Analysis of the doubly heavy baryon states and pentaquark states with QCD sum rules

In this article, we study the doubly heavy baryon states and pentaquark states with the QCD sum rules by carrying out the operator product expansion up to the vacuum condensates of dimension $7$ and $13$ respectively in a consistent way. In calculations, we separate the contributions of the negative parity and positive parity hadron states unambiguously, and study the masses and pole residues of the doubly heavy baryon states and pentaquark states in details. The present predictions can be confronted to the experimental data in the future.


Introduction
In 2017, the LHCb collaboration observed the doubly charmed baryon state Ξ ++ cc in the Λ + c K − π + π + mass spectrum, and obtained the mass M Ξ ++ cc = 3621.40 ± 0.72 ± 0.27 ± 0.14 MeV [1]. The observation of the Ξ ++ cc provides the crucial experimental input on the strong correlation between the two charm quarks, which may shed light on the spectroscopy of the doubly charmed baryon states, tetraquark states and pentaquark states. The attractive interaction induced by one-gluon exchange favors formation of the diquarks in color antitriplet [2], the favored configurations are the scalar (Cγ 5 ) and axialvector (Cγ µ ) diquark states from the QCD sum rules [3]. For the heavy-heavy quark systems QQ, only the axialvector diquarks ε ijk Q T j Cγ µ Q k and tensor diquarks ε ijk Q T j Cσ µν Q k survive due to the Fermi-Dirac statistics, the axialvector diquarks ε ijk Q T j Cγ µ Q k are more stable than the tensor diquarks ε ijk Q T j Cσ µν Q k , we can take the axialvector diquarks ε ijk Q T j Cγ µ Q k as basic constituents to study doubly heavy baryon states [4,5,6,7,8], tetraquark states [9,10,11,12] and pentaquark states with the QCD sum rules. The doubly heavy pentaquark states have not been studied with the QCD sum rules. In Ref. [13], the mass spectrum of the doubly heavy pentaquark states are studied in a color-magnetic interaction model.
In 2015, the LHCb collaboration studied the Λ 0 b → J/ψK − p decays, and performed the amplitude analysis on all relevant masses and decay angles of the six-dimensional data using the helicity formalism and Breit-Wigner amplitudes to describe all resonances, and observed two exotic states P c (4380) and P c (4450) in the J/ψp invariant mass distributions with the significances of more than 9 standard deviations [14]. The P c (4380) and P c (4450) are excellent candidates for the hidden-charm pentaquark states [15,16,17]. Up to now, no experimental candidates for the doubly charmed or doubly bottom pentaquark states have been observed.
In Refs. [6,7,8,18,19], we separate the contributions of the positive parity and negative parity baryon states explicitly, and study the heavy, doubly-heavy and triply-heavy baryon states with the QCD sum rules in a systematic way, the truncations of the operator product expansion are shown explicitly in Table 1. We carry out the operator product expansion up to the vacuum condensates of dimension 4 for the positive parity doubly heavy baryon states [6,7], another detailed studied including the contributions of the higher dimensional vacuum condensates are still needed. While in Ref. [4], the contributions of the positive parity and negative parity doubly heavy baryon states are not separated explicitly.
In Ref. [17], we construct the diquark-diquark-antiquark type interpolating currents to study the J P = 1 2 ± , 3 2 ± , 5 2 ± hidden-charm pentaquark states with the QCD sum rules in a systematic way by taking into account the vacuum condensates up to dimension 10 in the operator product 1 E-mail: zgwang@aliyun.com.
expansion and separating the contributions of the positive parity and negative parity pentaquark states explicitly. In calculations, we take the energy scale formula µ = M 2 P − (2M c ) 2 with the effective c-quark mass M c to determine the ideal energy scales of the QCD spectral densities. In Ref. [20], we construct the diquark-diquark-antiquark type current to study the ground state triply-charmed pentaquark states with the QCD sum rules by carrying out the operator product expansion up to the vacuum condensates of dimension 10, and take the energy scale formula µ = M 2 P − (3M c ) 2 to determine the optimal energy scales of the QCD spectral densities. In Ref. [21], we study the diquark-diquark-antiquark type charmed pentaquark states with J P = 3 2 ± with the QCD sum rules by carrying out the operator product expansion up to the vacuum condensates of dimension 13 in a consistent way to explore the possible assignments of the new excited Ω c states as the pentaquark states. The new excited Ω c states may also be the P-wave excitations of the ground state Ω c [22]. In Ref. [23], we study theDΣ * c ,D * Σ c andD * Σ * c pentaquark molecular states with the QCD sum rules by carrying out the operator product expansion up to the vacuum condensates of dimension 13 in a consistent way to explore the possible assignments of the P c (4380) and P c (4450) as the pentaquark molecular states. In calculations, we observe that the vacuum condensates of dimensions 11 and 13 play an important role in obtaining stable QCD sum rules [21,23].
