Spectrum of singly heavy baryons from a chiral effective theory of diquarks
SSpectrum of singly heavy baryons from a chiral effective theory of diquarks
Yonghee Kim, ∗ Emiko Hiyama,
1, 2, † Makoto Oka,
3, 2, ‡ and Kei Suzuki § Department of Physics, Kyushu University, Fukuoka 819-0395, Japan Nishina Center for Accelerator-Based Science, RIKEN, Wako 351-0198, Japan Advanced Science Research Center, Japan Atomic Energy Agency (JAEA), Tokai 319-1195, Japan (Dated: July 10, 2020)The mass spectra of singly charmed and bottom baryons, Λ c/b (1 / ± , / − ) and Ξ c/b (1 / ± , / − ),are investigated using a nonrelativistic potential model with a heavy quark and a light diquark.The masses of the scalar and pseudoscalar diquarks are taken from a chiral effective theory. Theeffect of U A (1) anomaly induces an inverse hierarchy between the masses of strange and nonstrangepseudoscalar diquarks, which leads to a similar inverse mass ordering in ρ -mode excitations of singlyheavy baryons. I. INTRODUCTION
Diquarks, strongly correlated two-quark states, havea long history in hadron physics since the 1960s [1–8](see Refs. [9, 10] for reviews). It is an important con-cept for understanding the various physics in quantumchromodynamics (QCD), such as the baryon (and alsoexotic-hadron) spectra as well as the color superconduct-ing phase. The properties of various diquarks, such asthe mass and size, have been studied by lattice QCDsimulations [11–17].A phenomenon related to diquark degrees of freedomis the spectrum of singly heavy baryons (
Qqq ), where abaryon contains two light (up, down, or strange) quarks( q = u, d, s ) and one heavy (charm or bottom) quark( Q = c, b ), so that the two light quarks ( qq ) might be wellapproximated as a diquark (for model studies about di-quarks in Qqq baryons, e.g. , see Refs. [18–29]). In partic-ular, the spectrum of singly heavy baryons is a promisingcandidate visibly affected by diquark degrees of freedom.For example, the P -wave excited states of singly heavybaryons are classified by λ modes (the orbital excitationsbetween the diquark and heavy quark) and ρ modes (theorbital excitations between two light quarks inside thediquarks) [30, 31].The chiral symmetry and U A (1) symmetry are funda-mental properties of light quarks in QCD, and in thelow-energy region of QCD they are broken by the chiralcondensates and U A (1) anomaly, respectively. Such sym-metry breaking effects should be related to the propertiesof diquarks [29, 32, 33]. In Ref. [29], a chiral effective the-ory based on the SU (3) R × SU (3) L chiral symmetry withthe scalar ( J P = 0 + , where J and P are the total an-gular momentum and parity, respectively) diquarks be-longing to the color antitriplet ¯3 and flavor antitriplet ¯3 ∗ [email protected] † [email protected] ‡ [email protected] § [email protected] channel and its pseudoscalar (0 − ) counterpart was con-structed. These are the following new (and interesting)suggestions:(i)
Chiral partner structures of diquarks —A scalar di-quark and its pseudoscalar partner belong to a chi-ral multiplet, which is the so-called chiral part-ner structure. This structure means that chi-ral partners are degenerate when the chiral sym-metry is completely restored. As a result, theyalso predicted a similar chiral partner structurefor charmed baryons such as Λ c (1 / + )-Λ c (1 / − )and Ξ c (1 / + )-Ξ c (1 / − ) (for similar studies, seeRefs. [35, 36]).(ii) Inverse hierarchy of diquark masses —The effectof the U A (1) anomaly leads to an inverse hierar-chy for the masses of the pseudoscalar diquarks: M ( us/ds, − ) < M ( ud, − ). This is contrary toan intuitive ordering M ( ud, − ) < M ( us/ds, − )expected from the larger constituent mass of the s quark than that of the u and d quarks. Asa result of the inverse hierarchy, they also pre-dicted a similar ordering for the charmed baryons: M (Ξ c , / − ) < M (Λ c (1 / − )).In this paper, we investigate the spectrum of singlyheavy baryons by using a “hybrid” approach with theconstituent diquarks based on the chiral effective the-ory [29] and nonrelativistic two-body potential model(sometimes simply called quark-diquark model ). Our ap-proach has the following advantages:(i) It can study the singly heavy-baryon spectrumbased on the chiral partner structures of diquarks.