One-Loop Feynman Diagram Zeta-Functions inT > 0 Field Theory

Abstract

The Matsubara sums of Euclidean finite-temperature (T>0) field theory, over discrete energy, express the nontrivial topology of cylindrical T>0 spacetime. We explore the possibility that an individual T>0 Feynman diagram can be identified as a special value of a ζ-function or other meromorphic function, whose existence depends on the cylindrical topology. (Here “meromorphic” refers to one or more complex variables in which analytic continuation is performed.) We prove that one-loop Feynman diagrams with 0, 1, 2, 3, … external lines do indeed have an associated ζ-function. This “Feynman diagram ζ-function” is by construction a convenient tool for the regularization (dimensional or analytic) of its diagram. But the true power of the diagram ζ-function approach shows up in the straightforward derivation they provide of exact high-T series expansions of the one-loop diagrams. All Feynman diagrams which are simple products of one-loop diagrams can be analysed conveniently in this way. Diagrams which involve overlapping loops are more difficult, and are only briefly touched upon here. Scalar fields are studied in this paper. Two sequels will deal with T>0 fields having spin, and field theory on toroidal spacetimes TN×En, which generalizes the considerations in this paper on T>0 spacetime S1×En.