IC/94/333

INTERNATIONAL CENTRE FORTHEORETICAL PHYSICS

UNITARIZATION OF KORNER-KURODA MODELOF ELECTROMAGNETIC STRUCTURE

OF OCTET 1/2+ BARYONS

INTERNATIONALATOMIC ENERGY

AGENCY

UNITED NATIONSEDUCATIONAL,

SCIENTIFICAND CULTURALORGANIZATION

Stanislav Dubnicka

Anna Zuzana Dubnickova

Jerzy Kraskiewicz

and

Ryszard Raczka

MIRAMARE-TRIESTE

IC/94/333

International Atomic Energy Agencyand

United Nations Educational Scientific and Cultural Organization

INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS

UNITARIZATION OF KORNER-KURODA MODELOP ELECTROMAGNETIC STRUCTURE OF OCTET 1/2+ BARYONS

Stanislav Dubnicka1,Internationa] Centre for Theoretical Physics, Trieste, Italy,

Anna Zuzana DubnickovaDepartment of Theoretical Physics, Comenius University,

Mlynska dolina, 842 15 Bratislava, Slovakia

and

Jerzy Kraskiewicz2 and Ryszard Raczka3

Scuola Internazionale Superiore di Studi Avanzati (SISSA), Trieste, Italy.

ABSTRACT

The Korner-Kuroda model of the electromagnetic structure of octet l /2 + baryons isrestored on a more topical physical basis. Electromagnetic radii of baryons under consid-eration are calculated and compared with other model predictions. By an incorporationof a two-cut approximation of correct form factor analytic properties and nonzero vector-meson widths, the Korner-Kuroda model is unitarized, providing in this manner imaginaryparts of the octet l /2+ baryon form factors to be nonzero just starting from a branchpoint corresponding to the lowest threshold.

MIRAMARE - TRIESTE

October 1994

'On leave of absence from: Institute of Physics, Slovak Academy of Sciences, Dubravskacesta, S42 15 Bratislava, Slovakia.

sOn leave of absence from: Maria Sklodowska-Curie University, ul. Radziszewskiego10, 20031 Lublin, Poland.

3On leave of absence from: Soltan Institute for Nuclear Studies, ul. Hoza 69, 00681Warsaw, Poland.

1 INTRODUCTIONThough there is approximately 40 years from the discovery of the electromagnetic (EM)structure of hadrons, a knowledge about the EM structure of octet l /2 + baryons is un-satisfactory up to now. Almost all experimental investigations are concentrated to theproton [1,2] and mainly in the space- like region [1]. Less precise information exists onneutron [1,3]. Concerning other members of the baryon 1/2+ octet, there is only one ex-perimental point on the total cross section of the AA production in e+e~ annihilation andan upper limit on the cross-section of the E°S° production at t=5.693 GeV2 [4]. There isno experimental information on the electromagnetic structure of E* and E-hyperons upto now.

Inspired by a good experimental situation, many more or less successful phenomeno-logical models [5-14] have been constructed for a global description of the nucleon EMstructure. However, missing are analogous attempts to predict the behaviour of EM formfactors (FF) of other baryons of the same octet. The main reason is that almost all con-structed models depend on some number of free parameters which, however, have to befixed in a comparison with experimental data. Possibly, exceptions are a calculation [15]of the EM FF's of the A0- and S°-hyperons in the space-like region using a relativisedconstituent quark model, which includes correction terms related to the potential actingbetween the quarks and to the center-of- mass motion during the photon baryon scat-tering. Then also a global reproduction [16] of yf-hyperon EM FF's was carried out inthe framework of the simplified unitary and analytic model and the Korner- Kuroda zerovector-meson width model [17] (and therefore real in both, the space-like and also in thetime-like region) of the EM structure of octet 1/2+ baryons with the correct asymptoticbehaviour does exists. The latter was used (see [17]) for a prediction of total cross-sectionsof electron-positron annihilation into baryon-antibaryon pair which depends on absolutevalues of EM FF's of baryons. However, for an investigation of polarization effects [18-20], where in the corresponding cross-sections real and imaginary part of a product ofelectric and magnetic FF's in ( > 4rr>g region appear, the Korner-Kuroda model is nomore applicable. The main reason is, that EM FF's in time-like region for t > t0, whereto is the lowest normal or anomalous threshold, are complex functions with a phase de-pendent on energy and the Korner-Kuroda model gives only the phase to be zero up to+oo. However, as the Korner-Kuroda model is one of the first models for EM FF's ofall members of l /2 + octet, by using an effective method proposed in the paper [14] weunitarize the Korner-Kuroda model, providing in such manner baryon FF's suitable foran investigation of polarization effects in annihilation channels.

