Establishing the first hidden-charm pentaquark with strangeness

We study the $P_{cs}(4459)^0$ recently observed by LHCb using the method of QCD sum rules. Our results support its interpretation as the $\bar D^* \Xi_c$ hadronic molecular state of either $J^P=1/2^-$ or $3/2^-$. Within the hadronic molecular picture, the three LHCb experiments observing $P_c$ and $P_{cs}$ states \cite{lhcb,Aaij:2015tga,Aaij:2019vzc} can be well understood as a whole. This strongly supports the existence of hadronic molecules, whose studies can significantly improve our understanding on the construction of the subatomic world. To verify this picture, we propose to further investigate the $P_{cs}(4459)^0$ to examine whether it can be separated into two states, and to search for the $\bar D \Xi_c$ molecular state of $J^P=1/2^-$.

Introduction.--Atomic nuclei are made of protons and neutrons, which are themselves composed of quarks and gluons. In the past century a huge number of subatomic particles, called hadrons, were discovered in particle experiments, whose properties are similar to the proton and neutron [4]. One naturally raises an interesting question: are there subatomic particles corresponding to the nucleus? Nowadays we call them "hadronic molecules", whose studies can significantly improve our understanding on the construction of the subatomic world, as illustrated in Fig. 1.
It is not so easy to answer the above question. Most of the experimentally observed hadrons can be described as qq mesons or qqq baryons in the conventional quark model, while there can also exist (compact) qqqq tetraquarks and qqqqq pentaquarks [5,6], etc. With the experimental progress on this issue over the past decade, dozens of XY Z charmoniumlike states were reported, providing us good opportunities to identify exotic hidden-charm tetraquarks [4]. However, these states may be explained as hadronic molecules, while they may also be explained as compact tetraquarks that are still hadrons. Fortunately, in the LHCb experiments performed in 2015 and 2019 [1,2], the famous hidden-charm pentaquark states, P c (4312), P c (4380), P c (4440), and P c (4457), were discovered. These four P c states contain at least five quarks ccuud, so they are perfect candidates of hidden-charm pentaquark states. Among them, the three narrow states P c (4312), P c (4440), and P c (4457) are just below theDΣ c andD * Σ c thresholds, so their natural interpretations are theDΣ c andD * Σ c hadronic molecular states [7,8,9,10,11], whose existence had been predicted in Refs. [12,13,14,15,16] before the LHCb experiment performed in 2015 [1]. However, there still exist other possible explanations [17,18,19,20,21,22], and we refer to the reviews [23,24,25,26,27,28] for detailed discussions.
It is natural to conjecture whether the hidden-charm pentaquark state with strangeness exists or not. Such a state is usually denoted as "P cs ", whose quark content is ccsqq (q = u/d). There have been some but not many theoretical studies on it [29,30,31,32,33,34,35,36,37]. Especially, we proposed in Ref. [38] to search for the P cs in the J/ψΛ invariant mass spectrum of the Ξ − b → J/ψK − Λ decays. Besides, in Ref. [39] the authors studied the P cs using the chiral effective field theory, and calculated the mass of the J P = 1/2 −D * Ξ c molecular state to be 4456.9 +3.2 −3.3 MeV. Note that both of these two references are based on the hadronic molecular picture.
Very recently, the LHCb Collaboration reported the evidence of a hidden-charm pentaquark state with strangeness, P cs (4459) 0 , in the J/ψΛ invariant mass spectrum of the [3]. Its mass and width were measured to be: MeV , while its spin-parity quantum number was not determined since the statistic is not enough. Note that the channel observing the P cs (4459) 0 is just the one proposed by us in Ref. [38], and the above mass value is almost identical to the mass of the J P = 1/2 −D * Ξ c molecular state predicted in Ref. [39]. Therefore, the present LHCb experiment [3] strongly supports the hadronic molecular picture once more.
Actually, as indicated by LHCb, the P cs (4459) 0 is about 19 MeV below theD * 0 Ξ 0 c threshold [3], so it is natural to interpret it as theD * Ξ c molecular state, with the spinparity quantum number J P = 1/2 − or 3/2 − . Accordingly, in this letter we shall study theD * Ξ c molecular states of J P = 1/2 − and 3/2 − using the method of QCD sum rules, and at the same time we shall also study theDΞ c molecular state of J P = 1/2 − . We calculate masses ofD * Ξ c molecular states to be 4.46 +0. 16 −0.14 GeV for the J P = 1/2 − one and 4.47 +0. 19 −0.15 GeV for the J P = 3/2 − one. These two values are both consistent with the experimental mass of the P cs (4459) 0 , supporting its interpretation as theD * Ξ c molecular state of either J P = 1/2 − or 3/2 − . We also calculate the mass of the J P = 1/2 −D Ξ c molecular state to be 4.29 +0.13 −0.12 GeV. To verify the hadronic molecular picture, we propose to further investigate the P cs (4459) 0 state in future experiments to examine whether it can be separated into two states, and to search for the J P = 1/2 −D Ξ c molecular state. If the hadronic molecular picture turns out to be correct, our understanding on the construction of the subatomic world would be significantly improved. Besides, these studies are helpful to improve our understanding on the non-perturbative behaviors of the strong interaction at the low energy region.
