# Cyclotomic expansion for the colored HOMFLY-PT invariants of double twist
knots

Qingtao Chen, Kefeng Liu and Shengmao Zhu  Division of Science
New York University Abu Dhabi
Abu Dhabi
United Arab Emirates<EMAIL_ADDRESS>Mathematical Science Research Center
Chongqing University of Technology
Chongqing, P. R. China.  Department of mathematics
University of California at Los Angeles, Box 951555
Los Angeles, CA, 90095-1555<EMAIL_ADDRESS>Department of Mathematics
Zhejiang International Studies University  Center of Mathematical Sciences
Zhejiang University, Box 310027
Hangzhou, P. R. China<EMAIL_ADDRESS>
###### Abstract.

In this short note, we prove the cyclotomic expansion formula for the colored
HOMFLY-PT invariant of double twist knots, it confirms the cyclotomic
expansion conjecture for $SU(N)$-invariants proposed in [2].

##  1\. Introduction

Colored HOMFLY-PT invariant of link is an important quantum invariant in
mathematics and physics, we refer to [8] and the references therein for a
recent review of the colored HOMFLY-PT invariant and related topics. During
the past several years, we spent a lot of efforts to study the general
structures for colored HOMFLY-PT invariants. Motivated by the congruence skein
relations discovered in [1] and the celebrated cyclotomic expansion formula
for colored Jones polynomial due to Habiro [3], we proposed the cyclotomic
expansion conjecture in [2] (see also Conjecture 2.4 in version2 of [7]) for
the colored $SU(n)$ invariants $J_{N}^{SU(n)}(\mathcal{K};q)$ of a knot
$\mathcal{K}$ which is defined as the $a=q^{n}$ specialization of the
normalized colored HOMFLY-PT invariant

(1.1)
$\displaystyle\mathscr{H}_{N}(\mathcal{K};q,a):=\frac{W_{(N)}(\mathcal{K};q,a)}{W_{(N)}(U;q,a)},$

where $W_{(N)}(\mathcal{K};q,a)$ denotes the colored HOMFLY-PT invariant of
$\mathcal{K}$ with the symmetric representation labeled by partition $(N)$,
and $U$ denotes the unknot. $\mathscr{H}_{N}(\mathcal{K};q,a)$ is normalized
so that it is to be $1$ for the unknot $U$.

######  Conjecture 1.1.

For any knot $\mathcal{K}$, there exist Laurent polynomials
$H_{k}^{(n)}(\mathcal{K})\in\mathbb{Z}[q,q^{-1}]$, independent of $N$ ($N\geq
0$). Such that

(1.2) $\displaystyle
J_{N}^{SU(n)}(\mathcal{K};q)=\sum_{k=0}^{N}C_{N+1,k}^{(n)}H_{k}^{(n)}(\mathcal{K}),$

where
$C_{N+1,k}^{(n)}=\\{N-(k-1)\\}\\{N-(k-2)\\}\cdots\\{N-1\\}\\{N\\}\\{N+n\\}\\{N+n+1\\}\cdots\\{N+n+(k-1)\\}$,
for $k=1,..,N$, and $C_{N+1,0}^{(n)}=1$. In particular,
$J_{0}^{SU(n)}(\mathcal{K};q)=H_{0}^{(n)}(\mathcal{K})=1$.

For a given knot or link, it is difficult to calculate its colored HOMFLY-PT
invariant in general. In [4], Kawagoe provided some formulae for the colored
HOMFLY-PT invariants with symmetric representations based on the HOMFLY skein
theory. Such formulae are useful to compute these invariants of the knots and
links with twisted strands of opposite orientations. Furthermore, in the
recent work [5], Kawagoe obtained a rigorous single sum formula for the
colored HOMFLY-PT polynomial $\mathscr{H}_{N}(\mathcal{K}_{p};q,a)$ of the
twist knot $\mathcal{K}_{p}$.

In this short note, based on Kawagoe’s recent results [5], we apply the
techniques in [6] to derive the following cyclomotic expansion formula for
$\mathscr{H}_{N}(\mathcal{K}_{p,s};q,a)$ of double twist knot
$\mathcal{K}_{p,s}$

Figure 1. Double twist knot $\mathcal{K}_{p,s}$

######  Theorem 1.2.

The colored HOMFLY-PT invariant of the double twist knot $\mathcal{K}_{p,s}$
is given by

(1.5)
$\displaystyle\mathscr{H}_{N}(\mathcal{K}_{p,s};q,a)=\sum_{k=0}^{N}(-1)^{k}(a^{k}q^{k(k-1)})^{2(p+s)}\tilde{C}_{k,k}^{(p)}\tilde{C}_{k,k}^{(s)}\left[\begin{array}[]{@{\,}c@{\,}}N\\\
k\end{array}\right]\\{N+k-1;a\\}_{k}\\{k-2;a\\}_{k},$

where the coefficient $\tilde{C}_{k,k}^{(p)}\in\mathbb{Z}[q,q^{-1}]$, whose
explicit expression is given by formula (3.21).

In particular, let $a=q^{n}$ in the above formula (1.5), we obtain

######  Corollary 1.3.

The Conjecture 1.1 holds for the $SU(n)$-invariant
$J_{N}^{SU(n)}(\mathcal{K}_{p,s};q)$ of the double twist knot
$\mathcal{K}_{p,s}$.

##  2\. Preliminaries

In this section, we briefly review Kawagoe’s recent work [5] and fix the
notations. Let $a$ and $q$ be two non-zero variables in $\mathbb{C}$. For an
integer $n$, we define the symbols by

$\displaystyle[n]$
$\displaystyle=\frac{q^{n}-q^{-n}}{q-q^{-1}},\qquad\\{n\\}=q^{n}-q^{-n},\qquad\\{n;a\\}=aq^{n}-a^{-1}q^{-n}.$

For integers $n>0,i\geq 0$, we introduce the products of $i$ terms of these
symbols by

$\displaystyle[n]_{i}$ $\displaystyle=[n][n-1]\cdots[n-i+1],$
$\displaystyle\\{n\\}_{i}$ $\displaystyle=\\{n\\}\\{n-1\\}\cdots\\{n-i+1\\},$
$\displaystyle\\{n;a\\}_{i}$
$\displaystyle=\\{n;a\\}\\{n-1;a\\}\cdots\\{n-i+1;a\\},$
$\displaystyle\\{-n;a\\}_{i}$
$\displaystyle=\\{-n;a\\}\\{-n+1;a\\}\cdots\\{-n+i-1;a\\},$

which are defined to be $1$ if $i=0$. Furthermore, we let

$\displaystyle[n]!=[n]_{n},\quad\\{n\\}!=\\{n\\}_{n},\quad\left[\begin{array}[]{@{\,}c@{\,}}n\\\
i\end{array}\right]=\frac{[n]!}{[i]![n-i]!}.$

The HOMFLY skein module $\mathcal{S}(M)$ of a oriented 3-manifold $M$ is the
free $\mathbb{C}$-module modulo the submodule generated by the following
HOMFLY-PT skein relations:

  1. (1)

$L\cup U=\displaystyle\frac{\\{0;a\\}}{\\{1\\}}L$, and $\varnothing=1$,

  2. (2)

$\,-\,$ $\,=(q-q^{-1})\,$ ,

  3. (3)

$\,=a\,$ $\,,\quad$ $\;=a^{-1}\,$ $\,.$

For an integer $n\geq 1$, we recursively define the $n$-th $q$-symmetrizer by
Figure 2, where the $q$-symmetrizer is denoted by a white rectangle. The
integer $n$ beside an arc means $n$ copies of the arc.

