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VOLUME 31, NUMBER 18 PHYSICAL REVIEW LETTERS 29 OCTOBER 1973 Scaling Laws at Large Transverse Momentum 51 Stanley J. Brodsky Stanford Linear Accelerator Center, Stanford University, Stanford, California 94305 and Glennys R. Farrar California Institute of Technology, Pasadena, California 91109 (Received 14 August 1973) The application of simple dimensional counting to bound states of pointlike particles en- ables us to derive scaling laws for the asymptotic energy dependence of electromagnetic and hadronic scattering at fixed cm. angle which only depend on the number of constitu- ent fields of the hadrons. Assuming quark constituents, some of the s—*oo, fixed-*/s pre- dictions are (da/dt)^^^^ s" s t {da/dt) pp ^pp^ s' 10 , (da/dt) yp ^ vp ~s' 1 , (dv/dt)y p ^yp ~s" 6 , F v (q 2 )~(q 2 )~^ 9 and F^(q 2 ) ~ (q 2 )' 2 . We show that such scaling laws are character- istic of renormalizable field theories satisfying certain conditions. Nature has presented us with a tantalizing glimpse of the hadrons. On one hand, they are evidently composite since the meson and nucleon form factors fall with increasing momentum transfer. On the other hand, the carriers of the currents within the hadron seem to be structure- less, as indicated by the apparent scaling behav- ior of the deep-inelastic structure functions. In this note we shall show that asymptotic proper- ties of electromagnetic form factors and large - angle exclusive and inclusive scattering ampli- tudes can also be understood in terms of renor- malizable field theories with certain conditions, namely, asymptotically scale-invariant interac- tions among the constituents, and hadronic wave functions which are finite at the origin, as dis- cussed below. Accordingly, the application of di- mensional counting to the minimum quark-field components of a hadron will be shown to account for many of the experimental consequences of its compositeness. Our central result for exclusive scattering 1 is (do/dt) AB ^ AB ~>CD n f(t/s) (1) (s -~°°, t/s fixed). Here n is the total number of leptons, photons, and quark components (i.e., elementary fields) of the initial and final states. This result follows heuristically if the only phys- ical dimensional quantities are particle masses and momenta. We begin by considering a world in which a hadron would become a collection of free quarks with equal momenta if the strong in- teractions were turned off. Note that the dimen- sion 2 of the connected invariant amplitude M n .^ n with n i + n f external lines is (length)"* +n /~ 4 . If all the invariants are large relative to the mass- es and are proportional to s, then M n .^ n If we now assume that the intro- duction of binding between the quarks of the ha- drons does not modify this basic scaling result (i.e., no fundamental scale is introduced), then Eq. (1) follows, using do/dt~s~ 2 \M I 2 . Multipar- ticle production is described by the obvious gen- eralization of Eq. (1), when all invariants are large: We obtain Ao .~ s -l-A^-2JV B a t 5 -oo ? for the exclusive cross section integrated over a (cm. angular) region where the p i pj /s are held fixed and N M and N B are the total number of ex- ternal mesons and baryons in the process. The validity of this heuristic argument for Eq. (1) can be examined in renormalizable field the- ories. First construct the minimum connected (Born) diagrams for the amplitude Af„._»„ with the quarks of each hadron carrying a finite frac- tion of the total momentum and having the appro- priate total spin. Typical graphs are shown in Figs. l(a)-l(c). In these simple graphs, the mass terms play no role in the asymptotic fixed- angle region and the total Born amplitude scales with 5 as argued above, since the coupling con- stant is dimensionless. 3 The essential assump- tion required to justify Eq. (1) is that the scaling behavior of the physical scattering amplitude (when s -* °°, t/s fixed) is the same as the scaling behavior for the free-quark amplitude in Born approximation. There are two classes of higher- order diagrams which give the full hadronic am- plitude 4 in terms of quark scattering and which could invalidate our result. (i) The irreducible diagrams [see Fig. 1(d)]: These are the loop corrections to the Born am- plitude M _> n which involve interactions between 1153

Scaling Laws at Large Transverse Momentum

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Page 1: Scaling Laws at Large Transverse Momentum

