Implications of Analyticity to Mass Gap, Color Confinement and Infrared Fixed Point in Yang–Mills theory

Analyticity of gluon and Faddeev–Popov ghost propagators and their form factors on the complex momentum-squared plane is exploited to continue analytically the ultraviolet asymptotic form calculable by perturbation theory into the infrared non-perturbative solution. We require the non-perturbative multiplicative renormalizability to write down the renormalization group equation. These requirements enable one to settle the value of the exponent characterizing the infrared asymptotic solution with power behavior which was originally predicted by Gribov and has recently been found as approximate solutions of the coupled truncated Schwinger–Dyson equations. For this purpose, we have obtained all the possible superconvergence relations for the propagators and form factors in both the generalized Lorentz gauge and the modified Maximal Abelian gauge. We show that the transverse gluon propagators are suppressed in the infrared region to be of the massive type irrespective of the gauge parameter, in agreement with the recent result of numerical simulations on a lattice. However, this method alone is not sufficient to specify some of the ghost propagators which play the crucial role in color confinement. Combining the above result with the renormalization group equation again, we find an infrared enhanced asymptotic solution for the ghost propagator. The coupled solutions fulfill the color confinement criterion due to Kugo and Ojima and also Nishijima, at least, in the Lorentz–Landau gauge. We also point out that the solution in compatible with color confinement leads to the existence of the infrared fixed point in pure Yang–Mills theory without dynamical quarks. Finally, the Maximal Abelian gauge is also examined in connection with quark confinement.