Baryon Resonances and Pentaquarks on the Lattice ✩

Nuclear Physics A 754 (2005) 248c–260c

Baryon resonances and pentaquarks on the lattice ✩

F.X. Lee a,b,∗, C. Bennhold a

a Center for Nuclear Studies, George Washington University, Washington, DC 20052, USA b Jefferson Lab, 12000 Jefferson Avenue, Newport News, VA 23606, USA Received 17 December 2004; accepted 22 December 2004 Available online 21 January 2005

1. QCD primer

Quantum Chromodynamics (QCD) is widely accepted as the fundamental theory of the strong interaction. The QCD Lagrangian density can be written down simply in one line (in Euclidean space) 1 L = Tr F F µν +¯q γ µD + m q, (1) QCD 2 µν µ q where Fµν = ∂Aµ − ∂Aν + g[Aµ,Aν] is the gluon ﬁeld strength tensor and Dµ = ∂µ + gAµ is the covariant derivative which provides the interaction between the gluon and quark terms. The action of QCD is the integral of the Lagrangian density over space– 4 time: SQCD = LQCD d x. QCD is a highly non-linear relativistic quantum ﬁeld theory. It is well known that the theory has chiral symmetry in the mq = 0 limit and the symmetry is spontaneously broken in the vacuum. At high energies, it exhibits asymptotic freedom,

✩ Based on plenary talk by F.X. Lee at HYP2003, JLab. * Corresponding author. E-mail address: [email protected] (F.X. Lee).

0375-9474/$ – see front matter  2004 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysa.2004.12.072 F.X. Lee, C. Bennhold / Nuclear Physics A 754 (2005) 248c–260c 249c while at low energies it has confinement. At the present, the only tool that provides a solu- tion to QCD with controlled systematic errors is lattice QCD which solves the theory on a discrete space–time lattice using numerical simulations. The basic building block for computing the spectrum is the fully-interacting quark propagator defined via the path integral = | [ ¯ | S(x,0) 0 T q(x)q(0) 0 − − − DADq Dq¯ [q(x)q(¯ )]e SQCD DAM 1 Me Sg = 0 = det − − , (2) DADq Dqe¯ SQCD DAdet Me Sg µ where M = γ Dµ + mq is the quark matrix. In the last step the quark fields are exactly integrated, resulting in an expression that resembles a statistical system with the weighting − factor det Me SG for which Monte-Carlo methods can be employed. In this sense the quark propagator is simply the expectation value of the inverse quark matrix S =M−1. The determinant det M proves costly to simulate so it is usually set to a constant, leading to savings of up to a factor of 100. This is called the quenched approximation which amounts to ignoring the quark–antiquark bubbles in the QCD vacuum.

2. Baryon resonances

The rich structure of the excited baryon spectrum, as documented by the particle data group [1], provides a fertile ground for exploring the nature of quark–quark interactions. Most of the spectrum, however, is poorly known. Traditionally, quark models have led the way in making sense of the spectrum. But many puzzles remain. What is the nature of the roper resonance, and the Λ(1405)? How to explain the inverted ordering of the lowest-lying states which has the order of positive and negative-parity excitations inverted between N, ∆ and Λ channels? Two contrasting views have emerged about the nature of the hyperfine splittings in the baryons. One is from the constituent quark model [2,3] which c · c · has the interaction dominated by one-gluon-exchange type, i.e., color–spin λ1 λ2σ1 σ2. f · f · The other is based on Goldstone-boson-exchange [4] which has flavor–color λ1 λ2 σ1 σ2 as the dominant part. Lattice QCD is perhaps the most desirable tool to adjudicate the theoretical controversy surrounding these issues. Evidence from valence QCD [5] favors the flavor–color picture. Lattice QCD has evolved to the point that the best quenched calculation, of the ground- state hadron spectrum, shown in Fig. 1(left), has reproduced the observed values to within 7%, with the remaining discrepancy attributed to the quenched approximation. This bodes well for the exploration of the excited sectors of the spectrum, even in the quenched approximation.

