Topics in lattice gauge theory: Staggered chiral perturbation theory and lattice gauge fixing

Quantum Chromodynamics (QCD) is the presently accepted theory of the strong interactions between quarks and gluons. Formulating QCD on a discrete spacetime lattice allows for a non-perturbatioe study of the theory, but gives rise to several complications. Quarks must be simulated at large, unphysical quark masses since the computational cost increases rapidly as we decrease the quark mass. We can use chiral perturbation theory (chiPT) to extrapolate to the physical quark masses, although there are discretization errors that arise which must be taken into account. We will formulate chiPT taking into account a specific discretization of lattice fermions---staggered fermions---in order to systematically extrapolate simultaneously to the continuum and to physical quark masses. Additionally we will study the violation of spectral positivity in the gauge-fixed lattice gluon propagator. In the lattice Landau gauge, the effective gluon mass is observed to rise rather than fall with increasing distance, which violates spectral positivity. We trace this violation to the use of quenched scalar fields in the lattice gauge fixing process, and show that violation of spectral positivity is a general feature of quenching.