Our paper is organized as follows. In the next section we restore the Korner- Kurodamodel on a more topical physical basis. Then, since much progress has been achieved inunderstanding the static EM properties of the octet l /2+ baryons, directly connected witha behaviour of EM FF's around t = 0, we calculate by means of the Korner-Kuroda zerovector-meson width model EM radii and compare them with predictions of other models,like quark models [15,21,22], Skyrme model [23], lattice QCD approach [24], naive VMDmodel [25] and extended VMD model constrained by QCD asymptotics [26]. The section

T I

3 is devoted to the linearization of the Korner-Kuroda model and to predictions of abehaviour of total cross-sections of electron-positron annihilation into baryon-antibaryonpair. In conclusions the main results of our paper are summarized.

2 KORNER-KURODA MODEL OF THE EM FORMFACTORS OF OCTET 1/2+ BARYONS

The EM structure of every baryon of l /2 + octet is completely described by two scalarfunctions depending on the four momentum transfer squared t = — Q3. In accordancewith [17] they are chosen in the form of the Sachs electric GE(1) and magnetic G M ( 0 FF 'S[27), which can be expressed through Dirac Fi(t) and Pauli F2(t) FF's as follows

(1)

= F,(f)

GK{t) = F,(0

where mB is the mass of a corresponding baryon. All those FF's are normalized in thefollowing way

GE(0) = QB;

(2)

where Qg and fiB are the charge and anomalous magnetic moment of the baryon, respec-tively.

Now, for a. decomposition of F\[t) and F2(i) into various vector-meson family con*tributions, we use the Korner-Kuroda tabulated values of relative p-, u>- and 4>- familycouplings to the baryon-antibaryon system [17]

Fin(t) = 4 F ' n

= F,"E+ + j F ^ t - 5F lE+

- 1 +

(3)

(4)

(5)

(6)

(7)

(8)

(-1

( -1 - ̂

which automatically respect the normalizations (2) provided

(9)

(10)

(11)

and also transformation properties of the baryon EM current under rotations in isotopicspace. Just in order to conserve the latter symmetry (a change of the sign in the p-contribution from p to n, from E+ to E~ and from 5° to E") we do not employ in thispaper experimental values of magnetic moments of baryons, but we take them to be SU(6)values related to the magnetic moment of the proton, like in [17].

It is well known that the perturbative QCD predicts [28,29] the asymptotic behaviourGE(<) and G M ( 0 to be (up to logarithmic corrections)

from where it is natural to deduce the asymptotic behavior of Fj(() and F2(t) as follows

In order to satisfy (13) we assume that all FfB and F2"g in (3)-(10) behave Sike

(13)

(14)

Then, parametrizing F"g and Fg"fl by means of the vector-meson- dominance (VMD)model, we are restricted always with two and three vector mesons, respectively, as follows

(IS)

where m; are masses of vector mesons with quantum numbers of the photon, v' and v"are the first and the second excited state of vector meson u, fwg is the coupling constantof the vector meson to baryon and f< is the so-called universal vector meson couplingconstant describing the transition between the virtual photon and the vector meson.

