We present recent updates for 𝜀 𝐾 determined directly from the standard model (SM) with lattice QCD inputs such as B𝐾 , |𝑉 𝑐𝑏 |, |𝑉 𝑢𝑠 |, 𝜉 0 , 𝜉 2 , 𝜉 LD , 𝑓 𝐾 , and 𝑚 𝑐 . We find that the standard model with exclusive |𝑉 𝑐𝑏 | and other lattice QCD inputs describes only 66% of the experimental value of |𝜀 𝐾 | and does not explain its remaining 34%, which leads to a strong tension in |𝜀 𝐾 | at the 4.5𝜎 ∼ 3.7𝜎 level between the SM theory and experiment. We also find that this tension disappears when we use the inclusive value of |𝑉 𝑐𝑏 | obtained using the heavy quark expansion based on the QCD sum rule approach.

We study the chirality of staggered quarks on the Dirac eigenvalue spectrum using deep learning (DL) techniques. The Kluberg-Stern method to construct staggered bilinear operators conserves continuum property such as recursion relations, uniqueness of chirality, and Ward identities, which leads to a unique and characteristic pattern (we call it "leakage pattern (LP)") in the matrix elements of the chirality operator sandwiched between two quark eigenstates of staggered Dirac operator. DL analysis gives 99.4(2)% accuracy on normal gauge configurations and 0.998 AUC (Area Under ROC Curve) for classifying non-zero mode octets in the Dirac eigenvalue spectrum. It confirms that the leakage pattern is universal on normal gauge configurations. The multi-layer perceptron (MLP) method turns out to be the best DL model for our study on the LP.

We study the chirality of staggered quarks on the Dirac eigenvalue spectrum using deep learning (DL) techniques. The Kluberg-Stern method to construct staggered bilinear operators conserves continuum property such as recursion relations, uniqueness of chirality, and Ward identities, which leads to a unique and characteristic pattern (we call it "leakage pattern (LP)") in the matrix elements of the chirality operator sandwiched between two quark eigenstates of staggered Dirac operator. DL analysis gives 99.4(2)% accuracy on normal gauge configurations and 0.998 AUC (Area Under ROC Curve) for classifying non-zero mode octets in the Dirac eigenvalue spectrum. It confirms that the leakage pattern is universal on normal gauge configurations. The multi-layer perceptron (MLP) method turns out to be the best DL model for our study on the LP.