The mass dependence of the renormalization constant for the vector current Zv extracted from lattice simulations near the strange quark mass. The two functional forms, shown as red and blue lines, differ by higher order discretization errors. By comparing with measurements at two much heavier values of quark mass, we are able to ascertain that the fit corresponding to the blue line works better when extrapolating up to the charm quark mass.

Research Objectives
Our goal is to calculate B_K, the matrix element that gives rise to charge parity (CP) violation in kaon decays, using three flavors of dynamical quarks. Our analyses show that this is best achieved using improved staggered fermions.

Computational Approach
Our lattice QCD simulations involve five steps: (1) Generation of a statistical sample of background gauge configurations which provides an accurate representation of the functional integral. This is done by Monte Carlo methods and using molecular dynamics evolution with the R algorithm. (2) Calculation of quark propagators by inversion of a very large but sparse matrix, using a BiCGstab iterative solver. (3) Calculation of correlation functions, which is done by tying together the gauge links and quark propagators. (4) Calculation of improvement constants and normalization factors. (5) Analysis of the correlation functions to extract desired masses of hadrons and matrix elements within these states.

Accomplishments
In FY 2001 we completed our study of O(a) improvement of quenched Wilson fermions. Calculations of the renormalization and improvement constants for quark bilinears were carried out at three values of the coupling, = 6.0, 6.2, 6.4. A scaling analyses of the residual discretization errors has been carried out. We have also completed the theoretical analyses required to extend the calculation to dynamical quarks. This analyses shows that all but a few unimportant constants can be determined, but the calculation will be extremely CPU time intensive. In view of the CPU requirements, and on comparing this cost against improved staggered fermions, we conclude that for the proposed calculation of B_K with three dynamical flavors of quarks, it is better to use improved staggered fermions. With this in view, we have already begun quenched staggered calculations to develop improved operators.

Significance
Our proposed calculation of B_K will have significant impact on the phenomenology of CP violation. Current phenomenology uses quenched lattice estimates of B_K, and we propose to remove the largest remaining uncertainty due to quenching. This is estimated to be a 5-10% effect, and removing it will pin down this important parameter, leading to a better estimate of the Cabibbo-Kobayashi-Maskawa matrix.

Publications
T. Bhattacharya, R. Gupta, W. Lee, and S. Sharpe, "Order a improved renormalization constants," Phys. Rev D 63, 074505 (2001).

T. Bhattacharya, S. Chandrasekharan, R. Gupta, W. Lee, and S. Sharpe, "Non-perturbative renormalization constants using Ward identities," Phys. Lett. B 461, 79 (1999).