Partially quenched gauge theories and an application to staggered fermions

We extend our Lagrangian technique for chiral perturbation theory for quenched QCD to include theories in which only some of the quarks are quenched. We discuss the relationship between the partially quenched theory and a theory in which only the unquenched quarks are present. We also investigate the peculiar infrared divergences associated with the ${\eta }^{\prime }$ in the quenched approximation, and find the conditions under which such divergences can appear in a partially quenched theory. We then apply our results to staggered fermion QCD in which the square root of the fermion determinant is taken, using the observation that this should correspond to a theory with four quarks, two of which are quenched.