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\begin{document}
\begin{titlepage}
\begin{flushright}
LU TP 94

May 1994
\end{flushright}

\begin{center}
\LARGE
{\bf AN EVENT GENERATOR FOR FIRECRACKER MODEL AND RESCATTERING IN HIGH ENERGY 
NUCLEUS-NUCLEUS COLLISION-LUCIAE VERSION 1.0}\\
\vspace{0.3in}
\large
Ben-Hao Sa \footnote{Permanent address: China Institute of Atomic Energy, P.O.Box 275, 
Beijing,102413 China.E-mail address: CIAEDNP\%BEPC2@SCS.SLAC.STANFORD.EDU} and An Tai
\footnote{taian@thep.lu.se} \\
\vspace{0.3in}
Department of Theoretical Physics, University of Lund, \\
S\"{o}lvegatan 14 A, S-223 62 Lund, Sweden

\vspace{0.3in}


\end{center}


\vspace{0.4in}
\large
Abstract: 
LUCIAE  is a Monte Carlo program that implements FIRECRACKER model ,a model of gluon emission from colour
field of multi-string and RESCATTERING dealing with interaction of final hadrons for nucleus-nucleus collisions.
This paper includes a brief presentation of the picture and physics in the program and a description of it.
\end{titlepage}
\large
\section{Introduction} 
The main interest both theoretically and experimentally to study heavy ion collision at high energy for many people nowadays 
is to search Quark-Gluon Plasma(QGP) and investigate properties of the new matter state.Apparently the study
is very important not only because it provides a test of QCD on a new natural scale(interacting system with high energy density),
but also it might improve our understanding to more fundamental concepts as natures of confinement and chiral symmetry.Many 
signals have been suggested theoretically to be possible evidence of QGP.So far,the energy density needed for QGP formation 
($\sim$ 1-2 GeV/fm$^{3}$) from QCD prediction is believed having been reached at
SPS in 200 AGeV/c S-Au collision,but no unambiguous signal of QGP formation has been found experimentally in the sense that 
most of predictions based on QGP,though seen in experiment can be also explained by non-QGP models.Many theoretical problems 
about natures and formation of QGP and its evolution are still largely unknown to us.The fact suggests
that we need not only study more carefully about QGP physics itself,but also build non-QGP models based on known physics. 
Those models will be used to investigate if we are able to explain experiment data in high energy heavy ion collision using 
conventional physics and they can serve to be non-QGP background as well in order to check possible QGP signals.\\

FRITIOF is such a non-QGP model which has succeeded in describing many experiment data in hadron-hadron collisions from 
low energies at ISR up to top of SPS energy \cite{B.N}\cite{H.P1}by
careful treatment of gluon radiation (Soft Radiation Model SRM) and hard parton scattering (Rutherford Paton Scattering 
RPS).In FRITIOF,during collision two hadrons are excited due to momentum transfers and the highly excited,string-like objects 
(Colour Dipole) are allowed to emit gluons until transverse momentum of emission gluon reaches a given minimal cut.Then the 
formed objects (Lund String) are fragmented into final hadrons.
The FRITIOF model has been also developed to describe hadron-nucleus and nucleus-nucleus collision by assuming that reaction
is a linear superposition of binary hadron-hadron collisions in which collision geometry of nucleus plays an important role.This 
picture is supported by the fact that the global features in heavy ion collisions are satisfactorily explained by collision geometry 
together with independent hadron-hadron collisions.\\

On the other hand,people have also realized that when both baryon density and energy density are  high enough in high energy 
heavy ion collision,a few new features appear compared with hadron-hadron collisions,e.g.high $P_{t}$ enhancement ,strangeness 
enhancement,anti-baryon production and so on.They reveal that heavy-ion collision may possess some kinds of unknown 
characteristics or collectivity which are
not able to be described by superposition of independent hadron-hadron collisions.Formation of QGP is then suggested to be 
responsible for those new features.\\

In the language of string we can imagine that there are generally many `` string'' formed through relativistic heavy-ion collisions.
Those strings can thus overlap each other and produce a heavily interacting system in which behaviour of individual ``string'' is 
affected by the presence of other 
``strings'' and ``string'' in this case can not be treated independently any longer.In paper \cite{B.A1}
a picture of interacting-string (Colour Rope) fragmentation has been developed based on fragmentation 
scheme of single string in Lund Model,in which there is besides the ``individual'' string energy density
(the string tension at rest $\kappa \simeq$ 1 GeV/fm),also some interaction energy in the overlapping 
region of strings.String interaction in DPM model has also discussed in \cite{C.P}.\\

In the paper we propose FIRECRACKER (FCM) model to discuss how gluons are emitted from a system contained several strings
formed during collision.In the model gluons can be emitted collectively from colour field of the multi-string system.
On the other hand a lot of particles are produced in high energy nucleus-nucleus collisions .They will scatter with each other 
and with spectator nucleons when going through interaction region.The scattering (rescattering as it is called) effect has been 
considered important when we come to heavy-ion collisions\cite{sa}.FIRECRACKER model and Rescattering is found to be able to describe 
high $P_{t}$  enhancement and increase of strangeness.


The paper is designed as follows.After the introduction in Sect.2 we will give a brief review of Soft Radiation Model in FRITIOF.
Physics of FIRECRACKER model will be described in Sect.3 .Sect.4-Sect.6 deals with rescattering of final particles.Description of the
program will be found in Sect.7.
\section{Formation of colour dipole and its radiation in FRITIOF}
In FRITIOF hadron behaves like a relativistic string with a confined colour field
similar to the vortex line in a type II superconductor embedded in a superconducting vacuum.During collision between two 
hadrons there are a large amount of momentum transfers,most of them are rather small and uncorrelated,but it is assumed 
that no net colour is exchanged
between hadrons despite of the momentum transfers.The result of many such momentum transfers $q_{j}$ is 
assumed to be essentially additive,i.e.the string reacts as if it had been worked upon by a single momentum transfer 
$Q=\sum q_{j}$.There are also situations when one or more of the $q_{j}$'s have a large
transverse component,which corresponds to Rutheford parton-parton scattering(RPS)
Through collisions hadron gets excited and a colour dipole is formed with the hadronic remainder of colour \={3} 
moving in a direction opposite the stuck quark in the rest frame of initial hadron.Then the dipole is allowed to emit gluons d
described by well-known dipole emission formula for bremsstrahlung\cite{L.L}
%
\begin{equation}
d\sigma=\frac{3\alpha_{s}}{4\pi^{2}}\frac{dk_{\perp}^2}{k_{\perp}^2}dy
\label{f1}
\end{equation}
%
where $k_{\perp}$ and $y$ are transverse momentum and rapidity of a emitted gluon
respectively.
Energy momentum conservation provides constraint
%
\begin{equation}
k_{\perp}\cosh(y)\leq \frac{M}{2}                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    
\label{f2}
\end{equation}
%
with $M$ the invariant mass of the emitting dipole.This relation can be approximated with 
%
\begin{equation}
k_{\perp} \lsim e^{|y|}M
\label{f3}
\end{equation}
which means kinematically allowed phase space of the emitted gluon corresponds to a 
triangular region in the $y-\ln k_{\perp}$ plane as shown in Fig.1a\\

