Testing the parton model in massive lepton pair production

Volume 39B, ;:umber r

PHYSICS

LETTERS

1 May 1972

T E S T I N G THE P A R T O N M O D E L IN M A S S I V E L E P T O N P A I R P R O D U C T I O N

S

M. G R O N A U

Cal~rnialnstitute ~ Technolo~,Pasadena, Cal~.91109, USA Received 21 December 1971 Revised m a n u s c r i p t r e c e i v e d 17 F e b r u a r y 1972

The parton model and the conjectured diffraction b e h a v i o r of F2~N(x ) at s m a l l x a r e used to e s t a b l i s h a r e l a t i o n s h i p between the deep inelastic electroproduction p r o c e s s and m a s s i v e lepton p a i r production. The r e s u l t is applied to p r e s e n t and future e x p e r i m e n t s .

Two d i f f e r e n t s c h e m e s , t h e p a r t o n m o d e l [1] a n d t h e l i g h t c o n e a n a l y s i s [2], h a v e b e e n v e r y u s e f u l i n d e s c r i b i n g t h e s c a l i n g p h e n o m e n a o b s e r v e d i n t h e d e e p i n e l a s t i c e l e c t r o n - n u c l e o n s c a t t e r i n g at S L A C [3]. T h e r e g i o n s of a p p l i c a b i l i t y f o r t h e two a p p r o a c h e s s e e m at t h e p r e s e n t t i m e to d i f f e r f r o m e a c h o t h e r [4] and i t c e r t a i n l y w o u l d be i m p o r t a n t to find s o m e e x p e r i m e n t a l t e s t s w h i c h c o u l d e n h a n c e o u r c o n f i d e n c e i n o n e of t h e two s c h e m e s * . A n o t h e r f u n d a m e n t a l q u e s t i o n i s n a t u r a l l y - w h a t a r e t h e i n t e r n a l q u a n t u m n u m b e r s of t h e p a r t o n s o r t h e b a s i c f i e l d s o u t of w h i c h c u r r e n t s a r e m a d e . S i n c e d e t a i l e d q u a n t i t a t i v e o b s e r v a t i o n s a r e s t i l l r e s t r i c t e d to t h e d e e p i n e l a s t i c eN s c a t t e r i n g p r o c e s s , n o t h i n g but t h e s p i n of t h e c o n s t i t u e n t s h a s b e e n a p p r o x i m a t e l y t e s t e d [3] ** In t h e p r e s e n t l e t t e r we s u g g e s t a c e r t a i n q u a n t i t a t i v e t e s t f o r b o t h the g e n e r a l i d e a of t h e p a r t o n m o d e l a s w e l l a s t h e c h a r g e s of t h e p a r t o n s . Before actually describing the test let us, however, emphasize that our argument will be critically b a s e d o n t h e f o l l o w i n g c o n j e c t u r e p r e v i o u s l y p r o p o s e d [6]: T h e b e h a v i o r of t h e d e e p i n e l a s t i c s t r u c t u r e f u n c t i o n s a t x ~ - q 2 / 2 M v ~ 0 i s g i v e n by t h e R e g g e b e h a v i o r of t h e c o r r e s p o n d i n g e x c h a n g e c h a n n e l * * * T h i s s u g g e s t i o n i s v e r i f i e d i n t h e S L A C - e x p e r i m e n t s by t h e o b s e r v a t i o n t h a t t h e n e u t r o n a n d p r o t o n s t r u c t u r e f u n c t i o n s b e c o m e e q u a l w h e n o n e a p p r o a c h e s t h e v a l u e x = 0. In t h e s p i r i t of t h e a b o v e c o n j e c t u r e t h i s i s a r e f l e c t i o n of t h e i d e a t h a t t h e P o m e r a n c h u k c o n t r i b u t i o n d o m i n a t e s i n e l a s t i c vN s c a t t e r i n g . A n o t h e r c o n s e q u e n c e of t h e s a m e i d e a w o u l d o b v i o u s l y b e t h a t F 2 ~ P ( x ) a p p r o a c h e s a c o n s t a n t w h e n x ~ 0, w h i c h i s n o t i n c o n s i s t e n t w i t h e x p e r i m e n t f o r x < ~ [3]. C o n s i d e r i n g t h e n e u t r o n a n d p r o t o n d a t a i t w o u l d n o t be u n r e a s o n a b l e to c o n c l u d e t h a t w i t h i n t h e B j o r k e n limi.t d o m i n a n c e by t h e P o m e r a n c h u k c o n t r i b u t i o n i s a c t u a l l y r e a c h e d to s a t i s f a c t o r y a c c u r a c y a t x p = ¼ S$. B a s i n g o u r a r g u m e n t s o n t h e a b o v e c o n j e c t u r e we w o u l d l i k e to c o n s i d e r t h e p r o c e s s of m a s s i v e l e p t o n p a i r p r o d u c t i o n i n h a d r o n i c c o l l i s i o n s [8]

