source: trunk/documents/UserDoc/DocBookUsersGuides/PhysicsReferenceManual/latex/hadronic/theory_driven/ChiralInvariantPhaseSpace/CHIPS.tex @ 1211

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\chapter{Chiral Invariant Phase Space Decay.}
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\newtheorem{theorem}{Theorem}
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\newtheorem{conclusion}[theorem]{Conclusion}
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% \title{Manual for the CHIPS event generator in GEANT4}
%\author{M.V.Kossov}
%\address{Mikhail.Kossov@itep.ru, Mikhail.Kossov@cern.ch, kossov@jlab.org,\\
%kossov@post.kek.jp}
% \date{\today}
% \maketitle

\section{Introduction}

\noindent \qquad  
The CHIPS computer code is a quark-level event generator for the 
fragmentation of hadronic systems into hadrons.  In contrast to other parton 
models \cite{Parton_Models} CHIPS is nonperturbative and
three-dimensional. It is based on the Chiral Invariant Phase Space
(ChIPS) model \cite{CHIPS1,CHIPS2,CHIPS3} which employs a
3D quark-level SU(3) approach. Thus Chiral Invariant Phase Space refers
to the phase space of massless partons and hence only light (u, d, s)
quarks can be considered. The c, b, and t quarks are not implemented
in the model directly, while they can be created in the model as a
result of the gluon-gluon or photo-gluon fusion. The main parameter of
the CHIPS model is the critical temperature $T_c\approx 200~MeV$. The
probability of finding a quark with energy $E$ drops with the energy
approximately as $e^{-E/T}$, which is why the heavy flavors of quarks
are suppressed in the Chiral Invariant Phase Space. The s quarks,
which have masses less then the critical temperature, have an
effective suppression factor in the model.

The critical temperature $T_c$ defines the number of 3D partons in
the hadronic system with total energy $W$. If masses of all partons
are zero then the number of partons can be found from the equation
$W^2=4T_c^2(n-1)n$. The mean squared total energy can be calculated
for any ``parton'' mass (partons are usually massless). The
corresponding formula can be found in \cite{hadronMasses}. In this
treatment the masses of light hadrons are fitted better than by the 
chiral bag model of hadrons~\cite{Chiral_Bag} with the same number of
parameters. In both models any hadron consists of a few quark-partons, 
but in the CHIPS model the critical temperature defines the mass of 
the hadron, consisting of $N$ quark-partons, while in the bag
model the hadronic mass is defined by the balance between the
quark-parton internal pressure (which according to the uncertainty
principle increases when the radius of the ``bag'' decreases) and the
external pressure ($B$) of the nonperturbative vacuum, which has
negative energy density.

In CHIPS the interactions between hadrons are defined by the Isgur
quark-exchange diagrams, and the decay of excited hadronic 
systems in vacuum is treated as the fusion of quark-antiquark or
quark-diquark partons. An important feature of the model is the
homogeneous distribution of asymptotically free quark-partons over the
invariant phase space, as applied to the fragmentation of various
types of excited hadronic systems. In this sense the CHIPS model may
be considered as a generalization of the well-known hadronic phase
space distribution \cite{GENBOD} approach, but it generates not only
angular and momentum distributions for a given set of hadrons, but
also the multiplicity distributions for different kinds of hadrons,
which is defined by the multistep energy dissipation (decay) process.  

The CHIPS event generator may be applied to nucleon excitations,
hadronic systems produced in $e^{+}e^{-}$ and $p\bar p$ annihilation,
and high energy nuclear excitations, among others. Despite its quark
nature, the nonperturbative CHIPS model can also be used successfully
at very low energies. It is valid for photon and hadron projectiles
and for hadron and nuclear targets. Exclusive event generation models
multiple hadron production, conserving energy, momentum, and other
quantum numbers. This generally results in a good description of
particle multiplicities, inclusive spectra, and kinematic correlations
in multihadron fragmentation processes. Thus, it is possible to use
the CHIPS event generator in exclusive modeling of hadron cascades in
materials.

In the CHIPS model, the result of a hadronic or nuclear interaction is
the creation of a quasmon which is essentially an intermediate state
of excited hadronic matter.  When the interaction occurs in vacuum the
quasmon can dissipate energy by radiating particles according to the
quark fusion mechanism~\cite{CHIPS1} described in section \ref{annil}.
When the interaction occurs in nuclear matter, the energy dissipation
of a quasmon can be the result of quark exchange with surrounding
nucleons or clusters of nucleons \cite{CHIPS2} (section \ref{picap}),
in addition to the vacuum quark fusion mechanism.

