There has been much speculation as to the origin of the $\Delta I=1/2$ rule ($\mathrm{Re}{A}_0/\mathrm{Re}{A}_2\simeq 22.5$). We find that the two dominant contributions to the $\Delta I=3/2$, $K\to \pi \pi$ correlation functions have opposite signs, leading to a significant cancelation. This partial cancelation occurs in our computation of $\mathrm{Re}{A}_2$ with physical quark masses and kinematics (where we reproduce the experimental value of $A_2$) and also for heavier pions at threshold. For $\mathrm{Re}{A}_0$, although we do not have results at physical kinematics, we do have results for pions at zero momentum with $m_{\pi }\simeq 420\mathrm{MeV}$ $[\mathrm{Re}{A}_0/\mathrm{Re}{A}_2=9.1(2.1)]$ and $m_{\pi }\simeq 330\mathrm{MeV}$ $[\mathrm{Re}{A}_0/\mathrm{Re}{A}_2=12.0(1.7)]$. The contributions which partially cancel in $\mathrm{Re}{A}_2$ are also the largest ones in $\mathrm{Re}{A}_0$, but now they have the same sign and so enhance this amplitude. The emerging explanation of the $\Delta I=1/2$ rule is a combination of the perturbative running to scales of $O(2\mathrm{GeV}$), a relative suppression of $\mathrm{Re}{A}_2$ through the cancelation of the two dominant contributions, and the corresponding enhancement of $\mathrm{Re}{A}_0$. QCD and electroweak penguin operators make only very small contributions at such scales.

• Received 11 December 2012

DOI:https://doi.org/10.1103/PhysRevLett.110.152001