Open Communications in Nonlinear Mathematical Physics ]ocnmp[ Vol.2 (2022) pp
1–References Letter

††footnotetext:  © The author(s). Distributed under a Creative Commons
Attribution 4.0 International License

Letter to the Editors

On differential equations invariant under two-variable Möbius transformations

M Euler 1, N Euler${}^{\,1\,*}$ and M C Nucci 2

1 Centro Internacional de Ciencias, Av. Universidad s/n, Colonia Chamilpa,
62210 Cuernavaca, Morelos, Mexico
2 Department of Mathematics, University of Bologna, Piazza di Porta San Donato
5, Bologna, Italy
${\ }^{*}$ Corresponding author’s email address<EMAIL_ADDRESS>
Received October 25, 2022; Accepted November 29, 2022

Abstract: We compute invariants for the two-variable Möbius transformation. In
particular we are interested in partial differential equations in two
dependent and two independent variables that are kept invariant under this
transformation.

##  1 Introduction: motivation from the one-variable Möbius transformation

In this section we discuss in some detail the invariants of the one-variable
Möbius transformation

$\displaystyle{\cal M}:\displaystyle{u\mapsto
v=\frac{\alpha_{1}u+\beta_{1}}{\alpha_{2}u+\beta_{2}}},$ (1.1)

for two cases, namely $u=u(x)$ and $u=u(x,t)$. This serves as a motivation for
the current study of the two-variable Möbius transformation for two dependent
variables $\\{u_{1},\,u_{2}\\}$ in two independent variables $\\{x,\,t\\}$. In
fact, we have reported results for the one independent variable case in [6],
and the current paper is a continuation of that study. In particular, we note
the importance of the Schwarzian derivative $S$ (see (1.24b)) which is an
invariant of the Möbius transformation (1.1). This is well known, but we use
the opportunity to discuss this here again in order to compare the results
with the case of the two-variable Möbius transformation that is introduced in
Section 2 (see transformation (2.1)). Instead of the 3rd-order invariant for
the one-variable Möbius transformation, namely the Schwarzian derivative
(1.24b), we obtain two 4th-order invariants for the two-variable Möbius
transformation. We consider this an interesting finding that could be of
importance in the study of integrable systems of nonlinear ordinary and
partial differential equations.

We recall that, under the condition
$\alpha_{1}\beta_{2}-\alpha_{2}\beta_{1}=1$, the Möbius transformation (1.1)
is associated with the Special Linear Transformation $SL(2,\mathbb{R})$, where

$\displaystyle\Phi=\left(\begin{array}[]{cc}\alpha_{1}&\beta_{1}\\\
\alpha_{2}&\beta_{2}\end{array}\right)\in
SL(2,\mathbb{R}),\qquad\mbox{det}\Phi=1.$ (1.4)

A basis for the corresponding 3-dimensional Lie algebra $sl(2,\mathbb{R})$ is
given by the three matrices

$\displaystyle X_{1}=\left(\begin{array}[]{cc}0&1\\\
0&0\end{array}\right),\qquad X_{2}=\left(\begin{array}[]{cr}1&0\\\
0&-1\end{array}\right),\qquad X_{3}=\left(\begin{array}[]{cr}0&0\\\
1&0\end{array}\right).$ (1.11)

We recall that

$\displaystyle\Phi=\exp(\epsilon X),\quad\mbox{where}\
\Phi\in\mbox{SL$(2,\mathbb{R})$ and}\ X\in\mbox{sl$(2,\mathbb{R})$},$ (1.12)

where $\epsilon$ is a small real parameter. Since
$\det\Phi=\exp(\epsilon\,\mbox{Tr}\,X)$ and $\det\Phi=1$, we have
Tr$\,(X_{j})=0$ for all $j=1,2,3$. For a given infinitesimal transformation

$\displaystyle u\mapsto\varphi(u;\epsilon)$ (1.13)

the Lie generator takes the form

$\displaystyle\left.Z=\left(\frac{\partial\varphi(u;\epsilon)}{\partial\epsilon}\right|_{\epsilon=0}\right)\frac{\partial\
}{\partial u}.$ (1.14)

This leads to the following set of basis Lie generators for
$sl(2,\mathbb{R})$:

$\displaystyle\\{\frac{\partial\ }{\partial u},\ u\frac{\partial\ }{\partial
u},\ u^{2}\frac{\partial\ }{\partial u}\\}.$ (1.15)

For more details we refer to [9] (see also the classical work [12])

In order to compute the invariants $I$ of order $p$ for the transformation
(1.1) one needs to solve the linear system of partial differential equations
given by the condition

$\displaystyle Z_{j}^{(p)}I=0,\quad j=1,2,3.$ (1.16)

Here $Z_{j}^{(p)}$ denotes the prolongation of the Lie generator $Z_{j}$ up to
order $p$ (see [9] for details).

###  1.1 One dependent variable and one independent variable: $u(x)$

The simplest case is given by the mapping of one dependent variable $u$ for
one independent variable $x$, i.e. $u=u(x)$. For the Möbius transformation
(1.1), we now have

$\displaystyle{\cal M}:\left\\{\begin{array}[]{l}\displaystyle{u({x})\mapsto
v(\bar{x})=\frac{\alpha_{1}u({x})+\beta_{1}}{\alpha_{2}u({x})+\beta_{2}}}\\\
\\\ \displaystyle{{x}\mapsto\bar{x}={x}}\\\ \end{array}\right.$ (1.20)

where

$\displaystyle\det\left(\begin{array}[]{cc}\alpha_{1}&\beta_{1}\\\
\alpha_{2}&\beta_{2}\end{array}\right)=1.$ (1.23)

Computing the invariants for this transformation up to order 3, we obtain

$\displaystyle\omega_{0}=f(x)$ (1.24a) $\displaystyle
S=\frac{u_{xxx}}{u_{x}}-\frac{3}{2}\frac{u_{xx}^{2}}{u_{x}^{2}},$ (1.24b)

where $f$ is an arbitrary smooth function and $S$ is the well-know Schwarzian
derivative [10]. In fact, all higher-order invariants consist of
$x$-derivatives of $S$. We can now state that any $n$th-order ordinary
differential equation of the form

$\displaystyle\Psi(x,S,S_{x},S_{xx},\ldots,S_{nx})=0$ (1.25)

is invariant under the given Möbius transformation for arbitrary smooth
$\Psi$. Here $S_{nx}$ denotes the $n$th derivative of $S$ with respect to $x$.

Example 1.1: We consider

$\displaystyle S_{x}=\Psi(x,S),$ (1.26)

which is the following 4th-order equation:

$\displaystyle
u_{4x}=\frac{4u_{xxx}u_{xx}}{u_{x}}-\frac{3u_{xx}^{3}}{u_{x}^{2}}+u_{x}\Psi(x,S).$
(1.27)

Equation (1.27) is clearly invariant under the current Möbius transformation
(1.20) and it is moreover the most general semilinear ordinary differential
equation of order four that admits the Lie symmetry algebra $sl(2,\mathbb{R})$
with basis (1.15). Equation (1.27) with $\Psi=\Psi(S)$ has been reported in
[2] for the 4-dimensional Lie symmetry algebra

$\displaystyle\\{\frac{\partial\ }{\partial x},\frac{\partial\ }{\partial
u},u\frac{\partial\ }{\partial u},u^{2}\frac{\partial\ }{\partial u}\\}.$
(1.28)

###  1.2 One dependent variable and two independent variables: $u(x,t)$

We compute all invariants $I$, up to order three, for the Möbius
transformation that maps $u(x,t)$. That is

$\displaystyle{\cal M}:\left\\{\begin{array}[]{l}\displaystyle{u({x,t})\mapsto
v(\bar{x},\bar{t})=\frac{\alpha_{1}u({x,t})+\beta_{1}}{\alpha_{2}u({x,t})+\beta_{2}}}\\\
\\\ \displaystyle{{x}\mapsto\bar{x}={x}}\\\ \\\
\displaystyle{{t}\mapsto\bar{t}={t}},\\\ \end{array}\right.$ (1.34)

where

$\displaystyle\det\left(\begin{array}[]{cc}\alpha_{1}&\beta_{1}\\\
\alpha_{2}&\beta_{2}\end{array}\right)=1.$ (1.37)

The general solution of

$\displaystyle Z_{j}^{(3)}I=0,\quad j=1,2,3,$ (1.38)

is

$\displaystyle
I=\Psi(\omega_{0},\omega_{1},\omega_{2},\omega_{3},\omega_{4},\omega_{5},\omega_{6},\omega_{7}),$
(1.39)

whereby $\omega_{0}=f(x,t)$ is a trivial invariant and $\omega_{k}$ are as
follows:

$\displaystyle\omega_{1}=\frac{u_{t}}{u_{x}}$ (1.40a)
$\displaystyle\omega_{2}=\omega_{1,x}$ (1.40b)
$\displaystyle\omega_{3}=\omega_{1,t}+\omega_{1}\omega_{2}$ (1.40c)
$\displaystyle\omega_{4}\equiv
S=\frac{u_{xxx}}{u_{x}}-\frac{3}{2}\frac{u_{xx}^{2}}{u_{x}^{2}}:\ \mbox{the
Schwarzian derivative}$ (1.40d)
$\displaystyle\omega_{5}=\frac{u_{xxt}}{u_{x}}-\frac{2u_{xx}u_{xt}}{u_{x}^{2}}+\frac{1}{2}\frac{u_{t}u_{xx}^{2}}{u_{x}^{3}}=\omega_{2,x}+\omega_{1}\omega_{4}$
(1.40e)
$\displaystyle\omega_{6}=\frac{u_{xtt}}{u_{x}}-\frac{u_{xx}u_{tt}}{u_{x}^{2}}-\frac{2u_{t}u_{xx}u_{xt}}{u_{x}^{3}}+\frac{3}{2}\frac{u_{t}^{2}u_{xx}}{u_{x}^{4}}=\omega_{3,x}+\omega_{1}^{2}\omega_{4}$
(1.40f)
$\displaystyle\omega_{7}=\frac{u_{ttt}}{u_{x}}-\frac{3u_{t}u_{xx}u_{tt}}{u_{x}^{3}}+\frac{3}{2}\frac{u_{t}^{3}u_{xx}^{2}}{u_{x}^{5}}=\omega_{3,t}+\omega_{2}\omega_{3}+\omega_{1}^{3}\omega_{4}.$
(1.40g)

From the set of invariants,
$\\{\omega_{0},\omega_{1},\omega_{2},\ldots,\omega_{7}\\}$ we note that two
stand out, namely $\omega_{1}$ and the Schwarzian derivative $\omega_{4}$,
which are fundamental for this transformation in the sense that all the other
invariants can be expressed as functions of $\omega_{1}$, $\omega_{4}$ and its
derivatives.

