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

Using chiral effective field theory, we predict that there must be isovector charmonium-like $D\bar D^*$ hadronic molecules with $J^{PC}=1^{++}$ 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 $J/\psi\rho^0$ 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 $D\bar D^*$ 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 $W_{c1}^0$. The pole positions are $3881.2^{+0.8}_{-0.0}- i 1.6^{+0.7}_{-0.9}$ MeV for $W_{c1}^0$ on the fourth Riemann sheet of the $D^0\bar D^{*0}$-$D^+D^{*-}$ coupled-channel system, and $3866.9^{+4.6}_{-7.7}- i (0.07\pm0.01)$ MeV for $W_{c1}^\pm$ on the second Riemann sheet of the $(D\bar D^*)^\pm$ single-channel system. The findings imply that the peak in the $J/\psi\pi^+\pi^-$ invariant mass distribution is not purely from the $X(3872)$ but contains contributions from $W_{c1}^0$ predicted here. The states should have isovector heavy quark spin partners with $J^{PC}=0^{++}$, $2^{++}$ and $1^{+-}$, with the last one corresponding to $Z_c$. We suggest to search for the charged $0^{++}$, $1^{++}$ and $2^{++}$ states in $J/\psi\pi^\pm \pi^0$.