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%% subsection 4.1 Relations between different formfactors [slac-pub-7064-0-0-4-1 in slac-pub-7064-0-0-4: ^slac-pub-7064-0-0-4 >slac-pub-7064-0-0-4-2]
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\sectionLink{slac-pub-7064-0-0-4}{slac-pub-7064-0-0-4}{Above: 4. Inclusive and diffractive structure functions}%
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\subsection{\usemenu{slac-pub-7064::context::slac-pub-7064-0-0-4-1}{Relations between different formfactors}}\label{subsection::slac-pub-7064-0-0-4-1}
Before discussing the actual functional form of $f,\, g_R,\, g_L$ and $h$,
introduced in Sect. 3.1, we shall derive some relations between
terms involving $g_R$ and $g_L$. For brevity, colour indices will be dropped
throughout this subsection.
As we shall see in the following subsection, the Fourier integrals
$\tilde{f}$, $\tilde{g}_{L,R}$ and $\tilde{h}$ receive negligible
contributions from the region where the spatial point $x$ lies inside
the proton. Therefore, one may use the approximation
\beq
g_{R\mu}(x) \simeq U_0(x)\partial_\mu V_0(x)\quad,\quad
g_{L\mu}(x) \simeq U_0(x)\pdl\!\!{}_\mu V_0(x)\, .
\eeq
Since $l$ is a light-like vector with negligible transverse components,
and $l^\mu g_\mu = 0$, Eq. (\docLink{slac-pub-7064-0-0-3.tcx}[a2]{43}) can be written as
\beq
A_2=-4\,\frac{1-\alpha}{\alpha}\sum_{i=1}^{2}|\tilde{g}_{Ri}(\Delta)|^2-4\,
\frac{\alpha}{1-\alpha}\sum_{i=1}^{2}|\tilde{g}_{Li}(\Delta)|^2-8\mbox{Re}
\tilde{f}(\Delta)^*\tilde{h}(\Delta)\, .\label{sum}
\eeq
In the following, we mean by $\tilde{g}_R$, $\tilde{g}_L$,
$\tilde{f}$ and $\tilde{h}$
the 3-dimensional Fourier transforms in the sense of Eq. (\docLink{slac-pub-7064-0-0-3.tcx}[cs]{58}) which
occur in the cross sections. Correspondingly, $A_i$, $i=0,1,2$, are the
expressions defined in Sect. 3.1 in terms of the 3-dimensional Fourier
transforms.
It is convenient to further specify the coordinate system. In addition
to choosing the $x^3$-axis anti-parallel to the photon momentum,
we assume the plane spanned by $\vec{l}$ and $\vec{l'}$
to be orthogonal to the $x^1$-axis, neglecting the small transverse
momentum transfer. Defining $\theta_1,\,\theta_2$ and $\theta$ to be
the angles between $\vec{l'}$
and $\vec{q}$, between $\vec{q}$ and $\vec{l}$ and between $\vec{l'}$ and
$\vec{l}$, respectively, and writing $-i(x_\p\Delta_++x_\perp\Delta_\perp)=ix
\Delta$ for brevity, one has
\bea\label{g2}
\hspace*{-.5cm}i\Delta_+\int e^{ix\Delta}U_0V_0&=&
\int e^{ix\Delta}\partial_3(U_0V_0)\nn\\
&=&\int e^{ix\Delta}\left(\theta_1U_0
\partial_2\!\!\!\!\!\raisebox{1.5ex}{$\leftarrow$}
V_0-\theta_2U_0\partial_2V_0
\right) \nn\\
&=&\int e^{ix\Delta}\left(-i\Delta_2\theta_1U_0V_0-
\theta_1U_0\partial_2V_0-\theta_2U_0\partial_2V_0\right)\nn\\
&=&-\int e^{ix\Delta}
\left(i\Delta_2\theta_1U_0V_0+\theta U_0\partial_2V_0\right)\, .
\eea
In order to obtain the second equality, one has to observe that in
our coordinate system, moving e.g. the quark-line, associated with $U_0$,
by some small amount $\epsilon$ in $x^3$-direction is equivalent to
moving it by the amount $\theta_1\epsilon$ in $x^2$-direction.
{}From Eq. (\docLink{slac-pub-7064-0-0-4.tcx}[g2]{61}) we obtain the relation
\beq
\left|\int e^{ix\Delta}U_0\partial_2V_0\right|=\frac{\Delta_+}{\theta}
|\tilde{f}|+\frac{\Delta_2\theta_1}{\theta}|\tilde{f}|\, .\label{d20}
\eeq
The first term on the r.h.s. of Eq. (\docLink{slac-pub-7064-0-0-4.tcx}[d20]{62}) gives a contribution of
the same order of magnitude as the formally leading terms $A_0$ and $A_1$.
