Analysis of the $\Lambda_c(2860)$, $\Lambda_c(2880)$, $\Xi_c(3055)$ and $\Xi_c(3080)$ as D-wave baryon states with QCD sum rules

In this article, we tentatively assign the $\Lambda_c(2860)$, $\Lambda_c(2880)$, $\Xi_c(3055)$ and $\Xi_c(3080)$ to be the D-wave baryon states with the spin-parity $J^P={\frac{3}{2}}^+$, ${\frac{5}{2}}^+$, ${\frac{3}{2}}^+$ and ${\frac{5}{2}}^+$, respectively, and study their masses and pole residues with the QCD sum rules in a systematic way by constructing three-types interpolating currents with the quantum numbers $(L_\rho,L_\lambda)=(0,2)$, $(2,0)$ and $(1,1)$, respectively. The present predictions favor assigning the $\Lambda_c(2860)$, $\Lambda_c(2880)$, $\Xi_c(3055)$ and $\Xi_c(3080)$ to be the D-wave baryon states with the quantum numbers $(L_\rho,L_\lambda)=(0,2)$ and $J^P={\frac{3}{2}}^+$, ${\frac{5}{2}}^+$, ${\frac{3}{2}}^+$ and ${\frac{5}{2}}^+$, respectively. While the predictions for the masses of the $(L_\rho,L_\lambda)=(2,0)$ and $(1,1)$ D-wave $\Lambda_c$ and $\Xi_c$ states can be confronted to the experimental data in the future.

In this article, we tentatively assign the Λ c (2860), Λ c (2880), Ξ c (3055) and Ξ c (3080) to be the D-wave charmed baryon states with the spin-parity J P = 3 2 + , 5 2 + , 3 2 + and 5 2 + , respectively, and study their masses and pole residues with the QCD sum rules in a systematic way. The QCD sum rules is a powerful theoretical approach in studying the ground state mass spectrum of the heavy baryon states, and has given many successful descriptions [8,14,15,16,17,18,19].
We can construct the interpolating currents without introducing the relative P-wave to study the negative parity heavy, doubly-heavy and triply-heavy baryon states [14,15,16], or introducing the relative P-wave explicitly to study the negative parity heavy, doublyheavy and triply-heavy baryon states [18,19]. For the D-wave heavy baryon states, it is better to introduce the relative D-wave explicitly to study them with the QCD sum rules [8]. In Ref. [8], Chen et al study the mass spectrum of the D-wave heavy baryon states with the QCD sum rules combined with the heavy quark effective theory in a systematic way. In this article, we study the Λ c (2860), Λ c (2880), Ξ c (3055) and Ξ c (3080) as the D-wave heavy baryon states with the full QCD sum rules by introducing the relative D-wave explicitly in constructing the interpolating currents, which differ from the currents constructed in Ref. [8].
The article is arranged as follows: we derive the QCD sum rules for the masses and pole residues of the D-wave 3 2 + and 5 2 + charmed baryon states in Sect.2; in Sect.3, we present the numerical results and discussions; and Sect.4 is reserved for our conclusion.
