Kaon B parameter from quenched Lattice QCD

We present results of a large-scale simulation for the Kaon B parameter $B_K$ in quenched lattice QCD with the Kogut-Susskind quark action. Calculating $B_K$ at 1% statistical accuracy for seven values of lattice spacing in the range $a\approx 0.24-0.04$ fm on lattices up to $56^3\times 96$, we verify a quadratic $a$ dependence of $B_K$ theoretically predicted. Strong indications are found that, with our level of accuracy, $\alpha_{\bar{MS}}(1/a)^2$ terms arising from our one-loop matching procedure have to be included in the continuum extrapolation. We present $B_K$(NDR, 2 GeV)=0.628(42) as our final value, as obtained by a fit including the $\alpha_{\bar{MS}}(1/a)^2$ term.Comment: 8 pages, Latex(revtex, epsf), 2 epsf figure

Evidence for hard chiral logarithms in quenched lattice QCD

We present the first direct evidence that quenched QCD differs from full QCD in the chiral ($m_q \rightarrow 0$) limit, as predicted by chiral perturbation theory, from our quenched lattice QCD simulations at $\beta = 6/g^2 = 6.0$. We measured the spectrum of light hadrons on $16^3 \times 64$, $24^3 \times 64$ and $32^3 \times 64$, using staggered quarks of masses $m_q=0.01$, $m_q=0.005$ and $m_q=0.0025$. The pion masses showed clear evidence for logarithmic violations of the PCAC relation $m_{\pi}^2 \propto m_q$, as predicted by quenched chiral perturbation theory. The dependence on spatial lattice volume precludes this being a finite size effect. No evidence was seen for such chiral logarithms in the behaviour of the chiral condensate $\langle\bar{\psi}\psi\rangle$.Comment: 10 pages, 4 figures, uuencoded compressed postscript fil

The Kaon B-parameter with the Wilson Quark Action using Chiral Ward Identities

We present a detailed description of the method and results of our calculation of the kaon B parameter using the Wilson quark action in quenched QCD at $\beta=5.9-6.5$. The mixing problem of the $\Delta s=2$ four-quark operators is solved non-perturbatively with full use of chiral Ward identities. We find $B_K(NDR, 2 GeV)=0.562(64)$ in the continuum limit, which agrees with the value obtained with the Kogut-Susskind quark action.Comment: 10 pages, latex source-file, 10 figures as epsf-file, uses espcrc2.sty. Talk given at International Workshop on Lattice QCD on Parallel Computers, 10-15 Mar 1997, Tsukuba, Japa

Matrix elements relevant for Delta I=1/2 rule and epsilon-prime from Lattice QCD with staggered fermions

We perform a study of matrix elements relevant for the Delta I=1/2 rule and the direct CP-violation parameter epsilon-prime from first principles by computer simulation in Lattice QCD. We use staggered (Kogut-Susskind) fermions, and employ the chiral perturbation theory method for studying K to 2 Pi decays. Having obtained a reasonable statistical accuracy, we observe an enhancement of the Delta I=1/2 amplitude, consistent with experiment within our large systematic errors. Finite volume and quenching effects have been studied and were found small compared to noise. The estimates of epsilon-prime are hindered by large uncertainties associated with operator matching. In this paper we explain the simulation method, present the results and address the systematic uncertainties.Comment: 40 pages, 17 figures, LATEX with epsf, to be submitted to Phys. Rev. D. Minor errors are corrected, some wording and notation change

The Kaon B-parameter with the Wilson Quark Action using Chiral Ward Identities

A lattice QCD calculation of the kaon $B$ parameter $B_K$ is carried out with the Wilson quark action in the quenched approximation at $\beta=6/g^2=5.9-6.5$. The mixing problem of the $\Delta s=2$ four-quark operators is solved non-perturbatively with full use of chiral Ward identities employing four external quarks with an equal off-shell momentum in the Landau gauge. This method, without invoking any effective theory, enables us to construct the weak four-quark operators exhibiting good chiral behavior. Our results for $B_K$ with the non-perturbative mixing coefficients show small scaling violation beyond the lattice cut-off $a^{-1}\sim 2.5$GeV. Our estimate concludes $B_K(NDR, 2 GeV)=0.69(7)$ at $a^{-1}=2.7-4.3$GeV, which agrees with the value obtained with the Kogut-Susskind quark action. For comparison we also calculate $B_K$ with one-loop perturbative mixing coefficients. While this yields incorrect values at finite lattice spacing, a linear extrapolation to the continuum limit as a function of $a$ leads to a result consistent with those obtained with the Ward identity method.Comment: 42 pages, 22 Postscript figures, added a new reference[26

