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The light baryon spectrum in a relativistic quark model with instanton-induced quark forces I. The non-strange baryon spectrum and ground-states

This is the second of a series of three papers treating light baryon resonances up to 3 GeV within a relativistically covariant quark model based on the three-fermion Bethe-Salpeter equation with instantaneous two- and three-body forces. In this paper we apply the covariant Salpeter framework (which we developed in the first paper) to specific quark model calculations. Quark confinement is realized by a linearly rising three-body string potential with appropriate spinorial structures in Dirac-space. To describe the hyperfine structure of the baryon spectrum we adopt 't Hooft's residual interaction based on QCD-instanton effects and demonstrate that the alternative one-gluon-exchange is disfavored phenomenological grounds. Our fully relativistic framework allows to investigate the effects of the full Dirac structures of residual and confinement forces on the structure of the mass spectrum. In the present paper we present a detailed analysis of the complete non-strange baryon spectrum and show that several prominent features of the nucleon spectrum such as e.g. the Roper resonance and approximate ''parity doublets'' can be uniformly explained due to a specific interplay of relativistic effects, the confinement potential and 't Hooft's force. The results for the spectrum of strange baryons will be discussed in a subsequent paper.


Introduction
In the previous paper [1] we analyzed the three-fermion Bethe-Salpeter equation with instantaneous two-and threebody interaction kernels. Without being too specific concerning the interaction kernels, we derived the three-fermion Salpeter equation. We now want to apply this covariant formalism to a system of three light quarks with flavors up, down and strange and thus use this fully relativistic framework as basis for a quark model of light baryons. In fact, the reduced Salpeter equation provides a suitable, fully relativistic framework, which nonetheless keeps as close as possible to the rather successful non-relativistic potential models: On the one hand, we found a one-to-one correspondence of the Salpeter amplitudes with the ordinary states of the non-relativistic quark model meditated by the embedding map of non-relativistic three-quark Pauli spinors to full three-quark Dirac spinors. On the other hand, this approach adopts the concept of constituent quarks with an effective mass where the underlying interactions are described by inter-quark potentials in the rest-frame of the baryons. Hence, we basically have the same input describing the quark dynamics as in non-relativistic quark models, in particular the number of parameters remains exactly the same as in a corresponding non-relativistic approach 1 . To be specific, we now have to fill in the details of the underlying quark interactions, i.e. we have to specify the three-and two-body potentials V (3) and V (2) , respectively, which we use as instantaneous interaction kernels. We then solve the resulting Salpeter equation numerically. The fact that this equation can be cast in Hamiltonian form allows for the use of a variational principle: We expand our wave functions in terms of harmonic oscillator functions and diagonalize the Hamiltonian with respect to a truncated wave function basis checking carefully numerical stabilities.
One of our aims is to extend a covariant quark model for mesons [10,11,12,13,14,15,16] which is based on the quark-antiquark Bethe-Salpeter equation with instantaneous two-body interaction kernels. In this model the interaction between quark and antiquark included a linearly rising (string-like) confinement potential provided with a suitable spinorial form (Dirac structure), which was combined with the effective residual interaction first computed by 't Hooft from instanton effects in QCD [6]. In fact it turned out that this relativistic approach, which employs 't Hooft's force as residual interaction, provides significant improvements with respect to other, non-relativistic or "relativized" approaches which in general use parts of the residual one-gluon-exchange in addition to a confining central potential.
In particular, this model allows for a consistent and complete description of the whole mesonic mass spectrum but at the same time also for the description of dynamical observables such as form factors, where a fully covariant treatment of the quark dynamics becomes particularly crucial. The results are rather encouraging to extend this approach to baryons. Thus, our main choice for V (2) is 't Hooft's instanton-induced quark interaction. However, in appendix A we will also show the result of calculations with a one-gluon-exchange potential, demonstrating that from a phenomenological point of view it should be discarded.
Similar to the Salpeter model for mesons, quarks in baryons shall be confined by a linearly rising string potential. Generalizing the linear quark-antiquark confinement for mesons our quark confinement for baryons will be produced by a three-quark string potential V (3) provided with an appropriate Dirac structure. As in the meson model [10,11,15,16] we use in addition 't Hooft's two-quark interaction mentioned before as residual two-body force V (2) . A non-relativistic version with these dynamics has been applied already by Blask et al. [17,18,19] for the calculation of baryon (and meson) mass spectra. We would like to note that this model could satisfactorily account for the gross features of the light baryon spectrum with only seven parameters. In particular, it was able to explain the sign and the rough size of hyperfine splittings of ground-state baryons as well as the right size of splittings of negative parity excited baryons. However, a closer look at the mass spectra reveals that special features, such as e.g. the conspicuously low-lying first scalar/isoscalar excitations of the octet ground-states (Roper resonances) or the highest members of Regge trajectories cannot be accounted for in the non-relativistic framework. In particular, these issues improve in the present relativistic approach. We will not always comment in detail on calculations in alternative baryon models; for an excellent review see ref. [51].
In this paper baryon resonances are treated as bound states and no calculation of widths is performed. This is certainly questionable, but so far is due to technical limitations. To improve this situation one has to specify the decay channels and perform at least a perturbative calculation of the decay widths [20]. Very often this is not sufficient since final state interactions may also change (nonperturbatively) the resonance positions appreciably [21,22,23,24,25,26]. As long as this shift is approximately uniform, it can be absorbed in the potential parameters of the quark model. We are aware of the fact that this is true in many, but not all cases [21,22,23,24,25,26].
Our paper is organized as follows: In section 2 we specify the explicit form of the instantaneous three-quark confinement kernel V (3) . We introduce two alternative confinement models (model A and model B) which essentially differ in the choice of the Dirac structures only. Both versions shall be tested in the subsequent investigations in a comparison to the experimental mass spectrum. In section 3 we introduce 't Hooft's instanton-induced residual twoquark interaction and discuss its specific structure. In the subsequent sections a detailed discussion of our results in comparison to the phenomenological non-strange baryon (and ground-state) spectrum is given. We start with some general comments concerning the parameter dependencies in section 4. In section 5 we investigate the results of our calculations for the ∆-spectrum where 't Hooft's interaction gives no contribution. These investigations constitute a first test of the confinement models considered. Section 6 is concerned with the hyperfine structure of the groundstate baryons and the role of the instanton-induced interaction for generating this structure. Section 7 is devoted to an extensive discussion of the excited nucleon spectrum. A principal objective of this discussion is to demonstrate how instanton-induced effects along with the ∆ − N hyperfine splitting simultaneously generate several prominent structures seen in the experimental nucleon spectrum such as e.g. the low position of the Roper resonance or the occurrence of approximate "parity doublets". Finally we give a summary and a conclusion in section 8. A detailed discussion of the corresponding results for the strange baryons will be presented in a subsequent paper [2].

Three-body confinement
It is well known that the global structure of the experimental baryon and meson mass spectrum suggests a linearly rising, flavor-independent confinement potential. The appearance of the experimentally observed baryon resonances as nearly degenerate, alternating even-and odd-parity shells motivates the picture that the quarks are moving in a local potential which roughly reflects harmonic forces between the quarks. This picture led to the naive quark oscillator shell model. Moreover, phenomenological analyses of the experimental baryon (and also meson) spectra up to highest orbital excitations show a remarkable empirical connection between the total spin J and the squared mass M 2 of the states, namely that certain states, within a so-called Chew-Frautschi plot of M 2 versus J, lie on linear Regge trajectories M 2 ∼ J. The interpretation of this empirical feature motivates a string picture for the confinement mechanism, where the quarks are connected by gluonic strings (flux tubes) such that the effective confining potential rises linearly with the string length for large distances of the quarks. A further confirmation of this scenario stems from lattice QCD calculations, which in fact indicate a string-like realization of the confinement force in the static limit of heavy quarks; for a review of these issues we refer to [27] and references therein. We therefore choose a three-body confinement force which in the rest-frame of the baryon takes the form: Here V conf (x 1 , conf (r 3q ) ∼ r 3q denotes the local three-quark potential whose radial dependence is assumed to be linearly increasing with some 'collective' radius r 3q = r 3q (x 1 , x 2 , x 3 ) (string picture). (As will be discussed below such a three-quark distance can be realized in various ways.) In contrast to non-relativistic quark models, in our fully relativistic framework the radial dependence has to be provided with an appropriate spinorial three-quark Dirac structure W. Thus, the most general ansatz for a flavor-independent confinement potential V (3) conf , which rises linearly with the inter-quark distance r 3q , is given by where a is an overall constant offset associated with a three-quark Dirac structure W off and b is the slope of the linearly rising part which is associated with the Dirac structure W str . In general the spin structures W off and W str of the constant and linear part can be chosen differently (see discussion below). As illustrated in fig. 1 there are several possibilities to define the linear inter-quark distance, which we refer to as Y-type, ∆-type and hyperspherical (H) string. The first type was proposed by Carlson et al. [28,29] and is intimately related to the color-SU (3) group: In a color singlet qqq system, each quark acts as a source of one flux tube. For SU (3) gauge fields the flux tubes can merge at a single point x 0 , which in the adiabatic approximation is chosen such that the energy and hence the length r 3q = r Y of the connecting path is minimized to Then the straight lines emanating from the quarks will meet at an angle 2π/3 at the central junction point x 0 (see fig. 1), unless one of the angles within the baryonic triangle exceeds the value 2π/3, in which case a linear geometry is preferred. An alternative three-quark distance r 3q = r ∆ is defined by the ∆-string configuration, in which case the string is formed by the sum of two-body strings between each quark pair: Rescaling this quark distance with a factor f ≃ 0.5493 [9,30,18] its length f ·r ∆ constitutes a fairly good approximation to the Y -type string potential Obviously, whenever the three quarks are collinear, the factor is exactly 1/2, while for the other extreme case of an equilateral triangle the factor is 1/ √ 3. The choice f = 0.5493 lies in between these two extremes. This particular ratio is chosen such that it minimizes the size of the expectation value of |r Y − f r ∆ | in the harmonic-oscillator basis [9,18] and moreover is also favored by some investigations of flux tubes on the lattice [30]. Only a few lattice results on static baryonic potentials (see [27] and references therein) exist so far with statistical errors too large to rule out either possibility. In a more recent study (see [27]), however, clear evidence in support of the ∆-configuration has been found. A further example is a hyperspherical ansatz [31,32], which increases linearly with the hyperradius r 3q (x 1 , x 2 , x 3 ) = r hyp := |ρ| 2 + |λ| 2 , where ρ := 1 We would like to note at this stage that we have tested the various radial dependencies (3), (4) and (6) in our Salpeter model. Our investigations, however, clearly showed that the structure of resulting spectra depends only slightly on the various radial dependencies chosen. It turned out that the slope parameter b can always be appropriately rescaled (as e.g. in eq. (5) with the factor f) to obtain almost the same spectrum for all three choices. We therefore prefer for our model the ∆-shape string potential rising linearly with r ∆ (x 1 , x 2 , x 3 ) = i<j |x i − x j | which, on the one hand, is favored by the most recent lattice studies anyway and, on the other hand, is also much easier to handle numerically. We found, however, that the structure of the resulting spectra depends much more on the Dirac structure chosen, which we shall consider next.
The gross features of the baryon resonances seem to indicate that the dominating confinement forces should be spin-independent, at least in the non-relativistic limit. This property can be realized exactly if and only if W off and W str have the form W str = α str 1I ⊗ 1I ⊗ 1I + β str γ 0 ⊗ γ 0 ⊗ 1I + cycl.perm.
which have in common that they give the same nonrelativistic limit and that they produce equivalent results in flavor-symmetric states. The offset constant a and the slope b enter as free parameters in our model.