PI + P2 ~ ~+ + ~- + anything .

(i)

For the purpose of illustration let us consider the experimentally most feasible process of two colliding protons with very high total energy ~-s in the center of mass frame. The produced lepton pair has a very high invariant (mass) 2, Q2, which is a finite fraction of s. Ideally we deal with the limiting process S Work supported in p a r t by the US Atomic Energy Commission. P r e p a r e d under Contract AT(11-1)-68 for the San F r a n c i s c o OPerations Office, US Atomic Energy Commission. * A somewhat different approach would be to extend both s c h e m e s to wider f r a m e w o r k s of a s s u m p t i o n s , in which the light cone analysis will be a p r e c i s e a b s t r a c t i o n of the p a t t o n model (R. P. F e y n m a n , private communication). ** If one admits including the not too high energy neutrino e x p e r i m e n t in the a n a l y s i s , some t e s t s of the c h a r g e s of the constituents can be made as well (ref. [5]). * * * This idea, which amounts to an a s s u m p t i o n about the interehangeability of the x "--~0 limit with the Bjorken limit, cannot be proved f r o m the light cone analysis. In the parton model it r e p r e s e n t s the existence of a smooth e x t r a p o l a t i o n f r o m the hard (finite x) parton region into the wee (xp ~ 1 GeV) domain (ref. [1]). $ Minor differences between the proton and neutron s t r u c t u r e functions at s m a l l values of x o b s e r v e d in r e c e n t deuterium data (ref. [7]) may be attributed to the too low values of Q2.

395

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Q2, s ~

PHYSICS

LETTERS

1 May 1972

1 > T ~ Q2/s = finite

(2)

to which the p a r t o n model may be applied [8]. The c r o s s section for the above p r o c e s s is given by:

d~

16n2E1E2 4~a2 f d 4 q 6 ( q 2 _ Q 2 ) f d 4 y e x p ( - i q . y ) < P l P 2 ] J e ~ m ( y ) J p e m ( 0 ) [ P 1 P 2>

dQ2 = -

s

(3)

3Q2

In the parton model the process may be essentially described in terms of parton-antiparton annihilation diagrams for which the cross-section can be written in the following form [9]: dcr _ 4~a 2 / dQ 2 3 ( V ~ 2) T

2 ' F~2(T/x) hi [F~(X) + (i ~ { ) ]

dx~ ~- i

(4)

where h i a r e the c h a r g e s of the p a r t o n s and Fg(x) a r e t h e i r m o m e n t u m d i s t r i b u t i o n functions in the p r o ton which d e t e r m i n e the ep s t r u c t u r e function:

