Spectrum of singly heavy baryons from a chiral effective theory of diquarks

The mass spectra of singly charmed and bottom baryons, $\Lambda_{c/b}(1/2^\pm,3/2^-)$ and $\Xi_{c/b}(1/2^\pm,3/2^-)$, 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. The effect of $U_A(1)$ anomaly induces an inverse hierarchy between the masses of strange and non-strange pseudoscalar diquarks, which leads to a similar inverse mass ordering in $\rho$-mode excitations of singly heavy baryons.

A phenomenon related to diquark degrees of freedom is the spectrum of singly heavy baryons (Qqq), where a baryon 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 well approximated as a diquark (for model studies about diquarks in Qqq baryons, e.g., see Refs. [18][19][20][21][22][23][24][25][26][27][28][29]). In particular, the spectrum of singly heavy baryons are promising candidates visibly affected by diquark degrees of freedom. For example, the P -wave excited states of singly heavy baryons are classified by λ modes (the orbital excitations between the diquark and heavy quark) and ρ modes (the orbital excitations between two light quarks inside the diquarks) [30,31].
The chiral symmetry and U A (1) symmetry are fundamental properties of light quarks in QCD, and in the low-energy region of QCD they are broken by the chiral condensates and U A (1) anomaly, respectively. Such symmetry breaking effects should be related to the properties of diquarks [29,32,33]. In Ref. [29], a chiral effective theory based on the SU (3) R ×SU (3) L chiral symmetry with the scalar (J P = 0 + , where J and P are the total angular momentum and parity, respectively) diquarks belonging * kimu.ryonhi@phys.kyushu-u.ac.jp † hiyama@riken.jp ‡ oka@post.j-parc.jp § k.suzuki.2010@th.phys.titech.ac.jp to the color antitriplet3 and flavor antitriplet3 channel and its pseudoscalar (0 − ) counterpart was constructed. 1 There the following new (and interesting) suggestions are given.
(i) Chiral partner structures of diquarks-A scalar diquark and its pseudoscalar partner belong to a chiral multiplet, which is the so-called chiral partner structure. This structure means that chiral partners are degenerate when the chiral symmetry is completely restored. As a result, they also predicted a similar chiral partner structure for charmed baryons such as Λ c (1/2 + )-Λ c (1/2 − ) and Ξ c (1/2 + )-Ξ c (1/2 − ) (for similar studies, see Refs. [35,36]).
(ii) Inverse hierarchy of diquark masses-The effect of the U A (1) anomaly leads to an inverse hierarchy for the masses of the pseudoscalar diquarks: M (us/ds, 0 − ) < M (ud, 0 − ). This is contrary to an intuitive ordering M (ud, 0 − ) < M (us/ds, 0 − ) expected from the larger constituent mass of the s quark than that of the u and d quarks. As a result of the inverse hierarchy, they also predicted a similar ordering for the charmed baryons: In this paper, we investigate the spectrum of singly heavy baryons by using a "hybrid" approach with the constituent diquarks based on the chiral effective theory [29] and nonrelativistic two-body potential model (sometimes simply called quark-diquark model). Our approach has the following advantages: (1) It can study the singly heavy-baryon spectrum based on the chiral partner structures of diquarks.
(2) It can introduce the inverse hierarchy of the pseudoscalar diquark masses originated from the U A (1) anomaly and examine its effects on the singly heavy baryons.
(3) It can take into account the contribution from the confining (linear and Coulomb) potential. This is an additional advantage missing in Ref. [29].
(4) It can predict λ-mode excited states of singly heavy baryons. This is more profitable than the approach in Ref. [29], where it will be difficult to calculate λmode excitations only by the effective Lagrangian though the ρ-mode states are naively estimated.
This paper is organized as follows. In Sec. II, we formulate the hybrid approach of the chiral effective theory and the potential model. In Sec. III, we show the numerical results. Section IV is devoted to our conclusion and outlook.

II. FORMALISM
In this section, we summarize the mass formulas of diquarks based on the chiral effective theory [29]. After that, we construct a nonrelativistic potential model for singly-heavy baryons composed of a heavy quark and a diquark.

A. Chiral effective Lagrangian
In this work, we concentrate on the scalar (0 + ) and pseudoscalar (0 − ) diquarks with color3 and flavor3. In the chiral effective theory of diquarks [29], we consider the 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) symmetry are broken, the mass terms for the diquarks are given by [29]  By diagonalizing the mass matrix (1) in R/L and flavor space, we obtain the mass formulas for the diquarks, M i (0 ± ) [29]: From Eqs. (2)-(5), we get From this relation with A > 1 and m 2 1 > m 2 2 , one finds the inverse mass hierarchy for the pseudoscalar diquarks: where the non-strange diquark (i = 3) is heavier than the strange diquark (i = 1, 2).