If we transform relations (15) in order to have common denominator, then requiringnormalizations (11) and the asymptotic behaviour (14), we come to the following twosystems of closed algebraic equations

•„.) = 1 (16)

(yi2fl//.) +

M = o

and by their solutions, to the expressions as follows

(17)

(18)

Finally, substituting relations (18) into (3)-(10) we find just the representation of theDirac and Pauli FF's of baryons to be identical with the form presented in [17].

As we have mentioned in the Introduction, much progress has been achieved in un-derstanding of the static EM properties of the octet l /2+ baryons, including the electro-magnetic radii. Therefore it is interesting to calculate the latter in the framework of theKorner-Kuroda zero vector-meson width model and to compare them with other modelpredictions.

The electric and magnetic mean-square radius of any baryon can be extracted fromthe corresponding electric and magnetic FF's with the standard small t expansion of theFourier transform of a charge or magnetic moment distribution by

at |, | , . o(19)

respectively. Applying them to the octet l /2+ baryon EM FF's one gets the electric andmagnetic mean-square radii of baryons in the following form

Vu) + (2 (-2

( - 2

{'•MA) = 2(£>iM - 0i# - D,

: 6Dip + 2(D,W - £>^) - (2 - 4^tp)D2p -

(f

{'•ME-) = - 6

(2 -2(1 - 2

(20)

(21)

(22)

(23)

(24)

(25)

- (4

Dlu -

D,« - 40,*(2 -

(4 +

where

(26)

(27)

(28)

In order to get numerical values from (20)-(27), unlike the paper [17], where the massesof excited states are determined by means of slopes of Regge trajectories, here we takethe existing experimental values of all the vector meson masses, given by the Rev. ofParticle Properties [30]. There is, however, an experimental confirmation only of one ofthe excited states of fli(1020)- meson. Therefore the mass of the second one was estimatedby means of the Gell-Mann-Okubo mass formula [31]

3ml (29)

considering an ideal <f> — u mixing i.e. taking cost) = W2/3 and sintf = y l / 3 -Here we would like to note, that just from (29) one can know the experimentally

confirmed ^(I680)-meson to be the second excited state of ^(1020)- meson and the firstone can be expected with the mass m^ ss 1.4 GeV.

The predicted values of electric and magnetic mean-square radii by the Korner-Kurodamodel are presented in Table 1 and Table 2, where they are compared also with othermodel predictions, like the lattice QCD [24], nonrelativistic [21,22] and relativistic [15]quark models, Skyrme model [23], naive VMD model combined with SU(3) symmetry[25] and the extended VMD model constrained by QCD asymptotics. They are quiteencouraging and at the same time they support reliability of the Korner- Kuroda model,at least around the point ( = 0.

3 UNITARIZATION OF KORNER-KURODA MODEL

The main aim of this paper is the unitarization of the Korner-Kuroda zero vector-mesonwidth model, restored in the previous section, in order to provide in such manner octetl /2+ baryon FF's suitable for an investigation of polarization effects in annihilation chan-nels. For the latter an effective method developed in paper [14] for nucleons will beused.

There is a general belief (in some cases proved in the framework of the axiomaticquantum field theory, but always restored from the Feynmann diagrams of a formal per-turbation expansion) that the octet l /2+ baryon EM FF's are analytic in the wholecomplex plane of t, besides the cut starting from the lowest branch point („ on the realaxis up to +oo. From the physical point of view, the unitarity condition requires theimaginary part of the octet l /2 + baryon FF's to be different from zero just above the

lowest branch point (o and, moreover, it determines a smoothly varying behaviour of theimaginary part in the ia < ( < oo region.

For ( < tv the baryon EM FF imaginary part is equal to zero as a consequence of thehermiticity of the EM current.

All these properties are not contained in the Korner-Kuroda zero vector- meson widthmodel given by (1), (3)-(10) and (18).

