Analysis of the D-wave Σ -type charmed baryon states with the QCD sum rules

We construct the Σ-type currents to investigate the D-wave charmed baryon states with the QCD sum rules systematically. The predicted masses M = 3 . 35 +0 . 13 − 0 . 18 GeV (3 . 33 +0 . 13 − 0 . 16 GeV), 3 . 34 +0 . 14 − 0 . 18 GeV (3 . 35 +0 . 13 − 0 . 16 GeV) and 3 . 35 +0 . 12 − 0 . 13 GeV (3 . 35 +0 . 12 − 0 . 14 GeV) for the Ω c (0 , 2 , 12+ ), Ω c (0 , 2 , 32+ ) and Ω c (0 , 2 , 52+ ) states are in excellent agreement with the experimental data 3327 . 1 ± 1 . 2 MeV from the LHCb collaboration, and support assigning the Ω c (3327) to be the Σ-type D-wave Ω c state with the spin-parity J P = 12+ , 32+ or 52+ . PACS number: 14.20.Lq, 14.20.Mr

In 2021, the LHCb collaboration observed the Ω − b → Ξ + c K − π − decay for the first time using the pp collision data at centre-of-mass energies of 7, 8 and 13 TeV, which corresponds to an integrated luminosity of 9 fb −1 , and confirmed the four excited Ω c states Ω c (3000), Ω c (3050), Ω c (3066) and Ω c (3090) in the Ξ + c K − mass projections with significances larger than 5σ [3].
The article is arranged in the form: we derive the QCD sum rules for the D-wave charmed baryon states in Sect.2; in Sect.3, we give the numerical results and discussions; and Sect.4 is hold for conclusions.

