Kaon decays in the standard model

A comprehensive overview of kaon decays is presented. The standard model predictions are discussed in detail, covering both the underlying short-distance electroweak dynamics and the important interplay of QCD at long distances. Chiral perturbation theory provides a universal framework for treating leptonic, semileptonic, and nonleptonic decays including rare and radiative modes. All allowed decay modes with branching ratios of at least ${10}^{-11}$ are analyzed. Some decays with even smaller rates are also included. Decays that are strictly forbidden in the standard model are not considered in this review. The present experimental status and the prospects for future improvements are reviewed.

Figure 1

Penguin and box topologies.

Figure 2

Compilation of values for $|V_{us}|f_{+}(0)$ extracted from all $K_{\ell 3}$ channels. The vertical band denotes the average. From 51.

Figure 3

Graphical representation of the current status of $|V_{ud}|$, $|V_{us}|$ and the corresponding CKM unitarity test. The horizontal band represents the constraint from $K_{\ell 3}$ decays, the thin vertical band represents the constraint from $0^{+}\to 0^{+}$ nuclear decays, the oblique band represents the constraint from $K_{\mu 2}/{\pi }_{\mu 2}$, and the ellipse is the $1\sigma$ fit region. From 215.

Figure 4

Diagrams describing the $K_{\ell 3\gamma }^0$ amplitude.

Figure 5

Measurements of the phase shift $\delta$ without (open circles) and with (full circles) isospin mass effect corrections from NA48/2 $K_{e4}$ data. From 84.

Figure 6

Fits of the NA48/2 $K_{e4}$ data in the $(a_0,a_2)$ plane without and with isospin mass effects. The wide band (UB) refers to the “universal band” [see 260 and 84 for details]. The large ellipses are 68% C.L. contours of the two-parameter fit leading to Eq. (4.89) and the circles the one-parameter fit of Eq. (4.91), imposing the CHPT constraint of Eq. (4.90). The small ellipse corresponds to Eq. (4.92). Adapted from 84.

Figure 7

Interference between tree and one-loop amplitudes generates a cusp at $M_{{\pi }^0{\pi }^0}^2=4M_{{\pi }^{+}}^2$ in $K\to 3\pi$ decays.

Figure 8

NA48/2 $K_{e4}$ and cusp results from two-parameter fits in the $(a_0,a_2)$ plane. The smallest contour corresponds to the combination of NA48/2 results. The crosshatched ellipse is the CHPT prediction (4.92) of 231, 232. The dash-dotted lines correspond to the recent result from DIRAC (14). From 84.

Figure 9

$Z$-penguin and box contributions to $K\to \pi \nu \overline{\nu }$.

Figure 10

Lowest-order [$\mathcal{O}(p^4)$] contributions to $K_S\to \gamma \gamma$.

Figure 11

Dominant contribution to $K_L\to \gamma \gamma$.

Figure 12

Two-photon contribution to $K^0\to {\ell }^{+}{\ell }^{-}$.

Figure 13

Unitarity contribution from $K^{+}\to {\pi }^{+}{\pi }^{+}{\pi }^{-}$ to $K^{+}\to {\pi }^{+}\gamma \gamma$.

Figure 14

Predicted spectra in the diphoton invariant mass at different values of $\widehat{c}$ for $K^{+}\to {\pi }^{+}\gamma \gamma$.

Figure 15

Spectrum for $K^{+}\to {\pi }^{+}\gamma \gamma$ in the diphoton invariant mass by NA48/2. From 421.

Figure 16

Comparison of the normalized spectra in the diphoton invariant mass for $K_L\to {\pi }^0\gamma \gamma$ at $\mathcal{O}(p^4)$ and $\mathcal{O}(p^6)$ in CHPT. NA48/2 data from 386.

Figure 17

Unitarity $K\to \pi \pi \pi$ contribution to $K^{+}$ and $K_S\to \pi {\gamma }^{*}$.

Figure 18

Spectra for $K^{+}\to {\pi }^{+}{e}^{+}{e}^{-}$ in the dilepton invariant mass. NA48/2 data from 81.

Figure 19

Spectra for $K^{+}\to {\pi }^{+}{\mu }^{+}{\mu }^{-}$ in the dilepton invariant mass. NA48/2 data from 85.

Figure 20

$CP$-conserving contribution to $K_L\to {\pi }^0{\ell }^{+}{\ell }^{-}$.

Figure 21

(a) Anomalous and resonance-dominated contributions to $K_L\to {\pi }^{+}{\pi }^{-}\gamma$. (b) A direct weak transition in terms of the couplings in Eq. (2.16).