Study of the<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msup><mml:mi>e</mml:mi><mml:mo>+</mml:mo></mml:msup><mml:msup><mml:mi>e</mml:mi><mml:mo>−</mml:mo></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:msup><mml:mi>π</mml:mi><mml:mn>0</mml:mn></mml:msup></mml:math>process using initial state radiation with<i>BABAR</i>

The process ${e}^{+}{e}^{\ensuremath{-}}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}{\ensuremath{\pi}}^{0}\ensuremath{\gamma}$ has been studied at a center-of-mass energy near the $\ensuremath{\Upsilon}(4S)$ resonance using a $89.3\text{ }\text{ }{\mathrm{f}\mathrm{b}}^{\ensuremath{-}1}$ data sample collected with the BABAR detector at the PEP-II collider. From the measured $3\ensuremath{\pi}$ mass spectrum we have obtained the products of branching fractions for the $\ensuremath{\omega}$ and $\ensuremath{\phi}$ mesons, $\mathcal{B}(\ensuremath{\omega}\ensuremath{\rightarrow}{e}^{+}{e}^{\ensuremath{-}})\mathcal{B}(\ensuremath{\omega}\ensuremath{\rightarrow}3\ensuremath{\pi})=(6.70\ifmmode\pm\else\textpm\fi{}0.06\ifmmode\pm\else\textpm\fi{}0.27)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}5}$ and $\mathcal{B}(\ensuremath{\phi}\ensuremath{\rightarrow}{e}^{+}{e}^{\ensuremath{-}})\mathcal{B}(\ensuremath{\phi}\ensuremath{\rightarrow}3\ensuremath{\pi})=(4.30\ifmmode\pm\else\textpm\fi{}0.08\ifmmode\pm\else\textpm\fi{}0.21)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}5}$ and evaluated the ${e}^{+}{e}^{\ensuremath{-}}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}{\ensuremath{\pi}}^{0}$ cross section for the ${e}^{+}{e}^{\ensuremath{-}}$ center-of-mass energy range 1.05 to 3.00 GeV. About 900 ${e}^{+}{e}^{\ensuremath{-}}\ensuremath{\rightarrow}J/\ensuremath{\psi}\ensuremath{\gamma}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}{\ensuremath{\pi}}^{0}\ensuremath{\gamma}$ events have been selected and the branching fraction $\mathcal{B}(J/\ensuremath{\psi}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}{\ensuremath{\pi}}^{0})=(2.18\ifmmode\pm\else\textpm\fi{}0.19)%$ has been measured.