$\displaystyle\raisebox{-14.0pt}{
\includegraphics[width=28.0pt]{onebox1.pdf}}=\raisebox{-1.0pt}{
\includegraphics[width=28.0pt]{zerobox1.pdf}}$

$\displaystyle\raisebox{-14.0pt}{
\includegraphics[width=28.0pt]{oneboxn.pdf}}=\frac{q^{-n+1}}{[n]}\raisebox{-14.0pt}{
\includegraphics[width=20.0pt]{oneboxonelinen-1.pdf}}+\frac{q^{[n-1]}}{[n]}\raisebox{-20.0pt}{
\includegraphics[width=80.0pt]{twoboxesn-11n-2n-11.pdf}}$

Figure 2. $n$-th $q$-symmetrizer

Using the $m$-th and $n$-th $q$-symmetrizers, Kawagoe introduced the
$(m,n)$-th $q$-symmetrizer by Figure 3, where $x_{m,n}^{i}$ is given by

$\displaystyle x_{m,n}^{i}=(-1)^{i}\left[\begin{array}[]{@{\,}c@{\,}}m\\\
i\end{array}\right]\left[\begin{array}[]{@{\,}c@{\,}}n\\\
i\end{array}\right]\frac{\\{i\\}!}{\\{m+n-2;a\\}_{i}}.$

$\displaystyle\raisebox{-25.0pt}{
\includegraphics[width=28.0pt]{mnmn.pdf}}=\sum_{i=0}^{\min{\\{m,n\\}}}x_{m,n}^{i}\raisebox{-25.0pt}{\includegraphics[width=55.0pt]{mnmnii.pdf}},$
Figure 3. $(m,n)$-th $q$-symmetrizer

According to their definitions, the $n$-th $q$-symmetrizer and $(m,n)$-th
$q$-symmetrizer carry the following properties

(2.1) $\displaystyle\raisebox{-14.0pt}{
\includegraphics[width=58.0pt]{onebox1twistn.pdf}}=\raisebox{-14.0pt}{
\includegraphics[width=58.0pt]{1twistoneboxn.pdf}}=q\raisebox{-14.0pt}{
\includegraphics[width=30.0pt]{oneboxn.pdf}}$ (2.2)
$\displaystyle\raisebox{-12.0pt}{
\includegraphics[width=28.0pt]{oneboxleftarrows.pdf}}=\raisebox{-12.0pt}{
\includegraphics[width=28.0pt]{oneboxrightarrows.pdf}}=\raisebox{-12.0pt}{
\includegraphics[width=28.0pt]{oneboxrightarrow.pdf}}$

(2.3) $\displaystyle\raisebox{-14.0pt}{
\includegraphics[width=32.0pt]{fourboxes.pdf}}=\raisebox{-14.0pt}{
\includegraphics[width=16.0pt]{twoboxes.pdf}}$

(2.4) $\displaystyle\raisebox{-14.0pt}{
\includegraphics[width=18.0pt]{twoboxeslefthalfcircle.pdf}}=\raisebox{-14.0pt}{
\includegraphics[width=18.0pt]{twoboxesrighthalfcircle.pdf}}=0$

We denote the skein module of the solid torus $S^{1}\times D^{2}$ by
$\mathcal{S}$. For a circle along $S^{1}$ of the solid torus, let
$H_{n}\in\mathcal{S}$ be the $n$ copies of the circles inserted by the $n$-th
$q$-symmetrizer, and $H_{n,n}$ be two copies of $H_{n}$, one is anticlockwise
and the other is clockwise. Let $D_{m,n}$ by $m$ copies of the anticlockwise
circle and $n$ copies of the clockwise inserted by the $(m,n)$-th
$q$-symmetrizer. Kawagoe introduced two submodule of $\mathcal{S}$,
$\mathcal{H}_{n,n}$ and $\mathcal{D}_{n,n}$, which are spanned by $H_{i,i}$
and ${D}_{i,i}$ for $i=0,\ldots,n$, respectively. Then, he proved that
$\mathcal{D}_{n,n}=\mathcal{H}_{n,n}$. Let $\langle\;\;\rangle$ be the linear
map on $\mathcal{S}$ defined by evaluating it in $S^{3}$.

As in [6], Kawagoe introduced the twist map $t:\mathcal{S}\to\mathcal{S}$
induced by one right-handed twist on the solid torus as shown in the right-
hand side of Figure 4, where the twist is acting on the bottom of the solid
torus. Similarly, let $t^{-1}$ be the twist map induced by one left-hand
twist. For $x\in\mathcal{S}$, let $e_{x}:\mathcal{S}\to\mathcal{S}$ be the
encircling map which encircles an element of $\mathcal{S}$ by $x$ as shown in
the left-hand side of Figure 4, where $x$ slides into the bottom of the solid
torus and encircles the element.

(2.5) $\displaystyle\raisebox{-10.0pt}{
\includegraphics[width=80.0pt]{linewithcircle.pdf}}=\raisebox{-5.0pt}{
\includegraphics[width=80.0pt]{linewithkink.pdf}}$ Figure 4. The encircling
map $e_{\omega}$ and a positive full twist

We need the following lemmas which are borrowed from [5] directly.

######  Lemma 2.1 (cf. Lemma 2.6 in [5] ).

For positive integers $m\geq n$, we have the following:

$\displaystyle\langle
D_{m,n}\rangle=\frac{\\{m+n-1;a\\}\\{m-2;a\\}_{m-1}\\{n-2;a\\}_{n-1}\\{-1;a\\}}{\\{m\\}!\\{n\\}!}.$

For $i=0,\ldots,n$, Kawagoe recursively defined elements $R_{i}\in\mathcal{S}$
as follows:

$\displaystyle R_{0}$ $\displaystyle=H_{0}=1,$ $\displaystyle R_{n}$
$\displaystyle=H_{n}-\sum_{i=0}^{n-1}\frac{\\{n-1+i;a\\}_{n-i}}{\\{n-i\\}!}R_{i},\
\text{for}\ n\geq 1.$

Conversely, each $H_{i}$ can be expressed by

$\displaystyle
H_{i}=\sum_{j=0}^{i}\frac{\\{i-1+j;a\\}_{i-j}}{\\{i-j\\}!}R_{j}.$

These elements $R_{i}$ play important role in calculating the colored HOMFLY-
PY polynomial of double twist knot. Actually, the behavior of $H_{n}$
concerning the encircling map is very complicated, so Kawagoe introduced
$R_{n}$ as a linear combination of $H_{i}(i=0,\ldots,n)$ which simplifies the
computations.

######  Lemma 2.2 (cf. Proposition 3.4 and Corollary 3.5 in [5]).

For an integer $i\geq 0$, we have $e_{R_{i}}(D_{n,n})=\theta_{n,i}D_{n,n}$,
where $\theta_{n,i}=\\{n\\}_{i}=\\{n\\}_{i}\\{n_{i}-2,a\\}_{i}$, and for
$i>n$, $e_{R_{i}}(D_{n,n})=0$.

######  Lemma 2.3 (cf. Lemma 2.1 in [5]).

We have the following formula

(2.6) $\displaystyle\raisebox{-25.0pt}{
\includegraphics[width=100.0pt]{mn2mn.pdf}}=\sum_{i=0}^{\min{\\{m,n\\}}}\alpha_{m,n}^{i}(q,a)\raisebox{-25.0pt}{\includegraphics[width=50.0pt]{mnmnii.pdf}},$

where

(2.7)
$\displaystyle\alpha_{m,n}^{i}(a,q)=(-a)^{-i}q^{-i(m+n)+\frac{i(i+3)}{2}}\frac{\\{m\\}_{i}\\{n\\}_{i}}{\\{i\\}!}.$

######  Remark 2.4.

In formula (2.6), when all the negative crossings change into positive
crossings, the coefficients $\alpha_{m,n}^{i}(q,a)$ will change into
$\alpha_{m,n}^{i}(q^{-1},a^{-1})$.

######  Lemma 2.5 (cf. Lemma 4.1 in [5]).

The following holds:

$\displaystyle\raisebox{-20.0pt}{
\includegraphics[width=18.0pt]{twoboxesmn.pdf}}=\sum_{i=0}^{\min{\\{m,n\\}}}y_{m,n}^{i}\raisebox{-22.0pt}{\includegraphics[width=50.0pt]{sixboxesmnm-
in-imnii.pdf}}$

where $y_{m,n}^{i}$ is defined by

$\displaystyle
y^{i}_{m,n}=\frac{\\{m\\}_{i}\\{n\\}_{i}}{\\{i\\}!\\{m+n-i-1;a\\}_{i}}.$

We set the element $\omega_{n}^{p}\in\mathcal{S}$ by

(2.8) $\displaystyle\omega_{n}^{p}=\sum_{i=0}^{n}t_{i,p}R_{i},$

for some $t_{i,p}\in\mathbb{C}$, Kawagoe determined the coefficients $t_{i,p}$
so that $e_{\omega_{n}^{p}}(x)=t^{p}(x)$ for $x\in\mathcal{D}_{n,n}$ as
follows.

