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

TL;DR

This paper reviews the status of the $ abla I=1/2$ rule in kaon decays using an analytic dual QCD approach, incorporating vector mesons and NLO corrections, and explores potential New Physics contributions to explain discrepancies.

Contribution

It updates the dual QCD framework with vector meson contributions and NLO Wilson coefficients, providing improved theoretical insights into the $ abla I=1/2$ rule and $ar B_K$, and discusses possible NP effects.

Findings

Achieves a ratio ${ m Re}A_0/{ m Re}A_2=16.0\\pm 1.5$, closer to experimental 22.3.

Incorporates vector mesons and NLO corrections, improving the matching of short- and long-distance effects.

Suggests tree-level $Z'$ or $G'$ exchanges could account for remaining discrepancies.

Abstract

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.