In this sense the CHIPS model can be a successful competitor of the
cascade models, because it does not break the projectile, instead it
captures it, creating a quasmon, and then decays the quasmon in
nuclear matter. The perturbative mechanisms in deep inelastic
scattering are in some sense similar to the cascade calculations,
while the parton splitting functions are used instead of
interactions. The nonperturbative CHIPS approach is making a ``short
cut'' for the perturbative calculations too. Similar to the time-like
$s=W^2$ evolution of the number of partons in the nonperturbative
chiral phase space (mentioned above) the space-like $Q^2$ evolution of
the number of partons is given by $N(Q^2)=n_V+\frac{1}{2\alpha_s(Q^2)}$, 
where $n_V$ is the number of valence quark-partons. The running
$\alpha_s(Q^2)$ value is calculated in CHIPS as
$\alpha_s(Q^2)=\frac{4\pi}{\beta_0ln(1+Q^2/T_c^2)}$, where
$\beta_0^{n_f=3)=9}$. In other words, the critical temperature $T_c$
plays the role of $\Lambda_QCD$ and still cuts out heavy flavors of
quark-partons and high orders of the QCD calculation (NLO, NNLO,
N$^3$LO, etc.), substituting for them the effective LO ``short cut''. 
This simple approximation of $\alpha_s$ fits all the present measurements
of this value (Fig.~\ref{alphas}).
It is very important that
$\alpha_s$ is defined in CHIPS for any $Q^2$, and that the number of partons
at $Q^2=0$ converges to the number of valence quarks.

\begin{figure}
% \centerline{\epsfig{file=hadronic/theory_driven/ChiralInvariantPhaseSpace/plots/mommul.eps, height=3.5in, width=4.5in}}
%  \resizebox{1.00\textwidth}{!}
%{
\includegraphics[angle=0,scale=0.6]{hadronic/theory_driven/ChiralInvariantPhaseSpace/plots/alpha.eps}
%\includegraphics[angle=0,scale=0.6]{plots/alpha.eps}
%}
\caption{The CHIPS fit of the $\alpha_s$ measurements.}
\label{alphas}
\end{figure}

The effective $\alpha_s$ is defined for all $Q^2$, but at $Q^2=0$ it
is infinite. In other words at $Q^2=0$ the number of the virtual
interacting partons goes to infinity.  This means that on the boundary
between perturbative and non-perturbative vacuums a virtual
``thermostate'' of gluons with an effective temperature $T_c$
exists. This ``virtual thermostate'' defines the phase space
distribution of partons, and the ``thermalization'' can happen very
quickly. On the other hand, the CHIPS nonperturbative approach can be used
below $Q^2~=~1~GeV^2$. This was done for the neutrino-nuclear
interactions  (section \ref{numunuc}).

\section{Fundamental Concepts}

The CHIPS model is an attempt to use a set of simple rules which govern 
microscopic quark-level behavior to model macroscopic hadronic systems with 
a large number of degrees of freedom. The invariant phase space distribution
as a paradigm of thermalized chaos is applied to quarks, and simple 
kinematic mechanisms are used to model the hadronization of quarks into 
hadrons. Along with relativistic kinematics and the conservation of quantum 
numbers, the following concepts are used:

\begin{itemize}
\item {\bf Quasmon:} in the CHIPS model, a quasmon is any excited hadronic 
system; it can be viewed as a continuous spectrum of a generalized
hadron.  At the constituent level, a quasmon may be thought of as a
bubble of quark-parton plasma in which the quarks are massless and the
quark-partons in the quasmon are homogeneously distributed over the
invariant phase space.  It may also be considered as a bubble of the
three-dimensional Feynman-Wilson \cite{Feynman-Wilson} parton gas. The
traditional hadron is a particle defined by quantum numbers and a
fixed mass or a mass with a width. The quark content of the hadron is
a secondary concept constrained by the quantum numbers. The quasmon,
however, is defined by its quark content and its mass, and the concept
of a well defined particle with quantum numbers (a discrete spectrum)
is of secondary importance. A given quasmon hadronic state with fixed
mass and quark content can be considered as a superposition of
traditional hadrons, with the quark content of the superimposed
hadrons being the same as the quark content of the quasmon.

\item {\bf Quark fusion:} the quark fusion hypothesis determines the rules 
of final state hadron production, with energy spectra reflecting the 
momentum distribution of the quarks in the system. Fusion occurs when a 
quark-parton in a quasmon joins with another quark-parton from the same 
quasmon and forms a new white hadron, which can be radiated. If a
neighboring nucleon (or the nuclear cluster) is present, quark-partons
may also be exchanged between the quasmon and the neighboring nucleon
(cluster). The kinematic condition applied to these mechanisms is that
the resulting hadrons are produced on their mass shells. The model
assumes that the u, d and s quarks are effectively massless, which
allows the integrals of the hadronization process to be done easily
and the modeling decay algorithm to be accelerated. The quark mass is
taken into account indirectly in the masses of outgoing hadrons. The
type of the outgoing hadron is selected using combinatoric and
kinematic factors consistent with conservation laws.  In the present
version of CHIPS all mesons with three-digit PDG Monte Carlo codes
\cite{CH.PDG} up to spin $4$, and all baryons with four-digit PDG
codes up to spin $\frac{7}{2}$ are implemented. 

