P H Y S I C A L R E V I E W D V O L U M E 1 5 , N U M B E R 7 1 A P R I L 1 9 7 7

Couplings of vector mesons to tensor mesons and the Pomeron*

Kashyap V. Vasavada Department of Physics and Lawrence Berkeley Laboratory, University of California, Berkeley, California 94720

and Department of Physics, Indiana -Purdur University, Indianapolis, Indiana 46205f (Received 23 September 1976)

We saturate the matrix elements of the stress tensor between vector-meson states with f, f', and the Pomeron, assuming that the coupling constants go like l /m, (m, = vector-meson mass) within the symmetry- group (SU,, SU,, etc.) multiplet apart from the Clebsch-Gordan coefficients. It is found that (i) nonet (mass)2 formulas are obtained, (ii) vector-meson-nucleon total cross sections are determined, with a I /mV dependence, and (iii) tensor-vector-vector couplings are obtained. All of these are in agreement with the experiments.

The idea of tensor-meson dominance of the energy-momentum tensor O,, h a s been explored for a number of years.' Recently, this concept h a s become even m o r e at t ract ive because of i t s possible connection with a gravitation-type theory of strongly interacting part ic les . In previous paperszu3 we have attempted t o build in the P o m e r - on (P) contribution along with that of f and f ' mesons in Q,,. In part icular , in Ref. 3 (I) matr ix elements of O,, between the octet baryon s ta tes were saturated with f , f ', and P assuming that the coupling constants go like l /m, (m, =baryon m a s s ) within the SU, multiplet, apar t f r o m the Clebsch-Gordan coefficients. We obtained the Gell-Mann-Okubo (GMO) m a s s formula, D / F ra t io fo r tensor-meson-baryon-baryon couplings, values of P- and f -nucleon-nucleon couplings, a l l in good agreement with the experiments. In this work we apply a s i m i l a r procedure t o the mat r ix elements of O,, between vector-meson s tates and find very interesting consequences fo r the tensor couplings and total c r o s s sect ions of the old (p ,K* , w , +) and the new ($, $', D*, etc.) vector mesons.

11. TENSOR-MESON AND POMERON DOMmlANCE

The mat r ix elements of Q,, between two identical vector-meson s ta tes contain s ix f o r m f a c t o ~ - s . ~ In the presen t work we will be concerned with only one of these:

where P = P, + P,, q = P1 - p,. el and E~ a r e the polarization vec tors of the vector mesons. The condition

gives Gl(0) = -1. Similarly the tensor-meson matr ix element be-

tween the vector-meson s ta tes is given by four coupling constant^.^ Again we will need only one of these:

(3) c P u and m T a r e respectively the polarization tensor and the m a s s of the tensor meson. m, is the m a s s of the vector meson and iii is s o m e average mass . The factor m2/mvZ is explicitly extracted f rom the coupling constants. gTvv will be assumed to obey SU, symmetry relations.= The tensor-meson cou- plings to O,, a r e defined by

(TI O,,,IO) =mT3gT~,,v. (4)

As in I and Ref. 2, we saturate the f o r m factor G,(O) with f , f ' (or singlet f, and octet f,), and P. Now the nature of the Pomeron has remained quite myster ious over the years . F o r our purpose we t rea t i t a s a factorizable SU,-singlet Regge pole with a linearly r is ing t rajectory (a, = 1 + akt), and the contribution t o G, is evaluatedby introducing a spin-2 part ic le with m a s s M , = 1 / a . The sing- le t nature and factorization a r e probably c o r r e c t within about 20%. T o obtain c r o s s sections, the spin-2 part ic le pole will be Reggeized, s o the actual existence of suchapar t ic le may not be crucial . The smal l r i s e in total c r o s s sect ions found a t labo- ra to ry energ ies g r e a t e r than about 200 GeV could be built in if necessary by taking cu,(O) to be slightly l a r g e r than 1. Such a procedure h a s been followed by s e v e r a l authors recently in their fits, and the fundamental problem remains common to many such approaches. At any r a t e we r e g a r d the r e - s u l t s obtained here and in I a s a posteriori justifi-

C O U P L I N G S O F V E C T O R M E S O N S T O T E N S O R M E S O N S ...

cation for such an effective parametrization of the Pomeron contribution.

