Strange results from chiral soliton models

0.15 MB·English

by  Aleksey Cherman

Strange results from chiral soliton models
Aleksey Cherman and Thomas D. Cohen
Department of Physics
University of Maryland
College Park, MD 20742
Thestandardcollectivequantizationtreatmentofthestrangenesscontentofthenucleoninchiral
solitonmodelssuchastheSkyrmionisshowntobeinconsistentwiththesemi-classicalexpansionon
7
whichthetreatmentisbased. Thestrangenesscontentvanishesatleadingorderinthesemi-classical
0
expansion. Collectivequantizationcorrectlydescribessomecontributionstothestrangenesscontent
0
at the first nonvanishing order in the expansion, but neglects others at the same order—namely,
2
thoseassociated with continuum modes. Moreover, thereare fundamental difficultiesin computing

n at aconstant orderin theexpansion duetothenon-renormalizable natureof chiral soliton models.
a Moreover,therearefundamentaldifficultiesincomputingataconstantorderintheexpansiondueto
J thenon-renormalizablenatureofchiralsolitonmodelsandtheabsenceofanyviablepowercounting

0 scheme. We show that the continuum mode contribution to the strangeness diverges, and as a
3 resultthecomputationofthestrangenesscontentatleadingnon-vanishingorderisnotawell-posed
mathematical problem in these models.

2
v
3 Theextractionofthe“strangenesscontentofthenucleon” tal difficulties intrinsic to any consistent description of the
4
has been the subject of considerable experimental activ- strangeness content.
1
ity over the past twenty years, largely involving parity vi- These models are only known to be meaningful in the
1
olating electron scattering[1]. One class of models studied context of a semi-classical expansion. All treatments of the
0
7extensively[2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, model begin with a classical solution which determines the
016, 17, 18, 19, 20, 21] for the purpose of understanding gross structure of the baryon. To proceed in a systematic
/the strangeness content (i.e. the matrix elements of strange fashion, one must assume that a semi-classical treatment is
h
quark bilinear operators in the nucleon) were chiral soliton valid. In this context, it will be shown that:
p
-models such as the Skyrmion[22]. The semi-classical treat-
pment of these models based on collective quantization has 1. Strange quark matrix elements of the nucleon neces-
e sarily occur at a subleading order in a semi-classical
providedareasonablygooddescriptionofmanynon-strange
h expansion.
properties of the nucleon[23]. It is interesting to consider
:
vhow well these models do in describing the strange matrix 2. Certain subleading effects—those associated with the
i
XelementsForexample,thestrangemagneticmomentinthese collective zero modes—are automatically included in
rmodels typically comes out in the range µs ∼ −0.5 in units treatments that use collective quantization. However,
aofthenuclearmagnetonwiththestrangenessradiusofabout other contributions which also occur at the first non-
rs2 ∼−0.1fm2 (with the exact values depending on the vari- vanishing order—those from the non-collective con-
ant of the model used). The experimental values for these tinuum modes—are not included. Since these effects
appear to be consistent with zero. A recent fit[24] to the have been neglected in typical computations of the
world’sdatayieldsµs =0.12±0.55andrss =0.01±0.95fm2. strangenesscontent,thecomputationsareinconsistent.
Thus the values predictedinthe models appearto be onthe
largesizebutdonotappeartobeinconsistentwiththedata.
3. The computation of the strangeness content at lead-
What is one to conclude from this situation? Before this
ing non-vanishing order starting from the Lagrangian
issue can be addressed, it is necessary to understand how
of the chiral soliton model in not a well-posed math-
these quantities were computed in these models. We note
ematical problem. Contributions from the continuum
that these calculations have been based directly on collec-
modes contribute at the leading non-vanishing order
tivequantization,orsomevariationofthissuchastheYabu-
andaredivergent. Themodelsarenotsystematiceffec-
Andomethod[25]whichreducestocollectivequantizationin
tive field theories with controlled power counting; the
the SU(3) flavor limit. As will be shown in this letter, cal-
divergenceswhichappearcannotbeabsorbedbyrenor-
culations of strange quark matrix elements in these models
malizing coefficients in the models as they are givenin
using collective quantization are inconsistent from the per-
their Lagrangians.
