Exact solution of Dyson-Schwinger equations for a scalar field theory

TL;DR

This paper provides an exact analytical solution to Dyson-Schwinger equations for a massless quartic scalar field theory, revealing its trivial fixed points and implications for related gauge theories.

Contribution

It offers the first exact solutions for n-point functions and the propagator in this scalar field theory, and explores its fixed point structure and connection to gauge theories.

Findings

The spectrum matches that of a harmonic oscillator.

The theory has only trivial fixed points.

The coupling vanishes at low energy and diverges at high energy.

Abstract

We exactly solve Dyson-Schwinger equations for a massless quartic scalar field theory. n-point functions are computed till n=4 and the exact propagator computed from the two-point function. The spectrum is so obtained, being the same of a harmonic oscillator. Callan-Symanzik equation for the two-point function gives the beta function. This gives the result that this theory has only trivial fixed points. In the low-energy limit the coupling goes to zero making the theory trivial and, at high energies, it reaches infinity. No Landau pole appears, rather this should be seen as a precursor, in a weak perturbation expansion, of the coupling reaching the trivial fixed point at infinity. Using a mapping theorem, recently proved, between massless quartic scalar field theory and gauge theories, we derive some properties of the latter.