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 $a\!\to\!0$. The current--potential functional, by contrast, produces a finite, mode--dependent energy shift for $l\!\ge\!1$ 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.