't Hooft's instanton induced interaction
As originally shown by 't Hooft [6] instantons lead to an effective contribution to the interaction of quarks. The corresponding effective Lagrangian ∆L eff which has been calculated by 't Hooft for the case of the SU (2) gauge group has been generalized to the case of the SU (3) gauge group by Shifman, Vainshtein and Zakharov [7]. They have shown that the contribution of a single instanton and antiinstanton configuration to the effective quark Lagrangian in the case of the SU (3) color group and three light quarks with flavors up (u), down (d) and strange (s) is given by [7] ∆L eff (y) = × m 0 s ρ − 4 3 π 2 ρ 3 (Ψ sR Ψ sL ) + 9 40 4 3 π 2 ρ 3 d abc (Ψ uR σ µν λ a Ψ uL )(Ψ dR σ µν λ b Ψ dL )(Ψ sR λ c Ψ sL ) + cycl. perm. of (uds) + 9 320 Here Ψ f L,R ≡ Ψ f L,R (y) are the projections on left and right handed components of the quark field operators: An important quantity entering eq. (11) as a generic weight is the reduced instanton density D(ρ) = d 0 (ρ)/ρ 5 which describes the vacuum-to-vacuum tunneling probability in the presence of an instanton with size ρ. In the case of three colors and three flavors d 0 (ρ) is given by [33] d 0 (ρ) = (3.63 10 −3 ) 8π 2 g 2 (ρ) 6 exp − 8π 2 g 2 (ρ) (14) The function g(ρ) denotes the ρ-dependent running coupling constant, which in two-loop accuracy reads 8π 2 g 2 (ρ) = 9 ln 1 ρ Λ QCD + 32 9 ln ln 1 ρ Λ QCD , where Λ QCD is the QCD scale parameter. Unfortunately, using the semi-classical 't Hooft formula (14), large size instantons make the integration diverge due to the power law behavior of the reduced instanton size distribution D(ρ) = d 0 (ρ)/ρ 5 . This known as the so-called 'infrared problem'. To cure this problem, we cut the ρ integral for the moment by hand [34], introducing a critical instanton size ρ c for which the ln ln-term of (15) is still reasonably small compared to the ln-term.
The interaction described before is chirally invariant if the current quark masses are strictly zero. As is well known it is no longer U A (1)-invariant and thus exhibits the characteristic breaking of this symmetry in QCD. In addition we must take the spontaneous breaking of chiral invariance into account which is most easily done by normal ordering ∆L eff with respect to a vacuum of massive quarks with constituent quark masses. The result is ∆L eff = ǫ + ∆L (1) eff + ∆L (2) eff + ∆L (3) eff .
1. Here ǫ is a constant (a shift in the vacuum energy). 2. ∆L (1) eff is a mass term: ∆L eff (1) = − ∆m n : Ψ u Ψ u : + : Ψ d Ψ d : − ∆m s : Ψ s Ψ s :, with where m n := m 0 n +∆m n and m s := m 0 s +∆m s are naturally identified with the non-strange and strange constituent quark masses (if possible contributions from the confining forces are excluded). In the following we will treat the effective constituent quark masses m n and m s as free parameters which we fit to the experimental baryon spectrum. 3. ∆L (2) eff is a two-body interaction which can be cast in the more convenient form ∆L (2) eff = −2 : Ψ ⊗ Ψ 1I ⊗ 1I + γ 5 ⊗ γ 5 P D S=0 ⊗ g nn P F A (nn) + g ns P F A (ns) ⊗ P C 3 Ψ ⊗ Ψ : − : Ψ ⊗ Ψ 1I ⊗ 1I + γ 5 ⊗ γ 5 P D S=1 ⊗ g nn P F A (nn) + g ns P F A (ns) ⊗ P C 6 Ψ ⊗ Ψ : .
where the effective coupling constants g nn := 3/8 g eff (s) and g ns := 3/8 g eff (n) are given by This expression in terms of two-quark projection operators in Dirac, flavor and color spaces allows an immediate identification of the diquark channels that in fact are affected by the interaction: In color space P C 3 and P C 6 denote the projectors onto color antitriplet and color sextet pairs. Since the three quarks within the baryon have to constitute a color singlet two quarks always have to be in a color antitriplet state. For this reason the second term of the Lagrangian (20), which affects color-sextet quark pairs only, does not contribute in lowest order to the baryon dynamics. In Dirac space, P D S=0 and P D S=1 are the projectors onto antisymmetric spin-singlet and symmetric spin-triplet states, respectively, defined by where σ i are the usual Pauli matrices. In flavor space the operators P F A (nn) and P F A (ns) denote the projectors onto flavor-antisymmetric quarks which either are non-strange (nn), i.e. with isospin zero, or non-strange-strange (ns): Here P F n = |u u| + |d d| and P F s = |s s| are the projectors for single quark flavors and P F A = 1 2 (1I F − P F 12 ) is the antisymmetrizer in the two-particle flavor space. Accordingly, 't Hooft's force acts exclusively on quark pairs which are antisymmetric in flavor and it distinguishes between diquarks of different flavor content (nn) and (ns) by means of the different effective couplings g nn and g ns . 4. ∆L (3) eff is just the original interaction but now with constituent quark field operators. It yields interaction vertices whose contribution within the Bethe-Salpeter equation vanishes for color-free three-quark states, and hence needs no further discussion.
The most interesting contribution stems from the two-body interaction Lagrangian. It is usually used in connection with the solution of the U A (1)-problem. In the framework of calculations for mesons [11,16] we have indeed shown that it yields the correct splitting of the lowest meson nonet and in general of all low-lying meson states. Hence we have good reasons to use it also for the calculation of baryon masses. As in the case of mesons we have to regularize the spatial δ-function dependence of this two-body Lagrangian by a form factor, which we give a Gaussian form: For the moment we make no effort to derive it and treat the effective range λ as well as the couplings g nn and g ns as phenomenological parameters which we fit to the experimental baryon spectrum. Our candidate for the instantaneous two-quark interaction kernel V (2) has then the following form:

Parameters and general comments
In this (and a consecutive [2]) paper we will present a detailed discussion of the resulting mass spectra of light baryons (with the light quark flavors 'up', 'down' and 'strange') calculated within our covariant Salpeter framework using the confinement models A and B and employing 't Hooft's instanton-induced force as residual interaction. The seven free parameters entering in the present calculations are listed in table 1. These parameters involve the effective non-strange and strange constituent quark masses m n and m s , and the confinement parameters a and b describing the off-set and the string tension of the three-body confinement potential. 't Hooft's instanton-induced interaction is determined by the couplings g nn and g ns and the effective range λ.

Parameter
Model Our detailed discussion of the results for the light flavor baryons will be organized according to the specific parameter dependence dictated by the simple characteristic and selective action of the instanton-induced interaction. Due to the simplicity of 't Hooft's force and the induced strong selection rules, a clear cut identification of the various effects of the long range confining potential and the short range residual interaction can be made by comparison with the experimentally observed baryon spectra. Accordingly, the discussion of our results in the subsequent sections will be organized as follows: As 't Hooft's force acts only on flavor-antisymmetric quark pairs, this residual interaction does not contribute to the non-strange spectrum of ∆-resonances due to their common totally symmetric flavor wave function with isospin T = 3/2. Therefore, in our approach (without any additional residual interactions apart from 't Hooft's force) the whole ∆-resonance 2 spectrum is determined by the confining three-body force and the relativistic dynamics alone! This feature thus provides a first test of our phenomenological ansatz for the confinement potential. Furthermore, we can fix the confinement parameters, i.e. the off-set parameter a and the slope b, as well as the non-strange quark mass m n by the positions of the experimentally best established ∆-resonances without any influence of the residual interaction. In this respect, the confining three-body interaction kernel, which is a string-like, i.e. linearly rising three-body potential with the two distinct Dirac structures of the models A and B, has to account for the correct description of the positive parity ∆-Regge trajectory M 2 ∼ J up to the highest spin known J = 15/2 + . At the same time, it should not induce too large spin-orbit effects, for which there is hardly any evidence in the experimental baryon spectrum. The results for the complete ∆-spectrum in both model versions will be discussed in sect. 5.
Like the ∆-resonances with the corresponding ground-state ∆(1232), also the remaining strange spin-3/2 + decuplet ground-states, i.e. the hyperons Σ * (1385), Ξ * (1583) and Ω(1672), are unaffected by 't Hooft's interaction. Hence, the position of these states then determines the strange constituent quark mass m s . In this way, all parameters except for those of the 't Hooft interaction are fixed, and in the second step we then can analyze the effect of the instanton-induced interaction on the structure of the remaining baryon spectra.
As already known from non-relativistic potential models the long range confinement potential cannot describe the splittings of the spin-1/2 octet and spin-3/2 decuplet ground-states. Thus, the first feature the 't Hooft interaction has to account for is the hyperfine structure of the ground states, i.e. the mass splittings ∆(1232) − N (939) of the nonstrange ground-states and the hyperon splittings Σ * (1385) − Σ(1193), Ξ * (1583) − Ξ(1318) and Σ(1193) − Λ(1116). In this respect we will show that the instanton-induced interaction indeed leads to an at least as satisfactory description as the short-range spin-spin part of the Fermi-Breit interaction, which is generated by the one-gluon exchange (OGE) and is commonly used in non-relativistic (or 'relativized') quark models [35,36,9]. In the non-strange sector the ∆ − N 2 Of course the same also applies to the whole Ω-spectrum. Unfortunately the only well-established Ω-state still is just the decuplet ground-state Ω 3 2 + (1672) mass splitting is explained by 't Hooft's force being attractive for the nucleon, where two (non-strange) quarks can be in a state with trivial spin and isospin. Thus the ∆ − N splitting fixes the coupling g nn . In a similar manner 't Hooft's force is attractive for Σ and Ξ with the strength g ns where a non-strange-strange (ns) quark pair can be in a state with trivial spin and antisymmetric flavor. In this way the splittings Σ * − Σ and Ξ * − Ξ determine the coupling g ns . At the same time, the position of Λ(1116), which contains both types (nn and ns) of quark pairs and thus is influenced by both couplings g nn and g ns , should then be properly described in order to get the right experimentally observed Σ − Λ mass difference. Generating the hyperfine structure of ground-state baryons by 't Hooft's force will be investigated in detail in sect. 6.
All the parameters being fixed, the calculation of all other baryon masses is then parameter-free, i.e. the majority of excited mass spectra of N -resonances, Λ-resonances, Σ-resonances, Ξ-resonances and the Ω baryons are thus predictions of our models A and B. In this paper we restrict our extensive discussion to the predictions for the nucleon sector (sect. 7). The parameter-free predictions for the complete spectrum of excited strange baryons will be discussed in a subsequent paper [2].
A principal objective of our discussion is to demonstrate that the instanton-induced interaction along with a fully relativistic treatment of the quark dynamics within our covariant Salpeter approach plays an important role in the description of the complete spectrum of light baryons, i.e. not only for the hyperfine structure of the ground-state baryons as e.g. the ∆ − N -splitting, but especially for distinctive features of the excited spectra. In both confinement models we will explore in detail how and to what extent the striking features of the excited spectra can really be understood by the effect of 't Hooft's force with its parameters g nn , g ns and λ fixed from the hyperfine structure of the octet and decuplet ground state baryons. Prominent features of the excited baryon spectra that shall be discussed in this context are for instance the conspicuously low positions of the first isoscalar/scalar excitations of octet ground-states, i.e. the Roper-resonance N 1 2 + (1440) and its strange counter parts Λ 1 2 + (1600) and Σ 1 2 + (1660), or the appearance of approximately degenerate states of the same spin but opposite parity (approximate "parity doublets"). Now let us turn to the discussion of the light baryon spectra. We begin with the discussion of the ∆-spectrum, which phenomenology reflects the role of the three-body confinement kernel in the Salpeter equation.
5 The ∆-spectrum -A first test of the confinement kernel

Introductory remarks
In this subsection we investigate and discuss the results of both confinement models A and B in the sector of ∆-baryons (with isospin T = 3 2 and strangeness S * = 0). We compare our results with the main features of the complete, hitherto measured ∆-resonance spectrum up to the highest orbital excitations with total spin J = 15 2 . As already mentioned, there is no effect of 't Hooft's force in this sector, because the flavor part of the amplitudes is given by the totally symmetric isospin-3/2 flavor function. Therefore, the internal quark dynamics of the ∆-baryons is described by the relativistic kinetic energy and the three-body confinement potential only. This offers the possibility to study the role of the confinement force only, i.e. without the influence of the residual instanton-induced interaction. Therefore, the calculation of the ∆-spectrum constitutes a first important test of the different three-body confinement kernels A and B. As mentioned before the phenomenology of Regge trajectories motivates the string picture for confinement, i.e. in our case a three-body potential, which rises linearly with the distance between the quarks. In contrast to the radial (stringlike) dependence of the confinement mechanism originating e.g. from the flux tube model [28,29], but also motivated by recent lattice QCD calculations [27], the spinorial (Dirac) structure of three-quark confinement lacks any clue from first principles or lattice QCD. Thus, an appropriate ansatz for the Dirac structure of the confinement kernel is motivated by phenomenological arguments only. Here the aim is to reproduce the global structure of the ∆-spectrum, especially the well-established positive parity ∆-Regge trajectory, with the empirical characteristic M 2 ∝ J. In particular, the slope of the trajectory should be quantitatively correctly described in order to reproduce the resonance positions even of the highest orbital excitations. At the same time the confinement force has to account for the correct energy gaps between the approximately degenerate bands, which are formed by the radially excited states with different total angular momenta J (corresponding to the even-parity 2hω, 4hω, 6hω and odd-parity 1hω, 3hω, 5hω 'oscillator shells' of the naive quark oscillator shell model 3 ). Moreover, the observed approximate degeneracy in these band structures indicates that spin-orbit effects are rather small. This is an additional experimental feature that we aim to describe in our models. Consequently, the Dirac structures chosen should be such that spin-orbit forces are not too large and compatible with the experimentally observed small intra-band splittings of the different shells. In this respect, the spin structures of model A and B, which in both cases consist of a special combination of a scalar (1I ⊗ 1I ⊗ 1I) and a time-like vector part (γ 0 ⊗ γ 0 ⊗ 1I + . . .), have already been selected from a wider class of possible Dirac structures that are all compatible with Lorentz covariance, time-reversal-, parity-and CPT -invariance, as well as the hermiticity requirements of the Salpeter equation. Here we want to verify, that both confinement versions can indeed account for the phenomenological features of the ∆-spectrum. Then, (due to the assumed flavor independence of the confinement force) these should be also appropriate for the other baryon sectors, where in addition instanton effects become substantial. The phenomenology of the ∆-spectrum already fixes the offset parameter a and the slope b of the confinement potential, as well as the non-strange quark mass m n . This is mainly done by fitting the positiveparity ∆-Regge trajectory, i.e. the ground-state ∆ 3 2 + (1232, ****), and its orbital excitations ∆ 7 2 + (1950, ****) and ∆ 11 2 + (2420, ****), which all are well-established four-star resonances with moderate uncertainties in the observed resonance positions. There is evidence of a fourth member of this trajectory, namely the next spin-15/2 resonance ∆ 15 2 + (2950, **), which, however, is only a two-star resonance with quite large experimental uncertainties. Therefore, this member is not used in the fit and thus will be a prediction. In order to achieve the correct gap between the positive-parity ground state ∆ 3 2 + (1232, ****) and the first negative parity (1hω) band, we additionally consider also the lowest two, well-established four-star resonances of negative parity, i.e. ∆ 1 2 − (1620, ****) and ∆ 3 2 − (1700, ****), for fixing the three parameters a, b and m n .

5.2
The positive-parity ∆-Regge trajectory Figure 2 shows the Chew-Frautschi plot (M 2 versus J) of the positive parity ∆-Regge trajectory as obtained in models A and B with the parameters a, b and m n given in table 1. Indeed, both models yield excellent Regge trajectories, which show the qualitatively correct linear characteristics.
In Comparing both models, we find model A yielding a slightly larger slope of the trajectory (see fig. 2) in somewhat better agreement with experiment than model B for the high-spin states. On the other hand model A predicts the position of the ∆-ground-state ∆ 3 2 + (1232, ****) slightly too high in comparison to the experimentally determined position, whereas model B fits this resonance position exactly. Altogether, we thus find both models to be of the same good quality. Note that the states ∆ 3 2 + , ∆ 7 2 + , ∆ 11 2 + , ∆ 15 2 + , . . ., belonging to the Regge trajectory, correspond to a sequence of baryons with increasingly large separations of the quarks. Accordingly, this result strongly supports our ansatz of the string-like confinement mechanism in both models. In the following a more detailed discussion of these models concerning the structure of the complete ∆-spectrum is presented. in a (finite N < Nmax) basis of oscillator states with given flavor, spin and parity, the states of all shells in general mix. On the other hand it is very likely that the dominant part of the corresponding wave function is made up of Nhω states.

