A Schwinger--Dyson Equation in the Borel Plane: singularities of the solution

TL;DR

This paper maps the Schwinger--Dyson and renormalization group equations into the Borel plane, revealing the singularity structure of the anomalous dimension and providing insights into the asymptotic behavior of solutions.

Contribution

It introduces a novel Borel plane approach to analyze the singularities of solutions to Schwinger--Dyson equations in the Wess--Zumino model, extending previous analytical methods.

Findings

Singularities of the anomalous dimension lie on the real line in the Borel plane.

Singularities are linked to the Mellin transform of the one-loop graph.

Preliminary numerical study suggests the Borel transform remains bounded at infinity.

Abstract

We map the Schwinger--Dyson equation and the renormalization group equation for the massless Wess--Zumino model in the Borel plane, where the product of functions get mapped to a convolution product. The two-point function can be expressed as a superposition of general powers of the external momentum. The singularities of the anomalous dimension are shown to lie on the real line in the Borel plane and to be linked to the singularities of the Mellin transform of the one-loop graph. This new approach allows us to enlarge the reach of previous studies on the expansions around those singularities. The asymptotic behavior at infinity of the Borel transform of the solution is beyond the reach of analytical methods and we do a preliminary numerical study, aiming to show that it should remain bounded.