The unitarization of the Korner-Kuroda model is achieved by the incorporation ofthe correct FF analytic properties and non-zero values of vector meson widths into theexpressions (18). Here we have to distinguish the isovector (p-meson) vector-meson familycontributions from the isoscalar (w- and <£-meson) vector meson contributions, becausein the isoscalar case the lowest normal threshold is given by (J = 9mJ and in the isovectorcase the lowest normal threshold is at (Jj = 4mJ, where m, is the pion mass. So, practicallythe unitarization of Korner-Kuroda model is realized (up to the two-cut approximation)by the following special nonlinear transformations

(30)

4 (*£ -° \\/w - w]2

applied to (18), where (J, tg and tfn = Am2B are square-root branch points. The latter is

transparent from the inverse transformation

ti ~ D ' / 2 + t'o - D 1 " -U !

and similarly for W(t).Practically, in the incorporation of the two-cut approximation of the octet l /2+ baryon

FF analytic properties into (18), besides (30), we also use expressions for vector mesonmasses squared

(32)

and identitiesA j f B _ fV

" ~ l0 ~~ 777T> 1/ n i " — vo ~~ n / u / 7T7 H ' 33 )

[1/Vjv — vjvj' [1/VVjv — l-v^J2

following from (30), where KJQ, WM are the zero-width (that is why they have a subindex0) VMD poles and VN, WN are the normalization points (corresponding to / = 0) in theV, W planes, respectively. Relations (30), (32) and (33) first transform every VMD termm1l(m\ — t) into the corresponding new variable as follows

^t^T = Vi -w%) ' (w - vn0){W + wl)(w-1/ivj(w + i / iv,0f ' ( 3 4 )

Then, utilizing the relation between complex and complex conjugate values of the corre-sponding zero-width VMD pole positions in the V, W planes

(35)

following from the fact that all (ml ~ Fj/4) < tfn, and subsequently incorporating thenon-zero values of vector meson width F ^ 0 in a correct way, one gets for every octetl /2+ baryon EM FF, one analytic function in the whole complex t-plane besides tworight-hand cuts of the following form

(37)

ft ^ (33)

fi +i

6 c"

2 (39)

(J + J,P) fi fl?+[^

FlSf(t) = 77

5" ii*> 8where

Each of these FF's is defined on a four sheeted Riemann surface in t-variable withthe first sheet to be a physical one. Poles corresponding to vector meson resonances areplaced on unphysicat sheets.

Since the Korner-Kuroda model contains no free parameters and its unitarizationproduces also no free parameters, provided the second branch point is placed just in thebaryon-antibaryon threshold, then one can predict the behaviour of all octet l /2 + baryonEM FF's as it is shown in Figs.1-8. The corresponding total cross-sections of electron-positron annihilation into baryon- antibaryon, obtained by means of the expression

BB) =2ml

(45)

are presented in Figs.9-16, where they are also compared with existing data.In Fig.9 a better agreement with low energy data on protons is achieved with uni-

tarized Korner-Kuroda model, however, the last point from the newest measurement bythe FENICE experiment in Prascati at (=5.95 GeVJ, equally the three points from theFERMILAB E760 experiment at large values of cm. energies, are described even worsethan with zero-width Korner-Kuroda model. But unitarization improved a description ofthe data on the cross-section of the e+e~ —> nh process, as it is shown in Fig. 10, and oneexperimental point on the e+e" —* AA process {see Fig. 11) is also described quite well.There are no data on the rest of baryons of the octet ]/2+ .

4 CONCLUSIONS AND SUMMARY

It is well known [18-20], that the investigation of polarization effects in electron-positronannihilation processes into baryon-antibaryon pairs, or in reverse processes, one can notdo without complex baryon EM FF's. However, they exist up to now only for micleons[12-14], and for the A- hyperon [16], to some extent. Therefore, in this paper, we havetried to remedy this shortage of the latter, at least in baryon l /2+ octet.

Our problem was simplified by an existence of the elegant zero vector-meson widthKorner-Kuroda model [17], which, moreover, is without any free parameters.

