BK with high precision

We calculated BK in partially quenched QCD using MILC gauge configurations. We tried both degenerate quark mass combinations and non-degenerate quark mass combinations. The results are analyzed using the results of partially quenched (PQ) chiral perturbation theory. The fitting of the data with non-degenerate quark masses to the prediction of PQ chiral perturbation theory illustrates that we can determine, in principle, all the low energy constants using only degenerate quark mass combinations. However, the quality of fitting was significantly better with non-degenerate quark mass combinations. We plan to extend this preliminary study to large scale numerical simulation such that we can fit the data to the form suggested by the staggered chiral perturbation theory in near future.

Hadron spectrum

We plan to measure staggered meson spectrum in quenched QCD in order to probe the improvement by HYP/Fat7 fat links. First, we will have a look at the pion multiplets to measure the splittings between the Goldstone and non-Goldstone pions. Second, we will investigate the rho, a1, a0, b1 spectrum to understand the effect of improvement. By using HYP/Fat7 fat links, we expect, the splitting would be reduced so significantly that the chiral-log corrections originating from the non-Goldstone pions become quite important. Hence, this research will provide a basic background work for other researches including weak matrix elements. We also plan to measure the staggered meson spectrum using MILC gauge configurations. The results will be used to analyze the data of weak matrix elements using the prediction of the staggered chiral perturbation theory.


We are calculating the size of the direct CP violation using lattice gauge theory. In particluar, we use staggered fermions which preserve enough chiral symmetry to calculate ε'/ε. At the early stage of this work, we calculated ε'/ε using unimproved staggered fermions. The results for B6 indicate a large quenching uncertainty. In addition, the operators constructed out of unimproved staggered fermions receive large perturbative corrections (~100%). Hence it was essential to reduce the large perturbative correction by improving staggered fermions. In order to find the best scheme, we calculated explicitly the one loop matching coefficients for various improvement schemes for the staggered fermion action and operators. Through this study, we figure out that the HYP and Fat7 schemes lead to the smallest one loop corrections. In order to fully match the lattice calculation to the continuum, we calculated the one loop renormalizations constants for the four-fermion operators. It turns out that the matching coefficients are reduced down to about 10% level using improved staggered fermions. We also performed a numerical study on ε'/ε using HYP staggered fermions in quenched QCD. We plan to calculate ε'/ε in partially quenched QCD using MILC gauge configurations in near future.

Perturbative calculation of matching factors

In order to convert the numerical data into physical quantity, we need to find the conversion formula named "matching factors" which are, in general, matrices. One possibility is to calculate the matching factors using one-loop perturbation theory. At present it is necessary to develop a method to obtain matching factors in the mixed action environment (HYP sea quarks and AsqTad valence quarks). We also plan to develop a method to evaluate the matching factor at the two-loop level. At the same time, we will also search for a new method to implement non-perturbative renormalization (NPR) method for staggered fermions.

Staggered chiral perturbation theory

In order to perform a data analysis as a function of quark masses, it is important to understand the chiral behavior of physical observables. The best tool to calculate the chiral behavior is the staggered chiral perturbation theory, which was invented originally by Weonjong Lee and Stephen Sharpe and developed later by Claude Bernard. So far, for most of the weak matrix elements, the chiral behavior is estimated based on the (partially) qenched chiral perturbation theory. Since we are using highly improved staggered fermions such as HYP or Fat7 types, the contribution from the non-Goldstone pions becomes so important that we need to include them into the data analysis. A systematic method to do this is staggered chiral perturbation theory. We plan to figure out the chiral behavior of hadron masses and weak matrices relevant to CP violations in near future.

Kl3 decay and form factors

We plan to calculate the vector form factor of the Kl3 decay channel in order to determine Vus very precisely. The results will be used to overconstrain the unitarity ansatz of the standard model.

Non-perturbative renormalization for staggered fermions

We plan to calculate the matching factors for the BK and ε'/ε using the NPR method. This method is based on the RI-MOM scheme. We will apply both exceptional momentum scheme and non-exceptional momentum scheme in order to control the systematic error. In the NPR, we trade the systematic error due to the truncated higher order with the statistical error.