1 Introduction

## Abstract

In this article, we tentatively assign the , , and to be the D-wave baryon states with the spin-parity , , and , 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 , and , respectively. The present predictions favor assigning the , , and to be the D-wave baryon states with the quantum numbers and , , and , respectively. While the predictions for the masses of the and D-wave and states can be confronted to the experimental data in the future.

Analysis of the , , and as D-wave baryon states with QCD sum rules

[2mm] Zhi-Gang Wang 1

Department of Physics, North China Electric Power University, Baoding 071003, P. R. China

PACS number: 14.20.Lq

Key words: Charmed baryon states, QCD sum rules

## 1 Introduction

Recently, the LHCb collaboration studied the mass spectrum of excited states that decay into , and observed a new resonance near threshold [1]. The measured masses, widths and quantum numbers of the , and states are

 Λc(2860):M=2856.1+2.0−1.7±0.5+1.1−5.6 MeV,Γ=67.6+10.1−8.1±1.4+5.9−20.0 MeV,JP=32+, Λc(2880):M=2881.75±0.29±0.07+0.14−0.20 MeV,Γ=5.43+0.77−0.71±0.29+0.75−0.00 % MeV,JP=52+, Λc(2940):M=2944.8+3.5−2.5±0.4+0.1−4.6 MeV,Γ=27.7+8.2−6.0±0.9+5.2−10.4 MeV,JP=32−, (1)

but other assignments with the spins to are not excluded for the [1]. The was first observed by the CLEO collaboration in the channel [2], confirmed by the BaBar collaboration in the channel [3] and the Belle collaboration in the channels [4]. The available experimental analysis indicates that the has the spin-parity . The theoretical predictions for the masses of the D-wave baryon states with and are about [5, 6, 7, 8, 9, 10]. The and can be assigned to be the D-wave charmed baryon states.

Their strange cousins and were observed in the channel by the Belle collaboration [11] and in the channels by the BaBar collaboration [12]. In 2016, the and were first observed by the Belle collaboration in the and channels, respectively [13], the measured masses and widths were

 Ξc(3055)+:M=3055.8±0.4±0.2 MeV,Γ=7.8±1.2±1.5 MeV, Ξc(3055)0:M=3059.0±0.5±0.6 MeV,Γ=6.4±2.1±1.1 MeV, (2)

furthermore, the Belle collaboration observed the first evidence for the with the estimated mass and width . The theoretical predictions of the masses of the D-wave baryon states with and are about [5, 6, 7, 8, 9, 10], the , and can be assigned to be the D-wave charmed baryon states.

In this article, we tentatively assign the , , and to be the D-wave charmed baryon states with the spin-parity , , and , 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, doubly-heavy 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 , , and 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 and charmed baryon states in Sect.2; in Sect.3, we present the numerical results and discussions; and Sect.4 is reserved for our conclusion.

## 2 QCD sum rules for the D-wave 32+ and 52+ charmed baryon states

Firstly, we write down the two-point correlation functions and in the QCD sum rules,

 Παβ(p) = i∫d4xeip⋅x⟨0|T{J/ηα(x)¯J/¯ηβ(0)}|0⟩, Παβμν(p) = i∫d4xeip⋅x⟨0|T{J/ηαβ(x)¯J/¯ημν(0)}|0⟩, (3)