2.1. Roper and S11

There exist a number of lattice studies of the excited baryon spectrum using a variety of actions [7–13]. The nucleon channel is the most-studied, focusing on two independent local ﬁelds: abc Ta b c χ1 = u Cγ5d u , (3) 250c F.X. Lee, C. Bennhold / Nuclear Physics A 754 (2005) 248c–260c

Fig. 1. Left: light hadron spectrum from quenched lattice QCD by the CP-PACS Collaboration [6]. Right: nucleon, 2 roper, and S11 masses as a function of mπ , using the standard nucleon interpolating ﬁeld χ1. The insert is the ratio of roper to nucleon mass. The experimental values are indicated by the corresponding open symbols. abc Ta b c χ2 = u Cd γ5u . (4)

χ1 is the standard nucleon operator, while χ2, which has a vanishing non-relativistic limit, is sometimes referred to as the ‘bad’ nucleon operator. Note that baryon interpolating fields couple to both positive- and negative-parity states, which can be separated by well- established parity-projection techniques. There are two problems facing these studies. First, they have not been able to probe the relevant low quark mass region while preserving chiral symmetry at finite lattice spacing (except Ref. [12] which uses the domain wall fermion). Since the controversy about the nature of roper hinges on chiral symmetry, it is essential to have a fermion action which explicitly exhibits the correct spontaneously broken chiral symmetry. Another difficulty of the calculation of the excited states in lattice QCD is that the conventional two-exponential fits are not reliable. Facing the uncertainty of the fitting procedure for the excited state, it has been suggested to use the non-standard nucleon interpolating field χ2, in the hope that it may have negligible overlap with the nucleon so that the roper state can be seen more readily. However, the lowest state calculated with this interpolation field (2.2 GeV) is much higher than the roper state. Employing the maximum entropy method allows one to study the nucleon and its radial excitation with the standard nucleon interpolation field [14]. However, with the pion mass at ∼600 MeV, the nucleon radial excitation is still too high (∼2 GeV). So the ordering of the nucleon, S11(1535) and the roper(1440) in these studies remains the same as that from quark models. In order to have a reliable prediction of the excited state in lattice QCD calculations by way of addressing the above mentioned difficulties, we aim to address both problems in our recent study [15]. First, we employ the overlap fermion [16] which has exact chiral symmetry even on a finite lattice, and on a large lattice which admits calculations with realistically small quark masses [17] to study the chiral region. Second, we implement constrained curve fitting based on an empirical Bayes method. It has been advocated [18,19] recently as a powerful tool which utilizes more data points while better controlling the systematics. The details of our implementation are discussed in [20]. The lattice size we F.X. Lee, C. Bennhold / Nuclear Physics A 754 (2005) 248c–260c 251c

3 use is 16 × 28 with the scale of a = 0.202(1) fm set from fπ . We consider a wide range of quark masses: 26 masses with the lowest mπ = 180 MeV (or mπ /mρ = 0.248), very close to the physical limit, and with 18 masses below the strange quark mass. We analyzed 80 configurations. Our final result is shown in Fig. 1(right). We see that for heavy quarks (mπ 800 MeV), the roper, S11, and nucleon splittings are like those of the heavy quarkonium. When the quark mass becomes lighter, the roper and S11 have a tendency to coincide and cross over around mπ = 220 MeV. More statistics are needed to clarify this point. However, from the insert in the figure, we can see that the ratio of roper to nucleon has a smaller error (by ∼40%) than the roper mass itself. It shows that the roper is consistent with the experimental value near the physical pion mass. We use the form + + 2 C0 C1/2mπ C1mπ to extrapolate the roper and S11 to the physical pion mass. The re- sultant nucleon mass of 928(56) MeV, roper at 1462(157) MeV, and S11 at 1554(65) MeV are consistent with the experimental values. Our result confirms the notion that the order reversal between the roper and S11(1535) compared to the heavy quark system is caused by the flavor–spin interaction between the quarks due to Goldstone boson exchanges [4]. It serves to verify that the roper (1440) is a radial excitation of the nucleon with three valence quarks. It also casts doubts on the viability of using the non-standard interpolation field for the roper. We further support the notion that there is a transition from heavy quarks (where the SU(6) symmetry supplemented with color–spin interaction for the valence quarks gives a reasonable description) to light quarks (where the dynamics is dictated by chiral symmetry). It is suggested that this transition occurs at mπ ∼ 400 MeV for the nucleon.

