  Motivated by the measured large branching ratio of $\bar{B}^{0}\to\pi^0\pi^0$
(the so-called "$\pi\pi$" puzzle), we investigate the effects of a family
non-universal $Z^{\prime}$ model on the tree-dominated $B\to\pi\pi$ decays. We
find that the $Z^{\prime}$ coupling parameter $\zeta_{d}^{LR}\sim0.05$ with a
nontrivial new weak phase $\phi_d^L\sim-50^{\circ}$, which is relevant to the
$Z^{\prime}$ contributions to the QCD penguin sector $\triangle C_5$, is needed
to reconcile the observed discrepancy. Combined with the recent fitting results
from $B\to\pi K$, $\pi K^{\ast}$ and $\rho K$ decays, the $Z^{\prime}$
parameter spaces are severely reduced but still not excluded entirely, implying
that both the "$\pi\pi$" and "$\pi K$" puzzles could be accommodated
simultaneously within such a family non-universal $Z^{\prime}$ model.

High Energy Physics - Phenomenology;High Energy Physics - Experiment