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Following our recent publication on direct CP violation and the ΔI = 1/2 rule in K → ππ decay, which was made with G-parity boundary conditions, we revisit this problem with a conventional lattice setup employing periodic boundary conditions and two lattice spacings to check our previous result and to improve the precision. We show that the physical amplitude, which corresponds to an excited state in this case, can be obtained reliably with the Generalized Eigenvalue Problem (GEVP) method. Not only are periodic boundary conditions cheaper and allow the use of existing ensembles to take the continuum limit, but they provide a straightforward path to introduce electromagnetism and strong isospin symmetry breaking, which will be needed in the near future. In this article we show our preliminary results and discuss the prospect of the high-precision calculation of K → ππ decay with periodic boundary conditions.

We present a lattice QCD calculation of the ΔI=1/2, K→ππ decay amplitude A0 and ϵ′, the measure of direct CP violation in K→ππ decay, improving our 2015 calculation [1] of these quantities. Both calculations were performed with physical kinematics on a 323×64 lattice with an inverse lattice spacing of a-1=1.3784(68) GeV. However, the current calculation includes nearly 4 times the statistics and numerous technical improvements allowing us to more reliably isolate the ππ ground state and more accurately relate the lattice operators to those defined in the standard model. We find Re(A0)=2.99(0.32)(0.59)×10-7 GeV and Im(A0)=-6.98(0.62)(1.44)×10-11 GeV, where the errors are statistical and systematic, respectively. The former agrees well with the experimental result Re(A0)=3.3201(18)×10-7 GeV. These results for A0 can be combined with our earlier lattice calculation of A2 [2] to obtain Re(ϵ′/ϵ)=21.7(2.6)(6.2)(5.0)×10-4, where the third error represents omitted isospin breaking effects, and Re(A0)/Re(A2)=19.9(2.3)(4.4). The first agrees well with the experimental result of Re(ϵ′/ϵ)=16.6(2.3)×10-4. A comparison of the second with the observed ratio Re(A0)/Re(A2)=22.45(6), demonstrates the standard model origin of this “ΔI=1/2 rule” enhancement.

We extend the position-space renormalization procedure, where renormalization factors are calculated from Green's functions in position space, by introducing a technique to take the average of Green's functions over spheres. In addition to reducing discretization errors, this technique enables the resulting position-space correlators to be evaluated at any physical distance, making them continuous functions similar to the $O(4)$-symmetric position-space Green's functions in the continuum theory but with a residual dependence on a regularization parameter, the lattice spacing $a$. We can then take the continuum limit of these renormalized quantities calculated at the same physical renormalization scale $|x|$ and investigate the resulting $|x|$-dependence to identify the appropriate renormalization window. As a numerical test of the spherical averaging technique, we determine the renormalized light and strange quark masses by renormalizing the scalar current. We see a substantial reduction of discretization effects on the scalar current correlator and an enhancement of the renormalization window. The numerical simulation is carried out with $2+1$-flavor domain-wall fermions at three lattice cutoffs in the range 1.79--3.15~GeV., Comment: 34 pages, 12 figures

We propose a strategy to non-perturbatively match the Wilson coefficients in the three- and four-flavor theories, which uses two-point Green's functions of the corresponding four-quark operators at long distances. The idea is refined by combining with the spherical averaging technique, which enables us to convert two-point functions calculated on the lattice into continuous functions of the distance $|x-y|$ between two operators. We also show the result for an exploratory calculation of two-point functions of the $\Delta S=1$ operators $Q_7$ and $Q_8$ that are in the $(8_L,8_R)$ representation of ${\rm SU(3)}_L\times{\rm SU(3)}_R$ and mix with each other., Comment: 7 pages, 2 figures, Proceedings of the 36th International Symposium on Lattice Field Theory (Lattice 2018), July 22-28, 2018, East Lancing, USA

We find that the quark propagator constructed from the domain-wall fermion operator has $L_s-1$ extra poles as well as the pole that realizes the physical quark in the continuum limit. We show the energy-momentum dispersion relation for the physical and unphysical poles of M\"obius domain-wall fermions in free field theory at finite $L_s$. The dependence of extra pole energies on the M\"obius parameter $b-c$ and on the domain-wall height $M_5$ is investigated. Our result suggests that small values of $b-c$ set a large lower bound on the unphysical pole masses and the contribution of these poles could be suppressed well by calculating with small $b-c$., Comment: 24 pages, 8 figures