We usually construct the heavy baryon states according to the light-diquark-heavyquark model. In the diquark-quark models, the angular momentum between the two light quarks is denoted by L ρ , while the angular momentum between the light diquark and the heavy quark is denoted by L λ . If the two light quarks in the diquark are in relative S-wave or L ρ = 0, then the baryons with the J P = 0 + and 1 + diquarks (the ground state diquarks) are called Λ-type and Σ-type baryons, respectively [22]. We can denote the Cγ 5 and Cγ µ diquarks as J P d = 0 + d and 1 + d , respectively, the relative P-wave and D-wave as J P ρ/λ = L P ρ/λ = 1 − ρ/λ and 2 + ρ/λ , respectively, the c-quark as J P c = 1 2 + c , then we construct the D-wave baryon states according to the routines, It is difficult or impossible to construct currents to interpolate all the D-wave baryon states with J P = 1  [5,6,7,8,9,10]. We can choose either the partial derivative ∂ µ or the covariant derivative D µ to construct the interpolating currents. The currents with the covariant derivative D µ are gauge invariant, but blur the physical interpretation of the the angular momentum. The currents with the partial derivative ∂ µ are not gauge invariant, but manifests the physical interpretation of the ∂ µ being the angular momentum. In Ref. [23], we study the masses and decay constants of the heavy tensor mesons D * 2 (2460), D * s2 (2573), B * 2 (5747) and B * s2 (5840) with the QCD sum rules. In calculations, we observe that the predictions based on the currents with the partial derivative and covariant derivative differ from each other about 1%, if the same parameters are chosen. If we refit the Borel parameters and threshold parameters, the differences about 1% can be reduced remarkably, so the currents with the partial derivative work well. In this article, we choose the partial derivative ∂ µ to construct the interpolating currents. Furthermore, from the Table 1 in Section 3, we can see that the dominant contributions come from the perturbative terms, so neglecting the contributions originate from the gluons in the covariant derivative D µ cannot change the conclusion.
For L ρ = 1 and L λ = 0, the light diquark state with J P = 1 − can be written as then we introduce an additional P-wave between the two quarks q and q ′ , and obtain the light diquark state with L ρ = 2, L λ = 0 and J P = 2 + , In the heavy quark limit, the c-quark is static, the ↔ ∂ µ is reduced to ← ∂ µ when operating on the c-quark field. For L ρ = 0 and L λ = 2, the light diquark state with J P = 2 + can be written as For L ρ = 1 and L λ = 1, the light diquark state with J P = 2 + can be written as We symmetrize the Lorentz indexes µ and ν, and obtain the light diquark state with L ρ = 1 and L λ = 1 in a more simple form, The light diquark states with J P = 2 + then combine with the c-quark to form J P = 3 2 + or 5 2 + baryon states, see Eqs.(9-10).
The interpolating currents can be classified by The currents J/η α (0) and J/η αβ (0) couple potentially to the 3 2 ± and 5 2 ± charmed baryon states B ± 3 2 and B ± 5 2 , respectively [17,24,25], which are supposed to be the excited Λ c or Ξ c states, where the λ ± 3 2 and λ ± 5 2 are the pole residues or the current-baryon coupling constants, the At the hadron side, we insert a complete set of intermediate charmed baryon states with the same quantum numbers as the current operators J/η α (x), iγ 5 J/η α (x), J/η αβ (x) and iγ 5 J/η αβ (x) into the correlation functions Π αβ (p) and Π αβµν (p) to obtain the hadronic representation [26,27]. We isolate the pole terms of the lowest charmed baryon states with positive parity and negative parity, and obtain the results: where g µν = g µν − pµpν p 2 . The currents J/η α (0), J/η αβ (0) also have non-vanishing couplings with the spin J = 1 2 and J = 1 2 , 3 2 charmed baryon states, respectively, we choose the tensor structures g αβ and g µα g νβ + g µβ g να for analysis, the baryon states with the spin 1 2 and 3 2 have no contaminations [25].
