Delta resonances, quark models, chiral symmetry and AdS/QCD
The mass spectrum of resonances is compared to
predictions based on three quark-model variants, to predictions
assuming that chiral symmetry is restored in high-mass baryon
resonances, and to predictions derived from AdS/QCD. The latter
approach yields a nearly perfect agreement when the confinement
property of QCD is modeled by a soft wall in AdS.
PACS: 11.25.Tq Gauge/string duality, 11.30.Rd Chiral symmetries, 12.40.Yx Hadron mass models and calculations, 14.20.Gk Baryon resonances with
The study of high-mass baryon resonances is a demanding task. From an experimental point of view, most existing information on baryon resonances (1) has been derived from experiments on pion (or Kaon) elastic scattering off polarized proton targets which were performed in the 70ties. Significant data at low pion energies were added by the pion factories LAMPF, TRIUMF, and PSI (former SIN). These data were not sufficient to construct the complex scattering amplitude; experiments at Gatchina have provided a few measurements of proton recoil polarization and spin rotation parameter. The data base (2) is, however, not complete; analysis and interpretation require further theoretical input from dispersion relations, analyticity and unitarity. Different analyses of these data lead to different conclusions concerning mass, width and the number of observed states. Even worse, the most recent analysis of Arndt et al. (3) – which included more and better data than all analyses before – found the smallest number of states.
From a theoretical side, very different concepts have been proposed to understand the baryon mass spectrum. Quark models – discussed here in three variants (4); (5); (6) – proved to be very successful in explaining the pattern of states with low-mass excitation energy. At high masses, above 1.8 GeV, there is the well known problem of the missing resonances: quark models predict many more states than have been observed experimentally. Possibly, these states have not yet been found as their conjectured small couplings prevented their identification in elastic scattering. However, they may not exist. Diquark effects are often invoked to explain the reduced number of observed states (7); (8); (9).
Glozman observed that in many cases, high-mass baryon resonances occur in parity doublets (10). He proposed that their valence quarks could have typical momenta larger than the chiral symmetry breaking scale and could decouple from the quark condensate. If the constituent quark mass originates from their coupling to the quark condensate, then their constituent (chiral symmetry breaking) mass should vanish, and chiral symmetry might be restored in high-mass resonances (11); (12). In this way, high-mass baryon resonances decouple from , in agreement with experimental observation.
A third approach is based on AdS/QCD, a new development to overcome the difficulties encountered when dealing with bound states in quantum chromodynamics. In AdS/QCD the conjectured gauge/gravity correspondence (13) is used to map string modes in a five-dimensional Anti-de-Sitter (AdS) space into interpolating ‘hadronic’ operators of a gauge theory defined on its four-dimensional boundary (14); (15). Recently, the baryon mass spectrum has been calculated in different variants of AdS/QCD. The variants differ in the way, conformal symmetry is broken: either explicitly by an infrared boundary in AdS (hard wall) (17), by introducing a non-conformal dilaton field (called dilation soft wall here), or by infrared deformation of the AdS metric (metric soft wall) (18); (19).
2 The mass spectrum and its interpretation
2.1 The mass spectrum
The lowest-mass state is of course the well known
having isospin (a property shared by all resonances), total angular momentum and positive parity. Decays to the ground state require orbital angular momentum between pion and nucleon (-wave); the quantum numbers are described by . In scattering, it is the first resonance region (which contains a small contribution as well). The second resonance region contains nucleon resonances only, while the third resonance region receives important contributions from resonances:
The Particle Data Group (PDG) lists four positive-parity and three negative-parity states in a narrow mass gap from 1900 to 1950 MeV. is the second resonance (after ) falling onto the main Regge trajectory comprising .
Intriguingly, , , and were not observed in the most recent partial wave analysis of data on elastic scattering by Arndt et al. (3)), while in the wave a state at 2233 MeV mass and with 773 MeV width was found which could be discussed as separate state belonging to the states (6). Thus different scenarios are possible which will be discussed at the end of the paper. For the moment we assume that the seven states (4) and (5) do exist and have masses of about 1.9 to 2.0 GeV.
The next states were reported with rather weak evidence only. The PDG lists a based on two observations giving masses at about MeV and one with 2200 MeV; we use the latter value and call it to avoid mix-up with . Surprisingly, is listed with two stars. In addition, there are two candidates with negative-parity. None of these states was observed by Arndt et al. (3).
Given the uncertainty of the mass determination, this resonance (if it exists at all) could as well be the missing state of a quartet of positive parity resonances below. Likewise, two negative-parity states (with quantum numbers and ) are missing to form a complete quartet.
