\clearpage
\vspace*{-2cm}
\begin{centering}
{\large
ABSTRACT \\
}
\vspace*{2ex}
{\Large
Virtual Compton Scattering at High Energy\\
}
\vspace*{2ex}
{\large
ZHANG CHEN\\
}
\vspace*{6ex}
\end{centering}

In this dissertation we develop a theoretical framework in the context
of perturbative QuantumChromoDynamics (pQCD) for studying non-forward 
scattering processes. In particular, we investigate a non-forward
unequal mass virtual Compton scattering amplitude by performing the 
general operator product expansion (OPE) and the formal
renormalization group (RG) analysis. 

We discuss the general tensorial decomposition of the amplitude to
obtain the invariant amplitudes in the non-forward kinematic region.
We study the OPE to identify the relevant operators and their reduced
matrix elements, as well as the corresponding Wilson coefficients. We 
find that the OPE now should be done in double moments with new moment 
variables. There are in the expansion new sets of leading twist 
operators which have overall derivatives. They mix under 
renormalization in a well-defined way. We compute the evolution 
kernels from which the anomalous dimensions for these operators can 
be extracted. We also obtain explicitly the lowest order Wilson coefficients. 

In the high energy limit we find the explicit form of the dominantly 
contributing anomalous dimensions. We are then able to solve the resulting 
renormalization group equations (RGE) and give a prediction of the
high energy behavior of the invariant amplitudes. We find that it is the 
same as is indicated by the conventional double leading 
logarithmic analysis. 
\addcontentsline{toc}{chapter}{Appendices}
 
\par
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\appendix

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%\input{appd/gg/gg}
%\input{appd/gq/gq}

\input{qg}
\input{gg}
\input{gq}
\addcontentsline{toc}{chapter}{Bibliography}
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\chapter{Forward Scattering}
\thispagestyle{myheadings}
\markright{}

In this chapter, we give a review of the traditional operator
product expansion (OPE) and renormalization group (RG) analysis for 
a deeply inelastic scattering (DIS) of lepton and proton (e-p), which 
serves to lay out the framework and conventions that will be used in 
later discussions of non-forward scatterings. This will be the forward 
case that we refer and compare to in those chapters.

We start with the parton model description of DIS to present
the problem and establish the kinematics. We proceed to decompose 
the amplitude into invariant amplitudes and define the 
structure functions. We then perform an OPE of the invariant
amplitude and discuss the RG equation and its solution of the operators 
and Wilson coefficients involved. To set the conventions, we
also include a review of the Light Cone(LC) gauge convention and
explicitly calculate the quark-quark splitting function in the LC gauge.

\nid Most results in this chapter concerning quantum field theory are
more or less standard material and can be found in, for example, 
\cite{Peskin}, \cite{Cheng&Li} or \cite{Collinsbook}, while more
specific details of DIS process can be found in for example,
\cite{DESYreview}. However, we will be presenting them in a different
way \cite{G8070}. We will generally not make explicit
references to the standard literature on well-known results. 


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Amplitude and Kinematics \label{sec:fw-kinematics}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 

The process we consider here is an electron-proton scattering at very
high energy, such as those that occur at DESY. Figure \ref{fig:DIS} depicts
one such process, where the initial proton has four momentum $p$ and
spin $\sigma$ while that of the initial electron is $k$ and $\lm$,
respectively. After the scattering, the electron has four momentum
$k'$ and spin $\lm'$ while the final state of the proton is
collectively labeled $n$.

\nid The exchanged virtual particle, with momentum $q = k-k'$, can 
be either a photon, a $Z$ or a $W$ boson (in figure \ref{fig:DIS} it
is a photon). For the purpose of our 
discussion, we will consider only one photon exchange, which is a 
good approximation at the leading level. In the kinematic region we 
will be working, that is, with $Q^2 = -q^2 \ge 1 \; GeV^2$ and energy 
of the electron at tens of $GeV$ (for example, DESY has a electron 
beam energy of about $30 \; GeV$), we can neglect the mass of the electron 
in the following discussion, which means $k^2 = k'^2 = 0$.

\subsection{Invariants of DIS \label{sec:fw-invariants}}

We first work in the rest frame of the proton to define the
kinematic variables used to describe such a process. We have
\beq
\label{eq:dis-ini}
\left\{
\begin{array}{ll}
p = (M,\!\!&\!\! 0, 0, 0) \\
k = (\,\eps, \!\!&\!\! 0,0, \eps)
\end{array} \right. \,.
\eeq
 
\nid If the scattering angle of the electron is $\theta$, then
\beq
k = (\,\eps', \eps'\sin\theta, 0, \eps'\cos\theta) \,,
\eeq

\nid thus the exchanged four momentum square is
\beq
Q^2 = -q^2 = - (k-k')^2 = 2 k \cdot k' - k^2 - k'^2
= 2 \eps \eps' (1 - \cos\theta) \,.
\eeq

\nid The first invariant of a DIS process, $Q^2$, is then given by
\beq
\label{eq:fw-Qsqyare}
Q^2 = 4 \eps \eps' \sin^2 {\theta \over 2} \,,
\eeq

\nid which indicates the hardness of the scattering. We can see that
large $Q^2$ needs large $\eps$, $\eps'$, and $\theta$ not
approaching zero.

A second invariant $\nu$ is defined by
\beq
M\nu \equiv p \cdot q = p \cdot (k-k') = M (\eps - \eps') \,,
\eeq

\nid which gives
\beq
\label{eq:fw-nu}
\nu = \eps - \eps' \,.
\eeq

\nid We can see that $\nu$ is the energy loss of the electron in the
proton rest frame.

A third invariant, the Bjorken-$x$ variable, is
more commonly used instead of $\nu$ and is defined as
\beq
\label{eq:x-bj}
x_{Bj} \equiv {1 \over \om} \equiv { Q^2 \over {2 M \nu}} 
\equiv {Q^2 \over {2 p \cdot q} }\,.
\eeq

\nid We note $0 \leq x_{Bj} \leq 1$.

$Q^2$ and $x_{Bj}$ are the usual variables used to describe
DIS, and sometimes we use equivalently $Q^2$ and $\nu$.  Although 
they are defined above in the photon rest frame, they are by
definition invariant when we go to any other frame.

\subsection{Scattering Amplitude and Differential Cross-section}

The scattering amplitude of the DIS process in figure \ref{fig:DIS}
can be written as
\beq
\label{eq:fw-t}
T^{(n)}_{\sigma \lm'\lm} = -ie \tilde{U}(k') \ga_\mu U(k)
{ {-i g_\mn} \over {q^2} } \bra{n^{(-)}} j_\nu (0) \ket{p\sigma}
i e \,,
\eeq

\nid with the minus sign on the $n$ state meaning outgoing states and
$j_\nu$ the electromagnetic current given by
\beq
\label{eq:emcurrent}
j_\nu = \sum_f e_f \tilde{q}_f \ga_\nu q_f \,.
\eeq

\nid The differential cross-section is then given by
\beq
\label{eq:fw-dsigmadef}
d \sigma = {1 \over 4} \sum_{\lm'\lm} \sum_{\sigma n}
|T^{(n)}_{\sigma \lm'\lm}|^2 (2\pi)^4 \de^4(p+q-p_n) 
{ {d^3 k'} \over {2 \eps 2 \eps'} } \,,
\eeq

\nid where the factor ${1 \over 4}$ comes from averaging over initial
proton and electron spin and the phase space and kinematic factor are
obtained from a plane wave normalization.

To simplify the expression of $d\sigma$ we define the electron factor
$l_\mn$ and the proton factor $L_\mn$ such that
\beq
\label{eq:fw-dsigma}
{ {d^3 \sigma} \over {d k'^3} } = {1 \over {2 \eps 2 \eps'}}
{ e^4 \over q^4} L^\mn l_\mn \,.
\eeq

\nid We have, for the electron part,
\beqs
\label{eq:efactor}
l_\mn &\equiv& \half \sum_{\lm'\lm} 
[\tilde{U}_{\lm'}(k') \ga_\mu U_\lm(k)]^*
[\tilde{U}_{\lm'}(k') \ga_\nu U_\lm(k)] \non \\
&=&  \half \sum_{\lm'\lm} \tilde{U}_\lm (k) \ga_\mu U_{\lm'}(k') 
\tilde{U}_{\lm'}(k') \ga_\nu U_\lm(k) \,.
\eeqs

\nid Because we know that
\beq
\sum_\lm U_{\lm,\al}(k) \tilde{U}_{\lm,\be}(k) 
= (\not{\!k} + m_e)_{\al\be} \,,
\eeq

\nid we arrive at
\beqs
l_\mn &=& \half Tr[\ga_\mu (\not{\!k}' + m_e) \ga_\nu (\not{\!k} + m_e)]
\non \\
&=& \half Tr[\ga_\mu \not{\!k}' \ga_\nu \not{\!k}] \,,
\eeqs

\nid where we have taken the limit $m_e = 0$. Thus we obtain
\beq
\label{eq:efactorfinal}
l_\mn = 2 (k'_\mu k_\nu + k_\mu k'_\nu - g_\mn k \cdot k') 
+ O(m_e^2) \,,
\eeq

\nid which is determined completely from the kinematics of the
process. 


On the other hand, for the proton part, we have
\beqs
\label{eq:pfactor}
L^\mn &=& \half \sum_\sigma \sum_n \{ 
\bra{n^{(-)}} j^\mu (0) \ket{p\sigma}^* 
\bra{n^{(-)}} j^\nu (0) \ket{p\sigma} (2 \pi)^4 \de^4(p+q-p_n) \} \non \\
&=& \half \sum_{\sigma,n} \bra{p\sigma} j^\mu (0) \ket{n^{(-)}}
\bra{n^{(-)}} j^\nu (0) \ket{p\sigma} (2 \pi)^4 \de^4(p+q-p_n) \,,
\eeqs

\nid where we have used the fact that $j_\mu$ is Hermitian.

\nid Because the presence of the $\de$-function, the sum of $n$ state
is not a complete set summation, but rather an on-shell summation. We
rewrite the $\de$-function and obtain
\beqs
L^\mn &=& \half \sum_{\sigma,n} \int d^4x e^{i (p+q-p_n) \cdot x}
\bra{p\sigma} j^\mu (0) \ket{n^{(-)}}
\bra{n^{(-)}} j^\nu (0) \ket{p\sigma} \non \\
&=& \half \sum_{\sigma,n} \int d^4x e^{i q \cdot x}
\bra{p\sigma} j^\mu (x) \ket{n^{(-)}}
\bra{n^{(-)}} j^\nu (0) \ket{p\sigma} \,,
\eeqs

\nid where we have used the translation relationship
\beq
\label{eq:translation}
j_\mu(x) = e^{i \hat{p} \cdot x} j_\mu(0) e^{-i \hat{p} \cdot x} \,.
\eeq

\nid This translation operation is exactly analogous to the spacial
translation by the three momentum operator $\hat{\vec{p}}$ or the time
translation by the Hamiltonian $\hat{H}$ in quantum mechanics (see,
for example, \cite{SakuraiQM}). In the above, $\hat{p}$ is the 
$4$-momentum operator where 
\beqs
\label{eq:p-hat}
\left\{
\begin{array}{l}
\hat{p} \cdot x = \hat{H} t - \hat{\vec{p}} \cdot \vec{x} \\
\hat{p} \ket{\, Vacuum \,} = 0 \\
\hat{p}_\mu \ket{n} = p_{n \mu} \ket{n} \,.
\end{array}
\right.
\eeqs

\nid The $n$ state summation is now of a complete set, thus by the
completeness relationship we arrive at the expression of the proton 
factor as
\beq
\label{eq:pfactordef}
L^\mn = \half \sum_\sigma \int d^4x e^{i q \cdot x}
\bra{p\sigma} j^\mu (x) j^\nu (0) \ket{p\sigma} \,.
\eeq



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Structure Function}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

The structure functions of the proton are defined from the invariant
amplitudes of the proton factor, which in turn are obtained by
a general tensorial decomposition.

\subsection{Tensorial Decomposition \label{sec:fw-decomp}}

\nid $L_\mn$ as a rank-$2$ tensor depends on the kinematic variables
$p$ and $q$ only. Thus the general tensorial decomposition of $L_\mn$
can be written as
\beq
\label{eq:fw-decomp}
L_\mn(p,q) = A g_\mn + B p_\mu p_\nu + C (p_\mu q_\nu + p_\nu q_\mu)
+ D q_\mu q_\nu + E (p_\mu q_\nu - p_\nu q_\mu)
+ F \veps_{\mu\nu\rho\sigma} q^\rho p^\sigma \,, 
\eeq

\nid where $A, B, ...$ are invariant amplitudes depending only on the
invariants of the process, that is, $A = A (p \cdot q, q^2)$ and so
on. We have deliberately grouped the terms into symmetric and
anti-symmetric parts to make the symmetry properties of $L\mn$ more
manifest. The last term does not occur for an electron scattering,
however, we do have to consider it if it is a neutrino scattering.

