Scaling laws for large-momentum-transfer processes

P H Y S I C A L R E V I E W D V O L U M E 11, N U M B E R 5 1 M A R C H 1 9 7 5

Scaling laws for large-momentum-transfer processes*

Stanley J. Brodsky Stanford Linear Accelerator Center, Stanford University, Stanford, California 94305

Glennys R. Farrar Calijornia Institute of Technology, Pasadena, California 91109

(Received 4 September 1974)

Dimensional scaling laws are developed as an approach to understanding the energy dependence of high-energy scattering processes at fixed center-of-mass angle. Given a reasonable assumption on the short-distance behavior of bound states, and the absence of an internal mass scale, we show that at large s and t , d o / d t ( A B -I C D ) - s -" "f (t /s); n is the total number of fields in A , B , C , and D which carry a finite fraction of the momentum. A similar scaling law is obtained for large-p - inclusive scattering. When the quark model is used to specify n , we find good agreement with experiments. For instance, this accounts naturally for the (q 2)-2 asymptotic behavior of the proton form factor. We examine in detail the field-theoretic founddtions of the scaling laws and the assumption which needs to be made about the short-distance and infrared behavior of a bound state.

I . INTRODUCTION

The dimensional scaling

for the asymptotic behavior of fixed-angle sca t te r - ing appear to compactly summar ize the resu l t s of a broad range of hadronic scat ter ing, photoproduction, and elast ic f o r m fac tor measurements . The integer n is given by the (minimum) total number of lepton, photon, and elementary quark f ie lds carrying a finite f ract ion of the momentum i n the part ic les A , B, C, and D. The scaling laws represent , i n the s imples t possible manner , the connection between the degree of complexity of a hadron and i t s dynamical behavior.

One of the mos t important consequences of E q . (1) is i t s application t o e las t i c electron-hadron scat ter ing. This rule immediately connects the asymptotic dependence of the (spin-averaged) electromagnetic f o r m factor to the minimum num- b e r s of f ie lds n, i n the hadron:

Thus, using the quark model, we have B ( t ) - t -' f o r mesons and F l ( t ) - t-' f o r baryons. We a l so find ( see Sec. I1 D 2 ) F2 - t -3, and thus G, - G , sca l - ing. All of these resu l t s a r e consistent with the asymptotic dependence indicated by present experiments . In Sec. I11 of th i s a r t i c le we survey presen t data which a r e relevant t o testing the sca l - ing laws, Eq. (1). They f a r e very well, since they a r e consistent in al l c a s e s and accurately verified f o r yp-i'rp and p p - p p . We predict &/dt - s-7 and s-lo, respectively, a t fixed c .m. angle; experiment gives s-7.3* 0.4 and ,-9.7*0.5 . A catalog of

predictions which can be tested i n the future is a l - s o given i n Sec. 111.

In fact , dimensional analysis and some simple assumptions immediately lead to the scaling law of Eq . (1). Imagine that a hadron is a bound s ta te of n, constituents, each of which c a r r i e s a finite f ract ion of the total hadron momentum. The amplitude f o r the scat ter ing of a system of hadrons (see Fig. 1 ) is therefore related to the amplitude f o r the scat ter ing of their constituents, integrated over possible constituent momenta with the constraint that the constituent momenta add up to the hadron momenta. If the total number of fields i n the initial and final s ta tes is n, the Feynman amplitude M, h a s dimension [length] " -4 when the conventional normaljza_tion of s ta tes is chosen [ (p Ip') = 2 ~ 6 ~ ( p - ~ ' ) ] . If, a t l a rge energy and momentum transfer , 6-' is the only length scale f o r the amplitude, M,, - (&)4-n f ( t /s) . Now i f each of the n, constituents of hadron H c a r r i e s a finite f rac - tion of the momentum, say, i n the hadron r e s t f rame, integrating over the possible momenta of the constituents can never introduce a dependence on s . Thus the amplitude, M, f o r the physical p rocess has the s a m e asymptotic behavior a s hl, . Equation (1) then follows immediately since

More generally f o r any exclusive process A + B ' H I + . . . H , ,

f o r l a rge s a t fixed invariant rat ios . Alternatively, using the quark model we have simply the scaling

1310 S T A N L E Y J . B R O D S K Y A N D G L E N N Y S R . F A R R A R 11

law

f o r the exclusive c r o s s section integrated over any fixed center-of-mass region when the ra t ios of invariants s tay finite. Here NM ( N B ) is the total number of mesons (baryons) i n the initial and final s ta te .

The s a m e model and dimensional counting ru les may be applied to inclusive processes A + B- C +X, where C is detected a t l a rge t ransverse momentum.' In th i s c a s e only a subset N of the constituent f ie lds need to participate in the l a rge- angle scat ter ing process ; the r e s t remain a s "spectators." The resul t f o r s >>M2, fixed in- var iant rat ios , is

where M 2 is the missing m a s s . The subset N which contributes to the scaling of a given hadronic p rocess can be model-dependent; we d i scuss this fu r ther in Sec. IV. Fur thermore , a s shown i n Ref. 3, the dimensional rule allows one t o predict the threshold dependence a t the exclusive bound- a r y [i.e., M2/s = (I-PC,, /p , , , ) - 0 a t fixed bc,m]:

where n i s the total number of spectators in A , B, and C . Thus, f o r example, in deep-inelastic e-p scat ter ing N = 4 (for eq- eq) and i = 2 (for the two spectator quarks i n the proton) and we recover both s c a l e invariance f o r vW, and the Dre l l - Yan-West behavior vW, - (1 - x ) ~ f o r x- 1 . Pos- s ible spin modifications a r e reviewed by Ezawa, Ref. 4. A s shown inRef. 1 , the inclusive-exclusive connection of Bjorken and Kogut gives the relation n = N + P + 1, where n is the number of f ie lds involved i n the exclusive scat ter ing [ see Eq. (311.

T h e above scaling laws s h a r e a significant fea- t u r e with the predictions of parton models: The c r o s s section multiplied by a power of s becomes a universal scale-independent function, dependent only on ra t ios of invariants . In contrast , if had- r o n s were homogeneous objects, no elementary

FIG. 1. n-point "decomposition" of A B - CD (in this example n = 10) .

constituent would c a r r y a finite f ract ion of the total momentum, and la rge- t ransverse -momentum hadron scat ter ing would take place via the cumulative effect of a n infinite number of soft interact ions. Exponential damping in t ransverse momentum is therefore t o be expected i n such a model.5

I t is evident that m o r e careful reasoning is nec- e s s a r y in o r d e r to establish that Eqs . (1)-(5) should be t rue. That is the purpose of Sec. I1 of th i s a r t i c le . The principal i s sue of course is whether m a s s e s o r binding energ ies could s e t the sca le r a t h e r than s . In the deep-inelastic c a s e that s e e m s not to happen when q2 and v become sufficiently l a rge . We a rgue in Sec. I1 that f o r exclusive scat ter ing when a l l kinematic variables a r e l a rge i t is likely that again only those la rge invariants s e t the sca le . Of course, purely dimensional reasoning cannot specify possible powers of logari thms. Hence al l the scaling laws discussed above mus t be regarded by the r e a d e r a s t r u e "modulo logs ." In Sec. I1 o u r analysis of renormalizable field theories will allow u s to be m o r e prec i se about logarithmic modifications to canonical power-law scaling.

The s imple model of the hadron in which i t s momentum is partitioned among i t s constituents s o that each quark has a finite f ract ion t u r n s out to be v e r y useful f o r a broad range of hadronic - scat ter ing calculations-especially those involving multiquark s ta tes . An application to effective Regge t ra jec tor ies and residue functions is given i n Section I1 D 2. Section I11 is devoted to the experimental situation and Sec. IV briefly deals with inclusive-reaction applications. Appendix A gives a detailed example of a Born amplitude calculation. Finally, Appendix B shows that the fixed- angle unitarity bound on the asymptotic behavior of scat ter ing amplitudes is the s a m e a s the dimensional scaling law /MN 1 - s4-".

11. EXCLUSIVE SCALING LAWS

The crucial s t e p s i n the dimensional analysis of the scaling laws given in the Introduction a r e (a) the effective replacement of the composite hadron by constituents carrying finite f ract ions of the hadronic momentum, and (b) the absence of any m a s s sca le in the amplitude M,, o r binding correct ions to i t . It is evident that a super - o r non-renormalizable field theory could not satisfy these conditions s ince such theories contain a fundamental length (in the coupling constant) which necessar i ly s e t s the scale . T h i s is in contrast with renormalizable perturbation theories in which m a s s s c a l e s enter through propagators and ex- t e rna l m a s s e s . In fact, the required conditions

11 - S C A L I N G L A W S F O R L A R G E - M O M E N T U M - T R A N S F E R P R O C E S S E S 1311

(a) and (b) a r e natural fea tures of renormalizable field theories , given cer tain dynamical assumptions concerning the nature of the Bethe-Salpeter wave function, the absence of infrared effects, and the accumulation of logari thms.

In this section we shal l systematical ly investi- gate the validity i n renormalizable field theor ies of the dimensional argument and i t s underlying assumptions. In o r d e r t o examine the effects of binding in bound-state scat ter ing (a) , we shal l use the Bethe-Salpeter f o r m a l i s m . ~ h i s enables one to separa te the behavior of individual graphs f rom that due t o the infinite summation required t o fo rm bound s ta tes . By definition the hadronic amplitude i s given by the convolution of the hadronic wave functions and a n n-particle amplitude 1% integrated over relative momenta ky ( see Fig. 2 ) :

We shal l d i scuss jb), the asymptotic behavior of ,If,, , by explicitly examining i t in perturbation theory. We shal l show that with the following th ree assumptions, the scaling laws [ E ~ s . (1)-(4) ] a r e cor rec t (modulo logari thms) in any renormalizable field theory:

(A) The physical mesons and baryons a r e s-wave Bethe-Salpeter bound s t a t e s of quark-antiquark and th ree quark fields, respectively, such that the wave functions a r e finite when the quarks have z e r o separat ion i n coordinate space and vanish f o r l a rge coordinate separation. Thus the la rge momentum components of the wave function a r e restr ic ted; e.g., f o r the mesons we have

Moreover , s ince the coordinate space extent of the wave function is bounded, the wave function is finite a t every point i n momentum space.

(B) The large-momentum-transfer interact ions of the constituents a r e asymptotically scale-in- var iant .

(C) Multiple ( L 2 2) scale-invariant interact ions

FIG. 2 . Schematic representation of the full meson- meson scattering amplitude in t e rms of Bethe-Salpeter wave functions $ and the irreducible amplitude I\?', .

between the constituents of different hadrons can be neglected.

Assumption A is necessary s o that binding c o r - rections a r e l imited. Then the computation of Mn involves the scat ter ing amplitude obtained by r e - placing each hadron by a collection of quarks of the appropriate spin, each constituent carrying a finite f ract ion of the hadron's momentum. Note that if we turn off the binding adiabatically, the constituent momenta a r e pi = (mi, 6 ) in the r e s t sys tem and p, =xipH, xi =nz,/mH i n a general f r a m e . Assumption (A) implies that t h e r e a r e no e le - mentary f ie lds with the quantum numbers of the hadrons. Because of assumption (A), the d 4 k , integrations a r e convergent in Eq . (7), and i t is easy to s e e that the scaling behavior a t fixed angle of M is given by the scal ing behavior of .Mn multiplied by finite coefficients of o r d e r $ (x=O) . Note that i n the c a s e of nonzero orbi tal angular momentum, helicit y constraints , o r quantum number restr ic t ions, the amplitude IVZ could fal l by additional powers of s.

