Physics Letters B 651 (2007) 39–43

www.elsevier.com/locate/physletb

Strange results from chiral soliton models

Aleksey Cherman ∗, Thomas D. Cohen

Department of Physics, University of Maryland, College Park, MD 20742, USA

Received 31 January 2007; received in revised form 8 May 2007; accepted 18 May 2007

Available online 24 May 2007

Editor: W. Haxton

Abstract

The standard collective quantization treatment of the strangeness content of the nucleon in chiral soliton models, for instance Skyrme-typemodels, is shown to be inconsistent with the semi-classical expansion on which the treatment is based. The strangeness content vanishes at leadingorder in the semi-classical expansion. Collective quantization correctly describes some contributions to the strangeness content at the first non-vanishing order in the expansion, but neglects others at the same order—namely, those associated with continuum modes. Moreover, there arefundamental difficulties in computing at a constant order in the expansion due to the non-renormalizable nature of chiral soliton models. Moreover,there are fundamental difficulties in computing at a constant order in the expansion due to the non-renormalizable nature of chiral soliton modelsand the absence of any viable power counting scheme. We show that the continuum mode contribution to the strangeness diverges, and as a resultthe computation of the strangeness content at leading non-vanishing order is not a well-posed mathematical problem in these models.© 2007 Elsevier B.V. All rights reserved.

The extraction of the “strangeness content of the nucleon”has been the subject of considerable experimental activity overthe past twenty years, largely involving parity violating electronscattering [1]. One class of models studied extensively [2–21]for the purpose of understanding the strangeness content (i.e.the matrix elements of strange quark bilinear operators in thenucleon) were chiral soliton models such as the skyrmion [22].The semi-classical treatment of these models based on collec-tive quantization has provided a reasonably good descriptionof many non-strange properties of the nucleon [23]. It is in-teresting to consider how well these models do in describingthe strange matrix elements. For example, the strange mag-netic moment in these models typically comes out to be aboutμs ∼ −0.5 in units of the nuclear magneton with the strange-ness radius of about r2

s ∼ −0.1 fm2 (with the exact valuesdepending on the variant of the model used). The experimen-tal values for these appear to be consistent with zero. A re-cent fit [24] to the world’s data yields μs = 0.12 ± 0.55 andrss = 0.01 ± 0.95 fm2. Thus the values predicted in the models

* Corresponding author.E-mail address: alekseyc@physics.umd.edu (A. Cherman).

0370-2693/$ – see front matter © 2007 Elsevier B.V. All rights reserved.doi:10.1016/j.physletb.2007.05.032

appear to be on the large size but do not appear to be inconsis-tent with the data.

What is one to conclude from this situation? Before thisissue can be addressed, it is necessary to understand howthese quantities were computed in these models. We note thatthese calculations have been based directly on collective quan-tization, or some variation of this such as the Yabu–Andomethod [25] which reduces to collective quantization in theSU(3) flavor limit. As will be shown in this Letter, calcula-tions of strange quark matrix elements in these models usingcollective quantization are inconsistent from the perspectiveof the semi-classical expansion, and hence should not be re-garded as true predictions from the models. Moreover, the non-renormalizability of such models creates fundamental difficul-ties intrinsic to any consistent description of the strangenesscontent.

These models are only known to be meaningful in the con-text of a semi-classical expansion. All treatments of the modelbegin with a classical solution which determines the gross struc-ture of the baryon. To proceed in a systematic fashion, one mustassume that a semi-classical treatment is valid. In this context,it will be shown that:

40 A. Cherman, T.D. Cohen / Physics Letters B 651 (2007) 39–43

1. Strange quark matrix elements of the nucleon necessarilyoccur at a subleading order in a semi-classical expansion.

2. Certain subleading effects—those associated with thecollective zero modes—are automatically included in treat-ments that use collective quantization. However, other contribu-tions which also occur at the first non-vanishing order—thosefrom the non-collective continuum modes—are not included.Since these effects have been neglected in typical computationsof the strangeness content, the computations are inconsistent.

3. The computation of the strangeness content at leadingnon-vanishing order starting from the Lagrangian of the chiralsoliton model in not a well-posed mathematical problem. Con-tributions from the continuum modes contribute at the leadingnon-vanishing order and are divergent. The models are not sys-tematic effective field theories with controlled power counting;the divergences which appear cannot be absorbed by renor-malizing coefficients in the models as they are given in theirLagrangians.