QCD sum rules for the Σ-type D-wave baryon states
Firstly, we write down the two-point correlation functions Π(p), Π αβ (p) and Π αβµν (p), where the interpolating currents, with i = 1, 2, 3, with q = u, d, the i, j, k in the ε ijk are color indexes, the C is the charge conjugation matrix.We choose the currents J(x), J α (x) and J αβ (x) to interpolate the spin-parity + and 5 2 + charmed baryon states, respectively.We tentatively assign the Ω c (3327) to be the D-wave Ω c state with the spin-parity + , the currents J 3 (x), η 3 (x), J 3 α (x), η 3 α (x), J 3 αβ (x) and η 3 αβ (x) maybe couple potentially to the Ω c (3327).In the Isospin limit, the uuc, udc and ddc baryon states have degenerated masses, while the usc and dsc baryon states have degenerated masses, we only study the udc and usc baryon states for simplicity.
Now we take a short digression to explain how to construct the currents in Eqs.( 4)- (10).We usually resort to the diquark-quark model to explore the baryon states.The attractive interaction of one-gluon exchange favors forming diquark correlations in color antitriplet 3 c [29,30].The diquarks ε ijk q T j CΓq ′ k (with q, q ′ = u, d or s) have five structures, where CΓ = Cγ 5 , C, Cγ µ γ 5 , Cγ µ and Cσ µν for the scalar, pseudoscalar, vector, axialvector and tensor diquarks, respectively.The structures Cγ µ and Cσ µν are symmetric (in other words, they are Σ-type states), while the structures Cγ 5 , C and Cγ µ γ 5 are antisymmetric (in other words, they are Λ-type states).The calculations based on the QCD sum rules lead to the conclusion that the favored configurations are the Cγ 5 and Cγ µ diquark states [31].For the qq diquark states with q = u, d or s, we have to resort to the Σ-type diquark states to satisfy the Fermi-Dirac statistic without introducing additional P-wave.In short, we prefer the Cγ µ diquark states in the present work.
In the diquark-quark models, we usually denote the angular momentum between the two light quarks by L ρ , and denote the angular momentum between the light diquark and heavy quark by L λ .In the case of L ρ = 0, we obtain the spin-parity J P = 0 + and 1 + diquarks, therefore the Λ-type and Σ-type baryons, respectively [32].While in the case of (L ρ , L λ ) = (2, 0), (0, 2) and (1, 1), we can obtain copious spectrum of the D-wave charmed baryon states.For L ρ = 2 and L λ = 0, the qq diquark states have the spin-parity however, the L ρ = 1 diquark states cannot exist due to the Fermi-Dirac statistic.In the heavy quark limit, the c/b-quark is static, the ∂ µ when it operates on the c/b-quark field.Therefore, for L ρ = 0 and L λ = 2, we acquire the qq diquark states with the spin-parity Then we classify the interpolating currents by the quantum numbers L ρ and L λ , where i = 1, 2, 3.
In fact, it is difficult or impossible to construct all currents to interpolate all the Dwave baryon states with the spin-parity J P = 1 In Ref. [26], we explore the Λ-type D-wave baryon states with the spin-parity J P = 3 2 + and 5 2 + in details, and explore the possible assignments of the Λ c (2860) + , Λ c (2880) + , Ξ c (3055) + , Ξ c (3055) 0 and Ξ c (3080) + .Experimentally, the Λ c (2860) + and Λ c (2880) + have been observed to have the spin-parity J P = 3 2 + and 5 2 + respectively by the LHCb collaboration [33].Now we study the Σ-type D-wave charmed baryon states with the spin-parity In general, we can choose either the partial derivative ∂ µ or covariant derivative D µ to construct the interpolating currents, see Eqs.( 4)- (10).The currents with covariant derivative D µ are gauge covariant/invariant, but hinders interpreting the as angular momentum.For example, under the gauge transformation U ii ′ (x) for the quark fields q i (x), the baryon currents without partial derivatives (or with covariant derivatives) undergo, where the i, j and k are color indexes, and we have neglected other indexes and matrixes.
The currents with partial derivative ∂ µ are not gauge covariant, but favors interpreting the as angular momentum, furthermore, the covariant derivative D µ leads to some hybrid components in meson or baryon states due to the gluon field G µ .For example, under the gauge transformation U ii ′ (x), the baryon currents with partial derivatives undergo, In this work, we present the results with both the partial derivatives ∂ α and covariant derivatives D α for completeness.
Now we obtain the hadronic spectral densities through dispersion relation, where j = 1 2 , 3 2 , 5 2 , we add the subscript H to stand for the hadron side, then we introduce the weight function exp − s T 2 to suppress the higher resonances (excited states) and continuum states to achieve the QCD sum rules at the hadron side, where the s 0 are the continuum thresholds and the T 2 are the Borel parameters [10,24,25,26,27,28,36,37,38].Because of the special combination √ sρ 1 j,H (s) + ρ 0 j,H (s), the negative parity charmed baryon states cannot contaminate the QCD sum rules, and they saturate other QCD sum rules unambiguously, At the QCD side, we calculate the correlation functions Π(p), Π αβ (p) and Π αβµν (p) with the full light quark propagators S ij (x), and full c-quark propagator C ij (x), , the λ n is the Gell-Mann matrix [41].In Eq.( 27), we adopt the qj σ µν q i comes from the Fierz transformation of the q i qj to absorb the gluons emitted from the other quark lines to extract the mixed condensate qg s σGq .Then we accomplish all the integrals in the coordinate and momentum spaces in sequence to achieve the QCD representation up to the vacuum condensates of dimension 10 in a consistent way [10,24,25,26,27,28,37], and achieve the QCD spectral densities through dispersion relation, where j = 1 2 , 3 2 , 5 2 .For simplicity, we give the explicit expressions of the QCD spectral densities in the Appendix.
In calculations, we carry out the operator product expansion by choosing the partial derivatives ∂ µ firstly, then take the simple replacement 2 in the vertexes to take account of the additional terms originate from the covariant derivatives.In Fig. 1, we show the additional Feynman diagrams (originate from the covariant derivatives) make contributions to the vacuum condensates qg s σGq , αsGG π , qq αsGG π , qg s σGq 2 and qq 2 αsGG π with q = u, d or s, which correspond to truncations of the quark-gluon operators of the orders O(α k s ) with k ≤ 1 (adopted in all our previous works) [10,24,25,26,27,28,37].We have chosen the fixed-point gauge G µ (x) = 1 2 x σ G σµ (0), and the vacuum condensates qg s σGq 2 and qq 2 αsGG π happen to have no contribution.In Fig. 2, we show the perturbative contributions originate from the gluons in the covariant derivatives, they are of the orders O(α s ) and O(α 2 s ) and are neglected, just like what have been done in the literatures [43].Furthermore, taking account of the diagrams in Fig. 2 amounts to introducing some valence gluon Fock components in the interpolating currents, while we choose the covariant derivative only for the sake of obtaining gauge covariant/invariant currents.
Now we suppose quark-hadron duality below the continuum thresholds s 0 , again we resort to the weight function exp − s T 2 to suppress the higher resonances (excited states) and continuum states to achieve the QCD sum rules: We differentiate Eq.( 31) in regard to 1 T 2 , then eliminate the pole residues λ + j through a fraction, and achieve the QCD sum rules for the Σ-type D-wave charmed baryon masses, For the Ioffe currents, we can obtain the relation, by neglecting the tiny u and d quark masses for the proton and neutron [41], which indicates that the ground state masses mainly originate from the quark condensates.Such simple relation does not exist in the present case due to the large c-quark mass and additional D-wave, the net effects of the perturbative terms, quark condensates, gluon condensates and mixed condensates lead to the excited baryon masses.