The conservation of the electromagnetic current gives 
(cf equation \ref{eq:currentconsv})
\beq
\label{eq:fw-crtcsv}
q^\mu L_\mn = q^\nu L_\mn = 0 \,.
\eeq

\nid Thus we have
\beq
q^\mu L_\mn = A q_\nu + B p \cdot q p_\nu 
+ C (p \cdot q q_\nu + q^2 p_\nu) + D q^2 q_\nu 
+ E (p \cdot q q_\nu - q^2 p_\nu) = 0 \,,
\eeq

\nid being true for any $p$ and $q$. We can vary $p$ and $q$ while
keeping $p \cdot q$ and $q^2$ constant and this means that the
coefficients of the individual $p_\nu$ and $q_\nu$ have to vanish. We
have
\beq
\label{eq:qmuresult}
\left\{
\begin{array}{l}
A + C p \cdot q + E p \cdot q + D q^2 = 0 \\
B p \cdot q + C q^2 - E q^2 = 0
\end{array}
\right. \,.
\eeq

\nid Similarly, from
\beq
q^\nu L_\mn =  A q_\mu + B p \cdot q p_\mu 
+ C (p \cdot q q_\mu + q^2 p_\mu) + D q^2 q_\mu 
+ E (q^2 p_\mu - p \cdot q q_\mu) = 0 
\eeq

\nid we have
\beq
\label{eq:qnuresult}
\left\{
\begin{array}{l}
A + C p \cdot q - E p \cdot q + D q^2 = 0 \\
B p \cdot q + C q^2 + E q^2 = 0
\end{array}
\right. \,.
\eeq

\nid Combine equations \ref{eq:qmuresult} and \ref{eq:qnuresult}
we arrive at
\beq
\left\{
\begin{array}{l}
E = 0 \\
A + C p \cdot q + D q^2 = 0 \\
B p \cdot q + C q^2 = 0
\end{array}
\right. \,,
\eeq

\nid which eventually leads to
\beq
\left\{
\begin{array}{l}
C = - { {p \cdot q} \over q^2 } B \\
D = - { 1 \over q^2} ( A - { {(p \cdot q)^2} \over q^2 } B) \\
E = 0
\end{array}
\right. \,.
\eeq

\nid Therefore, only two of the invariant amplitudes survive after
considering current conservation and we have
\beq
L_\mn = A g_\mn + B p_\mu p_\nu 
- B { {p \cdot q} \over q^2 } (p_\mu q_\nu + p_\nu q_\mu)
- { 1 \over q^2} ( A - { {(p \cdot q)^2} \over q^2 } B) q_\mu q_\nu 
\,,
\eeq

\nid or eventually, after collecting terms,
\beq
\label{eq:fw-decompfinal}
L_\mn = A (g_\mn - { {q_\mu q_\nu} \over q^2 })
+ B \left( p_\mu p_\nu 
         - { {p \cdot q} \over q^2 } (p_\mu q_\nu + p_\nu q_\mu)
         + { {(p \cdot q)^2} \over {(q^2)^2} } q_\mu q_\nu
    \right) \,.
\eeq

\subsection{The Structure Functions \label{sec:fw-strfcn}}

The traditional structure structure function $\W_\mn$ of the proton is indeed
just $L_\mn$ with kinematic factors. Explicitly (see \ref{eq:pfactordef}),
\beq
\label{eq:fw-w}
{1 \over {4 \pi^2}} {M \over E_p} \W_\mn (p,q)  \equiv L_\mn 
= \half \sum_\sigma \int d^4x e^{i q \cdot x}
\bra{p\sigma} j^\mu (x) j^\nu (0) \ket{p\sigma} \,.
\eeq

\nid And after the tensorial decomposition, the two proton structure
functions $\W_1$ and $\W_2$, which are only functions of the kinematic
invariants of the problem,  are defined by
\beqs
\label{eq:fw-w12def}
\W_\mn(p,q) &=&  \W_1(Q^2, x) \left( -g_\mn + {{q_\mu q_\nu} \over q^2}
\right)   \non \\
&+& \, { {\W_2(Q^2, x)} \over M^2 }
\left( p_\mu p_\nu 
         - { {p \cdot q} \over q^2 } (p_\mu q_\nu + p_\nu q_\mu)
         + { {(p \cdot q)^2} \over {(q^2)^2} } q_\mu q_\nu
    \right) \,.
\eeqs

\nid Now in terms of the structure functions, the differential
cross-section \ref{eq:fw-dsigma} becomes
\beq
\label{eq:fw-dsigmaW}
{ {d^3 \sigma} \over {d k'^3} } = {1 \over {4 \pi^2}} {M \over E_p}
 {1 \over {2 \eps 2 \eps'}} { e^4 \over Q^4} \W^\mn l_\mn \,.
\eeq

In calculating $\W^\mn l_\mn$ we note that 
$q_\mu l^\mn = 0 + O(m_e^2)$, so we can drop all terms having explicit
factor of $q_\mu$ or $q_\nu$ in $\W_\mn$. We get
\beqs
\label{eq:wl}
l_\mn \W^\mn &=& 2 (k'_\mu k_\nu + k_\mu k'_\nu - g_\mn k \cdot k')
\; (- \W_1 g^\mn + { \W_2 \over M^2 } p^\mu p^\nu) \non \\
&=& 4 \W_1 k \cdot k' + 2 { \W_2 \over M^2 } 
    (2 p \cdot k p \cdot k' - M^2 k \cdot k') \non \\
&=& 4 \eps \eps' (2 \W_1 \sin^2 {\theta \over 2}
+ \W_2 \cos^2 {\theta \over 2} ) \,.
\eeqs
 
\nid Thus the differential cross-section is given by
\beq
{ {d^3 \sigma} \over {d k'^3} } = 4 { {\al_{em}^2} \over Q^4}
{M \over E_p}
(2 \W_1 \sin^2 {\theta \over 2} + \W_2 \cos^2 {\theta \over 2} ) \,,
\eeq

\nid where $\al_{em} \equiv {e^2 \over {4 \pi}}$ is the fine structure
constant.

It is conventional to give $d \sigma /d Q^2 d \nu$ rather than
$d \sigma / d^3k'$, so we need the Jacobian of the variable
transformation. We have
\beqs
\label{eq:Jacobi}
d^3 k' &=& 2 \pi d \cos \theta (\eps')^2 d \eps' \non \\
&=& 2 \pi (\eps')^2 \left|
\begin{array}{cc}
\d{\cos \theta}/d {Q^2} & \d{\cos \theta}/d{\nu} \\
\d{\eps'}/d{Q^2} & \d{\eps'}/d{\nu}
\end{array}
\right| d \nu d Q^2 
= \pi (\eps')^2 d \nu d Q^2 \left|
\begin{array}{cc}
\d{Q^2}/d{\cos \theta} & \d{Q^2}/d{\eps'} \\
\d{\nu}/d{\cos \theta} & \d{\nu}/d{\eps'}
\end{array}
\right|^{-1} \non \\
&=& \pi {\eps' \over \eps} d\nu dQ^2 \,. 
\eeqs

\nid Therefore, the expression of the differential cross-section
eventually becomes
\beq
\label{eq:fw-dsigmaresult}
{ {d\sigma} \over {d\nu dQ^2} } = 4 \pi { {\al_{em}^2} \over Q^4}
{\eps' \over \eps} {M \over E_p}
(2 \W_1 \sin^2 {\theta \over 2} + \W_2 \cos^2 {\theta \over 2} ) \,.
\eeq

\nid In the laboratory frame where $\theta$ is the scattering angle,
it is possible to fix $x$ and $Q^2$ (or equivalently $Q^2$ and $\nu$) 
and vary $\theta$, thus experimentally $\W_1$ and $\W_2$ can be determined
separately. Since there are more small angle experimental data, $\W_2$
is better measured. 

Theoretically, what we usually do is to relate the cross-section, in
terms of the structure functions, to the imaginary part of a forward
scattering amplitude \cite{Christ&Mueller72} via an optical theorem. 

\nid To be more specific, define the forward scattering amplitude 
$\T_\mn$ as
\beq
\label{eq:fw-amp}
\T_\mn = i 4\pi^2 {E_p \over M} \half \sum_\sigma \int d^4x e^{i q \cdot x}
\bra{p\sigma} T j^\mu (x) j^\nu (0) \ket{p\sigma} \,,
\eeq

\nid we have (see \ref{eq:fw-w})
\beq
\label{eq:fw-optical}
\W_\mn = 2 Im \; \T_\mn \,.
\eeq

\nid The forward amplitude is shown in figure \ref{fig:fw-amp}(a) while
the structure function is shown in figure \ref{fig:fw-amp}(b), both
up to kinematic factors. The only difference in the graphs is the
cut in the middle in figure \ref{fig:fw-amp}(b), which means that all
the intermediate lines the cut goes through are put on the mass
shell. Mathematically this is equivalent to replacing the Feynman
propagators of all the intermediate particles with its corresponding
on-shell $\de$-function and the proper factors (see 
section \ref{sec:fw-gamma}).


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Operator Product Expansion Analysis}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

We will use a straight forward operator product expansion (OPE) analysis 
to discuss the deeply inelastic scattering (DIS) of previous sections. 
While similar and more complete discussions can be found in
 \cite{Collinsbook}, we will present ours in a slightly different
notation, adopted from \cite{G8070}.

\subsection{General Statement}

Let $\hat{O}_1(x)$ and $\hat{O}_2(x)$ be local operators built 
out of fundamental
fields like $q, \tilde{q}$ (quark and anti-quark fields) or $A_\mu$
(gauge fields) and finite number of (covariant) derivatives at one
space-time point, for example, 
\beq
\label{eq:opeg}
j_\mu^f(x) = \tilde{q}^f(x) \ga_\mu q(x) \;\;\;\;\;\;\;\;; 
\hat{O}_\mu(x) = \tilde{q}D_\mu(x) q(x) \;; ... \;.
\eeq

\nid We have, in the short distance limit,
\beq
\label{eq:fw-ope}
\hat{O}_1(x) \hat{O}_2(0) 
\stackrel{x \rightarrow 0}{\longrightarrow} \sum_{r=1}^N
\hat{O}_{\mu_1 \mu_2 ... \mu_{n_r}}^{(r)}(0) 
\; E_{\mu_1 \mu_2 ... \mu_{n_r}}^{(r)}(x) + Remainder \,.
\eeq

\nid $\hat{O}^{(r)}$s are local operators which can include the
identity operator $\hat{I}$ and $E^{(r)}(x)$ are the so-called Wilson
coefficients, $c$-number functions that are usually singular in the
limit of $x \rightarrow 0$.

By a dimensional analysis we can obtain the asymptotic behavior of 
the Wilson coefficients $E^{(r)}(x)$ at short distance. The convention
we use is such that mass and energy has dimension $+1$, i.e., 
$[E] = dim (E) = [M] = dim (M) = +1$, 
while coordinate has dimension of $-1$, ie, $[x] = -1$. Thus,
we have under our convention, $[q]  = - {3 \over 2}$,
$[A_\mu] = -1$, and so on.

\nid Now let $d_r$ be the naive dimension of $\hat{O}^{(r)}$, $d_1$ and
$d_2$ be those of $\hat{O}_1$ and $\hat{O}_2$, respectively, $E^{(r)}(x)$ has
an $x$ behavior/dependence, for small $x$, given by
$(\sqrt{x^2})^{-\lm}$, where $\lm = d_r - d_1 - d_2$ since we must
have $d_1+d_2 = d_r - \lm$. That is, we have
\beq
\label{eq:fw-asym}
E_{\mu_1 \mu_2 ... \mu_{n_r}}^{(r)}(x) 
\stackrel{x \rightarrow 0}{\sim}(\sqrt{x^2})^{ d_1+d_2 - d_r} \,.
\eeq

\nid When we increase $N$, we can only generate new operators by
adding new fields or more derivatives into the operators, 
these new operators must have more negative dimensions, that is, 
$d_r$ decreases with increasing $N$, therefore, we may choose $N$
large enough such that the remainder becomes as small as desired,
namely, $remainder \sim (x^2)^R$ for any desired positive value $R$ in
the small $x$ limit.

\subsection{Renormalization Group Equations}

We work in the zero quark mass limit. The operators under
renormalization behaves like
\beq
\label{eq:fw-oprenorm}
\hat{O}_i(x, \mu^2, \al_\mu) = Z_i^{-1}( {\mu^2 \over \mu_0^2}, \al_{\mu_0} ) 
\; \hat{O}_i(x, \mu_0^2, \al_{\mu_0}) \,,
\eeq

\nid where we have explicitly shown the dependence of the operators on
the renormalization scale. This is the so-called multiplicative
renormalization and $Z_i$ more generally can be a matrix when there is
mixing among the operators.

\nid Taking the logarithmic derivative of the scale $\mu^2$ on both
sides of \ref{eq:fw-oprenorm} we have
\beqs
\mu^2 {d \over {d \mu^2}} \hat{O}_i(x, \mu^2, \al_\mu) &=&
[ \mu^2 {d \over {d \mu^2}} Z_i^{-1}( {\mu^2 \over \mu_0^2}, \al_{\mu_0})] \;
\hat{O}_i(x, \mu_0^2, \al_{\mu_0}) \non \\
&=& - {1 \over {Z_i}} \mu^2 {{d Z_i} \over {d \mu^2}} \;\, 
Z_i^{-1} \; 
\hat{O}_i(x, \mu_0^2, \al_{\mu_0} ) \non \\
&\equiv& \ga_i(\al_\mu) \; \hat{O}_i(x, \mu^2, \al_\mu) \;,
\eeqs

\nid where in the last step we have defined the so-called anomalous
dimension $\ga_i$ of the operator $\hat{O}_i$ as
\beq
\label{eq:fw-gadef}
\ga_i(\al_\mu) = - {1 \over {Z_i}} \mu^2 {{d Z_i} \over {d \mu^2}} \,.
\eeq

\nid Note that $\ga_i$ is a function of $\al_\mu$ only. This is
because from \ref{eq:fw-gadef} $\ga_i$ can only be a function of
$\mu^2$ and $\al_\mu$ and is dimensionless. Because we have taken all
the quark masses to be zero, there is no scale to set $\mu$, thus
$\ga_i$ is a function of the coupling only. On the other hand, if 
$m_q \neq 0$, then $\ga_i \to \ga_i(\al_\mu, {\mu^2 \over m_q^2})$,
complications will arise. 