Using the two invariants (1.40a) and (1.40d), as well as their $x$ and
$t$-derivatives of any order, one can write partial differential equations of
the form

$\displaystyle\Psi(\omega_{0},\omega_{1},S,\omega_{1,x},\omega_{1,t},\omega_{1,xx},\ldots,S_{x},S_{t},S_{xx},\ldots)=0,$
(1.41)

all of which are invariant under the Möbius transformation (1.34) and all of
which admit (1.15) as its Lie symmetry algebra. Of particular interest are the
autonomous evolution equations

$\displaystyle u_{t}=u_{x}\Psi(S,S_{x},\ldots,S_{(m-3)x}),$ (1.42)

where $m\geq 3$. We discuss this in Example 1.2.

Example 1.2: An interesting question regarding nonlinear partial differential
equations of the class (1.42) to find those equations that are symmetry-
integrable. A symmetry-integrable equation is defined as an equations that
admits an infinite set of local generalized (or Lie-Bäcklund) symmetries. In
[4] we have addressed this problem and consequently obtained a class of
symmetry-integrable 3rd-order and 5th-order autonomous evolution equations of
the form (1.42). In particular, the 3rd-order evolution equations of the form

$\displaystyle u_{t}=u_{x}\Psi(S),$ (1.43)

contains the following class of equations that are symmetry-integrable [4]:

$\displaystyle u_{t}=u_{x}S\ :\quad\mbox{the Schwarzian KdV equation}$ (1.44a)
$\displaystyle u_{t}=-2\frac{u_{x}}{\sqrt{S}}$ (1.44b) $\displaystyle
u_{t}=\frac{u_{x}}{(b_{1}-S)^{2}}$ (1.44c) $\displaystyle
u_{t}=\frac{u_{x}}{S^{2}}$ (1.44d) $\displaystyle
u_{t}=u_{x}\left(\frac{a_{1}-S}{(a_{1}^{2}+3a_{2})(S^{2}-2a_{1}S-3a_{2})^{1/2}}\right).$
(1.44e)

Here $S$ is the Schwarzian derivative (1.24b) and the constants $a_{1},\
a_{2}$ and $b_{1}$ are arbitrary, except for the conditions
$a_{1}^{2}+3a_{2}\neq 0$ and $b_{1}\neq 0$. In [5] we propose a method by
which one can compute higher-order symmetry-integrable and Möbius-invariant
evolution equations from the above fully-nonlinear third-order equations
(1.44b) – (1.44e).

For the 5th-order semi-linear Möbius-invariant autonomous evolution equations

$\displaystyle u_{t}=u_{x}\Psi(S,S_{x},S_{xx}),$ (1.45)

the following were found to be symmetry-integrable [4]:

$\displaystyle u_{t}=u_{x}\left(S_{xx}+\frac{1}{4}S^{2}\right):\quad\mbox{the
Schwarzian Kupershmidt I equation;}$ (1.46a) $\displaystyle
u_{t}=u_{x}\left(S_{xx}+4S^{2}\right):\quad\mbox{the Schwarzian Kupershmidt II
equation;}$ (1.46b) $\displaystyle
u_{t}=u_{x}\left(S_{xx}+\frac{3}{2}S^{2}\right)\quad\mbox{the Schwarzian 5th-
order KdV equation.}$ (1.46c)

The recursion operators that generate the higher-order members in the
symmetry-integrable hierarchies of (1.46a), (1.46b) and (1.46c) are given in
[4]. The three Möbius-invariant equations (1.46a), (1.46b) and (1.46c) play a
central role for a large class of 5th-order symmetry-integrable equations that
are related to these equations in terms of nonlocal transformations by the so-
called multipotentialization procedure (see [3] for details).

######  Remark 1.

Replacing $t$ by $x$ for the invariants (1.40e), (1.40f) and (1.40e), results
in the Schwarzian derivative (1.40d). One could therefore think of (1.40e),
(1.40f) and (1.40e) as two-dimensional Schwarzian-type derivatives involving
two independent variables $x$ and $t$. Let us denote this two-dimensional
Schwarzian-type derivative defined by (1.40e), by the notation
$S_{1}^{[x,y]}$, i.e.

$\displaystyle
S_{1}^{[x,y]}u(x,t):=\frac{u_{xxt}}{u_{x}}-\frac{2u_{xx}u_{xt}}{u_{x}^{2}}+\frac{1}{2}\frac{u_{t}u_{xx}^{2}}{u_{x}^{3}}.$
(1.47)

Consider now a function $f(u(x,t))$. Using $S_{1}^{[x,y]}$ we have

$\displaystyle(S_{1}^{[x,y]}f)(u(x,t))=\frac{D_{xxt}f}{D_{x}f}-2\frac{(D_{xx}f)(D_{xt}f)}{(D_{x}f)^{2}}+\frac{1}{2}\frac{(D_{t}f)(D_{xx}f)^{2}}{(D_{x}f)^{3}}$
$\displaystyle\qquad\qquad\qquad\quad\
=\left(\frac{f_{uuu}}{f_{u}}-\frac{3}{2}\frac{f_{uu}^{2}}{f_{u}^{2}}\right)u_{x}u_{t}+S_{1}^{[x,y]}u(x,t),$
(1.48)

where $D$ denote the total derivative and the subscript $u$ the ordinary
derivative with respect to $u$. Additional two-dimensional Schwarzian-type
derivatives can be defined using the remaining two third-order invariants,
that is $S_{2}^{[x,y]}:=\omega_{6}$ and $S_{3}^{[x,y]}:=\omega_{7}$, where
$\omega_{6}$ and $\omega_{7}$ are given by (1.40f) and (1.40e), respectively.
One should remark that the invariants listed above are of course not unique,
as any function of the given invariants is an invariant for the same
transformation and, moreover, the roles of $x$ and $t$ can be exchanged in the
derivatives of the expressions (1.40a) to (1.40g) without affecting the
invariance under the Möbius transformation (1.34).

######  Remark 2.

It is interesting to note that the Bateman equation

$\displaystyle u_{tt}u_{x}^{2}-2u_{x}u_{t}u_{xt}+u_{xx}u_{t}^{2}=0$ (1.49)

can be presented in terms of the invariants (1.40a), (1.40b) and (1.40c),
namely as follows:

$\displaystyle(\omega_{3}-2\omega_{1}\omega_{2})u_{x}^{3}=0.$ (1.50)

Equation (1.49) plays an important role in the Painlevé analysis of PDEs (see
e.g. [11] and [7]). It is clear that
$u_{tt}u_{x}^{2}-2u_{x}u_{t}u_{xt}+u_{xx}u_{t}^{2}$ is not invariant under the
Möbius transformation (1.34). However, (1.49) does admit the Lie symmetry
algebra $sl(2,\mathbb{R})$ with basis generators (1.15). This can directly be
verified by the Lie symmetry condition

$\displaystyle\left.\vphantom{\frac{DA}{DB}}Z_{j}^{(2)}E\right|_{E=0}=0,\
j=1,2,3,$ (1.51)

where $E:=u_{tt}u_{x}^{2}-2u_{x}u_{t}u_{xt}+u_{xx}u_{t}^{2}$ and
$\\{Z^{(2)}_{1},Z^{(2)}_{2},Z^{(2)}_{3}\\}$ are the 2nd-prolongations of the
generators (1.15).

##  2 The Möbius transformation that maps two variables

We now consider the Möbius transformation

$\displaystyle{\cal M}:\displaystyle{{\bf u}\mapsto{\bf v}=\frac{A{\bf
u}+B}{C{\bf u}+\beta}},$ (2.1)

where ${\bf u}=(u_{1},u_{2})$, ${\bf v}=(v_{1},v_{2})$, $A$ is a constant
$2\times 2$ matrix,

$\displaystyle A=\left(\begin{array}[]{cc}a_{11}&a_{12}\\\
a_{21}&a_{22}\end{array}\right),$ (2.4)

$B$ is a constant $2\times 1$ matrix and $C$ a constant $1\times 2$ matrix,

$\displaystyle B=\left(\begin{array}[]{c}b_{11}\\\
b_{21}\end{array}\right),\qquad
C=\left(\begin{array}[]{cccc}c_{11}&c_{12}\end{array}\right),$ (2.8)

and $\beta$ is a constant, with the condition

$\displaystyle\mbox{Det}\,\left(\begin{array}[]{cc}A&B\\\
C&\beta\end{array}\right)=1.$ (2.11)

Now

$\displaystyle\Phi=\left(\begin{array}[]{cc}A&B\\\
C&\beta\end{array}\right)\in SL(3,\mathbb{R}).$ (2.14)

and $sl(3,\mathbb{R})$ is the Lie algebra of all real traceless $3\times 3$
matrices with dim$[sl(3,\mathbb{R})]=8$. A basis for the 8-dimensional Lie
algebra $sl(3,\mathbb{R})$ is given by the following eight matrices:

$\displaystyle X_{1}=\left(\begin{array}[]{ccc}0&0&1\\\ 0&0&0\\\
0&0&0\end{array}\right),\qquad\ \ X_{2}=\left(\begin{array}[]{ccc}0&0&0\\\
0&0&1\\\ 0&0&0\end{array}\right),\qquad
X_{3}=\left(\begin{array}[]{crc}1&0&0\\\ 0&-1&0\\\ 0&0&0\end{array}\right),$
(2.24) $\displaystyle X_{4}=\left(\begin{array}[]{ccr}0&0&0\\\ 0&1&0\\\
0&0&-1\end{array}\right),\qquad X_{5}=\left(\begin{array}[]{ccc}0&0&0\\\
1&0&0\\\ 0&0&0\end{array}\right),\qquad
X_{6}=\left(\begin{array}[]{ccc}0&1&0\\\ 0&0&0\\\ 0&0&0\end{array}\right),$
(2.34) $\displaystyle X_{7}=\left(\begin{array}[]{ccc}0&0&0\\\ 0&0&0\\\
1&0&0\end{array}\right),\qquad\ \ X_{8}=\left(\begin{array}[]{ccc}0&0&0\\\
0&0&0\\\ 0&1&0\end{array}\right).$ (2.41)