\item {\bf Critical temperature} the only non-kinematic concept of the model
is the hypothesis of the critical temperature of the quasmon. This has a 
40-year history, starting with Ref.~\cite{Hagedorn} and is based on the 
experimental observation of regularities in the inclusive spectra of hadrons
produced in different reactions at high energies.  Qualitatively, the 
hypothesis of a critical temperature assumes that the quark-gluon hadronic 
system (quasmon) cannot be heated above a certain temperature.  Adding more 
energy to the hadronic system increases only the number of constituent 
quark-partons while the temperature remains constant.  The critical 
temperature is the principal parameter of the model and is used to
calculate the number of quark-partons in a quasmon.  In an infinite
thermalized system, for example, the mean energy of partons is $2T$
per particle, the same as for the dark body radiation.  

\end{itemize}

\section{Code Development}

Because the CHIPS event generator was originally developed only for final 
state hadronic fragmentation, the initial interaction of projectiles with 
targets requires further development.  Hence, the first applications of 
CHIPS described interactions at rest, for which the interaction cross 
section is not important \cite{CHIPS1}, \cite{CHIPS2}, and low energy 
photonuclear reactions \cite{CHIPS3}, for which the interaction cross
section can be calculated easily \cite{photNuc}. With modification of
the first interaction algorithm the CHIPS event generator can be used
for all kinds of hadronic interaction. The Geant4 String Model
interface to the CHIPS generator \cite{GEANT4}, \cite{MC2000} also
makes it possible to use the CHIPS code for nuclear fragmentation at
extremely high energies.

In the first published versions of the CHIPS event generator the class
{\tt G4Quasmon} was the head of the model and all initial interactions
were hidden in its constructor.  More complicated applications of the
model such as anti-proton nuclear capture at rest and the Geant4
String Model interface to CHIPS led to the multi-quasmon version of
the model.  This required a change in the structure of the CHIPS event
generator classes.  In the case of at-rest anti-proton annihilation in
a nucleus, for example, the first interaction occurs on the nuclear
periphery. After this initial interaction, a fraction (defined by a
special parameter of the model) of the secondary mesons independently
penetrate the nucleus.  Each of these mesons can create a separate
quasmon in the interior of the nucleus. In this case the class {\tt
G4Quasmon} can no longer be the head of the model. A new head class,
{\tt G4QEnvironment}, was developed which can adopt a vector of
projectile hadrons ({\tt G4QHadronVector}) and create a vector of 
quasmons, {\tt G4QuasmonVector}. All newly created quasmons then begin
the energy dissipation process in parallel in the same nucleus. The
{\tt G4QEnvironment} instance can be used both for vacuum and for nuclear 
matter.  If {\tt G4QEnvironment} is created in vacuum, it is practically 
identical to the {\tt G4Quasmon} class, because in this case only one 
instance of {\tt G4Quasmon} is allowed.  This leaves the model unchanged 
for hadronic interactions.

The convention adopted for the CHIPS model requires all its class names 
to use the prefix {\tt G4Q} in order to distinguish them from other Geant4 
classes, most of which use the {\tt G4} prefix. The intent is that the 
{\tt G4Q} prefix will not be used by other Geant4 projects.

\section{Nucleon-Antinucleon Annihilation at Rest} \label{annil}

In order to generate hadron spectra from the annihilation of a proton
with an anti-proton at rest, the number of partons in the system must be 
found. For a finite system of $N$ partons with a total center-of-mass energy
$M$, the invariant phase space integral, $\Phi_N$, is proportional to 
$M^{2N-4}$.  According to the dimensional counting rule, $2N$ comes from
$\prod\limits_{i=1}^{N}\frac{d^{3}p_{i}}{E_{i}}$, and $4$ comes from
the energy and momentum conservation function, $\delta ^{4}($\b{P}$-\sum
$\b{p}$_{i})$.  At a temperature $T$ the statistical density of states is 
proportional to $e^{-\frac{M}{T}}$ so that the probability to find a system 
of $N$ quark-partons in a state with mass $M$ is $dW \propto
M^{2N-4}e^{-\frac{M}{T}}dM$.  For this kind of probability distribution the 
mean value of $M^{2}$ is
\begin{equation}
<M^{2}>=4N(N-1)\cdot T^{2}.  \label{temperature}
\end{equation}
When $N$ goes to infinity one obtains for massless particles the
well-known $<M>\equiv \sqrt{<M^{2}>}=2NT$ result. 

