P H Y S I C A L R E V I E W D V O L U h I E 2 0 , N U M B E R 9 1 N O V E M B E R 1 9 7 9

Constituent-i~iterchange model for inclusive vector-meson production

Kashyap V. Vasavada Department of Physics, Indiana-Purdue University, Indianapolis, Indiana 46205

(Received 14 July 1978)

The constituent-interchange model is applied to inclusive vector-meson productiori in proton-proton collisions. In particular, production of $, 3', and p mesons is considered and good agreement with all the distributions is obtained.

I. INTRODUCTION complete parton-model calculations have not ap- peared in the literature.16 In Sec. I1 we describe

Models based on quark-partons have been exten- the model. Numerical resul ts for distributions a r e sively considered recently for high-energy reac- given in Sec. III and coupling constants a r e dis- tions. In part icular , the constituent-interchange cussed in Sec. IV. Section V contains concluding model (CIM), in which a quark-parton from one remarks . par t ic le in te rac t s with a meson o r baryon consti- tuent of the other part ic le , has been very succes- 11. DETAILS OF THE MODEL sful in dealing with large-transverse-momentum (Q,) processes. ' In a recent p a p e 3 Blankenbecler, Duong-Van, and the author have applied this model to massive-lepton-pair production for the en t i re Q , range. I t was found that the CIM gives a cor - rec t gauge-invariant extension of the Drell-Yan model3 and automatically gives r i s e to large-Q, dileptons without assuming la rge t ransverse mo- mentum of the initial partons (KT). That work dealt mainly with the continuum and, although very good agreement was obtained for an extensive range of Q, and Q' (mass2 of dimuonl, the overal l normalization had to be readjusted for low Q2 and especially in the resonance region ($1. In another paper4 the author considered angular distributions of muons in various theoret ical models including CIM. The present work i s a natural extension of these ideas to vector-meson production which con- tains a large part of the dilepton production c r o s s section. Most of the t ime we will consider dimuon resonant production data. But it i s c l e a r that the cross-sect ion equations a r e readily applicable to the production of o ther decay products of the s a m e resonances when the corresponding branching r a - t ios a r e used. F o r the present we will consider production of $(3. I) , 4'(3.7), and p(0.770) mesons in proton-proton collisions. Production of other vector mesons and react ions involving different in- cident beams will be considered later . I t tu rns out that the model fits the extensive s e t of data ra ther well fo r the ent i re range of k i n e m a t ~ c variables in each case ,

Since the discovery of 4 production in hadronic collisions, a number of authors have considered

In the CIM picture [Fig. l ( a ) ] the hadrons A and B emi t partons8 (quark a and meson b and vice v e r s a ) which then undergo a hard scat ter ing to pro- duce the vector meson Q and the quark d. Fig-

various models. They include generalized Drel l - FIG. 1. (a) CIM diagram for dilepton production in the

Yan models o r quark-fusion models with charmed resonance region. (b) s-channel pole for the subprocess: quarks5 and normal quarkse and a l so gluon-fusion q u a r k + meson-quark+ vector meson (dilepton). (c) u- model^.^ F o r the p production, to our knowledge, channel pole for the s a m e subprocess.

2 0 - 2 3 0 4 @ 1979 The American Physical Society

20 - C O N S T I T U E N T - I N T E R C H A N G E M O D E L F O R I N C L U S I V E . . . 2305

u r e s l ( b ) and l ( c ) show the two leading s- and u- g ram involves quark-antiquark annihilation and is channel q- (q-) exchange d iagrams which would clear ly related t o the Drell-Yan model fo r the pro- make the amplitude gauge invariant when Q is a cess . The fully differential c r o s s section is given photon and b is a no meson. T h e u-channel dia- by

d 4 0 % ( ( A B - n - x ) = j1 dx, I1 dx,GalA(xl)G,/,(%) 7 (ah - I*i-d) . d & a , b , d 0 0 cl &

