Inclusive breakup of three-body projectiles: A unified four-body framework for pair-detected and single-particle observables

Abstract

Inclusive breakup of three-body projectiles $a=i+j+k$ on a target $A$ admits two distinct inclusive observables: detection of a correlated pair $b=(ij)$ with $k+A$ unresolved, and detection of a single particle $i$ with $jk+A$ unresolved. A four-body DWBA sum-rule framework is derived for both channels from a common Hamiltonian. For the pair-detected channel, the unresolved propagator remains the two-body $k+A$ Green function and all three-body projectile effects enter through a pair-projected source built from ${\Phi }_a$; a reference pair-target optical interaction splits this source into a target-elastic reference part and an explicit pair-target coupling part, yielding a state-resolved semi-inclusive coincidence observable and an amplitude-level diagnostic of the two-body cluster approximation. For the single-particle channel, the unresolved propagator is the three-body $jk+A$ resolvent, whose reference-channel Feshbach reduction reproduces the Carlson-Frederico-Hussein (CFH) absorptive kernel $W_j+W_k+W_{3B}$; the additional source $V_{iA}-U_{iA}$ drives target excitations, with its direct $g_Q$ component yielding target-excited CFH-like kernels under a diagonal-intermediate-states approximation. Prior forms are derived for both partitions, with reduced post-prior identities at the single-channel level (pair-detected) and at the CFH-optical level (single-particle). For ${}^6$Li$=\alpha +n+p$, the explicit deuteron-target coupling has an E1/E2/monopole tidal structure evaluated on the full three-body wave function. The framework is validated by recovery of the two-body IAV, CFH, and detected-cluster limits, and separates exact DWBA identities from later optical and diagonal-target approximations.