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The analyticity properties of the scattering amplitude in the nonforwarddirection are investigated for a field theory in the manifold$\mathbb{R}^{3,1}\times S^1$. A scalar field theory of mass $m_0$ is consideredin $D = 5$ Minkowski space to start with. Subsequently, one spatial dimensionis compactified to a circle. The mass spectrum of the resulting theory is: (a)a massive scalar of mass, $m_0$, same as the original five dimensional theoryand (b) a tower of massive Kaluza-Klein states. We derive nonforward dispersionrelations for scattering of the excited Kaluza-Klein states in theLehmann-Symanzik-Zimmermann formulation of the theory. In order to accomplishthis object, first we generalize the Jost-Lehmann-Dyson theorem for arelativistic field theory with a compact spatial dimension. Next, we show theexistence of the Lehmann-Martin ellipse inside which the partial wave expansionconverges. It is proved that the scattering amplitude satisfies fixed-$t$dispersion relations when $|t|$ lies within the Lehmann-Martin ellipse.

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