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\begin{document} \title{Multiple scattering in EPOS}
\author{K. Werner$^{(a)}$, T.$\,$Pierog$^{(b)}$, S. Porteboeuf$^{(a)}$}
\institute{$^{(a)}$SUBATECH, University of Nantes -- IN2P3/CNRS-- EMN, Nantes, France\\
$^{(b)}$ Forschungszentrum Karlsruhe, Institut fuer Kernphysik,
Karlsruhe, Germany}
\maketitle
\begin{abstract}
We discuss the multiple scattering approach in EPOS and its consequences
in particular for proton-proton scattering at the LHC.
\end{abstract}

\section{Introduction}

It has been known since a long time that very high energy hadrons
experience multiple scatterings when they hit protons or neutrons.
Concerning inclusive cross sections, the situation becomes quite simple
due to the fact that different multiple scattering contributions cancel
due to destructive interference (AGK cancellations). The corresponding
formulas are simple and can be expressed in terms of parton distributions
functions, based on evolutions equations (DGLAP, BFKL, BK).

To get more detailed information, one needs partial cross sections,
since individual hadronic interactions are of a particular multiple
scattering type (single, or double, or triple...) and contribute differently
to certain observables. Even if the inclusive cross sections were
perfectly known, one still would need addition information concerning
the treatment of multiple scattering. Here, Gribov-Regge theory provides
a solution, in particular when energy sharing is properly taken into
accound, as in the EPOS approach.

An important issue is the concept of remnants, based on the hypothesis
that in a hadron-hadron collision there are three sources of particle
production: (1) hadrons from partons which are due to the parton evolution,
(2) hadrons from projectile remant excitations, and (3) hadrons from
target remnants. Remnants are meant to be the spectator partons from
the incident hadrons, representing hadron excitations. In the language
of cut diagrams such contributions must exist. There is not much guidance
from theory, how to define a {}``remnant model''. However, we expect
the remants to be rather energy independent, so one may rely on the
wealth of data at relatively low energies ($\sqrt{s}\approx20-1800$GeV)
to test the model assumptions.

Concerning the partons from the parton evolution (source (1) in the
previous paragraph), we expect that low momentum fraction (low$\, x$)
partons do not simply evolve following linear evolution equations
(like DGLAP or BFKL). There are nonlinear effect becoming more and
more important (with decreasing $x$ and increasing nuclear mass number
in case of collisions with nuclei), finally leading to saturation.
Apart of the theoretical reasoning discussed earlier, one needs such
{}``nonlinear effects'' to tame the hadron-hadron cross sections
at very high energies (which would otherwise {}``explode''). So
any realistic model needs to deal with saturation, in a more or less
sophisticated way. 

Finally, if one wants to make precise predictions concerning the hadron
chemistry, a crucial ingredient is the fragmentation procedure. Concerning
the low transverse momentum hadrons (representing the overwhelming
majority of all particles), the preferred procedure is the string
approach. Using fragmentation functions is certainly a useful concept
for jet fragmentation, but not necessarily for soft particle production.


\section{Parton evolution in EPOS}

An elementary scattering in EPOS \cite{epos} is given by a so-called
{}``parton ladder'', see fig. \ref{cap:Elementary-interaction},
\textbf{\large }%
\begin{figure*}[htb]
\begin{center}\includegraphics[%
  scale=0.45]{ladderfluxaa.eps}\end{center}


\caption{Elementary interaction in the EPOS model.\label{cap:Elementary-interaction}}
\end{figure*}
 representing parton evolutions from the projectile and the target
side towards the center (small $x$). The evolution is gouverned by
an evolution equation, in the simplest case according to DGLAP. In
the following we will refer to these partons as {}``ladder partons'',
to be distinguished from {}``spectator partons'' to be discussed
later. It has been realized more than 20 years ago that such a parton
ladder may be considered as a longitudinal color field, conveniently
treated as a relativistic string when it comes to hadronization. The
intermediate gluons are treated as kink singularities in the language
of relativistic strings. A string decays via the production of quark-antiquark
pairs, creating in this way string fragments -- which are identified
with hadrons. Such a picture is also in qualitative agreement with
recent developments concerning the CGC. 