The is an obvious candidate to be a member of the leading Regge trajectory (with ).
It is degenerate in mass with the negative-parity state . A few further states with positive and negative parity are known at about the same mass.
Finally we list the two highest mass states, one with negative, one with positive parity. Likely, they are both members of a quartet of states. is the highest-mass state (listed in RPP) on the leading Regge trajectory.
Note the PDG star-rating of the overall status. Only three- and four-star resonances are considered as established.
2.2 Quark models
In quark models, baryons are described by the dynamics of a system composed of three (constituent) quarks. Assuming locally quadratic potentials between quarks, wave functions can, to first order in perturbation theory, be expressed by harmonic-oscillator wave functions in the two relative coordinates. The spatial wave functions are characterized by the two orbital angular momenta and and their radial excitation quantum numbers and . The vector sum defines the total quark angular momentum . Here, we introduce as scalar sum of and , , and as radial excitation quantum number. Resonances are assigned to a band. The band number gives a first estimate of the mass, approximately the relation holds. The band number is not identical to the radial excitation quantum number N, instead . Baryons are classified by the number of excitation quanta, the dimensionality of the SU(3) representation, the total angular momentum , the quark orbital angular momentum , the total quark spin , and by the parity . Of course, the assignment of intrinsic orbital and spin angular momenta to the 3 quark system is a non-relativistic concept. There is no real understanding why the non-relativistic quark model works so well, but it does work. It is hard to avoid the conclusion that the five low-mass negative-parity nucleon resonances between 1.5 and 1.7 GeV
represent a triplet of states with and a doublet with . Of course, mixing of states having identical external quantum numbers but having different spin-orbital angular momentum configuration is possible. However, the effect of mixing on the observed masses is small even though mixing may have a considerable effect on decays where amplitudes and not only probabilities are relevant. These conclusions are confirmed in explicit quark model calculations where mixing angles are mostly small ().
Quark model predictions and data are compared in Table 1. The states are assigned to bands. The ground state (1) has ; it belongs to a 56-plet in SU(6), total quark spin is , total quark angular momentum is , and parity is . It belongs to the ground state baryons, . A possible (small) admixture has attracted high interest (22) but does not interfere with our conclusions.
The two negative-parity states (2) belong to . The number of observed and expected states coincides. Two further states (3) similar in mass have positive parity. The plays the same role here as the Roper resonance in the nucleon excitation spectrum. The orbital angular momenta vanish and one of the oscillators is excited radially with . It is assigned to . In , both and are 1 and couple to (but ). For this state, . Both these states are approximately mass degenerate with the spin doublet and which are members of the first excitation band. The four states form two parity doublets.
The quartet of states (4) is readily understood as having a total quark spin and total quark angular momentum coupling to . The small mass splitting is then interpreted by nearly vanishing spin-orbit forces. The states are members of the supermultiplet. The number of expected states in the second band is 8 while 6 are observed. The supermultiplet is missing.
|(4)||1835 1875||1795 1915 1985||1910 1990||1940|
|(5),A||1866 1905||1810 1871 1950||1897 1985||1956|
|(5),B||1901 1928||1923 1946 1965||1916 1948||1912|
|(4)||2035 2140||2080 2145||2155 2165||2090|
|(5),A||2100 2141||2089 2156 2170||2170 2187||2181|
|2202||2218 2260||2210, 2290||2239|
|(5),B||2169 2182||2161 2177||2152 2179||2182|
|2252||2239 2253 2270||2230 2247||2220|
Likewise it is tempting to interpret the negative parity states (5) as states, coupling to , again with vanishing spin-orbit forces. states are symmetric in their spin and flavor wave function, antisymmetric in their color wave function, hence their spatial wave function must be symmetric. This is impossible for the ground state. For a spatially symmetric state, at least one of the two oscillators of the three-body system must be radially excited. If the states (5) are interpreted as spin triplet of resonances, they must have and belong to the supermultiplet. The positive and negative parity states are mass degenerate, they form three parity doublets; only is not accompanied by an odd-parity partner. In the band, 14 states are expected while at most 5 are observed. The missing supermultiplets have , , , and .
(6) is likely member of a = spin doublet with its companion missing. Its partner is the third resonance in this partial wave. While is rather low in mass, fit very well to quark models. The two states (6) are mass degenerate with (7). Quark models predict two states in the second excitation band but the – mass gap is much larger than expected. Hence we rather prefer to assign (at 2200 MeV) to the forth excitation band with , N=1.
