In this thesis we describe two studies concerting lattice quantum chromodynamics (LQCD): first, an analysis of the phase structure of Wilson and twisted-mass fermions with isospin breaking effects, second a computational study measuring non-perturbative Greens functions. We open with a brief overview of the formalism of QCD and LQCD, focusing on the aspects necessary for understanding how a lattice computation is performed and how discretization effects can be understood. Our work in Wilson and twisted-mass fermions investigates an increasingly relevant regime where lattice simulations are performed with quarks at or near their physical masses and both the mass difference of the up and down quarks and their dif- fering electric charges are included. Our computation of a non-perturbative Greens functions on the lattice serves as a first attempt to validate recent work by Dine et. al. [24] in which they calculate Greens functions which vanish in perturbation theory, yet have a contribution from the one instanton background. In chapter 2, 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 dis- cretization errors, using Wilson and twisted-mass chiral perturbation theory. 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 of the lattice theory with degen- erate quarks. We show that discretization effects can, in some cases, push simulations with physical masses closer to either the CP-violating phase or another phase not present in the continuum, so that at sufficiently large lattice spacings physical-point simulations could lie in one of these phases. In chapter 3, we extend the work in chapter 2 to include the effects of electromag- netism, so that it is applicable to recent simulations incorporating all sources of isospin breaking. For Wilson fermions, we find that the phase diagram is unaffected by the inclu- sion of electromagnetism--the only effect is to raise the charged pion masses. For maximally twisted fermions, we previously took the twist and isospin-breaking directions to be different, in order that the fermion determinant is real and positive. However, this is incompatible with electromagnetic gauge invariance, and so here we take the twist to be in the isospin-breaking direction, following the RM123 collaboration. We map out the phase diagram in this case, which has not previously been studied. The results differ from those obtained with different twist and isospin directions. One practical issue when including electromagnetism is that the critical masses for up and down quarks differ. We show that one of the criteria suggested to determine these critical masses does not work, and propose an alternative. In chapter 4, we delve deeper into the technical details of the analysis in chapter 3. 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]QCD. 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]QCD ([omega] = 0 or [pi]/2 and [Theta]QCD = 0 or [pi]) are there second-order transitions. In chapter 5 we move on to a new topic, working to make a first measurement of non- perturbative Greens functions which vanish in perturbation theory but have a non-vanishing one-instanton contribution, as suggested in recent work by Dine et. al. [24] using a semi- classical approach. This measurement was done using 163 × 48 configurations generated by the MILC collaboration, with coupling [Beta] = 6.572, light quark mass mla = 0.0097, strange quark mass msa = 0.0484, lattice spacing a ≈ 0.14 fm and pion mass m[pi]a = 0.2456. The analysis was done by separating the Green function of interest into pseudoscalar and scalar components. These are separately calculated on 440 configurations, using the Chroma software package. To improve statistics, we used the various reduction technique suggested in Ref. [13]. We subtracted out the long distance contributions from the pion, excited pion and a0 from the Green function, in the hope of obtaining the short distance form predicted by Ref. [24]. Unfortunately, after subtraction of the a0 and pion states only noise remained. While the results are not in themselves useful, we believe this approach will be worth repeating in the future with finer lattices with a fermion action with better chiral symmetry.

This book consists of a series of lectures to cover every facet of the modern version of lattice QCD. All the lectures are self-contained starting with the necessary background material and ending up with the latest development. Most of the lectures are given by pioneers in the field.This book may be useful as an advanced textbook for graduate students in particle physics and its modern and fascinating contents will inspire the interest of the non-experts.