Analyticity Property of Scattering Amplitude in Theories with Compactified Space Dimensions

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.