In calculations, we have used two summations over the polarizations s in the spinors and p 2 = M 2 ± on mass-shell. We obtain the hadronic spectral densities at the hadron side through dispersion relation, where j = 3 2 , 5 2 , the subscript H denotes the hadron side, then we introduce the exponential function exp − s T 2 to depress the continuum state contributions to obtain the QCD sum rules at the hadron side, where the s 0 are the continuum thresholds and the T 2 are the Borel parameters [25]. From Eq. (25) The contributions of the 3 2 ± and 5 2 ± charmed baryon states can be separated unambiguously. In this article, we will focus on the Λ c and Ξ c states with positive parity. At the QCD side, we calculate the light quark parts of the correlation functions Π αβ (p) and Π αβµν (p) with the full light quark propagators S ij (x) in the coordinate space and take the full c-quark propagator C ij (x) in the momentum space, q = u, d, s, t n = λ n 2 , the λ n is the Gell-Mann matrix [27]. In Eq. (27), we retain the term q j σ µν q i originates from the Fierz re-arrangement of the q iqj to absorb the gluons emitted from the other quark lines to form q j g s G a αβ t a mn σ µν q i to extract the mixed condensate qg s σGq . Then we compute the integrals both in the coordinate space and momentum space to obtain the correlation functions Π j (p 2 ), and obtain the QCD spectral densities through dispersion relation, where j = 3 2 , 5 2 , the explicit expressions of the QCD spectral densities ρ 1 j,QCD (s) and ρ 0 j,QCD (s) are shown in the Appendix. In this article, we carry out the operator product expansion up to the vacuum condensates of dimension 10 and take into account the condensates, which are vacuum expectations of the operators of order O(α k s ) with k ≤ 1, in a consistent way. In calculations, we observe that only the vacuum condensates qq , ss , αsGG π , qg s σGq , sg s σGs , qg s σGq 2 , qg s σGq sg s σGs have contributions. Once the analytical expressions of the QCD spectral densities ρ 1 j,QCD (s) and ρ 0 j,QCD (s) are obtained, we take the quark-hadron duality below the continuum thresholds s 0 and introduce the exponential function exp − s T 2 to depress the continuum state contributions to obtain the QCD sum rules: We derive Eq.(31) with respect to 1 T 2 , then eliminate the pole residues λ + j and obtain the QCD sum rules for the masses of the charmed baryon states with J P = 3 2 + and 5 2 + , 3 Numerical results and discussions  [30]. Furthermore, we set m u = m d = 0 due to the small current quark masses. We take into account the energy-scale dependence of the input parameters from the renormalization group equation, , ss (µ) = ss (Q) α s (Q) α s (µ) 4 9 , qg s σGq (µ) = qg s σGq (Q) α s (Q) α s (µ) 2 27 , sg s σGs (µ) = sg s σGs (Q) α s (Q) α s (µ) 2 27 , , Λ = 213 MeV, 296 MeV and 339 MeV for the flavors n f = 5, 4 and 3, respectively [30], and evolve all the input parameters to the optimal energy scales µ to extract the masses of the charmed baryon states.
In the heavy quark limit, the Q-quark serves as a static well potential and combines with a light quark q to form a heavy diquark in color antitriplet, or combines with a light antiquarkq to form a heavy meson in color singlet (meson-like state in color octet), or combines with a light diquark ε ijk q i q ′j to form a heavy baryon in color singlet (triquark in color triplet), where the i, j, k, l, m are color indexes, the λ a is Gell-Mann matrix. The Q-quark serves as another static well potential and has similar property. Then The three-quark systems qq ′ Q, four-quark systems qq ′ QQ, five-quark systems qq ′ q ′′ QQ are characterized by the effective heavy quark masses M Q (or constituent quark masses) (or bound energy not as robust), where the B denotes the conventional baryon states, the X, Y , Z denote the hidden-charm (bottom) tetraquark quark states, molecular states or moleculelike states, the P denotes the (molecular) pentaquark states. It is natural to take the energy scales of the QCD spectral densities to be µ = V .
In this article, we use the empirical formula µ = M 2 B − M 2 c to determine the ideal energy scales of the QCD spectral densities. If we take the updated value of the effective cquark mass M c = 1.82 GeV [36], then the optimal energy scales are µ = 2.2 GeV, 2.  Table 1. From the figure, we can see that the predicted masses depend on the energy scale µ slightly, the acceptable ranges of the energy scale are rather large, is not difficult to satisfy in the present case. On the other hand, the pole residues increase monotonously and quickly with increase of the energy scale, it is important to choose the ideal energy scales.
