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Journal Article

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


Maharana,  Jnanadeva
AEI-Golm, MPI for Gravitational Physics, Max Planck Society;

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Maharana, J. (2020). Proof of dispersion relations for the amplitude in theories with a compactified space dimension. Journal of High Energy Physics, 2020(6): 139. doi:10.1007/JHEP06(2020)139.

Cite as: https://hdl.handle.net/21.11116/0000-0006-A026-A
The analyticity properties of the scattering amplitude in the nonforward
direction 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 considered
in $D = 5$ Minkowski space to start with. Subsequently, one spatial dimension
is 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 theory
and (b) 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. It is proved that the scattering amplitude satisfies fixed-$t$
dispersion relations when $|t|$ lies within the Lehmann-Martin ellipse.