Heavy Baryons with Strangeness in a Soliton Model

We present results from a chiral soliton model calculation for the spectrum of baryons with a single heavy quark (charm or bottom) and non-zero strangeness. We treat the strange components within a three flavor collective coordinate quantization of the soliton that fully accounts for light flavor symmetry breaking. Heavy baryons emerge by binding a heavy meson to the soliton. The dynamics of this heavy meson is described by the heavy quark effective theory with finite mass effects included.

it was extended to heavier baryons [11]. In those studies the relevance of the heavy spin-flavor symmetry was not yet recognized. Subsequently, also heavy vector meson fields were included [12]. More or less at the same time investigations were performed in the heavy limit scenario [13,14]. Those heavy limit studies neither included corrections to the heavy spin-flavor symmetry from finite masses nor strangeness degrees of freedom. In that case, baryons like Ξ c cannot be addressed. Strangeness was indeed included in Ref. [15], however, light flavor symmetry breaking was treated in a perturbation expansion and finite mass effects were omitted. This does not distinguish between even and odd parity or charm and bottom baryons and typically overestimates the binding energy of the heavy meson [7]. Also the parameters of the final energy formula were fitted rather than calculated from a realistic soliton model. These widespread bound state studies derive a potential for the meson fields from the soliton that is fixed in position. We note that this picture is strictly valid only in the large number of colors limit when the soliton is more massive than the heavy meson. Though this approach is a systematic and consistent expansion in the number of colors, kinematical corrections should be expected in the real world with three colors.
Our soliton model calculation for the spectrum of heavy baryons will improve with regard to the following aspects: We take the parameters in the mass formula from an actual soliton model calculation (we allow for moderate adjustment of the light flavor symmetry breaking strength), go beyond the perturbation expansion in that symmetry breaking and construct the heavy meson bound state from a model that systematically incorporates finite mass corrections. Our model calculation will produce an extensive picture of baryons, from the nucleon up to the Ω b . We will not consider doubly-heavy baryons, though.
The spectrum of heavy baryons has been investigated in other approaches as well. A comprehensive account of the (non-relativistic) quark model approach is given in Ref. [16] with some newer results reported in Ref. [17].
Relativistic effects are incorporated within quark-diquark models [18]. QCD sum rules were not only used to obtain the spectrum [19], but also to extract the heavy quark mass poles [20]. Lattice QCD calculations can be traced from Ref. [21] that also studies baryons with more than one heavy quark. Finally, Ref. [22] contains comprehensive reviews on baryon spectroscopy that discuss a variety of approaches and may be consulted for further references.

II. THE SOLITON MODEL
In chiral Lagrangians the interaction terms are ordered by the number of derivatives acting on the pseudoscalar fields. The more derivatives there are, the more unknown parameters appear in the Lagrangian. Replacing these higher derivatives by resonance exchange term is advantageous because more information is available to determine the parameters. We thus consider a chiral soliton that is stabilized by vector mesons ρ and ω [23] as a refinement of the Skyrme model [4,9]. Other shortcomings of the pseudoscalar soliton, like the neutron proton mass difference or the axial singlet matrix element of the nucleon are also solved when including light vector mesons [10].
The basic building block of the model is the chiral field U = exp i 8 a=1 φ a (x)λ a /f a , i.e. the non-linear realization of the pseudoscalar octet field φ a (x). Here f a are the respective decay constants [f π = 93MeV (for a = 1, 2, 3), f k = 114MeV (for a = 4, . . . , 7). The case a = 8 requires additional input [24] but is not relevant here.] and λ a are the eight Gell-Mann matrices of SU (3). The static field configuration of the soliton is the hedgehog ansatz (1) The isovector τ = (λ 1 , λ 2 , λ 3 ) comprises the three Pauli matrices from the isospin subspace of flavor SU (3). The spatial components of the ω µ and the time components of the ρ µ fields are zero. For the latter, i is an isospin/flavor index and m = 1, 2, 3 labels its spatial components. The profile functions F (r), ω(r) and G(r) enter the classical energy functional, E cl . The profiles are determined by the minimization of E cl , subject to boundary conditions that ensure unit baryon number: All profile functions vanish asymptotically. Configurations that are suitable for quantization are obtained by introducing time dependent collective coordinates for the flavor orientation A(t) ∈ SU (3) In addition profile functions are induced for the spatial components of ω µ and the time components of ρ µ [25,26].
Defining eight angular velocities Ω a via the time derivative of the collective coordinates allows a compact presentation of the Lagrange function for the collective coordinates from the light meson fields It is obtained from the spatial integral over the Lagrange density with the above described field configuration substituted. Note that the collective coordinates only appear via the angular velocities; A does not appear explicitly.