Assumptions (B) and (C) a r e necessary to insure that Mn - (-6)4-n. Assumption (C) s e r v e s to e l im- inate asymptotic contributions to 121, f rom d is - connected graphs ( see Fig. 3) . Recently Land- shoff7 h a s made the important observation that ?.I, d ( G ) 4 - n i f hadrons can sca t te r a t l a rge angles by successive independent, near -mass -she l l e las - t ic scat ter ings of each constituent of one hadronoff a constituent of the other (as in Fig. 3 ) . In fact ( see Sec. I1 D 1 and Appendix A) i n this c a s e Nn - ( 6 i 4 - " (6)L-1, where L is the number of p a i r s of constituents f r o m different hadrons which have a large-angle scale-invariant interact ion. (It should be noted that whether o r not this p r o c e s s t akes place in hadron-hadron scat ter ing, there is no modification f rom such a phenomenon t o f o r m fac tors o r to fixed-angle p rocesses involving photons o r leptons.) However, t h e r e is both d i rec t and indirect evidence that Landshoff's mechanism is not physically important a t l eas t a t p resen t energies . The d i rec t evidence is that i n pp- pp,

YI p

Y* PI'

( I - Y , - Y * ) P ' / \

FIG. 3 . Example of a disconnected or nonplanar diagram forpp -pp wide-angle scattering.

1312 S T A N L E Y J . B R O D S K Y A N D G L E N N Y S R . F A R R A R - 11

&,,I& - s-9.7+ 0.3 f ( t / s ) ( see Ref. 8) r a t h e r than the

s-*, a s would be predicted if L = 3. More indirect but equally important i s the fact that no high-energy, fixed-angle scale-invariant interaction between quarks of different hadrons s e e m s to occur in nature. The best evidence of th i s is f rom high- p , inclusive pion production ( pp - n + X) whose c r o s s section fa l l s much f a s t e r than the a t fixed p , / 6 predicted if such a scale-invariant interaction between quarks of different hadrons did occur ( see Sec. IV f o r detai ls) . F o r these empir ical reasons, then assumption (C) is nec- e s s a r y .

The organization of this section is a s follows. We proceed by f i r s t (Sec. I1 A-C) giving the con- struction of an amplitude in t e r m s of Bethe-Sal- pe te r wave functions in the spinless-constituent case , using the meson form factor a s a n example (IIA). We show that with assumption (A) the asymptotic behavior of the full amplitude is the s a m e a s the behavior of the dominant i r reducible contribution, M, . The extension of the analysis to spin-a constituents is not difficult and is given in Sec. I1 B. We show that the meson form factor has the s a m e behavior in the spin-+ a s i n the spin- l e s s c a s e . T h i s is an appropriate point to ex- plain why, even though shor t dis tances a r e being probed, bound s ta tes of spin-0 f ie lds (whose dimension is [L] -') and spin-$ f ie lds (dimension [L] -312) have the s a m e behavior in large-momentum-transfer exclusive scat ter ing. In Sec. I1 C we confront the difficult question of the validity of our assumptions (A) and (B). Not very much is known about the short-dis tance and infrared behavior of Bethe-Salpeter wave functions. We r e - view what is known, give s o m e plausibility arguments in favor of o u r assumptions, and speculate on the situation in non-Abelian gauge theor ies . Modifications in our resu l t s when the wave function a t short dis tances is not finite a r e discussed.

Having established that assumption (A) reduces the problem to the behavior of i r reducible graphs, we examine the lowest-order i r reducible (Born) graphs f o r a number of interest ing processes in Sec. I1 D. We s t a r t (I1 D 1) with the Landshoff dia- g r a m s and show why they violate the dimensional resu l t . We presen t s o m e speculative arguments in favor of assumption (C), which allows us to neglect the i r contribution. Next (IID 2 ) we show

that the behavior of the connected Born d iagrams reproduces o u r scaling law independent of the spin of the constituents o r the detai ls of the interaction a s long a s the coupling constants a r e dimension- l e s s . While discussing the Born d iagrams we show that a model with spin-$ quarks gives G, /GM asymptotic scaling. We a l so obtain f r o m general arguments the asymptotic Regge t ra jec tor ies in the gluon-exchange and quark-interchange models. Finally, in Sec. I ID 3 we d iscuss the effects of higher-order correct ions to the Born d iagrams .

A . Spinless Bethe-Salpeter wave function and the meson form factor

The s imples t example, which i l lus t ra tes how to obtain the asymptotic behavior of a hadronic amplitude is the calculation of the f o r m factor of a meson, taken a s a bound s ta te of two s c a l a r f ie lds . The full Bethe-Salpeter wave function satisfies" ( see Fig. 4)

(9) (Note that the wave function d , , a s conventionally defined, includes the propagators f o r the constituent legs.) T h i s is the eigenvalue equation f o r the meson m a s s M 2 = p 2 . The kernel K(k, I , p) is the sum of a l l two-particle i r reducible d iagrams . F o r illustration we f i r s t consider a generalized ladder approximation

where we can choose the spec t ra l function a(h2) such that f o r l a rge 12, KL ( p , k , 2 ) s c a l e s a s (1')-'. F o r a super-renormalizable theory 6 i s positive; 43 theory corresponds to 6 = 1 . F o r renormalizable theories , e.g., 44, 6 =O. We shal l always compute with 6 > 0 a s a regulator , and take the l imi t 6- 0 a t the end of the c a l c ~ l a t i o n . ~ This is analogous to the generalized Feynman-Pauli-Villars method o r dimensional regularization in perturbation theory. Note that f o r 6> 0, we have 4, (k) - [k]-4-6 and y, ( x = O)< m . (The minimum condition f o r 3, ( X = O)< is (k) - [ k ] -4/[ln(k2)] lie with E > 0.)

The form factor in ladder approximation i s ( see Fig. 5)

FIG. 4. Schematic representation of the Bethe- Salpeter equation for a two-particle bound state.

FIG. 5 . The meson form factor in ladder approximation to the Bethe-Salpeter equation.


If additional kernels a r e introduced which contain correct ions f o r each loop a s 6 - 0. We re turn to internal chargedl ines, then additional contributions their contributions below. to the cur ren t in Eq. (10) a r e obtained consistent with By assumption (A) t h e wave functions a r e bounded gauge invariance ( see Fig. 6 . ) A comprehensive a t every point i n momentum space, s o that the t reatment of these contributions has been given by asymptotic behavior of F(q2) is controlled by two Mandelstam .6 The loop correct ions a r e non-lead- regions of the d4k integration: ing for qZ- i f 6 > 0, and can give logarithmic

(a ) k-xp; k2-O(m2), ( p - k ) 2 - 0 ( r t z 2 ) , ( k + q ) ' - ( 1 - x ) q 2 ,

where x is finite. More precisely, integration r e - (2xP + 9)" 1 Mg - gion ( a ) corresponds to k = xp + K, where K i s a (1 - x)q"(l - y ) ( l - x)q21"

spacelike vector orthogonal to p which is of bounded magnitude [by assumption (A) that the wave function is damped in l a rge relat ive momenta] . Thus k2 = 2 m 2 + K' and k - p = x m 2 a r e finite. (The v a r i - which is properly gauge-invarian~: qu.Lf: = 0. able x is s i m i l a r to the variable used in the Sud-

Asymptotically, then, akov and infinite-momentum f r a m e analyses.) I t is easi ly shown that, a f te r the dkZ integration is performed, only the region 0 a x 1 can contribute.

F ( ~ z ) - ( $ )" "n(q2jni2).

Region (b) corresponds to k = ( p + q) + ti', with ti'' ( p + q) = 0 and (K") bounded.

At th i s point we can re la te the asymptotic fal l - off of F ( q Z ) t o that of q2$(qZ), up to logari thms. However, f o r our purposes i t will be convenient to i t e ra te the equation of motion wherever l a rge relat ive momentum i s encountered. Thus we obtain f o r l a rge q2

with k = xp + K and 1 = y(p + q ) - K'. The integrations a r e limited to the dominant region of each wave function: K' and ~ ' ~ = 0 ( m ~ ) . Mg is the five-point connected scat ter ing amplitude i l lustrated in Fig. 7.

This is in fact the prototype of our general p ro- cedure. We employ the equation of motion f o r the wave function wherever i t involves l a rge relat ive momentum. In this manner we generate the connected amplitude Mg which represen ts the sca t te r - ing of the quark constituents, each with a finite f ract ion of the hadron momenta: P i =x, p H , z , x i = 1. Mg i s , of course, exactly the connected amplitude which occurs when the hadronic binding is turned off adiabatically, in which case x i - m , / M H .

Let us now d iscuss the asymptotic behavior of F(qZ) using Eq . (11). F o r l a rge q2 one can readily verify that

where one finds that the logarithm occurs because of the end point of the integration over ( 1 - x) - ' o r (1 - y)-'. Th is resu l t for 6 > Oagreeswith those of Ref. 9.

More generally, the higher-order kernels mod- ify this resu l t by ex t ra powers of ($)-"or 6> 0, and by possible logarithms f o r each additional loop if 6 = 0. However, if we assume that the t rue u l t ra - violet behavior of the theory is m o r e convergent than indicated by elementary gluon exchange ( a s i n asymptotically f r e e theories , f o r example), then the proper regularization of the renormalizable theory is 6 = 0' and additional logarithmic modifications a r e suppressed. T h i s is a consequence of our assumptions (A) and (B), that the accumulation of logari thms affects neither the asymptotic

FIG. 6 . Examples of nonladder kernels for the Bethe- Salpeter equation and the irreducible contributions to the current matrix element they engender by gauge invariance.

1314 S T A N L E Y J . B R O D S K Y A N D G L E N N Y S R . F A R R A R 11 -

behavior of the wave function nor the sca le in- var iance of the connected amplitude Af; . We thus have the prediction f o r a bound s ta te of two spin-0 part ic les:

In the case of an n-field bound state , we have simply (neglecting logs)

i .e . , F (q2) - (q2)'-" in the renormalizable limit, in agreement with Eq. (2). A detailed discussion of th i s resul t when n = 3 h a s been given by Alabiso and S c h i e r h ~ l z . ~ Note that the powers of (q2)- ' a r i s e f rom each off-shell constituent line and the (q2)-6 f rom each gluon l ine.

Thus, physically, one pays the penalty of one power of q2 f o r changing the direction of each constituent f rom along P to along p + q. The spin independence of this resul t i s discussed in Sec, 11 B.

B . Spin-% constituents

The calculations of the asymptotic behavior of f o r m fac tors become somewhat m o r e complicated when the effects of spin a r e included. The resu l t s in the renormalizable l imit , however, a r e effect-

ively the s a m e as the spinless resu l t s . We begin with the example of the fo rm fac tor of a meson which is a bound s t a t e of two spin-+ f ie lds . We assume, fo r the present , z e r o orbi tal angular momentum. The Bethe-Salpeter wave function sat isf ies ( see Fig. 4)

where K i s the full Bethe-Salpeter two-particle i r reducible kernel . Again, a s in the spinless case, we begin with the generalized ladder approximation form

The r(,, and q,, represen t the (momentum-independent) Dirac couplings of the gluons to constituent a and b , respect ively. Jus t a s in the spin- l e s s c a s e , we may choose o(h2) such that KL (k2i - (k2)-I-' (6> 0 , k2 - a), which gives assumption (A): q ( x = 0) = Jd4k 4, (k)< a. Renormalizable theories correspond to 6 - 0+, a t l eas t in the lad- d e r approximation. T h i s i s discussed fur ther in Sec. I11 C.

The form factor f o r the bound s ta te using the kernel [ E ~ . (13)] i s

Additional contributions required by gauge in- var iance a r e necessary in the presence of the higher-loop kernels , as in the spinless c a s e ( see Fig. 6 ) . These a r e of the s a m e o r d e r o r a r e nonleading for q2- a if 6 > 0 .

