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.

Highlights

  • Consider 2-2 scattering in quantum field theory

  • 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

Read more

Summary

Introduction

Consider 2-2 scattering in quantum field theory. Suppose for simplicity that the external particles are identical massive scalars of mass m. In [15], it was shown how there is a close connection between the bounds arising from the Bieberbach conjecture and the bounds on the Wilson coefficients, which follow from a careful examination of the crossing symmetric dispersion relation. We will make use of existing mathematical literature which deal with the Taylor coefficients of functions not just inside the disk and inside the annulus This enables us to include in our discussions bound states as well as a massless pole. After briefly reviewing the necessary results from the GFT of typically-real functions [22, 24, 28] we proceed to get bounds for the Wilson coefficients in the low-energy expansion of the amplitude and compare with several other results known in the literature.

Set-up
Dispersion relation in QFT
GFT: the need for typically real functions
Typically-real functions
Amplitude and typically real-ness
Scalar EFTs and GFT
Extremal functions
A toy problem
Wpq bounds
Why we get two-sided bounds: a general proof
Numerical tighter bounds
Comparsion with known results
2.10 Comments on low spin dominance and the d → ∞ limit
2.11 Bounds in presence of poles
2.11.1 Maximal supergravity g0 bound
2.11.2 Impact parameter representation
Two channel symmetric case
Amplitude and typical real-ness
Differences from the fully symmetric case
Numerical bounds
Comparison with known results
Bounds in the presence of poles
Non-linear constraints and the EFT-hedron
Toeplitz determinant conditions
Connection with the crossing symmetric EFT-hedron
Analysis of nonlinear constraints
Discussion
Fully crossing symmetric case
Two-channel symmetric case
Implementation of Nc
Findings
Fully-crossing symmetric case We shall provide the details of the n = 3, 4, 5 cases here
Full Text
Published version (Free)

Talk to us

Join us for a 30 min session where you can share your feedback and ask us any queries you have

Schedule a call