Pseudoscalar decay constant of the pseudoscalar meson is plotted as a function of the quark mass. The calculation is done on the quenched 204 lattice at lattice spacing a = 0.148 fm with the overlap fermion. At sufficiently small quark mass, the decay constant appears to diverge. This verifies the quenched chiral log behavior predicted from the quenched chiral perturbation theory.

Keh-Fei Liu, Terrence Draper, and Shao-Jing Dong, University of Kentucky

Research Objectives
The proposed numerical study of nucleon structure will produce results to be compared with experimental results already obtained and to be obtained from the DOE-supported nuclear and high energy labs.

Computational Approach
The new overlap fermion action involves a matrix sign function. We approximate the square root of the matrix by the optimal rational fraction approach, and we invert the matrix with conjugate gradient with multiple mass algorithm. To speed up the convergence, we project out some of the smallest eigenvalues and treat the sign function of these states exactly. The overall inversion of the quark matrix to obtain the quark propagator is also done with conjugate gradient with multiple quark masses.

Accomplishments
We have carried out a production run on a quenched 20⁴ lattice with lattice spacing at 0.15 fm. The size of each direction is thus 3 fm, which is about the largest volume that any lattice calculation has attempted. We have also pushed the pion mass to as low as ~ 200 MeV, which is also a record. We found many interesting features of chiral symmetry at such a low mass and large volume. First of all, we found that the zero mode contribution to the pion propagator is not negligible at small time region, and its contamination needs to be avoided when extracting pion properties such as the pion mass and decay constant. We found that after nonperturbative renormalization, the pseudoscalar meson decay constant is very constant in the range between the strange and up/down quark mass. In view of the fact that it has a very small statistical error (less than 2% with only 63 gauge configuration), it is a physical quantity free of quenched chiral log, and is thus an ideal quantity to set the lattice scale. On the other hand, we have clearly observed the quenched chiral log in the pseudoscalar matrix elements at small quark masses, and it is in agreement with the prediction based on the U(1) anomaly with the mass.

We have computed the a₀ (isovector-scalar) and a₁ (isovector-axial) masses. We found that the propagator of a₀ becomes negative in the large time separation when the quark mass becomes small and that of a₁ flattens off. We understand them as the quenched artifacts due to decays to the p and p respectively. We have calculated one- and two-loops in the effective meson theory to help fit them and extract the a₀ and a₁. This is clear and concrete evidence for the quenched artifacts which people have been expecting to see.

Significance
With the advent of Neuberger's overlap fermion, which has the promise of exact lattice chiral symmetry for finite lattice spacing, it is time to calculate the fundamental quantities such as the quark condensate, chiral logs in hadron masses, light quark masses, decay constants, and nucleon form factors with this new action. Besides the continuum and large volume limits, this new action allows an extrapolation and perhaps interpolation to the physical quark mass region, one of the last frontiers in lattice QCD in the quenched approximation. With the guidance of chiral symmetry, physical observables which are sensitive to this symmetry should be calculated much more reliably than before, and they can be compared with experiments more readily and directly.

Publications
K. F. Liu, S. J. Dong, F. X. Lee, and J. B. Zhang, "Overlap fermions on a 20⁴ lattice," Nucl. Phys. Proc. Suppl. 94, 752 (2001); hep-lat 0011072.

S. J. Dong, F. X. Lee, K. F. Liu, and J. B. Zhang, "Chiral symmetry, quark mass, and scaling of the overlap fermions," Phys. Rev. Lett. 85, 5051 (2000); hep-lat/0006004.

N. Mathur, S. J. Dong, K. F. Liu, L. Mankiewicz, and N. C. Mukhopadhyay, "Quark orbital angular momentum from lattice QCD," Phys. Rev. D 62, 114504 (2000); hep-lat/9912289.

http://www.pa.uky.edu/~liu