[Paper Review] Quarkonium correlators at finite temperature and potential models

Ágnes Mócsy, Péter Petreczky|ArXiv.org|Oct 27, 2007

High-Energy Particle Collisions Research6 references3 citations

TL;DR

This paper investigates quarkonium spectral functions and Euclidean time correlators in potential models using lattice-calibrated static quark-antiquark free energies. It finds that quarkonium state melting above $T_c$ causes minimal changes in correlators, while large changes in scalar/axial-vector channels arise primarily from zero-mode contributions, challenging prior interpretations of lattice MEM data as evidence for bound states at high temperature.

ABSTRACT

We discuss the calculations of quarkonium spectral functions in potential models and their implications for the interpretation of the lattice data on quarkonium correlators. In particular, we find that melting of different quarkonium states does not lead to significant change in the Euclidean time correlators. The large change of the quarkonium correlators above deconfinement observed in the scalar and axial-vector channels appears to be due to the zero mode contribution.

Motivation & Objective

• To reconcile discrepancies between lattice QCD calculations of quarkonium correlators and their interpretation via Maximum Entropy Method (MEM) at finite temperature.
• To investigate whether the observed temperature dependence in quarkonium correlators above $T_c$ reflects genuine melting of bound states or other effects.
• To assess the reliability of MEM in extracting spectral functions at finite temperature, particularly for charmonium and bottomonium.
• To determine the role of zero-mode contributions in scalar and axial-vector channel correlators, which dominate the observed temperature dependence.
• To compare model predictions of Euclidean time correlators with lattice data, using potential models based on lattice-calibrated static quark-antiquark free energies.

Proposed method

• Construct non-relativistic potential models using the Cornell potential with screening at large distances, calibrated to lattice data on static quark-antiquark free energy.
• Solve the Schrödinger equation numerically for $S$- and $P$-wave quarkonium states to compute spectral functions $\sigma(\omega)$ using the non-relativistic Green's function.
• Match non-relativistic spectral functions smoothly to perturbative QCD results at high energies to extend validity beyond the non-relativistic regime.
• Compute Euclidean time correlators via integral transform: $G(\tau,T) = \int_0^\infty d\omega \, \sigma(\omega,T) \, K(\omega,\tau,T)$, with kernel $K$ involving hyperbolic functions.
• Include zero-mode contributions $\sim \omega \delta(\omega)$ in spectral functions for vector, scalar, and axial-vector channels, which are absent in the pseudo-scalar channel.
• Compare model predictions of $G(\tau,T)$ and $G/G_{\text{rec}}$ with lattice QCD results, especially from isotropic and anisotropic lattices, to validate the model.

Experimental results

Research questions

• RQ1 To what extent does the melting of quarkonium states at $T > T_c$ lead to measurable changes in the Euclidean time correlators?
• RQ2 Why do scalar and axial-vector quarkonium correlators show large temperature dependence above $T_c$, while pseudo-scalar and vector channels do not?
• RQ3 Can the observed temperature dependence in scalar/axial-vector correlators be explained by zero-mode contributions rather than bound state melting?
• RQ4 How reliable is the Maximum Entropy Method (MEM) for extracting spectral functions from lattice correlators at finite temperature?
• RQ5 To what extent do potential models based on lattice-calibrated static quark-antiquark free energy reproduce lattice data on quarkonium correlators?

Key findings

• Melting of quarkonium states, including the $1S$ charmonium state at $1.2T_c$, causes only small changes in the Euclidean time correlators, contrary to expectations.
• The large temperature dependence observed in scalar and axial-vector channel correlators is primarily due to zero-mode contributions, not bound state dissociation.
• The spectral function bump observed in lattice MEM analyses at $1.5T_c$ may instead arise from a threshold enhancement rather than a surviving $1S$ resonance.
• For bottomonium, the $1S$ state remains bound until $\sim 2T_c$, but its binding energy drops significantly (from 1.570 GeV to 0.041 GeV at $2T_c$), leading to a large thermal width.
• The model reproduces lattice data for $G(\tau,T)$ and $G/G_{\text{rec}}$ in all channels, especially when zero-mode contributions are included in scalar/axial-vector channels.
• The ratio $G/G_{\text{rec}}$ in the scalar and axial-vector channels matches lattice results only when the zero-mode contribution is added, confirming its dominant role in the observed temperature dependence.