%Title: Charge Asymmetry of Heavy Quarks at Hadron Colliders
%Author: J.H. Kuehn and G. Rodrigo
%Published:     Proc. <EM>29th ICHEP' </EM> Vancouver, CA, 23-29 July 1998, edited by A. Astbury, D. Axen, Jacob Robinson (World Scientific, Singapore, 1999) Vol.2, 1117-1120. 
%UNIX --- UPDATED ON 14/8/97 
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\begin{document}

\title{
CHARGE ASYMMETRY OF HEAVY QUARKS AT HADRON COLLIDERS$^*$
}

\author{
J.~H.~K\"UHN$^{a)}$ and G.~RODRIGO$^{b)}$
}

\address{
$a)$ Institut f\"ur Theoretische Teilchenphysik,
       Universit\"at Karlsruhe, Germany \\ 
$b)$ INFN-Sezione di Firenze, Italy \\ 
}

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\twocolumn[\maketitle\abstracts{
A sizeable difference in the differential production cross section
of top and antitop quarks, respectively, is predicted for 
hadronically produced heavy quarks.
It is of order $\alpha_s$ and arises from the 
interference between charge odd and even amplitudes respectively.
For the TEVATRON it amounts up to 15\% for the differential 
distribution in suitable chosen kinematical regions.
The resulting integrated forward-backward asymmetry of 
4--5\% could be measured in the next round of experiments.
At the LHC the asymmetry can be studied by selecting 
appropriately chosen kinematical regions.
Furthermore, a slight preference at LHC for centrally 
produced antitop is predicted, with top quarks more abundant at 
large positive and negative rapidities.
}]

\footnotetext
{$^*$Presented by J.\ H.\ K\"uhn.}

Heavy flavor production at hadron colliders is one of the most
active fields of current theoretical and experimental studies.
Large event rates, combined with improved experimental
techniques, allow for detailed investigations of the 
properties of heavy quarks and their production mechanism
at the same time.
While charm production with a quark mass around $1.5$~GeV is 
barely accessible to perturbative QCD calculations, bottom and 
{\it a forteriori} top production should be well described by this 
approach.

Theoretical and experimental
results~\cite{Catani:1997rn,Tipton:1996}
for the cross section of hadronic top production are well
consistent with this expectation.
Obviously, in view of the large QCD coupling, the inclusion 
of higher order QCD corrections in these calculations is 
mandatory for a successful comparison. Recent studies have, 
to a large extent, concentrated on the predictions of the 
total cross section and a few selected one particle 
inclusive distributions.
In this paper a different issue of heavy flavor production 
is investigated, namely the charge asymmetry,
which is sensitive toward
a specific subclass of virtual and real radiative corrections.

Evaluated in Born approximation 
the lowest order processes relevant for heavy flavor 
production
\beq
q + \bar{q} \to Q + \bar{Q}~,
\label{eq:qqbar}
\eeq
\beq
g + g \to Q + \bar{Q}~,
\label{eq:gg}
\eeq
do not discriminate between the final quark and antiquark, thus  
predicting identical differential distributions  also for 
the hadronic production process.
However, radiative corrections involving either virtual or 
real gluon emission lead to a sizeable difference between 
the differential quark and antiquark production process 
and hence to a charge asymmetry~\cite{Kuhn:1998jr,Kuhn:1998jra} which could be well 
accessible experimentally.

This asymmetry has its origin in two different reactions:
radiative corrections to quark-antiquark fusion (Fig.~\ref{fig:qqbar})
and heavy flavor production involving interference terms
of different amplitudes contributing to 
gluon-quark scattering 
\beq
g+q \to Q + \bar{Q} + q~,
\label{eq:gq}
\eeq
a reaction intrinsically of order $\alpha_s^3$.
The contribution from quark gluon scattering to the asymmetry has 
been shown to be small~\cite{Kuhn:1998jra} and will be ignored in this review.
Gluon fusion remains of course charge symmetric.
In both reactions (\ref{eq:qqbar}) and (\ref{eq:gq})
the asymmetry can be traced to the 
interference between amplitudes which are relatively odd under the 
exchange of $Q$ and $\bar{Q}$. 
In fact, as shown below in detail, the asymmetry can be understood
in analogy to the corresponding one in QED reactions
and is proportional to the color factor $d_{abc}^2$.
In contrast, the non-Abelian contributions,
in particular those involving the triple gluon coupling, 
lead to symmetric pieces in the 
differential cross section.

