%% slacpub7204: page file slacpub7204003.tcx.
%% section 3 Quark pair production in a colour field [slacpub7204003 in slacpub7204003: slacpub7204004]
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\section{\usemenu{slacpub7204::context::slacpub7204003}{Quark pair production in a colour field}}\label{section::slacpub7204003}\label{qq}
The basic process in deep inelastic scattering, viewed in the proton
rest frame, is the dissociation of a virtual photon into a
quark antiquark pair which then interacts with the
proton. The corresponding scattering amplitude reads (cf.~\cite{8})
\beq\label{amp}
S_{\mu} = i e \int d^4x e^{iqx} \bar{\psi}_u(x)\gamma_{\mu}
\psi_v(x)\ ,
\eeq
where $\psi_u$ and $\psi_v$ are the wave functions of the outgoing
quark and antiquark depending on the momenta $p'$ and $l'$,
respectively, and $e$ is the electric charge (cf.~Fig.~\docLink{slacpub7204003.tcx}[dqq]{2}).
\begin{figure}[h]
\begin{center}
\parbox[b]{10cm}{\psfig{width=10cm,file=fig2.eps}}\\
\end{center}
\refstepcounter{figure}
\label{dqq}
{\bf Fig.\docLink{slacpub7204003.tcx}[dqq]{2}} Quark pair production in the colour field of the proton.
\end{figure}
In the semiclassical approximation the interaction with the proton is
treated as scattering in a classical colour field $G_{\mu}(x) = T^a
G_{\mu}^a(x),\,T^a = {1\over 2} \lambda^a$. The quark and antiquark wave
functions are solutions of the Dirac equation with the colour field. Hence,
they satisfy the integral equations
\bea\label{int}
\psi_v(x) &=& \psi^{(0)}_v (x)  \int d^4x' S_F(xx') \Gs(x')
\psi_v(x')\ ,\\ \bar{\psi}_u(x) &=& \bar{\psi}^{(0)}_u (x) \int d^4x'
\bar{\psi}_u(x') \Gs(x') S_F(x'x)\ .
\eea
Here $\psi^{(0)}_v$ and $\bar{\psi}^{(0)}_u$ are solutions of the
free Dirac equation. In the following we will explicitly consider
only those contributions to $S_{\mu}$ in which both the quark and
antiquark interact with the field $G_{\mu}$.
Inserting these equations into the amplitude (\docLink{slacpub7204003.tcx}[amp]{10}), using the
Fourier decomposition of the free propagator $S_F$ and performing the
integration over the position of the photon vertex, one obtains
\beq\label{amp2}
S_{\mu} = i e \int {d^4p\over (2\pi)^4} \int d^4x
\bar{\psi}_u(x)\Gs(x) e^{ipx} {1\over \psu  m} \gamma_{\mu}
{1\over \ls + m} \int d^4y e^{ily} \Gs(y) \psi_v(y)\ ,
\eeq
where $l=qp$. Explicit expressions for $\bar{\psi}_u$
and $\psi_v$ have been obtained in a highenergy expansion for
an arbitrary soft colour field \cite{8}.
We are interested in deep inelastic scattering at small $x$, where
quark and antiquark have large momenta in the proton rest frame.
Hence, the propagators in Eq.~(\docLink{slacpub7204003.tcx}[amp2]{13}) can be treated in a
highenergy approximation. It is convenient to introduce lightcone
variables, e.g.,
\beq
l_+=l^0+l^3\ ,\ l_=l^0l^3\ ,\ \bl_={l_{\perp}^2+m^2\over l_+}\ .
\eeq
Here $\bl_{\mu}$ denotes the momentum vector whose ``''component
satisfies the mass shell condition. The propagators in Eq.~(\docLink{slacpub7204003.tcx}[amp2]{13})
can be written as
\bea
{1\over \ls +m} &=& \frac{\sum_r v_r(\bl)\bar{v}_r(\bl)}
{l_+(l_\bl_) + i\epsilon} + \frac{\gamma_+}{2 l_+}\ ,\label{prop1}\\
{1\over \psu  m} &=& \frac{\sum_s u_s(\bp) \bar{u}_s(\bp)}
{p_+(p_\bp_) + i\epsilon} + \frac{\gamma_+}{2 p_+}\ .\label{prop2}
\eea
To obtain the first term in a high energy expansion
of the scattering amplitude $S_{\mu}$ one can drop the terms
proportional to $\gamma_+$ in Eqs.~(\docLink{slacpub7204003.tcx}[prop1]{15}),(\docLink{slacpub7204003.tcx}[prop2]{16}).