(ii) It can introduce the inverse hierarchy of the pseu-doscalar diquark masses originated from the U A (1) The diquark with the color ¯3 and flavor ¯3 is often referred to asthe “good” diquark [10, 34]. a r X i v : . [ h e p - ph ] J u l anomaly and examine its effects on the singly heavybaryons.(iii) It can take into account the contribution from theconfining (linear and Coulomb) potential. This isan additional advantage missing in Ref. [29].(iv) It can predict λ -mode excited states of singly heavybaryons. This is more profitable than the approachin Ref. [29], where it will be difficult to calculate λ -mode excitations only by the effective Lagrangianthough the ρ -mode states are naively estimated.It should be noted that the diquark-heavy-quark ap-proach can cover all the excitation modes that appear inthe conventional quark model. The orbital excitations ofa three-quark system consist of the λ -mode, in which thediquark is intact, and the ρ -mode, in which the diquarkis internally excited. The latter can be represented by anew type of diquark. In the present approach, we con-sider only the scalar and pseudoscalar diquarks, but thereare many other possible diquarks [29]. Among them thevector and axial-vector diquarks are known to be low-lying and play major roles in the flavor baryons, suchas Σ c and Ω c . The chiral effective theory for the vec-tor and axial-vector diquarks and their couplings to thescalar and pseudoscalar diquarks is being considered inthe forthcoming paper.This paper is organized as follows. In Sec. II, we for-mulate the hybrid approach of the chiral effective theoryand the potential model. In Sec. III, we show the numer-ical results. Section IV is devoted to our conclusion andoutlook. II. FORMALISM
In this section, we summarize the mass formulas ofdiquarks based on the chiral effective theory [29]. Afterthat, we construct a nonrelativistic potential model forsingly heavy baryons composed of a heavy quark and adiquark.
A. Chiral effective Lagrangian
In this work, we concentrate on the scalar (0 + ) andpseudoscalar (0 − ) diquarks with color ¯3 and flavor ¯3 . Inthe chiral effective theory of diquarks [29], we considerthe right-handed and left-handed diquark fields, d R,i and d L,i , where i is the flavor index of a diquark. The i = 1( ds ) and i = 2 ( su ) diquarks include one strange quark,while the i = 3 ( ud ) diquark has no strange quark.When the chiral symmetry and flavor SU (3) symmetryare broken, the mass terms for the diquarks are given by [29] L mass = − m ( d R,i d † R,i + d L,i d † L,i ) − ( m + Am )( d R, d † L, + d L, d † R, + d R, d † L, + d L, d † R, ) − ( Am + m )( d R, d † L, + d L, d † R, ) , (1)where m , m , and m are the model parameters. m iscalled the chiral invariant mass . The term with m satis-fies the chiral symmetry, while the terms with m and m break the chiral symmetry spontaneously and explicitly. m and m are the coefficients of the six-point quark (ordiquark-duquark-meson) interaction motivated by the U A (1) anomaly and the eight-point quark (or diquark-duquark-meson-meson) interaction which conserves the U A (1) symmetry, respectively. A ∼ / SU (3) symmetry breaking due to the quarkmass difference, m s > m u (cid:39) m d . B. Mass formulas of diquarks
By diagonalizing the mass matrix (1) in
R/L and fla-vor space, we obtain the mass formulas for the diquarks, M i (0 ± ) [29]: M (0 + ) = M (0 + ) = (cid:113) m − m − Am , (2) M (0 + ) = (cid:113) m − Am − m , (3) M (0 − ) = M (0 − ) = (cid:113) m + m + Am , (4) M (0 − ) = (cid:113) m + Am + m . (5)From Eqs. (2)-(5), we get (cid:2) M , (0 + ) (cid:3) − (cid:2) M (0 + ) (cid:3) = (cid:2) M (0 − ) (cid:3) − (cid:2) M , (0 − ) (cid:3) = ( A − m − m ) . (6)From this relation with A > m > m , one findsthe inverse mass hierarchy for the pseudoscalar diquarks: M (0 − ) > M , (0 − ), where the nonstrange diquark ( i =3) is heavier than the strange diquark ( i = 1 , C. Potential quark-diquark model
