BOUND VALENCE-QUARK CONTRIBUTIONS TO HADRON ...

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STANLEY J. BRODSKY .... where Bethe-Heitler pair production in the field of an atom does not give ..... even if the quark fields of the target are charge-less.
-

.-

SLAC-PUB-5169 June 1990

_-

P/E)

f

BOUND

VALENCE-QUARK HADRON

CONTRIBUTIONS

STRUCTURE

STANLEY

TO

FUNCTIONS*

J. BRODSKY

Stanford Linear Accelerator Center, Stanford University, Stanford, California 94309 and

IVAN

SCHMIDT

Stanford Linear Accelerator Center, Stanford University, Stanford, California 94309 and Universidad Federico Santa Mar& Casilla 1 iO- V, Valparako, Chile ABSTRACT The conventional separation of “valence” and “sea” quark distributions of a hadron implicitly assumes that the quark and anti-quark contributions to the sea are identical in shape, although this cannot be strictly correct due to the exclusion principle. A new sepnration of “bound va.lence” and “non-valence” quark distributions of a hadron is proposed which incorporates the Pauli principle and relates the valence component to the wave-functions of the bound-state valence constituents. With this new definition, the non-valence quark distributions correspond to structure functions which would be measured if the valence quarks of the target hadron were charge-less. The bound valence-quark distributions are not singular at small 2, thus allowing for the calculation of sum rules and expectation values which would otherwise be divergent. Submitted

to Physica. Review D

* Work supported in part by Department of Energy contract (SLAC), and by Fundacion Andes and FONDECYT, Chile.

DE-AC03-76SF00515

I -

1.

.-

INTRODUCTION

ic Deep inelastic the light-cone

lepton scattering

longitudinal

in hadrons through

and lepton-pair

momentum

distributions

production

experiments

x = (‘cz +k,“)/(&

measure

+pfr)

of quarks

the relation

FF(x, Q2) = c e~xGqpdx, 0%

(1)

P

FzH(x,Q2)

is the leading-twist

Q, Four-momentum .-

.

XBj

.

conservation

= Q2/2p * q. In p rinciple,

from

the bound

q;

(“i,Li’

function

functions

G,,H

For example,

of QCD.l

Xi> in the light-cone function

at the momentum

transfer

scale

at large Q2 then leads to the identification

the distribution

state solutions

write the distribution

-

structure

Fock expansion

could be computed

given the wavefunctions

of the hadronic

state, one can

in the form2

Gq,i&,Q2) = c /- n dx;;2il@$I (xi,i+ &)I2c S(xb - 4. n,X, each constituent,

= (kf + kf)/(pi

the electrons

momentum

fraction

of

n and helicities

Xi, integrated

over the unconstrained

momenta.

An important of “valence”

+ p&) is the light-cone

where Ci xi = 1 and Ci lcl, = 0 in each Fock state n. The sum

is over all Fock components constituent

(2)

b=q

i

Here xi = lc’/pi

z =

concept

constituents. beyond

Correspondingly,

in the description In atomic

of any bound

physics the term

state is the definition

“valence electrons”

the closed shells which give an atom its chemical

refers to

properties.

the term “valence quarks” refers to the quarks which give the bound

state hadron its global quantum

numbers.

In quantum 2

field theory, bound states of

-

.-

fixed particle

.T

definition _.

number do not exist; however, the expansion Eq. (2) allows a consistent

of the valence quarks of a hadron:

Fock state together component

with

any number

_-

tion G,,H G $H

.

structure

has been separated

+ GTH

7

of the valence quarks of the bound state

functions ? Traditionally,

into

definition,

and thus GiyH(x,Q2)

= Gq,H(xrQ2)

quark and anti-quark

sea distributions

symmetrization

of identical

and q sea contributions. where Bethe-Heitler electron and positron electron

from gluon splitting distributions;

- GuH(x,Q2).

=

(3)

The assumption

of identical

is clearly reasonable for the s and s quarks in to the sea, anti-

q

quarks in the higher Fock states implies non-identical

This is immediately

apparent in the case of atomic physics,

pair production

in the field of an atom does not give symmetric

distributions

since electron capture is blocked in states where

is already present. as in Fig.

contributions

Although

l(a)

Similarly

in QCD, the qij pairs which arise

d o not have identical

from interference

from the anti-symmetrization into account.

