$K_L-K_S$ Mass Difference from Lattice QCD

We report on the first complete calculation of the $K_L-K_S$ mass difference, $\mathrm{\Delta }{M}_K$, using lattice QCD. The calculation is performed on a $2+1$ flavor, domain wall fermion ensemble with a 330 MeV pion mass and a 575 MeV kaon mass. We use a quenched charm quark with a 949 MeV mass to implement Glashow-Iliopoulos-Maiani cancellation. For these heavier-than-physical particle masses, we obtain $\mathrm{\Delta }{M}_K=3.19(41)(96)\times {10}^{-12}\mathrm{MeV}$, quite similar to the experimental value. Here the first error is statistical, and the second is an estimate of the systematic discretization error. An interesting aspect of this calculation is the importance of the disconnected diagrams, a dramatic failure of the Okubo-Zweig-Iizuka rule.

Figure 1

One type of diagram contributing to $\mathcal{A}$ in Eq. (1). Here, $t_2$ and $t_1$ are integrated over the time interval $[t_a,t_b]$ represented by the shaded region.

Figure 2

The four types of diagrams contributing to the mass difference $\mathrm{\Delta }{M}_K$. The shaded circles are the $\mathrm{\Delta }S=1$ weak Hamiltonians. The black dots represent the kaon sources.

Figure 3

Integrated correlators for the three products of operators $Q_1{Q}_1$, $Q_1{Q}_2$, and $Q_2{Q}_2$. The three lines give the linear fits to the data in the time interval $[7,20]$.

Figure 4

Effective slope plots for the three products of operators $Q_1{Q}_1$, $Q_1{Q}_2$, and $Q_2{Q}_2$.