E - SLAC - Stanford University

-.

srslc-m-1473 CALT 68-441 August 1974

!,

._

T/E

Scaling

Laws for Large Momentum Transfer

Processes*

STANLEYJ. BRODSKY Stanford

Linear

Accelerator

Center,

Stanford

University,

Stanford,

California

94305

and GLENNYSR. FARRAR California

Institute

of Technology,

Pasadena, California

91109

ABSTRACT Dimensional

scaling

laws are developed

as an approach to understanding

the energy dependence of high energy scattering mass angle.

Given a reasonable

bound states, large

and the absence of an internal

s and t,

do/dt

(AB -t CD) Q s

in A,B,C and D which carry law is obtained

for

used to specify

n, we find

this

large

accounts naturally

factor.

-n-i-2

a finite pI

for

f(t/s);

distance

center-ofbehavior

we show that

n is the total

scattering.

number of fields scaling

When the quark model is For instance,

experiments.

theoretic

of

at

of the momentum. A similar

2 -2 the (q ) asymptotic the field

at fixed

mass scale,

fraction

inclusive

behavior

of the proton

foundations

which needs to be made about the short

form

of the scaling

distance

laws

and infrared

of a bound state.

(Submitted

*Work supported Contract

on the short

good agreement with

We examine in detail

and the assumption behavior

assumption

processes

in part

AT(_11-11-68

Energy Commission.‘

for

to Fhys. Rev.)

by the U, S, Atomic Energy Commission. the San Francisco

Operations

Office,

Prepared under U. S, Atomic

-2-

1.

Introduction The dimensional

laws 1,2

scaling

(AB + CD) Q s -n+2f(t/s)

$

(1)

-a iTj$$

for

the asymptotic

behavior

summarize the results and elastic total

angle scattering

of a broad range of hadronic

form factor

number of lepton,

fraction

of fixed

measurements. photon,

The integer

and elementary

of the momentum in the particles

represent,

in the simplest

of complexity

possible

of a hadronand

its

One of the most important elastic

electron-hadron

aymptotic

quark fields

carrying

a finite

The scaling

laws

between the degree

behavior.

consequences of Eq. (1) is its

scattering.

to the minimum numbers of fields ‘L t

photoproduction,

n is given by the (minimum>

manner, the connection

This rule

dependence of the (spin-averaged)

FHW

scattering,

A, B, C and D.

dynamical

. :$

appear to compactly

immediately

application

connects

electromagnetic

to

the

form factor

nB in the hadron:

l-nE (2)

Thus, using the quark model; we have F(t) for baryons. All

We also find

of these results

by present

(see Sec. II.B2)

are consistent

data which is relevant

to testing

well,

in all

being consistent

PP + PP*

We predict

experimentgives can be tested

with

In Section

experiments.

'L t

dajdt

III

F2%t

of this

the scaling

article

-7

and s-lo,

-2

dependence indicated we survey

law, Eq. (1).

cases and accurately

+- s

%t

-3 , and thus GE Q, GN scaling.

the asymptotic

s -7,3..+9. .4 and s-g*7 '"*3,v in the future

-1 for mesons and Fl(t)

verified

respectively,

fares

of predictions III.

very

for yp + up and at fixed

A catalog

is also given in Section

It

present

cm angle! which

-

-_

-3-

In fact,

dimensional

analysis

and some simple assumptions

lead

to the scaling

state

of

total

hadron momentum. The amplitude

hadrons

9%

constituents

(see Fig.

scattering

1.)

of their

momenta with

is therefore

constituents,

If

that

the total

at large

normalization

carries

a finite

integrating

fraction

the scattering

over possible

the constituent

number of fields

--?a & +J

the

constituent

in the initial [length]

and final n-4

when the

is chosen

4'n

angles by successive each constituent

to insure

Landshoff

to Mn from has made the

hadrons can scatter

near mass shell

a constituent

elastic

at large

scatterings

of the other

(as in Fig.

of 3).

(see Sec. II.Bl and Appendix A) in this case Mn Q (&) 4-n, (&)L-l where L is the number of pairs of constituents from different hadrons In fact

which have a large that whether

angle scale

or not this

there

is no modification

fixed

angle processes

both direct cally

inpp+pp:$ dicted

if

energy,

L-3. fixed

photons or leptons.)

evidence

s s -9.7

that

at present F0.3

More indirect

but equally

of this

(Section

I1.A)

Salpeter

wave functions

form factor asymptotic

giving

evidence

is the fact

reasons,

of this

such a scale

then,

constituent

We show that amplitude

that

no-high

is from high falls

much

invariant

(see SectionIV

assumption

C is necessary.

We proceed by first

of an amplitude

in the spinless

of the full

if

is as follows,

the construction

is that

between quarks of different

hadrons did occur

section

as an example (II.AlI_. behavior

important

predicted

is

the s -8 as would be pre-

The best evidence

p,/&

or. to there

(pp +- T +X). whose cross section

For these empirical

The organization

scattering,

However

The direct

interaction

between quarks of different

for details).

should be noted

mechanism is not physi-

rather.than

angle scale 'invariant

inclusive pion production pi faster than the p;" at fixed

Landshoff's

energies.

f(t/s)8

hadrons seems to occur in nature.

interaction

(It

takes place in hadron-hadron

involving

at least

dajdt

process

interaction.

from such a phenomenon to form factors

and indirect

important

invarkant

in terms of Bethe-

case, using with

assumption

the meson A the

is the same as the behavior

of

-.

-lO-

the dominant irreducible

contribution,

to spin-l/2

is not difficult

constituents

show that

the meson form factor

the spinless short

case.

distances

in large

momentum transfer question

wave fundt$ons. favor

of the validity

d&stances

and speculate

is not finite

are discussed.

of irreducible

that

I

graphs,

their

Connected~Bcrn

that obtain

spin-l/2

from general

order

in Section

I1.B.

order

the asymptotic

corrections

(Born)

We start

(II.Bl)

the dimensional

result.

gauge i

graphs with We

C, which allows the behavior

law independent

of the interaction

models.

in

at short

irreducible

us

of the

of the spin of

as long as the coupling the Born diagrams we show

quarks gives GE/GM asymptotic

exchange and quark interchange of higher

our scaling

While discussing

arguments

arguments

A reduces the problem to the behavior

we show that

are dimensionless.

Not very

of.Bethe-Salpeter

when the wave function

Next (II.B2)

or the details

3-t '* ."

we confront

in non-abelian

of assumption

diagrams reproduces

a model with

the effects

II.A3

behavior

arguments in favor

contribution,,

the constituents constants

processes

'.%a

(whose dimen-

A and B.

on the situation

diagrams and show why they violate

some speculative

to neglect

assumption

as in

have the same behavior

In Section

and infrared

we examine the lowest

a number of interesting

present

distance

in our:results

the Landshoff

fields

of our assumptions

Modifications

Having established

for

[L] -3'2)

We

why, even though

We review what is known, give some plausibibity

of our assumptions,

theories.

to explain

scattering.

II,A2.

in the spin&l/i!

of spin-0

(dimension

exclusive

much is known about the short

point

bound states

fields

of the analysis

and is given in Section

This is an appropriate

and spin-l/2

the difficult

The extension

has the same behavior

are being probed,

sion is IL]-')

Mn.

scaling.

Regge trajectories

Finally,

in Section

to the Born diagrams.

We also in gluon

II.B3

we discuss

._

-_

-ll-

II.Al

Spinless

Bethe-Salpeter

The simplest behavior

Wave Function

example, which illustrates

of an hadronic

amplitude

wave function

of two scalar (see Fig.

satisfies6

(k2-ma2)((p-k)2-mb2)$p(k)

(Note that

the wave function

propagators

for

irreducible

=

diagrams.

For illustration

~(p,W

I

The full

Bethe-

4)

A K(k,a,p)$p(R) s (2n) 4 defined,

.

includes

This is the eigenvalue

legs.)

TheFkernel

the asymptotic

of the form factor

fields.

Y, as conventionally

the constituent

the meson mass M2 = p2.

how to obtain

is the calculation

of a meson, taken as a bound state Salpeter

and the Meson Form Factor.