As we have described above a hadron in FRITIOF is like a vortex line,so many small momentum transfers are acted over an 
extended object.Therefore the source
responsible for gluon emission has certain finite size rather than being point-like as in $e^{+}e^{-}$ annihilation.It is 
assumed that,similar to photon emission from an electrical 
dipole,radiation with wavelengths short than the source size would be damped due to destructive interference.In a 
Lorenz frame where a gluon with transverse momentum $k_{\perp}$ is emitted at a
right angle to momenta of the string ends,the gluon wavelength is $\lambda \sim 1/k_{\perp}$.
Introducing a parameter $\mu_{0}$ to characterize the inverse of the source size,then only a piece of the source ,with energy fraction
\begin{equation}
\label{f4}
a(k_{\perp}) = \left({\mu_0\over k_{\perp}}\right)^\alpha
\end{equation}
can radiate coherently,where $\alpha$ is a parameter characterizing the dimensionality of the source.The available phase space 
represented by Eq.(\ref{f3}) in the rest frame of the string for the gluon emission is now further restricted into region 
%
\begin{equation}
k_{\perp} \lsim e^{|y|}a(k_{\perp})M
\label{f5}
\end{equation}
namely,gluon emission is suppressed,see Fig.1b.$\alpha$ has been set to unity, consistent with the assumption that the energy density 
has one-dimensional string-like shape and parameter $\mu_{0}$ is assumed the same for both ends of a dipole.\\

Then in Soft Radiation Model gluon emission from a colour dipole is repetitive application of Eq(\ref{f1}).The Original dipole splits into two 
dipoles after 
emitting one gluon,and the two dipoles go on to emit gluons in their own allowed 
phase space independently and more dipoles are formed to result in a partonic cascading process. Such a scheme has already been 
implemented with success in the ARIADNE Monte Carlo program \cite{ari}, 
which is then incorporated into FRITIOF program \cite{H.p12} to perform the gluon bremsstrahlung radiation.\\

For hadron-nucleus and nucleus-nucleus collision,the process can be regarded as incoherent collisions between their nucleons.Thus a 
nucleon from the projectile interacts independently with the
encountered target nucleons as it passes through the nucleus.The probability distribution for the number of the inelastic collision $\nu$ will 
be taken in accordance with geometric calculations.
Each of these sub-collision can be treated the same way as a usual
hadron-hadron collision.So in this picture a projectile nucleon passing through the nucleus
will exchange momentum each time it encounters
a target nucleon. If it interacts with $\nu$ nucleons in the target,
$\nu+1$ excited string states will be formed as a result.  These string
states (colour dipole) will then have associated bremsstrahlung radiations based on Soft Radiation Model described above independently and 
finally fragment into hadrons.\\
\section{Physics of Model}
\subsection{Gluon Emission from Colour Field of Multi-String}
In heavy-ion collision many strings can be formed in certain space-time volume in one event.When generalizing hadron-hadron process into 
hadron-nucleus and nucleus-nucleus collision in FRITIOF an assumption (or approximation) has been made that a formed string can emit gluons without 
noticing existence of other strings.The assumption would be questionable when density of string is large.
As we know in QED case spectrum of photon emission is strongly affected by the external field in which the emission occurs.When photon
 emitting from electron transition of an atom 
in external electromagnetic field we have $induced\ emission$ and $spontaneous\ emission$ which have different emission probability.
By analogy we think that gluon emission from string would be of different feature  under the field of other colour strings around.\\

Here we will consider a string dynamical scenarium.In particular we will extend and provide a detailed structure for the FCM 
for
collective multiple gluon emission\cite{fire}.The idea is that the QCD vacuum might work as a color superconductor type II.Then the basic 
objects
should be QCD force-fields similar to vortex-lines Such objects must have dynamical properties very similar to the mass-less relativistic 
string,
which is at the basis of The Lund Model.When many string-like QCD force fields are available in a small space-time region it is conceivable
 that
they interact.As we will discuss below there are some possible non-trivial collective bremsstrahlung emission phenomena in this situation.\\


For an electro-magnetic superconductor type II there are three regions of interest when one considers the magnetic field $\cal H$ as
the order parameter.When the field is below a certain number,$\cal H$$_{1}$ the Meissner effect works but in a region  
$\cal H$$_{1}$$\leq$
$\cal H$$\leq$$\cal H$$_{2}$ the quantised vortex lines start to penetrate the superconducting state and exhibit a repulsive interaction.
Above
$\cal H$$_{2}$ the field is so strong that the whole region "melts" into a state of the same character as the break-down of a superconductor 
type I.This is the usual intuitive picture of the QGP.

The FCM might work inside the corresponding intermediate region(with the energy density as the order parameter).This means that the
strings at this stage interact but still keep some individuality.\\

In high energy nuclear collissions there will in the FRITIOF Model be many strings stretched out and in case they work like vortex
lines there may be some interaction between the strings(the repulsive interaction between the "ordinary" electro-magnetic vortex
lines means the storing of some energy in between them).We can rewrite Eq(\ref{f5}) when $\alpha$ is unity
%
\begin{equation}
k_{\perp} \leq k_{\perp max} =\sqrt{\mu_{0} M}
\label{f5.1}
\end{equation}
We note that the factor $\mu_{0} M$ in Eq(\ref{f5.1}) effectively corresponds to an energy density over the region 1/$\mu_{0}$.\\

In the FCM it is assumed that in the very early stage of the collision when several ($n$) excited strings (masses $W_{j}$) are close by 
{\em it 
may be possible for them to emit gluons using their common total energy density $\mu_{0} W_{tot}$ }.\\

In order to estimate the effect we may for simplicity assume that the strings are at rest relative to each other with the same masses
$W_{j}=W$.This means that {\em each string may be able to take energy from the common pool nW.} Thus one may in principle be able to
emit $n$ gluons,each with a transverse momentum limited by $\sqrt{\mu_{0} nW}$!\\

In our model,a hadron is in analogy to vortex line in type II superconductor.When
excited hadrons come together  after collision between two nuclei we could have a bunch of vortex lines forming an interacting system in 
space-time 
as shown in Fig.2 with several colour-triplets(quarks) on one end of the system and the same amount of anti-colour-triplets(diquarks) 
on the other end.

