Vanishing of non-axisymmetric even-parity nonlinear Love number couplings

Determine whether all even-parity nonlinear Love number couplings for Schwarzschild black holes that are not fixed by axisymmetric matching—specifically, the couplings of worldline effective field theory operators built from the electric tidal tensors E_{i_L} whose contributions vanish in axisymmetric Einstein equations—also vanish when axisymmetry is relaxed, by analyzing static, parity-even perturbations without imposing axisymmetry and performing the corresponding matching beyond axisymmetric Weyl solutions.

Background

The paper matches axisymmetric Weyl solutions to the point-particle worldline effective field theory to define and compute nonlinear Love numbers for Schwarzschild black holes in general relativity. Using a carefully chosen operator basis, the authors prove that an infinite subset of even-parity Love number couplings vanish to all orders in perturbation theory by exploiting axisymmetric configurations.

However, some operator couplings—those arranged to give zero contribution in axisymmetric equations—remain unconstrained by this matching. The authors explicitly state they were not able to fix these remaining couplings under axisymmetry and point to the need to relax axisymmetry to determine whether they also vanish. This poses a concrete open question about the full, non-axisymmetric static tidal response in the even sector.