In the high energy expansion the leading term for the wave functions
$\psi_v$ and $\bar{\psi}_u$ is the product of a nonabelian eikonal factor
and a plane wave solution of the Dirac equation \cite{11}.
The scattering amplitude (\docLink{slacpub7204003.tcx}[amp2]{13}) then takes the
form of a product of the photonquarkantiquark vertex, propagator factors
and two matrix elements of an effective gluon vertex between onshell
spinors. These matrix elements describe the elastic scattering between
the high energy (anti)quark and the proton, as discussed in Sect.~2.
They are evaluated in Appendix A.
For a soft gluon field inside the proton one finds for the matrix element
of the antiquark
\bea
T_{r,r'}(l,l') &=& \int d^4y\ \bar{v}_r(\bar{l}) e^{ily}\Gs(y) \psi_v(y)\nn\\
&\simeq& 2\pi i\ 2 l_+ \delta_{rr'} \delta(l_+l_+')
\left(\tilde{F}(l_{\perp}l'_{\perp}) 
(2\pi)^2 \delta^2(l_{\perp}l'_{\perp})\right)\ ,
\eea
where
\bea
\tilde{F}(l_{\perp}l'_{\perp}) &=&
\int_{y_{\perp}}\ e^{i(l_{\perp}l'_{\perp})y_{\perp}} F(y_{\perp})\ , \nn\\
F(y_{\perp}) &=& P \exp{\left({i\over 2} \int_{\infty}^{\infty} dy_+
G_(y_+,y_,y_{\perp})\right)}\ .
\eea
$F(y_{\perp})$ is the eikonal factor of the antiquark trajectory.
If the colour field of the proton is `soft'
the dependence on $y_$ can be neglected. For the corresponding matrix
element of the quark one finds
\bea
\int d^4y\ \bar{\psi}_u(y) \Gs(y) u_s(\bp) e^{ipy}
&\!\!\simeq\!\!& 2\pi i\ 2 p_+ \delta_{s's} \delta(p_+'p_+)
\left(\tilde{F}^{\dagger}(p'_{\perp}  p_{\perp}) 
(2\pi)^2 \delta^2(p'_{\perp}p_{\perp})\right)\nn\\
&\!\!=\!\!& T^{\dagger}_{s's} (p',p)\ .
\eea
Inserting these matrix elements into Eq.~(\docLink{slacpub7204003.tcx}[amp2]{13}) and adding the
contributions where one of the particles is not scattered one obtains,
after performing the $p_$integration,
\bea\label{amp3}
\bar{S}_{\mu} &=& {e\over \pi}\ q_+ \delta(q_+  p_+'  l_+')
\int d^2l_{\perp}\ {\alpha(1\alpha)\over N^2 + l_{\perp}^2}
\ \bar{u}_{s'}(\bar{p})\gamma_{\mu}v_{r'}(\bar{l})\nn\\
&&\qquad\qquad\qquad \left(\tilde{F}^{\dagger}(p'_{\perp}p_{\perp})
\tilde{F}(l_{\perp}l'_{\perp})  (2\pi)^4 \delta^2(p'_{\perp}p_{\perp})
\delta^2(l_{\perp}l'_{\perp})\right)\nn\\
&\equiv&  4\pi\ \delta(q_+  p'_+  l'_+)\ T_{\mu}\ ,
\eea
where
\beq
p'_+ = (1\alpha)\ q_+\ ,\ l'_+ = \alpha\ q_+\ ,\
N^2 = \alpha (1  \alpha) Q^2 + m^2\ .
\eeq
In the following it will turn out to be useful to restrict the integrand
in Eq.~(\docLink{slacpub7204003.tcx}[amp3]{20}) to configurations with $\alpha < 1/2$,
where the quark is faster than the antiquark.
The nonperturbative interaction with the proton is contained in $\tilde{F}$,
the Fourier transform of the eikonal factor. The product of eikonal factors
appearing in Eq.~(\docLink{slacpub7204003.tcx}[amp3]{20}) may be expressed as Fourier transform with
respect to the transverse distance between quark and antiquark. Introducing
\beq\label{wdef}
W_{x_{\perp}}(y_{\perp}) = F^{\dagger}(x_{\perp})F(x_{\perp}+y_{\perp})  1\ ,
\eeq
one has
\bea
\tilde{F}^{\dagger}(p'_{\perp}p_{\perp}) \tilde{F}(l_{\perp}l'_{\perp})
&=& \int_{x_{\perp},z_{\perp}}\ e^{i(p_{\perp}p'_{\perp})x_{\perp}}
e^{i(l_{\perp}  l'_{\perp}) z_{\perp}}
F^{\dagger}(x_{\perp}) F(z_{\perp})\nn\\
&=& \int_{x_{\perp},y_{\perp}}\ e^{i\Delta_{\perp}x_{\perp}}
e^{i(l_{\perp}l'_{\perp})y_{\perp}}\
\left(W_{x_{\perp}}(y_{\perp})+1\right)\nn\\
&=& \int_{x_{\perp}}\ e^{i \Delta_{\perp}x_{\perp}}\
\left(\tilde{W}_{x_{\perp}}(l_{\perp}l'_{\perp}) +
(2\pi)^2 \delta^2(l_{\perp}l'_{\perp})\right)\ .