We illustrate in Fig.3 (in the case when the system is formed by three strings for simplicity)how gluons could be emitted 
from the interacting system.Let's assume that the system has invariant mass 
%
\begin{equation}
W_{tot}\equiv((\sum_{j} E_{j})^{2}-(\sum_{j} P_{j})^{2})^{1/2}
\label{f6}
\end{equation}               
%
with $E_{j}$ and $P_{j}$ being energy and momentum of individual string and gluons emitted in the same way as single 
excited string from the system.Then a gluon (firecracker gluon)could now be emitted from   
a phase space corresponding to the  largest triangle with cut-off expressed by 
Eq(\ref{f5}).\\

In the model  strings are not allowed to melt them  together and  distribute their energies uniformly into the system like the 
case in type I superconductor.Instead individual string still keeps its identity being a vortex line.
Therefore a gluon emitted from the interacting system is still regarded being owned by a single string,which means that an 
emitted gluon will be rejected if it does not fall
into kinematically allowed phase space (described by Eq(\ref{f3})) of an individual string .After firecracker gluon emission,
strings are considered independent and treated like a normal dipole.\\

One case needs to be mentioned here.In FRITIOF RPS gluon always corresponds to hardest gluon on a string.Therefore a RPS 
gluon will be drowned  if one string gets a firecracker gluon harder than the RPS gluon on the string as the case when a 
RPS gluon
is drowned by bremsstrahlung background \cite{H.p12}.And the firecracker gluon is now looked upon as a hard gluon.\\

\subsection{Clusters in Nucleus-Nucleus Collision}
In hadron-nucleus collision it is reasonable assuming that all the participant nucleons from target form a single interacting 
system after collision because the hadron mainly hits  nucleons aligned on its way out of the target nucleus and those 
participant nucleons should therefore be close to each other transversely.But it is not the case for Nucleus-Nucleus Collision 
when projectile nucleons have different
impact parameters with respect to target nucleus.Here we apply an algorithm introduced in \cite{S.C} to cluster participant 
nucleons.The point of the algorithm
is to make participants that have small relative transverse momenta into a group because it is those participants that could 
move together in space after collisions and form a parallelly stretched string system as shown in Fig.2.Let us recall algorithm 
briefly.\\

1)For every target nucleon(the same with projectile nucleons)$n_{k}$ and for every pair $n_{k},n_{l}$ one computers the 
corresponding value of resolution variable defined by
%
\begin{equation}
d_{kl}=\min(P_{tk},P_{tl}) \sqrt{(\eta_{k} - \eta_{l})^{2}+(\phi_{k} - \phi_{l})^{2} } 
\label{f7}
\end{equation}               
%
and find a pair $n_{i},n_{j}$ with minimum `distance' $d_{ij}$.Where $P_{tk}$($P_{tl}$) is transverse momentum of nucleon 
$n_{k}$($n_{l}$).$\eta_{k}$($\eta_{l}$) and $\phi_{k}$($\phi_{l}$) are pseudrapidity and azimuthal angle of nucleon 
$n_{k}$($n_{l}$).\\

2)If $d_{ij}$ is smaller than a given value $d_{cut}$,$n_{i}$ and $n_{j}$ have to be combined into a single particle 
(`pseudoparticle') with four-momentum given by sum of four-momentum of the two nucleons,i.e.,
%
\begin{equation}
p_{(ij)}=p_{i} + p_{j}
\label{f8}
\end{equation}               
%

3)Repeat the two steps above for all nucleons and pseudoparticles until all pairs of objects(nucleons and/or pseudoparticles) 
have $d_{kl}$ $>$ $d_{cut}$.Whatever objects remain at this stage are called $cluster$.\\

Note that in small-angle limit that is generally true for participants of high energy collision Eq.(\ref{f7}) can be replaced by 
$\min(E_{k},E_{l})\theta_{kl} $ with $\theta_{kl}$ being angle between the momenta of nucleons $n_{k}$ and $n_{l}$,
which 
is nothing but minimal relative transverse momentum of one nucleon with respect to the other,and the algorithm is thus 
longitudinal-boost-invariant.\\

In the model participant nucleons from target and projectile are clustered separately since they move away from each 
other after collisions.In calculation we have used mean transverse momentum of participant nucleons to be  $d_{cut}$.\\

\section{Initiation of rescattering transport process}
\subsection{Formation Time}
In FRITIOF a crucial assumption is made that on the time scale of the collision process,the excited nucleonic state does not
fragment in the intermediate stage and so there is no intra nuclear cascade.This assumption is not reasonable any more
when mass of target is big.Moreover reinteractions among produced hadrons are not included in FRITIOF,the process is 
also considered important in high energy heavy ion collisions.For describing those processes we should first know when a
hardon is formed.\\

In LUND fragmentation model probability distribution in invariant time (formation time of a hadron) $\tau$ 
of $q$$\bar{q}$ breakup
vertices has been given  in terms of $\Gamma=(\kappa \tau)^2$ as \cite{B.A2}
%
\begin{equation}
P(\Gamma)d\Gamma \propto \Gamma^a exp(-b\Gamma)d\Gamma
\label{f8}
\end{equation}               
%
with its mean value ,
%
\begin{equation}
<\Gamma> = \frac{a+1}{b}
\label{f8}
\end{equation}               
%
Here $a$ and $b$ are fragmentation parameters and $\kappa$ is energy density of string.They have been determined
through experimental data,which allow us to find mean formation time $<\tau>$ to be $1\sim 2$ fm.For simplicity,we
use the $<\tau>$ to be formation time of all hardrons produced from a string.In Lab frame,formation time of a hardon
is then $t=<\tau>E/m$.

\subsection{Initiation in spatial space}
Now fragmentation of strings formed in nucleus-nucleus occurs in a nuclear environment.For initiation in spatial space 
we remain assuming  participant-
spectator picture. Hadrons produced from 
fragmentation would scatter with each other as well as with spectator nucleons(rescattering transport process).
We assume that those produced
hadrons will distribute randomly in space in an overlap region between target and projectile nucleus sphere which is formed 
when projectile passes through target nucleus.And
target (projectile)spectator nucleons are arranged randomly outside the overlap region and inside target(projectile)
nucleus sphere and are given thermal momenta sampled from Boltzmann(Gaussian)distribution.Non-overlapping 
demand is required for nucleons here and the minimum allowed distance between nucleons is assumed to be equal to 0.5 fm.\\

\subsection{Initiation in momentum space}
It is obvious that primary produced particles have had their four momentum from 
FRITIOF properly. 
For projectile spectator nucleons,it is reasonable to assume that they share total longitudinal momentum of projectile 
spectator 
nucleons in  FRITIOF output and their transverse momenta are sampled based on two dimensional Gaussian distribution
\begin{equation}
f(p_t)dp_t = Cp_t\exp[-\frac{p_t^2}{<p_t^2>}]dp_t .                      
\end{equation}   
As for  $<p_t^2>$ it can be assumed to be a few tens of MeV and  transverse momentum of last projectile spectator nucleon should be 
decided by transverse momentum conservation. Here energy-momentum conservation is required.

