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%% subsection 3.1 Matrix elements [slac-pub-7064-0-0-3-1 in slac-pub-7064-0-0-3: ^slac-pub-7064-0-0-3 >slac-pub-7064-0-0-3-2]
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\sectionLink{slac-pub-7064-0-0-3}{slac-pub-7064-0-0-3}{Above: 3. Pair production in a colour field}%
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\subsection{\usemenu{slac-pub-7064::context::slac-pub-7064-0-0-3-1}{Matrix elements}}\label{subsection::slac-pub-7064-0-0-3-1}
We are interested in the diffractive and inclusive structure functions
of the proton at large momentum transfer of the electron and at small $x$,
\beq
-q^2 = Q^2 \gg \Lambda^2\quad,\qquad x = {Q^2\over 2m_pq_0}\ll 1\, ,
\eeq
where $\Lambda$ is the QCD scale parameter and $q_0$ is the photon energy in
the proton rest frame. Here, the photon energy is much larger than the QCD
scale $\Lambda$ and, at low $x$, the kinematically required momentum transfer
to the proton is very small.
Hence, it appears reasonable to describe the photon-proton
interaction as pair production in a classical colour field.
In case of the diffractive structure function, we shall
consider final states with invariant mass
\beq
M^2 = (l+l')^2 = O(Q^2)\, ,
\eeq
where $l'$ and $l$ are the 4-momenta of quark and antiquark, respectively
(cf. Fig. 1). Our goal is to calculate the leading term
of the cross section in the limit
$x\ll 1$ and $\Lambda^2/Q^2 \ll 1$.
The S-matrix element for pair production is given by the expression
\beq
S_{fi} = ie\epsilon_\mu\int d^4xe^{-iqx}\bar{\psi}_u\gamma^\mu\psi_v\equiv
ie \epsilon_\mu S^\mu\, .\label{sm}
\eeq
Here $\psi_u$ and $\psi_v$ are the wave functions of the produced quark
and antiquark, respectively. They have been calculated in the previous section
and they contain the dependence on the colour field.
To leading order one has (cf. eqs. (\docLink{slac-pub-7064-0-0-2.tcx}[antiq]{5}),(\docLink{slac-pub-7064-0-0-2.tcx}[quark]{6})),
\beq
S_{(0)ab}^\mu = \left(\int d^4xe^{i\Delta x}\bar{u}(l')
U_0(x)\gamma_\mu V_0(x)v(l)\right)_{ab}\, ,
\eeq
where
\beq
\Delta = l+l'-q\, ,
\eeq
and the indices $ab$ denote the colour of the created state.
For the evaluation of longitudinal and transverse structure functions the
combinations $S_\mu^*S^\mu$ and $S_0^*S^0$ will be needed.
Summing over the spins of quark and antiquark, one easily obtains,
\beq\label{S00}
S_{(0)}^\mu S_{(0)\mu}^* = -8(ll')|\tilde{f}(\Delta)_{ab}|^2
= -4M^2|\tilde{f}(\Delta)_{ab}|^2\, ,
\eeq
with
\beq\label{ft}
\tilde{f}(\Delta)_{ab}=\int d^4x e^{i\Delta x}f(x)_{ab}=\int d^4xe^{i\Delta x}
(U_0(x)V_0(x))_{ab}\, .
\eeq
Note, that here and below the dependence of the matrices $U$, $V$ and $f$
on $l'$ and $l$, as described in the previous section,
is not shown explicitly. As it is obvious
from Eq. (\docLink{slac-pub-7064-0-0-3.tcx}[S00]{30}), the leading term, which is formally of order $l_0l_0'
\sim (q_0)^2$ in the high energy limit, turns out to be only of order
$(q_0)^0$ due to the contraction of the 4-vectors $l$ and $l'$.
Therefore the formally
next-to-leading and next-to-next-to-leading contributions are competitive and
have to be calculated.
Hence, we start from the high energy expansion of $S^\mu$,
\beq
S^\mu=S_{(0)}^\mu+S_{(1)}^\mu+S_{(2)}^\mu+\cdots\, ,
\eeq
where
\begin{eqnarray}
S_{(1)}^\mu&=&\int d^4xe^{i\Delta x}\bar{u}(l')\left( U_1\gamma^\mu V_0+
U_0\gamma^\mu V_1\right)v(l)\\ \nonumber\\
S_{(2)}^\mu&=&\int d^4xe^{i\Delta x}\bar{u}(l')\left( U_1\gamma^\mu V_1+
U_2\gamma^\mu V_0+U_0\gamma^\mu V_2\right)v(l)\, .
\end{eqnarray}
The first three terms of the corresponding expansion of the squared
S-matrix element,
\beq
S^\mu S_\mu^*=A_0+A_1+A_2+\ldots\label{lc}
\eeq
have to be taken into account. $A_0$ is given by Eq. (\docLink{slac-pub-7064-0-0-3.tcx}[S00]{30}). The
two following terms are
\bea
A_1 &=& 2\mbox{Re}S_{(0)}^\mu S_{(1)\mu}^*\, ,\nn\\
A_2 &=& S_{(1)}^\mu S_{(1)\mu}^*+2\mbox{Re}S_{(0)}^\mu S_{(2)\mu}^*\, .
