Universal behavior of mass gaps existing in the single heavy baryon family
UUniversal behavior of mass gaps existing in the single heavy baryon family
Bing Chen , , ∗ Si-Qiang Luo , , † and Xiang Liu , , , ‡§ School of Electrical and Electronic Engineering, Anhui Science and Technology University, Fengyang 233100, China School of Physical Science and Technology, Lanzhou University, Lanzhou 730000, China Lanzhou Center for Theoretical Physics, Lanzhou University, Lanzhou, 730000, China Research Center for Hadron and CSR Physics, Lanzhou University & Institute of Modern Physics of CAS, Lanzhou 730000, China Key Laboratory of Theoretical Physics of Gansu Province, Lanzhou University, Lanzhou 730000, China (Dated: January 28, 2021)In this work, the mass gaps existing in the single heavy flavor baryon family are analyzed, which showsome universal behavior. Under the framework of a constituent quark model, we quantitatively explain whysuch interesting phenomenon for the single heavy flavor baryon happens. Due to this universal behavior ofthe discussed mass gaps, we may have three implications including the prediction of the masses of excited Ξ b baryons which are still missing in the experiment. I. INTRODUCTION
As an e ff ective approach to solve the nonperturbative prob-lem of strong interaction, studying hadron spectroscopy hasbecome an active research issue with the abundant observa-tions of new hadronic states in experiment (see Refs. [1–8]for the recent progress). Among di ff erent research aspects ofthe study of hadron spectroscopy, mass spectrum analysis is acrucial way to decode the property of hadronic states.Until now, there have been di ff erent methods to performmass spectrum analysis. If seriously depicting the mass spec-trum, we have various versions of potential model [9–16],the flux tube model [17, 18], the quantum chromodynam-ics (QCD) sum rule [19, 20], the lattice QCD [21–23], andso on. Besides, some semi-quantitative methods were exten-sively applied to the mass spectrum analysis. A typical exam-ple is the Regge trajectory analysis [24–27] which has beenadopted to investigate the mass spectrum of di ff erent kindsof hadrons [28–35]. Recently, Chen gave a mass formula forlight meson and baryon, and discussed its implication [36]. Ofcourse, some mass gap relations existing in the hadron massspectrum have been realized by the theorists, which can be-come the simple but e ff ective approach when scaling massspectrum. For illustrating this point, we review several rep-resentative recent work. As indicated in Ref. [37], the similardynamics describing ω and φ meson families requires the sim-ilarity between ω and φ meson families, where the mass gap of ω (782) and ω (1420) is similar to that of φ (1020) and φ (1680).Adopted this mass gap relation for higher states, the Y (1915)state as the partner of Y (2175) was predicted. In Ref. [38],Lanzhou group predicted the existence of a narrow charmo-nium ψ (4 S ) with the mass around 4.263 GeV, which is esti-mated by the mass gap between Υ (3 S ) and Υ (4 S ), if consid-ering the mass gap relation m ψ (4 S ) − m ψ (3 S ) = m Υ (4 S ) − m Υ (3 S ) .Usually, for charmonium and bottomonium families, there ex-ists a mass relation M + + M + + M + (cid:39) M + (cid:48) [39], where ‡ Corresponding author ∗ Electronic address: [email protected] † Electronic address: [email protected] § Electronic address: [email protected] the spin-parity quantum numbers are applied to distinguish thedi ff erent charmonium masses with the corresponding quan-tum number. In Ref. [40], Chang et al. suggested a newmass relation M + + M + = M + (cid:48) + M + ) for the P -wave B c mesons.In the following, we need to pay attention to the singleheavy flavor baryon. Among this fantastic hadron zoo, thesingle heavy flavor baryon family is being constructed stepby step, which is due to the big progress on the observationsof charm and bottom baryons, where the LHCb Collaborationhas played important role [41–51] in the past years. In Fig. 1,we list these reported single heavy flavor baryon states. Ob-viously, the present situation of single heavy flavor baryonsshows that it is a good opportunity to check whether thereexist some mass gap relations for single heavy flavor baryonfamily, which will be one of tasks of this work. We will an-alyze the mass gaps which are extracted by these measuredmass spectrum of single heavy flavor baryons, by which wemay find some universal behavior of mass gaps for singleheavy flavor baryons.Facing such interesting mass gap phenomenon, we wantto further explain why there exists this universal behavior ofmass gaps, which is involved in the dynamics of single heavyflavor baryon. In this work, we start with a constituent quarkmodel [52], which was successfully applied to depict the massspectrum of single heavy baryon in our previous works [53–56]. Here, two light quarks in single heavy flavor baryon aretreated as a cluster, and then the single heavy flavor baryon issimplified as a quasi-two-body system. Focusing on the uni-versal behavior of mass gaps, we perform a study of mass gapsof single heavy baryon under the framework of constituentquark model, which shows that this universal behavior of massgaps can be well understood.For phenomenological study, mass gap relation is always awelcome approach. Thus, in this work, we continue to applythis universal behavior of mass gaps to predict some highersingle heavy baryon states, which can be tested by the futureexperiments.The paper is organized as follows. After Introduction, weillustrate these mass gap relations by the measured mass spec-trum of single heavy flavor baryon in Sec. II. And we deducethese mass gap relations by a constituent quark model. In Sec.III, its application to predict some higher states in the single a r X i v : . [ h e p - ph ] J a n FIG. 1: All observed charm and bottom baryons [47, 48, 50, 59]. heavy flavor baryon family will be given. Finally, the paperends with a summary.
II. UNIVERSAL BEHAVIOR OF MASS GAPSA. Mass gaps of single heavy baryon
The single heavy baryon system, which contains one heavyquark ( c or b quark) and two light quarks ( u , d , or s quark), oc-cupies a particular place in the whole hadron family [57]. Theheavy quark in a single heavy flavor baryon system provides aquasi-static colour field for two surrounding light quarks [58].Based on the flavor SU(3) symmetry of light quark cluster, i.e.3 f ⊗ f = ¯3 f ⊕ f , the Λ Q and Ξ Q baryon states belong to the¯3 f multiplet, while the Σ Q , Ξ (cid:48) Q , and Ω Q baryon states can begrouped into the 6 f multiplet. Thus, in the following, we willgive the mass gasps of single heavy flavor baryons in ¯3 f and6 f multiplets, and show their universal behaviors.