Model B
Model A PDG 2000, [37] ∆ * *  [37]). Both models yield the correct linear Regge characteristic M 2 ∼ J in good agreement with experiment. See table 2 for numerical values. Figure 3 shows the resulting positions of positive and negative parity ∆-baryons (isospin T = 3 2 and strangeness S * = 0) with total angular momenta up to J = 15 2 obtained in model A. These are compared to all presently known resonances quoted by the Particle Data Group [37]. Likewise, figure 4 displays the corresponding results of model B. The resonances in each column are classified by their total spin J and parity π. On the left of each column at most the first ten calculated states are shown and can be compared with the experimentally observed positions on the right hand side of each column. The corresponding uncertainties in the measured resonance positions are indicated by the shaded areas. The status of each resonance is denoted by the corresponding number of stars following the notation of the Particle Data Group [37] and moreover, by the shading of the error box which is darker for better established resonances. In addition to figs. 3 and 4, the calculated masses of the excited states are summarized according to their assignment to a particular shell in one of the tables 3 -8. belong to the positive-parity ∆-Regge trajectory. The centroids of the various even-and odd-parity bands are in good agreement with the centroids of the corresponding experimentally observed band structures with the exception of the negative-parity states around 1900 MeV. Concerning the structures of the shells themselves it should be noted that model A exhibits larger intra-band splittings than model B due to slightly larger spin-orbit effects in model A. In this respect, the Dirac structure (consisting of the scalar and time-like vector part in both models) is combined in model B such that the spin-orbit forces of both parts cancel and thus these relativistic effects, which arise in connection with the embedding map in the Salpeter amplitudes, are really minimized. Nevertheless, also the intra-band splittings of model A are still moderate enough to be compatible with the central values for the experimentally observed resonance masses. They even generate the desired feature that the average mass values in each shell rise with total angular momentum J, in contrast to model B, where the states with different spin J are almost degenerate. This produces a slight tilt in the shells of model A (see e.g. the positive parity 2hω shell), which generally yields higher resonance positions for the states with maximum angular momentum, thus producing positive-and negative-parity Regge trajectories with a slightly bigger slope in model A in better agreement with the experiment. Now let us discuss the single resonance positions and the pattern of splittings in the positive-and negative-parity bands of the excited ∆-spectrum in some more detail.

States of the 1hω band
We start with the negative-parity 1hω band, where our models predict (as usual in constituent quark models) two states which can be uniquely assigned to the two well-established four-star resonances ∆ 1 2 − (1620, ****) and ∆ 3 2 − (1700, ****). Our results for these states are given explicitly in table 3.
Both models yield the correct center-of-gravity of this 1hω band. In particular, the position of ∆ 1 2 − (1620, ****) is fairly well reproduced within the experimental range: model A predicts 1654 MeV and model B gives 1625 MeV. Note that in quark models, which use the one-gluon exchange as a residual interaction, the ∆ ground-state is shifted upwards Exp. state PW J π Rating Mass range [MeV] Model state Model state [37] [ 37] in model A in model B ∆(1620) S31  Table 3. Calculated positions of ∆-states assigned to the negative parity 1hω shell in comparison to the corresponding experimental mass values taken from [37]. PW denotes the partial wave and the rating is given according to the PDG classification [37].
Here and throughout this work we use the notation [B J π ]n(M ) for the predicted model states in model A and B, respectively, where B denotes the baryon (i.e. the flavor), J π are spin and parity, and M is the predicted mass given in MeV. n = 1, 2, 3, . . . is the principal quantum number counting the states in each sector J π beginning with the lowest state.  2 + (2000, **) note, however, that their rating is only oneand two-star, respectively. On the other hand, the intra-band splittings in model A, which are induced by moderate spin-orbit effects, reproduce the experimentally indicated pattern of splittings in this shell quite well. Consequently, a tentative assignment of the states on the basis of their positions compared to the experimental values is less ambiguous. Let us discuss the intra-band pattern in some more detail (see also  In the spin-1/2 + sector models A and B describe the situation very similarly. In model A we find two close states at 1866 MeV and 1906 MeV, where the latter fits nicely the well-established four-star resonance ∆ 1 2 + (1910, ****) within the assigned errors, while the other one overestimates the position of the poorly determined one-star resonance ∆ 1 2 + (1750, *). In the sector with J π = 3/2 + model A generates the largest splittings within the 2hω band in at least qualitative agreement with the low position of the ∆ 3 2 + (1600, ***) resonance. Nevertheless, despite this improvement relative to model B, the role of this state remains still unclear also in model A, since the lowest calculated state in this sector at 1810 MeV is still far away from the PDG reported mass value centered around 1600 MeV. We would like to comment here that this state is often viewed as the analogue of the low-lying Roper resonance. Both states have in common that they are the first scalar excitation of the corresponding ground-state, positioned even below the lowest excitations of negative parity. However, in the spirit of our model, which in addition to a proper confinement force uses exclusively 't Hooft's instanton-induced force as residual interaction, the low positions of ∆ 3 2 + (1600, ***) and the Roper resonance cannot originate from the same dynamics. In this respect, we should anticipate already here that in the case of the Roper resonance and its strange counter parts (which all appear in the sectors of the spin-1 2 + octet ground-state baryons) the low positions can be nicely explained as an instanton effect due to the attractive action of 't Hooft's force in these sectors. Thus, quite in contrast to the N -sector we do not find a similar selective lowering for the ∆ 3 2 + (1600, ***). Models which use the OGE interaction [9], have similar problems to account for this resonance [51]. In other phenomenological approaches [47,48] the low position of the ∆ 3 2 + (1600, ***) is explained due to a flavor-dependent Goldstone-boson exchange interaction. It thus seems worthwhile to comment on the experimental situation of this somewhat puzzling resonance: Its position is extracted from tedious analyses of mostly older πN → πN scattering data. The different analyses do often not agree, but even exhibit very large discrepancies with a big range of possible values. The mass range quoted by the Particle Data Group [37] for this state is 1550 to 1700 MeV. In our opinion it is astonishing that the Particle Data Group [37] has given this state a three-star rating although the various analyses are not in good agreement. Thus, despite a three-star rating, the current experimental evidence concerning the position of ∆ 3 2 + (1600, ***) is in our opinion not compulsory but rather unclear. Hence it becomes questionable if our calculated mass value of 1810 MeV for this resonance really constitutes a serious discrepancy, especially in view of more recent analysis [38,39] that predict the resonance position at the upper end of this range at about 1700 MeV. In this respect we hope that with the new generation of experimental facilities and the corresponding new efforts in baryon spectroscopy, the situation for this state will soon be clarified. In agreement with other (non-relativistic or 'relativized') constituent quark models our framework predicts two further states with J π = 3 2 + in the 2hω shell. The state predicted at 1950 MeV in model A matches the observed three-star state ∆ 3 2 + (1920, ***) within its uncertainties. The other state, whose mass value is predicted at 1871 MeV lies near this mass range. In model B all three states are clustered in the mass range of the ∆ 3 2 + (1920, ***). In the spin-5/2 + sector two states are predicted within the 2hω band, as required by the experimental findings. The positions of the well-established four-star resonance ∆ 5 2 + (1905, ****) and the two-star state ∆ 5 2 + (2000, **) and accordingly also their mass difference are fairly well described by the predictions at 1897 MeV and 1985 MeV in model A. Again the situation is better than in model B, which cannot reproduce this splitting due its minimal relativistic spin-orbit effects. But we should not attach too much importance to the ∆ 5 2 + (2000, **), which has only a two-star rating and moreover reveals big uncertainties in its determined mass position. In the sector J π = 7 2 + the well-established four-star resonance ∆ 7 2 + (1950, ****) is the only state seen in this mass range of the F 37 partial wave, compatible with our prediction. The position of this state, which is a member of the positive-parity ∆-Regge trajectory discussed previously, is exactly reproduced by the prediction at 1956 MeV in model A, whereas model B, which yields the mass value 1912 MeV, slightly underestimates this position.

States of the 3hω band
Apart from the puzzling, low-lying ∆ 3 2 + (1600, ***) resonance discussed previously, the present experimental ∆spectrum also shows a curious structure in the negative parity sector, namely the three resonances ∆ 1 2 − (1900, **), shell. Taking these states seriously within a constituent quark model, they have to be naturally assigned to the 3hω band, since the two states predicted in the 1hω shell could already be uniquely assigned to the well-established resonances ∆ 1 2 − (1620, ****) and ∆ 1 2 − (1700, ****). As can be seen in figs. 3 and 4 neither model A nor model B can account for these rather low-lying states. In both models the rich structure of states assigned to the 3hω shell is spread around 2200 to 2300 MeV, which agrees with the center-of-gravity of the other one-and two-star resonances observed in the 3hω band; see also table 5, where the predicted masses for all ∆-states in the 3hω band are given explicitly. The only state of the 3hω band that can be uniquely assigned to an experimentally observed resonance is the single state in the ∆ 9   The lowest 3hω states calculated in our models lie at about 2100 MeV, rather far above the masses of the three conspicuously low-lying states. Several other constituent quark models also cannot account for this puzzling structure (e.g. models with OGE forces ??): in general quark model predictions for the masses of these states are consistently too high by about 150-250 MeV. As in the case of the ∆ 3 2 + (1600, ***), a comment concerning the experimental status of these mysterious resonances is necessary at this stage: Also here, the experimental situation is rather unclear and unsatisfactory. The existence of the one-star ∆ 3 2 − (1940, *) resonance anyway is questionable and concerning the twostar ∆ 1 2 − (1900, **) resonance it is worth emphasizing that in the 1998 edition of the 'Review of Particle Physics' [40] this resonance has already been downgraded from three stars to two due to its weak signal in speed plots. Moreover, it should be mentioned that ∆ 3 2 − (1940, *) and ∆ 1 2 − (1900, **) have not been seen in various partial wave analyses of πN → πN scattering data [41]. Furthermore, additional corrections to the existing partial wave solutions due to additional new data from recently measured spin rotation parameters in π + p elastic scattering at (ITEP)-PNPI [42] indicate that the signal of the ∆ 3 2 − (1940, *) resonance even completely disappears (see [41,43] and references therein).
Concerning the three-star resonance ∆ 5 2 − (1930, ***) the Particle Data Group [37] states that various analyses are not in good agreement. Due to this quite unclear experimental situation, we again do not pay too much attention to this puzzling structure when evaluating the quality of our confinement mechanisms. But we hope that also in this case the situation will soon be clarified by the new experimental investigations in baryon spectroscopy. Let us emphasize that the confirmation of these resonances would strongly disfavor our model as well as several other models [51].

Beyond the 3hω band
The high mass part of the experimental ∆-spectrum is still hardly explored. Explicit mass values for some of the states in model A and B assigned to the 4hω, 5hω and 6hω shell, are summarized in tables 6, 7 and 8, respectively. In the positive-parity 4hω band three resonances with spins J π = 7 2 + , 9 2 + and 11 2 + are found experimentally. In the    [37] in model A in model B I3 11 Table 7. Calculated positions of the negative-parity ∆-states in the 5hω shell with J ≥ 11 2 in comparison to the corresponding experimental mass values taken from [37]. Notation as in  Table 8. Calculated positions of the positive-parity ∆ states in the 6hω shell with J ≥ 13 2 in comparison to the corresponding experimental mass values taken from [37]. Notation as in table 3.

Summary for the ∆-spectrum
To summarize our discussion of the ∆-spectrum, we have presented the results for two different confinement models, which essentially differ only in the Dirac structure of their linearly rising part. In both models this part consists of a combination of a scalar and a time-like vector part. In model B this combination is such that relativistic spin-orbit forces of both parts almost cancel, whereas the combination in model A produces spin-orbit forces which are small enough to be still compatible with experimental findings. Once the Dirac structures are fixed, we obtain in both models a quite good description of the complete ∆-spectrum up to highest orbital excitations J ≤ 15 2 . This is achieved by adjusting only three (!) parameters: the non-strange quark mass m n , the confinement offset-parameter a and the slope b, see table 1.
-Both models yield excellent Regge trajectories with the correct phenomenological characteristic M 2 ∝ J and the right slope, indicating that baryons with increasingly large separations of the quarks are well described. This feature strongly supports our string-like ansatz for confinement. -In both variants A and B, the resulting positions of the even and odd parity shell structures agree fairly well with the centroids of the experimentally observed band structures. -Note that all of the seven well-established resonances of the ∆-spectrum which have a four-star rating [37] are very well described in both confinement models. -The results of the confinement models A and B mainly differ in the intra-band splittings. Due to the minimization of spin-orbit effects, model B shows hardly any splitting and the different states in each shell are nearly degenerate.
Model A, however, shows moderate intra-band splittings due to rather small relativistic spin-orbit effects. These splittings, e.g. in the 2hω band, even tend to agree with experimentally observed structures as far as they actually can be disentangled from the rather large experimental uncertainties.
Thus we have shown that the major, well-established structures of the complete ∆-resonance spectrum in fact can satisfactorily be reproduced by proper choices of the confinement potential alone, i.e. without any need of an additional residual interaction in this flavor sector. This is quite in the spirit of our model which uses the instanton-induced interaction as residual force. Moreover, it is worth to emphasize once more the economical simplicity of our model: with the spinorial Dirac structure of the confinement kernel fixed, the bulk of all structures is parameterized by three parameters only! However, some puzzling structures of the ∆-spectrum, such as the low-lying resonance band do not fit in. Despite the three-star rating of ∆ 3 2 + (1600, ***) and ∆ 5 2 − (1930, ***), we have the impression that the currently available experimental evidence for these structures from πN -phase shift analysis is not very convincing. Therefore a verification of the positions of these resonances in reactions complementary to π − N scattering such as the electro-production of mesons off nucleons is highly desirable to decide if the lack of these structures in our models (and more generally in all other quark models) really reflects a serious discrepancy. Unfortunately, the present data basis of the other states, which also is extracted almost entirely from partial-wave analyses of older πN → πN total, elastic and charge exchange scattering data, is partly of rather limited quality and hence does not allow to favor one of the models at this stage. Hence, both models A and B are of almost the same quality in describing the present experimental situation of the ∆-spectrum. 6 The hyperfine structure of the light ground-state baryons In the foregoing section, we focused on the role of the confinement force alone. We considered the ∆-spectrum which is not influenced by 't Hooft's force and thus we could fix in each model the three parameters a, b and m n to get a satisfactory description of the whole ∆-spectrum. In order to fix models A and B for those strange baryons that are likewise not influenced by 't Hooft's force either, it remains to determine the strange constituent quark mass m s . This is done by the positions of the other (strangeness S * = 0) spin-3/2 decuplet ground-states, i.e. the hyperons Σ * (1385), Ξ * (1583) and Ω(1672). In this respect, the heavier strange quark mass m s > m n then leads to the pattern of approximately equally spaced distances between decuplet ground-states differing by ∆S * = 1. Then all parameters in table 1 apart from those of the 't Hooft interaction are fixed for both models. This is a proper starting point for analyzing the effect of 't Hooft's instanton-induced interaction on the remaining states, when turning on the 't Hooft couplings g nn > 0 and g ns > 0.
We will start this discussion with the spin-1/2 octet ground-states N (939), Λ(1116), Σ(1193) and Ξ(1318): In this respect, the first prominent feature of the baryon spectrum the instanton-induced interaction has to account for is generating the hyperfine structure of the ground-states. In the present approach this means the lowering of the octet ground-states relative to the unaffected decuplet ground-states, in order to realize the mass differences ∆ − N , Σ * − Σ, Ξ * − Ξ and Σ − Λ. As in non-relativistic quark models, the confinement potentials of models A and B cannot describe this splitting and, without a residual interaction, the octet and decuplet states with the same strangeness content are nearly degenerate. In earlier (mostly non-relativistic) attempts, as initially suggested by De Rujula, Georgy and Glashow [44] and subsequently applied by Isgur et al. and others [35,36,9], this structure was explained quantitatively by the short-range spin-spin hyperfine part (Fermi contact term) of the Fermi-Breit interaction due to perturbative one-gluon-exchange. Despite the success in describing the ground-state hyperfine structure by this spin-dependent contact interaction, there remain a lot of objections, for instance the spin-orbit problem connected with additional strong spin-orbit forces of the Fermi-Breit interaction or inconsistently large values of the strong coupling constant α s . As mentioned before this calls into question the justification for applying perturbative one-gluon exchange at least for light quarks. The most convincing argument against one-gluon exchange and in favor of the non-perturbative and explicitly flavor dependent instanton-induced interaction is that the latter provides a natural solution of the π − η − η ′ puzzle, as well as an explanation of the η − η ′ mixing in the mesonic sector, see for instance [11]. In leading order the OGE, however, is flavor independent and thus yields degenerate π and η mesons in clear contradiction to experiment. Moreover, the instanton force does not have a (phenomenologically unwanted) large spin-orbit part. Ground-state baryon mass splittings with the alternative instanton-induced interactions alone have been first explored in a simple model by Shuryak and Rosner [45]. A first, more extensive study of the baryon (and meson) spectra in the framework of a non-relativistic quark model 4 using a string-based confining interaction and instanton-induced interactions was made by Blask et al. [17,46]. Both attempts have shown that 't Hooft's interaction (which explains the ρ − π, K * − K and π − η − η ′ mass splittings in the mesonic sector), also provides an appropriate description of the spin splittings in the octet and decuplet which is at least as good as that from the short-range spin-spin hyperfine part of the Fermi-Breit interaction.

