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%% section 2 QCD ON THE LIGHT CONE [slacpub7152002 in slacpub7152002: slacpub7152003]
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\section{\usemenu{slacpub7152::context::slacpub7152002}{QCD ON THE LIGHT CONE }}\label{section::slacpub7152002}
The bound state structure of hadrons plays a critical role in
virtually every area of particle physics phenomenology. For
example, in the case of the nucleon form factors, pion
electroproduction $ep \rightarrow e \pi^+n$, exclusive $B$ decays,
and open charm photoproduction $\gamma p\rightarrow D\Lambda_c$, the
cross sections depend not only on the nature of the quark currents,
but also on the coupling of the quarks to the initial and final
hadronic states. Exclusive decay amplitudes such as $B \rightarrow
K^*\gamma$, processes which will be studied intensively at $B$
factories, depend not only on the underlying weak transitions
between the quark flavors, but also the wavefunctions which describe
how the $B$ and $K^*$ mesons are assembled in terms of their
fundamental quark and gluon constituents. Unlike the leading twist
structure functions measured in deep inelastic scattering, such
exclusive channels are sensitive to the structure of the hadrons at
the amplitude level and to the coherence between the contributions
of the various quark currents and multiparton amplitudes.
The analytic problem of describing QCD bound states is compounded
not only by the physics of confinement, but also by the fact that
the wavefunction of a composite of relativistic constituents has to
describe systems of an arbitrary number of quanta with arbitrary
momenta and helicities. The conventional Fock state expansion based
on equaltime quantization quickly becomes intractable because of
the complexity of the vacuum in a relativistic quantum field theory.
Furthermore, boosting such a wavefunction from the hadron's rest
frame to a moving frame is as complex a problem as solving the bound
state problem itself. The BetheSalpeter bound state formalism,
although manifestly covariant, requires an infinite number of
irreducible kernels to compute the matrix element of the
electromagnetic current even in the limit where one constituent is
heavy.
Lightcone quantization (LCQ) is formally similar to equaltime
quantization (ETQ) apart from the choice of initialvalue surface.
In ETQ one chooses a surface of constant time in some Lorentz frame
on which to specify initial values for the fields. In quantum field
theory this corresponds to specifying commutation relations among
the fields at some fixed time. The equations of motion, or the
Heisenberg equations in the quantum theory, are then used to evolve
this initial data in time, filling out the solution at all spacetime
points.
In LCQ one chooses instead a hyperplane tangent to the light
coneproperly called a null plane or light frontas the
initialvalue surface. To be specific, we introduce LC coordinates
\begin{equation}
x^\pm \equiv x^0\pm x^3
\end{equation}
(and analogously for all other fourvectors). The selection of the
3 direction in this definition is of course arbitrary.
In terms of LC coordinates, a contraction of fourvectors decomposes
as
\begin{equation}
p\cdot x = \frac{1}{2}(p^+x^+p^x^+)p_\perp \cdot x_\perp\; ,
\end{equation}
from which we see that the momentum ``conjugate'' to $x^+$ is $p^$.
Thus the operator $P^$ plays the role of the Hamiltonian in this
scheme, generating evolution in $x^+$ according to an equation of
the form (in the Heisenberg picture)
\begin{equation}
[\phi,P^] = 2i\, \frac{\partial\phi}{\partial x^+}\; .
\end{equation}
As was first shown by Dirac \cite{1}, seven of the ten
Poincar\'e generators become kinematical on the LC , the maximum
number possible. The most important point is that these include
Lorentz boosts. Thus in the LC representation boosting states is
trivialthe generators are diagonal in the Fock representation so
that computing the necessary exponential is simple. One result of
this is that the LC theory can be formulated in a manifestly
frameindependent way, yielding wavefunctions that depend only on
momentum fractions and which are valid in any Lorentz frame. This
advantage is somewhat compensated for, however, in that certain
rotations become nontrivial in LCQ. Thus rotational invariance will
not be manifest in this approach.