\eea
Here $\Delta_{\perp}=p'_{\perp}+l'_{\perp}$ is the transverse momentum
transfer from the proton. We assume that the colour field of the proton is
smooth on a scale $\Lambda$, which implies that $\tilde{W}_{x_{\perp}}
(k_{\perp})$ falls off exponentially with increasing
$k_{\perp}^2/\Lambda^2$.
Note, that our treatment of the background field assumes the
factorization of soft gluonic physics associated with the proton state
and higher order $\alpha_S$corrections associated with the photon
wave function. This property is not proved in the present paper.
Nevertheless, the calculations of the following section illustrate for
the specific case of high$p_\perp$ jets in diffraction, how such
$\alpha_S$corrections can be implemented in the present approach.
Cross sections and structure functions can now be evaluated in the standard
manner (cf.~\cite{8}). The deep inelastic cross sections are given by
\beq\label{dsig}
d \sigma_{\mu\nu} = {2\pi\over q_+}\ T^*_{\mu}T_{\nu}\ d\PH^{(2)}\ ,
\eeq
where $d\PH^{(2)}$ is the phase space factor. Different projections with
respect to Lorentz and colour structure yield longitudinal and transverse,
diffractive and inclusive structure functions.
Consider first the inclusive longitudinal structure function $F_L$. We use
the conventional kinematic variables $\xi=x/\beta=x(Q^2+M^2)/Q^2$,
where $M^2$ is the invariant mass of the produced quarkantiquark pair and
$m=0$. A straightforward calculation, described in Appendix B,
yields
\bea
dF_L &=& {Q^2\over \pi e^2}\ d\sigma_L \nn\\
&=& {4\ Q^6\over (2\pi)^7 \beta}\ {d\xi\over \xi}\ d\alpha
(\alpha(1\alpha))^3\
\int_{x_{\perp}} \left\int d^2 l_{\perp}
\frac{\tilde{W}_{x_{\perp}}(l_{\perp}l'_{\perp})}{N^2 + l_{\perp}^2}
\right^2\ .
\eea
Note that only a single integration over transverse coordinates occurs.
This is a consequence of the $\delta$function induced by the phase space
integration over $\Delta_{\perp}$ in Eq.~(\docLink{slacpub7204003.tcx}[dsig]{24}). Since the form factor
$\tilde{W}_{y_{\perp}}(l_{\perp}l'_{\perp})$ falls off exponentially with
increasing momentum transfer, one can expand the integrand around
$l_{\perp}=l'_{\perp}$. This leads to the final result
\bea\label{fl}
F_L &=& {2\over \pi^3}\ \int_x^1{d\xi\over \xi}\
\int_0^{1/2} d\alpha\ \beta^2(1\beta)
\int_{x_{\perp}} \mid\partial_{\perp}W_{x_{\perp}}(0)\mid^2 \nn\\
&=& {1\over 6\pi^3}\ \int_{x_{\perp}}
\mid\partial_{\perp}W_{x_{\perp}}(0)\mid^2\ .
\eea
The inclusive transverse structure function $F_T$ can be evaluated in a
similar way. In the perturbative region $\alpha > \Lambda^2/Q^2$, where
the antiquark is sufficiently fast, one obtains
\bea\label{ft0}
dF_T &=& {Q^2\over \pi e^2}\ d\sigma_T = dF_2  dF_L \nn\\
&=& {1\over 8\pi^3}\ {d\xi\over \xi}\ d\alpha\
\frac{\alpha^2+(1  \alpha)^2}{\alpha(1\alpha)}\
\beta(\beta^2+(1\beta)^2)\
\int_{x_{\perp}} \mid\partial_{\perp}W_{x_{\perp}}(0)\mid^2\ .
\eea
The integration over $\alpha$ above the infrared cutoff
$\alpha_{min}=\Lambda^2/Q^2$ yields
\beq\label{infra}
\int_{\alpha_{min}}^{1/2}\ d\alpha\
\frac{\alpha^2+(1  \alpha)^2}{\alpha(1\alpha)}
\simeq \ln{{Q^2\over \Lambda^2}}  1\ .
