P H Y S I C A L R E V I E W D V O L U M E 1 6 , N U M B E R 1 1 J U L Y 1 9 i 7

Theoretical models for the angular distribution of massive muon pairs produced in hadronic collisions

Kashyap V. Vasavada Stanford Linear Accelerator Center, Stanford University, Stanford, California 94305 and Physics Department, Indiana-Purdue University, Indianapolis, Indiana 46205

(Received 24 January 1977)

We point out that accurate measurements of the angular distributions of direct muons produced in hadronic collisions will have important consequences for various theoretical models for such processes. A number of models based on concepts of quark-partons and gluons are discussed within the context of the preliminary data available at present.

Production of massive p p a i r s in hadronic col- lisions has been studied extensively recently both experimentally and theoretically. It has already provided insight into the dynamics of various theo- ret ical models based on the concepts of quark- partons and gluons. Ideally one would like to have data for @"(mass of d i m u ~ n ) ~ ] , Q,, (longitudinal momentum of the dimuon), and Q, ( t ransverse momentum) distributions along with the angular distribution of the muons in a range of values of each of these variables . It will be s o m e t ime in the future before such complete high-statistics data become available. But already some p r e - l iminary data for angular distributions averaged over the other variables have been obtained.' In the past , severa l authors have considered Q2, Q, , , and Q, distributions of the dimuon in various models, but the angular distributions of muons have not been considered. In the present work we discuss various theoret ical models for dimuon production, with special emphasis on the angular distributions. We show that even f rom such aver - aged data some interest ing conclusions follow. Q2, Q , and Q,distributions, fo r one of the models, a r e considered separately. ' We hope to present a more detailed analysis of angular distributions when complete data become available. One of the purposes of this work i s t o point out that accurate measurements of such distributions will have very important consequences fo r various theoretical models, s ince predictions differ considerably f r o m one model to another .

The prel iminary resu l t s which follow f rom a recent experiment' give the muon angular distribu- tion in the dimuon r e s t f r a m e a s 1 + a cos", where the angle 6 i s measured with respect t o the in- cident beam axis . There a r e considerable vari- ations in the measured a values. But they a r e found t o be consistent with 1 (although they could be different f r o m 1) for continuum p pa i r in the

Q = (1.9-2.3)-GeV region. In the 4 resonance r e - gion (Q =3.1 GeV), however, a i s found t o be con- sistent with z e r o (a! = -0.26k0.19 for incident pro- tons, (Y z 0 . 2 6 f 0 . 2 5 for incident pions). If future analyses uphold this conclusion, it will a l ready be a n indication that the production mechanism for continuum p p a i r s production i s ent i rely different f rom that of q! production.

The complete angular distribution of the muon is given by the expression3

where the p ' s a r e the density matr ix elements . The angles (6, r$) a r e measured with respec t t o some convenient axes . In the Gottfried-Jackson (GJ) f r a m e the z axis is along the direction of the incident beam in the dimuon r e s t f r a m e . In the helicity f r a m e it i s opposite t o the direction of the momentum of the recoiling missing m a s s in the dimuon r e s t f r a m e . Averaging over the azi- muthal angle 4 W ( 0 , r$) can be written a s

where we have used the normalization condition poo + 2p1, = 1. Hence

3 ~ " - 1 a = - 1 - Pll

In general p,, (and hence a ) will be a function of other kinematic variables such a s Q,,, Q,, e tc . At present , however, data only give the average value of a . If m,, m,, and Q a r e the m a s s e s of the incident par t ic le , target par t ic le , and dimuon, and m, i s the missing m a s s , the c ross ing angle x which re la tes the GJ f r a m e ( t -channel) to the s-channel helicity f r a m e i s given by

cosx = [(s +Q2 - m;)(t + Q 2 - m12) + 2 Q 2 A ] / ~ , (4)

where

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

and

D = [(Q +m,)' - t ]'I2[(Q - m1)2 - t ]1/2

x [s - (Q + m x ) 2 ] 1 / 2 [ ~ - (Q -mx2)z]1/2 (6)

Here s and t a r e the usual Mandelstam variables for the p rocess

m1 + m 2 - Q +mx . Then p::GJ) i s related t o pf, by

(1 - 2 ~ : l ) + p:,(l +cos2x) p:1= 2 2 , ( 7 )

assuming that p , , - I and p,,, a r e ze ro . Next we consider various quark-parton and gluon

models. Throughout this work we assume, fo r simplicity, that the quark-partons a r e on the m a s s shel l . Some r e m a r k s about this assumption a r e made in Sec. M.

