2025 update on $\varepsilon_K$ in the Standard Model with lattice QCD inputs

We present theoretical results for the indirect CP violation parameter, $|\varepsilon_K|$, evaluated directly within the Standard Model using lattice QCD inputs including $B_K$, $|V_{cb}|$, $|V_{us}|$, $|V_{ud}|$, $\xi_0$, $\xi_2$, $F_K$, and the charm quark mass $m_c$. Our analysis reveals a significant tension at the $\sim 5\sigma$ level ($4.6\sigma$ to $5.2\sigma$) between the Standard Model prediction and the experimental value of $|\varepsilon_K|$. The Standard Model prediction, informed by lattice QCD inputs, accounts for only approximately 65% of the experimental value, leaving a 35% discrepancy unexplained. Notably, this tension vanishes when using the inclusive determination of $|V_{cb}|$, derived from heavy quark expansion. The discrepancy is therefore tied to the well-known tension between the exclusive and inclusive determinations of $|V_{cb}|$. We further report results for $|\varepsilon_K|$ obtained using the Brod--Gorbahn--Stamou (BGS) method based on $u\text{--}t$ unitarity, which yields an even stronger discrepancy, in the range of $4.9\sigma$ to $5.5\sigma$, when combined with lattice QCD inputs.