Naive considerations
For the further discussion it is instructive to illustrate first in a simple (naive) non-relativistic first order perturbative calculation, how the octet-decuplet mass splittings is generated in principle. Therefore, consider the non-relativistic limit of the (unregularized) 't Hooft interaction (in lowest order ( p m ) 0 of a ( p m )-expansion) leading for color anti-triplet quark pairs to a spin-flavor dependent pairing force where P S S12=0 := 1 4 (1I ⊗ 1I − σ·⊗ σ) is the projector on quark pairs with trivial spin S 12 = 0, and P F A (nn) and P F A (ns) denote the flavor projectors on antisymmetric non-strange (nn) and non-strange-strange (ns) quark pairs, respectively. For simplicity we assume the normalized wave functions |B 8 of all octet ground-states B 8 = N , Λ, Σ and Ξ with total spin and parity J π = 1 2 + to be given approximately by the S 3 -invariant 56-plet states where the totally symmetric S-wave ground-state function |ψ L=0 + S in coordinate space is assumed to be the same for all states, i.e. distortions of the wave functions due to the heavier strange quark are ignored. |χ 1/2 M S/A denotes the mixed symmetric/antisymmetric spin-1/2 wave function and |φ B8 M S/A the mixed symmetric/antisymmetric flavor octet wave functions. Accordingly, we assume the wave functions |B 10 of all decuplet ground-states B 10 = ∆, Σ * , Ξ * and Ω with total spin and parity J π = 3 2 + to be given approximately by the S 3 -invariant 56-plet states where χ 3/2 S denotes the totally symmetric spin-3/2 wave function and |φ B10 S the totally symmetric flavor decuplet wave functions. Without 't Hooft's force, the octet and decuplet states with the same strangeness content are degenerate. Due to its spin-flavor projector structure, the pairing force causes no shifts of the masses of decuplet baryons: but it shifts the octet states downward as they contain spin-singlet, S-wave, flavor antisymmetric quark pairs ('scalar diquarks'). In this simplified picture a first order perturbative calculation yields the following decuplet-octet mass splittings: where C : -In a similar way the equally big downward mass shifts of Σ and Ξ relative to Σ * and Ξ * , respectively, are proportional to the strength g ns > 0, since both these octet states contain a scalar non-strange-strange diquark only (see C in fig. 5). -The Λ-hyperon, however, contains both types of diquarks, where the scalar non-strange diquark content is twice that of the scalar non-strange-strange diquark content. This implies: concerning the non-strange coupling g nn the mass shift of Λ is a factor 2/3 weaker than the nucleon mass shift and the lowering of the Λ mass due to the non-strange-strange coupling is only 1/3 of the corresponding lowering of Σ and Ξ, respectively (see B and C in fig. 5). In particular, we find the Σ − Λ mass splitting to be given by the difference of the couplings g nn and g ns : Note that in the limit g nn = g ns the 't Hooft interaction becomes flavor SU (3) symmetric, such that the decupletoctet mass splittings are then all the same. In particular, the Σ − Λ mass difference then vanishes. Consequently, the correct description of the phenomenological value and sign of the Σ − Λ mass difference within this simplified model requires the non-strange coupling g nn to be bigger than the non-strange-strange coupling g ns , thus implying a ratio g ns /g nn < 1. Using eqs. (31) and (32) the explicit flavor dependence of the pairing force leads to the simple relations between the ∆ − N mass splitting and the splittings Σ − Λ, Σ * − Σ and Ξ * − Ξ, respectively. This implies that all three relative mass differences have to be described by just one parameter, namely the ratio g ns /g nn < 1. Relating the experimental estimates of the Σ − Λ splitting (≃ 70 MeV) to the ∆ − N mass difference (≃ 300 MeV) due to the first equation yields a ratio of about g ns /g nn ≃ 2/3. This is consistent with the same ratio g ns /g nn = 2/3 that we get from the second relation by comparing the unequal mass shifts Σ * − Σ, Ξ * − Ξ (≃ 200 MeV) and ∆ − N (≃ 300 MeV). Consequently the instanton-induced, flavor-dependent pairing force indeed provides a consistent explanation of the ground-state pattern even in this crude naive model. As illustrated schematically in the last column C of fig. 5 the choice of the ratio g ns /g nn = 2/3 can indeed account for the correct level ordering of octet and decuplet ground-states in accordance with the experimental findings: Finally we want to add a remark here concerning the ratio g ns /g ns < 1. The fact that the instanton-induced attraction is weaker between a strange and a non-strange quark than between two non-strange quarks, is exactly what one anticipates from the normal ordering of the original three-flavor 't Hooft vertex (see sect. 3): The effective coupling g nn results from a Wick contraction of the strange quark fields with the heavier effective strange quark mass, whereas g ns is obtained by a contraction of a non-strange quark field with the lighter effective non-strange quark mass, thus in fact yielding g ns < g nn as required by the phenomenology of the hyperfine structure (see eq. (21)).

Generating the hyperfine splittings in models A and B
Now let us examine the instanton-induced hyperfine splittings with the full relativistic dynamics of our covariant Salpeter models. The influence of 't Hooft's force on the ground-state baryons in model A and model B is shown in figs. 6 and 7, respectively. Without any residual interaction, i.e. g nn = 0 and g ns = 0, all ground states are bound from the flavor independent three-body confinement potential only. Therefore, the octet ground-states indeed are nearly degenerate to the corresponding decuplet ground-states with the same number of strange quarks. Henceforth, the decuplet ground-states remain unaffected by 't Hooft's force. The decuplet masses calculated in models A and B with the confinement parameters and quark masses given in table 1 are explicitly shown in the upper part of table 9. As discussed in the previous section, the position of the ∆-resonance in model A turns out to be slightly too high compared to the experimental value. The same also applies to Σ * , whereas Ω comes out to be roughly 30 MeV too low compared to the observed  5). Of course, the two other octet states Σ and Ξ remained unaffected so far, as they do not contain quark pairs with trivial isospin (nn-diquark). We now turn to the g ns dependence of the ground-state spectrum as shown by the mass curves in the right part of figs. 6 and 7. With g nn fixed to reproduce the ∆ − N splitting in both models, an increasing coupling g ns to the scalar, flavor-antisymmetric non-strange-strange diquarks has to lower the calculated masses of the hyperons Λ, Σ and Ξ in such a way that they simultaneously fit the corresponding experimentally observed resonance positions, thus generating the correct Σ − Λ, Σ * − Σ and Ξ * − Ξ mass splittings. This fixes the value of g ns in both models. As shown in figs. 7 and 6, the dependence of the Λ, Σ and Ξ masses on the non-strange-strange coupling g ns exhibits again a qualitatively similar behavior as in the naive model discussed previously (see   The values of the hyperfine splittings ∆ − N , Σ * − Λ, Σ * − Σ, Ξ * − Ξ and Σ − Λ calculated in both models are explicitly given in table 10. The absolute values of the ground-state splittings can be nicely reproduced. Here the predictions of model B are slightly better than of model A, where all splittings turn out to be slightly too big. The small discrepancies in model A can be traced back to the decuplet states, whose calculated absolute positions came out to be slightly too high in model, as discussed before. Finally, we want to comment on the values of the 't Hooft couplings g nn and g ns which have been adjusted to the experimental octet ground-state positions. The two model variants A and B mainly differ in the form assumed for the spinorial Dirac structure of the linearly rising part of the three-body confinement potential, or more precisely in the way how the scalar and the time-like vector part are combined. In particular, both versions yield the same  non-relativistic limit. Although we are using for both models the same value λ = 0.4 fm for the effective range of the regularized instanton-induced interaction, the absolute values for the 't Hooft couplings g nn and g ns differ significantly: the couplings in model A are about 50% larger than in model B indicating a dependence of the action of 't Hooft's force on the structure of the confinement potential. Nevertheless, the ratio of the two couplings g nn and g ns turns out to be the same. Both models yield the ratio g ns /g nn = 0.69 which again is in qualitative accordance with the crude phenomenological estimate g ns /g nn = 2/3 from the previous naive consideration. Moreover, we should give a comment concerning the special choice for the effective range λ = 0.4 fm of the regularized instanton-induced four-fermion interaction. We note that there is some freedom to choose the value for λ. While a choice smaller than ∼ 0.3 fm spoils the agreement with experiment (and moreover is also much more expensive to handle numerically), for arbitrary values λ > 0.3 fm the couplings g nn and g ns can always be adjusted such that an equally good result for the hyperfine ground-splittings can be achieved. Recall from sect. 3 that 't Hooft's force describes the interaction of quark pairs via instantons up to a critical extension ρ c which is required to cure the infrared problem. Thus, we expect the effective range λ of 't Hooft's force roughly to correspond to the critical instanton size ρ c . The value for ρ c (and thus also λ) should not be larger than ∼ 0.5 fm to be still within the scope of the two-loop approximation of the ρ-dependent running coupling constant (15). Hence, a reasonable value λ is somewhere in between 0.3 and 0.5 fm and we choose λ = 0.4 fm.

Summary for the hyperfine structure of ground-states
In summary, the use of 't Hooft's residual instanton-induced interaction within our covariant framework provides in both confinement models A and B (with parameters being fixed to reproduce the linear ∆-Regge trajectory) a good and consistent description of the decuplet and octet ground-state baryons (see fig. 8 for a comprehensive presentation of our results). It turns out that also with the fully relativistic dynamics the ∆ − N , Σ * − Σ − Λ and Ξ * − Ξ hyperfine splittings arise due to the attractive correlation of quark pairs with trivial spin and antisymmetric flavor to 'scalar diquarks' in qualitatively the same manner as one already expects from a naive static, non-relativistic picture. The phenomenology of the hyperfine splittings requires a ratio g ns /g ns < 1 of the 't Hooft couplings which is compatible with the expectation from instanton physics. Our result is at least as satisfactory as other attempts (see e.g. ref. [9,51] and the discussion in appendix A) that explain the hyperfine structure by the short range spin-spin hyperfine part of the one-gluon-exchange.
Now all parameters of our model variants A and B are fixed to the values given in table 1. Calculations of all other resonance masses are parameter-free and thus constitute true predictions. In particular, we can investigate next, to what extent the instanton force can also account for the features of the non-strange and strange excited baryons. Therefore, we now turn to a detailed discussion of the complete excited N -baryon spectrum and especially analyze how instanton effects shape its characteristic structures. As already mentioned, the predictions for the strange excited baryons will be discussed in a separate paper [2].

The Nucleon spectrum
We now turn to the investigation of the complete spectrum of nucleon resonances with isospin T = 1 2 and strangeness S * = 0, where now in addition to the confinement force also the influence of the instanton-induced interaction in general plays an essential role even for the excited states.

