Calculations are of course done in the center-of-momentum system. If p_i and p_f are the initial and final COM momenta, with the corresponding energies being related by E_f = E_i + Q, then the momentum transfer involved in the reaction is clearly given by q² = p_i² + p_f² - 2p_ip_f cos θ, where θ is the center of momentum angle at which reaction product b is observed.  Of course nuclear reactions conserve total energy, total momentum, total angular momentum, charge, and baryon number.

The simplest possible analysis, dating from the early 1950s, takes the a + A and b + B channels to be described by plane waves, and takes the “stripping” to take place at a specific distance R from A. Then a remarkably simple (and successful!) result emerges, dσ/dΩ ∝ |j_ℓ(qR)|². Here j_ℓ(z) is the spherical Bessel function. This is the so-called Plane Wave Born Approximation for Direct Nuclear Reactions, and its unusual success, when digital computers became readily available,  prompted replacing the plane waves by numerical solutions to the Schrodinger equation with optical potentials for a + A and b + B, which then led to even greater success. This is called the Distorted Wave Born Approximation, DWBA.  Almost everything known about the energy levels of odd-A nuclei comes from such analyses of direct nuclear reactions!  [See G. R. Satchler, Reviews of Modern Physics, Vol. 50, No. 1, pp. 1 - 10 (1978).]

Before the late 1940s, it had been assumed that all nuclear reactions proceed by forming a “compound nucleus.” In other words, the incident particle is sucked into the nucleus, and the combined system reaches equilibrium. Because of the excitation energy, the compound system has a variety of channels through which it can decay, but the decay process is the decay of a system with no information about its formation. In such a case, the cross section is expected to be isotropic or symmetric about 90 degrees center-of-momentum angle, for obvious reasons. However, many processes showed a forward peak in the cross section, impossible if there were an equilibrium system decaying, since a compound system would have no memory of the projectile incident momentum direction. This was the birth of interest in “direct nuclear processes.”

The mistaken concept of nuclear reactions which completely misled physicists until the early 1950s.  The success of the optical model showed how wrong the idea was, in general.