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MAD/PH/797 \\
October 1993\\
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{\Large \bf Initial State Showering in Resolved Photon
Interactions}\footnote{Talk held at the {\it 23rd International Symposium on
Multiparticle Dynamics}, Aspen, Colo., Sep. 1993}\\
\vspace{5mm}
Manuel Drees\footnote{Heisenberg fellow}\\
{\em Physics Department, University of Wisconsin, 1150 University Ave,
Madison, WI 53706, USA}
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\begin{abstract}
\noindent
After a brief review of recent data that confirm qualitative and
quantitative predictions for resolved photon processes, a possible problem
with the implementation of initial state showering in existing Monte Carlo
event generators is pointed out. It is argued that this is responsible for
the rather poor description of the jet rapidity distribution measured by the
H1 collaboration at HERA.
\end{abstract}
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``Resolved photon interactions" are hard scattering processes (with momentum
transfer $Q^2 \gg \Lambda^2_{\rm QCD}$) involving the partonic structure
of the photon, i.e. the quarks and gluons ``in" the photon \cite{1}. The
existence of such processes was first pointed out almost 15 years ago
\cite{2}. These early papers already emphasized that these processes
contribute at the {\em same} order in coupling constants as the
corresponding ``direct" processes do, where the photon couples directly
to one of the partons participating in the hard scattering; the crucial
observation \cite{1} here is that the parton distribution functions in
the photon $\vec{q}^{\gamma} \equiv (q_i^{\gamma}, G^{\gamma})$ are
${\cal O}(\alpha_{em}/\alpha_s)$. Detailed predictions for resolved photon
contributions to $\gamma p$ \cite{3} and $\gamma \gamma$ scattering \cite{4}
followed. However, only in the last two years have these predictions [2--4]
actually been confronted with data. In addition to the seminal AMY study
\cite{5} which for the first time established the existence of resolved
photon contributions to jet production in $\gamma \gamma$ collisions, there
are now two published analyses from TOPAZ \cite{6} and ALEPH \cite{7} reaching
similar conclusions. Moreover, both H1 \cite{8} and ZEUS \cite{9} confirmed
predictions \cite{3} of a very large resolved photon conribution at HERA.

In both $\gamma \gamma$ and $\gamma p$ collisions the resolved photon
contribution is characterized by a $p_T$ spectrum which is harder than the
exponential spectrum of soft interactions (since this contribution is based on
hard partonic subprocesses), but softer than that of the direct process (since
partons ``in" the photon carry less energy than the photon itself). An even
more tell--tale sign of resolved photon contributions is the presence of
spectator or remnant jets from the photon, leading to large hadronic energy
flow around the direction of the electron beam. This has been seen directly
both in $\gamma p$ \cite{8} and $\gamma \gamma$ \cite{6} collisions.

These properties are characteristic not only for resolved photon process, but
for any (hard) reaction involving hadrons in the initial state. One might
therefore be led to the conclusion that resolved photons can be treated just
like ordinary hadrons; this is indeed the case in the default versions of the
HERWIG and PYTHIA photoproduction event generators \cite{10}. However, this
treatment ignores the important fact that the very existence of a nonvanishing
parton content of the photon is due to the \ver\ vertex, which allows a
$\gamma \rightarrow \bar{q} q$ splitting, see fig. 1. If the relative
transverse momentum of the quark and antiquark is small, of order
$\Lambda_{\rm QCD}$ or less, the $\bar{q} q$ system can indeed be expected to
evolve more or less like any real hadron (so--called ``hadronic component" of
the photon structure function). However, the pointlike nature of the \ver\
vertex allows the spectator quark to be produced with sizable transverse
momentum. This ``intrinsic $k_T$" distribution follows \cite{11} a $k_T^{-2}$
behaviour for $k_T^2 \gg \Lambda^2_{\rm QCD}$, in contrast to the
$\exp(-k_T^2/k_0^2)$ behaviour of real hadrons. Indeed, it is this
distribution which, after integration over $k_T$, leads to the (approximate)
logarithmic growth of $\vec{q}^{\gamma}$ with scale $Q^2$, as opposed to the
approximately constant (scaling) behaviour of hadronic parton densities.

