Transformations of nucleon-nucleon amplitudes from the center of mass frame to the proton-nucleus Breit frame are developed for use in multiple scattering analyses. As a by product, a matrix equation is given for the invariant NN amplitudes in Dirac spinor representation. Based on recent NN phase shifts, Gaussian parametrization of the pp and np amplitudes are calculated and some proton-nucleus calculations are presented to illustrate the effects of the Breit frame amplitudes. For~Ca, these effects are quite small. NUCLEAR REACTIONS Nucleon-nucleon amplitudes, impulse approximation, Breit frame, Wigner rotation, Dirac invariant amplitudes, proton-nucleus elastic scattering.Intermediate energy proton-nucleus elastic scattering is qualitatively explained by the impulse approximation, as has been shown in many analyses. ' In multiple scattering theory, the impulse approximation consists of the use of free nucleonnucleon (NN) scattering amplitudes without corrections for off-shell effects or for the influence of the nuclear medium. Gurvitz, Dedonder, and Amado * have defined an optimal impulse approximation in which leading order corrections due to Fermi motion are shown to vanish provided that the NN I amplitude commutes with the nuclear potential. This is expected to be a good approximation because the NN amplitudes which determine the single scattering optical potential do not involve the spin of nucleons in the nucleus and these amplitudes seem to be reasonably local (i.e. , they depend mainly on momentum transfer, q).Working in the proton-nucleus center of mass (c.m. ) system, the single scattering approximation to the proton-nucleus elastic scattering amplitude corresponds to the diagram of Fig. 1: F'"(k, k', TL, )= --g(2n. ) ' J d'pg, ' p+ -, q --k, 2&VI A X k, --qp+ -q --k, t TL -H k+ -, qp --q --k, p --, q --k, + where k, -:-, (k+ k') and q-:k -k' are the average of initial and final proton momenta and the momentum transfer, respectively. TL, is the laboratory proton energy, H represents the nuclear Hamiltonian, and vL is the laboratory velocity of the proton. The single-particle wave functions QJ depend on the struck nucleon momentum relative to the nuclear center of mass. In principle, off-shell XN t-matrix elements are needed to evaluate Eq. (1). However, as already noted, the important t-matrix elements seem reasonably local. Furthermore, the variation of the NN t matrix with momentum p in the integral of Eq. (1) is compensated to leading order in p/rn by appropriate choice of the energy parameter upon which the t matrix depends. This is the optimal impulse approximation of Refs. 3 and 4. The condition on the energy is that the NN t matrix be on shell when evaluated at p =0 in Fig. 1 or Eq. (1). We refer to the p =0 situation as the Breit frame [see Fig. 2(a)].

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We revisit the work of K. Goeke, M. Harvey, F. Gr\"ummer, and J. N. Urbano (Phys. Rev. {\bf D37}, 754 (1988)) who considered a chiral model for the nucleon based on the linear sigma model with scalar-isoscalar scalar-isovector mesons coupled to quarks and solved using the coherent-pair approximation. In this way the quantum pion field can be treated in a non-perturbative fashion. In this work we review this model and the coherent pair approximation correcting several errors in the earlier work. We minimize the expectation value of the chiral hamiltonian in the ansatz coherent-pair ground state configuration and solve the resulting equations for nucleon quantum numbers. We calculate the canonical set of nucleon observables and compare with the Hedgehog model and experiment. Using the corrected equations yield slightly different values for nucleon observables but do not correct the large virial deviation in the $\pi$-nucleon coupling. Our results therefore do not significantly alter the conclusions of Goeke, et al..Comment: RevTeX, 24 pages, 5 figure