Possible interpretations of the $P_c(4312)$, $P_c(4440)$, and $P_c(4457)$
Hua-Xing Chen Wei Chen Shi-Lin Zhu









hxchen@buaa.edu.cn

chenwei29@mail.sysu.edu.cn

zhusl@pku.edu.cn

^1School of Physics, Beihang University, Beijing 100191, China 

^2School of Physics, Sun Yat-Sen University, Guangzhou 510275, China 

^3School of Physics and State Key Laboratory of Nuclear Physics and Technology, Peking University, Beijing 100871, China 

^4Collaborative Innovation Center of Quantum Matter, Beijing 100871, China 

^5Center of High Energy Physics, Peking University, Beijing 100871, China




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 [Σ_c^++D̅^-] bound state with J^P = 1/2^-, while the P_c(4440) and P_c(4457) can be interpreted as  the [Σ_c^+D̅^0] bound state with J^P = 1/2^-, the [Σ_c^*++D̅^-] and [Σ_c^+D̅^*0] bound states with J^P = 3/2^-, or the [Σ_c^*+D̅^*0] bound state with J^P = 5/2^-. We propose to measure their spin-parity quantum numbers to verify these assignments.


12.39.Mk, 14.20.Lq, 12.38.Lg


Possible interpretations of the P_c(4312), P_c(4440), and P_c(4457)
    Shi-Lin Zhu^3,4,5
    December 30, 2023
===================================================================



arabic


Introduction.
Very recently, the LHCb Collaboration discovered a new enhancement, P_c(4312), in the J/ψ p invariant mass spectrum of the Λ_b→ J/ψ p K decays <cit.>. At the same time, they separated the P_c(4450) into two structures, P_c(4440) and P_c(4457). This experiment <cit.> was based on their previous one performed in 2015 <cit.>, 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 uud c c̅, so they are perfect candidates of pentaquark states. Together with many charmonium-like XYZ states <cit.>, their studies have improved our understanding of the non-perturbative behaviors of the strong interaction at the low energy region. However, there is still a long way to fully understand how the strong interaction binds quarks, gluons, and hadrons together, and exotic hadrons will continuously be one of the most intriguing research topics in hadron physics.

In the new LHCb experiment <cit.> the following resonance parameters were measured:

    P_c^+(4312):     M    =    4311.9 ±  0.7 ^+6.8_-0.6  ,
    
    Γ   =       9.8 ±  2.7 ^+3.7_-4.5  ,
    
      P_c^+(4440):     M    =    4440.3 ±  1.3 ^+4.1_-4.7  ,
    
    Γ   =      20.6 ±  4.9 ^+8.7_-10.1  ,
    
      P_c^+(4457):     M    =    4457.3 ±  0.6 ^+4.1_-1.7  ,
    
    Γ   =       6.4 ±  2.0 ^+5.7_-1.9  .

Hence, these three structures are quite narrow, and can be clearly seen in the J/ψ p invariant mass spectrum.
As discussed by LHCb <cit.>, the P_c(4312), P_c(4440), and P_c(4457) are just below the Σ_c^+ D̅^0 and Σ_c^+ D̅^*0 thresholds (4318 MeV and 4460 MeV <cit.>, respectively), so they can be naturally interpreted as the bound states composed of charmed baryons and anti-charmed mesons, whose existence had been predicted in several theoretical studies <cit.> before the LHCb-2015 experiment <cit.>; while after this experiment <cit.>, lots of theoretical studies were performed to explain the nature of the P_c(4380) and P_c(4450), such as meson-baryon molecules <cit.>, compact diquark-diquark-antiquark or diquark-triquark pentaquarks <cit.>, and kinematical effects related to the triangle singularity <cit.>, etc.

We have applied the method of QCD sum rules <cit.> to systematically study the hidden-charm pentaquarks in Refs. <cit.>. Based on the new experimental information <cit.> as well as our previous theoretical studies <cit.>, 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. <cit.> 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. <cit.>, where we systematically constructed all the possible local hidden-charm pentaquark currents with spin J = 12/32/52 and quark contents uud c c̅, and investigated them using the method of QCD sum rules.

Especially, in the abstract of Ref. <cit.> 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 hidden-charm 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. (<ref>), 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 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. <cit.> 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 <ref>. They are extracted using the following interpolating currents

    ξ_14   =    [ϵ^abc (u^T_a C γ_μ d_b) γ_μγ_5 c_c] [c̅_d γ_5 u_d]   ,
    
    ψ_2    =    [ϵ^abc (u^T_a C γ_μ u_b) γ_μγ_5 c_c] [c̅_d γ_5 d_d]   ,
    
    ξ_33μ   =    [ϵ^abc (u^T_a C γ_ν d_b) γ_νγ_5 c_c] [c̅_d γ_μ u_d]   ,
    
    ψ_2μ   =    [ϵ^abc (u^T_a C γ_μ u_b) c_c] [c̅_d γ_5 d_d]   ,
    
    ψ_9μ   =    [ϵ^abc (u^T_a C γ_ν u_b) γ_νγ_5 c_c] [c̅_d γ_μ d_d]   ,
    
    ξ_13μν   =    [ϵ^abc (u^T_a C γ_μ d_b) c_c] [c̅_d γ_ν u_d] + {μ↔ν}  ,

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 positive- and negative-parity pentaquark states, and we need to determine the parity of the state through the derived sum rule equations. See Refs. <cit.> for detailed analyses.