After a nucleon absorbs an incident quark-parton, such as a real or 
virtual photon, for example, the newly formed quasmon has a total of $N$ 
quark-partons, where $N$ is determined by Eq. \ref{temperature}.
Choosing one of these quark-partons with energy $k$ in the center of mass 
system (CMS) of $N$ partons, the spectrum of the remaining $N-1$ 
quark-partons is given by
\begin{equation}
\frac{dW}{kdk} \propto (M_{N-1})^{2N-6},
\end{equation}
where $M_{N-1}$ is the effective mass of the $N-1$ quark-partons. 
This result was obtained by applying the above phase-space relation 
($\Phi_N \propto M^{2N-4}$) to the residual $N-1$ quarks.  The effective 
mass is a function of the total mass $M$,
\begin{equation}
M_{N-1}^{2}=M^{2}-2kM ,  \label{m_n-1}
\end{equation}
so that the resulting equation for the quark-parton 
spectrum is:
\begin{equation}
\frac{dW}{kdk}\propto (1-\frac{2k}{M})^{N-3}.  \label{spectrum_1}
\end{equation}

\subsection{Meson Production}

In this section, only the quark fusion mechanism of hadronization is
considered. The quark exchange mechanism can take place only in
nuclear matter where a quasmon has neighboring nucleons.  In order to 
decompose a quasmon into an outgoing hadron and a residual quasmon, one
needs to calculate the probability of two quark-partons combining to 
produce the effective mass of the outgoing hadron.  This requires that 
the spectrum of the second quark-parton be calculated.  This is done by
following the same argument used to determine Eq.~\ref{spectrum_1}.
One quark-parton is chosen from the residual $N-1$.  It has an energy 
$q$ in the CMS of the $N-1$ quark-partons.  The spectrum is obtained by 
substituting $N-1$ for $N$ and $M_{N-1}$ for $M$ in 
Eq.~\ref{spectrum_1} and then using Eq.~\ref{m_n-1} to get 
\begin{equation}
\frac{dW}{q dq }\propto \left( 1-\frac{2q }{M\sqrt{1-
\frac{2k}{M}}}\right) ^{N-4}.  \label{spectrum_2}
\end{equation}

Next, one of the residual quark-partons must be selected from this spectrum
such that its fusion with the primary quark-parton makes a hadron of
mass $\mu$.  This selection is performed by the mass shell condition for 
the outgoing hadron,
\begin{equation}
\mu^2 = 2 \frac{k}{\sqrt{1-\frac{2k}{M}}} 
 \cdot q \cdot (1-\cos \theta ) .  \label{hadron}
\end{equation}
Here $\theta$ is the angle between the momenta, {\bf k} and {\bf q} of
the two quark-partons in the CMS of $N-1$ quarks.  Now the kinematic quark 
fusion probability can be calculated for any primary quark-parton with 
energy $k$:
\begin{eqnarray}
P(k,M,\mu )=&&\int \left( 1-\frac{2q }{M\sqrt{1-\frac{2k}{M}}}\right)
^{N-4} \nonumber\\
 && \times\  \delta \left( \mu ^{2}-\frac{2kq (1-\cos \theta )}{\sqrt{1-
\frac{2k}{M}}}\right) q dq d\cos \theta .\ \ \ \ 
\end{eqnarray}
Using the $\delta$-function\footnote{\protect{
If $g(x_0)$=0, $\int f(x)\delta\left[g(x)\right]dx = 
\int \frac{f(x)\delta\left[g(x)\right]}{g^\prime(x)} dg(x) = 
\frac{f(x_0)}{g^\prime(x_0)}$
}}
to perform the integration over $q$ one gets: 
\begin{eqnarray}
P(k,M,\mu )=&&\int \left( 1-\frac{\mu ^{2}}{Mk(1-\cos \theta )}\right)
^{N-4} \nonumber\\
 && \times\ \left( \frac{\mu ^{2}\sqrt{1-\frac{2k}{M}}}{2k(1-\cos \theta )}
\right)^{2}d\left(\frac{1-\cos \theta }{\mu ^{2}}\right) ,\ \ 
\end{eqnarray}
or 
\begin{eqnarray}
P(k,M,\mu )=&&\frac{M-2k}{4k}\int \left(1-\frac{\mu ^{2}}{Mk(1 - 
\cos\theta)}\right) ^{N-4} \nonumber\\
 && \times\ d\left(1-\frac{\mu ^{2}}{Mk(1-\cos \theta )}\right).
\end{eqnarray}
After the substitution
$z=1-\frac{2q }{M_{N-1}}=1-\frac{\mu ^{2}}{Mk(1-\cos \theta )}$, this
becomes
\begin{equation}
P(k,M,\mu ) = \frac{M-2k}{4k} \int z^{N-4} dz ,
\end{equation}
where the limits of integration are $0$ when 
$\cos\theta = 1 - \frac{\mu ^{2}}{M\cdot k}$, and 
\begin{equation}
z_{\max }=1-\frac{\mu^2}{2Mk}, \label{z_max}
\end{equation}
when $\cos \theta =-1$.  The resulting range of $\theta$\ is therefore
$-1<\cos \theta < 1-\frac{\mu ^{2}}{M\cdot k}$.  Integrating from $0$ to 
$z$ yields
\begin{equation}
\frac{M-2k}{4k\cdot (N-3)}\cdot z^{N-3},  \label{z_probab}
\end{equation}
and integrating from $0$ to $z_{max}$ yields the total kinematic 
probability for hadronization of a quark-parton with energy $k$ into a 
hadron with mass $\mu$:
\begin{equation}
\frac{M-2k}{4k \cdot (N-3)} \cdot z_{\max}^{N-3} . 
                                   \label{tot_kin_probab}
\end{equation}
The ratio of expressions \ref{z_probab} and \ref{tot_kin_probab} can be 
treated as a random number, $R$, uniformly distributed on the interval
[0,1].  Solving for $z$ then gives
\begin{equation}
z=\sqrt[N-3]{R}\cdot z_{\max }.  \label{z_random}
\end{equation}

