Semileptonic Decays: an Update Down Under

Heavy-meson semileptonic decays calculations on the lattice are reviewed. The focus is upon obtaining reliable matrix elements. Errors that depend upon the lattice spacing, $a$, are an important source of systematic error. Full $O(a)$ improvement of matrix elements for arbitrary-mass four-component quarks is discussed. With improvement, bottom-quark matrix elements can be calculated directly using current lattices. Momentum dependent errors for $O(a)$-improved quarks and statistical noise limit momenta to around 1 GeV/c with current lattices. Hence, maximum recoil momenta can be reached for $D$ decays while only a fraction of the maximum recoil momentum can be reliably studied for the light-meson decay modes of the $B$. Differential decay rates and partial widths are phenomenologically important quantities in $B$ decays that can be reliably determined with present lattices.Comment: 14 pages, 9 postscript figures, requires espcrc2.st

Chiral Perturbation Theory and Weak Matrix Elements

I describe recent developments in quenched chiral perturbation theory (QChPT) and the status of weak matrix elements involving light quarks. I illustrate how, with improved statistical errors, and with calculations of the masses of baryons containing non-degenerate quarks, there is now a clear need for extrapolations of higher order than linear in the quark mass. I describe how QChPT makes predictions for the functional forms to use in such extrapolations, and emphasize the distinction between contributions coming from chiral loops which are similar to those present in unquenched theories, and those from $\eta'$ loops which are pure quenched artifacts. I describe a fit to the baryon masses using the predictions of QChPT. I give a status report on the numerical evidence for $\eta'$ loops, concluding that they are likely present, and are characterized by a coupling $\delta=0.1-0.2$. I use the difference between chiral loops in QCD and quenched QCD to estimate the quenching errors in a variety of quantities. I then turn to results for matrix elements, largely from quenched simulations. Results for quenched decay constants cannot yet be reliably extrapolated to the continuum limit. By contrast, new results for $B_K$ suggest a continuum, ``quenched'' value of $B_K(NDR, 2 GeV) = 0.5977 \pm 0.0064 \pm 0.0166$, based on a quadratic extrapolation in $a$. The theoretical basis for using a quadratic extrapolation has been confirmed. For the first time there is significant evidence that unquenching changes $B_K$, and my estimate for the value in QCD is $B_K(NDR, 2 GeV) = 0.66 \pm 0.02 \pm 0.11$. Here the second error is a conservative estimate of the systematic error due to uncertainties in the effect of quenching. A less conservative viewpoint reduces $0.11$ to $0.03$.Comment: 16 pages, 11 figures, Latex using espcrc2.sty and psfig. Talk presented at LATTICE96(phenomenology

Semileptonic Form Factors

I report the current status of the heavy-light decay constants, the bag parameters and the semileptonic form factors. I compare the heavy-light decay constants with Wilson-Wilson and clover-clover fermions. Systematic errors such as scale setting and renormalization factors are also discussed. 1/M dependences for the heavy-light semileptonic form factors near $q^2 = q^2_max$ with clover-clover and NRQCD-Wilson fermions are found to be small.Comment: 12 pgs. 15 figures. Talk presented at LATTICE9

Heavy quark mass dependence of semileptonic form factors for B decays

We present our study of the dependence of the heavy-to-light semileptonic B decay form factors on the heavy-light meson mass $M_{PS}$. Simulations are made over a range of the heavy quark mass covering both the charm and bottom quarks using the $O(a)$-improved clover action at $\beta=5.9$ on a $16^3\times 40$ and $24^3\times 64$ lattice. We find that a weak dependence of form factors on $M_{PS}$ observed in previous studies in the region of charm quark persists up to the region of$b$ quark. The soft pion relation $f^0(q^2_{max})=f_B/f_\pi$ is examined and found to be largely violated.Comment: 3 pages, latex source-file, 4 figures as epsf-file, uses espcrc2.sty. Talk presented by S. Tominaga at Lattice 97, Edinburgh, Scotland, 22-26 Jul 199

Heavy Light Weak Matrix Elements

I review the status of lattice calculations of heavy-light weak matrix elements, concentrating on semileptonic B decays to light mesons, B -> K* gamma, the B meson decay constant, f_B, and the mixing parameter B_B.Comment: 12 pages, LaTeX2e with 6 postscript figures. Uses espcrc2.sty and epsf.sty (included). Talk presented at LATTICE96(heavy quarks

Status of Heavy Quark Physics on the Lattice

The status of lattice calculations of some phenomenology of heavy quarks is presented. Emphasis is on progress made in calculating those quantities relevant to estimating parameters of the quark mixing matrix, namely leptonic decay constants, the bag parameter of neutral $B$ mixing, and semileptonic form factors. New results from studies of quarkonia are highlighted.Comment: LATTICE98(Plenary Review Talk), to be published in Nucl. Phys. Proc. Suppl.; LaTeX, 15 pages, 9 PostScript figures, uses espcrc2.st

Renormalization of the effective theory for heavy quarks at small velocity

The slope of the Isgur-Wise function at the normalization point, $\xi^{(1)}(1)$,is one of the basic parameters for the extraction of the $CKM$ matrix element $V_{cb}$ from exclusive semileptonic decay data. A method for measuring this parameter on the lattice is the effective theory for heavy quarks at small velocity $v$. This theory is a variant of the heavy quark effective theory in which the motion of the quark is treated as a perturbation. In this work we study the lattice renormalization of the slow heavy quark effective theory. We show that the renormalization of $\xi^{(1)}(1)$ is not affected by ultraviolet power divergences, implying no need of difficult non-perturbative subtractions. A lattice computation of $\xi^{(1)}(1)$ with this method is therefore feasible in principle. The one-loop renormalization constants of the effective theory for slow heavy quarks are computed to order $v^2$ together with the lattice-continuum renormalization constant of $\xi^{(1)}(1)$ . We demonstrate that the expansion in the heavy-quark velocity reproduces correctly the infrared structure of the original (non-expanded) theory to every order. We compute also the one-loop renormalization constants of the slow heavy quark effective theory to higher orders in $v^2$ and the lattice-continuum renormalization constants of the higher derivatives of the $\xi$ function. Unfortunately, the renormalization constants of the higher derivatives are affected by ultraviolet power divergences, implying the necessity of numerical non-perturbative subtractions. The lattice computation of higher derivatives of the Isgur-Wise function seems therefore problematic.Comment: Latex, 43 pages, 5 figures available by fax upon request. To be published in Nucl. Phys

Tuning Fermilab Heavy Quarks in 2+1 Flavor Lattice QCD with Application to Hyperfine Splittings

We report the non-perturbative tuning of parameters--- kappa_c, kappa_b, and kappa_crit ---that determine the heavy-quark mass in the Fermilab action. This requires the computation of the masses of Ds^(*) and Bs^(*) mesons comprised of a Fermilab heavy quark and a staggered light quark. Additionally, we report the hyperfine splittings for Ds and Bs mesons as a cross-check of our simulation and analysis methods. We find a splitting of 145 +/- 15 MeV for the Ds system and 40 +/- 9 MeV for the Bs system. These are in good agreement with the Particle Data Group average values of 143.9 +/- 0.4 MeV and 46.1 +/- 1.5 MeV, respectively. The calculations are carried out with the MILC 2+1 flavor gauge configurations at three lattice spacings $a$ approximately 0.15, 0.12, and 0.09 fm.Comment: 34 pages, 8 figures, 26 tables; some sections rearranged for clarity; conclusions unchanged; version accepted by Phys. Rev.