For the infinitesimal transformation

$\displaystyle v_{1}=\varphi_{1}(u_{1},u_{2};\epsilon)$ (2.42a) $\displaystyle
v_{2}=\varphi_{2}(u_{1},u_{2};\epsilon)$ (2.42b)

the Lie generator takes the form

$\displaystyle\left.\left.Z=\left(\frac{\partial\varphi_{1}(u_{1},u_{2};\epsilon)}{\partial\epsilon}\right|_{\epsilon=0}\right)\frac{\partial\
}{\partial
u_{1}}+\left(\frac{\partial\varphi_{2}(u_{1},u_{2};\epsilon)}{\partial\epsilon}\right|_{\epsilon=0}\right)\frac{\partial\
}{\partial u_{2}}.$ (2.43)

This leads to the following basis set of Lie generators for
$sl(3,\mathbb{R})$:

$\displaystyle\\{Z_{1}=\frac{\partial\ }{\partial u_{1}},\
Z_{2}=\frac{\partial\ }{\partial u_{2}},\ Z_{3}=u_{1}\frac{\partial\
}{\partial u_{1}},\ Z_{4}=u_{2}\frac{\partial\ }{\partial u_{2}},\
Z_{5}=u_{1}\frac{\partial\ }{\partial u_{2}},\ Z_{6}=u_{2}\frac{\partial\
}{\partial u_{1}},$ $\displaystyle\ \ Z_{7}=u_{1}^{2}\frac{\partial\
}{\partial u_{1}}+u_{1}u_{2}\frac{\partial\ }{\partial u_{2}},\
Z_{8}=u_{1}u_{2}\frac{\partial\ }{\partial u_{1}}+u_{2}^{2}\frac{\partial\
}{\partial u_{2}}\\}.$ (2.44)

###  2.1 Two dependent and one independent variables: $\\{u_{1}(x),\
u_{2}(x)\\}$

We consider the Möbius transformation (2.1) for ${\bf
u}(x)=(u_{1}(x),u_{2}(x))$, that is

$\displaystyle{\cal
M}:\left\\{\begin{array}[]{l}\displaystyle{{u_{1}}({x})\mapsto{v_{1}}(\bar{x})=\frac{a_{11}u_{1}(x)+a_{12}u_{2}(x)+b_{11}}{c_{11}u_{1}(x)+c_{12}u_{2}(x)+\beta}}\\\
\\\
\displaystyle{{u_{2}}({x})\mapsto{v_{2}}(\bar{x})=\frac{a_{21}u_{1}(x)+a_{22}u_{2}(x)+b_{21}}{c_{11}u_{1}(x)+c_{12}u_{2}(x)+\beta}}\\\
\\\ \displaystyle{{x}\mapsto\bar{x}={x}}\\\ \end{array}\right.$ (2.50)

where

$\displaystyle\mbox{Det}\,\left(\begin{array}[]{ccc}a_{11}&a_{12}&b_{11}\\\
a_{21}&a_{22}&b_{21}\\\ c_{11}&c_{12}&\beta\end{array}\right)=1.$ (2.54)

This case has already been studied and is reported in [6]. Besides the obvious
invariant $\omega_{0}=f(x)$, (2.50) admits the following two invariants of
order four:

$\displaystyle\omega_{41}=-\frac{3u_{1,x}u_{2,x}(S_{1,x}-S_{2,x})}{u_{1,x}u_{2,xx}-u_{2,x}u_{1,xx}}-4\left(\frac{u_{1,x}u_{2,x}}{u_{1,x}u_{2,xx}-u_{2,x}u_{1,xx}}\right)^{2}(S_{1}-S_{2})^{2}$
$\displaystyle\qquad+6\left(\frac{u_{1,x}u_{2,xx}S_{1}-u_{2,x}u_{1,xx}S_{2}}{u_{1,x}u_{2,xx}-u_{2,x}u_{1,xx}}\right)$
(2.55a)
$\displaystyle\omega_{42}=\frac{u_{1,x}u_{2,xx}S_{2,x}-u_{2,x}u_{1,xx}S_{1,x}}{u_{1,x}u_{2,xx}-u_{2,x}u_{1,xx}}$
$\displaystyle\qquad+\frac{2}{3}\left(\frac{u_{1,x}u_{2,x}}{u_{1,x}u_{2,xx}-u_{2,x}u_{1,xx}}\right)^{2}(S_{1,x}-S_{2,x})(S_{1}-S_{2})$
$\displaystyle\qquad+\frac{2}{9}\left(\frac{3u_{1,xx}(S_{1}-S_{2})}{u_{1,x}u_{2,xx}-u_{2,x}u_{1,xx}}-\frac{2u_{2,x}u_{1,x}^{2}(S_{1}-S_{2})^{2}}{(u_{1,x}u_{2,xx}-u_{2,x}u_{1,xx})^{2}}\right)\times$
$\displaystyle\qquad\quad\times\left(3u_{2,xx}-\frac{2u_{1,x}u_{2,x}^{2}(S_{1}-S_{2})}{u_{1,x}u_{2,xx}-u_{2,x}u_{1,xx}}\right).$
(2.55b)

Here $S_{1}$ and $S_{2}$ denote the Schwarzian derivatives for $u_{1}(x)$ and
$u_{2}(x)$, respectively. That is

$\displaystyle
S_{1}=\frac{u_{1,xxx}}{u_{1,x}}-\frac{3}{2}\frac{u_{1,xx}^{2}}{u_{1,x}^{2}},\quad
S_{2}=\frac{u_{2,xxx}}{u_{2,x}}-\frac{3}{2}\frac{u_{2,xx}^{2}}{u_{2,x}^{2}}.$
(2.56)

Systems of ordinary differential equations invariant under (2.50) and related
to the above invariants (2.55a) and (2.55b) have been described and reported
in [6].

###  2.2 Two dependent and two independent variables: $\\{u_{1}(x,t),\
u_{2}(x,t)\\}$

Here we consider the Möbius transformation (2.1) for ${\bf
u}(x,t)=(u_{1}(x,t),u_{2}(x,t))$, that is

$\displaystyle{\cal
M}:\left\\{\begin{array}[]{l}\displaystyle{{u_{1}}({x,t})\mapsto{v_{1}}(\bar{x},\bar{t})=\frac{a_{11}u_{1}(x,t)+a_{12}u_{2}(x,t)+b_{11}}{c_{11}u_{1}(x,t)+c_{12}u_{2}(x,t)+\beta}}\\\
\\\
\displaystyle{{u_{2}}({x,t})\mapsto{v_{2}}(\bar{x},\bar{t})=\frac{a_{21}u_{1}(x,t)+a_{22}u_{2}(x,t)+b_{21}}{c_{11}u_{1}(x,t)+c_{12}u_{2}(x,t)+\beta}}\\\
\\\ \displaystyle{{x}\mapsto\bar{x}={x}}\\\ \\\
\displaystyle{{t}\mapsto\bar{t}={t}},\\\ \end{array}\right.$ (2.64)

where

$\displaystyle\mbox{Det}\,\left(\begin{array}[]{ccc}a_{11}&a_{12}&b_{11}\\\
a_{21}&a_{22}&b_{21}\\\ c_{11}&c_{12}&\beta\end{array}\right)=1.$ (2.68)

We compute all 3rd-order invariants for (2.64). For the general solution of

$\displaystyle Z_{j}^{(3)}I=0,\qquad j=1,2,\ldots,8,$ (2.69)

we obtain

$\displaystyle I=F(\omega_{0},\omega_{1},\omega_{2},\ldots,\omega_{12}),$
(2.70)

where $\omega_{0}=f(x,t)$ is the trivial invariant, and

$\displaystyle\omega_{1}=\frac{u_{2,x}u_{1,xx}-u_{1,x}u_{2,xx}}{u_{2,x}u_{1,t}-u_{1,x}u_{2,t}}$
(2.71a)
$\displaystyle\omega_{2}=\frac{u_{1,xx}u_{2,t}-u_{1,t}u_{2,xx}+2u_{2,x}u_{1,xt}-2u_{2,xt}u_{1,x}}{2u_{2,x}u_{1,t}-2u_{1,x}u_{2,t}}$
(2.71b)
$\displaystyle\omega_{3}=\frac{u_{1,tt}u_{2,t}-u_{1,t}u_{2,tt}}{u_{2,x}u_{1,t}-u_{1,x}u_{2,t}}$
(2.71c)
$\displaystyle\omega_{4}=\frac{u_{2,x}u_{1,tt}+2u_{1,xt}u_{2,t}-2u_{1,t}u_{2,xt}-u_{2,tt}u_{1,x}}{u_{2,x}u_{1,t}-u_{1,x}u_{2,t}}$
(2.71d)
$\displaystyle\omega_{5}=2(\omega_{2,x}-\omega_{1,t}-\omega_{1}\omega_{4}+\omega_{2}^{2})$
(2.71e) $\displaystyle\omega_{6}=\omega_{1,x}-2\omega_{1}\omega_{2}$ (2.71f)
$\displaystyle\omega_{7}=\frac{2}{3}\omega_{4,x}-\frac{2}{3}\omega_{2,t}-\omega_{1}\omega_{3}$
(2.71g)
$\displaystyle\omega_{8}=\omega_{1,t}+\omega_{1}\omega_{4}-2\omega_{2}^{2}$
(2.71h) $\displaystyle\omega_{9}=\omega_{3,x}$ (2.71i)
$\displaystyle\omega_{10}=-\frac{1}{3}\omega_{4,x}+\frac{4}{3}\omega_{2,t}+2\omega_{3}$
(2.71j) $\displaystyle\omega_{11}=\omega_{3,t}+\omega_{3}\omega_{4}$ (2.71k)
$\displaystyle\omega_{12}=-2\omega_{3,x}+\omega_{4,t}+4\omega_{2}\omega_{3}+\omega_{4}^{2}$
(2.71l)

Note that there exist no fundamental invariants of order three since all the
3rd-order invariants are combinations of the 2nd-order invariants and its
derivatives. In addition to the above listed invariants, we recall the 4th-
order invariants (2.55a) and (2.55b) that was obtained for the Möbius
transformation (2.50). These 4th-order invariants are of course also valid for
the Möbius transformation (2.64), whereby $u_{1}=u_{1}(x,t)$ and
$u_{2}=u_{2}(x,t)$ in (2.55a) and (2.55b). All these invariants can easily be
verified by checking condition (2.69).

