One-loop self-energy and Ward-consistent dynamical response

Derive the full frequency-dependent one-loop self-energy and the corresponding Ward-consistent one-loop dynamical response for the weakly broken dipole-conserving lattice model.

Background

The paper develops a Gaussian theory for topological pseudo-Goldstone modes arising from weakly broken dipole conservation. Its analysis establishes the quadratic Bogoliubov spectrum, mode topology, tensor-source response, finite-resolution spectroscopy, and lowest-order decay thresholds within that approximation.

The authors explicitly identify the full frequency-dependent one-loop self-energy and the associated Ward-consistent dynamical response as unresolved. Solving this problem would extend the Gaussian treatment to interaction-induced frequency dependence and ensure that the resulting response functions remain consistent with the torque/Ward identities of the explicitly nonconserving system.