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

Abstract

A direct calculation of the complex ∆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 π −π 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 × 32 × 16) to large (32 3 × 64 × 32). 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 ππ threshold decay using a 24 3 × 64 × 16 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.

Nonleptonic Kaon Decays from Lattice QCD

“…Recall that the ∆I = 1/2 rule refers to the large ratio found for the I = 0 and I = 2 amplitudes: Re(A 0 )/Re(A 2 ) = 22.4. For our two sets of unphysical, threshold kinematics [1,19,20] While not equal to the experimental value of 22.5, these numbers are substantially larger than the ratio of approximately 2 given by the Wilson coefficients alone. In addition, as we decrease the pion mass towards its physical value, the ratio is seen to increase.…”

“…In the first we locate the quark fields associated with each of the two pions on separate time slices, where a separation of four appears to work best. This allowed a calculation of the more difficult Im(A 0 ), again at threshold with unphysical masses, from 138 configurations [19,20]. This "split-source" method removes the terms where a quark-anti-quark pair, one from each pion, can immediately annihilate -behavior enhancing the ⟨ππ|vac⟩ overlap.…”

“…There we plot both the lattice results for Re(A 2 ) computed with essentially physical kinematics as well as results for Re(A 2 ) and Re(A 0 ) computed for a threshold π − π final state and a pion with mass 422 and 329 MeV, results presented in Refs. [1] and [19,20] respectively. We see that Re(A 2 ) decreases dramatically with increasingly physical kinematics and agrees well with the physical value at the physical point.…”

Contribution of orbital angular momentum to the nucleon spin

“…Also, they can be large, but have nodes such that their second moments become small. Lattice calculations [36] have verified that indeed the total anomalous gravitomagnetic moment of the nucleon is zero. A more Recent lattice calculation [47] have shown that B q,g = 0.00 (6).…”

Thermal Analysis and Simulation of the Superconducting Magnet in the SpinQuest Experiment at Fermilab

“…The recent experimental data from RHIC also showed that the sum of both quark and gluon spin contributions still cannot account for the rest of the proton spin [3]. The large missing fraction of the spin is likely to be carried by the orbital angular momentum (OAM) of the quarks and gluons [4]. One way to extract the quark OAM L q is through Transverse Momentum Distribution (TMD).…”