III. HEAVY MESON BOUND STATE
In effective meson theories, the heavy flavor enters via a heavy meson containing a single heavy quark (charm or bottom) of mass M . The dynamics of the heavy meson follows the heavy flavor effective theory [2] that treats the pseudoscalar (P ) and vector meson (Q µ ) components equivalently. That is, in the limit M → ∞ these components are part of a single multiplet (The constant four-velocity V µ characterizes the heavy quark rest frame.) The Lagrangian that describes the coupling of this multiplet to the light mesons including the vector mesons ρ and ω and respects the heavy spin-flavor symmetry is [28] 1 whereH = γ 0 H † γ 0 . We take the covariant derivative to be 4 D µ = ∂ µ + iv µ . The chiral currents of the light The heavy-light coupling constants d ≈ 0.53 and c ≈ 1.60 were determined from heavy meson decays. A field theory model that minimally extends to finite M and M * for the pseudoscalar and vector components, respectively, has also been constructed in Ref. [28] so that L H → L H in the heavy limit. Here Q µν and F µν are the field strength tensors of the heavy and light vector mesons, respectively. The central feature is that, through the coupling to the light meson soliton, solutions for the heavy meson fields emerge with energy 0 < ω < M , i.e. bound states. (Negative energy bound states are also possible. Eventually they build pentaquark baryons that will not be considered here.) The most strongly bound solution has P-wave structure in the pseudoscalar component: Here P and Q µ are three component spinors whose flavor content is parameterized by the (constant) spinor χ. construction, in particular with regard to finite M corrections, and their normalization to carry unit heavy charge is explained in Refs. [7] and [14], respectively. A heavy baryons is then a compound system of the soliton for the light flavors and the bound state of the heavy meson [8]. There are also bound states in the S-wave channel in which the heavy meson field is parameterized as (See Ref. [12] for parameterizations of higher angular momenta.) They combine with the soliton to form negative parity heavy baryons [7,14]. For convenience we have used equal symbols for the S and P -wave profile functions but, of course, they are different. The computation of the bound state energies ω from identifying localized solutions to the equations of motions that arise by substituting the parameterizations, Eqs. (9) and (10), into the Euler-Lagrange equations of Eq. (8) is detailed in appendix A of Ref. [7]. That reference also provides figures of the resulting profile functions.
The heavy meson fields must also account for the collective flavor rotation introduced in Eq. (3). This enforces the where the right hand sides contain the fields introduced in Eq. (9). This gives non-zero strange components of the heavy mesons and couples the heavy meson strange quark to that of the soliton. Substituting this flavor rotating configuration into the Lagrange density and integrating over space provides the collective coordinate Lagrange function from the heavy fields Again, the flavor rotation matrix A does not appear explicitly. With the time dependence of the collective coordinates, terms that involve 8 a=1 λ a Ω a enter. In the heavy meson sector the quadratic terms provide the bound state contributions to the moments of inertia α 2 and β 2 . Since the bound state wave-functions are strongly localized around the center of the soliton 5 the latter dominates the moments of inertia. It is thus safe to only retain the linear terms in Eq. (12). At that order only a = 1, 2, 3 and a = 8 survive because the bound states do not have any strangeness components. The normalization of the bound state wave-function dictates the coefficients in the first and second terms. The hyperfine splitting parameter ρ is a functional of all profile functions, including some of the induced light vector fields. Its explicit expression is given in Eqs. (B.1)-(B.4) of Ref. [14], where it is called χ P and χ S for P -and S-wave channels, respectively.

