A perturbative study of two four-quark operators in finite volume renormalization schemes

Starting from the QCD Schroedinger functional (SF), we define a family of renormalization schemes for two four-quark operators, which are, in the chiral limit, protected against mixing with other operators. With the appropriate flavour assignments these operators can be interpreted as part of either the $\Delta F=1$ or $\Delta F=2$ effective weak Hamiltonians. In view of lattice QCD with Wilson-type quarks, we focus on the parity odd components of the operators, since these are multiplicatively renormalized both on the lattice and in continuum schemes. We consider 9 different SF schemes and relate them to commonly used continuum schemes at one-loop order of perturbation theory. In this way the two-loop anomalous dimensions in the SF schemes can be inferred. As a by-product of our calculation we also obtain the one-loop cutoff effects in the step-scaling functions of the respective renormalization constants, for both O(a) improved and unimproved Wilson quarks. Our results will be needed in a separate study of the non-perturbative scale evolution of these operators.Comment: 22 pages, 4 figure

Investigation of the fine structure of antihydrogen.

At the historic Shelter Island Conference on the Foundations of Quantum Mechanics in 1947, Willis Lamb reported an unexpected feature in the fine structure of atomic hydrogen: a separation of the 2S1/2 and 2P1/2 states1. The observation of this separation, now known as the Lamb shift, marked an important event in the evolution of modern physics, inspiring others to develop the theory of quantum electrodynamics2-5. Quantum electrodynamics also describes antimatter, but it has only recently become possible to synthesize and trap atomic antimatter to probe its structure. Mirroring the historical development of quantum atomic physics in the twentieth century, modern measurements on anti-atoms represent a unique approach for testing quantum electrodynamics and the foundational symmetries of the standard model. Here we report measurements of the fine structure in the n = 2 states of antihydrogen, the antimatter counterpart of the hydrogen atom. Using optical excitation of the 1S-2P Lyman-α transitions in antihydrogen6, we determine their frequencies in a magnetic field of 1 tesla to a precision of 16 parts per billion. Assuming the standard Zeeman and hyperfine interactions, we infer the zero-field fine-structure splitting (2P1/2-2P3/2) in antihydrogen. The resulting value is consistent with the predictions of quantum electrodynamics to a precision of 2 per cent. Using our previously measured value of the 1S-2S transition frequency6,7, we find that the classic Lamb shift in antihydrogen (2S1/2-2P1/2 splitting at zero field) is consistent with theory at a level of 11 per cent. Our observations represent an important step towards precision measurements of the fine structure and the Lamb shift in the antihydrogen spectrum as tests of the charge-parity-time symmetry8 and towards the determination of other fundamental quantities, such as the antiproton charge radius9,10, in this antimatter system

Non-perturbative renormalization of the static axial current in two-flavour QCD

We perform the non-perturbative renormalization of matrix elements of the static-light axial current by a computation of its scale dependence in lattice QCD with two flavours of massless O(a) improved Wilson quarks. The regularization independent factor that relates any running renormalized matrix element of the axial current in the static effective theory to the renormalization group invariant one is evaluated in the Schroedinger functional scheme, where in this case we find a significant deviation of the non-perturbative running from the perturbative prediction. An important technical ingredient to improve the precision of the results consists in the use of modified discretizations of the static quark action introduced earlier by our collaboration. As an illustration how to apply the renormalization of the static axial current presented here, we connect the bare matrix element of the current to the B_s-meson decay constant in the static approximation for one value of the lattice spacing, a ~ 0.08 fm, employing large-volume N_f=2 data at beta=5.3.Comment: 33 pages including figures and tables, latex2e, uses JHEP3.cls; version published in JHEP, small additions, results unchange

Effective heavy-light meson energies in small-volume quenched QCD

We study effective energies of heavy-light meson correlation functions in lattice QCD and a small volume of (0.2 fm)^4 to non-perturbatively calculate their dependence on the heavy quark mass in the continuum limit. Our quenched results obtained here constitute an essential intermediate step of a first fully non-perturbative computation of the b-quark's mass in the static approximation that has recently been presented as an application of a new proposal to non-perturbatively renormalize the Heavy Quark Effective Theory. The renormalization constant and the improvement coefficients relating the renormalized current and subtracted quark mass are determined in the relevant parameter region at weak couplings, which allows to perform the numerical simulations at several, precisely fixed values of the renormalization group invariant heavy quark mass in a range from 3 GeV to 15 GeV.Comment: 24 pages including figures and tables, latex2e; version published in JHEP, small additions, results unchange