K(k,R,p)

is the sum of all

we first

consider

the

equation

for

two particle,

a generalized

ladder

approximat&on = g2

o(?)dh2 (&-k)2

where we can choose the spectral scales

2 -6 as (!L ) .

corresponds

-A2+iE

function

a(h2)

For a super-renormalizable

to 6 = 1.

always compute with of the calculation'.

For renormalizable

theory theories,

for

large

6 is positive; e.g.,

and take the l&it

6 > 0 as a regulator, This is analogous

such that

to the generalized

c$~, 6 = 0.

R2, s(p,k,R) Ip3 theory We shall

6 -+ PaC .Ghe. Wd Feynman Pauli-Villars

6 > 0, we have q,(k)

Note. that regularizat$on in perturbatjron theory. -4-6 @he m&-&mumcondi:t$on and JI,(x=O) 4 =. Q [kl

q~,(x=O) < 00 is $p(k)

'L [kl-4/[:log(k

method Or dimensional

2

)I

If&

with

E 4 a)

for for

-12-

The form factor

in ladder

0,

f,or q2+ 03 if

We return

By assumption space, SO that

internal

6

treatment

contributions

A the wave functions

the asymptotic

of these

The loop corrections

.

and can give logarithmic

to their

charged lines;

in Eq, (10) are obtained

A comprehensive

has been given by Mandelstam

loop as 6 -t 0.

5)

h’>

kema&s are introduced

with

is (see Fig.

Yd$k "r 04 (2k+q) ’ [ (p-k) 2-%2 1JI, 04 + (a-b > J (10) (zT)4 ?P+-4

c.

If additional

approximation

corrections

for

each

below.

are bounded at every point

behavior

are non-

of F(q2)

is controlled

in momentum

by two regions

of the d 4k integration: (a)

k 'L xp; k2 Q O(m2),

(b)

(k+q)

Q x(p+q)

xpttc, where

magnitude

It

K

Thus k2

=

x2m2

to the variable

is easily

shown that,

' (p+q) = 0 and

(K

A that i-

K

i

(p-Q2

Q, O(m2), k2 'L (l-x)q2

integration vector

orthogonal

the wave functllon and k

l

region

p

after

is damped in large

= xm2 are finite.

the dk2 integration

Region (b) corresponds

12) bounded.

(a) corresponds

to

to p which is of bounded

used in the Sudakov and infinite

0 5 x < 1 can contribute. K'

‘L Ob2),

is a spacelike

(by assumption

momenta). similar

; (k+qj2

(k+q>2 'L (l-ir>q2,

Q Ob2),

More precisely,

where x is finite. k=

(p-U2

relative

(The variable

x is

momentum frame anaj.yses.)

is done, only the regton to k = (p+q) + ~~.'.with

-13At this of q2$(q2>, venient

point

we can relate

the asymptotic

up to logarithms.

to iterate

However, for

the equation

is encountered.

fall-off

of F(q2> to that

our purposes it

of motion wherever--large

Thus we obtain

for

large

will

be con-

relative

momentum

q2 %L.

(2p+q)‘F(q2)

$5

]d!L

IT

with

k = xp + ~2 and R = yp +

of each wavefunction:

point

scattering

This is in fact

p+q

(R)M;(p,q,U+4Jp(k)

The integrations

K’,

dominant region connected

,$-

(2a)4

K

amplitude

the prototype

are limited

2 and K V2 = O(m2).

illustrated

in Fig.

of our general

equatPon of motion

for

the wave function

momentum.

In this

manner we generate

represents

the scattering

wherever

6.

to the

ME is the five-

prodedure.

We employ the

it

large

involves

the connected amplitude

of the quark constituents,

each with

Mg which a finite

of the hadron momenta: P =xPi H, 5: pi = 1. Mg is of course exactly 1 1 ._ connected amplitude which occurs when the'hadronic binding is turned off

adiabatically,

fraction the

in which case x

Let us now discuss large

relative

q2 one can readffly

+ m / i i"H* the asymptotic behavior verSfy-that 1

M1-I ,,, (2q+d

5

(1-x)q2

I B-r]&-(X-Yh1216

+ zy (p+q3 j&p

w-Y>S2 which is properly

1 1 ~l-y);2-(x-y)q2]6

gauge invariant:

F(q2)

where the logarithm (l-x) -I. or (l-y) -1 .

of F(q2) using Eq. (11).

(\I (+)1+6 4

q M' = 0. P5

Asymptotically,

then,

log q2/m2

occurs because of the endpoint This result

for

of the integration

S > 0 agrees with

those of Ref.

over 9.

For

-14-

More generally powers of (q2)+ loop

if

6 = 0.

of the theory

the higher for

we assume that

is more convergent

theories,

of the renormalizable

theory

behavior

the accumulation

MF.

particles

:

result

by extra

for each additional

the true ultraviolet by elementary

behavior gluon exchange

then the proper. regulari-

is 6 = O+ and additional

the-scale

We thus have the prediction

F(q2>2, +

logarithms

for example),

of logarithms

of the wavefunction.nor

amplitude

this

logarithmic

This is a consequence of our assumptions

are suppressed.

and B, that

modify

than indicated

free

modifications

kernels

6 > 0, and by possible

However, if

(as in asymptotically zation

order

affects

neither

invariance

A

the asymptotic

of the connected

for a bound state

-

of two spin-0

log q2/m2 .

Q Tn the case of an n-field

bound state,

we have simply

(neglecting

logs)

F(q2) -t (n211-n (q2Yn ’ i.e.,

F(s2)

A detailed

'L (q2)1-n discussion

and Schierholz', constituent

in the renormalizable of this

Note that

line

result

in agreement with

arise

from each off-shell

from each gluon line.

one pays the penalty

of one power of q2 for

the direction

of each constituent

from along p to along p-tq.

dence of this

result

in the next section.

is discussed

Eq (2).

when n = 3 has been given by Alabiso

the powers of (q2)-l

and the (q2)-"

Thus,physically,

limit

changing

The spin indepen-:

-15-

II.A2

Spin-l/2

Constituents.

The calculations

of the asymptotic

somewhat more complicated results'in

when the effects

the renormalizable

the spinless

results.

limit,

zero orbital

satisfies

(see Fig.

4)

become

are effectively

The the same as

the example of the form factor

of two spin-l/2

angular

of form factors

of spin are included.

however,

We begin with

meson which is a bound state present,

behavior

momentum.

fields.

of a

We assume, for

The Bethe-Salpeter

the

wave function

s.

s

4

@-ma)(d-&mb)+p

(k)==

id

K(k,R,p)&p

(R)

(12)

Cm4

where K is the full

Bethe-Salpeter

as in the spinless

two-particle

case, we begin with

irreducible

the generalized

kernel.

ladder

Again,

approximation

form (13)

The l'(a)

and I'(b)

represent

gluons to constituent may choose o(h2)

the (momentum-independent)

%(k2)

couplings

Just as in the spinless

a and b respectively,

such that

Dirac

Q (k2)'1-"(6

of the case, we

> 0, k2 + m), which gives

theories correspond to assumption A: $(x=0) = d4k qp(k) < a. Renarmalizable f 6+-O, + at least in the ladder approximation. This is discussed further in Section

11.A3.

The form factor

for

the bound state

using the kernel

(Eq. (13))

is

--

= (2pfq)VF(q2)

=e

a

-.id4:k 4 (2n)4Qp+q('k+q)yu (a) 'g-lpl-y$ *, 04

+ (a++@

.

(14)

-16-

Additional

contributions

presence of the higher

required

loop kernels,

These are of the same order As'in asymptotic evaluated

the spinless form factor at large

the wavefunction

by gauge-invariance

are necessary

as in the spinless

or are non-leading

calculation,

case (see Fig.

for q2 + co if

the important

in the

6 >o.

contributions

for

occur when only one of the two wave functions

relative at large

momentum. Again, relative

it

6).

the is

is convenient

to iterate

momentum using the equation

of motion

and we obtain d4ke @+qh(q2)

Q

-s (2Td4

s G-4

xg is the connected Feynman amplitude case the dominant contribution fractions more, if

of the longitudinal only the leading

spin structure structure

J,p+q (QM;:,rL,(k).

shown in Fig.

