  We present measurements of the branching fractions for the charmless two-body
decays $\Bz\to\pip\pim$ and $\Bz\to\Kp\pim$, and a search for the decay
$\Bz\to\Kp\Km$. We include the effects of final-state radiation from the
daughter mesons for the first time, and quote branching fractions for the
inclusive processes $\Bz\to h^+ h^{\prime -} n\gamma$, 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 \upsbb decays collected with the \babar\
detector at the \pep2 asymmetric-energy \epem collider at SLAC, we measure:
\begin{eqnarray*} {\cal B} (\Bz \to \pip\pim n\gamma;\: E_{\gamma}^{\rm
max}=150\mev) & = & (5.1\pm 0.4\pm 0.2)\times 10^{-6}, {\cal B} (\Bz \to
\Kp\pim n\gamma;\: E_{\gamma}^{\rm max}=105\mev) & = & (18.1\pm 0.6\pm
0.6)\times 10^{-6}, {\cal B} (\Bz \to \Kp\Km n\gamma;\: E_{\gamma}^{\rm
max}=59\mev) & < & 0.5 \times 10^{-6} (90% {\rm confidence level}),
\end{eqnarray*} where the first uncertainty is statistical and the second is
systematic. Theoretical calculations can be used to extrapolate from the above
measurements the non-radiative branching fractions, ${\cal B}^0$. Using one
such calculation, we find: \begin{eqnarray*} {\cal B}^0 (\Bz \to \pip\pim) & =
& (5.5\pm 0.4\pm 0.3)\times 10^{-6}, {\cal B}^0 (\Bz \to \Kp\pim) & = &
(19.1\pm 0.6\pm 0.6)\times 10^{-6}, {\cal B}^0 (\Bz \to \Kp\Km) & < & 0.5
\times 10^{-6} (90% {\rm confidence level}). \end{eqnarray*} Meaningful
comparison between theory and experiment, as well as combination of
measurements from different experiments, can be performed only in terms of
these non-radiative quantities.
