We review the present status of QCD corrections to weak decays beyond the leading-logarithmic approximation, including particle-antiparticle mixing and rare and $\mathrm{CP}$-violating decays. After presenting the basic formalism for these calculations we discuss in detail the effective Hamiltonians of all decays for which the next-to-leading-order corrections are known. Subsequently, we present the phenomenological implications of these calculations. The values of various parameters are updated, in particular the mass of the newly discovered top quark. One of the central issues in this review are the theoretical uncertainties related to renormalization-scale ambiguities, which are substantially reduced by including next-to-leading-order corrections. The impact of this theoretical improvement on the determination of the Cabibbo-Kobayashi-Maskawa matrix is then illustrated. [S0034-6861(96)00304-2]

Weak decays beyond leading logarithms

In view of the recent start of the NA62 experiment at CERN that is expected to measure the $$ {K}^{+}\to {\pi}^{+}\nu \overline{\nu} $$
 branching ratio with a precision of 10%, we summarise the present status of this promising decay within the Standard Model (SM). We do likewise for the closely related $$ {K}_L\to {\pi}^0\nu \overline{\nu} $$
 , which will be measured by the KOTO experiment around 2020. As the perturbative QCD and electroweak corrections in both decays are under full control, the dominant uncertainties within the SM presently originate from the CKM parameters |V
 cb
 |, |V
 ub
 | and γ. We show this dependence with the help of analytic expressions as well as accurate interpolating formulae. Unfortunately a clarification of the discrepancies between inclusive and exclusive determinations of |V
 cb
 | and |V
 ub
 | from tree-level decays will likely require results from the Belle II experiment available at the end of this decade. Thus we investigate whether higher precision on both branching ratios is achievable by determining |V
 cb
 |, |V
 ub
 | and γ by means of other observables that are already precisely measured. In this context e
 K
 and ΔM
 s,d
 , together with the expected progress in QCD lattice calculations will play a prominent role. We find $$ \mathrm{\mathcal{B}}\left({K}^{+}\to {\pi}^{+}\nu \overline{\nu}\right)=\left(9.11 \pm 0.72\right) \times 1{0}^{-11} $$
 and $$ \mathrm{\mathcal{B}}\left({K}_L\to {\pi}^0\nu \overline{\nu}\right)\Big) = \left(3.00 \pm 0.30\right) \times 1{0}^{-11} $$
 , which is more precise than using averages of the present tree-level values of |V
 cb
 |, |V
 ub
 | and γ. Furthermore, we point out the correlation between $$ \mathrm{\mathcal{B}}\left({K}^{+}\to {\pi}^{+}\nu \overline{\nu}\right),\ \overline{\mathrm{\mathcal{B}}}\left({B}_{\mathrm{s}}\to {\mu}^{+}{\mu}^{-}\right) $$
 and γ within the SM, that is only very weakly dependent on other CKM parameters. Finally, we update the correlation of $$ {K}_L\to {\pi}^0\nu \overline{\nu} $$
 with the ratio e′
 /e in the SM taking the recent progress on e′
 /e from lattice QCD and the large N approach into account.

/pdf/k-p-n-n-k-to-pi-nu-overline-nu-and-k-l-p-0-n-n-k-l-to-pi-0-17kzwih9zx.pdf

K + → π + ν ν ¯ $$ {K}^{+}\to {\pi}^{+}\nu \overline{\nu} $$ and K L → π 0 ν ν ¯ $$ {K}_L\to {\pi}^0\nu \overline{\nu} $$ in the Standard Model: status and perspectives

In view of the recent start of the NA62 experiment at CERN that is expected to measure the $K^+\to\pi^+\nu\bar\nu$ branching ratio with a precision of 10%, we summarise the present status of this promising decay within the Standard Model (SM). We do likewise for the closely related $K_L\to\pi^0\nu\bar\nu$, which will be measured by the KOTO experiment around 2020. As the perturbative QCD and electroweak corrections in both decays are under full control, the dominant uncertainties within the SM presently originate from the CKM parameters $V_{cb}$, $V_{ub}$ and $\gamma$. We show this dependence with the help of analytic expressions as well as accurate interpolating formulae. Unfortunately a clarification of the discrepancies between inclusive and exclusive determinations of $V_{cb}$ and $V_{ub}$ from tree-level decays will likely require results from the Belle II experiment available at the end of this decade. Thus we investigate whether higher precision on both branching ratios is achievable by determining $V_{cb}$, $V_{ub}$ and $\gamma$ by means of other observables that are already precisely measured. In this context $\varepsilon_K$ and $\Delta M_{s,d}$, together with the expected progress in QCD lattice calculations will play a prominent role. We find $\mathcal{B}(K^+\to\pi^+\nu\bar\nu) = (9.11\pm 0.72) \times 10^{-11}$ and $\mathcal{B}(K_L\to\pi^0\nu\bar\nu) = (3.00\pm 0.30) \times 10^{-11}$, which is more precise than using averages of the present tree-level values of $V_{cb}$, $V_{ub}$ and $\gamma$. Furthermore, we point out the correlation between $\mathcal{B}(K^+\to\pi^+\nu\bar\nu)$, $\overline{\mathcal{B}}(B_s\to\mu^+\mu^-)$ and $\gamma$ within the SM, that is only very weakly dependent on other CKM parameters. Finally, we also update the ratio $\varepsilon'/\varepsilon$ in the SM and present its correlation with $K_L\to\pi^0\nu\bar\nu$.