In order to calculate the spectrum of singly heavybaryons, we apply a nonrelativistic two-body potentialmodel with a single heavy quark and a diquark.The nonrelativistic two-body Hamiltonian is writtenas H = p Q M Q + p d M d + M Q + M d + V ( r ) , (7)where the indices Q and d denote the heavy quark anddiquark, respectively. p Q/d and M Q/d are the momen-tum and mass, respectively. r = r d − r Q is the relativecoordinate between the two particles. After subtract-ing the kinetic energy of the center of mass motion, theHamiltonian is reduced to H = p µ + M Q + M d + V ( r ) , (8)where p = M Q p d − M d p Q M d + M Q and µ = M d M Q M d + M Q are the relativemomentum and reduced mass, respectively.For the potential V ( r ), in this work, we apply threetypes of potentials constructed by Yoshida et al. [31],Silvestre-Brac [37], and Barnes et al. [38]. These poten-tials consist of the Coulomb term with the coefficient α and the linear term with λ , V ( r ) = − αr + λr + C, (9)where C in the last term is a “constant shift” of thepotential, which is a model parameter depending on thespecific system.Note that, only in Ref. [31], the coefficient α of theCoulomb term depends on 1 /µ . In other word, this isa “mass-dependent” Coulomb interaction, which is mo-tivated by the behavior of the potential obtained fromlattice QCD simulations [39]. On the other hand, theother potentials [37, 38] do not include such an effect.Such a difference between the potentials will lead to aquantitative difference also in singly heavy-baryon spec-tra.In this work, the charm quark mass M c , bottom quarkmass M b , α , and λ are fixed by the values estimatedin the previous studies [31, 37, 38], which are summa-rized in Table I. The other parameters are determined inSec. III A.In order to numerically solve the Schr¨odinger equation,we apply the Gaussian expansion method [40, 41]. III. NUMERICAL RESULTSA. Parameter determination
In this section, we determine the unknown model pa-rameters such as the diquark masses M i (0 ± ) and con-stant shifts, C c for charmed baryons and C b for bottombaryons. The procedure is as follows:(i) Determination of C c —By inputting the mass ofthe ud scalar diquark, M (0 + ), we determine theconstant shift C c so as to reproduce the ground-state mass of Λ c . As M (0 + ), we apply the valuemeasured from recent lattice QCD simulations with2 + 1 dynamical quarks [17]: M (0 + ) = 725 MeV.As the mass of Λ c , we use the experimental valuefrom PDG [42]: M (Λ c , / + ) = 2286 .
46 MeV.(ii)
Determination of M , (0 + ) and M (0 − )—After fix-ing C c , we next fix two diquark masses, M , (0 + )and M (0 − ). Here, we apply the following twomethods: Model I. —The first method is to input M , (0 + ) and M (0 − ) measured from recentlattice QCD simulations [17]: M , (0 + ) =906 MeV and M (0 − ) = 1265 MeV. We callthe choice of these parameters
Model I , whichis similar to Method I in Ref. [29].
Model II. —Another method is to determine M , (0 + ) and M (0 − ) from the potentialmodel and some known baryon masses, whichwe call Model II . After fixing C c , M , (0 + )and M (0 − ) are determined so as to repro-duce M (Ξ c , / + ) and M ρ (Λ c , / − ), respec-tively. As input parameters, we use the ex-perimental values of the ground-state Ξ c fromPDG [42]: M (Ξ c , / + ) = 2469 .
42 MeV. Forthe mass of the ρ mode of the negative-parityΛ c , we use the value predicted by a nonrel-ativistic three-body calculation in Ref. [31]: M ρ (Λ c , / − ) = 2890 MeV. (iii) Determination of M , (0 − )—Using the mass rela-tion (6) and our three diquark masses, M (0 + ), M , (0 + ), and M , (0 − ), we determine the massesof us/ds pseudoscalar diquarks, M , (0 − ). Here,we emphasize that the estimated M , (0 − ) reflectsthe inverse hierarchy for the diquark masses, whichhas never been considered in previous studies ex-cept for Ref. [29].(iv) Determination of C b —For singly bottom baryons,we also determine the constant shift C b by in-putting M (0 + ) = 725 MeV and reproducingthe mass of the ground state Λ b , M (Λ b , / + ) =5619 .
