%Title: Fixed-Order QCD Backgrounds to BFKL Dynamics         in Forward Jet Production
%Author: Erwin Mirkes and Dieter Zeppenfeld
%Published: <EM>  Proceedings of the 1996 HERA Physics Workshop </EM> 
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% -----------------
%Forward Jet Production at HERA in the Low $x$ Regime\\
%       in Next-to-Leading Order\\
%The NLO Inclusive Forward Jet Background to
%BFKL Dynamics at HERA\\

\begin{document}
\hbox to \hsize{
%\special{psfile=uwlogo.ps
%hscale=8000 vscale=8000
%hoffset=-12 voffset=-2}
%\hskip.5in \raise.1in\hbox{\fortssbx University of Wisconsin - Madison}
\hfill$\vtop{   \hbox{\bf MADPH-96-960\hspace{1cm}}
                \hbox{\bf TTP96-40 }
                \hbox{\bf hep-ph/9609255\\}
                \hbox{September 1996}}$ }

\vspace*{1cm}
\begin{center}  \begin{Large} \begin{bf}
Fixed-Order QCD Backgrounds to BFKL Dynamics\\[2mm]
 in Forward Jet Production
  \end{bf}  \end{Large}
\end{center}
  \vspace*{5mm}
\begin{center}
  \begin{large}
Erwin Mirkes$^a$ and Dieter Zeppenfeld$^b$\\ 
  \end{large}
\end{center}
$^a$Institut f\"ur Theoretische Teilchenphysik, 
         Universit\"at Karlsruhe, D-76128 Karlsruhe, Germany\\
$^b$Department of Physics, University of Wisconsin, Madison, WI 53706, USA\\
%
\begin{quotation}
\noindent
{\bf Abstract:}
The production of forward jets of transverse momentum $p_T(j)\approx Q$ and 
large momentum fraction $x_{jet}\gg x$ probes the onset of BFKL dynamics 
at HERA. A full ${\cal O}(\alpha_s^2)$ calculation of the inclusive forward
jet cross section is presented and compared to the expected BFKL
cross section. 
\end{quotation}

Deep-inelastic scattering (DIS) at HERA provides an ideal place to
probe strong interaction dynamics. One focus of interest has been the small
Bjorken-$x$ region, where one would like to distinguish 
BFKL evolution~\cite{bfkl}, which resums the leading $\alpha_s \ln 1/x$
terms, from the more traditional DGLAP evolution equation~\cite{dglap},
which resums leading $\alpha_s \ln Q^2$ terms.
Unfortunately, the  measurement of $F_2(x,Q^2)$ in the HERA range is probably 
too inclusive to discriminate between the two~\cite{viele}. 

A more sensitive test of BFKL dynamics at small $x$ is expected from deep 
inelastic scattering with a measured forward jet (in the proton direction) 
and  $p_T^2(j)\approx Q^2$~\cite{mueller}. The idea is to study DIS events 
which contain an identified jet of longitudinal momentum 
fraction $x_{jet}=p_z(jet)/E_{proton}$ which is large compared to Bjorken $x$. 
 When
tagging a forward jet with $p_T(j)\approx Q$ this leaves little room for
DGLAP evolution while the condition $x_{jet}\gg x$ leaves BFKL evolution 
active. This leads to an enhancement of the forward jet production cross 
section proportional to $(x_{jet}/x)^{\alpha_P -1}$ over the DGLAP 
expectation.

A conventional fixed order QCD calculation up to ${\cal O}(\alpha_s^2)$ 
does not yet contain any BFKL resummation and must be 
considered a background for its detection; one must search
for an enhancement in the forward jet production cross section above the 
expectation for two- and three-parton final states. 
In this contribution we perform a full next-to-leading order (NLO) analysis
of this ``fixed order'' background. Such an analysis has become possible with
the implementation of QCD radiative corrections to dijet production in DIS in
a fully flexible Monte Carlo program, MEPJET \cite{mepjet}.