In case of collider experiment initiation of spectator nucleons for both colliding nuclei can be done in the same way mentioned 
above. In case of fixed target experiment the value of three momentum of target spectator nucleon is sampled from Boltzmann 
distribution
\begin{eqnarray}
f(p)dp & = & C\frac{p^2}{T^{1.5}}\exp[-\frac{p^2}{2m_nT}]dp ,\\
<p^2> & = & 3m_nT ,                      
\end{eqnarray} 
where C is  normalization constant, $T$ (in GeV) stands for  temperature 
of spectator nucleons which can be assumed to be a few tens of MeV ,$m_n$ refers to  mass of nucleon. As for the direction of 
three 
momentum it is assumed to be 
isotropically distributed in 4$\pi$. Of course the three momentum of last target spectator nucleon is decided by  energy-
momentum 
conservation as well.\\


\section{Classical Motion of Particle and Collision Time}
All particles are assumed to be classical point like objects in rescattreing transport processes. Particle moves along Newton 
trajectory within two consecutive collisions
\begin{equation}
\vec{r}_i = \vec{R}_i + \vec{\beta}_i(t-T)
\end{equation}
where $\vec{R}_i$ and $T$ is  four coordinate of $i$-th particle at the moment of last collision.

In order to chose  next collision with least collision time one has to calculate  time lasted from last collision to next 
collision for all pairs of particles and then to make a comparison among them. 
Two particles will collide if their distance approaches $d_{min}$ and 
\begin{equation}
d_{min} \leq \sqrt{\frac{\sigma_{tot}}{\pi}},
\end{equation}
where $\sigma_{tot}$ is total cross section in fm$^2$. Since $d_{min}$ 
depends on  reference system it is necessary to view this problem in an inertia frame,where the distance of closest 
approach $d_{min}$ will be perpendicular to momenta of both particles \cite{inert}. CMS frame of two colliding particles is 
chosen 
here. Take particles $i$ and $j$ for example,their equations of motion are,respectively
\begin{eqnarray}
\vec{r}^{\ast}_i&=&\vec{R}^{\ast}_i + \vec{\beta}^{\ast}_i(t^{\ast}-T^{\ast}_i),\\
\vec{r}^{\ast}_j&=&\vec{R}^{\ast}_j + \vec{\beta}^{\ast}_j(t^{\ast}-T^{\ast}_j).
\end{eqnarray}
Thus one can calculate square of relative distance between $i$ and $j$, $d^2(t^{\ast})$.
On the other hand,from following condition  
\begin{equation}
\frac{\partial (d^2(t^{\ast}))}{\partial t^{\ast}} = 0 ,
\end{equation}
one can get  $d^2_{min}$ and  corresponding collision time of particles $i$ and $j$
\begin{equation}
t^{\ast}_{ij} =  [\frac{\Delta\vec{R}^{\ast}}{\Delta\vec{\beta}^{\ast}} + 
     \frac{1}{\Delta\vec{\beta}^{\ast}}(\vec{\beta}^{\ast}_jT^{\ast}_j -
     \vec{\beta}^{\ast}_iT^{\ast}_i)]\frac{\Delta\vec{\beta}^{\ast}}
     {\Delta\vec{\beta}^{\ast}} ,
\end{equation}
where
\begin{eqnarray}
\Delta\vec{R}^{\ast} &= &\vec{R}^{\ast}_j - \vec{R}^{\ast}_i ,\\
\Delta\vec{\beta}^{\ast} &= &\vec{\beta}^{\ast}_j - \vec{\beta}^{\ast}_i .
\end{eqnarray}
The d$^2_{min}$ should satisfy
\begin{equation}
d^2_{min} \leq \frac{\sigma_{tot}}{\pi}.
\end{equation}
and collision should happen in future thus
\begin{equation}
t^{\ast}_{ij} \geq Max(T^{\ast}_i,T^{\ast}_j) .
\end{equation}

When collision times of all pairs have been calculated in their own CMS one 
has to transfer them back to lab system. In lab system the collision time $t^L_i$ and $t^L_j$ of colliding 
particles $i$ and $j$ might not be equal and we should chose the earliest one as  collision time of particles $i$ and $j$ in lab
system as usual,i.e.
\begin{equation}
t^L_{ij} = Min(t^L_i,t^L_j).
\end{equation} 
%

\section{Particle-Particle Collision}
\subsection{Elastic scattering}
Particle-particle collision is performed in their CMS,no matter elastic or inelastic scattering. If total and elastic cross section 
satisfies 
\begin{equation}
\frac{\sigma_{el}}{\sigma_{tot}} \geq \xi
\end{equation}
where $\xi$ is a random number,then particles will scatter elastically.

For elastic scattering it is assumed,as usual,that  Mandelstam variable $t$ (i.e. negative of the momentum transfer) to be 
distributed as\cite{elas}
\begin{eqnarray}
\frac{d\sigma}{dt} &\sim &\exp(Bt) ,\\
B &= &\frac{\epsilon^6}{1+\epsilon^6}A ,\\
\epsilon &= &3.65(\sqrt{s}-m_i-m_j) ,\\
A &= &Min[10.3,(1.12<p_t>)^{-2}]  ,
\end{eqnarray}
where $<p_t>$ is average transverse momentum of rescattered particle. Thus $t$ is calculated 
\begin{eqnarray}
t  &=  &\frac{1}{B}\ln[\xi+(1-\xi)\exp{(Bt_0)}] ,\\
 t_0 &= &-4(p^\ast_i)^2 .
\end{eqnarray} 
Scattering pole angle is then calculated from
\begin{equation}
t = -2(p^\ast_i)^2(1-\cos(\theta^\ast)).
\end{equation}
As for azimuthal angle it is given by
\begin{equation}
\varphi{^\ast}=2\pi\xi.
\end{equation}

\subsection{Inelastic collision}
 It is impossible to include all inelastic channels in the program since there are so many possibilities.We have to restrain
ourselves to those channels that are of interest and importance.At the moment we are going to investigate production 
of strangeness particles 
K,$\Lambda$,$\bar{\Lambda}$ and $\Sigma$ from rescattering of final particles,so we focus on following two kinds of
reactions:\\

1) Reaction of strangeness production\\
$\pi$ + $\pi$$\rightarrow$ $\bar{K}$+$K $     \ ,\ \ $\pi + N$ $\rightarrow$$ K+Y $  \ ,\ \    $\pi$ + $\bar{N}$ $\rightarrow$ 
$\bar{K}$+$\bar{Y} $ \ \ \    

