Practical methods for a direct calculation of $\Delta I=1/2$ $K$ to $\pi\pi$ Decay

A direct calculation of the complex $\Delta I=1/2$ kaon decay amplitude is notoriously difficult because of the presence of disconnected graphs. Here we describe and demonstrate two practical methods to defeat this problem: the EigCG algorithm and the use of time-separated $\pi-\pi$ sources. With a fine tuned EigCG implementation for domain wall fermions, the calculation of light quark propagators is accelerated by a factor of 5-10 on a variety of lattices from small ($16^3\times32\times16$) to large ($32^3\times64\times32$). In addition, a substantial reduction in noise is achieved by separating each of the sources for the two pions in the time direction by 2-5 lattice spacings. These methods are combined in a calculation of $K$ to $\pi\pi$ threshold decay using a $24^3\times64\times16$ volume and 329 MeV pions. These methods result in non-zero signals for both Re($A_0$) and Im($A_0$) from 138 gauge configurations.