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\begin{document}
\title{Numeric exploration of Non-trivial emergent phenomena in Quark-Gluon Plasma}

\author{Nia O'Callaghan$^{1}$ and WA Horowitz$^{1,2}$}

\affil{$^1$ Department of Physics, University of Cape Town, Rondebosch 7701, South Africa}
\affil{$^2$ Department of Physics, New Mexico State University, Las Cruces, New Mexico, 88003, USA}

%only put the email of the main corresponding author, not of all authors.
\email{\url{OCLJON001@myuct.ac.za}}

\begin{abstract}
    We present here an analysis of probe quark behavior in a thermal Quark-Gluon Plasma via the AdS/CFT correspondence, modeling them as string endpoints with specific boundary conditions. We study both heavy (on-mass-shell) and light (off-mass-shell) quarks initially at rest, exploring analytic solutions where possible and using Wolfram's \emph{Mathematica} for numerical solutions elsewhere. This enables comparison and verification. Due to difficulties in obtaining these numeric string solutions, we also examine numerical solutions for a point particle in the same spacetime geometries as a simpler test case. Although not all objectives were achieved, we present our findings, discuss limitations, and outline future research directions.
\end{abstract}


\section{Introduction}\label{sec:intro}

The study of Quark-Gluon Plasma (QGP) advances our understanding of the quantum universe, especially the strong force and particle dynamics. Found in heavy ion collisions at RHIC and LHC, QGP is a complex, deconfined 
state of matter that challenges previous assumptions of a gas-like behavior, as evidenced by anisotropic momentum distribution (known as \emph{elliptic flow},~\cite{qgprhic,Heinz}). Both weak and strong coupling regimes are supported: perturbative QCD effectively predicts high transverse momentum observables~\cite{pQCD} assuming weak coupling, while near-ideal hydrodynamics describes low transverse momentum observables~\cite{Heinz2,gale} assuming strong coupling.

Our investigation focuses on probe quarks interacting with QGP, modeled via AdS/CFT as strings in a String Theory. We aim to solve the string equations of motions allowing us to analyze the mean and RMS displacements of 
string endpoints, reflecting probe quark motion. Due to limited analytic solutions, we develop numerical solutions in Wolfram's \emph{Mathematica}, validating with known cases. Additionally, we examine motion of point particles in the same spacetime geometries as a simplified approach to inform our methods and address challenges in the full string analysis.

\section{The String Case}
\subsection{Theory}
As established in Section~\ref{sec:intro}
, we are ultimately interested in solving the equations of motion for a string in Anti-de Sitter or \emph{AdS} spacetime for the purposes of connecting this motion to that of a probe quark via the AdS/CFT correspondence~\cite{Maldacena,Mes}. 
More specifically, we want to solve the equations of motion arising from the Polyakov~\cite{Tong,Polyakov,Zwieback,Mes} action,
\begin{equation}
    S_{P}=-\frac{T}{2} \int_{\mathcal{M}} d^{2}\sigma \, \sqrt{ -h } h^{ab}G_{\mu \nu} \partial_{a}X^{\mu}\partial_{b}X^{\nu} = -\frac{T}{2}\int_{\mathcal{M}} d^{2}\sigma \, \sqrt{ -h } h^{ab} \gamma_{ab} \label{eqn:P-action}\,,
\end{equation}
where $T$ is the string tension, $\mathcal{M}$ is the worldsheet of the string with coordinates $\sigma^a = (\tau,\sigma)^a$ and volume element 
$d^2\sigma = d\tau d\sigma$ (indexed with Latin
indices to distinguish them from target spacetime objects, which are indexed with Greek indices), $X^\mu = X^\mu(\tau,\sigma)$ are the coordinates of the string (to be understood as mapping functions 
from the string worldsheet to the target spacetime), $\gamma_{ab}=G_{\mu \nu}\partial_a X^\mu \partial_b X^\nu$ is the metric induced on the worldsheet by the pullback of the target spacetime metric $G_{\mu\nu}$ along the mapping functions $X^\mu$, $h_{ab}$ is the metric on the worldsheet itself (with inverse $h^{ab}$ and determinant $h$), and natural units
are used unless otherwise stated. The equations of motion resulting from the extremization of~\eqref{eqn:P-action} are
\begin{equation}
    \nabla_a \Pi^a_\mu - \Gamma^{\alpha}_{\mu\nu} \Pi^a_\alpha \partial_a X^\nu=0
    \label{eqn:gen-eom}
\end{equation}
where $\Pi^a_\mu = -T h^{ab}G_{\mu\nu}\partial_b X^\nu$
are the canonical momentum densities obtained from the action~\eqref{eqn:P-action} via the usual procedure, $\Gamma^{\alpha}_{\mu\nu}$ are the usual Christoffel symbols of the target spacetime metric $G_{\mu\nu}$, and 
$\nabla_a$ is the covariant derivative on the string worldsheet from the Levi-Civita connection associated with the worldsheet metric $h_{ab}$. 