Then, assuming a D-type SU,-symmetric cou- pling for the vector-vector-tensor-octet vertex, saturation of Gl (0) leads to the following relations:

Here glpP (g,) and g8,,(g8) are respectively the singlet and octet tensor-meson couplings to the p mesonT (8,"). w, and we are the singlet and the octet components of the w and 4 mesons. Note that Eqs. (5), (6), and (7) automatically satisfy the GMO mass formula

4rnK2 = 3mw82 +m p 2 . (10)

Of course, it should be stated that the (mass)2 in Eq. (10) ar ises essentially from the definitions of the coupling constants in Eq. (3).

Similar equations can be written for the diagonal representation with physical masses m$, mW2:

mw2/m2 = g p g p , w + g, g,,w + g,g,,w . (12)

The simplest assumption would be that the same matrix

which diagonalizes m2 in (7), (8), and (9) also dia- gonalizes all the couplings of P, l, and 8 sepa- rately. A more complicated possibility would be that the mixing angles are different and diagonal- ization i s achieved by conspiracy between different terms. We consider the first alternative. Then it readily follows that

We have used the fact that gpWlu8 =gl ,, =g8 w l = O . Then

mW12 +mu: =m$ +mw2 (16)

and

a s i t should be. We need one more input to solve the equations

for the couplings. In accordance with the Okubo- Zweig-Iizuka (OZI) rule8 we assume that gffPP =O. The OZI rule forbids couplings between particles which a re supposed to contain only strange (or charmed) quarks with the ones without these quarks. It seems to be approximately correct for couplings such as qpn, f 'nn, f'NN, etc. In the quark model nonvanishing values of these couplings (violation of the OZI rule) may just mean some small admixture of the other kind of quarks. Later we will consider deviation from the OZI rule. At the present time, we obtain

From Eqs. (5), (6), and (18) one finds

Now, a s inI, the principle of universality of scalar and tensor couplings to the s t r e s s tensors fixes gf, gf I, g, and hence gl and g, for a given mixing angle 8,. We take the values of these constants from I, so these a r e not new parameters here. gfp, i s then determined from (19) and g,,,, gp,,, g,,, from (51, (131, and (18).

Other couplings such a s gfww, gf,@@, etc. are now readily given by (14) and (15). Since gf tpp=O and g 8 W 8 W 8 = - g E P P we get

When 8, = 8, =35.3" (ideal), (20) and (21) reduce to the well-known relations gf ww =gf pp and gftem = -figf ,, . Now we consider some numerical val- ues of the coupling constants in this approach. Taking gf =0.076, gft =-0.064, and gp = 0 . 0 9 7 / a (a; in GeV-2) from paper I, we find, for ideal f -f' mixing 9, =35.3",

These are independent of the mixing angle 8,. As in I, however, the results a re sensitive to the value of 8,. This happens because of the large cancellation between gf and gft terms in the equa- tion for g8. As we have used the OZI rule, i t i s perhaps more appropriate (although not necessary)

1986 K A S H Y A P V . V A S A V A D A 15

t o use the ideal (canonica1)value for BT. But for the sake of completeness, we quote the values fo r 8, = 31" (which is in the range predicted by the m a s s formulas of the tensor mesons). We have

The estimation of deviation f r o m the OZI ru le will be very model-dependent. F o r the + meson the OZI suppression factor g,,,/g,,, is believed to b e typically about &. If, in the c a s e of f ', the re is a s imi la r factor , then gftPP/gf PP= &. Solv- ing the equations with th i s input, we find

In this way a l l the coupling constants can be de- termined in t e r m s of m a s s e s and a;. Next we d i scuss applications of these resu l t s .

