Lattice extraction of K→ππ K \to \pi \pi 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. 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.Comment: 54 pages, 3 figures; Important revisions: Corrections to formulas for K->pi pi with degenerate quark masses have been mad

Domain-wall fermions with U(1)U(1) dynamical gauge fields

We have carried out a numerical simulation of a domain-wall model in $(2+1)$-dimensions, in the presence of a dynamical gauge field only in an extra dimension, corresponding to the weak coupling limit of a ( 2-dimensional ) physical gauge coupling. Using a quenched approximation we have investigated this model at $\beta_{s} ( = 1 / g^{2}_{s} ) =$ 0.5 ( ``symmetric'' phase), 1.0, and 5.0 (``broken'' phase), where $g_s$ is the gauge coupling constant of the extra dimension. We have found that there exists a critical value of a domain-wall mass $m_{0}^{c}$ which separates a region with a fermionic zero mode on the domain-wall from the one without it, in both symmetric and broken phases. This result suggests that the domain-wall method may work for the construction of lattice chiral gauge theories.Comment: 27 pages (11 figures), latex (epsf style-file needed

Before sailing on a domain-wall sea

We discuss the very different roles of the valence-quark and the sea-quark residual masses ($m_{res}^v$ and $m_{res}^s$) in dynamical domain-wall fermions simulations. Focusing on matrix elements of the effective weak hamiltonian containing a power divergence, we find that $m_{res}^v$ can be a source of a much bigger systematic error. To keep all systematic errors due to residual masses at the 1% level, we estimate that one needs $a m_{res}^s \le 10^{-3}$ and $a m_{res}^v \le 10^{-5}$, at a lattice spacing $a\sim 0.1$ fm. The practical implications are that (1) optimal use of computer resources calls for a mixed scheme with different domain-wall fermion actions for the valence and sea quarks; (2) better domain-wall fermion actions are needed for both the sea and the valence sectors.Comment: latex, 25 pages. Improved discussion in appendix, including correction of some technical mistakes; ref. adde

Chiral Gauge Theory on Lattice with Domain Wall Fermions

We investigate a U(1) lattice chiral gauge theory with domain wall fermions and compact gauge fixing. In the reduced model limit, our perturbative and numerical investigations show that there exist no extra mirror chiral modes. The longitudinal gauge degrees of freedom have no effect on the free domain wall fermion spectrum consisting of opposite chiral modes at the domain wall and at the anti-domain wall which have an exponentially damped overlap.Comment: 16 pages revtex, 5 postscript figures, PRD versio

The Tunneling Hybrid Monte-Carlo algorithm

The hermitian Wilson kernel used in the construction of the domain-wall and overlap Dirac operators has exceptionally small eigenvalues that make it expensive to reach high-quality chiral symmetry for domain-wall fermions, or high precision in the case of the overlap operator. An efficient way of suppressing such eigenmodes consists of including a positive power of the determinant of the Wilson kernel in the Boltzmann weight, but doing this also suppresses tunneling between topological sectors. Here we propose a modification of the Hybrid Monte-Carlo algorithm which aims to restore tunneling between topological sectors by excluding the lowest eigenmodes of the Wilson kernel from the molecular-dynamics evolution, and correcting for this at the accept/reject step. We discuss the implications of this modification for the acceptance rate.Comment: improved discussion in appendix B, RevTeX, 19 page

Perturbative study for domain-wall fermions in 4+1 dimensions

We investigate a U(1) chiral gauge model in 4+1 dimensions formulated on the lattice via the domain-wall method. We calculate an effective action for smooth background gauge fields at a fermion one loop level. From this calculation we discuss properties of the resulting 4 dimensional theory, such as gauge invariance of 2 point functions, gauge anomalies and an anomaly in the fermion number current.Comment: 39 pages incl. 9 figures, REVTeX+epsf, uuencoded Z-compressed .tar fil

The perfect action for non-degenerate staggered fermions

The perfect action of free staggered fermions is calculated by blocking from the continuum for degenerate and non-degenerate flavor masses. The symmetry structure, connecting flavor transformations and translations, is explained directly from the blocking scheme. It is convenient to use a modified Fourier transformation, respecting this connection, to treat the spin-flavor structure of the blockspins. The perfect action remains local in the non-degenerate case; it is explicitly calculated in two dimensions. I finally comment on the relation of the blocking scheme to the transition from Dirac-K\"ahler fermions to staggered fermions.Comment: 14 pages, Latex2e, 1 Latex figure, some minor changes and two references adde

Mobility edge in lattice QCD

We determine the location $\lambda_c$ of the mobility edge in the spectrum of the hermitian Wilson operator on quenched ensembles. We confirm a theoretical picture of localization proposed for the Aoki phase diagram. When $\lambda_c>0$ we also determine some key properties of the localized eigenmodes with eigenvalues $|\lambda|<\lambda_c$. Our results lead to simple tests for the validity of simulations with overlap and domain-wall fermions.Comment: revtex, 4 pages, 1 figure, minor change

K-->pipi amplitudes from lattice QCD with a light charm quark

We compute the leading-order low-energy constants of the DeltaS=1 effective weak Hamiltonian in the quenched approximation of QCD with up, down, strange, and charm quarks degenerate and light. They are extracted by comparing the predictions of finite volume chiral perturbation theory with lattice QCD computations of suitable correlation functions carried out with quark masses ranging from a few MeV up to half of the physical strange mass. We observe a large DeltaI=1/2 enhancement in this corner of the parameter space of the theory. Although matching with the experimental result is not observed for the DeltaI=1/2 amplitude, our computation suggests large QCD contributions to the physical DeltaI=1/2 rule in the GIM limit, and represents the first step to quantify the role of the charm quark-mass in K-->pipi amplitudes.Comment: 4 pages, 1 figure. Minor modifications. Final version to appear on PR