Inclusive breakup reactions

Inclusive breakup reactions of the type $d+A\to p+\mathrm{anything}$ require sums over excited states of the unobserved $n+A$ system. We show that the summed distorted-wave Born approximation cross section can be expressed in closed form as an elastic distorted-waves expectation of an optical model propagator for the unobserved system. The contribution of fluctuations to the energy-averaged cross section is automatically contained in the derived identity. Previous results can be recovered from this identity if corresponding surface approximations are introduced. A new approximation simplifies the identity by omitting the imaginary part of the optical potential. This approximation selects the energy-shell part of the propagator and reduces the inclusive cross section to the same form obtained for the ground state reaction $A(d,pn)A$, with a modified (real) optical potential for the unobserved neutron.