Unitarity of the box diagram

Roji Pius,Ashoke Sen

doi:10.1007/jhep11(2018)094

Abstract

The complete proof of cutting rules needed for proving perturbative unitarity of quantum field theories usually employs the largest time equation or old fashioned perturbation theory. None of these can be generalized to string field theory that has non-local vertices. In arXiv:1604.01783 we gave a proof of cutting rules in string field theory, which also provides an alternative proof of cutting rules in ordinary quantum field theories. In this note we illustrate how this works for the box diagram of ϕ4 field theory, avoiding the contributions from anomalous thresholds.

Highlights

JHEP11(2018)094 box diagram for which there are many derivations
In this note we illustrate how this works for the box diagram of φ4 field theory, avoiding the contributions from anomalous thresholds
The singularities of a Feynman diagram are associated with Landau singularities where the integrand has poles due to certain number of internal propagators going on-shell and the integration contour over loop momenta are pinched, i.e. it is not possible to move away from the poles by deforming the integration contour in the complex loop momentum plane

Summary

The issues

We shall briefly discuss the issues that plague the proof of unitarity directly in momentum space. (We are assuming that the incoming particles come from the left and the outgoing particles move to the right.) Branch points associated with such singularities are known as normal thresholds In this case the discontinuity computed from Cutkosky’s formula can be regarded as a product of two on-shell amplitudes, integrated over the phase space of the intermediate states. In computing −i T † T we need to reverse the signs of i in the propagators of the amplitude to the right of the cut so that it represents a matrix element of T † This does not follow from Cutkosky’s formula for discontinuity. This translates to the cutting rules which tell us that D is given by the sum over all cuts of the box diagram, with the following rules for evaluating a cut diagram: 1. A cut must divide the diagram into the left half and the right half, with the convention that the incoming particles come from the left and the outgoing particles travel to the right

A cut propagator corresponds to the replacement:

Unitarity of tree diagrams