A s in the spinless calculation, the important contributions f o r the asymptotic form factor occur when only one of the two wave functions is evaluated a t l a rge relat ive momentum. Again, i t i s convenient to i t e ra te the wave function at large relat ive momentum using the equation of motion, and we obtain

iZ.I: i s the connected Feynman amplitude shown in Fig. 7 . A s in the spinless case, the dominant contribution comes when the constituents c a r r y finite

f ract ions of the longitudinal momenta and smal l t ransverse momenta. Fur thermore , if only the leading q2 dependence of the fo rm factor i s de- s i red , the spin s t ruc ture of the wave function simplifies enormously. In general the s t ruc ture i s

If we use the identities

and

Eq. (16) can be rewritten" in the fo rm


with the appropriate spin projections understood. In the zero-binding l imit th i s s t ruc ture must reduce to the product of two f r e e sp inors u, (k =xp)Z,@ - k = (1 - x)p); hence f o r our purposes we may use

The t e r m s thus neglected a r e a t l eas t l inear in ~ l - ' (the " t ransverse momentum" which cannot become la rge) and give a nonleading contribution t o F(q2) . That these t e r m s a r e proportional to the binding energy and may be therefore legitimately neglected is easi ly seen'' in t ime-ordered perturbation theory: They a r i s e f r o m the presence in the wave function of a n ex t ra qq pa i r .

With the simplification, Eq . (18), we find

where

is the connected amplitude evaluated between on- shel l spinors . Again note that f o r the connected t r e e graph we can neglect the K and K' components of k p and Z p , and except fo r par t icular helicity configurations we may drop a l l m a s s t e r m s i n the asymptotic l imit . T h i s is explicitly evident in the Brei t f r a m e (G2 = -q2):

* p =((m2 + G2 4)lI2, -q/2)

and

p + q = ( ( m 2 + ?/4)lI2, + fq '2) ,

where each component of p and p + q becomes la rge . Explicit calculation then shows that the behavior of ic': f o r l a rge q2 is (-q2)-'-" a s dic- tated by dimensional counting." Accordingly, a s i n the spinless calculation, we have F ( q 2 ) -(q2)-'-61n(q2/nt2), where the logarithm resu l t s f r o m the endpoint of the x o r y integration. The canonical renormalizable l imit is 6- 0'. As in the spinless calculation a factor (q2)-' is associated with each off-shell fermion propagator, and a factor (q2)-' with each gluon carrying la rge q2 in the Born diagram. The over-all sca le of the fo rm factor is determined by the value of 4;- (x= 0) = Jd4k 4 ; - ( k ) . A s we have noted, 4'-(x=O) i s finite fo r 6 > 0.

Note that the wave function 4d-(k) plays the s a m e role a s the spinless Bethe-Salpeter wave function. The equivalence of asymptotic behavior with spin- 0 o r spin-$ constituents follows s ince +L;-(x) ( see below) h a s the s a m e dimensions a s the spinless Bethe-Salpeter wave function, $;P'"'~'~(X). F o r spin- l e s s constituents, the Bethe-Salpeter amplitude

since the fermion field operator + ( X I h a s dimen- s ions L - ~ " . However, +;-(x) -[L] -' since the explicit spin dependence of 4, is removed; accordingly, the short-dis tance behavior of + i - ( x ) and +Y'"(.Y) i s the s a m e . In the language of operator- product expansions a t shor t dis tance the point is this : The amplitude f o r the large-q2 form factor of a meson composed of two quarks will involve an operator product such a s

+ x l ( x l ) ~ " l ( ~ , ) J , ( z ) w h 2 ( ~ 2 ) + 0 2 ' ~ 2 )

in a l imit such a s z = Y, = y,. A significant differ- ence f rom, say, the operator-product expansion of two cur ren ts Jp (x )J , (0) = y l ( ~ ) y ~ )$(0)yy + ( O ) a s x- 0 i s that in the l a t t e r case the spin of the f ie lds is summed leaving objects \ cur ren ts ) whose ent i re angular momentum content i s contained in the indices p and v . Therefore , the operator-product expansion can be made in t e r m s of Lorentz t ensors such a s 0 , " . The f o r m e r c a s e i s more subtle: Spinors a r e required to c a r r y the spin content of the product of f ie lds . T h i s means that the number of degrees of f reedom which determine the scaling behavior a t short dis tances i s the s a m e a s in the spin-zero case .

Multiparticle bound s ta tes can be t reated by a generalization of the Bethe-Salpeter techniques fo r two par t i c les . F o r instance, the proton wave function sat isf ies the relation shown in F i g . 8. This t ime the necessary kernel f o r the equation is three-part ic le- i rreducible . Each aspect of the three-part ic le bound-state analysis paral le ls the

f o r a meson in position space i s

+;P'"'""x) Z ' j O I T(O(.U)O(O))IP) -[L1 - I ,

"71 +k;it"r k - l + q k - l + q

where we are+usQg continuum normalization ( p ' / p ) =2Eb3(p -p ' ) and (010) = l . F o r the spinor P-k p+q-P P-k P + Q - P

case

Y;"'""' ( x ) = (0 l T(4J(x)q(O)) I P i -[LI - 2 , FIG 7 . Born diagram (five-point connected amplitude

'\I;) for the meson form factor


two-body analysis . The mos t important p a r t of the spin s t ruc ture of the wave function is that which reduces to a product of f r e e quark spinors:

Project ing spins correct ly , t h e r e a r e two contributions to the wave function: (1) when u and b a r e in an s s ta te , and ( 2 ) when b and c a r e in an s state . The left-over quark h a s the helicity of the hadron.

C. Short-distance behavior of the Bethe-Salpeter wave function

T h e essent ial ro le of the finiteness condition on 4(x = 0) f o r the derivation of the dimensional counting ru les is c l e a r . The condition allows us to compute the high-momentum-transfer l imit of exclusive amplitudes by i terat ing the kernel once and computing a minimal connected graph, thus accounting f o r the effects of l a rge momentum t rans fe r being routed through the wave function. Fur ther , the values of the +b(x = 0) (which have dimensions [mass ] "-' f o r n constituent f ie lds) de- t e rmine the normalization of the hadronic amplitude. In this section we will a d d r e s s the i m - portant question of whether the wave function condition is actually t r u e i n renormalizable field theor ies . In fact, a definitive answer has not yet been given and may well depend upon the theory. T o s e e what is involved, consider the full Bethe- Salpeter equation, e.g., f o r a pion in quark-vector gluon theory [ E ~ . (12) and F ig . 41 :

where K is the two-particle i r reducible kernel . In ladder approximation to a theory with gluon m a s s lV1,

If we suppose that K,d,,o gives the c o r r e c t asymptotic limit, then the Bethe-Salpeter equation i s singular and the power falloff of 4J depends on the coupling constant.

However, in the weak binding (g2- 0) case , when a l l components of the relat ive momentum g become large, 4 -g-4 modulo logg. T h i s resu l t was f i r s t obtained by Salpeterl0 f o r the instantan- eous (Coulomb) ladder approximation.

Serious objections can be ra i sed to the use of the ladder approximation in the strong-binding theory f o r determining the t r u e asymptotic behavior of the Bethe-Salpeter wave functions:

(1) The ladder approximation resul t f o r the

asymptotic l imit is unstable under the perturbation of adding additional kerne l s (e,g., the c rossed graphs, vacuum polarization, ver tex correct ions, e tc . ) . In each c a s e the power dependence on the relat ive momentum is changed.

(2) The behavior of 4 a t l a rge relat ive momentum determined f r o m ladder approximation is d i s - continuous a s the dimensional regularization (4 - d ) f o r loop integrations is taken to z e r o . If i t is argued that the physical solution f o r quantities such a s the asymptotic behavior of F ( q 2 ) must be analytic a s a function of 4 - d, then the wave func - tion at the origin is finite.

( 3 ) The ladder approximation can only be valid in a limited range of coupling constants. When g2 becomes too large, the energy eigenvalue becomes imaginary, indicating a non-Hermiticity of the equation. One can s e e this explicitly f r o m the Salpeter equation." The situation is analogous to the fami l ia r situation f o r the strong-binding l imit of the Dirac -Coulomb equation with V = -Za :/r. F o r the lowest eigenstate, we have

EIS = [ I - ( ~ a ! ) ~ ] 1'2M,

41s(r) - r -1+[1-(za)2]1/2 (for r - 0)

The singular equation has no physical solution for ZCY > 1; i t is undefined. Regularization of the potential f o r r - 0 is thus required. The s tandard procedure i s to make V l e s s s ingular than r-I f o r a region around Y - 0 (which, of course, o c c u r s physically due t o the finite m a s s of the source) . Then E , , is r e a l and $,, ( r = O ) is finite f o r a l l Za." F o r v e r y la rge Z a a multiparticle pa i r creat ion is required.

T h e r e is reason t o believe that in renormalizable theories the full kernel K i s , i n fact , more convergent asymptotically than K L . A s we conjectured in Ref. 1, asymptotically f r e e theories a r e likely t o give r i s e to a wave function whose singularity a t x = 0 is a t mos t a power of a logari thm. T h i s is heuristically plausible s ince a s the charac te r i s t i c momenta in a graph become l a r g e r and l a r g e r relat ive t o m a s s e s , the effective coupling con- s tants become s m a l l e r and s m a l l e r , so that the weak-binding resu l t holds. If i t is legitimate to imagine taking the limit q2-a before summing the perturbation s e r i e s then the wave function condition holds, modulo a power of a logarithm. Re-

FIG. 8. Schematic representation of the Bethe-Salpeter equation for a three-particle bound state.


cently Appelquist and PoggioI3 have shown that in a q3 theory in s i x dimensions, which is asymptotically f r e e , the renormalization group can be used to demonstrate that the wave function is finite a t the origin up to a calculable logartihm. How- ever , they emphasize that the infrared cor rec t ions to the interact ions among constituents may lead to fu r ther logarithmic correct ions. (See Sec . I1 D 3 f o r our discussion of this point.)

Even i n renormalizable theories which a r e not asymptotically f r e e , the ver tex and vacuum polarization correct ions to the gluon-exchange d iagrams in the complete kernel may well damp the asymptotic behavior of K, a s do the finite-size c o r r e c - tions to the Coulomb potential. Thus there a r e both physical and mathematical reasons to believe that the full kernel is sufficiently regular that the wave function a t s h o r t distance is finite o r a t wors t logarithmically s ingular . Clearly this problem d e s e r v e s fu r ther study.

Suppose that the wave function i s not finite a t x = 0. T h e r e a r e two possibilities: I t diverges o r is zero . If the wave function diverges with a power 6 i t modifies the effective n, associated with that hadron, so, e.g., f o r a meson n = 2 - 6 ra ther than 2 . Similarly, if i t diverges o r vanishes logarithmically i t induces logarithmic deviations f r o m perfect scaling.

If the wave function vanishes a s a power a t x = 0, the next-to-leading Born t e r m s will determine the asymptotic behavior of the mat r ix element . This is the c a s e when a bound s ta te h a s nonzero orbital angular momentum,14 a t l eas t if the wave function's dependence on energy and three-momentum fac- to r izes in the hadron r e s t f ram?. That i sLi f 4, (q) can be written a s Iji, (qO)g (q), then 4 (r) - @ t imes angular fac tors . Hence f o r a two-constituent bound s ta te of orbi tal angular momentum L, #(x - 0 ) - 2 -m-L when a l l components of x a r e proportional to each other and smal l . The resu l t is to cause fur ther damping of mat r ix elements . Hence the fo rm fac tor of an L = 1 s ta te should have the asymptotic behavior (q2)1-n-L. Effects of orbi t - a l angular momentum have been considered in more detail by Amati e t and Ciafaloni.15

In addition to the finiteness of the wave function

FIG. 9. Diagram for meson-baryon scattering with multiple scattering of near-mass-shell quarks of different hadrons .

a t the origin in coordinate space, assumption (A) included the requirement that i t is bounded a t every point in momentum space. Although this l a t t e r condition is difficult to prove direct ly by studying the Bethe-Salpeter equation, l3 the re can be litt le doubt about i t s validity a s long a s quarks a r e not observed experimentally (because if the wave function is bounded everywhere in coordinate space but of finite "volumeJ' then i t s F o u r i e r t r a n s - f o r m is necessar i ly f ini te) .

