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. We have shown the dependence of the dynamical co-
efficients c1 and c2 as a function of the band number N or alternatively of the
excitation energy for N = 1, 2, 3 and 4 bands.
Highly excited states of baryons in largeNc QCD 53
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