Measurement of cross sections for $e^{+}e^{-} \rightarrow \mu^+\mu^-$ at center-of-mass energies from 3.80 to 4.60 GeV

The observed cross sections for $e^+e^-\rightarrow \mu^+\mu^-$ at energies from 3.8 to 4.6 GeV are measured using data samples taken with the BESIII detector operated at the BEPCII collider. We measure the muonic widths and determine the branching fractions of the charmonium states $\psi(4040)$, $\psi(4160)$, and $\psi(4415)$ decaying to $\mu^+\mu^-$, as well as making a first determination of the phase of the amplitudes. In addition, we observe evidence for a structure in the dimuon cross section near 4.220 GeV/$c^2$, which we denote as $S(4220)$. Analyzing a coherent sum of amplitudes yields eight solutions, one of which gives a mass of ${M}_{S(4220)}=4216.7 \pm 8.9 \pm 4.1$~MeV/$c^2$, a total width of ${\rm \Gamma^{\rm tot}_{S(4220)}}=47.2 \pm 22.8 \pm 10.5$~MeV, and a muonic width of ${\rm \Gamma}^{\mu\mu}_{S(4220)}=1.53\pm1.26\pm0.54$~keV, where the first uncertainties are statistical and the second systematic. The eight solutions give the central values of the mass, total width, muonic width to be, respectively, in the range from 4212.8 to 4219.4 MeV/$c^2$, from 36.4 to 49.6 MeV, and from 1.09 to 1.53 keV. The statistical significance of the $S(4220)$ signal is $3.9\sigma$. Correcting the total dimuon cross section for radiative effects yields a statistical significance for this structure of more than $7\sigma$.