On the convergence of the polarization tensor in space-time of three dimensions

TL;DR

This paper investigates the convergence and regularization of the polarization tensor in three-dimensional space-time for graphene, clarifying its behavior at zero and finite temperatures using dimensional regularization.

Contribution

It provides a detailed analysis of the polarization tensor's convergence properties in 3D space-time, resolving contradictions in the literature with a consistent regularization approach.

Findings

Thermal correction to the polarization tensor is finite at any dimension.

Zero-temperature polarization tensor behaves differently in 3D and 4D.

Analytic continuation yields finite results without subtraction in 3D.

Abstract

In this paper, we consider the convergence properties of the polarization tensor of graphene obtained in the framework of thermal quantum field theory in three-dimensional space-time. During the last years, this problem attracted much attention in connection with calculation of the Casimir force in graphene systems and investigation of the electrical conductivity and reflectance of graphene sheets. There are contradictory statements in the literature, especially on whether this tensor has an ultraviolet divergence in three dimensions. Here, we analyze this problem using the well known method of dimensional regularization. It is shown that the thermal correction to the polarization tensor is finite at any $D$, whereas its zero-temperature part behaves differently for $D=3$ and 4. For $D=3$, it is obtained by the analytic continuation with no subtracting infinitely large terms. As to the space-time of $D=4$, the finite result for the polarization tensor at zero temperature is found after subtracting the pole term. Our results are in agreement with previous calculations of the polarization tensor at both zero and nonzero temperature. This opens possibility for a wider application of the quantum field theoretical approach in investigations of graphene and other two-dimensional novel materials.