D. Irreducible diagrams

An important conclusion of Sec. IIA-C i s that, given sufficient short-distance smoothness of the wave function [assumption (A)] , the i r reducible amplitudes (the M,, ) determine the high-energy, fixed-angle behavior of the full amplitude. T h i s section is devoted t o discussing the conditions under which the i r reducible amplitudes obey naive dimensional scaling, i .e. , hfn - ~ 7 ~ - ~ asymptot ical- ly.

1. Landsho ff diagrams

A s mentioned in the introduction to th i s section, ~ a n d s h o f f ~ h a s recently emphasized a c l a s s of d iagrams which, if present , would violate dimensional scaling. Such a diagram is shown in Fig. 9. It i s character ized by having fewer off-shell f e r - mions than s tandard d iagrams f o r the s a m e pro- c e s s such a s shown i n Fig. 10. In fact , f o r meson- meson and baryon-baryon scat ter ing no fe rmion line need be f a r off shel l . Physically, i t c o r r e - sponds to (say f o r meson-meson scat ter ing) the independent, e last ic , on-shell scat ter ing of p a i r s of constituents such that the final momenta a r e properly aligned. Landshoff calculated their asymptotic behavior and found, using a Sudakov parameter izat ion, that f o r meson-meson sca t te r - ing ~ V l - s - ~ ' ~ f ( t / s ) and f o r baryon-baryon sca t te r - ing M- ~ - ~ f ( t / s ) , i n contrast to the dimensional r e - sult M-s-' and s - ~ , respectively. We have verified h i s resu l t s using both Feynman parameterization16 and infinite-momentum-frame perturbation theory ( see Appendix A). The kinematic configuration

FIG. 10 . Typical Born diagrams for meson-baryon scattering via (a) gluon exchange and (b) quark exchange. The dots label quark lines which a re far off the mass shell. All gluons shown are far off the mass shell.

1318 S T A N L E Y J . B R O D S K Y A ND G L E N N Y S R . F A R R A R 11 -

( s e e F ig . 11) which gives the pinch has a l l the quarks near the m a s s shell so the two elementary quark-quark scat ter ings a t l a rge momentum t r a n s - f e r a r e dimensionless s ince n = 4 . The energy dependence of the full amplitude a r i s e s f r o m the condition that the final momenta be aligned properly. Referr ing to F ig . 11 we can s e e how sd3/' comes about; the center-of - m a s s f r a m e of the two mesons i s convenient f o r th i s purpose. F o r definiteness, label the momenta of the quarks s o that x, and x, a r e l e s s than 8. If we examine Fig. I l ( b ) , i t i s evident that in o r d e r f o r the final quarks to have nearly paral le l momenta (so the t ransverse momenta of the quarks in the final mesons a r e finite a s required by the wave function) it i s necessary that x, = .u2 + 0 ( 1 / P ) , where P i s the magnitude of the meson 3-momenta in the center-of - m a s s system ( P - 6). Assuming then that x, = s, we have the configuration shown in Fig. l l ( b ) . Imagine that the quarks with fraction x , = x, sca t te r by s o m e finite angle b. Momentum con- servat ion means that x, =x, = y, = y, within 0(1/'P). Let their plane of scat ter ing define the x-z plane. Now consider the scat ter ing of the quarks with fract ions (1 - x,) = ( I - x,) + O ( l / P ) . They in gen- e r a l sca t te r in some different direction having polar angle 6' and azimuthal ( relat ive to the xz plane) angle +. Since they c a r r y a v e r y l a r g e 3- momentum (1 - x,)P, they will c a r r y la rge momentum t r a n s v e r s e to the direction defined by the other s e t of quarks unless +' = 4 + 0 ( 1 / P ) . Thus t h e r e a r e th ree constraints on the kinematics, each requiring a parameter normally allowed some finite range to be res t r i c ted to a range 0(1/-6). Hence M- s - ~ " and & / d t - s-5 [ ra ther than s - ~ a s given by E q . ( I ) ] .

Technically, the near -mass -she l l d iagrams have a s t ronger asymptotic behavior than the scaling law because of pinch s ingular i t ies that a r i s e in the integrals over the constituent momenta. T h i s gives r i s e to an anomalous dependence on the quark m a s s not p resen t in d iagrams whose leading behavior resu l t s f r o m an end-point singularity in the Feynman integral . That i s , a l inear infrared di- vergence in the quark m a s s s e r v e s to define a fundamental length sca le : I/?%. To i l lustrate how this occurs in pract ice, we give in Appendix A a n infinite-momentum-frame calculation of the Land- shoff contribution to meson-meson scat ter ing. A s can be deduced f rom the above analysis , the re will be no modification of naive scaling by such a mechanism f o r lepton-hadron scat ter ing, Compton scat ter ing, o r photoproduction."

T h i s c l a s s of near -mass -she l l scat ter ing dia- g r a m s should evidently dominate the hadron sca t - ter ing amplitudes in the asymptotic limit." The i r existence implies a violation of Eq . (1) f o r meson-

baryon and baryon-baryon scat ter ing, giving in- s tead

where L i s the number of wide-angle, on-shell quark scattering; e .g . , L = 2 f o r meson-baryon scat ter ing and L = 3 f o r baryon-baryon scat ter ing. Experiment ( see Sec. I11 and Ref. 18) favors the dimensional scaling resul t of Eq . (1). F r o m th i s we lea rn a s t r iking fact about nature which i s in- corporated i n assumption (C); multiple n e a r - m a s s - shell scat ter ing i s no! important f o r present experiments .

T h e validity of assumption ( C ) is much m o r e difficult to understand theoretically than the validity of (A) and (B), although their ultimate explanations may well be connected. ~ol l i inghorne '%as recently advocated adoption of a somewhat s t ronger version of (C): that large-angle scat ter ing of near -mass -she l l quarks i s damped in energy. A possible mechanism f o r th i s damping i s a n a c - cumulation of logari thms in the correct ions to the quark-quark-gluon ver tex when both quarks a r e near the m a s s shell."

Another proposal, made e a r l i e r and f o r a iiif - ferent purpose by Blankenbecler, Brodsky, and Gunion,'l i s that gluons cannot be exchanged between quarks of different hadrons, o r e l se that the amplitude f o r gluon exchange is v e r y smal l . The i r analysis indicates that the constituent-interchange picture gives a good description of the angular dependence of exclusive scat ter ing with no gluon exchange required (in th i s connection s e e our discussion of asymptotic Regge t ra jec tor ies , Sec. I I D 3 ) . Since a field theory with quarks and gluons would in general have both quark in te r - change and gluon exchange, th i s indicates that gluon exchange is suppressed in nature. Fur ther evidence f o r th i s proposal i s given i n Secs . I I D 3 and IV. I t should be emphasized however, that this is essent ial ly a phenomenological rule; a

FIG. 11. The pinch in meson-meson scattering. (a) Feynman graph and (b) momentum-space picture of the scattering.


consistent descr ipt ion of p resen t wide-angle experiments can be made with no gluon exchange whatever. No compelling theoret ical reason f o r this rule has yet been advanced.

Yet another possibility i s that i n a model with permanently confined quarks such near -mass- shell p r o c e s s e s would be suppressed. I t can be heuristically argued that, s ince this contribution is proportional t o l/nz,, if the effective quark m a s s were v e r y la rge (perhaps comparable to 6) i t would be small . F r o m the t ime-ordered perturbation theory calculation of Appendix A it can be seen that the internal s ta te with near-mass-shel l quarks [which in a color SU(3) theoryzz h a s nonzero color sys tems propagating] ex i s t s f o r a finite time, long compared t o s-'I2. Such a s t a t e might be suppressed in the full amplitude.

2. The irreducible diagrams

Typical Born d iagrams f o r meson-baryon sca t - ter ing and meson photoproduction a r e shown in Figs. 10 and 12. The heavy dots indicate which fermion propagators a r e f a r off the m a s s shel l . With sca la r constituents having $ J ~ coupling, the wavy l ines just reduce to a point interaction. By applying the mnemonic of Secs. I I A and I1 B (s-' f o r each off-shell fermion) the d iagrams of Fig. 10 a r e immediately seen to have the asymptotic behavior M- (6)4-" f ( t / s ) in the l imit 6- O f . Hence they reproduce Eq. (1) . The only process which needs special discussion is photoproduction (Fig. 12).

According to dimensional counting, if the photon counts a s one field the photoproduction amplitude should -(v%)~-". Figure 12 shows that th ree f e r - mion propagators a r e l a rge (as in the meson- baryon c a s e of F ig . 10). However (as the reader can easi ly verify by direct calculation), the vector coupling of the photon introduces a numerator fac- t o r proportional t o 6, resulting in the expected behavior M, - s-'I2. This is charac te r i s t i c of the vector coupling, not the spin of the constituent and occurs f o r e i ther spin-0 o r -+ quarks . Had the quark-antiquark pair (if we look in the photon channel) been constrained to have limited relat ive momenta, which a hadronic wave function would

do, the photon coupling would not have generated a 6 s o that M - s d 3 . This i s the vector-meson- dominance contribution to photoproduction a t l a rge s and t .

Figure 13 shows typical Born d iagrams f o r the proton f o r m fac tor with spin-$ constituents. I t is readily seen that f o r e i ther s c a l a r - o r vector- gluon exchange (helicity f l ip o r nonflip, respectively, in the ze ro-quark-mass l imit) graphs of the kind shown in Fig. 13(b) give a leading contribution -(q2)-' to F1(qz). When q2>>m,' the dia- g ram of Fig. 13(a) cannot contribute f o r s c a l a r gluons. Explicit calculation shows that I;,(qz) is proportional to vnqz so that in general a three-quark model of the proton f rom fac tor will give Fz(yz) -F,(q2)/qz when qz becomes la rge . In t e r m s of the commonly used f o r m fac tors

and

where K is the anomalous magnetic moment, th i s corresponds toz3 constant G, G , a t l a rge qz, a s i s indicated experimentally (Sec . 111).

We noted above that meson-baryon scat ter ing, o r indeed any hadronic scat ter ing, could take place ei ther by gluon exchange between the hadrons o r by quark interchange [ ~ i ~ s . 10(b) and 10(c)] . Both give the s a m e scaling behavior but i n genera l they give r i s e to very different angular distributions [i .e. , the function f ( t s ! which we leave unspecified depends on the relat ive importance of the two types of d i a g r a m s ] . Although we do not attempt h e r e to deal with the problem of the complete angular distributions, we a r e able to make some statements about the kinematic region t v e r y large, s - .-. . When t and s a r e both la rge we know the over-al l power dependence of do/dt - ~ - " + ~ f ( t i ~ ) but not in general the function f ( t , ' s ) . However, if e i ther gluon exchange o r quark i n t e r - change is the dominant t-channel p rocess ( see F ig . 14) we can specify the f o r m of f ( t s) .

Using Regge terminology we wri te the amplitude a s M- s-".!~ ("$(t ) . A s i s well known, a spin-1 o r spin-0 gluon in the t channel gives r i s e to a constant aer, ( t ) = 1 o r 0, respectively, f o r a l l t . Multigluon exchanges only generate cut c o r r e c - -2~c J=o{ lJ=oJ=O{ ~h ) J = o

( a ) ( b i

FIG. 1 2 . Born diagram for photoproduction. The h e a ~ y dots represent far-off-shell quark lines. All F IG. 1 3 . Typical Born diagrams for baryon form fac- gluon lines shown a r e far off the mass shell. tors .


t ions. When quarks a r e interchanged t h e r e a r e two important regions in the integration over the intermediate quark momenta: k - xP, o r k - xP,. Consider the f o r m e r . In th i s c a s e the lower half of the diagram is purely a function of t = q 2 (just a s in a f o r m factor calculation). In fact , i t h a s the s a m e asymptotic t dependence a s the helicity- nonflip f o r m factor F ( t ) - t - " c t l . T h e upper par t of the diagram when s >> t is a l so s imple. It i s essent ial ly high-energy, quark, + A -quark, + B scat ter ing. An explicit t dependence comes f r o m that required by helicity considerationsz4: J i - Ih I+ hA-XB-h2 I . Hence f o r A = B = meson o r A = B =baryon the leading behavior is t-independent. On the other hand, f o r n photoproduction the upper blob -6. Thus we find that f o r meson-baryon scat ter ing M- ~ - ~ , f ( t / s ) - s - ~ F ( ~ ); t h e r e a r e two t e r m s (coming f r o m k-xP, and k -xP,) proportional to F, ( t ) and F,(t ), respect ively. The leading behavior in s comes f r o m F, s o we have the quark interchange - 1. Similarly = - 2 and (u:rMB=-1.25

According to Ref. 26 the best f i t s to ae,, a r e indicative that gluon exchange is negligible. Since a priori counting coupling constants (e.g., i n Fig. 10) indicates that gluon exchange and quark in te r - change should be equally important, th i s gives support t o the proposal of the previous section that gluon exchange between quarks of different hadrons i s anomalously smal l . Hence it provides additional support to our assumption (C).