The operators, at the same time, obey a renormalization group equation 
of the form
\beq
\label{eq:fw-rge}
\mu^2 {d \over {d \mu^2}} \hat{O}_i(x, \mu^2, \al_\mu) = 
\ga_i(\al_\mu) \; \hat{O}_i(x, \mu^2, \al_\mu) \;,
\eeq

\nid or more generally, when there is operator mixing,
\beq
\label{eq:fw-grge}
\mu^2 {d \over {d \mu^2}} \hat{O}_i(x, \mu^2, \al_\mu) = 
\sum_j \ga_{ij}(\al_\mu) \; \hat{O}_j(x, \mu^2, \al_\mu) \;.
\eeq

To establish the renormalization group equations of
the Wilson coefficients, let us look at a simple case where
only one term is kept in the operator product expansion, that is,
\beq
\hat{O}_1(x, \mu^2, \al_\mu) \hat{O}_2(0, \mu^2, \al_\mu) 
\stackrel{x \rightarrow 0}{\longrightarrow} 
\hat{O}^{(3)}(0, \mu^2, \al_\mu) \; E^{(3)}(x, \mu^2, \al_\mu) + Remainder \,.
\eeq

\nid Taking the logarithmic derivative of $\mu^2$ on both sides and 
omitting the contribution from the remainder, we have, suppressing the
$\mu^2$ and $\al_\mu$ dependence,
\beq
(\ga_1 + \ga_2) \hat{O}_1(x) \hat{O}_2(0) 
\stackrel{x \rightarrow 0}{\longrightarrow}
\ga_3 \hat{O}^{(3)}(0) E^{(3)} + \hat{O}^{(3)}(0) \mu^2 {d \over {d \mu^2}}
E^{(3)} \;.
\eeq

\nid Combined with the operator product expansion itself, we have
\beq
\mu^2 {d \over {d \mu^2}} E^{(3)}(x) \equiv 
(\mu^2 \d/d{\mu^2} + \be \d/d{\al_\mu})  E^{(3)}(x)
= (\ga_1 + \ga_2 - \ga_3) E^{(3)}(x) \;,
\eeq

\nid where $\be = \be(\al_\mu) = \mu^2 {{d \al_\mu} \over {d \mu^2}}$
is the QCD beta-function. 

\nid Therefore, the renormalization group equation for the Wilson
coefficient functions is
\beq
\label{eq:fw-wcrge}
\left(\mu^2 \d/d{\mu^2} + \be \d/d{\al_\mu} + \ga_3(\al_\mu)
- \ga_1(\al_\mu) - \ga_2(\al_\mu) \right) 
\; E^{(3)}(x, \mu^2, \al_\mu) = 0 \;.
\eeq

\subsection{OPE Description of DIS \label{sec:fw-ope}}

We now put the operator product expansion formalism to use on the deeply
inelastic scattering process in \ref{sec:fw-kinematics}. We work on
the forward amplitude $\T_\mn$ defined in \ref{eq:fw-amp} and the
structure functions (and thus the cross-sections) can be obtained
through \ref{eq:fw-optical}. As usual we work in the so-called
Bjorken limit where we have 
\beq
\label{eq:bjlimit}
\left\{
\begin{array}{lll}
-q^2 \!\!\!& = \, Q^2 \;\;\; &{\rm large} \non \\
m\nu \!\!\!& = \, p \cdot q \;\;\; &{\rm large} \\
x_{Bj} \!\!\!& = \, {1 \over \om} = { Q^2 \over {2 p \cdot q} } 
\;\;\; &{\rm fixed} \;. \non
\end{array}
\right.
\eeq

\subsubsection{Forward Operator Product Expansion}

We expand $T j_\mu(x) j_\nu(0)$ as $x_\mu \to 0$.
The tensor structure of $\T_\mn$ can be proven as \cite{G8070}
\beqs
\label{eq:fw-orig-ope}
T j_\mu(x) j_\nu(0) = \hat{A}_\mn E^{(0)}(x^2) \hat{I} 
+ \hat{A}_\mn \sum_{i,n} \, F_n^{(i)} (x^2)
\hat{O}_{\mu_1 ... \mu_n}^{(i,n)} (0) 
x^{\mu_1}x^{\mu_2} ...x^{\mu_n} 
\,\,\,\,\,\,\,\,\,\,\,\, \non \\ 
+ \,\, \hat{B}_{\mu \nu \al \be} \sum_{i,n} \, E_n^{(i)} (x^2)
\hat{O}_{\al \be;\mu_1 ... \mu_n}^{(i,n)} (0) 
x^{\mu_1}x^{\mu_2} ...x^{\mu_n} + Rem. \;,
\eeqs

\nid where $\hat{A}_{\mu \nu}$ and $\hat{B}_{\mu \nu \al \be}$ are 
conserved tensor structure operators and (see also section \ref{sec:ope})
\beqs
\label{eq:fw-op-ab}
\hat{A}_{\mu \nu} &=& g_{\mu \nu} \Box - \pdr_\mu \pdr_\nu \non \\
\hat{B}_{\mu \nu \al \be} &=& g_{\mu \al}g_{\nu \be} \Box 
+g_{\mu \nu} \partial_\al \partial_\be
-g_{\mu \al} \partial_\nu \partial_\be
-g_{\nu \be} \partial_\mu \partial_\al \;.
\eeqs

\nid We suppose symmetric combinations in $\mu$, $\nu$ (since
from section \ref{sec:fw-decomp}, $\T_\mn$, after all, is symmetric) 
and also we suppose that all indices in $\hat{O}$s are symmetrized 
(even $\al$, $\be$ with $\mu_i$s). We note that the $\hat{O}$s are the
same operator sets for both terms and that the label $n$ is actually
the angular momentum quantum number, or spin, of the corresponding
operator. The term {\sl twist} is defined as the difference between
the naive dimension of $\hat{O}$ and its spin, ie, {\sl twist} of 
$\hat{O}_n = [\hat{O}_n] - n$.

We need to evaluate the above OPE between
two symmetric external proton states (see \ref{eq:fw-amp}), however,
in practice in the kinematic region where high energy particle
experiments are conducted, especially in the small $x_{Bj}$ region where
most of our interest lies, we can put in some transverse momentum to make
the external states asymmetric and physics will not see the difference
at this level (note it is the main goal of this thesis to discuss what will
happen with asymmetric external states when we go to higher level of
accuracy). That is,
\beq
\bra{p}T j_\mu(x) j_\nu(0) \ket{p} = 
\displaystyle\lim_{r \to 0} \bra{p-r} T j_\mu(x) j_\nu(0) \ket{p} \,.
\eeq

\nid Therefore the identity operator $\hat{I}$ does not contribute to
$\T_\mn$ in the OPE. We need to work to the next terms in 
 \ref{eq:fw-orig-ope} and we have
\beqs
\label{eq:fw-matrixexp}
\bra{p}T j_\mu(x) j_\nu(0) \ket{p} 
&\stackrel{x \rightarrow 0}{\longrightarrow}&
(g_{\mu \nu} \Box - \pdr_\mu \pdr_\nu) 
\sum_{i,n} \, F_n^{(i)} (x^2) x^{\mu_1}x^{\mu_2} ...x^{\mu_n} 
\bra{p}\hat{O}_{\mu_1 ... \mu_n}^{(i,n)} (0) \ket{p} \non \\
&& + \; (g_{\mu \al}g_{\nu \be} \Box 
+g_{\mu \nu} \partial_\al \partial_\be
-g_{\mu \al} \partial_\nu \partial_\be
-g_{\nu \be} \partial_\mu \partial_\al) \non \\
&& \;\;\;\; \sum_{i,n} \, E_n^{(i)} (x^2)
x^{\mu_1}x^{\mu_2} ...x^{\mu_n}
\bra{p} \hat{O}_{\al \be;\mu_1 ... \mu_n}^{(i,n)} (0)\ket{p} \;.
\eeqs

For the matrix elements 
$\bra{p}\hat{O}_{\mu_1 ... \mu_n}^{(i,n)} (0) \ket{p}$, 
the indices can be made only from either $g_\mn$ or $p_\mu$ because $x_\mu$
is gone, however, the contribution of the two types of indices are
different. After the Fourier transformation, we know from dimensional
analysis that
\beq
x_{\mu_i} \Rightarrow { q_{\mu_i} \over {q}^2 } \,.
\eeq

\nid Thus we have, for the $g_\mn$ terms,
\beq
g_{\mu_i \mu_j} { q_{\mu_i} \over {q}^2 }{ q_{\mu_j} \over {q}^2 }
\sim {1 \over q^2} \,,
\eeq

\nid while at the same time,
\beq
p_{\mu_i} p_{\mu_j} { q_{\mu_i} \over {q}^2 }{ q_{\mu_j} \over {q}^2 }
= ( \frac {p \cdot q} {q^2} )^2 \sim {\rm O}(1) \,.
\eeq

\nid Therefore the $g_\mn$ contributions are small compared with those of
$p_\mu p_\nu$. So the only term we keep is the contribution from a
totally symmetric (in $\mu_i$) combination $p_{\mu_1} p_{\mu_2} ...
p_{\mu_n}$. It is clear that this approximation is good to 
$O({1 \over q^2})$. That is, we write
\beq
\label{eq:fw-reducedef}
\bra{p}\hat{O}_{\mu_1 ... \mu_n}^{(i,n)} (0) \ket{p}
= p_{\mu_1} p_{\mu_2} ... p_{\mu_n} \, \bra{p}|\hat{O}^{(i)}_n|\ket{p}
+ O({1 \over q^2}) \,,
\eeq

\nid where $\bra{p}|\hat{O}^{(i)}_n|\ket{p}$ is the so-called reduced
matrix element of the operator $\hat{O}_{\mu_1...\mu_n}^{(i,n)}$. 
Because the external states are on-shell protons,  
$\bra{p}|\hat{O}^{(i)}_n|\ket{p}$ does not depend on kinematic
variables and thus is a number depending only on $i$ and $n$, the
flavor and spin indices, respectively. 

Note indeed this approximation is the so-called leading twist
approximation because non-leading twist operators will have in their
contributions extra factors of ${1 \over q^2}$ following similar
discussion as the above. 

\nid We therefore obtain, after taking leading twist approximation, 
\beqs
\label{eq:fw-redmatrixexp}
&& \bra{p}T j_\mu(x) j_\nu(0) \ket{p} 
\stackrel{x \rightarrow 0}{\longrightarrow}
\sum_{i,n} \, \left\{ (g_{\mu \nu} \Box - \pdr_\mu \pdr_\nu)
(p \!\cdot\! x)^n  F_n^{(i)} (x^2) \right. \non \\
&& \;\;\;\;\;\;\;\;\;\;
+ \left. [g_\mn(p \!\cdot\! \pdr)^2 + p_\mn p_\nu \Box
     - (p_\mu \pdr_\nu + p_\nu \pdr_\mu) p \!\cdot\! \pdr]
    (p \!\cdot\! x)^{n\!-\!2}  E_{n\!-\!2}^{(i)} (x^2) \right\} \;.
\eeqs

The forward amplitude $\T_\mn$ is essentially the Fourier transform 
with momentum $q$ of the matrix elements (\ref{eq:fw-amp}). 
And under the Fourier transform, it is clear that 
$ x_\mu \Rightarrow - i \d{}/d{q_\mu}$. Thus we can rewrite the
part concerning the Fourier transform of the Wilson coefficients 
$E_n^{(i)} (x^2)$ in terms of logarithmic derivatives of $q^2$ as 
\beqs
\label{eq:fw-logderi}
\int d^4 x \, e^{i q \cdot x} (p \cdot x)^n E_n^{(i)}(x^2) 
&=& (- i p_\mu \d{}/d{q_\mu})^n \int d^4 x \, e^{i q \cdot x}  
E_n^{(i)}(x^2) \non \\
&=& ((-i 2 p \cdot q) \d{}/d{q^2})^n 
\; \tilde{e}_n^{(i)}(q^2) \non \\
&=& \left( { {-i 2 p \cdot q} \over {q^2} } \right)^n
\left(q^2 \d{}/d{q^2}\right)^n \tilde{e}_n^{(i)}(q^2) \,,
\eeqs

\nid where $\tilde{e}_n^{(i)}$ is the Fourier transform of 
$E_n^{(i)}$; and we have a similar set of equations for $F_n^{(i)}$. 
(Note in our notation, 
$\left(q^2 \d{}/d{q^2}\right)^n \tilde{e}_n^{(i)}(q^2)$ is exactly the
coefficient $\tilde{C}^{Ja}(Q)$ in \cite{Collinsbook} with the
replacement of $n \to J$ (spin index) and $i \to a$(flavor index). )

\nid We then define the Wilson coefficients in the momentum
space $\tilde{E}_n^{(i)}$ and $\tilde{F}_n^{(i)}$ as 
\beqs
\label{eq:fw-momwc}
\int d^4 x \, e^{i q \cdot x} (p \!\cdot\! x)^n E_n^{(i)}(x^2)
\!&\!=\!&\! { {\!-2i} \over {(Q^2)^2} } 
\left( { {2 p \!\cdot\! q} \over {Q^2} } \right)^n
\tilde{E}_n^{(i)} (Q^2) 
\equiv { {\!-2i} \over {Q^4} } \om^n \tilde{E}_n^{(i)} (Q^2)
\non \\
\int d^4 x \, e^{i q \cdot x} (p \!\cdot\! x)^n F_n^{(i)}(x^2)
\!&\!=\!&\! { i \over {Q^2} } 
\left( { {2 p \cdot q} \over {Q^2} } \right)^n
\tilde{F}_n^{(i)} (Q^2)  
\equiv { i \over {Q^2} }  \om^n \tilde{F}_n^{(i)} (Q^2)
\;,
\eeqs

