Global analysis of the negative parity non-strange baryons in the 1/Nc expansion

A global study of the negative parity non-strange baryon observables is performed in the frame- work of the 1/Nc expansion. Masses, partial decay widths and photo-couplings are simultaneously analyzed. A main objective is to determine the composition of the spin 1/2 and 3/2 nucleon states, which come in pairs and involve two mixing angles which can be determined and tested for consistency by the mentioned observables. The issue of the assignment of those nucleon states to the broken SU(4) x O(3) mixed-symmetry multiplet is studied in detail, with the conclusion that the assignment made in the old studies based on the non-relativistic quark model is the preferred one. In addition, the analysis involves an update of the input data with respect to previous works.


I. INTRODUCTION
The excited baryon states that correspond in the constituent quark model to the first radial and orbital excitations fit well into respectively a positive parity symmetric and a negative parity mixed-symmetric irreducible representation of the spin-flavor group SU (2N f ), where N f is the number of light flavors. For N f = 2 both representations are 20-dimensional and are respectively denoted by 20 and 20 . The negative parity non-strange excited baryons states are empirically the best known ones [1], and are organized in the [20 , = 1] representation of SU (4) × O(3) [2]. Their masses, partial decay widths and photo-couplings are known to a degree where detailed analyses as the one presented here can be carried out and lead to some clear conclusions. The states consist of two spin 1/2 nucleon states: N(1535) and N(1650), two spin 3/2 nucleon states: N(1520) and N(1700), one spin 5/2 nucleon state: N(1675) and two isospin multiplets: ∆(1620) and ∆(1700) of spin 1/2 and 3/2 respectively.
A very important issue, central to the study in this work, is the assignment of the physical states to the pairs of spin 1/2 and 3/2 nucleon states, and also the accurate determination of the corresponding mixing angles θ 2J=1 and θ 3 . In order to settle that issue in as model independent as possible fashion it is necessary to carry out a simultaneous analysis of masses, partial decays widths and photo-couplings, as presented in this work. The framework used here is the 1/N c expansion as presented elsewhere for masses [3,4], decays [5] and photo-couplings [6].
The advantage of using the 1/N c expansion [7] is that, in principle, the analysis is based on a framework which is true to QCD. It provides a justification for the approximate spin-flavor symmetry observed in baryons [8][9][10][11], which permits the implementation of the 1/N c starting from that symmetry limit. From multiple applications it transpires that the 1/N c expansion works well in the real world with N c = 3 [18]. In particular, its implementation in the analysis of excited baryons (see e.g. Refs. [3][4][5][6][12][13][14][15][16][17]), has shown to be phenomenologically consistent in the sense that the sub-leading corrections are of natural size. In the present work, the 1/N c expansion is implemented as an operator expansion, where the coefficients are determined by the QCD dynamics, and those coefficients are evaluated by fitting to the pertinent observables. In addition, since usually at the first sub-leading order the number of observable quantities exceeds the number of operators, there are relations independent of the coefficients, which are exact up to sub-sub-leading order corrections, and which should serve as a good test of the framework. The physics of excited baryons thus gets sorted out hierarchically in powers of 1/N c , helping in this way to define and organize the discussion of the different effects. It should be noted that the implementation for the negative parity baryons is carried out under the assumption that SU (4) × O(3) configuration mixings, i.e. mixings of different irreducible representations of that group, can be disregarded. A discussion of such effects was presented in [19]. In the 1/N c expansion, baryon observables are represented by effective operators which are themselves decomposed in an operator basis which is organized according to the 1/N c power counting of the operators. The coefficients of that decomposition are determined by the QCD dynamics. These are obtained by fitting the corresponding observables to experimental data.
In the case of the negative parity baryons, the analysis of the masses suffers from the four-fold ambiguity mentioned above in the assignment of the spin 1/2 and 3/2 nucleon physical states to the theoretical ones. It turns out that such ambiguity cannot be resolved by analyzing the masses alone, as acceptable fits, i.e. χ 2 dof ∼ 1 can be obtained for all four assignments. Because the strong decay partial widths with emission of a single pion, and the EM transitions are more sensitive to those assignments, they must be used to arrive at a definite conclusion. Historically, the first such analysis was carried out in the framework of the quark model [20]. In the 1/N c expansion the work of Pirjol and Schat [21] was the first to address the problem , and it was also piecemeal addressed in other works on strong decays [5] and EM transitions [6]. However, an exhaustive combined analysis was still lacking, which is now presented in this work.
The present work is organized as follows. Sec. II provides a brief review of the framework used to analyze the non-strange low-lying negative parity excited baryons in the framework of the 1/N c expansion of QCD. Some parameter independent relations valid to O(N 0 c ) are also presented and discussed in this section. Sec. III describes the procedure followed to perform the global fits of the empirically known baryon properties to O(1/N c ) and analyze the corresponding results. The main conclusions are presented in Sec. IV. Finally, includes two Appendices: Appendix I gives the lists of operators furnishing the bases for the analysis of the baryon properties considered in this work, while Appendix II gives some parameter independent relations satisfied at LO by the photo-couplings.