Using now the six invariants (2.71a) - (2.71d), (2.55a) and (2.55b), as well
as $x$ and $t$-derivatives of any order of those invariants, one can write
systems of partial differential equations, all of which are invariant under
the Möbius transformation (2.64) and all of which admit (2.44) as its Lie
symmetry algebra.

Example 2.1: A system of two partial differential equations of order two that
is invariant under the Möbius transformation (2.64) is the following:

$\displaystyle
u_{1,xx}=\frac{1}{u_{2,x}u_{1,t}-u_{1,x}u_{2,t}}\left[\vphantom{\frac{DA}{DB}}2u_{1,x}u_{2,x}u_{1,xt}-2u_{1,x}^{2}u_{2,xt}\right.$
$\displaystyle\qquad\left.\vphantom{\frac{DA}{DB}}+(u_{2,x}u_{1,t}^{2}-u_{1,x}u_{1,t}u_{2,t})F_{1}(\omega_{3},\omega_{4})+2(u_{1,x}^{2}u_{2,t}-u_{1,x}u_{1,t}u_{2,x})F_{2}(\omega_{3},\omega_{4})\right]$
(2.72a) $\displaystyle
u_{2,xx}=\frac{1}{u_{2,x}u_{1,t}-u_{1,x}u_{2,t}}\left[\vphantom{\frac{DA}{DB}}-2u_{1,x}u_{2,x}u_{2,xt}+2u_{2,x}^{2}u_{1,xt}\right.$
$\displaystyle\qquad\left.\vphantom{\frac{DA}{DB}}-(u_{1,x}u_{2,t}^{2}-u_{2,x}u_{1,t}u_{2,t})F_{1}(\omega_{3},\omega_{4})-2(u_{2,x}^{2}u_{1,t}-u_{1,x}u_{2,t}u_{2,x})F_{2}(\omega_{3},\omega_{4}).\right],$
(2.72b)

This is obtained by algebraically solving $u_{1,xx}$ and $u_{2,xx}$ from the
system

$\displaystyle\omega_{1}=F_{1}(\omega_{3},\omega_{4})$
$\displaystyle\omega_{2}=F_{2}(\omega_{3},\omega_{4}).$

Here $F_{1}$ and $F_{2}$ are arbitrary smooth functions of their arguments,
whereby $\omega_{1}$, $\omega_{2}$, $\omega_{3}$ and $\omega_{4}$ are given by
(2.71a), (2.71b), (2.71c) and (2.71d), respectively.

Example 2.2: A system of two partial differential equations of order four that
is invariant under the Möbius transformation (2.64) is obtained by
considering, for example,

$\displaystyle\omega_{1}=F_{1}(\omega_{2},\omega_{4},\omega_{41},\omega_{42})$
$\displaystyle\omega_{3}=F_{2}(\omega_{2},\omega_{4},\omega_{41},\omega_{42}),$

where $\omega_{1}$, $\omega_{2}$, $\omega_{3}$, $\omega_{4}$, $\omega_{41}$
and $\omega_{42}$ are given by (2.71a), (2.71b), (2.71c), (2.71d), (2.55a) and
(2.55b), respectively. If we drop $\omega_{2}$ and $\omega_{4}$ from the
previous system and let $F_{1}=\omega_{41}$, $F_{2}=\omega_{42}$, we obtain

$\displaystyle\frac{u_{2,x}u_{1,xx}-u_{1,x}u_{2,xx}}{u_{2,x}u_{1,t}-u_{1,x}u_{2,t}}=-\frac{3u_{1,x}u_{2,x}(S_{1,x}-S_{2,x})}{u_{1,x}u_{2,xx}-u_{2,x}u_{1,xx}}$
$\displaystyle\qquad\qquad\qquad\qquad\quad-4\left(\frac{u_{1,x}u_{2,x}}{u_{1,x}u_{2,xx}-u_{2,x}u_{1,xx}}\right)^{2}(S_{1}-S_{2})^{2}$
$\displaystyle\qquad\qquad\qquad\qquad\quad+6\left(\frac{u_{1,x}u_{2,xx}S_{1}-u_{2,x}u_{1,xx}S_{2}}{u_{1,x}u_{2,xx}-u_{2,x}u_{1,xx}}\right)$
(2.75a)
$\displaystyle\frac{u_{1,tt}u_{2,t}-u_{1,t}u_{2,tt}}{u_{2,x}u_{1,t}-u_{1,x}u_{2,t}}=\frac{u_{1,x}u_{2,xx}S_{2,x}-u_{2,x}u_{1,xx}S_{1,x}}{u_{1,x}u_{2,xx}-u_{2,x}u_{1,xx}}$
$\displaystyle\qquad\qquad\qquad\qquad\quad+\frac{2}{3}\left(\frac{u_{1,x}u_{2,x}}{u_{1,x}u_{2,xx}-u_{2,x}u_{1,xx}}\right)^{2}(S_{1,x}-S_{2,x})(S_{1}-S_{2})$
$\displaystyle\qquad\qquad\qquad\qquad\quad+\frac{2}{9}\left(\frac{3u_{1,xx}(S_{1}-S_{2})}{u_{1,x}u_{2,xx}-u_{2,x}u_{1,xx}}-\frac{2u_{2,x}u_{1,x}^{2}(S_{1}-S_{2})^{2}}{(u_{1,x}u_{2,xx}-u_{2,x}u_{1,xx})^{2}}\right)\times$
$\displaystyle\qquad\qquad\qquad\qquad\quad\times\left(3u_{2,xx}-\frac{2u_{1,x}u_{2,x}^{2}(S_{1}-S_{2})}{u_{1,x}u_{2,xx}-u_{2,x}u_{1,xx}}\right),$
(2.75b)

where $S_{1}$ and $S_{2}$ are the Schwarzian derivatives (2.56) in
$u_{1}(x,t)$ and $u_{2}(x,t)$, respectively.

##  3 Concluding remarks

Given the well-known importance of the one-variable Möbius transformation
(1.1) and its invariant, the Schwarzian derivative (1.24b), for partial
differential differential equations (see Example 1.2), we investigate here the
invariants of the two-variable Möbius transformation (2.64) and its relation
to systems of two partial differential equations. Two examples of systems that
are invariant under the Möbius transformation (2.64) are given in Example 2.1
and Example 2.2. It is clear from the results obtained here that there exists
no system of evolution equations that is kept invariant under (2.64), i.e. no
system of the form

$\displaystyle
u_{1,t}=F_{1}(x,t,u_{1}.u_{2},u_{1,x},u_{2,x},\ldots,u_{1,px},u_{2,qx})$
$\displaystyle
u_{2,t}=F_{2}(x,t,u_{1}.u_{2},u_{1,x},u_{2,x},\ldots,u_{1,px},u_{2,qx}).$

It could be of interest to study the systems of partial differential equations
that are invariant under (2.64) for further properties, for example to see
whether there exist any symmetry-integrable systems amongst those, similar to
the case of evolution equations for one dependent variable (see Example 1.2).

Let us finally point to the two invariants (2.55a) and (2.55b) of the Möbius
transformation (2.1). These 4th-order invariants consist of only
$x$-derivatives of the variables $u_{1}$ and $u_{2}$ and are, in a sense,
similar to the Schwarzian derivative which is a 3rd-order invariant for (1.1).
It remains to be seen whether or not (2.55a) and (2.55b) will indeed turn out
to be significant for the study of systems of differential equations and their
applications.

The referee kindly pointed out the papers [1] and [8] where related aspects of
differential invariants have been reported. A comparison of our results to
these works could be of interest.

## References

  * [1] Beffa G M and Olver P J, Differential invariants for parametrized projective surfaces, Communications in Analysis and Geometry 7 (4), 807–839, 1999.
  * [2] Cerquetelli T, Ciccoli N and Nucci M C, Four dimensional Lie symmetry algebras and fourth order ordinary differential equations, Journal of Nonlinear Mathematical Physics 9 Suppl. 2, 24–35, 2002.
  * [3] Euler M and Euler N, Nonlocal invariance of the multipotentialisations of the Kupershmidt equation and its higher-order hierarchies In: Nonlinear Systems and Their Remarkable Mathematical Structures, N Euler (ed), CRC Press, Boca Raton, 317–351, 2018.
  * [4] Euler M and Euler N, On Möbius-invariant and symmetry-integrable evolution equations and the Schwarzian derivative, Studies in Applied Mathematics, 143, 139–156, 2019.
  * [5] Euler M and Euler N, On the hierarchy of fully-nonlinear Möbius-invariant and symmetry-integrable equations of order three, Journal of Nonlinear Mathematical Physics, 27, 521–-528, 2021.
  * [6] Euler M, Euler N and Nucci MC, Ordinary differential equations invariant under two-variable Möbius transformations, Applied Mathematics Letters, 117, 107105, 2021.
  * [7] Euler N and Lindblom O, n-Dimensional Bateman equation and the Painlevé analysis of wave equations International Journal of Differential Equations and Applications 1, 205–223, 2000.
  * [8] Hubert E and Olver P J, Differential invariants of conformal and projective surfaces, Symmetry, Integrability and Geometry: Methods and Applications 3 Paper 097, 15 pages, 2007.
  * [9] Olver P J, Applications of Lie Groups to Differential Equations, Springer, New York, 1986.
  * [10] Ovsienko V and Tabachnikov S, What is … the Schwarzian derivative? Notices of the AMS, 56 nr. 2, 2009.
  * [11] Steeb W-H and Euler N, Nonlinear Evolution Equations and Painlevé Test, World Scientific, Singapore, 1988.
  * [12] Wilczynski E J, Projective Differential Geometry of Curves and Ruled Surfaces, B. G. Teubner, Leipzig, 1906.


