VOLUME 24, NUMBER 15 PHYSICAL R E V I E W LETTERS 13 APRIL 1970

FINAL-PARTICLE CORRELATIONS IN D E E P INELASTIC LEPTON PROCESSES*

Sidney D. Dre l l and Tung-Mow Yan Stanford Linear Accelerator Center, Stanford University, Stanford, California 94305

(Received 16 February 1970)

A study is made of the additional information learned from measurements of the cor re ­lation of two (or more) detected particles in the final states of deep inelastic lepton pro­cesses. Generalized Bjorken limits are derived in which the new structure functions de­pend only on the ratios of the kinematical variables available, just as do the more famil­iar structure functions Wt and vW2 for electron-proton scattering in the deep inelastic region. Experimental implications are discussed.

The deep inelastic lepton-hadron processes of

e + p - e' + "anything," (i)

e +e - hadron + "anything," (ii)

and

v + p - e + "anything" (iii)

have been studied using the formalism of canoni­cal field theory.1 Within this framework the "parton" model2 has been derived and the Bjor­ken limiting3 behavior established for the invari­ant structure functions. A simple physical pic­ture emerges for these processes in which only one final particle is detected, by making full use of unitarity and summation over all unobserved final states. For instance, when viewed in an in­finite-momentum frame of the target, (i) appears as an incoherent superposition of elastic scatter­ings from the virtual constituents of the proton. These constituents behave as if they were point­like, structureless particles and the strong-in­teraction dynamics is isolated solely in the de­scription of the proton structure in terms of these constituents. The particles present imme­diately after the scattering propagate freely and independently as if there is no interaction among them. This is the so-called parton model and is illustrated in Fig. 1(a). In actuality, of course, interactions do occur among the particles after the elementary scattering act of the virtual pho-

(a) :b)

FIG. 1. Deep inelastic scattering of an incident pro­ton by a current interacting at X. All the initial and final-state interactions illustrated by the blobs in dia­gram (b) in the Bjorken limit reduce to (a) in the par-ton model.

ton. For a given set of particles in the final state an actual scattering event looks like the one shown in Fig. 1(b). The blobs in the figure rep­resent all possible interactions. However, the two groups of particles, labeled (A) and (B) in Fig. 1(b), do not interact because they are sepa­rated by an asymptotically large transverse mo­mentum of magnitude Q2, where -Q 2 is the invar­iant momentum transfer squared from the virtual photon. Completeness of the final states and uni­tarity make it possible to obtain from the true picture, as given in Fig. 1(b), the simplified overall picture of the parton model as represent­ed in Fig. 1(a).

Similar results can be derived for (ii) and (iii) and the three processes can be interrelated through the formal field-theory framework. A basic ingredient in this work is the assumption that there exists an asymptotic region of large Q2 relative to the transverse momenta of the had­ron constituents, virtual or real, when viewed in the infinite-momentum frame. A momentum cut­off was introduced to ensure existence of this asymptotic region as discussed fully in earlier papers.1

In this Letter we report results of a study of the additional information learned from measure­ments of the correlation of two (or more) detect­ed particles in the final states. We find that there exists a generalized Bjorken limit in which the new structure functions depend only on the ratios of the kinematical variables available, just as do the more familiar structure functions Wx and vW2 for electron-proton scattering in the deep inelastic region. Characteristics of the an­gular correlation between the detected final par­ticles are also discussed0 Since these results are subject to direct experimental test we report them here reserving details for a more complete publication.

Present experimental data4 indicate that for Q2

>0.5 (GeV/c)2, Wx and vW2 are consistent with the Bjorken limiting behavior.3 It is of extreme

855

VOLUME 24, NUMBER 15 P H Y S I C A L R E V I E W L E T T E R S 13 APRIL 1970

theore t i ca l i n t e r e s t to see if the genera l ized Bjorken l imit d i scussed in th is Let ter wil l a l so be reached at such low values of Q2.

Fo r ine las t ic e l ec t ron sca t t e r ing with detection of the final e lec t ron plus one hadron t h e r e a r e four invariant v a r i a b l e s to desc r ibe the hadron s t r u c t u r e :

Q2 = - ^ > 0 , v=P-q/M, v^P^q/Mv

K^P'PJM, (1)

where P^M and Plii,M1 a r e the four -momentum and m a s s of the ini t ial proton and the detected hadron, respec t ive ly . The two new va r i ab l e s a r e i/1? the energy loss of the e lec t ron in the r e s t f r ame of the detec ted hadron, and Kly the energy of the detected hadron in the labora tory sy s t em. Averaging over al l sp ins , t he r e a r e four s t r u c ­tu re functions for the hadrons . However we shal l immedia te ly simplify to a study of the differen­t ial c r o s s sect ion a s a function of the four v a r i ­ab les (1) by integrat ing over the ex t r a az imuthal angle assoc ia ted with d3Px. This gives

d4a Ana2 / e ' dQ2dvdK1dvl {(^)2\e

— ] [W2 cos2(0/2> + 2V?X s in 2 (0/2)] , (2)

where € is the incident energy and e ' , 6 a r e the energy and angle of the sca t t e r ed e lec t ron in the labo­r a t o r y sy s t em. Initial spins a r e averaged and final ones summed over . The two s t r u c t u r e functions ^ 1 2 a r e defined by