GqjH

(0 < x < l),

However, in the case of the u and d quark contributions

the proton.

4

func-

one assumes

GyH(x, Q’) = GTH(x, Q2),

an atomic

the distribution

“valence” and “sea” contributions:

where, as an operational

.

pairs; each

numbers of the hadron.

the contribution

with the phenomenological

appear in each

of gluons and quark-anti-quark

thus has the global quantum

How can one identify

the valence quarks

quark and anti-quark

diagrams such as Fig. l(b),

of the higher Fock state wavefunctions,

the integral

of the conventional

correct charge sum rules, such as JO1dx( G,,H (x) - G,,H ( x)), reading of the actual momentum

distribution 3

sea

which arise

must be taken

valence distribution

gives

it can give a misleading

of the valence quarks.

-

The standard distributions nomenology

definition

also has the difficulty

are apparently indicates

singular in the limit

394

UIP

depend on the < l/z

> moment

Furthermore,

distribution

in the proton

of the valence distribution

diverge.

phe-

behaves

where I2!R x 0.5. 5 This implies that quantities

that

This is the

algebra and the J = 0 fixed pole in Compton

case for the “sigma term” in current scatteringk

the derived valence quark

x + 0. For example, standard

that the valence up-quark

as Gval N zmaYRfor small 2

that

it has been shown7 that the change in mass of the proton

when the quark mass is varied in the light-cone of the Feynman-Hellmann

Hamiltonian

is given by an extension

theorem: 1 aiq

(Q2)

dm;

=

9

G,/,

Q2)-

h

J 0

(4)

In principle, this formula allows one to compute the contribution to the protonneutron mass difference due to the up and down quark masses. However, again, with the standard -

definition

of the valence quark distribution,

at low x. Even more seriously, the expectation

the integration

value of the light-cone

is undefined kinetic energy

operator 1

J

dx < Icf ; + m2

(5)

G,/,(x,Q).

0

is infinite for valence quarks if one uses the traditional way of associating tion.*

this divergence of the kinetic

definition.

energy operator

Notice that a divergence at x = 0 is an ultraviolet

since it implies k+ = k” + k” = 0; i.e. k” t not be expected to have support Part of the difficulty structure

functions

with

-co.

for arbitrarily identifying

There is no apparent

infinity

with renormalizafor a massive quark,

A bound state wavefunction large momentum

components.

bound state contributions

is that many physical processes contribute 4

would

to the proton

to the deep inelastic

-

.-

lepton-proton

.z

frame, -.

cross section: From the perspective

the virtual

photon

physics photoelectric can interact

can scatter

of the laboratory

out a bound-state

quark

or center of mass as in the atomic

process, or the photon can first make a qa pair, either of which

in the target.

As we emphasize here, in such pair-production

one must take into account the Pauli principle

which forbids creation

processes, of a quark in

the same state as one already present in the bound state wavefunction.

_-

lepton

interacts

structure, collision -

with

and with itself.

intrinsic

quarks which are both

quarks which are extrinsic;

Note that such extrinsic

to the proton’s

Thus the bound-state

i.e. created in the electron-proton

processes would occur in electroproduction

even if the valence quarks had no charge.

Thus much of the phenomena

observed

.

in electroproduction associated with functions directly -

at small values of x, such as Regge behavior,

photon-gluon

should be identified

fusion processes, and shadowing with the extrinsic

connected with the proton’s

bound-state

In this paper we propose a definition tions that correctly the hadron

interactions,

with

of the target new definition

functions

2.

the bound valence-quark

CONSTRUCTION In order to construct

distribution

numbers.

func-

which give

In this new separation,

quark distributions

are iden-

which would be measured if the valence quarks

hadron had zero electro-weak

as expected for a bound-state

than processes

of the valence constituents

its flavor and other global quantum

the structure

rather

structure.

Gq/ptx7 92) = ~9B/::tx, Q2)+ G~(x, Q2),non-valence tified

in nuclear structure

of “bound valence-quark”

isolates the contribution

sea distributions

charge.

distributions

We shall prove that

with

GBV q,p (37 Q2>vanish at x +

this 0,

constituent.