In the following, we will wri te equations ex- plicitly fo r the c a s e where the p pa i r is produced through a n intermediate heavy photon. Equations for the case of $ can be readily obtained by using a Breit-Wigner fo rm instead of 1/Q2.

11. DRELLYAN (DY) MECHANISM WITH LIGHT (NONCHARMED) QUARKS AND POINT COUPLINGS

The Drell-Yan mechanism4 i s the most popular model fo r this p rocess . Here, a s shown in Fig. 1 , a quark (parton) f r o m one hadron annihilates a n antiquark (antiparton) f r o m the other hadr on. If neither beam nor t a rge t contains a valence anti- quark, the antiquark i s supposed t o come f rom the s e a of q q pa i r s . It is often s ta ted that the d i s t r i - bution is of the fo rm 1 +cos20. Here we consider some more exact expressions. Let k,, k,, p, , p,, and Q be the four-momenta of the q-ij and pt -p- p a i r s and the intermediate heavy photon or @ p a r - t ic le . Then the matr ix element for pointlike (y,) couplings i s given by

Squaring and summing over the spins, one finds

where

is the square of the velocity of the quarks in the c .m. sys tem. 9 i s the angle between El and p', in the s a m e s y s t e m . We have neglected the lepton m a s s e s but keep the quark m a s s (m) nonzero. Setting m = 0, one gets the well-known Drell-Yan

FIG. 1. Drell-Yan model.

resul t 1 +cos26. So for very light quarks one should have this distribution for relatively large values of Q2. This should be t r u e for both the heavy photon and the + meson if the production mechanism is the Drell-Yan process . F o r very low values of Q2, the value of cu will be sensitive to the quark m a s s m. If the magnetic moment of the quark i s assumed to be the s a m e a s that of the proton, a typical value of m=0.336 GeV i s ob- tained. In the p ( w ) m a s s region this gives

On the other hand, if m = m ,,,, cu = 0. So for light quarks the distribution should be essentially iso- t ropic . In fact , experimentally it has been found that, a t Fermi lab energies , fo r p ( o ) production the distribution is close t o being isotropic .5 We note in passing that, f o r pion exchange (for p pro- duction) and p exchange (for w production), the distribution should be sin29 and 1 +cos20, respect- ively. At low energies (11.2-GeV/c n beam) r e - sul ts consistent with one-pion exchange + absorp- tion were founds3 But a t higher energies the quark models presumably should work bet ter .

111. DY MODEL WITH HEAVY QUARKS

As we noted above, for m - i Q , a = p 2 = 0 . Now the m a s s of a charmed quark i s believed t o be about half the m a s s of the 4 meson. So, in such a case, one would automatically get an isotropic distribution in Iji production. Some authors have already considered the QZ, Q,,, and Q, distribution in such m o d e l s . V h e number of charmed-quark cc pa i r s in the s e a associated with nucleons and pions i s presumably v e r y smal l . But this is com- pensated by a l a rge coupling of cE to $. In the qq model, that coupling is forbidden by the Zweig rule and hence i t i s small. In the cF model one h a s to a r r a n g e carefully s o that excessive numbers of charmed part ic les a r e not produced (in agree- ment with experiments) and there a r e some dif- f icul t ies . But, on the whole, a t p resen t the model i s not ruled out. The isotropic angular distribu- t ion will give s t rong support t o such a model if other predicted distributions a g r e e with experi- ment.