The three positive-parity and two negative-parity states in (8) and (9) also fall into a narrow mass window. The two states and are best interpreted assuming and or , respectively. If they had total quark spin , they would need to have two more units of orbital angular momentum which would place them higher in mass. Of course, a small admixture of high angular momenta is possible but does not affect this discussion. We assign the states in (8) to the forth excitation band while those in (9) might belong to the fifth band. The number of expected states increases considerably; we refrain from assigning the few observed states to supermultiplets. The two final states (10,11) need both spin to form, with and , respectively, the observed total angular momentum.
The predictions of quark models (see Table 1) agree only partly with observations. First, the number of expected states is much larger than observed. This is the well known problem of missing resonances. The second discrepancy concerns the mass pattern. Quark models predict a clear separation of states belonging to different excitation bands. Instead, states belonging to different bands are often degenerate in mass. The states of negative parity in the first excitation band are mass degenerate with states listed in the first line of the second band (Table 1). The second line in the band is mass degenerate with the first line in the band, and this pattern continues even though with decreasing reliability of data and interpretation.
Disagreement is found whenever we have assigned a unit of radial excitation to a resonance. The negative parity states in the first excitation band (2) are mass degenerate with the positive parity states in (3) belonging to the second band. is a radial excitation, has intrinsically orbital excitations and should have higher mass, too. We have argued that the triplet of negative-parity states at about 1920 MeV must have and belong to the band; they are mass degenerate with states belonging to the second band. from the band is mass degenerate with of the band. In all cases, the problem arises since a radial excitation to corresponds to a change of the band number by two units, . Yet experimentally, states of opposite parities acquire the same mass which suggests . Two reasons have been proposed for the pattern. The first one which we discuss next is restoration of chiral symmetry in high-mass baryon resonances.
2.3 Parity doublets from chiral symmetry restoration
The parity doublets were interpreted by Glozman (10) as evidence for restoration of chiral symmetry in high-mass excitations. There is an abundant literature on this subject, see (24); (25) for two recent reviews. Depending on how the symmetry is realized in Nature, parity doublets must not interact by pion emission or absorption, a striking prediction that can be tested experimentally (11); (12). The weakness of the signals for high-mass baryons in elastic scattering may thus provide further evidence for restoration of chiral symmetry.
Table 2 presents high-mass and resonances (1). The PDG star-rating is also given. Among the 10 parity doublets there are just 2 doublets for which both partners can be considered as established (with both partners having 3 or 4 stars). Reliable information on all four states of a chiral multiplet exists in no case. Even worse, a recent analysis of Arndt et al. (3) implementing recent precise data on elastic scattering from meson factories did not find any of the states with 1 or 2 stars, and just one of the parity doublets survives.
It is obvious that a new experimental approach beyond elastic scattering is needed to explore the high-mass region of and resonances with such weak couplings to the channel. Photoproduction of multiparticle final states seems to be a good choice to avoid in both the initial and the final state. Recently, evidence for one full chiral multiplet was reported with all four states derived from photoproduction. The preceding paper described the reaction from which the existence of and was deduced. The masses were determined to () and () MeV, respectively. In (26), evidence was presented for , at MeV, from an analysis of a large variety of photo- and pion-induced reactions, in particular from the new CLAS measurements of double polarization observables for photoproduction of hyperons. The forth resonance, a , was discussed as SAPHIR resonance in the literature (27). In (28), its mass was determined to MeV. The four states can be interpreted as two parity doublets, a doublet at 1900 MeV and a doublet at 1980 MeV. As full chiral multiplet, mass breaking effects of the order of 80 MeV have to be accepted.
From the experimental side, there is one problem with the concept of chiral symmetry restoration in high-mass resonances which was first pointed out in (29): the absence of a near-by parity partner of states like . Quite in general, “stretched” states with the maximum angular momentum and even (, , , ) seem to have no parity partner. The need to invoke chiral symmetry restoration to explain parity doublets was thus disputed (30). In view of the weak status of most high-mass resonances, non-observation of “stretched” states is of course not a conclusive argument.
2.4 Baryon resonances from AdS/QCD
The degeneracy of positive and negative parity states in Table 1 is phenomenologically reproduced if were approximately correct, a relation which is not supported by quark models, as can be seen from Table 1. Radial excitations with are not found at masses corresponding to the harmonic-oscillator band but rather to a band defined by . This relation – the mass of excited baryons is a function of and not of – was recently derived within a new holographic approach which analytically solves an approximate version of QCD where the strong coupling is large and nearly constant. (In fact, the QCD coupling itself may become scale-independent if its renormalization group flow approaches a conformal fixed point in the infrared. Dimensional counting rules for the near-conformal power law fall-off (31) can be used to argue that at large distances, the strong coupling constant indeed approaches a constant (32).)
