Charmed baryon spectrum from lattice QCD near the physical point
Huseyin Bahtiyar, Kadir Utku Can, Guray Erkol, Philipp Gubler, Makoto Oka, Toru T. Takahashi
CCharmed baryon spectrum from lattice QCD near the physical point
H. Bahtiyar, K. U. Can,
2, 3
G. Erkol, P. Gubler, M. Oka, and T. T. Takahashi (TRJQCD Collaboration) Department of Physics, Mimar Sinan Fine Arts University, Bomonti 34380 Istanbul Turkey CSSM, Department of Physics, The University of Adelaide, Adelaide SA 5005, Australia RIKEN Nishina Center, RIKEN, Saitama 351-0198, Japan Department of Natural and Mathematical Sciences, Faculty of Engineering, Ozyegin University,Nisantepe Mah. Orman Sok. No:34-36, Alemdag 34794 Cekmekoy, Istanbul Turkey Advanced Science Research Center, Japan Atomic Energy Agency, Tokai, Ibaraki, 319-1195 Japan National Institute of Technology, Gunma College, Maebashi, Gunma 371-8530 Japan (Dated: April 21, 2020)We calculate the low-lying spectrum of charmed baryons in lattice QCD on the 32 × N f = 2+1PACS-CS gauge configurations at the almost physical pion mass of ∼
156 MeV/c . By employinga set of interpolating operators with different Dirac structures and quark-field smearings for thevariational analysis, we extract the ground and first few excited states of the spin-1 / / c –Ξ (cid:48) c mixing and theoperator dependence of the excited states in a variational approach. We identify several states thatlie close to the experimentally observed excited states of the Σ c , Ξ c and Ω c baryons, including someof the Ξ c states recently reported by LHCb. Our results for the doubly- and triply-charmed baryonsare suggestive for future experiments. PACS numbers: 14.20.Lq, 12.38.Gc, 13.40.GpKeywords: charmed baryons, spectrum, excited states, lattice QCD
I. INTRODUCTION
Recent experimental results from the LHCb Collabora-tion on the Ω c , Ξ c and the doubly charmed Ξ cc state haveput further emphasis on the relevance of the hadron spec-troscopy. There now exist 31 observed charmed baryons,25 of which are classified with at least three stars by theParticle Data Group (PDG) [1]. Charmed baryons pro-vide a unique laboratory to study the strong interactionand confinement dynamics due to the composition of thelight and charm quarks. Studying the excited states ofthe charmed baryons has the potential to reveal theirinternal dynamics and the nature of the excitation mech-anisms.Experimentally, the singly-charmed baryon sector ismost accessible. Within this sector, the Λ c channel ismost established. In addition to the ground state, thereare four excitations with total spin up to 5 /
2, althoughin need of a confirmation of the assigned quantum num-bers. Out of the three Σ c states that are listed by thePDG, two are the lowest J P = 1 / + and 3 / + states andΣ c (2800) is their only observed excitation. This statehas been detected in the Λ c π channel by the Belle [2]and the BABAR [3] Collaborations. Its quantum num-bers are not measured. In contrast, the Ξ c sector is quiterich since it can have flavor symmetric and antisymmet-ric wave functions. There are up to seven Ξ c excitationsobserved by the Belle [4–9], the BABAR [10, 11] andvery recently by the LHCb [12] Collaborations in the en-ergy range of 2920 to 3120 MeV/c . The PDG considersthe existence of three of them to be very likely or cer-tain while the confidence for the other two is smaller.LHCb states are not included in the review yet. These excited states appear in the invariant mass distributionsof several singly-charmed baryon B c + K or π channelsdepending on the strangeness number of the baryon andin the Λ D channel where the charm quark is confined inthe meson system. This unique behavior makes the Ξ c system a good laboratory to study the internal excita-tion dynamics of the charmed baryons and the diquarkcorrelations. The quantum numbers of these states re-main undetermined. The LHCb Collaboration has alsoreported the precise measurements of the masses and thedecay widths of five new Ω c states [13], which are ob-served in the Ξ c K channel in the energy range from 3000to 3120 MeV/c . Their spin-parity quantum numbersremain undetermined. There are several works in theliterature investigating the nature of these states and as-signing conflicting spin-parity quantum numbers. It isa triumph of the experiments to identify many states insuch narrow energy windows.The lowest-lying states of the singly-charmed baryonsare already established by experimental studies and thelattice QCD results agree well with those observations.The Ξ cc is the only observed doubly-charmed baryon forthe time being. It was first observed by the SELEX Col-laboration [14, 15] but its results were not confirmed byother experiments until the LHCb Collaboration has re-ported the same particle with a different mass [16]. Lat-tice QCD predictions for the mass of the Ξ cc lie abovethe SELEX reported value but agree very well with theLHCb value.From the theoretical side, it remains to be a remarkablechallenge to extract the spectrum and assign quantumnumbers to the observed charmed baryons. For a com-plete understanding of these states, one would in princi- a r X i v : . [ h e p - l a t ] A p r ple would need to study their decay widths as well. Spec-tra and properties of the heavy baryons have been studiedextensively via several naive and improved quark mod-els [17–36], the Feynman-Hellmann theorem [37], largeN QCD [38], QCD sum rules [39–50], chiral effective-field theory [51], chiral diquark effective theory [52, 53]and heavy-quark effective theory [54] approaches. Dis-cussions about the excited Λ c , Σ c and Ξ c states fromvarious models are reviewed in detail in Refs. [55, 56].Specifically, the excited Ω c system is studied in the con-text of the QCD sum rules [48–50], the constituent quarkmodel [30] and in a chiral quark-soliton model [31]. Cal-culations based on a quark-diquark bound state pictureare presented in Refs. [32–34] and arguments for a poten-tial molecular [35], or a compact pentaquark nature [36]for these states are given in other works. A dedicated lat-tice QCD study assigning quantum numbers is reportedby the Hadron Spectrum Collaboration [57].The lowest-lying charmed baryon states have beenstudied by various lattice groups as well. Early inves-tigations utilized the quenched approximation [58–62],while recent studies employ up to 2 + 1 + 1-flavor dynam-ical gauge configurations with several lattice spacings,volumes and light-quark masses to estimate the baryonmasses at the physical point [63–75]. We summarize therecent studies of several lattice groups in Table I.There is a remarkable agreement between the resultsof the different groups utilizing different types of quarkactions and approaches to the physical point. Most ofthose studies are motivated by the observation of theΞ cc baryon by LHCb and thus their focus has been onthe lowest-lying positive parity baryons. Extracting theexcited states, however, is a challenge compared to cal-culating the ground states. The majority of the atten-tion has been on the light-quark sector, especially on theRoper resonance and the Λ(1405), while there are justa few groups that have studied the excited states of thecharmed baryons.The RQCD Collaboration reported results for thesingly- and doubly-charmed baryons, including excitedstates [69]. They employ several 2+1-flavor gauge ensem-bles with a fixed lattice spacing but two different volumesand varying light-quark masses with the lightest one cor-responding to a pion mass of m π ∼
