Sliding window adaptive fast QR and QR-lattice algorithms

OpenAlex · İngilizce

Starting from a molecular picture for the X(3872) resonance, this state and its $$J^{PC}=2^{++}$$ heavy-quark spin symmetry partner $$[X_2(4012)]$$ are analyzed within a model which incorporates possible mixings with 2P charmonium ( $$c\bar{c}$$ ) states. Since it is reasonable to expect the bare $$\chi _{c1}(2P)$$ to be located above the $$D\bar{D}^*$$ 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 $$\chi _{c2}(2P)$$ should provide an effective small repulsion, because it is placed well below the $$D^*\bar{D}^*$$ threshold. We show how the $$1^{++}$$ and $$2^{++}$$ bare charmonium poles are modified due to the $$D^{(*)}\bar{D}^{(*)}$$ 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 $$X_2(4012)$$ 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 $$D^{(*)}\bar{D}^{(*)}$$ pairs. We also show that, for reasonable values of this coupling, leading to X(3872) molecular probabilities of around 70–90 %, the $$X_2$$ 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 $$D^{*}\bar{D}^{*}$$ 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 $$\psi (2S)\gamma $$ and $$J/\psi \gamma $$ . Finally, we qualitatively discuss within this scheme, the hidden bottom flavor sector, paying a special attention to the implications for the $$X_b$$ and $$X_{b2}$$ states, heavy-quark spin–flavor partners of the X(3872).