Numerical results below will be presented both for leading order (LO) and 
NLO simulations. The LO 1-jet and 2-jet results employ the LO parton 
distributions of Gl\"uck, Reya and Vogt~\cite{grv} together with the 
one-loop formula for the strong coupling constant. At ${\cal O}(\alpha_s^2)$
all cross sections are determined using the NLO GRV parton distribution 
functions $f(x_i,\mu_F^2)$
 and the two loop formula for $\alpha_s(\mu_R^2)$. With this 
procedure the 2-jet inclusive rate at NLO is simply given as the sum of 
the NLO 2-jet and the LO 3-jet exclusive cross sections. 
The value of $\alpha_s$ is matched at the thresholds $\mu_R=m_q$ and the
number of flavors is fixed to $n_f=5$ throughout, {\it i.e.} gluons are 
allowed to split into five flavors of massless quarks.

Unless otherwise stated, both the 
renormalization and the factorization scales are tied to the sum of 
parton $k_T$'s in the Breit frame,
%\bq
$
\mu_R = \mu_F = {1\over 2} \sum_i k_T^B(i) \; ,
$
%\eq
where $(k_T^B(i))^2=2E_i^2(1-\cos\theta_{ip})$. Here $\theta_{ip}$ is the 
angle between the parton and proton directions in the Breit 
frame. $\sum_i k_T^B(i)$ constitutes a natural scale for jet
production in DIS~\cite{rheinsberg} because it interpolates between $Q$,
in the naive parton model limit, and the sum of jet transverse momenta, when
$Q$ becomes negligible. 

We are interested in events with a forward jet with $p_T(j)\approx Q$ 
and $x_{jet}\gg x$ and impose kinematical cuts which closely model the 
H1 selection\cite{deroeck} of such events. Jets are defined in the cone 
scheme (in the laboratory frame) with $\Delta R = 1$ and $|\eta|<3.5$. 
Here $\eta=-\ln\tan(\theta/2)$ denotes the pseudorapidity of a jet.
Unless noted otherwise, all jets must have transverse momenta of at least 
4~GeV in both the laboratory and the Breit frames. Events are selected 
which contain a forward jet (denoted ``$j$'') in the pseudorapidity 
range $ 1.735< \eta(j)< 2.9$ (corresponding to $6.3^o < \theta(j) < 20^o$) 
and with transverse momentum $p_T^{lab}(j)>5$~GeV. This jet must satisfy
\ba
x_{jet} & = & p_z(j)/E_p\; > \; 0.05\;,  \label{eq:fja} \\
   0.5  & < & p_T^2(j)/Q^2\; < \; 4\; \label{eq:fjb}
\ea
in the laboratory frame. The condition $x_{jet}\gg x$ is satisfied by 
requiring $x<0.004$.
Additional selection cuts are $Q^2>8~$GeV$^2$, $0.1 < y < 1$, an energy 
cut of $E(l^\prime)>11$~GeV on the scattered lepton, and a cut on 
its pseudorapidity of $ -2.868 < \eta(l^\prime)< -1.735$ 
(corresponding to $160^o < \theta(l^\prime) < 173.5^o$).
The energies of the incoming electron and proton are set to 27.5~GeV
and 820~GeV, respectively.