: - ( ^ - ^ r ) ^ v, *i, vd +JF(pi*-^r«*) pv-^r<iv )^M\ », *i, *i),

where J^ is the hadronic e lec t romagnet ic cu r ren t opera to r ; \p) is a one-proton s ta te and [P^ is a s ta te of the one hadron being detected plus al l poss ible o thers with quantum numbers s u m m a ­r ized by n\ and spin ave rages a r e implici t in this definition. Equation (2) has the s a m e form as the corresponding express ion d2a/dQ2dv for (i) b e ­cause the re a r e no additional t e n s o r s from which to const ruct the cu r r en t - conse rv ing t enso r "W^ after we per form the azimuthal average indicated. T h e r e a r e now four s c a l a r va r i ab les q2, v, KX, and vx which satisfy the kinemat ical cons t ra in t s

(q+P)2>M2 o r 2 M I A - Q 2 > 0 ,

(3)

or

(q+P-P1)2>0

2MV-Q2-2MK1-2M1V1 +M2 +M2 > 0. (4)

As before we analyze (3) in the inf in i te-momen­tum c e n t e r - o f - m a s s f rame of the ini t ial e lec t ron and the init ial proton, "undres s ing" the cu r r en t by the uni tary t rans format ion U(t): Jll(t) = U~1(t) xJii{t)U(t). Only the good components (JLL = 0 , 3) of the hadronic e lec t romagnet ic cu r ren t a r e consid­e red so that the e a r l i e r d i scuss ion in t e r m s of good and bad ve r t i ce s can be imita ted in toto. The key to the analys is he re as in Ref. 1 leading to a par ton model is th i s : The final pa r t i c l e s a r e

divided into two we l l - sepa ra t ed , well- identif ied, and noninteract ing groups , (A) and (B) as labeled in Fig. 1(b). Moreover , the pa r t i c l e s in each of the two groups move close to each o ther , due to our assumed t r a n s v e r s e - m o m e n t u m cutoff. There fore the invar iant m a s s e s of the two groups (A) and (B) a r e separa te ly finite and negligible as compared with Q2 o r Mv and hence energy as well as momentum is effectively conserved in the Bjorken limit a c r o s s the pointlike e l ec t romagne t ­ic ve r tex j^(x). F r o m this d i scuss ion we come to a c l ea r specific exper imenta l predic t ion a l ready made in Ref. 1: In the labora tory sys tem any ob­se rved final hadron will e m e r g e within a cone of width ^ l m a x ~ 4 0 0 MeV along the momentum-t r ans fe r d i rect ion in group (B) of Fig. 1 and with a finite fraction of the la rge energy v, o r it will be "left behind" and emerge in group (A) with low energy and momentum ^ ± m a x ~ 4 0 0 MeV. Con­cerning the dependence of the s t r u c t u r e functions on the s c a l a r va r i ab les (1) we must cons ider these two poss ib i l i t ies separa te ly .

Consider f i r s t the case in which the detected hadron or ig ina tes from group (B) and e m e r g e s along the di rec t ion of q in the lab. Then the two new var iab les u1 and Ky in (1) a r e not independent but a r e kinematical ly re la ted in the Bjorken limit by co = 2Mv/Q2 = -MKJM^^ This re la t ion is

856

VOLUME 24, NUMBER 15 P H Y S I C A L R E V I E W L E T T E R S 13 APRIL 1970

readily derived by introducing P s , the momentum immediately after scattering of the charged con­stituent which interacts with the virtual photon. Neglecting the bounded, finite transverse momen­tum relative to q±, we have PB^ = {l/oo)P11 +q^ in the infinite-momentum frame where (l/oo)P is the longitudinal momentum of the charged constituent before scattering. This just states that both en­ergy and momentum are conserved to leading or­der in Q2 and Mv across the electromagnetic ver­tex. If r]1 denotes the fraction of the longitudinal momentum of the detected hadron with respect to the scattered charged constituent, we can write -Pi = 77iJDB,+kJL; k±*Pj?=0, l>?71>0. The same ap­proximation as above gives P / =rji^>Bli =771[(l/ I

o))Pp +q^]. From this and definition (1) it follows to leading order that MKX =PX 'P = r\1Mv\ Mxvx

