Revisiting hidden-charm pentaquarks from QCD sum rules

We revisit the hidden-charm pentaquark states $P_c(4380)$ and $P_c(4450)$ using the method of QCD sum rules by requiring the pole contribution to be larger than or around 30\% to better insure the one-pole parametrization to be valid. We find two mixing currents and our results suggest that the $P_c(4380)$ and $P_c(4450)$ can be identified as hidden-charm pentaquark states having $J^P=3/2^-$ and $5/2^+$, respectively, while there still exist other possible spin-parity assignments, such as $J^P=3/2^+$ and $J^P=5/2^-$, which needs to be clarified in further theoretical and experimental studies.


I. INTRODUCTION
Many exotic hadrons have been discovered in the past decade due to significant experimental progresses [1], such as the two hidden-charm pentaquark resonances P c (4380) and P c (4450) discovered by the LHCb Collaboration [2]. Besides them, more exotic hadrons are likely to be observed in the future BaBar, Belle, BESIII, CMS and LHCb experiments, etc. They are new blocks of QCD matter, providing important hints to deepen our understanding of the non-perturbative QCD, and their relevant theoretical and experimental studies are opening a new page for the hadron physics [3].
In the past year, the P c (4380) and P c (4450) have been studied by various methods and models in order to explain their nature. There are many possible interpretations, such as meson-baryon molecules [4], compact diquark-diquark-antiquark pentaquarks [5], compact diquark-triquark pentaquarks [6], genuine multiquark states other than molecules [7], and kinematical effects related to thresholds and triangle singularity [8], etc. Their productions and decay properties are also interesting [9]. More extensive discussions can be found in Refs. [10].
The preferred spin-parity assignments for the P c (4380) and P c (4450) states were suggested to be (3/2 − , 5/2 + ), while some other assignments were also suggested to be possible by the LHCb Collaboration, such as (3/2 + , 5/2 − ) and (5/2 + , 3/2 − ) [2]. It is useful to study all these possible assignments theoretically in order to better understand their properties.
In this paper we shall use the method of QCD sum rule to study the possible spin-parity assignments of * Electronic address: hxchen@buaa.edu.cn † Electronic address: chenwei29@mail.sysu.edu.cn ‡ Electronic address: zhusl@pku.edu.cn the P c (4380) and P c (4450). Before doing this, we shall reinvestigate our previous studies on the P c (4380) and P c (4450) [11] by requiring the pole contribution to be larger than or around 30% to better insure the one-pole parametrization to be valid. This paper is organized as follows: the above reinvestigation will be done in Sec. II, numerical analyses will be done in Sec. III, the investigation of hidden-charm pentaquark states of J P = 3/2 + and J P = 5/2 − will be done in Sec. IV, and the results will be discussed and summarized in Sec. V. This paper has a supplementary file "OPE.nb" containing all the spectral densities.

II. QCD SUM RULES ANALYSES
All the local hidden-charm pentaquark interpolating currents have been systematically constructed in Refs. [11], and some of them were selected to perform QCD sum rule analyses. The results suggest that the P c (4380) and P c (4450) can be interpreted as hidden-charm pentaquark states composed of anticharmed mesons and charmed baryons. However, the analyses therein use one criterion which is not optimized, that is to require the pole contribution to be larger than 10% to insure the one-pole parametrization to be valid. This value is not so significant, and accordingly, the question arises whether we can find a larger pole contribution to better insure the one-pole parametrization?
In the present study we try to answer this question in order to find better (more reliable) QCD sum rule results.
Especially, we find the following two mixing currents: where a · · · d are color indices; θ 1/2 are two mixing angles; J µ,3/2− and J µν,5/2+ have the spin-parity J P = 3/2 − and 5/2 + , respectively. The four single currents, ξ 36µ , ψ 9µ , ξ 15µν and ψ 4µν , were first constructed in Refs. [11]. We can verify: 4. The current ψ 4µν well couples to the S-wave [Σ * cD1 ] and P -wave [Σ * cD ] channels, etc. We shall use the above two mixing currents, J µ,3/2− and J µν,5/2+ , to perform QCD sum rule analyses, and the results will be given in the next section. Before doing that we briefly introduce our approach here, and we refer interested readers to read Refs. [11,12] for details. More sum rule studies based on the heavy quark effective theory (HQET) can be found in Refs. [13].
We note that if the physical state has the opposite parity, the γ 5 -coupling should be used [14], for example, if Hence, we can compare terms proportional to 1×g µν and q / × g µν to determine the parity of X 3/2± . Accordingly, in the present study we shall use the terms proportional to 1 × g µν and 1 × g µρ g νσ to evaluate masses of X's, which are then compared with those proportional to q /×g µν and q / × g µρ g νσ to determine their parity.
At the hadron level, we use the dispersion relation to rewrite the two-point correlation function as where s < is the physical threshold. Its imaginary part is defined as the spectral function, which can be evaluated by inserting intermediate hadron states n |n n|, but adopting the usual parametrization of one-pole dominance for the ground state X together with a continuum contribution: 10) At the quark and gluon level, we insert Eqs. (1-2) into the two-point correlation functions (5)(6), and calculate them using the method of operator product expansion (OPE). In the present study we evaluate ρ(s) at the leading order on α s and up to dimension eight. To do this we have calculated the perturbative term, the quark condensate qq , the gluon condensate g 2 s GG , the quark-gluon condensate g sq σGq , and their combinations qq 2 and qq g sq σGq . We find that the D = 4 term m c qq and the D = 6 term m c g sq σGq are important power corrections to the correlation functions. Finally, we perform the Borel transform at both the hadron and quark-gluon levels, and express the two-point correlation function as After assuming that the continuum contribution can be well approximated by the OPE spectral density above a threshold value s 0 , we obtain the sum rule relation We use the mixing current J µ,3/2− defined in Eq. (1) to perform sum rule analyses, and the terms proportional to 1 × g µν are shown in Eq. (13), where t 1 = cos θ 1 and t 2 = sin θ 1 . We do not list those proportional to q / × g µν for simplicity, but note that they are almost the same as the former ones, suggesting that the state coupled by J µ,3/2− has the spin-parity J P = 3/2 − . Similarly, we use J µν,5/2+ defined in Eq. (2) to perform sum rule analyses, and find its relevant state has the spin-parity J P = 5/2 + . We show the terms proportional to 1 × g µρ g νσ in the supplementary file "OPE.nb". These two sum rules will be used to perform numerical analyses in the next section.

