⚛️ phenomenology

Complex heavy-quarkonium potential in an anisotropic collisional quark-gluon plasma

This paper computes the complex heavy-quark potential in an anisotropic quark-gluon plasma using kinetic theory with a collision kernel, demonstrating that while collisions minimally affect the real part of the potential, they are crucial for regularizing Weibel-induced singularities and significantly enhancing the imaginary part, thereby increasing quarkonium thermal widths and dissociation rates.

Original authors: Manas Debnath, Lata Thakur, Najmul Haque

Published 2026-08-26

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Manas Debnath, Lata Thakur, Najmul Haque

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Deep within the heart of the universe, just fractions of a second after the Big Bang, matter existed in a state unlike anything we see today. It was a seething, super-hot soup of fundamental particles called a quark-gluon plasma. In this extreme environment, the building blocks of protons and neutrons—quarks—were not bound together but roamed freely, separated by a sea of force-carrying particles known as gluons. To understand how this primordial state behaves, physicists recreate it in massive laboratories by smashing heavy atomic nuclei together at nearly the speed of light. These collisions create tiny, fleeting droplets of this hot plasma, allowing scientists to study how matter transforms under the most intense conditions imaginable. A key question in this field is how heavy particles, specifically pairs of quarks bound together like a microscopic dumbbell, survive or fall apart in this chaotic environment. These pairs, known as quarkonia, act as sensitive probes; if they melt, it tells us about the temperature and density of the plasma. However, the plasma created in these collisions is not a calm, uniform fluid. It is a turbulent, expanding system where particles move faster in some directions than others, creating a lopsided, anisotropic environment that complicates the physics of how these heavy pairs interact.

A team of researchers has now taken a significant step toward untangling this complexity by calculating exactly how the forces between heavy quarks change when the plasma is both collisional and lopsided. In a standard, calm plasma, the force holding these quarks together is weakened by the surrounding heat, a process known as screening, which eventually causes the pair to separate. But in the real, messy conditions of a heavy-ion collision, the plasma particles are constantly bumping into one another, and the system is stretched out in specific directions. The researchers set out to determine how these two factors—frequent collisions and directional stretching—alter the invisible tether between the quarks. They built a detailed mathematical model using the principles of kinetic theory, which tracks how particles move and collide, to describe the plasma's behavior. By incorporating a specific method to account for particle collisions, they were able to derive a complete picture of the potential energy between the quarks, including both the part that holds them together and a new, subtle part that describes how the plasma causes the pair to lose energy and decay.

The study revealed a fascinating split in how these two factors affect the quark pair. When the researchers looked at the part of the force that keeps the quarks bound, they found that the frequency of collisions among plasma particles had almost no effect. Whether the particles were colliding frequently or rarely, the strength of the binding force remained largely the same. This suggests that the energy holding the quark pair together is primarily determined by the directional stretching of the plasma, not by how often the particles bump into each other. However, the story changed completely when they examined the part of the interaction that causes the pair to fall apart. Here, the collisions played a dramatic role. In a plasma that is stretched out in one direction, the physics can become unstable, leading to a mathematical breakdown where the prediction for how fast the pair decays becomes undefined. The researchers discovered that the very act of particles colliding fixes this problem. The collisions smooth out the instability, allowing for a clear, well-defined prediction of how quickly the quark pair will dissolve.

Once this mathematical hurdle was cleared, the results showed that collisions significantly speed up the decay of the quark pair. In the presence of frequent collisions, the imaginary part of the interaction—which represents the rate at which the pair loses its identity—became much stronger. This means that in a realistic, collisional plasma, heavy quark pairs are far more likely to melt apart quickly than they would be in a theoretical, collision-free environment. The researchers applied these findings to specific types of quark pairs, such as those made of charm quarks and those made of bottom quarks. They found that while both types are affected, the lighter charm pairs are much more vulnerable to this accelerated decay. This aligns with experimental observations where heavier, more tightly bound states survive longer than their lighter counterparts. The study suggests that the combination of the plasma's directional stretch and the constant collisions between its particles creates a perfect storm that rapidly dissolves these heavy pairs.

This work provides a clearer window into the early moments of the universe and the extreme conditions created in modern particle accelerators. By showing that collisions are essential for stabilizing the mathematical description of the plasma and that they significantly enhance the rate at which heavy particles dissolve, the researchers have refined our understanding of how matter behaves at the edge of existence. The findings indicate that the suppression of heavy quark pairs in these collisions is not just a result of heat, but a complex interplay of directional expansion and particle collisions. This deeper insight helps explain why certain heavy particles vanish almost instantly in these experiments while others linger, offering a more precise map of the forces that govern the fundamental building blocks of our universe.

