Title: Lattice calculation of the mass difference between the long- and short-lived K mesons for physical quark masses
Description: The two neutral kaon states in nature, the 𝘒_𝐿 (long-lived) and 𝘒_s (short-lived) mesons, are the two time-evolution eigenstates of the 𝘒⁰ - 𝘒̅⁰̅ mixing system. The prediction of their mass difference 𝚫m_𝘒 based on the standard model is an important goal of lattice QCD. Non-perturbative formalism has been developed to calculate 𝚫 m_𝘒 and the calculation has been extended from the first exploratory calculation with only connected diagrams to full calculations on near-physical[1] and physical ensembles[2]. In this work, we extend the calculation described in Reference [2] from 59 to 152 configurations and present a new analysis method employed to calculate 𝚫 m_𝘒 with better reduction of statistical error on this larger set of configurations. By using a free-field calculation, we will show that the four-point contractions in our calculation method yields results consistent with the Inami-Lim calculation[3] in the local limit. We also report a series of scaling tests performed on 24³ × 64 and 32³ × 64 lattice ensembles to estimate the size of the finite lattice spacing error in our 𝚫 m_K$ calculation.

We will present the 𝚫 m_𝘒 calculation on the ensemble of 64³ × 128 gauge configurations with inverse lattice spacing of 2.36 GeV and physical quark masses obtaining results coming from 2.5 times the Monte Carlo statistics used for the result in [2]. With the new analysis method and estimated finite lattice spacing error, we obtain 𝚫 m_𝘒 = 5.8(0.6)_stat(2.3)_sys × 10¯¹²MeV. Here the first error is statistical and the second is an estimate of largest systematic error due to the finite lattice spacing effects. The new results also imply the validity of the OZI rule for the case of physical kinematics in contrast to the previous calculation of 𝚫 m_𝘒 with unphysical kinematics[1], where contributions from diagrams with disconnected parts are almost half the size of the contributions from fully connected diagrams but with the opposite sign.