Abstract
In this article, we tentatively assign Pc(4312) to be the pentaquark molecular state with the spin-parity , and discuss the factorizable and non-factorizable contributions in the two-point QCD sum rules for the molecular state in detail to prove the reliability of the single pole approximation in the hadronic spectral density. We study its two-body strong decays with the QCD sum rules, and special attention is paid to match the hadron side with the QCD side of the correlation functions to obtain solid duality. We obtain the partial decay widths and , which are compatible with the experimental value of the total width, and support assigning to be the pentaquark molecular state.
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1. Introduction
In 2015, the LHCb collaboration observed two pentaquark candidates and in the mass spectrum in the decays [1]. Recently, the LHCb collaboration observed a new narrow pentaquark candidate, , in the mass spectrum with the statistical significance of ; furthermore, they confirmed the old pentaquark structure, which consists of two narrow overlapping peaks and , with the statistical significance of [2]. The masses and widths are
The can be assigned to be a pentaquark molecular state [3-18], a pentaquark state [19-23], and a hadrocharmonium pentaquark state [24].
The lies near the threshold, which leads to the molecule assignment naturally. In Ref. [18], we performed detailed studies of the , , and pentaquark molecular states with the QCD sum rules by carrying out the operator product expansion up to the vacuum condensates of dimension in a consistent way. The prediction for the molecular state supports assigning to be the pentaquark molecular state with the spin-parity . On the other hand, our studies based on the QCD sum rules indicate that the scalar-diquark-scalar-diquark-antiquark type pentaquark state with the spin-parity has a mass of , whereas the axialvector-diquark-axialvector-diquark-antiquark type pentaquark state with the spin-parity has a mass of , which support assigning to be a diquark-diquark-antiquark type pentaquark state [23, 25, 26]. may be a diquark-diquark-antiquark type pentaquark state, which has a strong coupling to the scattering states. The strong coupling induces some components [27]. Thus, we can reproduce the experimental value of the mass of in both the scenarios of the pentaquark state and pentaquark molecular state. In Ref. [28], we choose the type tetraquark current to study the strong decays of with the QCD sum rules based on solid quark-hadron duality. Our calculations showed that the hadronic coupling constant is consistent with the observation of in the mass spectrum, and that it favors the molecule assignment [29, 30]. A similar mechanism may exist for .
In this article, we tentatively assign to be the pentaquark molecular state with the spin-parity , and study its two-body strong decays with the QCD sum rules. In Ref. [31], we assigned to be the diquark-antidiquark type axialvector tetraquark state, and studied the hadronic coupling constants in the strong decays , , with the QCD sum rules based on the solid quark-hadron duality by taking into account both the connected and disconnected Feynman diagrams in the operator product expansion. The method works well for studying the two-body strong decays of , , and [31-34]. Now, we extend the method to study the two-body strong decays of the pentaquark molecular state by carrying out the operator product expansion up to the vacuum condensates of dimension .
The article is arranged as follows: in Sec. 2, we present the comments on the QCD sum rules for the pentaquark molecular state; in Sec. 3, we derive the QCD sum rules for the hadronic coupling constants in the strong decays , ; in Sec. 4, we present the numerical results and discussions; and we conclude our report in Sec. 5.
2. Comments on the QCD sum rules for the pentaquark molecular state
In this section, we present the two-point correlation function to study the mass and pole residue of the pentaquark molecular state with the QCD sum rules,
where the current ,
and i, j, k are color indices. We choose the color-singlet-color-singlet type (or meson-baryon type) current to interpolate the pentaquark molecular state with the spin-parity [18]. For the technical details and numerical results, one can consult Ref. [18]. In the present work, we will focus on the reliability of the single pole approximation in the hadronic spectral density.
At the QCD side, the correlation function can be written as
where , , and are the full u, d, and c quark propagators, respectively ( ),
, is the Gell-Mann matrix [35-37].
In Fig. 1, we plot two Feynman diagrams for the lowest order contributions, where the first diagram corresponds to the term with two Tr's and the second diagram corresponds to the term with one Tr in Eq. (4). The first Feynman diagram is factorizable and has the color factor , while the second Feynman diagram is non-factorizable and has the color factor . In the large limit , the contribution of the second Feynman diagram is greatly suppressed. In reality, the color number , the second Feynman diagram plays an important role.
In the second Feynman diagram, we can replace the lowest order heavy quark lines and (or) light quark lines with other terms in the full propagators in Eqs. (5), (6), and obtain other non-factorizable Feynman diagrams.
