Classification and counting of independent Love-number operator contractions in the EFT

Classify and count all inequivalent tensor contractions of multiple electric tidal tensors E_{i_L} in the point-particle worldline effective field theory for static tidal response of Schwarzschild black holes, and determine the corresponding independent Wilson coefficient couplings at each order in the number of fields and spatial derivatives using group-theoretic methods.

Background

In developing the worldline effective field theory for the static, parity-even tidal response, the authors organize operators built from the electric tidal tensors E_{i_L} and discuss how, starting at fourth order in the number of E fields, multiple inequivalent contractions can appear. While they construct a basis tailored to axisymmetric matching and prove vanishing of one operator per multiplet, they do not attempt a complete classification of all independent contractions and associated Wilson couplings.

They explicitly note that a systematic counting and classification can be obtained via group-theoretic methods but leave this task for future work. Completing this classification would clarify the full operator basis and the independent nonlinear Love number couplings beyond the subset fixed by axisymmetric matching.