Unitarity and Cutkosky Rules in Nonlocal Quantum Field Theory

In this section we discuss the unitarity of nonlocal quantum field theories. For simplicity, we focus our considerations on the case of a nonlocal scalar field, which encodes all the features of the nonlocality. We show that, for a specific class of nonlocal form factors, one can give a prescription for defining the scattering amplitudes, such that the theory is unitary. This prescription consists in defining the amplitudes in Euclidean signature, assuming that all the external and loop energies are purely imaginary. Then, physics is recovered by analytic continuation of such amplitudes to real and positive values of the external energies. We prove that the imaginary part of the analytically continued scattering amplitudes is given by the Cutkosky rules, which imply the unitarity of the theory. We conclude by showing that, if the nonlocal theory is defined directly in Minkowskian signature, assuming real external and loop energies, unitary is flawed.