Measurement of the $e^+e^-\to \pi^+\pi^-\pi^0$ cross section in the centre-of-mass range 0.62 to 3.5$~$GeV at Belle$~$II

We report a measurement of the $e^+e^-\to \pi^+\pi^-\pi^0$ cross section in the energy range from 0.62$~$GeV to$~$3.5 GeV using an initial-state radiation technique. We use an $e^+e^-$ data sample corresponding to 191$~$fb$^{-1}$ of integrated luminosity, collected at a centre-of-mass energy at or near the $\Upsilon(4S)$ resonance with the Belle II detector at the SuperKEKB collider. The uncertainty at the $\omega$ and $\phi$ resonances is 2.2%. The leading order hadronic vacuum polarisation contribution to the muon anomalous magnetic moment using this result is $a^{3\pi}_{\mu} = (48.91 \pm 0.23 \pm 1.07) \times 10^{-10}$. This result differs by $2.5$ standard deviations from the current most precise determination.