\nid where we have used equation \ref{eq:x-bj}. Substitute these
definitions into the Fourier transform of the matrix elements 
and note that using integration by parts we can show that 
the derivatives in the conserved tensor operators
(\ref{eq:fw-op-ab}) become factors of momentum $q$ in the fashion
$\pdr_\mu \to i q_\mu$, we arrive at
\beqs
\label{eq:fw-melemft}
\int d^4 x \, e^{i q \cdot x} \bra{p}T j_\mu(x) j_\nu(0) \ket{p}
= - \sum_{i,n} i \bra{p}|\hat{O}^{(i)}_n|\ket{p} \left\{
-\left(g_\mn - { {q_\mu q_\nu} \over {q^2} } \right)
\om^n \tilde{F}_n^{(i)} (Q^2) \right. \non \\
+ \left. {2 \over Q^2} \left( p_\mu p_\nu
- { {p_\mu q_\nu + p_\nu q_\mu} \over q^2} p \cdot q
+ g_\mn { {(p \!\cdot\!q)^2} \over q^2} \right) \om^{n\!-\!2}
\tilde{E}_{n\!-\!2}^{(i)} (Q^2) \right\} \;. \;\;\; 
\eeqs

\nid Rewriting the last term in the parenthesis in front of
$\tilde{E}_{n\!-\!2}^{(i)}$ as
\beq
g_\mn { {(p \!\cdot\!q)^2} \over q^2} 
= (g_\mn - {{q_\mu q_\nu} \over {q^2}} + {{q_\mu q_\nu} \over {q^2}})
{{(p \!\cdot\!q)^2} \over q^2} \,,
\eeq

\nid we obtain
\beqs
&&\int d^4 x \, e^{i q \cdot x} \bra{p}T j_\mu(x) j_\nu(0) \ket{p}
\non \\
&&= - \sum_{i,n} i \bra{p}|\hat{O}^{(i)}_n|\ket{p} \left\{
-\left(g_\mn - { {q_\mu q_\nu} \over {q^2} } \right) \om^n 
\left(\tilde{F}_n^{(i)} (Q^2) - {2 \over {Q^2 \om^2}} 
{{(p \!\cdot\!q)^2} \over q^2} \tilde{E}_{n\!-\!2}^{(i)} (Q^2)\right)
\right. \non \\
&& \;\;\;\;\;\;\;\;\;\;\;\;
+ \left. \left( p_\mu p_\nu 
- { {p_\mu q_\nu + p_\nu q_\mu} \over q^2} p \cdot q
+ {{q_\mu q_\nu} \over {q^2}} { {(p \!\cdot\!q)^2} \over q^2}
\right) {2 \over Q^2} \om^{n\!-\!2}  
\tilde{E}_{n\!-\!2}^{(i)} (Q^2) \right\} \;.
\eeqs

\nid From equation \ref{eq:x-bj} we have
\beq
- {2 \over {Q^2 \om^2}}{{(p \!\cdot\!q)^2} \over q^2} = \half \;,
\eeq

\nid thus
\beqs
\label{eq:fw-opefinal}
&&\int d^4 x \, e^{i q \cdot x} \bra{p}T j_\mu(x) j_\nu(0) \ket{p}
\non \\
&&= - \sum_{i,n} i \bra{p}|\hat{O}^{(i)}_n|\ket{p} \left\{
-\left(g_\mn - { {q_\mu q_\nu} \over {q^2} } \right) \om^n 
(\tilde{F}_n^{(i)} (Q^2) +\half \tilde{E}_{n\!-\!2}^{(i)} (Q^2) )
\right. \non \\
&& \;\;\;\;\;\;\;\;\;\;\;\;
+ \left. {1 \over {M \nu}} \left( p_\mu p_\nu 
- { {p \cdot q} \over q^2} (p_\mu q_\nu + p_\nu q_\mu)
+ { {(p \!\cdot\!q)^2} \over (q^2)^2} q_\mu q_\nu
\right) \om^{n\!-\!1}  
\tilde{E}_{n\!-\!2}^{(i)} (Q^2) \right\} \;. \;\;\;\;
\eeqs

Recall the definition of $\T_\mn$, equation \ref{eq:fw-amp}, and the
tensorial decomposition of the structure function, equation 
 \ref{eq:fw-w12def}, we obtain the tensorial decomposition of the
forward amplitude as
\beqs
\label{eq:fw-t12def}
\T_\mn &=& i 4\pi^2 {E_p \over M} \int d^4x e^{i q \cdot x}
\bra{p} T j^\mu (x) j^\nu (0) \ket{p} \non \\
&=& \left( -g_\mn + {{q_\mu q_\nu} \over q^2} \right) \T_1
+ {1 \over M^2} \left( p_\mu p_\nu 
         - { {p \cdot q} \over q^2 } (p_\mu q_\nu + p_\nu q_\mu)
         + { {(p \cdot q)^2} \over {(q^2)^2} } q_\mu q_\nu
    \right) \T_2 \,, \non \\
\eeqs

\nid where the forward invariant amplitude $\T_{1,2}$ are related to
the structure functions $\W_{1,2}$ by
\beq
\label{eq:fw-tandw}
\W_1 = 2 Im \, T_1 \;; \;\;\;\;\;\;\; \W_2 = 2 Im \, T_2 \;.
\eeq

\nid Comparing with the expression of the operator product expansion
of $\T_\mn$, equation \ref{eq:fw-opefinal}, we obtain the formula for
the invariant amplitudes as
\beqs
\label{eq:fw-invamp}
\T_1 &=\!&\! { {4 \pi^2 E_p} \over M } \sum_n \om^n
\sum_i \bra{p}|\hat{O}^{(i)}_n|\ket{p} 
(\tilde{F}_n^{(i)} (Q^2) +\half \tilde{E}_{n\!-\!2}^{(i)} (Q^2) )
\non \\
\nu \T_2 &=& 4 \pi^2 E_p \, \sum_n \om^{n\!-\!1}
\sum_i \; \bra{p}|\hat{O}^{(i)}_n|\ket{p} \; 
\tilde{E}_{n\!-\!2}^{(i)} (Q^2) \;.
\eeqs

\subsubsection{Dispersion Relation and Optical Theorem}

Equation \ref{eq:fw-invamp} is essentially a power series expansion
of the invariant amplitudes. To further the computation we need the
analyticity properties of $\T_1$ and $\T_2$

Explicitly write out the time ordered product in $\T_\mn$ 
(suppressing the spin average) by inserting a summation over arbitrary
intermediate states $\ket{r}$ we get
\beqs
\T_\mn &=& i 4\pi^2 {E_p \over M} \int d^4x e^{i q \cdot x}
\bra{p} T j^\mu (x) j^\nu (0) \ket{p} \non \\
&=& 4\pi^2 {E_p \over M} i \sum_r \int d^4 x \, e^{i q \cdot x} 
\{ \; \theta (x_0) \bra{p} j_{\mu}(x) \ket{r} \bra{r} j_{\nu}(0) \ket{p} 
\non \\
&& \;\;\;\;\;\;\;\;\;\;\;\;\; 
+ \; \theta (-x_0) \bra{p} j_{\nu}(0) \ket{r} \bra{r} j_{\mu}(x) \ket{p}
\; \} \;.
\eeqs

\nid By applying the translation operators (see equation
 \ref{eq:translation}) we have
\beqs
\label{eq:fw-texpansion}
\T_\mn \!&\!=\!&\! i 4\pi^2 {E_p \over M} 
i \sum_r \int d^4 x \, e^{i q \cdot x}
\{ \, \theta (x_0) e^{i (p-p_r) \cdot x}
\bra{p} j_{\mu}(0) \ket{r} \bra{r} j_{\nu} (0)  \ket{p}
\non \\
&& \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;
+ \; \theta (-x_0) e^{-i (p - p_r) \cdot x}
\bra{p} j_{\nu}(0) \ket{r} \bra{r} j_{\mu} (0)  \ket{p} \, \}
\non \\
\!&\!=\!&\! i 4 \pi^2 {E_p \over M} (2 \pi)^3 \sum_r \int d x_0
\{ \de^3(\vec{q} \!+\! \vec{p} \!-\! \vec{p}_r) 
e^{i (q_0 \!+\! {p}_0 \!-\! p_{r\!,\!0}) \cdot x_0} \theta (x_0)
\bra{p} j_{\mu}(0) \ket{r} \bra{r} j_{\nu} (0) \ket{p}  \non \\
&& \;\;\;\;\;\;\;\;\;\;\;\;\;
+ \; \de^3(\vec{q} - \vec{p} + \vec{p}_r)
e^{i (q_0 - p_0 + p_{r,0}) \cdot x_0} \theta (-x_0)
\bra{p} j_{\nu}(0) \ket{r} \bra{r} j_{\mu} (0)  \ket{p} \, \} \non \\
\!&\!=\!&\! - \,4\pi^2 {E_p \over M} (2 \pi)^3 \sum_r
\left( \; \frac {\de^3(\vec{q} + \vec{p} - \vec{p}_r)
           \bra{p} j_{\mu}(0) \ket{r} \bra{r} j_{\nu} (0)  \ket{p} }
   {q_0 + {p}_0 - p_{r,0} +i \eps} \right. \non \\
&& \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; 
\left. - \;\frac {\de^3(\vec{q} - \vec{p} + \vec{p}_r)
              \bra{p} j_{\nu}(0) \ket{r} \bra{r} j_{\mu}(0) \ket{p}}
   {q_0 - p_0 + p_{r,0} - i \eps} \; \right) \;,
\eeqs     

\nid where in the second step we have performed the spacial
integration that yields the $3$-d momentum $\de$-function. In the
last step when we perform the time integration we have put in the
proper damping factors $\pm \eps$ in the exponent to assure the
convergence at the boundaries in an adiabatic approximation. 

To visualize the analytic properties of $\T_\mn$, again we go to the
rest frame of the proton where $p \cdot q = M q_0 = M \nu$. The
independent variables $\T_\mn$ depends on are $p$ and $q$ while for
the invariant amplitudes the independent variables are actually $Q^2$
and $\om$, with now (see equation \ref{eq:x-bj})
\beq
\om = { {2 p \cdot q} \over {Q^2} } = { {2 M} \over Q^2 } q_0 \,.
\eeq

\nid Thus, for fixed $Q^2$, singularities in $\om$ are the same as
singularities in $q_0$. Note this result is covariant--it is just for
the purpose of visualization that we need to choose a frame.

From equation \ref{eq:fw-texpansion} we have singularities in $q_0$ at
$(i) \; q_0 + p_0 = p_{r,0} \equiv E_r$ and 
$(ii) \; q_0 - p_0 = - E_r$. We shall discuss them one by one. 

\begin{description}

\item{(i) } The first set of poles in $q_0$ occur when

\beq
(q_0 + {p}_0)^2 = E_r^2 = M_r^2 + \vec{p}_r^2 
= M_r^2 + (\vec{q} + \vec{p})^2 \,, \non
\eeq

\nid where in the last step we have used the spacial $\de$-function. 
$M_r$ is the invariant mass of the intermediate state $\ket{r}$. 
We therefore have
\beq
M_r^2 = (q + p)^2 = -Q^2 + 2 p \cdot q + M^2 \,, \non
\eeq

and the corresponding poles in $\om$ are
\beq
\om_r = 1 + \frac {M_r^2 - M^2} {Q^2} \,. \non
\eeq


\item {(ii) } The second set of poles in $q_0$ occur when, after
similar discussion as above,
\beq
M_r^2 = (q-p)^2 = q^2 -2 p \cdot q + M^2 \,, \non
\eeq

\nid and the $\om$ poles are at
\beq
\om_r = - 1 - \frac {M_r^2 - M^2 } {Q^2} \,.
\eeq

\end{description}

\nid The intermediate states $\ket{r}$ can be $\ket{p}$,
$\ket{p,\pi}$, $\ket{p,\pi,\pi}$, etc. Electro-magnetic current and
Baryon number conservation require that $M$, mass of the proton
($\ket{p}$ state), be the smallest invariant mass of all possible
states involving any baryon. This means we always have $M_r \ge M$. 
Therefore we can clearly visualize the analytic property of $\T_\mn$
on the complex $\om$-plane as shown in figure \ref{fig:T-analyticity}:
\begin{itemize}

\item $\T_\mn$ has a pole at $\om =1$ and another one at $\om = -1$. 
They correspond to an elastic scattering where $\ket{r} = \ket{p}$ and
thus $M_r = M$.

\item $\T_\mn$ also has two branch cuts. One on the positive real
$\om$-axis and starts from the point 
$\om = 1 + \frac { (M+m_\pi)^2 - M^2} {Q^2}$, which corresponds to the
lowest excited state $\ket{p,\pi}$ with a pion generated almost at
rest and as such $M_r = M + m_\pi$. It extends out to
infinity, or rather, as long as the collision center of mass energy is
big enough to generate the states. The other one is simply a mirror
reflection of this cut about the imaginary $\om$-axis.

\end{itemize}

\nid It is therefore obvious that $\T_\mn$ has an analytic circle of 
unit radius on the $\om$-plane and thus its Taylor expansion in $\om$
exists as long as we are inside the unit circle.

However, the region of $0 \le \om \le 1$ is not physical. The physics,
namely, deeply inelastic scattering in the Bjorken limit, is happening
in the kinematic region where $0 \le x_{Bj} \le 1$ (see 
section \ref{sec:fw-kinematics}) and corresponds to $\om \ge 1$. Thus the
operator product expansion of equation \ref{eq:fw-invamp}, although
called the OPE of the forward amplitude, can not be applied directly
as it is to the physical scattering, nor the computation of structure
functions and cross sections. 

It turns out \cite{Christ&Mueller72} 
that by deriving a dispersion relation for $\nu
\T_2$, we can indeed correctly apply the operator product expansion to
the physical situation and relate directly to the structure function
$\nu \W_2$. 

We start by writing a dispersion integral of $\nu \T_2$,
\beq
\nu \T_2 (\om, Q^2) = {1 \over {2 \pi i}} \int_c 
\frac {d \om'}{\om' - \om} \nu \T_2 (\om', Q^2) \,
\eeq

\nid where $c$ is any coutour enclosing the origin on $\om$-plane and
lying complete inside the unit circle (see figure
 \ref{fig:T-analyticity}). The analyticity of $\T_\mn$ within the unit
circle will guarantee its convergence.

