Emerging understanding of the ΔI = 1/2 rule from lattice QCD

Abstract

There has been much speculation as to the origin of the ΔI = 1/2 rule (ReA₀/ReA₂ ~ 22.5). We find that the two dominant contributions to the ΔI = 3/2, K → π π correlation functions have opposite signs, leading to a significant cancelation. This partial cancelation occurs in our computation of ReA₂ with physical quark masses and kinematics (where we reproduce the experimental value of A₂) and also for heavier pions at threshold. For ReA₀, although we do not have results at physical kinematics, we do have results for pions at zero momentum with m_π ~ 420 MeV [ReA₀/ReA₂ = 9.1 (2.1)] and m_π ~ 330 MeV [ReA₀/ReA₂ = 12.0 (1.7)]. The contributions which partially cancel in ReA₂ are also the largest ones in ReA₀, but now they have the same sign and so enhance this amplitude. The emerging explanation of the ΔI = 1/2 rule is a combination of the perturbative running to scales of O (2 GeV), a relative suppression of ReA₂ through the cancelation of the two dominant contributions, and the corresponding enhancement of ReA₀. QCD and electroweak penguin operators make only very small contributions at such scales.

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