We search for the ideal Borel parameters T 2 and continuum threshold parameters s 0 according to the four criteria: 1 · Pole dominance at the hadron side, the pole contributions are about (50 − 80)%; 2 · Convergence of the operator product expansion, the dominant contributions come from the perturbative terms; 3 · Appearance of the Borel platforms, the uncertainties δM/M originate from the Borel parameters are about (2 − 5)% in the Borel windows; 4 · Satisfying the energy scale formula. by try and error, and present the optimal energy scales µ, ideal Borel parameters T 2 , continuum threshold parameters s 0 , pole contributions and perturbative contributions in Table 1. In the QCD sum rules for the baryon states, the predicted masses usually increase monotonously but slowly with increase of the Borel parameters [37], there cannot appear platforms as flat as that appear in the case of the conventional mesons and tetraquark states [29,31]. In this article, we observe that the predicted masses also increase with increase of the Borel parameters, so we constrain the uncertainties δM/M originate from the Borel parameters will not exceed 5% in the Borel windows.
From Table 1, we can see that the pole dominance at the hadron side is well satisfied and the operator product expansion is well convergent, the criteria 1 and 2 (the basic criteria of the QCD sum rules) are satisfied, so we expect to make reliable predictions. In Ref. [8], Chen et al study the D-wave heavy baryon states with the QCD sum rules combined with the heavy quark effective theory, and extract the masses with the pole contributions ≤ 20%, while in the present work, the pole contributions are about (50 − 80)%. The QCD spectral densities have the terms m s qq , m s ss , m s qg s σGq , m s sg s σGs , which are greatly depressed by the small s-quark mass and are of minor importance, the dominant contributions come from the perturbative terms.    Table 1, where the A, B, C and D correspond to the charmed baryon states Ξ c 0, 2; 5 2 , Ξ c 0, 2; 3 2 , Λ c 0, 2; 5 2 and Λ c 0, 2; 3 2 , respectively.      We take into account all uncertainties of the input parameters, and obtain the masses and pole residues of the D-wave charmed baryon states Λ c and Ξ c , which are shown explicitly in Figs.2-7 and Table 2. In Figs.2-7, we plot the masses and pole residues with variations of the Borel parameters at much larger intervals than the Borel windows shown in Table 1. In the Borel windows, the uncertainties δM/M originate from the Borel parameters are very small, about (2 − 5)%, the Borel platforms exist approximately.
Furthermore, the energy scale formula µ = M 2 B − M 2 c is well satisfied. The criteria 3 and 4 are satisfied, now the four criteria are all satisfied.
In Fig.2 and Table 2, we also present the experimental values [1,30] and predictions from the QCD sum rules combined with the heavy quark effective theory [8]. The present predictions are consistent with the experimental values [1,30] and other QCD sum rules calculations [8], and support assigning the Λ c (2860), Λ c (2880), Ξ c (3055) and Ξ c (3080) to be the D-wave charmed baryon states with the quantum numbers (L ρ , L λ ) = (0, 2) and J P = 3

Conclusion
In this article, we tentatively assign the Λ c (2860), Λ c (2880), Ξ c (3055) and Ξ c (3080) to be the D-wave charmed baryon states with J P = 3 2 + , 5 2 + , 3 2 + and 5 2 + , respectively, and study their masses and pole residues with the QCD sum rules in a systematic way by constructing three-types interpolating currents with the quantum numbers (L ρ , L λ ) = (0, 2), (2, 0) and (1, 1), respectively. As the currents couple potentially to both the positive parity and negative parity baryon states, we separate the contributions of the 3 2 ± and 5 2 ± charmed baryon states unambiguously, and the QCD sum rules do not suffer from the contaminations of the charmed baryon states with negative parity. We carry out the operator product expansion up to the vacuum condensates of dimension 10 in a consistent way, and use the empirical energy scale formula to determine the optimal energy scales of the QCD spectral densities to extract the hadron masses. The present predictions support assigning the Λ c (2860), Λ c (2880), Ξ c (3055) and Ξ c (3080) to be the D-wave baryon states with the quantum numbers (L ρ , L λ ) = (0, 2) and J P = 3 2 + , 5 2 + , 3 2 + and 5 2 + , respectively.
The predictions for the masses of the (L ρ , L λ ) = (2, 0) and (1, 1) D-wave Λ c and Ξ c states can be confronted to the experimental data in the future.