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

IV. DY MODEL WITH STRUCTURE IN relevant mat r ix element i s given by THE qij COUPLINGS TO y AND $

The scaling observed in electroproduction puts s t rong restr ic t ions on the s t ruc ture one could al- low in the qq y vertex. However, as suggested by Drel l and Chanowitz7 and West,' one can intro- duce a up, t e r m with a fo rm factor to descr ibe the anomalous magnetic moment of the quark if it ex i s t s . It can be argued that even if the fundamen- t a l quark vertex is pointlike ( y , type), renormal - ization effects due to exchanges of gluons could produce a up , t e r m analogous t o the anomalous magnetic moment t e r m for the elementary elec- tron.' Now with two form fac tors F,(Q2) and F2(QZ) Eq. (8) becomes

The leptonic vertex has been kept y , type. F o r the q i j y case p , i s the anomalous magnetic mo- ment of the quark. Squaring and spin-averaging lead t o (for s m a l l m and la rge Q2)

+ FZz(QZ)pq2(1 - p2 cos2Q) . (12) This gives the distribution function

where we have se t F2(Q2)/Fl(Q2) = 1. Wests found that values of p, of 0.1 to 0.2 GeV-' a r e consistent with electroproduction, e'e- annihilation data, and a l s o with the quark-model assumption that the quark magnetic moment be the s a m e a s that of the proton (with quark m a s s = $ proton m a s s ) . Taking /3 = 1 , Q =3 .1 GeV, this gives a = 0.83 and 0.44for p, = 0 . l and0.2GeV-', respectively. a = O would requ i re p, =0.32 GeV-'. Also, the value of a will decrease as QZ increases . As Q2 in- c r e a s e s the distribution changes f rom 1 +cos2@ to 1 and then to 1 - cos29 . This i s fo r the c a s e of heavy photons. In principle the hadronic qij+ vertex could be completely different f r o m the q q y vertex, and it is possible that the qqy ver- tex may be pointlike and that the q q p vertex has a s t ruc ture . In the la t ter c a s e there i s e s - sentially no restr ic t ion on the value of p a . It s e e m s that such models may be a r ranged s o that they a r e not in contradiction to existing experi- ments and could even be forced on us by future experiments , although the elegance and simplicity of the parton picture will be somewhat lost.

V. DY MODEL WITH SPIN-0 PARTONS (OR QUARKS?)

This is not a very at t ract ive hypothesis, but we consider it for the sake of completeness. The

Then spin-averaging leads to

F o r light quarks, this gives the 1 - cos2Q dis tr ibu- tion, a s i s well known. It should be noted that electroproduction data do favor spin-; partons.

In a l l four DY models, the distribution has been calculated with respec t t o the direction of the quark-antiquark three-momenta in the c.m. sys - tem of the dimuon. The relationship of this axis to the beam axis will depend on the dynamical details of the model (i.e., probability functions fo r the quarks t o have different momenta). We consider two simple c a s e s in view of the crudeness of the p resen t data which a r e given with respec t to the incident beam ax is and a r e averaged over al l Q,, and Q, values of the dimuon. In the f i r s t c a s e we assume that the quark-antiquark have essent ial ly only longitudinal momenta; the t ransverse momen- t a a r e extremely small . Then the values of a a r e approximately the s a m e a s the ones given above, even with respec t t o the beam axis. In the second case , we assume that the q-?j a r e essent ial ly moving in the same direction a s the heavy photon (or I/, o r dimuon) they produce. Then we use Eqs . (3), (4), (5 ) , (6), and (7 ) t o obtain a relat ive to the beam axis. The resu l t s a r e averaged over Q,, and QT to obtain (Y. As suggested by the experi- mental fits,' the weight factor ( l - X , ) l e - 2 Q ~ ( ~ , = 2 ~ , , / & ) is used. It turns out that in this c a s e the value of a does change substantially by such averaging. F o r example, a = 1 leads to =0.36. On the other hand, however, i t s e e m s unlikely that any amount of averaging can lead f r o m a: = 1 t o - ff =o.

So f a r we have considered quark-antiquark anni- hilation models. Next in o rder of complexity i s the constituent-interchange model of Blankenbecler, Brodsky, and G ~ n i o n , ~ in which a quark f rom one hadron s c a t t e r s on a hadron (meson) emitted by the other hadron and produces a photon or a vector meson and anything else .