Hidden-charm pentaquark currents.--We use thec, c, s, u, and d quarks to construct hidden-charm pentaquark interpolating currents with strangeness. To studyD ( * ) Ξ c molecular states, we consider the following type of currents: where a · · · e are color indices, Γ 1/2/3 are Dirac matrices, and C = iγ 2 γ 0 is the charge-conjugation operator. The other type of currents: can be similarly studied, but they just lead to the same QCD sum rule results as the η(x) currents. Hence, the present study can not distinguish the isospin of P cs states. The η(x) currents can be constructed by combining charmed meson operators and charmed baryon fields. We need the charmed meson operators J D , which couple to the ground-state charmed mesons D =D 0 /D * 0 : We also need the charmed baryon field J B , which couples to the ground-state charmed baryon B = Ξ 0 c : There can be altogether threeD ( * ) Ξ c hadronic molecular states, that areDΞ c of J P = 1/2 − ,D * Ξ c of J P = 1/2 − , andD * Ξ c of J P = 3/2 − . Their relevant interpolating currents are: In the above expressions, we have used D and B to denote the charmed meson operators J D and the charmed baryon field J B ; P µν 3/2 is the spin-3/2 projection operator QCD sum rule studies.--We use the method of QCD sum rules [40,41] to studyD ( * ) Ξ c molecular states through the currents η 1,2 of J P = 1/2 − and η α 3 of J P = 3/2 − . Taking the current η 2 as an example, we assume it couples to theD * Ξ c molecular state of J P = 1/2 − through where u(p) is the Dirac spinor of this state, denoted as X for simplicity. The two-point correlation function extracted from η 2 can be written as: In QCD sum rule studies we calculate the two-point correlation function Π 0 q 2 at both hadron and quarkgluon levels. At the hadron level, we use the dispersion relation to write it as where s < is the physical threshold. We further define the imaginary part of the correlation function as the spectral density ρ(s), which is usually evaluated at the hadron level by inserting intermediate hadron states n |n n|: In the last step we have adopted the usual parametrization of one-pole dominance for the ground state X and a continuum contribution. At the quark-gluon level we calculate Π 0 q 2 using the method of operator product expansion (OPE), and derive its corresponding spectral density ρ OPE (s). After performing the Borel transformation to Eq. (11) at both hadron and quark-gluon levels, we can approximate the continuum using the spectral density above a threshold value s 0 (quark-hadron duality), and obtain the sum rule equation (13) It can be used to further calculate M X through In the present study we calculate OPEs at the leading order of α s and up to the D(imension) = 10 terms, including the perturbative term, the charm quark mass, the quark condensates qq / ss , the gluon condensate g 2 s GG , the quark-gluon mixed condensates g sq σGq / g ss σGs , and their combinations. We summarized the obtained spectral densities ρ 1···3 (s) in Appendix A, which are extracted from the currents η 1···3 , respectively. In the calculations we ignore the chirally suppressed terms with the up and down quark masses, and adopt the factorization assumption of vacuum saturation for higher dimensional condensates. We find that the D = 3 quark condensates qq / ss and the D = 5 mixed condensates g sq σGq / g ss σGs are multiplied by the charm quark mass, which are thus important power corrections.
Numerical analyses.---We still use the current η 2 as an example to perform numerical analyses, where we use the following values for various QCD sum rule parameters [4,42,44,45,46,47,48,49,43,50]: Here the running mass in the M S scheme is used for the charm quark.
There are two free parameters in Eqs. (14): the Borel mass M B and the threshold value s 0 . We use two criteria to constrain the Borel mass M B for a fixed s 0 . The first criterion is used to insure the convergence of the OPE series. It is done by requiring the D = 10 terms m c qq 3 and g sq σGq 2 to be less than 20%, which can determine the lower limit M min B : This criterion leads to M min Altogether we extract the working region of Borel mass to be 2.93 GeV 2 < M 2 B < 3.07 GeV 2 for the current η 2 with the threshold value s 0 = 25.8 GeV 2 . We show the variation of M X with respect to the Borel mass M B in Fig. 2 in a broader region 2.7 GeV 2 ≤ M 2 B ≤ 3.2 GeV 2 , and find it more stable inside the above Borel window.
