Abstract
The discrepancies among the measurements of branching ratios and CP asymmetries of $B$ arrow ${\pi}{\pi}$ decays such as large direct CP asymmetry for $B^0 $ arrow ${\pi}^+ {\pi}^-$ mode and large branching ratio for $B^0$ arrow ${\pi}^0 {\pi}^0$ mode originate the $B$ arrow ${\pi}{\pi}$ puzzle. According to diagrammatic approach of this $b$ arrow $d$ transition the small ratio of color-suppressed amplitude (C) to color-allowed one (T) contradicts between the standard model (SM) approach and the experimental values. To make out the ambiguity we need to scrutinize the decays with different topological amplitudes. In this paper, the decays are studied in the SM by taking different values of C/T as constraints. We find that larger ratio of C/T is explained successfully in the SM but the lower ratio is not for which new physics (NP) is needed. The NP contribution can be included in $B$ arrow ${\pi}{\pi}$ decays at tree level by Z' boson. Here, we find that Z' model can explain the puzzle by providing a good solution for lower ratio of C/T.
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