S-Matrix Equivalence Theorem Evasion and Dimensional Regularisation with the Canonical MHV Lagrangian

TL;DR

This paper explores how the canonical MHV Lagrangian's change of variables leads to contributions that evade the equivalence theorem, affecting scattering amplitude calculations, and introduces a dimensional regularisation scheme to handle these effects.

Contribution

It demonstrates the ET evasion in the canonical MHV Lagrangian and develops a D-dimensional regularisation method to accurately compute scattering amplitudes.

Findings

ET evasion contributes to tree-level and one-loop amplitudes.

Dimensional regularisation enables consistent quantum MHV calculations.

Rearrangement of light-cone Yang-Mills contributions matches ET-evading terms.

Abstract

We demonstrate that the canonical change of variables that yields the MHV lagrangian, also provides contributions to scattering amplitudes that evade the equivalence theorem. This `ET evasion' in particular provides the tree-level (-++) amplitude, which is non-vanishing off shell, or on shell with complex momenta or in (2,2) signature, and is missing from the MHV (aka CSW) rules. At one loop there are ET-evading diagrammatic contributions to the amplitudes with all positive helicities. We supply the necessary regularisation in order to define these contributions (and quantum MHV methods in general) by starting from the light-cone Yang-Mills lagrangian in D dimensions and making a canonical change of variables for all D-2 transverse degrees of freedom of the gauge field. In this way, we obtain dimensionally regularised three- and four-point MHV amplitudes. Returning to the one-loop (++++) amplitude, we demonstrate that its quadruple cut coincides with the known result, and show how the original light-cone Yang-Mills contributions can in fact be algebraically recovered from the ET-evading contributions. We conclude that the canonical MHV lagrangian, supplemented with the extra terms brought to correlation functions by the non-linear field transformation, provide contributions which are just a rearrangement of those from light-cone Yang-Mills and thus coincide with them both on and off shell.