BLED WORKSHOPS IN PHYSICS VOL. 13, NO. 1 p. 47 Proceedings of the Mini-Workshop Hadronic Resonances Bled, Slovenia, July 1 - 8, 2012 Highly excited states of baryons in largeNc QCD ⋆ N. Matagnea, Fl. Stancub a Service de Physique Nucléaire et Subnucléaire, University of Mons, Place du Parc, B-7000 Mons, Belgium b Institute of Physics, B5, University of Liège, Sart Tilman, B-4000 Liège 1, Belgium Abstract. The masses of highly excited negative parity baryons belonging to the N = 3 band are calculated in the 1/Nc expansion method of QCD. We use a procedure which allows to write the mass formula by using a small number of linearly independent oper- ators. The numerical fit of the dynamical coefficients in the mass formula show that the pure spin and pure flavor terms are dominant in the expansion, like for the N = 1 band. We present the trend of some important dynamical coefficients as a function of the band number N or alternatively of the excitation energy. 1 The status of the 1/Nc expansion method The large Nc QCD, or alternatively the 1/Nc expansion method, proposed by ’t Hooft [1] in 1974 and implemented by Witten in 1979 [2] became a valuable tool to study baryon properties in terms of the parameter 1/Nc where Nc is the number of colors. According toWitten’s intuitive picture, a baryon containingNc quarks is seen as a bound state in an average self-consistent potential of a Hartree type and the corrections to the Hartree approximation are of order 1/Nc. These corrections capture the key phenomenological features of the baryon structure. Ten years after ’t Hooft’s work, Gervais and Sakita [3] and independently Dashen and Manohar in 1993 [4] derived a set of consistency conditions for the pion-baryon coupling constants which imply that the large Nc limit of QCD has an exact contracted SU(2Nf)c symmetry when Nc → ∞,Nf being the number of flavors. For ground state baryons the SU(2Nf) symmetry is broken by corrections proportional to 1/Nc [5, 6]. Analogous to s-wave baryons, consistency conditions which constrain the strong couplings of excited baryons to pions were derived in Ref. [7]. These con- sistency conditions predict the equality between pion couplings to excited states and pion couplings to s-wave baryons. These predictions are consistent with the nonrelativistic quark model. A few years later, in the spirit of the Hartree approximation a procedure for constructing large Nc baryon wave functions with mixed symmetric spin-flavor parts has been proposed [8] and an operator analysis was performed for ℓ = 1 baryons [9]. It was proven that, for such states, the SU(2Nf) breaking occurs at ⋆ Talk delivered by Fl. Stancu 48 N. Matagne, Fl. Stancu order N0c, instead of 1/Nc, as it is the case for ground and also for symmetric excited states [56, ℓ+] (for the latter see Refs. [10,11]). This procedure has been ex- tended to positive parity nonstrange baryons belonging to the [70, ℓ+] multiplets with ℓ = 0 and 2 [12]. In addition, in Ref. [12], the dependence of the contribution of the linear term in Nc, of the spin-orbit and of the spin-spin terms in the mass formula was presented as a function of the excitation energy or alternatively in terms of the band number N. Based on this analysis an impressive global com- patibility between the 1/Nc expansion and the quark model results forN = 0, 1, 2 and 4 was found [13] (for a review see Ref. [14]). More recently the [70, 1−]multi- plet was reanalyzed by using an exact wave function, instead of the Hartree-type wave function, which allowed to keep control of the Pauli principle at any stage of the calculations [21]. The novelty was that the isospin term, neglected previ- ously [9] becomes as dominant in∆ resonances as the spin term inN∗ resonances. The purpose of this work is mainly to complete the analysis of the excited states by including theN = 3 band for which results were missing in the system- atic analysis of Ref. [12]. An incentive for studying highly excited states with ℓ = 3 has been given by a recent paper [15] where the compatibility between the two alternative pictures for baryon resonances namely the quark-shell picture and the meson-nucleon scattering picture defined in the framework of chiral soliton models [16,17] has been proven explicitly. This work was an extension of the anal- ysis made independently by Cohen and Lebed [18, 19] and Pirjol and Schat [20] for low excited states with ℓ = 1. As explained below, we shall analyze the resonances thought to belong to the N = 3 band by using the procedure we have proposed in Ref. [21] for the N = 1 band. Details can be found in Ref. [22]. 