7.

momenta and small transverse q2 dependence of the form factor simplifies

(15)

As in the spinless

comes when the constituents

of the wave function

enormously.

carry

finite

momenta.

Further?

is desired,

the

In general

the

is

IJ, Ck) = [fla(P,k)g

using

-

d4R

-t- f2a(P,khal[flb(P,k)

(-Ed+%) + f2b(P,k)%l

the identities

and

.

%g=-

1 spin

v&s)&4

,

l

(16)

10

Eq. (16) can be rewritten $p(k)

in the form

u,(k);,(p-k)$;-(k)

=

+ va(k)-$p-k)$+(k)

c -8, f va 04 ;b (p-k) 'i';- (k)

+ ua&>% (p-W$+O4

(17)

, I

with

the appropriate

this

structure

spin projections

understood.

must reduce to the product

In the zero bindfng

of two free

spinors

1imT.t

ua(k=xp)Gb(p-k=(l-x)p),

hence for our purposeswemsyruse $,

04

?i

ua

The terms thus neglected

(k)vb

(p-W;-

are at least

which cannot become large)

they arise

neglected

linear

in,k',

to the binding

is easily

from the presence

seen

10:.

Eq. (18), s

1 d4k klxp

(the "transversemomentum" contribution

.to F(q2),

energy and may be therefore

in time-ordered

in the wave function

With the simplificatton, (2p+q)'F,(q2)

(18)

l

and give a non-leading

That these terms are proportional legitimately

04

perturbation

of an extra

theory:

q< pa%r.

we find /d4Q

fi=Y (p+4 >

+J :1-I Y W5 q;-(k) p+q

)

(19)

where = Ua[JT(p~)lvb[(l-y)(p+q~)]M~(k=xp,~=y(pfq))

is the connected that

for

amplitude

the connected

and R', and except

tree

evaluated

between on-shell

graph we can neglect

for particular

.

helicity

the

spinors. K

configurations

Again note

and K' components of k' we may drop all

mass

4 _,.i

-18-. terms in the asymptotic

G2

limit.

This is explicitly

evident

in the Breit

frame

= -q2) :_ P-= (ks

2 - $12)

and

pi-q =,(~z-zFz

, + 32)

'

where each component of p and p+q becomeslarge. shows that

the behavior

of ig for

large

Explicit

q2 is (-q2j+",

calculation as dictated

then by

counting 11 . Accordingly, as in the spinless calculation, we have 2 -1-g ._ 2 .;.& / . . FCq21 'L '(q ) Rn(q ./rn ) where the logar9thti results from the endp&$

dimensional

of the x or y integration. As in the spinless fermion

calculation

propagator,

a factor

The overall = p4k

renormalizable

is 6 + Of. with

each gluon carrying

scale of the form factor

q;(k).

limit

(q 2 ) -1 is associated

2 -6 (q ) with

and a factor

the Born diagram. value of $:(x=0)

The canonical

each off-shell large

is determined

As we have noted,++-(x=0)

is finite

q2 in by the for

6 ' 0. Note that

the wave function

Bethe-Salpeter spin-0

wave function.

or spin-l/2

same dimensions For spinless

I$r(k)

plays

the same role

The equivalence

of asymptotic

constituents

follows

as the spinless

constituents,

since * ;-(xl

Bethe-Salpeter

the Bethe-Salpeter

as the spinless behavior

( see below)

wave function, amplitude

$

with

has the ;pinless(x).

_

for a meson in position

space is '4;pin1ess(x)

5

where we are using continuum For the spinor

normalization

'L [L]-~

. $i andz do +s -5 (rath er than s -6 as given by Eq. (1)).

than the scaling

that..& ,.::.

define

direction

set of quarks unless

the near-mass-shell

fraction

fractions

angle $.

carryrllarge

momenta

by the wavefunatfon$>

range to be restricted

Technically,

in order

Momentum conseava&ionmeans

in some different

on the kinematics,

that

the quarks with

of the quarks with

they*will

directgon

some finite

Imagine that

to the xz plane)

3i;momentum (l-xl)P,

allowed

as required

Assuming then that

Let their

the scattering

general

large

lib.

finite~+angl.e

+o+. They in

is evident

momenta (so the transverse

mesons are finite

shown in Fig.

Now consider

it

= x2 + O(i) where P is the magnitude

= y2 within

2=y1

lib,

parallel

of mass (P Q &).

x1 M x2 scatter:,bysomex;‘ ..F x

Examining Fig.

behavior

That is,a

results

linear

a fundamental

length

from an

infrared scale;

1

.

q we give in the Appendix Alan infinite contr2bution there will

to meson-meson scattering. be no modification

scattering,

of naive

Compton scattering

or

.-

-26-

This class of near-mass-shell the hadron scattering implies giving

scattering

diagrams should evidently dominate 18 . Their existence in the asymptotic limit

amplitudes

a violation

of Eq. (1) for meson-baryon

and baryon-baryon

scattering, "I

instead sL-l do --'L dt! - ) sn-2

f(t/s) _'X ,j where L is the number..of wide angle, e&g. , L = 2 for meson-baryon Experiment

(see Section

of Eq. (1). porated

From this

may well

and Ref.

C; multiple

of assumption

version

scattering.

scaling

result

about nature

which is incor-

scattering

is not important

of logarithms

of A and B 'although

of C:

quarks is damped in energy.

that

21 ,

hadrons,

or else that

analysis

indicates

cription

of the angular

required

(in this

that

ultimate

advocated

angle scattering

mechanism for this

explanations

adoption

of a

of near-mass-shell damping is an accumu-

to the quark-quark-gluon

and for a different

vertex

when

purpose by Blankenbecler,

for gluon exchange is very small.

the con&ituentL-interchange dependence of exclusive

connection

see our discussion theory with

and gluon exchange,

suppressed

Further

evidence

picture scattering

gives with

of asymptotic

Their

a good des-

no gluon exchange Regge trajectories,

quarks and gluons would in general

both quark interchange in nature.

their

gluons cannot be exchanged between quarks of different

the amplitude

Since a field

to understand

20 .

made earlier is that

has recently large

in the corrections

proposal,

Brodsky and Gunion

19

A possible

both quarks are near mass shell

Sec. II.B3).

fact

the dimensional

near-mass-shell

Polkinghorne

be connected.

Another

18) favors

C is much more difficult

than the validity

somewhat stronger

lation

quark scattering,

and L = 3 for baryon-baryon

a striking

>

experiments.

The validity theoretically

(21)

on-shell

scattering

we learn

in assumption

for present

III

,

for

this this

indicates proposal

that

have

gluon exchange is

is given in Sections

'-

-27-

II.B3

and IV.

It

should be emphasized however , that

phenomenological

rule:

can be made with

no gluon exchange whatever.

for this

a consistent

possibility

such near-mass-shell argued that,

since

the time-ordered

this

of present No compelling

wide'angle

a experiments

theoretical

reason -91

state

non-zero -l/2

.

with

color

in a model with

would be suppressed.

contribution

perturbation

the internal

compared to s

is that

processes

quark mass were very large

theoryf'has

is essentially

rL4.e has yet been advanced.

Yet another

that

description

this

(perhaps theory

is proportional comparable calculation

near-mass-shell

to &)

It

confined

quarks

might be suppressed

_

to 5

, if the effective 4 it would be small. From can be-seen

(which in a color for a finite in the full

SU(3)

time, amplitude.

%?j ;ty

quarks

can be heuristically

of the Appendix it

systems propagating)exists

Such a state

permanently

long

__

-28. II.B2

The Irreducible Typical

Diagrams

Born diagrams for meson-baryon

are shown in Figs.

lines

just

IUAland

4

According

(Fig.

amplitude

propagators

are large

the reader

can easily

photon introduces M.p 2, s

-5/Z .

by direct factor

if

4-n .I.; .Figj

limited

momenta, which a hadronic

Fig. constituents. (helicity

It flip

is readily

or non-flip

the kind shown in Fig.

respectively

13b'gfve

q2 >> rni the diagram of Fig. calculation

shows that

large.

