The mass eigenvalue problem for baryons in the nonlinear spinor theory

Abstract

In the past the determination of baryon masses in the framework of the nonlinear spinor theory of elementary particles was only carried out with a certain truncated form of the coupled equations of the lowest, nontrivial New-Tamm-Dancoff approximation, in which the mass eigenvalue determination reduced to a purely algebraical problem, limited to the spin-1/2 isospin-1/2 baryon case. In the present paper the nontruncated NTD-equations are studied, which have the form of a coupled system of integral equations for 1- and 3-point baryon wave functions. With respect to the earlier, simplified treatment there arise two complications: Firstly, a purely algebraical complication, connected with the cumbersome, relativistic angular momentum decomposition of the wave functions and the corresponding integral kernels, and secondly an analytical complication, connected with the solution of integral equations instead of algebraical equations. The angular-momentum decomposition of the equations can be carried out exactly by making use of graphical techniques presented in a separate paper (DW III). The kernels of the resulting integral equations for the partial-wave amplitudes are in general 16×16 matrices with respect to the spin-isospin variables and can be expressed compactly by a certain combination of Wigner's 15j-symbols involving spin, and 6j-symbols involving isospin. Regarding their analytical structure they reveal themselves as kernels arising from an interaction of π, η, ρ, ω bosons (being established in the same approximation asS-wave bound states of the “nucleon-antinucleon” system) with a “nucleon” in various angular momentum states. The interaction contains spin-flip and nonspin-flip normal and exchange type terms and spin-orbit coupling terms. In an approximation in which the contributions of the virtual spin 1 bosons are neglected, the integral equations become much simpler, and have roughly the structure of a Bethe-Salpeter equation with (forl>0) a pure masszero exchange force of the exchange type. The structure of these equations are investigated with respect to their isospin and angular momentum dependence. Since the solutions of a Bethe-Salpeter exchange equation (in contrast to the nonexchange type) with unequal masses of the constituents to our knowledge are not known as yet (for equal masses of the constituents they can be reduced to the Wick-Cutkosky nonexchange case) at present only qualitative conclusions can be drawn regarding the baryon spectrum. A simple study of the signs of the various interaction terms seems to indicate that the structure of the equation is rich enough to produce, in principle, all essential features reflected in the empirical mass spectrum of strangeness zero baryon resonances. Hence a more detailed investigation of our integral equation seems very promising.