= P1-4 = -2il1Q2or

Vi = *Jv= I / ^ I ; M\v\ = - ( lA^M/q (5)

Thus as K1 and v1 individually become infinite their ratio is finite and determined, as we claimed above. Furthermore, it follows from (4) that 0 < l / w 1 < l . For later convenience we also define

u1 = M1v1/Mv = -l/(jt)u)1. (6)

Next we turn to establishing the analog of Bjorken scaling behavior for the structure func­tions. Inserting (5) and (6) and "undressing" the current as in Ref. 1 we obtain from (3)

A 2EPM1

"" MM v i^+^)^f{dx)eiq'Xd3p^-^){up ihwfcfi) (7)

The delta function outside of the integral in (7) just expresses the physical relation (6) arising from the finite bound on the transverse momentum. Since the U matrix acts on the two groups of final par­ticles (A) and (B) independently and separately it can be removed from group (A) because we sum over all possible states in group (A) and the total probability for anything to happen is unity. Also, •taking into account the fact that the undressed current j^ (x) is a one-body operator which scatters a single charged constituent denoted by Xa to momentum VB as defined earlier, we write

A 2EpMx 1 5 (u^)f{dx)eiQXn £ssJ

d3p^-^) \ '""l/-' nA,nBX,,s,s'J l

:(Up\jli(x)\nA+\a>s)(PB,\as\U(P1nB))((nBP1)U~1\PB,Xa,s'\jv(0)\Up), (8)

where nA and nB denote the two distinct subsets of states in Fig. 1(b) over which we sum and a com­plete set of such states has been introduced.

We can now perform the final-state integral formally, writing

Mi M (9)

where x is a two-component Pauli spinor. Since VB is the only preferred direction, / x (o)x, a) can de­pend on spin only through the combination ( P B ' C T ) 2 = P B

2 , as dictated by rotational invariance and par­ity conservation. We conclude therefore that/Xa is spin independent and s=s'9 which leads to the second form. Furthermore, the left-hand side of (9) is independent of the scale of the infinite momen­tum PB, sofXa is a function of the ratio cox only. This gives finally

< w^ = ^ 6 ( w i + i ) S wnv(KVx*(«>i)>

where

Wllv(\a) = 47r*^f (dx)eic?x (Up\j,(x)jv(0)\up)} (10)

is the total contribution of Wilv for process (i) from a particular type of charged5 constituent \a. Equation (10) in conjunction with the scaling results already derived for ^ ^ ( A ) shows that Mi^^x

857

VOLUME 24, NUMBER 15 P H Y S I C A L R E V I E W L E T T E R S 13 APRIL 1970

and ^%^2 are functions of the ratios co and OJ1 in the Bjorken limit:

(11)

This result states that the overall probability is a sum of contributions from each parton. The con­tribution from an individual parton is the probability that is kicked out times the probability that one of its decay products carr ies a fraction l/o)1 of its energy. In that this latter probability is indepen­dent of the actual energy of the parton, this result is reminiscent of the fact that the fractional momen­tum distribution of partons in the original nucleon is independent of the nucleon's energy.

For spin-! current contributions, 3^ and £F2 have a fixed ratio independent of co1?

^(co, u)lf Uj) =icoOr2(a>, o)1?uj;

and for spin-0 current contributions, 9r1 vanishes,

3r1(co,w1,M1) = 0.

(12)

(13)

Equations (11)-(13) are our central result of generalized scaling. This is a nonvanishing and hence nontrivial result because a sum over all charged constituents and kinematic values of ul} u)l9 and co gives the total inelastic cross section as a lower bound. According to present experimental indica­tions4 the spin-f current contribution is dominant. Hence the group of energetic recoiling particles in (B) should include a baryon or antibaryon; one of the octet according to our model. We have no prediction in our model on the ratio of p 's to n's appearing, plus "anything else," in the final state. Moreover any individual channel will have a rapidly decreasing production cross section, and it is only the aggregate sum of all possible channels (inclusive2 measurements) that survive in the Bjorken limit. This is a crucial prediction of our model that can be checked. It is very different from Hara-r i ' s 6 prediction of large "diffraction" production of single pQ,s only by very virtual photons.

For the second case when the detected hadron originates from group (A) one learns little more than this: (a) It moves with finite momentum in the lab system, i.e., it is one of the constituents "left be­hind" after the impact by the virtual photon. In this case therefore the new ratio KJV vanishes in the Bjorken limit, (b) As in the previous case MveW1/v

2eW2 = oo/2 for a spin-! current playing the dominant role and vV?l = 0 for a spin-0 current.