VI. CONSTITUENT INTERCHANGE MODEL (CIM)

This model h a s been quite successful in the large-Q, region, but application to the en t i re (3, range i s only current ly being made.' F o r the sake of definiteness we assume that, a s in Fig. 2 , the dominating hard-scat ter ing subprocess i s the one in which a quark is sca t te red by a n meson to p ro- duce a quark and a dimuon. The final quark then combines with the r e s t of the hadronic s t a t e s which

T H E O R E T I C A L M O D E L S F O R T H E A N G U L A R D I S T R I B U T I O N . . . 149

FIG. 2. Constituent-interchange model (CIM)

a r e not detected. This p rocess is analogous to nN- yN o r pN. We consider the dominant contribu- tion t o be given by an e lec t r ic Born model. Our t reatment will be somewhat s imi la r t o the one given by Sachrajda and Blankenbecler, who consider a n s- channel pole t e r m for this p rocess .lo L e t p , q , k , and Q be the four-momenta of the initial quark, initial meson, f inal quark, and dimuon, respectively. Then s' = (p +q)', t' = (P - k)', u ' = (p - Q)'. In the gauge-invariant e lec t r ic Born model there a r e two quark ( s t and u ' ) pole t e r m s and one pion (t') pole. (See Fig. 3. ) The mat r ix element i s given by

FIG. 3. (a) s-channel quark pole for CIM. (b) u-chan- nel quark pole for CIM. (c) t-channel meson pole for CIM.

sideration. e i s the proton charge and g i s the qqa coupling constant. The values of (A, B, C) a r e easi ly found to be (in this o r d e r )

where n-u - yd: (4, -$ , l )f i .

I =A 2 € * k + ( r v ~ ) ( y . Q ) +B 2 e . P - (Y.Q)(Y.E) u and d r e f e r to the up and down quarks with S' - ma2 u ' - ma2 charges $ and -4, respectively. F r o m Eq. (16)

2 ~ ' q - E ' k one can read off the Ball amplitudes fo r the pro-

+ C t ' - m,' ; c e s s and obtain the s-channel helicity amplitudes H:,.,,~(S', t'). A , hj, and A, a r e the helicities of

ma and m', a r e the m a s s e s of the quark and the the off-shell photon (or $ meson) and final and ini- meson, respectively. The values of the constants t ia l quarks, respectively." The nonzero H : ~ , ~ , A, B, C depend on the charge s ta tes under con- a r e a s follows:

P+- (s ' , t l )=- - [ Bs' - At' - Bt' Ct'

a t ? , , 2 ( s r - ni:) - 2(u1- m:) 4(s1 - m:) 4 ( u r - m;) - -1,

By squaring these, the s-channel density matr ix In t e r m s of G's the GJ (t-channel) density matr ix elements a r e obtained. These will be useful in the elements a r e given by future. However, present data give a distribution with respec t to the beam axis (Gottfried-Jackson P ~ ~ ( S ' , ~ ' ) = I G , / ~ / ( I G ~ I ~ + l ~ ~ / ~ + I G - ~ I ~ ) ,

f rame) . Hence we obtain the amplitudes G by (22) crossing, pl1(sr , t ' ) = ( l ~ , l ~ + I G - ~ I ~ ) / ~ ( I G ~ I ~ +b1I2 + I G - , ~ ~ ) ,

G T ~ , ~ ~ ( s ~ , ~ ~ ) = C ~ ~ ~ ~ ( X ) H : ~ , ~ ~ ( ~ ~ , ~ ~ ) . etc . Then the angular distribution for y o r # - r*tp-

x (20) i s obtained f rom Eq. (2). Now pll i s a function of

s ' and t'. T o compare with experiment we have to dmxl(X) i s the usual d function of the y (4) crossing

average over these variables . In the CIM the bas ic angle x given by Eq. (4) with the replacement

ecluation for the total inclusive c r o s s section of y m l - m,, m2- ma, and m,-m,. (21) (or $) is given by

