  KLOE and Babar have an observed discrepancy of 2% to 5% in the invariant pion
pair production cross section. These measurements are based on approximate NLO
$ \mu^+ \mu^- \gamma $ cross section predictions of the Monte Carlo event
generator PHOKHARA7.0. In this article, the complete NLO radiative corrections
to $ \mu^+ \mu^- \gamma $ production are calculated and implemented in the
Monte Carlo event generator PHOKHARA9.0. Numerical reliability is guaranteed by
two independent approaches to the real and the virtual corrections. The novel
features include the contribution of pentagon diagrams in the virtual
corrections, which form a gauge-invariant set when combined with their box
diagram partners. They may contribute to certain distributions at the percent
level. Also the real emission was complemented with two-photon final state
emission contributions not included in the generator PHOKHARA7.0. We
demonstrate that the numerical influence reaches, for realistic charge-averaged
experimental setups, not more than 0.1% at KLOE and 0.3% at BaBar energies. As
a result, we exclude the approximations in earlier versions of PHOKHARA as
origin of the observed experimental discrepancy.