###### Abstract

A strongly interacting Quark-Gluon Plasma (sQGP) is created in the high energy
heavy ion collisions at RHIC and LHC. Our present understanding of sQGP as a
very good liquid with astonishingly low viscosity is reviewed. With the
arrival of the interesting results from LHC in high-energy p+p and p+A, a new
endeavour to characterize the transition from these small systems to heavy
ions (A+A) is now in place, since, even the small systems showed prominent
similarities to heavy ions in the rising multiplicity domains. An outlook of
future possibilities for better measurements is also made at the end of this
brief review.

Exploring sQGP and Small Systems

Debasish Dasa111email : debasish.das<EMAIL_ADDRESS>
aSaha Institute of Nuclear Physics, HBNI, 1/AF, Bidhannagar, Kolkata 700064,
India

##  1 Introduction

We all know that the normal nuclear matter is made up of protons and neutrons,
which in turn are made up of the quarks [1] and gluons [2]. The quarks and
gluons are confined inside the colorless particles called hadrons and free
colored particles do not occur. As explained by Quantum Chromo-Dynamics (QCD),
the strong interaction is the governing interaction in the subatomic world [3,
4]. One of the important experimental observations that QCD needs to decipher,
is the confinement of the quarks and gluons [5]. The confinement property is
yet not fully understood, even though qualitatively we know about the hadron
properties (mesons are bound states of a quark and anti-quark and baryons are
bound states of 3 quarks) from the quark model [6]. The refinements of the
quark model of hadrons and the development of QCD, naturally led to
expectations that matter at very high densities [7, 8, 9, 10] may exist in a
state of quasi-free quarks and gluons, the Quark-Gluon Plasma (QGP) [11, 12,
13].

The very early universe was different than the present times. It was too hot
and dense to allow the quarks and gluons to form hadrons and was apparently
filled with a thermalized plasma of deconfined quarks, anti-quarks, gluons and
leptons, which was the primordial QGP [14]. The universe may have left the QGP
phase after a few microseconds with the available quarks and gluons combining
towards the formation of the mesons and baryons [15]. In our laboratories, we
can probe the QGP [16, 17, 18, 19, 20, 21, 22] which is a deconfined system of
quarks and gluons, by colliding heavy nuclei at relativistic energies. Such
collisions, create QGP which can be characterized by colored partons as the
dynamic degrees of freedom [23]. Smashing heavy ions, typically Au or Pb ions,
at relativistic energies in the present accelerator facilities, such as the
Relativistic Heavy Ion Collider (RHIC) and the Large Hadron Collider (LHC),
can create the QGP. The dynamics of the early universe in terms of the “Big
Bang” can be studied experimentally by relativistic nucleus-nucleus collisions
at RHIC and LHC in terms of “little bangs” in the laboratory [14, 15, 23].

The main epochs [23] for such a “little bang” collision are : (1) the two
nuclei which are lorentz contracted and now disk-like approach each other and
collide with a very small traversal time ($\ll$ 1 fm/c). (2) The interactions
start developing when the two nuclei hit each other and after such an impact
the “hard” processes [11, 24, 25, 27] i.e those which comprise of relatively
large transferred momenta Q $\gg$ 1 GeV between the quarks, anti-quarks or
gluons (partons) inside the nucleons of the two nuclei produce secondary
partons with large transverse momenta $p_{T}$ [26]. During these times the
matter is out of equilibrium and hence will need some time to equilibrate
[15]. (3) The “soft” collisions or those with small momentum exchange Q $<$ 1
GeV cause copius production of particles after sometime and thermalize the QGP
after about 1 fm/c [28]. The QGP now expands hydrodynamically and then cools
down approximately adiabatically [13, 15]. (4) The QGP then converts to a gas
of hadrons and the hadrons continue to interact quasi-elastically, further
accelerating the expansion and cooling the fireball until thermal freeze-out (
after $\approx$ 5-10 fm/c) into thousands of hadrons. The unstable hadrons
decay and the stable decay products fly out to the large scale detectors
surrounding the interaction region. During the hadronization process the
chemical composition of the hadron gas is fixed and remains basically constant
afterwards [15, 23, 28].

By studying the behavior of the matter created in “little bangs” we can
explore the phase structure of the strongly interacting matter [14, 15, 23,
28]. The QGP reveals emerging collective behavior [28, 29, 30, 31] that
originates from the many-body interactions in QCD. The heavy-ion experiments
have explored the close to perfect fluidity aspects of QGP [32, 33, 34, 35],
precisely, with varied experimental observables [36]. The new state of
strongly interacting matter created in these collisions, have low shear
viscosity($\eta$) to entropy density($s$) ratio, $\eta/s$, which is close to a
nearly perfect fluid [15, 36, 37, 38, 39, 40, 41].

The paper is organised to start with a brief introduction of QGP and in
Section 2 we have a brief survey of the different avenues of the formation and
promulgation of the strongly coupled QGP. The term “strongly coupled QGP
(sQGP)” was coined [32, 42] as we have realised that QGP formed in
relativistic heavy ion collisions is not a weakly coupled gas but on the other
hand is more a strongly coupled liquid [33, 42, 43]. The realization that QGP
created at RHIC is not a weakly coupled gas but a strongly coupled liquid has
aroused a significant development in this research field. In Section 3 the
varied probes for this dense matter formed in our laboratories and their
inferences towards the understanding of the small systems like p+p and p+A
collisions are discussed. Without the critial understanding of such small
systems we cannot characterize the A+A collisions. Finally, we summarise by
looking into the future scope for such studies that lie ahread.

##  2 sQGP

The results from the relativistic heavy ion collision experiments have changed
the theoretical understanding of the properties of the QCD matter. Also
significant know-how has evolved regarding the deconfined QCD matter created
in the central interaction volume at such high energies. Previously QGP was
felt to be a weakly interacting system of quarks and gluons which might be
described by perturbative QCD (pQCD). However contrary to the expectations,
the experimental results from RHIC [16, 17, 18, 19], have shown that a hot,
strongly interacting, nearly perfect and almost opaque relativistic liquid,
also termed as the strongly coupled QGP was created in central Au+Au
collisions at the top RHIC energy regime [15, 32, 33, 34].

The comparative studies of the experimental data [16, 17, 18, 19], and
especially the elliptic flow ($v_{2}$) [35], in terms of the hydrodynamic
models showed the nearly perfect fluid behavior of QGP. Such inferences
indicate that its properties correspond to non-perturbative, strongly
interacting matter. RHIC results showed that the resulting plasma could be
well described by a hydrodynamic picture of a nearly ideal liquid, which show
very limited internal friction or in other words very small shear viscosity
($\eta$). The created medium in such relativistic collisions, can connect to
the pressure gradients by flowing apparently unobstructed [36, 44, 45].

Shear viscosity, $\eta$, is a characterizing parameter for fluids [44, 46, 47,
48] and can be defined in terms of the friction force $F$ per unit area $A$
produced by a shear flow with transverse flow gradient $\nabla_{\\!y}v_{x}$,

$\frac{F}{A}=\eta\,\nabla_{\\!y}v_{x}\,.$ (1)

Small shear viscosity is a benchmark for a good fluid.

Shear viscosity for a weakly coupled gas can be estimated as

$\eta=\frac{1}{3}\,np\lambda\,,$ (2)

where $n$ is the density, $p$ is the average momentum of the gas molecules,
and $\lambda$ is the mean free path. The mean free path can be expressed as
$\lambda=1/(n\sigma)$ where $\sigma$ is a preferable transport cross-section.
For relativistic fluids it is more natural to normalize $\eta$ to the entropy
density $s$ rather than the particle density $n$.

It has been observed that good fluids are characterized by $\eta/s$ $\sim$
$\hbar/k_{B}$ and this value is consistent with simple theoretical
propositions. For all fluids, the proposed lower bound based on the results
from string theory [49], is,

$\frac{\eta}{s}\geq\frac{\hbar}{4\pi k_{B}}.$ (3)

A “perfect fluid” saturates around this value by dissipating the smallest
possible amount of energy. A perfect fluid thus follows the laws of fluid
dynamics in the largest possible domain [47, 48, 49].

The experimental results from RHIC indicate that the matter produced in
nuclear reactions has a small ratio of $\eta/s$ [37, 38]. The discovery of
such a close perfect fluid nature established relativistic fluid dynamics as
the new frame-work for deciphering the bulk evolution of the system [36, 44].
The observations illustrate that QGP near $T_{c}$ is a strongly coupled one
with the properties of a liquid with very low viscosity rather than that of a
dilute gas [50, 51].

Analysis infers [41] that the averaged specific viscosity of the QGP produced
in LHC collisions is quite similar to that for the dense matter created in
RHIC energy domain. So, the domain in which matter produced at RHIC/LHC is,
$T_{c}$ $<$ $T$ $<$ $2T_{c}$, was renamed into a strongly coupled QGP or
“sQGP” in short [35, 52, 53, 54, 55, 56, 57]. On the other hand the low value
for $\eta/s$ could also result from an anomalous viscosity $\eta_{A}$,
originating from turbulent color magnetic and electric fields dynamically
produced in the expanding quark-gluon plasma [37, 51]. That is,

$1/\eta=1/\eta_{A}+1/\eta_{C},$ (4)

where $\eta_{A}$ subjugates over the collisional viscosity $\eta_{C}$. Such
arguments do not rule out a more complex structure of the gluonic component of
the matter produced in the relativistic collisions [58].

At LHC energies the inital energy density(at $\tau_{0}$ = 1 fm/c) is about 15
GeV $fm^{-3}$ [59]. It is approximately a factor of three higher than the
Au+Au collisions at the highest energy regime at RHIC. Some researchers
expected that the QGP produced at the LHC would turn back to the previous
picture, where quarks and gluons were more weakly coupled at higher
temperature. Then the mean free path of particles in the medium and the
viscosity will be significant. As a result the experimental signature will
emerge as smaller flow components($v_{n}$). But the ALICE elliptic flow
$v_{2}$ results [60] have clearly shown, the opposite. The dependence of
$v_{2}$ on transverse momentum is comparable with the RHIC measurements and
ALICE has also established that radial flow grows with energy.

Understanding sQGP was a challenge which we have researched from RHIC data.
However the LHC program has added a lot to our understanding, and the
paramount issues in the field now include a critical search to study the
evolution between p+p, p+A collisions which are known as “small systems” and
heavy ion A+A collisions, with an goal to understand “the smallest drops” of
the sQGP showing collective/hydrodynamics behavior [56]. Some of these
assumptions are getting tested and understood carefully both in RHIC [61] and
LHC [62, 63] experiments.