As the energy scales of the QCD spectral densities are concerned, we give some discussions.In the heavy quark limit, the Q-quark plays a role as a static well potential in the qq ′ Q, q q′ QQ, qq ′ q ′′ QQ and q q′ QQ systems, then we introduce the effective heavy quark masses M Q and divide the baryon/multiquark states into both the heavy and light degrees of freedom.If we neglect the tiny u and d quark masses, we acquire the heavy degrees of freedom M Q /2M Q and light degrees of freedom V = M 2 B/X/Y /Z/T /P − (M Q /2M Q ) 2 (or virtuality), then we set the V to be the energy scales µ of the QCD spectral densities, therefore achieve the energy scale formula, where the B, X, Y , Z, T and P stand for the traditional baryon states, tetraquark (molecular) states and pentaquark states, respectively [36,47].In addition, we take account of the light flavor SU f (3) breaking effects by introducing the effective s-quark mass M s to achieve the modified energy scale formula where k = 0, 1, 2, 3 count for the numbers of the s-quark, which works well [48,49].The effective quark masses M Q/s have universal values, we take the updated effective c-quark mass M c = 1.82 GeV [50] and effective s-quark mass M s = 0.20 GeV [48,49], then if we identify the Ω c (3327) as the traditional D-wave Σtype baryon state, we achieve the suitable energy scale µ = M 2 Ω − M 2 c − 2M s = 2.4 GeV for the QCD spectral densities.In calculations, we observe that the modified energy scale formula leads to the universal energy scales µ = 2.4 GeV (2.7 GeV) approximately for the (L ρ , L λ ) = (0, 2) ((2, 0)) Σ-type charmed baryon states.In fact, at the energy scales µ ≥ 2.0 GeV, the predicted baryon masses change very slowly with variations of the energy scales of the QCD spectral densities, it is reasonable to choose the energy scales µ = 2.4 GeV and 2.7 GeV.For example, in Fig. 3, we plot the mass of the Ω c (0, 2, 3 2 + ) state with variations of the energy scale µ for the central values of the iuput parameters (for the current with covariant derivatives, see Table 2).From the figure, we can see explicitly that the predicted mass decreases slowly with increase of the energy scale of the QCD spectral density, slightly larger or smaller energy scales cannot change the predictions remarkably.
We search for the ideal Borel parameters T 2 and continuum threshold parameters s 0 to satisfy the two basic criteria to achieve reliable QCD sum rules: firstly, pole dominance at the hadron side, we place universal restrictions on the pole contributions, about (40−85)%; secondly, convergence of the operator product expansion at the QCD side, as the dominant contributions come from the perturbative terms, such a criterion is easy to satisfy.In the present work, we calculate the vacuum condensates up to dimension 10 in a consistent way, the higher dimensional vacuum condensates play an important role in acquiring the Borel windows, while in the Borel windows, they play a minor role, for example, the contributions of the vacuum condensates of dimension 10 are about (1 − 3)%, (1 − 4)% and (1 − 2)% for the Ω c (0, 2, 1 2 + ), Ω c (0, 2, 3 2 + ) and Ω c (0, 2, 5 2 + ) states, respectively, for the currents with the covariant derivatives.
Finally, we acquire suitable Borel parameters T 2 , continuum threshold parameters s 0 , pole contributions and perturbative contributions, see Tables 1-2, where we choose uniform Borel windows, T 2 max − T 2 min = 0.6 GeV 2 , the subscripts "max" and "min" stand for the maximum and minimum values respectively.We take account of all uncertainties of the input parameters, and obtain the values of the masses and pole residues of the ground state D-wave charmed baryon states, which are shown in Tables 3-4.From Tables 1, 3 and Tables 2, 4, we can see clearly that the continuum threshold parameters and predicted baryon masses have the relation √ s 0 − M B = 0.6 ± 0.1 GeV, the s 0 are large enough to take account of all the ground state contributions sufficiently but small enough to suppress contaminations from the first radial excited states efficaciously [10,24,25,26,27,28,37].From Tables 3-4, we can see explicitly that for the central values of the baryon masses, the currents with partial derivatives and with covariant derivatives only make tiny differences, while for the central values of the pole residues, the currents with covariant derivatives lead to (slightly) larger values.In general, the currents with covariant derivatives lead to (slightly) smaller uncertainties comparing to the currents with partial derivatives.Therefore, we obtain the conclusion tentatively that the currents with the covariant derivatives are better, as the vecuum condensates make slightly larger contributions (see Tables 1-2) and therefore better QCD sum rules.Moreover, from Tables 3-4, we can see clearly that the uncertainties of the masses are rather small, as we obtain the baryon masses through a fraction, see Eq.( 32), the uncertainties originate from the uncertainty of a parameter in numerator and denominator are canceled out with each other to a large extent, so the net uncertainties of the masses are small.We can examine the present calculations by the experimental data in the future.On the other hand, such a cancellation does not occur for the pole residues, see Eq.( 31), and the uncertainties of the pole residues can be as large as 40%.In previous works [10,24,25,26,27,37], we have given several successful (or reasonable) descriptions of the P-wave and D-wave heavy baryon states with the same traditional error analysis.In those works, we chose the currents with the partial derivatives, after they were published, we rechecked the calculations by taking the covariant derivatives in stead of the partial derivatives, the predicted baryon masses were as before, while the pole residues were improved slightly, just like in the present work.
The predicted masses M = 3.35 +0.13 −0.18 GeV (3.33 +0.13 −0.16 GeV), 3.34 +0.14 −0.18 GeV (3.35 +0.13 −0.16 GeV) and 3.35 +0.12 −0.13 GeV (3.35 +0.12 −0.14 GeV) for the Ω c (0, 2, ) states for the currents with partial derivatives (covariant derivatives) are in excellent agreement with the experimental data 3327.1±1.2MeV from the LHCb collaboration [4], and support assigning the Ω c (3327) to be the Σ-type D-wave Ω c state with the spin-parity . Other predictions can be confronted to the experimental data in the future to diagnose the nature of the D-wave charmed baryon states.
As an example, in Fig. 4, we plot the predicted masses of the Ω c (0, 2, and Ω c (0, 2, 5 ) states with variations of the Borel parameter T 2 for the currents with partial derivatives and covariant derivatives, respectively.From the figure, we can see clearly that the predicted masses increase quickly or slowly with increase of the Borel parameters, in the Borel windows, the platforms are not flat enough.This maybe due to the fact that the perturbative contributions dominate the QCD sum rules and the vacuum The Borel parameters T 2 , continuum threshold parameters s 0 , pole contributions (from the ground states) and perturbative contributions for the D-wave (with partial derivatives) charmed baryon states.

