Pionic Complex Level Shifts and Optical Potential Parameters in Terms of the π-NScattering Length

Abstract

By a certain numerical method, independent of Ericson's procedure/'· 2' to solve a complex eigenvalue problem, the optical potential parameters8' in terms of the 1r-N scattering length are determined through the pionic complex level shift for light nuclei. We start from the Klein-Gordon equation of a pion bound by a nucleus, in the unit h=c=1, f7 2</J+ (EVc)2</J-p2</J=2pv,</J, (1) where E is a complex eigenvalue, Vc the Coulomb potential including nuclear finite extension, V, the optical potential in terms of the pion-nucleon interaction, and f.l the reduced pion mass. For solving Eq. (1) it is necessary to give the nuclear charge density included in the potential Vc. It can be obtained from elastic electron scattering and muonic atoms with quantumelectrodynamical corrections. Since the pionic complex level shifts of the s state depend strongly on the s-wave 11:-N interaction parameters b0 and B0,4'•5' we solve the Klein-Gordon equation by regarding these as free parameters, although these are ones in the sense that the Fermi motion and nucleon correlation are mainly included, however, both of which almost cancel each other. From the analysis of the reaction 7r+ +d~p+p at low energy, the magnitude of the real part of B0 is suggested to be nearly twice that of the imaginary part with opposite sign.6' So we assume Re B0 = -2 Im B0 (a) instead of Re B0 = Im B0 (b) by Ericson, 3' and further take Re B0=0(c) as a limiting case. Assuming Re B0 =-2 Im B0 and the additivity rule for nuclei, the·set of the bestfit effective parameters b0 and B0 obtained from bound pionic states for light nuclei are given in Table I, where the comparison is made with those obtained from the scattering or reaction process. The probability density of the pionic 1s state for 12C is shown in Fig. 1. In addition to the above cases, we illustrate the