2.2. The η ghost

The above result is obtained only after the special effects of the so-called η ghost are removed. In full QCD, the η meson contributes to the proton via vacuum polarizations, as shown in Fig. 2(top). Being a relatively heavy meson, its contribution is much smaller than that of the pion. However, in quenched QCD, the vacuum loops are suppressed, as shown in Fig. 2(bottom) (hairpin diagram), resulting in the following peculiar properties. First, it becomes a light degree of freedom, with a mass degenerate with that of the pion. Second, it is present in all hadron correlators. Third, it gives a negative-metric contribution to the correlation function. For these reasons, it is termed the η ghost: it is an unphysical state, and a pathology of the quenched approximation. The effects of η ghost were ﬁrst observed in the a0 meson channel [21], where the ghost S-wave η π state lies lower than a0 for small quark mass. The situation here is similar with the excited state of the nucleon where the P-wave ηN appears in the vicinity of the roper. Since this is not clearly exhibited in the nucleon correlator where the nucleon is the lowest state in the channel and dominates the long-time behavior of the correlator, we can look at 1/2− the parity partner of the nucleon (N or S11) with I = 1/2. There, the lowest S-wave ηN state with a mass close to the sum of the pion and nucleon masses can be lower than S11 for sufﬁciently low quark mass. Due to the negative-metric contribution of the hairpin diagram, one expects that the S11 correlator will turn negative at larger time separations as is in the case of the a0 [21]. In Fig. 3, we show the S11 correlators for 6 low quark cases with pion mass from mπ = 181(8) MeV to mπ = 342(6) MeV. We see that for pion mass lower than 248(7) MeV, the S11 correlator starts to develop a negative dip at time 252c F.X. Lee, C. Bennhold / Nuclear Physics A 754 (2005) 248c–260c

Fig. 2. Top: quark-line diagram for the η contribution to the proton in full QCD (left) and its hadronic represen- tation (right). Any number of gluon lines can be present in the quark-line diagram. Bottom: the η contribution to the proton in quenched QCD.

Fig. 3. Evidence for the η ghost in the S11 correlators for six low quark masses. mπ are in GeV. slices beyond 4, and it is progressively more negative for smaller quark masses. This is a clear indication that the ghost ηN state in the S-wave is dominating the correlator over the physical S11 which lies higher in mass. This is the first evidence of η ghost in a baryon channel. Using our constrained curve fitting algorithm, we are able to distinguish the physical roper and S11 from the ghost two-particle intermediate states (η N) by checking their volume dependence and their weights as a function of the pion mass. For details, see [15]. Our results demonstrate that the effects of η ghost must be reckoned with in the chiral region (below mπ ∼ 300 MeV) in all hadron channels in quenched QCD. Another progress is the separation of the two nearby state in the 1/2− nucleon channel: N(1535)1/2− and N(1620)1/2−. Conventional techniques have difficulty in separating the two on the lattice. This has been achieved in a recent study [22] by using multiple F.X. Lee, C. Bennhold / Nuclear Physics A 754 (2005) 248c–260c 253c

− − Fig. 4. Results in the nucleon channel in lattice units. The two nearby states N(1535)1/2 and N(1620)1/2 are clearly separated. Figure is taken from [22].

operators and variational analysis, as shown in Fig. 4. It would be interesting to see if this can be pushed to smaller quark masses after the ghost states are taken care of.