spectiveofthesemi-classicalexpansion,andhenceshouldnot
be regarded as true predictions from the models. Moreover, At first sight these results suggesta deficiency of Skyrme-
the non-renormalizability of such models creates fundamen- type models for these observables. However, they actually
2
highlight a strength: the ability of the models to encode the ofthesolitonmatchesthe1/N expansionofQCD.Oneneed
c
underlying quark-loop structure of QCD. The semi-classical notinvoke largeN as the ultimate justification ofthe semi-
c
expansion of the models corresponds to an expansion in the classical expansion. However, regardless of how of the ex-
number of closed quark loops. At the QCD level, strange pansion is justified, factors of 1/N may be used as markers
c
quark matrix elements come from quantum loops and thus tokeeptrackofordersinthesemi-classicalexpansion. Inthe
haveaqualitativelydifferentoriginthandonon-strangema- present context we work in the semi-classical analysis anal-
trix elements of the nucleon. It is a virtue of the semi- ogous to the standard large N limit of ’t Hooft[27], which
c
classical models that their structure forces one to impose is the appropriate limit when the usual O(N ) strength is
c
additional physics inputs in the form of new prescriptions taken for the WWZ term[28].
in order to describe the qualitatively distinct physics of the To illustrate the underlying issues in a relatively simple
strangeness content of the nucleon. context, we consider the strange scalar matrix element of
The frameworkof the analysisis the semi-classicalexpan- the nucleon in the chiral limit of the three flavor version
sion. Onenaturalwaytojustify itisviaWitten’s celebrated of Skyrme’s original model[29]. However, the conclusions
connectionofthesemi-classicaltreatmentofthesolitonwith depend on neither the choice of observable or model. The
the large N limit of QCD [26]; the semi-classical expansion action for the model is
c
f2 ǫ2 B
S = d4x πTr(LµLµ)+ Tr([Lµ,Lν]2)+ 0Tr M(U +U†−2) +NcSWWZ (1)
4 4 4
Z (cid:18) (cid:19)
(cid:0) (cid:1)
where the left chiral current L is given by L ≡ U ∂ U, light flavors;denote this ratio X :
µ µ † µ s
with U ∈ SU(3) [22, 30, 31]; S is the Witten-Wess-
f WWZ
Zumino (WWZ) term, whose inclusion is necessary for the hN|ss−hssivac|Ni
X ≡ (3)
s
Skyrme model to correctly encode the anomaly structure N uu+dd+ss−huu+dd+ssi N
vac
of QCD[28, 31]. The U field can be written as U =
(cid:10) (cid:12) (cid:12) (cid:11)
exp(i aλaπa/fπ) where the π are the Goldstone boson |Ni is the nu(cid:12)cleon state. In collective quant(cid:12)ization, the
meson fields and the λ are the Gell-Mann matrices; M is collective SU(3) rotation variables, A, act on the standard
a
the quPark mass matrix. For simplicity the present analysis classical static hedgehog: U = A†UhA with the hedgehog
will be done in the chiral limit; however, the mass term is Skyrmion defined as U ≡ exp(irˆ·~τf(r)) (where ~τ are the
h
included as an external source. Thus the matrix element firstthreeGell-Mannmatrices);f(r)isthestandardSkyrme
of interest—the strange scalar matrix element in the chiral profile function for states of baryon number B = 1. Eval-
limit—can be obtained by computing the nucleon mass for uating Xs using standard SU(3) collective quantization[33]
arbitrary values of the mass and then differentiating: yields
dM 1 1
hN|ss−hssivac|Ni= dmN (2) Xs = 3hN|1−D88|Ni= 3 dAψN∗ (A)(1−D88)ψN(A)
s Z
(4)
“vac” indicates a vacuum value. where dA stands for the Haar measure on SU(3), D =
88
1Tr λ Aλ A (which is an SU(3) Wigner D-matrix), and
2 8 8 †
At the QCD level, the strangeness content of the nucleon ψ (A) is the collective wave function for the nucleon—i.e.,
N
can only arise from quark loops. This already establishes an a(cid:2)ppropriate(cid:3)ly normalized SU(3) Wigner D-matrix. Eval-
point 1 above: there is a suppressionfactor of 1/Nc for each uating the expression using the collective wave function for
quark loop. Thus strange quark matrix elements are sub- the nucleon gives[32, 33] X =7/30≈.23. While this is rel-
s
leading in the semi-classical expansion. ativelysmallnumerically,itis non-zero. Notethatthis ratio
does not depend on the form of the Skyrme profile function
Considerthe computationofthe scalarstrangequarkma- f(r). This might suggest that it is a model-independent re-
trix element using collective quantization, as done originally sult, but as pointed out by Kaplan and Klebanov[34], this
by Donahue and Nappi[32]. The computation simplifies simpleresultisnotuniversalanddependsontheformofthe
somewhatifoneconsiderstheratioofthestrangescalarma- mass term in the Skyrme lagrangian.
