Scalar-fermion theories on the lattice

We study scalar-fermion models with Yukawa interaction on a space-time lat- tice. Such models can describe the Higgs sector of the Standard Model in the case when the Higgs particle is very heavy (few hundred GeV) and there are very heavy fermions whose masses are due to their Yukawa interactions with the Higgs field. We study a realistic model with four component scalar field as well as simplified models with one and two component scalar fields. We use a mean field approximation to calculate equations for critical lines in the large d (dimension of space-time) limit. These lines are in very good agreement with available Monte Carlo data for the models at d = 4. We calculate fermion correlation functions in the mean field and large d approximations to study properties of different phases in the lattice models. We find two distinct phases with vanishing expectation values of the scalar field. One (at small Yukawa coupling Y) contains massless fermions, while in the other (at large F) the fermions have masses larger than the scale given by the inverse lattice spacing. We find that in the latter phase fermions can form bosonic bound states. These states show up as poles in a four-fermion correlator. We discuss pos- sible continuum limits in the lattice scalar-fermion models. In particular, we show that a theory defined near the critical line separating the disordered phase from the phase with antiferromagnetic order is not unitary