Supported by National Natural Science Foundation of China (11722540, 11261130311), Fundamental Research Funds for the Central Universities, and Foundation for Young Talents in College of Anhui Province (gxyq2018103)

^{3}and published under licence by Chinese Physical Society and the Institute of High Energy Physics of the Chinese Academy of Sciences and the Institute of Modern Physics of the Chinese Academy of Sciences and IOP Publishing Ltd

We revisit hidden-charm pentaquark states

Article funded by SCOAP^{3}

Many exotic hadrons have been discovered in the past decade owing to significant experimental progresses [

In the past year, to investigate their nature,

The preferred spin-parity assignments for the

In this study, we use the method of QCD sum rule to study the possible spin-parity assignments of

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

All the local hidden-charm pentaquark interpolating currents have been systematically constructed in Refs. [

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:

where

1) The current

2) The current

3) The current

4) The current

We use the above two mixing currents,

First, we assume

and write the two-point correlation functions as

where

Note that if the physical state has the opposite parity, the

then

Hence, we can compare terms proportional to

At the hadron level, we use the dispersion relation to rewrite the two-point correlation function as

where

At the quark and gluon level, we substitute Eqs. (1-2) 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

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

We use the mixing current

In this section, we use the sum rules for

We also need the charm and bottom quark masses, for which we use the running mass in the

There are three free parameters in Eq. (12): the mixing angles

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 one-pole 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. [

3) The third criterion is that the dependence of both

We use the sum rules (13) for the current
^{2}, and show CVG as a function of
^{2}. We also show the relative contribution of each term in the middle panel of ^{2}. Next, we still fix
^{2}, and show PC is a function of
^{2}. Accordingly, we fix the Borel mass to
^{2} and choose 2.59 GeV^{2}<
^{2} as our working region. We show variations of
^{2}, as well as inside the Borel window 2.59 GeV^{2}<
^{2}.

The left panel shows CVG, defined in Eq. (19), as a function of Borel mass
^{2}.

Variations of
^{2}, respectively. In the middle figure, the curve is obtained with
^{2}. In the right figure, the curve is obtained for
^{2} and with

To use the third criterion to determine
^{2}; therefore, the
^{2}. Moreover, the
^{2} and choose 21 GeV^{2}^{2} as our working region.

Finally, we vary
^{2} and choosing

For current
^{2}^{2} and 2.59 GeV^{2}<
^{2}. We assume the uncertainty of

where the central value corresponds to
^{2}, and
^{2}. The mass uncertainty is due to the mixing angle

Similarly, we investigate the current
^{2}^{2} and 2.31 GeV^{2}<
^{2}. We show the variations of

Variations of
^{2}, respectively. In the middle figure, the curve is obtained with
^{2}. In the right figure, the curve is obtained for
^{2} and with

where the central value corresponds to
^{2}, and
^{2}. The above mass value is consistent with the experimental mass of the

In this section we follow the same approach to study the hidden-charm pentaquark states of

which have structures similar to

where

First, we study the current
^{2}^{2} and 2.58 GeV^{2}<
^{2}. We show the variations of

Variations of

Then, we study the current
^{2}^{2} and 2.20 GeV^{2}<
^{2}. We show vthe ariations of

The above two values are both consistent with the experimental masses of

In this study, we used the method of QCD sum rules to study the hidden-charm pentaquark states

These values are consistent with the experimental masses of

We follow the same approach to study the hidden-charm pentaquark states of

These values are also consistent with the experimental masses of

We have also investigated the bottom partners of

Variations of

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

Similarly, we obtain the following results for the other three mixing currents,

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.