OF BOUND

VALENCE-QUARK

the bound valence-quark 5

distributions,

DISTRIBUTIONS we imagine a gedanken

-

.-

QCD where, in addition

_-. c

to the usual set of quarks {q} = {u, d, s, c, b, t}, there is an-

other set {qo} = {u~,do,s~,co, bo,to} with the same spin, masses, flavor, color, and other quantum

-.

numbers, except that their electromagnetic

charges are zero.

Let us now consider replacing the target proton p in the lepton-proton

by a charge-less proton po which has valence quarks qo of zero electro-

experiment _ -

magnetic

scattering

charge. In this extended QCD the higher Fock wavefunctions

of the proton

p and the charge-less proton po both contain qfj and qoqo pairs. As far as the strong QCD interactions

are concerned,

the physical

proton

and the gedanken charge-less

proton are equivalent. .-

.

We define the bound valence-structure between scattering

on the physical

pro_ton, in analogy to an “empty

function

proton

of the proton from the difference

minus the scattering

on the charge-less

target” subtraction:

(6)

F,B’(x, Q2) G Fip(x, Q2) - Fipo(x, Q2).

The non-valence

is thus Fr’(z,

distribution

Q2) = Fipo(x, Q2). Here the Fi(x,Q’)

(i = 1,2, etc.) are the leading twist structure

functions.

The situation

is similar to the atomic physics case, where in order to correctly ing from a bound electron, one must subtract without

that bound electron

through

virtual

present.’

pair production,

total cross section.

just described

define photon scatter-

the cross section on the nucleus alone,

Physically

the nucleus can scatter

and this contribution

In QCD we cannot construct

has to be subtracted

protons without

photons from the

the valence quarks;

thus we need to consider hadrons with charge-less valence constituents. Notice that the cross section measured in deep inelastic is not zero.

This is because the incident

photon 6

lepton scattering

on po

(or vector boson) creates virtual

-

_ T _.

qtj pairs which scatter strongly

in the gluonic field of the charge-less proton

In fact at small x the inelastic

cross section is dominated

contributions,

and thus the structure

functions

target.

by J = 1 gluon exchange

of the physical and charge-less protons

become equal:

lii

[F;(x, Q2) - Fr(x,

Q2)] = 0.

(7)

_In order to isolate a specific bound valence-quark

distribution

lar quark flavor q, we consider the difference between scattering

H and scattering .-

.

.

on an almost identical

Gr,L

for a particu-

on the target hadron

hadron with that particular

valence quark

q replaced by qo. For example, the difference of cross sections cr [lp(uud) + [‘Xl a [lpo (uudo) + [‘Xl

defines the bound valence down-quark

distribution

in the proton:

(8)

ei x G,B;/(x, Q2) = FltUZLd)(x, Q2) - FpCuudo)(x, Q2).

-

-

Then

Remarkably,

as shown in Section 4, the bound valence-quark

GBV vanishes at x --+ 0; it has neither Pomeron x-r qIH Although

the gedanken subtraction

that, nevertheless, small

Xbja

This opens up the opportunity

measured distributions In the following paying particular

function

nor Reggeon xmaYRcontributions.

is impossible

the bound valence-distribution

distribution

in the real world, we will show

can be analytically

constrained

to extend present phenomenology

at

and relate

to true bound state wavefunctions. sections we will

attention

analyze both the atomic

to the high energy regime. 7

and hadronic

cases,

-

.-

3.

.-

ATOMIC

CASE

rz Since it contains

the essential features relevant for our discussion,

we will first

analyze photon scattering from an atomic target.

This problem contains an interesting

paradox which was first resolved by Goldberger

and Low in 1968.’

simple, but explicit,

derivation

The Kramers-Kronig

of the main result.

dispersion relation relates the forward

to the total photo-absorptive

Here we give a

Compton

amplitude

11

cross section

f(k) - f(o) = $ j dk’k,2:$‘m ie’

. ._ .

(10)

0

where k is the photon

energy.

tering on a bound electron diction. forward

One should be able to apply this formula

(eb) in an atom.

However, there is an apparent

On the one hand, one can explicitly amplitude:

compute

00, the electron

Thomson

term, f(k)

+

binding.”