%%%%%%%%%%%%%%%
\mafigura{8 cm}{FBasymetry.ps}{Origin of the QCD charge
asymmetry in hadroproduction of heavy quarks:
interference of final-state (a) with initial-state (b) gluon bremsstrahlung
plus interference of the box (c) with the Born diagram (d).
Only representative diagrams are shown.}
{fig:qqbar}
%%%%%%%%%%%%%%%

Let us briefly discuss a few important aspect of this 
calculation.  The box amplitude for $q \bar{q} \to Q \bar{Q}$
is ultraviolet finite and the asymmetric contribution to the cross section
of order $\alpha_s^3$ is therefore not affected 
by renormalization, an obvious consequence of the symmetry 
of the lowest order reaction.
The same line of reasoning explains the absence of initial 
state collinear singularities in the limit $m_q \rightarrow 0$
which would have to be absorbed into the (symmetric) lowest 
order cross section.
Infrared singularities require a more careful treatment. 
They are absent in the asymmetric piece of the process in \eq{eq:gq}.
However, real and virtual radiation (Fig.~\ref{fig:qqbar}), if 
considered separately, exhibit infrared divergences, which 
compensate in the sum, corresponding to the inclusive 
production cross section.

The charge asymmetry in the partonic reactions~(\ref{eq:qqbar})
and~(\ref{eq:gq}) implies for example a forward-backward asymmetry
of heavy flavor production in proton-antiproton collisions.
In particular, it leads to a sizeable forward-backward
asymmetry for top production which is dominated by
reaction~(\ref{eq:qqbar}), and can, furthermore, be
scrutinized by studying $t \bar{t}$ production at fixed 
longitudinal momenta and at various partonic energies $\hat{s}$.
However, the charge asymmetry can also be observed in proton-proton
collisions at high energies. In this case one has to 
reconstruct the $t \bar{t}$ restframe and select kinematic regions,
which are dominated by $q \bar{q}$ annihilation 
or flavor excitation $g q \to t \bar{t} X$.
Alternatively, one may also study the difference 
in the one-particle inclusive rapidity distribution 
of top versus antitop, which again integrates to zero.

The analysis of these effects allows to improve our 
understanding of the QCD production mechanism. At the same time 
it is important for the analysis of single top production 
through $W b$ fusion. This reaction is charge asymmetric 
as a consequence of weak interactions. Although the final 
states in single top production and hadronic $t \bar{t}$ production 
are different and should in principle be distinguishable,
it is nevertheless mandatory to control the charge asymmetry 
from both sources.

As shown in~\cite{Kuhn:1998jr}, the dominant contribution to the charge 
asymmetry originates from $q \bar{q}$ annihilation, namely from the 
asymmetric piece in the interference between the Born 
amplitude for $q \bar{q} \to Q \bar{Q}$ (Fig.~\ref{fig:qqbar}d) and the 
one loop corrections to this reaction (Fig.~\ref{fig:qqbar}c),
which must be combined with the interference term between
initial state and final state radiation
(Fig.~\ref{fig:qqbar}a,\ref{fig:qqbar}b).

However, only QED like terms
are relevant for the charge asymmetric piece~\cite{Kuhn:1998jra}.
The QCD asymmetry is thus obtained from the QED results by the 
replacement
\beq
\alpha_{QED} Q_q Q_Q \rightarrow \frac{d_{abc}^2}{16 N_C T_F C_F}
\alpha_s = \frac{5}{12} \alpha_s~.
\eeq
Let us note in passing that diagrams involving the 
triple gluon coupling 
lead to 
charge symmetric terms.

The differential charge asymmetry in the inclusive cross section
\beq
q + \bar{q} \to Q + X~,
\eeq
at the partonic level can 
then be defined through 
\beq
\hat{A}(\cos \hat{\theta}) = 
\frac{N_t(\cos \hat{\theta})-N_{\bar{t}}(\cos \hat{\theta})}
     {N_t(\cos \hat{\theta})+N_{\bar{t}}(\cos \hat{\theta})}~,
\eeq
where $\hat{\theta}$ denotes the top quark production angle in 
the $q \bar{q}$ restframe and $N(\cos \hat{\theta}) =
d\sigma/d\Omega (\cos \hat{\theta})$.
Since $N_{\bar{t}}(\cos \hat{\theta}) = N_t(-\cos \hat{\theta})$
as a consequence of charge conjugation symmetry, 
$\hat{A}(\cos \hat{\theta})$ can also be interpreted as
a forward-backward asymmetry of top quarks (Fig.~\ref{fig:sfix}).

