Lattice extraction of $ K \to ππ$ amplitudes to NLO in partially quenched and in full chiral perturbation theory

We show that it is possible to construct $\epsilon^\prime/\epsilon$ to NLO using partially quenched chiral perturbation theory (PQChPT) from amplitudes that are computable on the lattice. All of the low energy constants needed to construct $\epsilon^\prime/\epsilon$ can be obtained when one uses $K\to\pi\pi$ computations at the two unphysical kinematics allowed by the Maiani-Testa theorem, along with the usual (computable) two- and three-point functions, namely $K\to0$, and $K\to\pi$ (with 4-momentum insertion). We also demonstrate that none of the needed amplitudes require three-momentum on the lattice for either the full theory or the partially quenched theory; non-degenerate quark masses suffice. Direct calculations of $K\to\pi\pi$ at unphysical amplitudes are plagued with enhanced finite volume effects in the (partially) quenched theory, but in simulations when the sea quark mass is equal to the up and down quark mass the enhanced finite volume effects vanish to NLO in PQChPT, when the number of sea quarks is $\geq 1$. In embedding the QCD penguin left-right operator onto PQChPT an ambiguity arises, as first emphasized by Golterman and Pallante. With one version (the "PQS") of the QCD penguin, the inputs needed from the lattice for constructing $K\to\pi\pi$ at NLO in PQChPT coincide with those needed for the full theory. Explicit expressions for the finite logarithms emerging from our NLO analysis to the above amplitudes are also given.