1. The ¯3 f multiplet With the joint e ff ort of experimentalists and theorists, thefamily of single heavy baryons has been established stepby step. In 2017, a new charm baryon state Λ c (2860) + was discovered by the LHCb Collaboration in the process Λ b → Λ c (2860) + π − → D p π − [43]. Combining withthe previously observed Λ c (2286) + , Λ c (2595) + , Λ c (2625) + , Λ c (2880) + , Ξ c (2470), Ξ c (2790), Ξ c (2815), Ξ c (3055), and Ξ c (3080) states [59], all 1 S , 2 S , 1 P , 1 D states of Λ + c and Ξ , + c baryons have been discovered. In the past years, experi-ment has also made a big progress on searching for the excited Λ b states. In 2012, two narrow P -wave Λ b states, i.e., the Λ b (5912) and Λ b (5920) , were first reported by the LHCb Collaboration in the Λ b π + π − invariant mass spectrum [41],which were confirmed by the further measurements from theCDF, CMS, and LHCb collaborations [50, 60, 61]. Besides,the D -wave states Λ b (6146) and Λ b (6152) have also beenconstructed in the recent years [50, 51, 61]. In last year, the Λ b (6072) , which could be regarded as a good 2 S Λ b candi-date, was found by CMS [61] and LHCb [50]. And then, thelow-lying Λ + c , Ξ , + c , and Λ b baryons including the 1 S , 2 S , 1 P ,and 1 D states have been reported by experiment. When check-ing the masses of these known heavy baryons, the interestinguniversal behavior of mass gaps can be found. For presentingsuch behavior clearly, the masses of all known 1 S , 2 S , 1 P ,1 D Λ + c and Ξ + c baryons and the corresponding mass gaps arecollected in Table I. One notices that the obtained mass gapsfor the discussed Λ + c and Ξ + c states are nearly about 180 ∼ TABLE I: The measured masses of 1 S , 2 S , 1 P , 1 D Λ + c and Ξ + c baryons [59] and the mass gaps of corresponding states (in MeV). nL ( J P ) States Masses Mass gap1 S (1 / + ) Λ c (2286) + /Ξ c (2470) + / P (1 / − ) Λ c (2595) + /Ξ c (2790) + / P (3 / − ) Λ c (2625) + /Ξ c (2815) + / S (1 / + ) Λ c (2765) + /Ξ c (2970) + / D (3 / + ) Λ c (2860) + /Ξ c (3055) + / D (5 / + ) Λ c (2880) + /Ξ c (3080) + / We define the following ratios R = M S − M S ¯ M P − M S , R = ¯ M D − M S ¯ M P − M S . (1)Here, M S and M S refer to the masses of the S -wave groundstate and its first radial excitation, respectively, while ¯ M P and TABLE II: The extracted ratios of R and R (see Eq. (1)) for the Λ Q and Ξ Q baryons. The 2 S candidate of Λ b baryon with the massaround 6.07 GeV, which was found recently by the CMS [61] andLHCb [50] collaborations in the Λ b π + π − mass spectrum, is taken asan input.Ratios Λ + c Ξ c Λ b Ξ b R · · ·R · · · ¯ M D denote the spin average masses of 1 P and 1 D states. Asshown in Table II, the values of R and R are nearly universalfor the Λ + c and Ξ + c baryon systems. If further checking thebottom baryon case, the obtained R and R of Λ b baryons areclose to the corresponding values of the Λ c and Ξ c baryons, which means the universal behavior of R and R . To someextent, this novel phenomenon reflects the similar dynamicsof charm and bottom baryons.
2. The f multiplet Additionally, we extract the mass gaps according to the dataof the measured Σ Q , Ξ (cid:48) Q , and Ω Q baryons. Before dealingwith it, we should briefly introduce the experimental progresson the Σ Q , Ξ (cid:48) Q , and Ω Q baryons. As well known, the mea-sured mass gaps of the ground Σ Q , Ξ (cid:48) Q , and Ω Q states are about125 MeV (see Table III). In the past years, some P -wave Σ Q , Ξ (cid:48) Q , and Ω Q candidates, including the Ξ c (2923) , Ξ c (2939) , Ξ c (2965) , Ω c (3000) , Ω c (3050) , Ω c (3065) , Ω c (3090) , Σ b (6097) , Ξ (cid:48) b (6227) , Ω b (6316) − , Ω b (6330) − , Ω b (6340) − ,and Ω b (6350) − , have also been announced by experiment [44–49, 64, 65]. These observed charm baryons can be categorizedinto the 6 f representation. Among these observed low-lying6 f heavy baryon states, the Ξ c (2939) , Ω c (3065) , Σ b (6097) , Ξ (cid:48) b (6227) , and Ω b (6350) − are suggested to be the J P = / − or 5 / − states [53–55, 62, 66–76]. The Σ c (2800) ++ state,which was discovered previously by the Belle Collaborationin the e + e − collision [77], could also be regarded as a P -wave3 / − or 5 / − state [52, 78]. Under these assignments, weobtain the mass gaps relevant to Σ c (2800) ++ , Ξ c (2939) , and Ω c (3065) , which are about 130 MeV (see Table III for moredetails). Similar value for the mass gaps of the Σ b (6097) − , Ξ (cid:48) b (6227) − , and Ω b (6350) − states can be found. The last col-umn in Table III also reflects the universal behavior of massgaps of single heavy flavor baryons in 6 f multiplet.In the next section, we will explain the universal behaviorsof mass gaps of single heavy flavor baryons. We take the masses of the newly observed Λ b (6072) , Λ b (6146) , and Λ b (6152) states as input, where Λ b (6072) , Λ b (6146) , and Λ b (6152) are treated as the 2 S and 1 D states, respectively. Other assignments arealso allowed for the broad resonance Λ b (6072) [62, 63]. TABLE III: The observed Σ Q , Ξ (cid:48) Q , and Ω Q baryons [48, 59] and thecorresponding mass gaps involved in these states (in MeV). In thelast column, there are two values for each line, where the first valueis the mass gap of the first and the second states listed in the sec-ond column, and the second value denotes the mass di ff erence of thesecond and the third states. Here, Ω − b ( · · · ) denotes the absent 1 S Ω − b (3 / + ) state in the experiment. nL ( J P ) States Mass gap1 S (1 / + ) Σ c (2455) ++ / Ξ (cid:48) c (2570) + / Ω c (2695) / Σ b (5815) + / Ξ (cid:48) b (5935) − / Ω b (6046) − / S (3 / + ) Σ ∗ c (2520) ++ / Ξ ∗ c (2645) + / Ω c (2765) / Σ ∗ b (5835) + / Ξ ∗ b (5955) − / Ω − b ( · · · ) 125.0 / · · · P ( − or − ) Σ c (2800) ++ / Ξ (cid:48) c (2939) / Ω c (3065) / Σ b (6097) − / Ξ (cid:48) b (6227) − / Ω b (6350) − / B. Understanding the universal behavior of mass gaps by aconstituent quark model