13a cannot contribute

2 F (q ) is proportional 2

quark model of the proton

the proton

form factor

scalar

contribution

will

Had the

quarks.

to have

s and t. form factor or vector

in the zero-quark-mass

a leading

not the

This is the vector

at large

for either

in the expected

would do, the photon

a & so that Mw s -3 .

seen that

of the

been constrained

wave function

Born diagrams for

fermion

coupling

coupling,

or -l/2

the

However (as

to & resulting

spin-0

to photoproduction

13 shows typical

the vector

in the photon channel)

would not have generated

in the limit

three

10).

of the vector

and occurs for either

meson dominance contribution

case of Fig.

proportional

pair

coupling

f(t/s)

12 shows that

calculation)

quark anti-quark relative

M Q (&)4-n

10 are

the photon counts as one field

This is characteristic

(looking

the mnemonic of Section

the diagrams of Fig.

(as in the meson-baryon

spin of the constituent

"~I i '"‘q . li-4

the wavy

(p4 coupling,

12).

counting,

a numerator

propagators

The only process which needs special

should 2, (&)

verify

having

behavior

Eq. (1).

to dimensional

photoproduction

behavior

fermion)

seen to have the asymptotic

is photoproduction

which fermion

By applying

interaction.

Hence they reproduce

discussion

constituents

(s -1 for each off-shell

II.AZ

immediately 6+0.

With scalar

reduce to a point

and meson photoproduction

The heavy dots indicate

10 and 12.

are far off mass shell.

scattering

with

spin l/2

gluon exchange

limit)

graphs of

'L (q 2) -2 to F,(q')j. for scalar

gluons.

When

Explicit

to m2 so that in general a threeQ when q2 becomes give F2(q2) Q Fl(q2)/q2

In terms of the commonly used form factors

GE(q2) z F,(q2)

+ Iss_l F (4') 4M2 '

_

-29..

and GM(q2) : Fl(q2) corresponds (Section

to

23

f icF2,(q2) where

constant

change (Fig.

meson-baryon

diagrams).

Although

angular

l

scattering

or indeed any hadronic

Both give the same scaling

to very different

which we leave unspecified

f(t/s)

experimentally

scattering,

angular

behavior

distributions

depends on the relative

we do not attempt-here

distributions,

(i.e., importance

to deal with

_

but in general the function

f(t/s)

of the two types of

the problem of the com-

we are able to make some statements

about the

very large, s + 00. When t and s are both large we know the -n-i-Z f(t/s) but not in general the function power dependence of da/dt 'L s region

t-

However, if

t channel process

either

gluon exchange or quark interchange 14) we can specify

(see Fig.

Using Regge terminology

we write

as Mnr s -ceff(t)s(t).

the amplitude

known, a spin 1 or spin 0 gluon in the t channel gives

aeff(t)

= 1 or 0 respectively

corrections.

for

all

t.

Multigluon

When quarks are interchanged

the integration

over the intermediate

the former.

In this

t = q2 (just

as in a form factor

t dependence as the helicity

are two important

In fact

form factor

also simple.

It

to a constant

F(t)

regions

k Q, xPC or k Q xPA.

of the diagram is purely

calculation).

non-flip

of the diagram when s >> t'is

there

rise

As

exchanges only generate

quark momenta:

case the lower half

is the dominant

the form of f(t/s).

is well

cut in Consider

a function

of

it has the same asymptotic -nC+l . The upper part 'L t

is essentially

high energy,

quark1

+ A + quark2 + B a~atteriin'g,;-';:~~~explici~ !J; t dependence comes from that required 1 X1+XA-h&A2 1 considerations 24 :& by helicity . Hence for A = B = meson or A=B=

baryon the leading

T photoproduction scattering

sii =!d ;:

by gluon exchange between the hadrons or by quark inter-

lob and 10~).

they give rise

overall

q2 as is indicated

III).

could take place either

kinematic

is the anomalous magnetic moment, this

GE/GM at large

We noted above that

plete

K

:

behavior

is t independent.

the upper blob Q, A.

M Q s-3f(t/s)

CL,s-cF(t):

Thus we find

On the other that

hand, for

for meson-baryon

there are two terms (coming from

-

to Fp(t)

k Q xPC and k Q xPA) proportional behavior Similarly

in s comes from F

is negligible. indicates this

so we have with

BB-+BB = -2 and I:::? aeff

According

that

to Ref.

counting

to the proposal

to our assumption

to aeff

are indicative

coupling

of the previous

hadrons is C.

The leading

~-tMB=-l aeff

quark interchange

gluon exchange and quark interchange

between quarks of different

respectively.

.

= -125 .

26 the best fits

Since a priori.

gives support

support

and F*(t)

constants

that (e.g.,in

should be equally section

that

-3% ; 4i -$a _‘:?

gluon exchange Fig.

10)

important,

-

gluon exchange additional ional

-31'> II.B3

Higher

Order Corrections

to the Born Diagrams

In any given order of perturbation corrections

to the behavior

these logarithms simple scaling shownz7 that

infrared

corrections

amplitudes

interaction

theory

it

It

is possible

that

the nominal which,;will

region

be modified

infrared

exponentiation

to a bound state

lepton

scattering

power law.reflecting ,_ by.~soft _r/ : : interactions

due ~2 -

amplitude.

In a gauge

dangerous in this

a modification

case theseaftering

cutoff.

of logarithms

which is p.otentially

introduce

Inthat.

oricanonical

has been

(22)

the possible

they will

the form of Eq.".(ZZ),.:

the

in QED it

angle exclusive

and X an effective

corrections

is the infrared

to fixed

alter

,

where t is the momentum transfer

strong

For instance,

If

have the form

-o[!Ln(slX2)]2 e

We are concerned with

be logarithmic

combine they could conceivably

of the Born amplitude. radiative

and photon scattering

there will

of the Born diagrams due to loop integrals.

were to coherently behavior

theory

respect.

to these amplitudes

rules' (Eqs-.:(&4))

will

the compositeness.of,the among..the constituents,, -. ..

bound state First,

scattering

in a theory

are neutral,

i.e.,

amplitude

non-abelian

color

so that

momenta k < O(d -1 > where d is a length

characteristic

in a bound state

the constituents

are generally

hence the infrared

singularities

are "shielded."

off

region

effects

on a

their

twe~rePsons: states

is damped for

of the hadron. mass shells

but

nvmber of

than Eq.. (22.) for 22 gauge theory'the physical

the infrared

give

ff::$&&fwda-

should be much milder

such as a color singlets,

the infrared

just

hadron;

mental; s~ro~g,intera~tion,~o~pling.4~ actually~.wea~,,~S,~~~jectured,~~~~,a ,;*,, .., ^ ,/ 28 people , such a ~yi$ic~$igp wqyld nqt be sqg ct+$ yery large t. On the other hand, arguments can be made that

of

Second, and

-32To illustrate

the first

point,

consider

massless vector

mesons couple to a conserved

to be a neutral

state,

infrared

of the virtual

region

factor

gauge theory

current.

We shall

matrix

element of the form exp[-A

line

Considering

(n,m) of external

pairs

charged lines.

and finm is the relative

(24)

such as the form factor of external

lines.

It

of a "meson," is readily

a leading

involving

one initial

Let us label

particles

a and b whose.momentapare

and one final

= 0 (as in Section

II,Al).

may be checked by explicit 'a

particle.

we must

checked that

the

t dependence are those the two (say)

initial

pa = xpr+ K and pb x (1-x) pz# :-with i t: h By our assumption

expansion

= xp and pb = (l-x)p,with

ve&ocity

l

only terms in the sum (23) which can introduce

K*P

For

frame of either:

= (p~~m)2]1’2

now an amplitude

sum Eq. (23) over all

pairs

n = -i- 1 (-1)

n and m in the rest Bnm : F

(23)

nm

where the summation is over all

of particles

errors

A,

of the logarithm of only order

Ban and Bbn are equal if

n is not equal to a or b.

thus causes cancellation

of the infrared

l/s

corrections

K

is bounded so that

in Eq. (23)) at fixed

(as

we may take

angle.

Then

Summing over a and b in Eq. (23) of order

-.