BARYONS IN THE 1/N c EXPANSION
In the following a brief description of the 1/N c expansion excited baryon framework is presented.
The group SU (4) has 15 generators, namely, the spin, isospin and spin-isospin generators: S i ,T a ,G ia , with i = 1, 2, 3 and a = 1, 2, 3, normalized to fulfill the following commutation relations: The matrix elements of G ia between baryons with spin of order N 0 c are O (N c ). Introducing the spherical notation for the group generators and omitting the projections for simplicity, the generators are denoted by: . These states are constructed by coupling a fundamental SU (4) state to a totally symmetric state with N c − 1 indices, which is denoted as the core [4]: Denoting the core states by | S c , S c3 , I c3 , where these states belong to the symmetric SU (4) representation in which I c = S c is satisfied, the mixed-symmetric states of interest denoted by | S, S 3 ; I, I 3 are given by [4]: where Angles can be defined to be in the interval [0, π) by conveniently choosing the phase of the physical states.
As already mentioned there are four possible assignments of the states N J and N J to the physical ones. They are: It should be noted that in previous works [3,4,21] the mixing angles have been defined using some specific state assignment. For example, in Ref. [21] they are defined in their Eqs. (18) and (19), which only correspond to the definition Eq.
The 1/N c order of the composite operators is determined by the n-body character or degree of these, which is given by the number of quark fields necessary to built an operator with the same transformation properties [25], and by the N c order of the matrix elements of the tensor operators appearing as factors. Since an n-body operator requires the exchange of n − 1 gluons to be produced, there is an overall factor 1/N n−1 c , and thus the (naive) order in 1/N c of an n-body composite operator becomes: where κ is given by the order of the factors, and n satisfies n ≤ N c . The coefficients C (n) in Eq. (12) are the unknown dynamical coefficients of O (N 0 c ), which are determined by the QCD dynamics.
In the subspace of the MS states of interest, a basis of composite operators can be constructed by distinguishing two sets of generators according to whether they act on the core or the "excited quark" which corresponds to the spin-flavor index associated to the second row of the Young tableaux. Generators acting on the core will be denoted by S c and those acting on the excited quark by [1,0] , s [1,0] , t [0,1] , g [1,1] . Then, n-body operators can be those consisting of the product of n SU (4) generators acting only on the core, or n − 1 SU (4) generators acting on the core and at least one generator of SU (4) × O(3) acting on the excited quark. It should be noted that the order κ appearing in Eq. (13) takes the value κ = 0 for all the generators except the generator G c for which κ = 1.

C. Mass operators
The basis of mass operators up to order 1/N c is taken from that obtained in [4], and shown here in Table VII Table II of Ref. [4].
For the content of states for N c = 3 there are seven possible masses and two mixing angles, which means at most nine independent mass operators. Up to O (N 0 c ) there are only three operators, and thus there are six parameter independent relations. As noted in Ref. [21], two of these relations are the values of the mixing angles, which are θ 1 = cos −1 (− 2/3 ) = 2.526 and θ 3 = cos −1 (− 5/6 ) = 2.721 in the present convention. It should be noted that using the mass formulas up to, and including O (N 0 c ) corrections, one can identify with the nucleon states in the paper of Pirjol and Schat [21] in the following way: , where T indicates the tower to which the state belongs as explained in [21]. Thus, their four alternative results for the O (N 0 c ) mixing angles (see their The remaining four relations involve only the physical masses and are as follows: Numerically, these relations are satisfied best with Set 3 in agreement with the result of Ref. [21] where it was denoted "Assignment No.1". However, as observed later at O (1/N c ) and after performing the combined fits with decays and EM amplitudes, a different assignment will be favored. The hyperfine interaction plays the key role in modifying the LO values of the mixing angles and in permitting a different identification of states.
Up to O (1/N c ) there are eight operators, which means that there will be one mass parameter independent relation which can be checked. Expressed in terms of the physical masses and mixing angles this relation reads Fig. 1 shows the correlation between the angles if the assignment of states is the one of Set 1 in Eq. (11). For the other possible assignments one just shifts the angles by π 2 correspondingly.

