Properties of solutions of the "naive" functional Schroedinger equation for QCD

TL;DR

This paper analyzes the properties of solutions to a simplified functional Schrödinger equation in QCD, revealing complex singularity structures and their relation to multidimensional hypergeometric functions.

Contribution

It characterizes the singularity structure of solutions to the functional Schrödinger equation in QCD, extending the class of meromorphic functions to multiple dimensions.

Findings

Solutions have multivalued Taylor coefficients with rational and logarithmic branchings.

Singularities occur along loci defined by polynomial equations.

Perturbative solutions asymptotically resemble multidimensional confluent hypergeometric functions.

Abstract

In this paper we consider the simplest functional Schroedinger equation of a quantum field theory (in particular QCD) and study its solutions. We observe that the solutions to this equation must possess a number of properties. Its Taylor coefficients are multivalued functions with rational and logarithmic branchings and essential singularities of exponential type. These singularities occur along a locus defined by polynomial equations. The conditions we find define a class of functions that generalizes to multiple dimensions meromorphic functions with finite Nevanlinna type. We note that in perturbation theory these functions have local asymptotics that is given by multidimensional confluent hypergeometric functions in the sense of Gelfand-Kapranov-Zelevinsky.