The scaling law e x p r e s s e d in eq. (4) (i.e. the i n t e g r a l is a function of T alone) has been proposed [4, 10] as a test of the p a r t o n model. This r e s u l t may not be obtained through light cone a n a l y s i s u n l e s s some special a s s u m p t i o n s a r e made to r e l a t e the two proton m a t r i x e l e m e n t of the product of two c u r r e n t s to the c o r r e s p o n d i n g single proton m a t r i x e l e m e n t s . F r o m the knowledge of all the l e p t o n - n u c l e o n s t r u c t u r e functions, for which e x p r e s s i o n s s i m i l a r to eq. (5) exist, one may obtain in p r i n c i p l e an absolute value for the /~-pair production c r o s s - s e c t i o n . This would r e q u i r e , however, m e a s u r e m e n t of the s t r a n g e n e s s - c h a n g i n g s t r u c t u r e functions in deep i n e l a s t i c n e u t r i n o s c a t t e r i n g , which a r e s u p p r e s s e d by a factor sin 2 0 c r e l a t i v e to the s t r a n g e n e s s - c o n s e r v i n g ones and a r e therefore p r a c t i c a l l y u n m e a s u r a b l e . We would like to show now that introducing diffraction behavior into the s c a l i n g region e n a b l e s an absolute evaluation of the c r o s s section for ~ p a i r production for T < X~. Since by hypothesis the P o m e r a n c h u k contribution d o m i n a t e s the l e p t o n - n u c l e o n s t r u c t u r e functions at x < xp, one i m m e d i a t e l y concludes that: •

F~(x) =

(x) = C

forx
(6)

where C i s a constant independent of i : 2

c

xi .

(7)

In w r i t i n g eq• (6) we have a s s u m e d that the diffractive component of v i r t u a l photon nucleon s c a t t e r i n g i s an SU(3) singlet. Contributions to eqs. (4) and (5) f r o m the n o n - s i n g l e t p a r t of the P o m e r o n will be e s s e n t i a l l y s u p p r e s s e d by a factor k2trang e / ~ h 2 , which i s } in the conventional third i n t e g r a l p a r t o n quark model• F r o m eqs. (4) - (6) one finds : •

=c

1

I

r/Xp

Xp

_

1

--

f ~ ,~P(x)+c f dX ~- F~(x)

The explicit dependence of the cross-section on the scaling variable T is the following: 3(Q2)2 dcr

<

4~a 2

dQ 2

= 2c

/

dx

(8)

(9)

Xp

F r o m the electroproduction e x p e r i m e n t at SLAC [3] one finds 45: ~,4 Our subsequent analysis is not sensitive to pessibte errors (which are less than 10%) in these values. The experimental vatues were taken without their errors. 396

Volume 39B, number 3

PHYSICS LETTERS

1 May 1972

1 F2YP(x) = 0.35

for x < Xp ;

J xp

~F2~P(x)

= 0.31 .

(10)

Hence: d(~ d~

(T< Xp 2) -

4ua2 (0.073 - 0.061 l n l 6 T ) . 3(Q2)2 ~ ~./2

(11)