260 MeV/c . All thesea and valance quarks (including the charm quark) aretreated via a non-perturbatively improved stout-smearedClover action. The bare charm-quark mass is tuned to re-produce the 1 S spin-averaged charmonium mass. In ad-dition to spectrum calculations, they also investigate thelight-flavor dependence of the singly and doubly charmedstates. To this end, the operator set they use consists ofinterpolating fields based on SU (4) symmetry and heavyquark effective theory (HQET) pictures. In order to ac-cess the excited states, they perform a variational anal-ysis over a set of interpolating fields with three differentquark-field smearings. Their chiral extrapolations followa different approach compared to the other groups sincethey start from an SU (3) symmetric point for the light and strange quarks and vary their masses while keepingthe singlet quark mass fixed in their descent to the physi-cal point. This leads to fits based on Gell-Mann – Okuborelations for the charmed baryons. The lowest-lying ex-tracted states are in good agreement with the other lat-tice determinations and with experimental values whereavailable.The Hadron Spectrum Collaboration (HSC) extractsthe charmed baryon spectrum including positive and neg-ative parity baryons with total spin up to J = 7 /
2. Theyuse N f = 2 + 1 anisotropic lattices generated with a tree-level tadpole-improved Clover fermion action with a pionmass of m π = 391 MeV/c . The anisotropic Clover ac-tion is used for the charm quark as well with its massparameter tuned non-perturbatively so as to reproducethe dispersion relation for the η c meson. By using a largeset of continuum interpolating operators, including non-local covariant derivative operators, subduced to the irre-ducible representations of the cubic group, they form thebasis for the variational correlation matrix analysis andextract the spectrum of the singly-, doubly- and triply-charmed baryons [71–75]. Although the systematics areleft unchecked and the pion mass is unphysical, their pio-neering results provide valuable insight into the charmedbaryon spectrum.In this work, we follow a conventional approach by us-ing local operators only. Notable improvements of thisstudy compared to the previous works that extract theexcited baryon spectrum are the fully relativistic treat-ment of the charm quark, thus suppressing the O ( am Q )discretization errors, and working on gauge configura-tions with almost physical light quarks, hence eliminat-ing the chiral extrapolation systematics. We also performvariational analyses over sets of operators with differentDirac structures and quark smearings and their combi-nations. Preliminary results of this work have been pre-sented in Ref [76].This paper is structured as follows: we outline the ap-proach to extract the baryon energies and the formulationof the variational analysis in Section II. Details of our lat-tice setup, the heavy quark action that we employ, andthe choice of baryon operators are given in Section III.A detailed discussion on the variational analyses and thestates we extract are presented in Section IV. Section Vholds the summary of our findings. II. EXTRACTING EXCITED STATES
For a given interpolator, χ i , the two-point correlationfunction contains the contributions from all the statesthat couple to the corresponding quantum number, C ij ( t ) = (cid:104) χ i ( t ) ¯ χ j (0) (cid:105) = (cid:88) B (cid:104) | χ i |B(cid:105)(cid:104)B| ¯ χ j | (cid:105) e − E B t , (1)where ( E B ) B stands for the (energy of the) baryon state.The desired parity state can be isolated by applying theparity operator, P ± C ij ( t ) = (1 ± γ ) C ij ( t ). TABLE I. Simulation properties of previous lattice QCD calculations. We indicate the number of flavors ( N f ), latticespacing(s) ( a ), number of volumes ( n V ) and the relevant sea- and valance-quark actions (S) used in the studies. Additionally,whether a relativistic treatment (RT) applied ( (cid:51) ) to the charm quark or not ( (cid:55) ) is indicated, and, in the last column, thechiral extrapolation method is quoted where applicable. NA (not applicable) means those groups run their simulations at thephysical quark mass. Abbreviations are: highly-improved staggered quark (HISQ), relativistic heavy-quark action (RHQA),heavy-hadron chiral perturbation theory treatment (HH χ PT), and Gell-Mann – Okubo relation (GMO).Ref. N f a [fm] m π [MeV] n V S seau,d,s,c S valc RT ExtrapolationETM [63] 2 0.094 130 1 Twisted Mass Twisted Mass (cid:55)
NAD¨ur et al. [64] 2 0.073 280 1 Clover Brillouin (cid:51) (cid:55)
Brown et al. [65] 2 + 1 0.085 - 0.11 227 - 419 2 Domain Wall RHQA (cid:51) HH χ PTPACS-CS [66] 2 + 1 0.09 135 1 Clover RHQA (cid:51)
NATWQCD [67] 2 + 1 + 1 0.063 280 1 Domain Wall Domain Wall (cid:55) (cid:55)
Brice˜no et al. [68] 2 + 1 + 1 0.06 - 0.12 220 - 310 5 HISQ RHQA (cid:51) HH χ PTRQCD [69] 2 + 1 0.075 259 - 460 2 Clover Clover (cid:51)
GMOHSC [71–75] 2 + 1 0.035 390 1 Clover Clover (cid:51) (cid:55)
Using a set of operators that couple to the same quan-tum numbers, one can utilize a variational approach toextract the tower of states. One can form an N × N correlation function matrix, C ( t ) = C ( t ) C ( t ) · · · C ( t ) C ( t ) · · · ... ... . . . , (2)where each element, C ij ( t ), is an individual correlationfunction given in Equation (1). Then, by solving thegeneralized eigenvalue problem [77, 78], C ( t ) ψ α ( t ) = λ α ( t, t ) C ( t ) ψ α ( t ) ,φ α ( t ) C ( t ) = λ α ( t, t ) φ α ( t ) C ( t ) , (3)one extracts the left and right eigenvectors, ψ α and φ α ,and uses them to diagonalize the correlation-function ma-trix, φ α ( t (cid:48) ) C ( t ) ψ β ( t (cid:48) ) ≡ C α ( t )= δ αβ Z α ¯ Z β e − E α t (cid:0) O ( e − ∆ E α t ) (cid:1) , (4)to access the energies of the states, E α . One can al-ternatively utilize the individual eigenvalues, λ α ( t, t ) ∼ e − E α ( t − t ) (1 + O ( e − ∆ E α t )), of the left and right eigen-value equations given in Equation (3) to extract the en-ergies of the states. Both approaches give complemen-tary results with some caveats[79]. We prefer the methodoutlined above. Note that a suitable combination of thetime-slice t and the time slice of the eigenvectors, t (cid:48) , ischosen with respect to the quality and stability of the sig-nal. Additionally, t (cid:48) may or may not be chosen equal to t . Once the correlation function matrix is diagonalized,one can follow the standard techniques and perform aneffective mass analysis for each state, α , m α eff ( t ) = ln C α ( t ) C α ( t + 1) . (5) III. LATTICE SETUPA. Quark Actions
We employ the 32 ×
64, 2 + 1-flavor gauge config-urations that are generated by the PACS-CS Collab-oration [80]. These configurations are generated withthe Iwasaki gauge action ( β = 1 .
9) and with the non-perturbatively O ( a )-improved Wilson (Clover) action( c sw = 1 . κ sea ud = 0 . m π =156(9) MeV /c as measured by PACS-CS. The hop-ping parameter of the strange quark is fixed to κ sea s =0 . a = 0 . a − = 2 .
