Possible interpretations of the $P_c(4312)$, $P_c(4440)$, and $P_c(4457)$

Based on our previous QCD sum rule studies on hidden-charm pentaquark states, we discuss possible interpretations of the $P_c(4312)$, $P_c(4440)$, and $P_c(4457)$, which were recently observed by LHCb. Our results suggest that the $P_c(4312)$ can be well interpreted as the $[\Sigma_c^{++} \bar D^-]$ bound state with $J^P = 1/2^-$, and the $P_c(4440)$ and $P_c(4457)$ as the $[\Sigma_c^{*++} \bar D^{-}]$ and $[\Sigma_c^{+} \bar D^{*0}]$ bound states with $J^P = 3/2^-$, respectively. Our results also suggest that one of the $P_c(4440)$ and $P_c(4457)$ may be interpreted as the $[\Sigma_c^{+} \bar D^0]$ bound state with $J^P = 1/2^-$ or the $[\Sigma_c^{*+} \bar D^{*0}]$ bound state with $J^P = 5/2^-$. We propose to measure their spin-parity quantum numbers to verify these assignments.

Introduction. Very recently, the LHCb Collaboration discovered a new enhancement, P c (4312), in the J/ψp invariant mass spectrum of the Λ b → J/ψpK decays [1]. At the same time, they separated the P c (4450) into two structures, P c (4440) and P c (4457). This experiment [1] was based on their previous one performed in 2015 [2], where the famous hidden-charm pentaquark candidates, P c (4380) and P c (4450), were first observed. All the above structures contain at least five quarks uudcc, so they are perfect candidates of pentaquark states, whose study is one of the most intriguing research topics in hadron physics, and has significantly improved our understanding of the non-perturbative QCD.
We have applied the method of QCD sum rules [22,23] to systematically study the hidden-charm pentaquarks in Refs. [11,24]. Based on the new experimental information [1] as well as our previous theoretical studies [11,24], we shall discuss several possible interpretations of the P c (4312), P c (4440), and P c (4457) in this letter.
The first possible interpretation. In Ref. [11] we applied the method of QCD sum rules and studied the P c (4380) and P c (4450) as exotic hidden-charm pentaquarks composed of charmed baryons and anti-charmed mesons. This study was later expanded in Ref. [24], where we systematically constructed all the possible local hiddencharm pentaquark currents with spin J = 1 2 / 3 2 / 5 2 and quark contents uudcc, and investigated them using the method of QCD sum rules.
Especially, in the abstract of Ref. [24] we wrote that:"...we also find a) the lowest-lying hidden-charm pentaquark state of J P = 1/2 − has the mass 4.33 +0.17 −0.13 GeV, while the one of J P = 1/2 + is significantly higher, that is around 4.7 − 4.9 GeV; b) the lowest-lying hiddencharm pentaquark state of J P = 3/2 − has the mass 4.37 +0. 18 −0.13 GeV, consistent with the P c (4380) of J P = 3/2 − , while the one of J P = 3/2 + is also significantly higher, that is above 4.6 GeV; c) the hidden-charm pentaquark state of J P = 5/2 − has a mass around 4.5 − 4.6 GeV, slightly larger than the P c (4450) of J P = 5/2 + ." Comparing these values with Eqs. (1), we arrive at the first possible interpretation that (A) the P c (4312) is the hidden-charm pentaquark state with J P = 1/2 − , while the P c (4440) and P c (4457) may be the two with J P = 3/2 − or/and 5/2 − . However, this picture is quite rough and can not naturally explain the small mass difference between the P c (4440) and P c (4457), which is just about 17 MeV. To I: Mass predictions for the hidden-charm pentaquark states with spin J = 1 2 / 3 2 / 5 2 and quark contents uudcc, taken from Ref. [24]. We summarize here all the mass predictions that are extracted from single currents and less than 4.5 GeV.
Eq. (7) [Σ * + understand this mass splitting, we turn to carefully examine their internal structures. Actually, this can be well investigated and described by using hadronic interpolating currents within the method of QCD sum rules.
The second and third possible interpretations. In Ref. [24] we found that the internal structure of hidden-charm pentaquark states is quite complicated. We constructed hundreds of interpolating currents to reflect this, from which we derived some mass predictions. We collect all the mass predictions that are extracted from single currents and less than 4.5 GeV, and summarize them in Table I. They are extracted using the following interpolating currents where u, d, c represent the up, down, and charm quarks, respectively, and the subscripts a, b, c, d are color indices. The above currents have the negative parity, but their mirror currents with the positive parity (such as γ 5 ξ 14 , etc.) lead to the same QCD sum rule results. This is because each of them can couple to both the positiveand negative-parity pentaquark states, and we need to determine the parity of the state through the derived sum rule equations. See Refs. [11,[24][25][26][27] for detailed analyses.