Technical Summary: Complex Heavy-Quark Potential in an Anisotropic Collisional Quark-Gluon Plasma

Problem Statement
The formation and properties of the Quark-Gluon Plasma (QGP) are central to high-energy nuclear physics, with heavy quarkonium suppression serving as a primary signature of QGP formation. While earlier theoretical frameworks successfully modeled heavy quarkonium in isotropic media using Hard Thermal Loop (HTL) resummation, they often neglected two critical features of the early stages of heavy-ion collisions: momentum-space anisotropy (arising from rapid longitudinal expansion) and finite collision rates among plasma constituents. Previous studies of anisotropic QGP were largely restricted to collisionless scenarios. A significant gap existed in the literature regarding the derivation of the complex heavy-quark potential in a medium that is simultaneously anisotropic and collisional. Specifically, the derivation of the Feynman (symmetric) propagator required to determine the imaginary part of the potential in such a nonequilibrium, anisotropic collisional medium was absent. Furthermore, the presence of Weibel instabilities in anisotropic plasmas introduces pinch singularities that can render the imaginary potential ill-defined, necessitating a rigorous treatment of collision effects to regularize these singularities.

Methodology
The authors employ kinetic theory with a Bhatnagar-Gross-Krook (BGK) collision kernel to model the QGP. The momentum anisotropy is incorporated via the Romatschke-Strickland spheroidal deformation of the parton distribution functions. The study proceeds through the following steps:

Gluon Self-Energy and Propagator: The linearized Boltzmann-Vlasov transport equation is solved to derive the gluon self-energy ( $\Pi^{\mu\nu}$ ) in the presence of the BGK collision term. Unlike the collisionless case, the self-energy becomes asymmetric in Lorentz indices due to the lack of time-reversal symmetry. The authors utilize a general tensor basis with five structure functions ( $\alpha, \beta, \gamma, \delta, \rho$ ) to decompose the self-energy and subsequently derive the resummed gluon propagator in the temporal axial gauge.
Dielectric Response: The real part of the inverse dielectric function is obtained from the retarded/advanced gluon propagators. To determine the imaginary part, the authors derive the Feynman (symmetric) propagator for the anisotropic collisional medium. This derivation bypasses the limitations of the standard fluctuation-dissipation theorem (which breaks down in anisotropic, out-of-equilibrium systems) by employing a stochastic transport framework.
Potential Calculation: The in-medium complex heavy-quark potential is computed by correcting the vacuum Cornell potential using the medium's dielectric permittivity. The real part is derived from the retarded propagator, while the imaginary part is determined from the Feynman propagator.
Singularity Regularization: The authors analyze the integrand of the imaginary potential, identifying a pinch singularity associated with the Weibel instability. They demonstrate that finite collision rates ( $\nu$ ) regularize this singularity, defining a region in parameter space where the imaginary potential is well-defined.
Thermal Width: The thermal decay width of quarkonium states (specifically $J/\psi$ and $\Upsilon$ ) is calculated by treating the imaginary part of the potential as a perturbation to the vacuum potential, utilizing Coulombic ground-state wave functions.

Key Contributions

Derivation of the Feynman Propagator: The paper establishes, for the first time, the Feynman propagator for an anisotropic collisional medium, enabling the calculation of the imaginary part of the heavy-quark potential in a regime previously inaccessible.
Regularization of Weibel Instabilities: The study identifies that sufficiently large collision rates regularize the pinch singularity caused by Weibel instabilities in anisotropic plasmas, rendering the imaginary potential well-defined in specific regions of the anisotropy-collision parameter space.
Systematic Derivation of Complex Potential: The work provides a complete derivation of both the real and imaginary parts of the heavy-quark potential, accounting for general momentum anisotropy and finite collisional rates simultaneously.

Results

Real Part of the Potential: The collisional parameter ( $\nu$ ) has only a minimal impact on the real part of the potential. Consequently, the binding energy of quarkonium states is found to be primarily sensitive to momentum anisotropy rather than the collisional rate. The anisotropy introduces a directional dependence, modifying the screening behavior based on the angle between the quark-antiquark dipole and the anisotropy vector.
Imaginary Part of the Potential: In contrast to the real part, the imaginary component is significantly influenced by collisions. Within the well-defined regime (where the singularity is regularized), increasing the collision rate enhances the magnitude of the imaginary potential. The interplay between anisotropy and collisions modifies the thermal fluctuations, leading to larger imaginary potentials compared to collisionless scenarios.
Thermal Widths and Dissociation: The calculated thermal decay widths for charmonium ( $J/\psi$ ) and bottomonium ( $\Upsilon$ ) increase monotonically with both the temperature scale, the anisotropy parameter ( $\xi$ ), and the collision rate ( $\nu$ ). The results indicate that charmonium states, being less tightly bound, exhibit significantly larger decay widths and dissociate more rapidly in a collisional, anisotropic environment compared to bottomonium. This supports the phenomenon of sequential suppression observed in experiments.

Significance
The paper claims to provide critical insights into quarkonium suppression mechanisms in relativistic heavy-ion collisions by bridging the gap between collisionless anisotropic models and realistic collisional scenarios. By demonstrating that collisions significantly enhance the imaginary part of the potential and the resulting thermal widths, the work suggests that nonequilibrium effects (both anisotropy and collisions) play a crucial role in the dissociation of heavy quarkonia. The regularization of the Weibel instability singularity by collisions is presented as a necessary condition for obtaining physically meaningful results in anisotropic plasmas. The authors note that while the current study uses Coulombic wave functions for the decay width calculation, future quantitative improvements would require solving the Schrödinger equation with the full complex potential.

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