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Phase structure with nonzero $Θ_{\rm QCD}$ and twisted mass fermions

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 $Θ_{\rm QCD}$. 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 $Θ_{\rm QCD}$ ($ω=0$ or $π/2$ and $Θ_{\rm QCD}=0$ or $π$) are there second-order transitions.