%Title: Azimuthal Asymmetries in Hadronic Final States at HERA
%Author: M.Ahmed, T.Gehrmann
%Published: * Phys. Lett. * ** B465 ** (1999) 297-302.
%hep-ph/9906503
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\begin{document}
\begin{titlepage}
\vspace*{-1cm}
\begin{flushright}
DESY--99--077 \\
TTP99--29\\
June 1999 \\
\end{flushright}
\vskip 3.5cm
\begin{center}
{\Large\bf Azimuthal Asymmetries in}
\vskip 0.2cm
{\Large\bf Hadronic Final States at HERA}
\vskip 1.cm
{\large M.~Ahmed$^{a,b}$} and {\large T.~Gehrmann$^c$}
\vskip .4cm
{\it $^a$ II.~Institut f\"ur Theoretische Physik,
Universit\"at Hamburg, Luruper Chaussee 149, D-22761 Hamburg, Germany}
\vskip .2cm
{\it $^b$ University of the Punjab, Lahore, Pakistan}
\vskip .2cm
{\it $^c$ Institut f\"ur Theoretische Teilchenphysik, Universit\"at
Karlsruhe, D-76128 Karlsruhe, Germany}
\end{center}
\vskip 2.cm
\begin{abstract}
The distribution of
hadrons produced in deeply inelastic electron--proton collisions
depends on the azimuthal angle between lepton scattering plane and
hadron production plane in the photon-proton centre-of-mass frame.
In addition to the well known up-down
asymmetry induced by the azimuthal dependence
of the Born level subprocess, there is also a non-vanishing left-right
asymmetry, provided the incoming electron is polarized. This
asymmetry is time-reversal-odd and induced by absorptive corrections
to the Born level process. We investigate the numerical magnitude
of azimuthal
asymmetries in semi-inclusive hadron production
at HERA with particular emphasis on a possible determination of the
time-reversal-odd asymmetry.
\end{abstract}
\vfill
\end{titlepage}
\newpage
The increase in the statistical accuracy of deep inelastic scattering (DIS)
data at the HERA collider will soon allow to investigate hadronic final state
observables which go beyond hadron multiplicity
distributions~\cite{haddistr} and jet rates
that have been studied up to now. Observables of particular interest are
angular correlations between the lepton scattering plane, defined by
the incoming and outgoing lepton momenta
and the
hadron production plane, defined by incoming proton and outgoing hadron
momentum. These correlations probe the dynamics and colour flow of the
underlying partonic interaction at a detailed level, such that
they can be used to test the perturbative description of hadron
production via partonic fragmentation.
An accurate prediction of the
perturbatively induced asymmetries is in particular desirable since
azimuthal correlations in semi-inclusive DIS have been suggested as
probes of non-perturbative effects in various places in the
literature~\cite{nplit}.
In this paper, we estimate the magnitude of the different azimuthal
asymmetries for kinematical
conditions at the HERA collider, using parton model
expressions at leading order. Particular emphasis is put on
time-reversal-odd ($T$-odd)
asymmetries, resulting from absorptive contributions
to the parton level scattering amplitudes~\cite{ru}. These $T$-odd
effects manifest in left-right asymmetries of the hadron
distribution with respect to the lepton scattering plane~\cite{men,hag}.
Corresponding to antisymmetric contributions to the hadronic tensor,
their observation requires either the contribution from parity violating
weak interactions or polarization of the initial lepton beam. Given that
lepton beam polarization will soon be realized for the HERA collider
experiments, both cases shall be investigated below. Perturbative
$T$-odd effects have up to now only been studied experimentally
in polarized electron-positron
annihilation at SLAC~\cite{sld}, where the
expected asymmetries~\cite{bb} are however too
small to be measured directly, such that only upper limits
could be determined~\cite{sld}.