Conclusion
In this article, we construct the Σ-type interpolating currents with both the partial derivatives and covariant derivatives to explore the D-wave charmed baryon states via the QCD sum rules in a systematic way.We carry out the operator product expansion up to the vacuum condensates of dimension 10 consistently, and distinguish the contributions of the positive and negative parity baryon states unambiguously, then we investigate the masses and pole residues of the ground states in details, the predicted masses M = 3.35 +0.13  −0.18 GeV (3.33 ρ 0 j,Ωc (s) = 2ρ 0 j,Ξc (s) | ms→2ms, qq → ss , qgsσGq → sgsσGs , where s , and r = +1 and −1 for the currents with the quantum numbers (L ρ , L λ ) = (0, 2) and (2, 0), respectively.

2 +
in a systematic way.

Figure 1 :
Figure1: The additional Feynman diagrams (originate from the covariant derivatives) make contributions to the vacuum condensates, where the dashed (solid) lines denote the heavy (light) quark lines.

Figure 2 :
Figure 2: The additional Feynman diagrams (originate from the covariant derivatives) make contributions of the orders O(α s ) and O(α 2 s ), where the dashed (solid) lines denote the heavy (light) quark lines.

Figure 3 : 2 +
Figure 3: The mass of the Ω c (0, 2, 3 2 + ) state with variations of the energy scale µ for the central values of the iuput parameters (for the current with covariant derivatives).

Table 2 :
B(L ρ , L λ , J P ) T 2 (GeV 2 ) The Borel parameters T 2 , continuum threshold parameters s 0 , pole contributions (from the ground states) and perturbative contributions for the D-wave (with covariant derivatives) charmed baryon states.

Table 3 :
The masses and pole residues of the D-wave (with partial derivatives) charmed baryon states.

Table 4 :
The masses and pole residues of the D-wave (with covariant derivatives) charmed baryon states.