\nid We can continuously and analytically distort the contour $c$ 
to $c'$ as shown in figure \ref{fig:contour}, where $c'$ crosses no
cut, encloses no pole and has its boundary segments pushed to
infinity. At infinity  although actually $\nu \T_2$ approaches a constant
 \cite{G8070}, there are cancellations between $\pm \infty$
resulting in extra convergent factors and as such we can drop the
integral region of the two semi-circles at infinity and obtain
\beqs
\label{eq:fw-distortion}
\nu \T_2 (\om, Q^2) &=& {1 \over {2 \pi i}} \int_{c \to c'}
\frac {d \om'}{\om' - \om} \nu \T_2 (\om', Q^2) \non \\
&=& {1 \over {2 \pi i}} \int_{1^-}^\infty \frac {d \om'}{\om' - \om}
(\nu \T_2 (\om' + i\eps, Q^2) -  \nu \T_2 (\om' - i\eps, Q^2)) \non \\
&& + {1 \over {2 \pi i}} \int_{-1^+}^{-\infty} \frac {d \om'}{\om' - \om}
(\nu \T_2 (\om' - i\eps, Q^2) -  \nu \T_2 (\om' + i\eps, Q^2)) \,,
\eeqs

\nid where $\eps$ is a infinitesimal positive number.

To simplify the above expression, we need the symmetry property of
$\T_\mn$ in $\om$, or equivalently, in $q_0$. $\T_\mn$ as a
time-ordered product can be represented systematically by a set of 
Feynman diagrams. Equation \ref{eq:fw-texpansion} is for real $q_0$
values. If we take $q_0$ to be complex and forget $i \eps$, we have an
extended $\T_\mn^c (q_0^c)$ on a complex plane. The usual time-ordered
product is obviously obtained when we take $q_0 \to |q_0| +i \eps$ or
$q_0 \to -|q_0| -i \eps$. This extended $\T_\mn^c (q_0^c)$ clearly
obeys
\beq
\T_\mn(q_0) = \T_\mn (-q_0) \,,
\eeq

\nid which can be obtained directly from equation
 \ref{eq:fw-texpansion}. When we go to real values, we actually have
\beq
\T_2 (\om+i\eps,Q^2) = \T_2(-\om - i\eps, Q^2) \,,
\eeq

\nid which means that for $\nu \T_2$ we have
\beq
\nu \T_2 ( \om \pm i \eps, Q^2) = - \nu \T_2 (-\om \mp i \eps, Q^2)
\,.
\eeq

\nid Therefore, by setting $\om' \to -\om'$ in the second integral of
equation \ref{eq:fw-distortion} and use the above we obtain
\beqs
\nu \T_2 (\om, Q^2) &=& {1 \over {2 \pi i}} \int_{1^-}^\infty 
\frac {d \om'}{\om' - \om}
(\nu \T_2 (\om' + i\eps, Q^2) -  \nu \T_2 (\om' - i\eps, Q^2)) \non \\
&& \;\;\;\;\;
+ \; {1 \over {2 \pi i}} \int_{1^-}^{\infty} \frac {d \om'}{\om' + \om}
(\nu \T_2 (-\om' - i\eps, Q^2) -  \nu \T_2 (-\om' + i\eps, Q^2)) \non
\\
&=& {1 \over {2 \pi i}} \int_{1^-}^\infty 
\frac {d \om'}{\om' - \om}
(\nu \T_2 (\om' + i\eps, Q^2) -  \nu \T_2 (\om' - i\eps, Q^2)) \non \\
&& \;\;\;\;\;
+ \; {1 \over {2 \pi i}} \int_{1^-}^{\infty} \frac {d \om'}{\om' + \om}
(- \nu \T_2 (\om' + i\eps, Q^2) +  \nu \T_2 (\om' - i\eps, Q^2)) \non
\\
&=& {1 \over {2 \pi i}} \int_{1^-}^\infty d \om'
\left( {1 \over {\om' \!-\!\om}} \!-\! {1 \over {\om' \!+\! \om}} \right) 
[\nu \T_2 (\om' \!+\! i\eps, Q^2) -  \nu \T_2 (\om' \!-\! i\eps, Q^2)]
\,. \non \\
&=& {1 \over {2 \pi i}} \int_{1^-}^\infty d \om'
\frac {2 \om} {\om'^2 - \om^2} 
[\nu \T_2 (\om' \!+\! i\eps, Q^2) -  \nu \T_2^* (\om' \!+\! i\eps, Q^2)]
\non \\
&=& {\om \over {\pi i}} \int_{1^-}^\infty 
\frac {d \om'} {\om'^2 - \om^2} 2 i \, Im \, \nu \T_2 (\om'+ i\ eps, Q^2)
\,,
\eeqs

\nid where in the last step we have used the Hermiticity of the
electro-magnetic current. Recalling equation \ref{eq:fw-optical}, the 
optical theorem, we obtain the dispersion relation of $\nu \T_2$
as
\beq
\label{eq:fw-dispersion}
\nu \T_2 (\om, Q^2) = {\om \over \pi} \int_{1^-}^\infty 
\frac {d \om'} {\om'^2 - \om^2} \nu \W_2 (\om', Q^2) \,.
\eeq

\nid Note that in the above, $\om$ should still be seen as inside the
convergence circle (the unit circle) of $\nu \T_2$ while only $\om'$ 
has the physical meaning of the inverse of Bjorken-$x$. Therefore we 
can expand the denominator of the dispersion integrand and get
\beqs
\nu \T_2 (\om, Q^2) &=& {\om \over \pi} \int_{1^-}^\infty
\frac {d \om'} {\om'^2} \nu \W_2 (\om', Q^2)
\frac {1} {1 - {\om^2 \over {\om'^2}} } \non \\
&=& {\om \over \pi} \int_{1^-}^\infty
\frac {d \om'} {\om'^2} \nu \W_2 (\om', Q^2) 
\sum_{n=0}^\infty \left({\om^2 \over {\om'^2}}\right)^n \non \\
&=& \sum_{n=0}^\infty \frac {\om^{2n \!+\!1}} {\pi}
\int_{1^-}^\infty \frac {d \om'}{ (\om')^{2n \!+\!2} }
\nu \W_2 (\om', Q^2) \,.
\eeqs

\nid Change the integration variable from $\om'$ to $x_{Bj} =
\om'^{-1} \equiv x$ we have
\beqs
&& d x = - \frac {d \om'} { \om'^2} \non \\
&& \Rightarrow \frac {d \om'} { (\om')^{2n \!+\!2} }
 = - x^{2n} d x \,,
\eeqs

\nid and thus
\beqs
\label{eq:fw-Texpfinal}
\nu \T_2 (\om, Q^2) &\stackrel{\om \eps O(0)} {=}& 
\sum_{n=0}^\infty \om^{2n \!+\!1}
{1 \over \pi} \int_0^1 dx \; x^{2n} \nu \W_2 (x, Q^2) \non \\
&=& \sum_{n \, even}^\infty \om^{n+1}
{1 \over \pi} \int_0^1 dx \; x^n \nu \W_2 (x, Q^2) \,.
\eeqs

\nid Again we would like to stress that $\om$ and $x$ are completely
different quantities. $\om$ is essentially a mathematical qunatity
introduced to Taylor expand the (invariant) amplitude(s) around the
origin in an operator product expansion. It is the so-called {\sl
moment variable} and has its value limited between $0$ and $1$. On the
other hand, $x$ is the Bjorken-$x$, $x_{Bj}$, a kinematic variable of
the actual physical scattering (see \ref{eq:x-bj}) with its value
also limited between $0$ and $1$. The convolution of the structure
function $\nu \W_2$ with the $n\!+\!1$-th power of $x$ is called
taking the $n$-th moment of the structure function. We can see that 
because of the symmetry properties of $\nu \T_2$, only odd moments 
of the structure function enters the expansion of the invariant amplitude. 

Comparing equations \ref{eq:fw-Texpfinal} and \ref{eq:fw-invamp} we
can relate the operator product expansion of the forward amplitude
with the moments of the (physical) structure function and we get 
\beq
4 \pi^2 E_p \, \sum_i \; \bra{p}|\hat{O}^{(i)}_n|\ket{p} \; 
\tilde{E}_n^{(i)} (Q^2) = 
{1 \over \pi} \int_0^1 dx \; x^n \nu \W_2 (x, Q^2) \;, \;\;\;\;\;
n \; even \,.
\eeq

\nid That is, although the structure function $\nu \W_2$ itself
can not be related directly to an expansion of local operators and
their corresponding Wilson coefficients, its moments in $x_{Bj}$ can
indeed be expressed as a product of local operators and Wilson
coefficients by application of operator product expansion and
dispersion relations. The result, after a trivial rewriting, is
\beq
\label{eq:fw-operesult}
\int_0^1 dx \; x^n \nu \W_2 (x, Q^2) = \frac { (2\pi)^3 E_p } {2}
\sum_i \; \bra{p}|\hat{O}^{(i)}_n|\ket{p} \; \tilde{E}_n^{(i)} (Q^2)
\;, \;\;\;\;\; n \; even \;.
\eeq

\subsubsection{The Operators}

Before we discuss the momentum evolution and renormalization group
properties of the operators and their corresponding Wilson coefficients
in equation \ref{eq:fw-operesult}, let us identify the operators
themselves. 

By recalling equation \ref{eq:fw-reducedef} we know that the dominant
operators should have the smallest negative dimension since those with
larger negative ones will be accompanied by extra suppressing factors of
${1 \over Q^2}$ at high energy. In QCD, the leading twist operators
are the quark operators $\hat{O}_{\mu_1 ... \mu_n}^f$ and the gluon
operators $\hat{O}_{\mu_1 ... \mu_n}^G$ as the following
\beq
\label{eq:fw-ops}
\left\{
\begin{array}{l}
\hat{O}_{\mu_1 ... \mu_n}^f = 2 \; \tilde{q}_f \ga_{\mu_1}
D_{\mu_2} ... D_{\mu_n} q_f  \\
\hat{O}_{\mu_1 ... \mu_n}^G = -2 F^\al_{\mu_1}
\cd_{\mu_2} ... \cd_{\mu_{n\!-\!1}} F_{\al \mu_n} \,.
\end{array}
\right.
\eeq

\nid In the quark operator, $D_\mu$ is the covariant derivative 
and $D_\mu = \pdr_\mu - i g A_\mu$ with $g$ the QCD coupling 
$A_\mu = \sum_i A_\mu^i \lm^i/2$ where $\lm^i$ is the fundamental 
representation of the $SU(3)$ color group and $A_\mu^i$ the gauge 
field. $f$ is the flavor label and we can form singlets and octets 
out of the flavor indices. The Dirac indices are all symmetrized. 
The dimension of the quark operator is $[O^f_n]= -n -2 $.

\nid The dimension of the gluon operator is also $[O^G_n] = -n-2$. The
Dirac indices are also symmetrized. In detail, $O^G_n$ is
\beq
\hat{O}_{\mu_1 ... \mu_n}^G = -2 \sum_{i_1,...,i_{n\!-\!1}}
F^{i_1, \al_{\mu_1}} (\cd_{\mu_2})^{i_1 i_2} (\cd_{\mu_3})^{i_2 i_3} 
... (\cd_{\mu_{n\!-\!1}})^{i_{n\!-\!2} i_{n\!-\!1}} 
F_{\al \mu_n}^{i_{n\!-\!1}} \,,
\eeq

\nid where the $i$'s are color indices and $\cd_\mu^{ij} = \pdr_\mu
+ \sum_l g f_{ilj}A_\mu^l$ with $f_{ilj}$ the adjoint representation
of the $SU(3)$ color group.  

We will not give an explicit proof of equation \ref{eq:fw-ops}, but 
rather present some motivations and explanations.
\begin{description}

\item{$\hat{O}_{\mu_1 ... \mu_n}^f$} Because the Dirac indices are
symmetrized, only one gamma matrix can be used since two of them will
give $g_\mn$ terms. The other indices have to be covariant derivatives
not only to preserve gauge symmetry but also to have the most
efficient way to get indices except for gamma matrices. For example,
in the case of $n=2$, we can have 
$\tilde{q}_f \ga_{\mu_1}q_f \tilde{q}_f \ga_{\mu_2}q_f$, which has
dimension of $-6$, or $\tilde{q}_f \ga_{\mu_1} D_{\mu_2} q_f$, which
has a dimension of $-4$. The Wilson coefficient ($E$ function) of the
former will have an extra power of ${1 \over Q^2}$ and thus the latter
dominates.

\item{$\hat{O}_{\mu_1 ... \mu_n}^G$} The gluon field $F_\mn$ has two
indices and dimension of $[F] = -2$, however, they are
anti-symmetric. To symmetrize the $\mu_i$ indices we have to
contract one of the two indices in each $F_\mn$. Thus they are not as
efficient as the (gluonic) covariant derivative $\cd_\mu$'s.

\end{description}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Renormalization Group Analysis}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Recall that the operators need renormalization and they obey 
renormalization group equations \ref{eq:fw-rge} and \ref{eq:fw-grge}.  
Their corresponding Wilson coefficients obey a related equation
 \ref{eq:fw-wcrge} so that their products, which are the $n$ moments of
the physical structure function, do not depend on the renormalization
scale.

\subsection{Equations and Solutions}

We define the Parton distribution functions $x P^f(x,Q^2)$s by 
\beq
\label{eq:fw-pdfdef}
\sum_f e^2_f \!\int_0^1 \!d x \, x^n x P^f(x,Q^2) 
\equiv \int_0^1 \!dx \; x^n \nu \W_2 (x, Q^2) = \frac { (2\pi)^3 E_p } {2}
\sum_i \; \bra{p}|\hat{O}^{(i)}_n|\ket{p} \; \tilde{E}_n^{(i)} (Q^2)
\;.
\eeq

\nid At this point $x P^f$s seem more like mathematical objects rather
than physical ones. However, in the Parton Model of DIS, $x P^f$ are
indeed the (momentum fraction) distribution functions of quarks of
flavor $f$ inside the proton \cite{DESYreview}. 