Remarks -Implications of 't Hooft's force and the experimental situation
Let us begin this discussion with some general remarks concerning the action of 't Hooft's force in the nucleon sector. The influence of the instanton force on the nucleon states is related to the content of quark pairs with trivial spin and isospin. The positive and negative energy components of the Salpeter amplitude Φ N J π describing an excited flavor-octet nucleon state with spin and parity J π are obtained by the embedding map (see ref. [1]) of totally S 3 -symmetric Pauli spinors ϕ N J π and ϕ N J −π which generally can be decomposed into the following four different spin-flavor SU (6)-configurations: with Here ψ L ± RL , χ S RS and φ N RF are the spatial, spin and flavor wave functions with definite The sum runs over possible orbital angular momenta L that can be coupled with the internal spin S to the total spin J as denoted by the brackets [. . .] J . Due to the strong selection rule of the instanton-induced interaction, its action on the different states of the nucleon spectrum is qualitatively understandable from a simplified picture, in which one disregards the negative energy component and the relativistic effects from the embedding map of the Salpeter amplitudes (corresponding to the non-relativistic limit). Then we expect all those resonances, which in the pure confinement case (g nn = 0) are dominantly 4 8[70] and 2 8[20], to be hardly influenced by 't Hooft's force.
Consequently, these resonances should be determined mainly by the confining kernel alone, similar to the resonances in the ∆-spectrum. For 4 8 [70] states in eq. (36) this selection rule is apparent from the total symmetry of the spinfunction χ 3/2 S . The 2 8 [20] states possess a totally antisymmetric spatial wave function, which is not affected by a point-like interaction. Moreover its spin-flavor wave function, which decomposes into mixed symmetric spin functions χ 1/2 MS and mixed antisymmetric flavor functions φ N MA and vice versa, never contains quark pairs with trivial spin and trivial flavor simultaneously. However, resonances which are dominantly 2 8 [56], like e.g. the nucleon ground-state, are expected to exhibit quite strong effects induced by the instanton force. Furthermore, also the resonances which mainly consist of 2 8 [70] states should be affected by the 't Hooft interaction. Finally, it should be noted that 't Hooft's force in general mixes 2 8[56] and 2 8 [70] configurations. From these simple non-relativistic considerations we thus expect that the majority of states is described by the confinement force alone, while 't Hooft's force acts in a selective manner on a particular limited set of states.
However, one has to be careful with such naive non-relativistic considerations. Due to our fully covariant Salpeter approach, relativistic effects might become essential in two ways: 1. From the outset (without instanton force), the initial mixing of the four different spin-flavor SU (6) configurations (36) in (35), especially for the excited states, can strongly depend on the relativistic (spin-orbit) effects that emerge from the chosen confinement Dirac structure in connection with the embedding map (34) of the Salpeter amplitudes. In particular, distinct Dirac structures cause different initial intra-band splittings and level orderings. Consequently, the influence of the instanton-induced interaction on the excited nucleon spectrum may be different in our two model variants. As we will see in the following discussion, models A and B indeed show substantial differences concerning the effects of 't Hooft's force in the excited spectra, in contrast to the description of the ground-state hyperfine structure, where 't Hooft's force in both models works equally well. This dependence of the effects of the instanton force on the confinement mechanism in fact is a purely relativistic effect, since the expressions for the confinement kernels of both models lead to the same non-relativistic limit and thus in this limit also to the same results. 2. Furthermore, the fully relativistic treatment of 't Hooft's force within our covariant Salpeter approach leads already by itself to differences in the effects of this interaction as compared to a non-relativistic treatment. On the one hand the action on the Pauli amplitudes (36) is modified due to the embedding map for the Salpeter amplitudes and on the other hand the fully relativistic version of the instanton-induced interaction causes effects that are a priori absent in the non-relativistic limit. In particular, it also has repulsive components, unlike its non-relativistic version. In this respect, we should remark here that the projector structure of 't Hooft's interaction kernel actually decomposes into two different parts. On the one hand there is the projector onto scalar diquarks with trivial spin and antisymmetric flavor. This is the dominant attractive part, which survives in the non-relativistic limit. It is responsible for the strong attraction of diquarks in the octet ground-state baryons leading to the correct octetdecuplet ground-state splittings as discussed in the previous subsection. But, on the other hand, there is also a projector onto the orthogonal pseudo-scalar diquarks with trivial spin and antisymmetric flavor. In this channel the instanton force is repulsive. This part of the instanton force completely vanishes in the non-relativistic limit.
These issues in fact emphasize the importance of describing baryons in a fully relativistic framework and accordingly it is interesting to study to what extent our covariant approach leads to improvements in the description of the excited state spectrum as compared to its non-relativistic version described earlier in [17,46]. Moreover, the different action of 't Hooft's force in combination with the two different confinement Dirac structures of models A and B will offer an additional, indirect criterion to decide which version provides a more realistic confinement force.
In the following discussion we will present a detailed investigation of instanton-induced effects on the excited spectrum of the nucleon resonances in both models (subsect. 7.3). In subsect. 7.2 we shall start by comparing the complete presently known empirical N * -spectrum with our final resulting resonance positions obtained with the coupling g nn adjusted in the previous section to reproduce the experimentally measured position M N = 939 MeV of the nucleon ground state. In this respect, the most striking experimental features of the nucleon spectrum, that a realistic quark model should account for, are [37]: -The overlap of alternating even-and odd-parity bands and accordingly the striking appearance of approximate "parity doublets": The overlapping negative-parity 1hω and positive-parity 2hω shells reveal the parity doublets In the higher mass region we find e.g. the approximate doublets The splittings within the parity partners are mostly within the experimental uncertainties.
Another aspect, which is currently of high interest, is the question of the so-called "missing resonances", i.e. states that appear in quark models but which have not been seen in πN partial-wave analyses. As already observed in the ∆-sector, our Salpeter equation-based quark model (and constituent quark models for baryons in general) predicts a much richer resonance spectrum of states than has been observed so far in scattering experiments. Most of the resonance parameters of the N * and ∆ states listed in the 'Review of Particle properties' [37] stem largely from partial-wave analyses of mostly older πN → πN scattering data. Fortunately, the newly established experimental electron and photon facilities at CEBAF, ELSA, etc., make it possible to investigate additional mechanisms of nucleon resonance excitations with photons with considerably improved experimental accuracy. Assuming that the "missing" states couple only weakly to the formation channels in N π scattering [50,51] and thus escape from experimental observation, the investigation of these new complementary formation channels should lead to the discovery of some of these "missing" states in the near future. Indeed, in the sector of nucleon resonances considered here, there are already indications of three new states around 1900 MeV obtained from recent studies of photo-induced reactions with the SAPHIR detector at the ELSA electron accelerator in Bonn [52,53]. These allow a first glimpse of the high-mass spectrum of N * -resonances: η ′ photoproduction: A fit to the SAPHIR total and differential cross sections for the η ′ photoproduction obtained from the reaction chain γp → pη ′ → pπ + π − η → pπ + π − π + π − π 0 has been made [54] assuming resonance dominance and taking only S-and P-wave multipoles into account. The data indicate a coherent resonant production of two pη ′ partial waves, S 11 and P 11 . The extracted resonance parameters are γp→K + Λ: Recent measurements of the γp → K + Λ total cross sections from SAPHIR [55] indicate for the first time a broad structure around 1900 MeV. This structure could not be resolved before due to low quality of the old data. An analysis of these new and associated differential cross-section and recoil-polarization data in the framework of an isobar model [56] suggests the existence of a broad D 13 state, where the choice of a D 13 state is based on the agreement with quark-model predictions [57,58]. The fitted resonance parameters are:

372
The discovery of "missing" states and the measurement of their resonance positions with high accuracy provides a convincing test for the quality and the predictive power of various constituent quark models in order to distinguish between realistic and less realistic quark models for the description of baryon masses. In this respect, it is interesting if and how these indications for these three new states fit into our models A and B.
Before we begin our discussion organized according to the phenomenological issues of the excited N * -spectrum listed, it is worth to remark once more that there is no freedom 5 left to fit the excited nucleon states. With the five parameters a, b, m n and g nn , λ fixed from the ∆-spectrum and the ∆ − N splitting, all the excited resonances of the N * -spectrum are now true predictions. In the subsequent subsection 7.3 we will then illustrate in some more detail, how instanton-induced effects due to 't Hooft's quark-quark interaction are in fact responsible for the phenomenology of the N * -spectrum.  [37]. Again, the resonances in each column are classified by the total spin J and the parity π, where left in each column the results for at most ten excitations in model A or B are shown. In comparison the experimental positions [37] are displayed on the right in each column with the uncertainties of the resonance positions indicated by the shaded boxes and the rating of each resonance denoted by the corresponding number of stars and a different shading of the error box. In addition we also display the determined resonance positions of the three new states that have been recently discovered by the SAPHIR collaboration [54,56,52,53]. These states are indicated by the symbol 'S'. In the following, we turn to a shell-by-shell discussion of the complete nucleon spectrum. According to their assignment to a particular shell, we additionally summarized the explicit positions of the excited model states in tables 11, 12, 14, 15, 16 and 17. [37] (right part of each column). The resonances are classified by the total spin J and parity π. The experimental resonance position is indicated by a bar, the corresponding uncertainty by the shaded box, which is darker the better a resonance is established; the status of each resonance is additionally indicated by stars. The states labeled by 'S' belong to new SAPHIR results [54,56,52,53], see text. Concerning these states let us first focus on the results of model A. Figure 9 impressively shows that in model A these four separated states fit rather well into the experimentally observed pattern of splittings, thus yielding a unique one-to-one correspondence between our model states and the observed resonances. In particular, we can account for the puzzling low position of the Roper resonance We would like to mention that a selective lowering of these states is also found in a non-relativistic treatment, see [17,46]. However these calculations  could only qualitatively describe these splittings: the Roper resonance and the other three positive-parity excited states tend to be too massive by about 200-250 MeV. In this respect, we thus find the predictions of our fully covariant model A being of considerably better quality, emphasizing the necessity of a fully relativistic framework to describe this striking structure quantitatively. Looking at fig. 10, the corresponding predictions for these four states in model B confirm this statement: even though model B has the same non-relativistic limit as model A the fully relativistic treatment within our Salpeter framework yields a totally different result. Although we likewise observe in the N 1  It has its origin in the different mixing of the spin-flavor SU (6) contributions (36) to the embedded Pauli spinors and moreover in a different level ordering within the intra-band structure. This originates from various relativistic effects that are induced by the two confinement Dirac structures of model A and B. Also these features shall be clarified in some more detail in the course of the next subsect. 7.3. Here let us note that this striking difference between the two models concerning the Roper resonance strongly supports model A to provide the more realistic confinement version in combination with 't Hooft's force as residual interaction. As already mentioned, apart from these four separated, low-lying resonances, the remaining bulk of states predicted in the 2hω shell is clustered around 2 GeV. In both models their mean mass corresponds nicely with the ranges of possible values of the three two-star states 2 + (1720, ****) and N 3 2 + (1900, **) is fairly well reproduced in both models and none of the three remaining "missing" states of this shell lies in between this gap as partly predicted in other quark models, as e.g. in the collective quark model of Bijker, Iachello and Leviatan [59,60], which is based on a spectrum-generating algebra. In the N 1 2 + sector, the results of model A and B again differ significantly. In (the less realistic) model B, the two remaining states that complete the 2hω shell in this sector, are predicted at 1974 MeV and 2092 MeV. The first state agrees fairly well with the position of the new discovered resonance P 11 (1986) extracted by a recent analysis of η ′ -photoproduction at the SAPHIR detector [54]. The second state fits the average value of the weakly established one-star resonance N 1 2 + (2100, *). Model A, however, predicts a cluster of three close states at 1950 MeV, 1996 MeV and 2009 MeV. We should note that the first two predictions correspond to two (of four) remaining states that one usually expects to be assigned to the 2hω shell. The third one is lowered from the 4hω band into this resonance region due to a strong attraction of the instanton force for this state (for an illustration see fig. 15 in the next subsect. 7.3). All three predicted masses nicely fit within the uncertainty range of the recently discovered 'SAPHIR resonance' P 11 (1986), which overlaps with the quite large error range of the one-star N 1 2 + (2100, *). To decide which of these three model states has to be assigned to this new resonance due to possible different couplings of these states to the N η ′ decay channel would require the calculation of quasi-two-body decays of baryons into different meson-baryon final states within our covariant Bethe-Salpeter framework. Unfortunately this information is not yet available but it will be a principal objective of our investigations in the near future. For the moment it is interesting to emphasize that, similar to the N 3 2 + sector, there are no predicted "missing" states in between the new 'SAPHIR resonance' P 11 (1986) and the established N 1 2 + (1710, ***) resonance. Also here, our model A differs significantly from other constituent quark models [9,59,60]. In this respect, the possible discovery of new (so far "missing" and undiscovered) baryon states at CEBAF, ELSA and elsewhere just in this N * resonance region is highly interesting and will provide an additional helpful criterion to distinguish between the different constituent quark models presently discussed in the literature [51].

States of the 1hω band
We now turn to the description of those states which, in the language of the harmonic oscillator basis, belong to the negative-parity 1hω band. Both models predict (as usual in constituent quark models for baryons) five states with spins J π = 1 2 − , 3 2 − and 5 2 − that can be uniquely identified with the five observed, well-established four-and three-star resonances listed in the baryon summary table of the Particle Data Group [37]. A comparison of the predicted masses with the corresponding empirical resonance positions is given in table 12.

Exp. state PW J π Rating Mass range [MeV]
Model state Model state [37] [ 37] in model A in model B N (1535) S11  In both models, the single 1hω state predicted in the N 5 Again, the situation is better described in model A: although the position of the N 1 2 − (1535, ****) is still predicted 100 MeV too low, the predicted mass of the second excitation at 1660 MeV nicely agrees with the empirical resonance position of the N 1 2 − (1650, ****), even within its uncertainty range. Altogether, we thus obtain a rather well predicted pattern of hyperfine splittings of the 1hω band in model A, which thus again is of better quality than in model B.
As already observed in the positive-parity sector, the differences between model A and model B have their origin in a different influence of the instanton-induced interaction. Again this can be traced back to a difference in relativistic effects that stem from the embedding map of the Salpeter amplitudes in combination with the two distinct confinement Dirac structures. For a more detailed discussion of this issue we again refer to subsect. 7.3.

Relative arrangement of the 2hω and 1hω bands -approximate parity doublets in the second resonance region
After discussing the individual hyperfine structures of the positive-parity 2hω band and the negative-parity 1hω band separately, let us now compare the relative positions of the positive and negative parity states involved. We restrict this discussion to model A, which so far led to consistently better results, especially in the sectors with spin J = 1 2 . Due to the selective lowering of a particular set of states of the 2hω band relative to the other states we can indeed reproduce the striking overlap of states of the two shells with opposite parity. The situation is displayed in fig. 11, where now in contrast to fig. 9 the states with the same total spin, but opposite parity are directly displayed side by side.  As discussed, model A nicely accounts for the low position of the Roper resonance. Unfortunately, due to the slightly too low predicted position of the first negative-parity N 1 2 − state, which still appears below the calculated position of the Roper resonance, the striking inversion of the ordering of positive-and negative-parity states in the spin-1/2 sector cannot be achieved. On the other hand, figure 11 impressively demonstrates that our model A indeed reproduces the remarkable appearance of the lowest approximate "parity doublets" of the nucleon spectrum in the second resonance region around ∼ 1700 MeV (shaded regions in fig. 11) remarkably well; see also table 13. Apart from the slightly too low calculated position of the N 3 2 − (1700, ***) even the experimentally observed splittings between the parity partners in general are fairly well described.  Table 13. Approximate parity doublets in the second resonance region in model A.