trix element to the total scalar matrix elements of the three
3
Apparently the standard leading order collective quanti- to-leading order (i.e. leading non-vanishing) in the semi-
zation calculation used in the computation of strange quark classical expansion is:
matrix elements includes subleading effects in the semi-
classical expansion. As was noted in ref. [35] one must hN|ss−hssivac|NiNLO = (6)
include an explicit coefficient of N rather than three as
c 1 d
the coefficient of the WWZ term in order to make explicit (2g+1) ωdisc + ωdisc +
the counting and trace orders in the semi-classical expan- 2 g,L dms n Lg+;n n Lg−;n!
X X X
sion. The coefficient of the WWZ term constrains the al- 1 d dω
lowed SU(3) multiplets. Thus at arbitrary Nc the nu- 2 (2g+1) dms π δL′g+(ω)+δL′g−(ω) ω
cleon is in the generalized representation “8”, specified by g,L Z
X (cid:0) (cid:1)
(p,q) = 1,Nc 1 [35]. X can be computed at arbitrary
2− s where δ is the phase shift for given L, g-spin and
N [38] from Eq. (4) using standard group theoretical meth- Lg
c (cid:0) (cid:1) strangeness±±1 and ωdisc indicates the nth discrete fre-
odsandthe use ofSU(3) Clebsch-Gordancoefficientsappro- Lg ;n
quency with fixed grand ±spin and strangeness.
priate for the “8” representation[36]. The result[34, 37, 38]
is The frequency of the discrete modes and the phase-shifts
for the continuous modes can be computed from the equa-
hN|ss−hssi |Nicoll.quant. (5) tions of motion for the kaon fluctuation around the soli-
vac
2(N +4) ton may be derived in the manner of Callan and Kle-
c
= N2+10N +21 N uu+dd+ss−huu+dd+ssivac N banov [39, 40]. A compact form for these is given in
c c ref. [40]. The equations are naturally expressed in terms
= 2 +O 1/N2(cid:10) (cid:12)(cid:12)N uu+dd−huu+dd+i N .(cid:12)(cid:12) (cid:11) of dimensionless lengths and masses: r˜= fπr, ω˜ = ǫ ω and
(cid:18)Nc (cid:0) c(cid:1)(cid:19)(cid:10) (cid:12) vac(cid:12) (cid:11) m˜K = fǫπmK. (Note that the conventionǫs used hfeπre dif-
Clearly, hN|ss−hssi |N(cid:12)i is subleading in N(cid:12)—and, fer from ref. [40]: the symbol fπ here corresponds to fπ/2
vac c
in ref. [40] and ǫ corresponds 1 ). The equations for the
hence, in the semi-classical expansion—as compared to its 2√2e
non-strange analog. modes are
The collective quantizationmethod builds in some contri- y(r˜)ω˜2∓2λ(r˜)ω˜+Θ kω˜ (r˜)=0 (7)
l,g,
butionstohN|ss−hssivac|Niattheleadingnonvanishingor- ±
der in the semi-classical expansion (i.e., the first subleading with (cid:0) (cid:1)
order). The questionis whetherit capturesallofthem. The
answer is no: there are contributions to hN|ss−hssivac|Ni Θ ≡ r˜ 2∂ h(r˜)∂ −m˜2 −V (r˜), λ(r˜)≡−Ncf′sin(f),
at first subleading order which are not included in the col- − r˜ r˜ K eff 8π2ǫ2r˜2
lective quantization. These may be computed via Eq. (2): y(r˜) ≡ 1+2s(r˜)+d(r˜), h(r˜)≡1+2s(r˜),
onedifferentiatesthefirstsubleadingcontributiontothenu- d(r˜) ≡ f 2, s(r˜)≡sin2(f)/r˜2, c(r˜)≡sin2(f/2)
′
cleon mass and with respect to m . The procedure for im-
s
plementing the semi-classical expansion for the calculation
wheref is the Skyrme profile,the prime indicates differenti-
of the mass of a topological soliton in a bosonic theory is
ation with respect to r˜and
very well established[39, 40, 41]: The boson fields are ex-
panded aroundthe classicalsolution to quadradic order and d+2s
V = − −2s(s+2d)
andthenquantized. Thenext-to-leadingcontributiontothe eff 4
mass is simply the energy of the zero point motion of these (1+d+s)(L(L+1)+2c2+4cI~·L~)
harmonic modes. +
r˜2
Ingeneraltherearecontributionsfrombothdiscreteeigen- 6
+ s(c2+(2c−1)I~·L~)+∂ (c+I~·L~)f sin(f) .