C. Potential quark-diquark model
In order to calculate the spectrum of singly heavy baryons, we apply a nonrelativistic two-body potential model with a single heavy quark and a diquark.
The nonrelativistic two-body Hamiltonian is written as where the indices Q and d denote the heavy quark and diquark, respectively. p Q/d and M Q/d are the momentum and mass, respectively. r = r d − r Q is the relative coordinate between the two particles. After subtracting the kinetic energy of the center of mass motion, the Hamiltonian is reduced to M d +M Q is the relative momentum and reduced mass, respectively.
For the potentail V (r), in this work we apply three types of potentials constructed by Yoshida et al. [31], Silvestre-Brac [37], and Barnes et al. [38]. These potentials consist of the Coulomb term with the coefficient α and the linear term with λ, where C in the last term is a "constant shift" of the potential, which is a model parameter depending on the specific system. 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, Cc, C b , Mc, and M b are the coefficients of Coulomb and linear 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 Cc 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 . Note that, only in Ref. [31], the coefficient α of the Coulomb term depends on 1/µ. In other word, this is a "mass-dependent" Coulomb interaction, which is motivated by the behavior of the potential obtained from lattice QCD simulations [39]. On the other hand, the other potentials [37,38] do not include such an effect. Such a difference between the potentials will lead to a quantitative difference also in singly heavy-baryon spectra.
In this work, the charm quark mass M c , bottom quark mass M b , α, and λ are fixed by the values estimated in the previous studies [31,37,38], which are summarized in Table I. The other parameters are determined in Sec. III A.
In order to numerically solve the Schrödinger equation, we apply the Gaussian expansion method [40,41].

A. Parameter determination
In this section, we determine the unknown model parameters such as the diquark masses M i (0 ± ) and constant shifts, C c for charmed baryons and C b for bottom baryons. The procedure is as follows: 1. Determination of C c -By inputting the mass of the ud scalar diquark, M 3 (0 + ), we determine the constant shift C c so as to reproduce the groundstate mass of Λ c . As M 3 (0 + ), we apply the value measured from recent lattice QCD simulations with 2 + 1 dynamical quarks [17]: As the mass of Λ c , we use the experimental value from PDG [42]: M (Λ c , 1/2 + ) = 2286.46 MeV.
Model II.-Another method is to determine M 1,2 (0 + ) and M 3 (0 − ) from the potential model and some known baryon masses, which we call Model II. After fixing C c , M 1,2 (0 + ) and M 3 (0 − ) are determined so as to reproduce M (Ξ c , 1/2 + ) and M ρ (Λ c , 1/2 − ), respectively. As input parameters, we use the experimental values of the ground-state Ξ c from PDG [42]: M (Ξ c , 1/2 + ) = 2469.42 MeV. For the mass of the ρ mode of the negative-parity Λ c , we use the value predicted by a nonrelativistic three-body calculation in Ref. [31]: 3. Determination of M 1,2 (0 − )-Using the mass relation (6) and our three diquark masses, M 3 (0 + ), M 1,2 (0 + ), and M 1,2 (0 − ), we determine the masses of us/ds pseudosalar diquarks, M 1,2 (0 − ). Here, we emphasize that the estimated M 1,2 (0 − ) reflects the inverse hierarchy for the diquark masses, which has never been considered in previous studies except for Ref. [29]. The constant shifts, C c and C b , estimated by us are summarized in Table I. The diquark masses predicted by us are shown in Table II. By definition, the diquark 2 We use M 3 (0 + ) and M 1,2 (0 + ) in the chiral limit in Ref. [17].
The chiral extrapolation of M 3 (0 − ) is not shown in Ref. [17], so that we use M 3 (0 − ) at the lowest quark mass (see Table 8 of Ref. [17]). 3 Note that while the known experimental value of negative-parity Λc, M (Λc, 1/2 − ) = 2592.25 MeV [42], is expected to be that of the λ-mode excitation, the resonance corresponding to the ρ-mode has not been observed.  . We compare the results from 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 values from PDG [42]. The asterisk ( * ) denotes the input values.
Chiral EFT [29] Potential masses in Model I are the same as the values from Method I in Ref. [29]. Here we focus on the comparison of the prediction from Model IIY and that from Method II in Ref. [29]. In both the approaches, the input values of M (Ξ c , 1/2 + ) = 2469 MeV and M ρ (Λ c , 1/2 − ) = 2890 MeV are the same. Our prediction is M 1,2 (0 + ) = 942 MeV, which is larger than 906 MeV estimated in Ref. [29]. This difference is caused by the existence of the confining potential (particularly, linear potential) which is not considered in the estimate in Ref. [29]. This tendency does not change in the results using the other potentials. Similarly, for M 3 (0 − ), we obtain 1406 MeV, which is significantly larger than 1329 MeV in Ref. [29].
Next we focus on the ordering of the pseudosalar diquarks. We find the inverse hierarchy M 1,2 (0 − ) < M 3 (0 − ) in all the models, which is consistent with the prediction in Ref. [29]. We emphasize that the inverse mass hierarchy of diquarks does not suffer from the confining potential.
Furthermore, from the diquark masses and Eqs. (2)-(5), we can determine the unknown parameters of chiral effective Lagrangian, m 0 , m 1 , and m 2 , which is also summarized in Table II. By definition, the values in Model I are the same as those from Method I in Ref. [29]. Here we compare our estimate from Model II and a naive es-timate from Method II in Ref. [29]. From Models IIY, IIS, and IIB, we conclude that these parameters are insensitive to the choices of the quark model potential. We also see that inclusion of the confining potential does not alter the parameters qualitatively. Quantitatively, the magnitude of these parameters is larger than that from Method II in Ref. [29], which is expected to be improved by taking into account the confining potential.