Beyond the 2hω band
Finally let us have a look at the experimentally still rather poorly explored high energy part of the nucleon spectrum, i.e. at the states that belong to the 3hω, 4hω, 5hω and 6hω bands with observed total spins up to J = 13 2 .
We start with the states of the 3hω and 4hω bands and their relative alignments. The predicted positions in both models for the lightest few states assigned to these shells are summarized and compared to the experimental mass values in table 14 and 15, respectively.
The Particle Data Group [37] lists all in all five negative-parity resonances in the energy region between roughly 2000 and 2300 MeV with J π = 1 2 − to 9 2 − . In the notation of the oscillator shell model these should be assigned to the   − (2250, ****) it is worth noting that the PDG states in the corresponding sector with the same total spin J = 9 2 but with opposite positive parity the first excited resonance N 9 2 + (2220, ****) at almost the same mass value, i.e. nearly degenerate with the N 9 2 − (2250, ****). Due to its high spin J = 9 2 , this well-established four-star resonance in the positive-parity sector has to be a member of the 4hω shell. The appearance of this approximate parity doublet N 9 2 + (2220, ****) -N 9 2 − (2250, ****) is a very striking feature in the high energy part of the nucleon spectrum, and we should note here that (non-relativistic or 'relativized') constituent quarks models which use one-gluon-exchange as residual interaction [9] generally cannot account for this well-established structure. Before comparing these experimentally found structures with our predicted spectrum in this energy region in detail, it is instructive to discuss first the predicted intra-band structures of the 3hω shell and the corresponding implications of the instanton force in general. As can be seen in figs. 9 and 10, in both models the bulk of states appears in a region between ∼ 2100 and ∼ 2300 MeV, which fairly well agrees with the range of possible mass values of the resonances presently stated by the PDG. The predicted spectrum of states in this shell is even richer as that of 2hω band discussed before. Furthermore, in both models we again observe a particular set of states, which is selectively lowered relative to the rest of the states in the ∼ 2200 MeV region. Consequently, also the 3hω band splits into two well separated parts, in a similar manner as observed in the 2hω band: again, this lowering is due to the attractive action of the instanton-induced interaction in these states as will become more apparent by the more detailed investigation of these instanton-induced effects in the next subsection (see figs. 16 [37] in model A in model B S11(1897) S11   The arrangement of the 3hω and 2hω bands is quite similar to that of the 2hω and 1hω shells, leading there to the appearance of approximate parity doublets discussed above. In fact, similar predicted alignments even appear between the higher alternating even-and odd-parity bands, e.g. between the lowest states of the 4hω band and the upper part of the 3hω band. Now let us discuss in detail to what extent these predicted intra-band structures along with the corresponding positioning of the alternating even-and odd-parity bands (i.e. 4hω ↔ 3hω and 2hω ↔ 3hω) are indeed realized in    12. In this respect we should mention that in contrast to the negative-parity 3hω state this positive-parity 4hω state is strongly influenced by 't Hooft's force, since in both models its spin-flavor part is dominantly a 2 8 [56] configuration with an additional strong admixture of 2 8 [70]. Consequently, this state is significantly lowered (together with a group of other states belonging to the 4hω band). It is remarkable that the 't Hooft coupling g nn (as fixed by the ∆ − N -splitting) is just the right size to produce this (almost degenerate) parity doublet.
In the N 7 sector might become a sensitive test for our model A and especially for the residual 't Hooft interaction used in this model. In this respect, we should note here that the prediction of such well separated states in N 5 2 − and N 7 2 − seems to depend strongly on the residual quark-quark interaction employed: other quark models which use a different residual force (as for instance one-gluon-exchange [9]) do not predict such states. Moreover, it is worth to anticipate already here that the corresponding lowest Λ-state in the Λ 7 2 − sector (see our subsequent paper [2]) is strongly lowered by the same instanton-induced effect and thus nicely explains the comparatively low position of the Λ 7 2 − (2100, ****). Figure 12 shows the relative arrangements of parts of the negative-parity 3hω band and parts of the positive-parity 2hω and 4hω bands in model A. In contrast to fig. 9 the sectors with the same total spin J and opposite parity are now  Finally, let us compare our results with the highest mass states of the experimental nucleon spectrum, i.e. the lowest excitations in the sectors N 11 2 − and N 13 2 + reported by the Particle Data Group [37]. These are the three-and two-star resonances N 11 2 − (2600, ***) and N 13 2 + (2700, **), which in the notation of the oscillator model should belong to the negative-parity 5hω and the positive-parity 6hω shell, respectively. The positions predicted for the lightest few 5hω and 6hω states in both models are given in tables 16 and 17, respectively. In the N 11 2 − sector the situation is very similar to that found in the N 7 2 − sector. Again the lowest excitation is strongly influenced by 't Hooft's force and thus is strongly lowered relative to other states in this sector. Consequently, the first excited state is predicted roughly 160-175 MeV too low compared to the position of the resonance  To summarize our investigations concerning the relative alignment of the alternating even-and odd-parity bands in the excited nucleon spectrum, let us finally stress the most striking feature of our model that could be exposed in  [37] in model A in model B G19   the course of this discussion and becomes immediately evident from figs. 12 and 13. It is the systematical occurrence of approximately degenerate states with the same spin and opposite parity. On the one hand our models nicely reproduce the approximate doublet structures which in fact are observed experimentally. On the other hand, these doublets also appear, where the present experimental situation either seems to deviate from such a parity doublet structure, or members of these doublets are "missing". In fact, we predict for really all the lowest excitations with spins from J = 5 2 to J = 13 2 such a pattern of approximate parity doublets. As already partly indicated in the previous discussion, this systematics originates from 't Hooft's force: the two members of the doublet belong to two adjacent shells with opposite parity and 't Hooft's force lowers one of these strongly, whereas the other one remains totally unaffected. In this way, both states become approximately degenerate in energy. For a more detailed discussion of this scenario in the context of a detailed investigation of instanton-induced effects we refer to subsect. 7.3.    [37]). The upper curves show the trajectories without influence of 't Hooft's instanton-induced interaction. Note the remarkable fact that both models exhibit the correct linear Regge characteristic M 2 ∼ J with almost the same slope in both cases, i.e. without and even with the contribution of the instanton force. Hence, adjusting the 't Hooft coupling gnn to describe just the ground state correctly in both models, the whole model trajectory fits the empirical Regge behavior in excellent quantitative agreement. See both confinement variants yield the characteristic Regge behavior M 2 ∝ J, as one would expect from the results for the ∆-Regge trajectory discussed before. In particular, the slope is approximately the same as in the ∆-sector. It is quite astonishing that the effect of the instanton force on the states of the trajectory indeed is such that it maintains the linear Regge characteristic M 2 ∝ J with almost the same slope, in nice agreement with phenomenology. Hence, adjusting the effective 't Hooft coupling g nn to describe just the ground state (i.e. the ∆ − N splitting) correctly, the whole model trajectory shows an equally large downward shift in the square of the baryon masses M 2 thus leading to an excellent quantitative agreement with the empirical linear N -Regge trajectory. This nontrivial compatibility of the instanton-induced effects with the observed linear Regge characteristic in the nucleon spectrum is a quite remarkable and an interesting feature of 't Hooft's force. Moreover, figure 14 convincingly demonstrates once more that even in the highest mass regions of the nucleon spectrum, instanton-induced effects are crucial for the correct description of resonance positions.

A study of instanton effects in the excited nucleon spectrum
So far we presented a detailed discussion of the predicted nucleon spectrum in both confinement models A and B with the strength of the instanton-induced interaction fixed to match the observed position of the nucleon ground state N (939). We investigated the calculated structures of the single positive-and negative-parity shells and compared  Table 18. Position of states belonging to the positive-parity N -Regge trajectory calculated in the models A and B in comparison to the experimental resonance positions [37]. For a graphical presentation see fig. 14.
these to the hitherto observed experimental patterns. In fact, we found a remarkably good agreement between the predictions of our model version A and the empirical nucleon mass spectrum in the lower resonance regions as well as for the higher and highest mass regions including orbital excitations up to J = 13 2 (Regge trajectory). In particular, all the striking hyperfine splittings within the intra-band structures and accordingly the arrangements between the alternating even-and odd-parity shells could be nicely reproduced in at least qualitative but mostly even in completely quantitative agreement with the experimental findings. In particular, we clearly reproduced the important downshift of the Roper resonance and three other low-lying states of the 2hω shell around 1700 MeV, -the hyperfine structure of the five states in the 1hω band, -the fine structure of the 3hω band as far as it is indicated by the new SAPHIR photoproduction results, -the approximate parity doublets due to partial overlapping of shells: 1hω ↔ 2hω, 2hω ↔ 3hω, 3hω ↔ 4hω, 4hω ↔ 5hω and 5hω ↔ 6hω.
In the discussion of the foregoing subsection we already mentioned that in our fully covariant model A all these striking features of the excited nucleon spectrum arise due to 't Hooft's instanton-induced quark-quark interaction.
To supplement this discussion we shall now illustrate in some more detail that the effects of this residual interaction are in fact responsible for generating all these structures simultaneously. In this respect it is instructive to study how the 't Hooft interaction affects the energy levels when, starting from the case with confinement only, it is switched on and its strength is gradually increased. It is our aim to demonstrate in this way that prominent features of the spectrum like the ∆ − N ground-state splitting and the conspicuous low position of the Roper resonance can be put along with all the other observed phenomena into a somewhat wider perspective, i.e. that instanton-induced effects indeed dominate the fine structure of the whole nucleon spectrum.
Another issue that has to be clarified in this context is the difference between the predictions of model A and model B. Although model B could also account for the most features discussed, it strongly failed concerning the low position of the Roper resonance in the N 1 2 + sector and moreover predicted a much too large splitting of the two lowest S 11 resonances in the corresponding N 1 2 − sector with negative parity. As already mentioned, this indicates that the action of 't Hooft's force might strongly depend on the Dirac structures chosen for the three-body confinement kernel and on the corresponding relativistic effects induced in combination with the embedding map of the Salpeter amplitudes. Also this aspect shall be analyzed here in some more detail.
To study the effects of 't Hooft's force in model A we display the dependence of the nucleon masses on the effective 't Hooft coupling g nn in figs. 15 and 16 for the positive and negative parity sector, respectively. The corresponding results of model B are shown in fig. 21. In each sector with spin J and parity π the leftmost spectrum in each column shows the spectrum obtained with the three-body confinement kernel alone. The curves show the change of the spectrum as a function of the 't Hooft coupling g nn : Starting from the case with confinement only the coupling is gradually increased and finally is fixed to reproduce the ∆ − N splitting (right spectrum in each column). The rightmost spectrum shows for comparison the experimental resonance positions as before.

Pure confinement spectra
Let us first consider the spectra in figs. 15, 16 and 21 as they are determined by the two different three-body confinement kernels alone, i.e. in the case g nn = 0 (leftmost spectrum in each column): Without 't Hooft's force the linear confining interaction arranges the nucleon spectrum into a sequence of alternating even-and odd-parity shells. As one expects, the situation is quite similar to the spectrum of ∆-resonances. The states of these rather narrow bands are clustered around a common mean value, which, however, fairly agrees only with the upper parts of the rather wide spread band structures observed experimentally. In fact some of these experimentally observed resonances even are already well described by single model states in both confinement versions. This, for instance, is the case for the three resonances N 5 2 − (1675, ****), N 7 2 + (1990, **) and N 9 2 − (2250, ****). These are single states with maximum total spin J in the 1hω, 2hω and 3hω shell, respectively. In general, in the naive oscillator shell model, the involved relative orbital angular momenta in the Nhω shell can be maximally combined to a total orbital angular momentum L max = N . To achieve the maximal total spin J max (N ) in this shell, L max = N has to be coupled with the symmetric S = 3 2 internal spin-quartet function to get J max (N ) = N + 3 2 . Thus, we find for the different Nhω multiplets of the oscillator shell model the maximum total spins as displayed in table 19, where the corresponding states have internal spin S = 3 2 and are just the lowest excited states in the sectors J π max (N ).
Shell Nhω 1hω 2hω 3hω 4hω 5hω 6hω J π max (N ) = N + 3 2 − (2250, ****) in these sectors. On the one hand, this result indicates that the confinement force, whose parameters have been fixed on the phenomenology of the ∆-spectrum, works equally well also in the nucleon spectrum concerning the arrangement and positions of the shells. On the other hand it shows that the strong selection rules of 't Hooft's interaction are really consistent with the phenomenology of these three particular nucleon resonances. Concerning the structures of the shells in the pure confinement spectrum it should be noted that model A exhibits somewhat larger intra-band splittings than model B, due to slightly bigger relativistic spin-orbit effects in model A (here compare to the discussion of the ∆-spectrum). However, unlike the ∆-spectrum, the multiplet structure of alternating positive and negative parity bands is obviously strongly broken in the phenomenological nucleon spectrum, and the intra-band structures generated by the two different confinement kernels by no means can account for the quite large observed hyperfine splittings in the shells. This is most apparent from the experimentally rather well-established structure of the lower part of the 2hω shell, where (in case of the low-lying Roper resonance) the deviations between the pure confinement spectra and the empirical mass spectrum are even of the same order of magnitude as the ∆ − N ground-state splitting, which also cannot be accounted for in the case g nn = 0. Similar large deviations are also found in higher mass regions, as e.g. in case of the N 9 2 + (2220, ****). Thus, apart from the ∆ − N ground-state hyperfine splitting, also the shortcomings in the excited pure confinement spectrum strongly indicate missing residual spin-spin interactions in the nucleon sector. Hence, the question arises, to what extent 't Hooft's instanton-induced interaction, which already could nicely explain the octet-decuplet ground-state splittings, can simultaneously also account for the striking mass splittings observed in the excited nucleon spectrum. That this is indeed the case in our model version A is already evident from the previous discussion of the complete nucleon spectrum but will now be even more convincingly demonstrated when, starting from the case with confinement only, the 't Hooft coupling g nn is turned on and then is increased gradually until the model ground-state fits the experimental position of N 1 2 + (939). In this respect let us first focus on the effects in the positive parity spectrum of model A shown in fig. 15.