176 GeV).We use the Clover action for the valence u/d and s quarks. The hopping parameter of the valence lightquarks is set equal to those of sea quarks, κ val u/d = κ sea u/d .Due to an overestimation of the mass of the Ω − particlewith κ val s = κ sea s , however, we re-tune the hopping pa-rameter of the valence strange quark to κ val s = 0 . − mass on these config-urations. Details of this tuning are discussed in Ref. [81].We employ a relativistic heavy-quark action for thecharm quark, S Ψ = (cid:88) x,y ¯Ψ x D x,y Ψ y , (6)where the Ψs are the heavy quark spinors and the fermionmatrix is given as D x,y = δ xy − κ Q (cid:88) µ =1 [( r s − νγ µ ) U x,µ δ x +ˆ µ,y +( r s + νγ µ ) U † x,µ δ x,y +ˆ µ (cid:3) − κ Q (cid:104) (1 − γ ) U x, δ x +ˆ4 ,y +(1 + γ ) U † x, δ x,y +ˆ4 (cid:105) − κ Q (cid:34) c B (cid:88) µ,ν F µν ( x ) σ µν + c E (cid:88) µ F µ ( x ) σ µ (cid:35) δ xy , (7)with the free parameters r s , ν , c B and c E to be tuned inorder to remove the discretization errors appropriately.We adopt the perturbative estimates r s = 1 . c B = 1 . c E = 1 . ν = 1 . κ Q =0 . η c and the J/ψ are m η c = 2 . /c , m J/ψ =3 . /c . The hyperfine splitting is estimatedas ∆ E ( V − P S ) = 116(4) MeV /c , in agreement with itsexperimental value. Further details of our charm quarktuning can be found in Ref. [81]. B. Baryon operators
The baryon operators that we employ are tabulated inTable II in a shorthand notation while the explicit formsof the operators can be found in Table III. Note that wedo not distinguish between u and d quarks since they aredegenerate in our lattice setup. TABLE II. Types of the interpolating operators used for thecharmed baryons. Their quark contents are shown in the thirdcolumns. spin-1 / / q , q , q ) Baryon Operator ( q , q , q )Λ c Λ - like ( u, d, c ) Σ ∗ c ∆ + - like ( u/d, u/d, c )Σ c N - like ( u/d, c, u/d ) Ξ ∗ c ∆ + - like ( u/d, s, c )Ξ c N - like ( u/d, s, c ) Ω ∗ c ∆ + - like ( s, s, c )Ξ c Λ - like ( s, u/d, c )Ξ (cid:48) c Ξ (cid:48) c ( u/d, c, s ) Ξ ∗ cc ∆ + - like ( u/d, c, c )Ω c N - like ( s, c, s ) Ω ∗ cc ∆ + - like ( s, c, c )Ξ cc N - like ( c, u/d, c )Ω cc N - like ( c, s, c ) Ω ccc ∆ + - like ( c, c, c ) For the spin-1 / , Γ ] = [ γ , , γ ], and [ γ γ ,
1] (see Table III). An explicit example for the N -like operator is χ ( x ) = ε abc (cid:2) q T a ( x ) Cγ q b ( x ) (cid:3) q c ( x ) , (8) χ ( x ) = ε abc (cid:2) q T a ( x ) Cq b ( x ) (cid:3) γ q c ( x ) , (9) χ ( x ) = ε abc (cid:2) q T a ( x ) Cγ γ q b ( x ) (cid:3) q c ( x ) . (10)The χ -type operator with the Dirac structure [Γ , Γ ] =[ γ γ ,
1] corresponds to the time component of an op-erator with [Γ , Γ ] = [ γ γ µ , γ ], which couples to bothspin-1 / / / χ and the χ , and a term containing the χ opera-tor [83]. Furthermore, the χ -type operator is distinctfrom the χ and the χ from a chiral transformation per-spective [84], making it a viable choice for the basis set ofthe spin-1 / / , Γ ] = [ γ µ , c and Ξ (cid:48) c states deserve special atten-tion. The Ξ c (Ξ (cid:48) c ), which belongs to an SU (3) anti-triplet(sextet) is anti-symmetric (symmetric) with respect tothe exchange of s and u/d quarks, which should hold forthe respective operators. For Ξ c , this can be achievedby both N -like and Λ-like operators, which will both beused in this work. Note that our N -like Ξ c operator wasreferred to as “HQET” in Ref. [69]. For Ξ (cid:48) c , we employa different operator combination with the correct sym-metry properties as shown in Table III. While Ξ c and Ξ (cid:48) c states decouple in the SU (3) limit, they can in princi-ple mix in our setup due to the breaking of the SU (3)symmetry. This mixing can be studied by computingcross-correlators of Ξ c and Ξ (cid:48) c operators. The results ofsuch an analysis will be discussed in Section IV. C. Simulation details
Quark fields of the interpolating operators are Gaus-sian smeared in a gauge-invariant manner at the source,( x, y, z, t ) = (16 a, a, a, a ), for all the baryons withthree different sets of smearing parameters, correspond-ing to an rms radius of ∼ .
2, 0 . . / u -, d - and the s -quarks on an equal footing and con-sider them as light quarks in comparison to the charmquark. When the interpolating operator is formed by TABLE III. Interpolating operators with generic Dirac structures for spin-1 / / C = γ γ is the chargeconjugation operator. [Γ , Γ ] choices and the quark contents are given in the text and in Table II. Spin Baryon Operator1/2 N - like ε abc (cid:2) q T a ( x ) C Γ q b ( x ) (cid:3) Γ q c ( x )Λ - like √ ε abc (cid:0) (cid:2) q T a ( x ) C Γ q b ( x ) (cid:3) Γ q c ( x ) + (cid:2) q T a ( x ) C Γ q b ( x ) (cid:3) Γ q c ( x ) − (cid:2) q T a ( x ) C Γ q b ( x ) (cid:3) Γ q c ( x ) (cid:1) Ξ (cid:48) c √ ε abc (cid:0)(cid:2) q T a ( x ) C Γ q b ( x ) (cid:3) Γ q c ( x ) + (cid:2) q T a ( x ) C Γ q b ( x ) (cid:3) Γ q c ( x ) (cid:1) + - like √ ε abc (cid:0) (cid:2) q T a ( x ) Cγ µ q b ( x ) (cid:3) q c ( x ) + (cid:2) q T a ( x ) Cγ µ q b ( x ) (cid:3) q c ( x ) (cid:1) two light quarks and a charm quark, we fix the smearingof the charm quark to 0 . ( ∗ ) cc , inwhich case the smearing of the strange quark is fixed to0 . ccc , for which the treatment is the same as lightquarks. For the spin-3 / P ± ,to the individual correlation functions.We bin our data with a bin size of 15 measurements toaccount for the autocorrelations on this ensemble and es-timate the statistical errors via a single elimination jack-knife analysis. We performed our computations using amodified version of the Chroma software system [85] onCPU clusters along with the QUDA library [86, 87] forthe valence u − /d − and s -quark propagator inversions onGPUs. The charm quark inversions are done on CPUs. IV. RESULTS AND DISCUSSIONA. Variational analysis
To obtain the individual states from a set of opera-tors, one solves the generalized eigenvalue problem oneach time slice, t , against a reference time-slice, t , asdiscussed in Section II. To ensure the consistency of thisstep, it is necessary to check that the solutions are stablewith respect to t , since it can be chosen freely. Anotherconcern is associating the eigenvalues with the states.Eigenvalues are sorted in increasing order on each timeslice. However due to the faster deterioration of thehigher states’ signal, their eigenvalues fluctuate heavilyas time evolves and can sometimes be smaller than theeigenvalue associated with the lower state. This situationmight misguide the analysis if not addressed properly.In order to make sure that the eigenvalues are associ-ated with the correct states, we fix the time-slice of the eigenvectors, t (cid:48) , that is used to diagonalize the correla-tion function matrix, to a specific value. This procedure,however, introduces an extra parameter dependence tothe analysis. We check this dependence for each channelfor a range of t (cid:48) values. The dependencies on t and t (cid:48) can be tracked by investigating the respective eigenvec-tors, whose components should be stable when changingboth fictitious time parameters. We illustrate such a con-sistency check in Figure 1. We perform this check for eachchannel and select a ( t (cid:48) , t ) combination that optimizesthe signal quality. A common choice is t (cid:48) ≥ a .