B. Spectrum of singly charmed baryons
The values of masses of singly charmed baryons are summarized in Table II. M (Λ c , 1/2 + ), M (Ξ c , 1/2 + ), and M ρ (Λ c , 1/2 − ) are the input values. Similarly to the ordering of M 1,2 (0 − ) < M 3 (0 − ), we find the inverse hierarchy for the ρ-mode excitations of the singly charmed baryons: M ρ (Ξ c , 1/2 − ) < M ρ (Λ c , 1/2 − ). This is our main 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 confining potential, which is consistent with the naive estimate with the chiral effective theory [29].
The energy spectra for Λ c and Ξ c from Model IY and IIY are shown in the left panels of Fig. 1 and 2. Here we [GeV] FIG. 1. The energy spectra of singly charmed and bottom baryons from our numerical results using Model IY. [GeV] 2.9 Λ " ( ) emphasize the qualitative difference between the spectra of the negative-parity Λ c and Ξ c . In the Λ c spectrum, the ρ mode is heavier than the λ mode, which is consistent with the three-body calculation [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 the Ξ c spectrum is smaller than that in the Λ c spectrum: where we note that the 1/2 − and 3/2 − states for λ modes in our model are degenerate as discussed later. The significant difference between Model IY and IIY is caused by M ρ (Λ c , 1/2 − ) which is related to M 3 (0 − ). From the diquark mass relation (6), a larger M 3 (0 − ) leads to a larger M 1,2 (0 − ). Then a heavier M ρ (Λ c , 1/2 − ) leads to a heavier M ρ (Ξ c , 1/2 − ). As a result, M ρ (Ξ c , 1/2 − ) from Model IIY is heavier than M ρ (Ξ c , 1/2 − ) from Model IY. Next, we discuss the masses of the λ modes. The λ modes are the excited states with the orbital angular momentum between the heavy quark and diquark, so that their masses are higher than those of the ground states, which is the "P -wave" states in our two-body potential model. Also, in singly heavy-baryon spectra, the masses of the λ modes is usually lower than that of the ρ modes, as shown by the three-body calculation [31]. In Model IY and IIY, the excitation energy from the ground state, For the other potentials, it is more than 400 MeV. This difference is caused by the coefficients α of the Coulomb interaction. In the Yoshida potential used in Model IY and IIY, α is relatively small, so that its wave function is broader. As a result, the difference between the wave functions of the ground and excited states becomes smaller, and the excitation energy also decreases.
The is assigned to the ρ mode, it is much smaller than our prediction. For the negative-parity Ξ c , when the experimental value of M (Ξ c , 1/2 − , 3/2 − ) is assigned to the λ modes, the value is close to our results from Models IS, IB, and IIY within 50 MeV. When the experimental value of M (Ξ c , 1/2 − ) 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 reproduce the known experimental values in any case. In addition, when these experimental values are assigned to the λ modes, the excitation energy of the λ modes from the ground state is estimated to be about 330-340 MeV, which is consistent with the results from Model IY and IIY.
We comment on the possible splitting in the λ modes. The splitting between 1/2 − and 3/2 − states are caused by the spin-orbit (LS) coupling. In order to study this splitting within our model, we need to introduce the LS coupling between the orbital angular momentum and the heavy-quark spin. In the heavy quark limit (m c → ∞), the two states are degenerate due to the suppression of the LS coupling, so that they are called the heavy-quark spin (HQS) doublet.