where , ,

 J1α(x) = εijk[∂μ∂νqTi(x)Cγ5sj(x)+∂μqTi(x)Cγ5∂νsj(x)+∂νqTi(x)Cγ5∂μsj(x) +qTi(x)Cγ5∂μ∂νsj(x)]Γμναck(x), J2α(x) = εijk[∂μ∂νqTi(x)Cγ5sj(x)−∂μqTi(x)Cγ5∂νsj(x)−∂νqTi(x)Cγ5∂μsj(x) +qTi(x)Cγ5∂μ∂νsj(x)]Γμναck(x), J3α(x) = εijk[∂μ∂νqTi(x)Cγ5sj(x)−qTi(x)Cγ5∂μ∂νsj(x)]Γμναck(x), (4)
 η1α(x) = εijk[∂μ∂νqTi(x)Cγ5q′j(x)+∂μqTi(x)Cγ5∂νq′j(x)+∂νqTi(x)Cγ5∂μq′j(x) +qTi(x)Cγ5∂μ∂νq′j(x)]Γμναck(x), η2α(x) = εijk[∂μ∂νqTi(x)Cγ5q′j(x)−∂μqTi(x)Cγ5∂νq′j(x)−∂νqTi(x)Cγ5∂μq′j(x) +qTi(x)Cγ5∂μ∂νq′j(x)]Γμναck(x), η3α(x) = εijk[∂μ∂νqTi(x)Cγ5q′j(x)−qTi(x)Cγ5∂μ∂νq′j(x)]Γμναck(x), (5)
 J1αβ(x) = εijk[∂μ∂νqTi(x)Cγ5sj(x)+∂μqTi(x)Cγ5∂νsj(x)+∂νqTi(x)Cγ5∂μsj(x) +qTi(x)Cγ5∂μ∂νsj(x)]Γμναβck(x), J2αβ(x) = εijk[∂μ∂νqTi(x)Cγ5sj(x)−∂μqTi(x)Cγ5∂νsj(x)−∂νqTi(x)Cγ5∂μsj(x) +qTi(x)Cγ5∂μ∂νsj(x)]Γμναβck(x), J3αβ(x) = εijk[∂μ∂νqTi(x)Cγ5sj(x)−qTi(x)Cγ5∂μ∂νsj(x)]Γμναβck(x), (6)
 η1αβ(x) = εijk[∂μ∂νqTi(x)Cγ5q′j(x)+∂μqTi(x)Cγ5∂νq′j(x)+∂νqTi(x)Cγ5∂μq′j(x) +qTi(x)Cγ5∂μ∂νq′j(x)]Γμναβck(x), η2αβ(x) = εijk[∂μ∂νqTi(x)Cγ5q′j(x)−∂μqTi(x)Cγ5∂νq′j(x)−∂νqTi(x)Cγ5∂μq′j(x) +qTi(x)Cγ5∂μ∂νq′j(x)]Γμναβck(x), η3αβ(x) = εijk[∂μ∂νqTi(x)Cγ5q′j(x)−qTi(x)Cγ5∂μ∂νq′j(x)]Γμναβck(x), (7)

with

 Γμνα = (gμαgνβ+gμβgνα−12gμνgαβ)γβγ5, Γμναβ = gμαgνβ+gμβgνα−16gμνgαβ−14gμαγνγβ−14gμβγνγα−14gναγμγβ−14gνβγμγα (8) +124γμγαγνγβ+124γμγβγνγα+124γνγαγμγβ+124γνγβγμγα,

, the , , are color indices, the is the charge conjugation matrix. The currents satisfy the relations , , , where . We choose the currents and to interpolate the and charmed baryon states, respectively. In this article, we tentatively assign the , , and to be the D-wave charmed baryon states with the spin-parity , , and , respectively, the currents , , and may couple potentially to the , , and , respectively.

Now we take a short digression to illustrate how to construct the currents. The attractive interaction of one-gluon exchange favors formation of the diquarks in color antitriplet [20]. The color antitriplet diquarks have five structures in Dirac spinor space, where , , , and for the scalar, pseudoscalar, vector, axialvector and tensor diquarks, respectively. The structures and are symmetric, while the structures , and are antisymmetric. The calculations based on the QCD sum rules indicate that the favored configurations are the and diquark states, while the most favored configurations are the diquark states [21].

We usually construct the heavy baryon states according to the light-diquark-heavy-quark model. In the diquark-quark models, the angular momentum between the two light quarks is denoted by , while the angular momentum between the light diquark and the heavy quark is denoted by . If the two light quarks in the diquark are in relative S-wave or , then the baryons with the and diquarks (the ground state diquarks) are called -type and -type baryons, respectively [22]. We can denote the and diquarks as and , respectively, the relative P-wave and D-wave as and , respectively, the -quark as , then we construct the D-wave baryon states according to the routines,

 0+d⊗2+ρ/λ⊗12+c = 2+–––⊗12+c=32+⊕52+–––––––––––, (9) 0+d⊗1−ρ⊗1−λ⊗12+c = [0+⊕1+⊕2+–––]⊗12+c=12+⊕[12+⊕32+]⊕[32+⊕52+]–––––––––––––––, (10) 1+d⊗2+ρ/λ⊗12+c = [1+⊕2+⊕3+]⊗12+c=[12+⊕32+]⊕[32+⊕52+]⊕[52+⊕72+], 1+d⊗1−ρ⊗1−λ⊗12+c = [0−⊕1−⊕2−]⊗1−λ⊗12+c (11) = [1+⊕[0+⊕1+⊕2+]⊕[1+⊕2+⊕3+]]⊗12+c = [12+⊕32+]⊕12+⊕[12+⊕32+]⊕[32+⊕52+]⊕[12+⊕32+] ⊕[32+⊕52+]⊕[52+⊕72+].