/pdf/k-to-pi-nu-bar-nu-and-k-l-to-pi-0-nu-bar-nu-in-the-standard-55n36kslxh.pdf

$K^+\to\pi^+\nu\bar\nu$ and $K_L\to\pi^0\nu\bar\nu$ in the Standard Model: Status and Perspectives

One of the major challenges of particle physics has been to gain an in-depth understanding of the role of quark flavor and measurements and theoretical interpretations of their results have advanced tremendously: apart from masses and quantum numbers of flavor particles, there now exist detailed measurements of the characteristics of their interactions allowing stringent tests of Standard Model predictions. Among the most interesting phenomena of flavor physics is the violation of the CP symmetry that has been subtle and difficult to explore. Till early 1990s observations of CP violation were confined to neutral $K$ mesons, but since then a large number of CP-violating processes have been studied in detail in neutral $B$ mesons. In parallel, measurements of the couplings of the heavy quarks and the dynamics for their decays in large samples of $K, D$, and $B$ mesons have been greatly improved in accuracy and the results are being used as probes in the search for deviations from the Standard Model. In the near future, there will be a transition from the current to a new generation of experiments, thus a review of the status of quark flavor physics is timely. This report summarizes the results of the current generation of experiments that is about to be completed and it confronts these results with the theoretical understanding of the field.

Flavor Physics in the Quark Sector

We present new results for the amplitude $A_2$ for a kaon to decay into two pions with isospin $I=2$: Re$A_2 = 1.50(4)_\mathrm{stat}(14)_\mathrm{syst}\times 10^{-8}$ GeV; Im$A_2 = -6.99(20)_\mathrm{stat}(84)_\mathrm{syst}\times 10^{-13}$ GeV. These results were obtained from two ensembles generated at physical quark masses (in the isospin limit) with inverse lattice spacings $a^{-1}=1.728(4)$ GeV and $2.358(7)$ GeV. We are therefore able to perform a continuum extrapolation and hence largely to remove the dominant systematic uncertainty from our earlier results, that due to lattice artefacts. The only previous lattice computation of $K\to\pi\pi$ decays at physical kinematics was performed using an ensemble at a single, rather coarse, value of the lattice spacing ($a^{-1}\simeq 1.37(1)$ GeV). We confirm the observation that there is a significant cancellation between the two dominant contributions to Re$A_2$ which we suggest is an important ingredient in understanding the $\Delta I=1/2$ rule, Re$A_0$/Re$A_2\simeq 22.5$, where the subscript denotes the total isospin of the two-pion final state. Our result for $A_2$ implies that the electroweak penguin contribution to $\epsilon^\prime/\epsilon$ is Re($\epsilon^\prime/\epsilon)_\textrm{EWP}=-(6.6\pm 1.0)\times 10^{-4}$.

K →ππ Δ I=3/2 decay amplitude in the continuum limit

We review and update our results for K-> pipi decays and K^0-ar 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= 16.0pm 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=0.73pm 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.

/pdf/large-n-approach-to-kaon-decays-and-mixing-28-years-later-22yvd9yuka.pdf

Large $N$ approach to kaon decays and mixing 28 years later: $\Delta I = 1/2$ rule, $\hat B_K$ and $\Delta M_K$

The K(L) - K(S) mass difference

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.

Jean Marc Gerard

Papers

Large $N$ approach to kaon decays and mixing 28 years later: $\Delta I = 1/2$ rule, $\hat B_K$ and $\Delta M_K$

The K(L) - K(S) mass difference

Large N Approach to Kaon Decays and Mixing 28 Years Later: Delta I = 1/2 Rule, \hat B_K and Delta M_K