John Negele, Christoph Best, Patrick Dreher, Andrew Pochinsky, and Uwe-Jens Wiese, Massachusetts Institute of Technology
Robert Edwards, Thomas Jefferson National Accelerator Facility
Thomas Lippert, Hartmut Neff, and Klaus Schilling, University of Wuppertal

Because topological properties of QCD are associated with zero modes of the Dirac operator, properties such as the topological susceptibility and the mass of the meson can be calculated most efficiently by calculating the low eigenmodes of the Dirac operator. This figure shows how, in a space of 1.57 million states, the lowest 300 eigenmodes already give a precise determination of the topological charge of a gluon configuration. Using extensive calculations at NERSC, this truncated eigenmode expansion has recently been used to calculate the mass of the meson with unprecedented statistical precision.

Research Objectives
The major focus this work is on understanding the role of instantons and their associated quark zero modes in nucleon structure, and using the quark zero modes to calculate the sea quark content of the nucleon.

Computational Approach
We calculate the low eigenmodes of the Dirac operator using the Arnoldi method, which has compelling advantages for our work. One advantage is that since it works in a fixed dimension space, there is no degradation of orthogonality and corresponding loss or duplication of modes. A second advantage, when applied to the non-hermitian Dirac operator, is its insensitivity to the quark mass, which makes it extremely useful near the chiral limit of low pion mass. We have two complementary implementations. One is an exploratory code in which we can control the region of eigenvalues at will. The other uses the robust and well optimized PARPACK package from ORNL combined with Chebyshev acceleration.

Accomplishments
NERSC resources enabled us to calculate on 400 configurations the low eigenmodes of the hermitian Dirac operator, which is G₅ times the standard Dirac operator. This showed that, contrary to our original expectation, at the large quark masses relevant to current unquenched calculation, the eigenmode expansion of the hermitian Dirac operator has superior convergence properties to that of the standard operator. An unfortunate property of the hermitian Dirac operator, however, is the need to calculate new eigenmodes at each quark mass, in contrast to the standard operator, for which a single set of eigenmodes applies for all masses. Hence, the computational needs of the project have increased relative to our original expectation in order to calculate eigenmodes at several masses.

Significance
Ever since the discovery of quarks in the nucleon, tremendous experimental effort and resources have been devoted to the measurement of the detailed quark and gluon structure of the nucleon, and theorists have sought to understand this structure from first principles. In a clever and difficult series of experiments at Bates and Jefferson Lab which are now finally coming to fruition, experimentalists have used the interference between parity-conserving and parity-violating electron scattering amplitudes to measure the contributions of strange quarks to electric and magnetic form factors. Given the investment of effort and resources in these fundamental experiments, it is extremely important to develop the means to calculate the strange quark content of the nucleon reliably using lattice QCD. We are developing a new method which can attain a higher level of statistical accuracy than existing methods, and will provide the essential quark zero modes necessary for these calculations. In addition to elucidating the physics for timely parity-violating electron scattering experiments, this new method should also enable the evaluation of the disconnected diagrams encountered in deep inelastic electron scattering.

Publications
W. Detmold, W. Melnitchouk, J. W. Negele, D. B. Renner, and A. W. Thomas, "Chiral extrapolation of lattice moments of proton quark distributions," Phys. Rev. Lett. 87, 172001 (2001); hep-lat/0103006.

D. Dolgov, R. Brower, S. Capitani, J. W. Negele, A. Pochinsky, D. Renner, N. Eicker, T. Lippert, K. Schilling, R. G. Edwards, and U. M. Heller, "Moments of structure functions in full QCD," Nucl. Phys. Proc. Suppl. 94, 303 (2001); hep-lat/0011010.

H. Neff, N. Eicker, T. Lippert, J. W. Negele, and K. Schilling, "On the low fermionic eigenmode dominance in QCD on the lattice," Phys. Rev. D (in press): hep-lat/0106016.

http://www-ctp.mit.edu/~negele/