content_image/347852.jpg content_image/347853.jpg content_image/347854.jpg content_image/347851.jpg
Within SM K$_{L}$ → π $^{0}$ν ¯ ν decay, proceeds almost entirely through CP violation, is com- pletely dominated by short-distance loop diagrams with top quark exchanges, here the charm contribution can be fully neglected. 

The branching fraction of K$_{L}$ → π $^{0}$ν ¯ ν can be written as follows 

<img src='content_image/347852.jpg'>

with 

<img src='content_image/347853.jpg'>

κ$_{+}$ and r$_{K}$$_{L}$ = 0 . 944 summarizing isospin breaking corrections in relating K$_{L}$ → π $^{0}$ν ¯ ν to K + → π $^{0}$e$^{+}$ν . The current value of branching fraction for K$_{L}$ → π $^{0}$ν ¯ ν with SM is given by [112] 

<img src='content_image/347854.jpg'>

In Fig. (10) the variation of branching fraction Br ( K$_{L}$ → π $^{0}$ν ¯ ν ) with the phase φ ds t ′ is shown $^{4}$. We note that with the constraint on Br ( K + → π $^{+}$ν ¯ ν ) (Table. III), while, in principle Br ( K$_{L}$ → π $^{0}$ν ¯ ν ) could be enhanced as much as 1 . 2 × 10 − 9 (right panel Fig. 10), the expected 1 σ range in SM4 (left panel Fig. 10) is only to 7 × 10 − $^{11}$, however, at 95% CL the value could be enhanced to 8 × 10 − $^{10}$. The branching fraction has its maximum value when the phase φ ds t ′ has the value ± 90 ◦ and 270 ◦ since SM4 contribution picks up its maximum value at those points (eq. 102). 

The SM4 1 σ limit on Br ( K$_{L}$ → π $^{0}$ν ¯ ν ) is given by 

<img src='content_image/347851.jpg'>

the upper limits are consistent with the limit calculated in Ref. [110]. 

## J. CP violation in B → πK modes 

The observed data from the currently running two asymmetric B factories are almost consistent with the SM predictions and till now there is no compelling evidence for new 