111. DECAYS OF TENSOR MESONS

F r o m the above values of gf ,, and gfl ,,, a l l the other gTvv couplings a l so can be readi ly determined by using SU, symmetry. These couplings become relevant in the models of decays of tensor mesons. Since th i s involves a detailed program by itself, we hope to p resen t such calculations separately in the future. Here we just make a few r e m a r k s based on comparison with already existing works. In Ref. 4, Renner studied tensor dominance of vector mesons using f and f' intermediate s t a t e s and determined gf,,. Then using vector dominance he related the other th ree couplings to gf ,,. In our notation he found gf ,, = 9.44, which is s m a l l e r by a factor of about 2.5 to 4 when compared with our values [ E ~ s . (22), (23), and (24)]. He calculated radiative decay widths of f mesons, fo r which, unfortunately, a s yet no experimental data a r e available. But some t ime in the future the s i tua- tion could change. In a recent work Levy, Singer, and Toafflo consider dual amplitudes fo r pseudo- scalar-vector scat ter ing and obtain g,,, couplings. In spi te of the fact that the i r model i s ent i rely dif- fe ren t f rom that of Renner , the values of the domi- nant couplings obtained w e r e very close in the two models. I t appears that the different values obtained for gf ,, in the p resen t model s t e m s f r o m the breaking of duality by the introduction of the Pomeron. With the i r values of gf, (comparable t o Renner's) the authors of Ref. 1 0 calculate the decay width of f - pOrtn- to b e 0.49 MeV. The experimental decay width for f - n'n-n'n- is given t o be 6.2 i 1 . 5 MeV. Even granting that the l a t t e r width includes m o r e than just p n r events, the re is a discrepancy by a factor of about 6 to 10. I t is v e r y interesting that the l a r g e r value of gfpp r e - quired to remove this discrepancy is indeed pro-

vided by the presen t model. Similarly, Gheory fo r A,- wn'n- comes out to be smal le r than re,,, by a factor of about 4 in Ref. 10. Also, in another recen t work Novikov and Eidelman" es t imate gf pp by assuming that the ent i re f - 2 8 ' 2 ~ - decay pro- ceeds through pop0 intermediate s ta tes , and they obtain a value which i s about a factor of 2 l a r g e r than that of Renner. The motivation in Ref. 11 was t o es t imate the charge asymmetry of n mesons in the e'e- - r'r- reaction, fo r which gf,, is clear ly relevant. Thus the coupling constants g,,, ob- tained here , which a r e l a r g e r than the ones de te r - mined by dual o r f - f J tensor-dominance models, s e e m t o be required by the experimental data. Now we consider the vector-meson-nucleon total c r o s s sections.

IV. VECTOR-MESON-NUCLEON CROSS SECTIONS

The coupling constant gpvv is clear ly relevant t o the calculation of vector-meson-nucleon total c r o s s sections for which data a r e becoming avail- able. We evaluate the c r o s s sect ions in a very s imple model in which the spin-2 contribution is Reggeized by assuming s t ruc ture less residue functions. We consider only the asymptotic Pom- e r o n contribution, although the presen t model a l so makes definite predictions fo r the secondary contributions f rom f and f '.

Let A(s , t ) be the spin-averaged amplitude for V - N scat ter ing [?(PI, e l ) +N,(ql)- V,(P,, E , ) +N2(q2)] a s s - m. Only the f i r s t t e r m in the vertex function (3) contr ibutes a s t - 0 (0, =p2, el =c2). The PNN vertex is taken f rom (I) [Eq. (3)] ,

where Q =q, +q2. Then

Reggeizing

Now the optical theorem

gives

where we have used Mp2 =l/ak.

15 - C O U P L I N G S O F V E C T O R M E S O N S T O T E N S O R M E S O N S . . . 1987

Similarly, the nucleon-nucleon total c ross sec- tion i s given by

The g's have been combined with a since their product i s determined by our tensor-dominance relations. As usual s , = l GeVZ. All the coupling constants have been already determined, s o a; remains the only free parameter. This can be fixed from, say, a;,;. Then Eq. (28) gives absolute predictions for a:;.

