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11 September 1997

PHYSICS LETTERS B

Physics Letters B 408 (1997) 381-386

Strange vector currents and the OZI-rule Ulf-G. Meifiner a,l, V. Mull a,2, J. Spetha’3, J.W. Van Ordenbp4

a Forschungszentrum Jiilich, IKP (Theorie), D-52425 Jiilich, Germany b Department of Physics, Old Dominion University, Norfolk, VA 23529, USA

and Jefferson Lab., 12000 Jefferson Ave., Newport News, VA 23606, USA

Received 26 March 1997; revised manuscript received 16 June 1997 Editor: R. Gatto

Abstract

We investigate within a meson-exchange model the OZI allowed coupling of the 4 meson to the nucleon with the inclusion of kaon loops and hyperon excitations. All parameters of the model have previously been determined from a variety of hadronic reactions. A strong cancellation of the various contributions is observed which results in a small c$NN coupling. We also show that a realistic isoscalar spectral function including the correlated GT~ exchange leads to sizeably reduced strange vector form factors based on the dispersion-theoretical analysis of the nucleons’ electromagnetic form factors explaining our previous result. @ 1997 Published by Elsevier Science B.V.

Dedicated to the memory of Karl Ho&de

1. One of the outstanding problems in the under- standing of the nucleon structure concerns the strength of various strange operators in the proton. A dedicated program at Jefferson Laboratory preceded by experiments at BATES (MIT) and MAMI (Mainz) is aimed at measuring the form factors related to the strange vector current lyc,s in the nucleon. It was already pointed out a long time ago by Genz and H(ihler [ 11 that the dispersion-theoretical analysis of the nu- cleans’ electromagnetic form factors allows one to get bounds on the violation of the OZI rule, which leads one to expect that strange matrix elements should be small [ 2 J . This rule has, however, never firmly been rooted in QCD but can be understood qualitatively in

1 E-mail: [email protected]. ‘Present address: IBM Informationssysteme GmbH, Kiiln,

Germany. 3 E-mail: [email protected]. 4 E-mail: [email protected].

large NC (with NC the number of colors) [ 31. Jaffe [ 41 showed that under certain assumptions the information encoded in the isoscalar nucleon form factors can be used to extract strange matrix elements. Of particular importance for this type of analysis is the identification of the two lowest poles in the isoscalar spectral function with the w (782) and the c$( 1020) mesons. The corresponding strange form factors turn out to be rather large in magnitude, related to the strong coupling of the C$ to the nucleon found in the dispersion-theoretical analysis [ 51. This analysis was later updated and extended in [ 61 based on the novel form factor fits presented in [7]. Loosely spoken, such an analysis is based on a “maximal” violation of the OZI rule because the spectral function in the mass region of about 1 GeV is assumed to be given entirely by the &pole. Our aim is to investigate the OZI allowed couplings of the d-meson to the nucleon through K and K* meson loops and hyperon excitations. We find

0370-2693/97/$17.00 0 1997 Published by Elsevier Science B.V. All rights reserved. PII SO370-2693(97)00828-9

382 U.-G. Megner et a/./Physics Letters B 408 (1997) 381-386

strong cancellations of the various contributions which results in a restoration of the OZI-rule in apparent con- tradiction to the dispersion-theoretical analysis. On the other hand, the coupling of various mesons (like the w and the r) to the nucleons has been investigated in great detail in the framework of the Bonn-Jtilich meson exchange potential for the nucleon-nucleon interaction by Holinde and coworkers [ 81 . In particular, the correlated up [ 8,9] and rr7r exchange [ lo] has recently been included consistently. This leads to a more realistic microscopic picture of the isoscalar spectral function in the mass region of the 4 [ 111 which offers a possible explanation of the abovementioned discrepancy. In the following, we combine these novel results from the NN interaction with a fit to the nucleon electromagnetic form factors to elucidate the strength of the CpNN couplings, i.e. the violation of the OZI rule, and the consequences for the extraction of the strange form factors.

2. The $-meson is believed to be an almost pure Ss state whereas the nucleon consists, at least in the naive constituent quark model, of valence u and d quarks only. Therefore, the C#J cannot couple directly to the nucleon, at least in this scenario. This is in line with the phenomenological OZI-rule. On the other hand, it has been known for quite some time that C$ production is enhanced beyond expectation from the OZI-rule in various hadronic reactions [ 121. Such an enhancement could point towards the existence of resonant gluonic intermediate states (glueballs) or a sizeable Ss component in the nucleon [ 121. However, one should be aware that OZI-forbidden transitions can go via two- step processes in which each individual transition is OZI-allowed [ 131. In this way, the &meson can couple via non-glueball, i.e. hadronic intermediate states, to the nucleon.

