Collective Modes in Weyl Superconductors and the Axial Anomaly

Abstract

We develop a covariant Lagrangian formulation for a time reversal symmetry broken, minimally relativistic three dimensional Weyl superconductor that preserves continuous chiral invariance in the chiral limit. Within this framework, the FFLO pairing spontaneously breaks the axial $U(1)_A$ symmetry, giving rise to distinct collective excitations. Using the Nambu Jona Lasinio approach, we identify a pseudo scalar Nambu Goldstone mode (This mode is absent for conventional BCS internode pairing) that acquires a small mass under explicit chiral symmetry breaking and couples to gauge fields through the axial anomaly, closely analogous to the neutral pion decay in QCD. Although this anomaly induced decay is strongly suppressed in the bulk by the Meissner effect, it may occur via surface electromagnetic fields. Our analysis also predicts additional vector and axial vector collective modes associated with the broken $U(1)_A$ symmetry, providing a unified description of collective excitations in Weyl superconductors and their correspondence to mesonic modes in QCD.