Analyticity Properties Of Scattering Amplitude In Theories With Compactified Space Dimensions: The Proof Of Dispersion Relations

The analyticity properties of the scattering amplitude for a massive scalar field are reviewed in this paper where the space-time geometry is R-3,R-1 circle times S-1, i.e. one spatial dimension is compact. Khuri investigated the analyticity of scattering amplitude in a nonrelativistic 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 behavior 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 space-time in five dimensions. The Kaluza-Klein (KK) compactification is implemented on a circle. The resulting four-dimensional manifold is R-3,R-1 circle times 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.