Scattering length from holographic duality

TL;DR

This paper introduces a holographic duality-based method to compute scattering lengths in strongly coupled theories, exemplified by a hard wall model, revealing new constraints on operator dimensions.

Contribution

It develops a novel approach using holographic duality to calculate scattering lengths and derives bounds on scalar operator dimensions in large-N theories.

Findings

Computed scattering length in a hard wall model with quartic potential.

Found a constraint on scalar operator dimension: Δ > d/4.

Identified more restrictive bounds than unitarity for certain dimensions.

Abstract

Interesting theories with short range interactions include QCD in the hadronic phase and cold atom systems. The scattering length in two-to-two elastic scattering process captures the most elementary features of the interactions, such as whether they are attractive or repulsive. However, even this basic quantity is notoriously difficult to compute from first principles in strongly coupled theories. We present a method to compute the two-to-two amplitudes and the scattering length using the holographic duality. Our method is based on the identification of the residues of Green's functions in the gravity dual with the amplitudes in the field theory. To illustrate the method we compute a contribution to the scattering length in a hard wall model with a quartic potential and find a constraint on the scaling dimension of a scalar operator $\Delta > d/4$. For $d< 4$ this is more stringent than the unitarity constraint and may be applicable to an extended family of large-$N$ theories with a discrete spectrum of massive states. We also argue that for scalar potentials with polynomial terms of order $K$, a constraint more restrictive than the unitarity bound will appear for $d<2K/(K-2)$.