Since it was constructed more than one decade ago, first, we have restored the zerovector-meson width Korner-Kuroda model on a more topical physical basis. By meansof the latter we have calculated as a by-product the electromagnetic mean-square radiiof octet l /2+ baryons. Their comparison with other model predictions indicates on areliability of the Korner- - Kuroda model [17], which stimulated us to unitarize it.

The unitarization of the Korner-Kuroda model was carried out by an incorporationof a two-cut approximation of the correct analytic properties of the octet l /2+ baryonEM FF's, where the first branch point is identified with the lowest and the second onewith baryon-antibaryon threshold. Finally an instability of vector-mesons was taken intoaccount by incorporating nonzero values of vector-meson widths. In this manner everyEM FF of octet l /2+ baryons is denned on four sheeted Riemann surface with complexpoles, corresponding to vector-meson resonances, placed on unphysical sheets.

The calculated total cross-sections of electron-positron annihilation into baryon- an-tibaryon processes show, that undoubtful improvement in a description of two existingFENICE points [3] on neutrons is achieved. The same can be said about the low-energydata on protons, but E760 data from FERMILAB at high values of cm. energy are

10

not described neither with the zero vector-meson width Korner-Kuroda model, nor withthe unitarized one. A good description is achieved also for one existing point on theA-hyperon.

Acknowled gments

One of the authors {S.D.) would like to thank Professor Abdus Salain, the Interna-tional Atomic Energy Agency and UNESCO for hospitality at the International Centrefor Theoretical Physics in Trieste. J.K. would like to thank Professor S. Fantoni forhospitality at SISSA.

11

held in

G.Bard

References[1] A.Bodek: Talk presented at the VI-th Rencontres de Blois, "The Heart of the Matter"

Blois, France, June 20-25, 1994.

[2] A.Antoielli, R.Baldini et al: Preprint LNF-94/032 (P), Frascati (Roma) (1994),n et al: Nucl. Phys. B411 (1994) 3.

T.A.Arratronget al: Phys. Rev. Lett. 70 (1992) 1212.D.Biseto et al: Nucl. Phys. B224 (1983) 379.

[3] A.Antonelli et al: Phys. Lett. B313 (1993) 283.

[4] D.Bisello et al: Z. Phys. C48 (1990) 23.

[5] T.Massam and A.Zichichi: Nuovo Cimento 43 (1966) 1137.

[6[ I.Bender et al: Nuovo Cimento A16 (1973) 377.

[7] S.BIatnik and N.Zovko: Acta Phys. Austriaca 39 (1974) 62.

[8] G.Hohler et al: Nucl. Phys. B114 (1976) 505.

[9] S.Mehrotra and M.Roos: Phys. Scr. 13 (1976) 265.

[10] P.Cesseli, M.Nigro and C.Voci: Proc. of a Workshop on Physics at. Lear with Low-Energy Cooled Antiprotons, Erice, 19S2, p.365.

[11] M.Gari and W.Krumpelmann: Phys. Lett. 14IB (1984) 295.

[12] S.Dubnicka: Nuovo Cimento A100 (19S8) 1, A103 (1990) 1417.

[13] S.I.Bilenkaya, S.Dubnicka, A.Z.Dubnickova and P.Stn'zenec: Nuovo Cimento A105(1992) 1421.

[14] S.Dubnicka, A.Z.Dubnickova and P.Stn'zenec: Nuovo Cimento A106 (1993) 1253.

[15] M.Warns, W.Pfeil and H.Rollnik: Phys. Lett. 25SB (1991) 431.

[16] A.Z.Dubnickova, S.Dubnicka, M.E.Biagini and A.Castro: Czech. J. Phys. 41 (1991)1177.

[17] J.G.Korner and M.Kuroda: Phys. Rev. D16 (1977) 2165.

[18] A.Z.Dubnickova, S.Dubnicka and M.P.Rekalo: Preprint JINR, E2-92-522, Dubna(1992).

[19] S.M.Bilenky, G.Giunti and V.Wataghin: Z. Phys. C59 (1993) 475.