H e r e G's a r e the usual probability functions fo r finding a and b in the hadrons with four momenta -xlPA and GP,, respectively. T r a n s v e r s e momenta of a and b a r e regarded a s smal l and the dependence is sup- p ressed in the equation. A s in Ref. 2 an average value ( ( K , ) ) can be added l a t e r if desired. T h e different- ia l c r o s s section f o r the subprocess is readily obtained by summing over the Feynman d iagrams [Figs. l ( b ) and ( c ) i and is given by

d 4 0 R - (ah - I+l'd)= 2 2 9 d 4 Q ( @ - ? T ? ~ ~ ) ~ + l l ~ y r

where

and

h 2 ( a , b , c ) = a 2 + b 2 + 6 ? - 2 ( a b + b c + c a ) .

s ' , t', u' a r e the usual Mandelstam var iab les for the subprocess . h i s the meson-quark-antiquark cou- pling constant used in Ref. 2. I t s value is obtained by a CIM fit to the large-p, production of n me- sons. The l a t t e r fit gives the 90" meson-quark scat ter ing differential c r o s s section by

with c=1200 GeV4. h2 is then given by m. The uncertainties in this number a r e believed t o be of the o r d e r of a factor of 2 o r so. gv,+,-2 is obtained from the par t i a l width of v t o 19- channel:

m is the effective m a s s of the quark and is taken to be 1 GeV a s in Ref. 2 (except in the c a s e of p pro- duction a s will be discussed in the following). Hence the only unknown in the presen t work is the coupling of V to the quarks (gvQ;).

F o r comparison with experiments we do the Q2 integration over the Breit-Wigner shape. T h e total c r o s s section (into 1'1-) is obtained by d4Q inte- gration, which gives,g fo r -4 e B,

I

where

and

with

In the above, Po,$ and k,,k a r e , respectively, the energy and three-momentum of the initial meson and final quark in the c.m. sys tem of the two-body subprocess (a + b - Q + d). M I s the m a s s of the me- son (note that the notation has been slightly changed from Ref. 2).

The factor of 2 in Eq. (8) a r i s e s from the two d iagrams with a = meson and b =quark and vice versa . F o r A fB, one has to exchange a and b ex- plicitly. The summation sign o v e r a , b, d is dropped from now on. Different quarks will be taken into

2306 K A S H Y A P V V A S A V A D A

account by explicit multiplication of the G function by a numerical factor , and the different mesons will be included by using one effective G function for the en t i re c l a s s of mesons. d fo rms par t of "anything" with probability 1 in the inclusive c r o s s section.

We have the usual meaning for the other kinema- tic variables. w i s the overal l proton-proton c.m. energy ( s=w2) , y and X a r e , respectively, the c .m. rapidity and Feynman scaling variable Q,/Q,,,, . The distributions do/dy, c l o / d ~ , ~ ~ d ~ u / d ~ ~ , do/ dQT2 can be obtained by doing one of the x integra- tions with the help of the 6 function in (3) and then integrating over the other independent variablesr B will denote the branching rat io into p+p- ( p - e universality i s assumed).

T h e probability functions a r e taken to be a s fol- lowslO: F o r an up quark in the proton

The down quarks in the proton could have a differ- ent distribution, but the difference i s not crucial here . They can be simply included by considering the fact that there a r e two up quarks and one down quark in the proton. F o r a meson in the proton we have

F o r charmed quarks o r antiquarks we take the sea- type distribution given by the dimensional-counting ru les

GMlP(x) was determined in Ref. 2 by requiring i t to satisfy the sum rule

F o r I$ production we consider a ( o r b ) to be ei ther a normal (up or down) quark o r a charmed quark. F o r p production they will be taken to be normal quarks. In the next section we discuss various dis- tributions.

111. DISTRIBUTIONS

A s we have already mentioned, a l l the param- e t e r s , with the exception of g,,, have been al- ready determined in the previous work.' T h i s i s fixed by requir ing agreement with one data point. All the other distributions a r e then predictions. Numerical values of these coupling constants will be discussed in the next section.

A. IL(3.1) production

We have studied this p rocess in detail over a wide range of kinematic variables , Data a r e taken from Refs, 1 1 and 12. Some of the resu l t s a r e shown in Figs. 2-6, The solid curves represen t our calculation with normal quarks [ ~ ( x ) given by Eq. (12)]. The dotted curves a r e obtained when a o r b (and hence d) is a charmed quark o r anti- quark with ~ ( x ) given by Eq. (14), and the coupling constant is suitably adjusted. The dashed curves a r e obtained for normal quarks with a (1 - x ) ~ dis- tribution (discussed below).