One consequence of this is that the spectator jet from the photon can carry
substantial transverse momentum, bending it away from the beam pipe.
This effect has been incorporated as an option in standard PYTHIA \cite{10}.
Notice that (in leading logarithmic approximation) the maximal allowed
$k_T^2$ should be smaller than or of the order of the ``hardness" $Q^2$ of the
process, e.g. the $p_T^2$ of the produced high$-p_T$ jets. This scale is
rather low for existing $\gamma \gamma \rightarrow$ jets data, which might
explain why introducing this harder $k_T$ spectrum does not seem to have much
impact on the simulation of these data \cite{13}. At HERA the importance of
this effect might be washed out somewhat by the boost from the $\gamma p$ cms
to the lab frame, which leads to a large opening angle of the spectator jet;
this might explain why PYTHIA (which includes this effect) and HERWIG (which
does not) make very similar predictions \cite{14}.

However, the pointlike nature of the \ver\ vertex has another consequence:
It allows quarks ``in" the photon to be produced with sizable (spacelike)
virtuality $|t|$. These two effects are actually closely related. To see this,
consider the simple $\gamma q$ scattering diagram of fig. 1. Let $p_1 =
\frac{1}{2} \sqrt{s} (1,0,0,1)$ and $p_4 = \frac{1}{2} \sqrt{s} (1,0,0,-1)$
be the 4--momenta of the initial photon and quark, respectively. We want to
interpret this diagram as a resolved photon process, i.e. the quark 3 is
supposed to be ``in" the photon. The scaled momentum \xg\ of this quark is
defined by \be \label{e1}
\left( p_3 + p_4 \right)^2 \equiv \xg \left( p_1 + p_4 \right)^2
\equiv \xg s.
\ee
Introduce the virtualities $t_2 \equiv p_2^2 \ (\geq 0)$ and
$t_3 \equiv p_3^2 \ (\leq 0)$, and let $k_T$ be the $y-$component of $p_3$
(with $p_{3,x} = p_{2,x} = 0$). Then
\beq \label{e2}
t_3 &= \frac{1}{2} \left\{ t_2 - s (1-\xg) + \sqrt{ \left[ t_2 - s \left(1-\xg
\right) \right]^2 - 4 s \left(t_2 \xg + k_T^2 \right) } \right\} \nonumber \\
 &\simeq - \frac {t_2 \xg + k_T^2} {1-\xg},
\eeq
where the second, approximate equality holds for $s \left(1-\xg \right)
\gg t_2, k_T^2$. Indeed, it is this relation which implies $\frac{d \sigma}
{d k_T^2} \propto 1/k_T^2$, since $\frac {d \sigma} {d t_3} \propto
1/t_3$ if $|t_3| \ll \xg s$.

The main significance of eq.(\ref{e2}) is that it serves as an IR cut--off
for the initial state showering process. In order to see this it is helpful
to briefly describe the way initial state showering is implemented in HERWIG
and PYTHIA, the so--called ``backward shower" algorithm \cite{15,16}. One
starts from the hard $2 \rightarrow 2$ scattering process ($ 3 + 4 \rightarrow
5 + 6$ in fig. 1). In this step one defines the transverse momenta and
rapidities of the outgoing partons (in the cms of the incoming partons),
which determine the $x$ values of the incoming partons. The
virtualities $|t|$ of the incoming partons are then chosen randomly between
``zero" and some upper limit $|t_{\rm max}| \simeq 4 p_T^2$, where $p_T$ is
the transverse momentum of partons 5 and 6; the probability distribution for
$|t|$ is given by the QCD evolution equations.\footnote{The two algorithms
differ somewhat at this point. Sj\"ostrand \cite{16} uses the parametrization
of QCD scaling violations, i.e. the $|t|$ dependence of the parton
distribution functions, in the definition of the branching probabilities.
Gottschalk \cite{15} uses branching probabilities which {\em only} depend on
the QCD splitting functions, and weights the event with the parton density at
the {\em input} scale $|t_{\rm min}|$; this automatically includes certain
NLO corrections at least in the simple case of Drell--Yan production.} If
$|t|$ comes out smaller than some lower limit $|t_{\rm min}|$ the shower is
terminated. Note that each branching (radiation) {\em decreases} the
virtuality $|t|$ and {\em increases} the Bjorken$-x$ of the incoming parton.

