Non-relativistic or relativized models

Non-relativistic or relativized models²² are based on the assumption that the SU(6)⊗O(3) is an underlying symmetry (for constituent quarks is approximately fulfilled). The quarks are bound inside the nucleon by a confinement potential. Except for the confinement potential another term is needed to describe residual interaction. This can be realized in different ways.

Here a short characteristics of some models is given. An excellent review of such models with extensive references can be found in [61].

20Other quantum numbers, parity and spin, have not been extracted from experiments yet because of low statistics.

21Anti-decuplet means that there are only two states with hypercharge Y=1 with spin 1/2.

22Sometimes called symmetrical quark models due to restriction imposed on the SU(6)⊗O(3) states. Here the symmetries corresponds to spin-flavour and space components of the wave function. This part of wave function should be symmetric, the antisymmetry is carried by color part.

In the phenomenological model of Isgur and Karl [62, 63, 64] the proton is treated as a ”soft”

region and consists of three constituent quarks. The light u and d quarks²³ have masses around 250-300 MeV and the s quark is 150-200 MeV heavier. The gluon field defines the quark dynamics by confining potential between pairs of quarks for the large distances. The one-gluon exchange (OGE) provides a Coulomb potential and a spin-dependent potential.

In this model any gluonic excitations are neglected. In the model the stationary Schr¨odinger equation is solved for the three valence-quark system with a Hamiltonian

H =

i

m_i+ p²_i 2m_i

+

i<J

V^ij +H_hyp^ij

, (1.39)

where the spin-independent potential V^ij =C_qqq+br_ij−²_3r^α_ij^s, with r_ij =|r_i−r_j|. In fact the V^ij is often chosen as a harmonic-oscillator potential Kr_ij/2 plus unharmonicity U_ij which is treated as a perturbation. The hyperfine interaction is chosen in the following way

H_hyp^ij = 2α_s 3mimj{8π

3

S_i·S_jδ³(r_ij) + 1

r_ij³ [3(S_i·r_ij)(S_j ·r_ij)

r_ij³ −S_i·S_j]}. (1.40) Here the contact and tensor terms²⁴ are arising from the color magnetic dipole-magnetic in-teraction. In this model the spin-orbit forces coming from OGE and from Thomas precession of the quark spins in the confining potential are neglected. If they are taken into account the agreement with experimental spectra will be worse where the splittings tend to be too large.

The spin-independent and momentum-dependant terms like Darwin and orbit-orbit interac-tion are neglected as well in the model of Karl and Isgur. By moving to a Jacobi coordinate system the Hamiltonian can be separated into two independent three-dimensional oscillators whenU =H_hyp = 0. Therefore the spatial wave function can be written as sums of products of three-dimensional harmonic oscillator eigenstates with quantum numbers (n,l,m): number of radial nodes and the orbital angular momentum quantum numbers.

This model satisfactorily describes the baryon spectrum but it still has inconsistencies. For example, for bound systems of light quarks p/m 1, so the non-relativistic approximation is not justified. It is also inconsistent to neglect spin-orbit terms and motivate this by the cancelation with Thomas precession; more recent calculations show that this is not true. The model also has difficulties describing the Roper resonance meaning that the wave function should have a large anharmonic mixing with the ground states; therefore arises the question whether a first order perturbation theory can be applied.

In an extended potential model based on OGE physicists have tried to correct some of the earlier inconsistencies by introducing additional terms. The relativized quark model for mesons by Godfrey and Isgur [65], which was extended to baryons by Capstick, Isgur and Roberts [66, 67, 68, 69], introduces extra relativistic terms for quark energies and momen-tum dependency in the potential. The Schr¨odinger equation is solved in Hilbert space with Hamiltonian

23The ”dressed” valence quarks can be like extended objects. There is no partons as in DIS and the interactions is effective but QCD inspired.

24Their strengths are as determined from the expansion to O(p²/m²) (Breit-Fermi limit) of the OGE potential.

H=

i

p²_i +m²_i +V, (1.41)

whereV is a relative-position and -momentum dependant potential, consisting of a confining string potential, a pairwise Coulomb potential, a hyperfine potential, as well as spin-orbit po-tentials with OGE and Thomas precession in the confining potential. This model reproduces the pattern of splitting in the negative and positive-parity bands of excited non-strange states rather well, although the centers of the bands are missed by +50 and -50 MeV respectively.

The Roper resonance is 100-150 MeV too high but fits into the pattern. Negative-parity ∆^∗ states around the 1900 MeV band appear at 2.1 GeV. The inclusion of the spin-independent but momentum-dependant terms [70] presented in anO(p²/m²) reduction of the OGE poten-tial reduces the energy of certain positive-parity excited states and raises the negative-parity, therefore partially fixing 50 MeV displacement.

In the model of Glozman and Riska [71] Goldstone-bosons are playing the role of exchange particles. This model analyzes the baryon spectrum using exchange of the particles of a pseudoscalar octet only for the hyperfine interaction. The baryons are described as three quark system which couples to meson fields. It is argued that there is no evidence for OGE hyperfine interaction. In more recent works [72, 73] the model was extended to include the exchange of a nonet of vector mesons and a scalar meson. This model describes the baryon spectrum rather well because of a large number of free parameters fitted to the data. The description of the spectra is somewhat better than in the relativized model of Capstick and Isgur with OGE [66] which uses only 13 parameters to fit the non-strange sector and 8 of them are similar for the meson sector [65]. For example the Goldstone-boson exchange model needs twelve new parameters to describe only strange baryons. It is not clear from [72, 73]

how many parameters in total have been used. The model has also some difficulties with unification of the description of mesons and baryons by using similar parameters.

There are also models based on an algebraic approach. A collective model of baryon masses, electromagnetic couplings and strong decays based on a spectrum-generating algebra has been developed by Bijker, Iachello and Leviatan [74, 75]. The idea is to extend the alge-braic approach leading to the mass formulas based on spin-flavor symmetry (SU(6), sym-metric quark models), to the spatial structure of the states. The quantum numbers of the states are considered to be distributed spatially over a Y-shaped string-like configuration. To find the dynamic of the system, the bosonic quantization of the spatial degrees of freedom (Schr¨odinger-like equations) is used (e.g. two relative Jacobi coordinates for an oscillator).

This leads to six vector boson operators bilinear in the components of these coordinates and their conjugate momenta, plus an additional scalar boson, generating the Lie algebra U(7). U(7)⊗SU(3)_{f lavor}⊗SU(2)_spin⊗SU(3)_color is given as the spectrum-generating algebra for baryons. The further technical details of constructing such algebra and further references can be found in [74, 75]. This model describes spectra quite well and also predicts more

”missing” baryon states than the valence quark model.

1.2.6 Phenomenological description of meson and baryon spectra