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

We have neglected the transverse-momentum dis- tributions of the incoming quark and meson. G,/" and CTIH a re the probabilities of finding the quark and the a meson in the two hadrons with fractions x , and x, of the longitudinal momenta. Then s' =x,x,s, where s is the square of the total c.m. energy of the two incoming hadrons. In the quark- meson c.m. system we have

where / < I , 161, and 0, are , respectively, the in- coming-quark three-momentum, the y ( $ ) three- momentum, and the scattering angle in this sys- tem. To produce experimentally observed rapid falloff in Q,, we multiply the integrand by a phe- nomenological factor e - 2 0 ~ as before.' The G functions a re determined from electroproduction and neutrino data. For example, for the up quark in the proton, we have

r

For G,/, one can consider various cases. At one detail but merely point out that if the mesons a re extreme we can take it to be the same as the sea- spinless, the angular distribution will be similar quark distribution in proton, i.e., to the case (Sec. TV) considered above. Next we

GnIp(x) =O.2(1 - x ) ~ / x . (27) consider gluon models.

At the other extreme it can be taken to be the VIII. GLUON MODELS same as the one for the up quark in the proton. It should be emphasized that since we a re interested only in the averaged angular distribution, and not the actual magnitude of the cross section, such differences a r e not crucial here. Actually we have verified numerically that this is so. The average values of a obtained using Eq. (23) are given below with corresponding values of (A, B, C ) for different charged states

It i s also interesting to note that part of the u- channel diagram is the Drell-Yan qq annihilation process. In fact, it i s amusing to note that if we ignore gauge invariance and set s and t pole terms equal to zero the value of cu obtained is 0.35, which i s extremely close to the Drell-Yan averaged value 0.36.

The above results show the range of variation with different charged states. With a full isospin treatment of the CIM some average results in this range can be expected. So CIM does lead to values of cu smaller than 1.

VII. CIM WITH MESON-MESON SCATTERING SUBPROCESS

In this model, which was considered by Chu and Koplik,I2 a meson is emitted by each hadron and they produce a heavy photon in analogy with the DY process. We do not consider this mechanism in

Such a model for 4 production has been con- sidered by Ellis, Einhorn, and Quigg13 and Carlson and Suaya.14 In this model, a s shown in Fig. 4 , one gluon is emitted by each of the hadrons to pro- duce an even charge conjugation state X. This in turn decays into $ and y. $ can then decay into muons. These authors did not consider implica- tions for the angular distributions of the muons. It is clear that. if the intermediate state x has a spin 0 (as one of the x states should have), the decay products of $ cannot have any correlation with the gluon axis o r incident hadron axis. Thus an isotropic distribution will be obtained. If the intermediate state x has a spin different from 0 (e.g., 2), the muons will have some angular corre- lation with the gluon axis and hence the beam axis. On the basis of the charmonium model it can be argued that the spin-2 intermediate state will be relatively suppressed a s compared to the spin-0 intermediate state. Thus, if the value of a = 0 is confirmed, it will lend support to the model with the spin-0 x intermediate state. Of course the

FIG. 4. Gluon model with intermediate state X .

16 - T H E O R E T I C A L M O D E L S F O R T H E A N G U L A R D I S T R I B U T I O N . . . 151

overal l predictive power of the gluon models s e e m s to be somewhat l e s s than the quark-parton models.

IX. CONCLUDING REMARKS

In the above sect ions we have discussed various theoretical models fo r the angular distribution of muons produced in hadronic collisions. An im- portant assumption was that the constituents act like on-mass-shel l par t ic les when they sca t te r off each other . This is the basis of the p resen t quark-parton phenomenology. The s t ruc ture func- tions (e .g., G functions used in the text) a r e a s - sumed t o damp the off-shell behavior very s t rongly ( see , for example, Refs. 9 and 10). F o r off-shell constituents, the problem of violation of gauge invariance becomes a very se r ious one. In ad- dition, the off-shell extrapolation i s very a m - biguous. Thus, f o r phenomenological purposes it is convenient t o r e g a r d the constituents a s on- mass -she l l pa r t i c les with some effective m a s s e s . These may o r may not be the s a m e a s the spec t ro- scopic masses . In fact, in Ref. 2 , in o rder to f i t Q 2 , Q ,,, and Q, distributions we have to choose mq,,- 1 GeV. In the absence of a relat ivis t ic theory for quark binding even the spectroscopic m a s s e s a r e uncertain, although mquU, i s believed t o be about (nucleon m a s s ) f o r noncharmed and nonstrange quarks and about $ ($ m a s s ) fo r the charmed quarks. In any c a s e there i s a n unavoid- able ambiguity about the p rec i se value of the quark m a s s e s . Of course , the hope i s that quarks which can be regarded a s light o r heavy in spectroscopy remain effectively light o r heavy respectively in scat ter ing processes a l so . In various models

