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

Abstract

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. We 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. 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 NLO using only degenerate ($m_K=m_{\pi }$) $K\to \pi$ computations without momentum insertion. Issues pertaining to power divergent contributions, originating from mixing with lower dimensional operators, are addressed. Direct calculations of $K\to \pi \pi$ at unphysical kinematics 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. 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.

arXiv comment54 pages, 3 figures; Important revisions: Corrections to formulas for K->pi pi with degenerate quark masses have been made

References in corpus (40)

• Weak Decays Beyond Leading Logarithms
• Observation of Direct CP Violation in K_{S,L} to ππDecays
• Weak transition matrix elements from finite-volume correlation functions
• Chiral Perturbation Theory for the Quenched Approximation of QCD
• A new measurement of direct CP violation in two pion decays of the neutral kaon
• Partially Quenched Gauge Theories and a Application to Staggered Fermions
• A precision measurement of direct CP violation in the decay of neutral kaons into two pions
• Physical Results from Unphysical Simulations
• Measurements of Direct CP Violation, CPT Symmetry, and Other Parameters in the Neutral Kaon System
• Enhanced chiral logarithms in partially quenched QCD
• Kaon Matrix Elements and CP-violation from Quenched Lattice QCD: (I) the 3-flavor case
• An Upgraded Analysis of $\epsilon \prime /\epsilon$ at the Next-to-Leading Order
• Partially Quenched Chiral Perturbation Theory and the Replica Method
• The Standard Model Prediction for epsilon'/epsilon
• New lattice approaches to the $\Delta I=1/2$ rule
• Calculation of Non-Leptonic Kaon Decay Amplitudes from $K\to \pi$ Matrix Elements in Quenched Domain-Wall QCD
• Effects of Quenching and Partial Quenching on Penguin Matrix Elements
• Finite-volume two-pion energies and scattering in the quenched approximation
• Obtaining K to pi pi from Off-Shell K to pi Amplitudes
• Quenched Chiral Perturbation Theory to one loop
• Analysis of O(p^2) Corrections to <pi pi|Q_{7,8}|K>
• K^+ -> pi^+ pi^0 Decays on Finite Volumes and at Next-to-Leading Order in the Chiral Expansion
• On the Determination of Nonleptonic Kaon Decays from $K\to \pi$ Matrix Elements
• On lattice extraction of K --> pi pi amplitudes to O(p^4) in Chiral Perturbation Theory
• K \to ππdecay amplitudes from the lattice
• Dispersion relations and soft pion theorems for K -> pi pi
• Analysis of the DeltaI=1/2 Rule and e'/e with Overlap Fermions
• Finite-Volume Two-Pion Amplitudes in the I=0 Channel
• Matrix elements relevant for Delta I=1/2 rule and epsilon-prime from Lattice QCD with staggered fermions
• Finite-Volume Partially-Quenched Two-Pion Amplitudes in the I=0 Channel
• On the effects of (partial) quenching on penguin contributions to K-> pi pi
• A joint analysis of the S-wave in the pi+pi- and pi0pi0 data
• Comment on `Analysis of O(p^2) Corrections to <2pi|Q_{7,8}|K>'
• $K\to \pi \pi$ matrix elements beyond the leading-order chiral expansion
• Weak matrix elements for CP violation
• Finite Volume Effects in Weak Hadronic Decays
• Lattice QCD with dynamical domain wall quarks
• Chiral perturbation theory, dispersion relations and final state interactions in K -> pi pi
• Analytic estimates of quenched penguins
• A note on the power divergence in lattice calculations of $\Delta I=1/2$ $K\to \pi \pi$ amplitudes at $M_K=M_{\pi }$

Cited by in corpus (15)

• Multichannel one-to-two transition amplitudes in a finite volume
• Twisted mass QCD and lattice approaches to the $\Delta I=1/2$ rule
• A strategy to study the role of the charm quark in explaining the Delta{I}=1/2 rule
• On the foundations of partially quenched chiral perturbation theory
• Finite-Volume Partially-Quenched Two-Pion Amplitudes in the I=0 Channel
• Probing the chiral weak Hamiltonian at finite volumes
• Systematic effects of the quenched approximation on the strong penguin contribution to epsilon'/epsilon
• Quenched penguins and the Delta I=1/2 rule
• K to pi and K to 0 in 2+1 Flavor Partially Quenched Chiral Perturbation Theory
• $\Delta I=3/2$ and $\Delta I=1/2$ channels of $K\to \pi \pi$ decay at the physical point with periodic boundary conditions
• On-shell Delta I = 3/2 kaon weak matrix elements with non-zero total momentum
• Progress in kaon physics on the lattice
• Possible lattice approach to B to D pi (K) matrix elements
• A note on the power divergence in lattice calculations of $\Delta I=1/2$ $K\to \pi \pi$ amplitudes at $M_K=M_{\pi }$
• Non-degenerate quark mass effect on $B_K$ with a mixed action