1) Reaction of strangeness exchange\\
\ \ \ $\bar{K}$ + $N$ $\rightarrow$ $\pi+Y $ \ \ \   

Where $Y$=$\Lambda$ or $\Sigma$.The reaction $N+N $$\rightarrow$ $N+Y+K$ plays almost no role due to arguement of 
threshold,
aboundance and phase space \cite{stra}.Reaction$ K$ + $\bar{N}$ $\rightarrow$ $\pi$+$\bar{Y}$ are neglected because 
of low 
aboundance of  $\bar{N}$.And finally reaction $\bar{K}$ +$ N $$\rightarrow$ $\Xi$+$K$ is not included since its cross 
section is low and 
threshold energy is high in comparison with reaction $\bar{K}$ + $N $$\rightarrow$ $\pi$+$Y$.\\

By the consideration above only following inelastic reactions are included in the program:\\

\begin{enumerate}
\item $\pi^{+}\pi^{-}\rightarrow{K^+K^-}$ ,
\item $\pi^{+}\pi^{0}\rightarrow{K^+\bar{K}^0}$ ,
\item $\pi^{0}\pi^{0}\rightarrow{K^+K^-}$ ,
\item $\pi^{-}p\rightarrow{K^+\Sigma^-}$ ,
\item $\pi^{-}p\rightarrow{K^0\Lambda}$ ,
\item $\pi^{-}p\rightarrow{K^0\Sigma^0}$ ,   
\item $\pi^{+}p\rightarrow{K^+\Sigma^+}$ ,
\item $\pi^{+}n\rightarrow{K^+\Sigma^0}$ ,
\item $\pi^{+}n\rightarrow{K^+\Lambda}$ ,
\item $\pi^{0}p\rightarrow{K^+\Sigma^0}$ ,
\item $\pi^{0}p\rightarrow{K^+\Lambda}$ ,
\item $\pi^{0}n\rightarrow{K^+\Sigma^-}$ ,
\item $\pi^{-}\bar{n}\rightarrow{K^-\bar{\Lambda}}$ ,
\item $\pi^{0}\bar{p}\rightarrow{K^-\bar{\Lambda}}$ , 
\item $\pi^{+}\bar{p}\rightarrow{\bar{K}^0\bar{\Lambda}}$ ,
\item $K^-p\rightarrow{\pi^{0}\Lambda}$ ,
\item $K^-p\rightarrow{\pi^{0}\Sigma^0}$ ,
\item $K^-p\rightarrow{\pi^{-}\Sigma^+}$ ,
\item $K^-p\rightarrow{\pi^{+}\Sigma^-}$ ,
\item $K^-p\rightarrow{\bar{K}^0n}$ ,
\item $K^-n\rightarrow{\pi^{-}\Lambda}$ .
\end{enumerate}
If the reaction cross section of any one of processes above ($i$-th,for example) satisfies
\begin{equation}
\frac{\sigma^i_{reac}}{\sigma_{inel}} \geq \xi 
\end{equation}
(be careful threshold energy of a reaction here) corresponding inelastic process will happen,
otherwise the collision should be treated as 
elastic scattering. In equation above  $\sigma_{inel}$ refers to  total inelastic cross section.

Since lack of experimental data (for simplicity as well) we make constant assumption for  total,elastic and inelastic 
cross sections for $nn$,$\pi$$n,Kn$ and $\pi\pi$ interactions,and following values are  adopted: $\sigma^{NN}_{tot}$ = 40 
mb,$\sigma^{\pi{N}}_{tot}$ = 25 mb,$\sigma^{KN}_{tot}$ = 35 mb and $\sigma^{\pi\pi}_{tot}$ = 10mb. The ratio of inelastic 
to total cross section is assumed to be 0.75 $\sim$ 0.85. As for the cross section of $\sigma_{\pi\pi\rightarrow{KK}}$ 
it is assumed,as usual,to be constant and a value of 2.0 mb is used. An isospin averaged 
parameterization formulas of \cite{stra} are adopted for the cross section of $\pi N $$\rightarrow{K+Y}$ . The cross section 
of 
$\bar{K}$$N$ induced strangeness exchange reactions is assumed to be equal 
to ten times as much as the corresponding cross section of $\pi$$N$ induced strangeness production reaction \cite{stra}. As for the 
cross 
section of $\pi\bar{n}$ 
and $\pi\bar{p}$ interactions they are assumed to be equal to the corresponding cross section of $\pi p$ and $\pi n$ interactions,
respectively.\\

The cross sections of inelastic channels listed above given by the isospin averaged parameterization formulas of 
Ref.\cite{stra} for $\pi N$ $\rightarrow{K+Y}$ decrease exponentially with CMS energy of two colliding particles .But on the other 
hand we know that
total inelastic cross sections of  $\pi N$ are approximately energy-independent,which means that more  inelastic channels
would take place like $\pi^{-} p$ $\rightarrow K^{*0}+\Lambda$ $\rightarrow K+ \pi+\Lambda$ and   $\pi^{-} p$ 
$\rightarrow K^{*0}+\Sigma^{0}$ $\rightarrow K+ \pi+\Sigma^{0}$ etc.,when CMS energy of two colliding particles is
increasing.In some sense the 21 reactions we listed above should be looked upon as 'representive channels'
of certain reactions.Therefore instead of applying  energy-dependent inelastic cross section,we could alternatively give 
a constant 'effective cross section' to reactions $\pi N$ $\rightarrow{K+Y}$.

As usual, two body final state,i.e. energy and momentum of the scattered particles of the 
inelastic reactions above can be 
calculated by two-body decay formulas \cite{inel} in CMS of colliding particles,i.e.
\begin{eqnarray}
E^{\ast}_i &= &\frac{s-m^2_j+m^2_i}{2\sqrt{s}} ,\\
|\vec{p}^{\ast}_i| &= &\frac{[(s-(m_i+m_j)^2)(s-(m_i-m_j)^2)]^{0.5}}{2\sqrt{s}} . 
\end{eqnarray} 
The orientation of three momentum is isotropically sampled in 4$\pi$.