60 MeV [42]. For the diquark masses, M , (0 + ), M (0 − ), and M , (0 − ), we use the samevalues as the case of the charmed baryons.The constant shifts, C c and C b , estimated by us aresummarized in Table I. The diquark masses predictedby us are shown in Table II. By definition, the diquarkmasses in Model I are the same as the values from MethodI in Ref. [29]. Here we focus on the comparison ofthe prediction from Model IIY and that from MethodII in Ref. [29]. In both the approaches, the input val-ues of M (Ξ c , / + ) = 2469 MeV and M ρ (Λ c , / − ) =2890 MeV are the same. Our prediction is M , (0 + ) =942 MeV, which is larger than 906 MeV estimated inRef. [29]. This difference is caused by the existence of theconfining potential (particularly, linear potential) which We use M (0 + ) and M , (0 + ) in the chiral limit in Ref. [17].The chiral extrapolation of M (0 − ) is not shown in Ref. [17], sothat we use M (0 − ) at the lowest quark mass (see Table 8 ofRef. [17]). Note that while the known experimental value of negative-parityΛ c , M (Λ c , / − ) = 2592 .
25 MeV [42], is expected to be thatof the λ -mode excitation, the resonance corresponding to the ρ -mode has not been observed. TABLE I. Potential model parameters used in this work. We apply three types of potentials, Yoshida (Potential Y) [31],Silvestre-Brac (Potential S) [37], and Barnes (Potential B) [38]. α , λ , C c , C b , M c , and M b are the coefficients of Coulomb andlinear terms, constant shifts for charmed and bottom baryons, and masses of constituent charm and bottom quarks, respectively. µ is reduced mass of two-body systems. The values of C c and C b are fitted by our model. For Potential B, M b is not given [38],so that we do not fit C b . α λ (GeV ) C c (GeV) C b (GeV) M c (GeV) M b (GeV)Potential Y [31] (2 / × /µ . − . − .
819 1 .
750 5 . . . − . − .
696 1 .
836 5 . / × . . − . . . . . . . . TABLE II. List of numerical values of scalar [ M (0 + ), M , (0 + )] and pseudoscalar [ M (0 − ), M , (0 − )] diquark masses, massesof singly-heavy baryons [ M (Λ Q ), M (Ξ Q )], coefficients of chiral effective Lagrangian ( m , m , and m ). We compare the resultsfrom our approach using three potential and two parameters, Yoshida (denoted as IY and IIY), Silvestre-Brac (IS and IIS),and Barnes (BI and IIS) with a naive estimate in the chiral EFT [29] (Method I and Method II) and the experimental valuesfrom PDG [42]. The asterisk ( ∗ ) denotes the input values.Chiral EFT [29] Potential model (this work)Mass (MeV) Method I Method II IY IS IB IIY IIS IIB Experiment [42] M (0 + ) 725 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ M , (0 + ) 906 ∗
906 906 ∗ ∗ ∗
942 977 983 M (0 − ) 1265 ∗ ∗ ∗ ∗ M , (0 − ) 1142 1212 1142 1142 1142 1271 1331 1341 M (Λ c , / + ) 2286 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ . M (Ξ c , / + ) 2467 2469 ∗ ∗ ∗ ∗ . M ρ (Λ c , / − ) 2826 2890 ∗ ∗ ∗ ∗ . . .M ρ (Ξ c , / − ) 2704 2775 2647 2600 2594 2765 2758 2758 (2793 . M λ (Λ c , / − , / − ) . . . . . . . M λ (Ξ c , / − , / − ) . . . . . . . M (Λ b , / + ) . . . . . . . . . ∗ ∗ . . . . M (Ξ b , / + ) . . . . . . . . . . . . . M ρ (Λ b , / − ) . . . . . . . . . . . . (5912 . M ρ (Ξ b , / − ) . . . . . . . . . . . . . . .M λ (Λ b , / − , / − ) . . . . . . . . . . . . (5917 . M λ (Ξ b , / − , / − ) . . . . . . . . . . . . . . . Parameter (MeV ) m (1031) (1070) (1031) (1031) (1031) (1119) (1168) (1176) m (606) (632) (606) (606) (606) (690) (746) (754) m − (274) − (213) − (274) − (274) − (274) − (258) − (298) − (303) is not considered in the estimate in Ref. [29]. This ten-dency does not change in the results using the other po-tentials. Similarly, for M (0 − ), we obtain 1406 MeV,which is significantly larger than 1329 MeV in Ref. [29].Next, we focus on the ordering of the pseudoscalardiquarks. We find the inverse hierarchy M , (0 − ) B. Spectrum of singly charmed baryons The values of masses of singly charmed baryons