Although the universal behavior of mass gaps introduced inSec. II A has been mentioned for many years [66, 79, 80], ithas never been investigated seriously. In the work, we willgive a quantitative study of the mass gap of single heavy fla-vor baryons with the same nL ( J P ) quantum numbers but dif-ferent strangeness under the framework of a non-relativisticconstituent quark model. Here, n and L denote the radial andorbital angular quantum numbers of a baryon state, respec-tively. J P denotes its spin-parity.The obtained 180 ∼
200 MeV mass gaps (see Table I) ofthe ground and excited states of Λ Q and Ξ Q baryons in ¯3 f rep-resentation reflect a universal behavior of these mass gaps. Itimplies that the similarity of mass gaps may be resulted fromthe same dynamics mechanism. Furthermore, the di ff erencesof the excited energy E nL for Λ Q and Ξ Q baryons with same nL quantum number can be roughly ignored. Such conclusionshould also hold for the 6 f heavy baryons. Indeed, the massgaps shown in Sec. II A can be due to the mass di ff erences oflight quark cluster involved in the discussed heavy baryons,which will be proved by a non-relativistic constituent quarkmodel.If treating light quarks as a cluster, the single heavy baryonsystem can be simplified as a quasi-two-body system [52].And then, the Hamiltonian of heavy baryon system reads asˆ H = m Q + m cluster + ˆ p µ + V SI ( r ) + V SD , (2)where the reduced mass µ is defined as µ = m cluster m Q m cluster + m Q . Di ff er-ent types of potentials can be taken for V SI ( r ), which describesthe spin-independent interaction between the light quark clus-ter and the heavy quark. In Sec. II C, we will list four typesof expression of V SI ( r ). By solving the Schr¨odinger equationwith the concrete V SI ( r ), the excited energy E nL can be de-termined. The spin-dependent interactions V SD in Eq. (2)include the hyperfine interaction, the spin-orbit forces, andthe tensor force [52, 81]. Usually, the spin-dependent interac-tions are much weaker than the spin-independent interaction.Here, we may take P -wave charm baryon states Λ c (2595) + and Λ c (2625) + as an example to show it. The mass splittingof Λ c (2595) + and Λ c (2625) + is about 36 MeV, which is oneorder smaller than their excited energies. So, in the practicalcalculation, the spin-dependent interaction can be treated asthe leading-order perturbation contribution.In the present subsection, we first ignore the spin-dependentinteractions, and ascribe the mass gaps between the singlebaryons with the di ff erent strangeness but with same nL ( J P )quantum numbers to the mass di ff erence of light quark clus-ters. In the qusi-two-body picture, the spin average mass of a nL heavy baryon multiplet could be denoted as follows¯ M nL = m Q + m cluster + E nL . (3)Since two light quarks in the cluster are in the ground state,we take the chromomagnetic model [82–85] to parameterzethe mass of light quark cluster, where the mass of the lightquark cluster could be written as m cluster = m + m + A s · s m m (4)with A = π (cid:104) α s ( r ) δ ( r ) (cid:105) . (5)In the following analysis, assuming the coe ffi cient A to be apositive value for simplicity and combing with Eqs. (3) and(4), we have¯ M Ξ Q nL − ¯ M Λ Q nL = m s − m q + A m q (cid:32) m q − m s (cid:33) = δ m + ∆ , ¯ M Ξ (cid:48) Q nL − ¯ M Σ Q nL = m s − m q − A m q (cid:32) m q − m s (cid:33) = δ m − ∆ , ¯ M Ω Q nL − ¯ M Ξ (cid:48) Q nL = m s − m q − A m s (cid:32) m q − m s (cid:33) = δ m − ∆ (cid:48) . (6)Here, m q and m s denote the constituent masses of up / downand strange quarks, respectively, while δ m denotes their massdi ff erence. Under the situations m s > m q and A > δ m , ∆ ,and ∆ (cid:48) defined in Eqs. (6) must be positive. Thus, we obtainthe following relation¯ M Ξ Q nL − ¯ M Λ Q nL > ¯ M Ξ (cid:48) Q nL − ¯ M Σ Q nL , (7)which explain why the mass gaps of the baryons in the ¯3 f mul-tiplet is smaller than the 6 f multiplet well (see the comparisonof the mass gap values shown in Table I and Table III). Here,the measured mass gaps of the involved Λ Q and Ξ Q states areabout 190 MeV (see Table I), while the measured mass gapsof Σ Q and Ξ (cid:48) Q states are around 120 MeV (see Table III).Besides, we can do a further numerical analysis to illustratewhy mass gaps of the discussed Ω Q and Ξ (cid:48) Q states are around120 MeV. One takes the average values of mass gaps of Λ Q and Ξ Q baryon states with the same nL to be 195 MeV, andthe average values of the mass gaps of Σ Q and Ξ (cid:48) Q baryonswith the same nL to be 122 MeV, by which we may fix the values of δ m and ∆ in Eqs. (6), i.e., δ m = . ∆ = . m q =
280 MeV and m s =
420 MeV, the parameters A and ∆ (cid:48) can be fixed as 1.717 × MeV and 12.2 MeV, respectively.Finally, the mass gap between Ξ (cid:48) Q and Ω Q states is estimatedto be ¯ M Ω Q nL − ¯ M Ξ (cid:48) Q nL (cid:39)
