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The process $e^+e^- \to \pi^+\pi^-2\pi^0\gamma$ is investigated by means ofthe initial-state radiation technique, where a photon is emitted from theincoming electron or positron. Using $454.3 \rm{fb}^{-1}$ of data collectedaround a center-of-mass energy of $\sqrt{s} = 10.58 \rm{GeV}$ by the BaBarexperiment at SLAC, approximately 150000 signal events are obtained. Thecorresponding non-radiative cross section is measured with a relativeuncertainty of $3.6\%$ in the energy region around $1.5 \rm{GeV}$, surpassingall existing measurements in precision. Using this new result, the channel'scontribution to the leading order hadronic vacuum polarization contribution tothe anomalous magnetic moment of the muon is calculated as$(g_\mu^{\pi^+\pi^-2\pi^0}-2)/2 = (17.9 \pm 0.1_\mathrm{stat} \pm0.6_\mathrm{syst}) \times 10^{-10}$ in the energy range $0.85\rm{GeV} <E_\mathrm{CM} < 1.8\rm{GeV}$. In the same energy range, the impact on therunning of the fine structure constant at the $Z^0$-pole is determined as$\Delta\alpha^{\pi^+\pi^-2\pi^0}(M^2_\mathrm{Z}) = (4.44 \pm 0.02_\mathrm{stat}\pm 0.14_\mathrm{syst}) \times 10^{-4}$. Furthermore, intermediate resonancesare studied and especially the cross section of the process $e^+e^- \to\omega\pi^0 \to \pi^+\pi^-2\pi^0$ is measured.

Measurement of the $e^+e^-\to\pi^+\pi^-\pi^0\pi^0$ cross section using initial-state radiation at BaBar