In previous work it has been shown that the cross sections for the reactions pp --) iA and %Z as well as into RK and R* K can be well understood in the framework of meson-exchange models [ 141. In a more re- cent work it has also been shown that the sizeable cross sections for the annihilation reaction pp -+ qbq5 can be explained in a purely hadronic framework by two-step processes with &A and % intermediate states [ 151. Within the same meson-exchange model with the parameters used in the previous calculations and thus

LA

N

-\

‘1 @

./I K,K;*

Fig. 1. Hadronic model for the 4NN vertex consisting of Born terms and interaction diagrams (as indicated by the blobs).

these parameters being completelyfied, we now want to address the question of what is the effective coupling constant of the &meson to the nucleon. We have performed a calculation of the @lN vertex function in a hadronic picture, considering EK, l?K - EK”, AA

and %H intermediate states, as shown in Fig. 1. There are sizeable cancellations between the various contributions from graphs with intermediate K’s, K*‘s and diagrams with the direct hyperon interactions [ 1 I] leading to a very small 4 coupling,

2 WN - N 0.005 , 47r

K,$ fl! f0.2.

Here, K denotes the tensor-to-vector coupling ratio. The various contributions are tabulated in Table 1. The sign of the tensor coupling is very sensitive to the de- tails of the calculation. The smallness of these couplings amounts to a “resurrection” of the OZI rule. Note that such strong cancellations have also been observed in the quark model study of [ 161. Also given in that table are the coupling constants gBom and fe”. These are calculated within the Born approximation. In the final calculation, we consider the rescattering of the K-mesons within a T-matrix approach and the interaction between the hyperons by an optical potential. Both interactions have been used previously and

U.-G. Mei$ner et al. / Physics Letters B 408 11997) 381-386

Table 1 Various contributions to the 4NN vector (g) and tensor coupling (f) (compare Fig. 1). Given is also the sum of the individual contributions. To show the importance of the higher order effects of the coupled channel approach, the Born contributions nre given separately. In the case of the tensor coupling, the meson contributions can not simply be added since the spectral functions have different signs and the corresponding monopole fit to the sum of the spectral functions has a considerably harder form factor than the individual contributions. In contmst, for the vector coupling the monopole cut offs are all of comparable size and thus one can essentially add the individual contributions

i?K -0.18 -0.32 -0.08 -0.15 l?*K - RK* -0.34 -0.39 -0.17 -0.20 9*K* f0.19 f0.25 +0.29 +0.40

alI mesons -0.33 -0.47 +0.09 +0.14

SS+AlI +0.64 +0.23 -0.72 -0.19

sum +0.31 -0.24 -0.63 -0.05

are not adjusted to the present process. The Born results differ considerably from the full calculation, in the case of g even in sign. This clearly shows the importance of the final state interactions (or higher loops and unitarization in another language) in addition to the cancellation between the K and K* contributions.

3. We now return to a possible explanation of the obvious discrepancy between our result for the @4N coupling constant and the one deduced from the dispersion-theoretical analysis which may be given by the correlated Irp-exchange. To be specific, consider first the nucleon-nucleon interaction. Although QCD is believed to be the theory underlying the strong interactions, in the non-perturbative regime of low- and medium-energy physics, mesons and baryons have re- tained their importance as effective, collective degrees of freedom for a wide range of nuclear phenomena. This is most apparent for the NN system. Here, the interaction between the two nucleons is generated by meson exchange [ 171. Resulting potentials, e.g. the Paris [ 181, Nijmegen [ 19 ] or Bonn [20] potentials, are able to describe the NN data below pion threshold in a truly quantitative manner. The full Bonn potential contains apart from single-meson exchanges higher-order diagrams involving also the A-isobar. The strength of the various baryon-meson vertices

383

Wg. 2. Correlated rp exchange missing in the Bonn potential.