128 MeV . (8)This value is consistent with the measured mass gaps of Ω Q and Ξ (cid:48) Q baryons with the same nL ( J P ) quantum numbers, asshown in Table III.In a word, the universal behavior of mass gaps of sin-gle heavy baryons can be well understood by the constituentquark model. C. Further study of the equal mass splitting phenomenoninvolved in the excited ¯3 f baryons Our study already indicates that the nearly equal mass gapof Λ c and Ξ c baryons with the same nL can be reproduced. Inthis subsection, we discuss the case when the spin-dependentinteraction is included for the ¯3 f baryons.Firstly, we present the experimental data as shown in TableIV, where the Λ c (2595) / Ξ c (2790) with J P = / − and the Λ c (2625) / Ξ c (2815) with J P = / − are the 1 P states. Wemay find that the mass splitting of Λ c (2595) and Λ c (2625)is comparable with the splitting of Ξ c (2790) and Ξ c (2815).Similar phenomenon happens for D -wave Λ c and Ξ c baryonlisted in the second line of Table IV. This equal mass splittingphenomenon can also be understood, as shown below. TABLE IV: The measured mass splittings of the 1 P and 1 D statesof Λ c and Ξ c baryons due to the spin-orbit interaction. The corre-sponding spin-parity quantum numbers of these states can be foundin Table I. Λ Q states splitting Ξ Q states splitting Λ c (2595) + / Λ c (2625) + Ξ c (2790) + / Ξ c (2815) + Λ c (2860) + / Λ c (2880) + Ξ c (3055) + / Ξ c (3080) + For these Λ Q and Ξ Q baryons in ¯3 f representation, the ex-pression of the spin-dependent interaction is very simple sincethe involved spin of light quark cluster is zero. Thus, the onlyspin-dependent interaction is relevant to the following spin-orbit coupling V so = α s r m cluster m Q S Q · L . (9)Here, the second and higher order contributions of 1 / m Q areignored since the heavy quark mass in the baryon system ismuch heavier than the mass of the light quark cluster. Thespin-orbital interaction, which is given in Eq. (9), has beenadopted in our previous work [79] and successfully predictedthe mass splitting of D -wave Λ + c baryons.In fact, the equal mass splitting phenomenon shown in Ta-ble IV requires the same value of ξ Q = (cid:104) α s r m cluster m Q (cid:105) in Eq. (9)for the excited Λ Q and Ξ Q baryons. In the following, we willprove it.Here, we adopt the scaling technique [86] to deal withthe Schr¨odinger equation, which can help us to find howthe eigenvalues E nL and the expectation value (cid:104) r n (cid:105) vary withchanging the parameters of quark potential model.The Schr¨odinger equation for the single heavy flavorbaryon could be written as (cid:34) − ∇ µ + V SI ( r ) (cid:35) ψ mnL = E nL ψ mnL . (10)In practice, di ff erent kinds of potentials can be taken to depictthe e ff ective interaction between light quark cluster and theheavy quark. We will show that the relationship ξ Λ Q (cid:39) ξ Ξ Q may always stand for the usual e ff ective potentials. To il-lustrate this point, we employ four typical kinds of poten-tials for the e ff ective interaction V SI ( r ), i.e., the power-lawpotential [87], the Cornell potential [9], the logarithmic poten-tial [88, 89], and the Indiana potential [90]. Their expresssionsare V ( r ) = b r ν − V , V ( r ) = − α r + b r − V , V ( r ) = b ln (cid:32) r + tr (cid:33) , V ( r ) = b (1 − Λ r ) r ln r . (11)More details of these phenomenological potentials could befound in Refs. [86, 91].We should mention the nearly equal excited energies of Λ Q and Ξ Q states, i.e., E Λ Q nL (cid:39) E Ξ Q nL , (12)which was implied in the discussion in Sec. II B (refer toEq. (3)). In the following, we take the power-law potential asan example to show how to obtain the following relationship ξ Λ Q ξ Ξ Q = (cid:32) µ Λ Q µ Ξ Q (cid:33) / m [ qs ] m [ ud ] , (13)associated with the relation in Eq. (12). Here, the m [ ud ] and m [ qs ] denote the masses of light scalar quark clusters in the Λ Q and Ξ Q baryons, respectively.With the power-law potential, the radical part of Eq. (10)could be written as − d χ nl d r + (cid:34) µ b r ν + l ( l + r (cid:35) χ nl = µ ( E nL + V ) χ nl . (14)Next, we set z = η r and define u nl ( z ) ≡ χ nl ( r ). Then Eq. (14)can be translated into the following form − d u nl d z + (cid:34) µ b η ν + z ν + l ( l + z (cid:35) u nl = µ ( E nL + V ) η u nl . (15) When setting2 µ b η ν + = , ε nL = µ ( E nL + V ) η , (16)the following scaled Schr¨odinger equation is obtained − d u nl d z + (cid:34) z ν + l ( l + z (cid:35) u nl = ε nL u nl , (17)which can be solved by the numerical calculation or the ap-proximation method when the parameter ν is given. For singleheavy flavor baryon, the parameter ν is constrained to be about0.38 by the values of R and R listed in Table II. The otherparameters m cluster , m Q , b , and V can be further constrainedby fitting the masses of these known Λ Q and Ξ Q baryons.In the following discussion, however, we do not need toperform the detailed numerical calculation. As pointed above,the parameter ν could be regarded as a constant for both Λ Q and Ξ Q states. Then the eigenvalue ε nL and the expectationvalue (cid:104) z − (cid:105) given by Eq. (17) are same for the Λ Q and Ξ Q states. With Eq. (12), we get A = E nL − E n (cid:48) L (cid:48) ε nL − ε n (cid:48) L (cid:48) = b (2 b µ ) − νν + , (18)which can also be regarded as a constant for the Λ Q and Ξ Q states. Here, Eq. (18) can be derived with the help of Eq. (16).Then one can obtain the following expression ξ Λ Q ξ Ξ Q = (cid:104) r − (cid:105) Λ Q (cid:104) r − (cid:105) Ξ Q m [ qs ] m [ ud ] = (cid:32) b Λ Q µ Λ Q b Ξ Q µ Ξ Q (cid:33) ν + m [ qs ] m [ ud ] (19)with help of (cid:104) r − (cid:105) = η (cid:104) z − (cid:105) = (2 b µ ) / ( ν + (cid:104) z − (cid:105) . In addition,one deduces b by Eq. (18) as b = A ν + (2 µ ) ν/ . (20)Combing Eq. (19) and Eq. (20), the result of Eq. (13) can beobtained. The similar derivations involved in other three kindsof potentials are presented in Appendix. With the definition of µ , we further have the following relations ξ Λ Q ξ Ξ Q = (cid:32) m Q + m [ qs ] m Q + m [ ud ] (cid:33) / (cid:32) m [ ud ] m [ qs ] (cid:33) / . (21)When taking the following values m c = .
