On multi-particle functions and pion-decay constant in Nambu--Jona-Lasinio model

The system of second-order equations of mean-field expansion is considered for Nambu--Jona-Lasinio model with chiral symmetry of SU(2)-group. The system includes equations for four-particle and three-particle functions and also equations for next-to-leading two-particle function and next-to-next-to-leading quark propagator. Exact solutions for four-particle and three-particle equations are obtained. The solution for four-particle function is a disconnected combination of the leading-order two-particle functions. The connected part of three-particle function includes two-meson and three-meson contributions. The solution of three-particle equation permits to close the equation for next-to-leading two-particle function and to calculate a correction to pion-decay constant $f_\pi$. This correction increases from 14% to 28% when values of quark condensate varies from $-(210 MeV)^3$ to $-(250 MeV)^3$.