It should be mentioned that the orientation of scattered particle is relative 
to scattering particle (the same in elastic scattering). Thus one has to 
rotate it into  CMS frame of colliding particles before transforming back to 
lab system. 
\section{Description of the program}
LUCIAE1.0 is a subroutine package for simulating gluon emission in FIRECRA\-CKER model and the rescattering of  produced 
particles from FRITIOF in nuclear environment. The program is written in FORTRAN 77. It should be used together with 
FRITIOF7.02\cite{H.p12}, JETSET7.3\cite{jet}, PYTHIA 5.5\cite{py} and ARIADNE4.02\cite{ari}.A user should read 
paper \cite{H.p12} first before using LUCIAE1.0 if he is not familiar with FRITIOF7.02. For running together
with LUCIAE1.0,some modification has been made in FRITIOF7.02 and ARIADNE4.02 and they are called
FRITIOF7.02R and ARIADNE4.02R.One important thing that a user has to keep in mind is that dimension size of 
common block COMMON/LUJETS/ (KSZJ) and COMMON/FRPARAI (KSZ1) has been extended.Now KSZJ=20000 and KSZ1=30.One 
who
has used FRI\-TIOF7.02 before will find it is very easy to use FRITIOF7.02R which is linked to LUCIAE1.0 since input
and output structure of the two programs are completely same.One simply starts with subroutine FREVENT and ends with 
final state particles (after rescattering) 
recorded in event record LUJETS.That is all ! You need ,of course to write your own analyses program.\\

Rescattering part of LUCIAE1.0 is quite time consuming.Furthermore,time consumption increases 
rapidly with mass of nuclei and collision energy.From S+S at lab frame  200GeV/c to Au+Au at CMS 200GeV/c
time consumption in this part of the program increases comparing with FRITIOF from a factor of 0.6 to 6.0 ! If a user
is not interested in distributions highly affected by rescattering process,like variables related to strangeness
production he might like to run LUCIAE without rescattering.This can be done by setting KFR(21)=0.\\

\subsection{User interface}
This section contains the most essential information a user should know about the program,namely the common block
FRPARA1 and LUCIDAT2 for input parameters and switch controls.The input parameters are provided with sensible default 
values.Several options are also included to make the program more accessible to user.\\



\begin{itemize}
\item{PARAMETER (KSZ1=30) \\
      COMMON/FRPARA1/KFR(KSZ1), VFR(KSZ1)}\\
\begin{description}
\item[Note:] KFR(1-14) and VFR(1-16) are used in FRITIOF7.02.We still list them here since some small change has been made.
\item[KFR(1)] (D=1)
       Fragmentation
      \begin{description}
       \item[=0] Off.
       \item[=1] On.   
      \end{description}
\item[KFR(2)] (D=1) 
       Multiple gluon emission (dipole radiation) 
      \begin{description}
       \item[=0] Off.
       \item[=1] On.
      \end{description}
\item[KFR(3)] (D=0) Event selection for collisions with a nucleus 
      \begin{description}
       \item[=0] Generate minimum bias events (all interactions recorded).
       \item[=1] Generate only events with all projectile nucleons
                  participated.
       \item[=2] Generate only events with impact parameter between
                  $b_{min}$=VFR(1) and $b_{max}$=VFR(2).
       \item[=3] Apply both requirements in 1 and 2.
      \end{description}
\item[KFR(4)] (D=1)
       Fermi motion in nuclei 
      \begin{description}
       \item[=0] Neglected.
       \item[=1] Included.
      \end{description}
\item[KFR(5)] (D=0) 
       Nucleon-nucleon overlap function 
      \begin{description}
       \item[=0] Eikonal.
       \item[=1] Gaussian.
       \item[=2] Gray disc.
      \end{description}
\item[KFR(6)] (D=2)
       Target Nucleus deformation
      \begin{description}
       \item[=0] No deformation.
       \item[=1] Deformed target nucleus.
       \item[=2] Apply deformation only if the target atomic number $A\geq 80$.  
      \end{description}
\item[KFR(7)] (D=1) 
       Rutherford parton scattering processes
      \begin{description}
       \item[=0] Off.
       \item[=1] On. Here only the hardest RPS is used in FRITIOF.    
       \item[=2] On. The full multiple hard scattering scenario of PYTHIA
               is used.
      \end{description}
\item[KFR(8)] (D=1) 
      Hard gluons cause a corner (soft gluon kink) on the string
      \begin{description}
       \item[=0] No kink is formed. 
       \item[=1] Gluon kink is formed.
      \end{description}
\item[KFR(9)] (D=1)
        `Drowning' of Rutherford parton scattering 
      \begin{description}
       \item[=0] Off. Accept all RPS events.    
       \item[=-1] On. Throw away the drowned RPS event completely 
                  and replace it by a purely soft event.
       \item[=1] As in -1, but the 
            transverse momentum transfer of the soft collision is superimposed
            by the $q_T$ of the drowned RPS.  
      \end{description}
\item[KFR(10)] (D=1) 
        SRM parameters in RPS events: $\mu_1=\mu_0/r$, $\mu_2=\mu_0/(1-r)$
      \begin{description}
      \item[=0] $\mu$ remains the same as in a soft event: $\mu_1=\mu_2=\mu_0$.  
      \item[=1] $r$\,=\,VFR(16). 
      \item[=2] $r$ takes a uniform distribution in (0,1).
      \end{description}
\item[KFR(11)] (D=4)
      Write out of a message when the arguments in FREVENT is changed.  
      \begin{description}
      \item[=-1] Write it out every time the change occurs.    
      \item[=$n$ ($n\geq 0$)] The write out is limited to $n$ times.    
      \end{description}
\item[KFR(12)] (D=2) 
        Set up of the dipole cascade and string fragmentation parameters.
      \begin{description}
      \item[=0] No set up.  The default values are used. 
      \item[=1] Set to the values optimised by 
       OPAL collaboration \cite{opal}:
       PARA(1)=0.20, PARA(3)=1.0, PARJ(21)=0.37, PARJ(41)=0.18, PARJ(42)=0.34.  
      \item[=2] Set to the values optimised by DELPHI collaboration:
       PARA(1)=0.22, PARA(3)=0.6, PARJ(21)=0.405, PARJ(41)=0.23, PARJ(42)=0.34.  
       \end{description}
\item[KFR(13)] (D=0) 
       Compresses the event record to save space in LUJETS.  This switch
       is particularly needed for heavy ion collisions at high energy
       where LUJETS must be compressed before it gets overfilled.  
      \begin{description}
      \item[=0] Do not compress LUJETS.  
      \item[=1-3] LUEDIT(KFR(13)) is called and LUJETS is compressed. 
        Specifically, for KFR(13)=1 fragmented jets and 
        decayed particles are removed, for KFR(13)=2 neutrinos and
        unknown particles are also removed, and for KFR(13)=3
        neutral particles are further excluded.
      \item[=4] A dummy subroutine FREDITD() is provided as an interface
        in which a user may write his own special purpose codes to edit
        and compress LUJETS.    
      \end{description}
\item[KFR(14)] (D=0) 
       Checked for charge and energy-momentum  conservation
      \begin{description}
      \item[=1] The outcome of each event will be checked.
      \item[=0] No checked. 
       \end{description}       
\item[KFR(15)] (D=1) 
       FIRECRACKER model
      \begin{description}
      \item[=1] On.
      \item[=0] Off.
       \end{description}       
\item[KFR(16)] (D=1) 
       Firecracker gluon
      \begin{description}
      \item[=1] Accept firecracker gluons which are kinematically allowed.
      \item[=0] Reject all firecracker gluons.It is equivalent to KFR(15)=0.
       \end{description}     
\item[KFR(17)] (D=1) 
       'Drowning' of firecracker gluons by bremsstrahlung radiation
      \begin{description}
      \item[=1] No drowning.
      \item[=0] Drowning.
       \end{description}     
  \item[KFR(18)] (D=1) 
         Size of firecracker cluster 
      \begin{description}
      \item[=1] Size of firecracker cluster = VFR(17)$<P_{t}>$,
         with $<P_{t}>$ being average transverse momentum of nucleons in the cluster.
      \item[=0] size of firecracker cluster =VFR(18).
       \end{description}   
\item[KFR(19)] (D=0) 
       Form of output ,only used when KFR(21)=0
      \begin{description}
      \item[=1] Output in event record includes decayed strings.
      \item[=0] Decayed strings have been taken away in output.
       \end{description}       
\item[KFR(20)] (D=1) 
       Cross section for  $\pi N$ $\rightarrow{K+Y}$
      \begin{description}
      \item[=1] Constant cross section,=VFR(20).
      \item[=0] Energy-dependent cross section.
       \end{description}       
\item[KFR(21)] (D=1) 
       Rescattering
      \begin{description}
      \item[=1] On.
      \item[=0] Off.
       \end{description}     
\item[KFR(22)] (D=1) 
       Rescattering channel
      \begin{description}
      \item[=1] All channels are included.
      \item[=0] Only special channels are included.
       \end{description}     
\item[KFR(23)] (D=0) 
       Checked for charge and energy-momentum  conservation
        after rescattering
      \begin{description}
      \item[=1] The outcome of each event will be checked.
      \item[=0] No checked. 
       \end{description}     