55 GeV , m [ ud ] = .
71 GeV , m b = .
65 GeV , m [ us ] = .
90 GeV , (22)for the masses of light quark clusters and heavy quarks in thesingle heavy flavor baryon, we estimate the values of two ra-tios ξ Λ c ξ Ξ c (cid:39) . , ξ Λ b ξ Ξ b (cid:39) . , (23)which prove that ξ Q almost keeps same for the Λ Q and Ξ Q baryons. III. APPLICATION
This universal behavior of mass gaps discussed above cangive some valuable implications, especially for predicting themasses of some missing single heavy baryon states.(1) A neutral resonance was discovered by the BaBarCollaboration in the process B − → Σ c (2846) ¯ p → Λ + c π − ¯ p [92]. Its mass and decay width are m ( Σ c (2846) ) = ± ±
10 MeV , Γ ( Σ c (2846) ) = + − ±
12 MeV . (24)With the higher mass and the weak evidence of J = / Σ c (2846) to be di ff erentfrom the Σ c (2800) , + , ++ states which were discoveredby the Belle Collaboration via the e + e − collision [77].For the limited information of the J P quantum numbersand branching ratios, however, PDG temporarily treatedthe Σ c (2846) and Σ c (2800) , + , ++ as a same state [59].The universal behavior of mass gap discussed in thiswork may provide some valuable clues for clarify-ing the puzzle of Σ c (2846) and Σ c (2800) , + , ++ states.We find that the recently reported Ξ (cid:48) c (2965) [47] and Ω c (3090) [44, 65] could form the following chain Σ c (2846) ↔ Ξ (cid:48) c (2965) ↔ Ω c (3090) , (25)if adding Σ c (2846) , which satisfies the requirementfrom universal behavior of mass gaps. Additionally,the Σ c (2800) , + , ++ has been grouped into another chainwith Ξ (cid:48) c (2923) and Ω c (3050) , as shown in Table III.This analysis based on universal behavior of mass gapsuggests that the Σ c (2846) and Σ c (2800) , + , ++ baryonsshould be two di ff erent states.(2) As the second example, we will apply the universal be-havior of mass gap rule to predict the masses of 1 P , 1 D ,and 2 S Ξ b states. At present, only the ground states,i.e., the Ξ b (5792) and the Ξ b (5797) − , have been estab-lished. Since LHCb has shown its capability in accu-mulating the data sample of the excited bottom baryonresonances in the past years, we expect the followingpredicted masses of excited Ξ b baryons to be tested bythe LHCb Collaboration in the near future.The mass gap between Ξ b (5792) and Λ b (5620) stateis 172.3 MeV which is about 10 MeV smaller thanthe mass di ff erence of Λ c (2286) + and Ξ c (2470) + states.Comparing the results in Table I, we may take the massgaps as 185 MeV for the excited Λ b and Ξ b . Withthe measured masses of Λ b (6072) states, the mass of2 S Ξ b state could be predicted as 6257 MeV directly.We have shown that the mass splitting of nL Ξ Q statesis nearly equal to the corresponding Λ Q baryons (seeEq. (23)). Therefore, the masses of two 1 P Ξ b statesare predicted to be 6097 MeV and 6105 MeV, while themasses of two 1 D Ξ b states are about 6331 MeV and6337 MeV. (3) As the last example, we point out that two resonancestructures may exist in the previously observed signalsof Σ b (6097) ± [46] and Ξ (cid:48) b (6227) , − [45, 49]. Two nar-row bottom baryons Ω b (6340) − and Ω b (6350) − whichwere reported by LHCb in the Ξ b K − decay channel [48]could be regarded as the good candidates of P -wavestates with J P = / − and J P = / − , respectively.With this assignment, the mass splitting of these twostates can also be explained by the quark potentialmodel [15, 55] and the QCD sum rule [93]. Accordingto the universal behavior of mass gaps of single heavyflavor baryons, we could conjecture that the Σ b and Ξ (cid:48) b partners of Ω b (6340) − and Ω b (6350) − have been con-tained in the signals of Σ b (6097) ± and Ξ (cid:48) b (6227) , − . Thesituation of charmed baryons is alike. The observed Ξ c (2923) and Ω c (3050) can be explained as the P -wave states with J P = / − , while the Ξ c (2939) and Ω c (3065) could be regarded as the J P = / − partners.Thus, we may point out that the signal of Σ c (2800) , + , ++ states [77] may also contain the J P = / − and J P = / − states. With the higher statistical precision, we ex-pect the LHCb and Belle II experiments to distinguishthese two resonance structures in future. IV. SUMMARY
Until now, about fifty single heavy baryons have been ob-served by experiment [59], where their isospin partners havenot been counted. Facing such abundant experimental data ofsingle heavy flavor baryons, we have a good chance to checkthe mass gaps existing in the established mass spectrum ofsingle heavy flavor baryons.In this work, we find some universal behaviors of massgaps. By a constituent quark model, we try to reveal the un-derly mechanism behind this universal phenomenon, and findthat the universal behaviors of mass gaps for the ¯3 f and 6 f single heavy flavor baryons can be understood well. Addition-ally, we also illustrate why there exists the equal mass splittingfor the orbital excited baryons in the ¯3 f representation.Universal behaviors of mass gaps can be applied to themass spectrum analysis. In this work, we give three impli-cations: 1) We indicated that the neutral resonance Σ c (2846) reported from the BaBar Collaboration [92] is di ff erent fromthe Σ c (2800) , + , ++ states [77]. 2) The masses of 1 P , 1 D , and2 S Ξ b states were predicted. 3) We pointed out that two res-onance structures may exist in the previously observed sig-nals of Σ c (2800) , + , ++ [59], Σ b (6097) ± [46], and Ξ (cid:48) b (6227) , − [45, 49] states. These predictions based on the universal be-haviors of mass gaps could be tested by the LHCb and BelleII experiments in future. ACKNOWLEDGMENTS
X. L. is supported by the China National Funds for Dis-tinguished Young Scientists under Grant No. 11825503, Na-tional Key Research and Development Program of China un-der Contract No. 2020YFA0406400 and the 111 Project un-der Grant No. B20063. B. C. is partly supported by the Na-tional Natural Science Foundation of China under Grants No.11305003 and No. 11647301.