However, the contribution proportional to the transverse
momentum transfer $\Delta_2$ can be neglected. Obviously, as long as $\alpha$
and $1-\alpha$ are $O(1)$,
it is suppressed with respect to the leading contributions by
a factor $|\Delta_2|/Q \sim \Lambda/Q$. This suppression is not so
obvious if $\alpha\ll 1$, since then the prefactor $(1-\alpha)/\alpha$
in Eq. (\docLink{slac-pub-7064-0-0-4.tcx}[sum]{60}) becomes large. Yet this enhancement is completely
compensated by the factor $\theta_1/\theta$, since in this limit $\theta_1/
\theta=\theta_1/(\theta_1+\theta_2)\approx\theta_1/\theta_2\approx\alpha/(1-
\alpha)$. These relations can be read off from the vector parallelogram
corresponding to $\vec{l}+\vec{l'}=\vec{q}$. Hence, we obtain for the
leading contribution
\beq
|\tilde{g}_{R2}(\Delta)| \simeq \left|\int e^{ix\Delta}U_0\partial_2V_0\right|
\simeq \frac{\Delta_+}{\theta}|\tilde{f}|\label{d2}\, .
\eeq
The term in Eq. (\docLink{slac-pub-7064-0-0-4.tcx}[sum]{60}) involving $\tilde{g}_{R1}$ can not be directly
related to $\tilde{f}$. However, for very small and very large values of
$x_\p$, one expects
\beq
\left|\int e^{ix\Delta}U_0\partial_1V_0\right|\simeq
\left|\int e^{ix\Delta}U_0\partial_2V_0\right|\label{ffp}\, .
\eeq
For sufficiently small $x_\p$ one has $U_0 \simeq 1$, which restores
the rotational invariance in the transverse plane.
For sufficiently large $x_\p$ only the quark or the antiquark trajectory
penetrates the proton field. Hence, either $U_0=1$ or $V_0=1$,
thus again restoring rotational invariance.
The above considerations suggest to replace $\tilde{g}_{R1}$ by
a function $\tilde{f}'$, defined in analogy to Eq. (\docLink{slac-pub-7064-0-0-4.tcx}[d2]{63}),
\beq
|\tilde{g}_{R1}(\Delta)|=\left|\int e^{ix\Delta}U_0\partial_1V_0\right|=
\frac{\Delta_+}{\theta}|\tilde{f'}|\, .\label{d1}
\eeq
The functions $\tilde{f}$ and $\tilde{f}'$ are then expected to have
a similar asymptotic behaviour.
The arguments used above to rewrite the terms involving $\tilde{g}_R$ also
apply to the terms involving $\tilde{g}_L$.
Combining the results of Sect. 3.1 for $A_0$ and $A_1$ with Eqs.
(\docLink{slac-pub-7064-0-0-3.tcx}[angle]{52}), (\docLink{slac-pub-7064-0-0-4.tcx}[sum]{60}), (\docLink{slac-pub-7064-0-0-4.tcx}[d2]{63}) and (\docLink{slac-pub-7064-0-0-4.tcx}[d1]{65}), we obtain for the
leading contribution to $S^{\mu}S^*_{\mu}$,
\beq
A_0+A_1+A_2=Q^2\left[4|\tilde{f}|^2-\frac{1-2\alpha(1-\alpha)}
{\beta(1-\beta)}\left(|\tilde{f}|^2+|\tilde{f}'|^2\right)\right]
-8\mbox{Re}\tilde{f}^*\tilde{h}\, ,
\eeq
where
\beq
\beta=\frac{Q^2}{Q^2+M^2}
\eeq
is the parameter conventionally introduced in diffractive deep-inelastic
scattering. The corresponding expression for $S^0S^*_0$ is given in
Eq. (\docLink{slac-pub-7064-0-0-3.tcx}[s0m2]{46}). Inserting these expressions in Eq. (\docLink{slac-pub-7064-0-0-3.tcx}[cs]{58}) yields
the $\gamma^*p$ cross sections
\bea
d\sigma^\mu{}_\mu &=&\frac{\pi e^2}{|\vec{q}|}\frac{1}{4(2\pi)^5}\!
\int d^2\Delta_\perp d\Delta_+d\alpha \nn\\
&&\qquad\qquad\times \left\{Q^2\left[4|\tilde{f}|^2\!
-\!\frac{1-2\alpha(1-\alpha)}{\beta(1-\beta)}\left(|\tilde{f}|^2
+|\tilde{f}'|^2\right)\right] -\!8\mbox{Re}\tilde{f}^*
\tilde{h}\right\}\, ,\label{smm}\\
d\sigma_{00} &=&\frac{\pi e^2}{|\vec{q}|}\frac{2}{(2\pi)^5}\!
\int d^2\Delta_\perp d\Delta_+d\alpha
q_0^2|\tilde{f}|^2\alpha(1-\alpha)\, .\label{s00}
\eea
{}From these two cross sections one can obtain transverse and longitudinal
structure functions in the usual way.