%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\vskip 12pt

\item[VFR(1)] (D=0.0 fm)
       Minimum impact parameter for options KFR(3)=2 or 3.
\item[VFR(2)] (D=0.2 fm)
       Maximum impact parameter for options KFR(3)=2 or 3.
\item[VFR(3)] (D=0.8 fm)
       The minimum allowable distance $R_{min}$ between nucleons in a nucleus.  
\item[VFR(4-5)] (D=0.2, 0.1)
       Dipole and quadrupole deformation coefficients for deformed target
       nucleus.
\item[VFR(6)] (D=0.01 GeV$^2/c^2$)
       The $<Q_T^2>$ for the Gaussian distribution of soft transverse 
       momentum transfer.
\item[VFR(7)] (D=0.20 GeV$^2/c^2$)
       The $<Q^2_{2T}>$ for the Gaussian distribution of primordial transverse
       momenta on the string ends.       
\item[VFR(8)] (D=0.75 GeV)
       Soft radiation coherence parameter $\mu_0$ for projectile hadron or
       nucleon. 
\item[VFR(9)] (D=0.75 GeV)
       Soft radiation coherence parameter $\mu_0$ for target hadron or nucleon. 
\item[VFR(10-11)] (D=0.0, 0.0 mb)
       Projectile-target nucleon total and elastic cross 
       sections, respectively.
       By default, they are taken from the parametrization 
       of Block and Cahn \cite{block} (MSTP(31)=5 in PYTHIA).  
       The meson-nucleon cross sections are obtained
       simply by scaling down the Block-Cahn fit. The scale factor is
       $(2/3-a/\sqrt s)$, where $a=1.13$ GeV for pions and $a=3.27$ GeV for kaons
       are chosen to reproduce the low energy experimental data. For
       all the other baryons, it is treated as a pion if it is a meson
       and it is treated as a proton if it is a baryon.
       User may override the default by setting VFR(17-18) to positive values.
       However, the user assigned cross sections will only affect the
       N-N interaction probability in nucleus collisions. The
       probability for Rutherford parton scattering is not affected.  
\item[VFR(12)] (D=1.0 GeV/$c$)
       The $q_{Tmin}$ for Rutherford parton scattering.  
\item[VFR(13-15)] (D=1/6, 1/3, 1/2)
       The probabilities for assigning various spins and flavours to the diquark
       end of the string.  For example in a proton, VFR(13-15) are the
       probabilities of finding a $ud$ diquark of spin 1,
       a $uu$ diquark of spin 1, and a $ud$ diquark of spin 0, respectively. 
\item[VFR(16)] (D=0.5)
       The fraction $r$ in option KFR(10)=1. 
\item[VFR(17)] (D=0.75)
       Size of firecracker cluster with respect to average transverse momentum of nucleons 
in the cluster.
       See also KFR(18).
\item[VFR(18)] (D=0.35GeV/c)
       Size of firecracker cluster when KFR(18)=0
\item[VFR(19)] (D=6.5GeV)
       Minimal CMS energy for starting FIRECRACKER model.
\item[VFR(20)] (D=0.3mb)
      Cross section when KFR(20)=1.
\end{description}
\end{itemize}
\begin{itemize}
\item COMMON/LUCIDAT2/KFMAXT,PARAM(20)\\
\begin{description}
\item[KFMAXT] (D=22)
       Maximum number of particle species included in rescattering.
\item[PARAM(1)] (D=40.0mb)
       total cross-section of $N-N$ reaction.
\item[PARAM(2)] (D=25.0mb)
       total cross-section of $\pi-N$ reaction.   
\item[PARAM(3)] (D=35.0mb) 
       total cross-section of $K-N$ reaction.    
\item[PARAM(4)] (D=10.0mb) 
       total cross-section of $\pi$-$\pi$ reaction.    
\item[PARAM(5)] (D=2.0mb) 
       Cross section of $\pi+\pi \rightarrow K K$.
\item[PARAM(6)] (D=0.85) 
       Ratio  of inelastic cross-section to  total cross-section.
\item[PARAM(7)] (D=1.5fm) 
       Formation time of a hadron at its rest-frame.
\item[PARAM(8)] (D=0.02fm) 
       Minimal step of time elapsing.
\item[PARAM(9)] (D=0.1) 
       Accuracy of four-momentum conservation when sampling spectator nucleons.
\item[PARAM(10)] (D=1.0) 
       Size of effective rescattering region is PARAM(10)$R_{t}$ with $R_{t}$ being 
       radius of target.
\item[PARAM(11)] (D=0.16/fm$^3$) 
       Nucleon density of nucleus.
\item[PARAM(12)] (D=0.04 GeV$^2$$/c^2$) 
       The $<P_{t}^2>$ for the Gaussian distribution of spectator nucleons.
\end{description}
\end{itemize} 
\subsection{Other common blocks}
The following common blocks are used for transmitting data internally.While they
can be accessed to read out information,they should never be used to input data.
\vskip 12pt
\begin{itemize}
\item COMMON/FCRSOUT/MCL(10),INEL(40),NST(2,KSZ4),ACL(2,KSZ4) \\
\begin{description}
\item[MCL(1): ]  
       Number of cluster from projectile.
\item[MCL(2): ]  
       Number of cluster from target.
\item[MCL(3): ]  
       Number of accepted firecracker gluon.
\item[MCL(4): ]  
       Number of initial collision pairs of rescattering.
\item[INEL(I): ]  
       total number that I-th inelstic channel takes place in aggregate.Order corresponds
                                 to list in section6.2.
\item[NST(L,J): ] 
       Number of strings in J-th cluster .L=1,2 - index to label the projectile 
(L=1) and target (L=2).
\item[ACL(L,J): ]  
       Invariant mass of J-th cluster .Index L has the same meaning as 
      in NST(L,J)  
\end{description}
\end{itemize}
\begin{itemize}
\item PARAMETER (mcludi=20000) \\
   common/wz/c17(mcludi,3),ishp(mcludi),tp(mcludi)\\
\begin{description}
\item[c17(i,j):] 
       Coordinate of i-th particle,j=1,2,3 stands for x,y,z,respectively.The 
       origin of Coordinate is at center of target and z along beam direction.
\item[tp(i):]     
       Time of i-th particle 
\item[ishp(i):] 
       Status of i-th particle.=1 inside rescattering region and =0 outside.
          \end{description}
          \end{itemize} 

