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We present a lattice QCD study of charmonium resonances and bound states with$J^{PC}=1^{--}$ and $3^{--}$ near the open-charm threshold, taking into accounttheir strong transitions to $\bar DD$. Vector charmonia are the most abundantin the experimentally established charmonium spectrum, while recently LHCbreported also the first discovery of a charmonium with likely spin three. The$\bar DD$ scattering amplitudes for partial waves $l=1$ and $l=3$ are extractedon the lattice by means of the L\"uscher formalism, using multiple volumes andinertial frames. Parameterizations of the scattering amplitudes provide massesand widths of the resonances, as well as the masses of bound states. CLSensembles with 2+1 dynamical flavors of non-perturbatively $O(a)$ improvedWilson quarks are employed with $m_\pi\simeq 280$ MeV, a single lattice spacingof $a\simeq0.086$ fm and two lattice spatial extents of $L=24$ and $32$. Twovalues of the charm quark mass are considered to examine the influence of theposition of the $\bar{D}D$ threshold on the hadron masses. For the lightercharm quark mass we find the vector resonance $\psi(3770)$ with mass$m=3780(7)$ MeV and coupling $g=16.0(^{+2.1}_{-0.2})$ (related to the width),both consistent with their experimental values. The vector $\psi(2S)$ appearsas a bound state with $m=3666(10)$ MeV. The charmonium resonance with$J^{PC}=3^{--}$ is found at $m=3831(^{+10}_{-16})$ MeV, consistent with the$X(3842)$ recently discovered by LHCb. At our heavier charm-quark mass the$\psi(2S)$ as well as the $\psi(3770)$ are bound states and the $X(3842)$remains a resonance. We stress that all quoted uncertainties are onlystatistical, while lattice spacing effects and the approach to the physicalpoint still need to be explored. This study of conventional charmonia sets thestage for more challenging future studies of unconventional charmonium-likestates.

Charmonium resonances with $J^{PC}=1^{--}$ and $3^{--}$ from $\bar DD$ scattering on the lattice