39 [38, 124]. Also the measured values of direct CP asymmetry parameters A$_{π}$ $^{+}$π − and A$_{π}$ $^{0}$π0 are higher than the corresponding SM predictions [38]. Thus, the discrepancy between the theoretical and the measured quantities imply that there may also be some new physics effect in the b → d penguins as speculated in b → s penguins. Let us first write down the most general topological amplitudes for B → ππ modes as From the above relations it can be seen that if there will be additional new contribution to the penguin sector with other amplitudes as expected in SM4 then that may explain B → ππ observations. As discussed earlier, due to the presence of the additional generation of quarks the uni- tarity condition becomes λ$_{u}$ + λ$_{c}$ + λ$_{t}$ + λ$_{t}$ ′ = 0. Thus, including the new contributions one can symbolically represent these amplitudes as where b$_{1}$ = | λ d t ′ /λ d u | , φ$_{d}$ is the weak phase of λ ′ d t . r ′ 1 = | A$_{new}$/A d u | , and δ d 2 is the relative strong phases between A$_{new}$ and A d $_{u}$. Thus from the above amplitude one can obtain the CP averaged branching ratio, direct and mixing induced CP asymmetry parameters as where √$_{2}$$_{A}$$_{(}$$_{B}$ + → π $^{+}$π$^{0}$) = − ( T + C + P$_{ew}$ ) , A ( B 0 → π $^{+}$π − ) = − ( T + P ) , √$_{2(}$$_{B}$ 0 → π $^{0}$π$^{0}$) = − ( C − ( P − P$_{ew}$ )) . (120) Amp = λ d $_{u}$A d u + λ d $_{c}$A d c − λ d t ′ A$_{new}$ = λ d $_{u}$A d u [ 1 − r$_{1}$ a$_{1}$ e i ( δ d $_{1}$+ γ ) − r ′ 1 b$_{1}$ e i ( δ d $_{2}$+ φ$_{d}$ ) ] , (121) Br = | p$_{c.m}$ | τ$_{B}$ 8 πM 2 B [ R 1 − 2 r$_{1}$a$_{1}$ cos δ d 1 cos γ − 2 r ′ $_{1}$b$_{1}$ cos δ d 2 cos( φ$_{d}$ + γ ) +2 r$_{1}$r ′ $_{1}$a$_{1}$ b$_{1}$ cos( δ d 2 − δ d $_{1}$) cos φ$_{d}$ ] , A$_{ππ}$ = 2 [ r$_{1}$a$_{1}$ sin δ d 1 sin γ + r ′ $_{1}$b$_{1}$ sin δ d 2 sin( φ$_{d}$ + γ ) + r$_{1}$r ′ $_{1}$a$_{1}$ b$_{1}$ sin( δ d 1 − δ d $_{2}$) sin φ$_{d}$ ] [ R 1 − 2 r$_{1}$a$_{1}$ cos δ d 1 cos γ − 2 r ′ $_{1}$b$_{1}$ cos δ d 2 cos( φ$_{d}$ + γ ) + 2 r$_{1}$r ′ $_{1}$a$_{1}$ b$_{1}$ cos( δ d 2 − δ d $_{1}$) cos φ$_{d}$ ] , S$_{ππ}$ = X$_{1}$ [ R 1 − 2 r$_{1}$a$_{1}$ cos δ d 1 cos γ − 2 r ′ $_{1}$b$_{1}$ cos δ d 2 cos( φ$_{d}$ + γ ) + 2 r$_{1}$r ′ $_{1}$a$_{1}$ b$_{1}$ cos( δ d 2 − δ d $_{1}$) cos φ$_{d}$ ] , (122) X$_{1}$ = − [ sin(2 β + 2 γ ) − 2 r$_{1}$a$_{1}$ cos δ d 1 sin(2 β + γ ) + 2 r ′ $_{1}$b$_{1}$ cos δ d 2 sin( φ$_{d}$ − (2 β + γ )) + ( r$_{1}$a$_{1}$ ) $^{2}$sin(2 β ) + ( r ′ $_{1}$b$_{1}$ ) $^{2}$sin(2 φ$_{d}$ − 2 β ) − 2 r$_{1}$r ′ $_{1}$a$_{1}$ b$_{1}$ cos( δ d 1 − δ d $_{2}$) sin( φ$_{d}$ − 2 β ) ] . (123)