arXiv subjects
Publications and source records attributed to Stephen R. Sharpe.
We summarize the extent to which one can use Ward identities to non-perturbatively improve flavor singlet and non-singlet bilinears with three flavors of non-degenerate dynamical Wilson-like fermions.
It has been proposed that partially quenched chiral perturbation theory can be used together with partially quenched lattice QCD to determine the low-energy constants of real QCD. We point out a complication of this approach and show how it can be resolved. Partial quenching changes the chiral symmetry group and introduces a new independent four-derivative operator. We explore the effects of this new operator. We describe tree-level scattering processes to which it contributes, and show how they can be used, in principle, to deterimine its unknown coefficient. We also calculate its lowest-order (one-loop) contribution to charged meson masses and decay constants. In addition, we present the form of the analytic next-to-next-to-leading order meson mass and decay constant corrections, which are needed for lattice fits.
We calculate one-loop matching factors for bilinear operators composed of improved staggered fermions. We compare the results for different improvement schemes used in the recent literature, all of which involve the use of smeared links. These schemes aim to reduce, though not completely eliminate, O(a^2) discretization errors. We find that all these improvement schemes substantially reduce the size of matching factors compared to unimproved staggered fermions. The resulting corrections are comparable to, or smaller than, those found with Wilson and domain-wall fermions. In the best case (``Fat-7'' and mean-field improved HYP links) the corrections are 10 % or smaller at 1/a = 2 GeV.
This is an approximate reconstruction of the panel discussion on chiral extrapolation of physical observables. The session consisted of brief presentations from panelists, followed by responses from the panel, and concluded with questions and comments from the floor with answers from panelists. In the following, the panelists have summarized their statements, and the ensuing discussion has been approximately reconstructed from notes.
We calculate one-loop matching factors for bilinear operators composed of improved staggered fermions. We compare the results for different improvement schemes used in the recent literature, including the HYP action and an action close to the Asqtad action. We find that all improvement schemes substantially reduce the size of the one-loop contributions to matching factors. The resulting corrections are comparable to, or smaller than, those found with Wilson and domain-wall fermions.
Recently, a method for O(a) improvement of composite operators has been proposed which uses the large momentum behavior of fixed gauge quark and gluon correlation functions (G. Martinelli et al., hep-lat/0106003). A practical problem with this method is that a particular improvement coefficient, $c_{NGI}$, which has a gauge non-covariant form, is difficult to determine. Here I work out the size of the errors made in improvement coefficients and physical quantities if one does not include the $c_{NGI}$ term.
I discuss the properties of pions in ``partially quenched'' theories, i.e. those in which the valence and sea quark masses, $m_V$ and $m_S$, are different. I point out that for lattice fermions which retain some chiral symmetry on the lattice, e.g. staggered fermions, the leading order prediction of the chiral expansion is that the mass of the pion depends only on $m_V$, and is independent of $m_S$. This surprising result is shown to receive corrections from loop effects which are of relative size $m_S \ln m_V$, and which thus diverge when the valence quark mass vanishes. Using partially quenched chiral perturbation theory, I calculate the full one-loop correction to the mass and decay constant of pions composed of two non-degenerate quarks, and suggest various combinations for which the prediction is independent of the unknown coefficients of the analytic terms in the chiral Lagrangian. These results can also be tested with Wilson fermions if one uses a non-perturbative definition of the quark mass.
The Schwinger model is used to study the artifacts of quenching in a controlled way. The model is solved on a finite-temperature cylinder of circumference $β=1/T$ with bag-inspired local boundary conditions at the two ends $x^1=0$ and $x^1=L$ which break the $γ_5$-invariance and thus play the role of a small quark mass. The quenched chiral condensate is found to diverge exponentially as $L\to\infty$, and to diverge (rather than melt as for $N_{\rm f}\geq1$) if the high-temperature limit $β\to0$ is taken at finite box-length $L$. We comment on the generalization of our results to the massive quenched theory, arguing that the condensate is finite as $L\to\infty$ and proportional to $1/m$ up to logarithms.
I summarize recent progress in lattice gauge theory, with particular emphasis on results from numerical simulations. A major success has been the determination of the light hadron spectrum in the quenched approximation with sufficient accuracy to expose statistically significant disagreements with the experimental spectrum. The differences are, however, fairly small, $\sim 5-10%$. The data are also accurate enough to show evidence for artifacts of quenching predicted by chiral perturbation theory. I give an update on results for light quark masses, the kaon B-parameter, and the decay constants and B-parameters of heavy-light mesons. Most of these are known in the quenched approximation to $\sim 10%$ accuracy or better, and preliminary estimates of quenching errors are of comparable size. One exception is the light quark masses, for which the quenching errors appear to be larger. I discuss the computational requirements for simulations of QCD with all approximations controlled, and argue that they will likely begin once computers sustain about 10 Teraflops. This is 30-40 times faster than present state-of-the-art machines. This estimate assumes that improvements in the discretization of lattice fermions are sufficient to allow continuum extrapolations to be made with a minimum lattice spacing of $\approx 0.1 $fm. I review results obtained with improved discretizations and conclude that they satisfy this requirement in most cases. Examples of successful improvement include the calculation of the glueball spectrum and excited heavy-quark potentials in pure Yang-Mills theory. Finally, I discuss recent developments which may allow simulations of QCD with full chiral symmetry even at finite lattice spacing.
