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

Using chiral effective field theory, we predict that there must be isovector charmonium-like $D{\overline{D}}^{\ast }$ \( D{\overline{D}}^{\ast } \) 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/ψρ⁰ and J/ψω. 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 $D{\overline{D}}^{\ast }$ \( D{\overline{D}}^{\ast } \) 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 $W_{c1}^0$ \( {W}_{c1}^0 \) . The pole positions are ${3881.2}_{-0.0}^{+0.8}$ \( {3881.2}_{-0.0}^{+0.8} \) − $i{1.6}_{-0.9}^{+0.7}$ \( i{1.6}_{-0.9}^{+0.7} \) MeV for $W_{c1}^0$ \( {W}_{c1}^0 \) on the fourth Riemann sheet of the $D^0{\overline{D}}^{\ast 0}$ \( {D}^0{\overline{D}}^{\ast 0} \) -D⁺D^∗− coupled-channel system, and ${3866.9}_{-7.7}^{+4.6}$ \( {3866.9}_{-7.7}^{+4.6} \) − i(0.07 ± 0.01) MeV for $W_{c1}^{\pm }$ \( {W}_{c1}^{\pm } \) on the second Riemann sheet of the ${\left(D{\overline{D}}^{\ast }\right)}^{\pm }$ \( {\left(D{\overline{D}}^{\ast}\right)}^{\pm } \) single-channel system. The findings imply that the peak in the J/ψπ⁺π⁻ invariant mass distribution is not purely from the X(3872) but contains contributions from $W_{c1}^0$ \( {W}_{c1}^0 \) 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/ψπ^±π⁰.