\eeq
{}From the general expression for $F_T$ (cf. Appendix B) one can easily
see that there is also a nonperturbative contribution from the range
$\alpha \leq \Lambda^2/Q^2$ which, to leading order in $1/Q^2$, is given
by a function of $\beta$ only. We thus obtain the final result
\bea\label{ft}
F_T &=& {1\over 4\pi^3}\ \int_x^1{d\xi\over \xi}\
\left( \beta(\beta^2+(1\beta)^2)\
\left(\ln{{Q^2\over \Lambda^2}}  1\right)\
\int_{x_{\perp}} \mid\partial_{\perp}W_{x_\t}(0)\mid^2\
+ f(\beta)\right)\nn\\
&=& {1\over 6\pi^3}\ \ln{{Q^2\over \Lambda^2}}\
\int_{x_{\perp}} \mid\partial_{\perp}W_{x_\t}(0)\mid^2\ +\ C\ ,
\eea
where $C$ is an unknown constant. The lower region of the
$\alpha$integration, responsible for the constant $C$, is dominated by
$\alpha \sim \Lambda^2/Q^2 $ (see Eq.~(\docLink{slacpub7204005u2.tcx}[dft]{101}) of Appendix B).
In the small$x$ limit this means that in the dominant configurations
the `soft' antiquark will still be sufficiently fast for the eikonal
approximation to apply. Results for the production of
electronpositron pairs in an electromagnetic field, which are completely
analogous to Eqs.~(\docLink{slacpub7204003.tcx}[fl]{26}) and (\docLink{slacpub7204003.tcx}[ft]{29}), have previously been
obtained by Bjorken, Kogut and Soper \cite{10} using lightcone
quantization.
Let us finally evaluate the diffractive structure functions. In our approach
they are determined by the projection onto a colour singlet final state
which corresponds to the substitution
\beq
W_{x_{\perp}}(y_{\perp}) \rightarrow {1\over \sqrt{3}}\
\mbox{tr}[W_{x_{\perp}}(y_{\perp})]\ .
\eeq
There is no nonperturbative contribution to $F_L$ to leading order
in $1/Q^2$. Further, from the definition of the colour matrix $W_{x_\t}(y_\t)$
it is clear that
\beq
\partial_{\perp}\mbox{tr}[W_{y_{\perp}}(0)] = 0\ .
\eeq
This immediately implies
\beq
F_L^D(x,Q^2,\xi) = 0\ .
\eeq
Like the transverse inclusive structure function, the transverse diffractive
structure function also has a nonperturbative contribution. The result can
be written in the form
(cf. Appendix B)
\beq\label{f2d1}
F_2^D(x,Q^2,\xi) = {\beta\over \xi} \bar{F}(\beta)\ ,
\eeq
with
\beq\label{f2d2}
\bar{F}(\beta) = {4\over 3(2\pi)^7}\int d\rho \rho^3\ \int_{x_{\perp}}
\left\int d^2k_{\perp}\ \frac{k_{\perp} + e_{\perp}\rho\sqrt{1\beta}}
{\beta\rho^2 + (k_{\perp}+e_{\perp}\rho\sqrt{1\beta})^2}
\mbox{tr}[\tilde{W}_{x_{\perp}}(k_{\perp})]\right^2\ .
\eeq
Here $e_\t$ is an arbitrary unit vector whose direction is irrelevant
due to rotational invariance. The function $\bar{F}(\beta)$ approaches a
finite limit as $\beta \rightarrow 0$, as well as $\beta \rightarrow 1$.
The results of this section essentially coincide with \cite{8}. There
is, however, a difference concerning the $\beta$spectrum of the
diffractive contribution. In the spacetime picture of \cite{8} the
change of direction of the outgoing quarks by their interaction with
the proton was not treated sufficiently accurately. This oversimplification
led to the conclusion that the $\beta$spectrum depends on the proton
structure only via two unknown constants. The present calculation shows
that the $\beta$spectrum is entirely nonperturbative and can only be
given in terms of an integral which depends on the proton field.
The results for $F_2$ and $F_L$ correspond to the
perturbative contribution of photongluon fusion with a gluon density
$x g(x) =$ const., i.e. a classical bremsstrahl spectrum. The
diffractive structure function $F_2^D$ is due to purely nonperturbative
contributions, where either the quark or the antiquark are soft.
This is analogous to modern views of the alignedjet model
\cite{15,16,17,18}.
We also note that the slope of $F_2$ in $x$
is larger by one unit than the slope
of $F_2^D$ in $\xi$. This property of the structure functions has
previously been discussed in \cite{6,8,19}.
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