In addition to the kinematic selection of the two quark-partons in the
fusion process, the quark content of the quasmon and the spin of the 
candidate final hadron are used to determine the probability that a 
given type of hadron is produced.  Because only the relative hadron 
formation probabilities are necessary, overall normalization factors can
be dropped.  Hence the relative probability can 
be written as
\begin{equation}
P_h(k,M,\mu )=(2s_h+1)\cdot z_{\max }^{N-3}\cdot C_{Q}^{h} .  
                                         \label{rel_prob}
\end{equation}
Here, only the factor $z_{\max }^{N-3}$ is used since the other factors 
in equation \ref{tot_kin_probab} are constant for all candidates for the 
outgoing hadron.  The factor $2s_h+1$ counts the spin states of a 
candidate hadron of spin $s_h$, and $C_{Q}^{h}$ is the number of ways the 
candidate hadron can be formed from combinations of the quarks within the 
quasmon.  In making these combinations, the standard quark wave functions 
for pions and kaons were used.  For $\eta$ and $\eta^{\prime }$ mesons the 
quark wave functions 
$\eta=\frac{\bar{u}u+\bar{d}d}{2}-\frac{\bar{s}s}{\sqrt{2}}$ and
$\eta^{\prime }=\frac{\bar{u}u+\bar{d}d}{2}+\frac{\bar{s}s}{\sqrt{2}}$
were used.  No mixing was assumed for the $\omega $\ and $\phi $\ meson 
states, hence $\omega =\frac{ \bar{u}u+\bar{d}d}{\sqrt{2}}$ and 
$\varphi=\bar{s}s$.

A final model restriction is applied to the hadronization process:
after a hadron is emitted, the quark content of the residual quasmon 
must have a quark content corresponding to either one or two real 
hadrons.  When the quantum numbers of a quasmon, determined by its quark 
content, cannot be represented by the quantum numbers of a real hadron, 
the quasmon is considered to be a virtual hadronic molecule such as 
$\pi ^{+}\pi ^{+}$ or $K^{+}\pi ^{+}$, in which case it is defined in 
the CHIPS model to be a Chipolino pseudo-particle.

To fuse quark-partons and create the decay of a quasmon into a hadron and 
residual quasmon, one needs to generate randomly the residual quasmon mass 
$m$, which in fact is the mass of the residual $N-2$ quarks.  Using an 
equation similar to \ref{m_n-1}) one finds that
\begin{equation}
m^{2}=z\cdot (M^{2}-2kM).  \label{m(z)}
\end{equation}
Using Eqs. \ref{z_random} and \ref{z_max}, the mass of the residual 
quasmon can be expressed in terms of the random number $R$:
\begin{equation}
m^{2}=(M-2k)\cdot (M-\frac{\mu ^{2}}{2k})\cdot \sqrt[N-3]{R} .
\label{res_quasmon}
\end{equation}
At this point, the decay of the original quasmon into a final state 
hadron and a residual quasmon of mass $m$ has been simulated.  The process
may now be repeated on the residual quasmon.

This iterative hadronization process continues as long as the residual
quasmon mass remains greater than $m_{\min }$, whose value depends on the 
type of quasmon.  For hadron-type residual quasmons
\begin{equation}
 m_{\min }=m_{\min }^{QC}+m_{\pi ^{0}},  \label{m_min}
\end{equation}
where $m_{\min }^{QC}$ is the minimum hadron mass for the residual
quark content (QC).  For Chipolino-type residual quasmons consisting
of hadrons $h_1$ and $h_2$, 
\begin{equation}
 m_{\min }=m_{h_1}+m_{h_2}. \label{m_min_chipolino}
\end{equation}
These conditions insure that the quasmon always has enough energy to decay 
into at least two final state hadrons, conserving four-momentum and charge.