The AdS/QCD approach is based on a (conjectured) equivalence between a string theory defined in a spacetime which becomes Anti-de-Sitter (AdS) near its boundary, and a gauge theory defined on this boundary. The gauge/string correspondence then relates interpolating operators (including those which carry hadronic quantum numbers) of the boundary gauge theory (and their correlators) to the propagation of weakly coupled strings in the five- dimensional, asymptotically AdS space (14); (15). Hence AdS/QCD bares the promise to simultaneously describe the nucleon mass spectrum and the partonic degrees of freedom observed in deep inelastic scattering (33).
In AdS/QCD the confinement properties of QCD are linked to infrared modifications of the asymptotically AdS space. Several variants have been proposed which differ in how they implement confinement. In ‘hard wall’ models (see (34) and references therein) a cutoff is imposed on the fifth AdS dimension and generates spectra of the type (17); (32). In the ‘dilaton soft wall’ model  an additional bulk dilaton field breaks conformal symmetry and generates confinement effects in the meson sector. In the ‘metric soft wall’ model (16), finally, smooth IR deformations of the AdS metric itself implement confinement and generate baryon spectra of the form (18); (19). In refs. (17); (18) only three- and four-star resonances were used to compare the predictions with data.
In Fig. 1, adapted from (17), not only established resonances but also one- and two-star resonances are included. The main (solid) line represents resonances with intrinsic spin 3/2, the dashed line those having . The splitting between the two different spin configurations was seen to be required for nucleons where ‘good diquarks’ (with spin and isospin both equal to zero) make the resonance lighter compared to nucleons with ‘bad diquarks’ (with or ). In decuplet states there are, of course, only bad diquarks.
At low masses there are a few states falling onto the wrong trajectory. The two negative parity states and must have and should fall onto the dashed line; yet they have masses just on the solid line. Of the two positive-parity states, has and is placed on the dashed line; its should fall onto the solid line. has , should have a lower mass than and be found on the dashed line.
In (17), was interpreted as excitation. The new evidence for – which seems to be a natural spin partner of – suggests quantum numbers for both, and the two-star to be the natural third partner to complete a spin triplet. In the interpretation of (17), one could of course also argue that and have , and and a missing below 2 GeV are excitations.
At high masses, some problems remain. In particular is far from the solid line.
In conclusion, there are clear discrepancies between hard-wall AdS/QCD and data in the 1.7 GeV region. Above 1.8 GeV, some inconsistencies with the hard wall solution exist, in particular the existence of (20); (21) and the non-observation of a candidate with mass between 1.9 and 2 GeV are difficult to reconcile with hard-wall AdS/QCD. But overall, the trend of most established states is reasonably reproduced.
were derived. Using the slope of the trajectory, masses were calculated. They are plotted as a function of in Fig. 2. The two states indicated by arrows are those found in (20); (21). While the positive-parity has three stars in the PDG rating, the negative-parity had one star only. Both states were not observed in the latest analysis of Arndt et al. (3) on elastic scattering.
The four positive- and negative-parity states between 1.60 and
1.75 GeV (2,3) are predicted to have the same mass
An early interpretation of strings was proposed by Nambu (36). He assumed that the gluon flux between the two quarks is concentrated in a rotating flux tube or a rotating string with a homogeneous mass density. Nambu derived a linear relation between squared mass and orbital angular momentum, . This mechanical picture was further developed by Baker and Steinke (37) and by Baker (38) to a field theoretical approach. For mesons, the functional dependence () was derived.
The relation (A) between masses and and has been derived earlier in a phenomenological analysis of the baryon mass spectrum (35). For octet and singlet baryons, one term ascribed to instanton-induced interactions was required to reproduce the full mass spectrum of all baryon resonances having known spin and parity.
The striking agreement between the measured baryon excitation spectrum and the predictions (18); (19) based on AdS/QCD and the success of the phenomenological mass formula (35) pose new questions. In both cases, the baryon masses depend on the number of orbital and radial excitations just as mesons. But baryons have an extra degree of freedom. So, where are the hidden states and, why are these states not realized in Nature? Why do they not appear in predictions of AdS/QCD ? The active programs at several laboratories to pursuit baryon spectroscopy using photon beams of linear and circular polarization and polarized target underline the hope that these issues can be solved in future.