F igure 2 shows dependence of o on w . T h e agree- ment i s good for the normal quark model except for the very highest CERN ISR point. The thres- hold behavior is well reproduced. In addition, we found that the slope can be increased for high w by using a very la rge quark m a s s m. But, fo r the f igures shown, m has been taken to be 1 GeV a s in Ref. 2,

F igure 3 shows ~ d u / d y ( ,=, vs w . Here again the agreement i s good.

Figure 4 shows ~ d u / d ~ vs X for w =20.8 GeV, Here the solid curve (a) i s calculated with previous functions and parameters . At higher values of X

W (GeV) FIG. 2. Predicted total cross sections o($) a t various

c.m. energies with data taken f rom Ref. 11. The solid curve is for the normal quark model with x distribution given by Eq. (12), and the dashed curve is for the same model with (1 -d4 distribution explained in the text. The dotted curve i s for the charmed-quark model with sea distributions.

C O N S T I T U E N T - I N T E R C H A N G E M O D E L F O R I N C L U S I V E . . .

Cobb et al .

J . .ss.. et ., . 0 Snyder st al.

/ :i 0 Anderson et ol.

o Antipov et 01.

v Aubert et 0 1 .

FIG. 3. Predicted differential cross sections at y = 0 for various c.m. energies. Data a r e taken from Ref. 12. The solid curve i s for the normal quark model. The dotted curve is for the charmedpuark model with sea distributions.

FIG. 4. Predicted Feynman-X distributions. Curve (a) is obtained with standard functions and parameters. Curve Cb) i s obtained with GulO(x)= (1 -x)4 as x- 1. Curve (c) represents charmed-quark model with sea dis- tributions. Curve (d) i s obtained from (c) by readjusting the normalization at X = 0 independently of the other dis- tributions. Data are taken from Anderson et al., Ref. 12 (E,,,,= 150 GeV).

FIG. 5. Predicted inclusive cross sections for various transverse momenta. The solid curve is for the normal quark model and the dotted curve i s for the charmed- quark model. Data are from Snyder et al., Ref. 12 (E,,, = 4 0 0 GeV).

(say -0.6), predicted values a r e l a r g e r by about a factor of 3 o r s o . This impl ies tha t the G's should fal l slightly m o r e strongly at high X . So we changed (1 - x ) ~ in Eq. (12) to (1 - x ) ~ and obtained curve (b) which a g r e e s completely with data. T h e effect of this change on other f igures will be dis- cussed below. T h e dotted curve (c) is obtained for the charmed-quark model by taking GCl,(x) f rom Eq. (14). If we adjust the normalization of this curve a t X=O independently of the o ther f igures , then curve (d) is obtained. In ei ther c a s e these curves a r e just too steep. Even a t not too la rge X (around x - 0.6) they give ~ d o / d ~ smaller than the

FIG. 6. Predicted transverse-momentum distribu- tion. The solid curve is for the normal quark model with x distribution given by Eq. (12), and the dashed curve is for the same model with (1 -xl4 distribution. The dotted curve is for the charmed-quark model. Data are taken from Anderson et al., Ref. 12 (Eh, = 150 GeV).

2308 K A S H Y A P V V A S A V A D A

data by a factor of about 40 to 70. Such a disagree- ment has been a l so found previously in the Drel l - Yan o r quark-fusion models f o r $ production with charmed quarks having s e a distribution^.^ Such a model then should be ruled out completely regard- l e s s of any ambiguities in the coupling constants. Of course, if one gives charmed quarks valence- type distributions, agreement can be obtained with data s ince then there is very litt le difference from the normal quark model presented above. T h i s will be s imi la r to the quark fusion model of Don- nachie and L a n d ~ h o f f . ~ However, i t should be noted that recent large- QZ data on dilepton production13 give G(X) cc (1 - x ) ~ (with N ranging from 7 to 10 de- pending on various scale-breaking assumptions) fo r s e a quarks and hence flat s e a distributions a r e ruled out. Moreover , electroproduction dataI4 and ra t ios of par t ic les produced a t l a rge p, in p-p col- lisions15 a l s o tend to make flat s e a distributions un- acceptable.

