Proof of dispersion relations for the amplitude in theories with a compactified space dimension

The analyticity properties of the scattering amplitude in the nonforward direction are investigated for a field theory in the manifold R3,1 ⨂ S1. The theory is obtained from a massive, neutral scalar field theory of mass m0 defined in flat five dimensional spacetime upon compactification on a circle, S1. The resulting theory is endowed with a massive scalar field which has the lowest mass, m0, as of the original five dimensional theory and a tower of massive Kaluza-Klein states. We derive nonforward dispersion relations for scattering of the excited Kaluza-Klein states in the Lehmann-Symanzik-Zimmermann formulation of the theory. In order to accomplish this object, first we generalize the Jost-Lehmann-Dyson theorem for a relativistic field theory with a compact spatial dimension. Next, we show the existence of the Lehmann-Martin ellipse inside which the partial wave expansion converges. The scattering amplitude satisfies fixed-t dispersion relations when |t| lies within the Lehmann-Martin ellipse.