At first sight these results suggest a deficiency of Skyrme-type models for these observables. However, they actually high-light a strength: the ability of the models to encode the underly-ing quark-loop structure of QCD. The semi-classical expansionof the models corresponds to an expansion in the number ofclosed quark loops. At the QCD level, strange quark matrixelements come from quantum loops and thus have a qualita-tively different origin than do non-strange matrix elements ofthe nucleon. It is a virtue of the semi-classical models that theirstructure forces one to impose additional physics inputs in theform of new prescriptions in order to describe the qualitativelydistinct physics of the strangeness content of the nucleon.

The framework of the analysis is the semi-classical expan-sion. One natural way to justify it is via Witten’s celebratedconnection of the semi-classical treatment of the soliton withthe large Nc limit of QCD [26]; the semi-classical expansionof the soliton matches the 1/Nc expansion of QCD. One neednot invoke large Nc as the ultimate justification of the semi-classical expansion. However, regardless of how of the expan-sion is justified, factors of 1/Nc may be used as markers to keeptrack of orders in the semi-classical expansion. In the presentcontext we work in the semi-classical analysis analogous to thestandard large Nc limit of ’t Hooft [27], which is the appropri-ate limit when the usual O(Nc) strength is taken for the WWZterm [28].

To illustrate the underlying issues in a relatively simple con-text, we consider the strange scalar matrix element of the nu-cleon in the chiral limit of the three flavor version of Skyrme’soriginal model [29]. However, the conclusions depend on nei-ther the choice of observable or model. The action for the modelis

S =∫

d4x

(f 2

π

4Tr

(LμLμ

) + ε2

4Tr

([Lμ,Lν]2)

(1)+ B0

4Tr

(M

(U + U† − 2

))) + NcSWWZ

where the left chiral current Lμ is given by Lμ ≡ U†∂μU , with

U ∈ SU(3)f [22,30,31]; SWWZ is the Witten–Wess–Zumino(WWZ) term, whose inclusion is necessary for the Skyrmemodel to correctly encode the anomaly structure of QCD[28,31]. The U field can be written as U = exp(i

∑a λaπa/fπ)

where the π are the Goldstone boson meson fields and the λa

are the Gell-Mann matrices; M is the quark mass matrix. Forsimplicity the present analysis will be done in the chiral limit;however, the mass term is included as an external source. Thusthe matrix element of interest—the strange scalar matrix el-ement in the chiral limit—can be obtained by computing thenucleon mass for arbitrary values of the mass and then differ-entiating:

(2)〈N |ss − 〈ss〉vac|N〉 = dMN

dms

,

“vac” indicates a vacuum value.At the QCD level, the strangeness content of the nucleon can

only arise from quark loops. This already establishes point (1)above: there is a suppression factor of 1/Nc for each quark loop.Thus strange quark matrix elements are subleading in the semi-classical expansion.

Consider the computation of the scalar strange quark ma-trix element using collective quantization, as done originallyby Donahue and Nappi [32]. The computation simplifies some-what if one considers the ratio of the strange scalar matrixelement to the total scalar matrix elements of the three lightflavors; denote this ratio Xs :

(3)Xs ≡ 〈N |ss − 〈ss〉vac|N〉〈N |uu + dd + ss − 〈uu + dd + ss〉vac|N〉 ,

|N〉 is the nucleon state. In collective quantization, the collec-tive SU(3) rotation variables, A, act on the standard classicalstatic hedgehog: U = A†UhA with the hedgehog skyrmion de-fined as Uh ≡ exp(ir · �τf (r)) (where �τ are the first three Gell-Mann matrices); f (r) is the standard Skyrme profile functionfor states of baryon number B = 1. Evaluating Xs using stan-dard SU(3) collective quantization [33] yields

Xs = 1

3〈N |1 − D88|N〉

(4)= 1

3

∫dAψ∗

N(A)(1 − D88)ψN(A),

where dA stands for the Haar measure on SU(3), D88 =12 Tr[λ8Aλ8A

†] (which is an SU(3) Wigner D-matrix), andψN(A) is the collective wave function for the nucleon—i.e.,an appropriately normalized SU(3) Wigner D-matrix. Evalu-ating the expression using the collective wave function for thenucleon gives [32,33] Xs = 7/30 ≈ 0.23. While this is rela-tively small numerically, it is non-zero. Note that this ratio doesnot depend on the form of the Skyrme profile function f (r).This might suggest that it is a model-independent result, but aspointed out by Kaplan and Klebanov [34], this simple result isnot universal and depends on the form of the mass term in theSkyrme Lagrangian.

