Convergence of the polarization tensor in spacetime of three dimensions

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 spacetime. During the last years, this problem attracted much attention in connection with the calculation of the Casimir force in graphene systems and the 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 <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi>D</a:mi></a:math>, whereas its zero-temperature part behaves differently for <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:mi>D</c:mi><c:mo>=</c:mo><c:mn>3</c:mn></c:math> and 4. For <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:mi>D</e:mi><e:mo>=</e:mo><e:mn>3</e:mn></e:math>, it is obtained by analytic continuation with no subtracting of infinitely large terms. As for the spacetime of <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:mi>D</g:mi><g:mo>=</g:mo><g:mn>4</g:mn></g:math>, 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 the possibility for a wider application of the quantum-field-theoretical approach in investigations of graphene and other two-dimensional novel materials.