Instanton-induced effects in model A -positive-parity spectrum
In the 2hω band one finds a selective lowering of exactly four states relative to the other states as required by the experimental findings and it is quite impressive how the instanton-induced effects even shape the pattern of these four model states to come into considerably good agreement with the observed (well-established) pattern at the value of the coupling, where the ∆ − N splitting is reproduced. In addition to the graphical representation of this effect in fig. 15 we demonstrate the influence of 't Hooft's force on the 2hω states (in comparison to the ground-state) also in table 21 by displaying the explicit mass values as well as the configuration mixing of these states for the cases without (g nn = 0 MeV fm 3 ) and with (g nn = 136 MeV fm 3 ) 't Hooft's force. In fact, one of the states in the ground-state sector N 1 2 + , namely the lowest excitation in the pure confinement spectrum at 1830 MeV, is lowered even strongly enough (by the same amount of roughly 300 MeV like the ground-state) to match almost the conspicuous low position In each column the leftmost spectrum is determined by confinement alone. The curves illustrate the variation of the spectrum with increasing 't Hooft coupling gnn which finally is fixed to reproduce the correct N − ∆-splitting. As can be seen, the 't Hooft interaction can indeed account for essential structures (e.g. the Roper resonance) of the experimental spectrum, which is shown on the right in each column.
190 MeV somewhat smaller than the Roper resonance but quite similar to the second excitation in the N 1 2 + sector. Hence the equally big downward mass shift of all three states is just of the right size to reproduce the nearly degenerate structure of the three resonances Note however that the formation of such a well separated shell substructure, which is lowered relative to the bulk of the other states, is not just restricted to the 2hω band but in general can be found also in the other even parity shells. In the 4hω and 6hω band the downward shift of these structures amounts to roughly 180 MeV when g nn is fixed by the correct ground-state position. Again the states of these well separated sub-shells have dominant 2 8 [56] contributions (55% -70%) with additional (in general quite big) admixtures of 2 8 [70] configurations (30% -45%). In the 4hω shell this downward shift is just the right size to achieve the correct description of the wellestablished Regge state  . 14). In summary we thus find that along with the ∆ − N ground-state splitting really all essential hyperfine structures of the positive-parity N * -spectrum, which the confinement kernel of model A alone cannot account for, are in fact consistently and simultaneously generated by the instanton-induced 't Hooft interaction. Apart from the without 't Hooft's force with 't Hooft's force J  success of 't Hooft's force in generating the hyperfine structure of ground-state baryons, these results thus give a further strong evidence for our conjecture that the instanton-induced short distance interaction in fact might play the dominant role for the structure of the whole nucleon spectrum.

Instanton-induced effects in model A -negative-parity spectrum
We now turn to the instanton-induced effects in the negative parity nucleon spectrum predicted by model A. Figure  16 shows that the instanton force causes similar effects in the odd-parity 1hω, 3hω, 5hω bands. Again, in each shell a multiplet of particular states shows a downward level shift of roughly 180-200 MeV, thus generating substructures of nearly degenerate states that are well separated from the rest of the shell. Analyzing the spin-flavor content of the corresponding Salpeter amplitudes, we find that all these states have dominant contributions of 2 8 [70] configurations with additional large admixtures of 2 8 [56].
In the 1hω band this mechanism generates the level splittings of the five experimentally well-established states. The single state of this shell in the N 5 2 − sector, which is almost a pure 4 8[70] configuration, remains totally unaffected, as previously mentioned. Note that moderate spin-orbit effects of the confinement kernel A already break the degeneracy of the states in the pure confinement spectrum and cause a slight mixture of the 2       Note that the two lowest predicted states of N 7 2 − and N 11 2 − belong to the same negative parity Regge trajectory and the lowering of the states of this trajectory is quite analogous to the lowering of the positive-parity trajectory discussed previously. There it was just this instanton-induced downward mass shift which brought all states of this trajectory simultaneously into excellent agreement with the experiment.

Explanation for approximate parity doublets
Finally let us focus in the context of instanton effects (in model A) on the appearance of more or less mass degenerate nucleon states of equal total spin and opposite parity. Comparing the illustrations of instanton-induced effects for the positive and negative parity nucleon spectrum in figs. 15 and 16, respectively, we find in fact a simple explicit foundation of the occurrence of near parity doublets in the excited N * -spectrum due to 't Hooft's force. In this respect let us consider the global arrangements of the band structures in the nucleon spectrum formed within the framework of our model: Without the instanton-induced interaction, the three-body confinement kernel alone arranges the nucleon spectrum into even and odd parity shells, where the odd parity bands lie in between the even parity bands (and vice versa). As demonstrated, the instanton-induced interaction then splits each of these shells into two well separated parts due to its strong selection rules: the lower part, namely a particular subset of states, which are mixtures of dominantly 2 8[56] and 2 8[70] configurations, is significantly lowered with respect to the upper part, i.e. the remaining bulk of states which is hardly influenced by the residual interaction. This situation is schematically demonstrated in fig. 17.
Let us in general consider a Nhω shell with N ≥ 2 in the positive (N even) or negative (N odd) parity sector, respectively. When the 't Hooft coupling g nn is adjusted to fit the nucleon ground state, the lowering of the shell substructure is such that it comes to lie approximately in between the unshifted parts of the adjacent Nhω and (N − 2)hω bands with the same parity. Accordingly it becomes positioned nearly degenerate with the corresponding unshifted part of the (N − 1)hω shell with the opposite parity. This nearly degenerate arrangement of the upper/lower part of a shell with the lower/upper part of a shell with opposite parity consequently leads to the near parity doublets in our model, which we already discussed in detail in the foregoing subsection (see figs. 11 and 12). The positioning of the positive-parity 2hω shell and the negative-parity 1hω shell due to instanton effects, which leads to approximate doublets in the second resonance region of the excited nucleon spectrum around 1700 MeV, is once more explicitly illustrated in fig. 18; we therefore combined the columns of fig. 15 (positive parity) and fig.  16 (negative parity) corresponding to the spins J = 1 2 , 3 2 and 5 2 in such a way that the sectors with same spin and opposite parity are directly displayed side by side. We thus see that in the framework of our covariant quark model A the instanton-induced effects indeed suggest a quite simple and appealing explanation for these experimentally observed approximate parity doublets in the second resonance region. Moreover, this mechanism works even for the higher mass regions of the nucleon spectrum: The general arrangement of shell multiplets as indicated schematically in fig. 17 is most prominent for the lowest-lying 6 states of the sectors with J π from 5 2 ± to 13 2 ± (Regge-trajectory-type states): From table 19 we see that the lowest states of the sectors J π = 5 2 − , 7 2 + , 9 2 − , 11 2 + , and 13 2 − are just the states with maximum total spin J max (N ) = N + 3 2 of the Nhω shells with N =1, 2, 3, 4, and 5. As discussed earlier these states are not influenced by 't Hooft's force since they necessarily contain a totally symmetric spin-quartet (S = 3 2 ) wave function to achieve this maximal total spin. But in the corresponding sectors with the same total spin J but opposite parity we then always find exactly one state of the higher lying N + 1 band that is selectively lowered by 't Hooft's force and thus becomes the lowest excitation in this opposite-parity sector fairly isolated from the other states. The arrangement of these states in model A in comparison with the present experimental situation is shown in fig. 19, where again the sectors with the same spin and opposite parity are displayed side by side. It is quite astonishing that the strength g nn as fixed on the ground-state is just In the N 7 2 sector, however, the present experimental findings of the lowest excitations seem to deviate from such a parity doubling structure due to the rather high measured resonance position of the − state, which presently differs significantly from our prediction, as well as the discovery of the still "missing" first (two) excitation(s) in the sectors N 11 2 + and N 13 2 − is essential to decide whether this striking feature of systematical parity doublets induced by instanton effects is indeed a realistic global structure exhibited also by the experimental nucleon spectrum.
To conclude this discussion of approximate parity doublets, it is worthwhile to mention that a further test of this scenario for parity doublets based on instanton effects should be possible by the measurement of electromagnetic (pγ * → N * ) transition form factors into each member of the doublets shown in fig. 19. Such a measurement provides a deeper insight into the structure of the corresponding resonances. In this respect it is crucial that one member of each doublet, namely that which belongs to the maximum total spin J max (N ) = N + 3 2 of the corresponding Nhω shell, is by no means influenced by 't Hooft's force whereas its partner with opposite parity is strongly affected by this residual interaction. Therefore, both members of the doublet have significantly different internal structures that should be manifest in systematically different shape of the two corresponding magnetic multipole transition form factors: the member of each parity doublet affected by 't Hooft's force exhibits a rather strong scalar-diquark correlation and thus its structure should be more compact, in comparison to its unaffected doublet partner whose structure is expected to be rather soft. Consequently for larger momentum transfers Q 2 the transition form factors pγ * → N * for the lowest excitation in N * 5 Thus, apart from further rather small differences, the most distinct deviations between model A and B show up in the sectors with total spin J = 1 2 . Since the equally good results of both models concerning the description of the complete presently known ∆-spectrum did not allow to favor one of the two phenomenological confinement potentials, the quite different results concerning the Roper resonance provide a suitable indirect criterion that strongly supports version A to provide the more realistic confinement potential in combination with 't Hooft's force as residual interaction. Studying now the effects of 't Hooft's force on the energy levels of the N * spectrum in model B and comparing these with the corresponding results of model A presented in the foregoing subsection, we want to show that the differences between the models indeed have their origin in a different influence of instanton-induced effects due to different relativistic effects of the confinement kernels in both models: In both models A and B the three-quark confinement kernel is provided with a combination of scalar (1I ⊗ 1I ⊗ 1I) and time-like vector (1I ⊗ γ 0 ⊗ γ 0 + cycl. perm.) Dirac structures. Both versions just differ (apart from different model parameters) in the way how these two spinorial Dirac structures are combined in the linearly rising part of the confinement kernel. Since both models have the same non-relativistic limit and hence yield in this limit the same results, the large differences between model A and B concerning the Roper resonance in N * 1 2 + and the hyperfine splittings of the 1hω states in N * 1 2 + in fact can be considered as pure relativistic effects in our Salpeter equation-based quark model. This shall be clarified more precisely in the course of this discussion.
In analogy to figs. 15 and 16 of model A, fig. 21 shows the influence of 't Hooft's force on the energy levels of the excited positive-and negative-parity nucleon spectrum in model B. Before analyzing in detail the differences to model A let us briefly discuss those effects that both models have in common. Considering globally the influence of 't Hooft's force on the N * -spectrum in model B we indeed find a lot of effects similar to those in model A. In each shell we again find the systematics of well separated multiplets of states which show dominant contributions of the 2 8 [56] and 2 8 [70] configurations or their mixtures. These shell substructures are rather strongly lowered relative to the majority of states in the corresponding bands that are hardly affected by 't Hooft's force. As in model A this systematics leads again to the occurrence of more or less mass degenerate states of equal spin and opposite parity. We do not want to discuss this aspect again in detail here and just remark that this instanton-induced mechanism of generating approximate parity doublets works slightly better in model A. As before, the instanton interaction for several model states induces downward mass shifts again of the right size to reproduce the correct experimentally determined resonance positions, if the 't Hooft coupling g nn is fixed such that the model ground-state matches the experimental mass. In the positive parity spectrum these are for instance the states assigned to the Regge trajectory, i.e. besides the ground-state the three states  [37]. This is caused in the same way as in model A just because of the strong downward mass shift of the first excited states in these sectors by roughly 150 MeV.
To investigate the most striking differences in the influence of 't Hooft's force compared to model A let us now have a closer look at the negative-parity 1hω shell and the positive-parity 2hω shell, where, as mentioned before, the biggest deviations show up in the N 1 2 ± sectors. We start with the states of the 1hω shell. For this discussion we also show the explicit masses and the configuration mixing of the 1hω states for the cases without and with instanton force in     [70] configuration. Thus, in contrast to model B, the increasing 't Hooft coupling g nn leads in model A first to a level crossing cancelling the first confinement-induced splitting before a reversed level ordering is achieved. Hence the net effect is a weaker and thus improved hyperfine splitting in model A that agrees better with the experimental findings than in model B.
We now turn to the instanton effects in the 2hω shell that show up in model B. For this discussion see also table 24, which shows the explicit mass values and configuration mixings of the states without and with 't Hooft interaction. In figure 21 again a lowering of exactly four states of this shell can be observed, when the 't Hooft coupling g nn is gradually increased. In the N 1  [20] occurs and can be assigned to one of these states. Of course, in the pure confinement spectra of our Salpeter equation-based models A and B the degeneracy is lifted due to the anharmonicity of the linear confinement potential and still more importantly due to relativistic effects depending on the spinorial Dirac structure of the confining forces. Consequently, the two confinement versions induce different intra-band hyperfine structures of these states. Comparing the masses and the corresponding contributions of the spin-flavor SU (6) configurations to the embedded Pauli spinors of the Salpeter amplitudes (see table 21 and 24) one finds, apart from the different pattern of mass splittings, a totally different level ordering and configuration mixing of the states in both models. In model B we find rather pure SU (6)  (1440, ****) and the position of N 1 2 + (1710, ***). In model B, however, the initial reversed ordering of these two states leads to a level crossing, which mixes the two configurations. Consequently the net effect of the residual interaction is weakened and thus neither the low position of the Roper resonance nor the splitting between the Roper resonance and the N 1 2 + (1710, ***) can be explained.
To summarize, our discussion convincingly demonstrates the importance of relativistic effects as they are present in our fully covariant framework based on the Salpeter equation. To generate prominent hyperfine structures in the N * -    spectrum like the Roper resonance by 't Hooft's residual force, the interplay of the residual force and the confinement force due to relativistic effects of the confinement Dirac structure is very crucial. We have seen that two different confinement Dirac structures, which exhibit the same non-relativistic limit and work almost equally well for the description of the ∆-spectrum and the spectrum of the light ground-states, might cause a totally different behavior of particular excited N * states under the influence of the residual 't Hooft interaction. The different results of both models concerning the Roper resonance thus strongly favor the confinement version of model A to be the more realistic confinement force in combination with 't Hooft's residual force. Let us conclude this subsection by drawing the following general conclusions from this discussion: -The very different effects of 't Hooft's residual force on the Roper state in two models that have the same nonrelativistic limit, convincingly demonstrates the importance for describing baryons in a genuine fully relativistic framework in order to decide whether a possible choice for a residual interaction is a realistic candidate for explaining prominent hyperfine structures of the excited baryon spectrum. -In particular, the effect of a residual interaction cannot be considered independently from a suitable assumption on the spinorial Dirac structure of the confinement force. This is often done by non-relativistic or 'relativized' potential models which either completely neglect the resulting relativistic effects of the confinement Dirac structure by just using central confining potentials or which take only a part of the terms of the rather poorly converging (|p q |/m q ≈ 1) generalized Fermi-Breit expansion of the confinement force (and also of the residual interaction itself) into account. Relativistic effects of both, the residual force and the confinement force, must be consistently taken into account.   -From our point of view this implies that statements concerning the role of a special choice of the residual interaction for generating the hyperfine structures in the excited light baryon spectrum, which are based on 'relativized' or even non-relativistic potential models, are highly context dependent.