%n(t/m2> since

Q, = - QBThe infrared is generally difficulty itself

behavior

expected

of non-abelian

theories

with massless vector

to be much worse than in the abelian

of such theories emit a pair

'1 6sr;l. . p _I.

with

Q, Q,

8n2 c

(incoming)

take a hadron

Rn.A/h]

+f3, A=-111, rl, PiiEn e r 1

an outgoing

in which

: . : ; .: Q = C Qi = 0. Using Weinberg p s notation>the . gluoi corrections (X < lk,] < A) gives a correction

so that

to the scattering

an abelian

of soft,

is that

a soft

gluon emitted

colored

massless gluons,

case.

A central

from an external

ad infinitum.

gluons

line

However, in

can

-33-

a bound state

such catastrophic

the gluon constituents Furthermore, octet with

gluon emission

in the case of a color are infinitely

for k2 and p 2 2 log fe(rn -m )]. 81

l

Even if

as usually

singlets.

the logarithms

is no reason that

to the scaling

law

30

emission

theory

corrections

There is very little

(essentially

B) except what we said in Ref. 1: if

in deep inelastic these scaling

laws

scattering, 31

.

there will

22

color in mass

corrections

changes quantum will

exponentiate

.

as argued above, they can give logarithmic

assumption

giving

such logarithms

found in perturbation

we can say about this

of

case the infrared

k below rni - rn: is suppressed,

there

corrections

envisioned,

not degenerate

In this

Moreover because the color

numbers and is not soft, or cause large

theory

by virtue

off-mass-shell.

massive or at least

the usual hadrons which are color

of order

be regulated

of the hadrons being effectively

hadrons either

contribution

may well

probably

there

do not exponentiate, to the scaling

laws.

about the validity are modifications

be similar

modifications

of of scaling to

“t? i

-34-

III.

The Experimental In this

exclusive

section

processes.

Status-&f~?Exihlusive Stialing'. V.--s+,+._I_*Y ".. ,_ .". we are concerned with tests of the scaling law (Eq.(l))in Three questions

choice of quark model assignments 2) Are the scaling

scattering

for

laws actually

or "anomalous dimension"

are of particular

the number of fields

corrections?

3) Are there

contributions

of Eq.(l))'?

Of course,

the whole program is the self-consistency processes,

scattering.

This section

to experimentalists) this

we turn

to the more difficult

is quite

the large

of the scheme.

detailed,

1) Is our

in hadrons

exact or are they perhaps modified

(which may change the scaling

for exclusive

importance:

by logarithmic from on-shell

essential

Once that

question

-

to

is established

of inclusive

in the hope of emphasizing

amount of important

-a; & 5% _,Ta

acceptable?

(especially

work which needs to be done.in

field. The simplest

applications

of Eq. (1) are purely

+-l-p+p- or e+e- -t e e , ye -f ye, e e + yy, etc. we are wrong at a fundamental asymptotic

behavior

electrodynamics extent

that

havior

of purely

n = 1 for the highest +etc.) ee,

da/dt

level,

% s -2 f(t/s).'

(modulo logarithms

QED is correct

leptons

leptonic

eh + eh scattering

This is just

the prediction

corrections)

processes

n = 1 is correct

at very large

spin averaged electromagnetic

form factor

for

F(t)

to the

the scaling

with

be-

the assignment

experiment

++on e e -+vv,ee

up to +-

-+

leptons.

the differential

s and large

+

of quantum

so that for

are correct

s and t (CEA32 and SPEAR33 results

exchange approximation,

e'e-

these should have the

Since QED is in agreement with

we may assume that

In one-photon

experimentally

s and t our predictions

and photonic

e.g.,

In each of these cases n = 4 unless

from radiative

at large

and photons.

available

so that

electrodynamic,

cross section

for

t is given in terms of the hadron

-.

by

(25)

-35-

so that we have (Eq. (2))

the general

formula

(modulo logarithms)

1-nh Fh (t) for asymptotic contilude

for

by studying

the asymptotic

II.BZ,

one finds behavior

q2 is necessary

covering

the spacelike Fig.

Thus using n

given spin l/2

that

= 2 and n

P

= 3, we

Q (q21s2 can be

constituents

F2(q2) % (q2)-3

to test

IT

the dependence of ep + ep scattering

GE Q l/q4

of large

t.

2 -1 'L (4 > and Flp(q2)

q2, Fn(s2)

As remarked in Section gluon exchange,

or timelike

large

determined

purposes.

,

spacelike

that

separately

Q t

at large 4 23 .

and GM 2, l/q these laws,

with

vector

q4GM(q2) is consistent

with

that

a constant

Since a substantial

the experiment

experiment.

within

or scalar

2 q , so that we predict range et al. 34

of P. N. Kirk

range 1.0 2 -q2 ( 25.0 GeV2 , is most suitable

15 shows q4GM(q2) for

on q2.

for

our

For -q2 > 4 GeV',

errors.

This vindicates

our choice

= 3 as suggested by the naive quark model. Moreover the very fact that GM(qL) nP falls as a power of q2 is support for our scaling laws. (The data is not accurate enough to discuss determined

the question

of logarithms.)

GE and GM have been separately

only for -q2 -? to conclusively

that

GE could have

from the q2 dependence of the

37 .

of the proton

available3*

the proton

However,;!-it'is,improbable

Data on the pion form factor to that

Treating

with

timelike

since it

is not yet available

for

a q2 range comparable

comes from e+e--annihilation.

q2 < 9.0 GeV2.

It

is consistent

Presently with

a p pole,

data is so

-36--

that it

for q2 large,

Fnki2)

2 -1 is an acceptable ,-t, (q >

should be kept in mind that

should be considered. rule

If

out other behavior.

form factor present

we will

continue

According

and extending

ways to verify

to assume nlT = 2.

scaling

distances , photoproduction interesting

or destroy

if

when interacting

therphoson,

and an;elementa@*fPeld state

will

with

4 and 7.5 GeV.

with

$3 interactions

For the

its

form factor

if

a photon behaves as a single

at large

s and t follows

that

Again,

da/dt

it

in da/dt

region

(90")

photoproduction

studied

here the

The highest

energy

s s-7*3 ' Oo4, in agreement with

However, if

with

at large

higher

or pp stattefing

that

lowest

than choosing

energy point

at 90".

scattering, is less

the entire

some small

conclusive

that

importance. to behave

for our purposes

This is partly

ecm range with

is dropped,

the prediction

which we expect

angular

is only

Thus our provisional

energy data of great

experiments.

our

to extend the range of s, since

measurements support

s and t,

to date have focused on obtaining rather

Of.,course

to a t of about 3 GeV2 which we expect

data on meson-baryon

Q s-*f(t/s)

being

s s -7f(t/s>.

is most desirable

photoproduction

(yN -t KN) Q s -7f (t/s>, The present

meson- dominance

hadkons;~is..a~..-suparposPtian~~of.rap.vec~or:meson

can be said about the s dependence of da/dt is that

vector

da Q s dt -7 f(t/s).

of R. L. Anderson et al. 40 which covers the range Ey between

They find

of s -7 .

rules,

39 , then in:ther*k&nematic

at E = 4 GeV, 90" corresponds Y barely in the scaling region.

values

our ideas.

pp + ap and.hence -da 2, s -8 f(t/s). dt t.

dominate and result

data at 90' is that

as da/dt

"I

model in which the pion

process because strict

yp + rp behaves like

da/dt

to

measurements of the pion

In a ladder

constituents

would say that

conclusion

q2 2 4 GeV2

our picture,

Improving

of two spin-0

This is an especially

little

However,

range is used, the data is not adequate

to the dimensional

at short

prediction

16).

2 -2 as (q ) .

would fall

latter

test

(see Fig.

only that

is one of the surest

is a bound state

field

to conclusively

fit

distribution

than

because experiments at modest s

enough sensitivity

to go to

_

-37-

high values

of s.