3. Higher-order corrections to the Born diagrams

In any given o r d e r of perturbation theory t h e r e will be logarithmic correct ions t o the behavior of the Born d iagrams due to loop integrals . If these logari thms were to coherently combine they could conceivably a l t e r the s imple scaling behavior of the Born amplitude. F o r instance, in QED i t has been shown2' that infrared radiative correct ions to fixed-angle exclusive lepton and photon sca t te r - ing amplitudes have the f o r m

be modified by soft interact ions among the con- s t i tuents . If the fundamental strong-interaction coupling i s actually weak, a s conjectured by a number of people,28 such a modification would not be seen until ve ry la rge t .

On the other hand, arguments can be made that the infrared effects on a bound-state scat ter ing amplitude should be much mi lder than Eq. (22) f o r two reasons: F i r s t , in a theory such a s a color non-Abelian gauge theoryz2 the physical s t a t e s a r e neutral, i .e. , color singlets, s o that the in- f r a r e d region is damped f o r momenta k < O ( d - ' ) , where d is a length charac te r i s t i c of the hadron. Second, i n a bound s ta te the constituents a r e gen- e ra l ly off their m a s s shel ls and hence the infrared s ingular i t ies a r e "shielded."

T o i l lustrate the f i r s t point, consider an Abelian gauge theory in which m a s s l e s s vector mesons couple to a conserved cur ren t . We shal l take a hadron t o be a neutral s ta te , s o that Q =CiQI = 0. If we use Weinberg's'' notation, the infrared r e - gion of the vir tual gluon cor rec t ions ( A < / k , / < 11) gives a correct ion factor t o the scat ter ing mat r ix element of the f o r m exp[-A ln(A /A)] with

where the summation is over a l l p a i r s (n, rn) of external charged l ines . F o r an outgoing (incom- ing) line q = + 1 (-1) and /3,, is the relat ive velocity of par t ic les n and m in the r e s t f r a m e of ei ther :

Considering now a n amplitude such a s the fo rm fac tor of a "meson," we must sum E q . (23) over a l l p a i r s of external l ines . It is readily checked that the only t e r m s i n the sum (23) which can introduce a leading t dependence a r e those involving one initial and one final par t ic le . Le t us label the two (say) initial par t ic les u and b whose momenta a r e p a = @ + ~andp,=(l-x)p- with K ' ~ = O (as in Sec. I1 A) . By our assumption (A), K is bounded

e-a [ I ~ ( s / x ~ ) 1 2 ( 2 2 ) SO that [ a s may be checked by explicit expansion , of the logarithm in Eq. (23)] we may take pa = x p

where t is the momentum t rans fe r and h is an effective infrared cutoff.

We a r e concerned with the possible exponentia- P B = P A - q

tion of logari thms due to strong-interaction c o r - rect ions to a bound-state scat ter ing a m ~ l i t u d e . In - - a gauge theory i t is the infrared region which i s potentially dangerous in th i s respec t . I t is pos- s ible that they will introduce a modification t o

1 0 1 l b l these amplitudes of the fo rm of Eq . (22) . In that c a s e the scat ter ing ru les [ E ~ s . (1)-(4)] will just FIG. 14 . Hadron scattering for t large, s - a if (a) give the nominal o r canonical power law reflecting gluon exchange o r (b) quark interchange i s dominant in the compositeness of the hadron, but which will the t channel

11 - S C A L I N G L A W S F O R L A R G E - M O M E : N T U M - T R A N S F E R P R O C E S S E S 1321

and pb = (1 - x )p with e r r o r s of only o r d e r l/s a t fixed angle. Then Pa, and ir),, a r e equal if n is not equal to a o r b . Summing over u and b i n Eq . (23) thus causes cancellation of the infrared c o r r e c - tions of o rder ln(t/mz) s ince Q A = - Q B .

The infrared behavior of non-Abelian theories with m a s s l e s s vector gluons is generally expected to be much worse than in the Abelian case . A central difficulty of such theories is that a soft gluon emitted f r o m a n external line can itself emit a pa i r of soft, colored m a s s l e s s gluons, ad ~ n f z n ~ t z ~ ~ ~ z . However, in a bound s ta te such cat- astrophic gluon emission may well be regulated by vir tue of the gluon constituents of the hadrons being effectively off the m a s s shel l .

Fur thermore , i n the case of a color theory a s usually envisioned," color-octet hadrons ei ther a r e infinitely mass ive o r a t l eas t not degenerate i n m a s s with the usual hadrons which a r e color singlets. In th i s c a s e the infrared contribution f o r k2 and p . k below m: - m,' is suppressed, giving correct ions of o rder ln[ t / (mt - m12)]. Moreover , because the color emission changes quantum num- b e r s and is not soft, the re is no reason that such logarithms will exponentiate o r cause la rge c o r - rect ions to the scaling law?'

Even if the logari thms found in perturbation theory do not exponentiate, a s argued above, they can give logarithmic correct ions to the scaling laws. T h e r e is v e r y litt le we can say about th i s [essentially about the validity of assumption (B)] except what we said i n Ref. 1: If the re a r e modifications of scaling i n deep-inelastic scat ter ing, there will probably be s i m i l a r modifications to these scaling laws .31

111. THE EXPERIMENTAL STATUS OF EXCLUSlVE SCALING

In this section we a r e concerned with t e s t s of the scaling law [ E ~ . ( I ) ] in exclusive processes . Three questions a r e of par t icular importance: (1) I s our choice of quark-model assignments f o r the number of f ie lds in hadrons acceptable? (2 ) A r e the scaling laws actually exact o r a r e they per - haps modified by logarithmic o r "anomalous dimension" cor rec t ions? (3) A r e t h e r e contributions f rom on-shell scat ter ing [which may change the scaling of Eq . ( I ) ] ? Of course, essent ial to the whole program is the self-consistency of the scheme. Once that is established f o r exclusive processes , we turn to the m o r e difficult question of inclusive scat ter ing. T h i s section is quite detailed, i n the hope of emphasizing (especially to experimental is ts) the l a rge amount of important work which needs to be done i n th i s field.

The s implest applications of Eq. (1) a r e purely

electrodynamic, e.g., e t e - - p 'p - o r e i e - - e'e-, y e - ye, e'e-- y y , e tc . In each of these c a s e s n = 4 unless we a r e wrong a t a fundamental level, s o that experimentally these should have the a s - ymptotic behavior da /d t - s-'f ( t l s ) . T h i s is just the prediction of quantum electrodynamics (modulo logari thms f rom radiative correct ions) s o that to the extent that QED is c o r r e c t a t l a rge s and t our predictions f o r the scaling behavior of purely leptonic and photonic p r o c e s s e s a r e c o r r e c t with the assignment n = 1 f o r leptons and photons. Since QED is i n agreement with experiment up t o the highest available s and t (CEA 32 and S P X P 33

r esu l t s on e'e-- p 'p- , e t e - - e te - , e tc . ) we may assume that n = 1 is cor rec t f o r leptons.

In one-photon-exchange approximation, the dif - ferent ial c r o s s section f o r ek- e h scat ter ing a t very la rge s and la rge t is given in t e r m s of the hadron spin-averaged electromagnetic f o r m factor F ( t ) by

s o that we have [ ~ q . (2)] the general formula (modulo logari thms)

f o r asymptotic spacelike o r timelike t . Thus using n , = 2 and n, = 3, we conclude that f o r l a rge q2, F,(q2) - (q2)-I and 6, (q2) - (q2)-' can be separately determined by studying the dependence of ep- ep scat ter ing on q2. AS remarked in Sec. I1 D 2, given spin-', constituents with vec tor - o r sca la r -gluon exchange, one finds that F2(q2) - (q2)-3 a t l a rge q2, s o that we predict the asymptotic behavior G, - 1 /q4 and G , - 1 1q4.23 Since a substantial range of l a rge q2 i s necessary to t es t these laws, the experiment of Kirk el cover- ing the spacelike range 1.0 6 -q2 < 25.0 GeV2, i s mos t suitable f o r our purposes. Figure 15 shows q4G,v(q2) f o r that experiment . F o r -11' > 4 GeV2, q4Gjv(q2) is consistent with a constant within e r - r o r s . T h i s vindicates our choice n, = 3 a s sug- gested by the naive quark model. Moreover , the v e r y fact that G,(q2) fa l l s a s a power of q2 i s support f o r our scaling laws. {The data a r e not a c - cura te enough to allow discussion of the question of logari thms.) G, and G , have been separately determined only f o r -q2 5 3.75 G ~ v ' . ~ ' They a r e found to be consistent with G, = GM /i within e r - r o r s . T h e asymptotic falloff of the proton f o r m factor h a s proved v e r y difficult t o account f o r up to now. Bound-state models f o r simplicity have focused on two-particle bound s ta tes . Treat ing the proton a s a spin-0, spin-5 bound s ta te with spin-0 exchange gives36 (modulo logs) G , - (qZ)-' but G, - (q2)-'. Separate determination of G, and

S T A N L E Y J . B R O D S K Y A N D G L E N N Y S R . F A R R A R

G M a t l a r g e r q2 will be useful to conclusively d i s - tinguish th i s model f r o m o u r s . However, i t i s improbable that G, could have the behavior (q2)-' without that being evident f rom the q2 dependence of the ep- ep c r o s s section.37

Data on the pion form factor a r e not yet available f o r a q2 range comparable to that of the proton since they come f rom e'e- annihilation. Present ly, data a r e available3' f o r timelike q2 < 9.0 GeV2. They a r e consistent with a p pole, s o that f o r q2 la rge , F,(q" - (9')-' i s an acceptable fit ( see Fig. 16). However, i t should be kept in mind that to conclusively test our picture, q2 d 4 GeV2 should be considered. If only that range i s used, the data a r e not adequate to ru le out other behavior. Improving and extending measurements of the pion form factor is one of the s u r e s t ways to verify o r destroy our ideas . F o r the present we will continue to assume that n , = 2. In a ladder model in which the pion i s a bound s ta te of two spin-0 constituents with e3 interact ions i t s f o r m factor would fal l a s (9')-'.

According to the dimensional scaling rules , if a photon behaves a s a single field a t short d i s - tances, photoproduction a t l a rge s and t follows do/dt - ~ - ~ f ( t / s ) . Th is i s a n especially interest ing p r o c e s s because s t r i c t vector-meson dominance would say that yp - np behaves like pp - np and hence &/dt - s-'f ( t 1s ) . Of course if the photon, when interacting with hadrons, is a superposition of a vec tor meson and a n elemen- t a r y field,38 then i n the kinematic region being studied h e r e the l a t t e r s ta te will dominate and resul t in &/dt - sq7f ( t / s ) . The highest-energy data a t 90" a r e those of Anderson et ~ 1 . ~ ' which cover the range E y between 4 and 7 . 5 GeV. They find that do/dt (90") - s-7a3'0e4, in agreement with our prediction of s-'. Again, it i s most d e s i r - able to extend the range of s , since at E , = 4

FIG. 1 5 . ( q 2 ) 2 ~ M ( q 2 ) / p for the proton versus -q2; data from P. N. Kirk et a l . (Ref. 34) .