modes andfromcontinuummodes. The modes inthe SU(3) r˜2 r˜ ′
Skyrme model aroundthe standardhedgehog canbe broken (cid:16) (cid:16) (cid:17)(cid:17)
up into kaon modes and pion modes. From the structure In Eq. (7), the ∓ indicates the strangeness of the mode, g
of Eq. (1), it is clear that only the kaon modes depend on theg-spinandLtheorbitalangularmomentum. Phaseshifts
m and contribute to the strangeness content. The kaon maybeextractedfromthemodesbycomparingwithsolution
s
modes separate into modes carrying strangeness plus or mi- for f =0.
nus one, corresponding to kaons and anti-kaons. Moreover, It is known that there is only one discrete mode for this
eigenmodes carry good total orbital angular momentum L2 system [41]. The mode has L = 1, g = 1/2 and s = −1.
and good “grand spin”~g =I~+L~ with g =L±1/2 [39, 40]. Moreover, at m = 0 the mode is collective and associated
s
Thus, the contribution to the strangeness content at next- withflavorrotationsoutoftheSU(2)subspaceoftheoriginal
4
hedgehog: kdisc(r)=kdisc (r)=sin(f(r)/2)wherekdisc (r) potential barrier growswith L, at any fixed L we are free to
isthespatialLpgsrofileoft1h21e−mode. Byconstruction,at1m12−=0 chooseanarbitrarilyhighω. ThisallowstheWKBregionto
s
this collective mode is in a flat direction and has zero fre- penetrate close to the origin, where the sum of the in- and
quency. It makes a nonzero contribution in Eq. (7), how- out-going wave functions must vanish as a boundary condi-
ever, since the derivative of the frequency with respect to tion,justasitdoesforL=0. RecallingEq.(7)oneseesfrom
m isnonzeroatm =0. Thefrequencyofthismodecanbe the abovethatthe continuummode sumcontributionsinall
s s
expanded perturbatively in m2 from the underlying equa- channels diverge logarithmically in the ultraviolet limit in a
K
tions ofref. [40]; one finds ωdisc = 4m2Kfπ2 d3x(1−cos(f)) universalway. Thusthecontinuummodecontributiontothe
Nc strangeness content of the nucleon is divergent. This diver-
which in turn implies that the discrete mode contribution
R gence is not surprising: it reflects the one-loop divergences
to the strangeness content at first subleading order in the
present in the underlying mesonic theory.
semi-classical expansion is:
Tomakemeaningfulpredictionsfromthetheory,onemust
2 renderthis divergencefinite ina manner consistentwith the
hN|ss−hssi |Nidisc = hN uu+dd−huu+ddi Ni.
vac vac theory. If the theory were renormalizable this would be a
N
c
(cid:12) (cid:12)(8) well-defined task; any divergence which arises in the loops
(cid:12) (cid:12) couldbeabsorbedbyrenormalizationoftheconstantsinthe
As expected, the contribution from the discrete mode in
original theory. However, chiral soliton models such as the
Eq. (8)—associated with the collective motion in the flat
Skyrme model are not renormalizable. More significantly,
direction—exactlyreproducesthe resultofcollectivequanti-
chiralsolitonmodelsarenoteffectivefieldtheoriessincethey
zationgiveninEq.(6)atfirstnonvanishingorderinthesemi-
lack a systematic power counting scheme. Terms with any
classical (1/N ) expansion. However, the collective quanti-
c
numberofderivativesofthemesonfieldscontributeatevery
zation does not include the contributions from the contin-
order in the 1/N expansion. Thus one cannot use power
uum modes. These contributions are clearly both nonzero c
counting to restrict the number and type of counterterms
generically—the phase shifts are nonzero and dependent on
at next-to-leading order in the semi-classicalexpansion: one
m [40]—and are of the same order in the semi-classical ex-
s
needs an entirely ad hoc and uncontrolled prescription.