The kinematics of the semi-inclusive reaction
\begin{displaymath}
l(k) + p(p) \longrightarrow l'(k') + h(P) + X
\end{displaymath}
are described by the following invariant variables
\begin{eqnarray}
Q^2 & = & -q^2 = -(k-k')^2\;,\nonumber \\
x &=& \frac{Q^2}{2q\cdot p}\;, \nonumber \\
z &=& \frac{p\cdot P}{q\cdot p}\;,\nonumber\\
\kappa^2 &=& z^2\left(1-\frac{q\cdot P}{xp\cdot P}\right)
\end{eqnarray}
and the azimuthal angle $\phi$ between outgoing lepton direction and outgoing
hadron direction measured in the centre-of-mass frame of virtual gauge
boson and proton. The variable $\kappa$ relates to the transverse
momentum of the outgoing hadron in this frame by $\kappa^2=P_T^2/Q^2$.
The semi-inclusive scattering cross section can be decomposed according
to the dependence on $\phi$:
\begin{equation}
\frac{\d \sigma}{\d x \d Q^2 \d z \d \phi \d P_T^2} =
\frac{\alpha^2 \pi}{2 Q^6 z} \left( A + B \cos \phi + C \cos 2\phi + D
\sin \phi + E \sin 2\phi\right)\; .
\label{eq:master}
\end{equation}
Explicit parton model expressions for the coefficients $A$--$E$ can be
found in~\cite{men,hag}, the description of charged current (CC) interactions
requires the substitution $\alpha \to G_F Q^2 /(\sqrt{2} \pi)$.
It should be noted that the leading order
contribution to $A$ is ${\cal O}(1)$, corresponding to vanishing
transverse momentum of the outgoing hadron. The first contribution to
$A$ yielding $\kappa \neq 0$ is ${\cal O}(\alpha_s)$.
The leading order contributions
to $B$ and $C$ are ${\cal O}(\alpha_s)$ and to $D$ and $E$ are
${\cal O}(\alpha_s^2)$.
$D$ and $E$ are time-reversal-odd, they
are induced by absorptive one-loop corrections to the partonic
scattering amplitudes. They appear in the hadronic tensor with
asymmetric coefficients, which implies their vanishing for
purely electromagnetic interactions with unpolarized
beams. Non-vanishing $T$-odd asymmetries are obtained only for
weak interactions or for electromagnetic interactions with
a longitudinally polarized lepton beam.
In order to suppress the large ${\cal O}(1)$ contribution to the
$\phi$-independent coefficient $A$, it is appropriate to restrict
studies of angular asymmetries to hadrons produced at non-zero $p_T$.
To project out individual terms in (\ref{eq:master}), we define the
following average asymmetries, depending on $x$, $Q^2$ and $P_T$:
\begin{eqnarray}
\langle \sin (n\phi) \rangle (x,Q^2,P_T)
& = & \frac{\displaystyle \int \d z \d \phi \sin(n\phi)
\frac{\displaystyle \d \sigma}{\displaystyle \d x \d Q^2 \d z \d \phi
\d P_T^2}}{\displaystyle
\int \d z \d \phi \frac{\displaystyle
\d \sigma}{\displaystyle \d x \d Q^2 \d z \d \phi
\d P_T^2}}\; ,\nonumber \\
\langle \cos (n\phi) \rangle (x,Q^2,P_T)
& = & \frac{\displaystyle \int \d z \d \phi \cos(n\phi)
\frac{\displaystyle \d \sigma }{\displaystyle \d x \d Q^2 \d z \d \phi
\d P_T^2}}{\displaystyle \int \d z \d \phi \frac{\displaystyle \d \sigma}
{\displaystyle \d x \d Q^2 \d z \d \phi
\d P_T^2}}\; .
\label{eq:asydef}
\end{eqnarray}
The integration over the outgoing hadron momentum $z$ is a priori
bounded only by the kinematical requirement
\begin{displaymath}
\kappa^2 \leq \frac{1-x}{x}z(1-z),
\end{displaymath}
which, at the $x$-values probed at HERA, involves contributions from
very small $z\ll 0.1$, where partonic fragmentation functions into
hadrons are only poorly ($0.010.1$, where
the predictions are most reliable. $\langle\sin \phi\rangle$
turns out to be larger than in the neutral current case, and amounts
up to one per cent. $\langle\sin (2\phi)\rangle$ is at the level
of half a per cent.