GeV, 90" corresponds to a t of about 3 GeV2 which we expect i s only barely in the scaling region. However, if that lowest-energy point i s dropped, lit t le can be said about the 5 dependence of do 'dt a t 90". Thus our provisional conclusion is that photoproduction measurements support the p r e - diction that do/dt (yN-nN) - s- 'f(t/s), with higher- energy data of g rea t importance.

The presen t data on meson-baryon scat ter ing, which we expect to behave a s do ' d t - s-8f ( t j s ) a t l a rge s and t , a r e l e s s conclusive for our pur - poses than photoproduction o r pp scat ter ing experiments . This i s par t ly because experiments to date have focused on obtaining the ent i re angu- l a r distribution a t modest s values ra ther than choosing some smal l bL range with enough sensitivity to go to high values of s . T h i s poses the usual problem of not having an adequate range of high enough s and t to check f o r scal ing. A special problem in s o m e of these meson-baryon sca t te r - ings i s the existence of ( resonance?) s t ruc ture a t quite high-t values. F o r instance, in a-p-a-p a dip in do 'dt is found a t t = -3.8 (GeV 'c)' ( see Ref . 41), and t h e r e is evidence that both a l p have a dip at t = -4.8 (GeV/cY. F o r an unambiguous analysis , one needs f o r this p rocess t and u d 5 (GeVIc)' and a l a rge range of s . One experiment which has been analyzed to determine the nature of the wide-angle energy dependence m e a s u r e s the 90" differential c r o s s sect ions f o r ROP-TJ-AO, P p - n - z o , and K j p- K gp f o r incident momenta between 1.0 and 7 .5 G ~ V / C . ~ ' They find that their resu l t s can be equally well parameter ized by ( d ~ / d S 2 ) , , , - s - ~ o r (do. /dS2),,,- s - ~ * ( see Fig. 17) . If s and t a r e sufficiently large, do. 'dC2 -s(du/dt If(l/s) s o that we predict t ~ = 7 . They give f o r the th ree react ions ri1=7.4* 1.4, nL= 8.1 i 1.4, and nz=8.5* 1.2, respect ively. In short , our predictions a r e consistent with data on meson-

F I G . 1 6 . q2F,(q2) for the pion versus q 2 ; data from AT. Bernardini et n l . (Ref. 38).


baryon scat ter ing, but this only demonstrates that we a r e not wildly wrong. However, resu l t s f r o m experiments which only attempt to cover the l a rge- angle region, but with much higher sensitivity (so that a l a r g e r range of s can be studied), will be much more conclusive. It is enough to show that the c r o s s section integrated over a fixed c .m. region fa l l s with the power s-' ra ther than another power o r an exponential.

Not surpris ingly, pp-pp i s the most thoroughly s tudies elast ic p rocess a t l a rge s and t . Landshoff and Polkinghorne8 have plotted the data for s > 1 5 GeV2 and 0, ,, between 38- and 90' (see Fig. 18) . They find that it i s well fitted by the fo rm do 'dt

s-9.7i0.i f(0). This i s in v e r y good agreement with our prediction do dl - . ~ - ' ~ f ( 6 ) , ~ ~ and apparent- ly ru les out a significant contribution to the amplitude by on-shell quark scattering.' '

The experimental evidence on Eq. (1) i s thus

FIG. 17. du/dQ(9Oo c.m.1 versus s for several meson- baryon scattering reactions. This figure is from Ref. 42.

ve ry encouraging. However, with lit t le difficulty it could be considerably improved. As indicated above, bet ter data on meson-baryon scat ter ing and e'e- - n'n- should be available fair ly soon and will be a significant tes t of the scaling idea. We close this section with a l i s t of predictions of the scaling law of Eq. (1) f o r other react ions whose asymptotic behavior is interesting, albeit difficult to measure :

y p - yp , do 'dt - s-'jf ( t 1s)

) p - pp , do 'dt - s-'f ( t S )

y y - nn, do di - s-4f ( t 1s)

(the photons need not have z e r o m a s s a s long a s their m a s s e s a r e kept fixed and a r e smal l compared with s and t ) .

(or any o ther L = 1 meson with nonvanishing form factor) ,

e d - e d , Fd(q2) - (q2)- '

(this probably only sca les for q2 2 8 GeV2). Finally, the scaling law for multiparticle exclusive la rge- angle scat ter ing [Eq. (3)] can be tested. When n , = 2 and n, = 3 i t can be written in a compact form: Let Ao be the invariant c r o s s section integrated over some fixed c .m. angular region keeping al l ra t ios pi . p j / s (i * j ) fixed. Then when s - a, a s the r e a d e r can easi ly verify [ ~ q . (411,

Ao- s - " * ~ v ~ ~ 2 ' v ~ ' f ( p i * p j / s ) ,

where A', and N , a r e , respectively, the total num- b e r of mesons and baryons. Thus for high-energy BB - MBB when a l l c .m. angles a r e l a rge [ E n ( & / d l ) d 3 p n ] - ~ - ' ~ f ( f ? , ) which i s equivalent to do/dSlld02- s- lo . Data expected a t SPEAR and DORIS

FIG. 18. Log-log plot of ( d u / d t ) forpp -pp versus s at various c.m. angles. Only data for s > 15 G ~ V ~ and If / > 2 . 5 G ~ V ' a re included. This figure is from Ref. 8.

1324 S T A N L E Y J . B R O D S K Y A ND G L E N N Y S R . F A R R A R 11 - on the exclusive channels e'e- - n'n-n'n- and e'e- - 6 charged n's with fixed angles will eventually provide a t e s t of our predictions (ha - s - ~ and Ao -s-5 , respect ively) .

IV. INCLUSIVE REACTIONS AT HIGH TRANSVERSE MOMENTUM

Having developed in previous sections the ru les f o r determining the power dependence of exclusive scat ter ings a t l a rge energy and momentum t rans - f e r , we wish to explore their implications f o r high-pi inclusive scattering.44 It i s to be expected that, just a s in the exclusive case , the power dependence of inclusive high-p, c r o s s sect ions will depend on the number N of constituents par t ic ipat- ing in the high-p, reaction. Our task, then, i s to s e e what can be said about this number N.

In parton language, an exclusive process p ro- ceeds by the large-momentum scat ter ing of n constituents with the wee partons behaving a s "spectators." The condition on the bound-state wave function at x p = O s e r v e s two purposes: F i r s t , i t fixes the minimum number of constituents having finite fraction of the total momentum (and hence n). Second, it ensures that the remaining "wee p a r - tons" a r e t ruly "spectators," that is , their interactions do not build up additional power dependence on momentum t rans fe r . (This l a t t e r function i s modified if the wave function in nature proves to have a logarithmic o r power singularity a t the origin.)

In inclusive scat ter ing, constituents having finite f ract ions of the momenta of the incident par t ic les can be spec ta tors . A good example of th i s i s deep- inelastic scat ter ing in the parton model, shown in Fig. 19(a). In the absence of a requirement that no part ic les be found in the forward direction, it i s energetically favorable fo r those partons not de- flected by the cur ren t to continue without l a rge changes of their t ransverse o r longitudinal momenta. In this c a s e only the single parton with appropriate momentum fraction x = -q2/2mv need part ic ipate in a large-momentum-transfer colli- sion. Hence n = 4 (2 leptons and 2 quarks) s o that the mat r ix element i s dimensionless and the different ial c r o s s section E, d a / d 3 p , - ~ - 2 f ( q 2 / ~ , ~ 2 / ~ ) asymptotically (M i s the missing mass) . In the one -photon-exchange approximation, this differ - ential c r o s s section may be written in t e r m s of s t ruc ture functions W, and vW2 which a r e defined in such a way that they have no energy (v) dependence f o r fixed (q2/v) if the behavior of the differential c r o s s section i s l/s2 a s argued above. The famous scaling behavior of vW2 seen at SLAC, if it p e r s i s t s a t higher energies , i s evidence for the validity of this argument . A logarithmic modifica-

tion of scaling a s expected in asymptotically f r e e theories i s not dis t ressing; that i s just the effect in such theories of the interactions of the spec- t a to rs .

Motivated by the success of the scaling prediction in deep-inelastic scat ter ing, we abs t rac t the notion that any inclusive amplitude factor izes into a par t which involves only l a r g e momentum t rans - f e r s and p a r t s involving only low momentum t r a n s - f e r s . That par t depending on the l a r g e momentum t rans fe rs de te rmines the over-al l power dependence of the amplitude. Of course , the dependence of the c r o s s section on invariant ra t ios , say t /s and M2/s, i s in general dependent on the low- a s well a s high-momentum p a r t s of the scat ter ing process . Thus we can immediately wri te down [ ~ q . (5 ) ] the invariant c r o s s section for high-energy inclusive scat ter ing at fixed t/s and M 2 / s :

Edo 1 t M 2 --- d3p s W - z / ( T T ) j

with N defined a s the minimum of fields in the large-momentum-transfer par t of the amplitude.

Evidently, in o r d e r to make predictions f o r inclusive scat ter ing, one must make a dynamical s ta tement which s e r v e s to specify the number of fields N, just a s in the exclusive c a s e a specifica- tion of the number of nonwee constituents was r e - quired. The s implest ansatz fo r N i s the one made by Berman, Bjorken, and K o g ~ t ~ ~ in what was essentially the f i r s t parton-model work on high- t ransverse-momentum inclusive processes . They observed that a t o r d e r a2 hard parton-parton sca t - ter ing must take place via exchange of a f a r -off - shell photon just a s lepton-parton scat ter ing does in deep-inelastic scat ter ing (Fig. 19). Presumably, hadronic final s ta tes would be generated f r o m the quarks just a s in the lepton scat ter ing c a s e , i .e . , in a scale-invariant manner. If the large-mo-

FIG. 19. Parton-model picture for (a) the one-photon- exchange deep-inelastic scattering and (b) the colored- gluon-exchange contribution to high-pi inclusive scattering, showing the color index i i j ) of the quarks. The notation -i (-j) refers to the color of the rest of the hadron.


mentum-transfer p r o c e s s i s qq - qq o r qq-qq, ,V=4. and in fact Berman, Bjorken. and Kogut predicted

However, it was natural to imagine that such a 44 -4q scat ter ing at high p , could take place via the exchange of vector (or s c a l a r ) gluons a s well a s via a photon. In fact, in perturbation theory it i s hard to imagine that gluons could be responsible fo r binding of quarks and yet not give r i s e to wide- angle quark scat ter ing, leading to the scaling behavior Edo/d3p -s-' a t a level well above that due to electromagnet ism.

In an alternative approach, Blankenbecler, Brod- sky, and Gunion" explored a model in which hadrons sca t te r not by gluon exchange but by quark "interchange." They argue that the resulting angular distributions fo r wide-angle elast ic sca t - ter ing a r e in good agreement with data , with no gluon exchange necessary for their f i t ( see Sec. I1 D). In their model, ha rd parton-parton scat ter ing which takes place via gluon exchange i s not a l - lowed. The minimal large-p, p rocesses which can take place in this c a s e involve quark-hadron scat ter ing and thus N a 6 . In addition it makes quite detailed predictions on the function J( t /s , M2/s) of Eq. (5). Thus the detai ls of the model will be subject to many experimental checks.

Here ra ther than advocate any part icular model we categorize and discuss the possibilities. Our principal contribution to this subject is mere ly the observation that the essent ial element of any model of high-PI inclusive scat ter ing, a s f a r a s the over-al l power of s goes, is the number of large-), participants. If the minimal large-)_ reaction i s qq -qq o r qq- qq then N = 4 a s d i s - cussed above. If, fo r some reason, that i s not possible o r i s dominated by other p rocesses giving a large-p_ meson, such a s q ~ i - qv, qq - nqq, o r -

qq- nn, then N = 6 o r m o r e ( see Fig. 20). If the pro- c e s s of in te res t i s , say, y p - n +Xthen the high-p I reaction might be q + y - n + q o r q + y - n + q , giving y = 5 .