A strategy for implementing non-perturbative renormalisation of heavy-light four-quark operators in the static approximation

We discuss the renormalisation properties of the complete set of $\Delta B = 2$ four-quark operators with the heavy quark treated in the static approximation. We elucidate the role of heavy quark symmetry and other symmetry transformations in constraining their mixing under renormalisation. By employing the Schroedinger functional, a set of non-perturbative renormalisation conditions can be defined in terms of suitable correlation functions. As a first step in a fully non-perturbative determination of the scale-dependent renormalisation factors, we evaluate these conditions in lattice perturbation theory at one loop. Thereby we verify the expected mixing patterns and determine the anomalous dimensions of the operators at NLO in the Schroedinger functional scheme. Finally, by employing twisted-mass QCD it is shown how finite subtractions arising from explicit chiral symmetry breaking can be avoided completely.Comment: 41 pages, 6 figure

Lattice QCD and the Schwarz alternating procedure

A numerical simulation algorithm for lattice QCD is described, in which the short- and long-distance effects of the sea quarks are treated separately. The algorithm can be regarded, to some extent, as an implementation at the quantum level of the classical Schwarz alternating procedure for the solution of elliptic partial differential equations. No numerical tests are reported here, but theoretical arguments suggest that the algorithm should work well also at small quark masses.Comment: Plain TeX source, 20 pages, figures include

Chiral symmetry breaking and the Banks--Casher relation in lattice QCD with Wilson quarks

The Banks--Casher relation links the spontaneous breaking of chiral symmetry in QCD to the presence of a non-zero density of quark modes at the low end of the spectrum of the Dirac operator. Spectral observables like the number of modes in a given energy interval are renormalizable and can therefore be computed using the Wilson formulation of lattice QCD even though the latter violates chiral symmetry at energies on the order of the inverse lattice spacing. Using numerical simulations, we find (in two-flavour QCD) that the low quark modes do condense in the expected way. In particular, the chiral condensate can be accurately calculated simply by counting the low modes on large lattices. Other spectral observables can be considered as well and have a potentially wide range of uses.Comment: Plain TeX, 30 pages, 5 figure

The continuum limit of the quark mass step scaling function in quenched lattice QCD

The renormalisation group running of the quark mass is determined non-perturbatively for a large range of scales, by computing the step scaling function in the Schroedinger Functional formalism of quenched lattice QCD both with and without O(a) improvement. A one-loop perturbative calculation of the discretisation effects has been carried out for both the Wilson and the Clover-improved actions and for a large number of lattice resolutions. The non-perturbative computation yields continuum results which are regularisation independent, thus providing convincing evidence for the uniqueness of the continuum limit. As a byproduct, the ratio of the renormalisation group invariant quark mass to the quark mass, renormalised at a hadronic scale, is obtained with very high accuracy.Comment: 23 pages, 3 figures; minor changes, references adde

QCD with light Wilson quarks on fine lattices (II): DD-HMC simulations and data analysis

In this second report on our recent numerical simulations of two-flavour QCD, we provide further technical details on the simulations and describe the methods we used to extract the meson masses and decay constants from the generated ensembles of gauge fields. Among the topics covered are the choice of the DD-HMC parameters, the issue of stability, autocorrelations and the statistical error analysis. Extensive data tables are included as well as a short discussion of the quark-mass dependence in partially quenched QCD, supplementing the physics analysis that was presented in the first paper in this series.Comment: TeX source, 35 pages, figures include

A way to estimate the heavy quark thermalization rate from the lattice

The thermalization rate of a heavy quark is related to its momentum diffusion coefficient. Starting from a Kubo relation and using the framework of the heavy quark effective theory, we argue that in the large-mass limit the momentum diffusion coefficient can be defined through a certain Euclidean correlation function, involving color-electric fields along a Polyakov loop. Furthermore, carrying out a perturbative computation, we demonstrate that the spectral function corresponding to this correlator is relatively flat at small frequencies. Therefore, unlike in the case of several other transport coefficients, for which the narrowness of the transport peak makes analytic continuation from Euclidean lattice data susceptible to severe systematic uncertainties, it appears that the determination of the heavy quark thermalization rate could be relatively well under control.Comment: 17 pages. v2: clarifications and references added, published versio