By the idempotent and vanishing properties of $(n,n)$-the $q$-symmetrizer, the
left-hand side of Figure 5 is transformed into $D_{n,n}$ with the
multiplication of some $T_{n,p}\in\mathbb{C}$.

$\displaystyle\raisebox{-55.0pt}{
\includegraphics[width=95.0pt]{nn2pnnrotate.pdf}}=T_{n,p}\quad\raisebox{-15.0pt}{\includegraphics[width=55.0pt]{nnnnrotate.pdf}}$
Figure 5.

To determine $t_{n,p}$, Kawagoe calculated $T_{n,p}$ in two ways as in [6].
One is to make use of the $(n-i,n-i)$-th $q$-symmetrizer ($i=0,\ldots,n$), and
the other is to encircle $p$ full twists by $\omega_{n}^{p}$. Finally, he
obtained

(2.9) $\displaystyle
T_{n,p}=(a^{-n}q^{-n(n-1)})^{2p}\\{n\\}!\sum_{i=0}^{n}(-1)^{i}(a^{i}q^{i(i-1)})^{2p}\frac{\\{n\\}_{n-i}}{\\{n-i\\}!}\frac{\\{2i-1;a\\}}{\\{n+i-1;a\\}_{n+1}},$

and

(2.10) $\displaystyle
T_{n,p}=(-1)^{n}(a^{n}q^{n(n-1)})^{-2p}(\\{n\\}!)^{2}t_{n,p}.$

Hence,

(2.11) $\displaystyle
t_{n,p}=(-1)^{n}\sum_{i=0}^{n}(-1)^{i}\frac{(a^{i}q^{i(i-1)})^{2p}}{\\{i\\}!\\{n-i\\}!}.\frac{\\{2i-1;a\\}}{\\{n+i-1;a\\}_{n+1}}.$

##  3\. Proof of the Theorem 1.2

The crucial step to prove the Theorem 1.2 is to give another explicit formula
for $T_{n,p}$ which shows that $T_{n,p}$ is a Laurent polynomial divisible by
$\\{n\\}_{k}$.

First, we have the following lemma.

######  Lemma 3.1.

We have the expansion formula

(3.1) $\displaystyle\raisebox{-25.0pt}{
\includegraphics[width=120.0pt]{nn2pnn.pdf}}=\sum_{k=0}^{n}C_{n,k}^{(p)}\raisebox{-25.0pt}{\includegraphics[width=60.0pt]{nnnnkk.pdf}}$

where the coefficient $C_{n,k}^{(p)}$ is a Laurent polynomial divisible by
$\\{n\\}_{k}$.

###### Proof.

After applying Lemma 2.3, one term index by $k$ in sigma sum reads

(3.2) $\displaystyle=\raisebox{-20.0pt}{
\includegraphics[width=100.0pt]{nn22nnkk.pdf}}$
$\displaystyle=(a^{k}q^{k(k-1)})^{-2}\raisebox{-20.0pt}{\includegraphics[width=100.0pt]{nn22twistnnk.pdf}}$
$\displaystyle=(a^{k}q^{k(k-1)})^{-2}q^{-4k(n-k)}\raisebox{-20.0pt}{\includegraphics[width=100.0pt]{nn2nnkk.pdf}}$

where we have used the properties (2.1) and (2.2). Then, applying Lemma 2.3 to
the $n-k$ twisted strands in the last diagram of formula (3.2), we obtain the
expansion (3.1) for $p=2$, and

(3.3) $\displaystyle
C_{n,k}^{(2)}=\sum_{l_{1}+l_{2}=k}\alpha_{n,n}^{l_{1}}a^{-2l_{1}}q^{2l_{1}(l_{1}-2n+1)}\alpha_{n-l_{1},n-l_{1}}^{l_{2}}.$

The formula (2.7) for $\alpha_{n,n}^{i}$ implies that $C_{n,k}^{(2)}$ is a
Laurent polynomial divisible by $\\{n\\}_{k}$. The proof for general $p$ is
similar.

∎

######  Remark 3.2.

In formula (3.1), when all the negative crossings change into positive
crossings, the coefficients also change so that $a,q$ are replaced by
$a^{-1},q^{-1}$, i.e.

(3.4) $\displaystyle C_{n,k}^{(p)}(q,a)=C_{n,k}^{(-p)}(q^{-1},a^{-1}),\
\text{for}\ p<0.$

######  Lemma 3.3.

The following holds:

(3.5) $\displaystyle T_{n,p}=C_{n,n}^{(p)}.$

###### Proof.

By the idempotent and vanishing properties (2.3) and (2.4) for the $(n,n)$-th
$q$-symmetrizer, we have

(3.6) $\displaystyle\raisebox{-40.0pt}{
\includegraphics[width=85.0pt]{nn2pnnrotate.pdf}}=\sum_{k=0}^{n}C_{n,k}^{(p)}\quad\raisebox{-40.0pt}{\includegraphics[width=75.0pt]{nnnnkkrotate.pdf}}=C_{n,n}^{(p)}\quad\raisebox{-20.0pt}{\includegraphics[width=60.0pt]{nnnnrotate.pdf}}.$

Comparing to the Figure 5, we obtain $T_{n,p}=C_{n,n}^{(p)}.$ ∎

Actually, we can present the explicit formula for $C_{n,n}^{(p)}$ as follows.
Note that the formula (2.7) gives

(3.9)
$\displaystyle\alpha_{n,n}^{i}=(-a)^{-i}q^{-2ni+\frac{i(i+3)}{2}}\left[\begin{array}[]{@{\,}c@{\,}}n\\\
i\end{array}\right]\\{n\\}_{i}.$

When $p\geq 1$, we consider the sum over all multi-indices denoted by
$\underline{l}=(l_{1},...,l_{p})$ such that $l_{i}\geq 0$ for all $i$, and
$\sum_{i=1}^{p}l_{i}=n$. We put $s_{i}=l_{1}+\cdots+l_{i}$ for $i=1,...,p$ and
define

(3.10)
$\displaystyle\varphi(\underline{l})=\sum_{i=1}^{p-1}2s_{i}(s_{i}-2n+1)+\sum_{i=1}^{p}\frac{l_{i}(l_{i}+3)}{2}+2\sum_{i=1}^{p-1}s_{i}l_{i+1},$

and

(3.13) $\displaystyle\left[\begin{array}[]{@{\,}c@{\,}}n\\\
\underline{l}\end{array}\right]=\frac{[n]!}{[l_{1}]!\cdots[l_{p}]!}.$

Then

(3.14) $\displaystyle C_{n,n}^{(p)}$
$\displaystyle=\sum_{\underline{l}}\alpha_{n,n}^{l_{1}}a^{-2l_{1}}q^{2l_{1}(l_{1}-2n+1)}\alpha_{n-l_{1},n-l_{1}}^{l_{2}}a^{-2s_{2}}q^{2s_{2}(s_{2}-2n+1)}\cdots$
$\displaystyle\times\alpha_{n-s_{p-2},n-s_{p-2}}^{l_{p-1}}a^{-2s_{p-1}}q^{2s_{p-1}(s_{p-1}-2n+1)}\alpha_{n-s_{p-1},n-s_{p-1}}^{l_{p}}$
$\displaystyle=\sum_{\underline{l}}a^{-2\sum_{i=1}^{p-1}s_{i}}q^{\sum_{i=1}^{p-1}2s_{i}(s_{i}-2n+1)}\alpha_{n,n}^{l_{1}}\alpha_{n-s_{1},n-s_{1}}^{l_{2}}\cdots\alpha_{n-s_{p-1},n-s_{p-1}}^{l_{p}}$
(3.17)
$\displaystyle=(-1)^{n}a^{-n}q^{-2n^{2}}\\{n\\}!\sum_{\underline{l}}a^{-2\sum_{i=1}^{p-1}(p-i)l_{i}}q^{\varphi(\underline{l})}\left[\begin{array}[]{@{\,}c@{\,}}n\\\
\underline{l}\end{array}\right]$

For $p\leq-1$, by Remark 3.2, we can define

(3.18) $\displaystyle C_{n,n}^{(p)}(q,a)=C^{(-p)}_{n,n}(q^{-1},a^{-1}).$

Finally, we put

(3.21)
$\displaystyle\tilde{C}_{k,k}^{(p)}:=\frac{C_{k,k}^{(p)}}{\\{k\\}!}=(-1)^{k}a^{-k}q^{-2k^{2}}\sum_{\underline{l}}a^{-2\sum_{i=1}^{p-1}(p-i)l_{i}}q^{\varphi(\underline{l})}\left[\begin{array}[]{@{\,}c@{\,}}k\\\
\underline{l}\end{array}\right],$

then $\tilde{C}_{k,k}^{(p)}$ is Laurent polynomial in $\mathbb{Z}[q,q^{-1}]$.

