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

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2003.14330.pdf

(Preprint), 289KB

Maharana2020_Article_ProofOfDispersionRelationsForT.pdf

(Publisher version), 384KB

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##### Citation

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

##### Abstract

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.

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.