2.3. Hyperon resonances

Fig. 5(left) show the results in Λ channel. The lowest negative-parity state is the flavor- singlet Λ(1405)1/2−. The correct ordering between it and the lowest two octet Λ states is reproduced on the lattice. There is no level-crossing in this channel. An interesting puzzle is the ordering between N(1535)1/2− and Λ(1405)1/2−. Λ(1405)1/2− has the same spin– parity, but a heavier s quark its quark content (uds) than N(1535)1/2− which has quark content (uud). Yet it lies lower than N(1535)1/2−. This is correctly reproduced on the lattice, as shown in Fig. 5(right). The reason it can happen has to do with the different flavor structure of the Λ(1405)1/2−. This example shows the importance of flavor–spin interactions in the baryon spectrum.

2.4. Chiral dynamics

To understand the chiral dynamics taking place in the small quark mass region, we show quark-line diagrams that contribute to the meson cloud surrounding the nucleon. In the quenched approximation, the last two disconnected diagrams are suppressed, but the connected Z-type diagrams survives. These connected diagrams are responsible for the non-linear behavior (curvature) in the chiral region. Effective theories that incorporate these diagrams are likely to capture the chiral degrees of freedom of QCD. 254c F.X. Lee, C. Bennhold / Nuclear Physics A 754 (2005) 248c–260c

± Fig. 5. Left: preliminary results for the level-ordering in the Λ(1/2 ) channel. Right: preliminary results for the − − level-ordering between N(1535)1/2 and Λ(1405)1/2 .

3. Pentaquarks

A number of recent experiments have reported on the discovery of an exotic 5-quark resonance, named as Θ+(uudds)¯ , with a mass of about 1540 MeV and a narrow width of less than 20 MeV [23], and main decay modes of K+n or K0p. Its strangeness quantum number is S =+1, but its isospin and spin–parity assignments are undetermined by the experiments. Based just on the valence quark content, the isospin could be 0, 1, or 2. ± ± 1 3 The spin–parity could be 2 , 2 , or higher. The isospin would have to be established by discovering the other charge states, while the spin–parity assignment will have to await detailed measurements of decay angular distributions. The discovery has spawned intense interest on the theoretical side. The studies range from chiral soliton and large Nc models [24,25], quark models [26], KN phase shift analysis [27], QCD sum rules [28], and lattice studies [29–31]. Here we use [29] to recap the essential elements of a lattice calculation of the pentaquark, focusing on spin-1/2 and different isospins. Pentaquarks do not have a unique color structure aside from being a color singlet. For a pentaquark of the type uudds¯,we consider both isospin I = 0 and I = 1 states with the following interpolating field e e Ta b c χ = s¯ γ5d abc u Cγ5d u ∓[u ↔ d], (5) where the minus sign is for I = 1 and the plus sign for I = 0. The explicit spin–parity − 1 of this interpolating field is 2 , but it couples to both parities. Under the anti-periodic boundary condition used in this work, the positive-parity state propagates in the forward time direction in the lower component of the correlation function, while the negative-parity state propagates backward in the same lower component. The correlation function using the interpolating field in Eq. (5) has four terms. Due to isospin symmetry in the u and d quarks, the two diagonal terms are equal, and so are the two cross terms. The master formula, after contracting out all possible quark pairs, reads F.X. Lee, C. Bennhold / Nuclear Physics A 754 (2005) 248c–260c 255c