It is usually not easy to understand the QCD sum rule results for multiquark states, because we still do not well understand the relations between interpolating currents and their relevant hadron states. One can even use the Fierz transformation to write a local molecular current c e , and vice versa. However, this is an overall connection, i.e., a molecular current can be written as a combination of many diquark-diquarkantiquark currents. We recommend interested readers to Ref. [28], where we first pointed out such connection by systematically studying the relation among various tetraquark currents. Hence, we can still extract some useful information from the currents being used: • The P + c (4312) can be described by the current ψ 2 . The quark contents inside ψ 2 can be naturally separated into two color-singlet components, [ abc (u T a Cγ µ u b )γ µ γ 5 c c ] and [c d γ 5 d d ]. They are the two standard charmed baryon and charmed meson interpolating fields, which couple to Σ ++ c andD − , respectively. Accordingly, ψ 2 would couple to the bound state of [Σ ++ cD − ] with J P = 1/2 − , if it exists. Note that we made a typo in Ref. [24] to label this as [Σ * cD ]. Hence, our result suggests that the P + c (4312) can be well interpreted as the [Σ ++ cD − ] bound state with J P = 1/2 − .
• The P + c (4440) and P + c (4457) can be described by the currents ξ 33µ and ψ 2µ .
The quark contents inside ξ 33µ can be separated into • The current ξ 14 can also be used to describe one of the P + c (4440) and P + c (4457). Its quark contents can be separated into [ abc (u T a Cγ µ d b )γ µ γ 5 c c ] and [c d γ 5 u d ], coupling to Σ + c andD 0 , respectively. Hence, one of the P c (4440) and P c (4457) may be interpreted as the [Σ + cD 0 ] bound state with J P = 1/2 − . The current ξ 14 is similar to ψ 2 , but their extracted sum rule results are much different, simply because the two up quarks inside ξ 14 are located in both of the two color-singlet components, so that there can be up quark exchange between these two components, i.e., Feymann diagrams exchanging up quarks.
• The current ξ 14 can be used to roughly describe one of the P + c (4440) and P + c (4457). Its quark contents suggest that one of the P c (4440) and P c (4457) may be interpreted as the [Σ * + cD * 0 ] bound state with J P = 5/2 − .
• There is still a place for the P c (4380), that is to use ψ 9µ , whose quark contents can be separated , coupling to Σ ++ c andD * − , respectively. Again, its extracted sum rule result is much different from ξ 33µ , due to the locations of the two up quarks.
Mass splitting between the P c (4440) and P c (4457). Comparing the above three interpretations, the third one seems to explain the small mass difference between the P c (4440) and P c (4457) in a more natural way, since the two mass values listed in Table I, 4.45 GeV (3/2 − ) and 4.46 GeV (3/2 − ), are both for the J P = 3/2 − states. However, there is still a problem in this picture, that the two currents ξ 33µ and ψ 2µ can couple to the same physical state, although they have different internal structures.
To check whether ξ 33µ and ψ 2µ couple to the same state or not, we calculate their off-diagonal correlation function [30]: where Π ξ33µψ2µ q 2 is contributed by the spin 3/2 components of ξ 33µ and ψ 2µ , while contributions from their spin 1/2 components are all contained in · · · . If ξ 33µ and ψ 2µ do strongly couple to the same physical state P * c with the mass M * , we would have so that Π ξ33µψ2µ q 2 should be nonzero. However, our QCD sum rule calculation gives us that Therefore, ξ 33µ and ψ 2µ should couple to different states, and can be used to describe the P c (4440) and P c (4457) at the same time. Since the two QCD sum rule parameters, the threshold value s 0 and the Borel mass M B , are almost the same when investigating these two currents, we can extract their mass difference to be where the central value corresponding to s 0 = 22 GeV 2 and M B = 4.17 GeV 2 . The uncertainty comes from the Borel mass M B , the threshold value s 0 , the charm quark mass, and various condensates [24]. It is much smaller than those of the absolute mass values listed in Table I, although still significant. For completeness, we show ∆M as a function of M B in Fig. 1. Anyway, the above mass splitting is consistent with the mass difference between the P c (4440) and P c (4457), making the third interpretation slightly more natural than others. It also suggests that the P + c (4440) is preferred to be interpreted as the [Σ * ++ that we can extract two mass predictions from them for two states both having J P = 3/2 − , whose mass difference is extracted to be 8.1 +30.9 −18.9 MeV. Besides the above picture, one of the P c (4440) and P c (4457) may also be interpreted as the [Σ + cD 0 ] bound state with J P = 1/2 − or the [Σ * + cD * 0 ] bound state with J P = 5/2 − . There is still a place for the P c (4380), that is to be interpreted as the [Σ ++ cD * − ] bound state with J P = 3/2 − . Note that there exist more possible interpretations with the positive-parity assignments [11,31]. In the present QCD sum rule studies we intend to use various internal structures of hidden-charm pentaquark states to explain the P c (4312), P c (4440), and P c (4457) at the same time, while there are many other possible approaches. For example, in the molecular picture within the one-boson-exchange model, there can be Swave molecular pentaquarks as well as P -wave orbitally excited molecular pentaquarks [32,33].
We propose to measure the spin-parity quantum numbers of hidden-charm pentaquark states to verify whether the picture of the present study is correct or not. If it is correct, one would think that the internal structure of hadrons does influence their observed properties, and we might face the same situation as the light spectrum described by QED [34], so that lots of new exotic structures could be waiting to be discovered in the future. To end this letter, we would like to note that, together with many charmonium-like XY Z states [3], the hidden-charm pentaquarks are opening a new window for studying exotic hadronic matter and improving our understanding of QCD.