For examples,

(3.22) $\displaystyle\tilde{C}_{k,k}^{(1)}$
$\displaystyle=(-1)^{k}a^{-k}q^{-\frac{3k(k-1)}{2}},\ \
\tilde{C}_{k,k}^{(-1)}=a^{k}q^{\frac{3k(k-1)}{2}},$ (3.25)
$\displaystyle\tilde{C}_{k,k}^{(2)}$
$\displaystyle=(-1)^{k}a^{-k}q^{-\frac{3k(k-1)}{2}}\sum_{l=0}^{k}a^{-2l}q^{-3kl+l(l+2)}\left[\begin{array}[]{@{\,}c@{\,}}k\\\
l\end{array}\right],$ (3.28) $\displaystyle\tilde{C}_{k,k}^{(-2)}$
$\displaystyle=a^{k}q^{\frac{3k(k-1)}{2}}\sum_{l=0}^{k}a^{2l}q^{3kl-l(l+2)}\left[\begin{array}[]{@{\,}c@{\,}}k\\\
l\end{array}\right].$

Now, we are ready to calculate the colored HOMFLY-PT polynomial of the double
twist knots $\mathcal{K}_{p,s}$. By its definition, the $N$-th colored HOMFLY-
PT polynomial for a knot $\mathcal{K}$ is given by

(3.29)
$\displaystyle\mathscr{H}_{N}(\mathcal{K})=\frac{\langle\mathcal{K}(H_{N})\rangle}{\langle
U(H_{N})\rangle}=\frac{\\{N\\}!}{\\{N-1;a\\}_{N}}{\langle\mathcal{K}(H_{N})\rangle},$

where $\mathcal{K}(H_{N})$ denotes the knot $\mathcal{K}$ cabled by $H_{N}$
with compatible orientations. Note that if we assume that the framing of the
knot $\mathcal{K}$ is to be $0$, then this definition of
$\mathscr{H}_{N}(\mathcal{K})$ is equal to the formula $(\ref{formula-
normalizedhomfly})$.

For two integers $p,s$, the double twist knot $\mathcal{K}_{p,s}$ is described
in Figure 1. Note that $\mathcal{K}_{-p,-s}$ is the mirror of
$\mathcal{K}_{p,s}$, we will assume $p\in\mathbb{Z}$ and $s\geq 1$. From this,
in particular, we can see that $\mathcal{K}_{1,1}$ is a left-handed trefoil,
$\mathcal{K}_{-1,1}$ is a figure-eight knot and $\mathcal{K}_{2,1}$ is the
$5_{2}$ knot.

We observe that the techniques used in [6, 5] are also workable to compute the
double twist knot $\mathcal{K}_{p,s}$. First, by using the definition of the
encircling map and $\omega^{p}$, we have the following description for double
twist knot with framing zero.

Let us calculate $\langle\mathcal{K}_{p,s}(H_{N})\rangle$ by using the above
description for $\mathcal{K}_{p,s}$. Since $H_{N}$ has the following
presentation:

$\displaystyle
H_{N}=\sum_{k=0}^{N}\frac{\\{N+k-1;a\\}_{N-k}}{\\{N-k\\}!}R_{k},$

one can insert this presentation along the above description instead of
$H_{N}$. Consider a pair $R_{i}\in\mathcal{K}_{p,s}(H_{N})$ and $R_{j}$ from
the expression $\omega_{N}^{p}$ (cf. formula (2.8)), the key observation is
that this pair is a cabling of the (twisted) Whitehead link, of which one
component pierces the other twice in the opposite direction each other.
Moreover, $R_{i}$ and $R_{j}$ are a linear combination of $D_{k,k}$ with
$k\leq i$ and $k\leq j$, respectively. According to Lemma 2.2, this pair
vanishes if $i\neq j$. Therefore, we can calculate
$\langle\mathcal{K}_{p,s}(H_{N})\rangle$ as follows:

(3.30)
$\displaystyle\langle\mathcal{K}_{p,s}(H_{N})\rangle=\sum_{k=0}^{N}\frac{\\{N+k-1;a\\}}{\\{N-k\\}!}t_{k,p}\raisebox{-40.0pt}{
\includegraphics[width=100.0pt]{double-twist-2s-twist-RkRk.pdf}},$

and since $R_{k}-H_{k}$ is a linear combination of $R_{j}$ with $j<k$, we have

$\displaystyle=\raisebox{-40.0pt}{ \includegraphics[width=100.0pt]{double-
twist-2s-twist-RkHk.pdf}}\ \ \text{(by Lemma \ref{Coro:vanishi})}$
$\displaystyle=(a^{k}q^{k(k-1)})^{2s}\raisebox{-40.0pt}{
\includegraphics[width=100.0pt]{double-twist-2s-twist-with-boxes.pdf}}\ \
\text{(By Lemma \ref{lemma:ymn} and Lemma \ref{Coro:vanishi})}$
$\displaystyle=(a^{k}q^{k(k-1)})^{2s}C_{k,k}^{(s)}\raisebox{-40.0pt}{
\includegraphics[width=120.0pt]{double-twist-with-boxes.pdf}}\ \ \text{(By
formula (\ref{formula-Cnn}))}$
$\displaystyle=(a^{k}q^{k(k-1)})^{2s}C_{k,k}^{(s)}\theta_{k,k}\langle
D_{k,k}\rangle\ \ \text{(By Lemma \ref{Coro:vanishi} and Lemma
\ref{lemma:Dmn})}$
$\displaystyle=(a^{k}q^{k(k-1)})^{2s}C_{k,k}^{(s)}\frac{\\{2k-1\\}_{2k}\\{k-2;a\\}_{k}}{\\{k\\}!}$

Therefore,

(3.31) $\displaystyle\mathscr{H}_{N}(\mathcal{K}_{p,s})$
$\displaystyle=\frac{\langle K_{p,s}(H_{N})\rangle}{\langle U(H_{N})\rangle}$
$\displaystyle=\frac{\\{N\\}!}{\\{N-1;a\\}_{N}}\sum_{k=0}^{N}\frac{\\{N+k-1;a\\}_{N-k}}{\\{N-k\\}!}t_{k,p}(a^{k}q^{k(k-1)})^{2s}C_{k,k}^{(2s)}\frac{\\{2k-1\\}_{2k}\\{k-2;a\\}_{k}}{\\{k\\}!}$
(3.34)
$\displaystyle=\sum_{k=0}^{N}t_{k,p}(a^{k}q^{k(k-1)})^{2s}C_{k,k}^{(s)}\left[\begin{array}[]{@{\,}c@{\,}}N\\\
k\end{array}\right]\\{N+k-1;a\\}_{k}\\{k-2;a\\}_{k}.$

By using formula (2.10) and Lemma 3.3, we obtain

(3.35) $\displaystyle
t_{k,p}=(-1)^{k}(a^{k}q^{k(k-1)})^{2p}\frac{C_{k,k}^{(p)}}{\\{k\\}!^{2}}.$

Substituting it back to formula (3.31), we obtain Theorem 1.2.

As an application of our main Theorem 1.2, we present several concrete
calculations for the twist knots with small crossings.