Important in particular at moderate energies (RHIC): our {}``parton
ladder'' is meant to contain two parts \cite{nexus}: the hard one,
as discussed above (following an evolution equation), and a soft one,
which is a purely phenomenological object, parametrized in Regge pole
fashion. The soft part essentially compensates for the infrared cutoffs,
which have to be employed in the perturbative calculations. 

As discussed earlier, at high energies one needs to worry about non-linear
effects, due to the fact that the gluon densities get so high that
gluon fusion becomes important. In our language this means that two
partons ladders fuse (or split, if we look from inside to outside
\cite{epos}). Nonlinear effects could be taken into account by using
BK instead of DGLAP evolution. What we try to realize here is a phenomenological
approach, which (hopefully) grasps the main features of these non-linear
phenomena, and still remains technically doable (we should nor forget
that we finally have to generalize the treatment in order to take
into accound multiple scatterings, as discussed earlier).

Our phenomenological treatment is based on the fact that there are
two types of nonlinear effects: a simple elastic rescattering of a
ladder parton on a projectile or target nucleon (elastic ladder splitting),
or an inelastic rescattering (inelastic ladder splitting), see fig.
\ref{split}. The elastic process provides screening, therefore a
reduction of total and inelastic cross sections. The importance of
this effect should first increase with mass number (in case of nuclei
being involved), but finally saturate. The inelastic process will
affect particle production, in particular transverse momentum spectra,
strange over nonstrange particle ratios, etc. Both, elastic and inelastic
rescattering must be taken into account in order to obtain a realistic
picture.%
\begin{figure*}[htb]
\begin{center}\includegraphics[%
  scale=0.45]{laddersplitela.eps}$\qquad\qquad$ \includegraphics[%
  scale=0.45]{laddersplitine.eps}$\,$\end{center}


\caption{Elastic (left) and inelastic (right) {}``rescattering'' of a ladder
parton. We refer to (elastic and inelastic) parton ladder splitting.
\label{split}}
\end{figure*}


To include the effects of elastic rescattering, we first parameterize
a parton ladder (to be more precise: the imaginary part of the corresponding
amplitude in impact parameter space) computed on the basis of DGLAP.
We obtain an excellent fit of the form $\alpha(x^{+}x^{-})^{\beta}$,
where $x^{+}$ and $x^{-}$ are the momentum fractions of the {}``first''
ladder partons on respectively projectile and target side (which initiate
the parton evolutions). The parameters $\alpha$ and $\beta$ depend
on the cms energy $\sqrt{s}$ of the hadron-hadron collision. To mimick
the reduction of the increase of the expressions $\alpha(x^{+}x^{-})^{\beta}$
with enegy, we simply replace them by $\alpha(x^{+})^{\beta+\varepsilon_{P}}(x^{-})^{\beta+\varepsilon_{T}}$,
where the values of the positive numbers $\varepsilon_{P/T}$ will
increase with the nuclear mass number and $\log s$.

The inelastic rescatterings (ladder splittings, looking from insider
to outside) amount to providing several ladders close to projectile
(or target) side, which are close to each other in space. They cannot
be consider as independend color fields (strings), we should rather
think of a common color field built from several partons ladders.
In the string language one used the term {}``string fusion'', where
the fused string is still an one-dimensional longitudinal object,
but with a modified string tension $\kappa$. Also this string tension
is expected to increase with the nuclear mass number and $\log s$
(for more details see \cite{epos}). This affects hadronization, since
the flavor dependence of $q-\bar{q}$ string breaking is given by
the probabilities $\exp(-\pi m_{q}^{2}/\kappa$), with $m{}_{q}$
being the quark masses. Also mean transverse momenta are affected,
since they are proportional to $\sqrt{\kappa}$.


\section{Remnants in EPOS}

Still the picture is not complete, since so far we just considered
two interacting partons, one from the projectile and one from the
target. These partons leave behind a projectile and target remnant,
colored, so it is more complicated than simply projectile/target deceleration.
One may simply consider the remnants to be diquarks, providing a string
end, but this simple picture seems to be excluded from strange antibaryon
results at the SPS \cite{sbaryons}. We therefore adopt the following
picture: not only a quark, but a two-fold object takes directly part
in the interaction, namely a quark-antiquark or a quark-diquark pair,
leaving behind a colorless remnant, which is, however, in general
excited (off-shell). If the first ladder parton is a gluon or a seaquark,
we assume that there is an intermediate object between this gluon
and the projectile (target), referred to as soft Pomeron. And the
{}``initiator'' of the latter on is again the above-mentionned two-fold
object.

