Cutkosky rules and perturbative unitarity in Euclidean nonlocal quantum field theories

TL;DR

This paper proves the unitarity of Euclidean nonlocal scalar quantum field theories at all perturbative orders by demonstrating that their amplitudes satisfy Cutkosky rules and only include normal threshold contributions, with implications for gauge and gravity theories.

Contribution

It establishes the all-order perturbative unitarity of Euclidean nonlocal scalar field theories and extends the analysis to gauge and gravity theories using symmetry principles.

Findings

Amplitudes satisfy Cutkosky rules in Euclidean nonlocal theories.

Only normal threshold diagrams contribute to the imaginary part.

Unitarity is preserved in nonlocal gauge and gravity theories via symmetry constraints.

Abstract

We prove the unitarity of the Euclidean nonlocal scalar field theory to all perturbative orders in the loop expansion. The amplitudes in the Euclidean space are calculated assuming that all the particles have purely imaginary energies, and afterwards they are analytically continued to real energies. We show that such amplitudes satisfy the Cutkowsky rules and that only the cut diagrams corresponding to normal thresholds contribute to their imaginary part. This implies that the theory is unitary. This analysis is then exported to nonlocal gauge and gravity theories by means of Becchi-Rouet-Stora-Tyutin or diffeomorphism invariance, and Ward identities.