Signature of an $h_1$ state from $J/\psi \to \eta K^{*0}\bar{K}^{*0}$ and theoretical description of the $Z_c(3900)$ and $Z_c(4020)$ as $D \bar D^*$ and $D^* \bar D^*$ molecular states

In this talk we address two topics: The first one is an empirical explanation in terms of a new state $h_1$ of the peak in the $K^{*0}\bar{K}^{*0}$ invariant mass distribution close to threshold of this channel in the $J/\psi \to \eta K^{*0}\bar{K}^{*0}$ decay. The second one is a theoretical description of the isospin $I=1$ $Z_c(3900)$ and $Z_c(4020)$ states in terms of molecular states of $D \bar D^*+ cc$ and $D^* \bar D^*$.


Introduction
In this talk we will address two topics: In the first topic we addreess the enhancement of the K * 0K * 0 invariant mass distribution close to threshold in the J/ψ → ηK * 0K * 0 decay and interprete it in terms of a new state h 1 which has been theoretically predicted as a dynamically generated state from the vector-vector interaction. In the second one we use and extension of the local hidden gauge approach for the interaction of mesons and provide a theoretical description of the Z c (3900) and Z c (4020) in terms of molecular states of DD * and D * D * .

Experimental support for a new h 1 (1830) state
The decay J/ψ → ηK * 0K * 0 was measured by the BES Collaboration [1] searching for the Y(2175) resonance, however, no clear enhancement in the K * 0K * 0 mass distribution was found near 2.175 GeV. Instead an enhancement in the K * 0K * 0 mass distribution was found that passed unnoticed. In [2] this fact was used to claim support for an h 1 resonance that was predicted in [3] as dynamically generated from the interaction of vector mesons. The h 1 state has quantum numbers 0 − (1 +− ), and, since the J/ψ and the η mesons have quantum numbers 0 − (1 −− ) and 0 + (0 −+ ), respectively, the decay J/ψ → ηK * 0K * 0 constitutes the ideal reaction to look for an h 1 state, coupling to an s-wave K * K * pair. In Fig. 1 we show diagrammatically the process. The K * 0K * 0 state is produced and the two K * interact. This interaction is the one that leads to the h 1 resonance in [3]. It is interesting to note that this state is very peculiar. Because of its negative C-parity, the h 1 state cannot couple to two vector mesons of the type ρ, ω, φ. Also because of its positive parity it cannot couple to two pseudoscalar mesons, which would have to be in L=1 to match the spin of the h 1 and hence would have negative parity. The K * K * is then a unique state for both the construction and decay of this h 1 state. The scattering matrix for the K * K * interaction is given in terms of a potential V by where G(M 2 inv ) is the loop function for the K * K * pair. This function is divergent, and it can be regularized both with a cutoff or with dimensional regularization in terms of a subtraction constant. On the other hand the full amplitude for the process J/ψ → ηK * K * can be written, according to the diagrams in Fig. 1, as: where the last equality follows from Eq. (1). Our strategy to interpret the data of [1] is to use the potential V derived in [3] and fit the data of [1] by changing the subtraction constant in the G function in dimensional regularization a(μ), where μ is a regularization scale, taken here as μ = 1 GeV.
As seen in figure 2 a good agreement with the data is found for reasonable values of the parameter a(μ). The figure also shows the results with pure phase space normalized to the same area and we see that the disagreement with the data is quite obvious.
In order to see that with this parameter a(μ) one gets a resonance, in figure 3 we show the magnitude of |t| 2 and we see indeed that it has a resonant shape. We also perform an alternative fit to the data which could be carried on by someone not knowing the potential of [3]. In this case we can take a constant potential and fit it to the data. The results obtained are similar and the conclusion the same. Altogether we find from this analysis that there is a resonance h 1 with a mass M h 1 = 1830 ± 20 MeV and Γ h 1 = 110 ± 10 MeV, respectively.
We think that these data offer enough evidence for this new state. In [4] we have made predictions for two more reactions η c → φK * K * , or η c (2S ) → φK * K * , that should help establish this resonance on firmer grounds. In Fig. 4 we show the predictions for η c (2S ) → φK * K * comparing the results with the h 1 resonance and pure phase space. The differences are striking and we encourage such a measurement to give further support to our claims based on the J/ψ → ηK * 0K * 0 decay.   In the charm sector there is also a related h 1 hidden charm state stemming from the vector vector interaction in [13] at 3945 MeV, which is also found in [5]. In [6] a reaction is also proposed to find this state, looking at the invariant mass distribution of D * D * in the X(4660) → ηD * D * decay.
3 Theoretical description of the Z c (3900) and Z c (4020) as DD * and D * D *