On the other hand, the 0 (e2) cross section for the photoelectric

-e2/$,

where rn8 is the effective electron mass corrected for atomic

yet, -+ e’ behaves as OphOtoN l/k in Eq. (10) predicts logarithmic the explicit

contra-

the high energy yeb -+ yeb

value at k +

it tends to a constant

to scat-

calculation.

at high energies. But then the dispersion behavior for f(k)

Evidently

effect integral

at high energy in contradiction

other contributions

to the inelastic

to

cross-section

cannot resolve this conflict. This problem

was solved9

on a bound state electron.

defining

what one means by scattering

For both the elastic Compton

cross section one must subtract Coulomb

by carefully

the contribution

field of the nucleus (empty

and the inelastic

in which the photon scatters off the

target subtraction). 8

amplitude

Thus a(k) in Eq. (10) is

-

_z

.-

really the difference between the total atomic cross section G~~~~(IC) and the nuclear ’ h is ’ d ominated cross section gnucleus(k), w 1UC

-.

by pair production.

We will present a

simple proof that the high energy behavior N l/k

of the cross sections exactly cancels

in this difference, which is a necessary condition

for a consistent

The total cross section for photon main terms:

the photoelectric

duced electron atom. l2

contribution

going into a different

pair production

this cross section contains

a con-

electron goes into the same state as the bound state states.

These

cancel in the difference oatom - flnucleus. Thus the bound-state

tron photo-absorption -

with the pro-

in the field of the

electron of the atom, plus other terms in which it goes into different last-contributions

by two

state than the electron already present in the

by the Pauli principle;

where the produced

on the atom is dominated

and eSe- pair production,

On the other hand, in the subtraction,

nucleus is not restricted tribution

scattering

dispersion relation.

cross section is the difference between the photoelectric

section on the atom and the pair production

eleccross

capture cross section on the nucleus,

where the produced electron is captured in the same state as the original bound state electron:

ges =

apl~otoelectric

-

Ocapture.

This is depicted graphically

We next note that the squared amplitude equal, by charge conjugation, 3.) Furthermore,

this last process is equal to the (helicity e-2,

with pz and (-pAtom)

the Mandelstam photon

variables

the (helicity summed)

intercha.nged.

for yz + e-Atom.

e+Atom is (See Fig.

summed) squared amplitude

This is equivalent

for

for yAtom

-+

to the interchange

of

squared amplitude

s = (py + p~)~ and u = (pr - PAtom)2.

energies (where s N -u),

Fig. 2 cancel, consistent

for the capture process yZ t

to the squared amplitude

by crossing symmetry,

in Fig. 2.

Thus at high

the two cross sections gphotoelectric and ocapture of

with the Kramers-Kronig 9

relation.

In Regge language, the

I -

-.

part of the J = 0 Compton

imaginary

c

amplitude

The proof we have presented implicitly -.

the photoelectric is normally identical _ -

assumes the equality

process on the atom and the capture

a good approximation

since the atomic

of the flux factors for

process on the nucleus.

high energy. The difficulty

and nuclear masses are almost

target”

and the cross sections do not cancel at

in this case is that the nucleus does not provide the correct

subtraction.

However, we can extend the analysis to the general atomic problem atoms A0 consisting

hypothetical Coulomb

interactions

with

we c_onsider an extended

by considering

of null leptons !o with normal electromagnetic

the nucleus but with

zero external

charge.

U(1) x U( 1) g au g e interactions,

QED with

where the null

and the normal lepton and nucleus have charges (-1,

and (2, Z), respectively.]

The empty

the cross section on the normal

the null atom A0 = (Zlo). identical,

subtraction

atom A = (Ze) and the cross section on interactions

the matrix

element for the photoelectric

process on the

at high energies with the matrix

element for the

process on the null atom Ao. Note that in the computation

in the original

the presence of the spectator

quantum

of A and A0 are

flux factors are the same in both cases.

atom A becomes equal in modulus

process amplitude,

state (say 1s):

lepton lo is irrelevant

The required

matrix

of the capture since it remains

element of the current

is

(Alo(lS)L+(J’“[Ao)

-1)

is defined as the difference

Since the mass and binding

the photo-absorption

As in the earlier proof,

capture

target

and

[In effect,

lepton has charge (-l,O),

between

This

for Mz >> me. However, for finite mass systems such as muonic atoms, the

mass of the nucleus and atom are unequal,

“empty

is zero.

= (~e~(is)~+~~er”~~~~~(is)~z) 10

= (~t+(~fip).

