Framework for improved lattice calculations of epsilion'/epsilon

In this thesis we show that it is possible to construct epsilon '/epsilon to NLO using both full and partially quenched chiral perturbation theory (PQChPT) from amplitudes that are computable using numerical lattice gauge theory. We find that the electro-weak penguin (Delta I = 3/2 and 1/2) contributions to epsilon'/epsilon in PQChPT can be determined to NLO using only degenerate (mK = mpi) K → pi computations without momentum insertion. All one-loop formulas needed to extract the necessary NLO constants from the lattice are presented in this work. Issues pertaining to power divergent contributions, originating from mixing with lower dimensional operators in a lattice regularization, are addressed.; In embedding the QCD penguin left-right operator onto PQChPT an ambiguity arises when the number of light sea quarks is not the physical value of three, as first emphasized by Golterman and Pallante. In the quenched theory they have pointed out that there are additional effective operators that appear in the quenched chiral perturbation theory needed to make contact with K → pipi amplitudes at physical kinematics. They have also proposed a method for determining the leading order low-energy constant, aNSq , associated with the new operators. We show that their method has difficulties due to power divergent contributions and propose a new method to obtain this constant from the lattice which does not suffer from this problem. Using this alternative method, we obtain aNSq , and show that our value implies a large ambiguity in the quenched contribution of Q6 to epsilon'/epsilon.