1. Operator dependence a. Operator basis:
Having three operators with dif-fering Dirac structures, it is possible to analyze both thefull 3 × × × × / { χ , χ } and { χ , χ } , give two distinct second eigen-values for the positive parity states. The { χ , χ } setproduces similar results to that of { χ , χ } . For negativeparity, only the { χ , χ } combination yields mostly well-separated second eigenvalues, whereas the second eigen-values of the { χ , χ } and { χ , χ } bases lie closer tothe first eigenvalues. When we extend the operator ba-sis to the { χ , χ , χ } set and solve the corresponding3 × × c , Ω c , and Ξ cc baryons wherewe show the fit results from a plateau approach. Theserepresentative baryons are chosen such that they corre-spond to the different operator characteristics, i.e. Λ-like, singly-charmed N -like, and doubly-charmed N -like,respectively. b. N -like operators: Although we use the same N -like operators for the singly-charmed and the doubly-charmed spin-1 / | E i g e n v ec t o r t t' = t t' = t t' = t t' = t t' = t t' = t t' = t t' = t t' = t ψ α i ψ ψ ψ ψ t' = | E i g e n v ec t o r t t' = t t' = t t' = t t' = t t' = t t' = t t' = t t' = t t' = t ϕ α i ϕ ϕ ϕ ϕ t' = FIG. 1. Consistency check of the variational parameters for positive parity spin-1 / c with ∼ . ψ α and φ α for varying reference time, t , and the time-slice of the eigenvector, t (cid:48) .Association of the operators to the states flip when t > t (cid:48) . SU (4) 20-plet. Such a difference is evident when wecompare the solutions from the operator sets { χ , χ } , { χ , χ } , and { χ , χ } . The lower three sections, dividedby the solid lines, of the positive parity Ξ cc , and Ω c inFigures 2a and 2b show that different operators couple todifferent states. The different couplings can be trackedto the eigenvectors of each solution as shown in Figure 3.The χ operators couple only to the second states in theΞ cc channel while it couples to the third state only forthe Ω c . c. Λ -like operators: Λ c and Ξ c belong to the totallyflavor-antisymmetric SU (4) anti-quadruplet and henceare studied via the flavor-octet Λ-like operators. Thebehaviors of these operators depicted in Figures 2c and 3show similarities to the N -like Ξ cc case. It can naively beexpected that the first term of the Λ-like operator (seeTable III) would have the dominant contribution, whichwould mean that it is in essence the same as the N -likeoperator. Indeed, by rearranging the latter two termsof the Λ-like operator via Fierz transformations, one canshow that the coefficient of the (cid:2) q T a ( x ) Cγ q b ( x ) (cid:3) q c ( x )term of the operator is five times the other resultingterms. The same argument holds for the other Diracstructures as well. This dominance is realized in ourcomparisons of the Ξ c (
12 + ) results illustrated in Figure 4,where we have an almost identical signal for the groundstates calculated via the Λ-like and the N -like operator.Additionally, the flavor decomposition of the Λ c stud-ied in Ref. [88] by three of the present authors showsthat the negative parity Λ c baryon consists of a mixtureof flavor-singlet and flavor-octet wave-functions. Theflavor-octet interpolating operator that we employ forthe Λ c baryon may therefore be inadequate to resolve the lowest-lying negative parity state by itself. A simi-lar conclusion was reached in Ref. [89]. The first excitednegative parity state on the other hand, is dominated bya flavor-octet wave-function and it is possible that thisstate is contaminating our lowest Λ c ( − ) signal, whichcould be a plausible explanation of the apparent overes-timation of its mass (see Table IV and Figure 8).We analyze the Ξ c channel with two different typesof operators. One being the Λ-like, the other the N -like operator as given in Table III. We find that bothgive consistent results for the positive parity case whilethere is a difference for negative parity. As shown inFigure 2f, the N -like operator couples to a lower-lyingstate for the { χ , χ } basis. Similar differences betweenthese operators for the negative parity sector have beenreported by the RQCD Collaboration [69]. d. Ξ c − Ξ (cid:48) c mixing: We perform a correlation ma-trix analysis consisting of the Ξ (cid:48) c , and N -like and Λ-likeΞ c operators in order to investigate the possible mixingbetween these baryons. We construct the correlation-function matrices for this analysis in two steps. First, wesolve a variational system over the { χ , χ } basis for eachelement of the correlation matrix and take the lowest ly-ing state. We find that this approach helps to isolate theground states better. We then solve another 2 × c and Ξ (cid:48) c ground state operatorsto investigate the mixing effects.For positive parity Ξ c and Ξ (cid:48) c , we analyze the cross cor-relators between the flavor-octet SU (4) Ξ c - Ξ (cid:48) c , and the N -like Ξ c - Ξ (cid:48) c individually. We find that the Ξ c and Ξ (cid:48) c signals separate nicely, and the N -like and Λ-like Ξ c op-erators produce consistent signals with negligible mixing(see Figure 4). Magnitudes of the eigenvectors also con- ○○○○○○○○○ ◇◇◇□□□ □□□○○○○ ○○ □□□ □ □□ ◇◇◇◇ ◇◇○○ □□◇◇ ▽▽ Ξ cc + ○ α = □ α = ◇ α = ▽ α = { χ , χ } - { χ , χ } - { χ , χ } - { χ , χ } - { χ , χ } - { χ , χ } - { χ , χ } - { χ , χ } - { χ , χ } - { χ , χ , χ } - { χ , χ , χ } - { χ , χ , χ } - χ - {
10, 50, 150 } χ - {
10, 50, 150 } χ - {
10, 50, 150 }{ χ , χ } - {
10, 50 } χ - N iter M [ GeV ] (a) Ξ cc (cid:16)
12 + (cid:17) ○○○○○○○○○ □□□ ◇◇◇ ◇◇◇○○○ □□□ ◇◇◇ Ω c + ○ α = □ α = ◇ α = { χ , χ } - { χ , χ } - { χ , χ } - { χ , χ } - { χ , χ } - { χ , χ } - { χ , χ } - { χ , χ } - { χ , χ } - { χ , χ , χ } - { χ , χ , χ } - { χ , χ , χ } - χ - N iter M [ GeV ] (b) Ω c (cid:16)
12 + (cid:17) ○○○○○○○○○ ◇◇◇□□□ □□□○○○ □□□ ◇◇◇ Ξ c + ○ α = □ α = ◇ α = { χ , χ } - { χ , χ } - { χ , χ } - { χ , χ } - { χ , χ } - { χ , χ } - { χ , χ } - { χ , χ } - { χ , χ } - { χ , χ , χ } - { χ , χ , χ } - { χ , χ , χ } - χ - N iter M [ GeV ] (c) Ξ c (cid:16)
12 + (cid:17) ○○○ □ □□ ◇◇◇ ◇◇◇□□□ □□□○○○○○○ ○○○ Ξ cc - ○ α = □ α = ◇ α = { χ , χ } - { χ , χ } - { χ , χ } - { χ , χ } - { χ , χ } - { χ , χ } - { χ , χ } - { χ , χ } - { χ , χ } - { χ , χ , χ } - { χ , χ , χ } - { χ , χ , χ } - χ - N iter M [ GeV ] (d) Ξ cc (cid:16) − (cid:17) ○○ ○○○○○○○ ◇◇ □□□□□□ ○○○□□□ ◇◇◇ Ω c - ○ α = □ α = ◇ α = { χ , χ } - { χ , χ } - { χ , χ } - { χ , χ } - { χ , χ } - { χ , χ } - { χ , χ } - { χ , χ } - { χ , χ } - { χ , χ , χ } - { χ , χ , χ } - { χ , χ , χ } - χ - N iter M [ GeV ] (e) Ω c (cid:16) − (cid:17) ●● ●●●● ◆◆■■■■●● ■■ ◆◆ ○○○ ○○○ ○○○ ◇◇◇□□□ □□□○○○ □□□ ◇◇◇ Ξ c HQ - ● α = ■ α = ◆ α = Ξ c Λ - like - ○ α = □ α = ◇ α = { χ , χ } - { χ , χ } - { χ , χ } - { χ , χ } - { χ , χ } - { χ , χ } - { χ , χ } - { χ , χ } - { χ , χ } - { χ , χ , χ } - { χ , χ , χ } - { χ , χ , χ } - χ - N iter M [ GeV ] (f) Ξ c (cid:16) − (cid:17) FIG. 2. Operator dependence of the extracted states for representative baryon channels. Vertical axes label the operator basiswith respect to the Dirac structure, χ , and the smearing steps of the quark fields, N iter , where the iterations correspond toan rms radius of ∼ .