\begin{table}[t]
%\vspace{3mm}
\begin{tabular}{lccc}
        \hspace{0.8cm}
     &  \mbox{with forward jet} &
     &  \mbox{without forward jet } \\
     &  \mbox{$p_T^B,p_T^{lab}>4$~GeV}
     &  \mbox{$k_T^B>4$~GeV}
     &  \mbox{$p_T^B,p_T^{lab}>4$~GeV}\\
\hline\\[-3mm]
\mbox{${\cal O}(\alpha_s^0)$: 1 jet}
                    & 0    pb & 0    pb & 8630 pb    \\
\mbox{\mbox{${\cal O}(\alpha_s)$}: 2 jet }
                    & 18.9 pb & 22.4 pb & 2120 pb    \\
\mbox{${\cal O}(\alpha_s^2)$: 1 jet inclusive} 
                    & 100 pb & 100 pb &           \\
\mbox{\phantom{${\cal O}(\alpha_s^0)$:} 2 jet inclusive} 
                    & 83.8 pb & 98.3 pb & 2400 pb    \\
\mbox{\phantom{${\cal O}(\alpha_s^0)$:} 2 jet exclusive}  
                    & 69.0 pb & 66.8 pb & 2190 pb    \\
\mbox{\phantom{${\cal O}(\alpha_s^0)$:} 3 jet }   
                    & 14.8 pb & 31.5 pb & 210  pb    \\
\end{tabular}
\caption{Cross sections for $n$-jet events
in DIS at HERA at order $\alpha_s^0$, $\alpha_s$, and $\alpha_s^2$. 
The jet multiplicity includes the forward jet which, when
required, must satisfy $p_T(j)>5$~GeV and the cuts of 
Eqs.~(\protect\ref{eq:fja},\protect\ref{eq:fjb}). 
The transverse momenta of additional (non-forward)
jets must only exceed cuts of 4~GeV (first and third column). This 
requirement is replaced by the condition $k_T^B>4$~GeV in the second column.
No $p_T^B$ cut is imposed in the 1-jet case at ${\cal O}(\alpha_s^0)$
and the factorization scale is fixed to $Q$.
}\label{table1}
\vspace{2mm}
\end{table}
%

Numerical results for the multi-jet cross sections with (or without) a 
forward jet are shown in Table~\ref{table1}. 
Without the requirement of a forward jet, the cross 
sections show the typical decrease with increasing jet multiplicity which
is expected in a well-behaved QCD calculation. The 3-jet cross section
in the last column constitutes only about 10\% of the 2-jet cross section
and both rates are sizable.
The requirement of a forward 
jet with large longitudinal momentum fraction
($x_{jet}>0.05$) and restricted transverse momentum ($0.5<p_T^2(j)/Q^2<4$)
severely restricts the available phase space. In particular one finds that 
the 1-jet cross section 
vanishes at LO, due to the contradicting $x<0.004$ and $x_{jet}>0.05$ 
requirements: this forward jet kinematics is impossible for one single
massless parton in the final state. 

Suppose now that we had performed a full ${\cal O}(\alpha_s^2)$ calculation 
of the DIS cross section, which would contain 3-parton final states at tree 
level, 1-loop corrections to 2-parton final states and 2-loop corrections to
1-parton final states. These 2-loop contributions would vanish identically,
once $x\ll x_{jet}$ is imposed. The remaining 2-parton and 
3-parton differential cross sections, however, and the cancellation of
divergences between them, would be the same as those entering 
a calculation of 2-jet inclusive rates. These elements are already 
implemented in the MEPJET program which, therefore, can be used to determine 
the inclusive forward jet cross section, within the cuts discussed above. 
At ${\cal O}(\alpha_s^2)$ this 
cross section is obtained from the cross section for 2-jet inclusive events 
by integrating over the full phase space of the additional
jets, without any cuts on their transverse momenta or pseudorapidities. 
Numerical results are shown in the third row of Table~\ref{table1}. 

The table exhibits some other remarkable features of forward jet events:
the NLO 2-jet inclusive cross section exceeds the LO 2-jet cross section 
by more than a factor of four and the 3-jet rate at ${\cal O}(\alpha_s^2)$ is 
about as large as the 2-jet rate at ${\cal O}(\alpha_s)$. 
The smallness of the
LO 2-jet compared to the NLO 2-jet inclusive cross section means that
at least three final-state partons are required to access the relevant part
of the phase space. This three-parton cross section, however, has only been 
calculated at tree level and is subject to the typical scale uncertainties
of a tree level calculation. Thus, even though we have performed a full 
${\cal O}(\alpha_s^2)$ calculation of the forward jet cross section at HERA,
including all virtual effects, our calculation effectively only gives a LO
estimate of this cross section and large corrections may be expected from
higher order effects.