In HERWIG and the PYTHIA version used by HERA experiments to date a constant
$|t_{\rm min}| = {\cal O}(1)$ GeV$^2$ is chosen both for real hadrons and for
resolved photons. The point I wish to make is that this may be {\em
inconsistent} with the kinematic limit (\ref{e2}). In other words, for the
case of resolved photons the initial state shower should be terminated if
$|t|$ is less than $|t_{\rm min}| \simeq 1$ GeV$^2$ {\em or} less than $|t_3|$
of eq.({\ref{e2}). Notice that $k_T$ and $t_2$ have to be knwon {\em before}
the initial state shower can be constructed. A possible algorithm for the
implementation of initial state showering in resolved photon events might
therefore be:
\begin{itemize}
\item Choose $p_T$ and the Bjorken$-x$ variables of the hard $2 \rightarrow 2$
scattering process as usual.
\item Choose $k_T$, logarithmically equally distributed between some IR
cut--off and $k^2_{T,{\rm max}} = \min \left[ \frac{s}{4} (1-\xg),
|t_{\rm max}| \left( 1 - \frac {|t_{\rm max}|}{s (1-\xg)} \right) \right]$;
the upper bound follows from the requirement $|t_3| \leq |t_{\rm max}|$, which
is neccessary to interpret the event as a LO resolved (rather than NLO direct)
contribution. (Typically $|t_{\rm max}| = 4 p_T^2.$)
\item If one wants to start a final state shower from the ``spectator parton"
2 (which after all originates from a perturbative process here) one should
choose a value of $t_2$ according to standard final state shower algorithms
\cite{20}, with $t_2 \leq \frac{1}{\xg} \left[ |t_{\rm max}| (1-\xg) - k_T^2
\right]$ which again follows from $|t_3| \leq |t_{\rm max}|$.
\item Now $t_3$ is determined by eq.(\ref{e2}) and the initial state shower
can be implemented in the standard way, but with lower cut--off given by
$|t_3|$ if this is bigger than the IR cut--off $|t_{\rm min}|$.
\end{itemize}

This procedure only works properly for the pointlike (perturbative)
contribution to the parton content of the photon. One can attempt to describe
\cite{11} the nonperturbative component by the substitution $ 1/k_T^2
\rightarrow 1/(k_T^2 + k_0^2)$ in the $k_T$ distribution. Alternatively
\cite{17,17a} one can introduce a sharp $k_T$ cut--off, and model the
remaining,
nonperturbative part of the photon structure function like a real hadron.

There is some experimental evidence that initial state showering as
implemented in PYTHIA and HERWIG of ref.\cite{10} does not describe the data
very well. The ALEPH collaboration found \cite{7} that the TOPAZ event
generator, which uses string fragmentation but {\em no} showering, describes
their data {\em better} than standard PYTHIA does. Moreover, very recently the
H1 collaboration published results \cite{18} on the rapidity distribution of
jets in tagged photoproduction events; fig. 2 shows that these data are poorly
described by MC results for all existing parametrizations of
$\vec{q}^{\gamma}$. Notice in particular that this version of PYTHIA predicts
a large difference between the GRV parametrization \cite{19} with (solid) and
without (dotted) gluon contribution at {\em all} rapidities,even though this
parametrization assumes the gluon to be concentrated at small \xg; gluons
should therefore only contribute at positive (large) rapidities. Generally the
correlation between \xg\ and jet rapidities seems to be almost entirely washed
out once initial state showering is included \cite{14}. This is perhaps not
surprising since eq.(\ref{e2}) has not been implemented here, so that the
impact of showering has been over--estimated.

Let me demonstrate that the H1 jet rapidity distribution can quite easily be
brought into agreement with theoretical predictions. The key observation here
is that the ``plateau" of soft particles ``under" the jet seems to depend
quite strongly on the jet rapidity, see fig. 3. From this I estimate very
roughly \be \label{e3}
E_T({\rm plateau}) = 0.5 \ {\rm GeV} + \kappa \cdot (\eta + 1), \ee
with 0.6 GeV $\leq \kappa \leq$ 1.0 GeV. Figure 4 shows that a simple
parton level LO calculation, incorporating the H1 cuts on the energy and
virtuality of the incoming photon and assuming \be \label{e4}
E_T({\rm jet}) = p_T({\rm parton}) + E_T({\rm plateau}), \ee
can reproduce the data well.

