Bled Workshops in Physics Vol. 12, No. 1 p. 25

Spectroscopy of heavy baryons*
Joseph P. Day, Ki-Seok Choi, Willibald Plessas
Theoretical Physics, Institute of Physics, University of Graz, A-8010 Graz, Austria
Abstract. We report first results from a study of heavy-baryon spectroscopy within a relativistic constituent-quark model whose hyperfine interaction is based on Goldstone-boson-exchange dynamics.
1 Introduction
The relativistic constituent-quark model (RCQM) has become quite successful for the description of hadron properties at low energies. This is especially true for the RCQM based on Goldstone-boson-exchange (GBE) dynamics [1] with regard to baryons (for a short review see ref. [2]). So far the GBE RCQM has been restricted to baryons consisting of constituent quarks Q with flavors u, d, and s only, as it has been argued that their hyperfine interaction should be governed by GBE dynamics due to the spontaneous breaking of chiral symmetry (SBxS) of low-energy quantum chromodynamics (QCD) [3]. Regarding the other known baryons, i.e. the ones with flavors c and b, we still face the interesting questions after the light-heavy and heavy-heavy Q-Q interactions. It remains to be clarified, which kind of dynamics, gluon exchange and /or Goldstone-boson exchange, is dominant.
We have looked into these problems within the framework of the RCQM. Accepting the GBE RCQM in the SU(3)F sector, there are in principle three ways to add interactions for the light-heavy and heavy-heavy Q-Q interactions:
•	employ GBE dynamics throughout,
•	extend the SU(3)F GBE RCQM with one-gluon exchange (OGE) for the c and b flavors, and
•	use a superposition of both the GBE and OGE hyperfine interactions beyond
According to our present experience the best performance of a universal RCQM for all SU(5)f baryons is achieved by the first way [4]. Here, we thus report results of a SU(5)f RCQM that is based on GBE dynamics for baryons of all five quark flavors.
* Talk delivered by J. Day
SU(3)F.
2 Theory
Our theoretical framework is relativistic quantum mechanics (RQM), which assumes a fixed number of relevant degrees of freedom and relies on an invariant mass operator KX = KXfree + KXint fulfilling all symmetry requirements of the Poincare group. Here, the free and interaction parts of the mass operator are expressed in the rest frame of the baryon (i.e.

free
X vm2 + k2, MXint = X Vi, = £ j + Vf
(1)
i<,
i<,
where kt represent the three-momenta of the individual quarks with rest masses mi and the Q-Q potentials are composed of confinement and hyperfine interactions. By employing such a mass operator KX2 = PHPH, with baryon four-momentum PH = (ft, P), the Poincare algebra of all ten generators {ft, Pt,tt, Kt}, for i = 1,2,3,
[Pi, Pj] = 0, [K i,A]= iPi, Ti,Tj] = ieijkPk,
[Ti, A] = 0, t [Ti,T,] = ieijkTk, [K i, K,] = —ieijkJk,
[Pi, A] = [Ti,Kj] = [Ki,Pj]
0,
: ieijkKk, = iôijA.
is guaranteed.
3
3
3
3 The GBE RCQM
з.1	The SU(3)f Sector
The hyperfine interaction of the GBE RCQM for constituent quarks with flavors
и,	d, and s, confined by a linear potential Vi?nf(r) = Cr, reads
V (r)
V„(r) £ AtaAa + Vk (r) £ AtaAa + V (r) AtsAf + V ' (r) AOaO
a = 1
a=4
CTi • CT,
(2)
where r is the relative vector between constituent quarks i and j. The A? represent the SU(3)f Gell-Mann matrices of flavor a and the o"t the SU(2) Pauli spin matrices of the individual constituent quarks. The GBE is described by the exchange of the octet of pseudoscalar mesons n, K, and n, where due to the U(1) anomaly also the singlet exchange n' is added. The corresponding regularized meson-exchange potentials, derived in instantaneous approximation, are expressed by
Vy (r)
9Y
1
2n 12mi m,
e-Hyr
r
-ayt
AY
Y = n,K,n,n ', (3)
with gY the quark-meson coupling constant, |iy the exchanged meson mass, and AY a cut-off parameter. The complete parameterization of the GBE RCQM can be found in ref. [1]. An extended version of it, including beyond spin-spin forces also all other interaction components stemming from GBE dynamics, was published in ref. [5].
3
7
e
2
M
Y
r
3.2 Generalization to SU(5)F
In the spirit of the ansatz (3) we have generalized the GBE RCQM to SU(5)F in order to cover also heavy baryons, containing the flavors c and b, in a universal model. Keeping the confinement potential unaltered, the extended hyperfine interaction is proposed to be
V hf(r)
3	7	^
Vn(r) KK + VK(r) ^ KK + V„g (r)A?Af + -V,0 (r) +
a=1	a=4
12	14
VD(r)^ AtaA,a + VDs(r) £ AtaA,a + V^(r^A?5 +
a=9	a=13
19	21	23
VB(r) £ A?A? + Vbs (r) £ AtaA,a + Vbc(r) £ AtaA,a +
a=16	a=20	a=22
Vn24 (r)A?4A2^ at • a, .
It contains the GBE in SU(5)F, which is represented by the exchange of the 24-plet of pseudoscalar mesons plus the singlet n0. The various regularized mesonexchange potentials have the same functional dependence as in Eq. (3). The detailed parameterization is given in a forthcoming paper [6].
3.3 Consistency of the Universal GBE RCQM
Since SU(3) c SU(4) c SU(5), the generalized GBE RCQM should perform with similar or even better success as the corresponding SU(3)F model specifically for u-, d-, and s-flavor baryons. This is not immediately obvious, as the light-and strange-baryon sectors are now influenced by an altered singlet exchange, namely, no that corresponds to SU(5)F rather than to SU(3)F. In addition the exchanges of n15 and n24 come into play.
We thus present in Figs. 1 and 2 first a comparison of the spectroscopy of light and strange baryons, as yielded by the original SU(3)F and the extended SU(5)f GBE RCQMs. As becomes clearly evident, the SU(5)F GBE RCQM performs equally well, in some instances even better, than the original SU(3)F one. In particular, the new model also produces the right level orderings in the N and A excitation spectra due to the specific flavor dependence in the hyperfine interaction in Eq. (4).
3.4 Results for Heavy-Baryon Spectra
Next we present the predictions of the SU(5)F GBE RCQM for the spectra of c-and b-flavor baryons in comparison to experimental data available for states with at least 4- or 3-star status according to the PDG (see Fig. 3). It appears that all experimental results, for which also a definite JP is known, are reproduced quite well.
M
[MeV]
1800-_,_
17""-_____	=_
I'.....-
150^ - ^^ ==a""
1400-^^ 1300 -1200 -1100 -
1000 -
900	--- ---
1+ 1- 3- 5-	1- 3+ 3-
2 2 2 2 2 2 2
N	A
Fig. 1. N and A spectra of definite spin and parity JP produced by the extended SU(5)F GBE RCQM (left/red levels) in comparison to the ones of the original SU(3)F GBE RCQM [1,3] (right/blue levels) and to experimental data with their uncertainties (green boxes) from the PDG [7].
M
[MeV]
1800-
1+ 1- 3-
1 +
3+ 3-
1+ 3+ 3-
A	E
Fig. 2. Same as in Fig. 1 but for strange baryons.
3 + 2
Q
5 -
5 -
5 -
2
2
2
2
2
2
2
2
2
2
2
In Fig. 4 we also present the predictions of the SU(5)f GBE RCQM for double-charm baryons. Here, there is only one measurement reported by the PDG, namely, the ground state of ^cc. As can be seen from Fig. 4 and also the Table below, the theoretical level produced by the GBE RCQM remains at variance with the experimental data. For this comparison, however, one should bear in mind that the
M
[MeV]
3100-
3000 -
2900 -
2800 -
27002600 -
2500 -2400 -
2300 -
2200
1 +
2
3 -2
Ac
5 +
2
1 +
2
3 + 2
Sc
1 +
2
3 + 2