Applications of Partially Quenched Chiral Perturbation Theory

Partially quenched theories are theories in which the valence- and sea-quark masses are different. In this paper we calculate the nonanalytic one-loop corrections of some physical quantities: the chiral condensate, weak decay constants, Goldstone boson masses, B_K and the K+ to pi+ pi0 decay amplitude, using partially quenched chiral perturbation theory. Our results for weak decay constants and masses agree with, and generalize, results of previous work by Sharpe. We compare B_K and the K+ decay amplitude with their real-world values in some examples. For the latter quantity, two other systematic effects that plague lattice computations, namely, finite-volume effects and unphysical values of the quark masses and pion external momenta are also considered. We find that typical one-loop corrections can be substantial.Comment: 22 pages, TeX, refs. added, minor other changes, version to appear in Phys. Rev.

The Kaon BB-parameter with Wilson Fermions

We calculate the kaon $B$-parameter in quenched lattice QCD at $\beta=6.0$ using Wilson fermions at $\kappa=0.154$ and $0.155$. We use two kinds of non-local (``smeared'') sources for quark propagators to calculate the matrix elements between states of definite momentum. The use of smeared sources yields results with much smaller errors than obtained in previous calculations with Wilson fermions. By combining results for $\vec p =(0,0,0)$ and $\vec p =(0,0,1)$, we show that one can carry out the non-perturbative subtraction necessary to remove the dominant lattice artifacts induced by the chiral symmetry breaking term in the Wilson action. Our final results are in good agreement with those obtained using staggered fermions. We also present results for $B$-parameters of the $\Delta I = 3/2$ part of the electromagnetic penguin operators, and preliminary results for \bk\ in the presence of two flavors of dynamical quarks.Comment: 39 pages, including 9 PS figures (LA UR-91-3522

Hadron Spectrum with Wilson fermions

We present results of a high statistics study of the quenched spectrum using Wilson fermions at $\beta=6.0$ on $32^3 \times 64$ lattices. We calculate the masses of mesons and baryons composed of both degenerate and non-degenerate quarks. Using non-degenerate quark combinations allows us to study baryon mass splittings in detail. We find significant deviations from the lowest order chiral expansion, deviations that are consistent with the expectations of quenched chiral perturbation theory. We find that there is a $\sim 20$ systematic error in the extracted value of $m_s$, depending on the meson mass ratio used to set its value. Using the largest estimate of $m_s$ we find that the extrapolated octet mass-splittings are in agreement with the experimental values, as is $M_\Delta - M_N$, while the decuplet splittings are 30% smaller than experiment. Combining our results with data from the GF11 collaboration we find considerable ambiguity in the extrapolation to the continuum limit. Our preferred values are $M_N / M_\rho = 1.38(7)$ and $M_\Delta / M_\rho = 1.73(10)$, suggesting that the quenched approximation is good to only $\sim 10-15$. We also analyze the $O(ma)$ discretization errors in heavy quark masses.Comment: 52 pages. Tex. Modified "axis" source for figures also included. Needs macro packages lanlmac and epsf. Uses hyperbasics if available. Significant number of typographical errors correcte

One Spin Trace Formalism for BK B_K

It has been known for some time that there are two methods to calculate $B_K$ with staggered fermions: one is the two spin trace formalism and the other is the one spin trace formalism. Until now, the two spin trace formalism has been exclusively used for weak matrix element calculations with staggered fermions. Here, the one spin trace formalism to calculate $B_K$ with staggered fermions is explained. It is shown that the one spin trace operators require additional chiral partner operators in order to keep the continuum chiral behavior. The renormalization of the one spin trace operators is described and compared with the two spin trace formalism.Comment: 47 pages, latex, 4 figures are available on reques

Pion-Nucleon Sigma Term in Lattice QCD

We calculate both connected and disconnected contribution to the $\pi$-$N$ $\sigma$-term in quenched lattice QCD with Wilson quark action on a $12^3\times 20$ lattice at $\beta=5.7$. The latter is evaluated with the aid of the variant wall source method, which was previously applied successfully for extraction of $\pi$-$\pi$ scattering lengths and $\eta^\prime$ meson mass. We found that the disconnected contribution is about twice larger than the connected one. The value for the full $\pi$-$N$ $\sigma$ term $\sigma=40-60$ MeV is consistent with the experimental estimates. The nucleon matrix element of the strange quark density $\bar s s$ is fairly large in our result.Comment: 11 pages. Latex file. Figures are also included as ps file