C. Spectrum of singly bottom baryons
For the singly bottom baryons, the input value is only the mass of the ground-state Λ b (1/2 + ), and here we give predictions for the other states. For the ground state of Ξ b (1/2 + ), our prediction with Model IY and IIY is in good agreement with the the known mass M (Ξ b , 1/2 + ) = 5794.45 MeV [42]. This indicates that the quark-diquark picture is approximately good for Ξ b (1/2 + ).
The energy spectra for Λ b and Ξ b from Model IY and Model IIY are shown in the right panels of Fig. 1 and  2. Similarly to the charmed baryon spectra, we again emphasize the difference between the Λ b and Ξ b spectra. For the ρ modes, we also find the inverse mass hierarchy:  [42,43], but its spin and parity are not determined so far.

D. Root mean square distance
We summarize the root mean square (RMS) distance, √r 2 , between the diquark and the heavy quark in Table III. We find the RMS distance of the ρ mode is smaller than those of the ground states and the λ mode. This is because the pseudoscalar diquark is heavier than the scalar diquark, M (0 − ) > M (0 + ). Then the kinetic energy of the system with M (0 − ) is suppressed, and, as a result, the wave function shrinks compared to its ground state with M (0 + ). Due to the inverse hierarchy of the diquark masses, we find also the inverse hierarchy for the RMS distance: which is different from the standard hierarchy seen in the ground states, √r 2 (Λ c , 1/2 + ) > √r 2 (Ξ c , 1/2 + ). The λ modes are the P -wave excitations within a twobody quark-diquark model, so that their RMS distance is larger than those of the ground and ρ-mode states which is "S-wave" states within our model. The RMS distances in the bottom baryons are shorter than those of the charmed baryons because of the heavier bottom quark mass.
We find that the RMS distances from Model IY and IIY 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 Yoshida potential in Model IY and IIY has the relatively small α, so that its wave function and the RMS distance are larger than those from other models.
Note that the real wave function of a diquark must have a size which is the distance between a light quark and another light quark. In our approach, namely the quarkdiquark model, diquarks are treated as a point particle, so that such a size effect is neglected. To introduce such an effect would be important for improving our model. In particular, it would be interesting to investigate the form factors of singly heavy baryons with the negative parity by lattice QCD simulations and to compare it with our predictions.

IV. CONCLUSION AND OUTLOOK
In this paper, we investigated the spectrum of singly heavy baryons using the hybrid approach of the chiral effective theory of diquarks and nonrelativistic quarkdiquark potential model.
Our findings are as follows: • We found the inverse mass hierarchy in the ρ-mode excitations of singly heavy baryons: M (Ξ Q , 1/2 − ) < M (Λ Q , 1/2 − ), which is caused by the inverse mass hierarchy of the pseudosaclar diquarks M (us/ds, 0 − ) < M (ud, 0 − ). This conclusion is the same as the naive estimate in Ref. [29], but it is important to note that the effect from the confining potential between a heavy quark and a diquark does not change this conclusion.
• We found that the mass splitting between the ρ-and λ-mode excitations in the Ξ Q spectrum is smaller than that in the Λ Q spectrum: The inverse mass hierarchy in singly heavy baryons can be also investigated by future lattice QCD simulations, as studied with quenched simulations [44][45][46][47][48][49][50] as well as with dynamical quarks [51][52][53][54][55][56][57][58][59][60]. Although studying negativeparity baryons from lattice QCD is more difficult than the positive-parity states, there are a few works for singlyheavy baryons [50,58,60]. Our findings give a motivation to examine the excited-state spectra from lattice QCD simulations. Here, the careful treatment of the chiral and U A (1) symmetry on the lattice would be required.
In this paper, we focused only on the scalar diquark and its chiral partner. As another important channel, the chiral-partner structure of the axialvector (1 + ) diquarks with the color3 and flavor 6 (the so-called "bad" diquarks [10,34]) could be related to the spectra of Σ Q , Σ * Q , Ξ Q , Ξ * Q , Ω Q , and Ω * Q baryons. Furthermore, the diquark correlations at high temperature is expected to modify the production rate of singly heavy-baryons in high-energy collision experiments [70][71][72]. In extreme environments such as high temperature and/or density, chiral symmetry breaking should be also modified, and it would strongly affect the the chiral partner structures of diquarks and the related baryon spectra.