In this article, we extend our previous works [6,7,8,12,17,19,20,21,23] to study the doubly heavy baryon states and pentaquark states with the QCD sum rules by carrying out the operator product expansion up to the vacuum condensates of dimension 7 and 13 respectively in a consistent way.
The article is arranged as follows: we derive the QCD sum rules for the masses and pole residues of the doubly heavy baryon states and pentaquark states in Sect.2; in Sect.3, we present the numerical results and discussions; and Sect.4 is reserved for our conclusion.

QCD sum rules for the doubly heavy baryon states and pentaquark states
In the following, we write down the two-point correlation functions Π(p), Π µν (p) and Π µναβ (p) in the QCD sum rules, (p 2 ), this may be our next work.
In this article, we choose the tensor structures 1, g µν and g µα g νβ + g µβ g να for the correlation functions Π(p), Π µν (p) and Π µναβ (p) respectively to study the J P = Now we obtain the hadron spectral densities at phenomenological side through the dispersion relation, where j = 1 2 , 3 2 for the doubly heavy baryon states, and ImΠ j (s) where j = 1 2 , 3 2 , 5 2 for the doubly heavy pentaquark states, we introduce the subscript H to denote the hadron side. Then we introduce the weight function exp − s T 2 to obtain the QCD sum rules at the phenomenological side (or the hadron side), with j = 1 2 , 3 2 for the doubly heavy baryon states, with j = 1 2 , 3 2 , 5 2 for the doubly heavy pentaquark states, where the s 0 are the continuum threshold parameters and the T 2 are the Borel parameters. We separate the contributions of the negative parity hadron states from that of the positive parity hadron states unambiguously. In Eqs. (17)(18), we choose the special combinations introduced in Ref. [17] to obtain the QCD sum rules, which differ from the non-covariant approach in Refs. [6,7,8,18,19,26].
We carry out the operator product expansion for the correlation functions Π(p), Π µν (p) and Π µναβ (p) up to the vacuum condensates of dimension 7 for the doubly heavy baryon states and dimension 13 for the doubly heavy pentaquark states, and assume vacuum saturation for the higher dimensional vacuum condensates. In calculations, we take the full light quark and heavy quark propagators, and t n = λ n 2 , the λ n is the Gell-Mann matrix [28,29]. In Eqs. (19)(20), we retain the term q j σ µν q i ( s j σ µν s i ) comes from the Fierz re-arrangement of the q iqj ( s isj ) to absorb the gluons emitted from other quark lines to extract the mixed condensate qg s σGq ( sg s σGs ).
For the correlation functions Π(p) and Π µν (p) of the doubly heavy three-quark currents, there are two heavy quark propagators and a light quark propagator, if each heavy quark line emits a gluon and each light quark line contributes a quark pair, we obtain a operator GGqq (or GGss), which is of dimension 7, for example, with the simple replacements jj ′ , we should take into account the vacuum condensates at least up to dimension 7. For the correlation functions Π(p), Π µν (p) and Π µναβ (p) of the doubly heavy five-quark currents, there are two heavy quark propagators and three light quark propagators, if each heavy quark line emits a gluon and each light quark line contributes a quark pair, we obtain a operator GGūuddqq, which is of dimension 13, for example, we should take into account the vacuum condensates at least up to dimension 13. We can carry out the operator product expansion by taking into account the vacuum condensates beyond dimension 7 or 13, however, it is a very difficult work.
The higher dimensional vacuum condensates play an important role in determining the Borel windows, as there appear terms of the orders O 1 in the QCD spectral densities, which manifest themselves at small values of the Borel parameter T 2 , we have to choose large values of the T 2 to warrant convergence of the operator product expansion and appearance of the Borel platforms. In this article, we take the truncations n ≤ 7(13) and k ≤ 1 in a consistent way, the operators of the orders O(α k s ) with k > 1 are discarded. For technical details in calculations, one can consult Refs. [30,31].
Once the analytical QCD spectral densities are obtained, we can take the quark-hadron duality below the continuum thresholds s 0 and introduce the weight function exp − s T 2 to obtain the following QCD sum rules: where with j = 1 2 , 3 2 for the doubly heavy baryon states, for the doubly heavy pentaquark states, the explicit expressions of the QCD spectral densities are given in the appendix.
