Analyticity Property of Scattering Amplitude in Theories with Compactified Space Dimensions

TL;DR

This paper proves that in a five-dimensional scalar field theory with one compactified dimension, the elastic scattering amplitude maintains analyticity and satisfies dispersion relations, extending known properties to theories with extra compactified dimensions.

Contribution

It demonstrates that the forward elastic scattering amplitude in a compactified higher-dimensional theory retains analyticity and obeys dispersion relations within the LSZ framework.

Findings

The forward scattering amplitude satisfies dispersion relations.

Unitarity constrains the absorptive part of the amplitude.

Analyticity properties extend to theories with compactified extra dimensions.

Abstract

We consider a massive, neutral, scalar field theory of mass $m_0$ in a five dimensional flat spacetime. Subsequently, one spatial dimension is compactified on a circle, $S^1$, ofradius $R$. The resulting theory is defined in the manifold, $R^{3,1}\otimes S^1$. The mass spectrum is a state of lowest mass, $m_0$, and a tower of massive Kaluza-Klein states. The analyticity property of the elastic scattering amplitude is investigated in the Lehmann-Symanzik-Zimmermann (LSZ) formulation of this theory. In the context of nonrelativistic potential scattering, for the $R^3\otimes S^1$ spatial geometry, it was shown that the forward scattering amplitude does not satisfy analyticity properties in some cases for a class of potentials. If the same result is valid in relativistic quantum field theory then the consequences will be far reaching. We show that the forward elastic scattering amplitude of the theory, in the LSZ axiomatic approach, satisfies forward dispersion relations. The importance of the unitarity constraint on the S-matrix, is exhibited in displaying the properties of the absorptive part of the amplitude.