How to Accurately Extract the Running Coupling of QCD from Lattice Potential Data

By (a) using an expression for the LATTICE potential of QCD in terms of a CONTINUUM running coupling and (b) globally parameterizing this coupling to interpolate between 2- (or higher-) loop QCD in the UV and the flux tube prediction in the IR, we can perfectly fit lattice data for the potential down to ONE lattice spacing and at the same time extract the running coupling to high precision. This allows us to quantitatively check the accuracy of 2-loop evolution, compare with the Lepage-Mackenzie estimate of the coupling extracted from the plaquette, and determine the scale $r_0$ ten times more accurately than previously possible. For pure SU(3) we find that the coupling scales on the percent level for $\beta \geq 6$.Comment: 3 pages Latex incl. 2 figures, uses espcrc2.sty, contribution to LATTICE '9

Sine-Gordon =/= Massive Thirring, and Related Heresies

By viewing the Sine-Gordon and massive Thirring models as perturbed conformal field theories one sees that they are different (the difference being observable, for instance, in finite-volume energy levels). The UV limit of the former (SGM) is a gaussian model, that of the latter (MTM) a so-called {\it fermionic} gaussian model, the compactification radius of the boson underlying both theories depending on the SG/MT coupling. (These two families of conformal field theories are related by a ``twist''.) Corresponding SG and MT models contain a subset of fields with identical correlation functions, but each model also has fields the other one does not, e.g. the fermion fields of MTM are not contained in SGM, and the {\it bosonic} soliton fields of SGM are not in MTM. Our results imply, in particular, that the SGM at the so-called ``free-Dirac point'' $\beta^2 = 4\pi$ is actually a theory of two interacting bosons with diagonal S-matrix $S=-1$, and that for arbitrary couplings the overall sign of the accepted SG S-matrix in the soliton sector should be reversed. More generally, we draw attention to the existence of new classes of quantum field theories, analogs of the (perturbed) fermionic gaussian models, whose partition functions are invariant only under a subgroup of the modular group. One such class comprises ``fermionic versions'' of the Virasoro minimal models.Comment: 50 pages (harvmac unreduced), CLNS-92/1149, ITP-SB-92-3

Non-Perturbative Improvement of the Anisotropic Wilson QCD Action

We describe the first steps in the extension of the Symanzik O($a$) improvement program for Wilson-type quark actions to anisotropic lattices, with a temporal lattice spacing smaller than the spatial one. This provides a fully relativistic and computationally efficient framework for the study of heavy quarks. We illustrate our method with accurate results for the quenched charmonium spectrum.Comment: LATTICE98(improvement), 3 pages, 4 figure

Rg Flows in the DD-Series of Minimal Cfts

Using results of the thermodynamic Bethe Ansatz approach and conformal perturbation theory we argue that the $\phi_{1,3}$-perturbation of a unitary minimal $(1+1)$-dimensional conformal field theory (CFT) in the $D$-series of modular invariant partition functions induces a renormalization group (RG) flow to the next-lower model in the $D$-series. An exception is the first model in the series, the 3-state Potts CFT, which under the \ZZ_2-even $\phi_{1,3}$-perturbation flows to the tricritical Ising CFT, the second model in the $A$-series. We present arguments that in the $A$-series flow corresponding to this exceptional case, interpolating between the tetracritical and the tricritical Ising CFT, the IR fixed point is approached from ``exactly the opposite direction''. Our results indicate how (most of) the relevant conformal fields evolve from the UV to the IR CFT.Comment: 30 page

The Anisotropic Wilson Gauge Action

Anisotropic lattices, with a temporal lattice spacing smaller than the spatial one, allow precision Monte Carlo calculations of problems that are difficult to study otherwise: heavy quarks, glueballs, hybrids, and high temperature thermodynamics, for example. We here perform the first step required for such studies with the (quenched) Wilson gauge action, namely, the determination of the renormalized anisotropy $\xi$ as a function of the bare anisotropy $\xi_0$ and the coupling. By, essentially, comparing the finite-volume heavy quark potential where the quarks are separated along a spatial direction with that where they are separated along the time direction, we determine the relation between $\xi$ and $\xi_0$ to a fraction of 1% for weak and to 1% for strong coupling. We present a simple parameterization of this relation for $1\leq \xi \leq 6$ and $5.5 \leq \beta \leq \infty$, which incorporates the known one-loop result and reproduces our non-perturbative determinations within errors. Besides solving the problem of how to choose the bare anisotropies if one wants to take the continuum limit at fixed renormalized anisotropy, this parameterization also yields accurate estimates of the derivative $\partial\xi_0/\partial\xi$ needed in thermodynamic studies.Comment: 24 pages, LaTeX, 15 ps figures (added high statistics simulations confirming our results; to appear in Nucl. Phys. B

