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9 pages, 12 postscript figues, submitted to PRDWe present measurements of the branching fractions for the charmless two-body decays $B^0 \to \pi^+ \pi^-$ and $B^0 \to K^+ \pi^-$, and a search for the decay $B^0 \to K^+ K^-$. We include the effects of final-state radiation from the daughter mesons for the first time, and quote branching fractions for the inclusive processes $B^0 \to h^+ h^{\prime -} n\gamma$, where $h$ and $h^\prime$ are pions or kaons. The maximum value of the sum of the energies of the $n$ undetected photons, $E_\gamma^{\rm max}$, is mode-dependent. Using a data sample of approximately 227 million $\Upsilon (4S) \to B \bar B$ decays collected with the BaBar detector at the PEP-II asymmetric-energy $e^+ e^-$ collider at SLAC, we measure: BF (B^0 \to \pi^+ \pi^-, n\gamma;\: E_{\gamma}^{\rm max}=150 MeV) & = & (5.4\pm 0.4\pm 0.3) \times 10^{-6}, BF (B^0 \to K^+ \pi^-, n\gamma;\: E_{\gamma}^{\rm max}=105 MeV) & = & (18.6\pm 0.6\pm 0.6)\times 10^{-6}, BF (B^0 \to K^+ K^-, n\gamma;\: E_{\gamma}^{\rm max}=59 MeV) & < & 0.40\times 10^{-6} (90% {\rm confidence level}), where the first uncertainty is statistical and the second is systematic

physical review. d. particles, fields, gravitation, and cosmology/physical review. d, particles, fields, gravitation, and cosmology

Improved measurements of the branching fractions for<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msup><mml:mi>B</mml:mi><mml:mn>0</mml:mn></mml:msup><mml:mo>→</mml:mo><mml:msup><mml:mi>π</mml:mi><mml:mo>+</mml:mo></mml:msup><mml:msup><mml:mi>π</mml:mi><mml:mo>−</mml:mo></mml:msup></mml:math>and<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msup><mml:mi>B</mml:mi><mml:mn>0</mml:mn></mml:msup><mml:mo>→</mml:mo><mml:msup><mml:mi>K</…

Improved measurements of the branching fractions forB0→π+π−andB0→K