Quarkonium Contribution to Meson Molecules

Abstract

Starting from a molecular picture for the X(3872) resonance, this state and its JPC= 2 + + heavy-quark spin symmetry partner [ X2(4012) ] are analyzed within a model which incorporates possible mixings with 2P charmonium (cc¯) states. Since it is reasonable to expect the bare χc1(2 P) to be located above the DD¯ ∗ threshold, but relatively close to it, the presence of the charmonium state provides an effective attraction that will contribute to binding the X(3872), but it will not appear in the 2 + + sector. Indeed in the latter sector, the χc2(2 P) should provide an effective small repulsion, because it is placed well below the D∗D¯ ∗ threshold. We show how the 1 + + and 2 + + bare charmonium poles are modified due to the D(∗)D¯ (∗) loop effects, and the first one is moved to the complex plane. The meson loops produce, besides some shifts in the masses of the charmonia, a finite width for the 1 + + dressed charmonium state. On the other hand, X(3872) and X2(4012) start developing some charmonium content, which is estimated by means of the compositeness Weinberg sum rule. It turns out that in the heavy-quark limit, there is only one coupling between the 2P charmonia and the D(∗)D¯ (∗) pairs. We also show that, for reasonable values of this coupling, leading to X(3872) molecular probabilities of around 70–90 %, the X2 resonance destabilizes and disappears from the spectrum, becoming either a virtual state or one being located deep into the complex plane, with decreasing influence in the D∗D¯ ∗ scattering line. Moreover, we also discuss how around 10–30 % charmonium probability in the X(3872) might explain the ratio of radiative decays of this resonance into ψ(2 S) γ and J/ ψγ. Finally, we qualitatively discuss within this scheme, the hidden bottom flavor sector, paying a special attention to the implications for the Xb and Xb2 states, heavy-quark spin–flavor partners of the X(3872).