  We consider a model in which Goldstone bosons, described by a SU(N) chiral
nonlinear sigma model, are coupled to an N-plet of colored fermions by means of
a Yukawa interaction. We study the one-loop renormalization group flow and show
that the non-Gaussian UV fixed point, which is present in the purely bosonic
model, is lost because of fermion loop effects unless N is sufficiently large.
We then add four-fermion contact interactions to the lagrangian and show that
in this case there exist several non-Gaussian fixed points. The strength of the
contact interactions, predicted by the requirement that the theory flows
towards a fixed point in the UV, is compared to the current experimental
bounds. This toy model could provide an important building block of an
asymptotically safe model of the weak interactions.