At LHC since the collision energies increase, one expects a QGP which is
hotter. Such favourable high energy of LHC is more evident in the area of
parton energy loss analogous to the opaque nature of the sQGP where the
kinematic domain exceeds that of RHIC. The significant impact of this increase
of the collision energy is the huge excess of the rates of hard probes, such
as jets, electro-weak particles and heavy-flavors, including the full family
of quarkonia ( $c\bar{c}$ and $b\bar{b}$ bound states) [59, 64]. With a larger
in-elastic cross-section, the production of $b\bar{b}$ pairs will increase
more in LHC energies. The abundance of $b\bar{b}$ pairs enable the possibility
for bottom quark and anti-bottom quark pairs to recombine, following
bottomonium state breakup, or combination after the pair forms from the open
bottom states. The available high rates allow detail studies of the dense
medium using the interactions of these probes with the medium constituents
[64, 65, 66].

The elastic re-scattering of the heavy quarks in the sQGP is an important
element for the understanding of heavy-flavor and single-electron/muon
observables in heavy ion reactions at collider energies [26, 67]. The produced
heavy-flavor interacts with the dense medium by exchanging energy and
momentum. The ratio of the measured number of heavy-flavors in heavy ion (A+A)
collisions to the expected number in the absence of nuclear or partonic matter
i.e p+p collisions, is the definition of nuclear modification factor($R_{AA}$)
which is suppressed at high transverse momentum [66]. The elementary degrees
of freedom and basic forces at the shortest distances are understood via small
systems [15]. So a clear understanding of the small systems emerge as a
necessity. The small collision systems like p+p and p+A collisions at LHC
energies thus needs detail study to understand the initial and final state
effects in Cold Nuclear Matter(CNM), which can provide baseline for the
interpretation of heavy ion (A+A) results [28, 68].

##  3 Small Systems

Study of QGP requires reference measurements which is provided by the small
system (p+p and p+A) collisions [28, 68]. QGP is not expected to be formed in
small systems as the transverse size of the overlap region is comparable to
that of a single proton [20, 69, 70, 71]. Particle production in A+A and p+A,
as compared to p+p collisions, expressed as $R_{AA}$, is termed as the nuclear
modification factor. It has long been formulated to understand particle
production mechanisms [66]. The $R_{AA}$ of heavy-flavor is expected to be
less suppressed and elliptic flow $v_{2}$ of heavy-flavor is felt to be
smaller in comparison with the light hadrons. The experimental results from
ALICE, however, show that the suppression of heavy-flavor hadrons (D-meson) at
high transverse momentum ($p_{T}$) and its elliptic flow $v_{2}$ are
comparable to those of the light hadrons [72, 73], which needs to be
understood [74]. Hence looking into the p+A collisions is required [75], where
medium absence provides necessary conditions, to isolate the nuclear effects
from the initial hard-scattering processes which we often describe as CNM [76,
77, 78, 79].

Broadly the CNM effects emcompass : (i) initial-state nuclear effects on the
parton densities (i.e shadowing); (ii) coherent energy loss comprising of
initial-state parton energy loss and final-state energy loss; and (iii) the
final-state absorption by nucleons, which is expected to be negligible at LHC
energies. The CNM effects like the change of the Parton Distribution Functions
(PDFs) within the nucleons contained within the nuclei, as compared to the
unbound nucleons can modify the interaction and production cross-sections
[80]. That’s why the p+A collisions are important to decouple the effects of
QGP from those of CNM, and to provide very much required input to the
understanding of A+A collisions [76, 77, 78, 79]. The nuclear modification
factor of charged particles from CMS experiment [81] in p+Pb collisions, in
contrast to the Pb+Pb system at top LHC energies of $\sqrt{s}_{\rm NN}$=5.02
TeV, demonstrate no suppression in the 2-10 GeV/c $p_{T}$ region. However we
visualize a weak momentum dependence for $p_{T}$ $>$ 10 GeV/c in the p+Pb
system, since we observe a moderate excess above unity at high $p_{T}$ for
charged particles. Also for heavy-flavor(D-meson), the nuclear modification
factor, measured by ALICE experiment [82] in p+Pb collisions at same energies,
show no suppression within the uncertainties in the measured $p_{T}$ range of
1-24 GeV/c. The strong suppression of the D-meson yields for $p_{T}$ $>$ 3
GeV/c has been observed in central and semi-peripheral Pb+Pb collisions [83],
whereas, for the charged particles [81] we see for $p_{T}$ $<$ 2 GeV/c a
rising trend in both p+Pb and Pb+Pb systems. In the Pb+Pb collisions the
charged particles [81] then show a significant suppression in the
2$<$$p_{T}$$<$10 GeV/c region, and again a rising trend around 10 GeV/c to the
highest $p_{T}$. The p+Pb and Pb+Pb nuclear modification factors presented in
these papers [81, 82, 83], covering the light and heavy quarks respectively,
provide stringent constraints on cold and hot nuclear matter effects. They
also clearly establish why the CNM effects are of crucial importance for
accurate interpretation of the measurements in heavy ion collisions and in
turn advocate the necessity of studying the small system collisions.

But at LHC energies do we see any new features in p+p collisions? At LHC
energies the particle multiplicity is high and even reach values, which are of
the same order as those found in heavy ion collisions at lower energies, and
as a matter of fact, they are well above the ones observed at RHIC for
peripheral Cu+Cu collisions at $\sqrt{s_{\rm NN}}$ = 200 GeV [84]. When LHC
started with the p+p collisions, the high-multiplicity environment revealed a
“ridge” which was measured by CMS [85] while studying the long-range azimuthal
correlations for 2.0 $<$ $|\Delta\eta|$ $<$ 4.8. The first observation of a
long-range ridge-like structure at the near-side ($\Delta\phi$ $\approx$ 0)
was observed for 7 TeV p+p collisions. For the high multiplicity domain of
$N\approx$ 90 or higher, this notable feature is clearly observed for large
rapidity differences $|\Delta\eta|$ $>$ 2\. Also in the high-multiplicity p+Pb
collisions at $\sqrt{s_{\rm NN}}$=5.02 TeV, the azimuthal correlations for 2.0
$<$ $|\Delta\eta|$ $<$ 4.0 showed a qualitatively similar long-range structure
at the nearside $\Delta\phi$ $\approx$ 0\. Thus the long-range, near-side
angular correlations in particle production emerged in p+p and subsequently in
p+Pb collisions [86], which was further followed by an away-side structure,
located at $\Delta\phi$ $\approx$ $\pi$ and exceeding the away-side jet
contribution, in p+Pb collisions [87, 88].

In a typical p+p collision, a ridge correlation is not expected because the
system is too dilute to produce a fluid-like state. This paved the way to
encourage the researchers to look for a detailed investigation of the
existence of collective phenomena in p+p collisions which was known since long
in heavy ion collisions [89]. The strong evidence for the collective nature of
the long-range correlations was observed with the charged particles (light
quarks) [63] by CMS experiment at $\sqrt{s}$ = 13 TeV. Also the elliptic flow
($v_{2}$) coefficients for heavy-flavor decay muons was measured by ATLAS in
p+p collisions at same energy [90].

Since heavy quark yields in heavy ion collisions are expected to be modified
relative to minimum bias p+p collisions [66], the obvious question arises if
their production rates in high-multiplicity p+p collisions at LHC energies
show any effect like J/$\Psi$ suppression [24, 25]. A stronger than linear
rise of the relative production of J/$\Psi$ as a function of multiplicity was
observed for $p_{T}$-integrated yields and this increase is stronger for high-
pT J/$\Psi$ mesons which we see for p+p collisions at $\sqrt{s}$ = 13 TeV
[91]. An esclation of the relative J/$\Psi$ and $\Upsilon$ yields [92, 93, 94]
with the relative charged-particle multiplicity was observed in p+Pb
collisions at $\sqrt{s_{\rm NN}}$=8.16 TeV [95]. The results in p+A are very
similar to the results from p+p collisions [93, 94, 96]. The rise of the
J/$\Psi$ normalized yields are comparable to the increase observed for
D-mesons [97, 98] which indicate that a common mechanism may be at its origin.
A plethora of new, unexpected phenomena have been observed so far in small
system (p+p and p+A) collisions, which, produce remarkable similarities to
heavy ion phenomenology.

##  4 Summary and Outlook

The more-central Au+Au collisions at RHIC, on the basis of elliptic-flow
systematics, have been characterized in terms of a sQGP with small viscosity —
a “perfect liquid”. Crucial input to our comprehension of the sQGP were
inferred from the measurements of “collective flow”, which in other words is
the correlated emission of particles in azimuthal angle around the axis of the
colliding beams. Conventionally, we have diagnosed the effects of the sQGP on
the final-state particle production and correlations in A+A collisions, by
using the relative to baseline measurements of p+p and p+A collisions, and
thus assuming that in the smaller, and therefore shorter-lived systems, no QGP
effects can happen. At LHC we found new things and even the small systems
showed flow features in the rising multiplicity domains. With the increasing
multiplicity, the p+p and p+A collisions enter the stage where the macroscopic
description (thermodynamics and hydrodynamics) becomes applicable. While
hydrodynamic models, when applied to p+A data, can explain many of the
observed features, there are serious questions regarding their applicability
[99]. Thus, a very detailed description of a broad range of signatures, in an
even broader range of systems, will be required to finally demonstrate a full
understanding of these new discoveries.

Data which will be collected in Run-3 at the LHC, will be a significant
addition for such studies. Better picture will be also available with the
results from p+Pb collisions. Also exceptionally high-multiplicity p+p
collisions are expected in Run-3 and 4 at LHC [100]. The LHC delivered nearly
30 $fb^{-1}$ by the end of 2012 and propose to reach 300 $fb^{-1}$ in its
first 13-15 years of operation. The second long shutdown (LS-2) before Run-3
will consolidate the luminosity and reliability as well as the upgrading of
the LHC injectors. After LS-3, the machine will be in the High Luminosity
configuration. The High Luminosity LHC(HL-LHC) is an important and extremely
challenging, upgrade [101]. The large p+p collision data sets expected to be
collected at the HL-LHC will provide a compelling setting for these
investigations [100, 102]. Such higher multiplicities will help us to bridge
the gap between the p+p and heavy ion collisions, with better detector
upgrades in LHC experiments [103].

## References

  * [1] E. M. Riordan, Science 256, 1287-1293 (1992) doi:10.1126/science.256.5061.1287
  * [2] J. Ellis, Int. J. Mod. Phys. A 29, no.31, 1430072 (2014)
doi:10.1142/S0217751X14300725 [arXiv:1409.4232 [hep-ph]].

  * [3] W. J. Marciano and H. Pagels, Phys. Rept. 36, 137 (1978) doi:10.1016/0370-1573(78)90208-9
  * [4] W. J. Marciano and H. Pagels, Nature 279, 479-483 (1979) doi:10.1038/279479a0
  * [5] M. Bruno, M. Caselle, M. Panero and R. Pellegrini, JHEP 03, 057 (2015) doi:10.1007/JHEP03(2015)057 [arXiv:1409.8305 [hep-lat]].
  * [6] J. M. Richard, [arXiv:1205.4326 [hep-ph]].
  * [7] J. Berges, arXiv:hep-ph/9902419.
  * [8] J. B. Kogut, Nucl. Phys. B Proc. Suppl. 119, 210-221 (2003) doi:10.1016/S0920-5632(03)01508-1 [arXiv:hep-lat/0208077 [hep-lat]].
  * [9] K. Fukushima, J. Phys. G 39, 013101 (2012) doi:10.1088/0954-3899/39/1/013101 [arXiv:1108.2939 [hep-ph]].
  * [10] F. Wilczek, arXiv:hep-ph/0003183.
  * [11] L. S. Kisslinger and D. Das, Int. J. Mod. Phys. A 31, no. 07, 1630010 (2016) doi:10.1142/S0217751X16300106 [arXiv:1411.3680 [hep-ph]].
  * [12] E. V. Shuryak, Phys. Rept. 61, 71 (1980).
  * [13] S. A. Bass, M. Gyulassy, H. Stoecker and W. Greiner, J. Phys. G 25, R1 (1999) doi:10.1088/0954-3899/25/3/013 [hep-ph/9810281].
  * [14] J. R. Ellis, J. Phys. Conf. Ser. 50, 8-21 (2006) doi:10.1088/1742-6596/50/1/002 [arXiv:astro-ph/0504501 [astro-ph]].
  * [15] U. W. Heinz, J. Phys. A 42, 214003 (2009) doi:10.1088/1751-8113/42/21/214003 [arXiv:0810.5529 [nucl-th]].
  * [16] J. Adams et al. [STAR Collaboration], Nucl. Phys. A 757, 102 (2005) doi:10.1016/j.nuclphysa.2005.03.085 [nucl-ex/0501009].
  * [17] K. Adcox et al. [PHENIX Collaboration], Nucl. Phys. A 757, 184 (2005) doi:10.1016/j.nuclphysa.2005.03.086 [nucl-ex/0410003].
  * [18] B. B. Back et al., Nucl. Phys. A 757, 28 (2005) doi:10.1016/j.nuclphysa.2005.03.084 [nucl-ex/0410022].
  * [19] I. Arsene et al. [BRAHMS Collaboration], Nucl. Phys. A 757, 1 (2005) doi:10.1016/j.nuclphysa.2005.02.130 [nucl-ex/0410020].
  * [20] B. B. Abelev et al. [ALICE Collaboration], Int. J. Mod. Phys. A 29, 1430044 (2014) doi:10.1142/S0217751X14300440 [arXiv:1402.4476 [nucl-ex]].
  * [21] G. K. Krintiras [CMS], [arXiv:2006.05556 [nucl-ex]].
  * [22] N. Armesto and E. Scomparin, Eur. Phys. J. Plus 131, no. 3, 52 (2016) doi:10.1140/epjp/i2016-16052-4 [arXiv:1511.02151 [nucl-ex]].
  * [23] W. Busza, K. Rajagopal and W. van der Schee, Ann. Rev. Nucl. Part. Sci. 68, 339 (2018) doi:10.1146/annurev-nucl-101917-020852 [arXiv:1802.04801 [hep-ph]].
  * [24] T. Matsui and H. Satz, Phys. Lett. B 178 (1986) 416.
  * [25] H. Satz, Nucl. Phys. A 418, 447C (1984).
  * [26] M. He, R. J. Fries and R. Rapp, Phys. Lett. B 735, 445-450 (2014) doi:10.1016/j.physletb.2014.05.050 [arXiv:1401.3817 [nucl-th]].
  * [27] P. Foka and M. A. Janik, Rev. Phys. 1, 172-194 (2016) doi:10.1016/j.revip.2016.11.001 [arXiv:1702.07231 [hep-ex]].
  * [28] J. L. Nagle and W. A. Zajc, Ann. Rev. Nucl. Part. Sci. 68, 211 (2018) doi:10.1146/annurev-nucl-101916-123209 [arXiv:1801.03477 [nucl-ex]].
  * [29] S. A. Voloshin, A. M. Poskanzer and R. Snellings, arXiv:0809.2949 [nucl-ex]. Landolt-Bornstein 23 (2010) 293-333.
  * [30] R. Snellings, New J. Phys. 13, 055008 (2011) doi:10.1088/1367-2630/13/5/055008 [arXiv:1102.3010 [nucl-ex]].
  * [31] P. Foka and M. A. Janik, Rev. Phys. 1, 154 (2016) doi:10.1016/j.revip.2016.11.002 [arXiv:1702.07233 [hep-ex]].
  * [32] E. Shuryak, Prog. Part. Nucl. Phys. 53, 273-303 (2004) doi:10.1016/j.ppnp.2004.02.025 [arXiv:hep-ph/0312227 [hep-ph]].
  * [33] M. Gyulassy and L. McLerran, Nucl. Phys. A 750, 30-63 (2005) doi:10.1016/j.nuclphysa.2004.10.034 [arXiv:nucl-th/0405013 [nucl-th]].
  * [34] E. Shuryak, Prog. Part. Nucl. Phys. 62, 48 (2009) doi:10.1016/j.ppnp.2008.09.001 [arXiv:0807.3033 [hep-ph]].
  * [35] R. A. Lacey, Nucl. Phys. A 774, 199-214 (2006) doi:10.1016/j.nuclphysa.2006.06.041 [arXiv:nucl-ex/0510029 [nucl-ex]].
  * [36] F. Becattini, J. Liao and M. Lisa, [arXiv:2102.00933 [nucl-th]]. Part of the Lecture Notes in Physics book series (LNP, volume 987).
  * [37] R. A. Lacey, N. N. Ajitanand, J. M. Alexander, P. Chung, W. G. Holzmann, M. Issah, A. Taranenko, P. Danielewicz and H. Stoecker, Phys. Rev. Lett. 98, 092301 (2007) doi:10.1103/PhysRevLett.98.092301 [arXiv:nucl-ex/0609025 [nucl-ex]].
  * [38] R. A. Lacey, A. Taranenko and R. Wei, [arXiv:0905.4368 [nucl-ex]].
  * [39] A. Bagoly and M. Csanad, Int. J. Mod. Phys. A 31, no.28 $\&$ 29, 1645016 (2016) doi:10.1142/S0217751X16450160 [arXiv:1507.05005 [nucl-th]].
  * [40] J. Y. Jia, Nucl. Phys. A 834, 229C-236C (2010) doi:10.1016/j.nuclphysa.2009.12.047
  * [41] R. A. Lacey, A. Taranenko, N. N. Ajitanand and J. M. Alexander, Phys. Rev. C 83, 031901 (2011) doi:10.1103/PhysRevC.83.031901 [arXiv:1011.6328 [nucl-ex]].
  * [42] J. Liao and E. Shuryak, Phys. Rev. C 75, 054907 (2007)
doi:10.1103/PhysRevC.75.054907 [arXiv:hep-ph/0611131 [hep-ph]].

  * [43] J. L. Nagle, Eur. Phys. J. C 49, 275-279 (2007) doi:10.1140/epjc/s10052-006-0061-1 [arXiv:nucl-th/0608070 [nucl-th]].
  * [44] E. Shuryak, Nucl. Phys. A 774, 387-396 (2006) doi:10.1016/j.nuclphysa.2006.06.058 [arXiv:hep-ph/0510123 [hep-ph]].
  * [45] M. H. Thoma, J. Phys. G 31, L7 (2005) [erratum: J. Phys. G 31, 539 (2005)] doi:10.1088/0954-3899/31/1/L02 [arXiv:hep-ph/0503154 [hep-ph]].
  * [46] T. Hirano and M. Gyulassy, Nucl. Phys. A 769, 71-94 (2006) doi:10.1016/j.nuclphysa.2006.02.005 [arXiv:nucl-th/0506049 [nucl-th]].
  * [47] T. Schäfer and D. Teaney, Rept. Prog. Phys. 72, 126001 (2009) doi:10.1088/0034-4885/72/12/126001 [arXiv:0904.3107 [hep-ph]].
  * [48] T. Schäfer, AIP Conf. Proc. 1182, no.1, 755-758 (2009) doi:10.1063/1.3293917 [arXiv:0906.5399 [physics.flu-dyn]].
  * [49] P. Kovtun, D. T. Son and A. O. Starinets, Phys. Rev. Lett. 94, 111601 (2005) doi:10.1103/PhysRevLett.94.111601 [arXiv:hep-th/0405231 [hep-th]].
  * [50] H. Niemi, G. S. Denicol, P. Huovinen, E. Molnar and D. H. Rischke, Phys. Rev. Lett. 106, 212302 (2011) doi:10.1103/PhysRevLett.106.212302 [arXiv:1101.2442 [nucl-th]].
  * [51] M. Asakawa, S. A. Bass and B. Muller, Phys. Rev. Lett. 96, 252301 (2006) doi:10.1103/PhysRevLett.96.252301 [arXiv:hep-ph/0603092 [hep-ph]].
  * [52] H. Berrehrah, E. Bratkovskaya, W. Cassing and R. Marty, J. Phys. Conf. Ser. 612, no.1, 012050 (2015) doi:10.1088/1742-6596/612/1/012050 [arXiv:1412.1017 [hep-ph]].
  * [53] M. Csanad, EPJ Web Conf. 70, 00011 (2014) doi:10.1051/epjconf/20147000011 [arXiv:1208.4683 [nucl-th]].
  * [54] R. Rapp, D. Cabrera, V. Greco, M. Mannarelli and H. van Hees, [arXiv:0806.3341 [hep-ph]].
  * [55] F. Wang, Nucl. Phys. A 834, 223C-228C (2010) doi:10.1016/j.nuclphysa.2009.12.046 [arXiv:0910.2754 [nucl-ex]].
  * [56] E. Shuryak, Rev. Mod. Phys. 89, 035001 (2017) doi:10.1103/RevModPhys.89.035001 [arXiv:1412.8393 [hep-ph]].
  * [57] U. W. Heinz, [arXiv:nucl-th/0512051 [nucl-th]].
  * [58] J. Liao, J. Phys. Conf. Ser. 779, no.1, 012014 (2017) doi:10.1088/1742-6596/779/1/012014 [arXiv:1611.05391 [nucl-th]].
  * [59] B. Muller, J. Schukraft and B. Wyslouch, Ann. Rev. Nucl. Part. Sci. 62, 361-386 (2012) doi:10.1146/annurev-nucl-102711-094910 [arXiv:1202.3233 [hep-ex]].
  * [60] K. Aamodt et al. [ALICE], Phys. Rev. Lett. 105, 252302 (2010)
doi:10.1103/PhysRevLett.105.252302 [arXiv:1011.3914 [nucl-ex]].