So far we have considered only the well esta- blished SU, nonet of vector mesons @,K*, w, 4). Recently, however, some new vector mesons such a s $, D*, q ' , etc. have been discovered. All of these could be part of some SU, multiplets such as 1, 15, etc. One can extend the present formalism to such cases. However, in view of the current un- certainty in SU, mass formulas we limit our dis- cussion to some general remarks. By the OZI rule presumably f and f' will not contribute to the mass equation for, say, the $ meson, but the corresponding charmed tensor mesons will. Their contributions could be similarly evaluated. Now from (13) we see that the values of g,,,, g,,,, and g,, a r e the same in spite of mass breaking and I$-w mixing. So it i s plausible that gp*+ (or g,,*,t) will also have comparable value. The main difference in the effective coupling will then ar ise from the factor l/mv2. An immediate inter- esting consequence from Eq. (28) i s that a,, will go like l/mV2. This fact seems to be consistent with the experiments. Vector-meson-nucleon total c ross sections have been determined by studying the A dependence of the diffractive photoproduction of vector mesons on different nuclear targets. There a re a number of uncertainties, but the typi- cal current values are12

a,,=a,,-25-30 mb, a6,=13 mb ,

aJ , ,=3.5i0 .8 m b . (30)

Since m,2/m,z =0.97, rn pz/m$ =0.57, and mp2/m$2 =0.062, the agreement i s very good, considering the simplicity of the model. In particular, the present model does explain the smaller value of a, and a drastically smaller value for a+, a s compared to a,,. Similar arguments would lead to a,+,/u,, ~ 0 . 1 5 and a,,,/apN=0.044, assuming that P couplings without the mass factors a r e the same. It will be interesting to test these predic- tions when data become available.

While this work was being completed, we be- came aware of a recent work by Carlson and Freund13 in which such a l/rn ,' dependence i s ob- tained from the assumptions of a tensor-meson-

dominated Pomeron,14 exchange degeneracy, and equality of slopes of the vector-meson trajectories (a6 =a;= a;). However, in our model such a de- pendence comes from an entirely different assump- tion of quadratic mass formulas for vector me- sons. Also in the tensor-meson-dominated Pomer- on model, the following equality i s obtained:

Comparing the values of f and P couplings found in this paper with the values from I, we find that the equality i s satisfied very well by the couplings for any value of 8,. In our model, however, the couplings a r e determined by the masses. Looking at the equations, it i s seen that the result comes out because the experimental masses satisfy the relation m2/mp2 =%,/rn, to an extremely good accuracy [mz and m, are, respectively, the aver- age (mass)2 of the vector octet and the average mass of the baryon octet and not just the scale factors introduced in Eqs. (3) and (25)]. Thus, although the results a re similar, the two models obtain them in an entirely different manner. In the present model there i s no constraint on the Pomer- on couplings except the satisfaction of the mass relations.

Now, relations between cross sections a re given by models such a s the quark model. But the pres- ent approach also gives the absolute magnitude of the cross sections. To give some numerical exam- ples, let 8, =35.3" (gffP =gftNN =O). Then Eq. (19) of I gives gpNN = 21.6/&. Taking gppp from Eq. (22) of the present work we have

up, =438.3ak mb (a; in GeV-'),

a,, =428.5a; mb. (32)

Taking a,, to be 39 mb we have a; =0.09 GeV -'. Hence,

a,, = 39.9 mb,

a,, = 38.7 mb,

a$, = 22.7 mb,

On the other hand, if we have gf,PP/gfPP= &, while still taking gflNN=O and 8, =35.3", Eq. (24) gives

a,, =29.0 mb,

a,, =28.1 mb,

a,, = 16.5 mb,

a,, =1.8 mb.

