Subtleties in the calculation of correlation functions for hot and dense systems

TL;DR

This paper examines the mathematical subtleties involved in calculating correlation functions in hot and dense systems, emphasizing the importance of operation order in loop integrals and illustrating with examples from quantum field theories.

Contribution

It highlights the delicate nature of loop integral calculations in relativistic theories at finite temperature and density, providing general guidance and specific examples.

Findings

Order of operations affects loop integral results.

Zero-temperature limit is delicate near Fermi surfaces.

Concrete examples from Gross-Neveu-Yukawa and QED models.

Abstract

We discuss subtleties in the calculation of loop integrals in studies of hot and dense systems as they appear in both perturbative and non-perturbative approaches. To be specific, we address subtleties which appear in situations where the order of integration, differentiation, and limit processes plays a crucial role. For example, this applies to computations of the effective action and the computation of momentum-dependent correlation functions. In particular, the zero-temperature limit is delicate in systems with fermions because of the presence of discontinuities at the Fermi surface. We provide a general discussion of scenarios where the computation and evaluation of loop integrals in the context of relativistic theories requires particular attention as a change of the order of the involved mathematical operations may lead to a different result. Our general considerations are then illustrated with the aid of concrete examples, namely by the computation of masses from fully momentum-dependent correlation functions in the context of the Gross-Neveu-Yukawa model and quantum electrodynamics.