In leading order of the QCD coupling $\al(Q^2)$ (when $Q^2$ is large), 
only quark operators come in because photons only couple to quarks 
directly. Thus in equation \ref{eq:fw-pdfdef} the $\hat{O}$'s are
quark operators $\hat{O}^f$ from equation \ref{eq:fw-ops}. We have
chosen the normalization of $\hat{O}^f$ such that
\beq
(2 \pi)^3 E_p \bra{p}  \hat{O}_{\mu_1 ... \mu_n}^f \ket{p} 
= 2 p_{\mu_1 ... \mu_n} 
\eeq

\nid for quarks of flavor $f$ in free field theory. On the other hand,
in QCD $\hat{O}^f$ requires renormalization and so a scale $\mu$ has
to be introduced. There will in general be complicated $Q^2$ and $\mu^2$
dependence in the reduced matrix elements and Wilson coefficients. To
be more specific, if $\mu^2/Q^2 << 1$ then $\tilde{E}_n^f$ would have terms
involving $\mu^2$ as 
\beq
\tilde{E}_n^f (Q^2, \mu^2)= \tilde{E}((\al(\mu^2)\log Q^2/\mu^2)^n)
= \cc_f + O ((\al(\mu^2)\log Q^2/\mu^2)^n) \,.
\eeq

\nid If we simply choose $\mu^2 = Q^2$, then all the $Q^2$ dependence
is in the reduced matrix elements obtained by equation
 \ref{eq:fw-reducedef} and we have 
\beq
\label{eq:fw-wc}
\tilde{E}_n^f (Q^2)= e^2_f + O(\al(Q^2)) \,
\eeq

\nid under the normalization convention we use. Therefore we have, in 
leading order, at the renormalization scale $\mu^2 = Q^2$, 
\beq
\sum_f e^2_f \int_0^1 d x \, x^n x P^f(x,Q^2) =
\frac { (2\pi)^3 E_p } {2} \sum_f \; \bra{p}|\hat{O}^f_n|\ket{p}
e^2_f \,,
\eeq

\nid which means that up to kinematic factors the moments of the 
parton distribution functions are the reduced matrix elements in 
operator product expansion, or explicitly, 
\beq
\label{eq:fw-pdf-ops}
\int_0^1 d x \, x^n x P^f(x,Q^2) = \frac { (2\pi)^3 E_p } {2}
\bra{p}|\hat{O}^f_n|\ket{p}_{Q^2} \,.
\eeq

\nid The subscript $Q^2$ means that the reduced matrix elements are
renormalized at the scale $\mu^2 = Q^2$.

Recall that the operators obey the renormalization group equation 
(see equations \ref{eq:fw-rge} and \ref{eq:fw-grge}), at an
arbitrary scale $\mu$,
\beq
\label{eq:fw-erge}
(\mu^2 \d/d{\mu^2} + \be \d/d{\al_\mu}) \; \hat{O}^f_n(\mu^2, \al_\mu)
= \ga_n^f (\al_\mu) \, \hat{O}^f_n(\mu^2, \al_\mu) \,,
\eeq

\nid where $\ga_n^f$ is the anomalous dimension of the operator.

\nid The matrix elements of the operators will obey the same
equation. Furthermore, because the differences between the matrix
elements and the reduced matrix elements are only kinematic factors
(see equation \ref{eq:fw-reducedef}), the reduced matrix elements and
thus the $n$-moments of the parton distribution functions also obey
the same renormalization equation, namely, we have
\beq
\label{eq:fw-pdfrge}
(\mu^2 \d/d{\mu^2} + \be \d/d{\al_\mu}) \; 
\int_0^1 d x \, x^n x P^f(x,Q^2) = \ga_n^f (\al_\mu)
\int_0^1 d x \, x^n x P^f(x,Q^2) \,.
\eeq

\nid The solution to the renormalization group equation
 \ref{eq:fw-erge} is (formally when we have operator mixing and an
anomalous dimension matrix)
\beq
\label{eq:fw-rge-solu}
\hat{O}^f_n(\mu^2, \al_\mu) = e^{- \int_{\al_\mu}^{\al_{\mu_0}}
d \al' \frac {\ga_n^f(\al')} {\be(\al')}} 
\hat{O}^f_n(\mu^2_0, \al_{\mu_0}) \,,
\eeq

\nid and the moments of parton distribution functions will obey
exactly the same evolution in momentum scale.

\nid The anomalous dimensions are calculable in perturbation 
theory. A lowest order calculation gives the dominant large $\mu^2$
dependence.

\subsection{Relationship to DGLAP Evolution \label{sec:DGLAP}}

The DGLAP equations (Dokthitze, Gribov, Lipatov, Altarelli, Parisi)
(for example, see \cite{A&P77}) state that the momentum scale $Q^2$
evolution of the parton distribution functions $P^f(x, Q^2)$ obeys
\beq
\label{eq:fw-DGLAP}
Q^2 \frac {d} {d Q^2} \, x P^f(x,Q^2) = \frac {\al (Q^2)} {2 \pi}
\int_x^1 \frac {d x'}{x'} \, \ga^f ({x \over {x'}}) \; x' P^f(x',Q^2) \,,
\eeq

\nid where $\ga^f(x)$ is the so-called Altarelli-Parisi splitting
function of the corresponding parton distribution(s). $\ga^f(x)$ is 
defined to be zero outside the range $(0,1)$. Again in general
$\ga^f(x)$ can be a matrix due to mixing among the parton distribution
functions. 

The DGLAP evolution equations and the renormalization group equations
 \ref{eq:fw-pdfrge} are actually equivalent, with the proper
identification of the splitting function with the anomalous
dimension. We will not explicitly prove this (for that see, for
example, \cite{Peskin, Cheng&Li}), but rather show that given the
DGLAP equation \ref{eq:fw-DGLAP} we can obtain equation \ref{eq:fw-pdfrge} 
and illustrate the explicit relationship between the splitting function 
and the anomalous dimension (in the leading order).

Taking the $n+1$th moment (here we adopt the convention that
$n$th moment is take by convoluting with $x^{n\!-\!1}$) of both 
sides of equation \ref{eq:fw-DGLAP} we obtain
\beq
Q^2 \frac {d} {d Q^2} \int_0^1 d x \, x^n x P^f(x,Q^2) = 
\frac {\al (Q^2)} {2 \pi}  \int_0^1 d x \, x^n
\int_x^1 \frac {d x'}{x'} \ga^f ({x \over {x'}}) x' P^f(x',Q^2) \,.
\eeq
\nid Change the integration variables in the right-hand-side (rhs) 
from $x$ and $x'$ to $x'$ and $x/x'$ by $dx = x' \, d (x/x')$ and 
$dx \, dx'/x' = d(x/x') \,dx'$, so that 
\beq
\int_0^1 dx \int_x^1 \frac {d x'}{x'} 
= \int_0^1 dx' \int_0^1 d \left(\frac {x}{x'}\right) \,,
\eeq

\nid where we have also changed the integration limits appropriately
(see figure \ref{fig:x-x} for illustration). Rewrite $x^n$ as 
$(x/x')^n (x')^n$ we have
\beqs
rhs &=& \frac {\al (Q^2)} {2 \pi} \int_0^1 dx' 
\int_0^1 d( \frac {x}{x'}) x' P^f(x',Q^2)  (x')^n (\frac {x}{x'})^n
\ga^f ({x \over {x'}}) \non \\
&=& \frac {\al (Q^2)} {2 \pi} \int_0^1 dx' x'^n x' P^f(x',Q^2)
\int_0^1 d( \frac {x}{x'}) (\frac {x}{x'})^n \ga^f ({x \over {x'}})
\non \\
&\equiv& \frac {\al (Q^2)} {2 \pi} \int_0^1 dx \, x^n \,x P^f(x,Q^2)
\ga^f_n \,,
\eeqs

\nid where we have defined the $n$-moment of the splitting function as
\beq
\ga^f_n \equiv \int_0^1 dx \, x^n \ga^f(x) \,.
\eeq

\nid By the proper identification between the $n(+1)$th moment of the
splitting function and the anomalous dimension in leading order as
\beq
\ga^f_n(\al(Q^2)) = \frac {\al (Q^2)} {2 \pi} \ga_n^f \,,
\eeq

\nid and noting that at the normalization scale $\mu^2 = Q^2$ we have
$\al_\mu = \al(Q^2)$ and thus 
\beq
Q^2 \frac {d} {d Q^2} \equiv Q^2 \d/d{Q^2} + \be \d/d{\al}
\eeq

\nid we have indeed recovered equation \ref{eq:fw-pdfrge}.

\subsection{The Anomalous Dimensions and Splitting Functions 
\label{sec:fw-gamma}}

In terms of diagrams, the operator product expansion of the
DIS forward amplitude into the product of reduced matrix elements and
Wilson coefficients is shown in figure \ref{fig:fw-ope}. If we put a
cut through both sides, it becomes the OPE of the structure function
into Wilson coefficient and parton distribution function. Only the
lowest order contribution is shown where the connecting part on the
right hand side is a quark operator.

\nid The parton distribution on the bottom evolves in the momentum
scale via an anomalous dimension until it reaches the scale of the
hard scattering $Q^2$. We will continue the computation and discussion
in the light cone (LC) gauge.

\subsubsection{Review of the Light Cone Gauge}

\input{lc}

\subsubsection{The Quark-Quark Anomalous Dimension}

The diagram of the quark distribution function
$x P^f(x)$ is shown in figure \ref{fig:fw-pdf}, where we take
the lowest order quark-quark vertex function to be (\cite{G8070})
\beq
\label{eq:fw-ga0}
\Ga_+^{(0)} = \ga_+ \left( \frac {k_+}{p_+} \right)^{n\!-\!1} \,.
\eeq

Let us look at the first radiative corrections. The quark-quark
diagram is shown in figure \ref{fig:fw-qq}, where we have an extra
on-shell gluon line. 

\nid Using standard Feynman rules we can write the value of part of
the diagram within the dotted circle as
\beq
\label{eq:fw-ga1-def}
\Ga_+^{(1)} = C \int { {d^4 k} \over { (2 \pi)^4}}
\ga_\al {i \over \not{\!k}} \ga_+ 
\left( \frac {k_+}{p_+} \right)^{n\!-\!1} \!\!\!
(-2 \pi) \de^+[(k_1-k)^2] {i \over \not{\!k}} 
\ga_\be D^{\al \be} (k_1 -k) \,.
\eeq

\nid In the above, $C$ is the color factor and
\beq
C = (ig)^2 \sum_{i,b} (T^i_{ab}T^i_{ba'}) \,,
\eeq

\nid where $T^i_{ab} = \left(\frac {\lm^i} {2} \right)_{ab}$ is the
generator of the $SU(3)$ color group. Since we have \cite{Group}
\beq
\sum_i (T^i)^2 = C_F I \,,
\eeq

\nid where $C_F$ is the Casmir operator of the fundamental
representation of the gauge group and for $SU(N)$ 
\beq
C_F = \frac {N^2 -1} {2N} \,.
\eeq

\nid Defining the strong coupling constant $\al_s \equiv \al$ by 
$g^2 = 4 \pi \al$, we have
\beq
C = (ig)^2 C_F \de_{a'a} = - 4 \pi \al C_F \,,
\eeq

\nid where in the last step we have set $a = a'$. 

\nid In arriving at equation \ref{eq:fw-ga1-def} we have also used the
transition from the usual gluon propagator \ref{eq:LC-gluon} to the
on-shell gluon by
\beq
\frac {-i} {k^2 + i \eps} \to (-2 \pi) \de^+ [k^2] \,,
\eeq

\nid which is consistent with the optical theorem \ref{eq:fw-optical}.

