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

TL;DR

This paper generalizes the IAV sum rule formalism for inclusive breakup reactions to include non-spectator fragments, accounting for their internal degrees of freedom and providing a more accurate theoretical framework.

Contribution

It introduces a formal generalization of the IAV sum rule that removes the spectator approximation, incorporating internal states of the detected fragment in inclusive cross section calculations.

Findings

Standard IAV corresponds to total cross section over all internal states.

Non-spectator effects are significant at the nuclear surface.

Framework establishes operators for future numerical evaluation.

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.