######  Example 3.4.

For the trefoil knot $3_{1}$ and figure-8 knot $4_{1}$ which are
$\mathcal{K}_{1,1}$ and $\mathcal{K}_{-1,1}$ respectively,

(3.38)
$\displaystyle\mathscr{H}_{N}(3_{1};q,a)=\sum_{k=0}^{N}(-1)^{k}a^{2k}q^{k(k-1)}\left[\begin{array}[]{@{\,}c@{\,}}N\\\
k\end{array}\right]\\{N+k-1;a\\}_{k}\\{k-2;a\\}_{k}.$ (3.41)
$\displaystyle\mathscr{H}_{N}(4_{1};q,a)=\sum_{k=0}^{N}\left[\begin{array}[]{@{\,}c@{\,}}N\\\
k\end{array}\right]\\{N+k-1;a\\}_{k}\\{k-2;a\\}_{k}.$

which are the formulae given by Corollary 5.2 in [5].

######  Example 3.5.

For the $5_{2}$ knot $\mathcal{K}_{2,1}$, we have

(3.44) $\displaystyle\mathscr{H}_{N}(5_{2};q,a)=$
$\displaystyle\sum_{k=0}^{N}\left((-1)^{k}a^{4k}q^{3k(k-1)}\sum_{l=0}^{k}a^{-2l}q^{-3kl+l(l+2)}\left[\begin{array}[]{@{\,}c@{\,}}k\\\
l\end{array}\right]\right)$ (3.47)
$\displaystyle\cdot\left[\begin{array}[]{@{\,}c@{\,}}N\\\
k\end{array}\right]\\{N+k-1;a\\}_{k}\\{k-2;a\\}_{k}.$

In particular, for $a=q^{2}$, we have

$\left[\begin{array}[]{@{\,}c@{\,}}N\\\
k\end{array}\right]\\{N+k-1;a\\}_{k}\\{k-2;a\\}_{k}=\frac{\\{N+k+1\\}\\{N+k\\}\cdots\\{N-k+1\\}}{\\{N+1\\}}.$

Then

(3.50) $\displaystyle\mathscr{H}_{N}(5_{2};q,q^{2})=$
$\displaystyle\sum_{k=0}^{N}\left((-1)^{k}q^{3k^{2}+5k}\sum_{l=0}^{k}q^{-3kl+l(l-2)}\left[\begin{array}[]{@{\,}c@{\,}}k\\\
l\end{array}\right]\right)$
$\displaystyle\cdot\frac{\\{N+k+1\\}\\{N+k\\}\cdots\\{N-k+1\\}}{\\{N+1\\}}.$

which is equal to $J^{\prime}_{K_{2}}(N+1)$ in [6] ( cf. Theorem 5.1 in [6]
for $p=2$, where the variable $a$ is just the variable $q$ in our notation).

######  Example 3.6.

For the $6_{1}$ knot $\mathcal{K}_{-2,1}$, we have

(3.53) $\displaystyle\mathscr{H}_{N}(6_{1};q,a)=$
$\displaystyle\sum_{k=0}^{N}\left(a^{-2k}q^{-2k(k-1)}\sum_{l=0}^{k}a^{2l}q^{3kl-l(l+2)}\left[\begin{array}[]{@{\,}c@{\,}}k\\\
l\end{array}\right]\right)$ (3.56)
$\displaystyle\cdot\left[\begin{array}[]{@{\,}c@{\,}}N\\\
k\end{array}\right]\\{N+k-1;a\\}_{k}\\{k-2;a\\}_{k}.$

In particular, for $a=q^{2}$, we obtain

(3.59) $\displaystyle\mathscr{H}_{N}(6_{1};q,q^{2})=$
$\displaystyle\sum_{k=0}^{N}\left(q^{-2k(k+1)}\sum_{l=0}^{k}q^{3kl-l(l-2)}\left[\begin{array}[]{@{\,}c@{\,}}k\\\
l\end{array}\right]\right)$
$\displaystyle\cdot\frac{\\{N+k+1\\}\\{N+k\\}\cdots\\{N-k+1\\}}{\\{N+1\\}}.$

which is equal to $J^{\prime}_{K_{-2}}(N+1)$ in [6] ( cf. Theorem 5.1 in [6]
for $p=-2$, where the variable $a$ is just the variable $q$ in our notation).

## References

  * [1] Q. Chen, K. Liu, P. Peng and S. Zhu, _Congruent skein relations for colored HOMFLY-PT invariants and colored Jones polynomials_ , arXiv: 1402.3571, submitted.
  * [2] Q. Chen, K. Liu and S. Zhu, Volume conjecture for $SU(n)$-invariants, arXiv:1511.00658.
  * [3] K. Habiro, _A unified Witten-Reshetikhin-Turaev invariant for integral homology spheres_ , Invent. Math. 171 (2008), no. 1, 1-81.
  * [4] K. Kawagoe. On the formulae for the colored HOMFLY polynomials. J. Geom. Phys. 106 (2016), 143-154.
  * [5] K. Kawagoe, The colored HOMFLY-PT polynomials of the trefoil knot, the figure-eight knot and twist knots. arXiv:2107.08678.
  * [6] G. Masbaum, Skein-theoretical derivation of some formulas of Habiro. Algebraic & Geometric Topology, 3 (2003), 537-556.
  * [7] S. Nawata and A. Oblomkov, Lectures on knot homology, arXiv: 1510.01795.
  * [8] S. Zhu, New structures for colored HOMFLY-PT invariants, arXiv:2105.02037.


# Higher order dynamical charge fluctuations in heavy-ion collisions

Bhanu Sharma and P. K. Sahu  Institute of Physics, HBNI, Bhubaneswar 751005,
India

###### Abstract

The event-by-event charge fluctuation measurements are proposed to provide the
signature of quark-gluon plasma (QGP) in heavy-ion collisions. Measure of
dynamical charge fluctuations is expected to carry information of initial
fractional charge of the QGP phase at the final state. We propose the higher
order charge fluctuation measurement to study the QGP signal in heavy-ion
collisions. This higher order charge fluctuation observable can amplify the
signature of QGP. Also, the SMASH model is used to study the behavior of these
observable in heavy-ion collisions at center of mass energies accessible in
the STAR beam energy scan program.

##  1 Introduction

Shortly after the Big Bang, the deconfined state of Quantum Chromodynamics
(QCD) matter–known as the Quark-Gluon Plasma (QGP)–filled a few microsecond
old universe. The dedicated facilities at Large Hadron Collider (LHC) and the
Relativistic Heavy Ion Collider (RHIC) have been trying to recreate the
conditions very similar to the early universe by making head-on collisions
between heavy-ions [1, 2, 3, 4, 5].

In heavy-ion collisions, the transition from the QGP to normal hadron gas
occurs at chemical freeze out in a short time scale. However, the QGP phase
carries fractional charge due to the presence of quarks and the hadron
resonance gas (HRG) phase contains the integral charge of hadrons. Hence, it
is expected that the charge fluctuations measurement may carry the information
of initial fractional charge to the final state and can be measured at the
detectors.

The main idea is that the charge fluctuations are directly proportional to the
square of charges present in the system. Due to the difference between the
charges carried by quarks in QGP as compared to hadrons in hadronic phase, a
dramatic reduction of event-by-event fluctuations of the charge has been
predicted by various theoretical calculations [6, 7, 8, 9, 10, 12].

As of now, the charge fluctuations have been analysed by various experiments.
The first measurements were done by PHENIX [13] and STAR [14] for Au+Au
collisions at $\sqrt{s_{NN}}$ = 130 GeV. The charge fluctuations by PHENIX
experiment were measured (observable in [13] ) and the results are in
qualitative agreement with HRG calculation and far away from the plasma. The
measurements reported by STAR experiment were in terms of a robust variable,
known as dynamical charge fluctuations. The behaviour of dynamical charge
fluctuations was studied [15] with varying beam energy and different collision
system size like Au+Au and Cu+Cu collisions. However, the measurements were in
agreement with the previous measurements. The charge fluctuations measured by
ALICE experiment [16] for Pb+Pb collisions at $\sqrt{s_{NN}}$ = 2.76 TeV for
$\Delta\eta$ = 1.0 are closer to the theoretical predictions for the QGP
formation. Recently, a similar measurement by the CMS experiment has been
reported for charge fluctuations in Pb+Pb collisions at $\sqrt{s_{NN}}$ = 5.02
TeV for a wide pseudorapidity range $\Delta\eta$ = 4.8 [17]. The measured
value reaches the QGP predictions even without considering the global charge
conservation in account. This indicates that to measure a possible signal for
QGP formation, we need to measure at very high energies and large acceptance
range.