2 Mixed symmetric baryon states If an excited baryon belongs to a symmetric SU(6) multiplet the Nc-quark sys- tem can be treated similarly to the ground state in the flavour-spin degrees of freedom, but one has to take into account the presence of an orbital excitation in the space part of the wave function [10, 11]. If the baryon state is described by a mixed symmetric representation of SU(6) , the [70] at Nc = 3, the treatment becomes more complicated. In particular, the resonances up to about 2 GeV are thought to belong to [70, 1−], [70, 0+] or [70, 2+] multiplets and beyond to 2 GeV to [70, 3−], [70, 5−], etc. There are two ways of studying mixed symmetric multiplets. The standard one is inspired by the Hartree approximation [8] where an excited baryon is described by a symmetric core plus an excited quark coupled to this core, see e.g. [9, 12, 23, 24]. The core is treated in a way similar to that of the ground state. In this method each SU(2Nf) × O(3) generator is separated into two parts Si = si + Sic; T a = ta + Tac ; G ia = gia +Giac ; ℓ i = ℓiq + ℓ i c, (1) where si, ta, gia and ℓiq are the excited quark operators and S i c, T a c , G ia c and ℓ i c the corresponding core operators. Highly excited states of baryons in largeNc QCD 49 As an alternative, we have proposed a method where all identical quarks are treated on the same footing and we have an exact wave function in the orbital- flavor-spin space. The procedure has been successfully applied to theN = 1 band [21, 25, 26]. In the following we shall adopt this procedure to analyze the N = 3 band. 3 The mass operator When hyperons are included in the analysis, the SU(3) symmetry must be broken and the mass operator takes the following general form [27] M = ∑ i ciOi + ∑ i diBi. (2) The formula contains two types of operators. The first type are the operators Oi, which are invariant under SU(Nf) and are defined as Oi = 1 Nn−1c O (k) ℓ ·O (k) SF , (3) where O (k) ℓ is a k-rank tensor in SO(3) and O (k) SF a k-rank tensor in SU(2)-spin. Thus Oi are rotational invariant. For the ground state one has k = 0. The excited states also require k = 1 and k = 2 terms. The rank k = 2 tensor operator of SO(3) is L(2)ij = 1 2 { Li, Lj } − 1 3 δi,−jL · L, (4) which we choose to act on the orbital wave function |ℓmℓ〉 of the whole system of Nc quarks (see Ref. [12] for the normalization of L(2)ij). The second type are the operators Bi which are SU(3) breaking and are defined to have zero expec- tation values for non-strange baryons. Due to the scarcity of data in the N = 3 band hyperons, here we consider only one four-star hyperon Λ(2100)7/2− and accordingly include only one of these operators, namely B1 = −S where S is the strangeness. The values of the coefficients ci and di which encode the QCD dynamics are determined from numerical fits to data. Table 1 gives the list of Oi and Bi opera- tors together with their coefficients, which we believe to be the most relevant for the present study. The choice is based on our previous experience with theN = 1 band [26]. In this table the first nontrivial operator is the spin-orbit operator O2. In the spirit of the Hartree picture [2] we identify the spin-orbit operator with the single-particle operator ℓ · s = Nc∑ i=1 ℓ(i) · s(i), (5) the matrix elements of which are of order N0c. For simplicity we ignore the two- body part of the spin-orbit operator, denoted by 1/Nc (ℓ · Sc) in Ref. [9], as being of a lower order (we remind that the lower case operators ℓ(i) act on the excited quark and Sc is the core spin operator). 50 N. Matagne, Fl. Stancu Table 1.Operators and their coefficients in the mass formula obtained from numerical fits. The values of ci and di are indicated under the heading Fit n (n = 1, 2, 3, 4) from Ref. [22]. Operator Fit 1 (MeV) Fit 2 (MeV) Fit 3 (MeV) Fit 4 (MeV) O1 = Nc l1 c1 = 672 ± 8 c1 = 673 ± 7 c1 = 672 ± 8 c1 = 673 ± 7 O2 = ℓ isi c2 = 18 ± 19 c2 = 17 ± 18 c2 = 19 ± 9 c2 = 20 ± 9 O3 = 1 Nc SiSi c3 = 121 ± 59 c3 = 115 ± 46 c3 = 120 ± 58 c3 = 112 ± 42 O4 = 1 Nc [TaTa − 1 12 Nc(Nc + 6) ] c4 = 202 ± 41 c4 = 200 ± 40 c4 = 205 ± 27 c4 = 205 ± 27 O5 = 3 Nc LiTaGia c5 = 1 ± 13 c5 = 2± 12 O6 = 15 Nc L(2)ijGiaGja c6 = 1± 6 c6 = 1 ± 5 B1 = −S d1 = 108 ± 93 d1 = 108 ± 92 d1 = 109 ± 93 d1 = 108 ± 92 χ2dof 1.23 0.93 0.93 0.75 The spin operator O3 and the flavor operator O4 are two-body and linearly independent. The expectation values ofO3 are simply equal to 1 Nc S(S+ 1)where S is the spin of the whole system. For nonstrange baryons the eigenvalue of O4 is 1 Nc I(I + 1) where I is the isospin. For the flavor singlet Λ the eigenvalue is −(2Nc + 3)/4Nc, favourably negative, as shown in Ref. [22]. Note that the definition of the operator O4, indicated in Table 1, is such as to recover the matrix elements of the usual 1/Nc(TaTa) in SU(4), by subtracting Nc(Nc + 6)/12. This is understood by using Eq. (30) of Ref. [25] for the matrix elements of 1/Nc(TaTa) extended to SU(6). Then, it turns out that the expectation values of O4 are positive for octets and decuplets and of order N−1c , as in SU(4), and negative and of orderN0c for flavor singlets. The operators O5 and O6 are also two-body, which means that they carry a factor 