\nid Therefore we obtain
\beq
\label{eq:fw-ga1}
\Ga_+^{(1)} = \frac {\al C_F} {2 \pi^2} 
\int \frac {d^4k \de^+[(k_1-k)^2]} { (k^2 + i \eps)^2}
\left( \frac {k_+}{p_+} \right)^{n\!-\!1} \!\!\!
\ga_\al \not{\!k} \ga_+ \not{\!k} \ga_\be D^{\al \be} (k_1 -k) \,.
\eeq

\nid Note in the light cone gauge, there is only one solution to 
$\de (k^2)$ because of the linearization of the light cone variable. 
We write the integral over $d^4k$ in light cone variables
\beq
d^4k = d k_0 d k_1 dk_2 dk_3 \equiv dk_+ dk_- d^2 \perb{k} \,,
\eeq

\nid and integrate over $k_-$ with the $\de$-function to get
\beq
d k_- \de [(k_1\!-\!k)^2] 
= d k_- \de [2 (k_1\!-\!k)_+(k_1 \!-\!k)_- \!-\! (\perb{k}_1 \!-\! \perb{k})^2]
= {1 \over {2 (k_1 \!-\! k)_+} } \,.
\eeq

\nid Define three momentum ratio variables
\beq
x = \frac {k_+}{p_+} \,, \;\;\;\;\;
x_1 = \frac {k_{1+}} {p_+} \,, \;\;\;\;\;
\om = \frac {k_+} {k_{1+}} = {x \over x_1} \,,
\eeq

\nid we evaluate $k^2$ with $(k_1 -k)^2 = 0$ and get
\beqs
k^2 &=& 2 k_+ k_- - \perb{k}^2 \non \\
&=& 2 k_+ ( k_{1-} - \frac {(\perb{k}_1 \!-\! \perb{k})^2}
{2 (k_1 \!-\! k)_+} ) - \perb{k}^2 \non \\
&=& \frac {k_+}{k_{1+}} (k_1^2 + \perb{k}_1^2) - \perb{k}^2 
- \frac {k_+}{(k_1 \!-\! k)_+} (\perb{k}_1 \!-\! \perb{k})^2 \,,
\eeqs

\nid where in the last step we have used
\beq
2 k_+ k_{1-} = \frac {k_+}{k_{1+}} 2 k_{1+}k_{1-} 
= \frac {k_+}{k_{1+}} (k_1^2 + \perb{k}_1^2) \,.
\eeq

\nid Expressing things in terms of $\om$ we have
\beqs
k^2 &=& \om k_1^2 + \om \perb{k}_1^2 - \perb{k}^2 
- \frac {\om} {1 \!-\! \om} (\perb{k}_1 \!-\! \perb{k})^2 \non \\
&=& - {1 \over {1 \!-\! \om}} [ -\om (1 \!-\! \om)k_1^2 
+ \om (\perb{k}_1 \!-\! \perb{k})^2 - \om (1 \!-\! \om) \perb{k}_1^2
+ (1 \!-\! \om) \perb{k}^2 \non \\
&=& - {1 \over {1 \!-\! \om}} [ ( \perb{k} - \om \perb{k}_1)^2
- \om (1 \!-\! \om)k_1^2] \,.
\eeqs

Our goal is not to calculate $\Ga_+$ completely, but only to calculate
the ultraviolet divergent part of it. In evaluating the numerator of 
 \ref{eq:fw-ga1} we keep only the terms of the highest $\perb{k}$
power, which turns out to be quadratic. Note because of the
$\de$-function, $(k_1 -k)_-$ also has a $\perb{k}^2$ contribution. 
The denominator is simple after we take the leading contribution, that
is,
\beq
k^2 = - \frac {\perb{k}^2} {1 \!-\! \om} \,.
\eeq

\nid As for the numerator, we now need
\beqs
\ga^\al \!\not{\!k} \ga_+ \!\not{\!k} \ga^\be D_{\al \be} &=&
\ga^\al (\{\not{\!k}, \ga_+ \} - \ga_+ \!\not{\!k} ) 
\not{\!k} \ga^\be D_{\al \be} \non \\
&=& 2 k_+ \ga^\al \! \not{\!k} \ga^\be D_{\al \be} 
- k^2 \ga^\al \ga_+ \ga^\be D_{\al \be} \,.
\eeqs

\nid The second term of the above is
\beqs
- k^2 \ga^\al \ga_+ \ga^\be \left( g_{\al \be}
- { {n_\al (k_1 \!-\!k)_\be + (k_1\!-\!k)_\al n_\be} 
\over {n \cdot (k_1 - k)}} \right) &=&
- k^2 \ga^\al \ga_+ \ga_\al + 0 \non \\
&=& 2 k^2 \ga_+ = - 2 \ga_+ \frac {\perb{k}^2} {1 \!-\! \om} \,, 
\;\;\;\;\;\;\;\;\;
\eeqs

\nid where in the first step we used the facts that $n \cdot \ga =
\ga_+$ and $\ga_+^2 = 0$ while in the last step we used the
anti-commuting relation \ref{eq:anticommudef}. The first term of
the numerator, on the other hand, can be expanded to be
\beq
2k_+ \{ \ga_\al \!\not{\!k} \ga^\al 
- { 1 \over { (k_1 \!-\!k)_+}} [ \ga_+ \!\not{\!k} 
(\not{\!k}_1-\!\not{\!k}) + (\not{\!k}_1-\!\not{\!k}) \!\not{\!k} 
\ga_+]\} \,.
\eeq

\nid We have from the $\de$-function
\beq
k_- \sim \frac {\perb{k}^2} {2 (k_1-k)_+} \,,
\eeq

\nid and thus
\beq
\ga_\al \!\not{\!k} \ga^\al = - 2 \!\not{\!k} = -2 \ga_+ k_-
= \ga_+ \frac {\perb{k}^2} { (k_1 \!-\!k)_+} \,.
\eeq

\nid At the same time,
\beqs
\ga_+ \!\not{\!k}(\not{\!k}_1-\!\not{\!k}) &=& 
- \ga_+ \!\not{\!k} \!\not{\!k} = - \ga_+ k^2 \non \\
&=& \ga_+ \frac {\perb{k}^2} {1 \!-\! \om} \,,
\eeqs

\nid while similarly
\beq
(\not{\!k}_1-\!\not{\!k}) \!\not{\!k} \ga_+ = 
\ga_+ \frac {\perb{k}^2} {1 \!-\! \om} \,.
\eeq

\nid Therefore the first term of the numerator is actually equal to
\beq
2 \ga \perb{k}^2 \frac {\om}{1 \!-\! \om}
\{ 1- \frac {2} {1 \!-\! \om} \} 
= - 2 \ga \perb{k}^2 \frac {\om}{1 \!-\! \om}
\frac {1 \!+\! \om} {1 \!-\! \om} \,.
\eeq

\nid Combined with the second term, we find that the divergent
contribution to the numerator is
\beq
-2 \ga_+ \frac {\perb{k}^2} {(1 \!-\! \om)^2}
[ \om (1 \!+\! \om) + (1 \!-\! \om)] 
= -2 \ga_+ \perb{k}^2 \frac {1 \!+\! \om^2} {(1 \!-\! \om)^2} \,.
\eeq

Rewrite the $\perb{k}$ integral as
\beq
\int d^2 \perb{k} = 2 \pi \int \perb{k} d\perb{k} = \pi \int d
\perb{k}^2 \,,
\eeq

\nid we arrive at the expression of the divergent part of
$\Ga_+^{(1)}$ as
\beqs
\Ga_+^{(1)} &=& \frac {\al C_F} {2 \pi} 
\int \frac {d k_+} {2 (k_1-k)_+} 
\left( \frac {k_+}{p_+} \right)^{n\!-\!1} 
\int \frac {d \perb{k}^2} {\perb{k}^2} \ga_+  
\left( \frac {k_+}{k_{1+}} \right)^{n\!-\!1}
\left( \frac {k_{1+}}{p_+} \right)^{n\!-\!1} 
2 (1 + \om^2) \non \\
&=& \ga_+ \left( \frac {k_{1+}}{p_+} \right)^{n\!-\!1} 
 \frac {\al C_F} {2 \pi} \int \frac {d \perb{k}^2} {\perb{k}^2}
\int d \om \, \om^{n\!-\!1} \frac {1 \!+\! \om^2} {1 \!-\! \om} \,.
\eeqs

Our original graph, one that is figure \ref{fig:fw-amp}(b) plus an
on-shell gluon running from the incoming quark to the outgoing quark,
is not divergent in $\perb{k}^2$. The divergence we encountered arises
because we let $Q^2$ become very large for fixed $\perb{k}^2$. This
means that the cutoff for the logarithmic $\perb{k}^2$ integral is at
$Q^2$.  That is, explicitly,
\beq
\Ga_+^{(1)} = \ga_+ \left( \frac {k_{1+}}{p_+} \right)^{n\!-\!1}
\frac {\al C_F} {2 \pi} \int^{Q^2} \frac {d \perb{k}^2} {\perb{k}^2}
\int_0^1 d \om \, \om^{n\!-\!1} \frac {1 \!+\! \om^2} {1 \!-\! \om}
\,,
\eeq

\nid where we have also explicitly written out the limit on the $\om$
integration. 

Taking the logarithmic derivative with respect to $Q^2$, and
recall equation \ref{eq:fw-ga0}, we obviously have
\beq
Q^2 {d \over {d Q^2}} \Ga_+^{(1)}(x,Q^2) 
= \frac {\al C_F} {2 \pi} 
\int_0^1 d \om \, \om^{n\!-\!1} \frac {1 \!+\! \om^2} {1 \!-\! \om}
\Ga_+^{(0)}(x, Q^2) \,.
\eeq

\nid Remember the graphic definition of the parton distribution
function and $\Ga_+^{(0),(1)}$ as the lowest and first order factors,
we realize that we indeed have
\beq
Q^2 {d \over {d Q^2}} \int_0^1 x^{n\!-\!1} x P^f(x, Q^2) 
= \frac {\al C_F} {2 \pi} \int^1_0 x^{n\!-\!1} x P^f(x, Q^2) 
\int_0^1 d \om \, \om^{n\!-\!1} \frac {1 \!+\! \om^2} {1 \!-\! \om} \,.
\eeq

\nid By exactly reversing the argument in section \ref{sec:DGLAP}
(with $x'$ now being $x_1$), it is straight forward to show that we 
indeed have the DGLAP evolution equation for the quark distribution 
function
\beq
\label{eq:fw-DGLAPqq}
Q^2 {d \over {d Q^2}} x P^f(x,Q^2) = \frac {\al C_F} {2 \pi}
\int_x^1 \frac {d x_1}{x_1} \ga ( {x \over x_1} )
x P^f (x_1, Q^2) \,,
\eeq

\nid where the Altarelli-Parisi splitting function is
\beq
\label{eq:fw-spltqq1}
\ga(y) = \frac {1 + y^2} {1 - y} \,.
\eeq

The integral in equation \ref{eq:fw-DGLAPqq} is actually divergent at
the end point when $x_1 \to x$. This is an infrared divergence due to
emission of soft gluons with $(k_1 - k)_+ \to 0$. This divergence
should not be present in physical processes and indeed it is removed
once we add the quark self-energy graphs shown in figure 
 \ref{fig:fw-selfE}.

\nid After virtual corrections are included, the full quark-quark
splitting function that is free of singularities is 
\beq
\label{eq:fw-qqfinal}
\ga(x) = C_F [ \frac {1 + x^2} {(1 -x)_+} + {3 \over 2} \de (1-x)] \,,
\eeq

\nid where the special function $(1-x)_+^{-1}$ is defined by
\beq
\label{eq:fw-plusfcn}
\int_0^1 \frac {f(x) dx} {(1 -x)_+} 
= \int_0^1 dx  \frac {f(x) - f(1)} {(1 -x)} 
\eeq

\nid for any function $f(x)$ that has reasonable behavior.


\subsubsection{The Mixing of Evolution}

There are additional order $\al$ terms in the $Q^2$-evolution equation
due to operator mixing, for example, see figure \ref{fig:fw-qg}. 
We thus have the necessity of considering both quark and gluon distributions.

To organize, define the flavor singlet quark distribution as
\beq
\label{eq:fw-singlet}
\Sigma (x,Q^2) = x \sum_f [ P^f (x, Q^2) + P^{\bar{f}} (x, Q^2)] \,,
\eeq

\nid while the flavor octet distribution is
\beq
\label{eq:fw-octet}
\Delta^{ff'}(x, Q^2) = x [ P^f (x, Q^2) -  P^{f'}(x, Q^2)] \,,
\eeq

\nid and a similarly defined function $\Delta^{\bar{ff'}}$ for
anti-quarks. 

\nid The octet distributions obey a generic DGLAP evolution equation
\beq
\label{eq:fw-DGLAPoct}
Q^2 {d \over {d Q^2}} \Delta^{ff'}(x, Q^2)
= \frac {\al(Q^2)} {2 \pi} \int_x^1 \frac {d x_1}{x_1} 
\ga_{qq'} \left( {x \over x_1} \right) \Delta^{ff'}(x_1, Q^2) 
\;+\; O(\al^2) \,.
\eeq

\nid However, the singlet distribution will mix under renormalization
with the gluon distribution $G (x, Q^2)$. The DGLAP equation has a
matrix form and explicitly
\beq
\label{eq:fw-DGLAPsin}
Q^2 {d \over {d Q^2}} \left(
\begin{array}{c}
\Sigma (x,Q^2) \\
G (x, Q^2)
\end{array} \right)
= \frac {\al(Q^2)}{2 \pi}
\int_x^1 \frac {d x_1}{x_1}
\left(
\begin{array}{cc}
\ga_{qq'} (x/x_1) & \ga_{qg} (x/x_1) \\
\ga_{gq} (x/x_1) & \ga_{gg} (x/x_1)
\end{array} \right)
\left(
\begin{array}{c}
\Sigma (x_1,Q^2) \\
G (x_1, Q^2)
\end{array} \right)
\;.
\eeq

\nid Let $C_F$ again be the Casmir operator in the fundamental
representation with $C_F = 4/3$ for $SU(3)$ color group, and let $C_A$
be the Casmir operator in the adjoint representation with $C_A = N_c =
3$ for the color group, we have the explicit expression of the four
splitting functions as (eg, see \cite{Peskin})
\beqs
\label{eq:fw-spltfcnfinal}
%\left(
%\begin{array}{rcl}
\ga_{qq}(x) \!&=&\! C_F \, \left[ \frac {1+x^2}{(1-x)_+} + 
{3 \over2} \de (1-x)\right] \\
\ga_{gq}(x) \!&=&\! C_F \; \frac { 1 + (1-x)^2 } {x} \\
\ga_{qg}(x) \!&=&\! \half \, \left[ x^2 + (1-x)^2 \right] \\
\ga_{gg}(x) \!&=&\! 2C_A \, \left[ \frac {x}{(1\!-\!x)_+} 
+ \frac {1\!-\!x}{x} + x(1\!-\!x) \right] 
+ \frac {11 C_A \!-\! 2 n_f}{6} \, \de (1\!-\!x)
%\end{array}
%\right.
\eeqs

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{High Energy Behavior}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

It is more convenient to discuss the high energy behavior of the
parton distribution functions when we go to the $n$-moment space by
taking the $n$-th moment of the DGLAP evolution equation, or
equivalently directly using the renormalization group equation from
operator product expansion. 

