Predicting isovector charmonium-like states from X(3872) properties

Abstract: Using chiral effective field theory, we predict that there must be isovector charmonium-like $D\bar D<sup>*$ hadronic molecules with $J<sup>{PC}=1<sup>{++}$ denoted as $W_{c1}$. The inputs are the properties of the $X(3872)$, including its mass and the ratio of its branching fractions of decays into $J/\psi\rho<sup>0$ and $J/\psi\omega$. The predicted states are virtual state poles of the scattering matrix, pointing at a molecular nature of the $X(3872)$ as well as its spin partners. They should show up as either a mild cusp or dip at the $D\bar D<sup>*$ thresholds, explaining why they are elusive in experiments. The so far negative observation also indicates that the $X(3872)$ is either a bound state with non-vanishing binding energy or a virtual state, only in these cases the $X(3872)$ signal dominates over that from the $W_{c1}<sup>0$. The pole positions are $$3881.2^{{+0.8}_{-0.0}-} i 1.6<sup>{+0.7}_{-0.9}$$ MeV for $W_{c1}<sup>0$ on the fourth Riemann sheet of the $D^0\bar D<sup>{<em>0}$-$D^{+D^{}</em>-}$ coupled-channel system, and $3866.9^{{+4.6}_{-7.7}-} i (0.07\pm0.01)$ MeV for $W_{c1}<sup>\pm$ on the second Riemann sheet of the $(D\bar D<sup>*)<sup>\pm$ single-channel system. The findings imply that the peak in the $J/\psi\pi<sup>+\pi<sup>-$ invariant mass distribution is not purely from the $X(3872)$ but contains contributions from $W_{c1}<sup>0$ predicted here. The states should have isovector heavy quark spin partners with $J<sup>{PC}=0<sup>{++}$, $2<sup>{++}$ and $1<sup>{+-}$, with the last one corresponding to $Z_c$. We suggest to search for the charged $0<sup>{++}$, $1<sup>{++}$ and $2<sup>{++}$ states in $J/\psi\pi^\pm \pi<sup>0$.