D. Strong decays
The partial decay widths for the strong decays of the negative parity excited baryons via the emission of a π or η meson are given by [5]: where the asterisk refers to the excited baryon, P = 0 or 2 is the angular momentum of the meson and I P its isospin, and Λ is an arbitrary scale introduced for convenience so that all quantities in the sum are dimensionless. By choice in the calculations, Λ = 200 MeV will be taken. B q ( P , I P , S, I, J * , I * , S * ) are reduced matrix elements defined via the Wigner-Eckart theorem as follows: The formulas above hold if the physical states have well defined quark-spin S * . However, the physical states are a mix of the S * = J * ± 1 2 quark spin states, and thus they need to be modified in the obvious way via the corresponding mixing of the reduced matrix elements.
This of course only happens for the case of the N 1 2 and N 3 2 states, and thus the dependence of these partial widths on the angles θ 1 and θ 3 . Note that the analysis is restricted to the S and D-wave decays, because the only possible G-wave transition in the decay N 5 2 → ∆π is not yet empirically sorted out, and is also irrelevant to the main discussion of this work.
The corresponding bases of baryon decay operators and their associated matrix elements were obtained in Ref. [5]. For convenience those operator bases are depicted in Table VIII of Appendix I.

E. Photoproduction helicity amplitudes
The multipole components of the helicity amplitudes for photo-production of the [20 , 1 − ] states can expressed in terms of the matrix elements of effective operators as follows [6]: where M L and EL indicates the respective multipoles. Note that, due to parity conservation, only E1, M 2 and E3 multipoles are allowed. Moreover, λ = 1 2 , 3 2 is the helicity defined along theẑ-axis which coincides with the photon momentum, and ω = (M 2 B * − M 2 N )/2M B * is the photon energy in the rest frame of the excited baryon B * . These amplitudes correspond to the standard definition as used by the Particle Data Group [1], which includes a sign factor η(B * ) that stems from the strong decay amplitude of the excited baryon to a πN state making them independent of the phase conventions used to define the excited states.
The sign factors are on the other hand convention dependent. The sum over n is over all available basis operators with the given [L, I] quantum numbers. The factor √ N c appears as usual for transition matrix elements between excited and ground state baryons [19]. In the electric multipoles there is a combination of the coefficients g n,L are going to be determined by fits to the empirical helicity amplitudes. In order to streamline the notation the coefficients will then be denoted by: XL (I) n , X = E or M , and I = 0, 1. As in the case of decays, one must take into account the effects of state mixing for the physical amplitudes.
The corresponding bases of operators and their associated matrix elements were obtained in Ref. [6]. For convenience the list is depicted in Table IX The minimization is carried out using the MINUIT program. Using the procedure mentioned above, fits for the four possible assignments Eq. (11) were performed. It is found that in all cases it is possible to obtain values of χ 2 dof < 1. However, these fits have qualitatively very different characteristics. This can be observed in Figs. 2 and 3, which depict the coefficients associated with mass and strong decay operators and with the EM amplitude operators, respectively.
In order to decide which assignments are acceptable from a physical point of view one needs to consider a certain number of criteria. The first one has to do with the naturalness of the coefficients. In fact, if the 1/N c expansion is assumed to be an appropriate framework, all the coefficients should be within a natural range in magnitude. It should be noted that in the case of the strong decays and helicity amplitudes this criterion should be applied to each partial wave separately. A second criterion is related to the assumption that in the strong and EM amplitudes, when 1-body operators can contribute they should not be suppressed with respect to 2-body operators. This assumption is based on the fact that the quark model description of these amplitudes, where it is assumed by construction the dominance 1-body operators, is quite successful. Finally, the third criterion takes into account the importance of the hyperfine interaction in the description of the masses. In fact, this interaction plays a crucial role in explaining the lightness of the singlet Λ(1405) and Λ(1520), whose masses are pushed down by the hyperfine interaction as shown in [14], and as realized in the early paper by Isgur [26]. The lightness of those Λ baryons is also observed in recent lattice QCD calculation of the baryon spectrum [28], where due to the larger quark masses and the possible suppression of the finite width effects due to the implementation of the lattice calculation, the hyperfine interaction is clearly driving the downward shift of the masses.
The result of the application of these criteria to the results shown in Figs. 2 and 3 is summarized in Table II. From that table one can conclude that the preferred assignment of states is that of Set 1. This assignment is the one that had been adopted in the early works with the quark model [20,26,27] and in the various analysis in the 1/N c expansion, [3-6, 12, 21]. The detailed analysis presented here further establishes that scheme.
The numerical results for the operator coefficients corresponding to the favored Set 1 are given in Table IV. Two possibilities have been considered: Fit 1 includes all the independent operators appearing in the basis of each observable, while in Fit 2 those operators whose coefficients are compatible with zero have been excluded. More specifically, Fit 2 was obtained by taking the result of Fit 1 and removing those operators whose coefficients are compatible with zero in a subsequent way. In the case that at certain step there were more than one operator in this situation, those with higher relative error were removed.
It is interesting to compare the operator coefficients obtained in the present global analysis with those reported in Refs. [4][5][6] where fits were made independently for each baryon property. In making this comparison one should keep in mind that some new empirical val-ues became available since those analysis were performed. In the case of strong decays it is observed that except for a few exceptions the coefficients obtained in global and non-global analysis are quite similar when error bars are taken into account. One of the exceptions concerns the coefficients C 7,8 associated to 3-body operators which were not taken into account in Ref. [5]. While the present analysis indicates that C  π∆. These can be checked using the Tables III, IV and V of Ref. [5]. Since the mixing angle θ 1 changes by more than π/2 from LO to NLO, one expects that for the J = 1/2 nucleons the selection rules will be poorly satisfied. → π∆ is quite suppressed with respect to the S-wave one, N 1 2 → πN , but this is simply because of being a higher partial wave. The large change in the angle θ 1 from LO to NLO is due to the fact that the mass operators of O (N 0 c ) are quite weak and the NLO hyperfine mass operator is very strong, leading to a large rearrangement in the composition of the J = 1/2 nucleon states. Finally the suppression of the S-wave decay N3 2 → π∆ should be well fulfilled because the angle θ 3 does not change too much from LO to NLO. This seems to be indeed the case as the D-wave decay N3 2 → π∆ has similar rate to that S-wave decay.

