Analyticity Properties of Scattering Amplitude in Theories with Compactified Space Dimensions: The Proof of Dispersion Relations

TL;DR

This paper proves dispersion relations and bounds for scattering amplitudes in theories with one compactified spatial dimension, providing insights into how extra dimensions could be detected through high-energy scattering experiments.

Contribution

It establishes the analyticity properties and dispersion relations for scalar field scattering in a spacetime with a compactified dimension, extending quantum field theory results to higher-dimensional models.

Findings

Proved nonforward dispersion relation in compactified spacetime.

Established analogs of Froissart-Martin bounds in this setting.

Found no evidence of bound violation at LHC energies.

Abstract

The analyticity properties of the scattering amplitude for a massive scalar field is reviewed in this article where the spacetime geometry is $R^{3,1}\otimes S^1$ i.e. one spatial dimension is compact. Khuri investigated the analyticity of scattering amplitude in a nonrelativitstic potential model in three dimensions with an additional compact dimension. He showed that, under certain circumstances, the forward amplitude is nonanalytic. He argued that in high energy scattering if such a behaviour persists it would be in conflicts with the established results of quantum field theory and LHC might observe such behaviors. We envisage a real scalar massive field in flat Minkowski spacetime in five dimensions. The Kaluza-Klein (KK) compactification is implemented on a circle. The resulting four dimensional manifold is $R^{3,1}\otimes S^1$. The LSZ formalism is adopted to study the analyticity of the scattering amplitude. The nonforward dispersion relation is proved. In addition the Jin-Martin bound and an analog of the Froissart-Martin bound are proved. A novel proposal is presented to look for evidence of the large-radius-compactification scenario. A seemingly violation of Froissart-Martin bound at LHC energy might hint that an extra dimension might be decompactified. However, we find no evidence for violation of the bound in our analysis.