By charge conjugation

and crossing this is equal in modulus

to

-. It

(ze-lJ’LIA),

-. the corresponding

.-

photoelectric

matrix

element with s --+ u. Final-state

can only affect the phase at high energies. Thus we obtain _-

electric

and capture

dispersion

relation

cancellation

cross sections at high energies, and verify for Compton

scattering

REGGEON

4.

interactions of the photo-

the Kramers-Kronig

on leptons bound to finite mass nuclei.

CANCELLATIONS

IN QCD

..-

.

We now return proton.

to the analysis of the “bound

According

valence-quark

to the discussion

distribution

valence-quark

of Section

requires an “empty

distributions”

2, the measurement

target”

of the

of the bound

subtraction:

a(T*P + x> - a(Y*Po ---f X).

For example,

in order to specifically

the proton p(uud),

we subtract

isolate the bound valence d-quark

(Both

the scattering

of gluons,

In this section we shall prove that

the Reggeon terms also cancel, and thus the resulting

distribution

Q2) = [ [5’$Uud)(~, Q2) - ~~(UUdo)(zr, Q2)]/ez

As in the atomic

number

It is clear that the terms associated with J M 1 Pomeron behavior

due to gluon exchange cancel in the difference.

quarks Gfi(z,

but does not carry electric

p and pa contain higher Fock states with arbitrary

qij, and qo@ pairs.)

of

the deep inelastic cross section on the system po(uudo)

in which the do valence quark has normal QCD interactions charge.

distribution

of bound valence d

zr] vanishes as 2 + 0.

case, we now proceed to describe the leading contributions

of a photon

from both the proton 11

to

p and the state po. The high Q2

-

virtual

t

photo-absorption

of terms: -.

cross section on the proton

contributions

in which a quark in p absorbs the momentum

and terms in which a qa pair is created, but the produced

photon; quantum

state than the quarks already present in the hadron.

the cross section for scattering contributions

_ -

virtual

(lab frame) contains two types

that

photon

of the virtual

q is in a different

On the other hand,

photon from the state po(uudo) contains

of the virtual

differ from the p(uud)

case in two important

aspects:

first the

can be absorbed only by charged quarks; and in dd pair production

on the null proton po, the d quark can be produced in any state. Thus the difference between the cross sections off p and po equals a term analogous to Crphoto&ctric, in .-

.

which a d quark in p absorbs the photon contribution

on po analogous to gcapture, in which the produced

the same quantum graphically

state as the d quark in the original

to the appearance of a spectrum

cross section can be understood

as due

The absorptive

ladder diagrams is depicted in Fig. 5(a). The

of such diagrams leads to Reggeon behavior of the deep inelastic structure at small x.14

distance proportional distance contributes A corresponding term indicated radiation,

proton state p. This is shown

of bound qij states in the t-channel.

cross section associated with t-channel

functions

d quark ends up in

in Fig. 4.13

Reggeon behavior in the electroproduction

summation

minus a dd pair production

momentum,

In the rest system, the virtual

to l/x

before the target.

photon creates a dd pair at a

The radiation

which occurs over this

to the physics of the Reggeon behavior. Reggeon contribution

in Fig.

5(b).

at low 2 also occurs in the subtraction

In th e case of the proton

cannot appear in the quantum

the proton because of the Pauli principle.

target,

the d-quark,

after

state already occupied by the d-quark

in

However, the corresponding

is

12

contribution

-

.-

allowed on the po target:

in effect, the d-quark

replaces the do-quark

and is captured

-. t

into a proton.

d*p d&?15 As in the corresponding

-.

._

_-

for y*po +

The capture cross section is computed from the amplitude atomic physics analysis, the spectator

do quark

in the null target po is inert and cancels out from the amplitude.

Thus we only need

to consider effectively

the (helicity

for r*(uu)

However, as illustrated

in Fig. 6 this amplitude,

s -+ u, is equal to the (helicity

summed)

summed)

squared amplitude

after charge conjugation

y*p + d*(uu)

-+ d*p.

and crossing

squared amplitude

at small

x. The flux factors for the proton and null proton target are equal. If we write

0 then the subtraction net virtual

as a sum of Regge terms of the form ,BR\s\“~, where CYR>

SOphotoelectric

of the capture cross section on the null proton

photo-absorption

cross section as a sum of terms S(TBV=

1~1~~). If we ignore mass corrections

contribution

for oR = l/2 for non-singlet

The bound valence-quark

CR PR(IsY~-

in leading twist, then s N Q2(1 - X)/X and u 21

-Q 2 /x. Thus for small x every Regge term is multiplied For example,

will give the

by a factor KR = (-CXE)X.