Except information which is stored in particle list (COMMON/JETSETS/),other important information
for proceeding rescattering is stored in collision list.The Variables relevant to the collision list are:

\begin{description}
\item[lc(i,j):]  i indicates the order number in collision list,\\

j=1 and 2 give the order number of colliding particles in particle list, respectively,\\
j=3 and 4 give the flavor code of scattered particles,respectively, \\ 
j=5 indicates which kind of inelastic scattering is selected, \\
\item[tc(i):]   The collision time ,\\ 
\item[tw(i):]   The ratio of reaction cross section of selected inelastic process to total .
\end{description}
\subsection{The main subroutine}
Following subroutines build main framework of the program.It might be worthwhile to know.
\begin{itemize}
\item SUBROUTINE  FRCLUST\\
to cluster wounded nucleons by their relative transverse momenta.
\item SUBROUTINE  FRARCLU \\
to emit gluons from FIRECRACKER model.
\item SUBROUTINE  FRCRGLU \\
 to assign firecracker gluons to strings.
\item SUBROUTINE FRINGES(KT,PT,VT,KT0,PT0,NB,N0) \\
 to emit bremsstrahlung gluons from strings without RPS.
\item SUBROUTINE FRINGEH \\
 to emit bremsstrahlung gluons from strings with RPS.
\item SUBROUTINE FRRESEXE \\
 to administrate rescattering of final particles.
\item SUBROUTINE even$_{-}$init(lc,tc,tw,i) \\
 to proceed initiation of rescattering transport process.
\item SUBROUTINE rescat(i,ij) \\
 to proceed rescattering.
\item SUBROUTINE ctlcre(lc,tc,tw,n)\\
 to create initial collision list.
\item SUBROUTINE his(time,lc,tc,tw,istop \\
 to perform classical Newton motion
\item SUBROUTINE coinel(l,l1,ss,b,pi,pj,icp,pii,pjj,lc,tc,tw,winel,ik3,ik4) \\
 to check if an inelastic reaction could take place.Go head if one channel
       is chosen.
\item SUBROUTINE coelas(l,l1,ss,pi,pj) \\
 to perform elastic scattering.
\item SUBROUTINE updatl(l,l1,time,lc,tc,tw,winel,n) \\
 to update collision list after rescattering of a pair of particles. 
\end{itemize}
\subsection{Logical flow of the program}
\begin{enumerate}
\item Cluster formation:After wounded nucleons are produced from FRITIOF those participant nucleons from
projectile and target are clustered,separatively.
\item Emission of firecracker gluons:Gluons are emitted from those clusters and shared by strings.
\item Bremsstrahlung radiation:\\
\ \ \ \ \ a) The strings without RPS gluon are sent into Ariadne for bremsstrahlung radiation.\\
\ \ \ \ \ b) A string with RPS gluon will first compare hardness of its RPS gluon with that of 
         firecracker 
      gluon if it happens to have one and the RPS gluon would be drowned and replaced by the
     firecracker gluon if the firecracker gluon is harder.Then the string is sent into  Ariadne for 
     bremsstrahlung radiation.
\item Fragmentation:Strings after bremsstrahlung radiation will go into Jetset for fragmentation.
\item Formation of initial collision list: Rescattering transport process starts by creation 
of initial collision list.
\item Find next collision: From the collision list the collision with earliest collision time is selected to perform collision.
\item Move along Newton trajectory: Each particle is moved along its own trajectory from moment of last collision to current one.
\item Adjudge and perform the collision: Adjudge what kind of scattering (elastic or inelastic) will take place and perform the 
corresponding scattering.
\item Update collision list: All collisions,in the collision list,involving 
scattering particles i or/and j should be removed away from the list. Collision list is updated.
\item Loop: Loop over until no collision pair is left in the collision list.
\end{enumerate}
\begin{itemize} 
\begin{description}
\item [Remark:] If a user wishes to include more rescattering channels besides the 21 channels in present program,he could
       change the criteria in subroutine 'ctlcre' and corresponding one in subroutine 'updatl'.At the same time he should also add
       relevant cross sections and reactions in subroutine 'prod'.
\end{description}
\end{itemize} 


\section{Acknowledgments}

One of authers,A.Tai would like to thank Prof. B.Andersson for initiating the idea of the FIRECRACKER 
model and constant instruction of the work.Ben-Hao Sa would like to thank Prof. B. Andersson for kind 
invitation. Both of us would like to thank Prof.G.Gustafson for helpful discussions with respect to 
rescattering.Special thanks also go to Dr.Hong Pi,from whom we have benefited a lot through 
discussions with respect to FRITIOF program.




\newpage
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   An updated version of this paper, ``A Manual to The Lund Monte Carlo
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\end{thebibliography}

 \newpage
\parskip 6ex




\begin {center}

\epsfig{file=plotfr1.eps,height=7cm,width=8cm}
\mbox{ }

Fig.1a:The phase space limit for emission of a gluon.



\epsfig{file=plotfr2.eps,height=7cm,width=8cm} 
\mbox{ }

Fig.1b:The available phase space for gluon emission from a extended colour dipole.

\newpage
\parskip 6ex
\epsfig{file=plotfr3.eps,height=3cm,width=13cm}

\mbox{ }
Fig.2:A cluster of nucleons after collisions.



\epsfig{file=plotfr4.eps,height=8cm,width=9cm} 

\mbox{ }
Fig.3:The available phase space for firecracker gluon emission. 

\end{center}                       


\end{document}

