Misc,
\$\Delta I=1/2\$ Rule and \$B\_K\$ : 2014
(Aug 20, 2014)
Abstract
I summarize the status of the \$\Delta I=1/2\$ rule in \$K\to\pi\pi\$ decays
within an 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 Gérard and myself identified already 28 years ago 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 \$ReA\_2\$ amplitude and \$ReA\_0/\rm
ReA\_2=16.01.5\$ 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 \$B\_K\$ that
enters the evaluation of \$\varepsilon\_K\$. In contrast to the \$\Delta I=1/2\$
rule the chapter on \$B\_K\$ in QCD appears to be basically closed.
