Bled Workshops in Physics Vol. 17, No. 1 p. 25
A
Proceedings of the Mini-Workshop
Quarks, Hadrons, Matter
Bled, Slovenia, July 3 - 10, 2016

Excited hyperons of the N = 2 band in the 1/Nc expansion
Fl. Stancu
University of Liege, Institute of Physics B5, Sart Tilman, B-4000 Liege 1, Belgium
Abstract. The spectrum of excited baryons in the N =2 band is reanalyzed in the 1/Nc expansion method, with emphasis on hyperons. Predictions are made for the classification of these excited baryons into SU(3) singlets, octets and decuplets.
1 Introduction
The 1/Nc expansion method [1,2] where Nc is the number of colors, is a powerful and systematic tool for baryon spectroscopy. For Nf flavors, the ground state baryons display an exact contracted SU(2Nf) spin-flavor symmetry in the large Nc limit of QCD [3,4]. The Skyrme model, the strong coupling theory and the static quark model share a common underlying symmetry with QCD baryons in the large Nc limit [5].
The method has been successfully applied to ground state baryons (N = 0 band), in the symmetric representation 56 of SU(6) [4,6-9]. At Nc —» oo the ground state baryons are degenerate. At large, but finite Nc, the mass splitting starts at order 1/Nc as first observed in Ref. [5].
The extension of the 1/Nc expansion method to excited states requires the symmetry group SU(2Nf) x O(3) [10], in order to introduce orbital excitations. It happens that the experimentally observed resonances can approximately be classified as SU(2Nf) x O(3) multiplets, grouped into excitation bands, N = 1, 2, 3, ... , each band containing a number of SU(6) x O(3) multiplets.
The situation is technically more complicated for mixed symmetric states than for symmetric states. Two approaches have been proposed so far. The first one is based on the Hartree approximation and describes the Nc quark system as a ground state symmetric core of Nc — 1 quarks and an excited quark [11].
The second procedure, where the Pauli principle is implemented to all Nc identical quarks has been proposed in Refs. [12,13]. There is no physical reason to separate the excited quark from the rest of the system. The method can straightforwardly be applied to all excitation bands. It requires the knowledge of the matrix elements of all the SU(2Nf) generators acting on mixed symmetric states described by the partition (Nc — 1,1). In both cases the mass splitting starts at order N0. The latest achievements for the ground state and the current status of large Nc QCD excited baryons (N = 1, 2, 3, 4) can be found in Ref. [14]. The N
26 Fl. Stancu
= 1 band is the most studied. The N =2 band received considerable attention too. Here we reanalyze the results of Ref. [15] for N =2. The reason is that in a few octets an anomalous situation appeared where the hyperons A or I (presently degenerate) appeared slightly lighter than the nucleon in the same octet.
Here we use the data of the 2014 Particle Data Group [16] which includes changes due to a more complex analysis of all major photo-production of mesons in a coupled-channel partial wave analysis.
2 The Mass Operator
The general form of the mass operator, where the SU(3) symmetry is broken, has first been proposed in Ref. [9] as
M = ^ CiOi + ^ diBi.	(1)
i i
The operators Oi are defined as the scalar products
Oi = —^ O™ • oSk,	(2)
1 Ni1-1 1 SF
where O^k) is a k-rank tensor in SO(3) and O^F a k-rank tensor in SU(2)-spin, but invariant in SU(Nf). Thus Oi is rotational invariant. For the ground state one has k = 0. The excited states also require k = 1 and k = 2 terms. The k = 1 tensor has three components, which are the generators Li of SO(3). The components of the k = 2 tensor operator of SO(3) read
L(2)ij = 2 {L\Lj} - lSi,HL • L.	(3)
The operators oSF are expressed in terms of the SU(Nf) generators S1, Ta and Gia.
The operators Bi break the SU(3) flavor symmetry and are defined to have zero expectation values for nonstrange baryons. The coefficients ci encode the quark dynamics and di measure the SU(3) breaking. They are obtained from a numerical fit. The most dominant operators considered in the mass formula together with the fitted coefficients are presented in Table 1.
For the [56]-plets the spin-orbit operator O2 is defined in terms of angular momentum L1 components acting on the whole system as in Ref. [17] and is order O(1/Nc)
O2 = NcL •S'	(4)
while for the [70]-plets it is defined as a single-particle operator I • s of order O(N0).
Nc
O2 = I • s = ^ £(i) • s(i).	(5)
i=1
Excited hyperons of the N = 2 band in the 1/Nc expansion 27
3 Matrix elements
The matrix elements of the [56,2+] multiplet were derived in Ref. [17]. Details of the derivation of the matrix elements of Ot for [70,£+], as a function of Nc, can be found in Ref. [18]. Note that in the case of mixed symmetric states the matrix elements of O6 are O(N0), in contrast to the symmetric case where they are O(N-1), and non-vanishing only for octets, while for the symmetric case they are non-vanishing for decuplets. Thus, at large Nc the splitting starts at order O(N0) for mixed symmetric states due both to O2 and O6.