If the remaining CMS energy of the residual quasmon falls below $m_{\min}$, 
then the hadronization process terminates with a final two-particle decay.  
If the parent quasmon is a Chipolino consisting of hadrons $h_1$ and $h_2$, 
then a binary decay of the parent quasmon into $m_{h_1}$ and $m_{h_2}$ 
takes place.  If the parent quasmon is not a Chipolino then a decay into 
$m_{\min}^{QC}$ and $m_h$ takes place.  The decay into $m_{\min}^{QC}$ and 
$m_\pi^0$ is always possible in this case because of condition \ref{m_min}.

If the residual quasmon is not Chipolino-type, and $m>m_{\min}$, the
hadronization loop can still be finished by the resonance production
mechanism, which is modeled following the concept of parton-hadron 
duality \cite{Duality}.  If the residual quasmon has a mass in the vicinity 
of a resonance with the same quark content ($\rho$ or $K^{\ast}$ for 
example), there is a probability for the residual quasmon to convert to 
this resonance.\footnote{When comparing quark contents, the quark content 
of the quasmon is reduced by canceling quark-antiquark pairs of the same 
flavor.}
In the present version of the CHIPS event generator the probability of
convert to the resonance is given by
\begin{equation}
P_{\rm{res}}=\frac{m_{\min }^{2}}{m^{2}}.  \label{res_probab}
\end{equation}
Hence the resonance with the mass-squared value $m_{r}^{2}$ closest to 
$m^{2}$ is selected, and the binary decay of the quasmon into $m_{h}$ 
and $m_{r}$ takes place.

With more detailed experimental data, it will be possible to take into 
account angular momentum conservation, as well as $C$-, $P$- and 
$G$-parity conservation.  In the present version of the generator, $\eta$ 
and $\eta ^{\prime }$ are suppressed by a factor of $0.3$.  This factor 
was tuned using data from experiments on antiproton annihilation at rest 
in liquid hydrogen and can be different for other hadronic reactions.  It 
is possible to vary it when describing other reactions.

Another parameter, $s/u$, controls the suppression of heavy quark 
production \cite{JETSET}.  For proton-antiproton annihilation at rest the 
strange quark-antiquark sea was found to be suppressed by the factor 
$s/u = 0.1$.  In the JETSET \cite{JETSET} event generator, the default 
value for this parameter is $s/u = 0.3$.  The lower value may be due to 
quarks and anti-quarks of colliding hadrons initially forming a non-strange 
sea, with the strange sea suppressed by the OZI rule \cite{OZI}.  This 
question is still under discussion \cite{OZI_violation} and demands further 
experimental measurements.  The $s/u$ parameter may differ for other 
reactions.  In particular, for e$^{+}$e$^{-}$ reactions it can be closer to 
0.3.

Finally, the temperature parameter has been fixed at $T=180$ MeV.  In
earlier versions of the model it was found that this value successfully 
reproduced spectra of outgoing hadrons in different types of medium-energy 
reactions.

\begin{figure}
% \centerline{\epsfig{file=hadronic/theory_driven/ChiralInvariantPhaseSpace/plots/mommul.eps, height=3.5in, width=4.5in}}
%  \resizebox{1.00\textwidth}{!}
%{
\includegraphics[angle=0,scale=0.6]{hadronic/theory_driven/ChiralInvariantPhaseSpace/plots/mommul.eps}
%\includegraphics[angle=0,scale=0.6]{plots/mommul.eps}
%}
\caption{(a) (left): momentum distribution of charged pions produced in 
proton-antiproton annihilation at rest.  The experimental data are from 
\protect\cite{pispectrum}, and the histogram was produced by the CHIPS 
Monte Carlo.  The experimental spectrum is normalized to the measured 
average charged pion multiplicity, 3.0. (b) (right): pion multiplicity
distribution.  Data points were taken from compilations of experimental 
data \protect\cite{pap_exdata}, and the histogram was produced by the 
CHIPS Monte Carlo.  The number of events with kaons in the final state is 
shown in pion multiplicity bin 9, where no real 9-pion events are 
generated or observed experimentally.  In the model, the percentage of 
annihilation events with kaons is close to the experimental value of 
6\% \cite{pap_exdata}.
}
\label{mommul}
\end{figure}

\begin{figure}
% \centerline{\epsfig{file=hadronic/theory_driven/ChiralInvariantPhaseSpace/plots/channels.eps, height=3.5in, width=4.5in}}
%  \resizebox{1.00\textwidth}{!}
%{
\includegraphics[angle=0,scale=0.6]{hadronic/theory_driven/ChiralInvariantPhaseSpace/plots/channels.eps}
%\includegraphics[angle=0,scale=0.6]{plots/channels.eps}
%}
\caption{Branching probabilities for different channels in
proton-antiproton annihilation at rest. The experimental data are from
\protect\cite {pap_exdata}, and the histogram was produced by the CHIPS 
Monte Carlo. }
\label{channels}
\end{figure}