Apparently the standard leading order collective quantiza-tion calculation used in the computation of strange quark matrixelements includes subleading effects in the semi-classical ex-pansion. As was noted in Ref. [35] one must include an explicit

A. Cherman, T.D. Cohen / Physics Letters B 651 (2007) 39–43 41

coefficient of Nc rather than three as the coefficient of the WWZterm in order to make explicit the counting and trace orders inthe semi-classical expansion. The coefficient of the WWZ termconstrains the allowed SU(3) multiplets. Thus at arbitrary Nc

the nucleon is in the generalized representation “8”, specifiedby (p, q) = (1, Nc−1

2 ) [35]. Xs can be computed at arbitraryNc [38] from Eq. (4) using standard group theoretical methodsand the use of SU(3) Clebsch–Gordan coefficients appropriatefor the “8” representation [36]. The result [34,37,38] is

〈N |ss − 〈ss〉vac|N〉coll.quant.

= 2(Nc + 4)

N2c + 10Nc + 21

〈N |uu + dd + ss

− 〈uu + dd + ss〉vac|N〉

(5)

=(

2

Nc

+O(1/N2

c

))〈N |uu + dd − 〈uu + dd〉vac|N〉.

Clearly, 〈N |ss − 〈ss〉vac|N〉 is subleading in Nc—and, hence,in the semi-classical expansion—as compared to its non-strangeanalog.

The collective quantization method builds in some contribu-tions to 〈N |ss − 〈ss〉vac|N〉 at the leading non-vanishing orderin the semi-classical expansion (i.e., the first subleading order).The question is whether it captures all of them. The answer isno: there are contributions to 〈N |ss − 〈ss〉vac|N〉 at first sub-leading order which are not included in the collective quantiza-tion. These may be computed via Eq. (2): one differentiates thefirst subleading contribution to the nucleon mass with respectto ms . The procedure for implementing the semi-classical ex-pansion for the calculation of the mass of a topological solitonin a bosonic theory is very well established [39–41]: The bosonfields are expanded around the classical solution to quadradicorder and then quantized. The next-to-leading contribution tothe mass is simply the energy of the zero point motion of theseharmonic modes.

In general there are contributions from both discrete eigen-modes and from continuum modes. The modes in the SU(3)

Skyrme model around the standard hedgehog can be broken upinto kaon modes and pion modes. From the structure of Eq. (1),it is clear that only the kaon modes depend on ms and contributeto the strangeness content. The kaon modes separate into modescarrying strangeness plus or minus one, corresponding to kaonsand anti-kaons. Moreover, eigenmodes carry good total orbitalangular momentum L2 and good “grand spin” �g = �I + �L withg = L ± 1/2 [39,40]. Thus, the contribution to the strangenesscontent at next-to-leading order (i.e., leading non-vanishing) inthe semi-classical expansion is:

〈N |ss − 〈ss〉vac|N〉NLO

= 1

2

∑g,L

(2g + 1)d

dms

(∑n

ωdiscLg+;n +

∑n

ωdiscLg−;n

)

(6)+ 1

2

∑g,L

(2g + 1)d

dms

∫dω

π

(δ′Lg+(ω) + δ′

Lg−(ω))ω,

where δLg± is the phase shift for given L, g-spin and strange-ness ±1 and ωdisc

Lg±;n indicates the nth discrete frequency withfixed grand spin and strangeness.

The frequency of the discrete modes and the phase-shifts forthe continuous modes can be computed from the equations ofmotion for the kaon fluctuation around the soliton may be de-rived in the manner of Callan and Klebanov [39,40]. A compactform for these is given in Ref. [40]. The equations are natu-rally expressed in terms of dimensionless lengths and masses:r = fπ

εr , ω = ε

fπω and mK = ε

fπmK . (Note that the conven-

tions used here differ from Ref. [40]: the symbol fπ here cor-responds to fπ/2 in Ref. [40] and ε corresponds 1