This poses the usual problem of not having

of high enough s and t to check for scaling. meson-baryon

scatterings

is the existence

A special

an adequate range

problem in some of these

of (resonance?)

structure

at quite

For instance in IT-P + IT-P a dip in d$/dt is found at t = high t values. 2 41 - 3.8(GeV/c) and there is evidence that both $p have a dip at t = - 4.8 ceV/c)2 For an unambiguous analysis, a large

range of s !

nature

one needs for

One experiment

this

process

which has been analyzed

meterized

the three

reactions

In short,

baryon scattering, results

region, studied)

their

results

but it

will

much higher

another

power or an exponential.

Not surprisingly, s and t.

are consistent

sensitivity

over a fixed

our prediction

da/dt

s and t

m = 7.

They

and m = 8.5 + 1.2 data on meson-

to cover the large a larger

falls

with

and Polkinghorne'

the power s

studied

have plotted 18).

How-

angle

range of s can be

is enough to show that

s > 15 GeV2 and 0 cm between 38" and 90" (see Fig. by the form da/dt

If

17).

with

pp + pp is the most thoroughly Landshoff

para-

that we are not wildly'wrong.

(so that It

cm region

1.4,

cross

well

so that we predict

which only attempt

be much more conclusive.

can be equally

m = 8.12

only demonstrates

from experiments

but with

f(t/s)

our predictions

integrated

bution

the

momenta between

QJ's'~~ (see Fig.

m = 7.4 + 1.4,

section

at large

that

da/da Q s da/dt

large,

respectively.

ever,

They find

by (da/dQ)goO Q smrn or (da/dfi)goo

are sufficiently give for

incident

for Fop -t IT+A', K" + a+C" and

1.0 and 7.5 GeV/c42.

fit

to determine

of the wide angle energy dependence measures the 90' differential

sections

and

t and u >, 5(GeV/c)2

the cross -1

rather

elastic

than

process

the data for

They find

that

it

is well

-9.7 2 O.Sf@)* This is in very good agreement with 'L s -10 f(e), and $parently rules out a significant contri'L s

to the amplitude

by on-shell

quark scattering

1%.

"1; iyi$

-38..

The experimental with

little

evidence

difficulty

it

could be considerably

better

data on meson-baryon

fairly

soon and will

section

with

of predictions

behavior

improved.

and e+e- +lrlT+-

scattering

be a significant

a list

whose asymptotic

on Eq. (1) is thus very encouraging.

test

is interesting,

albeit

difficult

YP + PP

da/dt

Q sW7f(t/s>

YY + Trv

daldt

2, s-4f(t/s)

probably

particle

+ a’)

da/dt

andn

B = 3 it

non-vanishing

form factor)

for q2L8GeV2).

integrated

fixed.

Then when s + = as the reader

and NB are,

over some fixed

cm angular

the scaling

law for multi-

can be tested.

When nM = 2

Let ha be the invariant region

can easily

keeping

verify

all

ratios

cross PI'pj

(i+j)

S

(Eq. (4))

-(+IM+2BB)

respectively,

for high energy BB -t MBBwhen all which is equivalent

(Eq. (3))

in a compact form:

section

where s

Finally,

angle scattering

can be written

Aa 'L s

Q sB3f(t/s>

Fd(n2) 2, (q2 >-5

only scales large

masses are kept fixed

2 -2 FA (q2) s (4 > 2

+.A; A;

L = 1 meson with

exclusive

y> i-

other‘reactions

s and t)

ed +- ed (this

"xi

to measure:

Q sM6f(t/s>

+ee

We close this

law of Eq. (1);

da/dt

ey + ek’fy*y

(or any,-other

idea.

YP + YP

compared with

above,

should be available

(the photons need not have zero mass as long as their and are small

However

As indicated

of the scaling

of the scaling

:

the total cm.angles

to da/dRldQ2 Q s

-10

.

number of mesons and baryons. are large

(Er da/dt

Thus

d3pT) Q s-12f(ei)

Data expected at SPEAR and DORIS

_

-39-. on the exclusive angles will respectively).

channels

eventually

+-

+-+-

e e +lTlT-TTr

provide

a test

and e+e- + 6 charged T'S with of our predictions

fixed

(Aa Q s -3 and Aa % s -5

-4oIV.

Inclusive

Reactions

Having developed

at High Transverse

in previous

dependence of exclusive wish to explore

their

to be expected

that,

sections

scatterings

First,

energy and momentum transfer,

we

for high p inclusive scattering 44 . It is i as in the exclusive case, the power depsndence of

just

language,

an exclusive

of n constituents

it

the power

depend on the number N of constituents Our task,

then,

is to see what can be

number N.

In parton

The condition

for determining

implications

high p cross sections will 1 participating in the high p1 reaction.

scattering

the rules

at large

inclusive

said about this

Momentum

with

on the bound state

fixes

process proceeds by the large

the wee partons wave function

behaving

momentum (and hence n).

are truly

"spectatorsT1'

that

is,

interactions

power dependence on momentum transfer. the wave function

in nature

hav&ng finite

Second, it‘ensures their

as "spectators.."

at x = 0 serves two purposes:

the minimum number of constituents

the total

momentum

that

fraction

the remaining

do not build

(This latter

function

proves to have a logarithmic

of "wees"

up additional

is modified

if

or power singularity

at the origin). In inclusive

scattering,

momenta of the incident is deep inelastic

particles

scattering

absence of a requirement it

constituents

In this

without

that

large

finite

can be spectators.

in the parton model,

is renerget~oal?&.y~ favorable

to continue

having

no particles for

x = -q2/2mv need participate

parton

those partons

with

in a large

A good example of this shown in Fig.

not deflected

transverse

199. In the

appropriate

direction,

by the current

or longitudinal

momenta.

momentum fraction

momentum transfer

n = 4 (2,lleptons

and 2 quarks)

so that

the differential

cross

Eeda/d3pe 't s -2 f(q2/s,M2/s)

section

of the 'L

be found in the forward

changes of their

case only the single

fractions

collision.

Hence

the mat.rix element is dimensionless asymptotically

and

“?1 & %ij ,-+

-41(M is the missing differential

mass).

In the one photon exchange approximation,

cross section

may be written

Wl and vW2 which are defined dependence for is l/s2 if

it

persists

tically

if

energies,

A logarithmic

theories

theories

scattering,

the notion

that

only large

the overall

as expected

is just

in asympto-

the effect

in such

on invariant

scattering

process.

Thus we can immediately

for high energy inclusive

2 f(-r,s 8

prediction

in deep inelastic amplitude

factorizes

and parts

involving

momentum

power dependence of the amplitude.

on the low as well

N defined

section

of this

depending on the large

dependent

with

the validity

any inclusive

in general

E da -L 'L 3-2 d3p

cross

of vW2 seen at ST.&Z,

momentum transfers

That part

the dependence of the cross section

cross section

for

(v)

of the !'spectators'."

only low momentum transfers, determines

behavior

of scaling

by the success of the scaling

a part which involves

transfers

of the differential

is not distressing:that

we abstract

functions

they have no energy

is evidence

modffication

of the interactions

Motivated

the behavior

The famous scaling

at higher

free

in such a way that

(q'/v)

as argued above.

argument.

into

fixed

in terms of structure

this

ratios,

say ;i and "", of the

down (Eq. (5))

scattering

is

8

as high momentumrparts write

Of course

at fixed

t/s

the invariant and H2/s:

)

as the minimum of fields

in the large

momentum transfer

part

of the amplitude. Evidently,

in order

must make a dynamical just tuents

statement

as in the exclusive was required.

to make predictions

for

innlusive

which serves to specify

case a specification

The simplest

scattering,

one

the number of fields

of the number of non-wee consti-

ansatz for N is the one made by Berman,

N,

-42-

Bjorken

and Kogut4'

high transverse

in what was essentially

momentum inclusive

a2 hard parton-parton shell

photon just

scattering

scattering

.19:).

from the quarks just invariant

manner.

d3p

In fact

quark scattering, well

4

above that

such a qq + qq scattering

it

(or scalar)

to the scaling

at high pL

gluons as well

is hard to imagAne that

as via a

gluons to wide angle

E da/d3p Q s-2 at a level

behavior

approach,

the resulting

Blankenbecler,

(see Section

which can take place

and thus N 2 6.