pansion as the discrete mode contribution encoded in the
collectivequantization. Thus,calculationsofthestrangeness Ofcourse,the actofbuilding a chiralsolitonmodelin the
content which neglect these are unjustified from the per- first place required making a similarly bold prescription: of
spective of the semi-classical expansion. Since these are ne- theinfinitenumberoftermswhichcouldbeincludedatlead-
glected by those calculations on the market which purport ing order in the 1/Nc expansion only a very few are kept.
to compute the strangeness in chiral solitons[2, 3, 4, 5, 6, The troubling issue here, however, is that unlike for lead-
7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], one ing order observables, the initial prescription used to set up
mustregardthesecalculationsasbeinginconsistentwiththe the model is not sufficient to compute strange quark ma-
semi-classical expansion[42]. This establishes point 2. We trix elements; an additional prescription is needed. Since
note that this point is implicit in the work of Kaplan and the computation of all strange quark matrix elements com-
Klebanov[34]. pletely depends on the prescription used at next-to-leading
order, the initial model given by the lagrangian,on its own,
Thecureforthisproblemseemsobvious: oneoughttosim-
has no predictive power for these matrix elements.
ply include these continuum mode contributions in the cal-
This implies that the problem of computing the
culation. Unfortunately, the contribution from these modes
strangeness content from chiral soliton models at the first
diverges.
nonvanishingorderinthesemi-classicalapproximationisnot
This divergencecan be seen fromthe formof Eq.(7). For
well-posed. To proceed, one must make some prescription
large ω and fixed L the system is in the WKB regime. At
not fixed from the Lagrangian of the original model. This
L=0andhighω,asimpleWKBcalculationatlowestorder
is highly problematic in that result for the strange content
in ω 1 yields that
− is not fixed by the original model. This establishes our fi-
nal point. As we noted above, this is unsurprising: the
∂δ α
= +O(ω−3) (9) strangeness content arises from quark loops. This is qual-
∂m ω
s(cid:12)(ms=0) itatively different from the dominant origin of non-strange
(cid:12)
(cid:12) ǫ dm2 r˜ matrix elements and thus new physical inputs are required.
w(cid:12)ith α = K dr˜ .
fπ dms 2 h(r˜)y(r˜) It should be clear that the conclusions in this letter are
Z
quite general. Although we have focused on the problem
Infact,Eq.(9)holdsforallpartialwavesp(althoughthevalue of computing the scalar matrix element at zero momentum
of ω at which the asymptotic regime sets in grows with L). transfer at the chiral limit of the Skyrme model, the struc-
The reason for this is that although the angular momentum ture of the argument holds quite generally. The argument
5
that the contributions come from quark loops and must be not. The need for a prescription not contained in the orig-
subleadinginthesemi-classicalexpansionholdsgenerallyfor inal model to compute at the leading nonvanishing order in
anystrangeoperatorinanymodelandregardlessofwhether the semi-classicalexpansionalsoappliesto allstrangequark
the system is in the chiral limit. We have shown this ex- observables in any non-renormalizable chiral soliton model.
plicitly for the scalar matrix elements in the Skyrme model
by demonstrating that the collective quantization leads to
contributions which are subleading in 1/N (and hence in
c
the semi-classical expansion). We have explicitly verified
Acknowledgments
that the same thing occurs for the case of the strange elec-
tric form factor—as it must. The general argument that
the quantum fluctuations of all modes, collective and non- ThisworkissupportedbytheU.S.DepartmentofEnergy
collective alike, contribute at the lowest nonvanishing order under grantnumber DE-FG02-93ER-40762. We aregrateful
in the semi-classical expansion again holds for any strange to DavidKaplan,IgorKlebanov,andVictor Kopeliovichfor
matrix element in any model whether in the chiral limit or helpful discussions.
[1] For recent reviews see A. Acha et al., HAPPEX collabo- [21] C. W. Wong, D. Vuong and K. c. Chu, Nucl. Phys. A 515,
ration, arXiv:nucl-ex/0609002 ; E.J. Beise, M.L. Pitt and 686 (1990).
D.T. Spayde, Prog. Part. Nucl. Phys. 54, 289 (2005); [22] T. H.Skyrme,Proc. Roy.Soc. A260, 127 (1961).
D.H. Beck and R.D. McKeown Annu. Rev. NuclP˙art. Sci. [23] I. Zahed and G. E. Brown, Phys.Rept. 142, 1 (1986).
51,189(2001);D.H.BeckandB.R.Holstein,Int.Jour.Mod. [24] R. D. Young, J. Roche, R. D. Carlini and A. W. Thomas,
Phys.E10, 1 (2001). Phys. Rev.Lett. 97, 102002 (2006).