In this calculation, we propose the higher orders of dynamical charge
fluctuations measure in heavy-ion collisions that can amplify the signature of
QGP within limited kinematic phase space. The paper is organized into various
sections. In Section 2, the derivation of the observable and methodology are
introduced. In Section 3, the event-generation model description is mentioned
that is used to study these observables. In Section 4, simulation inputs and
analysis details are discussed. Section 5 presents the results with
discussion. Finally, Section 6 summarizes and outlooks this study.

##  2 Motivation and obvervables

The observable of dynamical charge fluctuations, $\nu_{\pm,dyn}$, has already
been introduced and measured in heavy-ion experiments [14, 15, 16, 17]. A
detailed discussion on this observable is mentioned in Sec. 2.1. However, one
can readily extend this observable to their $m^{\rm th}$ order as,

$\nu_{\pm}^{\mathrm{m}}=\Bigg{\langle}\Bigg{(}\frac{N_{+}}{\langle
N_{+}\rangle}-\frac{N_{-}}{\langle N_{-}\rangle}\Bigg{)}^{m}\Bigg{\rangle}.$
(1)

Here, $m$ is an positive integer. $N_{\pm}$ are the event-by-event number of
positive or negative charged particles. The $<\dots>$ represents ensemble
average. By definition, the 1st order of this observable vanishes, whereas
higher order observables contain higher order correlations terms between
positive and negative charged particles per event. The $2^{\rm nd}$, $3^{\rm
rd}$, and $4^{\rm th}$ order of these observables are discussed in the
following subsections.

###  2.1 Second order dynamical charge fluctuations

The $2^{\rm nd}$ order of Eq. 1 can be written as,

$\begin{split}\nu_{\pm}^{\mathrm{2nd}}&=\Bigg{\langle}\Bigg{(}\frac{N_{+}}{\langle
N_{+}\rangle}-\frac{N_{-}}{\langle N_{-}\rangle}\Bigg{)}^{2}\Bigg{\rangle}\\\
&=\frac{\langle N_{+}^{2}\rangle}{\langle N_{+}\rangle^{2}}+\frac{\langle
N_{-}^{2}\rangle}{\langle N_{-}\rangle^{2}}-2\frac{\langle
N_{+}N_{-}\rangle}{\langle N_{+}\rangle\langle N_{-}\rangle}.\end{split}$ (2)

The above equation contains cross-correlation between $N_{+}$ and $N_{-}$.

For independent particle production, the cross-correlation becomes
uncorrelated and the statistical contribution becomes :

$\nu_{\pm,\mathrm{stat}}^{\mathrm{2nd}}=\frac{1}{\langle
N_{+}\rangle}+\frac{1}{\langle N_{-}\rangle}.$ (3)

Hence, the dynamical charge fluctuations of $2^{\rm nd}$ order between $N_{+}$
and $N_{-}$ can be expressed using Eq. 2 and 3 as follows,

$\begin{split}\nu_{\pm,dyn}^{\mathrm{2nd}}&=\nu_{\pm}^{\mathrm{2nd}}-\nu_{\pm,\mathrm{stat}}^{\mathrm{2nd}}\\\
&=\frac{\langle N_{+}(N_{+}-1)\rangle}{\langle N_{+}\rangle^{2}}+\frac{\langle
N_{-}(N_{-}-1)\rangle}{\langle N_{-}\rangle^{2}}-2\frac{\langle
N_{+}N_{-}\rangle}{\langle N_{+}\rangle\langle N_{-}\rangle}.\end{split}$ (4)

One can express the above equation in terms of factorial moments; in factorial
moments the detection efficiencies of $N_{\pm}$ are factorized with an ansatz
of the Binomial detector response. The $\nu_{\pm,dyn}^{\mathrm{2nd}}$ is a
robust variable as it has no detector efficiency effect [18].

###  2.2 Third order dynamical charge fluctuations

The $3^{\rm rd}$ order of Eq. 1 can be written as,

$\begin{split}\nu_{\pm}^{\mathrm{3rd}}&=\Bigg{\langle}\Bigg{(}\frac{N_{+}}{\langle
N_{+}\rangle}-\frac{N_{-}}{\langle N_{-}\rangle}\Bigg{)}^{3}\Bigg{\rangle}\\\
&=\frac{\langle N_{+}^{3}\rangle}{\langle N_{+}\rangle^{3}}-\frac{\langle
N_{-}^{3}\rangle}{\langle N_{-}\rangle^{3}}-3\frac{\langle
N_{+}^{2}N_{-}\rangle}{\langle N_{+}\rangle^{2}\langle
N_{-}\rangle}+3\frac{\langle N_{+}N_{-}^{2}\rangle}{\langle
N_{+}\rangle\langle N_{-}\rangle^{2}}.\end{split}$ (5)

For independent particle production and considering both $N_{\pm}$
multiplicity distributions are the Poisson distributions, the statistical
contribution of $3^{\rm rd}$ order becomes :

$\nu_{\pm,\mathrm{stat}}^{\mathrm{3rd}}=\frac{1}{\langle
N_{+}\rangle^{2}}-\frac{1}{\langle N_{-}\rangle^{2}}.$ (6)

Using above equations, the dynamical charge fluctuation of $3^{\rm rd}$ order
can be expressed as :

$\begin{split}\nu_{\pm,dyn}^{\mathrm{3rd}}&=\nu_{\pm}^{\mathrm{3rd}}-\nu_{\pm,\mathrm{stat}}^{\mathrm{3rd}}\\\
&=\frac{\langle N_{+}^{3}\rangle}{\langle N_{+}\rangle^{3}}-\frac{\langle
N_{-}^{3}\rangle}{\langle N_{-}\rangle^{3}}-3\frac{\langle
N_{+}^{2}N_{-}\rangle}{\langle N_{+}\rangle^{2}\langle
N_{-}\rangle}+3\frac{\langle N_{+}N_{-}^{2}\rangle}{\langle
N_{+}\rangle\langle N_{-}\rangle^{2}}-\frac{1}{\langle
N_{+}\rangle^{2}}+\frac{1}{\langle N_{-}\rangle^{2}}.\end{split}$ (7)

Here it is worth to note here that $\nu_{\pm,dyn}^{\mathrm{3rd}}$ contains
higher order cross-correlations between $N_{+}$ and $N_{-}$ and also the last
two terms just remains unfactorized in terms of factorial moments. Hence,
$\nu_{\pm,dyn}^{\mathrm{3rd}}$ is not a robust observable for detector
efficiency effect in the experiment.