2√

2e.) The

equations for the modes are

(7)(y(r)ω2 ∓ 2λ(r)ω + Θ

)kωl,g,±(r) = 0

with

Θ ≡ r−2∂rh(r)∂r − m2K − Veff(r), λ(r) ≡ −Ncf

′ sin(f )

8π2ε2r2,

y(r) ≡ 1 + 2s(r) + d(r), h(r) ≡ 1 + 2s(r),

d(r) ≡ f ′2, s(r) ≡ sin2(f )/r2, c(r) ≡ sin2(f/2)

where f is the Skyrme profile, the prime indicates differentia-tion with respect to r and

Veff = −d + 2s

4− 2s(s + 2d)

+ (1 + d + s)(L(L + 1) + 2c2 + 4c �I · �L)

r2

+ 6

r2

(s(c2 + (2c − 1) �I · �L) + ∂r

((c + �I · �L)f ′sin(f )

)).

In Eq. (7), the ∓ indicates the strangeness of the mode, g theg-spin and L the orbital angular momentum. Phase shifts maybe extracted from the modes by comparing with the solution forf = 0.

It is known that near ms = 0 there is only one bound dis-crete mode for this system [41]. (The state associated withthe Λ(1405), which also appears as a discrete mode for real-istic values of ms , is not bound at ms = 0.) The bound dis-crete mode has L = 1, g = 1/2 and s = −1. Moreover, atms = 0 the mode is collective and associated with flavor ro-tations out of the SU(2) subspace of the original hedgehog:kdiscLgs(r) = kdisc

1 12 −(r) = sin(f (r)/2) where kdisc

1 12 −(r) is the spatial

profile of the mode. By construction, at ms = 0 this collectivemode is in a flat direction and has zero frequency. It makesa non-zero contribution in Eq. (7), however, since the deriva-tive of the frequency with respect to ms is non-zero at ms = 0.The frequency of this mode can be expanded perturbativelyin m2

K from the underlying equations of Ref. [40]; one finds

ωdisc = 4m2Kf 2

π

Nc

∫d3x (1 − cos(f )) which in turn implies that

the discrete mode contribution to the strangeness content at firstsubleading order in the semi-classical expansion is:

(8)

〈N |ss − 〈ss〉vac|N〉disc = 2

Nc

〈N |uu + dd − 〈uu + dd〉vac|N〉.

42 A. Cherman, T.D. Cohen / Physics Letters B 651 (2007) 39–43

As expected, the contribution from the discrete mode inEq. (8)—associated with the collective motion in the flatdirection—exactly reproduces the result of collective quantiza-tion given in Eq. (6) at first non-vanishing order in the semiclas-sical (1/Nc) expansion. However, the collective quantizationdoes not include the contributions from the continuum modes.These contributions are clearly both non-zero generically—the phase shifts are non-zero and dependent on ms [40]—andare of the same order in the semi-classical expansion as thediscrete mode contribution encoded in the collective quanti-zation. Thus, calculations of the strangeness content whichneglect these contributions are unjustified from the perspec-tive of the semi-classical expansion. Since these are neglectedby those calculations on the market which purport to computethe strangeness in chiral solitons [2–21], one must regard thesecalculations as being inconsistent with the semi-classical ex-pansion.1 This establishes point 2. We note that this point isimplicit in the work of Kaplan and Klebanov [34].

The cure for this problem seems obvious: one ought to sim-ply include these continuum mode contributions in the calcu-lation. Unfortunately, the contribution from these modes di-verges.

This divergence can be seen from the form of Eq. (7). Forlarge ω and fixed L the system is in the WKB regime. At L = 0and high ω, a simple WKB calculation at lowest order in ω−1

yields that

∂δ

∂ms

∣∣∣∣(ms=0)

= α

ω+O

(ω−3),

(9)with α = ε

fπ

dm2K

dms

∫dr

r

2√

h(r)y(r).

In fact, Eq. (9) holds for all partial waves (although the valueof ω at which the asymptotic regime sets in grows with L). Thereason for this is that although the angular momentum poten-tial barrier grows with L, at any fixed L we are free to choosean arbitrarily high ω. This allows the WKB region to penetrateclose to the origin, where the sum of the in- and out-going wavefunctions must vanish as a boundary condition, just as it doesfor L = 0. Recalling Eq. (7) one sees from the above that thecontinuum mode sum contributions in all channels diverge log-arithmically in the ultraviolet limit in a universal way. Thusthe continuum mode contribution to the strangeness contentof the nucleon is divergent. This divergence is not surprising:it reflects the one-loop divergences present in the underlyingmesonic theory.