The above parameters were used to fit not only the spectrum of pions
Fig.~\ref{mommul},a and the multiplicity distribution for pions
Fig.~\ref{mommul},b but also branching ratios of various measured
\cite{pispectrum,pap_exdata} exclusive channels as shown in Figs.
~\ref{channels},~\ref{threechan},~\ref{twochan}.  In Fig.~\ref{twochan} 
one can see many decay channels with higher meson resonances.  The 
relative contribution of events with meson resonances produced in the 
final state is 30 - 40 percent, roughly in agreement with experiment. The 
agreement between the model and experiment for particular decay modes is 
within a factor of 2-3 except for the branching ratios to higher 
resonances.  In these cases it is not completely clear how the resonance 
is defined in a concrete experiment.  In particular, for the 
$a_{2}\omega $ channel the mass sum of final hadrons is 2100 MeV with a 
full width of about 110 MeV while the total initial energy of the p\={p} 
annihilation reaction is only 1876.5 MeV.  This decay channel can be 
formally simulated by an event generator using the tail of the Breit-Wigner 
distribution for the $a_{2}$ resonance, but it is difficult to imagine how 
the $a_{2}$ resonance can be experimentally identified $2\Gamma $ away 
from its mean mass value. 

\subsection{Baryon Production}

To model fragmentation into baryons the POPCORN idea \cite{POPCORN} was
used, which assumes the existence of diquark-partons.  The assumption of 
massless diquarks is somewhat inconsistent at low energies, as is the 
assumption of massless s-quarks, but it is simple and it helps to generate 
baryons in the same way as mesons.

Baryons are heavy, and the baryon production in $p\bar p$ annihilation
reactions at medium energies is very sensitive to the value of the 
temperature. If the temperature is low, the baryon yield is small, and
the mean multiplicity of pions increases very noticeably with center-of-mass
energy as seen in Fig.~\ref{apcmul}.  For higher temperature values the baryon 
yield reduces the pion multiplicity at higher energies.  The existing
experimental data \cite{Energy_Dep}, shown in Fig.~\ref{apcmul}, can be
considered as a kind of ``thermometer'' for the model.  This thermometer 
confirms that the critical temperature is about 200 MeV.

\begin{figure}
% \centerline{\epsfig{file=hadronic/theory_driven/ChiralInvariantPhaseSpace/plots/threechn.eps, height=4.5in, width=4.5in}}
%  \resizebox{1.00\textwidth}{!}
%{
\includegraphics[angle=0,scale=0.6]{hadronic/theory_driven/ChiralInvariantPhaseSpace/plots/threechn.eps}
%\includegraphics[angle=0,scale=0.6]{plots/threechn.eps}
%}
\caption{Branching probabilities for different channels with
three-particle final states in proton-antiproton annihilation at
rest.  The points are experimental data \protect\cite{pap_exdata} and the
histogram is from the CHIPS Monte Carlo. }
\label{threechan}
\end{figure}
\begin{figure}
% \centerline{\epsfig{file=hadronic/theory_driven/ChiralInvariantPhaseSpace/plots/twochn.eps, height=4.5in, width=4.5in}}
%  \resizebox{1.00\textwidth}{!}
%{
\includegraphics[angle=0,scale=0.6]{hadronic/theory_driven/ChiralInvariantPhaseSpace/plots/twochn.eps}
%\includegraphics[angle=0,scale=0.6]{plots/twochn.eps}
%}
\caption{Branching probabilities for different channels with
two-particle final states in proton-antiproton annihilation at
rest. The points are experimental data \protect\cite{pap_exdata} and the
histogram is from the CHIPS Monte Carlo. }
\label{twochan}
\end{figure}
\begin{figure}
% \centerline{\epsfig{file=hadronic/theory_driven/ChiralInvariantPhaseSpace/plots/apcmul.eps, height=4.5in, width=4.5in}}
%  \resizebox{1.00\textwidth}{!}
%{
\includegraphics[angle=0,scale=0.6]{hadronic/theory_driven/ChiralInvariantPhaseSpace/plots/apcmul.eps}
%\includegraphics[angle=0,scale=0.6]{plots/apcmul.eps}
%}
\caption{Average meson multiplicities in proton-antiproton and in
electron-positron annihilation, as a function of the center-of-mass energy of 
the interacting hadronic system.  The points are experimental data
\protect\cite {Energy_Dep} and the lines are CHIPS Monte Carlo calculations
at several values of the critical temperature parameter $T$. }
\label{apcmul}
\end{figure}

It can be used as a tool for the Monte Carlo simulation of a wide variety
of hadronic reactions.  The CHIPS event generator can be used not only for 
``phase-space background'' calculations in place of the standard GENBOD 
routine \cite{GENBOD}, but even for taking into account the reflection of 
resonances in connected final hadron combinations.  Thus it can be useful 
for physics analysis too, even though its main range of application is the 
simulation of the evolution of hadronic and electromagnetic showers in 
matter at medium energies.

