ABSTRACT
This Chapter addresses the problem of the dominance of Isospin=1/2 over Isospin=3/2 amplitudes in non-leptonic weak decays of strange particles, the so-called https://www.w3.org/1998/Math/MathML" display="inline"> Δ I = 1 / 2 rule. In 1969, K. Wilson had suggested the enhancement to be due to a different strong interaction renormalisation of the https://www.w3.org/1998/Math/MathML" display="inline"> Δ I = 1 / 2 and 3 / 2 products of strange and non strange Cabibbo's weak currents but, before the advent of a computable strong interaction theory, there was no way to test the hypothesis. The discovery of asymptotically free QCD provided the opportunity. In the text, it is shown that coloured quarks lead to the existence of two 4-fermion operators responsible for non leptonic strange decays, called https://www.w3.org/1998/Math/MathML" display="inline"> O - and O + , with Isospin=1/2 and a mixture of 1/2 and 3/2, respectively. The anomalous dimensions of https://www.w3.org/1998/Math/MathML" display="inline"> O ± are computed in QCD (one loop approximation). Assuming free-field normalisation at the scale of the Intermediate Boson mass ( https://www.w3.org/1998/Math/MathML" display="inline"> ≃ 80 GeV ), the normalisation at the strange particle mass scale ( https://www.w3.org/1998/Math/MathML" display="inline"> ≃ 1 GeV ) is computed by solving the renormalisation group (Callan-Szymazik) equation. The result: https://www.w3.org/1998/Math/MathML" display="inline"> O - ( 1 GeV ) / O + ( 1 GeV ) ≃ 5 vs the experimental value https://www.w3.org/1998/Math/MathML" display="inline"> Γ ( K S 0 → π + π - ) Γ ( K + → π + π 0 ) = | A 0 A 2 | ≃ 21 has provided an encouraging hint that indeed QCD may provide a reasonable explanation of the dominance of the https://www.w3.org/1998/Math/MathML" display="inline"> Δ I = 1 / 2 operator https://www.w3.org/1998/Math/MathML" display="inline"> O - , once non-perturbative corrections at the lower end of the integration region are accounted. The final part of the chapter reports the most recent results of non-perturbative, lattice QCD determinations of the https://www.w3.org/1998/Math/MathML" display="inline"> A 0 / A 2 ratio, with the very encouraging result: https://www.w3.org/1998/Math/MathML" display="inline"> ( A 0 / A 2 ) l a t t i c e Q C D = 22.45 ( 6 ) . Quoting the Authors: if confirmed by independent calculations, this extraordinary result would close a long story of attempts to demonstrate the standard model origin of the https://www.w3.org/1998/Math/MathML" display="inline"> Δ I = 1 / 2 enhancement.
