Analysis of the [56,2^+] Baryon Masses in the 1/Nc Expansion

The mass spectrum of the positive parity [56,2^+] baryons is studied in the 1/Nc expansion up to and including O(1/Nc) effects with SU(3) symmetry breaking implemented to first order. A total of eighteen mass relations result, several of which are tested with the available data. The breaking of spin-flavor symmetry is dominated by the hyperfine interactions, while spin-orbit effects are found to be small.

approach that has turned out to be very successful in baryon phenomenology.
The 1/N c expansion has been applied to the ground state baryons [4,5,6,7,8,9,10], and to excited baryons, where the masses and decays of the negative parity spin-flavor 70plet [11,12,13,14,15] and the positive parity Roper 56-plet [16] have been analyzed. Two frameworks have been used in implementing the 1/N c expansion for baryons. One framework is based on the contracted spin-flavor SU(2N f ) c symmetry, N f being the number of light flavors, which is a symmetry of QCD in the N c → ∞ limit [4,12,17]. In this framework commutation relations of operators like axial currents and hadron masses are constrained by consistency relations. The observed baryons at N c = 3 are identified with the low lying spin states of an infinite representation of the contracted symmetry. The second framework makes use of the spin-flavor SU(2N f ) algebra, with an explicit representation of operators that act on a space of states constructed as tensor products of N c valence quarks [7]. Both approaches are consistent and deliver equivalent results order by order in the 1/N c expansion.
From the practical point of view, however, the second one is easier to work with, especially at subleading orders in 1/N c , and for this reason it has been chosen in most analyses. Another advantage in this approach, is the possibility of using the language of the constituent quark model, as applied to the spin-flavor degrees of freedom, without any loss of generality.
The study of excited baryons is not free of difficulties. Although a significant amount of symmetry in the form of a contracted SU(2N f ) c is always present in the N c → ∞ limit [12,18], there is no strict spin-flavor symmetry in that limit. Indeed, as it was shown in [11], spin-orbit interactions break spin-flavor symmetry at O(N 0 c ) in states belonging to mixed symmetric spin-flavor representations, and configuration mixing, i.e., mixing of states belonging to different spin-flavor multiplets in general occurs at O(N 0 c ) as well. The use of spin-flavor symmetry as a zeroth order approximation is therefore not warranted for excited baryons. However, a phenomenological fact is that spin-orbit interactions are very small (in the real world with N c = 3 they have a magnitude expected for O(N −2 c ) effects), and since all sources of O(N 0 c ) spin-flavor breaking, including the configuration mixing, requires such orbital interactions, it is justified to treat them in practice as subleading. Thanks to this observation, the usage of spin-flavor SU(2N f ) as the zeroth order symmetry is justified. A second problem is posed by the fact that excited baryons have finite widths. The impact of this on the analyses of the masses is not fully clarified yet. One likely possibility is that their effects are included in the effective parameters that determine the masses' 1/N c expansion. This is an issue that has been recently considered in Ref. [19].
The analysis of the [56, 2 + ] masses is made along the lines established in previous investigations of the [70, 1 − ] baryons [11,13,15]. The [56, 2 + ] multiplet contains two SU(3) octets with total angular momentum J = 3/2 and 5/2, and four decuplets with J = 1/2, 3/2, 5/2 and 7/2, as listed in Table  The mass operator can be expressed as a string of terms expanded in 1/N c : where the operators O i are SU(3) singlets and the operatorsB i provide SU(3) breaking and are defined to have vanishing matrix elements between non-strange states. The effective coefficients c i and b i are reduced matrix elements that encode the QCD dynamics and they are determined by a fit to the empirically known masses.
The operators O i andB i can be expressed as positive parity and rotationally invariant products of generators of SU(6) ⊗ O(3) as it has been explained elsewhere [11]. A generic n-body operator has the structure where the factors O ℓ and O SF can be expressed in terms of products of generators of the orbital group O(3) (ℓ i ), and of the spin-flavor group SU(6) (S i , T a and G ia ), respectively.
The explicit 1/N c factors originate in the n − 1 gluon exchanges required to give rise to an nbody operator. The matrix elements of operators may also carry a nontrivial N c dependence due to coherence effects [4]: for the states considered, G ia (a = 1, 2, 3) and T 8 Table I. This can be shown using reductions, valid for the symmetric representation, of matrix elements involving excited quark and/or core operators, such as: where S = s + S c , s being the spin operator acting only on one quark (the excited one for instance), and S c acts on the remaining (N c − 1) core quarks. Similarly, relations for two-body operators can also be derived, e.g.: For N c = 3, and ǫ ∼ 1/3, the ratios associated with the relations (1) to (8) in Table IV are estimated to be of the order of 4%. The ratios obtained with the physical masses are listed in the last column of Table IV, and they are within that estimated theoretical range. It is important to emphasize that all these empirically verified relations represent a genuine test of spin-flavor symmetry and its breaking according to the 1/N c expansion, as pointed out above. The fact that they are all verified to the expected accuracy is remarkable and gives strong support to the analysis based on the premises of this work.
The fit to the available data, where states with three or more stars in the Particle Data Listings are included, leads to the effective constants c i and b i shown in Table I and the results for the masses shown in Table V, where fourteen of them are predictions. The χ 2 dof of the fit is 0.7, where the number of degrees of freedom (dof) is equal to four. The errors shown for the predictions in Table V are obtained propagating the errors of the coefficients in Table   I. There is also a systematic error O(N −2 c ), resulting from having included only operators up to O(N −1 c ) in the analysis, which can be roughly estimated to be around 30 MeV. In Table V  The better established Λ − Σ splitting in the J = 5/2 octet is almost 100 MeV, while the other splitting in the J = 1/2 octet is small and slightly negative. The latter one involves however the one star state Σ(1840), which might also be assigned to the radially excited 56 ′ . The large N c analysis implies that these splittings are O(ǫ/N c ), and are produced only by the operatorsB 2 andB 3 . The result from the fit indicates that the Λ 5/2 − Σ 5/2 receives a contribution of 63 MeV fromB 3 and 40 MeV fromB 2 . It is interesting to observe that several mass splitting differences receive only contributions fromB 2 as it is obvious from Table III. These involve the splittings in the octets (Λ 5/2 − N 5/2 ) − (Λ 3/2 − N 3/2 ), (Σ 5/2 − N 5/2 ) − (Σ 3/2 − N 3/2 ) and (Ξ 5/2 − N 5/2 ) − (Ξ 3/2 − N 3/2 ), and the decuplet splittings information on these splittings would allow to pin down with better confidence the relevance ofB 2 . The fit implies for instance that the contribution ofB 2 to the Ω 1/2 − Ω 7/2 splitting is about 225 ± 100 MeV, a rather large effect. The operatorB 2 involves the orbital angular momentum operator, and since in all other known cases where orbital couplings occur their effects are suppressed, the same would be expected here. The naive expectation is that the coefficient ofB 2 would be of order 2 √ 3 ǫ times the coefficient of O 2 . It is in fact substantially larger. However, this result is not very conclusive, because b 2 is largely determined by only a few inputs resulting in a rather large relative error for this parameter. Related to this, the sign of the coefficient ofB 2 determines the ascending or descending ordering of the masses of strange states in the decuplet as J increases. In the present analysis the higher J states are lighter. However, the structure of SU(3) breaking splittings cannot be established better because of the rather small number of strange states available for the fit. This is perhaps the most important motivation for further experimental and lattice QCD study of the still non-observed states.
It is of interest to draw some comparisons among the analyses carried out in previous works, that include the ground state baryons [8,9], the [70, 1 − ] baryons [13,15],