APPENDIX
In this section, we follow the similar derivations presentedin Sec. II C and show how to obtain the relation in Eq. (13) bythe Cornell potential, the logarithmic potential, and the Indi-ana potential.For the Cornell potential, the Schr¨odinger equation is − d χ nl d r + (cid:34) µ (cid:32) − α r + b r (cid:33) + l ( l + r (cid:35) χ nl = µ ( E nL + V ) χ nl . (26)When we set z = η r and define u nl ( z ) ≡ χ nl ( r ), Eq. (26) can betranslated into the following form − d u nl d z + (cid:34)(cid:32) µ b η z − αµ η z (cid:33) + l ( l + z (cid:35) u nl = µ ( E nL + V ) η u nl . (27)We further set αµ η = µ b η = λ , and ε nL = µ ( E nL + V ) η , thescaled Schr¨odinger equation for the Cornell potential is givenas − d u nl d z + (cid:34)(cid:32) − z + λ z (cid:33) + l ( l + z (cid:35) u nl = ε nL u nl . (28)The only parameter λ in Eq. (28) could be fixed as λ (cid:39) . R and R (see Eq. (1) and Table II) for Λ Q and Ξ Q baryons. With the variable substitutions above, we have r = αµ z , b = µ α λ, E nL = µα ε nL − V . Due to the condition of Eq. (12), the following ratio A = E nL − E n (cid:48) L (cid:48) ε nL − ε n (cid:48) L (cid:48) = µα . (29)could be regarded as a constant for the Λ Q and Ξ Q baryons.For the Cornell potential, we have ξ Λ Q ξ Ξ Q = (cid:104) r − (cid:105) Λ Q (cid:104) r − (cid:105) Ξ Q m [ qs ] m [ ud ] = (cid:32) µ Ξ Q α Ξ Q µ Λ Q α Λ Q (cid:33) m [ qs ] m [ ud ] . (30)Then Eq. (13) can be obtained directly by considering the re-lationship α = (cid:113) A µ .For the logarithmic potential, the Schr¨odinger equation isgiven as − d χ nl d r + (cid:34) µ b ln (cid:32) r + tr (cid:33) + l ( l + r (cid:35) χ nl = µ E nL χ nl . (31) Next, we set z = η r and define u nl ( z ) ≡ χ nl ( r ). Then Eq. (31)can be translated into the following form − d u nl d z + (cid:34) ln ( z + z ) + l ( l + z (cid:35) u nl = ε nL u nl . (32)where we have done the following variable substitutions η = (cid:112) b µ, z = (cid:112) b µ t , E nL = b (cid:104) ε nL − ln (cid:16) (cid:112) b µ r (cid:17)(cid:105) . For the Λ Q and Ξ Q baryons, the parameter z in the scaledSchr¨odinger equation of Eq. (32) can be fixed as z (cid:39) . R and R . Similar to the Eq. (18), one could treat theparameter b as a constant for the Λ Q and Ξ Q baryons. Thus,Eq. (13) can be obtained directly.Finally, the Schr¨odinger equation of the Indiana potential isgiven as − d χ nl d r + (cid:34) µ b (1 − Λ r ) r ln( Λ r ) + l ( l + r (cid:35) χ nl = µ E nL χ nl . (33)In the same way, we set z = η r and define u nl ( z ) ≡ χ nl ( r ).Eq. (33) becomes as − d u nl d z + µ b η (1 − Λ η z ) z ln( Λ η z ) + l ( l + z u nl = µ E nL η u nl . (34)When one sets η = Λ and does the following variable substi-tutions κ = µ b Λ , E nL = Λ µ ε nL , (35)the scaled Schr¨odinger equation for the Cornell potential isgiven as − d u nl d z + (cid:34) κ (1 − z ) z ln z + l ( l + z (cid:35) u nl = ε nL u nl . (36)For the Λ Q and Ξ Q baryons, the parameter κ in the scaledSchr¨odinger equation above should be fixed as κ (cid:39) .
80 bythe R and R . Similar to the Eq. (18), the following ratio A = E nL − E n (cid:48) L (cid:48) ε nL − ε n (cid:48) L (cid:48) = Λ µ (37)could be treated as a constant for the Λ Q and Ξ Q baryons.With the η = Λ = (cid:112) A µ , Eq. (13) could be obtained for theIndiana potential. [1] H. X. Chen, W. Chen, X. Liu and S. L. Zhu, The hidden-charmpentaquark and tetraquark states, Phys. Rept. , 1 (2016).[2] A. Ali, J. S. Lange and S. Stone, Exotics: Heavy Pentaquarksand Tetraquarks, Prog. Part. Nucl. Phys. , 123 (2017).[3] A. Esposito, A. Pilloni and A. D. Polosa, Multiquark Reso-nances, Phys. Rept. , 1 (2017).[4] R. F. Lebed, R. E. Mitchell and E. S. Swanson, Heavy-QuarkQCD Exotica, Prog. Part. Nucl. Phys. , 143 (2017).[5] S. L. Olsen, T. Skwarnicki and D. Zieminska, Nonstandardheavy mesons and baryons: Experimental evidence, Rev. Mod.Phys. , 015003 (2018).[6] F. K. Guo, C. Hanhart, U. G. Meißner, Q. Wang, Q. Zhao andB. S. Zou, Hadronic molecules, Rev. Mod. Phys. , 015004(2018).[7] Y. R. Liu, H. X. Chen, W. Chen, X. Liu and S. L. Zhu, Pen-taquark and Tetraquark states, Prog. Part. Nucl. Phys. , 237(2019).[8] N. Brambilla, S. Eidelman, C. Hanhart, A. Nefediev, C. P. Shen,C. E. Thomas, A. Vairo and C. Z. Yuan, The XYZ states: ex-perimental and theoretical status and perspectives, Phys. Rept. , 1 (2020).[9] E. Eichten, K. Gottfried, T. Kinoshita, K. D. Lane andT. M. Yan, Charmonium: The Model, Phys. Rev. D , 3090(1978), Erratum: , 313(E) (1980)].[10] S. Godfrey and N. Isgur, Mesons in a Relativized Quark Modelwith Chromodynamics, Phys. Rev. D , 189 (1985).[11] S. Capstick and N. Isgur, Baryons in a Relativized Quark Modelwith Chromodynamics, Phys. Rev. D , 267 (1985).[12] J. Vijande, F. Fernandez and A. Valcarce, Constituent quarkmodel study of the meson spectra, J. Phys. G , 481 (2005).[13] D. Ebert, R. N. Faustov and V. O. Galkin, Mass spectra andRegge trajectories of light mesons in the relativistic quarkmodel, Phys. Rev. D , 114029 (2009).[14] D. Ebert, R. N. Faustov and V. O. Galkin, Heavy-light me-son spectroscopy and Regge trajectories in the relativistic quarkmodel, Eur. Phys. J. C , 197 (2010).