We determine the phase diagram and chiral condensate for lattice QCD with two flavors of twisted-mass fermions in the presence of nondegenerate up and down quarks, discretization errors and a nonzero value of $Theta_{rm QCD}$. Although such a theory
has a complex action and cannot, at present, be simulated, the results are needed to understand how to tune to maximal twist in the presence of electromagnetism, a topic discussed in a companion paper. We find that, in general, the only phase structure is a first-order transition of finite length. Pion masses are nonvanishing throughout the phase plane except at the endpoints of the first-order line. Only for extremal values of the twist angle and $Theta_{rm QCD}$ ($omega=0$ or $pi/2$ and $Theta_{rm QCD}=0$ or $pi$) are there second-order transitions.
In this talk we determine the phase diagram and pion spectrum for Wilson and twisted-mass fermions in the presence of non-degeneracy between the up and down quark and discretization errors. We find that the CP-violating phase of the continuum theory,
which occurs for sufficiently large non-degeneracy, is continuously connected to the Aoki phase found in the lattice theory with degenerate quarks. Both for the Aoki and first-order scenarios, this results in a critical surface along which at least one of the pions is massless. In the pion spectrum, we focus mainly on the untwisted case, where there is competition between the effects of non-degeneracy and discretization errors. A more extensive analysis can be found in our recent paper [11].
We apply non-perturbative renormalization to bilinears composed of improved staggered fermions. We explain how to generalize the method to staggered fermions in a way which is consistent with the lattice symmetries, and introduce a new type of lattic
e bilinear which transforms covariantly and avoids mixing. We derive the consequences of lattice symmetries for the propagator and vertices. We implement the method numerically for hypercubic-smeared (HYP) and asqtad valence fermion actions, using lattices with asqtad sea quarks generated by the MILC collaboration. We compare the non-perturbative results so obtained to those from perturbation theory, using both scale-independent ratios of bilinears (of which we calculate 26), and the scale-dependent bilinears themselves. Overall, we find that one-loop perturbation theory provides a successful description of the results for HYP-fermions if we allow for a truncation error of roughly the size of the square of the one-loop term (for ratios) or of size O(1) times alpha^2 (for the bilinears themselves). Perturbation theory is, however, less successful at describing the non-perturbative asqtad results.