There are two reasons why $E_T$(plateau) should increase with increasing
pseudorapidity $\eta$. First of all, one comes closer to the proton remnant
jet, which carries a lot of energy. Secondly, large (positive) $\eta$
correspond to small \xg, which in turn allows for a larger initial state
shower. Recall that each branching (radiation) in the shower increases \xg,
so that a smaller starting value allows for a longer cascade; moreover,
$|t_3|$ of eq.(\ref{e2}) decreases with decreasing \xg.\footnote{Of course,
increasing $\eta$ increases the distance to the photonic spectator jet.
However, $\eta_{\rm lab} = 1.5$, the largest value in the H1 data sample,
corresponds to $\eta_{ep\ cms} = - 0.2$ and $-0.9 \leq
\eta_{\gamma p \ cms} \leq -0.35$ (for massless jets) for the H1 cuts on
$E_{\gamma,{\rm lab}}$. The {\em whole} data set is therefore in the
forward hemisphere of the photon. In addition, initial state showering can
give a transverse ``kick" to the entire hard scattering system. This is not
related to the plateau of soft particles, but also increases $E_T$(jet)
compared to a simple LO parton--level calculation without showering.}

I should emphasize here that PYTHIA has recently been modified \cite{17a} to
include the effects discussed here at least in an approximate fashion. The
relation between the $k_T$ of the spectator jet and the virtuality $t_3$ at
the beginning of the initial state shower has been recognized, but the
\xg\ dependence in eq.(\ref{e2}) has been ignored and $t_2$ has been set to
zero; this is the version of PYTHIA recommended by the authors as standard for
analyses of photoproduction data. Inclusion of the \xg\ dependence of $t_3$
strictly speaking goes beyond the simple leading--log approximation. However,
a similar $x$ dependence is routinely taken into account in final state
showering (in the argument of $\alpha_s$) \cite{20}; it remains to be seen
whether the simpler scheme of ref.\cite{17a} can describe the H1 data. In any
case, fig. 4 makes it clear that a good understanding of initial state
showering, and of possible other contributions to $E_T$(plateau), is mandatory
before these data can be used to test models \cite{21,19,17a} of the photon
structure functions.

\subsection*{Acknowledgements}
I thank Gunnar Ingelman and Andrej Rostovtsev for very helpful discussions on
the interpretation of the H1 results, as well as Torbjorn Sj\"ostrand for a
critical reading of the manuscript. This work was supported in part by the
U.S. Department of Energy under contract No. DE-AC02-76ER00881, in part by the
Wisconsin Research Committee with funds granted by the Wisconsin Alumni
Research Foundation, by the Texas National Research Laboratory Commission
under grant RGFY93--221, as well as by a grant from the Deutsche
Forschungsgemeinschaft under the Heisenberg program.

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\clearpage
\section*{Figure Captions}
\renewcommand{\labelenumi}{Fig.\arabic{enumi}}
\begin{enumerate}

\item   %Fig. 1
Typical resolved photon $\gamma q$ scattering diagram leading to
two high$-p_T$ partons (5 and 6) and one spectator parton (2). \

\vspace*{5mm}

\item  %Fig. 2
Pseudorapidity distribution of jets with $E_T \geq 7$ GeV in tagged
photoproduction events at HERA \protect\cite{18}, compared to PYTHIA
predictions using various parametrizations for the parton content of the
photon: Solid -- GRV; dotted -- GRV ($G^{\gamma} = 0$); dot-dshed -- LAC2;
dashed -- LAC3. \

\vspace*{5mm}

\item   %Fig. 3
Transverse energy flow around the jets, for -1 $\leq \eta \leq 0.5$
(left), and 0.5 $\leq \eta \leq$ 1.5 (right) \protect\cite{18}.

\vspace*{5mm}

\item   %Fig. 4
LO QCD predictions for the rapidity distribution if the $E_T$ of
the jets is related to the partonic $p_T$ by eqs.(3),(4). The same cuts on the
incoming photon as in ref.\protect\cite{18} have been applied.

\end{enumerate}
\end{document}

