$\Delta I=1/2$ Rule and $\hat B_K$ : 2014

I summarize the status of the $\Delta I=1/2$ rule in $K\to\pi\pi$ decays within an {\it analytic} approach based on the dual representation of QCD as a theory of weakly interacting mesons for large $N$, where $N$ is the number of colours. This approximate approach, developed in the 1980s by William Bardeen, Jean-Marc G\'erard and myself, allowed us already 28 years ago to identify the dominant dynamics behind the $\Delta I=1/2$ rule. However, the recent inclusion of lowest-lying vector meson contributions in addition to the pseudoscalar ones to hadronic matrix elements of current-current operators and the calculation of the corresponding Wilson coefficients in a momentum scheme at the NLO improved significantly the matching between quark-gluon short distance contributions and meson long distance contributions over our results in 1986. We obtain satisfactory description of the ${\rm Re}A_2$ amplitude and ${\rm Re}A_0/{\rm Re}A_2=16.0\pm 1.5$ to be compared with its experimental value of $22.3$. While this difference could be the result of present theoretical uncertainties in our approach, it cannot be excluded that New Physics (NP) is here at work. The analysis by Fulvia De Fazio, Jennifer Girrbach-Noe and myself shows that indeed a tree-level $Z^\prime$ or $G^\prime$ exchanges with masses in the reach of the LHC and special couplings to quarks can significantly improve the theoretical status of the $\Delta I=1/2$ rule. I stress that our approach allows to understand the physics behind recent numerical results obtained in lattice QCD not only for the $\Delta I=1/2$ rule but also for the parameter $\hat B_K$ that enters the evaluation of $\varepsilon_K$. In contrast to the $\Delta I=1/2$ rule the chapter on $\hat B_K$ in QCD appears to be basically closed.