Covariance fitting of highly-correlated data in lattice QCD

Abstract

We address a frequently-asked question on the covariance fitting of highly-correlated data such as our B _K data based on the SU(2) staggered chiral perturbation theory. Basically, the essence of the problem is that we do not have a fitting function accurate enough to fit extremely precise data. When eigenvalues of the covariance matrix are small, even a tiny error in the fitting function yields a large chi-square value and spoils the fitting procedure. We have applied a number of prescriptions available in the market, such as the cut-off method, modified covariance matrix method, and Bayesian method. We also propose a brand new method, the eigenmode shift (ES) method, which allows a full covariance fitting without modifying the covariance matrix at all. We provide a pedagogical example of data analysis in which the cut-off method manifestly fails in fitting, but the rest work well. In our case of the B _K fitting, the diagonal approximation, the cut-off method, the ES method, and the Bayesian method work reasonably well in an engineering sense. However, interpreting the meaning of χ ² is easier in the case of the ES method and the Bayesian method in a theoretical sense aesthetically. Hence, the ES method can be a useful alternative optional tool to check the systematic error caused by the covariance fitting procedure.