Symanzik's Improvement Program Research Articles

Bs and Ds decay constants in three-flavor lattice QCD.

Capitalizing on recent advances in lattice QCD, we present a calculation of the leptonic decay constants f(B(s)) and f(D(s)) that includes effects of one strange sea quark and two light sea quarks via an improved staggered action. By shedding the quenched approximation and the associated lattice scale uncertainty, lattice QCD greatly increases its predictive power. Nonrelativistic QCD is used to simulate heavy quarks with masses between 1.5m(c) and m(b). We arrive at the following results: f(B(s))=260+/-7+/-26+/-8+/-5 and f(D(s))=290+/-20+/-29+/-29+/-6 MeV. The first quoted error is the statistical uncertainty, and the rest estimate the sizes of higher order terms neglected in this calculation. All of these uncertainties are systematically improvable by including another order in the weak coupling expansion, the nonrelativistic expansion, or the Symanzik improvement program.

On-shell-improved lattice QCD with staggered fermions

By using Symanzik's improvement program, we study on-shell improved lattice QCD with staggered fermions. We find that there are as many as 15 independent lattice operators of dimension of six~(including both gauge and fermion operators) which must be added to the unimproved action to form an $O(a^2)$ improved action. Among them, the total number of dimension-6 gauge operators and fermion bilinears is 5. The other 10 terms are four-fermion operators. At the tree-level and tadpole-improved tree-level, all the 10 four fermion operators are absent.

Perturbative improvement for lattice QCD: An update

Recent developments in the Symanzik improvement program for lattice QCD are reviewed.

QCD thermodynamics with an improved lattice action

We have investigated QCD with two flavors of degenerate fermions using the Symanzik-improvement program for both the gauge and fermion actions. Our study focuses on the deconfinement transition on an ${N}_{t}=4$ lattice. Having located the thermal transition, we performed zero temperature simulations nearby in order to compute hadronic masses and the static quark potential. We find that this action reduces lattice artifacts present in thermodynamics with the standard Wilson (gauge and fermion) actions. However, comparisons between improved Wilson and Kogut-Susskind actions show some disagreement.

Improving lattice quark actions

We explore the first stage of the Symanzik improvement program for lattice Dirac fermions, namely the construction of doubler-free, highly improved classical actions on isotropic as well as anisotropic lattices (where the temporal lattice spacing, a t , is smaller than the spatial one). Using field transformations to eliminate doublers, we derive the previously presented isotropic D234 action with O(a 3) errors, as well as anisotropic D234 actions with O(a 4) or O(a 3,a 4) errors. Besides allowing the simulation of heavy quarks within a relativistic framework, anisotropic lattices alleviate potential problems due to unphysical branches of the quark dispersion relation (which are generic to improved actions), facilitate studies of lattice thermodynamics, and allow accurate mass determinations for particles with bad signal/noise properties, like glueballs and P-state mesons. We also show how field transformations can be used to completely eliminate unphysical branches of the dispersion relation. Finally, we briefly discuss future steps in the improvement program.

Staggered fermions and their O( a) improvements

Expanding upon the arguments of Sharpe, we explicitly implement the Symanzik improvement program demonstrating the absence of order a terms in the staggered fermion action. We propose a general program to improve fermion operators to remove O( a) corrections from their matrix elements, and demonstrate this program for the examples of matrix elements of fermion bilinears and B K . We also determine the additional operators which must be added to improve the standard staggered fermion currents.

Implementation of Symanzik's improvement program for simulations of dynamical Wilson fermions in lattice QCD

We discuss the implementation of a Sheikholeslami-Wohlert term for simulations of lattice QCD with dynamical Wilson fermions as required by Symanzik's improvement program. We show that for the Hybrid Monte Carlo or Kramers equation algorithm standard even-odd preconditioning can be maintained. We design tests of the implementation using analytically and numerically computed cumulant expansions. We find that, for situations where the average number of Conjugate Gradient iterations exceeds 200, the overhead is only about 20%.

Improvement of the staggered fermion operators

We present a complete and detailed derivation of the finite lattice spacing corrections to staggered fermion matrix elements. Expanding upon arguments of Sharpe, we explicitly implement the Symanzik improvement program demonstrating the absence of order $a$ terms in the Symanzik improved action. We propose a general program to improve fermion operators to remove $O(a)$ corrections from their matrix elements, and demonstrate this program for the examples of matrix elements of fermion bilinears and $B_K$. We find the former does have $O(a)$ corrections while the latter does not.