To make meaningful predictions from the theory, one mustrender this divergence finite in a manner consistent with thetheory. If the theory were renormalizable this would be a well-defined task; any divergence which arises in the loops could beabsorbed by renormalization of the constants in the original the-ory. However, chiral soliton models such as the Skyrme modelare not renormalizable. More significantly, chiral soliton mod-

1 Ref. [21] is an exception and does discuss the mode sums. However it doesso in the context of the kaon number, an unphysical quantity, which is relatedto the strangeness content via an uncontrolled approximation.

els are not effective field theories since they lack a systematicpower counting scheme. Terms with any number of derivativesof the meson fields contribute at every order in the 1/Nc ex-pansion. Thus one cannot use power counting to restrict thenumber and type of counterterms at next-to-leading order in thesemi-classical expansion: one needs an entirely ad hoc and un-controlled prescription.

Of course, the act of building a chiral soliton model in thefirst place required making a similarly bold prescription: of theinfinite number of terms which could be included at leadingorder in the 1/Nc expansion only a very few are kept. Thetroubling issue here, however, is that unlike for leading orderobservables, the initial prescription used to set up the modelis not sufficient to compute strange quark matrix elements; anadditional prescription is needed. Since the computation of allstrange quark matrix elements completely depends on the pre-scription used at next-to-leading order, the initial model givenby the Lagrangian, on its own, has no predictive power for thesematrix elements.

One might argue that the divergence could be removed byimposing a cut-off on the integral in Eq. (9). Unfortunately,there is no systematic prescription for choosing a cut-off inthe original model, and any predictions obtained will be de-pendent on the (arbitrary) cut-off. This is a simple illustrationof the problem described above: the strange quark matrix ele-ments depend on the scheme used to regulate the behavior atnext-to-leading order (in this case, a cut-off).

This implies that the problem of computing the strangenesscontent from chiral soliton models at the first non-vanishing or-der in the semi-classical approximation is not well-posed. Toproceed, one must make some prescription not fixed from theLagrangian of the original model. This is highly problematicin that result for the strange content is not fixed by the origi-nal model. This establishes our final point. As we noted above,this is unsurprising: the strangeness content arises from quarkloops. This is qualitatively different from the dominant originof non-strange matrix elements and thus new physical inputsare required.

It should be clear that the conclusions in this Letter are quitegeneral. Although we have focused on the problem of com-puting the scalar matrix element at zero momentum transfer atthe chiral limit of the Skyrme model, the structure of the argu-ment holds quite generally. The argument that the contributionscome from quark loops and must be subleading in the semi-classical expansion holds generally for any strange operator inany model and regardless of whether the system is in the chi-ral limit. We have shown this explicitly for the scalar matrixelements in the Skyrme model by demonstrating that the col-lective quantization leads to contributions which are subleadingin 1/Nc (and hence in the semi-classical expansion). We haveexplicitly verified that the same thing occurs for the case of thestrange electric form factor—as it must. The general argumentthat the quantum fluctuations of all modes, collective and non-collective alike, contribute at the lowest non-vanishing order inthe semi-classical expansion again holds for any strange ma-trix element in any model whether in the chiral limit or not.The need for a prescription not contained in the original model

A. Cherman, T.D. Cohen / Physics Letters B 651 (2007) 39–43 43

to compute at the leading non-vanishing order in the semi-classical expansion also applies to all strange quark observablesin any non-renormalizable chiral soliton model.

Acknowledgements

This work is supported by the US Department of Energy un-der grant number DE-FG02-93ER-40762. We are grateful toDavid Kaplan, Igor Klebanov, and Victor Kopeliovich for help-ful discussions.

References

[1] For recent reviews, see A. Acha, et al., HAPPEX Collaboration, nucl-ex/0609002;E.J. Beise, M.L. Pitt, D.T. Spayde, Prog. Part. Nucl. Phys. 54 (2005) 289;D.H. Beck, R.D. McKeown, Annu. Rev. Nucl. Sci. 51 (2001) 189;D.H. Beck, B.R. Holstein, Int. J. Mod. Phys. E 10 (2001) 1.