Fig. 6. Quark skeleton diagrams that constitute the meson cloud in the nucleon channel. Any number of gluons can contribute in these diagrams. ∗ ¯ = abc abc aa eb T ef bf T cc χ(x)χ(0) 2 Su Cγ5 Sd Ss Sd Cγ5Su ∗ + aa eb cc T bf ef T Su Tr Sd Cγ5 Sd Cγ5Sd Ss − † aa bb T cc Trspin,color Sd Ss Su Cγ5 Sd Cγ5Su − † aa bb cc T Trspin,color Sd Ss Su Tr Su Cγ5 Sd Cγ5 ∗ ∓ abc abc − cf ef T ec aa bb T 2 Sd Ss Su Tr Sd Cγ5 Su Cγ5 ∗ − cb ba T af ef T ec Sd Cγ5 Su Cγ5 Sd Ss Su ∗ cf ef T ea ab T bc − S Ss S Cγ5 S Cγ5 S d u d u ∗ − cb ea T ef af T bc Sd Cγ5 Su Ss Sd Cγ5Su . (6)

In the above expression, Sq (x, 0) is the fully-interacting quark propagator; summation over all color indices is assumed; the transpose and trace are on the spin unless otherwise noted. 256c F.X. Lee, C. Bennhold / Nuclear Physics A 754 (2005) 248c–260c

+ Fig. 7. Left: the computed mass of the pentaquark state (uudds¯) with isospin 1 and spin–parity 1/2 as a function 2 = 2 + 2 + 2 + 2 of mπ . The curve is the KN two-particle energy in the p-wave EKN mK p mN p where p is the − smallest nonzero momentum on the lattice. Right: spin–parity 1/2 . The curve is the KN two-particle energy in the s-wave EKN = mK + mN .

Fig. 8. Same as Fig. 7, but for isospin 0.

The first four terms are the diagonal correlations, and the last four terms are cross correlations. The minus sign is for I = 1 and the plus sign for I = 0. Our results below were obtained on a 123 × 28 lattice using the Iwasaki gauge action and the overlap fermion action. The lattice spacing of a = 0.200(3) fm was determined from fπ [32]. Our quenched quark propagators cover a wide range of quark masses: 26 masses ranging from pion mass of 930 MeV to 184 MeV (or mπ /mρ = 0.248), reaching deep into the chiral regime. Our strange quark mass was set by the φ meson, corresponding ∼ P = 1 + to mπ 762 MeV. We analyzed 80 configurations. The fitted mass in the I(J ) 0( 2 ) 2 ∼ channel is shown in Fig. 7(left) as a function of mπ . Our fit stopped at about mπ MeV due to loss of signal in the remaining masses. Judging by the small curvature in the last few masses, the final mass appears to approach a value above 2.0 GeV at the chiral limit (we did not attempt a chiral extrapolation). This value is quite a bit higher than the experimental value of 1.54 GeV. The curve is the KN two-particle energy in the p-wave. On our lattice p F.X. Lee, C. Bennhold / Nuclear Physics A 754 (2005) 248c–260c 257c

