Angular--Momentum--Resolved Aharonov--Bohm Coupling Energy

We present an angular–momentum–resolved energetic formulation of the Aharonov–Bohm (AB) response for a confined Dirac electron based on two gauge–invariant interaction functionals: a magnetization–field functional and a current–potential functional. Using exact Dirac eigenmodes in a cylindrical cavity threaded by a solenoidal flux, we show that the magnetization–field functional yields a core–localized interaction energy restricted to the $l=0$ channel, with all higher angular–momentum contributions suppressed and vanishing entirely in the limit $a\text{\hspace{0.17em}⁣}\to \text{\hspace{0.17em}⁣}0$. The current–potential functional, by contrast, produces a finite, mode–dependent energy shift for $l\text{\hspace{0.17em}⁣}\ge \text{\hspace{0.17em}⁣}1$ in the same limit, arising from a local interaction between the solenoidal vector potential and the spatially distributed Dirac current, and explicitly encoding the geometric and topological structure of the coupling energy.