###  2.3 Fourth order dynamical charge fluctuations

The $4^{\rm th}$ order of Eq. 1 can be written as,

$\begin{split}\nu_{\pm}^{\mathrm{4th}}&=\Bigg{\langle}\Bigg{(}\frac{N_{+}}{\langle
N_{+}\rangle}-\frac{N_{-}}{\langle N_{-}\rangle}\Bigg{)}^{4}\Bigg{\rangle}\\\
&=\frac{\langle N_{+}^{4}\rangle}{\langle N_{+}\rangle^{4}}+\frac{\langle
N_{-}^{4}\rangle}{\langle N_{-}\rangle^{4}}-4\frac{\langle
N_{+}^{3}N_{-}\rangle}{\langle N_{+}\rangle^{3}\langle
N_{-}\rangle}-4\frac{\langle N_{+}N_{-}^{3}\rangle}{\langle
N_{+}\rangle\langle N_{-}\rangle^{3}}+6\frac{\langle
N_{+}^{2}N_{-}^{2}\rangle}{\langle N_{+}\rangle^{2}\langle
N_{-}\rangle^{2}}.\end{split}$ (8)

For independent particle production and considering both $N_{\pm}$
multiplicity distributions are the Poisson distributions, the statistical
contribution of $4^{\rm th}$ order becomes :

$\nu_{\pm,\mathrm{stat}}^{\mathrm{4th}}=\frac{1}{\langle
N_{+}\rangle^{3}}+\frac{1}{\langle N_{-}\rangle^{3}}+\frac{3}{\langle
N_{+}\rangle^{2}}+\frac{3}{\langle N_{-}\rangle^{2}}+\frac{6}{\langle
N_{+}\rangle\langle N_{-}\rangle}.$ (9)

Using above equations, the dynamical charge fluctuation of $4^{\rm th}$ order
can be expressed as :

$\begin{split}\nu_{\pm,dyn}^{4th}&=\nu_{\pm}^{\mathrm{4th}}-\nu_{\pm,\mathrm{stat}}^{\mathrm{4th}}\\\
&=\\!\begin{multlined}\frac{\langle N_{+}^{4}\rangle}{\langle
N_{+}\rangle^{4}}+\frac{\langle N_{-}^{4}\rangle}{\langle
N_{-}\rangle^{4}}-4\frac{\langle N_{+}^{3}N_{-}\rangle}{\langle
N_{+}\rangle^{3}\langle N_{-}\rangle}-4\frac{\langle
N_{+}N_{-}^{3}\rangle}{\langle N_{+}\rangle\langle
N_{-}\rangle^{3}}+6\frac{\langle N_{+}^{2}N_{-}^{2}\rangle}{\langle
N_{+}\rangle^{2}\langle N_{-}\rangle^{2}}\\\ -\frac{1}{\langle
N_{+}\rangle^{3}}-\frac{1}{\langle N_{-}\rangle^{3}}-\frac{3}{\langle
N_{+}\rangle^{2}}-\frac{3}{\langle N_{-}\rangle^{2}}-\frac{6}{\langle
N_{+}\rangle\langle N_{-}\rangle}.\end{multlined}\frac{\langle
N_{+}^{4}\rangle}{\langle N_{+}\rangle^{4}}+\frac{\langle
N_{-}^{4}\rangle}{\langle N_{-}\rangle^{4}}-4\frac{\langle
N_{+}^{3}N_{-}\rangle}{\langle N_{+}\rangle^{3}\langle
N_{-}\rangle}-4\frac{\langle N_{+}N_{-}^{3}\rangle}{\langle
N_{+}\rangle\langle N_{-}\rangle^{3}}+6\frac{\langle
N_{+}^{2}N_{-}^{2}\rangle}{\langle N_{+}\rangle^{2}\langle
N_{-}\rangle^{2}}\\\ -\frac{1}{\langle N_{+}\rangle^{3}}-\frac{1}{\langle
N_{-}\rangle^{3}}-\frac{3}{\langle N_{+}\rangle^{2}}-\frac{3}{\langle
N_{-}\rangle^{2}}-\frac{6}{\langle N_{+}\rangle\langle
N_{-}\rangle}.\end{split}$ (10)

Here the $\nu_{\pm,dyn}^{\mathrm{4th}}$ contains higher order cross-
correlations between $N_{+}$ and $N_{-}$ and also several terms just remains
unfactorized in terms of factorial moments. However, unlike
$\nu_{\pm,dyn}^{\mathrm{2nd}}$, the $\nu_{\pm,dyn}^{\mathrm{3rd}}$ and
$\nu_{\pm,dyn}^{\mathrm{4th}}$ are not robust observables for detector
efficiency effect in the experiment and hence the
$\nu_{\pm,dyn}^{\mathrm{3rd}}$ and $\nu_{\pm,dyn}^{\mathrm{4th}}$ need
additional detector effect correction in experiment.

As the $\nu_{\pm,dyn}^{\mathrm{3rd}}$ and $\nu_{\pm,dyn}^{\mathrm{4th}}$
contain higher order cross-correlations between $N_{+}$ and $N_{-}$, hence
they are expected to amplify the dynamical charge fluctuation signal as
compared to $\nu_{\pm,dyn}^{\mathrm{2nd}}$. Moreover, these observables are
insensitive to statistical fluctuations due to the subtraction of the
statistical contributions in Eq. 4, 7, and 10. To understand the behaviour of
these observables, a simulation study has been performed using an event
generator known as Simulating Many Accelerated Strong-interacting Hadrons
(SMASH). The detailed description and simulation results are discussed in the
following sections.

##  3 SMASH model description

In this paper, a new hadronic transport approach, SMASH [19], has been applied
to study the dynamical charge fluctuations in heavy-ion collisions at center
of mass energies available at RHIC. This new microscopic transport approach is
relevant to provide a better understanding of the late stage evolution of
resonance excitations and decays with vacuum properties in heavy-ion
collisions. In this model, the conditions for collision geometry are employed
as in the UrQMD (Ultra-relativistic Quantum Molecular Dynamics) approach [20].
For the initial conditions for heavy-ion collisions, the Woods-Saxon
distributions of nuclei are considered in coordinate space. The most well-
established hadronic states with their corresponding decays and cross sections
are implemented in this model. The detailed discussion and validation of this
model with different experimental data can be found in [19]. The simulation
and analysis details are discussed in Sec. 4.

##  4 Simulation and Analysis details

In this simulation, the Au+Au collisions with approximately two millions
events at center of masss energy, $\sqrt{s_{NN}}$ = 200, 62.4 , 27 , 19.6 ,
14.5, 11.5, 9.2 and 7.7 GeV are generated within impact parameter range
between 0 and 16 fm. A fixed time steps option and with the end time at 30
fm/$\it{c}$ are used to simulated these events. The above collisions energies
are the part of RHIC beam energy scan (BES) program including phase-I and II.

###  4.1 Analysis details

This study is performed based on the procedure applied in the STAR experiment
at RHIC. In heavy-ion experiment, the centrality definition is done based on
the charged particles multiplicity distribution at mid rapidity. In this
study, the charged particles multiplicity distribution within pesudorapidity
($\eta$) range -1.5 < $\eta$ < 1.5 is used to determine the centrality in
Au+Au collisions. These centrality classes are determined for different
collision energies. The nine centrality bins based on the fraction of
generated events are selected for this study and those are 0-5%, 5-10% 10-20%,
20-30%, 30-40%, 40-50%, 50-60%, 60-70% and 70-80%.

The event-by-event positive and negative charged particles are counted within
0.2 GeV/$\it c$ < $p_{\rm T}$ < 5.0 GeV/$\it c$ and $|\eta|$ < 1.5. All the
stable hadrons in the PDG have been selected in this simulation.

To avoid the dependence on the central bin width, the value of the observable
is determined using unit bin method. In this method, value of the observable
($\nu$) for each multiplicity bin in centrality class is calculated and then
averaged over the width of particular centrality bin with the weights
corresponding to relative cross section. The weighted average for the
observables are calculated as:

$\nu(J_{min}<j<J_{max})=\frac{\sum\nu(j)w(j)}{\sum w(j)}.$ (11)

Here, $w(j)$ is the weight of $j^{th}$ multiplicity bin; the $J_{min}$ and
$J_{max}$ represent the minimum and maximum range of multiplicity of each
centrality class. The $\nu$ observable can be of any order as mentioned in Eq.
4, 7, and 10.

###  4.2 Statistical uncertainty calculation

As discussed in Sec. 2, the different orders of dynamical charge fluctuations
observable are less sensitive to statistical fluctuation, hence it is expect
to have significantly smaller statistical uncertainty due to the presence of
some residual statistical fluctuations. To calculate the statistical
uncertainty, the Bootstrap method is used and this method is widely used in
the event-by-event fluctuation measurements in heavy-ion collisions [21, 22,
23]. In this method, seventy Bootstrap samples are used to calculate the
standard deviation.

##  5 Results and Discussions

The higher order dynamical charge fluctuations measures, $\nu_{\pm,dyn}^{m}$,
provide information about the strength of higher order particle correlations
which is subjected to change with the total multiplicity of particles in an
event ensemble. These observables can be used to study the charge fluctuations
developed at the time of phase transition in the heavy-ion collisions. In this
section, the behaviour of these observables, in Eq. 4, 7, and 10, with
centrality and at different collision energies have been discussed.