We study the vector and axial-vector current correlators in perturbative and non-perturbative regimes of QCD. The correlators in Euclidean coordinate space are calculated on the lattice using the M\"obius domain-wall fermion formulation at three lattice spacings covering 0.044--0.080~fm. The dynamical quark effects of $2+1$ light flavors are included. The sum $V+A$ and the difference $V-A$ of the vector ($V$) and axial-vector ($A$) current correlators calculated on the lattice after extrapolating to the physical point agree with those converted from the ALEPH experimental data of hadronic $\tau$ decays. The level of the agreement in the $V+A$ channel is about $1.3\sigma$ or smaller in the region of $|x|\ge0.4$~fm, while that in the $V-A$ channel is about $1.8\sigma$ at $|x|=0.74$~fm and smaller at other distances. We also extract the hiral condensate from the short-distance correlators on the lattice using the PCAC relation. Its result extrapolated to the chiral and continuum limit is compatible with other estimates at low energies., Comment: 32 pages, 19 figures

We calculate the vector and axial-vector current correlators in the coordinate space and compare them with the experimental information obtained through the spectral functions of hadronic tau decays measured by ALEPH. Lattice data are obtained with 2+1 Mobius domain-wall fermions at three lattice spacings 0.044, 0.055 and 0.080 fm and the continuum limit is taken. The correlators calculated on the lattice after extrapolating to the physical point agree with those converted from the ALEPH data., Comment: 7 pages, 7 figures; talk presented at Lattice 2016 -- 34th International Symposium on Lattice Field Theory, 24-30 July 2016, University of Southampton, UK

We determine the renormalization constants for flavor non-singlet fermion bilinear operators of M\"obius domain-wall fermions. The renormalization condition is imposed on the correlation functions in the coordinate space, such that the non-perturbative lattice calculation reproduces the perturbatively calculated counterpart at short distances. The perturbative expansion is precise as the coefficients are available up to $O(\alpha_s^4)$. We employ $2+1$-flavor lattice ensembles at three lattice spacings in the range 0.044--0.080~fm., Comment: 36 pages, 20 figures

The electroweak symmetry breaking and origin of masses may be attributed to the breakdown of chiral symmetry due to a strong gauge dynamics. Among several candidates of such gauge systems, we focus on two-color QCD with Nf = 6 massless Dirac fermions in the fundamental representation, and study on whether the dynamics of this gauge system trigger chiral symmetry breaking or not by simulating with Wilson fermions on lattices up to L/a = 32. We show the result for the quark mass dependence and the volume dependence of some quantities such as the mass of the lightest pseudoscalar meson, decay constant, and give the three evidences supporting the absence of chiral symmetry breaking in the six-flavor the ory.

We study the non-perturbative determination of the renormalization constants of flavor non-singlet quark bilinear operators on the lattice. The renormalization condition is imposed on correlation functions of bilinear operators in the coordinate space. The results are converted to the value at 2 GeV in the $\rm\overline{MS}$ scheme by a perturbative matching. The calculation is carried out on gauge configurations generated with the Mobius domain-wall fermions at two lattice spacings $a^{-1} = 2.4$ GeV and $a^{-1} = 3.6$ GeV., Comment: 7 pages, 7 figures; talk presented at Lattice 2014 -- 32nd International Symposium on Lattice Field Theory, 23-28 June, 2014, Columbia University New York, NY

We investigate the chiral properties of SU(2) gauge theory with six flavors, i.e. six light Dirac fermions in the fundamental representations by lattice simulation, and point out that the spontaneous breakdown of chiral symmetry does not occur in this system. The quark mass dependence of the mesonic spectrum provides an evidence for such a possibility. The decay constant tends to be increased by the finite size effect, which is opposite to the behavior predicted by chiral perturbation theory and indicates that the long distance dynamics in the six-flavor theory could be different from the theory with chiral symmetry breaking. The subtracted chiral condensate, whose utility is demonstrated by the simulation of two-flavor theory, is shown to vanish in the chiral limit within the precision of available data., Comment: 53 pages, 33 figures, 17 tables, typos corrected, references changed

We report results of masses and decay constants of light and charmed pseudo-scalar mesons using lattice QCD with M\"obius domain-wall fermions. Using this formulation we are able to compute pseudo-scalar decay constants through the pseudo-scalar density operator as well as with the axial-vector current. Results are shown from several lattice spacings and pion masses between 230 MeV and 500 MeV. We present an analysis of these results at different quark masses to show the chiral properties of the light mesons masses and decay constants., Comment: 7 pages, 4 figures, Proceedings of the 33rd International Symposium on Lattice Field Theory (Lattice 2015)