Unitarity of Minkowski nonlocal theories made explicit

In this work we explicitly show that the perturbative unitarity of analytic infinite derivative (AID) scalar field theories can be achieved using a modified prescription for computing scattering amplitudes. The crux new is continuation result obtained in Euclidean signature to Minkowski external momenta. We intensively elaborate an example non-local $\phi^4$ model various operators. General UV properties amplitudes are discussed.