fO-47 D E C A Y I N A R E L A T I V I S T I C Q U A R K M O D E L 3887

13strictly speaking, contributions due to other possible adopted in the present investigation. Moreover, mechanisms such as f O -pp - (TIT) (871) , f - EE - (88) (88) , Banyai and Rittenberg (Ref. 7) have shown that the t being the I= 0 scalar meson, etc., should be taken pp model of Ascoli et u l . (Ref. 3) is in some sense com- into account in the calculation of the decay rate for plementary to their own model. Our result [ ~ q . (12)1, theprocess f O - 4 ~ . Unfortunately at this stage the therefore, admits of appropriate corrections due to coupling structures for these channels are unknown, at contributions from these mechanisms. least in the context of the quark-model formulation

PHYSICAL R E V I E W D VOLUME 1 0 , NUMBER 11 1 D E C E M B E R 1 9 7 4

Relation among structure functions for massive currents*

Myron Bander and Mythili Rangachari Department of Physics, University of California, Irvine, Iwine, California 92664

(Received 18 March 1974)

A relation among structure functions for processes involving massive photons (deep-inelastic electron scattering, e + e annihilation into hadrons and massive lepton pair production) is presented. This relation is valid in a variety of models. A rough comparison with experimental data shows this relation to be grossly violated. Some speculations on this violation are presented.

Reactions involving c u r r e n t s with l a r g e momen- tum squared a r e believed t o probe fundamental con- stituents of hadronic physics. Among these we find not only deep-inelastic e lectron scat ter ing, but likewise e + + e - annihilation into hadrons and the production of mass ive lepton pa i r s . Th is pointlike constituent s t ruc ture supposedly manifests itself in scal ing behavior of the r a t e s for these process- e s .

Though the parton picture, ' o r to a l e s s e r extent the operator production expansion on the light cone,' may be mos t directly tied to the concept of fundamental point constituents, the hypothesized scal ing behavior i s consistent with m o r e prosaic views of hadronic reactions: mult iper ipheral , Regge-pole exchange, etc. In this note we obtain a relation between the above-mentioned c u r r e n t pro- c e s s e s that, under specific assumptions, may be common to a l l models.

At this juncture we m u s t point out that the en t i re scaling s t ruc ture may be suspect in light of recent experiments on e + + e - a n n i h i l a t i ~ n , ~ which (at l e a s t a t p resen t energies) do not indicate the des i red scaling. We shal l have m o r e to say on this l a te r ; f o r the moment l e t u s proceed on the assumption that a l l react ions will scale .

Before s tat ing the relation cen t ra l to this a r t i c le we review and establ ish the necessary notation. e + + e - annihilation into hadrons i s descr ibed in t e r m s of R(q2) , the rat io of the c r o s s section for this p r o c e s s to the c r o s s section for e + + e - - p + + p- . In the scal ing l imi t R(q2) i s taken to be independent of q 2 . Deep-inelastic e lectron sca t - ter ing is presented in t e r m s of the usual s t ruc ture

functions mw,(v, q2) , V W ~ ( Y , q 2 ) which a r e to ap- proach the scaling l imi t s ~ , ( x ) , F2(x) with x = q '/ 2mv. The third p rocess we study i s inclusive lepton pa i r production, specifically p + p - p + + p - + .. . . Let q p be the four-momentum of the lepton pa i r and p,, p2 the momenta of the incident p ro- tons. This p r o c e s s i s governed by the s t ruc ture function

Letting xi =q2/2p, .q, s =4p1.pZ, we obtain

with

In the scaling l imit V(s; x,, x,) c e a s e s to depend on S.

The following relation between the above s t ruc- tu re functions i s proposed:

l im [2xlFl(x,) + F,(x,)][ 2x2Fl(x2) +Fz(xz)] =R .

x 1 , x 2 - o 8V(xlx2)

(4)

We shal l briefly indicate, in two c l a s s e s of models, the conditions necessary f o r the validity of th i s relation:

3888 M . B A N D E R A N D M . R A N G A C H A R I

(i) .Pavtou ~~zode ls .* In the zaee parton region aLI partons and antipartons a r e p resen t with identical distributions.

(ii) MuLtiperifilze?,al and Regg-e-,tfnelLev models. The amplitude for any of these mass ive c u r r e n t p rocesses factor izes into a hadronic component and a cur ren t component. The c u r r e n t component extends in rapidity of the produced hadrons over a range h y =In& to ei ther s ide of the cur ren t rapid- it^.^ It i s the factorization assumption, ra ther .than details of the models, that leads to Eq. (4). Details of such models a r e presented in a separa te publication.'

