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\newcommand{\f}[2]{\frac{#1}{#2}}
\newcommand{\dd}{\displaystyle}
\newcommand{\nn}{\nonumber}
\def\a{\alpha}
\def\ap{\alpha^\prime}
\def\kkk{{\bf k}_\perp}
\def\ppp{{\bf p}^\prime_\perp}
\def\d{\partial}
\def\bk{\!\!\!\!}
\def\eps{\epsilon}
\def\pts{(\phi^3)_6}
\def\xb{{\bar x}}
\def\p{\phi}
\def\l{\f{\lambda^2}{(4\pi)^3}}
\def\mq{\left(\f{4\pi\mu^2}{Q^2}\right)^\epsilon}
\def\mt{\left(\f{4\pi\mu^2}{t}\right)^\epsilon}
\def\cc{\f{\lambda^2}{t^2}x} 
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\begin{document}
\thispagestyle{empty}
\begin{flushright}
UPRF-97-012
\end{flushright}
\vskip 1cm
\begin{center}
\begin{large}
{\bf Semi-Inclusive DIS: an explicit calculation in the
Target Fragmentation Region}\footnote{To appear in the Proceedings of
{\em QCD Euroconference 97},
Montpellier (France) 3-9 July 1997} \\
\end{large}
\vskip 1cm
{\bf M. Grazzini}\\

\vskip .5 cm

{\em Dipartimento di Fisica, Universit\`a di Parma\\ 
and I.N.F.N., Gruppo Collegato di Parma,\\
Viale delle Scienze, I-43100 Parma, Italy}\\
\end{center}

\vskip 2cm

 \begin{abstract}
 I present a calculation of the
 one particle deep inelastic cross section in the target fragmentation region
 in $\pts$.
 The renormalized cross
 section gets a large logarithmic correction whose
 coefficient is precisely the scalar DGLAP kernel.
 The result is found to be consistent
 with an extended factorization hypothesis and with infrared power counting.
 \end{abstract}


\newpage       

\section{INTRODUCTION}

Semi-inclusive
deep inelastic scattering has
been successfully studied
in the framework of perturbative QCD \cite{aemp},
at least in the case in which the transverse momentum 
of the produced hadron is of order of the hard scale $Q^2$.

In the last few years a new attention
has been devoted to this process
in the limit in which the transverse momentum, or
equivalently the momentum
transfer $t=-(p-p^\prime)^2$ between the incoming and outgoing hadron,
is very small with respect to $Q^2$. In this limit
the process is dominated by the target fragmentation
mechanism and, for this reason,
a new approach in terms of the so called {\em fracture functions} has been
proposed \cite{tv}, and developed \cite{grau,arg}.


In this talk I present a calculation \cite{gr1}
of the semi-inclusive cross section in the target fragmentation region ($t\ll Q^2$) in $\pts$ model field theory.
This model
has revealed
itself
a nice laboratory to study
strong interactions at short distances, since it is asymptotically free
and it has a much milder structure of infrared singularities with
respect to QCD \cite{scalar,kub}. In fact there are no soft but only
collinear singularities
and so factorization becomes simpler to deal with \cite{css}.
%
%The calculation
%I present here
%shows that the cross section gets a large
%logarithmic correction as one expects in a two scale regime.
%Moreover there is only a limited class of diagrams
%which give contribution at leading power in
%$t/Q^2$.  
%By using this fact and infrared power counting
%\cite{css,sterman}
%I will pursue the result of this calculation at all orders
%in perturbation theory.
%($Q^2=-q^2\gg
%\Lambda_{QCD}^2$)
%in which $t=-(p-p^\prime)^2 \sim Q^2$ \cite{aemp}.
%\cite{tv,grau}
%\cite{tv}.
%\cite{diff}