VOLUME 31, NUMBER 18 P H Y S I C A L R E V I E W L E T T E R S 29 OCTOBER 1973

Scaling Laws at Large Transverse Momentum51

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

and

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

(Received 14 August 1973)

The application of simple dimensional counting to bound states of pointlike particles en­ables us to derive scaling laws for the asymptotic energy dependence of electromagnetic and hadronic scattering at fixed cm. angle which only depend on the number of constitu­ent fields of the hadrons. Assuming quark constituents, some of the s—*oo, fixed-*/s pre­dictions are (da/dt)^^^^ s"s

t {da/dt)pp^pp^ s'10, (da/dt)yp^ vp~s'1, (dv/dt)yp^yp ~s" 6 , Fv(q

2)~(q2)~^9 and F^(q2) ~ (q2)'2. We show that such scaling laws are character­istic of renormalizable field theories satisfying certain conditions.

Nature has presented us with a tantalizing glimpse of the hadrons. On one hand, they are evidently composite since the meson and nucleon form factors fall with increasing momentum transfer. On the other hand, the carr iers of the currents within the hadron seem to be structure­less, as indicated by the apparent scaling behav­ior of the deep-inelastic structure functions. In this note we shall show that asymptotic proper­ties of electromagnetic form factors and large -angle exclusive and inclusive scattering ampli­tudes can also be understood in terms of renor­malizable field theories with certain conditions, namely, asymptotically scale-invariant interac­tions among the constituents, and hadronic wave functions which are finite at the origin, as dis­cussed below. Accordingly, the application of di­mensional counting to the minimum quark-field components of a hadron will be shown to account for many of the experimental consequences of its compositeness.

Our central result for exclusive scattering1 is

(do/dt)AB^ AB ~>CD nf(t/s) (1)

(s -~°°, t/s fixed). Here n is the total number of leptons, photons, and quark components (i.e., elementary fields) of the initial and final states. This result follows heuristically if the only phys­ical dimensional quantities are particle masses and momenta. We begin by considering a world in which a hadron would become a collection of free quarks with equal momenta if the strong in­teractions were turned off. Note that the dimen­sion2 of the connected invariant amplitude Mn.^n

with ni + nf external lines is (length)"*+n/~4. If all the invariants are large relative to the mass­es and are proportional to s, then Mn.^n

If we now assume that the intro­duction of binding between the quarks of the ha­drons does not modify this basic scaling result (i.e., no fundamental scale is introduced), then Eq. (1) follows, using do/dt~s~2\M I2. Multipar-ticle production is described by the obvious gen­eralization of Eq. (1), when all invariants are large: We obtain

A o .~ s - l -A^-2JV B a t 5 - o o ?

for the exclusive cross section integrated over a ( c m . angular) region where the pi• pj /s are held fixed and NM and NB are the total number of ex­ternal mesons and baryons in the process .

The validity of this heuristic argument for Eq. (1) can be examined in renormalizable field the­ories . First construct the minimum connected (Born) diagrams for the amplitude Af„._»„ with the quarks of each hadron carrying a finite frac­tion of the total momentum and having the appro­priate total spin. Typical graphs are shown in Figs. l (a ) - l (c) . In these simple graphs, the mass terms play no role in the asymptotic fixed-angle region and the total Born amplitude scales with 5 as argued above, since the coupling con­stant is dimensionless.3 The essential assump­tion required to justify Eq. (1) is that the scaling behavior of the physical scattering amplitude (when s -* °°, t/s fixed) is the same as the scaling behavior for the free-quark amplitude in Born approximation. There are two classes of higher-order diagrams which give the full hadronic am­plitude4 in terms of quark scattering and which could invalidate our result.