is parametrized by coupling constants. In addition, form factors with cut-off masses A, are included as additional parameters, they take into account the corresponding vertex extensions. However, there are some longstanding conceptual problems hidden in the choice of parameters which have been resolved in the past years by Holinde and coworkers. First, the fictitious scalar-isoscalar meson COBE, which is needed to provide the intermediate-range attraction, has been replaced by correlated 2r-exchange [ lo]. A second longstanding discrepancy existed for the cut-off A,NN, which is rather large in the present-day potential models (N 1.3 GeV) compared to the information from other sources, like e.g. aN scattering. In [ 81 it has been shown that the interaction between a 7r and a p meson (correlated np exchange) has a strong influence on the NN-potential in the pion channel. It provides a sizeable contribution with a peak around 1.1 GeV. Due to this additional r-like contribution one is able to RdUCe the cut-off A&,N, which is now in much better agreement with information from other sources. The third well-known discrepancy is the wNN coupling constant, which in most of the NN potentials is three times bigger than predicted by SU( 3) symmetry 5. It has also been shown in [9] that in the w-channel the correlated ?rp

exchange gives a sizeable contribution which allows the choice of a value for ~“NN which is in reasonable agreement with the SU(3) prediction. The process missing in the original Bonn potential is depicted in Fig. 2. It has been analyzed in detail in [9]. The re

5 Notice that in the dispersion-theoreticaI analyses of the nucleons electromagnetic form factors even larger values for &NN are found, see [ $71.

384 U.-G. MeiJner et al./Physics Letters B 408 (1997) 38I-386

sult is given in terms of a dispersion integral, which for simplicity can be represented by an effective one-boson exchange, denoted as w’, in the w-channel,

1 O”

s

2 &‘NN - s--------. (2)

IT t-M;, (&+MpP

The spectral function p: is again peaked around 1.1 GeV. The pole fit, Eq. (2)) gives the following w’ parameters: M,J = 1.12GeV, g2,,,/4r = 8.5 for the vector and f$NN/4vr = 1.5 for the tensor COUpling. Moreover, it turns out that go/NN < 0 and K& = f~',&&,'N,', > 0. All these coupling constants are given at the on mass-shell values. In what follows, we use this effective pole instead of the full spectral function. One might argue that this effective w’ pole is at variance with the data from e+e- annihilation into vector mesons, i.e. one should see a third peak besides the w and the 4 in the O- l-- channel. We argue, however, that this approach gives rise to microscopic description of the wNN and #lN couplings and can only be unraveled in the presence of nucleons (like it is the case with the well-known two-pion threshold enhancement in the isovector channel due to unitarity and the analytic structure of the nucleon pole terms). To make this statement more transparent, we note that in e+e- annihilation one only sees the p at its mass of 770 MeV and with its canonical width. On the other hand, in the isovector nucleon form factors, one does not just see the p but also a strong enhancement on its left wing related to correlated two-pion exchange. This effect is of utmost importance for a precise determination of the nucleon radii. It has its origin, however, in a second sheet singularity of the pion-nucleon scattering amplitude and thus only appears in the presence of nucleons. Similarly, the strong 7rp correlations discussed here are related to the hadronic reaction TN -+ pN which make them irrelevant for the annihilation process (much like the rr correlations are not seen in case of the p). Notice furthermore that the width of this w’ is very large (-300 MeV) and is thus smeared out in the continuum.

4. The structure of the nucleon as probed with virtual photons is parametrized in terms of the SO-

called Dirac (Fl) and Pauli (Fz) form factors.

These form factors have been measured over a wide range of space-like momentum transfer squared, t = o... - 35GeV2 but also in the time-like region either in pp annihilation or in e+e- --f pp, fin col- lisions. The tool to analyze these data in a largely model-independent fashion is dispersion theory [ 51. We therefore briefly review the dispersion-theoretical formalism developed in [ 71 and discuss the pertinent modifications due to the constraints from the NN interaction described before. Assuming the validity of unsubtracted dispersion relations for the four form factors F,(i=““)( t) 6, one separates the spectral functions of the pertinent form factors into a hadronic (meson pole) and a quark (pQCD) component as follows:

F!‘)(t) = W(t)L(t) * I

= [ q(t) 611 + c aj’) L-l (Mf,))

x [In ($y)j M2

(1) - t 1

--Y

(3)

where Fr ( t) = ty( t) L(t) -’ parameterizes the isovector (I = 1) two-pion contribution (including the one from the p) in terms of the pion form factor and the P-wave rrrl?Npartial wave amplitudes in a parameter-free manner. In addition, we have three isovector poles, the masses of the first two can be identified with physical ones, i.e. M,,I = 1.45 GeV and Mp,, = 1.65 GeV. In the isoscalar channel (I = 0), we have the poles representing the w, the (6, the w’ (parametrizing the correlated 7zp exchange) and a fourth pole (denoted 5). In what follows, we will assume that from these only the 4 and the S couple to strangeness. Notice that it has recently been shown that there is no enhancement close to threshold of the isoscalar spectral function due to pion loops [ 2 11. Furthermore, A N 10 GeV’ [ 71 separates the hadronic from the quark contributions, Qa is related to AQCD and y is the anomalous dimension,

Fi(t) -+ (-t)-(‘f’) ln -’ [( >I -Y

2 ’ QO

6 Our notation is identical to the one of [ 71, i.e. the isoscalar (I = 0) and isovector (I = 1) form factors are given as half the

sum and the difference of the proton and neutron ones, respectively.