C e r t a i n r e m a r k s have to be made c o n c e r n i n g the l a s t equation, which i s the c e n t r a l r e s u l t of this letter: 1) our r e s u l t has been obtained f r o m the b a s i c p a r t o n f o r m u l a , eq. (4) which was d e r i v e d [9] in the l i m i t (2). In fact, eq.(4) can be shown to be an a p p r o x i m a t e consequence of the p a r t o n model as long as ~- > 2M/~-s where M ~ 1 GeV~. Moreover, one may be willing to accept this r e s u l t for lower values of s J'~. This c o n j e c t u r e i s based on the fact that scaling was observed in the e l e c t r o p r o d u c t i o n e x p e r i m e n t to o c c u r at r a t h e r low v a l u e s of v and q2, although it could be proved to hold only in the Bjorken limit. 2) Accepting the l a t t e r c o n j e c t u r e one may be tempted to test the fit of the low ~" data obtained in the Y - p a i r p r o d u c t i o n e x p e r i m e n t to our r e s u l t , eq. (11). In the conventional p a r / o n - q u a r k model (~i)t~2 = 2) one finds that the di'fferential c r o s s - s e c t i o n d(~/d ¢ ~ should d e c r e a s e f r o m 4.0 × 10 -32 c m 2 / ( G e V / c 2) to 3.0 × 10 -33 c m 2 / ( G e V / c 2) while v a r y i n g Q2 from 1 (GeV)2 to 3.5 (GeV) 2 (the energy for the p r o c e s s being given by s = 56 (GeV)2). After c o r r e c t i n g these v a l u e s for the ¢ x p e r i m e n t a l r e s o l u t i o n functions [9], one finds a v a r i a t i o n which i s too slow (i.e. behaves like (_Q2)-3/2)to fit the^experimental findings [8] in which the c r o s s - s e c t i o n v a r i e s from 10 -32 cm2/(GeV/c 2) to 5 × 10 -34 cmZ/(GeV/c ~) in the above r e g i o n (i.e. has a (Q2)-5/2 behavior). This d i s c r e p a n c y is not unexpected, however, since we would not actually t r u s t eq. i l l ) for as low v a l u e s of s as the above one. 3) Scaling in ~ p a i r p r o d u c t i o n will c e r t a i n l y be tested at the CERN ISR J'J"~. Fig. 1 d e s c r i b e s the d i f f e r e n t i a l c r o s s s e c t i o n for p r o d u c t i o n of a ~ p a i r with i n v a r i a n t m a s s M .... = ~-Q2, as obtained f r o m eq. i l l ) for s = 3000 (GeV) 2. The value ~ i h / 2 = I was used as input to fix t ~ ' s c a l e . The behavior of the e x p e r i m e n t a l c r o s s - s e c t i o n for 10Ge V < M ~ < 14Ge V will be a c r u c i a l test of the p a r t o n model. If the predcited behavior will indeed be o b s e r v e d , the o v e r a l l scale may p r o v i d e evidence for the actual c h a r g e s of the p a r t o n s . Fig. 1 also i n d i c a t e s the predicted c r o s s - s e c t i o n for production of 8 GeV p a i r s by 500 GeV p r o t o n s at NAL. 4) one may find u p p e r and lower bounds to the p r o d u c t i o n of ~v p a i r s in pp s c a t t e r i n g by r e l a t i n g this p r o c e s s to the e l e c t r o p r o d u c t i o n data in a way s i m i l a r to the one i l l u s t r a t e d above for the bt p a i r p r o duction. In the a p p r o x i m a t i o n in which the weak hadronic c u r r e n t i s c o n s e r v e d (i.e. in the l i m i t in which both the pion m a s s as well as the Cabibbo angle vanish) one finds; d(fl"tv dcr (~t+v and /~- v) = dQ 2 --- dQ 2

G26~~Mx~--Q 2) flr dx [F2P(x)F2n(~/x) "

MW2.2

-

+F2n(x)F2P(T/x)+F2P(x)F2n(~-/x) + F2n(x) F2P('r/x)] "

(12)

where p and n (~ and H) denote the n o n s t r a n g e q u a r k - p a r t o n s (antipartons). Using the proton and n e u t r o n e l e c t r o p r o d u c t i o n data one finds: 0.045

G2 "

MW2

x2

dcr ( r < X p 2) ~<

--

(13)

The upper l i m i t i s actually r e a c h e d in q u a r k - p a r t o n m o d e l s in which the s t r a n g e quark has zero c h a r g e (as in the Sakata model for example). Notice that the lower bound i s independent of Q2, when MW >> Qz. The bounds on the gv p r o d u c t i o n c r o s s - s e c t i o n at the ISR (and NAL) a r e shown in fig. 2 for 8 GeV < roll v < 15 GeV. The c u r r e n t - c u r r e n t theory of the weak i n t e r a c t i o n s (or MW >> 15 GeV) and the This condition corresponds to hard partons in the center of mass frame. "~f Drell and Yan [9] analyze the results of the experiment described in ref. [8] in terms of eq. (4), aithough the constraint ~- > 2M/x/s is not obeyed in the experiment. ~ " An ISR experiment to search for /.t pairs and the W-boson is presently in operation. 397

Volume 39B, n u m b e r 3

PHYSICS

LETTERS

1 May 1972 5.0

4O + 2~5 ~ - 30

\

>

E

~.

2.0

::L :t E 20

I IE=L 1.5

o

0

f

Io

1.0 I

r

I

I

I

I

I

I

8

9

po

Jr

12

13

t4

15

I

i

/

J

i

I

m,,,, [GeV/C z]

m~,~ (GeV/C 2)

Fig. 2. Bounds on dtT/dmlL ~ obtained f r o m eq. (13) for the CERN ISR (shaded arena) and NAL (vertical line).