III. NUMERICAL ANALYSES
In this section we use the sum rules for J µ,3/2− and J µν,5/2+ to perform numerical analyses. Various condensates inside these equations take the following values [1,15]: We also need the charm and bottom quark masses, for which we use the running mass in the M S scheme [1,15]: There are altogether three free parameters in Eq. (12): the mixing angles θ 1/2 , the Borel mass M B , and the threshold value s 0 . We find that after fine-tuning the two mixing angles to be θ 1 = −42 • and θ 2 = −45 • , the following three criteria can be satisfied so that reliable sum rule results can be achieved: 1. The first criterion is used to insure the convergence of the OPE series, i.e., we require the dimension eight to be less than 10%, which can be used to determine the lower limit of the Borel mass: 2. The second criterion is used to insure the one-pole parametrization to be valid, i.e., we require the pole contribution (PC) to be larger than or around 30%, which can be used to determine the upper limit of the Borel mass: This criterion better insure the one-pole parametrization than that used in Refs. [11] which only requires PC≥ 10%.
3. The third criterion is to require that both the s 0 and the M B dependence of the mass prediction be the weakest in order to obtain reliable mass predictions.
We use the sum rules (13) for the current J µ,3/2− as an example. Firstly, we fix θ 1 = −42 • and s 0 = 23 GeV 2 , and show CVG as a function of M B in the left panel of Fig. 1. We find that the OPE convergence improves with the increase of M B , and the first criterion requires that M 2 B ≥ 2.89 GeV 2 . We also show the relative contribution of each term in the middle panel of Fig. 1, again we find that a good convergence can be achieved in the same region M 2 B ≥ 2.89 GeV 2 . Secondly, we still fix θ 1 = −42 • and s 0 = 23 GeV 2 , and show PC as a function of M B in the right panel of Fig. 1 To use the third criterion to determine s 0 , we show variations of M X with respect to s 0 in the middle panel of Fig. 2 when fixing θ 1 = −42 • . The mass curves have a minimum against s 0 when s 0 is around 17 GeV 2 , so the s 0 dependence of the mass prediction is the weakest at this point. However, the pole contribution at this point is quite small (just 8%). We find that PC = 32% at s 0 = 23 GeV 2 . Moreover, the M B dependence is the weakest at this point. Accordingly, we fix the threshold value to be s 0 = 23 GeV 2 and choose 21 GeV 2 ≤ s 0 ≤ 25 GeV 2 as our working region.
Finally, we change θ 1 and redo the above processes. We show variations of M X with respect to θ 1 in the right panel of Fig. 2 when fixing s 0 = 23 GeV 2 and choosing M B to satisfy CVG= 10%. We find that the θ 1 dependence of the mass prediction is weak when θ 1 ≥ 40 • . Accordingly, we fix the mixing angle θ 1 to be −42 • and choose θ 1 = −42 ± 5 • as our working region.
Altogether for the current J µ,3/2− , we fine-tune the mixing angle θ 1 to be −42 • , and the working regions are found to be 21 GeV 2 ≤ s 0 ≤ 25 GeV 2 and 2.59 GeV 2 < M 2 B < 3.19 GeV 2 . We assume the uncertainty of θ 1 to be −42 ± 5 • , and obtain the following numerical results: f 3/2 − = (2.4 +0.9 −0.9 ) × 10 −5 GeV 6 , where the central value corresponds to θ 1 = −42 • , s 0 = 23 GeV 2 and M 2 B = 2.89 GeV 2 . The mass uncertainty is due to the mixing angle θ 1 , the Borel mass M B , the threshold value s 0 , the charm quark mass m c , and various condensates [1,15]. We note that: a) when calculating the mass uncertainty due to the mixing angle θ 1 , we have fixed s 0 and M B ; and b) when plotting the mass variation as a function of θ 1 as shown in the right panel of Fig. 2, we have fixed s 0 but choosing M B to satisfy CVG= 10%. The above mass value is consistent with the experimental mass of the P c (4380) [2], and supports it to be a hidden-charm pentaquark having J P = 3/2 − . The current J µ,3/2− consists of ξ 36µ and ψ 9µν , suggesting that the P c (4380) may contain the S-wave [Λ c (1P ) Similarly, we investigate the current J µν,5/2+ of J P = 5/2 + . We fine-tune the mixing angle θ 2 to be −45 ± 5 • , and the working regions are found to be 21 GeV 2 ≤ s 0 ≤ 25 GeV 2 and 2.31 GeV 2 < M 2 B < 2.91 GeV 2 . We show variations of M X with respect to M B , s 0 , and θ 2 in Fig. 3, and obtain the following numerical results: where the central value corresponds to θ 2 = −45 • , s 0 = 23 GeV 2 and M 2 B = 2.61 GeV 2 . The above mass value is consistent with the experimental mass of the P c (4450) [2], and supports it to be a hidden-charm pentaquark having J P = 5/2 + . The current J µν,5/2+ consists of ξ 15µ and ψ 4µν , suggesting that the P c (4450) may contain the S-

