arXiv subjects
Publications and source records attributed to T. R. Klassen.
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.
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 $ξ$ as a function of the bare anisotropy $ξ_0$ 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 $ξ$ and $ξ_0$ to a fraction of 1% for weak and to 1% for strong coupling. We present a simple parameterization of this relation for $1\leq ξ\leq 6$ and $5.5 \leq β\leq \infty$, 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 $\partialξ_0/\partialξ$ needed in thermodynamic studies.
The ALPHA collaboration has determined the O(a) improved Wilson quark action for lattice spacings $a\leq 0.1$ fm, in the quenched approximation. We extend this result to coarser lattices, $a\leq 0.17$ fm, and calculate the hadron spectrum on them. The large range of lattice spacings obtained by combining our results with earlier ones on finer lattices, allow us to present a convincing demonstration of the efficiency of non-perturbative O(a) improvement. We find that scaling violations of the hadron masses studied drop from 30-40% for the unimproved Wilson action on the coarsest lattice to only 2-3%.
Accurate determinations of the physical scale of a lattice action are required to check scaling and take the continuum limit. We present a high statistics study of the static potential for the SU(3) Wilson gauge action on coarse lattices ($5.54 \leq β\leq 6.0$). Using an improved analysis procedure we determine the string tension and the Sommer scale $r_0$ (and related quantities) to 1% accuracy, including all systematic errors. Combining our results with earlier ones on finer lattices, we present parameterizations of these quantities that should be accurate to about 1% for $5.6 \leq β\leq 6.5$. We estimate the $\La$-parameter of quenched QCD to be $\La_\MSb = 247(16)$ MeV.
After describing the Schrödinger functional for standard and improved gluon and quark actions we present results for the non-perturbative clover coefficients of the SW quark action coupled to the Wilson plaquette action for $β\geq 5.7$, as well as the Lüscher-Weisz one-loop tadpole improved gauge action, both in the quenched approximation.
The Schrödinger 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.
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_t^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.
I present a brief summary of the status and prospects of improved Wilson-type quark actions for coarse lattice simulations. My conclusions are optimistic.
We describe two ideas useful in the construction of highly improved quark actions for simulations on coarse lattices: (1) Field transformations to solve the doubler problem without destroying tree-level improvement for on- or off-shell quantities. (2) Anisotropic lattices with smaller temporal than spatial lattice spacing. We present first simulation results for the quenched light hadron and charmonium spectra obtained with a D234 action on anisotropic lattices.
We present sum representations for all characters of the unitary Virasoro minimal models. They can be viewed as fermionic companions of the Rocha-Caridi sum representations, the latter related to the (bosonic) Feigin-Fuchs-Felder construction. We also give fermionic representations for certain characters of the general $(G^{(1)})_k \times (G^{(1)})_l \over (G^{(1)})_{k+l}}$ coset conformal field theories, the non-unitary minimal models ${\cal M}(p,p+2)$ and ${\cal M}(p,kp+1)$, the $N$=2 superconformal series, and the $\ZZ_N$-parafermion theories, and relate the $q\to 1$ behaviour of all these fermionic sum representations to the thermodynamic Bethe Ansatz.
We review recent results concerning the representation of conformal field theory characters in terms of fermionic quasi-particle excitations, and describe in detail their construction in the case of the integrable three-state Potts chain. These fermionic representations are $q$-series which are generalizations of the sums occurring in the Rogers-Ramanujan identities. (To appear in the proceedings of ``Yang-Baxter Equations in Paris'', July 1992, J.-M.~Maillard (ed.).)
We present fermionic quasi-particle sum representations for some of the characters (or branching functions) of ~${(G^{(1)})_1 \times (G^{(1)})_1 ø(G^{(1)})_2}$ ~for all simply-laced Lie algebras $G$. For given $G$ the characters are written as the partition function of a set of rank~$G$ types of massless quasi-particles in certain charge sectors, with nontrivial lower bounds on the one-particle momenta. We discuss the non-uniqueness of the representations for the identity character of the critical Ising model, which arises in both the $A_1$ and $E_8$ cases.