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

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

Quark bilinear step scaling functions and their continuum limit extrapolation

Some new results on nonperturbative renormalisation of quark bilinears in quenched QCD with Schroedinger Functional techniques are presented. Special emphasis is put on a study of the universality of the continuum limit for step scaling functions computed with different levels of O(a) improvement.Comment: Lattice2003(improve), 3 pages, 3 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

Non-perturbative scale evolution of four-fermion operators in two-flavour QCD

We apply finite-size recursion techniques based on the Schrodinger functional formalism to determine the renormalization group running of four-fermion operators which appear in the Delta S=2 effective weak Hamiltonian of the Standard Model. Our calculations are done using O(a) improved Wilson fermions with N_f=2 dynamical flavours. Preliminary results are presented for the four-fermion operator which determines the B_K parameter in tmQCD.Comment: 7 pages, 2 figures, talk presented at Lattice2006 (Renormalization

Non-perturbative renormalisation and running of BSM four-quark operators in Nf=2 QCD

We perform a non-perturbative study of the scale-dependent renormalisation factors of a complete set of dimension-six four-fermion operators without power subtractions. The renormalisation-group (RG) running is determined in the continuum limit for a specific Schrödinger Functional (SF) renormalisation scheme in the framework of lattice QCD with two dynamical flavours (Nf= 2). The theory is regularised on a lattice with a plaquette Wilson action and O(a)-improved Wilson fermions. For one of these operators, the computation had been performed in Dimopoulos et al. (JHEP 0805, 065 (2008). arXiv:0712.2429); the present work completes the study for the rest of the operator basis, on the same simulations (configuration ensembles). The related weak matrix elements arise in several operator product expansions; in Δ F= 2 transitions they contain the QCD long-distance effects, including contributions from beyond-Standard Model (BSM) processes. Some of these operators mix under renormalisation and their RG-running is governed by anomalous dimension matrices. In Papinutto et al. (Eur Phys J C 77(6), 376 (2017). arXiv:1612.06461) the RG formalism for the operator basis has been worked out in full generality and the anomalous dimension matrix has been calculated in NLO perturbation theory. Here the discussion is extended to the matrix step-scaling functions, which are used in finite-size recursive techniques. We rely on these matrix-SSFs to obtain non-perturbative estimates of the operator anomalous dimensions for scales ranging from O(Λ QCD) to O(MW)

Inflation Expectations, Wealth Perception, and Consumption Expenditure

The literature on wealth perception has been focused on the tax discounting of government bonds and, to a lesser extent, the Pesek-Saving effect. The authors consider here, in addition, the effects of expected inflation on wealth perception. In the resulting broadened framework, they find empirically that there is overwhelming expected-inflation discounting of money, but little or no tax discounting of bonds. This has far-reaching policy implications that are contrary to conventional wisdom. Based on an examination of equilibrium consumption, bond-financed budget deficits are, surprisingly, found to be more stimulative than money-financed deficits. More importantly, open-market operations not only turn out to be the least potent, but can in fact produce perverse effects.

The Perception of Government Bonds and Money as Net Wealth: An Integrated Approach

Although much work examines whether government bonds constitute net wealth, little attention focuses on whether government money does. Most analysts merely assert that government money is net wealth. In an inflationary environment, however, money experiences "expected-inflation discounting" just as bonds experience "tax discounting." Indeed, Chiang and Miller (1988) find empirical evidence suggesting that the private sector discounts money more heavily than bonds. This paper provides the theoretical underpinnings for the two types of discounting in an integrated approach, where both new money and new bonds can finance the interest on outstanding bonds.Government Bonds; Money

Twisted mass QCD and lattice approaches to the ΔI=1/2\Delta I = 1/2 rule

Twisted mass lattice QCD (tmQCD), generalised to four Wilson quark flavours, can be used for the computation of some weak matrix elements related to $\Delta I=1/2$ transitions. Besides eliminating unphysical zero modes, tmQCD may alleviate long-standing renormalisation problems of the four-quark operators which contribute to CP-conserving $K\to\pi$ transitions. With an active charm quark, the renormalisation of the $K\to\pi$ matrix elements requires at most the subtraction of a linearly divergent counterterm. Furthermore, in the (partially) quenched approximation the twist angles can be chosen so that only a finite counterterm needs to be subtracted.Comment: 35 pages;final version with references adde