IV. OTHER SPIN-PARITY ASSIGNMENTS
In this section we follow the same approach to study the hidden-charm pentaquark states of J P = 3/2 + and J P = 5/2 − . We find the following two currents which have structures similar to J µ,3/2− and J µν,5/2+ , respectively. The extracted spectral densities are also similar to previous ones: where ρ pert 3/2− (s) and others have been given in the sum rules (13) as well as the supplementary file "OPE.nb".
Firstly, we study the current J µ,3/2+ of J P = 3/2+. With the same mixing angle as θ 1 , i.e., θ 3 = θ 1 = −42 ± 5 • , the working regions are found to be 21 GeV 2 ≤ s 0 ≤ 25 GeV 2 and 2.58 GeV 2 < M 2 B < 3.18 GeV 2 . We show variations of M X with respect to s 0 in the left panel of Fig. 4 with θ 3 = −42 • , where the mass is extracted to be Then we study the current J µν,5/2− of J P = 5/2−. With the same mixing angle as θ 2 , i.e., θ 4 = θ 2 = −45 ± 5 • , the working regions are found to be 21 GeV 2 ≤ s 0 ≤ 25 GeV 2 and 2.20 GeV 2 < M 2 B < 2.80 GeV 2 . We show variations of M X with respect to s 0 in the right panel of Fig. 4 with θ 4 = −45 • , where the mass is extracted to be The above two values are both consistent with the experimental masses of the P c (4380) and P c (4450) [2], suggesting that their spin-parity assignments can be different from J P = 3/2 − and 5/2 + , and further theoretical and experimental efforts are required to clarify their properties.

V. RESULTS AND DISCUSSIONS
In this paper we use the method of QCD sum rules to study the hidden-charm pentaquark states P c (4380) and P c (4450). We achieve better QCD sum rule results by requiring the pole contribution to be larger than or around 30% to insure the one-pole parametrization to be valid, which criterion is more strict than that used in our previous studies [11]. We find two mixing currents, J µ,3/2− of J P = 3/2 − and J µν,5/2+ of J P = 5/2 + . We use them to perform sum rule analyses, and the masses are extracted to be We follow the same approach to study the hiddencharm pentaquark states of J P = 3/2 + and J P = 5/2 − , and extract their masses to be M 3/2 + = 4.40 +0.14 −0.16 GeV , M 5/2 − = 4.43 +0.26 −0.28 GeV .
These values are also consistent with the experimental masses of the P c (4380) and P c (4450) [2], suggesting that there still exist other possible spin-parity assignments for them, which needs to be clarified in further theoretical and experimental studies.
We propose to search for them in the future LHCb and BelleII experiments.