\section{DIS IN $\pts$}

I will start recalling some results one gets for inclusive DIS.
%As a first step I will focus on
%the deep inelastic inclusive cross section.
Let us consider the process
$p+J(q)\to X$ where $J=\f{1}{2} \p^2$. 
%is a scalar operator which plays the
%role of the electromagnetic current.
We define as usual
\begin{equation}
Q^2=-q^2~~~~~~~~~~~x=\f{Q^2}{2pq}.
\end{equation}
The structure function can be defined as
%is
\begin{equation}
W(x,Q^2)=\f{Q^2}{2\pi} \int d^6y e^{iqy} <\! p|J(y)J(0)|p\! >.
\end{equation}
It is easy
to 
calculate the parton-current cross section $w(x,Q^2)$
%at $p^2=0$
in dimensional regularization ($D=6-2\eps$).
At
lowest order we get
%Born level we have
(see Fig. \ref{dis0})
%%====================================
\begin{figure}[htb]
\begin{center}
\begin{tabular}{c}
\epsfxsize=4truecm
\epsffile{dis0.eps}\\
\end{tabular}
\end{center}
\caption{\label{dis0} {\small Lowest order contribution to the deep inelastic
cross section}}
\end{figure}
%%====================================
\begin{equation}
w_0(x,Q^2)=\f{Q^2}{2\pi} 2\pi \delta((p+q)^2)=\delta(1-x).
\end{equation}
The first order corrections are shown in Fig. \ref{dis}.
%and give
%\begin{equation}
%w_{1a}(x,Q^2)=\l x(1-x)
%\left( -\f{1}{\epsilon}\right)\left(\f{4\pi\mu^2}{Q^2}\right)^\epsilon
%\end{equation}
%\begin{equation}
%\label{eq2}
%w_{1b}(x,Q^2)=2\times\f{1}{2}\l \f{1}{\epsilon}\mq\delta(1-x)
%\end{equation}
%\begin{equation}
%w_{1c}(x,Q^2)=-\f{1}{12}\l \f{1}{\epsilon}\mq\delta(1-x)
%\end{equation}
%The factor $2$ in eq. (\ref{eq2}) takes into account the
%contribution from the symmetric diagram.
%The diagram in Fig. \ref{dis} (d) gives only a finite correction.
External self energies
are not taken into account
since we work at $p^2=0$.
%\vspace{-5mm}
%Usually one has not to worry about current renormalization,
%because the electromagnetic current is not renormalized by
%strong interactions. However in our simplified model the operator
%We must take into account the renormalization of the operator
%$J=\f{1}{2} \p^2$
%is 
%must be
%renormalized
%by the interaction.
%The renormalization constant $Z_J$ is defined by
%\begin{equation}
%J=Z_J^{-1} J_0
%\end{equation}
%We get, in the MS scheme
%\begin{equation}
%Z_J=1+\f{5}{12} \l\f{1}{\epsilon}
%\end{equation}
%As a matter of fact, being the renormalization scale dependent,
%the coupling to the current
%becomes scale dependent, so it turns out more convenient to define a $Q^2$
%dependent renormalization
In order to take into account the renormalization of the operator $J$
one has to multiply the total contribution by $Z_J^{-2}(Q^2)$ where
\cite{gr1}                                                                   
\begin{equation}
Z_J(Q^2)=1+\f{5}{12} \l\f{1}{\epsilon}\mq.
\end{equation}
%%====================================
\begin{figure}[htb]
\begin{center}
\begin{tabular}{c}
\epsfxsize=10truecm
\epsffile{dis.eps}\\
\end{tabular}
\end{center}
\caption{\label{dis} {\small One loop corrections to the
deep inelastic cross section}}
\end{figure}
%%====================================
%which takes into account this effect. 
%Therefore the cross section is obtained summing up all the contributions
%and multiplying by $Z_J^{-2}(Q^2)$.
Up to finite corrections we get
\begin{equation}
\label{ris}
w(x,Q^2) =\delta(1-x)
+\l P(x)\left(-\f{1}{\eps}\right)\mq
\end{equation}
where
\begin{equation} 
P(x)=x(1-x)-\f{1}{12}\delta(1-x)
\end{equation}
is the DGLAP kernel for our model.
%We see that this result has the same structure one gets in QCD.
The contribution to the structure function is obtained as a convolution
with a bare parton density $f_0(x)$
\begin{equation}
W(x,Q^2)=\int_x^1 \f{du}{u}f_0(u) w(x/u,Q^2).
\end{equation}
%The collinear divergence in $w(x,Q^2)$
%can be
%is
%absorbed as usual by
%defining a $Q^2$ dependent scalar parton density $f(x,Q^2)$
%by
%means of the equation
%\begin{eqnarray}
%\label{pd}
%f_0(x)\bk &=&\bk \int_x^1 \f{du}{u} \Big[\delta(1-u)\nn\\
%&+&\bk\l P(u)\f{1}{\eps}\mq\Big]
%f(x/u,Q^2)
%\end{eqnarray}
The collinear divergence in $w(x,Q^2)$ can
be lumped as usual in a $Q^2$ dependent parton density
by
means of the equation
\begin{equation}
\label{pd}
f_0(x) =\int_x^1 \f{du}{u} \Big[\delta(1-u)
+\l P(u)\f{1}{\eps}\mq\Big]
f(x/u,Q^2).
\end{equation}
The scale dependent parton density $f(x,Q^2)$
obeys the DGLAP evolution equation
%This renormalized scalar parton density
%obeys the DGLAP evolution equation
\begin{equation}
Q^2\f{\d}{\d Q^2} f(x,Q^2)=\int_x^1 \f{du}{u} P(u) f(x/u,Q^2).
\end{equation}
For the process $J(q)\to p+X$ with
$q$ timelike a fragmentation function $d(x,Q^2)$
can be defined in the same way
and it obeys the same DGLAP
evolution equation.
%Thanks to the Gribov-Lipatov reciprocity relation \cite{gl}
At one loop level the timelike DGLAP kernel is the
same as in the spacelike case, but this relation is broken at
two loops \cite{kub}.