aresummarized in Table II. M (Λ c , / + ), M (Ξ c , / + ), and M ρ (Λ c , / − ) are the input values. Similarly to the or-dering of M , (0 − ) < M (0 − ), we find the inverse hier-archy for the ρ -mode excitations of the singly charmedbaryons: M ρ (Ξ c , / − ) < M ρ (Λ c , / − ). This is ourmain conclusion: the inverse mass hierarchy between the ρ mode of Λ c (without a strange quark) and that of Ξ c (with a strange quark) is realized even with the confin-ing potential , which is consistent with the naive estimatewith the chiral effective theory [29].The energy spectra for Λ c and Ξ c from Models IYand IIY are shown in the left panels of Figs. 1 and2. Here we emphasize the qualitative difference be-tween the spectra of the negative-parity Λ c and Ξ c . Inthe Λ c spectrum, the ρ mode is heavier than the λ mode, which is consistent with the three-body calcula-tion [31]. On the other hand, in the Ξ c spectrum, the ρ and λ modes are close to each other. As a result,the mass splitting between the ρ and λ modes in theΞ c spectrum is smaller than that in the Λ c spectrum: | M ρ (Ξ c , / − ) − M λ (Ξ c , / − , / − ) | < | M ρ (Λ c , / − ) − M λ (Λ c , / − , / − ) | , where we note that the 1 / − and3 / − states for λ modes in our model are degenerate asdiscussed later. The significant difference between Mod-els IY and IIY is caused by M ρ (Λ c , / − ) which is re-lated to M (0 − ). From the diquark mass relation (6), alarger M (0 − ) leads to a larger M , (0 − ). Then a heav-ier M ρ (Λ c , / − ) leads to a heavier M ρ (Ξ c , / − ). Asa result, M ρ (Ξ c , / − ) from Model IIY is heavier than M ρ (Ξ c , / − ) from Model IY.Next, we discuss the masses of the λ modes. The λ modes are the excited states with the orbital angular mo-mentum between the heavy quark and diquark, so thattheir masses are higher than those of the ground states,which is the “ P -wave” states in our two-body potentialmodel. Also, in singly heavy-baryon spectra, the massesof the λ modes are usually lower than that of the ρ modes,as shown by the three-body calculation [31]. In ModelsIY and IIY, the excitation energy from the ground state, M λ (Λ c , / − , / − ) − M (Λ c , / + ), is about 300 MeV.For the other potentials, it is more than 400 MeV. Thisdifference is caused by the coefficients α of the Coulombinteraction. In the Yoshida potential used in ModelsIY and IIY, α is relatively small, so that its wave func-tion is broader. As a result, the difference between thewave functions of the ground and excited states becomessmaller, and the excitation energy also decreases.The known experimental values of the negative-parity Λ c and Ξ c are M (Λ c , / − ) = 2592 . 25 MeV, M (Λ c , / − ) = 2628 . 11 MeV, M (Ξ c , / − ) =2793 . 25 MeV, and M (Ξ c , / − ) = 2818 . 45 MeV [42].The λ modes in our results correspond to the spin averageof 1 / − and 3 / − . The spin averages of the experimen-tal values are M (Λ c , / − , / − ) = 2616 . 16 MeV and M (Ξ c , / − , / − ) = 2810 . 05 MeV. For the negative-parity Λ c , the experimental value of M (Λ c , / − , / − )is expected to be λ modes. Then our predictions fromModels IY and IIY are in good agreement with the exper-imental value. If the experimental value of M (Λ c , / − )is assigned to the ρ mode, it is much smaller than our prediction. For the negative-parity Ξ c , when the exper-imental value of M (Ξ c , / − , / − ) is assigned to the λ modes, the value is close to our results from Mod-els IS, IB, and IIY within 50 MeV. When the experi-mental value of M (Ξ c , / − ) is assigned to the ρ mode,the value is close to our results from Models IIY, IIS,and IIB within 50 MeV. Thus, Model IIY can repro-duce the known experimental values in any case. In ad-dition, when these experimental values are assigned tothe λ modes, the excitation energy of the λ modes fromthe ground state is estimated to be about 330-340 MeV,which is consistent with the results from Models IY andIIY.We