Our goal is to solve these equations of motion for two cases of open strings, the first with both endpoints fixed (corresponding to a heavy, on-mass-shell probe quark) and the second with 
one endpoint fixed and the other free (corresponding to a light, off-mass-shell probe quark), and we will refer to these two cases as the ``heavy'' and ``light'' cases respectively~\cite{Mes,limp_noodle}. Ultimately we want to consider 
the $AdS_5 \times S^5$ spacetime geometry as this is dual to $\mathcal{N}=4$ Super Yang-Mills CFT\footnote{In the appropriate limiting cases, this field theory provides a good approximation for QCD~\cite{Maldacena,hooft}}, however no analytic 
solutions exist in this case. We will therefore consider the simpler spacetime geometry of \adsS, where analytic solutions to equation~\eqref{eqn:gen-eom} are present\footnote{Though these analytic solutions 
are derived by making use of the static ($X^0(\tau,\sigma)=\tau$) and conformal ($h_{ab}=\eta_{ab}=\diag(-1,1)$) gauges.}. This will allow us to compare the obtained numeric solutions to these analytic solutions in order to 
establish the validity of our numeric solutions before proceeding to the more complex spacetime geometry. 

\subsection{Numeric Results}
Making use of the \emph{NDSolve} function build into Wolfram's \emph{Mathematica} to numerically solve equation~\eqref{eqn:gen-eom} in the above established spacetime geometry, we are able to obtain numeric solutions
in both the heavy and light cases. Figure~\ref{fig:stringSols} shows parametric plots of the radial component $X^1(\tau,\sigma)=r(t,\sigma)$ of these solutions in the static gauge, along with plots of the corresponding analytic solutions.
\begin{figure}[htb]
\centering
\begin{subfigure}{0.49\textwidth}
    \centering
    \includegraphics[width=\textwidth]{anaHQA1}
    \caption{Analytic String solution for the heavy Case}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
    \includegraphics[width=\textwidth]{anaLQA1}
    \caption{\label{fig:analightPara}Analytic String solution for the light Case}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
    \includegraphics[width=\textwidth]{numHQA1}
    \caption{Numeric String solution for the heavy Case}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
    \includegraphics[width=\textwidth]{numLQA1}
    \caption{\label{fig:numlightPara}Numeric String solution for the light Case}
\end{subfigure}
\caption{\label{fig:stringSols} Parametric plots of the analytic (\emph{top}) and numeric (\emph{bottom}) string solutions in both the heavy (\emph{left}) and light (\emph{right}) cases. These plots show the full spacial extent of the string on the vertical 
axis, plotted over time on the horizontal.}
\end{figure}

The plots in Figure~\ref{fig:stringSols} show good agreement for the heavy case but significant deviation for the light case. Notably, the light case exhibits non-physical behavior: the shaded region in plot~\ref{fig:numlightPara} indicates the string endpoint folding over itself, leading to that 
endpoint falling inwards faster than the analytic solution's inward fall at the local speed of light. This implies the numeric endpoint exceeds the local speed of light. The deviations are around one-tenth of the string length --- much larger than numerical error alone could account for.

Due to these issues with obtaining a numeric solution for the string that agrees with the analytic solution, and being unable to remedy this issue via the usual approaches\footnote{Specifically, neither re-forming the 
equations of motion nor adjusting the Working Precision, Accuracy Goal, Precision Goal, and/or Grid Spacing options supplied to \emph{NDSolve} are able to resolve the numeric problems.}, we will instead consider the simpler 
problem of numerically solving for the motion of a point particle in the same spacetime geometry.

