Analysis of the $\Lambda_c(2625)$ and $\Xi_c(2815)$ with QCD sum rules

In this article, we study the charmed baryon states $\Lambda_c(2625)$ and $\Xi_c(2815)$ with the spin-parity ${3\over 2}^-$ by subtracting the contributions from the corresponding charmed baryon states with the spin-parity ${3\over 2}^+$ using the QCD sum rules, and suggest a formula $ \mu=\sqrt{M_{\Lambda_c/\Xi_c}^2-{\mathbb{M}}_c^2}$ with the effective mass ${\mathbb{M}}_c=1.8\,\rm{GeV}$ to determine the energy scales of the QCD spectral densities, and make reasonable predictions for the masses and pole residues. The numerical results indicate that the $\Lambda_c(2625)$ and $\Xi_c(2815)$ have at least two remarkable under-structures.

possible assignments of the Λ c (2765) and Ξ c (2815) to be the Λ-type baryon states. In previous work, we take the Ξ c (2815) to be the Σ-type baryon state [6].
We usually resort to the diquark-quark model to construct the baryon currents. Without introducing additional P-wave, the ground state quarks have the spin-parity 1 2 + , two quarks can form a scalar diquark or an axialvector diquark with the spin-parity 0 + or 1 + , the diquark then combines with a third quark to form a positive parity baryon, for example, the Λ-type currents η Λ , the Σ-type currents η Σ and η Σ µ , which have positive parity, where the a, b and c are color indexes. Multiplying iγ 5 to the currents η Λ , η Σ and η Σ µ changes their parity, the currents iγ 5 η Λ , iγ 5 η Σ and iγ 5 η Σ µ couple potentially to the negative parity heavy baryons. In Refs. [4,6,8], we take the currents without introducing partial (or P-wave) to study the negative parity heavy, doubly-heavy and triply-heavy baryon states, and obtain satisfactory results.
If there exists a relative P-wave (which can be denoted as 1 − ) between the diquark and the third quark or between the two quarks in the diquark, we have the following two routines to construct the negative parity baryons, Recently, Chen et al introduce the relative P-wave explicitly, and study the negative parity charmed baryon states with the QCD sum rules combined with the heavy quark effective theory [19]. The baryons have complicated structures, more than one currents can couple potentially to a special baryon. In this article, we construct the interplaiting currents by introducing the relative P-wave explicitly, and study the negative parity charmed baryon states Λ c (2765) and Ξ c (2815) with the full QCD sum rules. In Ref. [20], Jido, Kodama and Oka suggest a novel method to separate the contribution of the negative-parity baryon N (1535) from that of the positive-parity baryon p, because the interpolating currents maybe couple potentially to both the negative-and positiveparity baryon states [21], which impairs the predictive power. Again, we follow this novel method to study the negative-parity baryon states Λ c (2765) and Ξ c (2815) by separating the contributions of the positive-parity baryon states explicitly. In the heavy quark limit, Bagan et al separate the contributions of the positive-and negative-parity heavy baryon states unambiguously [22].
The article is arranged as follows: we derive the QCD sum rules for the masses and pole residues of the Λ c (2765) and Ξ c (2815) in Sect.2; in Sect.3, we present the numerical results and discussions; and Sect.4 is reserved for our conclusions.
2 QCD sum rules for the Λ c (2625) and Ξ c (2815) In the following, we write down the two-point correlation functions Π αβ (p) in the QCD sum rules, where J α (x) = J 1 α (x), J 2 α (x), g µν = g µν − 1 4 γ µ γ ν , the i, j, k are color indexes, the C is the charge conjugation matrix. The currents J α (x) have negative parity.
The currents J α (0) couple potentially to the 3 2 − charmed baryon states B − , the spinor U − α (p, s) satisfies the Rarita-Schwinger equation ( p − M − )U − α (p) = 0 and the relations γ α U − α (p, s) = 0, p α U − α (p, s) = 0. The currents also satisfy the relation γ α J α (x) = 0, which is consistent with Eq.(9). On the other hand, the currents also couple to the positive parity baryon states B + , the spinors U ± α (p, s) have analogous properties and λ + = 0. We insert a complete set of intermediate baryon states with the same quantum numbers as the current operators J α (x) and iγ 5 J α (x) into the correlation functions Π αβ (p) to obtain the hadronic representation [23,24]. After isolating the pole terms of the lowest states of the charmed baryons, we obtain the following results: where the M ± are the masses of the lowest states with the parity ± respectively, and the λ ± are the corresponding pole residues (or couplings). In this article, we choose the tensor structure g µν for analysis. If we take p = 0, then where the A(p 0 ) + B(p 0 ) and A(p 0 ) − B(p 0 ) contain the contributions from the negative-and positive-parity baryon states, respectively [20]. We calculate the light quark parts of the correlation functions Π αβ (p) in the coordinate space and use the momentum space expression for the c-quark propagator, and t n = λ n 2 , the λ n is the Gell-Mann matrix [24], then compute the integrals both in the coordinate and momentum spaces, and obtain the correlation functions Π αβ (p) therefore the QCD spectral densities through dispersion relation, the explicit expression are give in the appendix. In Eq. (14), we retain the term s j σ µν s i originate from the Fierz rearrangement of the s isj to absorb the gluons emitted from the other quark lines to form s j g s G a αβ t a mn σ µν s i so as to extract the mixed condensate sg s σGs . Finally we introduce the weight functions exp − T 2 , and obtain the following QCD sum rules, where the s 0 are the continuum threshold parameters and the T 2 are the Borel parameters. The QCD spectral densities ρ A (p 0 ) and ρ B (p 0 ) are given explicitly in the Appendix.