\begin{figure}[bht]
\vspace*{0.5in}            
\begin{picture}(0,0)(0,0)
\special{psfile=recoil.ps
          voffset=-300 hoffset=540 hscale=70 vscale=70 angle=90}
%\special{psfile=recoil.ps
%          voffset=-260 hoffset=510 hscale=60 vscale=60 angle=90}
\end{picture}
%\vspace{7.6cm}
\vspace{8.5cm}
\caption{
Characteristics of the highest transverse momentum ``jet'' in the recoil 
system, i.e. excluding the forward jet. Distributions shown are  
(a) $d\sigma/dp_T$ in the lab frame (solid line) and $d\sigma/dk_T$ in the 
Breit frame (dashed line) and (b) the jets pseudorapidity distribution in 
the laboratory frame. All distributions are calculated at order $\alpha_s^2$.
Jet transverse momentum cuts have been relaxed to $p_T^{lab},p_T^B>1$~GeV.
\label{fig:recoil}
}
%\vspace*{-0.1in}
\end{figure}

The characteristics of forward jet events are demonstrated in 
Fig.~\ref{fig:recoil} where the transverse momentum and the pseudorapidity 
distributions of the recoil jet with the 
highest $p_T^{lab}$ are shown, subject only to a nominal requirement 
of $p_T^{lab},p_T^B>1$~GeV. Here the recoil system is defined as the 
complement of the forward jet, in the final state which arises in the 
photon-parton collision. Almost all forward jet events contain at
least one additional jet in the recoil system, 
with $p_T^{lab}\roughly>4$~GeV and, 
typically, in the central part of the detector. 

In the usual cone scheme final-state collinear singularities are 
regulated by the $\Delta R$ separation cut while infrared singularities
and initial state collinear emission are regulated by the $p_T$ cut.
In $\gamma^* p$ collisions the photon virtuality, $Q^2$, eliminates
any collinear singularities for initial state emission in the 
electron direction and therefore a large $k_T$ is as good a criterion 
to define a cluster of hadrons as a jet as its $p_T$.  
The dashed line in
Fig.~\ref{fig:recoil}(a) shows the $k_T$ distribution in the Breit frame 
of the recoil jet candidate with the largest $k_T^B$. Basically all 
forward jet events in this NLO analysis possess a recoil ``jet'' 
with $k_T^B>4$~GeV and would thus be classified as 2-jet inclusive events
in a variant of the cone scheme where the $p_T>4$~GeV condition is 
replaced by a $k_T^B>4$~GeV cut. This observation makes intuitively 
clear why we are able to calculate the 1-jet inclusive forward jet cross 
section with a program designed for the 2-jet inclusive cross
section at NLO: there exists a jet definition scheme in which all forward
jet events contain at least one additional hard jet. 

An estimate for higher order corrections
may be obtained by comparing to BFKL 
calculations or to existing experimental results. The H1 Collaboration has 
published such a measurement which was made during the 1993 HERA run with 
incident electron and proton energies of $E_e=26.7$~GeV 
and $E_p=820$~GeV~\cite{H1result}. The acceptance cuts used for this 
measurement differed somewhat from the ones described before.  Because 
of the lower luminosity in this early HERA run the $x_{jet}$ cut on the 
forward jet was lowered to 0.025 and defined in terms of the jet energy as 
opposed to the longitudinal momentum of the jet in the proton direction,
\bq
x_{jet}=E(j)/E_p > 0.025\;,  \label{eq:H1cuta}
\eq
and the pseudorapidity range of the forward jet was chosen slightly 
larger, $ 1.735< \eta(j)< 2.949$ (corresponding to $6^o < \theta(j) < 20^o$).
Scattered electrons were selected with an energy of $E(l^\prime)>12$~GeV and 
in the pseudorapidity range $ -2.794 < \eta(l^\prime)< -1.735$ 
(corresponding to $160^o < \theta(l^\prime) < 173^o$). Finally the 
Bjorken-$x$ and $Q^2$ ranges were chosen as $0.0002<x<0.002$ and 
5~GeV$^2<Q^2<100$~GeV$^2$. Within these cuts H1 has measured cross sections
of $709\pm 42\pm 166$~pb for $0.0002<x<0.001$ and $475\pm 39\pm 110$~pb 
for $0.001<x<0.002$. These two data points, normalized to bin sizes of 0.0002,
are shown as diamonds with error bars in Fig.~\ref{fig:h1comp}. Also included
(dashed histogram) is a recent calculation of the BFKL cross 
section~\cite{bartelsH1}.