Also, because the ratios in (31) remain equal a s

1988 K A S H Y A P V V A S A V A D A 15 -

we change OT, o /a,, h a s the s a m e value for 0, =35.s0 and 31'; the only difference is that a,, =39 mb would need ak=0.48 GeV-2 in the second case.

In the c a s e of OZI violation, even if gftPP/gf PP

=gflNN/gfNN, o p N will be s m a l l e r than a,, if the value of a; f o r the vector c a s e is s m a l l e r than that in the nucleon case. Although this i s not p a r - t icular ly appealing f o r the interpretat ion of a Porneron t rajectory, it should be noted that, in data fitting,'' the a; obtained f r o m photoproduction of vector mesons is found to be s m a l l e r than the one obtained f r o m the nucleon-nucleon scat ter ing. In addition, some deviations f r o m the universal values of gf, gff, g,, and F, (see I) will a l so resu l t in changes in absolute magnitudes of the c r o s s s e c - tions.

It is a l so interesting to note that the l inear m a s s formulas do not work. If we use the factor (%/m,) in Eq. (3) instead of (%/mv)2, we find that

which is i n c lea r contradiction with the data. In addition, the ra t io apN/aNN comes out to be 0.3, which is a l so completely off. Thus we can r e g a r d the presen t resu l t s a s supporting the c a s e fo r quadrat ic m a s s formulas. Because of the w - @ mixing problem, f r o m m a s s e s alone one cannot distinguish between the l inear and the quadratic fo rms .

In the above we have considered a,, and a,,. However, recent ly, some data on C-IV and Z - N scat ter ing have a l s o become a ~ a i l a b l e . ' ~ '16 The model in I gives f o r the asymptotic ( ~ o m e r o n ) con- tributions

The current ly available data15'16 indicate that

=0.85-0.87 and 5 -0.7. "PP OPP

(3 7)

Thus the data a r e quite consistent with our p re - diction that a,, < a,, < a,,, and even the l /m, dependence a l so could be approximately t rue .

V. CONCLUDING REMARKS

I t might be argued that, because of the relat ive crudeness with which the Pomeron contribution h a s been handled, the s t r iking numerical resu l t s may not have part icular significance. Even if this is t rue , s o m e genera l resu l t s should remain valid. In part icular , the l / m v 2 behavior of a,, and the fact that o,, and uvN come out to have reasonable magnitude with the usually accepted s m a l l values

of a; (<0.5 GeV'2) should be regarded a s suc- c e s s e s of this approach. Another significant p r e - diction l i es in the l a r g e r value of gTvv obtained in the p resen t work, a s compared to the dual o r non-Pomeron tensor-dominance models. As we have mentioned in the text, the data on tensor- meson decays s e e m t o requ i re such l a r g e r values. Also, a s mentioned in Sec. IV, a , ,a l /m, (baryon m a s s ) may be a l so approximately t rue . The r e - su l t s of this paper taken together with paper I lead us to conclude that the dominant mat r ix e le - ments of e,, between vector mesons and baryons can be r a t h e r well saturated with the known tensor mesons and the Pomeron without requiring any subtraction o r s o m e hypothetical mesons. Fur - thermore , this procedure does provide insight into the dynamics of symmetry breaking in m a s s e s and leads t o a number of experimentally verifiable predictions. As we have already remarked in I , the saturat ion procedure does not work in the c a s e of pseudoscalar mesons because of the large amount of symmetry breaking in masses . The m a s s relat ions probably need subtraction con- s tants o r e x t r a tensor mesons. In Ref. 2, the m a s s e s were absorbed in the definition of the cou- pling constants and the satisfaction of m a s s fo r - mulas was not required. Then, using nonuniversal values fo r gf, g,, etc . consistent solutions were obtained. In Ref. 3 (I) and h e r e satisfaction of the m a s s fo rmulas has been the s tar t ing point. I t should be noted that s imi la r difficulty fo r the pseudoscalar c a s e appears in the study of sca la r dominance.17