Summary for the Nucleon spectrum
After this rather lengthy and detailed discussion of the excited nucleon spectrum, which emphasized the role of 't Hooft's instanton-induced interaction for explaining all the various, striking features of the mass spectrum, it is worth to summarize our main results and statements.
With the confinement parameters and the non-strange quark mass fixed by the phenomenology of the global ∆spectrum and the 't Hooft coupling g nn chosen to fit the ∆ − N splitting we subsequently calculated the complete excited nucleon mass spectrum. All states of the N * -spectrum thus were real predictions within the models A and B.
In a first detailed investigation of the predicted structures in the positive and negative parity shells we indeed found excellent agreement between the empirical nucleon mass spectrum and the predictions of our model A. Both the lower resonance regions and the higher (and even highest) mass regions could be uniformly described: -As in the ∆ sector, we found an excellent description of the positive parity Regge trajectory In particular, the model yields the correct phenomenological characteristic M 2 ∝ J with the right slope of the trajectory. We pointed out that similar to the ground-state all excited states of the trajectory likewise are lowered by 't Hooft's force showing the nontrivial property of the instanton-induced force to be compatible with the observed linear Regge characteristics.
-All striking hyperfine intra-band structures in the even-and odd-parity shells can be nicely reproduced in at least qualitative (but mostly even in completely quantitative) agreement with the hitherto established, experimentally observed patterns. As a particular nice feature of model A we found that it is able to explain the conspicuous low position of the Roper resonance   -Comparing the relative arrangement of the different even-and odd-parity bands we indeed found overlapping substructures of shells with positive and negative parity leading to the occurrence of more or less degenerate states with the same spin and opposite parity. Hence, our model is able to account for the striking observed pattern of approximate parity doublets in the excited nucleon spectrum as e.g. The corresponding discussion of model B has shown that it works less well and in particular strongly fails in describing one of the most prominent features, namely the low position of the Roper resonance. Hence we favor model A as the more realistic model. In the subsequent, second part of our discussion we then analyzed in some detail the instanton-induced effects in the excited nucleon spectrum of model A in order to clarify to what extent 't Hooft's force actually is responsible for generating all these striking features of the N * -spectrum simultaneously. Starting from the case with confinement only we investigated how 't Hooft's force affects the energy levels when its strength is gradually increased from zero until the correct ∆ − N splitting is achieved. Due to the selection rules of 't Hooft's force we observed in each shell a systematic lowering of those states which exhibit a dominant 2 8 [56] or 2 8[70] spin-flavor SU (6) configuration. In fact, we found that really all the prominent features of the N * -spectrum like e.g the Roper resonance or the patterns of approximate parity doublets are simultaneously generated along with the ∆ − N splitting in this way. Accordingly, we thus could convincingly demonstrate that the instanton-induced interaction indeed provides a consistent, systematic and uniform explanation not only for the ground-state splitting between the ∆ and the nucleon, but really for all the observed phenomena of the complete N * -spectrum listed above. All prominent features of the nucleon spectrum can be remarkably well described in our fully relativistic model A.
To understand the shortcomings of model B in the J = 1 2 ± sectors, we finally analyzed the effects of 't Hooft's force also in model B. Comparing the instanton-induced effects in both models in the N 1 2 + sector, we demonstrated that the failure of model B in describing the low position of the Roper resonance indeed is caused by a quite different influence of the instanton induced interaction on the excited N 1 2 + states of the 2hω shell. We found that the different instanton-induced effects have their origin in a different initial level ordering and a different initial configuration mixing of the embedded Pauli spinors in the pure confinement case. This is caused by the different relativistic effects that are induced by the two distinct confinement Dirac structures of both models in combination with the embedding map of the Salpeter amplitudes. Hence we found the interplay of the residual force with the relativistic effects of the confinement Dirac structure to be very crucial. This clearly shows the importance to describe the light baryons in a fully relativistic framework.
The discussion of the whole non-strange baryon mass spectrum is thus completed and we have shown that with our model version A a simultaneous, consistent description of the complete ∆ and nucleon mass spectra can indeed be achieved with only five parameters.

Summary and conclusion
In this paper we have computed the non-strange baryon spectrum on the basis of the three-particle Bethe-Salpeter equation with instantaneous interactions kernels. More precisely we used a string-like three-body confinement force and a regularized version of 't Hooft's instanton-induced interaction. We could construct a more or less unique model concerning the best fit of the model parameters. The following points (concerning the favored model A) are particularly remarkable in comparison with the nonrelativistic or relativized calculations. Regge trajectories up to the highest observed angular momenta could be reproduced. The hyperfine structure of the mass spectrum and in particular the Roper resonance and the parity doublets found a natural explanation. The lowering of the Roper resonance was due to a specific interplay of relativistic effects, the confinement potential (in model A) and 't Hooft's force. This lowering was not isolated to the Roper resonance alone but could be identified also for higher resonance states. The question arises whether these achievements are purely accidental. Of course a clear answer can not be given for any possible phenomenological interaction, but it can be answered at least for another QCD-inspired candidate, namely one-gluonexchange. From phenomenological grounds this alternative can be discarded for light flavors: We have demonstrated this by a Bethe-Salpeter calculation in the same spirit as described in this paper, see appendix A for a short description.
The way in which we have introduced 't Hooft's interaction in our spectroscopic calculation may look a bit too ad hoc for experts. Indeed more elaborate treatments of the characteristic instanton effects exist in the literature [63,64,65,66,67,68,69]. The result is however identical in form with the final two-body force used in this paper. The difference shows up mainly in the form factor of the force which we determined in a purely phenomenological way anyhow.
It remains to compute the spectrum of strange baryons in order to complete our unified description of the light baryon masses. This will be done in a separate paper [2]. the experimental mass spectrum of excited baryon resonances indicates that such a strong spin-orbit force should not exist between quarks. In order to obtain a reasonable description of the spectra these approaches are therefore forced to remove the spin-orbit components of the OGE by hand and to include as spin-dependent part the color magnetic hyperfine interaction only. However, leaving these interactions out is rather unsatisfactory and inconsistent. Isgur argued heuristically [35,51] that spin-orbit forces might cancel with the Thomas term from a confinement force with scalar Dirac structure (see also ref. [70] and references therein). But in view of several terms of the Breit-Fermi expansion that are left out anyway, this explanation is not really convincing and thus still remains rather ad hoc.
In order to correct the flaws in the non-relativistic model, other potential model calculations, which retain the one-gluon-exchange picture of the quark interaction, have gone beyond the original model of Isgur et al.. The so-called 'relativized' extension has been investigated by Godfrey, Capstic and Isgur for mesons [8] and subsequently also for baryons [9]. Also in this attempt, which still is based on the ordinary Schrödinger equation (with the kinetic energy replaced by its relativistic expression), the usual terms of the Fermi-Breit reduction of the OGE (including spin-orbit forces) as in the non-relativistic approach have been used. But compared to the non-relativistic version the expressions have been modified in order to qualitatively parameterize the momentum dependence of the relativistic corrections away from the |p q |/m q → 0 limit. However, this effective parameterization of relativistic effects is at the cost of introducing several new, non-fundamental parameters which were not derived from first principles. Using the new freedom to fit the additional parameters, the spin-orbit interactions could indeed be suppressed relative to the contact interaction. These effects, along with a partial cancellation of OGE-induced spin-orbit effects with those of a scalar three-body confinement (calculated within a two-body approximation) could finally reduce the size of spin-orbit effects to an acceptable level. Nonetheless, in view of the crude method of "relativizing" the non-relativistic quark model, the spin-orbit puzzle connected with the employment of OGE as residual interaction is in fact unsolved.
More reliable investigations ultimately require the description of baryons in a fully relativistic framework, where relativistic effects of the residual force and the confinement force are consistently and fully taken into account. In this respect, a suitable approach to investigate this problem in fact is given by our covariant Salpeter framework. We therefore used this framework to verify the statements of the relativized quark model [9] concerning the spin-orbit problem in an analogous, but fully covariant model which we shall discuss here briefly. Recall that our approach does not introduce any additional parameters; all relativistic effects are uniquely and fully determined by means of the embedding map in the Salpeter amplitudes.
According to the assumptions of the relativized quark model [9] we parameterize confinement by a ∆-type threebody string potential which is assumed to have a scalar Dirac structure: In view of the instantaneous treatment of the OGE the natural gauge for the gluon propagator is the Coulomb gauge [62], which will be applied in the following. This specific gauge has the advantage that the gluon propagator given by γ µ D µν γ ν = 4π γ 0 ⊗ γ 0 |q| 2 + γ ·⊗ γ − (γ ·q) ⊗ (γ ·q) q 2 + iǫ (38) withq := q/|q| is already instantaneous in its component D 00 (q) which describes the ordinary coulomb potential.
In the instantaneous approximation we substitute q 2 in the second term by −|q| 2 . The two-quark OGE kernel in coordinate space then reads [62] wherex := x/|x|. Here α s is the running strong coupling constant, which in momentum space is assumed to saturate at a maximal value for q 2 → 0. For the sake of simplicity we shall treat the coupling α s as a constant. This coupling together with the other parameters listed in table 25 has been fixed by a common fit to the spin-3/2 decuplet and spin-1/2 octet ground-states baryons as well as to the states ∆ 7 2 + (1950, ****) and ∆ 11 2 + (2420, ****) of the ∆-Regge trajectory in order to obtain also the correct positions of the higher band structures. The Salpeter equation has been solved by expanding the states in a rather large harmonic oscillator basis up to N max = 14 for the spin- 1 2 baryons and N max = 12 for all other spins. Figure 23 shows the resulting masses of the octet and decuplet ground-state baryons in comparison with the experimentally observed positions [37]. In addition, also the results of the relativized model [9]   The resulting spin 1/2 octet and spin 3/2 decuplet ground-state baryons calculated with a scalar three-body confinement and residual OGE interaction (on the left in each column). For comparison the results of the 'relativized' model [9] are shown on the right in each column. In the middle of each column the experimental masses [37] are displayed.
ground-states are dominantly S-wave states the spin-orbit forces are irrelevant. As can be seen also a fully relativistic treatment of the one-gluon-exchange interaction allows a reasonable description of the hyperfine splittings ∆ − N , Σ * − Σ and Ξ − Ξ * which is comparable to the mass splittings generated by the spin-dependent Fermi contact interaction in the non-relativistic and relativized approach. Note however, that the Σ − Λ splitting turns out much too small in comparison to the observed splitting of roughly 75 MeV. It is worth to mention that the value of the strong coupling constant α s = 1.1 in our fully relativistic approach is not reduced relative to the usual values α s ≃ 1.0 determined in nonrelativistic quark models. Treating α s really as running coupling constant would require a saturated value even bigger than 1.1. This is quite in contrast to the result of the relativized model [9], where this value became significantly smaller, namely α s = 0.6. Concerning the spin-orbit problem, which in fact is our major interest here, let us as an example consider the spectrum of the excited non-strange baryons calculated with the parameter set of table 25. Our results for the ∆ and N sector up to 2600 MeV are depicted in fig. 24, once again in comparison with the currently experimentally known resonance spectrum reported by the Particle Data Group [37] and the corresponding results of the relativized quark model [9,50]. In each column the resonances are classified by the total spin and parity J π and in each sector from 1 2 ± to 11 2 ± at most the first ten predicted excitations are displayed. Altogether, the structure of the ∆ spectrum can still be reasonably well accounted for. The spin-orbit effects are moderate enough to be compatible with the uncertainties of the experimentally observed ∆ resonances in each band. Our calculation roughly agrees with that of the relativized quark model. For instance the predictions for the two lowest excited ∆ 1 2 − and ∆ 3 2 − negative parity states are quite the same.
Notice however that both models predict their centroid too low with respect to the observed states ∆ 1 2 − (1620, ****) and ∆ 3 2 − (1700, ****). This is due to the fact that the OGE interaction shifts the ∆ ground-state upwards relative to these two states.
Much bigger discrepancies between the model predictions and the phenomenological spectrum emerge in the nucleon sector. First of all the predicted intra-band structures do not well agree with prominent empirical structures such as e.g. the four comparatively low lying states of the positive-parity 2hω band (including the Roper resonance) or the hyperfine splittings in the negative-parity 1hω band. In fact, the most striking effect of the spin-orbit interaction is seen for the two lowest excitations of the negative parity N 1 2 − sector. The center of gravity of these states is lowered by roughly 300 MeV with respect to the N 5 2 − state. Hence the predictions are far below the empirical positions of N 1 2 − (1535, ****) and N 1 2 − (1650, ****). We should note that this effect, although less pronounced, is likewise indicated in the relativized model [9], and also Isgur [35] predicted within a perturbative calculation of OGE-induced spin-orbit effects a quite large downward mass shift of roughly 500 MeV relative to the N 5 2 − . However, we obviously cannot confirm the conjecture that these effects might cancel against corresponding equally large spin-orbit effects stemming from the scalar confinement force. We should remark that similar results are obtained also for the corresponding strange sectors. We tried to cure this problem by using other spin structures than scalar. In fact it turned out that hyperfine structures of excited states strongly depend on the Dirac structure chosen. But so far no appropriate choice has been found that could suppress the large OGE-induced spin-orbit effects in the J π = 1 2 − sectors. In our opinion a solution of the spin-orbit puzzle connected with the residual OGE interaction thus seems highly questionable even in fully relativistic quark model. Our results concerning OGE in the Bethe-Salpeter framework indicate that it can be discarded for light flavors from phenomenological grounds.
In fact, apart from the spin-orbit problem of the residual OGE force, there are even more basic objections against the perturbative OGE interaction. Actually, it is only valid in the asymptotic-free domain of QCD, where the strong coupling constant is small and perturbation theory is expected to work. However, baryon spectroscopy obviously belongs to the domain, where the strong coupling is large such that perturbation theory fails and therefore complicated higher order multi-gluon-exchange contributions should be of roughly the same order of magnitude as the lowest order OGE contribution itself. The strength of OGE determined in an empirical way by a fit to the hyperfine splittings turned out to be roughly α s ≈ 1 which indeed makes it hard to treat it as a perturbative effect. Furthermore, the OGE is explicitly flavor independent and one thus obtains in the mesonic spectrum degenerate π and η mesons in clear contradiction to experiment. In order to cure this discrepancy one in fact would have to take into account higher order QCD diagrams [8]. All these arguments thus call into question the justification for applying the perturbative OGE in a nonperturbative sector of QCD. It can at best be added with a small coupling to the stronger force which we favor in this paper and would only modify the results slightly.