Revisiting hidden-charm pentaquarks from QCD sum rules

We revisit hidden-charm pentaquark states and using the method of QCD sum rules by requiring the pole contribution to be greater than or equal to 30% in order to better that the one-pole parametrization is valid. We find two mixing currents, and our results suggest that and can be identified as hidden-charm pentaquark states having and , respectively. However, there still exist other possible spin-parity assignments, such as and , which must be clarified in further theoretical and experimental studies.


Introduction
Many exotic hadrons have been discovered in the past decade owing to significant experimental progresses [1], such as the two hidden-charm pentaquark resonances and discovered by the LHCb Collaboration [2][3][4][5]. More exotic hadrons are likely to be observed in the future by BaBar, Belle, BESIII, CMS, and LHCb experiments, etc. They are new blocks of QCD matter, providing insights to deepen our understanding of the non-perturbative QCD, and their relevant theoretical and experimental studies have opened a new page for hadron physics [6][7][8][9][10][11]. In the past year, to investigate their nature, and have been studied using various methods and models. There are many possible interpretations, such as meson-baryon molecules [12][13][14][15][16][17][18][19][20][21][22][23], compact diquarkdiquark-antiquark pentaquarks [24][25][26][27], compact diquarktriquark pentaquarks [28,29], genuine multiquark states other than molecules [30][31][32][33][34][35], and kinematical effects re-lated to thresholds and triangle singularity [36][37][38][39][40]. Their productions and decay properties are also interesting [41][42][43][44][45][46][47][48][49][50][51][52][53]. More extensive discussions can be found in Refs. [54][55][56]. The preferred spin-parity assignments for the and states were suggested to be ; however, some other assignments, such as and , have also been suggested by the LHCb Collaboration [2]. It is useful to theoretically study all possible assignments to better understand their properties. In this study, we use the method of QCD sum rule to study the possible spin-parity assignments of and . However, first, we reinvestigate our previous studies on and [57,58] by requiring the pole contribution (PC) to be greater than or equal to 30% in order to ensure that the one-pole parametrization is valid; this value was just 10% in our previous studies [57,58]. Note that there have been some experimental data on exotic hadrons; however, they are not sufficient, and more experimental results are necessary to The remainder of this paper is organized as follows: the above reinvestigation is presented in Section 2, numerical analyses are presented in Section 3, the investigation of hidden-charm pentaquark states of and are provided in Section 4, and the results will be discussed and summarized in Section 5.

QCD sum rules analyses
All the local hidden-charm pentaquark interpolating currents have been systematically constructed in Refs. [57,58]. Some of these currents were selected to perform QCD sum rule analyses. The results suggest that and can be interpreted as hidden-charm pentaquark states composed of anti-charmed mesons and charmed baryons. However, the analyses therein used one criterion, which was not optimized. The condition was that the PC should be greater than 10% to ensure that the one-pole parametrization was valid. This value is not so significant, and accordingly, the question arises whether we can find a larger PC to better ensure one-pole parametrization In the present study, we try to answer this question to find better (more reliable) QCD sum rule results. In particular, we find the following two mixing currents: a · · · d θ where are color indices; are two mixing angles; and have the spin-parity and , respectively. The four single currents, , , , and , were first constructed in Refs. [57,58]. We can verify: The current well couples to the S-wave , P-wave , P-wave , D-wave channels, etc. Here, the denotes the of and of . 2) The current well couples to the S-wave channel, etc.
3) The current well couples to the S-wave , P-wave channels, etc.
First, we assume and couple to physical states through and write the two-point correlation functions as where contains non-relevant spin components.
Note that if the physical state has the opposite parity, the -coupling should be used [65][66][67][68]. For example, if Hence, we can compare terms proportional to and to determine the parity of . Accordingly, in the present study, we use terms proportional to and to evaluate masses of X's, which are then compared with those proportional to and to determine their parity.
At the hadron level, we use the dispersion relation to rewrite the two-point correlation function as where is the physical threshold. Its imaginary part is defined as the spectral function, which can be evaluated by inserting the intermediate hadron states , but adopting the usual parametrization of one-pole dominance for ground state X along with a continuum contribution: in the two-point correlation functions (5)(6),and calculate them using the method of operator product expansion (OPE). In the present study, we evaluate at the leading order on , up to eight dimensions. For this, we calculated the perturbative term, quark condensate , gluon condensate , quark-gluon condensate , and their combinations and . We find that the D = 4 term and the D = 6 term are important power corrections to the correlation functions. Note that we assumed the vacuum saturation for higher dimensional operators such as , and this can lead to some systematic uncertainties.
Finally, we perform the Borel transform at both the hadron and quark-gluon levels, and express the two-point correlation function as s 0 After assuming that the continuum contribution can be well approximated by the OPE spectral density above a threshold value , we obtain the sum rule relation We use the mixing current defined in Eq. (1) to perform sum rule analyses, and the terms proportional to are given in Eq. (13), where and . The terms proportional to are listed in Eq. (14), which are almost the same as the former ones, suggesting that the state coupled by has the spinparity . Similarly, we use defined in Eq. (2) to perform sum rule analyses, and the terms proportional to and are listed in Eqs. (15) and (16), respectively. We find that its relevant state has the spin-parity . These two sum rules will be used to perform numerical analyses in the next section.