Fig. 1. dG/d_m/.t~ obtained from eq. (11) for the CERN ISR (solid [ine) and NAL (cross).

c o n v e n t i o n a l p a r t o n - q u a r k m o d e l h a v e b e e n u s e d . T h e e x i s t e n c e of a W - b o s o n w i t h a m a s s i n t h e v i c i n i t y of the a b o v e - d o m a i n w o u l d o b v i o u s l y e n h a n c e the c r o s s - s e c t i o n . A deviation from the limits desc r i b e d i n fig. 2, w h i c h m a y b e o b s e r v e d i n f u t u r e e x p e r i m e n t s , c o u l d h e l p i n d e t e c t i n g t h e W - b o s o n a n d determining its mass. In c o n c l u s i o n , t h r o u g h t h e i n t r o d u c t i o n of d i f f r a c t i o n b e h a v i o r i n t o t h e s c a l i n g r e g i o n o n e a t t r i b u t e s to t h e p a r t o n m o d e l a q u i t e s t r o n g p r e d i c t i v e p o w e r in t h e p r o c e s s e s of l e p t o n p a i r p r o d u c t i o n . In t h e s p i r i t of the c o n j e c t u r e d b e h a v i o r o n e s h o u l d e x p e c t to f i n d i n the d e e p i n e l a s t i c n e u t r i n o e x p e r i m e n t s at NAL the value F2uP(x)

= F 2 v n ( x ) = 2F2YP(0) / ~ ~/2

f o r x < Xp .

T h e a u t h o r w o u l d l i k e to t h a n k D r s . R. P . F e y n m a n , discussions.

S.D. Drell,

R. L. J a f f e a n d Y. Z a r m i f o r s t i m u l a t i n g

.References [1] H. P. Feynman, Proc. Third High energy collision Conf. at State University of New York, Stony Brook (Gordon B r e a c h 1970) ; J. D. Bjorken and E. A. P a s c h o s , Phys. Rev. 185 (1969) 1975. [2] H. F r i t z s c h and M. GeH-Mann, talk presented at Coral Gables Conf. on Fundamentat i n t e r a c t i o n s at high energy, University of Miami, Coral Cables, Florida 1971. [3] E. D. Bloom et a[., SLAC PUB 796, p r e s e n t e d at the Fifteenth ]ntern. Conf° of High e n e r g y physics, Kiev, USSR; J . I. F r i e d m a n e t a [ . , SLAC PUB 906 (1971). [4] S. D. Drel[, R a p p o r t e u r ' s Report at A m s t e r d a m Intern. Conf. on E l e m e n t a r y p a r t i c l e s , A m s t e r d a m , 1971. [5] O. Nachtmann, Orsay P r e p r i n t LPTHE 71/29. [6] H. D. 1. Abarbanei, M. L. Gotdberger and S. B. T r e i m a n , Phys. Rev. L e t t e r s 22 (1969) 500; H. H a r a r i , Phys. Rev. L e t t e r s 22 (1969) 1078. [7] H. Kendall, talk p r e s e n t e d at Fifth Intern. Symp. on Electron and photon i n t e r a c t i o n s at high e n e r g i e s , Cornell, 1971. [8] This p r o c e s s has been o b s e r v e d by J. H. C h r i s t e n s e n et a[,, Phys. Rev. L e t t e r s 25 (1970) 1523. It has been d i s cussed both in the p a t t o n model (S. D. Dre[l and T. M, Yah, Phys. Rev. L e t t e r s 25 (1970) 316) as well as in the [ight-cone-expans$on-approach (G. Aitare[[i, R.A. Brandt and G. P r e p a r a t a , Phys. Rev. L e t t e r s 26 (1971) 42). [9] S.D. DrelI and T.M. Yan, ref. [8] and Ann. Phys. (N.Y.) 66 (1971) 55. [10] R.L. Jaffe, Phys. L e t t e r s 37B (1971) 517. 398

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