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}| $, $ ξ_0 $, $ ξ_2 $, $ F_K $, and the charm quark mass $ m_c $. Our analysis reveals a significant tension at the $ \sim 5σ$ level ($ 4.6σ$ to $ 5.2σ$) 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 $ 4.9σ$ to $ 5.5σ$, when combined with lattice QCD inputs.