\section{SEMI-INCLUSIVE DIS}

In the semi-inclusive case a new structure function can be defined as
\begin{equation}
W(p,p^\prime,q)=\f{Q^2}{2\pi} \sum_X\int d^6x e^{iqx} 
<\! p|J(x)|p^\prime X\! >
<\! X p^\prime|J(0)|p\! >.
\end{equation}
We 
%will 
have calculated \cite{gr1} the partonic cross section in the limit
$t\ll Q^2$
at leading power, by keeping only divergent terms and possible $\log Q^2/t$
contributions.
%It turns out that
%the approximation (\ref{a}) selects a
%special class of diagrams, those in which the produced particle 
%is radiated by the incoming one.
As expected, the cross section is dominated by target fragmentation.
%As a matter of fact at the lowest order in $\lambda$ there is no contribution in the region
%$t\ll Q^2$.
The first diagram which give contribution
is the one in Fig. \ref{sdis0}.

%%====================================
\begin{figure}[htb]
\begin{center}
\begin{tabular}{c}
\epsfxsize=4.5truecm
\epsffile{sdis0.eps}\\
\end{tabular}
\end{center}
\caption{\label{sdis0} {\small Leading order contribution to one particle
deep inelastic cross section in the region $t\ll Q^2$}}
\end{figure}
%%====================================

It gives
\begin{equation}
w_1(x,z,t,Q^2)=\f{\lambda_0^2}{t^2} x \delta(1-x-z)
\end{equation}
where $\lambda_0$ is the bare coupling constant
and
\begin{equation}
z=\f{p^\prime q}{pq}.
\end{equation}
It turns out that the relevant one loop corrections
come from the diagrams in Fig. \ref{sdis1}.
The other diagrams in fact either give
finite contributions or are suppressed by powers of $t/Q^2$.

%%====================================
\begin{figure}[htb]
\begin{center}
\begin{tabular}{c}
\epsfxsize=9truecm
\epsffile{sdis1.eps}\\
\end{tabular}
\end{center}
\caption{\label{sdis1} {\small One loop leading contributions to the one
particle deep inelastic cross section}}
\end{figure}
%%====================================

