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

Renormalized quark mass in the scheme vs. the bare quark mass on the three lattices with the same physical volume but different lattice spacings, calculated with the overlap fermion action, which is known to possess lattice chiral symmetry. The renormalized quark mass does not have an additive part due to chiral symmetry. In fact, the linear fit including the smallest 5, 7, and 8 quark masses for = 5.7, 5.85, and 6.0, respectively, shows that the intercepts are of the order of 10-3 and are consistent with zero. This is proof that the overlap fermion action resembles the continuum physics and overcomes the difficulty faced by the previous Wilson-type fermion actions

Research Objectives
We plan to study chiral condensate, decay constants, hadron and quark masses, chiral logs, nucleon form factors, and the sea quark contributions, such as the strangeness content in the nucleon. Our goal is to push the calculation of various fundamental physical quantities to the continuum limit, the chiral limit, and the large-volume limit within the quenched approach.

Computational Approach
We have implemented Neuberger's overlap fermion to test chiral symmetry and scaling via the calculation of hadron masses, quark masses, and the chiral condensate. The new overlap fermion action involves a matrix sign function. We approximate the square root of the matrix by the optimal rational polynomial approach, and we invert the matrix with conjugate gradient with a 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
Our calculation on small volumes and three different lattice spacings yields encouraging results. The chiral symmetry breaking due to numerical implementation is limited to less than 1% for the smallest quark mass, and the scaling of hadron masses shows that there is no O(a) error, and even the O(a²) error is small. We have implemented the overlap fermion on large quenched lattices (20⁴ with a = 0.15 F) with 24 different quark masses, with the smallest one close to the physical quark mass. This requires a delicate balance between projecting enough small eigenvalues for chiral symmetry and faster convergence in the matrix inversion, and not exceeding the memory on 64 nodes.

We have accumulated data on 40 gauge configurations. We have fairly accurate results on the pion mass, and from it we have extracted the chiral log reasonably reliably. The other hadron masses are still noisy, but we begin to see a trend that the isovector scalar meson mass and that of the axial-vector meson cross over for light quark masses. This is consistent with experiments and is now seen on the lattice for the first time.

Significance
Chiral symmetry is a fundamental symmetry in QCD that governs low-energy hadron structure and dynamics. The lack of lattice formulation of this symmetry has so far hindered reliable extrapolation of lattice results to the physical pion mass region. With the advent of Neuberger's overlap fermion, physical observables sensitive to this symmetry should be calculated more reliably, and they can be compared with experiments more readily and directly.

Publications
J. Christensen, T. Draper, and Craig McNeile, "Renormalization of the lattice heavy quark effective theory Isgur-Wise function," Phys. Rev. D 62, 114006 (2000); hep-lat/9912046.

L. Lin, K. F. Liu, and J. Sloan, "A noisy Monte Carlo algorithm," Phys. Rev. D 61, 74505 (2000).

K. F. Liu, S. J. Dong, T. Draper, D. Leinweber, J. Sloan, W. Wilcox, and R. M. Woloshyn, "Valence QCD and quark model," Phys. Rev. D 59, 112001 (1999).

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