Fig. 9. Left: the computed mass of the Ξ-type pentaquark state (uussd¯) with isospin 3/2andspin1/2asa 2 function of mπ . The solid symbols are for negative parity, while the empty symbols are for positive parity. The solid line is the KΣ p-wave energy and the dashed line the s-wave. Right: the continuum extrapolated mass ratios P m5q /(mN + mK ) for the lowest mass pentaquark states in the various I channels. The horizontal line shows + the experimentally known mass of the Θ . Figure taken from [30]. is about 520 MeV which raises the threshold from 1.43 GeV to 1.79 GeV, but still below 2 GeV.So the computed mass is most likely a KN scattering state with a repulsive interaction. P = 1 − Fig. 7(right) shows the results in the I(J ) 0( 2 ) channel. The mass appears to approach the s-wave KN energy from above. The dominant state in this channel is the KN state with a repulsive interaction. The results in the I = 0 sector are given in Figs. 8. P = 1 + ThemassintheI(J ) 1( 2 ) channel exhibits an interesting down-turn in the small = 3 1 mass region. The results for the Ξ-type pentaquark in the I(J) 2 ( 2 ) channel are shown ¯e e Ta b c in Fig. 9(left), using the interpolating field χ = (d γ5u ) abc(s Cγ5u )s . The signal in this channel is good to the smallest quark mass. Our preliminary results on the 123 × 28 (a = 0.2 fm) lattice reveal no evidence for a ¯ P = 1 + pentaquark state of the type uudds with the quantum numbers I(J ) 0( 2 ) near a mass of 1540 MeV. Instead, the correlation function are dominated by KN scattering states (our results are consistent with the known features of the KN scattering phase shifts [33]). Analysis is under way on two larger lattices: 163 × 28 a = 0.2 fm and 203 × 32 a = 0.15 fm, to check finite-volume and scaling effects. Some comments are in order concerning other recent lattice studies of the pentaquark. The work in [30] considered two kinds of operators, one given in Eq. (5) and one with a mixed color contraction between the N and K states, and employed a 2 × 2 correlation matrix to separate the lowest two states. Their final result is given in Fig. 9(right) in terms of the mass ratio of the pentaquark and the N + K threshold of 1408 MeV. In the 0− channel, they isolated two states, one with a ratio of 0.994 which they identify as the S- wave scattering state, one with 1.074 which they identify as the pentaquark state. The result in the 0+ channel has a rather high ratio of about 2 which is ruled out as a candidate. The work has reasonable statistics, has finite-size effects under control by exploring several lattices with varying spacing and volume, and performs a continuum extrapolation. The main concern here is the separation of two nearby states, the S-wave scattering state and the pentaquark state, that lie within 150 MeV of each other. It relies on the two operators and 258c F.X. Lee, C. Bennhold / Nuclear Physics A 754 (2005) 248c–260c

Fig. 10. Left: an example of the effective mass for the positive parity Θ(uudds)¯ state at κ = 0.1506 for up and down quarks and the strange quark κs = 0.1515. The solid lines represent fitted mass and its statistical error. The dashed line corresponds to the total energy of the non-interacting KN state with the smallest possible nonzero lattice momentum. Figure taken from [31]. Right: for negative parity. The dashed line corresponds to the total energy of the non-interacting KN state with zero momentum. Figure taken from [31]. the use of the variational method of the 2 × 2 correlator matrix. We know from variational principle that the larger the basis, the more reliable the prediction for lowest-lying states. A secondary concern is that the two operators are not truly independent: they are related by a Fiertz transform. So it is desirable to study the dependence of the 2nd lowest state on the operator basis in this approach. It is expected that to make reliable predictions for the two lowest states requires the use of at least three independent operators, especially when the two states are close to each other. In [31], three new operators are proposed for the pentaquark which are based on an exotic description of the antidecuplet as diquark–diquark–antiquark. The main result using + one of the operators are given in Fig. 10(left) for the 1 channel, and Fig. 10(right) for the − 2 1 channel. The main claim is that two states are observed in the same correlation function 2 − 1 in the 2 channel, one of which is identified as the S-wave scattering state, the other as the candidate for pentaquark. The two states are fairly close to each other (on the order of 100 MeV) with the 2nd state having a larger amplitude than the first state. Our experience with fitting excited states is that it requires extraordinary statistics and lattice resolution to isolate two close-by states with the parameters given in the work. It is stressed by the author that it has to do with special nature of the operator having a large overlap with the pentaquark state. This operator, however, can be shown to be partially related to the KN operator (Eq. (5)) by a Fiertz transform.