It is difficult or impossible to construct currents to interpolate all the D-wave baryon states with , , and in a systematic way. In this article, we study the underlined D-wave baryon states with and in details based on the most favored configurations [21]. Experimentally, the measured quantum numbers of the and are and respectively from the LHCb collaboration [1], while the masses of the , and are consistent with the theoretical predictions of the D-wave baryon states with and [5, 6, 7, 8, 9, 10].

We can choose either the partial derivative or the covariant derivative to construct the interpolating currents. The currents with the covariant derivative are gauge invariant, but blur the physical interpretation of the being the angular momentum. The currents with the partial derivative are not gauge invariant, but manifests the physical interpretation of the being the angular momentum. In Ref.[23], we study the masses and decay constants of the heavy tensor mesons , , and 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 , if the same parameters are chosen. If we refit the Borel parameters and threshold parameters, the differences about can be reduced remarkably, so the currents with the partial derivative work well. In this article, we choose the partial derivative 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 cannot change the conclusion.

For and , the light diquark state with can be written as

 εijk[∂νqTi(x)Cγ5q′j(x)−qTi(x)Cγ5∂νq′j(x)], (12)

then we introduce an additional P-wave between the two quarks and , and obtain the light diquark state with , and ,

 εijk{[∂μ∂νqTi(x)Cγ5q′j(x)−∂νqTi(x)Cγ5∂μq′j(x)]−[∂μqTi(x)Cγ5∂νq′j(x) −qTi(x)Cγ5∂μ∂νq′j(x)]}. (13)

In the heavy quark limit, the -quark is static, the is reduced to when operating on the -quark field. For and , the light diquark state with can be written as

 ∂μ∂ν[εijkqTi(x)Cγ5q′j(x)] = εijk[∂μ∂νqTi(x)Cγ5q′j(x)+∂μqTi(x)Cγ5∂νq′j(x) (14) +∂νqTi(x)Cγ5∂μq′j(x)+qTi(x)Cγ5∂μ∂νq′j(x)].

For and , the light diquark state with can be written as

 ∂μεijk[∂νqTi(x)Cγ5q′j(x)−qTi(x)Cγ5∂νq′j(x)] =εijk[∂μ∂νqTi(x)Cγ5q′j(x)+∂νqTi(x)Cγ5∂μq′j(x)−∂μqTi(x)Cγ5∂νq′j(x) −qTi(x)Cγ5∂μ∂νq′j(x)]. (15)

We symmetrize the Lorentz indexes and , and obtain the light diquark state with and in a more simple form,

 εijk[∂μ∂νqTi(x)Cγ5q′j(x)−qTi(x)Cγ5∂μ∂νq′j(x)]. (16)

The light diquark states with then combine with the -quark to form or baryon states, see Eqs.(9-10).

The interpolating currents can be classified by

 (Lρ,Lλ)=(0,2) for J/η1α(x),J/η1αβ(x), (Lρ,Lλ)=(2,0) for J/η2α(x),J/η2αβ(x), (Lρ,Lλ)=(1,1) for J/η3α(x),J/η3αβ(x). (17)

The currents and couple potentially to the and charmed baryon states and , respectively [17, 24, 25], which are supposed to be the excited or states,

 ⟨0|J/ηα(0)|B+32(p)⟩ = λ+32U+α(p,s), ⟨0|J/ηαβ(0)|B+52(p)⟩ = λ+52U+αβ(p,s), (18) ⟨0|J/ηα(0)|B−32(p)⟩ = λ−32iγ5U−α(p,s), ⟨0|J/ηαβ(0)|B−52(p)⟩ = λ−52iγ5U−αβ(p,s), (19)

where the and are the pole residues or the current-baryon coupling constants, the spinors and satisfy the Rarita-Schwinger equations and , and the relations , , , , , which are consistent with relations