content_image/339051.jpg content_image/339050.jpg content_image/339049.jpg
with average 

<img src='content_image/339051.jpg'>

As seen from (110), the observed value of S$_{π}$ $^{0}$K$_{s}$ is found to be smaller than the present world average value of sin 2 β = 0 . 672 ± 0 . 024 measured in b → c ¯ cs transitions [52] by nearly 1 σ and the deviation from the SM expectation given above is possibly even larger. This deviation which is opposite to the SM expectation, implies the possible presence of new physics in the B 0 → K $^{0}$π0 decay amplitude. In the SM, this decay mode receives contributions from QCD penguin ( P ), electroweak penguin ( P$_{EW}$ ) and color suppressed tree ( C ) diagrams, which follow the hierarchical pattern P : P$_{EW}$ : C = 1 : λ : λ $^{2}$, where λ ≈ 0 . 2257 is the Wolfenstein expansion parameter. Thus, accepting the above discrepancy seriously one can see that the electroweak penguin sector is the best place to search for new physics. 

To account for these discrepancies here we consider the effect of sequential fourth gener- ation quarks [20, 42–45]. In the SM, the relevant effective Hamiltonian describing the decay modes B → πK is given by 

<img src='content_image/339050.jpg'>

With a sequential fourth generation, the Wilson coefficients C$_{i}$ ’s will be modified due to the new contributions from t ′ quark in the loop. Furthermore, due to the presence of the t ′ quark the unitarity condition becomes λ$_{u}$ + λ$_{c}$ + λ$_{t}$ + λ$_{t}$ ′ = 0, where λ$_{q}$ = V$_{qb}$V ∗ $_{qs}$. 

Thus, including the fourth generation and replacing λ$_{t}$ = − ( λ$_{u}$ + λ$_{c}$ + λ$_{t}$ ′ ), the modified Hamiltonian becomes 

<img src='content_image/339049.jpg'>

where Δ C$_{i}$ ’s are the effective (t subtracted) t ′ contributions. 

Thus, one can obtain the transition amplitudes in the QCD factorization approach as 