IV. QUANTIZATION IN SU(3), SYMMETRY BREAKING AND HYPERFINE SPLITTING
Before we construct a Hamilton operator for the collective coordinates via Legendre transformation of the Lagrangian L l + L h we recall that the rotations introduced in Eq. (3) are not exact zero modes in any sensitive model.
The reason is that SU (3) flavor symmetry is explicitly broken by different (current) quark masses. This breaking is measured by the ratio where the m q are the current quark masses of the respective quarks. It can be estimated from meson data [24,29,30].
In early soliton model studies this ratio was considered to be quite large, x ≈ 30 [24], or even bigger [31]. This was accompanied by sizable symmetry breaking among the hyperons [26]. Later this ratio was re-evaluated and found to be somewhat smaller: 20 ≤ x ≤ 25 [30]. Thus it is appropriate to consider this ratio for the (light) flavor symmetry breaking as a tunable parameter. Then symmetry breaking adds to the collective coordinate Lagrangian where D ab = 1 2 tr λ a Aλ b A † parameterizes the adjoint representation of the collective rotations. The coefficient γ is again a functional of the profile functions and acquires its main contribution from the classical fields, Eq. (1). It can be computed in any soliton model. (In the literature γ = x γ is typically used.) The heavy mesons also contribute to the symmetry breaking parameter by appropriately substituting mass matrices in Eq. (8). For example, for the charm heavy meson in the P -wave channel we have Numerically this contribution is small and can easily be compensated by a slight change of x. 5 Their asymptotic behavior is e −|ω|r ∼ e −M r compared to e −mπ r of the chiral field.
We have now collected all terms for the collective coordinate Lagrangian L(A, Ω) = L l (Ω) + L h (Ω) + L sb (A) and can construct the Hamilton operator by Legendre transformation, where R a = ∂L ∂Ωa defines the said Legendre transformation. The R a are the right generators of SU (3) since [A, R a ] = A(λ a /2) upon canonical quantization. The spinors χ contain annihilation and creation operators for the heavy meson bound state. They are quantized as ordinary harmonic oscillators. In particular χ † χ is the number operator for the heavy meson bound state. Since we are considering hadrons with a single heavy quark, contributions that are quartic in χ have been omitted for consistency. (In the square a term that is explicitly of quartic order is maintained because it cancels a similar term in a R 2 a , cf. subsection below.) The hyperfine splitting part, H hf , that emerges from the last term in Eq. (12), will be discussed later.

A. SU(3) diagonalization
The Hamiltonian, Eq. (16) is not complete without the constraint that arises from the terms linear in Ω 8 in Eqs. (5) and (12). Thus the heavy baryons have right hypercharge 2 /3.
Since the zero strangeness components of any SU (3) representation has equal hypercharge and right hypercharge the SU (3) coordinates must be quantized as diquarks for heavy baryons [15]. The most relevant diquark representations are the antisymmetric anti-triplet and the symmetric sextet.
When symmetry breaking is included, elements of higher dimensional representations with the same flavor and R 1,2,3 quantum numbers are admixed. We first determine the quantum number r in the intrinsic spin 3 i=1 R 2 i = r(r + 1): In addition to its dimensionality, an SU (3) representation is characterized by two sets of quantum numbers (I, I 3 , Y ) for the flavor and (r, r 3 , Y R ) for the R a degrees of freedom, respectively. The flavor generators are L a = 8 b=1 D ab R b with L 1,2,3 = I 1,2,3 and Y = 2 for a heavy baryon whose diquark component builds up from the anti-triplet and sextet, respectively. The admixture of higher dimensional representations has been estimated in a perturbation expansion for hyperons [32] and heavy baryons [15]. It can also be done exactly within the so-called Yabu-Ando approach [5]. The starting point is an Euler angle representation of the collective coordinates A in which the conjugate momenta R a are differential operators.
Then the eigenvalue equation is cast into a set of coupled ordinary second order differential equations. The single variable is the strangeness changing angle in A. The particular setting of the differential equations depends on the considered flavor quantum numbers.
For ordinary baryons (Y R = 1) this treatment is reviewed in Ref. [10] and the results for diquark wave-functions that enter the heavy baryon wave-functions (Y R = 2 /3) are reported in Ref. [33]. Having obtained the SU (3)-flavor eigenvalue ǫ from the differential equations we simplify the SU (3) part and write The dependence of the eigenvalues ǫ on x varies with spin and isospin. Hence there is implicit hyperfine splitting, however, it also appears explicitly as we discuss next.
Calling j the spin of the considered baryon this implies R · τ = j(j + 1) − r(r + 1) − 3 4 ∼ j(j + 1) − r(r + 1), where the expectation value refers to the heavy meson bound state. In the approximation we have again omitted terms that formally are quartic in χ. This scalar product appears in the Legendre transformation with respect to Ω, Collecting pieces we get the mass formula where N = 0, 1 counts the number of heavy valence quarks contained in the considered baryon. It has been included in the hyperfine splitting term since ordinary baryons have r = j. We have collected the leading contributions to the baryon energy in the large number of colors (N C ) expansion. However, a contribution O(N 0 C ) is missing, the vacuum polarization energy E vac . It is the quantum correction to the classical energy E cl that cannot be rigorously computed because the theory is not renormalizable. Estimates in the Skyrme model suggest that E vac considerably reduces E cl [35]. We circumvent this limitation by only considering mass difference for which E cl and E vac cancel and consequently omit these terms from Eq. (22). This quantization scheme predicts two heavy Ξ baryons with spin j = 1 /2: one has r = 0 and the other r = 1. In an SU (3) symmetric world the former would be an anti-triplet state and the latter a sextet state. There is no mixing between these baryons because H, R 2 = 0. For j = 3 /2 only one heavy Ξ baryon emerges in this scheme since then r = 1 is required. For the Ξ hyperon there is also only a single option with j = 1 /2 that is build from the octet state.
This counting suggests to relate r to the intermediate spin J m defined in Ref. [11].

