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April 1997

Quantum Chromodynamics and

Other Field Theories on the Light Cone �

Stanley J. Brodsky

Stanford Linear Accelerator Center

Stanford University, Stanford, California 94309

Hans-Christian Pauli

Max-Planck-Institut f�ur Kernphysik

D-69029 Heidelberg, Germany

Stephen S. Pinsky

Ohio State University

Columbus, Ohio 43210

Submitted to Physics Reports.

�Work supported in part by the Department of Energy, contract DE{AC03{76SF00515.

Page 2

Quantum Chromodynamics

and Other Field Theories

on the Light Cone

Stanley J. Brodsky,

Stanford Linear Accelerator Center

Stanford University, Stanford,California 94309

Hans-Christian Pauli

Max-Planck-Institut f�ur Kernphysik

D-69029 Heidelberg

Stephen S. Pinsky

Ohio State University

Columbus, Ohio 43210

28 April 1997

Preprint MPIH-V1-1997


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recurring parton substructure, both from the infrared region (from soft gluons) and the
ultraviolet regime (from QCD evolution to high momentum). In fact, there is no con
Because of coherent color-screening in the color-singlet hadrons, the infrared gluons with
wavelength longer than the hadron size decouple from the hadron wave function.

The question of parton substructure is related to the resolution scale or ultraviolet cut-
o� of the theory. Any renormalizable theory must be de�ned by imposing an ultraviolet
cuto� � on the momenta occurring in theory. The scale � is usually chosen to be much
larger than the physical scales � of interest; however it is usually more useful to choose
a smaller value for �, but at the expense of introducing new higher-twist terms in an
e�ective Lagrangian: [297]

L(�) = L(�)0 (�s(�);m(�)) +


�L(�)n (�s(�);m(�)) +O




L(�)0 = �

F (�)a��F

n +

i 6 D(�) �m(�)

(�) : (5:11)

The neglected physics of parton momenta and substructure beyond the cuto� scale has
the e�ect of renormalizing the values of the input coupling constant g(�2) and the input
mass parameter m(�2) of the quark partons in the Lagrangian.

One clearly should choose � large enough to avoid large contributions from the higher-
twist terms in the e�ective Lagrangian, but small enough so that the Fock space domain
is minimized. Thus if � is chosen of order 5 to 10 times the typical QCD momentum scale,
then it is reasonable to hope that the mass, magnetic moment and other low momentum
properties of the hadron could be well-described on a Fock basis of limited size. Further-
more, by iterating the equations of motion, one can construct a relativistic Schr�odinger
equation with an e�ective potential acting on the valence lowest-particle number state
wave function [293, 294]. Such a picture would explain the apparent success of con-
stituent quark models for explaining the hadronic spectrum and low energy properties of

It should be emphasized that in�nitely-growing parton content of hadrons due to the
evolution of the deep inelastic structure functions at increasing momentum transfer, is
associated with the renormalization group substructure of the quarks themselves, rather
than the \intrinsic" structure of the bound state wave function [62, 64]. The fact that the

light-cone kinetic energy

~k 2


of the constituents in the bound state is bounded by �2

excludes singular behavior of the Fock wavefunctions at x! 0: There are several examples
where the light-cone Fock structure of the bound state solutions is known. In the case of
the super-renormalizable gauge theory, QED(1+1); the probability of having non-valence
states in the light-cone expansion of the lowest lying meson and baryon eigenstates to be
less than 10�3, even at very strong coupling [224]. In the case of QED(3+1), the lowest
state of positronium can be well described on a light-cone basis with two to four particles,
je+e�i ; je+e�
i ; je+e�

i ; and je+e�e+e�i ; in particular, the description of the Lamb-
shift in positronium requires the coupling of the system to light-cone Fock states with two
photons \in
ight" in light-cone gauge. The ultraviolet cut-o� scale � only needs to be
taken large compared to the electron mass. On the other hand, a charged particle such

Page 104


as the electron does not have a �nite Fock decomposition, unless one imposes an arti�cial
infrared cut-o�.

We thus expect that a limited light-cone Fock basis should be su�cient to represent
bound color-singlet states of heavy quarks in QCD(3+1) because of the coherent color
cancelations and the suppressed amplitude for transversely-polarized gluon emission by
heavy quarks. However, the description of light hadrons is undoubtedly much more com-
plex due to the likely in
uence of chiral symmetry breaking and zero-mode gluons in the
light-cone vacuum. We return to this problem later.

Even without solving the QCD light-cone equations of motion, we can anticipate some
general features of the behavior of the light-cone wavefunctions. Each Fock component
describes a system of free particles with kinematic invariant mass squared:

M2 =

~k 2?i +m

; (5:12)

On general dynamical grounds, we can expect that states with very high M2 are sup-
pressed in physical hadrons, with the highest mass con�gurations computable from per-

turbative considerations. We also note that `n xi = `n
(P 0+P z)

= yi � yP is the rapidity
di�erence between the constituent with light-cone fraction xi and the rapidity of the
hadron itself. Since correlations between particles rarely extend over two units of rapidity
in hadron physics, this argues that constituents which are correlated with the hadron's
quantum numbers are primarily found with x > 0:2:

The limit x! 0 is normally an ultraviolet limit in a light-cone wave function. Recall,
that in any Lorentz frame, the light-cone fraction is x = k+=p+ = (k0 + kz)=(P 0 + P z):
Thus in a frame where the bound state is moving in�nitely fast in the positive z direction
(\the in�nite momentum frame"), the light-cone fraction becomes the momentum fraction

x! kz=pz : However, in the rest frame �!P = �!0 ; x = (k0 + kz)=M: Thus x! 0 generally
implies very large constituent momentum kz ! �k0 ! �1 in the rest frame; it is
excluded by the ultraviolet regulation of the theory |unless the particle has strictly zero
mass and transverse momentum.

If a particle has non-relativistic momentum in the bound state, then we can iden-
tify kz � xM � m: This correspondence is useful when one matches physics at the
relativistic/non-relativistic interface. In fact, any non-relativistic solution to the Schr�o-
dinger equation can be immediately written in light-cone form by identifying the two
forms of coordinates. For example, the Schr�odinger solution for particles bound in a har-
monic oscillator potential can be taken as a model for the light-cone wave function for
quarks in a con�ning linear potential: [295]

(xi; ~k?i) = A exp(�bM2) = exp�



k2?i +m


: (5:13)

This form exhibits the strong fall-o� at large relative transverse momentum and at the
x ! 0 and x ! 1 endpoints expected for soft non-perturbative solutions in QCD. The
perturbative corrections due to hard gluon exchange give amplitudes suppressed only
by power laws and thus will eventually dominate wave function behavior over the soft
contributions in these regions. This ansatz is the central assumption required to derive

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