Inclusive breakup reactions with non-spectator fragments: Generalization of the IAV sum rules

Abstract

The Ichimura-Austern-Vincent (IAV) sum rule formalism for inclusive breakup reactions $a + A \to b + \mathrm{anything}$ treats the detected fragment $b$ as a spectator by replacing its interaction with the target by an optical potential. This assumption becomes questionable when $b$ is a loosely bound composite particle such as a deuteron. I derive a generalization that removes the spectator approximation and retains $b$'s internal degrees of freedom, providing state-resolved inclusive cross sections. Within the DWBA, all non-spectator effects enter through the source function via the operator $V_{bA} - U_{bA}$. The exact sum rule involves the full $x + A$ resolvent $(E_{x,0}^+ - H_{xA})^{-1}$, while a single-channel IAV-like expression is recovered only when the explicit target dependence of $V_{bA}$ is neglected; post-prior equivalence is preserved in both cases. A key conceptual finding is that the standard IAV result for structureless $b$ corresponds, under closure, to the \emph{total} inclusive cross section summed over all of $b$'s internal states, rather than the cross section for $b$ in a specific state. An operator-level estimate for $b = d$ on ${}^{208}\mathrm{Pb}$ shows that the non-spectator correction is not a small perturbation at the nuclear surface. The present work is purely formal: it establishes the theoretical framework and identifies the relevant operators, while quantitative assessment of the cross-section impact awaits a full numerical evaluation of the source integrals.