In Figs. 5 and 6 we show B Q , ~ % / ~ ~ Q vs QT and B ~ I J / ~ Q , ~ vs QT2. The agreement is again good. Some deviation at smal l Q, i s presumably due to the neglect of smal l kT fluctuations in the initial state. These fluctuations can be seen to lower the theoret ical curve, in agreement with the data

The changes brought about by taking (1 - x)* in- stead of (1 - x ) ~ distribution in Eq. (12) fo r our model a r e shown by dashed curves. I t can be seen that there a r e smal l changes in Fig. 2 (o v s w ) and Fig. 6 (do/d~, ' v s QT2). The corresponding dashed curves obtained for Fig. 3 , and Fig. 5 a r e indistinguishable from the solid curves and hence a r e not shown. Thus for a l l the distributions the agreement remains reasonable even with the mod- ified (1 - x ) ~ behavior.

B. iL'(3.7) production

T h e $' production data a r e s t i l l very meager and the e r r o r s a r e large. We adjust the coupling con- stant g*,,; by requiring agreement with ~ d o / d y / ,=, for Ei,, =400-GeV data (Snyder et al., Ref. 12). Other c r o s s sections a r e then predicted. F o r ex- ample, the predicted and experimental values of ~ o ( $ ' ) a t E ,,, =400 GeV a r e 0.184*0.092 and 0.191 i 0.142, respectively. Hence the agreement i s good. F r o m da ta by Branson e t al. (Ref. 12) f o r E,,, =225 GeV, we have

while the predicted values a r e 0 . 0 1 5 ~ 0 . 0 1 1 . So again the predicted value a g r e e s with the experi- mental value within uncertainties.

C. p (0.770) production

Extensive p production da ta have become avail- able r e ~ e n t l ~ . ' ~ " ~ F igures 7-11 show resu l t s of our

0 I 1 I !L 10 10' 10'

I ( G ~ v ' )

FIG. 7. Predicted total cross section u(p ) for various s . The solid curve i s for the quark mass m = 0 . 4 7 GeV The dotted curve is for m = 1 GeV. Data are taken from the compilation by Bockmann, Ref. 16.

0 200 GeV

A 24 GeV

- I 0 I 2 Y

FIG. 8, Predicted c.m. rapidity distribution. The top and the bottom curves are for E,,, = 200 GeV and 24 GeV , respectively. The data a re from the compilation by Bockmann, Ref. 16.

C O N S T I T U E N T - I N T E R C H A N G E M O D E L F O R I N C L U S I V E .

16' '0 0.4 0 8 1.2 1.6 2

a, ( G e v )

FIG. 9. Predicted inclusive p production cross section for various transverse momenta. Data are taken from Anderson et al., Ref. 12 (Erne= 150 GeV).

calculation. If we keep the effective quark m a s s m a s 1 GeV a s in the + production, the dotted curve in Fig. 7 (a. v s s ) is obtained by normalizing at s =46.9 GeV2. Thus the energy dependence i s c lea r - ly wrong. Hence the effective m a s s was changed to 0.47 GeV and the solid curve was obtained. The question of quark m a s s will be discussed shortly. I n the meantime note that a l l the r e s t of the curves w e r e obtained with the s a m e quark mass , and the coupling constant gm; was obtained by requir ing agreement a t s =46.9 GeV2 f o r a . F igures 8-11 show do/dy v s y, B ~ ~ d ~ a / d ~ & vs Q,, ~ Q , d o / d x v s X, and ( ~ / ~ , ) ( d u / d Q , ) v s Q,. T h e agreement is reasonable. All the curves a r e shown for the normal quark distribution G u l p ( x ) f rom Eq. (12).

Now, the question of quark m a s s e s i s a subtle one. One problem in extending models like CIM f rom high-&,, high-Q2 to low-Q, and/or low-Q2 region is that the resu l t s do become sensi t ive to the masses . In fact, f o r z e r o m a s s (or even smal l m a s s in some cases ) poles s t a r t appearing i n the physical region. In the $-$' c a s e the intermediate and final quark m a s s e s in the Feynman d iagrams w e r e taken a s 1 GeV from our previous work2 on continuum p p a i r production a t l a r g e Q ~ . T h i s is

FIG. 10. Predicted Feynman-X distribution. Data a re from Anderson et a l . , Ref. 12 (Ehc= 150 G ~ v ) .

FIG. 11. Predicted transverse-momentum distribu- tion. Data a re taken from Anderson et al., Ref. 12 (Eh,=150 G ~ V ) .