So we have finally three {}``objects'', all of them being white:
the two off-shell remnants, and the parton ladder in between. Whereas
the remnants contribute mainly to particle production in the fragmentation
regions, the ladders contribute preferentially at central rapidities. 

We showed in ref. \cite{nex-bar} that this {}``three object picture''
can solve the {}``multi-strange baryon problem'' of ref. \cite{sbaryons}.
In addition, we assembled all available data on particle production
in pp and pA collisions between 100 GeV (lab) up to Tevatron, in order
to test our approach. Large rapidity (fragmentation region) data are
mainly accessible at lower energies, but we believe that the remnant
properties do not change much with energy, apart of the fact that
projectile and target fragmentation regions are more or less separated
in rapidity. But even at RHIC, there are remnant contribution at rapidity
zero, for example the baryon/antibaryon ratios are significantly different
from unity, in agreement with our remnant implementation. So even
central rapidity RHIC data allow to confirm out remnant picture.


\section{Factorization and Multiple Scattering}

An inclusive cross section is one of the simplest quantities to characterize
particle production. Often one need much more information, for example
when trigger conditions play a role. Also in case of shower simulations
one needs information about exclusive cross sections (the widely used
pQCD generators are not event generators in this sense, they are generators
of inclusive spectra, and a Monte Carlo event is not a physical event).
As discussed earlier, inclusive cross section are particulary simple,
quantum interference helps to provide simple formulas referred to
a {}``factorization''. Although factorization is widely used, strict
mathematical profs exist only in very special cases, and certainly
not for hadron production in $pp$ scattering.

To go beyond factorization and to formulate a consistent multiple
scattering theory is difficult. A possible solution is Gribov's Pomeron
calculus, which can be adapted to our language by identifying Pomeron
and parton ladder. Multiple scattering means that one has contributions
with several parton ladders in parallel. This formulation is equivalent
to using the eikonal formula to obtain total cross sections from the
knowledge of the inclusive one.

We indicated several years ago inconsistencies in this approach, proposing
an {}``energy conserving multiple scattering treatment'' \cite{nexus}.
The main idea is simple: in case of multiple scattering, when it comes
to calculating partial cross sections for double, triple ... scattering,
one has to explicitly care about the fact that the total energy has
to be shared among the individual elementary interactions.\textbf{\large }%
\begin{figure}[htb]
\begin{center}\includegraphics[%
  scale=0.4]{laddermult.eps}\end{center}


\caption{Multiple scattering with energy sharing.\label{cap:Energy}}
\end{figure}
 In other words, the partons ladders which happen to be parallel to
each other share the collision energy, see fig. \ref{cap:Energy}.

A consistent quantum mechanical formulation of the multiple scattering
requires not only the consideration of the usual (open) parton ladders,
discussed so far, but also of closed ladders, representing elastic
scattering. These are the same closed ladders which we introduced
earlier in connection with elastic rescatterings. The closed ladders
do not contribute to particle production, but they are crucial since
they affect substantially the calculations of partial cross sections.
Actually, the closed ladders simply lead to large numbers of interfering
contributions for the same final state, all of which have to be summed
up to obtain the corresponding partial cross sections. It is a unique
feature of our approach to consider explicitly energy-momentum sharing
at this level (the {}``E'' in the name EPOS). For more details see
\cite{nexus}. 


\section{Hadronization}

As mentionned already, the fragmentation procedure is a crucial ingredient
of our model. Here, we employ the string approach. Using fragmentation
functions is certainly a useful concept for jet fragmentation, but
not necessarily for soft particle production.

We will identify parton ladders with classical strings. Here, we consider
only strings $x$ with piecewise constant intial conditions $v(\sigma)\equiv\partial x/\partial\tau(\sigma,\tau=0)$,
which are called kinky strings. So the string is characterized by
a sequence of $\sigma$ intervals $[\sigma_{k},\sigma_{k+1}]$, and
the corresponding velocities $v_{k}$. Such an interval with the corresponding
constant value of $v$ is referred to as {}``kink''. Now we are
in a position to map partons onto strings: we identify the ladder
partons with the kinks of a kinky string, such that the length of
the $\sigma$-interval is given by the parton energies, and the kink
velocities are just the parton velocities. The string evolution is
then completely given by these initial conditions, expressed in terms
of parton momenta. Hadron production is finally realized via string
breaking, such that string fragments are identified with hadrons.
Here, we employ the so-called area law hypothesis: the string breaks
within an infinitesimal area $dA$ on its surface with a probability
which is proportional to this area, $dP=p_{B}\, dA,$where $p_{B}$
is the fundamental parameter of the procedure. 