IV. CONCLUSIONS
The determination of the spin-flavor structure of the negative parity baryons is of fundamental importance in the study of excited baryons, as it serves to unravel the QCD dynamics responsible for their properties. Of particular importance is the structure of the J = 1/2 and 3/2 nucleons, which coming in pairs can mix. The mixing is a sensitive measure of that dynamics. The connection between the theoretical states and the observed ones is a priori a four-fold undetermined problem. The physics is definitely different in each case, as it has been shown in this work through the global analysis of observables. The study comes to the conclusion that there is a favored assignment, based on the three criteria outlined in section III. Set 1 is acceptable on all counts, while Set 2 comes second, failing only in one criterion, namely that 2-body operators dominate over 1-body ones in the strong decays. Set 1 is moreover the favored one in the simplest versions of the quark model with suppressed spin-orbit interaction, which indicates that the seemingly simplest dynamics is the most realistic one. Concerning the mixing angles, this work finds (θ 1 , θ 3 ) = (0.49 ± 0.29, 3.01 ± 0.17) and (0.40 ± 0.13, 2.96 ± 0.05) for Fits 1 and 2 respectively. For comparison in a previous non-global analysis of the strong decays using the 1/N c [5], the value θ 1 = 0.39 ± 0.11 was obtained and θ 3 exhibited a two-fold degeneracy θ 3 = (2.38, 2.82) ± 0.11. In this sense the present global analysis is important to remove such degeneracy in the values for θ 3 . It is useful to compare the results obtained in the present analysis with old results obtained with analyses based on the quark model and of the SU (6) W approach. The present values are in good agreement with the earlier quark model determination (θ 1 , θ 3 ) = (0.55, 3.03), obtained from the analysis of strong decays alone [27]. The present results for θ 3 agree with old results in the SU (6) W approach [30], while for θ 1 those references disagreed with each other, for the most part because the input partial widths for the N1 2 baryons had changed in the meantime, with the latest agreeing with the present result. Finally, it is worthwhile to stress that further improvement in the empirical inputs for partial decay widths and helicity amplitudes is necessary in order to reduce the mixing angles uncertainties and to establish with an even stronger emphasis Set 1 as the correct assignment of spin-flavor states.

APPENDIX I: BASES OF OPERATORS
For the reader's convenience the operator bases are shown: Table VII gives the mass operators, Table VIII operators for the strong decays and Table IX operators for the helicity amplitudes. They were respectively obtained in Refs. [4][5][6], where the corresponding matrix elements relevant for the present work can be also found. Other LO relations involving simultaneously neutral and charged baryons are not given here, but can be easily obtained. In the following the notation s i ≡ sin θ i , c i ≡ cos θ i is used.

Charged baryon relations:
Neutral baryon relations: