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

Logical Specification and Analysis of Fault Tolerant Systems through Partial Model Checking

This paper presents a framework for a logical characterisation of fault tolerance and its formal analysis based on partial model checking techniques. The framework requires a fault tolerant system to be modelled using a formal calculus, here the CCS process algebra. To this aim we propose a uniform modelling scheme in which to specify a formal model of the system, its failing behaviour and possibly its fault-recovering procedures. Once a formal model is provided into our scheme, fault tolerance - with respect to a given property - can be formalized as an equational µ-calculus formula. This formula expresses in a logic formalism, all the fault scenarios satisfying that fault tolerance property. Such a characterisation understands the analysis of fault tolerance as a form of analysis of open systems and thank to partial model checking strategies, it can be made independent on any particular fault assumption. Moreover this logical characterisation makes possible the fault-tolerance verification problem be expressed as a general µ-calculus validation problem, for solving which many theorem proof techniques and tools are available. We present several analysis methods showing the flexibility of our approach

Lattice computation of structure functions

Recent lattice calculations of hadron structure functions are described.Comment: Plenary talk presented at LATTICE96, LaTeX, 7 pages, 5 figures, espcrc2.sty and epsfig.sty include

A High-Statistics Lattice Calculation of λ1\lambda_1 and λ2\lambda_2 in the BB meson

We present a high-statistics lattice calculation of the kinetic energy $-\lambda_1/2 m_b$ of the heavy quark inside the $B$-meson and of the chromo-magnetic term $\lambda_2$, related to the $B^*$--$B$ mass splitting, performed in the HQET. Our results have been obtained from a numerical simulation based on 600 gauge field configurations generated at $\beta=6.0$, on a lattice volume $24^3 \times 40$ and using, for the meson correlators, the results obtained with the SW-Clover $O(a)$ improved lattice action for the light quarks. For the kinetic energy we found $-\lambda_1=\langle B \vert \bar h (i\vec{D})^{2} h \vert B \rangle /(2 M_B )=-(0.09 \pm 0.14)$~GeV$^2$, which is interesting for phenomenological applications. We also find $\lambda_2= 0.07 \pm 0.01$ GeV$^2$, corresponding to $M^2_{B^*}-M^2_B= 4 \lambda_2= 0.280 \pm 0.060$ GeV$^2$, which is about one half of the experimental value. The origin of the discrepancy with the experimental number needs to be clarified.Comment: 26 pages, latex, 5 figure

DEPENDENCE OF THE CURRENT RENORMALISATION CONSTANTS ON THE QUARK MASS

We study the behaviour of the vector and axial current renormalisation constants $Z_V$ and $Z_A$ as a function of the quark mass, $m_q$. We show that sizeable $O(am_q)$ and $O(g_0^2 a m_q)$ systematic effects are present in the Wilson and Clover cases respectively. We find that the prescription of Kronfeld, Lepage and Mackenzie for correcting these artefacts is not always successful.Comment: Contribution to Lattice'94, 3 pages PostScript, uuencoded compressed

Hermite Calculus

We develop a new method of umbral nature to treat blocks of Her mite and of Hermite like poly- nomials as independent algebraic quantities. The Calculus we propose allows the formulation of a number of ”practical rules” allowing significant simplific ations in computational problem

Computing the Slope of the Isgur-Wise Function

We propose a method for evaluating the slope (and higher derivatives) of the Isgur-Wise function at the zero recoil point using lattice simulations. These derivatives are required for the extrapolation of the experimental data for $B\rightarrow D^*l\bar\nu$ decays to the zero recoil point, from which the $V_{cb}$ element of the CKM-matrix can be determined.Comment: Latex File Southampton Preprint 93/94-07; Rome Preprint 93/98

Non-perturbative renormalization in kaon decays

We discuss the application of the MPSTV non-perturbative method \cite{NPM} to the operators relevant to kaon decays. This enables us to reappraise the long-standing question of the $\Delta I=1/2$ rule, which involves power-divergent subtractions that cannot be evaluated in perturbation theory. We also study the mixing with dimension-six operators and discuss its implications to the chiral behaviour of the $B_K$ parameter.Comment: Talk presented at LATTICE96(improvement), LaTeX 3 pages, uses espcrc2, 2 postscript figure