Finally, as f o r the ro le of the Pomeron in the p resen t saturat ion scheme, one can s imply r e - g a r d i t a s the contribution which remains af ter the usual tensor-meson-pole contributions a r e taken out f r o m the particle-antiparticle c r o s s - channel amplitude. It would then include continu- um also. Of course, factorization would be diffi- cult t o understand in such a case. However, pos- sibly an effective pole could somehow take c a r e of such contributions. In such a scheme, then, m a s s e s of the part ic les could be consistently ex- p r e s s e d a s dynamical quantities related t o the tensor-meson and Pomeron-exchange forces.

ACKNOWLEDGMENTS

The author wishes to thank Professor S. Mandel- s t a m and Professor J. D. Jackson f o r their kind hospitality a t the Physics department and the Lawrence Berkeley Laboratory of University of California a t Berkeley, where this work was done. He is a l so thankful to Professor M. Suzuki f o r helpful discussions and to the NSF f o r support.

15 - C O U P L I N G S O F V E C T O R M E S O N S T O T E N S O R M E S O N S ...

APPENDIX

H e r e w e give the comple te m a t r i x e l e m e n t s f o r O,, [Eq. (I)] and T [Eq. (3)]. A f a c t o r of 2 is int roduced f o r convenience. For e,, w e have

~ ( V , ( P ~ ) I e , , I v , ( ~ ~ ) ) = c 1 ( q 2 ) ~ , . E , P , P ~ + G 2 ( q 2 ) ( ~ 1 .P) (% .P )P ,PV

+ G , ( ~ ~ ) [ ( E , . P ) E ~ , , P , +(e l ~ P ) E , . P ~ + ( E ~ - P ) E ~ , , P ~ + ( E ~ . P ) E ~ ~ P , , I

+ G ~ ( ~ ~ ) [ ( E ~ . ~ ) E ~ , Q ~ + ( ~ 1 ' q ) ~ 2 ~ 9 ~ + ( e 2 ' q ) ~ l p q v + ( € I ' q ) ~ i v q p

- 2 ( ~ ~ . q ) ( e z ' q ) g ~ ~ - q 2 ( ~ l p ~ 2 v + E Z ~ E ~ U ) I + G , ( q 2 ) ( ~ , . c2)(q,,qu -42&U)+Gg(92)(e1 ' P ) ( ~ ~ ' p ) ( q p q v -qzgpv) -

T h e matrix e l e m e n t f o r T is given by

( V ( P , ) ~ T I V ( P ~ ) ) = ~ E " A , , ,

w h e r e

In the p r e s e n t p a p e r only gg;, is needed and the s u p e r s c r i p t is dropped.

*This work has been supported in par t by the National Science Foundation.

t Permanent address. On sabbatical leave during fall semester 1976.

IP. G . 0. Freund, Phys. Lett. 2, 136 (1962); R. Del- bourgo, A. Salam, and J. Strathdee, Xuovo Cimento 49A, 593 (1967); H. Pagels, Phys. Rev. 144, 1250 - (1966); Univ. of North Carolina technical report (un- published); B. Renner, Phys. Lett. e, 599 (1970); K . Raman, Phys. Rev. D 3, 2900 (1971).

'L. R. Ram IvIohan and K. V. Vasavada, Phys. Rev. D 2, 2627 (1974).

3 ~ . V. Vasavada, Phys. Rev. D 12, 2918 (1975). This paper i s referred to a s I in the text.

4 ~ e e , for example, B. Renner, Nucl. Phys. g, 634 (1971). The complete matrix element i s given in the Appendix.

he complete matrix element for this case i s also given in the Appendix.