4. Conclusion

It appears that the ordering of low-lying baryons can be reproduced on the lattice with standard interpolating fields built from three quarks. We observed the cross-over of the roper and S11 in the region of pion mass 300 MeV. This shows the importance of pushing into the light quark region mass where chiral dynamics dominates. Our results support the notion that there is a transition from color–spin to flavor–spin in the hyperfine interaction from heavy to light quark masses. However, additional complications arise due to the η ghost states in the light mass region in the quenched approximation. This was clearly ex- F.X. Lee, C. Bennhold / Nuclear Physics A 754 (2005) 248c–260c 259c posed in the S11 channel. More advanced fitting algorithm that incorporates these ghost states has to be used. As long as the ghost states are properly dealt with, our results show that the quenched approximation can be used to explore the baryon spectrum deep in the chiral region. + On the pentaquarks, the results on 1 from all three calculations are consistent: there 2 + 1 is no evidence for a pentaquark of 1540 MeV in the 2 channel. The observed state is all high above it, around 2 GeV. This channel is ideal for identifying the pentaquark in the following sense: the threshold of KN scattering state in the p-wave is artificially raised by the smallest available momentum on the lattice so that a state appearing below it is − 1 more easily identified with a bound pentaquark. In the 2 channel, there are two claims of observing the pentaquark in the vicinity of 1540 MeV, one using a correlator matrix of two operators [30], the other using an exotic operator of diquark–diquark–antiquark [31]. Both claims should be taken with caution. The central issue is how to reliably separate a genuine pentaquark very close to the KN threshold in the S-wave. One may need more interpolation fields and different sinks to check if indeed there are near degenerate states at the KN threshold. It is extremely difficult to disentangle them in one correlation function given the resolution of current lattice simulations, unless it happens with exotic operators as described in [31]. There is one relatively easy test that can check whether a state is a genuine pentaquark or a KN scattering state. That is the volume dependence of spectral weight w as in G(t) = we−mt . For a one-particle state, there is almost not volume dependence for w.Butfora two-particle state, there is an inverse volume dependence w ∼ 1/V. It would be interesting to see results of such a test in the existing calculations.

Acknowledgements

This work is supported in part by US Department of Energy under grants DE-FG02- 95ER40907 and DE-FG05-84ER40154. The computing resources at NERSC (operated by DOE under DE-AC03-76SF00098) are also acknowledged. Collaboration with K.F. Liu, N. Mathur, Y. Chen, S.J. Dong, T. Draper, Horvath and J.B. Zhang is gratefully acknowledged.

References

[1] Particle Data Group, Eur. Phys. J. C 15 (2000) 1. [2] N. Isgur, G. Karl, Phys. Rev. D 18 (1978) 4187. [3] S. Capstick, N. Isgur, Phys. Rev. D 34 (1986) 2809. [4] L.Ya. Glozman, D.O. Riska, Phys. Rep. 268 (1996) 263; L.Ya. Glozman, et al., Phys. Rev. D 58 (1998) 0903. [5] K.F. Liu, et al., Phys. Rev. D 59 (1999) 112001. [6] CP-PACS Collaboration, Phys. Rev. Lett. 84 (2000) 238. [7] D.B. Leinweber, Phys. Rev. D 51 (1995) 6383. [8] F.X. Lee, D.B. Leinweber, Nucl. Phys. B (Proc. Suppl.) 73 (1999) 258; F.X. Lee, Nucl. Phys. B (Proc. Suppl.) 94 (2001) 251; 260c F.X. Lee, C. Bennhold / Nuclear Physics A 754 (2005) 248c–260c