2.5 Alternative interpretations
The experimental situation is, unfortunately, unsettled. The spectrum in the 1900 MeV region plays a decisive role in interpretation. At the end, we discuss briefly a few alternative scenarios which could be true. We discuss the possibility that 1., does not exist, or 2., does not exist or 3., that exists in this mass range but has not yet been found.
The two resonances and exist at about their nominal masses but the has a mass of about 2233 MeV as found in (3).
In this case, and would form a super-doublet of (dominantly) resonances and are likely radial excitations with resonances and could belong to one of the two supermultiplets. The masses do not agree well with quark model calculations (4); (5) (see Table 1) which predict a MeV mass gap between the negative-parity and the positive-parity states. However, given the experimental uncertainties, the experimental masses are not completely incompatible with the quark models. The falls into the mass range where it is predicted in all three quark models. With its large width, it may comprise several resonances. This scenario is in mild disagreement with quark models, disagrees with hard-wall and supports the metric soft-wall version of AdS/QCD.
The may not exist at about its nominal mass but both, and exist.
In this case, is unlikely. The two states could indicate a spin doublet with ; the two intrinsic oscillators could have orbital angular momenta and coupling to . Such configurations have never been observed so far and thus, this possibility would be very exciting. It would be a striking confirmation of quark models.
Alternatively, the could be partner of a spin quartet with which would include and two unobserved states with and . The large mass difference makes this possibility rather unlikely.
The three resonances , , exist at about their nominal masses. In addition, a exists in this mass range but has not yet been found.
The four resonances could form two doublets of (dominantly) and . Their masses are uncomfortably low when compared to quark models. The four negative-parity resonances would be accompanied by four positive-parity partners, , , , and . This scenario would be a striking confirmation of the scenario of chiral symmetry restoration in which parity doublets are predicted for all high-mass resonances.
The confirmation of two resonances (20); (21) of doubtful existence has initiated a comparison of the experimental mass spectrum with model predictions. Quark models predict a much larger number of states than observed in experiments; this is the well known problem of missing resonances. Quark models also predict radial excitations to have higher masses than observed experimentally, the Roper (in the nucleon excitation spectrum) is the best known example. A large number of states exist which can be grouped pairwise into parity doublets. The two states confirmed in (20); (21) form a parity doublet as well. Chiral symmetry restoration predicts the existence of further states.
The coincidence between masses (and abundance) of known resonances and very simple mass relations derived in AdS/QCD is intriguing. In particular when the confinement of QCD is modeled by a soft infrared deformation of the AdS metric, there is striking agreement between data and the prediction. The masses of all 23 resonances are well reproduced by just one single parameter, the slope of the Regge trajectory.
Does this success imply that AdS/QCD and the string picture are right and quark models and the concept of chiral symmetry restoration are wrong ? We do not believe so. AdS/QCD and the string picture pick up an important aspect of the baryon spectrum, the treatment of confinement. In AdS/QCD, confinement is parameterized as a ‘soft’ (or ‘hard’) limit for the off-shell structure of quark dynamics. This seems to work better than using a linear confinement potential used in quark models. But AdS/QCD does not tell us why we have five low-mass negative-parity resonances of the nucleon (12,13) and just two of the (2). And there could be a connection between two experimental observations: the ‘stretched’ states with are those which are best seen in pion elastic scattering experiments and they are those which miss a parity partner. Hence possibly, chiral doublets develop only for states weakly coupled ton . Likely, different models pick up different aspects of the baryon spectrum. Certainly, we are still far from a complete understanding of the dynamics of the formation of baryon resonances.
More data are needed to confirm or to disprove the present findings; to arrive at a solid understanding of the complicated pattern of highly excited and resonances, intense efforts are mandatory, in experiments, partial wave analyses and in theoretical foundations.
This paper was initiated by new results of the CBELSA collaboration; grateful thanks go to the collaboration for many fruitful discussions. In particular I would like to thank B. Krusche for a critical reading of the manuscript. I would like to thank S.J. Brodsky and G.F. de Teramond for illuminating discussions on AdS/QCD and its relation to baryon spectroscopy. I had the opportunity to discuss the different AdS/QCD approaches with H. Forkel; clarifying discussions and suggestions for the correct wording to describe AdS/QCD are gratefully acknowledged. I am indebted to B. Metsch and H. Petry for numerous discussions on the quark model.
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