1/Nc in the definition. However, as G ia sums coherently, it introduces an extra factor Nc and makes all the matrix elements of O6 of order N0c [25]. These matrix elements are obtained from the formulas (B2) and (B4) of Ref. [26] where the multiplet [70, 1−] has been discussed. Interestingly, when Nc = 3, the contribution of O5 cancels out for flavor singlets, like for ℓ = 1 [26]. This property follows from the analytic form of the isoscalar factors given in Ref. [26]. We remind that the SU(6) generators Si, Ta and Gia and the O(3) generators Li of Eq. (4) act on the total wave function of theNc system of quarks as proposed in Refs. [21], [25] and [26]. The advantage of this procedure over the standard one, where the system is separated into a ground state core + an excited quark, is that the number of relevant operators needed in the fit is usually smaller than the number of data and it allows a better understanding of their role in the mass formula, in particular the role of the isospin operator O4 which has always been omitted in the symmetric core + excited quark procedure.We should alsomention that in our approach the permutation symmetry is exact [21]. Highly excited states of baryons in largeNc QCD 51 Among the operators containing angular momentum components, besides the spin-orbit, we have included the operators O5 and O6, to check whether or not they bring feeble contributions, as it was the case in the N = 1 band. From Table 1 one can see that their coefficients are indeed negligible either included together as in Fit 1 or separately as in Fit 2 and 3. Thus in the expansion series, besides O1, proportional to Nc, the most dominant operators are the pure spin O3 and the pure isospin O4. 400 450 500 550 600 650 700 750 800 850 0 1 2 3 4 5 c1 (MeV) N • •• • • Fig. 1. The coefficient c1 as a function of the band number N: N = 1 Ref. [26], N = 2 Ref. [10] for [56, 2+] and Ref. [12] for [70, ℓ+], N = 3 Ref. [22], N = 4 Ref. [11]. The straight line is drawn to guide the eye. 4 Global results The above analysis helps us to complete previous results for N = 1, 2 and 4 with the values of ci obtained forN = 3. Thereforewe can drawnow a complete picture of the dependence of the coefficients c1 and c2 onN in analogy to Ref. [12] where results for N = 3 were missing. The new pictures are shown in Figs. 1 and 2. One can see that the values of c1 follow nearly a straight line which can give rise to a Regge trajectory. Remember that c1 describes the bulk content of the baryon mass, c1Nc being the most dominant mass term. In a quark model language it represents the kinetic plus the confinement energy. As as discussed in Refs. [13, 14] the band number N also emerges from the spin independent part of a semi- relativistic quark model. If this part contributes to the total mass by a quantity denoted byM0, then one can make the identification c21 =M 2 0/9 (6) when Nc = 3. In this way one can compare the Regge trajectory obtainable from the above results with that of a standard constituent quark model. It turns out 52 N. Matagne, Fl. Stancu -100 -50 0 50 100 150 0 1 2 3 4 5 c2 (MeV) N • • • • • Fig. 2. Same as Figure 1 but for the coefficient c2. that they are close to each other [13,14]. and the value obtained here for c1 atN = 3, missing in the previous work, is entirely compatible with the previous picture. The behaviour of c2 shows that the spin-orbit operator contributes very little to the mass, at all energies, in agreement to quark models, where it is usually neglected. Note that the behaviour of c2 in Fig. 2 is slightly different from that of [12], because we presently take the value of c2 at N = 1 from Ref. [26] (Fit 3 giving the lowest χ2dof) for consistency with our treatment, instead of that of Ref. [9], based on the ground state core + excited quark, the only available at the time the paper [12] was published. We refrain ourselves from presenting the global picture of c3, the spin term coefficient, because the results for positive parity mixed symmetric states are ob- tained on the one hand in the core + excited quark approach, where the isospin term is missing and on the other hand, for negative parity states where it is present, our approach is used. This term competes with the spin term. We plan to reanalyze the [70, ℓ+] multiplets before drawing a complete picture of c3. 5 Conclusions We have used a procedure which allows to write the mass formula by using a small number of linearly independent operators for spin-flavour mixed symmet- ric states of SU(6). The numerical fits of the dynamical coefficients in the mass formula forN = 3 band resonances show that the pure spin and pure flavor terms are dominant in the 1/Nc expansion, like for N = 1 resonances. This proves that the isospin term cannot be neglected, as it was the case in the ground state + excited quark procedure. 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