U.-G. MeiJner et al./Physics Letters B 408 (1997) 381-386 385

for r -+ -cm and p is the one loop QCD /?-function. In fact, the fits performed in [7] are rather insensi- tive to the explicit form of the asymptotic form of the spectral functions. To be specific, the additional factor L(r) in Eq. (3) contributes to the spectral functions for t > A*, i.e. in some sense parameter&s the intermediate states in the QCD regime, above the region of the vector mesons. The particular logarithmic form has been chosen for convenience. Obviously, the asymptotic behaviour is obtained by choosing the residues of the vector meson pole terms such that the leading terms in the l/r-expansion cancel. In practice, the additional logarithmic factor is of minor importance for the fit to the existing data. The number of isoscalar and isovector poles in Eq. (3) is determined by the stability criterion discussed in detail in [ 5,7]. In short, we take the minimum number of poles necessary to fit the data. Specifically, we have four isoscalar and three isovector poles. This fourth isoscalar pole is necessary since most isoscalar couplings are fixed (as described above) and otherwise we would not be able to ful- fill the various normalization and superconvergence relations. We are left with three fit parameters, these are the masses of the third isovector and the fourth isoscalar pole as well as the residuum a?.

5. The spectral functions of the isoscalar form factors 8’:,:) encode information about the strange vector current since the photon couples to a certain extent via mesons with strangeness (here the 4 and the S) to the nucleon. Assuming that the strange form factors have the same large momentum fall-off as the isoscalar electromagnetic ones [4,6] and neglecting the small w - 4 mixing, it is straightforward to extract the strange Dirac and Pauli form factors following the formalism outlined in [ 4,6]

M2,-Mi F;(f) = tL(t) 4;’ (t _ M$)(t _ MZ,) ’ (5)

F;(t)=L(t)u$L-l M;-M2,

6 (t-M;)(r-My (6)

with L;' = l/L( MS). Clearly, the size of these strange form factors is given by the strength of the #- nucleon couplings (as encoded in the residua at,). In

t~.~.l....l....r....I.,..~ 0 1

-‘t Ge”29

4 5

Fig. 3. Strange form factors F/S’(t) (solid line) and FiS’(t) (dashed line).

particular, we notice that the sign of the strange radius r:,, is determined from the sign of at whereas the

sign of the strange magnetic moment, p., = F:‘)(O),

is fixed by the sign of the tensor coupling N a$. A best fit to the available data as compiled in [ 221

is obtained with MS! = 1.63 GeV, iUptt~ = 1.72GeV and a? = 0.677 (for &NN = -0.24 and u,p = 0.2). The X*/datum of the fit is 1.02. All constraints are fulfilled to high numerical accuracy. A detailed account of these results is given in [23]. The corresponding strange form factors are shown in Fig. 3. Notice that F:“’ (t) varies very weakly between t = -l...- 10 GeV*. Furthermore, the strange magnetic moment and radius are

ps = 0.003 n.m., f-t = 0.002 fm* , (7)

respectively. These are orders of magnitude smaller than in previous analysis [ 4,6] where the &pole sub- sumed the non-strange physics of the isoscalar spectral function in the mass region of about 1 GeV, i.e. the sizeable effect of the np correlations. We note that the inclusion of w - # mixing, which is at the heart of Jaffe’s analysis, would increase the size of the strange matrix elements, but not dramatically. The main reason for the suppression observed here are the small @JN couplings which are independent of the precise parameters entering the meson-exchange model. To make this statement more transparent, we give in Table 2 the &couplings extracted from the nucleon isoscalar form factors of Refs. [ 5,6], which do not contain the up

continuum, compared the values used here. As already

386 U-G. MeiJner et al. /Physics Letters B 408 (I 997) 381-386

Table 2 Vector coupling (g) and tensor-to-vector coupling ratio (K) of the +-meson to nucleon% Note that the numbers for Ref. [5] have been updated with the latest value of f(4 -+ e+e-) and thus differ from what is quoted in that paper

Ref.