A verification of Eq. (4) based on presently avail- able data i s difficult on two grounds. As mentioned e a r l i e r , R is growing with q 2 . Second, mass ive lepton p a i r production data exis t f o r a uranium ta rge t , and the extraction of the required proton data i s ~ u s p e c t . ~ Taking this extraction a t face value, we s t i l l have a difficulty, a s only do/dq2 h a s been measured. However, with some addition- a l assumptions we may determine V(0,O) from the data. Assuming Regge-pole dominance, we expect V(x,, x2) to have the expansion8

leading to

In obtaining the above we have s e t q, = O in a l l ki- nematical calculations. Good data for s / q 2 la rge should yield V(0,O). In the c a s e of deep-inelastic electroproduction we assumed o,/o, = 0 (Ref. 91,

and thus Eq. (4) reduces to

F o r s - 58 GeV2, 1.5 < q < 2.1 GeV, the data of Ref. 7 and Eq. (6) yield a value of V(0,O) between 0.025 and 0.04. F2(0), on the other hand, v a r i e s between 0.2 and 0.3, froni which we obtain

Should R become constant in q 2 , a t i t s p resen t l a rge value of -6, i t would s t i l l be in disagree- ment with Eq. (4), even within the crude es t imates we have made. One may take a m o r e optimistic view and hope that the p resen t experimental trend in R will r e v e r s e and R will decrease before level- ing off a t a value consistent with Eq. (8). In this case the value of R i s consistent with those pre- dicted by quark-parton models with R = $ o r R = 2 .

However, if scaling does break down f o r reac- tions involving timelike q 2 , we might envisage that this could be due to a parton form factor,'' f (q2) . [,f(q2) - 1 f o r q 2 spacelike.] In such a situation 6 q . (4) may be t ransformed into a measurement of S t 4 2):

[ 2 x , ~ , ( x , ) + F , ( x , ) ] [ ~ x ~ F , ( ~ z ) + F z ( ~ z ) ] l im

X I , x2 -0 8 ~ ( 4 ')V(q 2; XI, x2)

Of course, fo r Eq. (9) to make sense R/V would have to scale .

One of us (M. B.) wishes to thank D r . M. Einhorn for stimulating discussion.

*Work supported in par t by the National Science Founda- tion.

'R. Feynman, in High Energy Collisions, edited by C. N . Yang, et a l . (Gordon and Breach, New York, 1969) ; 5. Bjorken and E . Paschos, Phys. Rev. 185, 1975 (1969).

'H. Fri tzsch and %I. Gell-Nann, in Proceedings of XVI Inte~xational Conference on High Encvgy Physics, Chicago-Batavia, Ill., 2972, edited by J. D. Jackson and A. Roberts WAL, Batavia, Ill., 1973), Vol. 2 , p. 135.

3 ~ . Litlce et al . , Phys. Rev. Lett. g, 1189 (1973); B. Richter, report presented to the Irvine Conference on Lepton induced Reactions, 1973 (unpublished).

4 ~ . Drell and T.-M. Yan, Phys. Rev. Lett. 25, 316 (1970). Relations analogous to Eq. (4) based on the parton model have been obtained by M. Gronau [Phys. Lett.

39B, 395 (1972)l and H. Paar and E . Paschos [Phys. Rev. D 2, 1502 (1974). We wish to thank Dr Paschos for pointing out the ear l ie r reference to us.

5 ~ . Bjorken, Phys. Rev. D 7, 282 (1973) ; R. Cahn and E. Colglazier, ibid. 8, 3019 (1973).

G ~ . Bander and Mythili Rangachari, Phys. Rev. D 10, 3081 (1974).

'3. Christenson et a l . , Phys. Rev. D 8-, 2016 (1973). 'K. Subbarao, Phys. Rev. D 8, 1498 (1973). 'H. Kendall, in Proceedings of the 1971 International

Symposium on Electron and Photon Interactions at High Energzes, edited by N. B. Mistry (Laboratory of Nuclear Studies, Cornell Univ., Ithaca, N. Y. , 1972), p. 247.

'OM. Chanowitz and S. Drell, Phys. Rev. Lett. 30, 807 (1973).