\eea
$A_1$ can be easily evaluated as follows. Ordinary
Dirac algebra manipulations give
\beq
S^{\mu*}_{(0)}S_{(1)\mu}=-2\tilde{f}(\Delta)_{ab}^*\int d^4xe^{i\Delta x}
\mbox{tr}\left[\ls\ls'(U_0V_1+U_1V_0)_{ab}\right]\, .
\eeq
Using Eq. (\docLink{slac-pub-7064-0-0-2.tcx}[Vn]{14}) for $V_1$ together with the easily derived relation
\beq
\Ds P_+\Ds P_+V_0=\Ds\,{}^2P_+V_0 \, ,
\eeq
and the corresponding expressions for $U_1$ and $U_0$, one obtains
\begin{eqnarray}
\mbox{tr}\left[\ls\ls'(U_0V_1+U_1V_0)\right]&=&\frac{i}{2}U_0\,\, \mbox{tr}
\left[\ls\ls'\left(\frac{1}{lD}\Ds\,{}^2P_++P_-\Dls\,{}^2\frac{1}{l\Dl}
\right)\right]V_0 \nn\\
&=&2iU_0\left(-l'D-l\Dl+M^2\cal{O}(\frac{D}{l})\right)V_0 \nn\\
&=&-2i(l+l')^\mu\partial_\mu(U_0V_0)+M^2\cal{O}(\frac{D}{l})\, .
\end{eqnarray}
Here the colour indices have been dropped for brevity. Note, that the inverse
differential operator has disappeared. After performing a
partial integration the leading order term $\sim (q_0)^0$ reads
\beq
A_1=4(Q^2+M^2)|\tilde{f}(\Delta)_{ab}|^2\, ,
\eeq
which is very similar to the expression (\docLink{slac-pub-7064-0-0-3.tcx}[S00]{30}) for $A_0$.
The calculation of $A_2$ can be carried out using the same technique and the
formulae of the last section. It turns out to be convenient to introduce
the total momentum $L\equiv l+l'$
and the momentum fraction $\alpha$,
\beq
l\equiv\alpha L+a\quad,\quad l'\equiv(1-\alpha)L-a\quad,\quad a_0\equiv 0\, .
\label{alpha}
\eeq
$\alpha$ and the four-vector $a$ are defined by Eq. (\docLink{slac-pub-7064-0-0-3.tcx}[alpha]{41}). From the
relation $\vec{a}^2=\alpha(1-\alpha)M^2$ it is clear that to leading
order in our high energy expansion
\beq\label{apprl}
l\approx\alpha L\quad,\quad l'\approx(1-\alpha)L\, .
\eeq
Since $A_2$ is formally of order $(q_0)^0$ its leading part is already
sufficient for our purposes. Therefore, the approximation eqs. (\docLink{slac-pub-7064-0-0-3.tcx}[apprl]{42})
may be used. A straightforward calculation yields the result
\beq
A_2=4\left[\frac{1-\alpha}{\alpha}\tilde{g}_{R\mu}(\Delta)_{ab}^*
\tilde{g}^\mu_R(\Delta)_{ab}+\frac{\alpha}{1-\alpha}
\tilde{g}_{L\mu}(\Delta)_{ab}^*\tilde{g}^\mu_{L}(\Delta)_{ab}\right]
-8\mbox{Re}\tilde{f}(\Delta)_{ab}^*
\tilde{h}(\Delta)_{ab}\, ,\label{a2}
\eeq
where $\tilde{g}_{R\mu}\, ,\, \tilde{g}_{L\mu}$ and $\tilde{h}$ are the
Fourier transforms of (cf. eq. (\docLink{slac-pub-7064-0-0-3.tcx}[ft]{31}))
\beq
g_{R\mu}(x)_{ab}=(U_0(x)D_\mu V_0(x))_{ab}\quad ,\quad g_{L\mu}(x)_{ab}=
(U_0(x)\Dl\!{}_\mu V_0(x))_{ab}\label{gdef}
\eeq
and
\beq
h(x)_{ab}=(U_0(x)\Dl\!{}_\mu D^\mu V_0(x))_{ab}\, .\label{hdef}
\eeq
Now the leading contribution to $S^\mu S^*_\mu$ is explicitly given by Eq.
(\docLink{slac-pub-7064-0-0-3.tcx}[lc]{35}).
The calculation of $S^0S^*_0$ is much simpler. Here, contrary to
$S^\mu S^*_\mu$, the leading term is of order $(q_0)^2$ and there is
no suppression after a contraction of Lorentz indices.
One finds,
\beq\label{s0m2}
S^0S^*_0=8q_0^2|\tilde{f}(\Delta)_{ab}|^2\alpha(1-\alpha)\, .\label{sl}
\eeq