As shown before, the MEPJET program allows to calculate the full 1-jet 
inclusive forward jet cross section\footnote{We have checked that 
also for the kinematical region considered now almost all forward jet 
events  contain at least one second jet with $p_T^{lab}>4$ GeV 
and $k_T^B>4$ GeV.}
for $x\ll x_{jet}$. 
The LO result is shown 
as the dash-dotted histogram in Fig.~\ref{fig:h1comp} and the NLO result
is shown as the solid histogram. The shaded area corresponds to a scale 
variation
%\bq
$
\mu_R^2 = \mu_F^2 = \xi\;{1\over 4} \left( \sum_i k_T^B(i) \right) \; ,
$
%\nonumber
%\eq
from $\xi=0.1$ to $\xi=10$, and indicates a range of ``reasonable'' 
expectations for the forward jet cross section at ${\cal O}(\alpha_s^2)$. 

While the BFKL results~\cite{bartelsH1} agree well with the H1 data, the 
fixed-order perturbative QCD calculations clearly fall well below the 
measured cross section, even when accounting for variations of the 
factorization and renormalization scales. The measured cross section is
%
\begin{figure}[t]
\epsfxsize=5.1in
\epsfysize=4.0in
\begin{center}
\hspace*{0in}
\epsffile{h1comp.eps}
\vspace*{0.5cm}
\caption{
Forward jet cross section at HERA as a function of Bjorken $x$ within the H1 
acceptance cuts~\protect\cite{H1result} (see text). 
The solid (dash-dotted) histogram gives the NLO (LO) MEPJET result for  
the scale choice $\mu_R^2=\mu_F^2=\xi(0.5\sum k_T)^2$ with $\xi=1$. 
The shaded area shows the uncertainty of the NLO prediction, corresponding
to a variation of $\xi$ between 0.1 and 10. The BFKL result of 
Bartels et al.~\protect\cite{bartelsH1} is shown as the dashed 
histogram. The two data points with error bars correspond to the H1 
measurement~\protect\cite{H1result}.
\label{fig:h1comp}
}
\vspace*{-0.1in}
\end{center}
\end{figure}
%
a factor of 4 above the NLO expectation. The shape of the NLO prediction, 
on the other hand, is perfectly compatible with the H1 results, and not very 
different from the BFKL curve in Fig.~\ref{fig:h1comp}. At LO
a marked shape difference is still observed, which can be traced directly 
to  kinematical arguments given in Ref.~\cite{forward}.
Additional details, including a study of the NLO scale dependence
of the forward jet cross section, can be found there.
First NLO studies for forward jet production have been presented
in Ref.~\cite{mz2}.
For a study of forward jet cross sections
with the ZEUS detector, see Ref.~\cite{zeus}.

We conclude that the existing H1 data show evidence for BFKL dynamics in 
forward jet events via an enhancement in the observed forward jet cross 
section above NLO expectations. The variation of the cross section with $x$, 
on the other hand, is perfectly compatible with either BFKL dynamics or 
NLO QCD. Since MEPJET provides a full NLO prediction of the 1-jet inclusive
forward jet cross section for arbitrary cuts and jet definition schemes,
more decisive shape tests may be possible as additional data become available.


%%%%%%%%%%%%%%%%%%%%% REFERENCES %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
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\end{document}

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