Large-N_f behaviour of the Yukawa model: analytic results

We investigate the Yukawa model in which N_f fermions are coupled with a scalar field through a Yukawa interaction. The phase diagram is rather well understood. If the fermions are massless, there is a chiral transition at T_c: for T < T_c chiral symmetry is spontaneously broken. At N_f=infty the transition is mean-field like, while, for any finite N_f, standard arguments predict Ising behavior. This apparent contradiction has been explained by Kogut et al., who showed by scaling arguments and Monte Carlo simulations that in the large-N_f limit the width of the Ising critical region scales as a power of 1/N_f, so that only mean-field behavior is observed for N_f strictly equal to infinity. We will show how the results of Kogut et al. can be recovered analytically in the framework of a generalized 1/N_f expansion. The method we use is a simple generalization of the method we have recently applied to a two-dimensional generalized Heisenberg model.