From Table <ref>, we find four mass predictions, 4.33 GeV (1/2^-), 4.45 GeV (1/2^-), 4.45 GeV (3/2^-), and 4.46 GeV (3/2^-), which are almost the same as those listed in Eqs. (<ref>). Accordingly, we arrive at the second and third possible interpretations that
(B) the P_c(4312) and P_c(4440) are the two hidden-charm pentaquark states with J^P = 1/2^-, while the P_c(4457) is the one with J^P = 3/2^-;
(C) the P_c(4312) is the hidden-charm pentaquark state with J^P = 1/2^-, while the P_c(4440) and P_c(4457) are the two with J^P = 3/2^-.

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. We also refer to Refs. <cit.>, which generally investigate how to apply the method of QCD sum rules and the large N_c approximation to study multiquark states. One can use the Fierz transformation to write a local “molecular” current [ϵ^abc u_a d_b c_c] [c̅_d u_d] as a combination of local diquark-diquark-antiquark currents ϵ^fge [ϵ^abf u_a d_b] [ϵ^cdg c_c u_d] c̅_e, and vice versa. However, this is an overall connection, i.e., a “molecular” current can be written as a combination of many diquark-diquark-antiquark currents. We recommend interested readers to Ref. <cit.>, where we first pointed out such connection by systematically studying the relation among various tetraquark currents. Take ψ_2μ defined in Eq. (<ref>) as an example, we can transform it to be

    ψ_2μ   =    [ϵ^abc (u^T_a C γ_μ u_b) c_c] [δ^dec̅_e γ_5 d_d]
    
       =   18 ϵ^fge [ϵ^abg u^T_a C γ_μ u_b] [ϵ^cdf d_c^T C c_d] [γ_5 C c̅_e^T]
    
       +   18 ϵ^fge [ϵ^abg u^T_a C γ_μ u_b] [ϵ^cdf d_c^T C γ_5 c_d] [C c̅_e^T]
    
       -   18 ϵ^fge [ϵ^abg u^T_a C γ_μ u_b] [ϵ^cdf d_c^T C γ_ν c_d] [γ^νγ_5 C c̅_e^T]
    
       +   18 ϵ^fge [ϵ^abg u^T_a C γ_μ u_b] [ϵ^cdf d_c^T C γ_νγ_5 c_d] [γ^ν C c̅_e^T]
    
       +   116 ϵ^fge [ϵ^abg u^T_a C γ_μ u_b] [ϵ^cdf d_c^T C σ_νν^' c_d] [σ^νν^'γ_5 C c̅_e^T]
    
       +   132 ϵ^fge [ϵ^acg u^T_a C γ_μ c_c] [ϵ^dbf d_d^T C u_b] [γ_5 C c̅_e^T]
    + ⋯
       +   132 ϵ^fge [ϵ^adg u^T_a C γ_μ d_d] [ϵ^cbf c_c^T C u_b] [γ_5 C c̅_e^T]
    + ⋯  ,

where ⋯ are other ϵ^fge [ϵ^acg u^T_a C Γ_i c_c] [ϵ^dbf d_d^T C Γ_j u_b] [Γ_k C c̅_e^T] and ϵ^fge [ϵ^adg u^T_a C Γ_l d_d] [ϵ^cbf c_c^T C Γ_m u_b] [Γ_n C c̅_e^T] components (this equation needs to be further simplified, but we shall not do this in the present study). This complicated relation suggests that although ψ_2μ can still be interpreted as a diquark-diquark-antiquark current containing many diquark-diquark-antiquark components, it seems much more natural to simply describe it as a “molecular” current.




In this sense, we can extract some useful information from the “molecular” 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, [ϵ^abc (u^T_a C γ_μ u_b) γ_μγ_5 c_c] and [c̅_d γ_5 d_d]. They are the two standard charmed baryon and charmed meson interpolating fields, which couple to Σ_c^++ and D̅^-, respectively. Accordingly, ψ_2 would couple to the bound state of [Σ_c^++D̅^-] with J^P = 1/2^-, if it exists.
    Note that we made a typo in Ref. <cit.> to label this as [Σ_c^* D̅]. Hence, our result suggests that the P_c^+(4312) can be well interpreted as the [Σ_c^++D̅^-] 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 [ϵ^abc (u^T_a C γ_ν d_b) γ_νγ_5 c_c] and [c̅_d γ_μ u_d], coupling to Σ_c^+ and D̅^*0, respectively; while those inside ψ_2μ can be separated into [ϵ^abc (u^T_a C γ_μ u_b) c_c] and [c̅_d γ_5 d_d], coupling to Σ_c^*++ and D̅^-, respectively (the standard interpolating current coupling to Σ_c^*++ is [ϵ^abc (u^T_a C γ_ν u_b) (g^μν - γ^μγ^ν/4 ) c_c] <cit.>). Hence, our result suggests that the P_c^+(4440) and P_c^+(4457) can be well interpreted as [Σ_c^+D̅^*0] and [Σ_c^*++D̅^-] bound states with J^P = 3/2^-.