F o r baryons observed in the final s ta te , say , pp -p +X or irp -/I + X , there a r e severa l possible N > 4 react ions: yq - f iq (B=6) , qp-qp(N=8) , qq-pp(ifr = 8 ) , e tc . The lack of an N =.6 process with a produced involving only quarks in the ini- t ia l s ta te would predict that the c r o s s section f o r pp -J; +S should be much s m a l l e r than in pp -P +X at large DL. This i s only a sample of the r ich phenomenology available when looking a t inclusive processes f rom this point of view.

The experimental resu l t s on large-momentum- t rans fe r ~ c a t t e r i n g ~ ~ have one remarkable fea ture

in common. They a r e not consistent with the power dependence E da/d3p - s-' a t fixed p /dz and d = 90°, i .e. , p,-". Rather , they favor much m o r e rapid falloff i n p , with a power between -8 and -11. The detai ls of how their resu l t s a r e described in t e r m s of the phenomenology outlined above a r e given in Ref. 44 . What i s of g rea tes t importance h e r e i s that the absence of p,-" behavior i s s t rong phenomenological support for the absence of sca le - invariant scat ter ing between quarks of different hadrons and hence for our assumption (C).

Equation (5) suggests a new way of anaIyzing the data on deep-inelastic scat ter ing. The s tandard method i s to take the observed experimental c r o s s section. assume that the one-photon-exchange approximation i s good (this has been cross-checked) , apply radiative correct ions, and ex t rac t the s t r u c - tu re functions vW2 and W, (and Uj fo r v scattering). Scaling then implies that the s t ruc ture functions a r e independent of any variables other than x = -q2/ 2mv. However, Eq. (5) with N = 4 can be checked directly. Examining direct ly the dependence of E da/d3p on p, fo r fixed rat ios of invariants would be most interesting. Even though radiative c o r - rect ions (and possibly s t rong interactions in the hadronic wave function) sure ly generate logarithmic modifications of perfect scal ing, such an analysis would provide a useful d i rec t tes t of scaling.

Of course , we find the parton-model resul t that pp -p-p- + X o r rrp - pap- +X can go via q q - pap- and hence i s scale- invariant . Sirnilarly, if the parton version of e'e- annihilation i s taken, that it goes via e'e- - qq-hadrons, these counting ru les give a scale-invariant minimal scat ter ing and hence <<::- - 1 / s .

V . SUMklARY AND CONCLUSIONS

In this paper we have examined the conditions under which the s imple dimensional analysis used to derive Eq. (1) can be valid in renormalizable field theories . The scalicg laws a r e consistent with finite Bethe-Salpeter hadronic wave functions and the scaling behavior of s imple planar Born d iagrams for the vz-particle scat ter ing amplitude.

( 0 ) ( b )

F I G . 20 . Possible minimal large-p_ scat ter ings if gluon exchange 1s not allowed: (a, y U r " -qk and ( b ~ YY - "VY


The question of the short-dis tance regular i ty of the Bethe-Salpeter wave function h a s now been partially set t led for asymptotic freec@m theories by Appelquist and Poggio.13 They show, a s conjectured in Ref. 1, that the i r reducible kernel i s m o r e convergent in the ultraviolet than indicated by s imple ladder approximation. We have given arguments why such behavior can be expected even in Abelian theories , and why infrared correct ions to the scaling laws a r e likely to be unimportant for color singlet o r "neutral" hadronic s tates .

In a general renormalizable perturbation theory, the scaling laws can be violated by nonplanar dia- g r a m s involving multiple on-shell quark-quark scat ter ing. The fact that such d iagrams do not s e e m to be important empir ical ly , together with the absence of scale- invariant l a rge- t ransverse - momentum inclusive react ions, plus the fact that effective Regge t ra jec tor ies ff,fr become negative a t l a rge t , evidently imply the suppression of scale-invariant (single o r multiple) gluon-exchange interactions between quarks of the scat ter ing hadrons . This remarkable empir ical fact has yet to be explained, but undoubtedly has important implications fo r the detailed s t ruc ture of the underlying quark theory.

Our general approach is different f rom what has been done previously. Starting with Wu and Yang, a number of authors have observed that given the asymptotic falloff of the hadronic fo rm fac tors , predictions can be made about the l a rge-s and -t behavior of e last ic scattering.*' Horn and Moshe4' proposed that c r o s s sect ions should have the f o r m of Eq. (I), without specifying n, and showed that it provides a good fit to the data. In contrast , we look directly at the underlying shor t - distance s t ruc ture which we abs t rac t f rom p e r - turbation theory. This enables us to predict the behavior of the form factor (and automatically gives the relat ions between form factor and wide- angle scat ter ing obtained by previous author^).^^'^^ On the other hand, without making m o r e detailed dynamical assumptions we cannot make the p r e - dictions that they make on the angular behavior of 2 - 2 scat ter ing. The next logical s tep i s to choose a theory which may actually be c o r r e c t (e.g., a color gauge theory) and s e e whether quali- tative features such a s the angular dependence, the minimal high-p, par t of an inclusive scat ter ing, e tc . , can be obtained without actually solving the theory. The techniques used in this paper , e s - pecially the simplified approach to bound-state scat ter ing which circumvents the complexities of the full Bethe-Salpeter analyses, should be very useful toward this goal.

In summary , the dimensional scaling laws for fixed-angle scat ter ing a s S - rn ,

give a fundamental connection between the degree of complexity of hadrons and the power-law behavior of c r o s s sections and f o r m factors . Al- though much more experimental information i s required, the present resu l t s support a composite representat ion of the hadrons based on quark de- g r e e s of freedom. Thus hadron scat ter ing a t l a rge t ransverse momenta implies something of fundamental importance: Quarks not only have a mathematical existence, giving cur ren t a lgebra, Bjorken scaling, and the hadron spec t rum, but a dynamical existence a s well. That i s not to say that quarks must be observed a s f r e e part ic les . However, bound-state models should be built f rom quarks to incorporate the c o r r e c t short-distance s t ruc ture of hadrons.

ACKNOWLEDGMENTS

We have benefited f rom conversations with many colleagues at SLAC and Caltech and elsewhere, especially T . Appelquist, J . D. Bjorken, R. Blan- kenbecler, J. Bronzan, P. CvitanoviE, S. D. Drel l , R. P . Feynman, Y. Fr i shman, M. Gell-Mann, J . Gunion, J . Kiskis, R. Sugar, and T.-M. Yan.

APPENDIX .4: DOUBLE-SCATTERING GRAPHS USING THE INFINITE-MOMENTUM METHOD

The physics of the Landshoff multiple-scattering d iagrams turns out to be part icular ly t ransparent using t ime-ordered perturbation theory in the infinite-momentum f r a m e . A full description of this method may be found in Ref. 51.

The prototype calculation which most simply contains the Landshoff contribution i s given in the case of n-T scat ter ing where we take anfrp4 theory for the constituents and ~ c # I ~ ver t ices to represen t the bound s ta tes . Generalizations to other c a s e s will be straightforward.

We choose the following reference f rame:

with

* * Note that f = q2 = - q L 2 , and 14 = v 2 = - r _ 2 . We a lso - - have q . - r = 0 if MA2 + .LIB2 = IZIC + >I/Ig2.

The contributing t ime-ordered diagrams which


a r e equivalent to the Feynman graph. Fig. 21(a). time orderings in which the vertex A o c c u r s be- a r e shown in Figs. 21 (b)-21 (e) . All other t ime fore and a f te r B, and C occurs before and af ter D. orderings vanish in the P- ffi f rame. It i s easy to The total amplitude for M ( h ) and M ( c ) i s then given check that d iagrams (d) and (e) do not contribute by the following t h i ~ e - l o o p expression for t ime- to leading o r d e r in the asymptotic fixed-angle l imit . o rdered perturbation theory: Diagrams (b) and (c) include the summation over

Three-momentum i s conserved a t each vertex. The range of the x, i s 0 to 1. The amplitudes M ( , , and M,,, a r e the scat ter ing amplitudes fo r a + b - c + d a n d a ' + b ' - c ' + d f , respectively. In t h e f a " theory Mc ,, = JQ', ,, =f. The energy denominators for d iagrams (b) and (c) a r e

where the Ei a r e the relat ivis t ic "kinetic energies"

In addition, there a r e two energy denominators coming before the scat ter ings MI and M2 and two af terward. When these a r e summed over the orderings A before and af ter B , C before and af ter D, they generate the product of wave functions:

e tc . Notice that because of the choice of var iables [c ,=,ya(p+G+F) +&, , I the wave functions do not depend explicitly on q o r r . The natural domain of the wave function i s thus G,, f inite. Dropping t e r m s of o r d e r .M2 temporari ly , we have

- 2 - - 2 -2 D,,,g ( q + F ) 2 - ~ C r ~ - (1 - - . ~ ~ ) ( q + r ) -xdq + -.

- E , , * r + 2 k L a . ( G + F ) - 2 i ; , d . G .. kt ld " L*2 G i c 2 kia2

.vc 1-s , Sd l - Y b ,

If we introduce the scaled variables - m 3 ( , v a - x d ) I q 1 +2(i;-o-i;ll) . q ^ ,

&1=(xa-xc) /;I+2(cL0 - E L d ) . ? ,

I

and

Bib,= a l q l + P l r l + ~ ( m ) + i c .

Changing variables f rom x, and .r, to a and 13 thus gives for diagram (b)

I V T ( ~ , - .I- da dB

t I F a~q ' l+ i - i iF i -o(nz) + i t (A2

multiplied by a finite integral over dx,, d 'k_a, d2k,,, d2k_, which i s independent of s, I , and 11.

The imaginary par t of (A2) gives

(as easily seen in polar coordinates) and the r e a l par t cancels against lll,c,.

(01 Feynmon

( e l then for cr and P of o r d e r 1 and k,:, kAb2, and k,' T ~ m e - Ordered Dtagrarns

of o r d e r ?n2, we have

E lb2 - 0 ( m 2 ) FIG 2 1 (a ) A Feynman d lagram tor ;7ri - r r r r and

(h)-(e) t ime-ordered d iagrams corresponding to it

1328 S T A N L E Y J . B R O D S K Y A N D G L E N N Y S R . F A R R A R

We should note the following a t this point. The n-rr amplitude i s imaginary, reflecting the Glauber- type r e a l pole intermediate s ta tes which contribute. The constituent scat ter ing amplitudes .Z/I(,) and .2.I(,, in (A3) correspond to near -mass -she l l scat ter ings of the constituents of different hadrons with fractional momenta s, and 1 - x , , , respect ively. Note that the wave function dependence i s not relevant a s long a s the d2k , integrals converge because if the .W,(.Y) a r e scale- invariant they a r e actually independent of r, e.g. , for vector-gluon exchange ~Vl(.l(s)- (xs - su) ,ut = (s - u ) t . The generalization to more real is t ic scat ter ings i s imme- diate . Each additional pa i r beyond the f i r s t two gives a factor in the amplitude of o rder

thus giving the resul t , Eq. (21) of Sec. 11 D 1.

APPENDIX B : FIXED -ANGLE UNITARITY

It i s interesting that the behavior

i s actually the fixed-angle unitarity bound for n part ic les in the external s ta te when the square of the amplitude i s averaged over a fixed angular region in the center-of-mass system. A convenient proof has been given by Bardeen5' a s follows: Consider a two-particle s ta te with continuum normalization

By smearing with an angular function

we have -

Unitarity then gives

and hence

Averaging / N) over c . m . angles thus gives in the high-energy l imit

whcre N i s the number of par t ic les in the final s ta te . By crossing, the proof holds f o r any num- b e r of initial par t i c les and ( B l ) follows.