In the first Feynman diagram, we can also replace the lowest order heavy quark lines and (or) light quark lines with other terms in the full propagators in Eqs. (5), (6), and obtain other factorizable Feynman diagrams. There are non-factorizable Feynman diagrams besides the factorizable Feynman diagrams, see Fig. 2. In Fig. 2, we plot the Feynman diagrams contributing to the vacuum condensates , which are the vacuum expectations of the quark-gluon operators of the order and not of the order . In Fig. 3, we plot the non-factorizable Feynman diagrams of the order from the terms with two Tr's in Eq. (4), where the first, second, third, and fourth diagrams are non-planar Feynman diagrams, while the fifth, and sixth diagrams are planar Feynman diagrams. The first and the second Feynman diagrams are suppressed by a factor in the large limit compared to the first Feynman diagram in Fig. 1, while the third, fourth, fifth, and sixth diagrams are suppressed by a factor . In reality, for the color number , the Feynman diagrams in Fig. 3 are suppressed by a factor , and play a minor role.
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Standard imageIn Fig. 4, we plot the non-factorizable Feynman diagrams contributing to the vacuum condensates for the meson-meson type currents. From the figure, we can see that the non-factorizable contributions begin at the order rather than at the order , as argued in Ref. [38]. For the nonperturbative contributions, we absorb the strong coupling constant into the vacuum condensates and count them as of the order .
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Standard imageWe insist on the viewpoint that the factorizable Feynman diagrams correspond to the two-particle reducible contributions, irrespective of the baryon-meson pair or the meson-meson pair, and give the masses of the two constituent particles. Then, the attractive interactions which originate from (or are embodied in) the non-factorizable Feynman diagrams attract the two constituent particles to form the molecular states. The non-factorizable Feynman diagrams are suppressed in the large limit, which is consistent with the small bound energies of the pentaquark molecular states. The baryon-meson type or the color-singlet-color-singlet type currents couple potentially to the pentaquark molecular states.
On the other hand, the baryon-meson type currents also couple to the baryon-meson pairs besides the molecular states as there exist two-particle reducible contributions. The intermediate baryon-meson loops contribute a finite imaginary part to modify the dispersion relation at the hadron side [18]. Through calculations, we observe that the zero width approximation works well, and the couplings to the baryon-meson pairs can be neglected safely.
If we only take into account the non-factorizable Feynman diagrams shown in Figs. 1 and 2, even if we obtain the stable QCD sum rules, we cannot distinguish the diquark-diquark-antiquark type substructure or the baryon-meson type substructure, and cannot select the color-singlet-color-singlet type substructure and refer to it as the molecular state. We simply obtain a hidden-charm five-quark state with the spin-parity . If we insist that it is a molecular state, then which diagram contributes to the masses of the baryon and meson constituents? In Ref. [39], the factorizable Feynman diagrams corresponding to the two-particle reducible contributions were subtracted, and only the non-factorizable Feynman diagrams were taken into account to study the pentaquark states. We do not agree with that approach.
3. QCD sum rules for the decays as a pentaquark molecular state
Here, we write down the three-point correlation functions and in the QCD sum rules,
where
and i, j, k are color indices. We choose the currents , , , and to interpolate , , p, and , respectively. Thereafter we will denote the proton p as N to avoid confusion due to the four-momentum .
At the hadron side, we insert a complete set of intermediate hadron states with the same quantum numbers as the current operators , , , and into the correlation functions and to obtain the hadronic representation [35, 40, 41]. After isolating the pole terms of the ground states, we obtain the following results:
where we have used the definitions
Here, the , , and are the hadronic coupling constants, and are the Dirac spinors, and are the pole residues, and are the decay constants, and is the polarization vector of .
It is important to choose the pertinent structures to study the hadronic coupling constants. If and , we expect that the two relations and also exist, where the subscripts H and denote the hadron side and QCD side of the correlation functions, respectively. and are some Dirac -matrixes.
In this article, we choose , , , ,
and choose the tensor structures , , and to study the hadronic coupling constants , and , respectively, where is a four-vector.
Now we write down the components , , , and explicitly,
where we introduce the formal functions , , , , , , , , , , , , , , , and to parameterize the transitions between the ground states and the excited states. , , , and are the threshold parameters for the radial excited states.
Now we smear the indexes , A, B, etc, and rewrite (components of) the correlation functions at the hadron side as
through dispersion relation, and take , for simplicity, where represents the hadronic spectral densities.
We carry out the operator product expansion at the QCD side, and write (components of) the correlation functions as
through dispersion relation, where the represents the QCD spectral densities. As the QCD spectral density does not exist,
we can write the QCD spectral density as for simplicity.
Now we match the hadron side with the QCD side of the correlation functions, and carry out the integration over first, to obtain the solid duality [31],
It is impossible to carry out the integration over explicitly due to the unknown functions , , , , , , , and . Now we introduce the parameters , , , and to parameterize the net effects,
Here, we write down the quark-hadron duality explicitly,
We set and perform a double Borel transform with respect to the variables and , respectively, to obtain the QCD sum rules,
where , , , and , when the function appears.
In this study, we carried out the operator product expansion to the vacuum condensates up to dimension 10, and assumed vacuum saturation for the higher dimension vacuum condensates. As the vacuum condensates are vacuum expectations of the quark-gluon operators, we take the truncations and in a consistent way, where the operators of the orders with are neglected. Furthermore, we set the two Borel parameters to be for simplicity. If we take and as two independent parameters, it is difficult to obtain stable QCD sum rules. In numerical calculations, we take , , , and as free parameters and choose the suitable values to obtain the stable QCD sum rules.