Some phase transitions of cosmological interest may be weakly first-order and cannot be analyzed by a simple perturbative expansion around mean field theory. We propose a simple two-scalar model--the cubic anisotropy model--as a foil for theoretical techniques to study such transitions, and we review its similarities and dissimilarities to the electroweak phase transition in the early universe. We present numerical Monte Carlo results for various discontinuities across very weakly first-order transitions in this model and, as an example, compare them to epsilon-expansion results. For this purpose, we have computed through next-to-next-to-leading order in epsilon.
I describe recent developments in quenched chiral perturbation theory (QChPT) and the status of weak matrix elements involving light quarks. I illustrate how, with improved statistical errors, and with calculations of the masses of baryons containing non-degenerate quarks, there is now a clear need for extrapolations of higher order than linear in the quark mass. I describe how QChPT makes predictions for the functional forms to use in such extrapolations, and emphasize the distinction between contributions coming from chiral loops which are similar to those present in unquenched theories, and those from $η'$ loops which are pure quenched artifacts. I describe a fit to the baryon masses using the predictions of QChPT. I give a status report on the numerical evidence for $η'$ loops, concluding that they are likely present, and are characterized by a coupling $δ=0.1-0.2$. I use the difference between chiral loops in QCD and quenched QCD to estimate the quenching errors in a variety of quantities. I then turn to results for matrix elements, largely from quenched simulations. Results for quenched decay constants cannot yet be reliably extrapolated to the continuum limit. By contrast, new results for $B_K$ suggest a continuum, ``quenched'' value of $B_K(NDR, 2 GeV) = 0.5977 \pm 0.0064 \pm 0.0166$, based on a quadratic extrapolation in $a$. The theoretical basis for using a quadratic extrapolation has been confirmed. For the first time there is significant evidence that unquenching changes $B_K$, and my estimate for the value in QCD is $B_K(NDR, 2 GeV) = 0.66 \pm 0.02 \pm 0.11$. Here the second error is a conservative estimate of the systematic error due to uncertainties in the effect of quenching. A less conservative viewpoint reduces $0.11$ to $0.03$.
We develop chiral perturbation theory for baryons in quenched QCD. Quenching (the elimination of diagrams containing virtual quark loops) is achieved by extending the Lagrangian method of Bernard and Golterman, and is implemented in a theory where baryons are treated as fixed velocity sources. Our method requires that the octet baryons be represented by a three index tensor rather than by the usual matrix field. We calculate the leading non-analytic corrections to the masses of octet and decuplet baryons. In QCD these are proportional to $M_π^3$. We find that quenching alters the $M_π^3$ terms, but does not completely remove them. In addition, we find non-analytic contributions to baryon masses proportional to $M_π$ and $M_π^2 \log M_π$. These terms, which are artifacts of quenching, dominate over the $M_π^3$ terms for sufficiently small quark masses. We also point out various pecularities of the quenched theory, most notably that the $Δ$ baryon can decay (if kinematically allowed), in the sense that its two point function will be dominated at long Euclidean times by a nucleon plus pion intermediate state.
We formulate quenched chiral perturbation theory for heavy-light mesons coupled to pions, and calculate the one-loop chiral logarithmic corrections to $f_B$, $f_{B_{s}}$, $B_B$ and $B_{B_{s}}$. We also calculate these corrections for ``partially quenched'' theories. In both theories, the chiral logarithms diverge in the chiral limit, indicating that (partially) quenched theories should not be used to study this limit. Comparing the chiral logarithms to those in QCD, we estimate the errors caused by (partial) quenching. By forming suitable ratios, we can reduce the uncertainties in our estimates.
This is the written version of four lectures given at the 1994 TASI. My aim is to explain the essentials of lattice calculations, give an update on (though not a review of) the present status of calculations of phenomenologically interesting quantities, and to provide an understanding of the various sources of uncertainty in the results. I illustrate the important issues using the examples of the kaon B-parameter ($B_K$) and various quantities related to $B$-meson physics. Contents: 1. Why do we need lattice QCD? 2. Basics of Euclidean Lattice Field Theory. 3. Discretizing QCD. 4. Simulations. 5. Numerical Results from quenched QCD. 6. Anatomy of a calculation: $B_K$. 7. Heavy mesons on the lattice. 8. A final flourish
We develop quenched chiral perturbation theory for baryons using the graded-symmetry formalism of Bernard and Golterman and calculate non-analytic contributions to the baryon masses coming from quenched chiral loops. The usual term proportional to $m_{q}^{3/2}$ is substantially altered due to the cancellation of diagrams with internal quark loops. In addition, the $η'$ ``hairpin'' vertex leads to a new correction, proportional to $m_{q}^{1/2}$. We compare our results to numerical lattice data and use them to estimate the size of the quenching error in the octet baryon masses.