The SU(3) flavor breaking operators Bt have the same definition for both the symmetric and mixed symmetric multiplets. The matrix elements of B2 and B3 for [70, Í+] were first calculated in Ref. [15]. For practical purposes we have summarized these results by two simple analytic formulas valid at Nc =3. The diagonal matrix elements of B2 take the following form
B2 — W ,	(6)
where ns is the number of strange quarks and (L • S) is the expectation value of the spin-orbit operator acting on the whole system. Similarly the diagonal matrix elements of B3 take the simple analytic form
r S(S +1)	^
B3 = -ns "67!", (7)
where S is the total spin. The contribution of B3 is always negative, otherwise vanishing for nonstrange baryons. These formulas can be applied to 28j, 48j, 210j and 211/2 baryons of the [70, £+] multiplet. Presently the SU(3) breaking operators B2 and B3 are included in the analysis of the [70, £+] multiplet, first considered in Ref. [15].
4 Fit and discussion
We have performed a consistent analysis of the experimentally known resonances supposed to belong either to the symmetric [56,2+] multiplet or to the mixed symmetric multiplet [70, £+] with Í = 0 or 2, by using the same operator basis. Results of the fitted coefficients et and dt are exhibited in Table 1 together with the values of Xdof for each multiplet.
The spin and flavor operators O3 and O4 are the dominant two-body operators and bring important 1/Nc corrections to the masses. The sum of c3 and e4 of [70, £+] is comparable to the value of c3 in [56,2+] where the equal contribution of O3 and O4 is included in c3. The contribution of the operator O6 containing an SO(3) tensor is important especially for [70, £+] multiplet. Together with the spin-orbit it may lead to the mixing of doublets and quartets to be considered in further studies when the accuracy of data will increase. The incorporation of B2 and B3 in the mass formula of the [70,£+] multiplet brings more insight into the SU(6) multiplet classification of excited baryons in the N = 2 band.
28 Fl. Stancu
Table 1. List of the dominant operators and their coefficients (MeV) from the mass formula (1) obtained in numerical fit for [56,2+] in column 2 and for [70, £+] in column 3. The spinorbit operator O2 is defined by Eq. (4) for [56,2+] and by Eq.(5) for [70, «+].
Operator
01	= Nc 1
02	spin-orbit
° = NCCsv
° = NC [T aT 0
O6 = —L(2'ij GiaGja
_Nc_
J^Nc (Nc + 6)
Bi = ns
B2 = nnc (LiGi8
b3 = nnc (siGi8

2
Xdof
2^3
-Ws1 )
2y3
[56,2+]	[70,£+]
542 ± 2	631 ± 10
7 ±10	62 ± 26
233 ± 11	91 ± 31
	112 ± 22
6 ±19	137 ± 55
205 ± 14	35 ± 33
97 ± 40	- 38 ± 121
197 ± 69	46 ± 159
1.63	1.67
4.1	The multiplet [56,2+]
The partial contribution and the calculated total mass obtained from the fit were presented in Table VI of Ref. [15] which we do not repeat here. The experimental masses were taken from the 2014 version of the Review of Particle Properties (PDG) [16], except for A(1905)5/2+ where we used the mass of Ref. [17] which gives a smaller Xdof, but does not much change the fitted values of ct and dt. As expected, the most important sub-leading contribution comes from the spin operator O3. The contributions of the angular momentum-dependent operators O2 and O6 are comparable, but small. Among the SU(3) breaking terms, Bt is dominant. An important remark is that in the [56,2+] multiplet B2 and B3 lift the degeneracy of A and I baryons in the octets, which is not the case for the [70, £+] multiplet.
4.2	The multiplet [70, £+]
As compared to Ref. [18] where only 11 resonances have been included in the numerical fit, here we consider 16 resonances, having a status of three, two or one star. This means that we have tentatively added the resonances E(2120)??*, 1(2070)5/2+*, I(1940)??*, E(1950)??*** and 1(2080)3/2+**. The masses and the error bars considered in the fit correspond to averages over data from the particle listings, except for a few which favor specific experimental values cited in the headings of Table 2.
We have ignored the N (1710) 1/2+*** and the 1(1770)1/2+* resonances, the theoretical argument being that their masses are too low, leading to unnatural sizes for the coefficients ct or dt [19]. On the experimental side one can justify the removal of the controversial N (1710) 1/2+*** resonance due to the latest GWU analysis of Arndt et al. [20] where it has not been seen. We have also ignored the
Excited hyperons of the N = 2 band in the 1/Nc expansion 
29
A(1750)1/2+* resonance, because neither Arndt et al. [20] nor Anisovich et al. [21] find evidence for it.
Table 2. Partial contribution and the total mass (MeV) predicted by the 1/N c expansion. The last two columns give the empirically known masses and status from the 2014 Review of Particles Properties [16] unless specified by (A) from [21], (L) from [22], (Z) from [23], (G1) from [24], (B) from [25], (AB) from [26], (G2) from [27],.