  * [61] C. Aidala et al. [PHENIX], Nature Phys. 15, no.3, 214-220 (2019)
doi:10.1038/s41567-018-0360-0 [arXiv:1805.02973 [nucl-ex]].

  * [62] V. Khachatryan et al. [CMS], Phys. Rev. Lett. 115, no.1, 012301 (2015) doi:10.1103/PhysRevLett.115.012301 [arXiv:1502.05382 [nucl-ex]].
  * [63] V. Khachatryan et al. [CMS Collaboration], Phys. Lett. B 765, 193 (2017) doi:10.1016/j.physletb.2016.12.009 [arXiv:1606.06198 [nucl-ex]].
  * [64] M. Cacciari, Nuovo Cim. C 035N1, 23-28 (2012) doi:10.1393/ncc/i2012-11112-2 [arXiv:1109.1500 [hep-ph]].
  * [65] D. Das and N. Dutta, Int. J. Mod. Phys. A 33, no. 16, 1850092 (2018) doi:10.1142/S0217751X18500926 [arXiv:1802.00414 [nucl-ex]].
  * [66] A. Andronic et al., Eur. Phys. J. C 76, no. 3, 107 (2016) doi:10.1140/epjc/s10052-015-3819-5 [arXiv:1506.03981 [nucl-ex]].
  * [67] D. Das [ALICE Collaboration],
Nucl. Phys. A 862-863, 223 (2011) doi:10.1016/j.nuclphysa.2011.05.044
[arXiv:1102.2071 [nucl-ex]].

  * [68] D. Das, Nucl. Phys. A 1007, 122132 (2021) doi:10.1016/j.nuclphysa.2020.122132
  * [69] B. Abelev et al. [ALICE Collaboration], Phys. Rev. Lett. 109, 112301 (2012) doi:10.1103/PhysRevLett.109.112301 [arXiv:1205.6443 [hep-ex]].
  * [70] H. Ni [CMS Collaboration], J. Phys. Conf. Ser. 1070, no. 1, 012009 (2018). doi:10.1088/1742-6596/1070/1/012009
  * [71] D. Moreira De Godoy [ALICE Collaboration], Nucl. Phys. A 967, 636 (2017) doi:10.1016/j.nuclphysa.2017.05.050 [arXiv:1705.02800 [nucl-ex]].
  * [72] B. Abelev et al. [ALICE], JHEP 09, 112 (2012) doi:10.1007/JHEP09(2012)112 [arXiv:1203.2160 [nucl-ex]].
  * [73] B. Abelev et al. [ALICE], Phys. Rev. Lett. 111, 102301 (2013)
doi:10.1103/PhysRevLett.111.102301 [arXiv:1305.2707 [nucl-ex]].

  * [74] X. Dong, Y. J. Lee and R. Rapp, Ann. Rev. Nucl. Part. Sci. 69, 417-445 (2019) doi:10.1146/annurev-nucl-101918-023806 [arXiv:1903.07709 [nucl-ex]].
  * [75] S. Acharya et al. [ALICE Collaboration], Eur. Phys. J. C 78, no. 6, 466 (2018) doi:10.1140/epjc/s10052-018-5881-2 [arXiv:1802.00765 [nucl-ex]].
  * [76] R. Vogt, Phys. Rev. C 81, 044903 (2010) doi:10.1103/PhysRevC.81.044903 [arXiv:1003.3497 [hep-ph]].
  * [77] H. Fujii and K. Watanabe,
Nucl. Phys. A 915, 1 (2013)
doi:10.1016/j.nuclphysa.2013.06.011 [arXiv:1304.2221 [hep-ph]].

  * [78] R. Vogt, [arXiv:1508.01286 [hep-ph]].
  * [79] J. L. Albacete, F. Arleo, G. G. Barnaföldi, G. Bíró, D. d’Enterria, B. Ducloué, K. J. Eskola, E. G. Ferreiro, M. Gyulassy and S. M. Harangozó, et al. Nucl. Phys. A 972, 18-85 (2018) doi:10.1016/j.nuclphysa.2017.11.015 [arXiv:1707.09973 [hep-ph]].
  * [80] N. Armesto, EPJ Web Conf. 171 (2018), 11001 doi:10.1051/epjconf/201817111001
  * [81] V. Khachatryan et al. [CMS], JHEP 04, 039 (2017) doi:10.1007/JHEP04(2017)039 [arXiv:1611.01664 [nucl-ex]].
  * [82] B. B. Abelev et al. [ALICE], Phys. Rev. Lett. 113, no.23, 232301 (2014) doi:10.1103/PhysRevLett.113.232301 [arXiv:1405.3452 [nucl-ex]].
  * [83] J. Adam et al. [ALICE], JHEP 03, 081 (2016) doi:10.1007/JHEP03(2016)081 [arXiv:1509.06888 [nucl-ex]].
  * [84] B. Alver et al. [PHOBOS], Phys. Rev. C 83, 024913 (2011)
doi:10.1103/PhysRevC.83.024913 [arXiv:1011.1940 [nucl-ex]].

  * [85] V. Khachatryan et al. [CMS], JHEP 09, 091 (2010)
doi:10.1007/JHEP09(2010)091 [arXiv:1009.4122 [hep-ex]].

  * [86] S. Chatrchyan et al. [CMS], Phys. Lett. B 718, 795-814 (2013)
doi:10.1016/j.physletb.2012.11.025 [arXiv:1210.5482 [nucl-ex]].

  * [87] B. Abelev et al. [ALICE], Phys. Lett. B 719, 29-41 (2013)
doi:10.1016/j.physletb.2013.01.012 [arXiv:1212.2001 [nucl-ex]].

  * [88] G. Aad et al. [ATLAS], Phys. Rev. Lett. 110, no.18, 182302 (2013) doi:10.1103/PhysRevLett.110.182302 [arXiv:1212.5198 [hep-ex]].
  * [89] B. I. Abelev et al. [STAR], Phys. Rev. C 80, 064912 (2009) doi:10.1103/PhysRevC.80.064912 [arXiv:0909.0191 [nucl-ex]].
  * [90] G. Aad et al. [ATLAS], Phys. Rev. Lett. 124, no.8, 082301 (2020) doi:10.1103/PhysRevLett.124.082301 [arXiv:1909.01650 [nucl-ex]].
  * [91] S. Acharya et al. [ALICE], Phys. Lett. B 810, 135758 (2020) doi:10.1016/j.physletb.2020.135758 [arXiv:2005.11123 [nucl-ex]].
  * [92] D. Adamová et al. [ALICE], Phys. Lett. B 776, 91-104 (2018)
doi:10.1016/j.physletb.2017.11.008 [arXiv:1704.00274 [nucl-ex]].

  * [93] S. Chatrchyan et al. [CMS], JHEP 04, 103 (2014) doi:10.1007/JHEP04(2014)103 [arXiv:1312.6300 [nucl-ex]].
  * [94] M. Aaboud et al. [ATLAS], Eur. Phys. J. C 78, no.3, 171 (2018) doi:10.1140/epjc/s10052-018-5624-4 [arXiv:1709.03089 [nucl-ex]].
  * [95] S. Acharya et al. [ALICE], JHEP 09, 162 (2020) doi:10.1007/JHEP09(2020)162 [arXiv:2004.12673 [nucl-ex]].
  * [96] B. Abelev et al. [ALICE], Phys. Lett. B 712, 165-175 (2012) doi:10.1016/j.physletb.2012.04.052 [arXiv:1202.2816 [hep-ex]].
  * [97] J. Adam et al. [ALICE], JHEP 09, 148 (2015) doi:10.1007/JHEP09(2015)148 [arXiv:1505.00664 [nucl-ex]].
  * [98] J. Adam et al. [ALICE], JHEP 08, 078 (2016) doi:10.1007/JHEP08(2016)078 [arXiv:1602.07240 [nucl-ex]].
  * [99] E. Shuryak and I. Zahed, Phys. Rev. C 88 (2013) no.4, 044915 doi:10.1103/PhysRevC.88.044915 [arXiv:1301.4470 [hep-ph]].
  * [100] F. Noferini [ALICE], J. Phys. Conf. Ser. 1014, no.1, 012010 (2018) doi:10.1088/1742-6596/1014/1/012010
  * [101] G. Apollinari, O. Brüning, T. Nakamoto and L. Rossi, doi:10.5170/CERN-2015-005.1 [arXiv:1705.08830 [physics.acc-ph]].
  * [102] E. Chapon, D. d’Enterria, B. Ducloue, M. G. Echevarria, P. B. Gossiaux, V. Kartvelishvili, T. Kasemets, J. P. Lansberg, R. McNulty and D. D. Price, et al. [arXiv:2012.14161 [hep-ph]].
  * [103] Z. Citron et al., CERN Yellow Rep. Monogr. , 1159 (2019) doi:10.23731/CYRM-2019-007.1159 [arXiv:1812.06772 [hep-ph]].