The details of the calculation are presented in Ref. \cite{gr1}.
Summing up all the contributions, multiplying by $Z_J^{-2}(Q^2)$,
introducing the running coupling constant
%\begin{eqnarray}
%\lambda^2(t)\bk&=&\bk\f{\lambda^2}{1+\beta_0 \lambda^2 \log t/4\pi\mu^2}\nn\\
%&\simeq&\bk\lambda^2\left(1-\f{3}{4}\f{1}{(4\pi)^3}\lambda^2\log\f{t}{4\pi\mu^2}\right)
%\end{eqnarray}
%and using
%\begin{equation}
%\lambda^2_0=\lambda^2 \left(1-\f{3}{4}\l\f{1}{\eps}\right)
%&=&\bk\lambda^2(t) \left(1-\f{3}{4}\l\f{1}{\eps}\mt\right) 
%\end{equation}
we finally get
\begin{align}
\label{sris}
w(x,z,t,Q^2)&=\f{\lambda^2(t)}{t^2}x
\Big(\delta(1-x-z)+
\l\f{1}{\eps}\mt\Big(\f{1}{6}\delta(1-x-z)\nn\\
&-\f{1-x-z}{(x+z)^2}-\f{1-x-z}{(1-x)^2}\Big)
+\f{1}{x} P\left(\f{x}{1-z}\right)\l\log\f{Q^2}{t}\Big).
\end{align}

The structure function is obtained as a convolution with the bare
parton density and fragmentation function.
By using 
eq. (\ref{pd})
and the corresponding definition for
the fragmentation function
we get
%eventually we find
\begin{align}
\label{final}
W(x,z,t,Q^2)&=
\int_{x+z}^1 \f{du}{u}\int_{\f{z}{u-x}}^1\f{dv}{v^4}
f(u,t)
\f{\lambda^2(t)}{t^2}\f{v^2}{u^2}
\Big[\delta\left(1-\f{x/u}{1-z/uv}\right)\nn\\
&+\l P\left(\f{x/u}{1-z/uv}\right)\log\f{Q^2}{t}\Big] d(v,t)
\end{align}
where again only leading $\log Q^2/t$  terms have been considered
and the integration limits are derived using
momentum conservation.

From eq. (\ref{final}) it appears that
the renormalized hard cross section gets a
large $\log Q^2/t $ correction whose coefficient is
the scalar DGLAP kernel.
Such correction, if not properly resummed,
can spoil perturbative calculations in the region $t\ll Q^2$.

Eq. (\ref{final}) shows a new singularity,
which corresponds to the configuration
in which $p^\prime$ becomes parallel to $p$.
%As pointed out
%in Ref.\cite{grau}, 
When we integrate over $t$,
in order to absorb such singularity,
%such singularity can't be absorbed in ordinary parton densities
%and fragmentation functions and 
the introduction
of a new phenomenological distribution, the fracture function \cite{tv}
becomes necessary \cite{grau}. 
Eq. (\ref{final}) can also be rewritten in the following form
\begin{align}
\label{eqjet}
W(x,z,t,Q^2)&=\f{\lambda^2(t)}{zt^2}
\int_x^{1-z}\f{dr}{r}\int_{z+r}^1 \f{du}{u(u-r)} 
{\hat P}
\left(\f{r}{u}\right) f(u,t)\Big[\delta\left(1-\f{x}{r}\right)\nn\\
&+\l P\left(\f{x}{r}\right)\log\f{Q^2}{t}\Big] d\left(\f{z}{u-r},t\right)
\end{align}
where we have defined the A-P real scalar vertex ${\hat P}(x)=x(1-x)$.
The function
\begin{equation}
E^{(1)}(x,Q^2/Q^2_0)=\delta(1-x)+\l P(x) \log \f{Q^2}{Q^2_0}
\end{equation}
appears to be the first order approximation of the evolution kernel $E(x,Q^2/Q^2_0)$
which resums the leading logarithmic series \cite{jet}.
This fact suggests that
an interpretation of eq. (\ref{eqjet}) can be given in terms of
Jet Calculus \cite{jet}.

\section{FACTORIZATION IN TERMS OF CUT VERTICES}

Cut vertices are a generalization of matrix elements of local operators
originally
proposed by Mueller in Ref.\cite{mueller}.
They can be very useful
to give
an interpretation of the results obtained in the previous sections.

