Improved measurements of the branching fractions for $B^0\to {\pi }^{+}{\pi }^{-}$ and $B^0\to K^{+}{\pi }^{-}$, and a search for $B^0\to K^{+}{K}^{-}$

We 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, $E_{\gamma }^{\max }$, is mode-dependent. Using a data sample of approximately $227\times {10}^6$ $\mathit{{\rm Y}}(4S)\to B\overline{B}$ decays collected with the BABAR detector at the PEP-II asymmetric-energy $e^{+}{e}^{-}$ collider at SLAC, we measure: $\mathcal{B}(B^0\to {\pi }^{+}{\pi }^{-}n\gamma ;E_{\gamma }^{\max }=150\mathrm{MeV})=(5.1\pm 0.4\pm 0.2)\times {10}^{-6}$, $\mathcal{B}(B^0\to K^{+}{\pi }^{-}n\gamma ;E_{\gamma }^{\max }=105\mathrm{MeV})=(18.1\pm 0.6\pm 0.6)\times {10}^{-6}$, $\mathcal{B}(B^0\to K^{+}{K}^{-}n\gamma ;E_{\gamma }^{\max }=59\mathrm{MeV})<0.5\times {10}^{-6}(90\%\mathrm{\text{confidence level}})$, where the first uncertainty is statistical and the second is systematic. Theoretical calculations can be used to extrapolate from the above measurements the nonradiative branching fractions, ${\mathcal{B}}^0$. Using one such calculation, we find: ${\mathcal{B}}^0(B^0\to {\pi }^{+}{\pi }^{-})=(5.5\pm 0.4\pm 0.3)\times {10}^{-6}$, ${\mathcal{B}}^0(B^0\to K^{+}{\pi }^{-})=(19.1\pm 0.6\pm 0.6)\times {10}^{-6}$, ${\mathcal{B}}^0(B^0\to K^{+}{K}^{-})<0.5\times {10}^{-6}(90\%\mathrm{\text{confidence level}})$. Meaningful comparison between theory and experiment, as well as combination of measurements from different experiments, can be performed only in terms of these nonradiative quantities.