  We search for annihilation decay modes of neutral $b$ mesons into pairs of
charmless charged hadrons with the upgraded Collider Detector at the Fermilab
Tevatron. Using a data sample corresponding to 6 fb$^{-1}$ of integrated
luminosity, we obtain the first evidence for the $B^0_s \to \pi^+\pi^-$ decay,
with a significance of $3.7\sigma$, and a measured branching ratio
$\mathcal{B}(B^0_s \to \pi^+\pi^-)= (0.57 \pm 0.15 (stat) \pm 0.10
(syst))\times 10^{-6}$. A search for the $B^0 \to K^+K^-$ mode in the same
sample yields a significance of $2.0\sigma$, and a central value estimate
$\mathcal{B}(B^0 \to K^+K^-)= (0.23 \pm 0.10 (stat) \pm 0.10 (syst))\times
10^{-6}$.