\section{The Point Particle Case}
\subsection{Theory}
The trajectory of a point particle is known as a geodesic of the spacetime it exists within. These geodesics can be understood as generalisations of the notation of a straight line\cite{Carroll,tongGR}; they are the paths that extremize the invariant spacetime interval between two events, analogous to how straight lines extremize the Euclidean distance between two points. Following this logic, we extremize the length functional
\begin{equation}
    I=\int d\lambda \, \sqrt{- G_{\mu \nu}\frac{dx^{\mu}}{d\lambda} \frac{dx^{\nu}}{d\lambda} }=\int d\lambda \, \sqrt{ -\dot{x}^{2} },\label{eqn:lengthfunc1}
\end{equation}
where $\dot{x}^{\mu}=\frac{dx^{\mu}}{d\lambda}$, and the square of a vector is 
shorthand for the contraction of the vector with itself by the metric.
From the functional~\eqref{eqn:lengthfunc1} we derive the geodesic equation~\cite{Carroll}
\begin{equation}\label{eqn:geod}
    \frac{d^{2}x^{\mu}}{d\lambda^{2}} +\Gamma^{\mu}_{\nu \rho}\frac{dx^{\nu}}{d\lambda} \frac{dx^{\rho}}{d\lambda} =0, 
\end{equation}
for a trajectory $x^{\mu}(\lambda)$ parameterised by an arbitrary (affine) parameter $\lambda$. In deriving equation~\eqref{eqn:geod}, we have considered the parameterised path $x^{\mu}(\lambda)$ as an object in its own right, rather than as the physical trajectory of a particle, but the switch to the latter is simple enough: We can take the action for a massive, relativistic point particle as simply being proportional\footnote{The proportionality constant must clearly have dimensions of energy to ensure that this new functional~\eqref{eqn:Sm} does actually have 
the dimensions of action, and what better to use than the mass of the particle being a constant property of it. The negative sign is then needed to ensure we recover the correct non-relativistic limit to the action.} 
to the length of its worldline~\cite{Carroll,tongGR,Zwieback},
\begin{equation}
    S_{m} = -m \int ds \, =-m\int d\lambda \, \sqrt{ -G_{\mu \nu}\frac{dx^{\mu}}{d\lambda} \frac{dx^{\nu}}{d\lambda} }.\label{eqn:Sm}
\end{equation}
This action~\eqref{eqn:Sm} is the point particle analogue to the Nambu-Goto 
action\footnote{$S_{NG}=-T\int d^2\sigma \,\sqrt{-\det(\gamma_{ab})}$, where $\gamma_{ab}$ is the induced metric on the string worldsheet, as in equation~\eqref{eqn:P-action}. This action
is proportional to the area traced out by the string worldsheet.} for the string.