[15] D. Ebert, R. N. Faustov and V. O. Galkin, Spectroscopy andRegge trajectories of heavy baryons in the relativistic quark-diquark picture, Phys. Rev. D , 014025 (2011).[16] D. Ebert, R. N. Faustov and V. O. Galkin, Spectroscopy andRegge trajectories of heavy quarkonia and B c mesons, Eur.Phys. J. C , 1825 (2011).[17] N. Isgur and J. E. Paton, A Flux Tube Model for Hadrons, Phys.Lett. B , 247 (1983).[18] N. Isgur and J. E. Paton, A Flux Tube Model for Hadrons inQCD, Phys. Rev. D , 2910 (1985).[19] M. A. Shifman, A. I. Vainshtein and V. I. Zakharov, QCDand Resonance Physics. Theoretical Foundations, Nucl. Phys.B , 385 (1979).[20] M. A. Shifman, A. I. Vainshtein and V. I. Zakharov, QCDand Resonance Physics: Applications, Nucl. Phys. B , 448(1979).[21] J. J. Dudek, R. G. Edwards, M. J. Peardon, D. G. Richards andC. E. Thomas, Toward the excited meson spectrum of dynami-cal QCD, Phys. Rev. D , 034508 (2010).[22] J. J. Dudek, R. G. Edwards, B. Joo, M. J. Peardon,D. G. Richards and C. E. Thomas, Isoscalar meson spec-troscopy from lattice QCD, Phys. Rev. D , 111502 (2011).[23] R. G. Edwards, J. J. Dudek, D. G. Richards and S. J. Wallace,Excited state baryon spectroscopy from lattice QCD, Phys. Rev.D , 074508 (2011). [24] T. Regge, Introduction to complex orbital momenta, NuovoCim. , 951 (1959).[25] T. Regge, Bound states, shadow states and Mandelstam repre-sentation, Nuovo Cim. , 947 (1960).[26] G. F. Chew and S. C. Frautschi, Principle of Equivalence for AllStrongly Interacting Particles Within the S Matrix Framework,Phys. Rev. Lett. , 394 (1961).[27] G. F. Chew and S. C. Frautschi, Regge Trajectories and the Prin-ciple of Maximum Strength for Strong Interactions, Phys. Rev.Lett. , 41 (1962).[28] A. V. Anisovich, V. V. Anisovich and A. V. Sarantsev, System-atics of q ¯ q states in the ( n , M ) and ( J , M ) planes, Phys. Rev.D , 051502 (2000).[29] M. M. Brisudova, L. Burakovsky, J. T. Goldman andA. Szczepaniak, Nonlinear Regge trajectories and glueballs,Phys. Rev. D , 094016 (2003).[30] A. Zhang, Regge trajectories analysis to D (cid:63) sJ (2317) ± , D sJ (2460) ± and D sJ (2632) + mesons, Phys. Rev. D ,017902 (2005).[31] D. M. Li, B. Ma and Y. H. Liu, Understanding masses of c ¯ s states in Regge phenomenology, Eur. Phys. J. C , 359 (2007).[32] K. W. Wei and X. H. Guo, Mass spectra of doubly heavymesons in Regge phenomenology, Phys. Rev. D , 076005(2010).[33] X. H. Guo, K. W. Wei and X. H. Wu, Some mass relations formesons and baryons in Regge phenomenology, Phys. Rev. D , 056005 (2008).[34] K. W. Wei, B. Chen and X. H. Guo, Masses of doubly and triplycharmed baryons, Phys. Rev. D , 076008 (2015).[35] K. W. Wei, B. Chen, N. Liu, Q. Q. Wang and X. H. Guo, Spec-troscopy of singly, doubly, and triply bottom baryons, Phys.Rev. D , 116005 (2017).[36] Y. Chen, The Origin of Hadron Masses, arXiv:2010.00900[hep-ph].[37] X. Wang, Z. F. Sun, D. Y. Chen, X. Liu and T. Matsuki, Non-strange partner of strangeonium-like state Y (2175), Phys. Rev.D , 074024 (2012).[38] L. P. He, D. Y. Chen, X. Liu and T. Matsuki, Prediction of amissing higher charmonium around 4.26 GeV in J /ψ family,Eur. Phys. J. C , 3208 (2014).[39] T. J. Burns, How the small hyperfine splitting of P -wave mesonsevades large loop corrections, Phys. Rev. D , 034021 (2011).[40] L. Chang, M. Chen, X. q. Li, Y. x. Liu and K. Raya, Can the Hy-perfine Mass Splitting Formula in Heavy Quarkonia be Appliedto the B c System?, Few Body Syst. , 4 (2021).[41] R. Aaij et al. (LHCb Collaboration), Observation of excited Λ b baryons, Phys. Rev. Lett. , 172003 (2012).[42] R. Aaij et al. (LHCb Collaboration), Observation of two new Ξ − b baryon resonances, Phys. Rev. Lett. , 062004 (2015).[43] R. Aaij et al. (LHCb Collaboration), Study of the D p ampli-tude in Λ b → D p π − decays, J. High Energy Phys. , 030(2017).[44] R. Aaij et al. (LHCb Collaboration), Observation of five newnarrow Ω c states decaying to Ξ + c K − , Phys. Rev. Lett. ,182001 (2017).[45] R. Aaij et al. (LHCb Collaboration), Observation of a new Ξ − b resonance, Phys. Rev. Lett. , 072002 (2018).[46] R. Aaij et al. (LHCb Collaboration), Observation of Two Res-onances in the Λ b π ± Systems and Precise Measurement of Σ ± b and Σ ∗± b properties,” Phys. Rev. Lett. , 012001 (2019)[47] R. Aaij et al. (LHCb Collaboration), Observation of New Ξ c Baryons Decaying to Λ + c K − , Phys. Rev. Lett. , 222001(2020).[48] R. Aaij et al. (LHCb Collaboration), First observation of excited Ω − b states, Phys. Rev. Lett. , 082002 (2020).[49] R. Aaij et al. (LHCb Collaboration), Observation of a new Ξ b state, Phys. Rev. D , 012004 (2021).[50] R. Aaij et al. (LHCb Collaboration), Observation of a newbaryon state in the Λ π + π − mass spectrum, J. High EnergyPhys. , 136 (2020).[51] R. Aaij et al. (LHCb Collaboration), Observation of New Res-onances in the Λ b π + π − System, Phys. Rev. Lett. , 152001(2019).[52] B. Chen, K. W. Wei, X. Liu and T. Matsuki, Low-lying charmedand charmed-strange baryon states, Eur. Phys. J. C , 154(2017).[53] B. Chen and X. Liu, New Ω c baryons discovered by LHCb asthe members of 1 P and 2 S states, Phys. Rev. D , 094015(2017).