[2] V.B. Kopeliovich, J. Exp. Theor. Phys. 85 (1997) 1060, hep-th/9707067.[3] H. Weigel, UNITU-THEP-5-1997, presented at International Workshop

on Strange Structure of the Nucleon, Geneva, Switzerland, 11–15 March1997.

[4] M. Wakamatsu, Phys. Rev. D 54 (1996) 2161.[5] C.V. Christov, et al., Prog. Part. Nucl. Phys. 37 (1996) 91, hep-

ph/9604441.[6] J.R. Ellis, M. Karliner, hep-ph/9601280.[7] H. Weigel, Int. J. Mod. Phys. A 11 (1996) 2419, hep-ph/9509398.[8] H.C. Kim, A. Blotz, C. Schneider, K. Goeke, Nucl. Phys. A 596 (1996)

415, hep-ph/9508299.[9] J. Kroll, B. Schwesinger, Phys. Lett. B 334 (1994) 287, hep-ph/9405267.

[10] H. Weigel, hep-ph/9211259.[11] M. Karliner, TAUP-1932-91, invited talk at International Symposium on

πN Physics and the Structure of the Nucleon, Bad Honnef, Germany, Sep-tember 9–13, 1991.

[12] N.W. Park, H. Weigel, Nucl. Phys. A 541 (1992) 453.[13] M. Praszalowicz, Nuovo Cimento A 102 (1989) 39.[14] D. Klabucar, I. Picek, Nucl. Phys. A 514 (1990) 689.[15] N.W. Park, J. Schechter, H. Weigel, Phys. Rev. D 41 (1990) 2836.[16] H. Weigel, J. Schechter, N.W. Park, U.G. Meissner, Phys. Rev. D 42 (1990)

3177.[17] N.W. Park, J. Schechter, H. Weigel, Phys. Lett. B 224 (1989) 171.[18] V. Bernard, U.G. Meissner, Phys. Lett. B 216 (1989) 392.[19] V.M. Khatsimovsky, I.B. Khriplovich, A.R. Zhitnitsky, Z. Phys. C 36

(1987) 455.[20] S.J. Brodsky, J.R. Ellis, M. Karliner, Phys. Lett. B 206 (1988) 309.[21] C.W. Wong, D. Vuong, K.c. Chu, Nucl. Phys. A 515 (1990) 686.[22] T.H. Skyrme, Proc. R. Soc. A 260 (1961) 127.[23] I. Zahed, G.E. Brown, Phys. Rep. 142 (1986) 1.[24] R.D. Young, J. Roche, R.D. Carlini, A.W. Thomas, Phys. Rev. Lett. 97

(2006) 102002.[25] H. Yabu, K. Ando, Nucl. Phys. B 301 (1988) 601.[26] E. Witten, Nucl. Phys. B 160 (1979) 57.[27] G. ’t Hooft, Nucl. Phys. B 72 (1974) 461.[28] E. Witten, Nucl. Phys. B 223 (1983) 422.[29] E. Guadagnini, Nucl. Phys. B 236 (1984) 35.[30] G.S. Adkins, C.R. Nappi, E. Witten, Nucl. Phys. B 228 (1983) 552.[31] E. Witten, Nucl. Phys. B 223 (1983) 433.[32] J.F. Donoghue, C.R. Nappi, Phys. Lett. B 168 (1986) 105.[33] J. Schechter, H. Weigel, hep-ph/9907554.[34] D.B. Kaplan, I.R. Klebanov, Nucl. Phys. B 335 (1990) 45.[35] T.D. Cohen, Phys. Lett. B 581 (2004) 175;

T.D. Cohen, Phys. Rev. D 70 (2004) 014011.[36] T.D. Cohen, R.F. Lebed, Phys. Rev. D 70 (2004) 096015.[37] V.B. Kopeliovich, Phys. Part. Nucl. 37 (2006) 623, hep-ph/0507028.[38] A. Cherman, T.D. Cohen, Phys. Lett. B 641 (2006) 401, hep-th/0607110.[39] C.G. Callan, I.R. Klebanov, Nucl. Phys. B 262 (1985) 365.[40] I.R. Klebanov, PUPT-1158, lectures given at NATO ASI on Hadron and

Hadronic Matter, Cargese, France, 8–18 August 1989.[41] N. Itzhaki, I.R. Klebanov, P. Ouyang, L. Rastelli, Nucl. Phys. B 684 (2004)

264, hep-ph/0309305.