Kinks in Finite Volume

A (1+1)-dimensional quantum field theory with a degenerate vacuum (in infinite volume) can contain particles, known as kinks, which interpolate between different vacua and have nontrivial restrictions on their multi-particle Hilbert space. Assuming such a theory to be integrable, we show how to calculate the multi-kink energy levels in finite volume given its factorizable $S$-matrix. In massive theories this can be done exactly up to contributions due to off-shell and tunneling effects that fall off exponentially with volume. As a first application we compare our analytical predictions for the kink scattering theories conjectured to describe the subleading thermal and magnetic perturbations of the tricritical Ising model with numerical results from the truncated conformal space approach. In particular, for the subleading magnetic perturbation our results allow us to decide between the two different $S$-matrices proposed by Smirnov and Zamolodchikov.Comment: 48/28 pages + 10 figs, 4 in pictex, the rest in postscript files attached at the en

A quark action for very coarse lattices

We investigate a tree-level O(a^3)-accurate action, D234c, on coarse lattices. For the improvement terms we use tadpole-improved coefficients, with the tadpole contribution measured by the mean link in Landau gauge. We measure the hadron spectrum for quark masses near that of the strange quark. We find that D234c shows much better rotational invariance than the Sheikholeslami-Wohlert action, and that mean-link tadpole improvement leads to smaller finite-lattice-spacing errors than plaquette tadpole improvement. We obtain accurate ratios of lattice spacings using a convenient ``Galilean quarkonium'' method. We explore the effects of possible O(alpha_s) changes to the improvement coefficients, and find that the two leading coefficients can be independently tuned: hadron masses are most sensitive to the clover coefficient, while hadron dispersion relations are most sensitive to the third derivative coefficient C_3. Preliminary non-perturbative tuning of these coefficients yields values that are consistent with the expected size of perturbative corrections.Comment: 22 pages, LaTe

The Schr\"odinger Functional for Improved Gluon and Quark Actions

The Schr\"odinger Functional (quantum/lattice field theory with Dirichlet boundary conditions) is a powerful tool in the non-perturbative improvement and for the study of other aspects of lattice QCD. Here we adapt it to improved gluon and quark actions, on isotropic as well as anisotropic lattices. Specifically, we describe the structure of the boundary layers, obtain the exact form of the classically improved gauge action, and outline the modifications necessary on the quantum level. The projector structure of Wilson-type quark actions determines which field components can be specified at the boundaries. We derive the form of O(a) improved quark actions and describe how the coefficients can be tuned non-perturbatively. There is one coefficient to be tuned for an isotropic lattice, three in the anisotropic case. Our ultimate aim is the construction of actions that allow accurate simulations of all aspects of QCD on coarse lattices.Comment: 39 pages, LaTeX, 11 embedded eps file

The (LATTICE) QCD Potential and Running Coupling: How to Accurately Interpolate between Multi-Loop QCD and the String Picture

We present a simple parameterization of a running coupling constant, defined via the static potential, that interpolates between 2-loop QCD in the UV and the string prediction in the IR. Besides the usual \Lam-parameter and the string tension, the coupling depends on one dimensionless parameter, determining how fast the crossover from UV to IR behavior occurs (in principle we know how to take into account any number of loops by adding more parameters). Using a new Ansatz for the LATTICE potential in terms of the continuum coupling, we can fit quenched and unquenched Monte Carlo results for the potential down to ONE lattice spacing, and at the same time extract the running coupling to high precision. We compare our Ansatz with 1-loop results for the lattice potential, and use the coupling from our fits to quantitatively check the accuracy of 2-loop evolution, compare with the Lepage-Mackenzie estimate of the coupling extracted from the plaquette, and determine Sommer's scale $r_0$ much more accurately than previously possible. For pure SU(3) we find that the coupling scales on the percent level for $\beta\geq 6$.Comment: 47 pages, incl. 4 figures in LaTeX [Added remarks on correlated vs. uncorrelated fits in sect. 4; corrected misprints; updated references.