[54] B. Chen, K. W. Wei, X. Liu and A. Zhang, Role of newly dis-covered Ξ b (6227) − for constructing excited bottom baryon fam-ily, Phys. Rev. D , 031502 (2018).[55] B. Chen and X. Liu, Assigning the newly reported Σ b (6097) asa P -wave excited state and predicting its partners, Phys. Rev. D , 074032 (2018).[56] B. Chen, S. Q. Luo, X. Liu and T. Matsuki, Interpretation ofthe observed Λ b (6146) and Λ b (6152) states as 1 D bottombaryons, Phys. Rev. D , 094032 (2019).[57] W. Roberts and M. Pervin, Heavy baryons in a quark model,Int. J. Mod. Phys. A , 2817 (2008).[58] J. G. K¨orner, M. Kr¨amer and D. Pirjol, Heavy baryons, Prog.Part. Nucl. Phys. , 787 (1994).[59] P. A. Zyla et al. (Particle Data Group), Review of ParticlePhysics, Prog. Theor. Exp. Phys. , 083C01 (2020)[60] T. A. Aaltonen et al. (CDF Collaboration), Evidence for a Bot-tom Baryon Resonance Λ ∗ b in CDF Data, Phys. Rev. D ,071101 (2013).[61] A. M. Sirunyan et al. (CMS Collaboration), Study of excited Λ states decaying to Λ π + π − in proton-proton collisions at √ s =
13 TeV, Phys. Lett. B , 135345 (2020).[62] L. Y. Xiao and X. H. Zhong, Toward establishing the low-lying P -wave Σ b states, Phys. Rev. D , 014009 (2020)[63] W. Liang and Q. F. L¨u, The newly observed Λ b (6072) structureand its ρ − mode nonstrange partners, Eur. Phys. J. C , 690(2020).[64] Y. B. Li et al. (Belle Collaboration), Observation of Ξ c (2930) and updated measurement of B − → K − Λ + c ¯ Λ − c at Belle, Eur.Phys. J. C , 252 (2018).[65] J. Yelton et al. (Belle Collaboration), Observation of Excited Ω c Charmed Baryons in e + e − Collisions, Phys. Rev. D , 051102(2018).[66] H. Y. Cheng and C. K. Chua, Strong Decays of CharmedBaryons in Heavy Hadron Chiral Perturbation Theory: An Up-date, Phys. Rev. D , 074014 (2015).[67] H. Y. Cheng and C. W. Chiang, Quantum numbers of Ω c statesand other charmed baryons, Phys. Rev. D , 094018 (2017).[68] K. Thakkar, Z. Shah, A. K. Rai and P. C. Vinodkumar, ExcitedState Mass spectra and Regge trajectories of Bottom Baryons,Nucl. Phys. A , 57 (2017).[69] K. Chen, Y. Dong, X. Liu, Q. F. L¨u and T. Matsuki, Regge-likerelation and a universal description of heavy–light systems, Eur.Phys. J. C , 20 (2018).[70] Z. Zhao, D. D. Ye and A. Zhang, Hadronic decay properties of newly observed Ω c baryons, Phys. Rev. D , 114024 (2017).[71] D. D. Ye, Z. Zhao and A. Zhang, Study of P -wave excitationsof observed charmed strange baryons, Phys. Rev. D , 114009(2017).[72] K. L. Wang, Q. F. L¨u and X. H. Zhong, Interpretation of thenewly observed Σ b (6097) ± and Ξ b (6227) − states as the P -wavebottom baryons, Phys. Rev. D , 014011 (2019).[73] T. M. Aliev, K. Azizi, Y. Sarac and H. Sundu, Structure of the Ξ b (6227) − resonance, Phys. Rev. D , 094014 (2018).[74] T. M. Aliev, K. Azizi, Y. Sarac and H. Sundu, Determinationof the quantum numbers of Σ b (6097) ± via their strong decays,Phys. Rev. D , 094003 (2019).[75] Q. F. L¨u, Canonical interpretations of the newly observed Ξ c (2923) , Ξ c (2939) , and Ξ c (2965) resonances, Eur. Phys. J.C , 921 (2020).[76] Z. G. Wang, Analysis of the Ω b (6316), Ω b (6330), Ω b (6340)and Ω b (6350) with QCD sum rules, Int. J. Mod. Phys. A ,2050043 (2020).[77] R. Mizuk et al. (Belle Collaboration), Observation of anisotriplet of excited charmed baryons decaying to Λ + c π , Phys.Rev. Lett. , 122002 (2005).[78] K. L. Wang, Y. X. Yao, X. H. Zhong and Q. Zhao, Strong andradiative decays of the low-lying S - and P -wave singly heavybaryons, Phys. Rev. D , 116016 (2017).[79] B. Chen, K. W. Wei and A. Zhang, Investigation of Λ Q and Ξ Q baryons in the heavy quark-light diquark picture, Eur. Phys. J.A , 82 (2015).[80] H. Y. Cheng, Charmed baryons circa 2015, Front. Phys. ,101406 (2015).[81] M. Karliner and J. L. Rosner, Very narrow excited Ω c baryons,Phys. Rev. D , 114012 (2017).[82] A. De Rujula, H. Georgi and S. L. Glashow, Hadron Masses ina Gauge Theory, Phys. Rev. D , 147 (1975).[83] R. L. Ja ff e, Multi-Quark Hadrons. II. Methods, Phys. Rev. D , 281 (1977)[84] L. Maiani, F. Piccinini, A. D. Polosa and V. Riquer, Diquark-antidiquarks with hidden or open charm and the nature of X (3872), Phys. Rev. D , 014028 (2005).[85] H. Hogaasen, J. M. Richard and P. Sorba, A Chromomagneticmechanism for the X (3872) resonance, Phys. Rev. D , 054013(2006).[86] C. Quigg and J. L. Rosner, Quantum Mechanics with Applica-tions to Quarkonium, Phys. Rept. , 167 (1979).[87] A. Martin, A Fit of Upsilon and Charmonium Spectra, Phys.Lett. B , 338 (1980).[88] C. Quigg and J. L. Rosner, Quarkonium Level Spacings, Phys.Lett. B , 153 (1977).[89] S. n. Jena, Fit of charmonium and upsilon spectra by a commonpotential V + A log(1 + r ), Phys. Lett. B , 445 (1983).[90] G. Fogleman, D. B. Lichtenberg and J. G. Wills, Heavy Me-son Spectra Calculated With a One Parameter Potential, Lett.Nuovo Cim. , 369 (1979).[91] W. Lucha, F. F. Schoberl and D. Gromes, Bound states ofquarks, Phys. Rept. , 127 (1991).[92] B. Aubert et al. (BaBar Collaboration), Measurements of B ( ¯ B → Λ + c ¯ p ) and B ( B − → Λ + c ¯ p π − ) and studies of Λ + c π − reso-nances, Phys. Rev. D , 112003 (2008).[93] H. M. Yang and H. X. Chen, P -wave bottom baryons of the S U (3) flavor F , Phys. Rev. D , 114013 (2020), Erratum:Phys. Rev. D102