We can additionally consider the einbein action~\cite{Tong}, 
\begin{equation}
    S_{e}= \frac{1}{2} \int d\lambda \, \left( \frac{\dot{x}^{2}}{e}-em^{2} \right), \label{eqn:Se}
\end{equation}
where $e=e(\lambda)$ is this new einbein object, an additional degree of freedom introduced in analogy to the intrinsic worldsheet metric\footnote{This analogy runs deeper than it at first appears, as equation~\eqref{eqn:Se} 
can be recast to show that the $e$ acts as a one-dimensional metric on the worldline of the particle~\cite{Tong}.} introduced in equation~\eqref{eqn:P-action}. 
This action~\eqref{eqn:Se} maintains reparameterisation invariance without the square root of equation~\eqref{eqn:Sm} and applies to both massive and massless particles, unlike the $S_m$ action which only works for massive particles. Additionally, a third point particle action can be derived by simplifying the length functional~\eqref{eqn:lengthfunc1}: by taking
advantage of the monotonicity of the square root function we can, by ``squaring'' the functional $I$, obtain a new length functional $I_2$ and an associated action $S_2$~\cite{Carroll},
\begin{equation}
    I_{2}=\int d\lambda \; G_{\mu \nu}\frac{dx^{\mu}}{d\lambda} \frac{dx^{\nu}}{d\lambda} = \int d\lambda \, \dot{x}^{2}\implies 
    S_{2}=m^{2}\int d\lambda \; G_{\mu \nu}\frac{dx^{\mu}}{d\lambda} \frac{dx^{\nu}}{d\lambda} = m^{2}\int d\lambda \, \dot{x}^{2}, \label{eqn:S2}
\end{equation}
completely analogously to how the action $S_m$ was obtained from the functional $I$. This action~\eqref{eqn:S2} is mathematically simpler, but this simplicity comes at the cost of reparameterisation invariance; we have effectively
fixed the proper time parameterisation for our worldline in using the $S_2$ action~\cite{Carroll}. Additionally, this action is also only valid for massive particles, much like the $S_m$ action. For the massless case, we do have one additional 
method to solve for the trajectory of the particle aside from the extremization of equation~\eqref{eqn:Se} or the direct use of the geodesic equation~\eqref{eqn:geod}, and this is to make use of the fact that massless particles
must travel on lightlike or null trajectories. These trajectories satisfy $\dot{x}^2=0$ which, by making use of the static gauge $x^0(\lambda)=t(\lambda)=\lambda$, gives us a first order differential equation that can be solved
to find $x^1(\lambda)=r(\lambda)$. 

The equations of motion arising from all three actions\footnote{That is, from $S_m$ in equation~\eqref{eqn:Sm}, from $S_e$ in equation~\eqref{eqn:Se}, and from $S_2$ in equation~\eqref{eqn:S2}.} can be written as

\begin{equation}
    \frac{D}{d\lambda} \pi^{(i)}_{\mu}=\dot{x}^{\nu}\nabla_{\nu}\pi^{(i)}_{\mu}=0,\label{eqn:geodparlMom}
\end{equation}
where $i\in\{m,e,2\}$ is the same label as the subscript on the associated action, $\frac{D}{d\lambda}=\dot{x}^\nu\nabla_\nu$ is the parallel transport operator~\cite{Carroll}, and $\pi^{(i)}_{\mu}$ is the canonical 
momentum obtained from the corresponding action via the usual procedure. All the difference arising from the specifics of the different actions
are contained within the different canonical momenta. 
In the case of the einbein action, we additionally need to consider the equations of motion for the einbein itself, which come out to be a constraint equation. These canonical momenta
and this constraint equation are given in equations~\eqref{eqn:moms} and~\eqref{eqn:einc} respectively.
\begin{align}\label{eqn:moms}
    \pi^{(m)}_{\mu} &= \frac{m\dot{x}_{\mu}}{\sqrt{ -\dot{x}^{2} }};\;\;\pi^{(2)}_{\mu}= 2 m^{2}\dot{x}_{\mu};\;\;
\pi^{(e)}_{\mu}= e^{-1} \dot{x}_{\mu};\\
0&=\frac{\dot{x}^{2}}{e^{2}}+m^{2}.\label{eqn:einc}
\end{align}