In addition

distributions data,

In their

in this it

and Gunion 46 explored

not by gluon exchange but by quark "interchange.'"

angular

1I.C).

Brodsky,

for w&de angle elastic

with no gluon exchange necessary model, hard parton-parton

which takes place via gluon exchange is not allowed, processes

is qq + qq or qi + qi,

of quarks and yet not give rise

are in good agreement with fit

in a scale

due to electromagnetism.

They argue that

their

that

theory

a model in which hadrons scatter

for

would be generated

case, i.e., process

1

_

the exchange of vector

for binding

In an alternative

scattering

scattering

states

off-

and Kogut predicted

s

to imagine

leading

final

momentum transfer

in perturbation

could be responsible

at order

does in deep inelastic

hadronic

Berman, Bjorken,

could take place via photon.

the large

was natural

model work on

They observed that

scattering

as in the lepton

2 "Lf(i,-) s2

Eda

parton

must take place via exchange of a far

Presumably,

If

N= 4, and in fact

However, it

processes.

as lepton-parton

(Fig.

the first

case involve

makes quite

detailed

The minimal quark-hadron predictions

scattering

large

pI.

scattering on the

2 function

f(i,

5

) of Eq. (4).

to many experimental

checks.

Thus the details

of the model will

be subject

._

-43Here rather

than advocate

the possibilities,

Our principle

observation

that

scattering,

as far

participants. N-

any particular

If

dominated by other

the minimal If

large for

processes-giving

then the high pI reaction observed

are several

possible

etc.

The experimental

results

scattering. section,

state,

that

falloff results

processes

say yp + VT+ X

The standard

N = 5.

a p produced involving the cross section

from this

p /&

and 6 = 90',

with in p i are described

for

away from p I-' phenomenology

point

of view. 47 have scattering

pL

the power -4

.

Rather

in terms of the phenomenology

of 44.

What is of greatest

scattering

our assumption

i.e.,

with

a power between -8 and -11.

the absence of pL -4 behavior

(5) suggests

giving

qp + qp(N=8),

momentum transfer

in the references

hadrons and hence for

qq + ?'rqq

p + X or np + p + X,

qq +- pq'(Ng6),

the absence of scale invarfant

assume that

say pp+

-3 & "W .d _,:

than in pp + p + X at large

on large

-2 at fixed

here is that

Equation

is,

in common. They are not consistent

feature

above is given

for

of interest

would predict

at inclusive

of how their

importance

the

or is

q?'r-+ qr,

This is only a sample of the rich

much more rapid

The details

different

the process

state

dependence E da/d3p Q s

support

If

is not possible

The lack of an N - 6 process with

when looking

they favor

is merely

p I is qq + qq or qi + qi then

pLmeson, su&as

N > 4 reactions:

the edge of phase space.

outlined

a large

should be much smaller

one remarkable

p reaction I some reason that

in the final

only quarks in the initial

available

subject

might be q-+ y + T + q.or i + y + K +'s

For baryons

PP -t 5 + X

to this

element of any model of high p inclusive L as the overall power of s goes, is the number of large

or qq -t aIT, then,N=6 ormore.

q4 + pp(N=*),

contribution

and discuss

the essential

4 as disc?ussed above.

there

model we categorize

fs strong

phenomenological

between quarks of

C.

a new way of analyzfng

the data on deep inelastic

method is to take the observed experAmenta1 cross

the one-photon-exchange

approximation

#s good (th$s has been

._

-44-

cross-checked),

apply radiative

-.

Scaling

vW2 and Wl (and W3 for v scattering). functions

are independent

Eq (:S) with

and extract

corrections,

of any variables

then implies

other

N = 4 can be checked directly.

in the hadronic

of perfect

scaling.,

wave function)

such an analyses

that

surely

functions

the structure

than ,x = -q2/2mv.

Examining directly

of E do/d3p o-n p for fixed ratios of invariants I, corrections interesting-, j ( Even though radiative actions

the structure

However -24

the dependence

%. -f

would be most (and possibly

generate

would provide

strong

logarithmic

a useful

direct

inter-

_

modifications test

of scaling.

Of course we find

the parton model result that pp + p+u-+ X or +Similarly and hence is scale invariant. rp + u+u- + X can go via qi -tlJF! . is taken, that it goes via e'eif the parton version of e'e- annihilation hadrons, ee+sTot

these counting

1 Q, ;

l

rules

give a scale invariant

minimal

scattering

-t q;i -f

and hence

-45-

v.

Summary and Conclusions In this

paper we have examined the conditions

dimensional

analysis

theories.

used to derive

The scaling

wavefunctions n-particle

scattering

that

the irreducible

kernel

laws are likely

by non-planar

The fact

that

and why infrared

for

evidently

that

effective

empirical

fact

Starting

with

asymptotic the large

multiple

of the hadronic

it

of elastic

a good fit

states.

scattering.

together

momentum inclusive

but undoubtedly

reactions, at large

or multiple)

t,

gluon

This remarkable

has important

implications

quark theory.

from what has been done previously.

form factors scattering

have observed that , predictions 48 .

should have the form of Eq. (l),

provides

quark-quark

hadrons.

can

laws can be

empirically,

(single

(1)

to the scaling

become negative

of scale-invariant

of the underlying

s and t behavior

in Ref.

hadronic

the scaling

on-shell

transverse

Wu and Yang, a number of authors

cross sections

showed that

theory,

Regge trajectories,&eff

approach is different

falloff

asymptotic

than indicated

or "neutral"

between quarks of the scattering

structure

Our general

singlet

large

has yet to be explained

for the detailed

for

corrections

do not seem to be important

imply the suppression

exchange interactions

color

perturbation

the absence of scale-invariant

plus the fact

that

theories,

-

regularity

They show, as conjectured in the ultraviolet

the

We have given arguments why such behavior

diagrams involving

such diagrams

hadronic

distance settled

‘es

field

Born diagrams for

of the short

is more convergent

renormalizable

violated

Bethe-Salpeter

of simple planar

and Poggio, 13 .

to be unimportant

In a general

finite

in renormalizable

has now been partially

approximation.

be expected even in abelian

with

The question

wavefunction

by Appelquist

by simple ladder

with

behavior

amplitude,

of the Bethe-Salpeter freedom theories

Eq. (1) can be valid

laws are consistent

and the scaling

under which the simple

to the data.

given

the

can be made about

Born and Moshe49 proposed without

In contrast,

specifying

n, and

we look directly

at

-46-

the underlying

short

distance

This enables us to predict gives

the relations

previous

the behavior

The next logical

be correct

features

such as the angular

scattering,

etc.,

(e.g.,

a color

techniques

used in this

paper,

scattering

which circumvents

making more detailed

actually

"54 i 4 _,T

dynamical behavior

toward this

qualitative

high p1 part solving

the simplified

the complexities

scaling

and see whether

the minimal

without

In summary,' the dimensional

by

they make on the angular

gauge theory)

especially

should be very useful

obtained

step is to choose a theory which may

dependence,

can be obtained

that

theory.

(and automatically

and wide angle scattering

On the other hand, without

actually

from perturbation

of the form factor

we cannot make the predictions

of 2 -t 2 scattering.

analyses,

which we abstract

between form factor

authors) 48,50 .

assumptions

structure

of an inclusive

the theory.

The

approach to bound state

of the full

Bethe-Salpeter

goal.

laws for

fixed

angle scattering

as s + ~0

2-nA-nB-nC-nD + CtD) -f s give a fundamental

connection

the power..law behavior experimental

information

representation scattering

between the'degree

of cross sections is required,

at large

transverse

the present

of hadrons and

Although

results

support

existence,

something giving

but a dynamical

of hadrons.

from quarks to incorporate

much more

a composite Thus hadron

of fundamental

current

existence

quarks must be observed as free particles.

models should be built structure

of complexity

and form factors.

momenta implies

and the hadron spectrum,

not to say that

(t/s>,

of the hadrons based on quark degrees of freedom.

