On the colour confinement and the minimal surface

Abstract

In the analysis of the energy of the four-quark system obtained in the SU(2) lattice Monte Carlo, the f-model in which the transition potential is expressed in the form $f=f_c exp(-k_A b_s A-k_P\sqrt{b_s}P)$, where A is the area and P is the perimeter of the Wilson loop, was successful in the simple configurations of the four quarks. In the case of tetrahedral geometry, an estimation of the minimal surface whose contours run the positions of the four quarks is necessary. We show that the regular surface approximation whose area can be calculated analytically, is a good approximation for evaluating the minimal surface. The numerical value of the coefficient $k_A b_s$ is close to $2 fm^{-2}$ which is the density of the $Z_2$ vortex in the SU(2) lattice Monte Carlo.