32 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 with κ$_{+}$ and r$_{K}$$_{L}$ = 0 . 944 summarizing isospin breaking corrections in relating K$_{L}$ → π $^{0}$ν ¯ ν to K + → π $^{0}$e$^{+}$ν . The current value of branching fraction for K$_{L}$ → π $^{0}$ν ¯ ν with SM is given by [112] In Fig. (10) the variation of branching fraction Br ( K$_{L}$ → π $^{0}$ν ¯ ν ) with the phase φ ds t ′ is shown $^{4}$. We note that with the constraint on Br ( K + → π $^{+}$ν ¯ ν ) (Table. III), while, in principle Br ( K$_{L}$ → π $^{0}$ν ¯ ν ) could be enhanced as much as 1 . 2 × 10 − 9 (right panel Fig. 10), the expected 1 σ range in SM4 (left panel Fig. 10) is only to 7 × 10 − $^{11}$, however, at 95% CL the value could be enhanced to 8 × 10 − $^{10}$. The branching fraction has its maximum value when the phase φ ds t ′ has the value ± 90 ◦ and 270 ◦ since SM4 contribution picks up its maximum value at those points (eq. 102). The SM4 1 σ limit on Br ( K$_{L}$ → π $^{0}$ν ¯ ν ) is given by the upper limits are consistent with the limit calculated in Ref. [110]. The observed data from the currently running two asymmetric B factories are almost consistent with the SM predictions and till now there is no compelling evidence for new 4 Right panel is added in our revised version to facilitate direct comparision with [70]. Br ( K$_{L}$ → π $^{0}$ν ¯ ν ) = (1 . 0 → 5 . 2) × 10 − $^{11}$; m$_{t}$ ′ = 400 GeV , Br ( K$_{L}$ → π $^{0}$ν ¯ ν ) = (1 . 0 → 6 . 2) × 10 − $^{11}$; m$_{t}$ ′ = 600 GeV , (105) Br ( K$_{L}$ → π $^{0}$ν ¯ ν ) = κ$_{L}$. [ ( Imλ$_{t}$ λ 5 X ( x$_{t}$ ) + Imλ$_{t}$ ′ λ 5 X ( x$_{t}$ ′ ) ) 2 ] , (102) κ$_{L}$ = r$_{K}$$_{L}$ r$_{K}$ + τ ( K$_{L}$ ) τ ( K $^{+}$) κ$_{+}$ = 1 . 80 × 10 − $^{10}$, (103) Br ( K$_{L}$ → π $^{0}$ν ¯ ν ) = (2 . 76 ± 0 . 40) × 10 − $^{11}$. (104) J. CP violation in B → πK modes