[20] J.Kraskiewicz and R.Raczka: Proc. of the 2-nd Adriatic Research Conf., "Polariza-tion Dynamics in Nuclear and Particle Physics", Ed. A.Barut et al., World. Sci. Co.,Singapore (1992).

12

[21] P.Leal Ferreira, J.A.Halayel and N.Zagury: NLIOVO Cimento A55 (1980) 215.

[22] N.Batik, S.N.Jena and D.P.Rath: Phys. Rev. D41 (1990) 1568.

[23] J.Kunz and P.J.Mulder: Phys. Rev. D41 (1990) 1578.

[24] D.B.Leinweber, R.M.Woloshyn and T.Drapper: Phys. Rev. D43 (1991) 1659.

[25] A.Z.Dubnickova, S.Dubnicka and P.Stnzenec: Czech. J. Phys. 43 (1993) 1177.

[26] S.Dubnicka, A.Z.Dubnickova and V.Wataghin: Proc. of Hadron Structure'94 Conf.,September 19-23, 1994, Kosice, Slovakia.

[27] R.G.Sachs: Phys. Rev. 126 (1962) 2256.

[28] G.P.Lepage and S.J.Biwlsky: Phys. Rev. D22 (1980) 2157.

[29] Ch.-R.Ji, A.F.Sill and It.M.Lombard-Nelsen: Phys. Rev. D36 (1987) 165.

[30] Review of Particle Properties: Phys. Rev. D45, Part 2, (June 1992).

[31] M.Gourdin: Unitary Symmetries, North-Holland Publ. C , Amsterdam (1967).

[32] O.Dumbrajs et al: Niicl. Phys. B216 (1983) 277.

13

Table 1. The electric mean-square radii of octet l / 2 + baryons predicted by the Korner-Kuroda (KK) model and their comparison with predictions following from the latticeQCD [24], t l l e constituent quark models [21,22], the Skyrme model [23], the naiveVMD model combined with SU(3) symmetry [25], the relavitised quark model [15]and the extended VMD model constrained by QCD asymptotics [26],

B

Pn

AE+

Eu

S"

[24]

0.426< 0> 0

0.532> 0

-0.332> 0

-0.262

[21]

0.8700

0.1200.9900.120•0.7500.240-0.630

(4)[22]

0.6640

0.0910.7530.091-0.5700.190-0.475

[fma

[23]

0.775-0.3080.1070.9640.107-0.7510.221-0.261

[25]

0.508-0.1330.1790.4180.185

- 0.0490.678-0.325

[15]

-0.0810.120

0.083

[26]

0.623-0.128- 0.0290.56 30.026-0.5110.117

- 0.334

KK

0.623•0.063

0.0090.6320.091-0.4490. 044-0.396

ExP.[32] |

0.743-0.119

14

Table 2. The magnetic mean-square radii of octet l /2 + baryons predicted by the Korner-

Kuroda (KK) model and their comparison with predictions following from the lattice

QCD [24], the Skyrme model [23], the naive VMD model combined with SU(3)

symmetry [25], the relativised quark model [15] and the extended VMD model

constrained by QCD asymptotics [26].

B

Pn

A£+S°£-

= • '

[24]

0.832

-0.451

-0.076

0.778

0.187

-0.416

-0.237

-0.009

[23]

1.241

-0.842

-0.116

1.401

0.305

-0.790

- 0.6050.162

<^>[25]

1.099

-0.763

0.175

0.016

0.152

0.2S7

0.967

- 0.625

[fm2

[15]

-1.094

-0.218

0.712

[26]

1.553

-1.122

-0.421

1.400

0.432

- 0.535

-0.35)00.042

KK

1.553

-1.092

- 0.400

1.508

0.498

-0.511

- 0.895

-0.269

Exp.[32]

1.986

-1.458

15

fit

Figure captionsFig.l. A prediction of the proton electric and magnetic FF's by means of the Militarized

Korner-Kuroda (UKK) model and their comparison with a prediction of the zerovector-meson width Korner-Kuroda (KK) model.

