B Physics on the Lattice: Λ‾\overline{\Lambda}, λ1\lambda_{1}, m‾b(m‾b)\overline{m}_{b}(\overline{m}_{b}), λ2\lambda_2, B0−Bˉ0B^{0}-\bar{B}^{0} mixing, \fb and all that

We present a short review of our most recent high statistics lattice determinations in the HQET of the following important parameters in B physics: the B--meson binding energy, $\overline{\Lambda}$ and the kinetic energy of the b quark in the B meson, $\lambda_1$, which due to the presence of power divergences require a non--perturbative renormalization to be defined; the $\overline{MS}$ running mass of the b quark, $\overline{m}_{b}(\overline{m}_{b})$; the $B^{*}$--$B$ mass splitting, whose value in the HQET is determined by the matrix element of the chromo--magnetic operator between B meson states, $\lambda_2$; the B parameter of the $B^{0}$--$\bar{B}^{0}$ mixing, $B_{B}$, and the decay constant of the B meson, $f_{B}$. All these quantities have been computed using a sample of $600$ gauge field configurations on a $24^{3}\times 40$ lattice at $\beta=6.0$. For $\overline{\Lambda}$ and $\overline{m}_{b}(\overline{m}_{b})$, we obtain our estimates by combining results from three independent lattice simulations at $\beta=6.0$, $6.2$ and $6.4$ on the same volume.Comment: 3 latex pages, uses espcrc2.sty (included). Talk presented at LATTICE96(heavy quarks

Results from a Non-Perturbative Renormalization of Lattice Operators

We propose a general renormalization method, which avoids completely the use of lattice perturbation theory. We present the results from its numerical applications to two-fermion operators on a $16^3 \times 32$ lattice, at $\beta=6.0$.Comment: 3 pages postscript file. Contribution to Lattice '9

A Numerical Study of Partially Twisted Boundary Conditions

We investigate the use of partially twisted boundary conditions in a lattice simulation with two degenerate flavours of improved Wilson sea quarks. The use of twisted boundary conditions on a cubic volume (L^3) gives access to components of hadronic momenta other than integer multiples of 2*pi/L. Partial twisting avoids the need for new gluon configurations for every choice of momentum, while, as recently demonstrated, keeping the finite-volume errors exponentially small for the physical quantities investigated in this letter. In this study we focus on the spectrum of pseudo scalar and vector mesons, on their leptonic decay constants and on Z_P, the matrix element of the pseudo scalar density between the pseudo scalar meson and the vacuum. The results confirm the momentum shift imposed by these boundary conditions and in addition demonstrate that they do not introduce any appreciable noise. We therefore advocate the use of partially twisted boundary conditions in applications where good momentum resolution is necessary.Comment: 10 pages, 3 figure

Non-Perturbative Renormalisation of Composite Operators

It is shown that the renormalisation constants of two quark operators can be accurately determined (to a precision of a few per-cent using 18 gluon configurations) using Chiral Ward identities. A method for computing renormalisation constants of generic composite operators without the use of lattice perturbation theory is proposed.Comment: 3 pages, uuencoded compressed postscript file, to appear in the Proceedings of the International Symposium on Lattice Field Theory, Dallas, Texas, 12-17 October 1993, Southampton Preprint 93/94-0

Computation of the b-quark Mass with Perturbative Matching at the Next-to-Next-to-Leading Order

We compute the two-loop term in the perturbation series for the quark-mass in the lattice Heavy Quark Effective Theory. This is an ingredient in the matching factor required to obtain the $b$-quark mass from lattice simulations of the HQET. Combining our calculations with numerical results from the APE collaboration, we find, at two-loop order, $\bar m_b \equiv m_b^{\bar{MS}}(m_b^{\bar{MS}}) =4.41\pm 0.05\pm 0.10$GeV. It was expected that the two-loop term would have a significant effect and this is indeed what we find. Depending on the choice of "reasonable" coupling constant in the one-loop estimates the result for $\bar m_b$ can change by several hundred MeV when the two-loop terms are included.Comment: 24 Pages, 1 Figur

Finite-Volume Two-Pion Amplitudes in the I=0 Channel

We perform a calculation in one-loop chiral perturbation theory of the two-pion matrix elements and correlation functions of an I=0 scalar operator, in finite and infinite volumes for both full and quenched QCD. We show that major difficulties arise in the quenched theory due to the lack of unitarity. Similar problems are expected for quenched lattice calculations of $K \to \pi \pi$ amplitudes with $\Delta I=1/2$. Our results raise the important question of whether it is consistent to study $K\to\pi\pi$ amplitudes beyond leading order in chiral perturbation theory in quenched or partially quenched QCD.Comment: Version to appear on Phys. Lett. B, with only very minor and stylistic change

A High Statistics Lattice Calculation of The B-meson Binding Energy

We present a high statistics lattice calculation of the B--meson binding energy $\overline{\Lambda}$ of the heavy--quark inside the pseudoscalar B--meson. Our numerical results have been obtained from several independent numerical simulations at $\beta=6.0$, $6.2$ and $6.4$, and using, for the meson correlators, the results obtained by the APE group at the same values of $\beta$. Our best estimate, obtained by combining results at different values of $\beta$, is $\overline{\Lambda}=180^{+30}_{-20}$ MeV. For the $\overline{MS}$ running mass, we obtain $\overline{m}_{b}(\overline{m}_{b})=4.15 \pm 0.05 \pm 0.20$ GeV, in reasonable agreement with previous determinations. The systematic error is the truncation of the perturbative series in the matching condition of the relevant operator of the Heavy Quark Effective Theory.Comment: Latex, 13 pages, 1 figure appended in uuencoded gzip.tar.fil

Future Perspectives in Lattice Field Theory

I review some of the contributions which lattice simulations are likely to make during the next five years or so to the development of our understanding of particle physics. Particular emphasis is given to the evaluation of non-perturbative QCD effects in experimentally measurable amplitudes, and the corresponding extraction of fundamental parameters.Comment: 10 Pages, Latex, 3 .eps figures and espcrc2.sty files. To be published in the proceedings of the XVIIth International Symposium on Lattice Field Theory, \textit{Lattice 99}, Pisa, Italy, June~28th - July~3rd 199

Renormalization of the Lattice HQET Isgur-Wise Function

We compute the perturbative renormalization factors required to match to the continuum Isgur-Wise function, calculated using lattice Heavy Quark Effective Theory. The velocity, mass, wavefunction and current renormalizations are calculated for both the forward difference and backward difference actions for a variety of velocities. Subtleties are clarified regarding tadpole improvement, regulating divergences, and variations of techniques used in these renormalizations.Comment: 28 pages, 0 figures, LaTeX. Final version accepted for publication in Phys. Rev. D. (Minor changes.