Let us go back to Sect.2 and
set $p^2<0$ with $p=(p_+,{\bf 0},p_-)$.
If we choose a frame in which $p_+\gg p_-$ we can write 
for the parton-current cross section \cite{mueller}
\begin{equation}
w(p,q)=\int \f{du}{u} v(p^2,u)C(x/u,Q^2)
\end{equation} 
where $v(p^2,x)$
%\begin{equation}
%\label{cut}
%v(p^2,x)=\int V(p,k)x\delta\left(x-\f{k_+}{p_+}\right)\f{d^6k}{(2\pi)^6}
%\end{equation}
is a spacelike cut vertex
with $C(x,Q^2)$ the corresponding coefficient function.
%In (\ref{cut})
%$V(p,k)$ is the discontinuity of the four point amplitude in the channel
%$(p-k)^2$ and the integration over $k$ is properly renormalized.

%The results of the previous sections can be easily recast in the
%form of cut vertex expansion.
If we define
\begin{equation}
v(x,\eps)=\delta(1-x)+\l P(x)\left(-\f{1}{\eps}\right)
\end{equation}
\begin{equation}
C(x,Q^2)=\delta(1-x)+\l P(x) \log\f{Q^2}{4\pi\mu^2}
\end{equation}
we can write eq. (\ref{ris}) in the form
\begin{equation}
w(x,Q^2)=\int_x^1 \f{du}{u} v(u,\eps)C(x/u,Q^2).
\end{equation}
Here $v(x,\eps)$ is a spacelike cut vertex
defined at $p^2=0$ whose mass divergence is regularized dimensionally.

A similar interpretation can be given of eq. (\ref{sris}).
We define
\begin{equation}
\xb=\f{x}{1-z} 
\end{equation}
and
\begin{align}
v(\xb,z,t,\eps)&=\f{\lambda^2(t)}{t^2}\Big[\delta(1-\xb)+
\l\f{1}{\eps}\mt
\Big(\hs\f{1}{6}\delta(1-\xb)\nn\\
&+\f{(1-z)^2\xb(1-\xb)}{(\xb(1-z)+z)^2}
+\f{(1-z)^2\xb(1-\xb)}{(1-\xb(1-z))^2}\hs\Big)
+P(\xb)\l\log\f{4\pi\mu^2}{t}\Big]
\end{align}
as a {\em generalized cut vertex} \cite{new} which contains all the leading
mass singularities of the cross section.
We can write up to ${\cal O}(t/Q^2)$ corrections
%(see Fig. \ref{nuovo})
%%====================================
%\begin{figure}[htb]
%\begin{center}
%\vspace{-10mm}
%\begin{tabular}{c}
%\epsfxsize=6truecm
%\epsffile{nuovo.eps}\\
%\end{tabular}
%\end{center}
%\vspace{-10mm}
%\caption{\label{nuovo} Factorization of the semi-inclusive structure function}
%\end{figure}
%%====================================
\begin{equation}
\label{fact}
w(\xb,z,t,Q^2,\eps)\!=\!\!\int_\xb^1 \!\f{du}{u}
v(u,z,t,\eps)C(\xb/u,Q^2)
\end{equation}
where the coefficient function is the same which occurs in
inclusive DIS.

The validity of this factorization relies on the fact that
diagrams with more than two legs connecting the soft
to the hard part are suppressed
by powers of $t/Q^2$ \cite{gr1}.
This is a
%general
result which can be
%pursued
generalized
at all orders
by using 
%infrared power counting
the ideas of Ref. \cite{css,sterman}.

%%====================================
\begin{figure}[htb]
\begin{center}
\begin{tabular}{c}
\epsfxsize=6truecm
\epsffile{leading.eps}\\
\end{tabular}
\end{center}
\caption{\label{leading} {\small Leading contributions to the semi-inclusive
structure function in $\pts$}}
\end{figure}
%%====================================