In the operator product expansion for the correlation functions and , if we take into account the finite spatial separation between the clusters and in the current operator , the current is modified to
By adding a small four-vector , the Feynman diagrams for the decays into the charmonium states are made non-factorizable; see the first Feynman diagram in Fig. 5, where we split the point into two points to site the baryon and meson clusters. In the limit , the lowest order Feynman diagrams for the decays into the charmonium states are factorizable; see the second Feynman diagram in Fig. 5. We observed in our calculations that there are both connected and disconnected Feynman diagrams contributing to the decays. The non-factorizable contributions begin at the order due to the quark-gluon operators , while at the order of the quark-gluon operators, the non-factorizable contributions are of the forms and . We absorb the strong coupling constant into the vacuum condensates and count them as of the order . In Fig. 6, we draw the non-factorizable Feynman diagrams contributing to the gluon condensate as an example. Although the correlation functions can be written as
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at the QCD side; there are both factorizable and non-factorizable contributions.
In previous section, we have proved that the current operator couples potentially to the molecular state, which receives both factorizable and non-factorizable contributions, while the couplings to the baryon-meson scattering states can be neglected. From Eqs. (15)-(18), we can see that there is a pole term at the hadron side, which should have originated at the QCD side. However, at the QCD side, there is no singular term with respect to the variable ; see Eq. (21). It does not imply that there is no contribution from or that the current-molecule coupling is zero, it just means that may not be on the mass-shell. If we set to obtain the QCD sum rules, the terms and at the hadron side cannot be singular simultaneously. A reasonable explanation is that the current operator in the three-point correlation functions and couples potentially to the molecular state or . However, the may not be on the mass-shell, which facilitates setting .
4. Numerical results and discussions
At the hadron side, we take the hadronic parameters as , , , , , [42], [2], , [43], [44], and [18].
At the QCD side, we take the standard values of the vacuum condensates , , , and at the energy scale [35, 40, 41, 45], and choose the mass from the Particle Data Group [42]. Moreover, we take into account the energy-scale dependence of the parameters,
where ; ; ; ; and , , for the flavors , , and , respectively [42, 46], and evolve all the parameters to the ideal energy scale with to extract the hadronic coupling constants , , and .
In the QCD sum rules for the mass of the pentaquark molecular state with the spin-parity or , the ideal energy scale of the QCD spectral density is [18], which is determined by the energy scale formula with the effective c-quark mass [47]. The energy scale is too large for N, , and . In this study, we take the energy scales of the QCD spectral densities to be , which is acceptable for the charmonium states [37].
We choose the values of the free parameters as , , , and to obtain flat platforms in the Borel windows , , , and for the hadronic coupling constants , , and , respectively. We fit the free parameters , , , and to obtain the same intervals of flat platforms , where and denote the maximum and minimum values of the Borel parameters, respectively.
We take into account the uncertainties of the input parameters, and obtain the values of the hadronic coupling constants , , and , which are shown in Fig. 7,
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where we have redefined the hadronic coupling constants in Eq. (13) with a simple replacement, , as the central values of are negative from the QCD sum rules in Eq. (30).
Now it is straightforward to calculate the partial decay widths of the decays , ,
where
and ,
where
The partial decay width is extremely small, and the total width can be saturated with the strong decay . The predicted width is compatible with the experimental data from the LHCb collaboration [2]. The present calculations support assigning to be the pentaquark molecular state with the spin-parity . We can search for in the mass spectrum, and measure the branching fraction , which may shed light on the nature of and test the predictions of the QCD sum rules.
The thresholds of and are and , respectively, and the decays into the final states and are kinematically allowed. At the quark level, the decays of the pentaquark molecular state to the and states take place through dissolving of the -type diquark states to form the -type diquark states by emitting an isospin quark-antiquark pair. At the hadron level, the decay can take place through the process with the subprocesses and ; the partial decay width may be as large as [48]. Direct calculations of these partial decay widths with the QCD sum rules are necessary to make a definite conclusion, which we aim to address in our next work.
5. Conclusion
In this article, we tentatively assign to be the pentaquark molecular state with the spin-parity , and discuss the factorizable and non-factorizable contributions in the two-point QCD sum rules for the molecular state in detail to prove the reliability of the single pole approximation in the hadronic spectral density. We study its two-body strong decays with the QCD sum rules by carrying out the operator product expansion up to the vacuum condensates of dimension . In calculations, special attention was paid to match the hadron side with the QCD side of the correlation functions to obtain solid duality. We obtain the partial decay widths and , which are compatible with the experimental data from the LHCb collaboration. The present calculations support assigning to be the pentaquark molecular state with the spin-parity . We can search for the decay to diagnose the nature of .
Footnotes
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Supported by National Natural Science Foundation (11775079)