(i) The irreducible diagrams [see Fig. 1(d)]: These are the loop corrections to the Born am­plitude M _>n which involve interactions between

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VOLUME 31, NUMBER 18 P H Y S I C A L R E V I E W L E T T E R S 29 OCTOBER 1973

(p+q + r)/2

r (p+q + r ) / 2 |

p/3 P/3 | p/3

(a)

e/

e /

P/2 \

ZZ-I "P/2

1 |

V i \

(p+r)/2

(P+r)/2

|(P + q)/3 (p+q)/3

(P+q)/3

(P + q)/2

(P+q)/2

(p+q+0/2 (P+r)/2 ? 1

(P+q+r)/2^

p/3 p/3 p/3

e

+ P/2

P/2

(c)

_L

iS

\ fTp+n/2

^(p+q)/3

| (p+q)/3

(P+q)/3

(b)

/

\ (P + Ql/2

5 (P+q)/2

>^ |

, / %, ..

(d) (e)

FIG. 1. Typical Born diagrams for large-momentum-transfer elastic scattering in the quark picture, (a) irp -*irp (quark scattering), (b) irp-^irp (quark interchange), (c) eir-^eir, (d) an irreducible loop diagram, (e) a r e ­ducible loop diagram.

the quarks of different hadrons and are not two-(or three-) particle reducible in the meson (or baryon) legs. We will assume that the sum of all irreducible corrections to the n-particle scatter­ing amplitude has the same scaling behavior, up to a finite number of logarithms, as the Born amplitude. The scaling of deep-inelastic scatter­ing depends on the behavior of diagrams of this type, so if anomalous dimensions or an infinite number of logarithms turn out to be necessary to understand deep-inelastic scattering, they will probably be necessary here.

(ii) The reducible diagrams [see Fig. 1(e)]: These are higher-order diagrams involving in­teraction between the quarks of the same hadron, and they are automatically summed if the total irreducible amplitude is convoluted with the full meson and baryon Bethe-Salpeter wave functions. If the wave functions are finite for zero quark separation, i.e.,

fd*kipM(k) = Ux = 0)<°° (2)

and similarly for baryons (integrating over both relative momenta), then the full hadronic ampli­tude has the same scaling behavior in s as the irreducible n-point amplitude. Although Eq. (2) has not been proved in renormalized field theory, it is very reasonable physically. It requires that the effective Bethe-Salpeter kernel is slightly less singular at short distances than indicated by

perturbation theory.5 Breaking of (2) by a finite number of logarithms modifies Eq. (1) by a finite number of logarithms. The values of the ^ ( ^ = 0) determine the absolute normalization of the a s ­ymptotic cross sections.

Some experimental consequences of Eq. (1) for specific high-energy processes at fixed t/s, using minimal quark representations for mesons and baryons, are (a) dv/dt~s~8 for meson-baryon scattering (up -^up, Kp-»ir3Z, etc.), (b) do/dt ~s "10 for baryon-baryon scattering (pp *pp, pp —pN*, etc.), (c) do/dt~s~7 for meson photopro-duction (yp -*up, yp—pp, etc.), (d) dv/dt~s~6

for Compton scattering (yp — yp). All these p re ­dictions should hold when s and t are much larger than the masses of the particles involved. The results (a) and (c) are in excellent agreement with recent experiments.6 For pp elastic scatter­ing the fitted power-law exponent is between - 10 and - 12.7 The result (d) is in agreement with dominance of a J=0 fixed pole in the Compton amplitude with form-factor residue.8

In the case of electron scattering and e +e' an­nihilation, we shall define the effective electro­magnetic elastic (or transition) form factors by (at fixed t/s)

dv/dtozr2\FHH,(t)\2.

Then, from (1), we predict for large t \FHH,(t)\ -t1'"*, where nH is the number of quark fields in hadron H. This holds for multiple as well as single photon exchange. In renormalizable spin-i theories, F2p~t~3 and thus we obtain GE/GM

asymptotic scaling for the nucleon, consistent with experiment.9 The meson results are in agreement with recent Frascati data10 for e *e " *u+u~,K+K~. The results for the transition form factors agree with Bloom-Gilman scaling.11 We predict the asymptotic power T5 for the spin-averaged deuteron form factor.