We derive Eqs. (24)(25) with respect to τ = 1 T 2 , then eliminate the pole residues λ ± j and obtain the QCD sum rules for the masses of the doubly heavy baryon states and pentaquark states,
In the article, we study the doubly heavy baryon states and pentaquark states, the two heavy quarks form a diquark state ε ijk Q T j Cγ µ Q k , which serves as a static well potential and combines with a light quark state in color triplet to form a compact baryon state or combine with a light diquark and a light antiquark in color antitriplet to form a compact pentaquark state. While in the hidden-charm or hidden-bottom pentaquark states, the heavy quark Q serves as a static well potential and combines with the light quark to form a heavy diquark in color antitriplet, the heavy antiquarkQ serves as another static well potential and combines with the light diquark in color antitriplet to form a heavy triquark in color triplet, then the heavy diquark and heavy triquark combine together to form a hidden-charm or hidden-bottom tetraquark state. The quark structures of the doubly heavy pentaquark states and hidden-charm or hidden-bottom pentaquark states are quite different.
The doubly heavy (or hidden-charm, hidden-bottom) tetraquark states X, Y , Z and pentaquark states P are characterized by the effective heavy quark masses M Q and the virtuality In Refs. [30,31], we study the acceptable energy scales of the QCD spectral densities for the hidden-charm (or hidden-bottom) tetraquark states and molecular states in the QCD sum rules in details for the first time, and suggest an energy scale formula to determine the optimal energy scales. The energy scale formula also works well in studying the hidden-charm pentaquark states [17]. The updated values are M c = 1.82 GeV and M b = 5.17 GeV for the hidden-charm and hidden-bottom tetraquark states, respectively [35]. It is not necessary for the M Q in the doubly heavy tetraquark states and pentaquark states to have the same values as the ones in the hidden-charm or hidden-bottom tetraquark states and pentaquark states. In Ref. [12], we observe that if we choose a slightly different value M c = 1.84 GeV for the doubly charmed tetraquark states, the criteria of the QCD sum rules can be satisfied more easily, while the value M b = 5.17 GeV survives for the doubly bottom tetraquark states. In this article, we take the energy scale formula as a constraint to study the doubly heavy pentaquark states.
At the phenomenological side, we exclude the contaminations of the higher resonances by setting in the truncations s 0 , At the QCD side, there are terms of the Dirac δ function type, δ(s − m 2 Q ) and δ(s − m 2 Q ), which are associated with the higher dimensional vacuum condensates, the upper bounds of the integrals are arbitrary for getting the same values, there may be some uncertainties originating from the higher resonances, as the truncations s 0 cannot exclude the contaminations of the higher resonances rigourously. Firstly, we need good convergent behaviors in the operator product expansion to obtain solid predictions. Now we write down the definition for the contributions of the different terms in the operator product expansion, in stead of where the ρ n (s) are the QCD spectral densities for the vacuum condensates of dimension n, and the total spectral densities ρ(s) = √ sρ 1 QCD (s) + ρ 0 QCD (s). The definition in Eq.(29) warrants the contributions of the higher dimensional vacuum condensates play a less important role if the operator product expansion is well convergent.
The contributions of the perturbative terms D(0) are usually small for the tetraquark states and pentaquark states, we approximate the continuum contributions as ρ(s)Θ(s − s 0 ), and define the pole contributions (PC) or ground state contributions as If the pole dominance is satisfied at the phenomenological side, the uncertainties originate from the higher dimensional vacuum condensates are greatly suppressed. So the QCD sum rules must  ds ρ QCD (s) exp − s T 2 for the conventional bottomonium states. In this article, we choose µ = 2.2 GeV for the doubly bottom baryon states Ξ bb , Ξ * bb , Ω bb and Ω * bb , which works well.
In Refs. [6,7,8,17,18,19], we separate the contributions of the positive parity and negative parity hadron states explicitly, and study the heavy, doubly-heavy, triply-heavy baryon states and the hidden-charm pentaquark states with the QCD sum rules in a systematic way. In calculations, we observe that the continuum threshold parameters √ s 0 = M gr +(0.6−0.8) GeV works well, where the subscript gr denotes the ground states. In this article, we can take the continuum threshold parameters as √ s 0 < M B/P + 0.8 GeV. In this article, we choose the Borel parameters T 2 and continuum threshold parameters s 0 to satisfy the following four criteria: C1. Pole dominance at the phenomenological side; C2. Convergence of the operator product expansion; C3. Appearance of the Borel platforms; C4. Satisfying the energy scale formula only for the doubly heavy pentaquark states, by try and error.