Part. contrib. (MeV)
Total(MeV) Experiment(MeV) Name, status
C1O1 C2O2 C3O3 C4O4 C6Û6 d1B1 d2B2 d3B3
4N[70,2+
4A[70,2+
4i
[70,2
1892 62 113 28 -22 0 0 35 11 70 22
0 2073 ± 38 2060 ± 65(A) N(1990)7/2+** -17 2102 ± 19 2100 ± 30(L) A(2020)7/2+* -34 2131 ± 8 2130 ± 8 !E(2120)??*
4N[70,2+ 4A[70,2+
2 5+
1892 -10 113 28 57 0 0
35 -2
0 2080 ± 32 2000 ± 50 N(2000)5/2+** -17 2096 ± 10 2100 ± 10 A(2110)5/2+***
4N[70, 2+ 4A[70, 2+
3 + 2
3 + 2_
1892 -62 113 28 0 0 0 0 1972 ±29
0 35 -11 -17 1979 ±39
4N[70, 2+ 4A[70, 2+
1892 -93 113 28 -80 0 0 0 1861 ± 33 1870 ± 35(A) N(1880)1/2+**
35 -16 -16 1869 ±79
2N[70, 2+ 21 [70,2+
1892 21 23 28 0 0 0 0 35 4
0 1964 ± 29
1860 ± ¿20(A)
-3 2000 ± 18 2051 ± 25(G1)
N(1860)5/2+** 1(2070)5/2+*
2
N[70, 2
2
I [70,2+ 2 S [70,2+
2
3 + 2
3 + 2
1892 -31 23 28 0 0 0 0 1912 ± 21 1905 ± 30(A) 0 35 -6 -3 1 938 ±10 1941 ±18
N(1900)3/2+*** I(1940)??*
0 70 -11 -7 1964 ± 7 1967 ± 7(B) E(1950)??
4N[70, 0+ 4I[70,0+
1892 0 113 28 0 0 0 0 2033 ± 18 2040 ± 28(AB) 35 0 -16 2052 ± 21 2100 ± 69
N (2040)3/2+* 1(2080)3/2+**
2
A[70,2+]- 1892 -21 23 140 0 0 0 0 2034 ± 31
1962 ± 139 A(2000)5/2+
1+
2I* [70,0+] ^ 1892 0
23 140 0 35 0 -3 2087 ± 30 1902 ± 96 1(1880)1/2+
1+
2 A '[70,0+]- 1890 0
23 -84 0 35 0
-3 1863 ± 19 1853 ± 20(G2) A(1810)1/2+***
+
7
2
+
5
2
+
+
5
2
+
3
A-A-A-
+
3
2
+
30 Fl. Stancu
The partial contributions and the calculated total masses obtained from the fit are presented in Table 2. Regarding the contribution of various operators we note that the good fit for N (1880) 1/2+** was due to contribution of the spin-orbit operator O2 of -93 MeV and of the operator O6 which contributed with -80 MeV. The good fit also suggests that I(1940)??* and E(1950)??*** assigned by us to the 2 [70,2+]3/2+ multiplet is reasonable, thus these resonances may have JP = 3/2+, to be experimentally confirmed in the future.
The 1/Nc expansion is based on the SU(6) symmetry which naturally allows a classification of excited baryons into octets, decuplets and singlets. In Table 2 the experimentally known resonances are presented. In addition some predictions are made for unknown resonances. Many of the partners in a given SU(3) multiplet are not known. Note that A and I are degenerate in our approach. Although the operators B2 and B3 have different analytic forms at arbitrary Nc [15] they become identical at Nc =3 for A and I in octets, thus they cannot lift the degeneracy between these hyperons, contrary to the [56,2+] multiplet.
The present findings can be compared to the suggestions for assignments in the [70, £+] multiplet made in Ref. [28] as educated guesses. The assignment of 1(1880)1/2+** as a [70,0+] 1/2+ decuplet resonance is confirmed as well as the assignment of A(1810) 1/2+*** as a flavor singlet. We agree with Ref. [28] regarding A(2110)5/2+*** as a partner of N (2000)5/2+** in a spin quartet, contrary to our previous work [15] where A(2110)5/2+*** was a member of a spin doublet, together with N(1860)5/2+** and 1(2070)5/2+*. This helps to restore the correct hierarchy of masses in all octets. However we disagree with Ref. [28] that N(1900)3/2+*** is a member of a spin quartet. We propose it as a partner of I(1940)??* and E(1950)??*** in a spin doublet.
The problem of assignment is not trivial. Within the 1/Nc expansion method Ref. [17] suggests that 1(2080)3/2+** and 1(2070)5/2+* could be members of two distinct decuplets in the [56, 2+] multiplet.
Here the important result is that the hierarchy of masses as a function of the strangeness is correct for all multiplets.
Acknowledgments
Support from the Fonds de la Recherche Scientifique - FNRS under the Grant No. 4.4501.05 is gratefully acknowledged.
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