Real and imaginary-time $Q\bar{Q}$ correlators in a thermal medium

TL;DR

This paper analyzes heavy fermion pairs in a thermal medium using simplified models, deriving exact results for infinite mass cases, and exploring the real and imaginary parts of the potential affecting their behavior.

Contribution

It provides an exact Schrödinger equation for heavy fermion correlators in a thermal medium and links real-time potentials to Euclidean free energies.

Findings

Imaginary part of the potential arises from medium-induced collisions.

Real part of the potential corresponds to the free energy in Euclidean formalism.

Damping rates of heavy fermions are determined by collision processes.

Abstract

We investigate the behavior of a pair of heavy fermions, denoted by $Q$ and $\bar{Q}$, in a hot/dense medium. Although we have in mind the situation where $Q$ and $\bar{Q}$ denote heavy quarks, our treatment will be limited to simplified models, which bear only some general similarities with QCD. We study in particular the limiting case where the mass of the heavy fermions is infinite. Then a number of results can be derived exactly: a Schr\"odinger equation can be established for the correlator of the heavy quarks; the interaction effects exponentiate, leading to a simple instantaneous effective potential for this Schr\"odinger equation. We consider simple models for the medium in which the $Q\bar Q$ pair propagates. In the case where the medium is a plasma of photons and light charged fermions, an imaginary part develops in this effective potential. We discuss the physical interpretation of this imaginary part in terms of the collisions between the heavy particles and the light fermions of the medium; the same collisions also determine the damping rate of the heavy fermions. Finally we study the connection between the real-time propagator of the heavy fermion pair and its Euclidean counterpart, and show that the real part of the potential entering the Schr\"odinger equation for the real-time propagator is the free energy calculated in the imaginary-time formalism.