The bound ( B l ) i s a l so often imposed on amplitudes in each o r d e r of perturbation theory a s a necessary condition for renormalizability.

In our work we show that the bound ( B l ) f o r the fixed-angle amplitude i s saturated in any theory without an inherent short-distance sca le . A c ru- cial point must be noted, however. In the hadronic amplitude, the constituent par t i c les fo r each hadron a r e not a t fixed angles relat ive to each other; thus the bound ( ~ 1 ) fo r the required amplitude M , can, in principle, fail due to singular behavior when the external l ines become paral le l . Th is i s in fact what occurs in the mult iscat ter ing diagram considered by Landshoff. (See Sec. I ID 1 and Appendix A .)

*Work supported in part by the U . S. Atomic Energy Commission. Prepared under Contract No. AT(11-1)- 68 for the San Francisco Operations Office, C. S. Atomic Energy Commission.

'A brief description of the results of this article was given in S. J. Brodsky and G . R . Farrar , Phys. Rev. Lett. 31, 1153 (1973).

'v. Matveev, R . Muradyan, and A. Tavkhelidze, Xuovo Cimento Lett. 1, 719 (1973).

3 ~ h e initial application of this result for antiquark distributions was given by G. R . Farrar , Nucl. Phys. m, 429 (1974). The general result [(Eq. (6)l and further applications are discussed by R . Blankenbecler and S. J . Brodsky, Phys. Rev. D 10, 2973 (1974), and J. Gunion, Phys. Rev. D E , 242 (1974).

4 ~ . F. Ezawa, Nuovo Cimento G, 271 (1974).

'see, for instance, J . Harte, Phys. Rev. 16j, 1157 (1968); 171, 1825 (1968).

6 ~ . E . Salpeter and H . A. Bethe, Phys. Rev. 2, 1232 (1951). A complete discussion of scattering amplitudes in the Bethe-Salpeter formalism is given by S. Mandel- stam, Proc. R. Soc. m, 248 (1953).

'P. V . Landshoff, Phys. Rev. D 10. 1024 (1974). 8 ~ . T.'. Landshoff and J. C. Polkinghorne, Phys. Lett.

44B, 293 (1973). - 9~ similar analysis has been given recently by Ezawa,

Ref. 4, and C. Alabiso and G . Schierholz, Phys. Rev. D l(r, 960 (1974).

'Osee, e.g., E. E . Salpeter, Phys. Rev. 87, 328 (19.52).

"A useful trick to perform the calculation is to make use of the identity x , , u ( k , s ~ F ( k , - s ) = I# + m ) y , and its


general izat ions to unequal momenta. The expression for iV1t i s then seen to be just a n ordinary t race of y m a t r i c e s .

l2I?or a review s e e S. J. Brodsky, Comments At. Mol. PhyS. E, 109 (1973).

1 3 ~ . Appelquist and E. Poggio, Phys. Rev. D 10, 3280 (1974).

''we a r e indebted to L. Caneschi and M . Gell-Mann for raising the question of the effect of o rb i ta l angular momentum on the scaling laws.

ISM. Ciafaloni, Phys. Rev. 176, 1898 (1968). 161n addition, a very elegant Feynman parameter calcu-

lation has been performed by P. cvitanovic, Phys. Rev. D 10, 338 (1974).

" ~ t can be shown using the Coleman-Korton ru les , a s presented in Phys. Rev. 125, 1422 (19621. that the only infrared l inearly divergent d iagrams a r e of the Land- shoff type. S. Coleman (private communication).

he prediction of the Landshoff d iagrams for pp -pp and a number of other processes which a r e measurab le have been obtained in a colored-quark and vector-gluon theory by G. R. F a r r a r and C.-C. Wu, Caltech Report No. CALT-68-455 (unpublished). They verify that for this , a s well a s other renormalizable theories. there i s no "accidental" cancellation of these leading dia- g r a m s . Comparing the angular dependence of pp - pp implied by the Landshoff model with the da ta , they find that i t provides considerably more convincing evidence against the existence of such a process than the power falloff alone.

'$J. C. Polkinghorne, Phys. Lett . G, 277 (1974). ,Vote added in p o o f . Recently Z. Ezawa and J. C. Polkinghorne have shown that the the dimensional counting r u l e s a r e rigorously t r u e for a cer ta in c l a s s of asymptotically scale-free kerne ls [Cambridge Univ. Report No. DAMTP 74/20 (unpublished)].

"T. Appelquist and E . Poggio (private communicationi; Ref. 13.

'IR. Blankenbecler, S. J. Brodsky, and J . F. Gunion, Phys. Lett . %, 649 (1972); Phys. Rev. D 8, 187 (1973).

"H. Fr i tzsch and &I. Gell-Mann, in Proceedings of the XL'I Internutional Conference on High Energy Phys ics , Chicago-Bataviu, Ill. , 1972, edited by J . D. Jackson and A. Roberts (NAL, Batavia, Ill. , 19731, Vol. 2, p . 135.

2 3 ~ e u s e the notation of Bjorken and Drell : (p + y lJ,(O)lp) = C(p - q)[y,Fi (q2) - ( u , , ~ ~ ~ ' / ~ I ~ ~ I K E ~ ( ~ ~ ) ~ u (P), where K i s the anomalous magnetic moment. The commonly used "electric" and "magnetic" f o r m fac tors a r e re - lated to F1 and FZ by ~ , ( q ' ) = F i (qZ) - ( K ~ ~ / ~ M ~ ) F , ( ~ ~ ) and ~ , ( q ' ) = F1 (q2) T K F ~ ( ~ ~ ) .

2 4 ~ e thank G. C. Fox for a helpful discussion on this point. It has been pointed out by J. Gunion that in s imple Born d iagrams there may be additional t dependence.

2 5 ~ h i s provides a somewhat m o r e genera l derivation of the cue,, than was obtained by d i r e c t calculation in the interchange model, however, the physics i s the same. The resu l t s a g r e e except for a","fffPp, which differs due to the fact that the original interchange model assumed a "quark-core" representat ion for the proton which led to the prediction du/dt (pp -pp) -s-I2 ra ther than the dimensional scal ing s-IO.

2 6 ~ . Blankenbecler, D. Coon, J . Gunion, and T . Than-

Van, SLAC Report (unpublished). "D. R. Yennie, S. C. Frau tsch i , and H. Suura, Ann.

Phys. (K.Y.) 13, 379 (1961); N . Meis te r and D. R. Yennie, Phys. Rev. 130, 1210 (1963), and re fe rences therein.

his i s natural for models which a i m to unify strong, weak, and electromagnetic interact ions and has been especially emphasized by H . Fr i tzsch and P . Minkowski, Phys. Lett. s, 462 (19741, and by H . D. Pol i tzer , Phys. Rev. D 9, 2174 (1974).

'$5. Weinberg, Phys. Rev. 140, B516 (1965). 3 0 ~ h e summation of leading logari thms in the infrared-

protected pseudoscalar theory gives a correct ion of o rder [lnt13/Lo to the fermion form factor; s e e T . Appel- quist and J. R. Pr imack . Phys. Rev. D 1, 1144 (19701, and re fe rences therein. \Ve would like to thank T.-M. T a n and P . Cvitanovi; for helpful conlsersations on the infrared problem.

3 1 ~ o r instance. A. De Rujula. Phys. Rev. Le t t . 32, 1143 (19741, and D. G r o s s and S. A. Treiman. ibid. 32. 1145 (1974). have an indirect argument that the logarithmic modifications of perfect scaling for the proton form factor [Eq. (211 should be of the ra ther mild f o r m (1nq?ilnq2,

3 2 ~ . Rladaras at a1 ., Phys. Rev. Le t t . 30. 507 (1973); H. Xewman et a1 . , ibid. 32. 483 (1974).

3 3 ~ a l k presented by A. Boyarski at the Lliashington APS Meeting, 1974 (unpublished,; B. L. Beron et a l . . Stanford Report No. HEPL No. 734 (unpublished) ; J.-E. Augustin et ul ., Phys. Rev. Lett. 34, 233 (1975).

3 4 ~ . N . Kirk et a1 ., Phys . Rev. D 8. 63 (1973 1 .

3 5 ~ . Litt et a1 ., Phys. Lett . s. -10 (19701. 3 6 ~ . S. Ball and F. Zachariasen. Phys. Rev. 170. 1541

(1968); D . Amati. R . Jengo. H . Rubinstein. G. Venezi- ano, and M. Virasoro, Phys . Le t t . D. 38 (19681; D. Amati, L . Caneschi, and R. Jengo, Nuoro Cimento 58A, 783 (1968); RI. Ciafaloni and D. Xlenotti, Phys. - Rev. 173, 1573 (1968); S. D. Drell and T . D. Lee , Phys. Rev. D 3, 1738 (1972).

3 7 ~ e wish to thanlc W. Atwood for a discussion of this point.

38iM. Bernardini et a1 ., Phys. Lett . e, 261 11973). 3 9 ~ h e r e i s already evidence that this may be the c a s e

f rom photoproduction off nuclei which shows l e s s shad- owing than expected f rom pure vector dominance. See, e .g . , K. Gottfried, in Proceedings of' the 1971 Interna- tional Symposiunz O?I Electron and Photon Iizteractions a t High Energ ies , edited by K. B. Mistry (Laboratory of Nuclear Studies, Cornell University, Ithaca, K.T., 1972).

"R. L . A n d e r s o n e t u l . , Phys. Rev. Le t t . 3, 627 (1973). "V. Chaband et ul .. Phys. Lett . E, 441 (1972). "G. W. Brandenburg et a1 ., Phys. Le t t . e, 305 (19731. I 3 ~ e c e n t l y da ta on p j -p$ at Piab = 5 and 6.2 Gel'/c h a w

been obtained by T. Buran el a1 ., CERN repor t (unpuh- l ished); however, the e r r o r s a r e s o la rge that it i s consistent with both a constant and s-lo falloff.

" ~ o s t of these resu l t s need l i t t le explanation beyond what was given in Ref. 1 so this analysis will be brief and principally for completeness. F o r a detailed review of recent developments. experimental and theoretical, s e e R. Blankenbecler and S . J . Brodsky, Ref. 3 , and R. Blankenbecler, SLAC Report Nc. SLAC- PUB-1438 (unpublished).

%. Berman, J . D. Bjorken, and J . I<ogut, Phys. Rev.

1330 S T A N L E Y J . B R O D S K Y A N D G L E N N Y S R . F A R R A R

D 4, 2381 (1971). "R. Blankenbecler, S. J. Brodsky, and J. Gunion, Phys.

Lett. a, 461 (1972). "I?. W. Biisser e t a l . , Phys. Lett. s, 471 (1973);

J. W. Cronin et a1 ., Phys. Rev. Lett. 31, 1426 (1973). See also "Large Transverse Momentum Phenomena, report of an ISR discussion meeting" (unpublished) for further relevant experiments.

a ~ . T, Wu and C . N . Yang, Phys. Rev. 137, B708 (1965); H . D. I . Abarbanel, S. D. Drell, and F. J . Gilman, Phys. Rev. Lett. 20, 280 (1968); T. K. Gaisser, Phys. Rev. D 3, 1337 (1970), and Ref. 21, among others.

"D. Horn and AT. Moshe, Nucl. Phys. E, 557 (1972);

B57, 139 (1973). 'QTR. Theis, Nucl. Phys. Lett. E, 246 (1972); gives

a connection between form factors and wide-angle scattering by assigning "anomalous dimensions" to the hadron fields. However, since all the hadrons a r e not probed at short distances or on the light cone a consistent analysis of this type does not seem possible in a general theory.

5 1 ~ . D . Bjorken, J. Kogut, and D. Soper, Phys. Rev. D 3 , 1382 (1970); S. Brodsky, R . Roskies, and R. Suaya, ibid. 8, 4574 (1973).

5 2 ~ . Bardeen (private communicationi .

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Scaling laws for large-momentum-transfer processes