⚛️ phenomenology

Complex-energy spectrum and dissociation of heavy quarkonia in an anisotropic collisional quark-gluon plasma

This paper investigates the dissociation of heavy quarkonia in an anisotropic, collisional quark-gluon plasma by solving the Schrödinger equation with a full complex potential, revealing that increased anisotropy raises dissociation temperatures and that thermal widths derived from complex eigenvalues differ significantly from first-order estimates.

Original authors: Najmul Haque

Published 2026-09-15

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: 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, in the first fractions of a second after the Big Bang, matter did not exist as the solid atoms we know today. Instead, it was a seething, super-hot soup of fundamental particles called quarks and gluons, known as a quark-gluon plasma. In our everyday world, quarks are permanently locked inside particles like protons and neutrons, never allowed to wander free. But in this primordial plasma, the heat is so intense that the forces holding them together are weakened, allowing them to roam. To understand how this extreme state of matter behaves, physicists often look at "heavy quarkonia," which are pairs of heavy quarks bound together. These pairs act like tiny, delicate probes that can survive briefly in the plasma, and by studying how they break apart, scientists can learn about the invisible forces and the temperature of this exotic environment.

A new study by physicist Najmul Haque at the National Institute of Science Education and Research in India takes a closer look at how these heavy quark pairs behave when the plasma around them is not perfectly uniform. In many theoretical models, scientists assume the plasma expands evenly in all directions, like a balloon inflating. However, in the violent collisions of heavy ions that create this plasma in particle accelerators, the expansion is often stretched and uneven, creating a state of "anisotropy" where the environment looks different depending on which way you look. Haque's work solves a complex mathematical puzzle to see how this unevenness changes the fate of the heavy quark pairs, specifically focusing on the most stable versions of these pairs, known as the 1S states of charmonium and bottomonium.

To do this, the researcher used a sophisticated version of the Schrödinger equation, a fundamental tool in physics that describes how quantum particles move and interact. Unlike previous studies that treated the messy, chaotic interactions of the plasma as a small, simple correction, this work treated the full complexity of the environment from the start. The potential energy holding the quark pair together was modeled as having two parts: a real part that keeps them bound, and an imaginary part that represents the thermal noise and collisions that try to tear them apart. By solving the equation with both parts active simultaneously, the study calculated exactly how much energy binds the pair and how quickly it would decay, or "dissociate," into the surrounding soup.

The results reveal a surprising resilience. As the anisotropy of the plasma increases—meaning the environment becomes more stretched and uneven—the temperature required to break apart these heavy quark pairs actually goes up. For the charmonium state, the temperature needed to dissociate it rises in a uniform plasma to a higher value in a highly stretched one. For the heavier bottomonium state, the dissociation temperature climbs from a lower value to a higher one. This suggests that the very distortion of the plasma that might seem like it would make things more chaotic actually helps hold these specific heavy particles together a little longer against the heat.

The study also examined the shape of these particles as they exist in the plasma. Even though the environment is stretched, the heavy quark pairs remain remarkably spherical in their internal structure. The researchers found that the pairs are still overwhelmingly dominated by their simplest, roundest shape, with only tiny, almost negligible amounts of distortion into more complex shapes. This stability is crucial because it means the simple models used to describe them are still largely valid, even in these extreme conditions. Furthermore, the study compared its calculated rates of decay with data from supercomputer simulations known as lattice QCD. While the numbers for the heavier bottomonium particles were somewhat higher than those simulations, the results for the lighter charmonium particles were in the same general range, offering a useful cross-check for our understanding of the quark-gluon plasma.

Finally, the researchers used these findings to estimate the "survival probability" of these particles. They calculated the chance that a heavy quark pair would remain intact if it were held in this hot, anisotropic environment for a specific, tiny slice of time. The results showed that while the heavier bottomonium pairs have a decent chance of surviving, the lighter charmonium pairs are much more likely to be destroyed, with only about a 13 to 20 percent chance of remaining intact under the conditions studied. This work provides a clearer, more consistent picture of how heavy particles interact with a distorted, collision-filled plasma, moving beyond simple approximations to capture the full, complex reality of the forces at play in the early universe and in modern particle colliders.

Technical Summary: Complex-energy spectrum and dissociation of heavy quarkonia in an anisotropic collisional quark-gluon plasma

Problem Statement
Heavy quarkonia serve as critical probes for deconfined QCD matter. While potential models have traditionally described the in-medium heavy-quark interaction, previous approaches often treated the imaginary part of the complex potential (responsible for thermal width) perturbatively or assumed an isotropic, collisionless plasma. In realistic heavy-ion collisions, the Quark-Gluon Plasma (QGP) exhibits momentum-space anisotropy due to rapid longitudinal expansion and finite collision rates. The author addresses the need to solve the bound-state problem for heavy quarkonia using the full complex potential of a momentum-anisotropic, collisional QGP without treating the imaginary component as a perturbation. This requires handling a noncentral, axially symmetric interaction that couples different partial waves.