6 ~ s was emphasized in I , the question a s to which coupling constants obey SU3 symmetry is entirely a dynamical one and cannot be settled a p r w r i , without appealing to the experiments. The definition (3) will give r i se to GMO mass formulas for (mass)' of the vector mesons. Previous authors such a s Renner in Ref. 4 did not define their couplings in this manner and accordingly did not get the mass formulas. If we pull out ( ~ / m , ) , l inear mass formulas will be obtained. This fact has nothing to do with the kind of normalization one uses for the states. As for the "folklore" that SUB i s broken only by masses and the coupling constants a r e SU3-invariant, we have to say

that it has been verified in very few cases and then only very approximately. The l i terature i s full of cases where such mass factors have to be provided by hand. In addition, in many cases such a s the present one the Lagrangian one writes down in a ''natural" way has dimensional coupling constants, which need mass fac- tors to make them dimensionless. Thus there i s an unavoidable ambiguity which can only be settled dy- namically. In our case after extracting my2 we have to introduce F', which we take to be 0.728 G ~ V ~ (average massZ of the vector octet). However, it can be chosen as completely arbitrary since only the pro- ducts of g's with Gi2 will appear in any application.

'we use the convention f =f icosOT+ f,sinOT, f'=-f , s W T + f8cosOT, w =w,cosBy +@,sin$, I$ =-wtsinf+, +w,cos$.

's. Okubo, Phys. Lett. 5, 165 (1963); G. Zweig, CERN Report No. 8419/TH 412, 1964 (unpublished); J . Iizuka, Suppl. Prog. Theor. Phys. 37-38, 21 (1966).

'P. Nath, R. Arnowitt, and M . H. Friedman, Phys. Rev. D 2, 1572 (1972); Phys. Lett. s, 361 (1972).

'ON. Levy, P . Singer, and S. Toaff, Phys. Rev. D 13, 2662 (1976).

"v. N. Novikov and S. I. Eidelman, Yad. Fiz. 21, 1029 (1975) [Sov. J. Nucl. Phys. 21, 529 (1976)l.

"A. Silverman, in Proceedings of the 1975 International Symposium on Lepton and Photon Interactions at High Energies, Stanford, California, edited by W. T . Kirk (SLAC, Stanford, 1976); R. L. Andersonet d., Phys. Rev. Lett. 2, 263 (1977).

I3c. E. Carlson and P . G. 0. Freund, Phys. Rev. D 11, 2453 (1975); C. E. Carlson and P. G. 0. Freund, Phys. Lett. s, 349 (1972). A number of other authors have

1990 K A S H Y A P V . V A S A V A D A

also used s imi lar models to explain suppression of udN and u+N relative to upN. Some of these are: Chan Hong-&To, J . Kwiecinski, and R. G . Roberts, Phys. Lett. E, 367 (1976); C. Rosenzweig and G. F. Chew, ibid. E, 93 (1975); T. Inami, ibid. E, 291 (1975); N . Papadopoulos, C . Schmid, C . Sorensen, and D. M. Webber, Nucl. Phys. w, 189 (1975). It i s also interesting to note that an entirely different model, the generalized vector-dominance model, also gives l/mv2 dependence of ufi (transverse part) when scaling behavior i s imposed on the e'e- annihilation c ros s section and the transverse virtual-photon total c ros s section. See, for example, S. Chavin and J. D. Sullivan, Phys. Rev. D 13, 2990 (1976), which gives ear l ie r references.

1 4 ~ . Carli tz, hZ. B. Green, and A . Zee, Phys. Rev. D 4, 3439 (1971).

1 5 ~ . Badier e t a1 ., Phys. Lett. s, 387 (1972). 1 6 ~ . Majka e t a1 ., Phys. Rev. Lett. 37, 413 (1976). Al-

though the data sample for E - i s too smal l , a rough value for u&lp can be determined by using the value of bs-p and the optical-model relation

given in this reference. This relation has been shown to be consistent with the data on ZT scattering.

"see, for example, hI. Gell-Mann, in Proceedings of the Third Hawaii Topical Conference on Particle Phys ics , edited by S. F. Tuan (Western Periodicals, North IIollywood, Calif., 1970); P . Carruthers , Phys. Rev. D 2, 2265 (1970); 3, 959 (1971).