2, 0 . . N iter . Data points in each section, divided by dashed or solid lines, areshifted for clarity. Filled symbols in (f) correspond to the states extracted via an N -like operator basis given in Table III. Notethat we only have two smearings for that case. State numbering, α = 1 −
3, follows the notation of the 3 × × t | E i g e n v e c t o r s u α i u u u u u u u u u t' = (a) Ξ cc (cid:16)
12 + (cid:17) t | E i g e n v e c t o r s u α i u u u u u u u u u t' = (b) Ω c (cid:16)
12 + (cid:17) t | E i g e n v e c t o r s ψ α i ψ ψ ψ ψ ψ ψ ψ ψ ψ t' = (c) Ξ c (cid:16)
12 + (cid:17)
FIG. 3. Eigenvectors from 3 × cc , (b) Ω c and (c) Ξ c channels. ψ αi is the right eigenvectorwhere α is the state and the index i stands for the individual operator in the operator basis i = { χ , χ , χ } . For instance ψ corresponds to the contribution of i = χ to the second state. t / a E ff . M a ss / a Ξ c ( Λ - like ) Ξ c ( N - like ) Ξ c t / a E ff . M a ss / a | E i g e n v e c t o r s i = { Ξ c ( Λ - like ) , Ξ c } ψ α i ψ ψ ψ ψ | E i g e n v e c t o r s i = { Ξ c ( N - like ) , Ξ c } ψ α i ψ ψ ψ ψ | E i g e n v e c t o r s i = { Ξ c ( Λ - like ) , Ξ c } ψ α i ψ ψ ψ ψ | E i g e n v e c t o r s i = { Ξ c ( N - like ) , Ξ c } ψ α i ψ ψ ψ ψ FIG. 4. Ground state signals for the Ξ c (cid:16) ± (cid:17) and Ξ (cid:48) c (cid:16) ± (cid:17) channels, the former obtained via Λ-like and N -like operators.The left (right) three panels show the positive (negative) parity results. Eigenvectors for variational solutions of the i = { Ξ c (Λ-like) , Ξ (cid:48) c } (filled) and i = { Ξ c ( N -like) , Ξ (cid:48) c } (hollow) operator sets for both positive and negative parity channels aregiven to show the strength of the mixing between operators. The state index α follows the order of i and is directly related tothe signals in the upper effective mass plots. For instance, the filled green diamond ψ in the lower leftmost eigenvector plotindicates the Ξ c (Λ-like) signal associated with the Ξ c (Λ-like) operator. The hollow red square ψ of the lower rightmost plotis the Ξ (cid:48) c contribution to the Ξ c ( N -like) signal. firm that the Ξ c and Ξ (cid:48) c states have distinct signals. Incase of negative parity, there appears to be non-negligiblemixing between the two states dependent on the varia-tional parameters. Specifically, the Λ-like Ξ c has a neg-ligible Ξ (cid:48) c component, while the N -like Ξ c state has upto a 10% Ξ (cid:48) c mixing although the effect seems to dependon the variational parameters. The reason why the neg-ative parity Λ-like operator gives signals close to the Ξ (cid:48) c is understood to be related to the overestimation of themass obtained for that operator rather than a mixing ef-fect. The Ξ (cid:48) c appears to have at most a mixing of 5%with the N -like Ξ c . In all, we see that for negative parity the mixing is not completely negligible, but neverthelessquite small.
2. Smearing dependence a. Spin- / baryons: We observe that, evidently,the ground state signals remain stable with respect to thesmearing radius. The excited-state signals on the otherhand show a clear dependence to the smearing radius ofthe source quark fields. This is readily visible for everycase given in Figure 2. For both positive and negativeparity, states that are clearly separated from the groundstate tend to decrease as the smearing radius increaseswith no apparent plateau behavior. Note that all theenergies are extracted via a plateau approach, which aredependent on the choice of the fit windows. Extractingthe energies from two-exponential fits are more reliablefor the ∼ . . ∼ . × { χ , χ } − { , } row of Figure 2a. A similar behavioris seen for other combinations of operators and smearingsas well. Investigations of the eigenvectors show that allstates, e.g. ground or excited states, couple to the op-erators with the wider quark sources. This is confirmedindependently if we compare the higher states in the rows { χ , χ } −
50 and { χ , χ } − { , } of Figure 2a, wherethe extracted values coincide. We find this to be true forany variational analysis over multiple smearings. b. Spin- / baryons: We find that solving a 3 × × B. Charmed baryon spectrum
The energy levels from the diagonalized correlationfunctions are extracted by fitting the data to the formgiven in Equation (4). Additional exponential terms areemployed to stabilize the fits against excited-state con-tributions. In most of the cases, where the signal formsa plateau in the effective-mass plots, masses of the low-est states extracted from the one-exponential fits agreewith the multi-exponential fit results within their errorbars. Yet, a two-exponential form stabilizes the fits andimproves the accuracy of the results. This is especiallytrue when analyzing the widest smearing case. The ex-tracted energies are compiled in Table IV. Since we areat the isospin-symmetric point, m u = m d , our resultsshould be understood as the isospin averaged masses ofthe respective states.As we have discussed in Section IV A 2, a variationalanalysis over a set of different smearings for a fixed op-erator returns solution eigenvectors that couple to thewidest smearing. Therefore, we always use an operatorbasis with quark smearings fixed to the widest one. Forthe spin-1 / × { χ , χ , χ } and extract signals of three states for each channel. Thethird energy level with largest energy is however usu-ally lost to noise already at relatively early time slices ordecays to the ground states due to inaccuracies in the di-agonalization procedure of Equations (3) and (4). For in-stance, in case of the positive parity spin-1 / c baryons,we find that the state dominantly coupling to the χ op-erator decays to the ground state signal before showinga plateau that may be a candidate signal for an excitedstate (blue rectangles in the top left plot of Figure 5).Signals of possible third states for the spin-1 /
2, positiveparity Σ c , Ξ (cid:48) c and Ω c channels emerge in early time slicesof effective mass analyses but are quickly lost to noise. Itis usually possible to identify a fit region of 2-3 points forthe narrowest smearing but we find the energy extractedvia this approach to be unreliable, since the fit windowis very small and the smearing dependency of the statecannot be established. Positive parity spin-1 / cc andΩ cc signals mimic the behavior of Ξ c , where there appearsignals one could potentially identify as distinct states.However we find that those states are rather unstableunder the change of variational parameters. In addition,extracted energies are highly dependent on the extrac-tion method – plateau approach or a two-exponential fit.Therefore, even though we show their signals in the plots,we do not extract or report any corresponding energy val-ues.In general, we find that the negative parity sector ap-pears to be richer in comparison to the positive paritycase. Indeed, we could identify three distinct states formost of the negative parity spin-1 / a. Mass differences: Hyperfine splittings, the massdifferences between the spin-3 / / c , Ξ c , and Ω c channels are reproduced in goodagreement with the experimental values. Mass differ-ences between the positive and negative parity states alsoagree well with the available experimental results. Thefirst excited states of the positive parity baryons lie quitehigh, 400 MeV to 1 GeV, above the ground states. Acommon pattern is that, more than one negative par-ity state for the singly- and doubly-charmed spin-1 / c , Ξ (cid:48) c , Ω c , Ξ cc , and Ω cc channels lie close to eachother. The splittings between those states are smallerfor the Ω c and Ω cc baryons compared to those of Σ c , Ξ (cid:48) c ,and Ξ cc . The situation is different for the Λ c and the Ξ c baryons where the negative parity states are roughly 300MeV apart. b. Scattering states: It is essential to examine therelevant thresholds for the negative parity states in order0