comment on the possible splitting in the λ modes.The splitting between 1 / − and 3 / − states is causedby the spin-orbit (LS) coupling. In order to study thissplitting within our model, we need to introduce the LScoupling between the orbital angular momentum and theheavy-quark spin. In the heavy-quark limit ( m c → ∞ ),the two states are degenerate due to the suppression ofthe LS coupling, so that they are called the heavy-quarkspin doublet . C. Spectrum of singly bottom baryons For the singly bottom baryons, the input value is onlythe mass of the ground-state Λ b (1 / + ), and here we givepredictions for the other states. For the ground stateof Ξ b (1 / + ), our prediction with Models IY and IIY isin good agreement with the known mass M (Ξ b , / + ) =5794 . 45 MeV [42]. This indicates that the quark-diquarkpicture is approximately good for Ξ b (1 / + ).The energy spectra for Λ b and Ξ b from Models IY andIIY are shown in the right panels of Figs. 1 and 2. Sim-ilarly to the charmed baryon spectra, we again empha-size the difference between the Λ b and Ξ b spectra. Forthe ρ modes, we also find the inverse mass hierarchy: M ρ (Ξ b , / − ) < M ρ (Λ b , / − ). The difference betweenModels IY and IIY is similar to the charmed baryons.The known experimental values of negative-parity Λ b are M (Λ b , / − ) = 5912 . 20 MeV and M (Λ b , / − ) = 5919 . 92 MeV [42], and their spin averageis M (Λ b , / − , / − ) = 5917 . 35 MeV. Whether thesestates are the ρ mode or λ mode is not determinedyet. When the experimental value of M (Λ b , / − , / − )is assigned to the λ modes, the value is in agreementwith the results from Models IY and IIY. On the otherhand, when the experimental value of M (Λ b , / − ) isassigned to the ρ mode, it is quite smaller than ourprediction. This fact indicates that the experimentalvalues correspond to λ modes. The negative-parityΞ b is still not observed experimentally. In 2018, aheavier state Ξ b (6227) with M (Ξ b ) = 6226 . TABLE III. Rms distance √ ˆ r between a heavy quark and a diquark.Rms distance (fm) IY IS IB IIY IIS IIB √ ˆ r (Λ c , / + ) 0 . 587 0 . 512 0 . 506 0 . 587 0 . 512 0 . √ ˆ r (Ξ c , / + ) 0 . 559 0 . 476 0 . 469 0 . 555 0 . 466 0 . √ ˆ r ρ (Λ c , / − ) 0 . 523 0 . 431 0 . 421 0 . 513 0 . 412 0 . √ ˆ r ρ (Ξ c , / − ) 0 . 534 0 . 444 0 . 435 0 . 523 0 . 425 0 . √ ˆ r λ (Λ c , / − , / − ) 0 . 832 0 . 783 0 . 814 0 . 832 0 . 783 0 . √ ˆ r λ (Ξ c , / − , / − ) 0 . 792 0 . 738 0 . 767 0 . 785 0 . 724 0 . √ ˆ r (Λ b , / + ) 0 . 548 0 . . . . . 548 0 . . . . √ ˆ r (Ξ b , / + ) 0 . 515 0 . . . . . 510 0 . . . . √ ˆ r ρ (Λ b , / − ) 0 . 471 0 . . . . . 459 0 . . . . √ ˆ r ρ (Ξ b , / − ) 0 . 484 0 . . . . . 471 0 . . . . √ ˆ r λ (Λ b , / − , / − ) 0 . 776 0 . . . . . 776 0 . . . . √ ˆ r λ (Ξ b , / − , / − ) 0 . 728 0 . . . . . 720 0 . . . . D. Root-mean-square distance We summarize the root-mean-square (rms) distance, √ ˆ r , between the diquark and the heavy quark in Ta-ble III. We find the rms distance of the ρ mode is smallerthan those of the ground states and the λ mode. Thisis because the pseudoscalar diquark is heavier than thescalar diquark, M (0 − ) > M (0 + ). Then the kineticenergy of the system with M (0 − ) is suppressed, and,as a result, the wave function shrinks compared to itsground state with M (0 + ). Due to the inverse hierarchyof the diquark masses, we find also the inverse hierarchyfor the rms distance, √ ˆ r ρ (Λ c , / − ) < √ ˆ r ρ (Ξ c , / − ),which is different from the standard hierarchy seen in theground states, √ ˆ r (Λ c , / + ) > √ ˆ r (Ξ c , / + ).The λ modes are the P -wave excitations within a two-body quark-diquark model, so that their rms distanceis larger than those of the ground and ρ -mode stateswhich is “ S -wave” states within our model. The rmsdistances in the bottom baryons are shorter than thoseof the charmed baryons because of the heavier bottomquark mass.We find that the rms distances from Models IY