Abstract
The newly observed Pc(4312), Pc(4440) and Pc(4457) at the LHCb experiment are very close to the {varSigma}_coverline{D} and {varSigma}_c{overline{D}}^{ast } thresholds. In this work, we perform a systematic study and give a complete picture on the interactions between the {varSigma}_c^{left(ast right)} and {overline{D}}^{left(ast right)} systems in the framework of heavy hadron chiral effective field theory, where the short-range contact interaction, long-range one-pion-exchange contribution, and intermediate-range two-pion-exchange loop diagrams are all considered. We first investigate the three Pc states without and with considering the Λc contribution in the loop diagrams. It is difficult to simultaneously reproduce the three Pcs unless the Λc is included. The coupling between the {varSigma}_c^{left(ast right)}{overline{D}}^{left(ast right)} and {Lambda}_c{overline{D}}^{left(ast right)} channels is crucial for the formation of these Pcs. Our calculation supports the Pc(4312), Pc(4440) and Pc(4457) to be the S-wave hidden-charm {left[{varSigma}_coverline{D}right]}_{J=1/2}^{I=1/2},{left[{varSigma}_c{overline{D}}^{ast}right]}_{J=1/2}^{I=1/2} and {left[{varSigma}_c{overline{D}}^{ast}right]}_{J=3/2}^{I=1/2} molecular pentaquarks, respectively. Our calculation disfavors the spin assignment {J}^P=frac{1^{-}}{2} for Pc(4457) and {J}^P=frac{3^{-}}{2} for Pc(4440), because the excessively enhanced spin-spin interaction is unreasonable in the present case. We obtain the complete mass spectra of the {left[{varSigma}_c^{left(ast right)}{overline{D}}^{left(ast right)}right]}_J systems with the fixed low energy constants. Our result indicates the existence of the {left[{varSigma}_c^{ast }{overline{D}}^{ast}right]}_Jleft(J=frac{1}{2},frac{3}{2},frac{5}{2}right) hadronic molecules. The previously reported Pc(4380) might be a deeper bound one. Additionally, we also study the hidden-bottom {varSigma}_b^{left(ast right)}{B}^{left(ast right)} systems, and predict seven bound molecular states, which could serve as a guidance for future experiments. Furthermore, we also examine the heavy quark symmetry breaking effect in the hidden-charm and hidden-bottom systems by taking into account the mass splittings in the propagators of the intermediate states. As expected, the heavy quark symmetry in the bottom cases is better than that in the charmed sectors. We notice that the heavy quark symmetry in the {varSigma}_coverline{D} and {varSigma}_c^{ast}overline{D} systems is much worse for some fortuitous reasons. The heavy quark symmetry breaking effect is nonnegligible in predicting the effective potentials between the charmed hadrons.
Highlights
Pc(4440), because the excessively enhanced spin-spin interaction is unreasonable in the present case
The heavy quark symmetry (HQS) breaking effect expounded above issues from the loop diagrams, which is the quantum physics of the light degrees of freedom at the low energy, and cannot be modified by any unknown physics that happens at the high energy
They were subsequently interpreted as the molecular states by many theoretical works [11,12,13,14,15] due to the proximities to the ΣcDand ΣcD ∗ thresholds
Summary
In the framework of heavy hadron chiral perturbation theory, the scattering amplitudes of the Σ(c∗)D (∗) systems can be expanded order by order in powers of a small parameter ε = q/Λχ, where q is either the momentum of Goldstone bosons or the residual momentum of heavy hadrons, and Λχ represents either the chiral breaking scale or the mass of a heavy hadron. The expansion is organized by the power counting rule [38, 39]. Where L and En represent the number of loops and external lines of the matter field. Vi denote the number of the type-i vertex with the order ∆i. Di and ni stand for the number of derivatives (or mπ factors) and external lines of the matter field in a type-i vertex Vi denote the number of the type-i vertex with the order ∆i. di and ni stand for the number of derivatives (or mπ factors) and external lines of the matter field in a type-i vertex
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