  * 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 [ϵ^abc (u^T_a C γ_μ d_b) γ_μγ_5 c_c] and [c̅_d γ_5 u_d], coupling to Σ_c^+ and D̅^0, respectively. Hence, one of the P_c(4440) and P_c(4457) may be interpreted as the [Σ_c^+D̅^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_c^*+D̅^*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 into

 [ϵ^abc (u^T_a C γ_ν u_b) γ_νγ_5 c_c] and [c̅_d γ_μ d_d], coupling to Σ_c^++ and D̅^*-, 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).
Among the above three interpretations, there is still a problem in the third interpretation, that the two mass values extracted from the two currents ξ_33μ and ψ_2μ, both of J^P = 3/2^-, are very close to each other (4.45 GeV and 4.46 GeV), so they 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 <cit.>:



    Π^ξ_33μψ_2μ_μν(q^2)    ≡    i ∫ d^4x e^iqx⟨ 0 | T ξ_33μ(x) ψ_2ν^† (0) | 0 ⟩
       =   (q_μ q_ν/q^2-g_μν) (q + M^*) Π^ξ_33μψ_2μ(q^2)
    
                                           + ⋯  ,



where Π^ξ_33μψ_2μ(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

    ⟨ 0 | ξ_33μ(x) | P_c^* ⟩⟨ P_c^* | ψ_2ν^† (0) | 0 ⟩≠ 0   ,

so that Π^ξ_33μψ_2μ(q^2) should be nonzero. However, our QCD sum rule calculation gives us that

    Π^ξ_33μψ_2μ(q^2) = 0   .

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

    Δ M = M_ξ_33μ - M_ψ_2μ = 8.1 ^+30.9_-18.9  MeV  ,

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 <cit.>. It is much smaller than those of the absolute mass values listed in Table <ref>, although still significant. For completeness, we show Δ M as a function of M_B in Fig. <ref>.




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_c^*++D̅^-] bound state with J^P = 3/2^-, while the P_c^+(4457) as the [Σ_c^+D̅^*0] bound state with J^P = 3/2^-.


Conclusion.
In summary, after the discovery of the P_c(4380) and P_c(4450) by LHCb in 2015 <cit.>, the P_c(4312), P_c(4440), and P_c(4457) observed in the new LHCb experiment <cit.> brought us a great surprise once more.
The coincidence of their measured masses with our previous theoretical predictions <cit.> drives us to the “molecular” picture that the P_c(4312) is the [Σ_c^++D̅^-] bound state with J^P = 1/2^-, and the P_c(4440) and P_c(4457) are the [Σ_c^*++D̅^-] and [Σ_c^+D̅^*0] bound states with J^P = 3/2^-, respectively. In this letter we further calculate the off-diagonal correlation function between ξ_33μ and ψ_2μ, and their non-correlation confirms 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) can also be interpreted as the [Σ_c^+D̅^0] bound state with J^P = 1/2^- or the [Σ_c^*+D̅^*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_c^++D̅^*-] bound state with J^P = 3/2^-. There exist more possible interpretations with the positive-parity assignments <cit.>. To clearly understand their nature, one still needs further experimental information.

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, a beautiful picture is to interpret them as loosely bound Σ_c D̅ molecular state with J^P = 1/2^-, Σ_c D̅^* with J^P = 1/2^- and Σ_c D̅^* with J^P = 3/2^-, respective <cit.>. However, the [Σ_c D̅^*] bound state with J^P = 1/2^- was not investigated in the present study, and we shall study this possibility in the near future. At the same time, we shall study the [Σ_c^* D̅] bound state with J^P = 1/2^- and the [Σ_c^* D̅^*] bound states with J^P = 1/2^- and 3/2^-, which were not investigated in the present study neither. We believe that a systematical QCD sum rule study might help to better understand these structures as well as this method itself.












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 <cit.>, 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 XYZ states <cit.>, the hidden-charm pentaquarks are opening a new window for studying exotic hadronic matter and improving our understanding of QCD.



§ ACKNOWLEDGMENTS


This project is supported by
the National Natural Science Foundation of China under Grants No. 11575008, No. 11722540, No. 11621131001,
the 973 program,
the Fundamental Research Funds for the Central Universities,
and
the Chinese National Youth Thousand Talents Program.



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