andIIY are larger than those from the other models IS, IB,IIS, and IIB. This difference is caused by the coefficient α of the attractive Coulomb interaction. The Yoshidapotential in Models IY and IIY has the relatively small α , so that its wave function and the rms distance arelarger than those from other models.Note that the real wave function of a diquark must havea size which is the distance between a light quark andanother light quark. In our approach, namely, the quark-diquark model, diquarks are treated as a point particle,so that such a size effect is neglected. To introduce suchan effect would be important for improving our model. Inparticular, it would be interesting to investigate the formfactors of singly heavy baryons with the negative parityby lattice QCD simulations and to compare it with ourpredictions. IV. CONCLUSION AND OUTLOOK In this paper, we investigated the spectrum of singlyheavy baryons using the hybrid approach of the chiraleffective theory of diquarks and nonrelativistic quark-diquark potential model.Our findings are as follows:(i) We found the inverse mass hierarchy in the ρ -mode excitations of singly heavy baryons, M (Ξ Q , / − ) < M (Λ Q , / − ), which is caused bythe inverse mass hierarchy of the pseudoscalar di-quarks M ( us/ds, − ) < M ( ud, − ). This conclu-sion is the same as the naive estimate in Ref. [29],but it is important to note that the effect from theconfining potential between a heavy quark and adiquark does not change this conclusion.(ii) We found that the mass splitting between the ρ - and λ -mode excitations in the Ξ Q spec-trum is smaller than that in the Λ Q spec-trum: | M ρ (Ξ Q , / − ) − M λ (Ξ Q , / − , / − ) | < | M ρ (Λ Q , / − ) − M λ (Λ Q , / − , / − ) | .The inverse mass hierarchy in singly heavy baryons canbe also investigated by future lattice QCD simulations,as studied with quenched simulations [44–50], as wellas with dynamical quarks [51–61]. Although studyingnegative-parity baryons from lattice QCD is more diffi-cult than the positive-parity states, there are a few worksfor singly heavy baryons [50, 58, 60, 61]. Our findings givea motivation to examine the excited-state spectra fromlattice QCD simulations. Here, the careful treatment ofthe chiral and U A (1) symmetry on the lattice would berequired. Furthermore, the chiral effective Lagrangianfor singly heavy baryons, as formulated in Sec. III-F ofRef. [29], is a useful approach for analytically studyingthe inverse mass hierarchy of heavy baryons. In this La-grangian, the assignment of the chiral partners for heavybaryons is the same as that for the diquarks in this work,so that we can obtain a similar spectrum.The internal structures of excited states, such as ρ and λ modes, can significantly modify their decay proper-ties [62–70], and to study the decay processes taking intoaccount the inverse hierarchy will be important.In this paper, we focused only on the scalar diquarkand its chiral partner. As another important channel,the chiral-partner structure of the axial-vector (1 + ) di-quarks with the color ¯3 and flavor (the so-called “bad”diquarks [10, 34]) could be related to the spectra of Σ Q ,Σ ∗ Q , Ξ (cid:48) Q , Ξ ∗ Q , Ω Q , and Ω ∗ Q baryons.Furthermore, the diquark correlations at high temper-ature are expected to modify the production rate of singly heavy baryons in high-energy collision experiments [71–73]. In extreme environments, such as high temperatureand/or density, chiral symmetry breaking should be alsomodified, and it would strongly affect the chiral partnerstructures of diquarks and the related baryon spectra. ACKNOWLEDGMENTS We thank Masayasu Harada and Yan-Rui Liu for use-ful discussions. This work was supported in part byJSPS KAKENHI Grants No. JP18H05407 (E.H.), No.JP19H05159 (M.O.), and No. JP17K14277 (K.S.). [1] M. Gell-Mann, “A schematic model of baryons andmesons,” Phys. 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