2310 K A S H Y A P V V A S A V A D A 20 -

just regarded a s some effective quark mass . In absence of an understanding of quark binding, one can hardly insis t that the effective m a s s must be the same a s the spectroscopic quark mass . In the p-meson case, m = 1 GeV did not give the c o r r e c t energy dependence but m = 0.47 GeV gave i t very well. Hence f o r a l l the p distributions the l a t t e r value was chosen. In the +-meson c a s e m= 0.47 GeV, however, does give r i s e to wrong shapes of the curves. Although this was forced on us by the data, i t does not seem surpr i s ing that the effective quark m a s s in the p production comes out t o be s m a l l e r than that in the c a s e of heavier $-$I pro- duction. Heavier s ta tes can be m o r e easily formed by heavier effective constituent masses . I t i s likely that the quark m a s s e s in o u r calculation play a role s i m i l a r to the cutoffs in field theories. A s a mat te r of fact, models like quantum chromo- dynamics do need cutoffs when they a r e ex- tended to low Q,. I t would be indeed meaningless i f i t were necessary t o have different effective m a s s e s f o r different kinematic regions of produc- tion of the s a m e states . But this was certainly not the case. F o r the p resen t then, we regard the necessity of heavier effective m a s s e s f o r produc- tion of heavier s ta tes a s intuitively acceptable, al- though deeper understanding will be desirable. In the following section we d i scuss the numerical val- ues of the coupling constants required by the data.

IV. COUPLING CONSTANTS

We have not considered charges o r co lors ex- plicitly. So our couplings gmF2 will be averages over these quantum numbers . F o r $ production with normal quarks, we find f rom the distributions,

with about 40% uncertainty. One way to get an ideaof the s i z e of this coupling is t o look a t the rat io of nor- mal hadrons to muons produced in e+-e- collisions in the $ region. We have

where

r ($ - normal hadrons) _ , R = r ( + - ~ + u - )

T h i s gives

Thus the value obtained in the p resen t work is l a r g e r than the value given by e'e- data by a factor of about 80. T h e constants of Ref. 2 can, however, be increased by a factor of 2 o r s o and s t i l l pro-

vide a good fit to the continuum data. So the dis- crepancy could be reduced to a factor of about 40. In addi.tion, we have noticed that in the p resen t model, a s m a l l e r quark m a s s i s generally ac- companied by a s m a l l e r coupling constant to fit the data. F o r example, reducing m to 0.35 GeV would reduce the required coupling gkF2 by a factor of 12. In any case , there is some discrepancy in the val- ues of the couplings obtained in the two processes .

Some previous authors5 who considered $ produc- tion in Drell-Yan o r quark-fusion models have been led to re jec t the ro le of normal quarks be- cause of such a smal l s ize of the coupling constant. However, i t s e e m s to us that this may be prema- ture in the absence of good understanding of quark confinement and the mechanism of the Okubo- Zweig-Iizuka (OZI) rule violation. A s i s generally believed, $ and $' have charmed quarks a s the i r main components. Hence 4-z)' couplings to normal quarks a r e OZI-rule-violating couplings. The ma- jor contribution in the + production process comes from the u-channel graph [Fig. l ( c ) ] in which at l eas t one of the quarks in the +qq vertex i s highly off m a s s shel l (in fact, spacelike). F o r the $ decay into hadrons in e+e- annihilation, the quarks will be timelike. The OZI rule and the magnitude of OZI-rule-violating couplings a r e s t i l l controversi- a l and ill understood."' Hence it i s possible that there is a l a r g e variation in going from timelike to spaceiike region.'* Moreover, our $qq vertex probably already contains renormalization effects due to gluon exchanges. T h e s e correct ions could depend on the off-mass-shell o r on-mass-shel l nature of the part ic les involved. Such effects could a l so account fo r the different effective quark mas- s e s found in fitting the $ and the p production. In cur ren t algebra, difficulties with such la rge ex- trapolations a r e well known. In connection with Regge pole couplings a l so la rge variations in going from spacelike to timelike regions a r e often found,'' So in the p resen t context such a variat ion cannot be ruled out a t present. Note that a l l the distributions look very s i m i l a r to the continuum 1~ p a i r production. T h i s fact would hint a t a s i m i l a r dynamics. Because of these reasons, we regard the normal quark production of 4 a s s t i l l a viable model. The s ize of the coupling would remain a s an outstanding problem.