TABLE IV. Extracted baryon masses in units of GeV.Baryon J P M M J P M M M Λ c
12 + − c
12 + − c
12 + (Λ-like) 2.474(11) 3.301(33) − ( N -like) 2.770(67) 3.059(10) 3.390(76)Ξ (cid:48) c
12 + − c
12 + − ∗ c
32 + − ∗ c
32 + − ∗ c
32 + − cc
12 + − cc
12 + − ∗ cc
32 + − ∗ cc
32 + − ccc
32 + − to check if they could correspond to scattering states. Itis possible for the negative parity ground states to cou-ple to the S - or D -wave scattering states of a positiveparity baryon and a negative parity meson. The rel-evant thresholds which respect to isospin, spin, parity,strangeness and charm quantum numbers are,Λ c → Σ c + π, Σ c → Λ c + π, Σ c → Σ c + π, Σ ∗ c → Σ ∗ c + π, Ξ c → Ξ c + π, Ξ c → Ξ (cid:48) c + π, Ξ c → Ξ ∗ c + π, Ξ c → Λ c + K, Ξ c → Σ c + K, Ξ c → Λ + D, Ξ (cid:48) c → Ξ c + π, Ξ (cid:48) c → Ξ (cid:48) c + π, Ξ (cid:48) c → Ξ ∗ c + π, Ξ (cid:48) c → Λ c + K, Ξ (cid:48) c → Σ c + K, Ξ ∗ c → Ξ ∗ c + π, Ω c → Ξ c + K, Ω c → Ξ (cid:48) c + K, Ω ∗ c → Ξ ∗ c + K, Ξ cc → Ξ cc + π, Ξ ∗ cc → Ξ ∗ cc + π, Ω cc → Ξ cc + K, Ω ∗ cc → Ξ ∗ cc + K. We plot the above two-particle thresholds together withthe extracted negative parity energies in Figure 6. Thetwo-particle scattering energies are calculated via E = (cid:112) M + p + (cid:112) M + p , where M i is the mass of theparticle and p i = 2 π n /L the lattice momentum. Weuse the π mass quoted in the PACS-CS paper [80] andthe experimental K mass, since we use a strange quarkmass re-tuned to its physical value via the K mass in-put [89], along with the positive parity baryon massesfrom Table IV of this work in calculating the thresholdenergies. The Λ + D threshold has to be estimated dif-ferently since we do not calculate the Λ baryon or the D meson in this work. In estimating the threshold, wetake the experimental Λ mass and multiply it by a cor-rection factor, Λ ourc / Λ expc , due to our overestimation ofthe Λ c mass. The uncertainty of this value is assumed tobe same as that of Λ ourc . The D meson mass is taken tobe its experimental value with its uncertainty neglected. The momenta p and p are set to zero. An inspectionof Figure 6 shows that some of the Ξ c baryon signals maycontain scattering states because of their vicinity to var-ious thresholds. Indeed, M [Ξ c ( − )], M , [Ξ (cid:48) c ( − )], and M [Ξ ∗ c ( − )] lie close to at least one related threshold.We also find some states that lie above the thresholds tobe close to their respective boosted ( n >
0) thresholds. c. Negative parity Σ c states: Three of our negativeparity Σ c states lie close to the PDG-listed Σ c (2800)baryon. BABAR reports a direct mass measurement ofthe Σ c state as M [Σ c (2800)] = 2846 ±
18 MeV/c . Belle,on the other hand, identifies the Σ c (2800) state from thesignals seen in the distribution of the mass difference,∆ M (Λ + c π ) ≡ M (Λ + c π ) − M (Λ + c ). The correspondingΣ c (2800) mass reported in the PDG based on this mea-sured difference is M [Σ c (2800)] = 2806 +5 − MeV/c , 40MeV/c lower than that of BABAR. It is noted in thePDG listings that the state that has been observed byBABAR might be a different Σ c excitation.Given that these states have been seen in the Λ c π in-variant mass spectra, a straightforward assignment forthe quantum numbers would be J P = 1 / − . From aquark model perspective (see paragraph f. ), there arethree possible low-lying negative parity spin-1/2 Σ c ex-citations. Two λ -modes with diquark spin j = 0 and j = 1, and a ρ -mode with diquark spin j = 1. In theheavy quark limit, the S -wave Σ c (2800) → Λ c π tran-sitions of the j = 1 λ - and ρ -modes would be forbid-den due to the violation of the spin-parity conservationof the light-quark degrees of freedom. A heavy quarkeffective theory calculation estimates a very large de-cay width, of the order of 885 MeV, for the j = 0 λ -mode [56], which rules out the 1 / − quantum numberfor Σ c (2800). On the other hand, a D -wave transitionis possible and points to the J P = 3 / − , / − possibil-1 ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇□ □ □ □ □ □ □ □ □ □ □ □ □ □○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ Ξ c + ○ α = □ α = ◇ α = t / a M e ff / a ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆● ● ● ● ● ● ● ● ● ● ● ● ● ●■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ Ξ c HQ - ● α = ◆ α = ■ α = t / a M e ff / a ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇□ □ □ □ □ □ □ □ □ □ □ □ □ □○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ Ξ cc + ○ α = □ α = ◇ α = t / a M e ff / a ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇□ □ □ □ □ □ □ □ □ □ □ □○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ Ξ cc - ○ α = □ α = ◇ α = t / a M e ff / a ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇□ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ Ω c + ○ α = □ α = ◇ α = ◇ α = t / a M e ff / a ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇□ □ □ □ □ □ □ □ □ □ □ □ □ □○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ Ω c - ○ α = □ α = ◇ α = t / a M e ff / a FIG. 5. Effective mass plots for representative baryons. Colored curves show the weighted two-exponential fits to the centralpoints. Bands spanning the plots show the energy levels and their 1 σ uncertainties extracted via configuration-by-configurationtwo-exponential fits. Plateau approach fit windows (colored rectangles) are shown for comparison only, as the two-exponentialfit is our preferred method of choice. We note that, although there appears a “bump” around t/a = 10, blue data points of theΞ c (1 / + ) and Ξ cc (1 / + ) plots show a decreasing trend (see the discussion in Section IV B). The Ω c (1 / + ) plot shows the signalsthat correspond to the narrowest and the widest smearings of the third state in relation to the discussion in Section IV B. ities. The lowest-lying Σ ∗ c ( − ) state we extract with amass of M = 2797 ±
38 MeV/c , might therefore be abetter suited candidate, which is situated in the vicin-ity of the masses M [Σ ++ c (2800)] = 2801 +4 − MeV/c and M [Σ + c (2800)] = 2792 +14 − MeV/c reported by the PDGbased on Belle’s measurements [2]. Additionally, the twolowest states that we extract for the Σ c ( − ) with masses M = 2814 ±
20 MeV/c and M = 2854 ±
17 MeV/c ,might be candidates for yet unobserved Σ c excitations.Note that the three extracted negative parity Σ c statesare well above their respective two-particle thresholds sothat the two-particle contribution to the signals should be suppressed. d. Excited Ξ c and Ξ (cid:48) c states: The experimental spec-trum of the Ξ c and Ξ (cid:48) c channels consists first of the re-spective J P = 1 / + ground states, and the first Ξ c ( − )excited state, which are all experimentally well estab-lished and which we reproduce well in our work. Theenergy levels above the lowest three are less well es-tablished, both experimentally and theoretically. Above2.9 GeV/c , the PDG reports the five states Ξ c (2930),Ξ c (2970), Ξ c (3055), Ξ c (3080) and Ξ c (3123), for none ofwhich the spin and parity quantum numbers have beenmeasured. Very recently, the spectrum of these stateshas received an update by a new measurement of the2 Λ c . . . . . . . . E [ G e V ] Σ c + π Σ c Λ c + π Σ c + π Ξ c Ξ c + π Ξ c + π Λ c + KΣ c + KΛ + D Ξ c Ξ c + π Ξ c + π Λ c + KΣ c + KΞ ∗ c + π Ω c Ξ c + KΞ c + K Σ ∗ c Σ ∗ c + π Ξ ∗ c Ξ c + π Ξ ∗ c + π Ω ∗ c Ξ ∗ c + K Ξ c + K Ξ cc . . . . . Ξ cc + π Ω cc Ξ cc + K Ξ ∗ cc Ξ ∗ cc + π Ω ∗ cc Ξ ∗ cc + K FIG. 6. The S -wave scattering thresholds for each charmed baryon channel. Open