Fig. 1 shows the different order of statistical fluctuations contributing to
their respective dynamical charge fluctuations at $\sqrt{s_{NN}}$=200 GeV in
Au+Au collisions. The expressions of $\nu_{\pm,\rm stat}^{\rm 2nd}$,
$\nu_{\pm,\rm stat}^{\rm 3rd}$, and $\nu_{\pm,\rm stat}^{\rm 4th}$ can be
found in Sec. 2. It shows $\nu_{\pm,\rm stat}^{\rm 2nd}$ and $\nu_{\pm,\rm
stat}^{\rm 4th}$ are positive and behave the same way unlike $\nu_{\pm,\rm
stat}^{\rm 3rd}$. The $\nu_{\pm,\rm stat}^{\rm 3rd}$ is negative as the total
number of positively charged particles larger than the negatively charged
particles in heavy-ion collisions. The values are significantly changed at
peripheral collisions as the $N_{+}$ and $N_{-}$ change rapidly. These
statistical contributions are subtracted in order to get the dynamical charge
fluctuations as discussed below.

Figure 1:  The $\nu_{\pm,\rm stat}^{\rm 2nd}$(left) , $\nu_{\pm,\rm stat}^{\rm
3rd}$(middle), $\nu_{\pm,\rm stat}^{\rm 4th}$(right) for the charge
fluctuations in pseudo-rapidity range $|\eta|<1.5$ for Au+Au collisions at
$\sqrt{s_{NN}}$ = 200 GeV as a function of number of participating nucleons.

Figure 2: The dynamical charge fluctuations, $\nu_{\pm,dyn}^{\rm 2nd}$(left) ,
$\nu_{\pm,dyn}^{\rm 3rd}$(middle), $\nu_{\pm,dyn}^{\rm 4th}$(right) for the
charged particles measured in pseudo-rapidity range $|\eta|<1.5$ for Au+Au
collisions at $\sqrt{s_{NN}}$ = 200 GeV as a function of number of
participating nucleons.

Fig. 2 shows the value of $\nu_{\pm,dyn}^{\rm 2nd}$, $\nu_{\pm,dyn}^{\rm
3rd}$, and $\nu_{\pm,dyn}^{\rm 4th}$for different centrality at
$\sqrt{s_{NN}}$= 200 GeV in Au+Au collisions. The statistical uncertainties
are smaller than the marker size. The values of $\nu_{\pm,dyn}^{\rm 2nd}$ and
$\nu_{\pm,dyn}^{\rm 4th}$ are negative whereas $\nu_{\pm,dyn}^{\rm 3rd}$ is
positive finite at all centrality. In the higher orders $\nu_{\pm,dyn}^{m}$ ,
the higher order correlation terms dominate as discussed in Sec. 2.

Fig. 3 shows these three observables for the eight collisions energies between
$\sqrt{s_{NN}}$= 200 to 7.7 GeV in Au+Au collisions. At a given centrality,
they show a clear energy dependence. The values of $\nu_{\pm,dyn}^{\rm 3rd}$
are very small and show no significant difference between different collisions
energies. This could be due to $\nu_{\pm,dyn}^{\rm 3rd}$ is less sensitive to
charge fluctuations in heavy-ion collisions in the SMASH model. This would be
important to investigate this feature in the data. On the other hand, the
$\nu_{\pm,dyn}^{\rm 4th}$ shows a noticeable difference at different collision
energies relative to its lower order. The fourth order $\nu_{\pm,dyn}^{\rm
4th}$ is relatively more sensitive to charge fluctuations due to presence of
higher order correlation terms. Therefore, in heavy-ion experiments it can
provide vital information about the phase transition.

Figure 3: The dynamical charge fluctuations, $\nu_{\pm,dyn}^{\rm 2nd}$(left) ,
$\nu_{\pm,dyn}^{\rm 3rd}$(middle), $\nu_{\pm,dyn}^{\rm 4th}$(right) for the
charged particles measured in pseudo-rapidity range $|\eta|<1.5$ for Au+Au
collisions at $\sqrt{s_{NN}}$ = 200 GeV to 7.7 GeV as a function of number of
participating nucleons.

Fig. 4 shows the collision energy dependence of the $\nu_{\pm,dyn}^{\rm 2nd}$,
$\nu_{\pm,dyn}^{\rm 3rd}$, $\nu_{\pm,dyn}^{\rm 4th}$ in 0-5% central Au+Au
collisions using the SMASH model. The $\nu_{\pm,dyn}^{\rm 2nd}$ shows a
increasing trend with increasing collision energy whereas $\nu_{\pm,dyn}^{\rm
3rd}$ and $\nu_{\pm,dyn}^{\rm 4th}$ show a decreasing trend with increasing
collision energy.

Figure 4:  The dynamical charge fluctuations, $\nu_{\pm,dyn}^{\rm 2nd}$(left)
, $\nu_{\pm,dyn}^{\rm 3rd}$(middle), $\nu_{\pm,dyn}^{\rm 4th}$(right) as a
function beam energy for 0-5% centrality in Au+Au collisions.

These higher order dynamical charge fluctuation measures can shed light on the
parton-to-hadron phase transition in heavy-ion experiments. These observables
are measured in finite acceptance and centrality, so the volume fluctuations
could play a significant role. The effect of global charge conservation in a
finite acceptance also show an importance as discussed in [11]. Furthermore,
the resonance decay contribution could contribute the higher order
$\nu_{\pm,dyn}^{m}$. It needs more study of these effects and possible impact
on these observables in heavy-ion experiments and beyond the scope of this
paper. Although similar discussion can be found for $\nu_{\pm,dyn}^{\rm 2nd}$
in [6, 12]. This is an attempt to explore new observable that is sensitive to
charge fluctuations and amplify the signal in heavy-ion collisions experiments
at RHIC and the LHC. A detailed theoretical work on these observables is
needed to understand the hardon gas and QGP limit.

##  6 Summary and outlook

This is an attempt to introduce the higher order charge fluctuation observable
those are sensitive to the quark-to-hadron phase transition and can amplify
the signal in heavy-ion collisions. This work is augmentation of previous work
on the dynamical charge fluctuation measure that is second order,
$\nu_{\pm,dyn}^{\rm 2nd}$[11]. In this paper we introduce the third and fourth
order dynamical charge fluctuations and provide the expression of
$\nu_{\pm,dyn}^{\rm 3rd}$ and $\nu_{\pm,dyn}^{\rm 4th}$. These observables
contain the higher order cross-correlation terms, hence they can amplify the
signal in heavy-ion collisions.

The SMASH model is used to study the behavior of the $\nu_{\pm,dyn}^{\rm
2nd}$, $\nu_{\pm,dyn}^{\rm 3rd}$, and $\nu_{\pm,dyn}^{\rm 4th}$ as function of
centrality and collision energies. It is observed that the $\nu_{\pm,dyn}^{\rm
4th}$ can amplify the signal compared to $\nu_{\pm,dyn}^{\rm 2nd}$ and
$\nu_{\pm,dyn}^{\rm 3rd}$. It is also found that the $\nu_{\pm,dyn}^{\rm 3rd}$
and $\nu_{\pm,dyn}^{\rm 4th}$ are sensitive to detector effects unlike the
$\nu_{\pm,dyn}^{\rm 2nd}$.

Various effects like the volume fluctuations, the global charge conservation,
and the resonance effects may play a role which need to be studied as an
outlook of this work. Furthermore, theoretical calculation pertaining to these
observables are required to set the limit of hadron gas and partonic phase of
matter. The measurements at the LHC and RHIC would be interesting to see the
signature of phase transtion in heavy-ion experiments using these observables.

### Acknowledgement

BS is thankful to Nihar Ranjan Sahoo for helpful discussions and suggestions
that initiated this work. She is also very grateful to Tapan Kumar Nayak,
Bedangadas Mohanty and Subhasish Chattopadhyay for their support. This
research used resources of the LHC grid computing center at the Variable
Energy Cyclotron Center, India. The simulations and computations were also
supported in part by the SAMKHYA: High Performance Computing Facility provided
by the Institute of Physics, Bhubaneswar.

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