[2] V. B. Kopeliovich, J. Exp. Theor. Phys. 85, 1060 (1997) [25] H. Yabuand K.Ando, Nucl.Phys. B 301, 601 (1988).
[arXiv:hep-th/9707067]. [26] E. Witten,Nucl. Phys.B 160, 57 (1979).
[3] H.Weigel,UNITU-THEP-5-1997,presented at International [27] G. ’t Hooft, Nucl. Phys.B 72, 461 (1974).
Workshop on Strange Structure of the Nucleon, Geneva, [28] E. Witten,Nucl. Phys.B 223, 422 (1983).
Switzerland, 11-15 Mar 1997 [29] E. Guadagnini, Nucl. Phys.B 236, 35 (1984).
[4] M. Wakamatsu, Phys.Rev.D 54, 2161 (1996). [30] G.S.Adkins,C.R.NappiandE.Witten,Nucl.Phys.B228
[5] C. V. Christov et al., Prog. Part. Nucl. Phys. 37, 91 (1996) (1983) 552.
[arXiv:hep-ph/9604441]. [31] E. Witten,Nucl. Phys.B 223, 433 (1983).
[6] J. R. Ellis and M. Karliner, arXiv:hep-ph/9601280. [32] J.F. Donoghue and C. R. Nappi, Phys. Lett. 168B, 105
[7] H. Weigel, Int. J. Mod. Phys. A 11, 2419 (1996) (1986).
[arXiv:hep-ph/9509398]. [33] J. Schechterand H.Weigel, arXiv:hep-ph/9907554.
[8] H.C.Kim,A.Blotz,C.SchneiderandK.Goeke,Nucl.Phys. [34] D. B. Kaplan and I. R. Klebanov, Nucl. Phys. B 335, 45
A 596, 415 (1996) [arXiv:hep-ph/9508299]. (1990).
[9] J. Kroll and B. Schwesinger, Phys. Lett. B 334, 287 (1994) [35] T.D.Cohen,Phys.Lett.581B,175(2004); Phys.Rev.D70,
[arXiv:hep-ph/9405267]. 014011 (2004).
[10] H.Weigel, arXiv:hep-ph/9211259. [36] T. D. Cohen and R. F. Lebed, Phys. Rev. D 70, 096015
[11] M. Karliner, TAUP-1932-91, invited talk at Int. Symp. on (2004).
pi N Physics and the Structure of the Nucleon, Bad Honnef, [37] V. B. Kopeliovich, Phys. Part. Nucl. 37, 623 (2006)
Germany, Sep 9-13, 1991 [arXiv:hep-ph/0507028].
[12] N.W. Park and H. Weigel, Nucl. Phys. A 541, 453 (1992). [38] A.ChermanandT.D.Cohen,Phys.Lett.B641,401(2006)
[13] M. Praszalowicz, NuovoCim. A 102, 39 (1989). [arXiv:hep-th/0607110].
[14] D.Klabucar and I.Picek, Nucl.Phys. A 514, 689 (1990). [39] C. G. . Callan and I. R. Klebanov, Nucl. Phys. B 262, 365
[15] N. W. Park, J. Schechter and H. Weigel, Phys. Rev. D 41, (1985).
2836 (1990). [40] I.R.Klebanov,PUPT-1158, lectures given at NATO ASI on
[16] H. Weigel, J. Schechter, N. W. Park and U. G. Meissner, Hadron and Hadronic Matter, Cargese, France, Aug 8-18,
Phys.Rev.D 42, 3177 (1990). 1989
[17] N.W. Park, J. Schechterand H.Weigel, Phys.Lett. B 224, [41] N.Itzhaki, I.R.Klebanov,P.Ouyangand L.Rastelli, Nucl.
171 (1989). Phys. B 684, 264 (2004) [arXiv:hep-ph/0309305].
[18] V. Bernard and U. G. Meissner, Phys. Lett. B 216, 392 [42] Ref. [21] is an exception and does discuss the mode sums.
(1989). Howeveritdoessointhecontextofthekaonnumber,anun-
[19] V.M.Khatsimovsky,I.B.KhriplovichandA.R.Zhitnitsky, physicalquantity,whichisrelatedtothestrangenesscontent
Z. Phys.C 36, 455 (1987). via an uncontrolled approximation.
[20] S.J.Brodsky,J.R.EllisandM.Karliner,Phys.Lett.B206,
309 (1988).