Improved actions, the perfect action, and scaling by perturbation theory in Wilson's renormalization group: the two-dimensional O( N)-invariant non-linear σ-model in the hierarchical approximation

We propose a method using perturbation theory in the running coupling constant and the idea of scaling to determine improved actions for lattice field theories combining Wilson's renormalization group with Symanzik's improvement program. The method is based on the analysis of a single renormalization group transformation. We test it on the hierarchical O( N)-invariant σ-model in two dimensions.

Nonperturbative nature of the renormalization group.

By considering Symanzik's improvement program of the continuum limit of scalar lattice field theory as a tool to make contact between the perturbative renormalization and Wilson's theory, we show the nonperturbative nature of the renormalization. As a consequence we justify the conjecture of Parisi that the critical exponents may be estimated by field-theoretical methods without recourse to the \ensuremath{\epsilon} expansion.

Symanzik's improved actions from the viewpoint of the renormalization group

We investigate Symanzik's improvement program in a four-dimensional Euclidean scalar field theory with smooth momentum space cutoff. We use Wilson's renormalization group transformation to define the improved actions as a sequence of initial data for the effective action at the fundamental cutoff. This leads to a sequence of solutions to the renormalization group equation. We define the parameters of the improved actions implicitly by conditions on the effective action at a renormalization scale. The improved actions are close approximations to the continuum effective action. We prove their existence to every order of improvement and to every order of renormalized perturbation theory.

An analytic approach to the non-universal quantities in the critical domain: The lattice O( N) Heisenberg model for d = 3 and N → ∞

The O( N) Heisenberg model on a three-dimensional lattice is expanded in powers of N −1. The inverse correlation length and the magnetic field as funetions of the coupling constant (temperature) and magnetization are studied in the critical domain. A method to extract analytic expressions for non-universal quantities is developed. It is based on a systematic expansion in powers of the infrared cutoff mass around the continuum approximation. In particular our method is independent of the specific model studied here and can be applied to quite a general class of problems, e.g., Symanzik's improvement program.

Symanzik's improvement program

We review the theoretical basis and the possible applications of Symanzik's improvement program. We discuss also how the program may be extended in such a way to become compatible with the Osterwalder-Schrader positivity.

Symanzik's improvement program applied to the Gross-Neveu model at large N

Symanzik's improvement program for lattice actions is applied to the large- N limit of the Gross-Neveu model in the σ-field formulation. We construct a series of actions which suppress scaling violations up to order a 2 at tree level, the l-loop level, and to all orders in the large- N coupling λ = g 2 N. The fermions are treated by Wilson's method. We investigate the gap equation for the dynamical fermion mass. For this nonperturbative quantity it is shown that the correlation between the order of improvement and the order to which powers of ln a are supressed follows the pattern expected from perturbation theory, i.e. tree improvement removes the leading logarithm, l-loop improvement removes the next leading logarithm, etc. This property is also verified for related quantities such as the mass defined by the fall-off behavior of the fermion correlation function and the fermion-antifermion scattering amplitude in the singlet channel. As far as scaling is concerned, we observe that beyond tree improvement the β-function contains non-universal terms implying a difference between scaling and asymptotic scaling which increases with the order of improvement. Comparing for the dynamical mass the gain by supressing scaling violations versus the loss in asymptotic scale behavior, we argue that in a numerical approach the tree-improved action would yield the best estimate for the dynamical mass.

Perturbative construction and rigorous proof of locality of effective lattice actions for continuum theories

We construct an effective lattice action for a continuum theory, by fixing a set of collective coordinates which play the role of lattice variables. As opposed to Symanzik's improvement program our method involves no expansion in powers of the lattice spacing; in other words it simultaneously yields all “irrelevant” operators generated by the renormalization group to a given order in the continuum coupling constant. We are thus able to rigorously establish that the effective lattice action, for both smooth and singular collective coordinates, is local in the sense that long-range couplings decay exponentially over a distance independent of the mass gap of the theory; for asymtotically free theories this is interpreted as an existence proof of Wilson's infrared stable trajectories. Our methods are for convenience described in the context of dimensional φ 4, but can be easily extended to any theory with a set of collective coordinates which (i) are renormalizable and (ii) provide an infrared cutoff. Application to the 2-dimensional O( N) σ-model is, in particular, discussed; the technical problems of renormalization posed by gauge invariance are, on the other hand, not dealt with in this paper, although our treatment of singular coordinates is meant as a prelude to them. A by-product of our proofs is the derivation of an interesting factorization property of Zimmermann-subtracted diagrams.