Calculation of nonleptonic kaon decay amplitudes fromK→πmatrix elements in quenched domain-wall QCD

Abstract

We explore application of the domain wall fermion formalism of lattice QCD to calculate the K → ππ decay amplitudes in terms of the K + → π + and K 0 → 0 hadronic matrix elements through relations derived in chiral perturbation theory. Numerical simulations are carried out in quenched QCD using domain-wall fermion action for quarks and an RG-improved gauge action for gluons on a 16 3 × 32 × 16 and 24 3 × 32 × 16 lattice at β = 2.6 corresponding to the lattice spacing 1/a ≈ 2 GeV. Quark loop contractions which appear in Penguin diagrams are calculated by the random noise method, and the ∆I = 1/2 matrix elements which require subtractions with the quark loop contractions are obtained with a statistical accuracy of about 10%. We investigate the chiral properties required of the K + → π + matrix elements. Matching the lattice matrix elements to those in the continuum at µ = 1/a using the perturbative renormalization factor to one loop order, and running to the scale µ = mc = 1.3 GeV with the renormalization group for N f = 3 flavors, we calculate all the matrix elements needed for the decay amplitudes.With these matrix elements, the ∆I = 3/2 decay amplitude ReA2 shows a good agreement with experiment after an extrapolation to the chiral limit. The ∆I = 1/2 amplitude ReA0, on the other hand, is about 50-60% of the experimental one even after chiral extrapolation. In view of the insufficient enhancement of the ∆I = 1/2 contribution, we employ the experimental values for the real parts of the decay amplitudes in our calculation of ε ′ /ε. The central values of our result indicate that the ∆I = 3/2 contribution is larger than the ∆I = 1/2 contribution so that ε ′ /ε is negative and has a magnitude of order 10 −4 . We discuss in detail possible systematic uncertainties, which seem too large for a definite conclusion on the value of ε ′ /ε.

Flavor physics in the quark sector

“…On the lattice, usually an indirect approach [889] is pursued, which measures K → π and K → vacuum operators instead and relates those via chiral perturbation to the wanted K → (π π ) operators. For quenched studies see [890,891], for recent work using dynamical domain-wall fermions see [892]. The latter work raises some doubts about the applicability of this approach, which is based on SU(3)-ChPT to be valid around the strange quark mass.…”

Matching factors forΔS=1four-quark operators in RI/SMOM schemes

“…We give results only for the chiral basis which does not contain linearly dependent operators. Since mixing only occurs within the blocks of the (27, 1), (8,1), and (8,8) operators, we renormalize each block separately.…”

“…3. The (27, 1) operator Q ′ 1 only mixes with itself, so we only need to provide a single projector P / q (27,1) . We adopt the RI/SMOM( / q, / q) and RI/SMOM( / q, γ µ ) schemes defined in Ref.…”

Scientific Challenges for Understanding the Quantum Universe

“…The first two difficulties have been studied extensively in calculations of the easier K to vacuum and K to pion matrix elements (Blum et al 2003, Noaki 2003. With sufficient statistics, these two obstacles are expected to contribute errors that can be reduced to the level of a few percent.…”