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
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dimensions
diagonalizing
the light-cone
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and H.-C. -
for meson, baryon,
Pauli,
Hornbostel,
SLAC-PUB-4678
SLAC-0333,
and nuclear
bound
have been computed See K. Hornbostel,
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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). -
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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