(which is the leading even charge-conjugation isospin structure non-singlet

functions),

(I = 1) distribution

Reggeon

E$UUd) - ~~(uudo) N x3i2. thus has leading behavior

GBV N x112 and vanishes for z -+ 0. q/H We can also understand from crossing symmetry charge-conjugation valence-quark

this result from symmetry Gqlp(x, Q2) - G,,,,(z,

Reggeon and Pomeron

considerations.

We have shown

Q2) +

0 at low 2. Thus the even

contributions

decouple from the bound

distributions.

5. The observation

CONCLUSIONS

that the deep inelastic

lepton-proton

cross section is non-zero,

even when the quarks in the target hadron carry no charge, implies that we should 13

-

.-

distinguish

two separate contributions

to deep inelastic

lepton scattering:

intrinsic

-. c

(bound-state)

and ex t &sic

(non-bound)

tions are created by the virtual

-.

structure

strong interactions

functions.

The extrinsic

contribu-

of the lepton itself, and are present

even if the quark fields of the target are charge-less. The bound valence-quark butions,

defined by subtracting

the distributions

charge-less valence quarks, correctly

_-

to the bound-state

structure

leading Reggeon contributions

photon-hadron

scattering

contributions

are thus associated with

reaction,

intrinsic

As we have shown, both the Pomeron and

are absent in the bound valence-quark

The leading Regge contributions -

for a gedanken “null” hadron with

isolates the valence-quark

of the target.

distri-

processes extrinsic

distributions.

particles

created by the

to the bound state physics of

.

the target hadron itself.

The bound valence-quark

distributions

are in principle

com-

putable by solving the bound state problem in QCD. Sum rules for the proton derived from properties

of the hadronic wavefunction

thus apply to the bound valence-quark

contributions. -

The essential reason why the new definition bution

differs from the conventional

principle: contain

the anti-symmetrization quarks of identical

role at low 5, eliminating butions.

In the atomic

analogous application Kramers-Kronig

flavor.

definition

of the bound valence-quark of valence distributions

of the bound state wavefunction

distri-

is the Pauli

for states which

As we have shown, this effect plays a dynamical

leading Regge behavior in the bound valence-quark

distri-

physics case, where there is no leading Regge behavior, of the Pauli principle

dispersion relation

leads to analytic

for Compton

14

scattering

consistency

with

on a bound electron.

the the

Acknowledgements

_rz We wish to thank discussions. ._

J. Bjorken,

J. Collins,

D. Issler, and M. Burkardt

IS also thanks the kind hospitality

of R. Blankenbecler

for helpful

and the theory

group at SLAC.

REFERENCES

1. Structure

functions

in one-space

and one-time

dimensions

diagonalizing

the light-cone

Hamiltonian.

and H.-C. -

for meson, baryon,

Pauli,

Hornbostel,

SLAC-PUB-4678

SLAC-0333,

and nuclear

bound

have been computed See K. Hornbostel,

states in QCD numerically

by

S. J. Brodsky,

and Phys. Rev. D (1990) (in proof);

Ph. D. thesis, unpublished

K.

(1988); and M. Burkardt,

Nucl. Phys. A504, 762 (1989). -

2. S. J. Brodsky

a.nd G. P. Lepage, Phys. Rev. D22, 2157 (1980). Equation

derived in light-cone 3. J. Kuti

gauge A+ = 0.

and V. F. Weisskopf,

4. F. E. Close, An Introduction 5. The position amplitude

a~ of J-plane

Phys. Rev. D4, 3418 (1971).

to Qua&

xl-~

at z t

in the forward

and thus the non-singlet 0 must be unaffected

thank J. Collins for conversations

6. S. J. Brodsky,

and Partons, Academic

singularities

are Q2-independent,

F,Ns(x, Q2) -

(2) is

Press (1979).

virtual

Compton

Reggeon behavior

by QCD evolution.

on this point.

F. E. Cl ose and J. F. Gunion, 15

Phys. Rev. D5, 1384 (1972).