symbols are the extracted energies ofthe negative parity baryons, given in Table IV, that lie close to the thresholds. Horizontal lines with the shaded regions arethe calculated threshold energies with the statistical errors associated with the baryon energies’ uncertainties only. See the Scattering states part of Section IV B for our treatment of the Λ + D threshold. LHCb Collaboration [12] in the Λ + c K − channel. Accord-ing to this measurement, the Ξ c (2930) (observed earlierby the Belle [4] and the BABAR [10] Collaborations inthe same channel) should be considered to be a previ-ously unresolved combination of two independent statesΞ c (2923) and Ξ c (2939). The third observed state inRef. [12], Ξ c (2965), corresponds either to the already seenΞ c (2970), or is another entirely new resonance.Let us discuss potential interpretations of our findingswith regard to this rather rich experimental spectrum.We find two negative parity spin-1 / (cid:48) c states in thevicinity of the lowest three (or four) states above 2.9GeV/c , Ξ c (2923), Ξ c (2939), Ξ c (2965) and potentiallyΞ c (2970), which suggests that such quantum numberscan be assigned to at least two of these states. While ournumerical results are not precise enough to draw any firmconclusions, our obtained spectrum is most naturally in-terpreted as either Ξ c (2923) or Ξ c (2939) and similarlyΞ c (2965) or Ξ c (2970) being a Ξ (cid:48) c ( − ) state.The already known Ξ c (2970) state has been observedin the Λ c Kπ channel – also proceeding approximatelyhalf of the time via the intermediate Σ c (2455) K chan-nel – and in the Ξ (cid:48) c π , and Ξ c (2645) π channels by theBelle [5–7] and BABAR [11] Collaborations. These de-cay channels imply several possible quantum numbers, J P = (1 / ± , / ± , / ± ), for this state, which is not incontradiction with the above potential assignment.For the energy levels above 3.0 GeV/c , we obtain twostates in the region of the states Ξ c (3055), Ξ c (3080) andΞ c (3123), one Ξ c ( − ) and one Ξ (cid:48) c (
12 + ) state, respectively.Again, the uncertainties of the numerical results are toolarge for definite assignments, but point to the possibilitythat one of the three measured states is either a Ξ c ( − ) and a Ξ (cid:48) c (
12 + ) state.The Ξ c (3055) was observed by the Belle and theBABAR Collaborations in the Σ c K channel [8, 11]and in the Λ D channel only by the Belle Collabora-tion [9]. Masses reported by the Belle Collaboration are M [Ξ c (3055)] = 3059 . ± . and M [Ξ + c (3055)] =3055 . ± . , which are close to our secondΞ c ( − ) which lies above all the relevant lattice thresholdsand the physical Λ D threshold.Finally, the Ξ c (3080) was reported by the Belle Col-laboration [9] in the Σ c K , Σ ∗ c K , and Λ D channels andby the BABAR Collaboration [11] in the Λ c Kπ channelvia the Σ c (2455) K channel. Similar to the Ξ c (2970) case,these decay channels suggest several quantum numbers,such as J P = (1 / ± , / ± , / ± ). Our second Ξ (cid:48) c (
12 + )state appears to be the most probable candidate for thisresonance. e. Excited Ω c states: The five new excited Ω c statesreported by the LHCb Collaboration [13] were seen inthe Ξ c K channel. One would hence naively expect thesestates to have negative parity. A first dedicated latticeQCD calculation has confirmed this expectation by as-signing negative parity to these states [57], with totalspin ranging from J = 1 / /
2. The two Ω c ( − )states and the lowest-lying Ω ∗ c ( − ) state that we extractlie in the vicinity of these excited Ω c baryons observed bythe LHCb Collaboration. The pattern depicted in Fig-ure 6 matches that of the experimental spectrum wherethere are two states closer to the Ξ (cid:48) c K thresholds and onecoinciding with the Ξ (cid:48) c K . The second Ω c ( − ) and thelowest-lying Ω ∗ c ( − ) states are close to the Ξ (cid:48) c K thresh-old. The statistical error of the M [Ω ∗ c ( − )] state spans3most of the energy region of the LHCb states. It there-fore at this stage is rather futile to draw any definiteconclusions.We should reiterate that since we only employ localthree-quark operators, we are limited in our ability toresolve all molecular, radial or orbital excitation modesof the higher lying states. Our results should hence beconsidered as indicative in identifying potential compactthree-quark states among the experimentally observedenergy levels in the Ξ c and the Ω c channels. Conversely,the levels that we are not able to reproduce, could becandidates for molecular or orbitally excited states. Itis however at present too early to assign definite quan-tum numbers without a through scattering state analysissince some of our negative parity states lie close to thethresholds.The values in Table IV are illustrated in Figure 7 to-gether with the relevant experimental results. The latestΞ c results from the LHCb Collaboration are shown aswell. The similarities between the Λ c and Ξ c , and Σ c , Ξ (cid:48) c and Ω c are evident as expected from their flavor struc-tures. f. Interpretation from a quark model perspective The quark model (QM) has has been useful in givinga pictorial and intuitive interpretation of the mass spec-trum obtained by lattice QCD computations. The QMderives the energy and structure of a system by consid-ering constituent valence quarks and their interactions.For the excited states, in particular, it can clarify whatthe essential degrees of freedom in a specific excitationare.For heavy-quark baryons, the heavy-quark spin sym-metry plays an important role. As the coupling of a heavyquark to the magnetic component of gluons is suppressedby a 1 /m Q factor, the heavy quark spin is approximatelyconserved. For singly charmed baryons, this symmetry ismanifested by the appearance of heavy-quark spin dou-blets, in which spin ( j − / , j +1 /
2) pair states approacheach other with increasing quark masses. Here, j repre-sents the total spin minus the heavy quark spin of theconsidered baryon.We will here briefly compare the present lattice QCDresults with the QM predictions and study how the es-sential excitation modes arise in the spectrum. Quiteremarkably, multiple features of the QM predictions areconfirmed in the obtained lattice QCD spectrum of thecharmed baryons.1. Our lattice QCD results for the positive parity“ground” states agree completely with the QM as-signments, i.e. , the spin, parity, isospin and flavorrepresentation, and the mass orderings are consis-tent. The QM predictions for the splitting betweenthe spin 1 / / c is mostinteresting, because it contains three different va-lence quarks, c , s , and u/d . In the QM, the total spin of s and u/d can take either S = 0 (Ξ c ), or 1(Ξ (cid:48) c ). The existence of two low-lying positive par-ity states is indeed realized in lattice QCD as wellas in experiment. In the QM, the distinction of Ξ c and Ξ (cid:48) c is guaranteed by the flavor SU (3) symme-try, while the SU (3) breaking with m s (cid:54) = m u/d willmix the two Ξ c ’s. The QM predicts, however, thatthe mixing is suppressed for the ground state due tothe heavy quark spin symmetry, which is confirmedin our lattice QCD results.3. Low-lying negative parity singly charmed baryonsare described in the QM as orbital P -wave excita-tions. They are categorized in two classes, λ -modeand ρ -mode [17, 28]. The λ -mode is characterizedby the P -wave excitation between the charm quarkand the center of mass of the light quarks, while the ρ -mode is given by the excitation between the lightquarks. The QM predicts that the λ modes arelighter than the ρ -modes for singly heavy baryons.The QM spectrum depends on the flavor struc-ture: For the flavor anti-triplet Λ c and Ξ c , we finda set of (1 / − , 3 / − ) states in the λ -mode, and(1 / − ), (1 / − , 3 / − ) and (3 / − , 5 / − ) states inthe ρ -mode. Thus, among the three 1 / − states,the QM predicts that one λ -mode state is lighterthan the other two. This structure is indeed seenin the Λ c and Ξ c spectrum given in Table IV andFigure 7. The next 1 / − state is about 300 MeVhigher, which can be regarded as the mass splittingbetween the λ - and ρ -mode states.On the other hand, the flavor baryons, Σ c , Ξ (cid:48) c and Ω c , have two λ -mode 1 / − states, one of thembeing accompanied by a 3 / − state. In terms ofthe heavy-quark spin symmetry, we have a (1 / − ,3 / − ) spin doublet and an isolated singlet 1 / − .The lower two λ -mode states come close in energy,but can be distinguished by the total angular mo-mentum of the light-quark system. Thus we expecttwo 1 / − and one 3 / − states as the lowest nega-tive parity excitations for Σ c , Ξ (cid:48) c and Ω c . One seesthat, indeed, these three states turn out to be al-most degenerate in the lattice QCD spectrum ofthese channels in Table IV and Figure 7. Otherstates are much higher in energy, which again con-firms the predicted QM assignments.In all, the low-lying spectra of both the positive and neg-ative parity charmed baryons confirm the effectiveness ofthe QM in assigning the quantum numbers and symme-try properties of heavy baryons. g. Comparison to other lattice results: We compareour results to other lattice determinations and experi-mental values in Figure 8. Our positive parity groundstates are in good agreement with the experimental re-sults and the calculations of the other lattice groups withthe exception of the Λ c , which is overestimated in ourwork. Taken altogether, this is a good indication that we4 FIG. 7. Our results from Table IV laid over related experimental results. Boxes indicate the statistical uncertainties. Close bystates are shifted for clarity. All black and cyan lines are experimental results, solid (dashed) for the states with (un)determinedquantum numbers. Recent LHCb results [12] (cyan dashed) in the Ξ ( (cid:48) , ∗ ) c channels are included as well. are close to the physical point. The first excited positiveparity states also mostly agree with the predictions ofthe HSC [71, 72] and the RQCD Collaboration [69]. Fornegative parity, there are notable differences between ourand RQCD’s results, especially for the doubly-charmedbaryons. For the excited states of the Ξ cc and Ω cc , thereare discrepancies between our extracted spectrum andthat of RQCD, while our results are similar to those ob-tained by the HSC [72]. Although we do not show thecorresponding HSC spectrum in Figure 8, the patternthey extract in their preliminary studies for the negativeparity spin-1 / V. SUMMARY AND CONCLUSIONS
We have calculated the ground and the first few excitedstates of the charmed baryons on 2+1-flavor gauge config-urations with a pion mass of ∼
156 MeV/c . The charmquark is treated relativistically by employing a relativis-tic heavy-quark action to remove O ( am Q ) discretizationerrors. The states are extracted via a variational ap-proach over a set of interpolating fields with differentDirac structures and quark-field smearings. By perform-ing separate variational analyses with multiple subsets ofthe operator basis, we have studied the Dirac-structureand smearing dependence of the excited states. Our re-sults indicate that the excited-state signals are highlysusceptible to the width of the quark smearing. Addi-tionally, solutions of a variational analysis over a set of smeared operators with fixed Dirac structure couple dom-inantly to the operator that is smeared the widest withinour employed smearing parameter range. These resultshighlight the importance of forming the variational basisfrom different Dirac structures since relying on smearedoperators only might miss some parts of the spectrum.In comparing the operator dependence of the extractedpositive and negative parity states, we have extended the SU (4) operator basis of the Ξ c baryons to include notonly Λ-like, but also N -like operators. Both operatorsgive consistent results for the positive parity case whilethere appears a difference for the negative parity states.We have also investigated the Ξ c –Ξ (cid:48) c mixing by studyingthe cross-correlators of this system.Our masses of the low-lying states agree well with theavailable experimental results and previous lattice deter-minations. Consequently, the hyperfine splittings and themass differences between the positive and negative paritystates are reproduced, which is a good check of the rel-ativistic action we employ for the charm quark. Excitedstates in the positive parity channel lie 400 MeV to 1GeV above the ground states depending on the quantumnumbers. One or more negative parity states appear inbetween. This pattern is consonant with the QM expec-tations. Although we identify several states that are closeto observed excited Σ c , Ξ c and Ω c baryons, mostly in thenegative parity channels, some of the signals are in closeproximity to the related two-particle thresholds. With-out a thorough scattering state analysis with multiplevolumes and two-particle operators, the contaminationfrom the thresholds remain unidentified.From a qualitative point of view, the spectrum we ex-tract is similar to what has been reported by the HadronSpectrum Collaboration (HSC). This is quite encourag-5 M [ G e V ] 𝜴𝛬 𝛴 𝛯 𝛯 𝜴 𝛴 𝛯 c c c c c c c c ‘ ** * M [ G e V ] This work PosThis work NegTWQCD PosTWQCD NegETMCBrown et al.Briceno et al.Durr et al.PACS-CSRQCD PosRQCD NegHSC PosHSC Neg
𝜴𝛯 𝛯𝜴 𝜴 cc cc cc ccccc * *
Exp.
FIG. 8. Our results in comparison with the determinations of the ETMC [63], D¨urr et al. [64], Brown et al. [65], PACS-CS [66],TWQCD [67], Brice˜no et al. [68], RQCD [69], and HSC [71, 72]. Note that the lowest two data points of the HSC for theΩ cc ( − ) baryon are almost on top of each other. Error bars are smaller than the symbols for some points. Only the lowest-lyingexperimental values are shown. ing since the HSC employs a large operator basis includ-ing nonlocal operators. The qualitative agreement in-dicates the practicality of using local operators to probethe low-lying excitations, even though further work espe-cially regarding the proper treatment of scattering statesis still needed. ACKNOWLEDGMENTS
K.U.C. thanks M. Padmanath for discussions andsharing the Hadron Spectrum Collaboration’s doubly-charmed baryon results. The unquenched gauge con-figurations employed in our analysis were generated bythe PACS-CS Collaboration [80]. We have downloaded the publicly available configurations via the ILDG/JLDGnetwork [90, 91]. This work is supported in part byThe Scientific and Technological Research Council ofTurkey (TUBITAK) under project number 114F261 andin part by KAKENHI under Contract Nos. JP18K13542,JP19H05159, JP20K03940 and JP20K03959. K.U.C issupported in part by the Special Postdoctoral Researcher(SPDR) program of RIKEN and in part by the Aus-tralian Research Council Grant DP190100297 during thecourse of this work. H.B. acknowledges financial supportfrom the Scientific and Technological Research Council ofTurkey (TUBITAK) BIDEB-2219 Postdoctoral ResearchProgramme. P.G. is supported by the Leading Initiativefor Excellent Young Researchers (LEADER) of the JapanSociety for the Promotion of Science (JSPS). [1] M. Tanabashi et al. (Particle Data Group), Phys. Rev.D , 030001 (2018).[2] R. Mizuk et al. (Belle), Phys. Rev. Lett. , 122002(2005), arXiv:hep-ex/0412069 [hep-ex].[3] B. Aubert et al. (BaBar), Phys. Rev. D78 , 112003(2008), arXiv:0807.4974 [hep-ex].[4] Y. B. Li et al. (Belle), Eur. Phys. J.
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