_{i}^{n_{bins}}(\frac{N_{i}^{TOT}-N_{i}^{B}}{N_{i}^{B}\Delta_{i}})^{2}$
(14)

Here, $N_{i}^{TOT}$ and $N_{i}^{B}$ are the number of events of total
contributions (anomalous couplings and SM) and relevant backgrounds,
respectively. On the other hand, $\Delta_{i}=\sqrt{1/N_{i}^{B}}$ is the
statistical uncertainty for every bin. The limits at 95$\%$ C.L. on anomalous
$C_{BB}/{\Lambda^{4}}$, $C_{BW}/{\Lambda^{4}}$,
$C_{\widetilde{B}W}/{\Lambda^{4}}$, and $C_{WW}/{\Lambda^{4}}$ couplings via
the process $e^{-}e^{+}\,\rightarrow\,ZZ$ with various electron polarization
with the corresponding integrated luminosities due to the CLIC program are
given in Table 3. Our best sensitivities are given as follows

$\displaystyle C_{BB}/{\Lambda^{4}}=[-3.40;3.49]\times
10^{-2}\,\text{TeV}^{-4}\,,$ (15) $\displaystyle
C_{BW}/{\Lambda^{4}}=[-8.66;8.45]\times 10^{-2}\,\text{TeV}^{-4}\,,$ (16)
$\displaystyle C_{\widetilde{B}W}/{\Lambda^{4}}=[-2.74;3.66]\times
10^{-2}\,\text{TeV}^{-4}\,,$ (17) $\displaystyle
C_{WW}/{\Lambda^{4}}=[-7.74;7.62]\times 10^{-2}\,\text{TeV}^{-4}\,.$ (18)

It can be seen that the most sensitive coupling is
$C_{\widetilde{B}W}/{\Lambda^{4}}$, with a value of $[-2.74;3.66]\times
10^{-2}$ TeV-4. The sensitivities of the anomalous
$C_{\widetilde{B}W}/{\Lambda^{4}}$ and $C_{WW}/{\Lambda^{4}}$ couplings is
improved with -$80\%$ polarized electron beams compared to unpolarized and
$80\%$ polarized beams. However, the $C_{BB}/{\Lambda^{4}}$ and
$C_{BW}/{\Lambda^{4}}$ couplings have the best sensitivity with unpolarized
beams. The obtained sensitivities are compared with the latest experimental
results using bar-chart graphics in Figs. 11-14 for unpolarized and polarized
beams with different integrated luminosities. Our results show that the most
sensitive limits in Eqs. (15)-(18) are approximately 15-75 times better than
the experimental limits in Ref. Sirunyan:2021edk .

Table 3: Sensitivities on aNTGCs $C_{BB}/\Lambda^{4}$, $C_{BW}/\Lambda^{4}$, $C_{\widetilde{B}W}/\Lambda^{4}$ and $C_{WW}/\Lambda^{4}$ through the process $e^{+}e^{-}\to ZZ$ at CLIC.  $P_{e^{-}}$ | $0\%$ | $-80\%$ | $80\%$
---|---|---|---
Couplings (TeV-4) |  ${\cal L}=5$ ab-1 |  ${\cal L}=4$ ab-1 |  ${\cal L}=1$ ab-1
$C_{BB}/\Lambda^{4}$ | $[-0.0340;0.0349]$ | $[-0.0543;0.0537]$ | $[-0.0402;0.0414]$
$C_{BW}/\Lambda^{4}$ | $[-0.0866;0.0845]$ | $[-0.1902;0.1854]$ | $[-0.0907;0.0919]$
$C_{\widetilde{B}W}/\Lambda^{4}$ | $[-0.0297;0.0292]$ | $[-0.0274;0.0366]$ | $[-0.0431;0.0396]$
$C_{WW}/\Lambda^{4}$ | $[-0.0922;0.0937]$ | $[-0.0774;0.0762]$ | $[-0.3343;0.3742]$

Figure 11: Comparison of the experimental limits and obtained sensitivity on
the anomalous $C_{BB}/\Lambda^{4}$ coupling for various luminosities for every
polarization states of electron beam.

Figure 12: Same as Fig. 11 but for the anomalous $C_{BW}/\Lambda^{4}$
coupling.

Figure 13: Same as Fig. 11 but for the anomalous
$C_{\widetilde{B}W}/\Lambda^{4}$ coupling.

Figure 14: Same as Fig. 11 but for the anomalous $C_{WW}/\Lambda^{4}$
coupling.

Our most sensitive limits in Eqs. (15)-(18) can be compared to
phenomenological studies for the processes $pp\rightarrow ZZ\rightarrow 4\ell$
Yilmaz:2020ser  and $pp\rightarrow ZZ\rightarrow 2\ell 2\nu$ Yilmaz:2021dbm
in the FCC-hh collider with 100 TeV. Although the FCC-hh collider has very
high center-of-mass energy compared to the CLIC, the obtained limits of each
coupling are 4 times more sensitive than Ref. Yilmaz:2020ser . In addition,
when the limits in Ref. Yilmaz:2020ser  are examined, it is seen that the
limits of $C_{BB}$ and $C_{\widetilde{B}W}$ couplings are better in our paper.

##  IV Conclusions

The LHC offers exciting possibilities for new discoveries and important
insights at high-energy scale. However, it is widely recognized that linear
colliders, with their clean experimental conditions, are the ideal environment
for exploring new physics beyond the SM. The ability to use polarized electron
beams in linear colliders enhances the physics program, allowing for precise
testing of the SM and exploring new physics beyond the SM. The CLIC, with its
multi-TeV energy and multiple electron polarization options, is a particularly
promising linear collider for these studies. In this study, we investigate the
process $e^{+}e^{-}\to ZZ$ using unpolarized and polarized beams to search for
effects of anomalous $ZZZ$ and $ZZ\gamma$ couplings at the CLIC. To
investigate the process $e^{+}e^{-}\to ZZ$ at the CLIC with a center-of-mass
energy of 3 TeV, a cut-based method is performed with the detector response
using Delphes card. The final state particles consists of two charged leptons
and pair neutrinos coming from $Z$ bosons, and the relevant backgrounds are
also analyzed. The results show that the transverse momentum $p^{\ell}_{T}$
and pseudo-rapidity $\eta^{\ell}$ of the charged leptons, $p_{T}$ balance,
$a_{T}^{\ell\ell}$, missing energy transverse $E_{T}^{miss}$, angular
separation $\Delta{R}({\ell_{1},\ell_{2}})$ and invariant mass of the final
state charged leptons $m_{\ell\ell}$ are the key parameters in differentiating
the signal from the background. After applying the selected cuts, the
sensitivities of each anomalous coupling are obtained via $\chi^{2}$ method
using the transverse momentum of the dilepton system at CLIC for unpolarized
and polarized beams with $\sqrt{s}$=3 TeV. The findings of this study show
that the CLIC can improve the sensitivity limits on
$C_{\widetilde{B}W}/{\Lambda^{4}}$, $C_{BB}/{\Lambda^{4}}$,
$C_{BW}/{\Lambda^{4}}$ and $C_{WW}/{\Lambda^{4}}$ parameters defining the
aNTGC $ZZZ$ and $ZZ\gamma$ with respect to the experimental limits of LHC.

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