M
[MeV]
6400 -
6300 -
6200 -
6100-
6000 -
5900 -5800 -
57005600 -
5500 -
1 +
2
Ab
1	+
2
3 + 2
Sb
1 +
2
fib
Fig. 3. Heavy-baryon spectra of definite JP as produced by the extended SU(5)F GBE RCQM (solid/red levels) in comparison to experimental data with their uncertainties (dotted/green levels resp. boxes) reported by the PDG [7].
lowest Ecc state with Jp = j+ is only rated by 1 star by the PDG. Its measurement was only made once in 2002 by the SELEX collaboration [8], and since then has never been reproduced independently. In view of other theoretical works having investigated double-charm baryons, one may have some doubt about the measured mass of Ecc. As is evident from the comparison in the Table below, for instance, the theoretical results from the RCQM of the Bonn group [11] and also the ones from the Bhaduri-Cohler-Nogami one-gluon-exchange model [9], reported in 2005 by Stancu and Richard [10] at the Bled Workshop, give mass values for
the Ecc ground state quite similar to the one we have achieved. Further measurements of double-charm baryons would thus be highly welcome.
M
[MeV]
440043004200410040003900 380037003600-
3500'
1 +
2
3 + 2
1 +
2
3 + 2
a
Fig.4. Scc and flcc spectra as produced by the extended SU(5)f GBE RCQM (solid/red levels) in comparison to experimental data reported only for the Scc ground state (dotted/green level/box) [7,8].
Baryon JP	Theory	Experiment [8]
Ref. [10] Ref. [11] GBE RCQM
3673 3518.9 ± 0.9 3711	-
3919	-
3919	-
Table 1. Comparison of the predictions by the GBE RCQM and other theoretical models for double-charm Scc ground and excited states vis-a-vis the experimental measurement reported by the SELEX collaboration.
1 +
2
3 + 2
J_-
2
32
3643 3724
3642 3723 3920 3920
— cc
—cc
—cc
-cc
4 Conclusion
We have constructed a universal RCQM for all baryons with flavors u, d, s, c, and b. It is based on a relativistically invariant mass operator describing systems
of three constituent quarks, confined by a linear potential according to QCD and interacting through hyperfine forces derived from GBE. This RCQM extends the previous GBE RCQM beyond SU(3)F and reproduces the phenomenologically known spectra with reasonable accuracy. For definitely pinning down the type of hyperfine interaction especially for light-heavy and heavy-heavy Q-Q subsystems more data in the sector of c- and b-flavor baryons would be highly desirable.
In future it will be very interesting, if the universal GBE RCQM discussed here will be able to describe also reactions involving heavy baryons with a similar good performance as has previously been found for the SU(3) GBE RCQM in the cases of light and strange baryons.
Acknowledgments This work was supported by the Austrian Science Fund, FWF, through the Doctoral Program on Hadrons in Vacuum, Nuclei, and Stars (FWF DK W1203-N16). J.P.D. would like to thank Profs. Ica Stancu and Veljko Dmitrasinovic for valuable discussions during the Workshop, giving him further insights into heavy-baryon spectroscopy.
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