The large $Q^2$ limit
of the semi-inclusive cross section can be studied by
looking at the singularities in the limit $p^2$, $p^{\prime 2}$, $t\to 0$.
The strength of such
singularities can be predicted
by using infrared power counting \cite{new}.
Starting from a given diagram, its {\em reduced} form in the large $Q$ limit
is constructed by
simply contracting to a point all the lines whose momenta are not on shell.
In $\pts$ the general leading diagrams in the large $Q^2$ limit
for the process under study involve a jet subdiagram $J$,
composed by on shell lines collinear to the incoming particle,
from which the detected particle emerges in the forward
direction and a hard subgraph $H$ in which momenta
of order $Q$ circulate, which is connected to the jet by the minimum number
of collinear lines.
Additional lines connecting $J$ to $H$
as well as soft lines connecting them
are suppressed by power counting.
So one can say that in $\pts$ the leading diagrams are of the form
depicted in Fig. \ref{leading} and this means that in this model
eq. (\ref{fact}) holds at all orders \cite{new}.

\section{SUMMARY}

In this talk I have presented
%the results of 
an explicit calculation of
the one particle
%semi-inclusive
deep inelastic cross section
in the target fragmentation region 
within $\pts$ model field theory.
%We have shown that 
The renormalized hard cross
section gets a large $\log Q^2/t$
correction as expected in a two scale regime and
%Furthermore
%we have found that 
the coefficient driving this logarithmic correction is precisely
the scalar DGLAP kernel.
%This result suggests that the $Q^2$ dependence of the cross section
%in such processes at fixed $z$ and $t$
%is driven by the same anomalous dimension which controls the inclusive DIS,
%as proposed in Ref. \cite{new}, and in
%in the context of diffraction in Ref.\cite{diff}.

Furthermore the result obtained fits
within an extended factorization hypothesis \cite{new}.
In fact
the partonic semi-inclusive cross section factorizes into a convolution of a
new object, a generalized cut vertex $v(p,p^\prime,\xb)$ \cite{new},
with four rather than two external legs,
and a coefficient function $C(\xb,Q^2)$ which
is the same as the one of inclusive DIS.
Infrared power counting applied to this process allows to say
that this last result holds in $\pts$ at all orders.

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D. Graudenz, Nucl. Phys. {B432} (1994) 351.
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D. De Florian and R. Sassot, Phys. Rev. D56 (1997) 426.
%\bibitem{diff}
%A. Berera and D.E. Soper, Phys. Rev. {D53} (1996) 6162;
%Z. Kunszt and W.J. Stirling, .
\bibitem{gr1}
M. Grazzini, UPRF-97-09, .
\bibitem{scalar}
J.C. Taylor, Phys. Lett. {B73} (1978) 85;
%\bibitem{kazama}
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Phys. Rev. {D19} (1979) 3111;
%\bibitem{bfk}
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\bibitem{kub}
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\bibitem{new}
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%\bibitem{graz}
%M. Grazzini, to appear.
%\bibitem{trenta}
%L. Trentadue, these proceedings
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\end{thebibliography}
%\vspace*{.4cm}
%\begin{large}
%\noindent
%{\bf Discussions}\\
%\end{large}

%\noindent
%K. Chetyrkin, Max Planck Institute,\\
%\begin{em}
%Can you say that the results and conclusions made by you on the basis of specific calculations can be generalized to all orders
%within the $\pts$ scalar model ?\\
%\end{em}

%\noindent
%M. Grazzini\\
%\begin{em}
%Yes, by using the concept of infrared power counting one can show, within $\pts$, that the leading contributions to the semi-inclusive structure function
%are given by decompositions with only two collinear lines connecting the hard to the jet subdiagram and soft connections are suppressed.
%One can say that factorization works in this case as
%it works in inclusive DIS.\\
%\end{em}

%\noindent
%L.N. Lipatov, S. Petersburg N. P. Institute,\\
%\begin{em}
%As it is known, the splitting kernels for parton distribution functions and for fragmentation functions coincide only in LLA. Because fracture functions are generalizations of above functions, what you expect for splitting kernels in evolution equations for fracture functions in the NLLA ?\\
%\end{em}

%\noindent
%M. Grazzini\\
%\begin{em}
%I believe that the anomalous dimension driving the inclusive
%and semi-inclusive structure function in the target fragmentation region are the same at all
%orders. Of course, by going in the timelike region I expect anomalous
%dimensions to be different beyond one loop.\\
%\end{em}
\end{document}