In all cases, we will use the initial conditions $x^\mu(0)=(t(0),r(0))^\mu=(0,1)^\mu$ and $\dot{t}(0)=1$. In the massive case, we will then use $\dot{r}(0)=0$ and solve equation~\eqref{eqn:einc} to obtain possible 
values for $e(0)$, while for the massless case we will obtain $\dot{r}(0)$ from the other
initial conditions to place the particle on a null trajectory at $\lambda=0$. We will then attempt to numerically solve all versions of equation~\eqref{eqn:geodparlMom} from the different actions, variations on
reforming these equation, variations on the form of equation~\eqref{eqn:einc} in the einbein case as well as its derivative, and with the options of $-1,0,\text{ or }1$ for $\dot{e}(0)$ in either case and $e(0)$ in the massless
case. This leads to a huge number of different potential configurations for the numeric solutions which will be run via \emph{NDSolve} and, once non-physical solutions have been ruled out\footnote{Solutions
are deemed non-physical and ruled out if they either do not satisfy at least one of the initial conditions laid out above, or else if the solution for the time coordinate has $\dot{t}(\lambda)<0$, as time
must be an increasing function.}, these solutions will be 
compared. Table~\eqref{tab:solCount} shows the final counts of the number of solution configurations, the number of non-physical solutions, the number of solutions that were not able to be solved, and the final count of remaining
numeric solutions both in Minkowski spacetime and in two coordinate systems\footnote{Those being the usual $(t,r)$ radial coordinate system, which we will refer to as ``AdS'' coordinates, and the
tortoise coordinates $(t,r_\ast)$ which transform the radial direction to put the metric into a conformally flat form where it is proportional to the Minkowski metric, and which we will refer to as such.} 
for \adsS{} spacetime~\cite{Mes,limp_noodle}. In all cases, we will take the metric to have the form $ds^2 = -f(r)dt^2 + g(r)dr^2$, 
which allows us to recover all three cases by the forms of $f(r),g(r)$\footnote{Minkowski spacetime corresponds to $f(r)=g(r)=1$, \adsS{} spacetime in AdS coordinates by $f(r)=\frac{1}{g(r)}=f_\text{AdS}(r)$ and in 
tortoise coordinates with $f(r)=g(r)=f_\text{Tort}(r)$; where $f_\text{AdS}(r)$ and $f_\text{Tort}(r)$ are the usual \adsS{} blackening factor in AdS and tortoise coordinates respectively}.

\begin{table}[htb]
\centering % centering table
\setlength{\tabcolsep}{0.5em} % for the horizontal padding
{\renewcommand{\arraystretch}{1.2}% for the vertical padding
    \begin{tabular}{ccc|cc|cc|ccc} % creating 10 columns
\hline\hline 
                    & \multicolumn{2}{c|}{Minkowski} & \multicolumn{2}{c|}{AdS} & \multicolumn{2}{c|}{Tortoise}   & \multicolumn{3}{c}{Final Count}         \\
                    & \multicolumn{1}{c|}{M}  & Ml & \multicolumn{1}{c|}{M}  & Ml          & \multicolumn{1}{c|}{M} & Ml & M & Ml              & Total \\ \hline 
Initially           & 12&18&180&244&180&244&372&506&878\\
Failed to run       &0&2&128&125&115&238&243&365&608\\
Non-physical        &0&0&34&39&40&3&74&42&116\\ \hline
Finally             &12&16&18&80&25&3&55& \multicolumn{1}{c|}{99} &154      
\end{tabular}}
\caption{\label{tab:solCount}Counts of the number of solutions available, computed, and ruled out, in Minkowski spacetime and in AdS and tortoise coordinates for \adsS{} spacetime.
    The column labels ``M'' and ``Ml'' refer to the massive and massless particle cases respectively.}
\end{table}

\subsection{Numeric Results}\label{sec:partNumRes}
Due to the large number of variations on the solutions, it is not possible to generate
a meaningful plot showing all the solutions and how they compare with each other. We have therefore instead opted to consider comparisons of individual solutions against one particular selected solution. 
Figure~\ref{fig:partComps} shows 
this comparison between the solution obtained from the geodesic equation\footnote{For numeric reasons, the geodesic equation with lower $\mu$ index is preferred}
and a sample solution\footnote{In both cases, the derivative of the einbein constraint equation~\eqref{eqn:einc} has been used rather than the constraint equation itself,
    and the same form of this equation is used in both cases. No initial conditions for $\dot{e}$ were given, and the initial condition $e(0)=\frac{\sqrt{f(r(0))}}{m}$ was used in the massive case,
    while the initial condition $e(0)=0$ was used in the massless case. }
generated using the equations of motion arising from the einbein action~\eqref{eqn:Se}, in the form of a ratio\footnote{As shown in the axes labels of figure~\ref{fig:partComps}, it is not just the ratio of the two solutions that are plotted but the difference between this ratio and one, so that perfect agreement is given by zero.} between the two solutions.

\begin{figure}[htb]
\centering
\begin{subfigure}{\textwidth}
    \includegraphics[width=0.49\textwidth]{pics/einbM}
    \includegraphics[width=0.49\textwidth]{pics/einbMl}
\end{subfigure}
\caption{\label{fig:partComps} Ratio plots showing the comparison between the geodesic equation and a specific solution of the einbein equation of motion in AdS coordinates, which
we will take as a representative example of the comparisons between solutions.}
\end{figure}

From the plots shown in figure~\ref{fig:partComps}, we see good agreement between the solutions in both cases --- the plots have the same scale on the vertical which is of the order $\sim 10^{-7}$, much smaller than any length 
scale present in the problem (given by $r_H=0.3$ and $r(0)=1$).