Another way t o get s o m e idea of the magnitude of Vqq o r VcZ coupling i s through the vector-dom- inance model of Fig. 12. Assuming saturation one finds

e2 g vaa -=g v = - 9

gv ,,I

with a s i m i l a r equation for cc. Applying the satu- ration procedure to +qq, one finds g+,F2/4n = 11.7 using Eq. (7) fo r r ( + - p + p - ) . F o r +qq, however, this

20 - C O N S T I T U E N T . I N T E R C H A N G E M O D E L F O R I N C L U S I V E

saturation is unjustified since other low-mass vec- to r mesons like p will surely contribute to Fig. 12(a). On theother hand, fo r $cF, the saturat ion may be c o r r e c t s ince presumably no other low- m a s s vector meson has appreciable coupling to cc. Then we have

Now our f i ts in the l a s t section require ~ & ~ ~ ~ / 4 n =0.044. Not much is known about G, fo r charmed quarks in the nucleon. So g,,F2/4r can be made to agree with the vector-dominance value in (20) by choosing Go to be 0.0037. Various authors have chosen values of Go ranging from very smal l t o quite large.5 But, a s we have already mentioned, the charmed-quark model (with sea-type distribu- t ions) has difficulties more se r ious than the cou- pling constants. One problem, not mentioned be- fore, is the associated production of charmed par- ticles. Experimentally th i s s e e m s to be sup- pressedz0 ( 0 , ~ ~ / 4 l r < I%), and, unless Go i s ex- t remely smal l , U , , ~ E would come out to be relative- ly l a r g e in the cF model. This , however, is a highly model-dependent question.

Turning now t o the $' coupling, we find that our f i ts requ i re

with about 70% uncertainty, while a calculation in- volving the ra t io of normal hadrons to k + ~ - in $' region of e+e- annihilation gives

So, although there is some discrepancy, curiously, i t is much smal le r than in the $ case. T h i s cou- pling is a l so OZI-rule violating, and the s a m e r e - m a r k s as in the c a s e of $ apply here.

A s for the p coupling, the f i ts require (with m =0.47 GeV)

with about 1500 uncertainty. Applying vector dom- inance we find

and hence

T h i s is a l so approximately consistent with the val- ue obtained from r (p - g'y -). T h u s the p coupling is in good agreement with the vector-dominance value. T h i s may be a reflection of the fact that

I;TG. 12. Vector-dominance model for coupling con- stants.

pqq is allowed by OZI rule , and variation f rom timelike t o spacelike region may be moderate.

V. CONCLUDING REMARKS

In the presen t work we extended our previous CIM for continuum g pa i r production to vector- meson production in proton-proton collisions. F o r p production the resu l t s were very good. Agree- ment with nearly al l distributions was obtained with an OZI-rule-allowed coupling constant g,,; consistent with the vector-meson-dominance mod- el. In fact, considering that there could be many other d iagrams , it is somewhat surpr i s ing that the CIM does well in widely different kinematic r e - gions. Presumably many of such effects a r e auto- matically included. F o r $ production good f i t s to the distributions were obtained but with a coupling constant l a r g e r than that suggested by the decay of $ into hadrons. Plausible reasons w e r e discussed i n the previous section. Difficulties with the charmed-quark model fo r $ production were a l so discussed.

In view of the simplicity of the model detailed x2 f i ts were not attempted. In both # and p produc- tion, the couplings were adjusted such that agree- ment was obtained with one data point i n each case. These data values themselves have uncertaint ies of a t l eas t 30 to 4%. In addition, i t should be kept in mind that the experiments themselves have s o m e ambiguities, var ious cu t s and acceptance prob- l ems . T h u s , on the whole, the agreement should be regarded a s good.

Several authors7 have considered gluon fusion to produce x which subsequently decays into I j , and y . Gluon distributions a r e not a priori known. In fact, some authors have used $ production t o de-

2312 K A S H Y A P V

te rmine gluon distributions." Thus, in general, the re a r e many more parameters than in the p res - ent c a s e even when one u s e s models like quantum chromodynamics. Also, to o u r knowledge, com- plete calculations considering al l the distributions i n the gluon-fusion model have not appeared in the l i terature. T h i s fact, of course, does not ru le out that model. But, a good point about the p resen t model is that most of the input came from the con- tinuum analysis and there were very few param- e te rs . I t i s , of course, possible that a combina- tion of various models could be the c o r r e c t ex- planation.