F.X. Lee, et al., Nucl. Phys. B (Proc. Suppl.) 106 (2002) 248. [9] S. Sasaki, Nucl. Phys. B (Proc. Suppl.) 83 (2000) 206, hep-ph/0004252; T. Blum, S. Sasaki, hep-lat/0002019; S. Sasaki, T. Blum, S. Ohta, Phys. Rev. D 65 (2002) 074503. [10] D. Richards, Nucl. Phys. B (Proc. Suppl.) 94 (2001) 269; M. Göckeler, et al., Phys. Lett. B 532 (2002) 63–70. [11] W. Melnitchouk, et al., J.M. Zanotti, J.B. Zhang, Phys. Rev. D 67 (2003) 114506. [12] S. Sasaki, T. Blum, S. Ohta, Phys. Rev. D 65 (2002) 074503. [13] D.G. Richards, et al., Nucl. Phys. B (Proc. Suppl.) B 109 (2002) 89. [14] S. Sasaki, K. Sasaki, T. Hatsuda, M. Asakawa, hep-lat/0209059; S. Sasaki, Prog. Theor. Phys. Suppl. 151 (2003) 143. [15] S.J. Dong, T. Draper, I. Horváth, F.X. Lee, K.F. Liu, N. Mathur, J.B. Zhang, hep-ph/0306199. [16] H. Neuberger, Phys. Lett. B 417 (1998) 141. [17] S.J. Dong, F.X. Lee, K.F. Liu, J.B. Zhang, Phys. Rev. Lett. 85 (2000) 5051. [18] G.P. Lepage, et al., Nucl. Phys. B (Proc. Suppl.) 106 (2002) 12. [19] C. Morningstar, Nucl. Phys. B (Proc. Suppl.) 109 (2002) 185. [20] Y. Chen, S.J. Dong, T. Draper, I. Horváth, F.X. Lee, K.F. Liu, N. Mathur, C. Srinivasan, S. Tamhankar, J.B. Zhang, hep-lat/0405001. [21] W. Bardeen, A. Duncan, E. Eichten, N. Isgur, H. Thacker, Phys. Rev. D 65 (2002) 014509. [22] BGR Collaboration, hep-lat/0309036. [23] T. Nakano, et al., LEPS Collaboration, Phys. Rev. Lett. 91 (2003) 012002; S. Stepanyan, et al., CLAS Collaboration, Phys. Rev. Lett. 91 (2003) 252001; V.V. Barmin, et al., DIANA Collaboration, Phys. At. Nucl. 66 (2003) 1715; J. Barth, et al., SAPHIR Collaboration, Phys. Lett. B 572 (2003) 127. [24] D. Diakonov, V. Petrov, M.V. Polyakov, Z. Phys. A 359 (1997) 305. [25] D. Diakonov, V. Petrov, Phys. Rev. D 69 (2004) 056002; T. Cohen, Phys. Lett. B 581 (2004) 175; T. Cohen, R. Lebed, Phys. Lett. B 578 (2004) 150. [26] R. Jaffe, F. Wilczek, Phys. Rev. Lett. 91 (2003) 232003; Fl. Stancu, D.O. Riska, Phys. Lett. B 575 (2003) 242; L.Y. Glozman, Phys. Lett. B 575 (2003) 18; S. Capstick, P.R. Page, W. Roberts, Phys. Lett. B 570 (2003) 185; C.E. Carlson, C.D. Carone, H.J. Kwee, V. Nazaryan, Phys. Lett. B 573 (2003) 101. [27] R.A. Arndt, I.I. Strakovsky, R.L. Workman, Phys. Rev. C 68 (2003) 042201; R.A. Arndt, I.I. Strakovsky, R.L. Workman, Phys. Rev. C 69 (2004) 019901, Erratum; B.K. Jennings, K. Maltman, Phys. Rev. D 69 (2004) 094020. [28] S.L. Zhu, Phys. Rev. Lett. 91 (2003) 232002; J. Sugiyama, T. Doi, M. Oka, Phys. Lett. B 581 (2004) 167. [29] F.X. Lee et al., presentation (unpublished) at Lattice03 and Cairns Workshop in summer 2003. [30] F. Csikor, Z. Fodor, S.D. Katz, T.G. Kovács, JHEP 0311 (2003) 070. [31] S. Sasaki, Phys. Rev. Lett. 93 (2004) 152001. [32] S.J. Dong, T. Draper, I. Horvath, F.X. Lee, K.F. Liu, J.B. Zhang, Phys. Rev. D 65 (2002) 054507; S.J. Dong, T. Draper, I. Horvath, F.X. Lee, K.F. Liu, J.B. Zhang, hep-ph/0306199. [33] J.S. Hyslop, R.A. Arndt, L.D. Roper, R.L. Workman, Phys. Rev. D 46 (1992) 961.