Analyticity Properties and Asymptotic Behavior of Scattering Amplitude in Higher Dimensional Theories

TL;DR

This paper investigates the analyticity and asymptotic behavior of scattering amplitudes in higher-dimensional field theories, extending classical results and deriving bounds on cross sections using axiomatic and unitarity principles.

Contribution

It generalizes the Lehmann-Jost-Dyson representation and Martin's theorem to higher dimensions, establishing analyticity domains and dispersion relations for D-dimensional theories.

Findings

Derived the higher-dimensional analog of the Lehmann ellipse.

Proved the existence of a finite subtracted dispersion relation in higher dimensions.

Established bounds on total cross sections from axiomatic principles.

Abstract

The properties of the high energy behavior of the scattering amplitude of massive, neutral and spinless particles in higher dimensional field theories are investigated. The axiomatic formulation of Lehmann, Symanzik and Zimmermann is adopted. The analyticity properties of the causal, the retarded and the advanced functions associated with the four point elastic amplitudes are studied. The analog of the Lehmann-Jost-Dyson representation is obtained in higher dimensional field theories. The generalized J-L-D representation is utilized to derive the t-plane analyticity property of the amplitude. The existence of an ellipse analogous to the Lehmann ellipse is demonstrated. Thus a fixed-t dispersion relation can be written down with finite number of subtractions due to the temperedness of the amplitudes. The domain of analyticity of scattering amplitude in $s$ and $t$ variables is extended by imposing unitarity constraints. A generalized version of Martin's theorem is derived to prove the existence of such a domain in D-dimensional field theories. It is shown that the amplitude can be expanded in a power series in t which converges for |t|<R; R being s-independent. The positivity properties of absorptive amplitudes are derived to prove the $t$-plane analyticity of amplitude. In the extended analyticity domain dispersion relations are written with two subtractions. The bound on the total cross section is derived from LSZ axioms without any extra ad hoc assumptions.