Enumerating higher-dimensional operators with on-shell amplitudes

TL;DR

This paper presents a systematic method for enumerating higher-dimensional operators in effective field theories using on-shell amplitude techniques, simplifying the classification of operators up to dimension eight.

Contribution

It introduces a simple formula for minimal operator dimension and an algorithmic approach to enumerate independent operators via on-shell amplitudes, including reduction procedures.

Findings

Systematic enumeration of operators up to dimension eight.

Complete set of spinor structures for various spins.

Application to GRSMEFT and standard-model operators.

Abstract

We establish a simple formula for the minimal dimension of operators leading to any helicity amplitude. It eases the systematic enumeration of independent operators from the construction of massless non-factorizable on-shell amplitudes. Little-group constraints can then be solved algorithmically for each helicity configuration to extract a complete set of spinor structures with lowest dimension. Occasionally, further reduction using momentum conservation, on-shell conditions and Schouten identities is required. A systematic procedure to account for the latter is presented. Dressing spinor structures with dot products of momenta finally yields the independent Lorentz structures for each helicity amplitude. We apply these procedures to amplitudes involving particles of spins 0,1/2,1,2. Spin statistics and elementary selection rules due to gauge symmetry lead to an enumeration of operators involving gravitons and standard-model particles, in the effective field theory denoted GRSMEFT. We also list the independent spinor structures generated by operators involving standard-model particles only. In both cases, we cover operators of dimension up to eight.