A specific model which gives many of the above predictions for hadron scattering is the inter­change model.12 Although the electromagnetic form factors of the hadrons involved must be in­put from experiment, the interchange model p re ­dicts the detailed angular dependence of the dif­ferential cross section, f(t/s), as well as its falloff in s. The results are consistent with (1) for processes involving photons and mesons. However, because the calculations of Ref. 12 treat all hadrons as a two-component 'fcarton-core" system, those results differ from the ap­proach here for processes involving only bar ­yons. Namely, we have (dv/dt), where-

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VOLUME 31, NUMBER 18 P H Y S I C A L R E V I E W L E T T E R S 29 OCTOBER 1973

a s the in terchange predic t ion i s {do/dt)pp^pp

~ s " 1 2 . Thus it i s pa r t i cu l a r ly useful to do th is exper imen t carefully to see if the proton i s be t t e r desc r ibed a t shor t d i s tances a s a par ton plus core or a s th ree q u a r k s .

Turning now to scal ing laws for inclusive p r o ­c e s s e s at la rge t r a n s v e r s e momentum and m i s s ­ing m a s s , we see that we cannot p roceed with­out additional assumpt ions . In cont ras t to exclu­s ive sca t t e r ing , the specification of the observed pa r t i c l e momenta in an inclusive p r o c e s s (away from the exclusive edge of phase space) does not uniquely de te rmine the number of e lementary fields which in terac t at shor t d i s tances s ince not al l the invar ian ts (of unobserved pa r t i c l e s ) need be la rge . However, in the absence of the accu­mulat ion of logar i thms , i . e . , the validity of the d imensional sca l ing a rgumen t s , the most i m p o r ­tant mechanism for producing high-/> x inclusive events involves the l a r g e - s and -t s ca t t e r ing of a subset n of the const i tuents (giving ?ftl ~s 2 ~" / 2 ) , plus many sca le - invar i an t l o w - m o m e n t u m - t r a n s ­fer sca t t e r ings of spec ta to r s . Models based on r enorma l i zab le field theory , however, do fall in­to two dist inct ca tegor ies with r e spec t to the min­imum numbers of in terac t ing fields and hence the i r asymptot ic behavior (see Fig. 2).

(I) Quark-quark scattering.—If a d i rec t , h a r d sca t t e r ing of quarks which a r e const i tuents of different hadrons i s allowed, then all inclusive reac t ions a r e asymptot ical ly sca le invar iant :

Eda/d3p~s~Nf(t/s,M2/s), (3)

with N= n - 2 = 2 for s — <», when t/s and M2/s a r e fixed. In such models 1 3 a quark from A and one from B s c a t t e r to la rge angles "for f r e e " and then, by (scale invariant) f ragmentat ion, produce the observed la rge t r a n s v e r s e momentum p a r t i ­cle C. Exper imenta l evidence a l ready excludes Eq. (3) with N=2 f o r p p - T T ° X , 1 4 so that we should take ser ious ly the o ther category of s ca l e l e s s models .

(II) Quark-hadron scattering,—If e lementa ry

( a ) (b )

FIG. 2. Typical diagrams for irp-^irX inclusive scat­tering at large momentum transfer and large missing mass, (a) Quark scattering, (b) quark interchange. In­teractions among the spectator quarks are not shown.

pointlike o r gluon in te rac t ions between qua rks of different hadrons a r e suppressed , 1 2 e i ther by a select ion ru le 1 5 or dynamical ly , then the num­b e r of e lementary fields which a r e r equ i red to r ece ive la rge momentum t r a n s f e r s will depend on the na ture of the pa r t i c l e s involved. F o r ex­ample , in pp (or irp)-~irX, the minimal l a r g e - s and -t exclusive sca t t e r ing s u b p r o c e s s , a s s u m ­ing q u a r k - p a r t o n s , would be qcf^ irir, qir-^q-n, o r possibly qq-*irqq, giving n=6 and N=4 in Eq. (3) a s predic ted by the in terchange model.1 6 This r e ­sult for pp — TT°X is in ag reement with recent da ­ta.1 4 S imi lar ly , for pp (or tip)-*pX, the min imal exclusive sca t t e r ing p r o c e s s could be qq —p$,17

again giving N-A. However, if, as in the i n t e r ­change model of Ref. 16, the subp roces s qq-^pq were suppressed , then one would cons ider the s u b p r o c e s s e s , qp-~qp, qp, qir^Pffl, o r qc[-~pp, giving n = 8, N=6. Fo r yp-^irX we find N= 3, based on the minimal p r o c e s s yq -~ irq. Deep in­e las t ic p r o c e s s e s ep — eX, yp — yX, and pp — nX a r e sca le invariant s ince n = 4.