ISI [71 this work

4.4 -0.3 6.7 -0.22 0.005 +0.2

stated, the c$NN tensor coupling could change sign, leading to a small and negative strange magnetic moment. The value for pu, found here is sizeably smaller than the experimental value reported in Ref. [24] but within the uncertainty, (&(Q2 = 0.1 GeV*) = +0.23 f 0.37 f 0.15 f 0.19 n.m. The theoretical uncertainty on the numbers given in Eq. (7) is certainly of the same size as the given numbers. A more precise estimate of these error bars can only be given when the meson-exchange potential has been fine-tuned to include effects like e.g. w - &mixing.

6. To summarize, we have presented a meson- exchange model calculation for the OZI allowed q5- meson couplings to the nucleon. The model includes kaon loops and hyperon excitations. It is important to stress that the various model parameters are tightly constrained since the model has been used to success- fully describe a large body of hadronic reactions, like

pp -+ iA, %Z., KK, PK as well as pp --+ qbq5. The resulting @IN coupling turns out to be very small due to large cancellations between individual terms. We have then given a possible explanation for this small @lN coupling based on the correlated rp exchange, which contributes significantly to the spectral function in the mass region about 1 GeV Parametrizing the correlated rp exchange by a single pole, we have repeated the dispersion-theoretical analysis of the nu- cleans’ electromagnetic formfactors and shown that the inclusion of this effect leads to a sizeable reduc- tion of the strange matrix elements as anticipated from the OZI rule. Of course, the analysis presented here can be sharpened by including the effects of o - 4 mixing and by studying the dependence on the large-r behaviour of the strange form factors. This will, however, not change the main conclusion of our work, i.e. the smallness of the matrix-elements related to the

strange vector currents has to be considered a genuine result. Detailed experimental information concerning the strange form factors is thus eagerly awaited.

We thank Nathan Isgur and Bob Jaffe for useful comments.

References

[ I] H. Genz and G. Hiihler, Phys. Lett. B 61 (1976) 389. [Z] S. Okubo, Phys. Lett. 5 (1963) 165;

Cl. Zweig, CERN Reports TH-401, TH-412 (1964), unpublished; J. Iiuzuka, Prog. Theor. Part. Sci. 37 ( 1966) 21.

[3] E. Witten, Nucl. Phys. B 160 (1979) 57. [4] R.L. Jaffe, Phys. Lett. B 229 (1989) 275. [5] Cl. Hiihler et al., Nucl. Phys. B 114 (1976) 505. [6] H.-W. Hammer, U.-G. MeiSner and D. Drechsel, Phys. Lett.

B 367 (1996) 323. [7] P. Mergell, U.-G. MeiSner and D. Drechsel, Nucl. Phys. A

ISI

[91

LlOl

Ill1

[I21

[I31

I141

1151

11’51 [I71

1181

[I91

[201

[211

1221

~231

~41

596 (1996) 367. G. Jan&m, K. Holinde and J. Speth, Phys. Rev. Lett. 73 (1994) 1332. G. JanBen, Dissertation, University of Bonn (1995), published in Berichte des PZ Jtllich, No. 3065; K. Holinde, Prog. Part. Nucl. Phys. 66 ( 1996) 311. H.C. Kim, J.W. Durso and K. Holinde, Phys. Rev. C 49 (1994) 2355. V. Mull, K. Holinde and J. Speth, Annual Report 1995 of the Institute of Nuclear Physics, published in Berichte des PZ Jtllich, No. 3200. J. Ellis, E. Gabathuler and M. Karliner, Phys. Lett. B 217 (1989) 173. H.J. Lipkin, Nucl. Phys. B 244 (1984) 147; B 291 (1987) 729. J. Haidenbauer, K. Holinde and J. Speth, Nucl. Phys. A 562 (1993) 317. V. Mull, K. Holinde and J. Speth, Phys. Lett. B 334 (1994) 295. N. Isgur and P. Geiger, Phys. Rev. D 55 ( 1997) 299. G.E. Brown and A.D. Jackson, The Nucleon-Nucleon interaction (North-Holland, Amsterdam, 1979). M. Lacombe et al., Phys. Rev. D 12 (1975) 1495; Phys. Rev. C 21 (1980) 861. M.M. Nagels, T.A. Rijken and J.J. de Swart, Phys. Rev. D 17 (1978) 768. R. Machleidt, K. Holinde and Ch. Elster, Phys. Rep. 149 (1987) 1. V. Bernard, N. Kaiser and U.-G. MeiRner. Nucl. Phys. A 611 (1996) 429. H.-W. Hammer, U.-G. MeiSner and D. Dtechsel. Phys. Lett. B 385 (1996) 343. U.-G. MeiSner, V. Mull, J. Speth and J.W. Van Orden, in preparation. B. Mueller et al. (SAMPLE collaboration), Phys. Rev. Lett. 78 (1997) 3824.