QFT, EFT and GFT

TL;DR

This paper investigates the connection between geometric function theory and quantum field theory, using crossing symmetric dispersion relations to derive bounds on Wilson coefficients in effective field theories, with numerical and comparative analysis.

Contribution

It introduces a systematic approach using GFT constraints to bound Wilson coefficients in crossing-symmetric EFTs, including numerical methods and analysis of bound state and massless poles.

Findings

Two-sided bounds on Wilson coefficients are generally guaranteed.

GFT techniques effectively bound Wilson coefficients in 2-2 scattering.

Comparison shows GFT bounds align with or improve upon existing results.

Abstract

We explore the correspondence between geometric function theory (GFT) and quantum field theory (QFT). The crossing symmetric dispersion relation provides the necessary tool to examine the connection between GFT, QFT, and effective field theories (EFTs), enabling us to connect with the crossing-symmetric EFT-hedron. Several existing mathematical bounds on the Taylor coefficients of Typically Real functions are summarized and shown to be of enormous use in bounding Wilson coefficients in the context of 2-2 scattering. We prove that two-sided bounds on Wilson coefficients are guaranteed to exist quite generally for the fully crossing symmetric situation. Numerical implementation of the GFT constraints (Bieberbach-Rogosinski inequalities) is straightforward and allows a systematic exploration. A comparison of our findings obtained using GFT techniques and other results in the literature is made. We study both the three-channel as well as the two-channel crossing-symmetric cases, the latter having some crucial differences. We also consider bound state poles as well as massless poles in EFTs. Finally, we consider nonlinear constraints arising from the positivity of certain Toeplitz determinants, which occur in the trigonometric moment problem.