Enumerating higher-dimensional operators with on-shell amplitudes

TL;DR

The paper develops a massless on-shell amplitude framework to systematically enumerate EFT operators generating given helicity amplitudes, up to dimension eight. It derives a minimal operator-dimension formula, solves little-group constraints to obtain minimal spinor structures, and uses momentum conservation and Schouten identities to achieve independent structure bases. It then dresses these bases with momentum dot products to reach higher dimensions, applying the method to GRSMEFT and SMEFT and enforcing gauge symmetry and statistics to produce complete operator sets. The approach avoids field-redefinition redundancies and provides explicit spinor-structure enumerations, yielding practical operator bases for gravitational and SM interactions in EFTs. This has direct implications for organizing GRSMEFT/SMEFT operator bases and guiding phenomenological analyses of beyond-Standard-Model effects.

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.