Dipole ghosts and spontaneous symmetry breaking in higher-order Chern-Simons theory

In this work, we investigate the emergence of dipole ghost structures in gauge theories at both the classical and quantum levels. Traditionally, dipole ghosts are introduced through an auxiliary field $\chi$ satisfying ${\mathrm{\square }}^2\chi =0$, whose presence is reflected in the appearance of double poles in the propagator. We show that such dipole ghost sectors can instead be understood as a consequence of the multiplicity of solutions of the classical equations of motion. To establish this connection, we develop a constructive method and first apply it to covariant Maxwell theory in $(2+1)$ dimensions, where the essential ingredients can be identified in a transparent manner. We then extend the analysis to the constrained higher-order Chern-Simons theory, demonstrating that the same mechanism gives rise to dipole ghost sectors and associated double poles in the propagator. Our results provide a unified perspective on the origin of dipole ghosts, relating them directly to the degeneracy structure of the underlying classical dynamics. As a physical application, we investigate spontaneous symmetry breaking and compute the one-loop effective potential. We show that symmetry breaking removes the degeneracy underlying the dipole ghost sector, leading to a spectrum of ordinary massive gauge excitations that determine the quantum vacuum structure of the theory.