Fig.2. A prediction of the neutron electric and magnetic FF's by means of the unitarizedKorner-Kuroda (UKK) model and their comparison with a prediction of the zerovector-meson width Korner-Kuroda (KK) model

Fig.3. A prediction of the A-hyperon electric and magnetic FF's by means of the unitarizedKorner-Kuroda (UKK) model and their comparison with a prediction of the zerovector-meson width Korner-Kuroda (KK) model.

Fig.4. A prediction of the E+-hyperon electric and magnetic FF's by means of the unita-rized Korner-Kuroda (UKK) model and their comparison with a prediction of thezero vector-meson width Korner-Kuroda (KK) model.

Fig.5. A prediction of the Ea-hyperon electric and magnetic FF's by means of tilt1 unitarizedKorner-Kuroda (UKK) model and their comparison with a prediction of the zerovector-meson width Korner-Kuroda (KK) model.

Fig.6. A prediction of the E"-hyperon electric and magnetic FF's by means of the unita-rized Korner-Kuroda (UKK) model and their comparison with a prediction of thezero vector-meson width Korner-Kuroda (KK) model.

Fig.7. A prediction of the E°-hyperon electric and magnetic FF's by means of the unitarizedKorner-Kuroda (UKK) model and their comparison with a prediction of the zerovector-meson width Korner-Kuroda (KK) model.

Fig.8. A prediction of the S~-hyperon electric and magnetic FF's by means of tlie uuita-rized Korner-Kuroda (UKK) model and their comparison with a prediction of thezero vector-meson width Korner-Kuroda (KK) model.

Fig.9. A prediction of the total cross-section of the e+e~ —> pp process by means of theunitarized Korner-Kuroda (UKK) model and its comparison with a prediction ofthe zero vector-meson width Korner-Kuroda (KK) model and existing data. Thelast three points are from the E760 experiment in FERMILAB.

Fig.10. A prediction of the total cross-section of the e+e~ —* nn process by means ofthe unitarized Korner-Kuroda (UKK) model and its comparison with a predictionof the zero vector-meson width Korner-Kuroda (KK) model and the existing twoexperirnental points from the FENICE experiment in Frascati.

Fig.l 1. A prediction of the total cross-section of the e+e~ —» AA process by means ofthe unitarized Korner-Kuroda (UKK) model and its comparison with a predictionof the zero vector-meson width Korner-Kuroda (KK) model and the existing oneexperirnental point obtained by DM2 detector in ORSAY.

16

Fig.12. A prediction of the total cross-section of the e+e~ -> E+E+ process by means ofthe unitarized Korner-Kuroda (UKK) model and its comparison with a predictionof the aero vector-meson width Korner-Kuroda (KK) model.

Fig.13. A prediction of the total cross-section of the e+e" -> £°£° process by means ofthe unitarized Korner-Kuroda (UKK) model and its comparison with a predictionof the zero vector-meson width Korner-Kuroda (KK) model and the existing upperbound obtained by DM2 detector in ORSAY.

Fig. 14. A prediction of the total cross-section of the e+e* -> E"£" process by means ofthe unitarized Korner-Kuroda (UKK) model and its comparison with a predictionof the zero vector-meson width Korner-Kuroda (KK) model. There is almost nodifference between predictions of both models.

Fig.15. A prediction of the total cross-section of the e+e" -> S°HD process by means of theunitarized Korner-Kuroda (UKK) model and its comparison with a prediction ofthe zero vector-meson width Korner-Kuroda (KK) model.

Fig.16. A prediction of the total cross-section of the e+e~ -» E~E~ process by means ofthe unitarized Korner-Kuroda (UKK) model and its comparison with a predictionof the zero vector-meson width Korner-Kuroda (KK) model. Here is also almost nodifference between predictions of both models.

i . . | . r • |<

3 ;

- t i

"EEEE

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