Quarks not only have a mathematical scaling,

fAwD

the correct

algebra, as well.

importance: Bjorken That is

However, bound state short

distance

'-

-47-

Acknowledgments We have benefitted Caltech

and elsewhere,

J. Bronzan, J. Gunion,

from conversations especially

P. Cvitanovic/, J. Kiskis,

with

many colleagues

T. Appelquist,

J. D. Bjorken,

S. D. Drell,

at SLAC and R. Blankenbecler,

R. P. Feynman, Y. Frishman,

R. Sugar and T.-M. Yan.

M. Gell-Mann,

-PI a' '$4

-48-

Appendix A Double-Scattering

Graphs Using the Infinite-Momentum

The physics particularly

of the Landshoff

transparent A full

momentum frame. The prototype bution

multiple-scattering

using time-ordered description

calculation

of this

3

other

cases will

to represent

turns

theory

out to be

in the infinite

method may be found in Ref.

which most simply

and g+ vertices

diagrams

perturbation

contains

51.

the Landshoff

contri-

where we take a f$4 theory

is given in the case of IT-IT scattering

constituents

Method

the bound states.

for

the

Generalizations

to

be straightforward.

We choose the following

reference

frame

with

Q=-( 2P'Q' +- -% 1 rzzz,:L ( 2P We also have &

l

graph,

Fig.

Mi + M2 =M;+%. B time-ordered diagrams which are equivalent

21a, are shown in Fig. It

leading

in the asymptotic

21b-e.

is easy to check that fixed

the summation over time-orderings-in B, and C occurs before

and after

then given by the following theory:

.

if

P -+ Q) frame. order

1

, and u = r2 = - rl

g~ = 0

The contributing

-y’p, 2P +2

t = q2 = - “;

Note that

'

All

other

diagrams angle limit.

three-loop

The total

time-orderings

vanish

in the

(d) and (e) do not contribute

A occurs before

amplitude

expression

to

Diagrams (b) and (c) include

which the vertex D.

to the Feynman

and after

for M(b) and M(c) is

.from time-ordered

perturbation

.-

,

-49-

Three momentum is conserved amplitudes

M(lj

at each vertex.

respectively.

energy denominators

for

diagrams

-E C

c'

where the Ei are the relativistic ii&

there

etc.

Notice

the wavefunction

-E

a'

-E

a

"kinetic

X.1

The

-E

d

-E b' + ie

d'

- Eb + is

energies"

3

are two energy denominators

and after

coming before

the scatterings

When these are summed over the orderings D, they generate

a that because of the choice

the wavefunctions

= f*

+ rdf

Ml and M2 and two afterward. B, C before

M (1) = Mw

(b) and (c) are

Dw ='EA+EB-E Ei =

is Otol.The i for a + b + c + d and

amplitudes

In the f$4 theory

DW=EAfEB-E

and after

The range of the x

and M(2j are the scattering

a' + b' +- c' + d',

In addition

6 (Xa+x.g-x~1

d2kidx. . ;? ..? 2xi(I-xi)(23T)3

M=

do not depend explicitly is thus $Aa finite.

the product

a of?variables on q or r..

9,

A before

of wavefunctions:

= xa ($+:j-;> The natural

+ XL,)

domain of

-_

-5o-. Dropping

terms of order E2 temporarily

DW

g

(;4;)2

-

l$

qc

+ 2Zia

-3-2::

If we introduce

kLa 1-xa

C

- (1-xa)&)2

l

+2 kLd Xd

+2

kJ.c -v-----X

- x,:I"

we have

(i$:)

-

+-2 - Xdq

2Zid

+2 kLb l-xb

l

l

;I

'

the scaled variables orn :E (xa-xd) Bm E (x

then

amXc)

for cx and g of order

DW= Changing variables

1 and kf

cxl$l + Bli![ + O(m) + is

from xd and x 1

M(b) 'L multiplied

43

by a finite

of s, t and u.

2 a' kfb and kfc of order m , we have

13

J

integral

The imaginary

C

.

to a and B thus gives for diagram

(b)

dcidf3 cxl;j + /3liSl + O(m) + is over dx a, d2kla,

part

d2k+

WI

d2kkd which is independent

of (A2) gives

i M(b) rl, m&K (as easily

seen in polar

(A3)

coordinates)

and the real

part

cancels

against

M cc>

We should note the following imaginary ) reflecting tribute.

the Glauber-like

The constituent

to - near -mass-shell

at this

scattering

scatterings

real

point.

The ITT-~amplitude

pole intermediate

amplitudes

states

l

is which con-

M in (A3) correspond (1) and M(2> of the constituents of different hadrons with

-5l-

fractional

momenta x a and l-x

dependence is not relevant if

the Mi(x)

for vector realistic two givesa

thus giving

are scale

a

, respectively.

factor

invariant

the wavefunction

integrals converge because I they are actually independent of x, e.g.,

is immediate.

in the amplitude

the result16Eq.

Note that

as long as the d2k

gluon exchange M(x) Q 7 scatterings

-.

= 7

. The generalization

Each additional of order

(21) of Section

II .Bl.

pair

to more

beyond the first

-+i 4 3;:

-52-

Appendix B Fixed Angle Unitarity It

is interesting

that

the behavior -a

(Bl)

1Mn12 Q, s4-n is actually

the fixed

angle unitarity

when the square of the amplitude center

of mass.

Consider

A convenient

a two-particle

=By smearing with

with

continuum normalization

an angular

63(;,,-$,')

6(n-i?a')64(pa+pb

Pcm

- Pa'-Pb')

'

function

(2) = Jl2)an

f(n),

1 [f(C)12dQ

= 1

we have

s4(pati,, - P,’ Unitarity

-pb’)

l

then gives (z[Z')

= (Zls%l2)

N 3 d Pi fiz, (Nlsl~') i i=l

= ~j(@lN) N

and hence 4& -= Pcm Averaging

IN) over cm angles

J

N d3p. e i i=l

1(ZITIN)(~~~ N

daj(~\TjN)j2

(21~1~

64(Patib

-

fpn) l

i-1

thus gives in the high energy limit 2 CN s2-N

where N is the number of particles

state

angular region in the 52 as follows: has been given by Bardeen

= 2Ea &3(;a-$a"2Eb

(21 mcrn

in the external

is averaged over a fixed

proof

state

bound for n particles

in the final

state.

By crossing,

the

i 4 ;t:

-53-

proof

holds

for any number of initial

The bound (Bl) perturbation

theory

is also often

is saturated point

in any theory

must be-noted,

particles

for

behavior

the required

when the external

occurs in the multiscattering and Appendix A.)

for

In the hadronic

amplitude lines

follows. in each order

of

for renormalizability.

an inherent

for each hadron are not at fixed

the bound (Bl) singular

condition

the bound (Bl)

without

however.

and (Bl)

imposed on amplitudes

as a necessary

In our work we show that

particles

the fixed

short

distance

amplitude,

angles relative Mn can,.jin

scale.

A crucial

the constituent to each other;

principle,

become parallel.

diagram considered

angle amplitude

fail

thus

due to

This is in fact-what

by Landshoff.

(See Sec. II.131

-54Footnotes 1.

A brief

description

and References

of the results

S. J. Brodsky and G. R. Farrar,

of this

Phys. Rev. Lett.

2.

V. Matveev,

R. Muradyan and A. Tavkhelidze,

3.

The initial

application

by G. R. Farrar, and further

of this

Z. F. Ezawa, Cambridge preprint

5.

See, for instance,

6.

E. E. Salpeter

and H. A. Bethe,

P. V. Landshoff,

8.

P. V. Landshoff

9.

A similar

10.

See, e.g.,

11.

A useful

Rev.=,

analysis

amplitudes

'"( &i 4 ":-

was given result

(6)).

(Eq.

and S, J, Brodsky,

preprint,

PITT-122.

1157 (1968) and 171, 1825 (1968).

in the Bethe-Salpeter

DAMTP 73/36,

A complete formalism

is given

E. E. Salpeter, to perform

to be published.

Phys. Lett.

has been given recently

to be published

C u(k,s)t(k,-s)

The general

Phys. Rev. 84, 1232 (1951).

and J. C. Polkinghorne,

trick

distributions

by R. Blankenbecler

of Pittsburgh

Cambridge preprint,

G. Schierholz,

Nuovo Cim. L, 719 (1973).

Proc. Roy. Sot.