The ratio between $\langle\cos (n\phi)\rangle$ and
$\langle\sin (n\phi)\rangle$ is therefore more favourable in the charged
current case, and the $\langle\sin (n\phi)\rangle$ are also
larger. Despite the significantly smaller cross section, a measurement of
$T$-odd asymmetries in charged current
DIS might therefore be competitive to the measurement in the neutral
current case.
In summary, we have investigated the numerical magnitude of
various asymmetries
in the angular distribution of hadrons in the final state of
deep inelastic scattering, as determined by parton model
expressions. The resulting estimates for neutral current deep inelastic
scattering show that the
$\langle \cos(n\phi)\rangle(P_T)$ asymmetries
are typically of the order of a few per cent,
and should thus be easily measurable. The time-reversal-odd
asymmetry $\langle \sin\phi\rangle(P_T)$ does hardly exceed $10^{-3}$
in neutral current interactions and $10^{-2}$ in charged current
processes,
an experimental determination of it is therefore a challenging task. If a
substantially larger $\langle \sin\phi\rangle(P_T)$ should be
observed at HERA, it would be a clear indication for large
non-perturbative $T$-odd effects, as suggested in the
literature~\cite{nplit}.
\section*{Acknowledgements}
\noindent
The work of M.A.~was supported by DAAD. The authors would like to thank
G.~Kramer for several discussions throughout the project.
\goodbreak
\begin{thebibliography}{99}
\bibitem{haddistr}
ZEUS collaboration: M.~Derrick et al., Z.~Phys. {\bf C70} (1996) 1;\\
H1 collaboration: C.~Adloff et al., Nucl. Phys. {\bf B485} (1997) 3.
\bibitem{nplit}
J.~Levelt and P.~Mulders, Phys.~Lett. {\bf B338} (1994) 357;\\
P.~Mulders and R.D.~Tangerman, Nucl. Phys. {\bf B461} (1996) 197;\\
K.A.~Oganesian et al, Eur. Phys. J. {\bf C5} (1998) 681;\\
K.A.~Oganesian et al, preprint hep-ph/9808368;\\
M.~Boglione and P.J.~Mulders, preprint VU-TH 99-03
(hep-ph/9903354).
\bibitem{ru}
A.~De R\'{u}jula, J.~M.~Kaplan and E.~de Rafael, Nucl.~Phys. {\bf B35}
(1971) 365.
\bibitem{sld}
SLD collaboration: K.~Abe et al., Phys.~Rev.~Lett. {\bf 75} (1995)
4173.
\bibitem{bb}
A.~Brandenburg, L.~Dixon and Y.~Shadmi, Phys.~Rev. {\bf D53} (1996)
1264.
\bibitem{men}
A.~Mendez, Nucl. Phys. {\bf B145} (1978) 199.
\bibitem{hag}
K.~Hagiwara, K.~Hikasa and N.~Kai, Phys.~Rev. {\bf D27} (1983) 84.
\bibitem{cteq}
H.~Lai et al., Phys. Rev. {\bf D55} (1997) 1280.
\bibitem{bkk}
J.~Binnewies, B.A.~Kniehl and G.~Kramer, Phys.~Rev. {\bf D52} (1995)
4947.
\bibitem{kramer}
C.~Rumpf, G.~Kramer and J.~Willrodt, Z. Phys. {\bf C7} (1981) 337;\\
A.S.~Joshipura and G.~Kramer, J. Phys. {\bf G8} (1982) 209.
\end{thebibliography}
\newpage
\begin{figure}
\begin{center}
~ \epsfig{file=plot1a.eps,width=4cm}
~ \epsfig{file=plot1b.eps,width=4cm}
~ \epsfig{file=plot1c.eps,width=4cm}
\end{center}
\caption{The asymmetry \protect{$\langle \cos \phi\rangle(P_T)$}
in neutral current charged hadron production.
Solid line: \protect{$0.005 < z < 0.01$},
dashed line: \protect{$0.01 < z < 0.05$},
dotted line: \protect{$0.05< z < 0.1$},
short dot-dashed line: \protect{$0.1