Finally we turn to the annihilation process e +e-*H1+H2 + "anything," where H denotes an arbitrary (anti-)hadron. Defining the variables as in (1) and doing the angular average over P 2 with fixed vx and KV we find the differential cross section, similar in form to dv/dEd cosO for (ii),

da dEdcosOdK^dv (q2)

4ua2 M2v ( #2_\1/2

~W " — 2Mv A q2

2 ^ + -^(l-f2-vV?2

2M sin20 •]• (14)

where E and 0 are the energy and angle with respect to the e +e * colliding beam of the_hadron with four-momentum P p . The relation of Wto V? in (3) is similar to that between W and W in Ref. 1 for integrated cross sections. Although the four-momentum P^ of the first hadron is fully specified, only the two invariant conbinations KX and vx for the second hadron are given. No information is lost in this angular averaging over the second hadron, in the absence of spin or polarization information, because in the Bjorken limit the two hadrons are approximately parallel or antiparallel to one another.

For the case that the second hadron emerges predominantly back to back relative to the first had­ron, an analysis similar to the one leading to (11)-(13) shows that for the annihilation process we have

w w i / x

limBj ^3W2 = ff2(u)9 W l , ux) = 6 M l _ — £ F 2 X M / X K ) , (15)

858

VOLUME 24, NUMBER 15 P H Y S I C A L R E V I E W L E T T E R S 13 APRIL 1970

where ux and coj a r e defined a s in (5) and (6), and

3^(0), wl9 ux) = ((JO/2)$2(<JO, d)l9 ut) ( s p i n - i cu r ren t ) ,

^ ( o ; , w19 ux) = 0 (spin-0 cu r r en t ) . (16)

In (15), Flt2X(u>) i s the total contribution to F12(u)jm (ii) from a charged consti tuent of type A. It has been verif ied explici t ly1 in III for our model that Flt2(w) i s the j same function o£ co a s Fh2(u)) continued from a>>l to co<l . This appl ies sepa ra te ly for each A, i . e . , FlX(u)=FlX(u); F2X((JO)=F2X(OO). F u r t h e r ­m o r e , /x(<^i) in (15) i s the s a m e function a s the one which appeared in (11). No continuation i s n e c e s ­s a r y h e r e , s ince a^ in both ca se s v a r i e s between the s a m e range 0 and 1. Equations (15), (16), and (11) imply

^i(w, colf uj = ff^co, u)l9 -uj; 52(u>, wl9 uj = JF2(w, ul9 - n j . (17)

On the r ight -hand s ide, the functions $lt2(w, < i, ~ui) a r e continued from co> 1 to ct><l. The continuation in the var iab le ux i s t r iv ia l s ince the ux dependence on both s ides i s explicit ly known in (15) and (11). Similar r u l e s can be der ived for the second case of pa ra l l e l hadrons .

When m o r e than two final p a r t i c l e s a r e detected the c r o s s sect ions have the s ame form a s in (2) and (14), a s suming that a l l az imuthal ave rages a r e taken, and the sca l ing p r o p e r t i e s (11) and (15) can be genera l ized in t e r m s of the s c a l a r s vt and Kt for a l l detected p a r t i c l e s . A m o r e deta i led d i scuss ion of these r e s u l t s will be published e l sewhere .

*Work supported by the U. S. Atomic Energy Commission. 1S. D. Drell, D. J. Levy, and T. M. Yan, Phys. Rev. Letters 2£, 744 (1969), and Phys. Rev. JL87, 2159 (1969),

and SLAC Reports No. SLAC-PUB-645, 1969 (Phys. Rev., to be published), and No. SLAOPUB-685, 1969 (Phys. Rev., to be published); T. M. Yan and S. D. Drell, SLAC Report No. SLAC-PUB-692, 1969 (Phys. Rev., to be pub­lished). The last four papers will be referred to as Papers I, II, IE, and IV, respectively.

2R. P. Feynman, Phys. Rev. Letters 23j 1415 (1969); see also J. D. Bjorken, in Selected Topics in Particle Physics, International School of Physics "Enrico Fermi," Course XLI, edited by J. Steinberger (Academic, New York, 1967).

3J. D. Bjorken, Phys. Rev. 1£9, 1547 (1969). 4E. Bloom et al. , Phys. Rev. Letters 23, 930 (1969); M. Breidenbach et al. , Phys. Rev. Letters 23, 935 (1969);

R. Taylor, SLAC Report No. SLAC-PUB-677, 1969 (to be published). 5In the field-theory model of Ref. 1, Xa can b e a ^ , />, or p. If their separate contributions can be untangled in

processes (i), (ii), and (iii), the oo dependence in (10) will be determined and the correlation measurements will yield fXa(ut) directly.

6H. Harari, Phys. Rev. Letters 24, 286 (1970).

859