Methodology
The study solves the Schrödinger equation for heavy quarkonia ( $c\bar{c}$ and $b\bar{b}$ ) using a complex potential $V(r; T, \xi, c) = V_R + iV_I$ derived for a spheroidally anisotropic plasma with a Bhatnagar–Gross–Krook (BGK) collision kernel.

Medium Inputs: The calculation employs a one-loop running coupling and a leading-order Debye mass. The collision rate $\nu$ is tied to the running coupling, with a parameter $c$ adjusted to ensure the system remains in a "pinch-free" domain, avoiding singularities in the imaginary potential associated with Weibel instabilities.
Potential Construction: The complex potential is constructed from static mass scales and a fluctuation function derived from the anisotropic medium. The real part includes a Cornell-like input ( $-\alpha/r + \sigma r$ ) with ultraviolet subtraction, while the imaginary part arises from Landau damping and scattering processes.
Numerical Solution: Because the anisotropic potential depends on both the separation $r$ and the angle $\Theta$ relative to the anisotropy axis, the Schrödinger equation is an axisymmetric two-dimensional problem. The author expands the potential in Legendre multipoles and the wavefunction in spherical harmonics. This transforms the problem into a set of coupled radial equations for a fixed magnetic quantum number $m$ . The noncentral interaction mixes partial waves (e.g., S-D-G mixing), which is solved as a non-Hermitian block eigenvalue problem.
Dissociation Criterion: The dissociation temperature $T_d$ is defined by the condition $\Gamma(T_d) = 2E_B(T_d)$ , where $\Gamma$ is the thermal width and $E_B$ is the binding energy. Auxiliary criteria include $E_B=0$ and $\Gamma = E_B$ .

Key Contributions

Non-perturbative Treatment of Imaginary Potential: Unlike previous studies (e.g., Ref. [11]) that used perturbative estimates of the width based on real-potential wavefunctions, this work diagonalizes the full complex Hamiltonian ( $H = V_R + iV_I$ ) from the outset. This yields the binding energy, width, and medium-modified wavefunction self-consistently from a single complex eigenvalue problem.
Anisotropic Coupled-Channel Formalism: The paper implements a spectral representation of the anisotropic Schrödinger equation, retaining angular kinetic energy and partial-wave mixing. This allows for the calculation of complex poles and wavefunctions in a single calculation rather than relying on angular averages.
Pinch-Free Parameter Domain: The author explicitly maps the "pinch-free" domain for the collision rate parameter $c$ as a function of anisotropy $\xi$ and temperature $T$ , ensuring the imaginary potential remains well-defined.

Results

Dissociation Temperatures: For both $1S$ charmonium ( $J/\psi$ ) and bottomonium ( $\Upsilon$ ) states, the dissociation temperature $T_d$ increases with increasing momentum-space anisotropy $\xi$ along the central pinch-safe trajectory. This trend is driven by both the direct anisotropy dependence of the potential and the required increase in the collision rate to avoid singularities.
Thermal Widths: The full complex-pole widths differ appreciably from first-order perturbative estimates ( $\Gamma_{PT}$ ). The ratio $\Gamma_{full}/\Gamma_{PT}$ is generally less than 1, indicating that non-perturbative treatment reduces the width, particularly for charmonium near dissociation.
Wavefunction Structure: Despite the noncentral interaction, the $1S$ eigenfunctions remain overwhelmingly dominated by the leading S-wave component. Even at high anisotropy ( $\xi=10$ ), the D-wave admixture remains below $8.3 \times 10^{-4}$ , and G-wave contributions are negligible.
Comparison with Lattice QCD: The calculated isotropic widths are compared with HotQCD and BNL lattice results. The charmonium widths are comparable in scale but do not uniformly reproduce the lattice temperature dependence. The calculated bottomonium widths tend to be larger than lattice determinations across the common temperature range. These comparisons are used for validation, not for tuning the collision parameter.
Survival Probability: A static survival probability $S_n$ is illustrated for a fixed time interval ( $\Delta \tau = 1$ fm/c) at $T=0.20$ GeV. The results show that bottomonium has a higher survival probability than charmonium due to its smaller width, and survival decreases as anisotropy increases along the pinch-safe trajectories.

Significance and Claims
The paper claims to provide a consistent framework for calculating the complex energy spectrum of heavy quarkonia in an evolving, anisotropic QGP. By treating the imaginary potential non-perturbatively within a coupled partial-wave basis, the author demonstrates that the dissociation temperature is sensitive to anisotropy and that the full complex pole yields widths distinct from perturbative approximations.

The author remains modest regarding the phenomenological application of these results. They explicitly state that the calculated static survival probabilities are illustrative benchmarks rather than predictions for the nuclear modification factor ( $R_{AA}$ ). A full $R_{AA}$ calculation would require a separate dynamical treatment incorporating time-dependent media, production geometry, feed-down, and regeneration, which are outside the scope of this static spectral analysis. Furthermore, the study is restricted to $1S$ states; extending the analysis to physical P-wave quarkonia would require including spin-dependent interactions and their in-medium modifications, which are not included in the current Hamiltonian.

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