Investigation of baryon-like bound states of nonrelativistic quarks in the self-consistent field approximation

The $1/N$ expansion is used to investigate the nonrelativistic model of baryons proposed by Witten: $N$ quarks ($N\gg1$) of one flavor in the same spin state bound by two-particle attractive forces determined by a potential $V(r)$. The coordinate part of the wave function of the $N$ quarks forming the bound state is represented in the form $\psi(\mathbf{x}_1,\dots,\mathbf{x}_N)= \prod\limits_{i=1}^N\varphi(\mathbf{x}_i)$. The resulting integrodifferential spectral problem is solved by reduction to nonlinear differential equations of higher order. The following potentials are considered: 1) $V(r)=-g^2r^{-1}$, 2) $V(r)=g^2\alpha^2r$, 3) $V(r)=g^2(-r^{-1}+\alpha^2r)$. A computer was used to find the characteristics of the corresponding baryon-like bound states of $N$ quarks.