An important set of data which will help in ruling out some of the models i s the accurate determina- tion of the angular distribution of the muons with respec t to some suitable axis. T h i s was dis- cussed in Ref. 4 by the present author. In particu- l a r , angular distribution of the fo rm 1 + acos28 with v e r y smal l a (in the $ region) will strongly rule out a Drell-Yan model with light s t ruc ture less quarks. CIM does give r i s e to s m a l l e r values of a and hence can be consistent, depending on the value of a. Another c a s e discussed in Ref. 4 was the $qTj vertex with s t ruc ture (both y, and o,, t e r m s ) . T h i s can produce a s m a l l o r z e r o value of a for cer tain values of the anomalous magnetic-moment- type te rm. T h e s e possible s t ruc ture effects could have relevance a l so for the fact that we had to take la rge value of the pointlike coupling g*,; (for spin-

'R. Blankenbecler, S. J. Brodsky, and J. F. Gunion, Phys. Rev. D 5, 2652 (1972); D. Sivers, S. J. Brodsky, and R. Blankenbecler, Phys. Rep. &, 1 (1976).

'M. Duong-Van, K. V. Vasavada, and R. Blakenbecler, Phys. Rev. D 16, 1389 (1977).

's. D. Drell and T. M. Yan, Phys. Rev. Lett. 25, 3i6 (1970).

4 ~ . V. Vasavada, Phys. Rev. D 16, 146 (1977). 5 ~ . F. Gunion, Phys. Rev. D 11, 1796 (1975); 12, 1345

(1976); M. B. Green, M. Jacob, and P. V. Landshoff, Nuovo Cimento s, 123 (1975); A. Donnachie and P. V. Landshoff, Nucl. Phys. m, 233 (1976); H. Komatsu, Prog. Theor. Phys. 56, 988 (1976); D. I. Sivers, Kucl. Phys. -, 95 (1976).

6 ~ . Kinoshita and K. Kinoshita, Univ. of Bielefeld Re- port No. BI-TP 77/11, (unpublished); H. Pritzsch. Phys. Lett. @, 217 (1977).

'M. B. Einhorn and S. D. Ellis , Phys. Rev. D 12, 2007 (1975); S. D. Ellis, M. B. Eihnorn, and C. Quigg, Phys. Rev. Lett. 2, 1263 (1976); C. E. Carlson and R. Suaya, Phys. Rev. D E, 3115 (1976); 2, 1416 (1977); S. Nandi and H. R. Schneider, ibid. 18, 756 (1978); M. Gluck, J. F. Owens, and E. Reya, ibid. 17, 2324 (1978).

Quarks a r e considered spinless in Ref. 2 and here. A recent paper deals with CIM with spin-4 quarks, and results similar to Ref. 2 a r e obtained. [See C. M. De-

l e s s quarks) in the present case. A s for the gluon- fusion model of q3 production through spin-0 X , i t was noted in Ref. 4 that the model does lead to iso- tropic distribution ( a = 0). The present experi- mental s ta tus of CY for $ production i s s t i l l uncer- tain1204 although a- 0 may be roughly consistent with data. In the c a s e of p production, a s i m i l a r experimental situation prevails.12

Some important fur ther t es t s of this model will be in the p, 4 production with different incident beams such a s s, p, y , etc. and also production of o, @, K*, etc. In the present paper , the proton- proton c a s e was considered separately because of the following reasons. F i r s t of all , the most ex- tensive se t of data ex i s t s for dilepton production in the proton-proton case. Data on o ther beams a r e only gradually becoming available. Fur thermore , f rom the point of view of the CIM, the present c a s e is the cleanest one in that only a limited number of d iagrams en te r the picture. F o r other beams there a r e other d iagrams which could make the answers more model dependent. Finally, the probability functions (G's) a r e well determined for protons from electroproduction, neutrino data, etc. Thus, although it i s desirable t o extend the model to other beams, agreement with such an ex- tensive s e t of data in the proton-proton react ions is by itself an indication of the validity of the mod- el. Other p rocesses a r e being considered, and we hope to d i scuss them later .

beau and D. Silverman, Phys. Rev. D 18, 2435 (1978). 1 Thus a t least for the present purpose, the spinor nature of quarks i s not essential. Spin considerations are , however, crucial for the angular distribution problem discussed in Ref, 4.

his expression for n i s exact. An approximate expres- sion valid for large s and Q~ was given in Eq. (7) oC Ref. 2. There i s a typographical e r r o r in that expres- sion. The right-hand side in the equation for r ( s , Q') should be divided by s.