Figure~\ref{fig:partComps} compares a small subset of solutions; while most align on similar scales, some pairs show large discrepancies, indicating at least one solution is incorrect. By scoring each solution, we identify patterns and conditions under which solutions perform best; taken here to mean those that agree\footnote{Where we are taking agreement to mean no deviations larger than
the arbitrarily chosen cutoff value of $10^{-5}$} with the most others. In the massive case, einbein equations of motion (using the derivative of the constraint and initial conditions for $e$) perform best, with some variations in the exact form of the equations and initial conditions across coordinate systems. Notably, solutions from the massive actions ($S_m$ and $S_2$) all fail or are physically invalid, suggesting the solver struggles with these equations or that they contain subtle issues. In the massless case, null methods excel, followed by direct use of the geodesic equation. In tortoise coordinates, einbein solutions fail or are invalid, but in AdS coordinates, they yield physically meaningful results---particularly when configurations mirror those successful in the massive case---highlighting the potential significance of this specific configuration choice.

\section{Conclusions and Outlook}
From the analyses presented in Section~\ref{sec:partNumRes},
we can see that there is no choice of action for which the resulting equations of motion numerically perform best in all cases. Instead, the closest we have is that the geodesic equation~\eqref{eqn:geod} arising from the 
extremization of the length functional~\eqref{eqn:S2} seems to show good results in all cases, although it is not best performing in any of them. We also see that the usual massive point particle actions~\eqref{eqn:Sm} 
and~\eqref{eqn:S2} do not result in equations that are numerically solvable; and while the einbein action~\eqref{eqn:Se} results in equations that are numerically solvable in \emph{most} cases, they are not in all cases and
even in the cases where they are, the solutions can be incredibly sensitive to the exact form of the constraint equation (or its derivative) and the choice of initial conditions for $e$ and/or $\dot{e}$ used. Potential 
directions to pursue in the future include a more detailed examination of the reasons for why certain actions do not result in numerically solvable equations of motion, and why this sometimes depends on the coordinate system
used. Additionally, there is also the question of why the einbein solutions are so sensitive to initial conditions of the non-physical degree of freedom and the form of the constraint equation used for this degree of freedom, as
well as if this sensitivity or the issues in obtaining numeric solutions are resolved by changing the precision or accuracy parameters of \emph{NDSolve}.
Finally then, since the geodesic equation shows good numerical solutions in all cases, it is potentially worth exploring a new comparison methodology of comparing all solutions to the geodesic equation rather than to all others, and
if it is possible to obtain an analytic solution to this geodesic equation to obtain more detailed comparisons.

Returning to the string case, we can also use the insights obtained above in the point particle case to illuminate potential directions to proceed here. All string solutions presented here were obtained in the conformal gauge, but
in the particle case we observe that the equations including an additional degree of freedom (there in the form of the einbein) generally performed better --- by analogy, perhaps including additional non-physical degrees of freedom
to be solved for in the string case will lead to better and more stable solutions. There is also the option of exploring the equations of motion obtained from the Nambu-Goto action, although this action
is analogous to the massive action $S_m$ in the particle case, indicating that we are unlikely to obtain useful solutions from this action unless we can determine the reason that the $S_m$ and $S_2$ actions fail in the particle
case and show those do not apply here. Finally then, we also have the option of using the obtained geodesic solutions from the point particle as seeds for the string solutions.
\section*{Acknowledgements}
NO and WAH
thank the National Research Foundation (NRF), the National Institute for Theoretical and Computational Physics (NITheCS), and the SA-CERN collaboration for their generous financial support
during the course of this work.%, and in particular their support surrounding the SAIP conference.

%%Bibliography
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\bibliographystyle{IEEEtran}
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\end{document}