V. NUMERICAL RESULTS
As mentioned above, we consider mass differences, because the model predictions for the absolute masses are subject to uncontrollable quantum contributions.
We find the energy eigenvalue ǫ in Eq. (18) for all baryons and then compute their energies according to Eq. (22).
We adopt the SU (3) parameters from Ref. [26]: α 2 = 5.144/GeV, β 2 = 4.302/GeV and 6 γ = 47MeV. For the heavy sector, the same soliton model was used in Ref. [14] to compute the bound state energies ω and hyperfine parameters ρ for both the P -and S-wave channels. From the model calculation described in section III the following bound state parameters were obtained ω P = 1326MeV , ρ P = 0.140 , ω S = 1572MeV , ρ S = 0.181 (23) and ω P = 4494MeV , ρ P = 0.053 , ω S = 4663MeV , ρ S = 0.046 (24) in the charm and bottom sector, respectively (Ref. [14] lists the binding energies ω P,S − M D and ω P,S − M B .). Then we are left with a single parameter, the effective symmetry breaking x defined in Eq. (13), that is not fully determined.
We list our results for the charm and bottom baryon spectra in table I, that also contains the data for experimentally observed candidates [34]. We note that most of the quantum numbers listed in Ref. [34] are adapted from the quark model and stress that r is not a physical observable. Hence assigning the experimental results for Ξ type baryons to a particular r value is a prediction. Ref. [34] furthermore lists Λ c (2625) and Λ b (5920) with spin j = 3 /2 that are not contained in our approach: We require |j − r| = 1 /2 but the Λ's have neither strangeness nor isospin so they must have r = 0 and j = 1 /2. We complete the picture by including the corresponding results for the low-lying non-heavy baryons in table II.   Finally we discuss our results for the masses of those strange heavy baryons that have previously not been considered in a heavy meson soliton model with realistic heavy meson masses: the Ξ's and Ω's. For the positive parity heavy strange baryons we again observe that the mass splittings within a heavy multiplet are overestimated. A moderate reduction of the symmetry breaking ratio x would be sufficient to match the experimental data. For the negative parity Ξ c with j = 1 /2 the too large binding of the S-wave reverses this picture. This is not the case for its spin 3 /2 counterpart. Interestingly enough, Ref. [34] assigns the quantum number of this resonance by assuming it to join an SU (4) multiplet with the negative parity Λ c (j = 3 /2). We have argued above that this Λ c is not contained in our approach but should be associated with a D-wave heavy meson. Thus, as indicated in table I, it is questionable to identify (I, j, r, p) = ( 1 /2, 3 /2, 1, −) with Ξ c (2815). Rather it is a prediction for an even heavier resonance like the observed Ξ c (2930) or Ξ c (2980) whose quantum numbers still need to be determined [34].

VI. CONCLUSION
We have presented a model calculation for the baryon spectrum that comprises light and heavy flavors. In particular we have focused on the role of light flavor symmetry breaking which is manifested by the strange quark being neither light nor heavy. When quantizing the flavor degrees of freedom, the corresponding deviations from the up-down sector are handled (numerically) exactly. In the heavy flavor sector the model is inspired by the heavy flavor symmetry, with subleading effects arising from finite masses included. The approach also includes the hyperfine splitting for the heavy baryons; a moderate effect that vanishes in the heavy limit. The model calculation is all-embracing as it contains spin 1 /2 and 3 /2 baryons starting from the lightest baryon (nucleon), including hyperons and extending to heavy baryons of either parity that have the valence quark content strange-strange-bottom. The spectrum is computed from a single mass formula where essentially all parameters are determined using data from the baryon number zero sector. We have also calculated masses for heavy baryons that are yet to be observed. Though we can only provide an estimate for their masses, we find a realistic indication for their positions relative to observed baryons.
The overall agreement with data is as expected for chiral soliton model estimates. As known from earlier studies, the mass predictions for the heavy baryons are on the low side when compared to the nucleon. Within a heavy baryon multiplet the computed mass differences are larger than the experimental data. This appears to be caused by too strong a remnant of the heavy spin-flavor symmetry in the approach. An understanding that goes beyond adopting reduced masses in the bound state approach is required. Furthermore the fine-tuning of the symmetry breaking ratio x as well as other model parameters that influence the soliton properties appears as an obvious endeavor. Actually, a complete analysis within a vector meson soliton model (but also a chiral quark model as the Nambu-Jona-Lasino model [36]) shows that additional symmetry breaking operators such as 3 i=1 D 8i R i [26] arise in the Hamiltonian, Eq. (16). Their effects on the heavy baryon spectrum will be reported in a forthcoming paper.