'OM. Duong-Van, Phys. Lett. m, 287 (1976) and private communication. The distribution has be modified a t small x to agree with recent large-energy muon scat- tering data.

"A. Bamberger et al. , Nucl. Phys. M, 1 (1 978). This paper gives compilation of data on ~ ( 9 ) vs ul.

"5. J. Aubert et al. , Phys. Rev. Lett. 3, 1404 (1974); Y. M. Antipov et al., Phys. Lett. s, 309 (1976); K. J. Anderson et al. , Phys. Rev. Lett. 37, 799 (1976), and contribution to Tbilisi Conference, 1976 (unpublished); F. W. Bcsser e t al., Phys. Lett. s, 482 (1975); J. M. Cobb et al. , ibid. %, 96 (1977); H. D. Snyder e t al. , Phys. Rev. Lett. 36, 1415 (1976); B. C. Brown et al. , Fermilab Report No. 77/54, 1977 (unpublished); J. G. Branson e t al., Phys. Rev. Lett. 38, 1331 (1977); M. Binkley et al., 8 i d . E, 574 (1976).

1 3 ~ . M. Kaplan et al. , Phys. Rev. Lett. 40, 435 (1978);

20 - C O N S T I T U E N T - I N T E R C H A N G E M O D E L F O R I N C L U S I V E . . . 2313

Washington-Tufts-Michigan-Northeastern Univ. Col- laboration experiment (private communication from Dr. J. Rutherfoord); V . Barger and R. J. N. Phillips, Phys. Lett. E, 91 (1978); E. L. Berger, Argonne Report No. ANL-HEP-PR-78-12 (unpublished).

1 4 ~ . R. F a r r a r , Nucl. Phys. E, 429 (1974); V . Barger, T . Weiler, and R. J. N. Phillips, ibid. m, 439 (1976).

1 5 ~ . Antreasyen e t al., Phys. Rev. Lett. 38, 112 (1977); 38, 115 (1977).

l6K Bockmann, invited talk a t the Conference on Multi- particle Production Processes and Inclusive Reactions a t High Energies, Serpukhov, 1976 (unpublished). This paper give a compilation of data on O ( ~ O ) vs s and du(p)/dy vs y and empirical fits to various distri- butions. S. Nandi, V. Rittenberg, and R. Schneider [Phys. Rev. D 17, 1336 (1978)l give quark-fusion-mo- del fits for small-QT spectra in the case of p produc- tion with n, K, and y beams.

"H. J. Lipkin, Fermilab-Conf-76/98-THY, 1976 (un- published).

' a~oss ibi l i ty of such variation has been also mentioned by Y. Kinoshita and K. Kinoshita (Ref. 6 ) and Donnachie and Landshoff (Ref. 5) in a somewhat similar context. The authors of the f i rs t paper in Ref. 6 give large-k, fluctuations to the incoming partons, consider them highly spacelike and argue for such a change in the coupling. In our CIM model the incoming partons a r e regarded essentially on the mass shell, and large k T fluctuations a re not allowed. However, the reaction proceeds dominantly through the u-channel exchange which makes atleast one of the quarks in the t/q? ver- tex spacelike even if the two-body subprocess is re- garded on the mass shell. Fritzsch (Ref. 6) has argued for circumventing the difficulty with the OZI rule by adopting a model where qq produce a highly virtual gluon which subsequently gives r i s e to a cF pair of $I.

''see, for example, V. Barger and D. B. Cline, Phen- omenological Theories of High Enevgy Scattering (Benjamin, New York, 1969), p. 132.

2 0 ~ . G. Branson et al., Phys. Rev. Lett. 38, 580 (1977); M. Binkley et al., ibid. 37, 578 (1976).