Waiting for precise measurements of $K^{+}\to {\pi }^{+}\nu \overline{\nu }$ and $K_L\to {\pi }^0\nu \overline{\nu }$

In view of future plans for accurate measurements of the theoretically clean branching ratios $\mathrm{Br}(K^{+}\to {\pi }^{+}\nu \overline{\nu })$ and $\mathrm{Br}(K_L\to {\pi }^0\nu \overline{\nu })$, which should occur in the next decade, the relevant formulas for quantities of interest are collected and their theoretical and parametric uncertainties are analyzed. In addition to the angle $\beta$ in the unitarity triangle (UT), the angle $\gamma$ can also be determined from these decays with respectable precision and in this context the importance of the recent NNLO QCD calculation on the charm contribution to $K^{+}\to {\pi }^{+}\nu \overline{\nu }$ and of the improved estimate on the long-distance contribution by means of chiral perturbation theory are presented. In addition to known expressions, several new ones that should allow transparent tests of the standard model (SM) and of its extensions are presented. While the review is centered around the SM, models with minimal flavor violation and scenarios with new complex phases in decay amplitudes and meson mixing are also discussed. A review of existing results within specific extensions of the SM, in particular the littlest Higgs model with $T$-parity, $Z^{\prime }$ models, the MSSM, and a model with one universal extra dimension are given. A new “golden” relation between $B$ and $K$ systems is derived that involves $(\beta ,\gamma )$ and $\mathrm{Br}(K_L\to {\pi }^0\nu \overline{\nu })$, and the virtues of $(R_t,\beta )$, $(R_b,\gamma )$, $(\beta ,\gamma )$, and $(\overline{\eta },\gamma )$ strategies for the UT in the context of $K\to \pi \nu \overline{\nu }$ decays with the goal of testing the SM and its extensions are investigated.

Figure 1

The penguin and box diagrams contributing to $K^{+}\to {\pi }^{+}\nu \overline{\nu }$. For $K_L\to {\pi }^0\nu \overline{\nu }$, only the spectator quark is changed from $u$ to $d$.

Figure 2

Unitarity triangle.

Figure 3

Unitarity triangle from $K\to \pi \nu \overline{\nu }$.

Figure 4

$\mathrm{Br}(K^{+}\to {\pi }^{+}\nu \overline{\nu })$ as a function of $\gamma$ for different values of $\beta$ and $\mid V_{cb}\mid$.

Figure 5

The UT from $K\to \pi \nu \overline{\nu }$ in scenario I of Table . Lines corresponding to several values of $\mathrm{Br}(K^{+}\to {\pi }^{+}\nu \overline{\nu })$ and $\mathrm{Br}(K_L\to {\pi }^0\nu \overline{\nu })$ (in units of ${10}^{-11}$) are also shown.

Figure 6

$\mathrm{Br}(K^{+}\to {\pi }^{+}\nu \overline{\nu })$ as a function of $\mathrm{Br}(K_L\to {\pi }^0\nu \overline{\nu })$ for several values of $a_{\psi K_S}$ in the case of $\mathrm{sgn}\left(X\right)=1$.

Figure 7

(Color online) $\mathrm{Br}(K^{+}\to {\pi }^{+}\nu \overline{\nu })$ as a function of $\mathrm{Br}(K_L\to {\pi }^0\nu \overline{\nu })$ for various values of ${\beta }_X$ (96). Dotted horizontal lines indicate the lower part of the experimental range (1.4) and the gray area the SM prediction. Also shown is the bound in Eq. (6.3).

Figure 8

The ratio of the $K\to \pi \nu \overline{\nu }$ branching ratios as a function of ${\beta }_X$ for $\mid X\mid =1.25$, 1.5, and 2.0. The horizontal line is the bound in Eq. (6.3).

Figure 9

Fake unitarity triangles compared to the reference triangle, $\Delta {\beta }_X=-{\theta }_X$.

Figure 10

(Color online) $\mathrm{Br}(K_L\to {\pi }^0\nu \overline{\nu })$ as a function of $\mathrm{Br}(K^{+}\to {\pi }^{+}\nu \overline{\nu })$ in the LHT model. The shaded area represents the experimental $1\sigma$ range for $\mathrm{Br}(K^{+}\to {\pi }^{+}\nu \overline{\nu })$. The GN bound is displayed by the dotted line, while the solid line separates the two areas where $\mathrm{Br}(K_L\to {\pi }^0\nu \overline{\nu })$ is larger or smaller than $\mathrm{Br}(K^{+}\to {\pi }^{+}\nu \overline{\nu })$.

Figure 11

(Color online) $\mathrm{Br}(K_L\to {\pi }^0\nu \overline{\nu })∕\mathrm{Br}(K^{+}\to {\pi }^{+}\nu \overline{\nu })$ in the LHT model as a function of ${\beta }_X^K$. The dashed line represents the GN bound.

Figure 12

(Color online) $\sin 2{\beta }_X^K∕\sin (2\beta +2{\phi }_{B_d})$ as a function of ${\delta }_{13}^d$ in the LHT model.

Figure 13

(Color online) $\mathrm{Br}(K_L\to {\pi }^0{e}^{+}{e}^{-})$ (upper curve) and $\mathrm{Br}(K_L\to {\pi }^0{\mu }^{+}{\mu }^{-})$ (lower curve) as functions of $\mathrm{Br}(K_L\to {\pi }^0\nu \overline{\nu })$ in the LHT model. The corresponding SM predictions are represented by dark points.

Figure 14

(Color online) A projection onto the $K_L\to {\pi }^0\nu \overline{\nu }-K^{+}\to {\pi }^{+}\nu \overline{\nu }$ plane including the upper bounds from $\Delta M_K$ and ${ϵ}_K$ for $M_{Z^{\prime }}=5$ TeV and $1\mathrm{TeV}$.

Figure 15

(Color online) Effect of the constraints from ${\epsilon }_K$, $K^{+}\to {\pi }^{+}\nu \overline{\nu }$, $K_L\to {\mu }^{+}{\mu }^{-}$, and ${\epsilon }^{\prime }∕\epsilon$ on the $U_{sd}$ FCNC coupling in the case of an extra isosinglet down quark (31).

Figure 16

(Color online) Supersymmetric contributions to $K\to \pi \nu \overline{\nu }$. Sensitivity to ${\left({\delta }_{RR}^d\right)}_{23}{\left({\delta }_{RR}^d\right)}_{31}$ of various rare $K$ and $B$ decays as a function of $M_{H^{+}}$, setting $\tan \beta =50$, $\mu <0$, and assuming almost degenerate superparteners [the bounds from the two $K\to \pi \nu \overline{\nu }$ modes are obtained assuming a 10% measurement of their branching ratios while, the $B_{s,d}\to {\mu }^{+}{\mu }^{-}$ bounds refer to the present experimental limits (214)].

Figure 17

One-loop diagrams with light quarks that generate higher-dimensional operators From 213.

Figure 18

Leading-order chiral perturbation theory diagrams contributing to a $K^{+}\to {\pi }^{+}{Z}^0$ vertex [from 235]. Dashed lines denote the pion and kaon, while the wavy line denotes the $Z^0$, and the dot indicates the insertion of a flavor-changing effective vertex.