%%%%%%%%%%%%%%%
\mafigura{8 cm}{sfix.ps}{Differential charge asymmetry 
in top quark pair production for fixed partonic center of
mass energies $\sqrt{\hat{s}}=400$ GeV (solid),
$600$ GeV (dashed) and $1$ TeV (dotted).
We also plot the differential asymmetry for b-quarks 
with $\sqrt{\hat{s}}=400$ GeV (dashed-dotted).}
{fig:sfix}
%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%
\mafigura{8 cm}{sdistr.ps}{Integrated charge
asymmetry as a function of the partonic center of mass energy
for top and bottom quark pair production.}
{fig:sdistr}
%%%%%%%%%%%%%%%


The integrated charge asymmetry
\beq
\bar{\hat{A}} = 
\frac{N_t(\cos \hat{\theta} \geq 0)
    - N_{\bar{t}}(\cos \hat{\theta} \geq 0)}
     {N_t(\cos \hat{\theta} \geq 0)
    + N_{\bar{t}}(\cos \hat{\theta} \geq 0)}~,
\eeq
is shown in Fig.~\ref{fig:sdistr} as a function of $\sqrt{\hat{s}}$.
With a typical value around $6-8.5 \%$  it should be well 
accessible in the next run of the TEVATRON.

In addition to the pure QCD amplitudes also a mixed QCD-electroweak 
interference term will lead to an asymmetric contribution
to the $q\bar{q}$ process~\cite{Kuhn:1998jr,Kuhn:1998jra}. 
This leads to an increase of the asymmetry as given by
pure QCD by a factor $1.09$. 

The asymmetry can in principle be studied experimentally in 
the partonic restframe, as a function of $\hat{s}$, by 
measuring the invariant mass of the $t \bar{t}$ system 
plus an eventually radiated gluon. 
It is, however, also instructive
to study the asymmetry in the laboratory frame by folding 
the angular distribution with the structure
functions~\cite{Martin:1996as,Lai:1997mg}.
For proton-antiproton collisions it is convenient to consider 
the forward-backward asymmetry as function of the 
production angle in the center of mass system.
The differential asymmetry for $\sqrt{s}=2$~TeV
is shown in Fig.~\ref{fig:PDFdistr}
which displays separately the contribution from 
$q\bar{q}$ and $qg$ (plus $\bar{q}g$) initiated reactions.
The denominator includes both $q\bar{q}$ and $gg$ initiated
processes in lowest order.
The numerator is evidently dominated by quark-antiquark annihilation. 

At this point we have to emphasize that both numerator and
denominator are evaluated in leading order (LO).
The next-to-leading (NLO) corrections to the $t\bar{t}$
production cross section are known to be large~\cite{Bonciani:1998vc},
around $30\%$ or even more.
In the absence of NLO corrections for the numerator we nevertheless
stay with the LO approximation in both numerator and denominator,
expecting the dominant corrections from collinear emission to cancel.
However, from a more conservative point of view an uncertainty of 
around $30\%$ has to be assigned to the prediction for the asymmetry.

For the total charge asymmetry  at $\sqrt{s}=1.8$~TeV we predict 
\beq
\bar{A} = 
\frac{N_t(\cos \theta \geq 0)
    - N_{\bar{t}}(\cos \theta \geq 0)}
     {N_t(\cos \theta \geq 0)
    + N_{\bar{t}}(\cos \theta \geq 0)} = 4.8 - 5.8 \%~,
\eeq
where different choices of the structure function and different
choices of the factorization and renormalization scale,
$\mu = m_t/2$ and $\mu = 2 m_t$,
have been considered and the factor $1.09$ is included.
An increase in the center of mass energy to $2$~TeV leads to a slight 
decrease of our prediction to $4.6-5.5 \%$.

%%%%%%%%%%%%%%%
\mafigura{8 cm}{cos_cteq1_2tev.ps}
{Differential charge asymmetry in the proton-antiproton restframe,
$\sqrt{s}=2$~TeV, using the CTEQ-1 structure function, $\mu=m_t$. 
The contributions from $q\bar{q}$ and $qg$ (plus $\bar{q}g$) 
initiated processes are shown separately.}
{fig:PDFdistr}
%%%%%%%%%%%%%%%

Top-antitop production in proton-proton collisions at the 
LHC is, as a consequence of charge conjugation symmetry,
forward-backward symmetric if the laboratory frame 
is chosen as the reference system. However, by selecting
the invariant mass of the $t \bar{t} (+g)$ system and its
longitudinal momentum appropriately, one can easily constrain
the parton momenta such that a preferred direction is 
generated for quark-antiquark reactions.