\subsection{Anomalous Dimension Matrix}

\nid To the leading order in $\al$ we obtain, after taking the moments
of the splitting functions, the anomalous dimension matrix for the
singlet-gluon mixing to be
\beq
\label{eq:fw-gamatrix}
\ga_n (\al (Q^2)) = \frac {\al(Q^2)} {2 \pi}
\left(
\begin{array}{cc}
\ga_n^{qq} & \ga_n^{qg} \\
\ga_n^{gq} & \ga_n^{gg}
\end{array}
\right)
\eeq

\nid where $\ga_n$ is the $n$-th moment of the corresponding $\ga(x)$ 
in equation \ref{eq:fw-spltfcnfinal} and explicitly
\beqs
\label{eq:fw-nmoments}
\ga_n^{qq} &=& -{2 \over 3} \left[1 + 4 \sum_2^n {1 \over j} 
- {2 \over {n(n+1)}} \right] \\
\ga_n^{qg} &=& 2 \frac {n^2+n+2}{n (n+1)(n+2)} \\
\ga_n^{gq} &=& {8 \over 3} \frac {n^2+n+2} {n (n^2-1)} \\
\ga_n^{gg} &=& - 3 \left[ {1 \over 3} + {2 \over 9} \,n_f 
+ 4 \sum_2^n {1 \over j} - \frac {4}{n(n-1)}
- \frac {4} {(n+1)(n+2)} \right] \,.
\eeqs

\subsection{Dominant Moment Contribution}

The anomalous dimension matrix elements are usually singular at some
values of the moment index $n$ (see equation
 \ref{eq:fw-nmoments}). In the high energy limit, the dominant
contribution at small $x$ is from the right most pole of the anomalous
dimension. We will illustrate this by looking again at the DIS
structure function $\nu \W_2$.

\nid Adopting the common terminology let us now call $\nu \W_2$
structure function $F_2$. The $n$-th moment of $F_2$ is defined as
\beq
\label{eq:fw-F2momdef}
F_2^{(n)}(Q^2) = \int_0^1 dx \, x^{n\!-\!1} F_2 (x, Q^2) \,.
\eeq

\nid As we already established $F_2^{(n)}$ can be factorized as the
product of parton distribution functions and perturbatively calculable
Wilson coefficients with well defined $Q^2$ evolution. To predict
cross-section, however, we do need directly $F_2(x,Q^2)$ itself. We
can reconstruct $F_2$ from its moments by a Mellin transformation
\beq
\label{eq:fw-mellin}
F_2(x,Q^2) = \int_c \frac {dn}{2 \pi i} e^{n \log {1 \over x}}
F_2^{(n)}(Q^2) \,,
\eeq

\nid where $c$ is a contour lies in the complex $n$ plane parallel to
the imaginary axis and to the right of all singularities.

Let us first check to see whether this is a self-consistent
definition. Substitute \ref{eq:fw-F2momdef} into equation
 \ref{eq:fw-mellin} we obtain for the right hand side (rhs)
\beqs
rhs &=& \int_c \frac {dn}{2 \pi i} \int_0^1 dx' x'^{n\!-\!1} 
F_2 (x',Q^2) e^{n \log {1 \over x}} \non \\
&=& \int_c \frac {dn}{2 \pi i} \int_0^1 \frac {dx'}{x'}
e^{n \log {1 \over x} - n \log {1 \over {x'}}} F_2 (x',Q^2) \non \\
&=& \int_0^1 \frac {dx'}{x'} F_2 (x',Q^2) \int_c \frac {dn}{2 \pi i}
e^{n [ \log {1 \over x} - \log {1 \over {x'}}]} \non \\
&=& \int_0^1 d \log x' \, F_2 (x',Q^2) \,
\de \left( \log {1 \over x} - \log {1 \over {x'}} \right) \non \\
&=& F_2 (x, Q^2) = lhs \,,
\eeqs

\nid where in the fourth step we have used the mathematical identity
\beq
\int_{L- i \infty}^{L+i \infty} \frac {dn}{2 \pi i} e^{(n-c)z} 
= \de (z) \,.
\eeq

\nid We can always distort the contour $c$ to the left (but not crossing
any poles) so that it'll pick out the residues of the $n$ poles. Each
pole $n_p$ will eventually translate to a power factor of the
Bjorken-$x$ as $(x)^{-n_p}$. Therefore, at high energy and small-$x$
the dominant contribution comes from the right most $n$ pole of the
anomalous dimension on the complex $n$-plane.

\subsection{Double Leading Logarithmic Approximation \label{sec:fw-DLLA}}

To illustrate the point that leading (right most) pole in moment space of the
anomalous dimension dominates in the high energy limit, we will
explicitly compute the high energy behavior of the structure function
$F_2$ of deeply inelastic scattering.

According to previous analysis, the moments of the structure
function obey the momentum evolution equation (see, eg,
 \ref{eq:fw-rge-solu}) 
\beq
F_2^{(n)}(Q^2) = F_2^{(n)}(Q_0^2) e^{\int_{Q_0^2}^{Q^2} 
\frac {d \lm^2}{\lm^2} \ga_n (\al(\lm^2))} \,.
\eeq

\nid Thus the structure function itself is given by
\beq
F_2 (x, Q^2) = \int_c \frac {dn}{2 \pi i} F_2^{(n)}(Q_0^2)
e^{n \log {1 \over x} + \int_{Q_0^2}^{Q^2}  \frac {d \lm^2}{\lm^2} 
\ga_n (\al(\lm^2))} \,.
\eeq

\nid From equation \ref{eq:fw-nmoments} it is obvious that the
gluon-gluon anomalous dimension has the right most pole among others
at $n=1$. Using the lowest order running coupling we can write the
leading contributing term of the anomalous dimension
\beq
\ga_n(\al(\lm^2)) = \frac {C_A}{\pi (n-1)} \frac{1}{b_0 \log
\lm^2/\Lm^2} \,,
\eeq

\nid where $b_0$ is the first coefficient of the QCD $\be$-function. 

\nid By substitution we have
\beqs
F_2 (x, Q^2) &=& \int_c \frac {dn}{2 \pi i} F_2^{(n)}(Q_0^2)
e^{n \log {1 \over x} + \frac {C_A}{\pi b_0}
\log \frac {\log Q^2/\Lm^2} {\log Q_0^2 /\Lm^2} \frac {1} {n\!-\!1}} 
\non \\
&\equiv& \int_c \frac {dn}{2 \pi i} F_2^{(n)}(Q_0^2)
e^{an+ \frac {b}{n-1}} \,,
\eeqs

\nid where we have defined at high energy,
\beqs
a &\equiv& \log {1 \over x} >> 1 \non \\
b &\equiv& \frac {C_A}{\pi b_0} \log 
\frac {\log Q^2/\Lm^2} {\log Q_0^2 /\Lm^2}
\eeqs

The high energy limit corresponds to $n\to 1$ and $a >>1$ so we can
make a saddle point approximation for the exponent function $f(n) = a n +
b {1 \over {n-1}}$. We have
\beq
f'(n) = a - \frac {b} {(n-1)^2} = 0 \Rightarrow
n_0 = 1 + \sqrt{{b \over a}} \,.
\eeq

\nid Thus at the saddle point $n_0$,
\beqs
f(n_0) = a (\sqrt{{b \over a}} +1 ) + {b \over \sqrt{{b \over a}}}
= a + 2 \sqrt{ab} \non \\
f''(n_0) = \frac {2b} {(\sqrt{b/a})^3} = \frac {2 a^{3/2}} 
{b^{1/2}} > 0 \,,
\eeqs

\nid which means $c$ is a fine contour. Let $i\nu = n-1$ we obtain
finally,
\beqs
\label{eq:fw-sad-hie}
F_2 (x, Q^2) &=& F_2^{(n)}(Q_0^2) \int_{-\infty}^{\infty} 
\frac {d \nu} {2 \pi} e^{(a + 2 \sqrt{ab})}
e^{- \frac {2 a^{3/2}} {b^{1/2}} \nu^2} \non \\
&=&  F_2^{(n)}(Q_0^2) e^{n \log {1 \over x}}
e^{2 \sqrt{\log {1 \over x} \frac {C_A}{\pi b_0}
\log \frac {\log Q^2/\Lm^2} {\log Q_0^2 /\Lm^2}}}
\sqrt{ \frac { \pi b^{1/2}} {a^{3/2}}} \non \\
&=&  F_2^{(n)}(Q_0^2) \left( {1 \over x} \right)^1
e^{2 \sqrt{\log {1 \over x} \frac {C_A}{\pi b_0}
\log \frac {\log Q^2/\Lm^2} {\log Q_0^2 /\Lm^2}}}
\sqrt{ \frac { \pi \sqrt{\frac {C_A}{\pi b_0}
\log \frac {\log Q^2/\Lm^2} {\log Q_0^2 /\Lm^2}} }
{(\log {1 \over x})^{3/2}}} \,. \non \\
\eeqs

This is the high energy limit of the forward DIS amplitude/structure
function. The dominate small-$x$ behavior is a power dependence on 
$\left({1 \over x}\right)^{n_0}$ with $n_0$ given by a saddle point 
approximation. It is obvious that the value of $n_0$, which determines
the leading high energy and small-$x$ behavior, is dictated by the
right most pole on the complex $n$ plane. In this forward case, it is
at $n=1$ and thus we arrive at the $x^{-1}$ leading behavior of the
gluon distribution under DLLA. 






\chapter*{Figures}
\addcontentsline{toc}{chapter}{Figures}
\thispagestyle{myheadings}
\markright{}

\centering
%\input{figure/intro.fig}
%\input{figure/c1.fig}
%\input{figure/c2.fig}
%\input{figure/c3.fig}
%\input{figure/c4.fig}
%\input{figure/appd.fig}

\input{intro.fig}
\input{c1.fig}
\input{c2.fig}
\input{c3.fig}
\input{c4.fig}
\input{appd.fig}
We take the convention that the $0$ component is the time component of
a four vector, and the $1,2,3$ components are the (spacial) $x, y, z$
components, respectively.

For any four vector $v$, we define the light cone components
\beq
v_{\pm} \equiv {1 \over \sqrt{2}} (v_0 \pm v_3) , \,\,\,\, 
\perb{v} \equiv \left(
\begin{array}{cc}
v_1 \\
v_2 
\end{array} \right) .
\eeq

\nid This means 
\beq
v^2 \equiv v \cdot v = 2 v_+ v_- - \perb{v}^2,
\eeq

\nid while for any two $4$-vectors $v_1$ and $v_2$,
\beq
v_1 \cdot v_2 = v_{1+}v_{2-} + v_{1-}v_{2+} - \perb{v}_1 \cdot \perb{v}_2.
\eeq


\nid We can then define the LC metric $g_{\al \be}$ by
\beqs
g_{+-} = g_{-+} = 1, \,\,\,\,\, 
g_{++} = g_{--} = 0, \,\,\,\,\, g_{\bot \bot} = -1 , \non \\
g_{+ \bot} = g_{\bot +} = g_{- \bot} = g_{\bot -} = 0 .\non
\eeqs

\nid such that for $\al, \be = +, -, \bot$
\beq 
v_1 \cdot v_2 = \sum_{\al, \be} g_{\al \be}
v_{1\al}v_{2\be}.
\eeq

The Dirac matrices $\ga_{\al}$ with $\al = 0, 1, 2, 3$ can be
viewed as a four vector and we can similarly define
\beq
\ga_{\pm} \equiv {1 \over \sqrt{2}} (\ga_0 \pm \ga_3), \,\,\,\, 
\perb{\ga} \equiv \left(
\begin{array}{cc}
\ga_1 \\
\ga_2 
\end{array} \right) .
\eeq

\nid It can be easily verified that the light cone components obey
the anti-commutation relationship
\beq
\label{eq:anticommudef}
\{\ga_{\al},\ga_{\be}\} = 2 g_{\al \be} \,.
\eeq

\nid In particular, we have
\beqs
\label{eq:gammasquare}
\ga_+^2 = \ga_-^2 = 0 \non \\
\ga_+ \ga_- + \ga_- \ga_+ = 2  \\
\ga_\pm \ga_{1,2} + \ga_{1,2} \ga_\pm = 0 \non
\eeqs

\nid and therefore,
\beqs
\label{eq:anticommu}
\ga_+ \ga_- \ga_+ = 2 \ga_+ \non \\
\ga_+ \! \not{\!v} \ga_+ = 2 v_+ \ga_+  
\eeqs

Introducing a vector $n_\mu$ such that for any $4$-vector $v$
\beq
%\label{eq:n-def}
n \cdot v = v_+ \,.
\eeq

\nid This means
\beq
\label{eq:n-def}
n_- = 1 \,, \;\;\;\;\; n_+ = n_\bot = 0 \,, \;\;\;\;\;
\& \;\, n^2 = n_\mu n^\mu = 0 \,.
\eeq

\nid $n_\mu$ is the so called LC null vector.

The light cone (LC) gauge is defined by requiring for the gauge
field $A$,
\beq
\label{eq:LC-def}
n \cdot A = A_+ = 0 \,.
\eeq
 
\nid The gluon propagator in the light cone gauge is 
\beq
\label{eq:LC-gluon}
\cd_\mn(k) = \frac {-i} {k^2 + i \eps} \left( g_\mn - { {n_\mu k_\nu + k_\mu
n_\nu} \over {n \cdot k}} \right)  
\equiv \frac {-i} {k^2 + i \eps}  D_\mn(k) \,,
\eeq

\nid where $D_\mn(k)$ is the so-called light cone gluon projector. We have
\beq
n_\mu \cd^\mn = n_\nu \cd^\mn = 0 \,.
\eeq

\nid The light cone gauge is also a {\sl ghostless} gauge.





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\begin{centering}
\huge \bf Virtual Compton Scattering \\
at High Energy \\
\vspace{1.0in}
\Large Zhang Chen \\
\vspace{1.0in}
{\em Advisor:} Professor Alfred H.~Mueller\\
\vspace{1.0in}
\large
Submitted in partial fulfillment of the \\
requirements for the degree \\
of Doctor of Philosophy \\
in the Graduate School of Arts and Sciences \\
\vspace{0.5in}
COLUMBIA UNIVERSITY \\
1999 \\
\end{centering}
\clearpage