MS
We also need the charm and bottom quark masses, for which we use the running mass in the scheme [1,[69][70][71][72][73][74][75][76]:  There are three free parameters in Eq. (12): the mixing angles , Borel mass , and threshold value . After fine-tuning, we obtain the two mixing angles as and . The following three criteria can be satisfied so that reliable sum rule results can be achieved: 1) The first criterion is used to ensure 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 ensure that the onepole parametrization is valid, i.e., we require the PC to be greater than or equal to 30%, which can be used to determine the upper limit of the Borel mass: This criterion better ensures the one-pole parametrization than the criterion used in Refs. [57,58] which only requires PC≥10%. 3) The third criterion is that the dependence of both and dependence of the mass prediction be the weakest in order to obtain reliable mass predictions.
We use the sum rules (13) for the current as an example. Firstly, we fix and GeV 2 , and show CVG as a function of in the left panel of Fig. 1. We find that the OPE convergence improves with an increase in , and the first criterion requires that GeV 2 . We also show the relative contribution of each term in the middle panel of Fig. 1. We find that a good convergence can be achieved in the same region, GeV 2 . Next, we still fix and GeV 2 , and show PC is a function of in the right panel of Fig. 1  To use the third criterion to determine , we show variations of with respect to in the middle panel of Fig. 2, with . The mass curves have a minimum against when is approximately 17 GeV 2 ; therefore, the dependence of the mass prediction is the weakest at this point. However, the PC at this point is significantly small (only 8%). We find that the PC = 32% at GeV 2 . Moreover, the dependence is the weakest at this point. Accordingly, we fix the threshold value to be Finally, we vary and repeat the above processes. We show variations of with respect to in the right panel of Fig. 2 with GeV 2 and choosing to satisfy CVG = 10%. We find that the -dependence of the mass prediction is weak when . Accordingly, we fix the mixing angle to be -42° and choose as our working region.
For current , we fine-tune the mixing angle to be −42°, and the working regions are found to be 21 GeV 2 25 GeV 2 and 2.59 GeV 2 < <3.19 GeV 2 . We assume the uncertainty of to be -42±5°, and we obtain the following numerical results: where the central value corresponds to , GeV 2 , and GeV 2 . The mass uncertainty is due to the mixing angle , Borel mass , threshold value , charm quark mass , and various condensates [1,[69][70][71][72][73][74][75][76]. We note the following: a) when calculating the mass uncertainty due to the mixing angle , we have fixed and ; and b) when plotting the mass variation as a function of , as shown in the right panel of Fig. 2, we have fixed , but while choosing to satisfy CVG = 10%. The above mass value is consistent with the experimental mass of the [2], and supports it to be a hidden-charm pentaquark having . The current consists of and , suggesting that the may contain the S-wave , P-wave , P-wave , D-wave , S-wave components, etc.
Similarly, we investigate the current of . We fine-tune the mixing angle to be -45±5°, and the working regions are found to be 21 GeV 2 25 GeV 2 and 2.31 GeV 2 < <2.91 GeV 2 . We show the variations of with respect to , , and in Fig. 3, and we obtain the following numerical results: where the central value corresponds to , GeV 2 , and GeV 2 . The above mass value is consistent with the experimental mass of the [2], and supports it to be a hidden-charm pentaquark having . The current consists of and , suggesting that the may contain the S-wave , P-wave , S-wave , P-wave components, etc.

Other spin-parity assignments
In this section we follow the same approach to study the hidden-charm pentaquark states of and . We find the following two currents which have structures similar to and , respectively. The extracted spectral densities are also similar to previous results: where , , and others have been defined in Eqs. (13) and (15).
First, we study the current of . With the same mixing angle , i.e., , the working regions are found to be 21 GeV In the right figure, the curve is obtained for GeV 2 and with satisfying CVG=10%.
The above two values are both consistent with the experimental masses of and [2], suggesting that their spin-parity assignments can be different from and , and further theoretical and experimental efforts are required to clarify their properties.

Results and discussions
In this study, we used the method of QCD sum rules to study the hidden-charm pentaquark states and . We achieved better QCD sum rule results by requiring the PC to be greater than or equal to 30% in order to ensure that the one-pole parametrization was valid; this criterion is stricter than the one used in our previous studies [57,58]. We found two mixing currents, of and of . We used them to perform the sum rule analyses, and the masses were extracted to be These values are consistent with the experimental masses of and , suggesting that they can be identified as hidden-charm pentaquark states composed of anti-charmed mesons and charmed baryons: has and may contain the S-wave , Pwave , P-wave , D-wave , S-wave components, etc. has and may contain the S-wave , P-wave , S-wave , P-wave components, etc. J P = 3/2 + J P = 5/2 − We follow the same approach to study the hiddencharm pentaquark states of and , and extract their masses to be  These values are also consistent with the experimental masses of and [2], suggesting that there still exist other possible spin-parity assignments, which should be clarified in further theoretical and exper-imental studies.
We propose to search for them in the future LHCb and BelleII experiments. In conclusion, we note that there are a considerable systematical uncertainties that are not considered in the present study, such as the vacuum saturation for higher dimensional operators, which is used to calculate the OPE . Moreover, in this study, we used the running charm and bottom quark masses in the scheme, while sometimes their pole masses were used. Consider thee current as an example: a) if we use , we would obtain =4.34 GeV~4.48 GeV (other uncertainties are not included); b) if we use the pole charm mass GeV [1], we would have to shift the mixing angle to be approximately to arrive at the similar mass GeV. Combining the previous uncertainties in Eqs. (21), (22), (27), and (28) The above (systematical) uncertainties are significant, suggesting that we still know little about exotic hadrons, and further experimental and theoretical studies are necessary to understand them well. We thank Professor Nikolai Kochelev for helpful discussions.