%For some of the more extreme kinematic regions a sizeable
%difference between top and antitop production can be 
%observed at the LHC (Fig.~\ref{fig:LHC_CTEQ1_cm}).
%The production cross section {\it per se}, which is decisive 
%for the possibility of measuring the asymmetry in 
%these regions is displayed in Fig.~\ref{}.
%In Fig.~\ref{fig:LHC_CTEQ1_cm} the contribution from $q\bar{q}$
%and $qg$ induced reactions are displayed separately. 
%In practice, only the region with $\hat{s}$ below $2$~TeV will 
%be observable, in particular at large $x$.

For some of the more extreme kinematic regions,
namely large $x$ and/or large $\hat{s}$, a sizeable difference
between top and antitop production can be observed at the LHC~\cite{Kuhn:1998jra}.
From this it may 
seem that the reconstruction of both $t$, $\bar{t}$ and even the 
gluon is required for the study of the charge asymmetry in $pp$
collisions. However, also the difference between 
the single particle inclusive distribution of $t$ and $\bar{t}$
respectively may provide evidence for the charge asymmetry. 
Production of $t\bar{t}(g)$ with negative $x$ is dominated 
by initial $\bar{q}$ with small $x_1$ and $q$ with large $x_2$.
The charge asymmetry implies that $Q(\bar{Q})$ is 
preferentially emitted into the direction of $q(\bar{q})$.
The same line of reasoning is applicable for 
positive $x$, with $Q(\bar{Q})$ again preferentially emitted in 
the direction of $q(\bar{q})$, and the role of $x_1$ and 
$x_2$ reversed. In total this leads to a slight preference 
for centrally  produced antiquarks and quarks slightly  
dominant in the forward and backward direction, 
i.e., at large positive and negative rapidities. 

%%%%%%%%%%%%%%%
\mafigura{8 cm}{singleinclusive_cteq1.ps}
{Rapidity distribution of charge asymmetry (a) and
total cross section at Born order (b) of top quark 
production in proton-proton collisions, $\sqrt{s}=14$~TeV and
$\mu=m_t$. Contributions from $q\bar{q}$ fusion 
and flavor excitation, $qg$($\bar{q}g$), are shown 
separately. Laboratory frame (CTEQ-1).}
{fig:singleinclusive}
%%%%%%%%%%%%%%%

The differential charge asymmetry 
\beq
A_{pp}(y) = \frac{\frac{\displaystyle dN(Q)}{\displaystyle dy}
- \frac{\displaystyle dN(\bar{Q})}{\displaystyle dy}}
{\frac{\displaystyle dN(Q)}{\displaystyle dy} 
+ \frac{\displaystyle dN(\bar{Q})}{\displaystyle dy}}~,
\eeq
is shown in Fig.~\ref{fig:singleinclusive}a for top quark
production at the LHC ($\sqrt{s}=14$~TeV).
As expected, a sizeable charge asymmetry is predicted 
in the region of large rapidity. It remains to be seen, 
if the low event rates in these extreme regions will permit 
the observation of this effect. The quark-gluon process is 
again negligible. 

\begin{thebibliography}{10}

\bibitem{Catani:1997rn}
S.~Catani, ``{QCD} at high-energies,'' {{\tt hep-ph/9712442}},
and references therein.

\bibitem{Tipton:1996}
P.~Tipton, ``Experimental top quark physics,'' {\em Proceedings of the ICHEP
  96, Warsaw, Poland} (World Scientific, Singapore, 1996), pg.123.

\bibitem{Kuhn:1998jr}
J.~H. K{\"u}hn and G.~Rodrigo {\em Phys. Rev. Lett.} {\bf 81}
  (1998) 49.

\bibitem{Kuhn:1998jra}
J.~H. K{\"u}hn and G.~Rodrigo, {{\tt hep-ph/9807420}}.

\bibitem{Martin:1996as}
A.~D. Martin, R.~G. Roberts, and W.~J. Stirling {\em Phys. Lett.} {\bf B387}
  (1996) 419, {{\tt hep-ph/9606345}}.

\bibitem{Lai:1997mg}
H.~L. Lai {\em et.~al.} {\em Phys. Rev.} {\bf D55} (1997) 1280,
{\tt hep-ph/9606399}.

\bibitem{Bonciani:1998vc}
R.~Bonciani, S.~Catani, M.~L.~Mangano and P.~Nason,
``NLL resummation of the heavy quark hadroproduction cross-section'', 
{{\tt hep-ph/9801375}}.



\end{thebibliography}

\end{document}

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