Near the exclusive edge of phase space the r e ­su l t s for inclusive (M2 la rge) s ca t t e r ing must match smoothly onto the cor responding exclusive (M2 smal l ) formula.1 1 Compar ing Eqs . (1) and (3), th is impl ies that for sma l l M 2 , f(t/s,M2/s) = (M2/ s)pf(t/s), withp=n-N-3.

If c l a s s - I I t heor i e s a r e c o r r e c t , the absence of ha rd qua rk -qua rk col l is ions between hadrons has s t r ik ing implicat ions for the t/s dependence of exclusive p r o c e s s e s . Let us cons ider the region s » \t\ with \t\ l a rge and p a r a m e t r i z e \M(AB -A'B'^-s^Pit), o r for exotic channels , ~ua(t)

*P(t). As shown in Ref. 12, in the absence of s c a l a r or vec tor gluon exchange between hadrons , P(t)~FBB*(t) [or FAA,(t), whichever falls f a s t e r ] . Then from Eq. (1) we find that the leading effec­tive t ra jec tory for very la rge \t\ i s a(t) = l - 1 x(nA + nA,). These r e s u l t s , fff^--l, <*pp-+pp — - 2, and ayp^yp-~ 0, a r e consis tent with expe r ­iment.1 8 '1 9

The approach p resen ted he re 2 0 i s quite r e m a r k ­able for so simply explaining such a l a rge num­be r of observa t ions . We have shown that expe r i ­ment just if ies the application of par ton sca l ing ideas in a l a r g e r r eg ime than previously thought. Even when the par ton which r ece ive s a l a rge m o ­mentum t r ans fe r r e m a i n s bound within a hadron, the sys tem behaves like a collect ion of f ree qua rks . It i s an in te res t ing possibi l i ty that the r e s u l t s given h e r e can be obtained by applying the t ech ­nique of ope ra to r -p roduc t expansions on the light cone and at shor t d i s tances to bound-s ta te p r o b -

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V O L U M E 31, N U M B E R 18 PHYSICAL REVIEW LETTERS 29 OCTOBER 1973

l e m s .

W e w i s h t o t h a n k o u r c o l l e a g u e s , e s p e c i a l l y

J . B j o r k e n , R. B l a n k e n b e c l e r , S. D r e l l , J . G u n -

i o n , Y. F r i s h m a n , and J . K i s k i s , f o r he lp fu l c o m ­

m e n t s , and one of u s ( G . R . F . ) w i s h e s t o t h a n k

t h e S t a n f o r d L i n e a r A c c e l e r a t o r C e n t e r fo r a

s t i m u l a t i n g and e n j o y a b l e s t a y w h i l e t h i s w o r k

w a s b e i n g d o n e ,

*Work supported by the U.S. Atomic Energy Commis ­sion.

l rThis r e su l t for e las t ic sca t te r ing has been obtained independently by V. Matveev, R. Muradyan, and A. Tav-khel idze, Joint Insti tute for Nuclear Resea rch Report No. D2-7110, 1973 (to be published). We thank J . Kis ­k is for bringing th is work to our attention.

2Spinors a r e normal ized to u^u^2E. 3It is in genera l c ruc ia l to include all Born d i ag rams

since (e .g. , in r enorma l i zab le Yang-Mills theor ies) ca ­nonical scaling is t r ue only for the total Born amplitude and not neces sa r i l y for each d iag ram separa te ly .

4A complete d iscuss ion of sca t t e r ing ampli tudes in the Bethe-Salpe ter fo rma l i sm is given by S. Mandel-s t a m , P r o c . Roy. S o c , Ser A _233, 248 (1953).