\section{Collective expansion}

Recent developments in EPOS concern the hydrodynamic expansion of
matter in case of heavy ion collisions -- or high multiplicity events
in very high energy proton-proton scattering, for example at the LHC.

The elementary scatterings as discussed above lead to the formation
of strings, which break into segments, which are usually identified
with hadrons. When it comes to high multiplicity events in very high
energy proton-proton scattering, the procedure is modified: one considers
the situation at an early proper time $\tau_{0}$, long before the
hadrons are formed: one distinguishes between string segments in dense
areas (more than some critical density $\rho_{0}$ of segments per
unit volume), from those in low density areas. The high density areas
are referred to as core, the low density areas as corona \cite{core}.
Let us consider the core part. It is important to note that initial
conditions from EPOS are based on strings, not on partons. Based on
the four-momenta of the string segments which constitute the core,
we compute the energy density $\varepsilon(\tau_{0},\vec{x})$ and
the flow velocity $\vec{v}(\tau_{0},\vec{x})$. 

Having fixed the initial conditions, the system evolves according
the equations of ideal hydrodynamics, see fig. \ref{cap:Hydrodynamic-evolution-in},
until the energy density reaches some critical value (usually expressed
in terms of a critical temperature). In the simplest case, particles
freeze out immediately at this freeze out hypersurface, based on the
Cooper-Frye prescription.

%
\begin{figure}[htb]
\begin{center}\includegraphics[%
  scale=0.48]{space3.eps}\end{center}


\caption{Sketch of a hydrodynamic evolution in space time, starting from the
hyperbola representing the initial proper time.\label{cap:Hydrodynamic-evolution-in}}
\end{figure}


%
\begin{figure}[htb]
\vspace{-0.5sp}
\begin{center}\includegraphics[%
  scale=0.3,
  angle=270]{z-selpp-2.ps}\end{center}
\vspace{-1cm}
\caption{The mean transverse momentum of (from top to bottom) lambdas, kaons,
and pions, in pp collisions at 1800 GeV. A ``hydro-inspired''
EPOS simulation is compared to data from CDF.\label{cap:The-mean-transverse}}
\end{figure}
The interesting question arises whether such {}``collective expansion
effects'' matter for pp. There are several signs which suggest this,
for example the increase of the mean transverse momentum of hadrons
in pp collisons observed at the Tevatron collider \cite{cdf}, see fig. \ref{cap:The-mean-transverse}.
Here, one sees the typical {}``flow pattern'', namely a considerably
larger increase of the mean pt's in case of heavier hadrons. The EPOS
calculations are, however, not (yet) based on a hydrodynamical evolution,
they are based on a statistical hadronization with imposed collective
flow, the latter one introduced by hand. Real hydrodynamical calculations
will be performed soon.


\section{Summary}

To summarize: we have discussed multiple scattering as realized by
the EPOS model, which is expected to be a very important issue for
proton-proton scattering at the LHC.

\begin{thebibliography}{1}
\bibitem{epos}Klaus Werner, Fu-Ming Liu, Tanguy Pierog, Phys. Rev. C 74, 044902
(2006), arXiv: hep-ph/0506232
\bibitem{nexus}H. J. Drescher, M. Hladik, S. Ostapchenko, T. Pierog and K. Werner,
Phys. Rept. 350, 93, 2001
\bibitem{sbaryons}M. Bleicher, F. M. Liu, A. Ker�nen, J. Aichelin, S.A. Bass, F. Becattini,
K. Redlich, and K. Werner, Phys.Rev.Lett.88, 202501, 2002.
\bibitem{nex-bar}F.M. Liu, J.Aichelin, M.Bleicher, H.J. Drescher, S. Ostapchenko, T.
Pierog, and K. Werner, Phys. Rev. D67, 034011, 2003
\bibitem{core}K. Werner, Phys. Rev. Lett. 98, 152301 (2007)
\bibitem{cdf}D. Acosta, Phys. Rev. D 65, 072005 (2002)\end{thebibliography}

\end{document}
