Abstract:
We review and update our results for K-> pipi decays and K^0-\bar K^0 mixing obtained by us in the 1980s within an approach based on the dual representation of QCD as a theory of weakly interacting mesons for large N colours. In our analytic approach the dynamics behind the enhancement of ReA_0 and suppression of ReA_2, the so-called Delta I = 1/2 rule for K-> pi pi decays, has a simple structure: the usual octet enhancement through quark-gluon renormalization group evolution down to the scales O(1 GeV) is continued as a meson evolution down to zero momentum scales at which the factorization of hadronic matrix elements is at work. The inclusion of lowest-lying vector meson contributions in addition to the pseudoscalar ones and of Wilson coefficients in a momentum scheme improves significantly the matching between quark-gluon and meson evolutions. In particular, the anomalous dimension matrix governing the meson evolution exhibits the structure of the known anomalous dimension matrix in the quark-gluon evolution. The recent results on ReA_2 and ReA_0 from the RBC-UKQC collaboration give support for our approach. In particular, the signs of the two main contractions found numerically by these authors follow uniquely from our analytic approach. At NLO in 1/N we obtain ReA_0/ReA_2= 16.0\pm 1.5 which amounts to an order of magnitude enhancement over the strict large N limit value sqrt{2}. QCD penguins contribute at 15% level to this result. We also find \hat B_K=0.73\pm 0.02, with the smallness of 1/N corrections to the large N value \hat B_K=3/4 resulting within our approach from an approximate cancellation between pseudoscalar and vector meson one-loop contributions. We summarize the status of Delta M_K in this approach.

Abstract:
We calculate the hadronic matrix elements to $O(p^4)$ in the chiral expansion for the ($\Delta S =1$) $K^0 \to 2 \pi$ decays and the ($\Delta S=2$) $\bar K^0$-$K^0$ oscillation. This is done within the framework of the chiral quark model. The chiral coefficients thus determined depend on the values of the quark and gluon condensates and the constituent quark mass. We show that it is possible to fit the $\Delta I =1/2$ rule of kaon decays with values of the condensates close to those obtained by QCD sum rules. The renormalization invariant amplitudes are obtained by matching the hadronic matrix elements and their chiral corrections to the short-distance NLO Wilson coefficients. For the same input values, we study the parameter $\hat B_K$ of kaon oscillation and find $\hat B_K = 1.1 \pm 0.2$. As an independent check, we determine $\hat B_K$ from the experimental value of the $K_L$-$K_S$ mass difference by using our calculation of the long-distance contributions. The destructive interplay between the short- and long-distance amplitudes yields $\hat B_K = 1.2 \pm 0.1$, in agreement with the direct calculation.

Abstract:
We discuss the application of the MPSTV non-perturbative method \cite{NPM} to the operators relevant to kaon decays. This enables us to reappraise the long-standing question of the $\Delta I=1/2$ rule, which involves power-divergent subtractions that cannot be evaluated in perturbation theory. We also study the mixing with dimension-six operators and discuss its implications to the chiral behaviour of the $B_K$ parameter.

Abstract:
A consistent census of penguins in the Delta I = 1/2 rule is taken from the eta0 pole contribution to the radiative KL to gamma gamma, KS to pi0 gamma gamma and K+ to pi+ gamma gamma decay modes. We briefly comment on its impact for KL to pi0 pi0 gamma gamma, KL to pi+ pi- gamma and check its compatibility with the KL - KS mass difference and the CP violating epsilon-prime / epsilon parameter.

Abstract:
The $K\to\pi\pi$ decay amplitudes are studied within the framework of generalized factorization in which the effective Wilson coefficients are gauge-invariant, renormalization-scale and -scheme independent while factorization is applied to the tree-level hadronic matrix elements. Nonfactorized contributions to the hadronic matrix elements of (V-A)(V-A) four-quark operators, which are needed to account for the suppression of the $\Delta I=3/2 K\to\pi\pi$ amplitude $A_2$ and the enhancement of the $\Delta I=1/2 A_0$ amplitude, are phenomenologically extracted from the measured $K^+\to\pi^+\pi^0$ decay and found to be large. The $A_0/A_2$ ratio is predicted to lie in the range 15-17 for $m_s(1 GeV)=(127-150)$ MeV. Vertex and penguin-type radiative corrections to the matrix elements of four-quark operators and nonfactorized effects due to soft-gluon exchange account for the bulk of the $\Delta I=1/2$ rule. Comparison of the present analysis with the chiral-loop approach is given.