5An in teres t ing possibi l i ty is that such a condition may be satisfied in non-Abelian gauge t h e o r i e s . See a lso S. D. Drel l and T. D. Lee , Phys . Rev. D j>, 1738 (1972) and re fe rences the re in for d iscuss ion of the va l ­idity of Eq. (2).

GIf the data a r e pa rame t r i zed at fixed c m . angle as du/dt = A/sn, then R, Anderson et ah, SLAC Report No. SLAC-PUB-1178 (unpublished), fit n(yN~~ ir+N) = 7.3 ± 0 . 4 ; G. Brandenberg et al,9 SLAC Report No. SLAC-PUB-1203 (unpublished), fit nJJKL°p~*Ks°p) = 8.5± 1.4, n(K°p~~ir+A°) = 7 .4±1.4 and n(K°p — TT+2°) = 8.1± 1.4.

7 For r e f e r e n c e s , see R, Blankenbecler , S. Brodsky, and J . Gunion, Phys . Let t . 39B, 649 (1972), and Phys . Rev. D 8, 187 (1973).

8S. Brodsky, F . Close , and J . Gunion, Phys . Rev, D (3, 177 (1972).

9 P . Kirk et al., Phys . Rev. D 8, 63 (1973). Although the data a r e not consis tent with a dipole fit over the en­t i r e q2 r a n g e , they a r e consis tent with our predict ion

of a pure power, (q2)~2, when \q2\^ 3 GeV2. 10See N. Si lves t r in i , in Proceedings of the Sixteenth

International Conference on High Energy Physics, The University of Chicago and National Accelerator Labora -tory, 1972, edited by J . D. Jackson and A. Rober ts (Na­t ional Acce le ra to r Labora to ry , Batavia, HI., 1973), Vol. 4, A fit to the combined 7r,i£ points gives \F{t)\ = A/tn with » = 1.08± 0.15.

U E . Bloom and F . Gilman, Phys . Rev. Let t . 2J3, 1140 (1970); J . Bjorken and J . Kogut, SLAC Report No. SLAC-PUB-1213 (unpublished).

1 2Blankenbecler , Brodsky, and Gunion, Ref. 7. See a lso P . V. Landshoff and J . C. Polkinghorne, Phys . Rev. DJ3, 927 (1973).

1 3Examples a r e S. Be rman , J . Bjorken, and J . Kogut, Phys . Rev. D 4, 3388 (1971); S. Be rman and M. Jacob , Phys . Rev. Let t . 25 , 1683 (1970); D. Horn and F . Moshe, Nucl. Phys . 48B, 557 (1972). This mechanism will a l ­ways occur on the e lec t romagnet ic level .

1 4B. Blumenfeld et al., to be published. 5This would happen in a model with fictitious colored

quarks and colored gluons as d i scussed by H. F r i t z sch and M. Gell-Mann, in Proceedings of the Sixteenth In­ternational Conference on High Energy Physics, The University of Chicago and National Accelerator Labora­tory, 1972, edited by J . D. Jackson and A. Rober ts (National Acce le ra to r Labora tory , Batavia, 111., 1973), Vol. 2, and could happen in a model with r e a l colored quarks and gluons as d iscussed by Y. Nambu, in Pre­ludes in Theoretical Physics, edited by A. de-Shal i t , H. Feshbach, and L. Van Hove (North-Holland, A m s t e r ­dam, 1966), and recen t ly advocated by H. Lipkin, Weizmann Institute of Science Report No. WIS 73 /13 , 1973 (to be published).

1 6These a r e the inclusive p r o c e s s predic t ions of the in terchange model . See R. Blankenbecler , S. Brodsky, and J . Gunion, Phys . Rev. D 6, 2652 (1972), and Phys . Lett , 42B, 461 (1972).

17We thank J . D. Bjorken for suggesting th is possibi l i ty . 18Except for the value for appi which depends on the de ­

ta i l s of the proton wave function, these r e s u l t s a r e con­s is tent with those of Ref. 12.

19D. Coon, J . T ran Thanh Van, J . Gunion, and R. Blank­enbec le r , to be published.

20A detailed r e p o r t of this work i s in p repara t ion .

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