Redoing the same procedures by changing s 0 , we find that there are non-vanishing Borel windows as long as s 0 ≥ s min 0 = 24.8 GeV 2 . Accordingly, we choose the threshold value s 0 to be about 1.0 GeV larger with the uncertainty ±1.0 GeV, i.e., 24.8 GeV 2 ≤ s 0 ≤ 26.8 GeV 2 . Altogether, our working regions for the current η 2 are determined to be 24.8 GeV 2 ≤ s 0 ≤ 26.8 GeV 2 and 2.93 GeV 2 < M 2 B < 3.07 GeV 2 , where the mass is extracted to be: MD * Ξc;1/2 − = 4.46 +0. 16 −0.14 GeV . Here the central value corresponds to M 2 B = 3.00 GeV 2 and s 0 = 25.8 GeV 2 . Its uncertainty comes from the Borel mass M B , the threshold value s 0 , the charm quark mass m c , and various QCD sum rule parameters listed in Eqs. (15). This mass value is consistent with the experimental mass of the P cs (4459) 0 , supporting its interpretation as theD * Ξ c molecular state of J P = 1/2 − . Similarly, we use the currents η 1 and η α 3 to perform numerical analyses, and extract masses of the J P = 1/2 − DΞ c molecular state and the J P = 3/2 −D * Ξ c molecular state to be: Hence, our results also support the interpretation of the P cs (4459) 0 as theD * Ξ c molecular state of J P = 3/2 − . We summarize all the above results in Table 1. Generally speaking, understanding the nature of exotic hadrons is a complicated topic, since different structures with the same quantum numbers can contribute to the same state, and different structures may have similar masses. However, these different structures may lead to different decay processes. Therefore, to determine whether the P cs (4459) 0 is theD * Ξ c molecular state of J P = 1/2 − or the one of J P = 3/2 − in our framework, we shall further calculate its width in our future study, to be compared with its experimental value Γ Pcs(4459) 0 = 17.3 ± 6.5 +8.0 −5.7 MeV [3].
Summary and Discussions.--Very recently, the LHCb Collaboration reported the evidence of a hidden-charm pentaquark state with strangeness, P cs (4459) 0 , in the J/ψΛ invariant mass spectrum of the Ξ − b → J/ψK − Λ decays [3]. This state contains at least five quarksccsqq (q = u/d), with one of them the strange quark. This LHCb experiment indicates that there probably exist many more exotic hadrons with strangeness to be discovered in the near future, so a new hadron spectrum is waiting to be constructed.
Hence, our QCD sum rule results support the interpretation of the P cs (4459) 0 as theD * Ξ c molecular state of either J P = 1/2 − or 3/2 − . Within the hadronic molecular picture, the three LHCb experiments observing P c and P cs states [3,1,2] can be well understood as a whole: -The transition of Ξ − b → J/ψK − Λ is dominated by the Cabibbo-favored weak decay of b → c +cs via the V -A current. This leads to an intuitive expectation that the s and d pair in Ξ − b may exist as a spectator, so that their total spin S = 0 is conserved. As a consequence, if this sd pair is to be combined with a c quark to form a charmed baryon, it will favor the Ξ c instead of the Ξ ′ c and Ξ * c , of which the total spin of the sd pair is S = 1. Accordingly, the P cs (4459) 0 possibly as theD * Ξ c molecular state was observed in this channel by LHCb [3], other than those possibly existingD ( * ) Ξ ′ c andD ( * ) Ξ * c molecular states. We illustrate this process in Fig. 3(a).
-The transition of Λ 0 b → J/ψK − p can be similarly analysed, which may favor the Λ c instead of the Σ c and Σ * c . However, in this case the Λ c is probably not bounded with the charmed mesons due to the lack of π exchanges. It is only because of the significantly larger data sample (680k for the Λ 0 b → J/ψK − p decays and only 4k for the Ξ − b → J/ψK − Λ decays [3]), that the P c (4312), P c (4440), and P c (4457) possibly as theDΣ c andD * Σ c molecular states were observed in this channel by LHCb [1,2]. We illustrate the relevant two processes in Fig. 3(b,c). Therefore, the three LHCb experiments observing P c and P cs states [3,1,2] strongly support the hadronic molecular picture and the existence of "hadronic molecules".
To further verify the above hadronic molecular picture, we propose to investigate the P cs (4459) 0 in future experiments to examine whether it can be separated into two states. We also propose to search for theDΞ c molecular state of J P = 1/2 − , whose mass is predicted to be 4.29 +0.13 −0.12 GeV. If the hadronic molecular picture turns out to be correct, our understanding on the construction of the subatomic world be significantly improved, and our understanding on the non-perturbative behaviors of the strong interaction at the low energy region would also be significantly improved.

A Spectral densities
In this appendix we list the spectral densities ρ 1···3 (s) extracted for the currents η 1···3 . In the following expressions, , and β max = 1 − α. In the calculations we take into account both the m s and m 2 s terms, but here we list only the m s terms.
The spectral density ρ 1 (s) extracted for the current η 1 is where ρ pert The spectral density ρ 2 (s) extracted for the current η 2 is where ρ pert