Quarkonium Contribution to Meson Molecules

Abstract

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 (cc) states. Since it is reasonable to expect the bare χ c1 (2P) 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 χ c2 (2P) 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 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 ( * )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 * 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(2S)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 X b and X b2 states, heavy-quark spin-flavor partners of the X (3872).a

Different pole structures in line shapes of the X(3872)

“…[59]. Among the many theoretical approaches [1][2][3][4][5][6]33,[51][52][53][54][55][56][60][61][62][63], we have paid special attention to the applicability of the popular ERE approximation up to and including the effective-range contribution to study near-threshold states like X (3872). We have elaborated about the fact that the ERE convergence radius might be severely limited due to the presence of nearthreshold zeros of the partial wave, the so-called CastillejoDalitz-Dyson poles.…”

“…one could think of the conventional χ c1 (2P) as cc, tetraquarks, hybrids, etc. [4,33,[51][52][53][54][55][56]. These facts about the smallness of the imaginary part of the two near-threshold poles and the small compositeness for the bound state can be understood in algebraic terms in the limit |E R |/ → 0 as shown in Appendix C; cf.…”

Hadronic molecules with hidden charm and bottom

“…The complexity on the X 2 due to the coupling of bare quark model cc poles was recently examined in Ref. [35].…”

An Effective Theory approach to $\bar{B}_s$ mesons involving $SU(3)$ heavy meson symmetry and constituent quark-model states

“…We evaluate such interplay in an effective theory framework, where the degrees of freedom are the heavyB,B * mesons coupled to kaons in S-wave, together with scalar and axial CQM bs states. On the one hand, the coupling ofB ( * ) K channels is performed using a Heavy Meson Chiral Lagrangian [5], on the other hand the 0 + and 1 + CQM states are coupled to the two-meson channels in a HQSS invariant formalism [6]. There is no experimental evidence of the (B * s0 ,B s1 ) jq = 1/2 doublet, contrary to the case of jq = 3/2 (B * s1 (5830),B * s2 (5840)) doublet.…”