The resulting Borel parameters or Borel windows T 2 , continuum threshold parameters s 0 , energy scales of the QCD spectral densities, pole contributions of the ground states and contributions of the highest dimensional vacuum condensates are shown explicitly in Table 2. From the table, we can see that the pole dominance at the phenomenological side and the convergence of the operator product expansion at the QCD side are satisfied, or the criteria C1 and C2 are satisfied.
In Figs.1-3, we plot the contributions of the vacuum condensates of dimension 7 and 13 with variations of the Borel parameters T 2 for the doubly heavy baryon states and pentaquark states, respectively. From the figures, we can see explicitly that the contributions of the highest dimen-    sional vacuum condensates are tiny in the Borel windows for the doubly heavy baryon states and doubly-charmed pentaquark states, while the contributions of the vacuum condensates of dimension 13 for the doubly bottom pentaquark states are somewhat larger, the smallest contributions are about (5 ∼ 7)% in the Borel windows, the operator product expansion is still convergent. In fact, for the doubly bottom pentaquark states, the D(13) decrease monotonously and quickly with the increase of the Borel parameter T 2 , a slight larger Borel parameter can lead to much smaller contribution. In calculations, we observe that the predicted doubly heavy pentaquark masses are rather stable with variations of the Borel parameters at the region T 2 ≥ T 2 min , where the min denotes the minimal values, the predictions survive for larger Borel parameters, the somewhat large contributions D(13) cannot impair the predictive ability.
We take into account all uncertainties of the input parameters, and obtain the masses and pole residues of the doubly heavy baryon states and pentaquark states, which are shown explicitly in Table 3 and Figs.4-7. From Table 3, we can see that the criterion C4 is satisfied for the doubly heavy pentaquark states. In this article, we choose the effective heavy quark masses M c = 1.84 GeV and M b = 5.17 GeV for the doubly heavy tetraquark states [12], if we choose slightly larger mass M b = 5.18 GeV, the energy scale formula is satisfied even better.
In Figs.4-7, we plot the masses and pole residues at much larger ranges of the Borel parameters than the Borel windows. From the figures, we can see that there appear Borel platforms in the Borel windows, the criterion C3 is also satisfied. In Figs.4-5, we also plot the masses with the truncations of the operator product expansion up to the vacuum condensates of dimension 4 for the doubly heavy baryon states and of dimension 10 for the doubly heavy pentaquark states. From the figures, we can see that without including the vacuum condensates of dimensions 5, 7 and 11, 13 for the doubly heavy baryon states and pentaquark states, respectively, we cannot obtain very stable QCD sum rules with respect to variations of the Borel parameters, the higher dimensional vacuum condensates play an important role in determining the Borel platforms.
For the doubly heavy baryon states, the criteria C1, C2 and C3 are satisfied, for the doubly heavy pentaquark states, the criteria C1, C2, C3 and C4 are satisfied, we expect to make reliable predictions, which can be confronted to the experimental data in the future.
In the present work, we obtain the mass M = 4.21 +0.10

Conclusion
In this article, we study the doubly heavy baryon states and pentaquark states with the QCD sum rules by carrying out the operator product expansion up to the vacuum condensates of dimension 7 and 13 respectively in a consistent way. In calculations, we separate the contributions of the negative parity and positive parity hadron states unambiguously, and study the masses and pole residues of the doubly heavy baryon states and pentaquark states in details, and obtain very stable QCD sum rules in the Borel windows. The prediction M Ξcc = 3.63 +0.08 −0.07 GeV is in excellent agrement with the LHCb data M Ξ ++ cc = 3621.40 ± 0.72 ± 0.27 ± 0.14 MeV, other predictions can be confronted to the experimental data in the future.    Figure 5: The masses of the doubly heavy pentaquark states with variations of the Borel parameters T 2 , where the A, B, C, D, E and F denote the P cc, 1 2 , P cc, 3 2 , P cc, 5 2 , P bb, 1 2 , P bb, 3 2 and P bb, 5 2 , respectively, the D = 10 denotes the predictions based on the truncations of the operator product expansion up to the vacuum condensates of dimension 10.    Error bounds Figure 7: The pole residues of the doubly heavy pentaquark states with variations of the Borel parameters T 2 , where the A, B, C, D, E and F denote the P cc, 1 2 , P cc, 3 2 , P cc, 5 2 , P bb, 1 2 , P bb, 3 2 and P bb, 5 2 , respectively.

Appendix
The explicit expressions of the QCD spectral densities: For the Ω QQ states, For the Ω * QQ states, For the QQcdq states with J P = 1 2 − ,