We

-

.-

7. W. Weisberger,

Phys. Rev. D5, 2600 (1972); S. J. Brodsky,

F. E. Close and

.z

J. F. Gunion, unpublished.

_.

Phys. Rev. D8, 3678 (1973); S. J. Brodsky

This result assumes that the valence quark mass only appears in

the light-cone

kinetic energy C(kt

8. One can identify

the ultraviolet

+ m2)/x.

divergences at x + 0 of Eqs. (4) and (5) for

sea quarks with the renormalization

_ -

and I. Schmidt,

the proton-neutron

of the gluon propagator

(which cancels in

mass difference.)

9. M. L. Goldberger

and F. E. Low, Phys. Rev. 176, 1778 (1968);

see also:

T. Erber, Ann. Phys. -6, 319 (1959).

..

.

10. We neglect here the 0 (e6) light-by-light Compton lr.

scattering

contributions

to the forward

amplitude.

For discussion and references see M. Damashek and F. J. Gilman,

Phys. Rev.

Dl, 319 (1970). -

12. To leading order in a we can ignore higher order processes in the total cross section such as Compton

scattering.

13. We neglect processes higher order in electromagnetic 14. For a derivation functions,

CYQED

such as pair production

in the

field of the quark d. of the Reggeon contribution

see P. V. Landshoff,

to the leading twist

J. C. Polkinghorne

B28, 224 (1971), and S. J. Brodsky,

structure

and R. Short, Nucl. Phys.

F. E. Close, and J. F. Gunion,

Phys. Rev.

D8, 3678 (1973). 15. As usual we neglect hadronization

effects and other final-state

high energies.

16

interactions

at

-

.-

Figure

.Fig. 1 Structure

function

contributions

Captions from the three-quark

plus one pair Fock state

-. of the proton.

The ddpair

in diagram (a) contributes

diagram

(b) d ue to anti-symmetrization

uniquely

into “valence” versus “sea” parts.

Fig. 2 The bound-electron

photo-absorption

to the sea distribution,

of the d-quarks

but

cannot be separated

cross section gyeb is defined as the differ-

ence of y - Atom and y - Nucleus cross sections. This can also be expressed as the difference between the atomic “photoelectric” .-

duction

“capture”

cross section on the nucleus, but with the produced electron

going into the same atomic state as the original Fig. 3 The helicity-summed

squared amplitude

equal, by charge conjugation, y,?J + -

e-Atom,

cross section and the pair pro-

bound state electron.

for the process yZ

to the helicity-summed

+

e+Atom

squared amplitude

is for

up to a phase. This is also equal by crossing to the helicity-

summed squared amplitude

for the process yAtom

+

e-2,

but with

s and u

interchanged. Fig. 4 The bound valence-quark difference

distribution

of quark d can be calculated

between (a) the cross section on the state p in which

photon momentum

the produced

same state as the d quark in the original

proton state p.

describing

Reggeon behavior

and (b) in the subtraction Fig. 6 The helicity-summed

the virtual

is absorbed by the quark d, and (b) the dd pair production

cross section in the field of po, but with

Fig. 5 Amplitudes

from the

d quark ending in the

at small x (a) in electroproduction,

term of Fig. 4(b).

squared amplitude 17

for (a) y*p t

d(uu) is equal, by charge

-

_-. c

.-

conjugation, y*p +

-.

to the helicity-summed

squared amplitude

for the process (b)

&GE), up to a phase. This is also equal, by crossing symmetry,

helicity-summed

squared amplitude

for (c) Y*(W)

+

changed. Thus at high energies the Reggeon contribution term of Fig. 5(b) cancels the Reggeon contribution

-

18

to the

dp, with s and u interfrom the subtraction

of Fig. 5(a).

-

.-

c

-.

P

. ,

6-90

(W

FIGURE 1

6554A

1

-

-

2

Y

2

Y

e-

7+

\,

z

Z

-

Photoelectric

Capture

FIGURE 2

6-90 6554A2

-

.-

.-

.

6-90

6554A3

FIGURE 3

-

Y”

Y*

2

2

.-

. /

P 6-90

(a)

(W FIGURE 4

6554A4

-

.-

.-

..

d-*

FIGURE 5

6 -90

6554A6

-

FIGURE 6