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We show that it is possible to construct $\epsilon^\prime/\epsilon$ to NLOusing partially quenched chiral perturbation theory (PQChPT) from amplitudesthat are computable on the lattice. We demonstrate that none of the neededamplitudes require three-momentum on the lattice for either the full theory orthe partially quenched theory; non-degenerate quark masses suffice.Furthermore, we find that the electro-weak penguin ($\Delta I=3/2$ and 1/2)contributions to $\epsilon^\prime/\epsilon$ in PQChPT can be determined to NLOusing only degenerate ($m_K=m_\pi$) $K\to\pi$ computations without momentuminsertion. Issues pertaining to power divergent contributions, originating frommixing with lower dimensional operators, are addressed. Direct calculations of$K\to\pi\pi$ at unphysical kinematics are plagued with enhanced finite volumeeffects in the (partially) quenched theory, but in simulations when the seaquark mass is equal to the up and down quark mass the enhanced finite volumeeffects vanish to NLO in PQChPT. In embedding the QCD penguin left-rightoperator onto PQChPT an ambiguity arises, as first emphasized by Golterman andPallante. With one version (the "PQS") of the QCD penguin, the inputs neededfrom the lattice for constructing $K\to\pi\pi$ at NLO in PQChPT coincide withthose needed for the full theory. Explicit expressions for the finitelogarithms emerging from our NLO analysis to the above amplitudes are alsogiven.Comment: 54 pages, 3 figures; Important revisions: Corrections to formulas for  K->pi pi with degenerate quark masses have been mad

physical review. d. particles, fields, gravitation, and cosmology/physical review. d, particles, fields, gravitation, and cosmology

Lattice extraction of<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>K</mml:mi><mml:mo>→</mml:mo><mml:mi>π</mml:mi><mml:mi>π</mml:mi></mml:math>amplitudes to next-to-leading order in partially quenched and in full chiral perturbation theory

Lattice extraction ofK→ππamplitudes to next-to-leading order in partially quenched and in full chiral perturbation theory