(especially the Drell-Yan model) a n important rat io determining the angular distribution i s mqU,,,Z/QL. In the off-shell c a s e this can be roughly replaced by k2/QZ (k =four-momentum of the quark) . If kZ i s not too different f rom ni ,,,,,,,v~ o r if Q L i s much l a r g e r than ei ther of them, the r e - su l t s will not change very much. A careful angular distribution analysis for var ious Q' and incident beam energies will shed considerable light on such effective m a s s e s involved in the models.

Different models discussed here differ consider- ably in their predictions. Hence in the future, good angular distribution data in both helicity and Gottfried-Jackson f r a m e s in different Q 2 . Q,,, and Q , bins will give c r i t i ca l t e s t s fo r various models. It will be a l s o interesting to s e e if the angular distributions of the muons depend on the nature of the incident beam. Simultaneous analyses of a l l these distributions (density mat r ices , Q Z , Q,, , Q , . etc.) will give constraints on various models and will have important consequences for understanding the role of the constituents in the scat ter ing of hadrons .

ACKNOWLEDGMENTS

I wish t o thank S. Drel l and S. Brodsky for the i r w a r m hospitality a t SLAC. Special thanks a r e due to R. Blankenbecler, S. Brodsky, and F. J . Gil- man for a number of discussions and suggestions and J. Sapirs tein fo r help with the SLAC computer sys tem. Par t of this work was initiated in a con- versation with B. Humpert. Conversations with J . M . Weiss , S. El l is , M . Duong-van and J . Pilcher were a l s o very useful.

*Work supported in p a r t by the U . S. Energy R e s e a r c h and Development Administration.

? p r e s e n t and permanent address . 'K. J. Anderson et d ., paper submit ted to the XVII In-

ternat ional Conference on High Energy Physics , Tbili- s i , U.S.S.R., 1976 (unpublished).

2 ~ . Duong-van, K. V. Vasavada, and R. Blankenbecler, (unpublished).

3 ~ . C. C. Ting, in Proceedings of the Third International Symposium on Electron and Photon Interactions at High Energies, Stanford Linear Accelerator Center, 1967 (Clear ing House of Federa l Scientific and Technical Information, Washington, D.C., 1968), p. 452.

4 ~ . D. Drel l and T.-M. Yan, Phys. Rev. Let t . 2, 316 (1970); Ann. Phys. (N.P.) 66, 578 (1971).

'M. Binkley et uL. , Phys. Rev. Let t . 37, 574 (19761. 6 ~ . Blankenbecler et d., SLAC Report No. SLAC-PUB-

1531, 1975 (unpublished); A. Donnachie and P. V.

Landshoff, Nucl. Phys. u, 233 (1976); J. F Gunion, Phys. Rev. D G , 1796 (1975).

'M. Chanowitz and S. D. Drell, Phys. Rev. D 2, 2078 (1974).

'G. B. West, Phys. Rev. D g , 320 (1974). or a review s e e D. S ivers , S. J. Brodsky, and 13. Blan-

kenbecler , Phys. Rep. 3, 1 (19761. 'OC. T . Sachrajda and R. Blankenbecler, Phys. Iiev. D

12, 3624 (1975). "See, f o r example, C. '3. Cho, Phys . Rev. D 4, 194

(1971); K. V. Vasavada and L. J. Gutay, zbid. 1, 2563 (1974).

"G. Chu and J. Koplik, Phys. Rev. D g , 3134 (1975). 13s. D. Ell is , M. B. Einhorn, and C. Quigg, Phys. Rev.

Let t . 36, 1263 (1976); M. B. Einhorn and S. D. E l l i s , Phys. Rev. D G , 2007 (1975).

j4c . E . Carlson and R. Suaya, Phys. Rev. D 2, 3115 (1976).