molecular states
One of the interesting recent surprises in hadronic physics has been the discovery of states that challenge the standard qq structure of the mesons. In this sense, charmonium states of isospin I=1 are of this type since cc has necessarily I=0. One state called Z c (3900) has been reported by the BESIII collaboration [9] and reconfirmed in other experiments. Similarly, in [10] another Z c (4020) has been reported, also reconfirmed in other experiments. It is unclear whether this state has something to do 01011-p.3 with the peak in the D * D * invariant mass distribution at 4025 MeV in the e + e − → (D * D * ) ± π ∓ reaction studied in [11]. Indeed, it was shown in [12] that this peak could be interpreted in terms of a resonance at much lower energy that was predicted in [13] from the D * D * interaction.
In [7,8] we have looked into the possibility of creating states from the interaction of DD * + cc and D * D * with isospin 1. In order to study the interaction of pseudoscalar of vector mesons we use the local hidden gauge approach [14], which generalizes the chiral Lagrangians to include vector mesons. The approach produces the interaction by means of the exchange or vector mesons and several states can be produced with I=0 in the charm sector [13], or in the bottom sector [15]. Yet, the I=1 states are more difficult to obtain. The reason can be devised by looking at fig. 5. We can see that if we try to exchange a light qq state it is OZI forbidden since we would need to exchange a dd state for the upper line and the lower line demands a uū one. In view of this one has to exchange a heavy vector cc, but one pays a penalty in this mechanism since the propagator of the heavy vector is much suppressed. This makes the interaction weaker and the possibilities of binding smaller. What we say about the exchange of a light vector can also be said about the exchange of pseudoscalar mesons and in [7,8] it was shown that in the limit of equal masses for the pseudoscalar mesons, the contribution of the π, η, η together vanished. In the case when different masses are taken into account then the cancellation is partial but the mechanism is also quite suppressed.
In view of this, in [7,8] we also studied the exchange of two pions, either correlated to give the f 0 (500) or σ meson through their interaction, or uncorrelated. They also were found to be relatively small. Yet, the strength of the heavy vector exchange, even if very much weakened, is enough to create some structure, pole or cusp, depending on the strength of the interaction which is governed by the cut

Conclusion
We have reported work on two topics. In the first one we showed how the data of the J/ψ → ηK * 0K * 0 decay, by looking at the invariant mass distribution of the K * 0K * 0 , provided some evidence for a new 01011-p.5 h 1 (1830) state which is not catalogued in the PDG. We also suggested performing a measurement of two more reactions, η c → φK * K * , or η c (2S ) → φK * K * , that should help establish this resonance on firmer grounds. In the second part we explored the possibility of forming molecular states of DD * + cc and D * D * . We stressed the observation that from the point of view of meson exchange, the exchange of light vector mesons or pseudoscalar mesons is OZI forbidden. Then we explored the exchange of heavy vectors and two pion exchange, correlated and uncorrelated. We still found the exchange of vector mesons bigger than the two pion exchange, and enough to create some molecular structure very close to the threshold of the DD * + cc and D * D * channels. We also included other pseudoscalar-vector and vector-vector states in a coupled channels formalism, which were open for the decay of the states found and they gave rise to a width in qualitative agreement with experiment, thus giving support for the Z c (3900) and Z c (4020) as weakly bound molecular states of DD * + cc and D * D * respectively.