Another advantage of going to the LC is even more striking: the
vacuum state seems to be much simpler in the LC representation than
in ETQ. Note that the longitudinal momentum $p^+$ is conserved in
interactions. For particles, however, this quantity is strictly
positive,
\begin{equation}
p^+=\left(p_3^2+p_\perp^2+m^2\right)^{\frac{1}{2}} + p^3 > 0\; .
\end{equation}
Thus the Fock vacuum is the only state in the theory with $p^+=0$,
and so it must be an exact eigenstate of the full interacting
Hamiltonian. Stated more dramatically, the Fock vacuum in the LC
representation is the {\em physical} vacuum state. To the extent
that this is really true, it represents a tremendous simplification,
as attempts to compute the spectrum and wavefunctions of some
physical state are not complicated by the need to recreate a ground
state in which processes occur at unrelated \hbox{locations} and energy
scales. Furthermore, it immediately gives a constituent picture;
all the quanta in a hadron's wavefunction are directly connected to
that hadron. This allows a precise definition of the partonic
content of hadrons and makes interpretation of the LC wavefunctions
unambiguous. It also raises the question, however, of whether LC
field theory can be equivalent in all respects to field theories
quantized at equal times, where nonperturbative effects often lead
to nontrivial vacuum structure. In QCD, for example, there is an
infinity of possible vacua labelled by a continuous parameter
$\theta$, and chiral symmetry is spontaneously broken. The question
is how it is possible to identify and incorporate such phenomena
into a formalism in which the vacuum state is apparently simple.
The description of relativistic composite systems using lightcone
quantization \cite{1} thus appears to be remarkably simple.
The Heisenberg problem for QCD can be written in the form
\begin{equation}
H_{LC }\vert H\rangle = M_H^2 \vert H\rangle\; ,
\end{equation}
where $H_{LC}=P^+ P^  P_\perp^2$ is the mass operator. The
operator $P^=P^0P^3$ is the generator of translations in the
lightcone time $x^+=x^0+x^3.$ The quantities $P^+=P^0+P^3$ and
$P_\perp$ play the role of the conserved threemomentum. Each
hadronic eigenstate $\vert H\rangle$ of the QCD lightcone
Hamiltonian can be expanded on the complete set of eigenstates
$\{\vert n\rangle\} $ of the free Hamiltonian which have the same
global quantum numbers: $\vert H\rangle=\sum\psi^H_n(x_i, k_{\perp
i}, \lambda_i) \vert n\rangle.$ In the case of the proton, the Fock
expansion begins with the color singlet state $\vert u u d \rangle $
of free quarks, and continues with $\vert u u d g \rangle $ and the
other quark and gluon states that span the degrees of freedom of the
proton in QCD. The Fock states $\{\vert n\rangle \}$ are built on
the free vacuum by applying the free lightcone creation operators.
The summation is over all momenta $(x_i, k_{\perp i})$ and
helicities $\lambda_i$ satisfying momentum conservation $\sum^n_i
x_i = 1$ and $\sum^n_i k_{\perp i}=0$ and conservation of the
projection $J^3$ of angular momentum.
The wavefunction $\psi^p_n(x_i, k_{\perp i},\lambda_i)$ describes
the probability amplitude that a proton of momentum $P^+= P^0+P^3$
and transverse momentum $P_\perp$ consists of $n$ quarks and gluons
with helicities $\lambda_i$ and physical momenta $p^+_i= x_i P^+$
and $p_{\perp i} = x_i P_\perp + k_{\perp i}$. The wavefunctions
$\{\psi^p_n(x_i, k_{\perp i},\lambda_i)\},n=3,\dots$ thus describe
the proton in an arbitrary moving frame. The variables $(x_i,
k_{\perp i})$ are internal relative momentum coordinates. The
fractions $x_i = p^+_i/P^+ = (p^0_i+p^3_i)/(P^0+P^3)$, $0