Abstract:
We present a lattice calculation of the $K\to\pi$ and $K\to 0$ matrix elements of the $\Delta S=1$ effective weak Hamiltonian which can be used to determine $\epsilon^\prime/\epsilon$ and the $\Delta I=1/2$ rule for $K$ decays in the Standard Model. The matrix elements for $K\to\pi\pi$ decays are related to $K\to\pi$ and $K\to 0$ using lowest order chiral perturbation theory. We also present results for the kaon $B$ parameter, $B_K$. Our quenched domain wall fermion simulation was done at $\beta=6.0$ ($a^{-1}\approx 2$ GeV), lattice size $16^3\times 32\times 16$, and domain wall height $M_5=1.8$.

Abstract:
We consider the possibility of violations of the selection rule $\Delta S=\Delta Q$ at an appreciable level in {\it exclusive} semi-leptonic decays of Kaons. At $\Phi$-Factories, intense Kaon beams will be available and will probe among others, the semi-leptonic decays $K_{l4}$ and $K_{l3\gamma}$ in addition to $K_{l3}$ and could provide novel testing grounds for the $\Delta S=\Delta Q$ rule. In particular, the branching ratio of $K_{l3\gamma}$ is non-negligible and could be used to probe new phenomena associated with the violation of this selection rule. Furthermore, we modify certain di-lepton event rate ratios and asymmetries and time asymmetries that have been constructed by Dass and Sarma for di-lepton events from Beon decays to test the $\Delta B=\Delta Q$ at the $\Upsilon (4S)$, to the Kaon system at the $\phi(1020)$. We find that the large width of the $K_S$ relative to that of $K_L$ plays an important role in enhancing some of the time asymmetries.

Abstract:
The $\Delta I=1/2$ rule and direct CP violation $\epsilon'/\epsilon$ in kaon decays are studied within the framework of the effective Hamiltonian approach in conjunction with generalized factorization for hadronic matrix elements. We identify two principal sources responsible for the enhancement of A0/A_2: the vertex-type as well as penguin-type corrections to the matrix elements of four-quark operators, which render the physical amplitude renormalization-scale and -scheme independent, and the nonfactorized effect due to soft-gluon exchange, which is needed to suppress the $\Delta I=3/2$ $K\to\pi\pi$ amplitude. Contrary to the chiral approach which is limited to light meson decays and fails to reproduce the A2 amplitude, the aforementioned approach for dealing with scheme and scale issues is applicable to heavy meson decays. We obtain A0/A2=13-15 if $m_s$(1 GeV) lies in the range (125-175)MeV. The bag parameters $B_i$, which are often employed to parametrize the scale and scheme dependence of hadronic matrix elements, are calculated in two different renormalization scehemes. It is found that $B_8^{(2)}$ and $ B_6^{(0)}$, both of order 1.5 at $\mu=1$ GeV, are nearly $\gamma_5$ scheme independent, whereas $B^{(0)}_{3,5,7}$ as well as $B_7^{(2)}$ show a sizable scheme dependence. Moreover, only $B_{1,3,4}^{(0)}$ exhibit a significant $m_s$ dependence, while the other $B$-parameters are almost $m_s$ independent. For direct CP violation, we obtain $\epsilon'/\epsilon=(0.7-1.1)\times 10^{-3}$ if $m_s(1 {\rm GeV})=150$ MeV and $\epsilon'/\epsilon=(1.0-1.6)\times 10^{-3}$ if $m_s$ is as small as indicated by some recent lattice calculations.

Abstract:
The $\Delta I=1/2$ rule and direct CP violation $\epsilon'/\epsilon$ in kaon decays are studied within the framework of the effective Hamiltonian approach in conjunction with generalized factorization for hadronic matrix elements.

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.