Numerical results and discussions
The  In the article, we take the M S masses m c (m c ) = (1.275 ± 0.025) GeV and m s (µ = 2 GeV) = (0.095 ± 0.005) GeV from the particle data group [1], and take into account the energy-scale dependence of the M S masses from the renormalization group equation, In Refs. [26,27], we study the acceptable energy scales of the QCD spectral densities for the hidden charmed (bottom) tetraquark states and molecular (and molecule-like) states in the QCD sum rules in details for the first time, and suggest a formula µ = M 2 X/Y /Z − (2M Q ) 2 to determine the energy scales, where the X, Y , Z denote the fourquark systems, and the M Q is the effective heavy quark mass. We can describe the system QQq ′q by a double-well potential with two light quarks q ′q lying in the two wells respectively. In the heavy quark limit, the Q-quark serves as a static well potential and bounds the light quark q ′ to form a diquark in the color antitriplet channel or binds the light antiquarkq to form a meson (or meson-like) in the color singlet (or octet) channel. Then , the effective mass M c = 1.8 GeV is the optimal value for the diquark-antidiquark type tetraquark states [26]. In this article, we use the diquark-quark model to construct the interpolating currents, and take the analogous formula, with M c = 1.8 GeV to determine the energy scales of the QCD spectral densities. Then we obtain the values µ = 1.9 GeV and µ = 2.2 GeV for the Λ c (2625) and Ξ c (2815), respectively.
In the conventional QCD sum rules [23,24], we usually use two criteria (pole dominance and convergence of the operator product expansion) to choose the Borel parameters T 2 and continuum threshold parameters s 0 . In Refs. [3,4,5,6,7,8], we study the 2) MeV from the particle data group [1]. In this article, we take the values √ s 0 ≈ M gr + (0.6 − 0.8) GeV, the two criteria of the QCD sum rules are also satisfied, see Table 1. In the table, we present the values of the Borel parameters T 2 , continuum threshold parameters s 0 , the pole contributions and the perturbative contributions explicitly. Taking into account all uncertainties of the revelent parameters, we can obtain the values of the masses and pole residues of the Λ c (2625) and Ξ c (2815), which are shown in Figs.1-2 and Table 2. From the table, we can see that the values of the masses M Λc(2625) and M Ξc(2815) can reproduce the experimental data for all the currents J 1 α and J 2 α . The angular momentums of the light diquarks are 1 and 2 in the currents J 1 α and J 2 α , respectively, they all couple potentially to the baryons Λ c (2625) and Ξ c (2815), so the Λ c (2625) and Ξ c (2815) have at least two remarkable under-structures.
In previous work [6], we take the Ξ c (2815) to be the Σ-type baryon state, and study the Ξ c (2815) with the interplaiting current J Ξ α (x) = ǫ ijk q i (x)Cγ α s j (x)c k (x) or J Ξ α (x) = ǫ ijk q i (x)Cγ β s j (x) g αβ c k (x), and obtain the value M Ξc(2815) = (2.86 ± 0.17) GeV, which is also consistent with the experimental data. If the prediction is robust, now the Ξ c (2815)

Conclusion
In this article, we study the charmed baryon states Λ c (2625) and Ξ c (2815) with the spinparity 3 2 − by subtracting the contributions from the corresponding charmed baryon states with the spin-parity 3 2 + using the QCD sum rules, and suggest an energy scale formula to determine the energy scales of the QCD spectral densities, and make reasonable predictions for their masses and pole residues. The numerical results indicate that the Λ c (2625) and Ξ c (2815) at least have two remarkable under-structures. We can take pole residues as basic input parameters and study the revelent hadronic processes with the QCD sum rules in further investigations of the under-structures of the Λ c (2625) and Ξ c (2815).