Relativistic Corrections for High-Energy p−p Scattering

It is shown that the treatment of the collision of two charged particles by means of a first-order Born approximation and M\o{}ller's matrix element involves an inconsistency connected with the infinite cross section for small angle scattering. It is then shown that an energy formula derived for the two-body interaction by means of an early form of the Heisenberg-Pauli quantum electrodynamics makes it possible to construct a relativistic two-body extension of the nonrelativistic one-body Mott-Gordon solution. This extension is good only to order ${e}^{2}$ but arguments are given for believing that the angle-dependent and ${e}^{2}$-containing factors are partially correct for the more important terms. The Gordon sphere construction naturally leads to such factors and the consideration of small angle collisions in the laboratory system leads to a similar result. The latter suggests the possible existence of correction terms. The explicit superposition of partial waves is avoided by noting a formal similarity of the relativistic and nonrelativistic problems for principal non-spin-dependent terms. Contributions of the spin-dependent terms are worked out, also avoiding explicit summation by employing a momentum space representation and noting that once the main terms are taken care of by the Gordon sphere construction, the spin-dependent terms can be treated as a perturbation on account of their more rapid fall-off with distance.The possibility of dealing with first-order phase shifts by means of a phase shift matrix is discussed in connection with Eqs. (17)-(17.2\ensuremath{'}) and the coupling of states with different $L$ but the same $J$ in relation with Eq. (9.8) and Eqs. (17.1)-(17.6). The definition of phase shifts in the relativistic problem, neglecting meson production, is discussed in connection with Eq. (8).