\section[Nuclear Pion Capture Below Delta(3,3)]{Nuclear Pion Capture at Rest and Photonuclear Reactions Below the Delta(3,3) Resonance} \label{picap}

When compared with the first ``in vacuum'' version of the model, described 
in Section \ref{annil}, modeling hadronic fragmentation in nuclear matter 
is more complicated because of the much greater number of possible 
secondary fragments.  However, the hadronization process itself
is simpler in a way.  In vacuum, the quark-fusion mechanism requires a 
quark-parton partner from the external (as in
JETSET \cite{JETSET}) or internal (the quasmon itself, Section \ref{annil})
quark-antiquark sea.  In nuclear matter, there is a second possibility: 
quark exchange with a neighboring hadronic system, which could be a nucleon 
or multinucleon cluster.  

In nuclear matter the spectra of secondary hadrons and nuclear fragments
reflect the quark-parton energy spectrum within a quasmon.  In the case of 
inclusive spectra that are decreasing steeply with energy, and 
correspondingly steeply decreasing spectra of the quark-partons in a quasmon,
only those secondary hadrons which get the maximum energy from the primary 
quark-parton of energy $k$ contribute to the inclusive spectra.  This 
extreme situation requires the exchanged quark-parton with energy $q$, 
coming toward the quasmon from the cluster, to move in a direction 
opposite to that of the primary quark-parton.  As a result the 
hadronization quark exchange process becomes one-dimensional along the 
direction of $k$.  If a neighboring nucleon or nucleon cluster with bound 
mass $\tilde{\mu}$ absorbs the primary quark-parton and radiates the 
exchanged quark-parton in the opposite direction, then the energy of the 
outgoing fragment is $E=\tilde{\mu}+k-q$, and the momentum is $p=k+q$. 
Both the energy and the momentum of the outgoing nuclear fragment are known, 
as is the mass $\tilde{\mu}$ of the nuclear fragment in nuclear matter, so 
the momentum of the primary quark-parton can be reconstructed using the 
approximate relation
\begin{equation}
k=\frac{p+E-B\cdot m_{N}}{2} .  \label{k}
\end{equation}
Here $B$ is the baryon number of the outgoing fragment 
($\tilde{\mu}\approx B\cdot m_{N}$) and $m_N$ is the nucleon mass.  In 
Ref.~\cite{K_parameter} it was shown that the invariant inclusive spectra of 
pions, protons, deuterons, and tritons in proton-nucleus reactions at 
400~GeV \cite{FNAL} not only have the same exponential slope but almost 
coincide when they are plotted as a function of $k=\frac{p+E_{\rm{kin}}}{2}$.
Using data at 10~GeV \cite{FAS}, it was shown that the parameter $k$, defined
by Eq.~\ref{k}, is also appropriate for the description of secondary 
anti-protons produced in high energy nuclear reactions.  This means that the 
extreme assumption of one-dimensional hadronization is a good approximation. 

The same approximation is also valid for the quark fusion mechanism.  In 
the one-dimensional case, assuming that $q$ is the momentum of the second 
quark fusing with the primary quark-parton of energy $k$, the total energy 
of the outgoing hadron is $E=q+k$ and the momentum is $p=k-q$.  In the 
one-dimensional case the secondary quark-parton must move in the opposite 
direction with respect to the primary quark-parton, otherwise the mass of 
the outgoing hadron would be zero.  So, for mesons $k=\frac{p+E}{2}$, in
accordance with Eq.~\ref{k}.  In the case of anti-proton radiation, the
baryon number of the quasmon is increased by one, and the primary
antiquark-parton will spend its energy to build up the mass of the
antiproton by picking up an anti-diquark.  Thus, the energy conservation 
law for antiproton radiation looks like $E+m_{N}=q+k$ and 
$k=\frac{p+E+m_{N}}{2}$, which is again in accordance with Eq.~\ref{k}.

The one-dimensional quark exchange mechanism was proposed in 1984
\cite{K_parameter}.  Even in its approximate form it was useful in the
analysis of inclusive spectra of hadrons and nuclear fragments in
hadron-nuclear reactions at high energies.  Later the same approach was
used in the analysis of nuclear fragmentation in electro-nuclear
reactions \cite{TPC}.  Also in 1984 the quark-exchange mechanism developed 
in the framework of the non-relativistic quark model was found to be 
important for the explanation of the short distance features of $NN$ 
interactions \cite{NN QEX}.  Later it was successfully applied to 
$K^{-}p$ interactions \cite{Kp QUEX}.  The idea of the quark exchange 
mechanism between nucleons was useful even for the explanation of the EMC 
effect \cite{EMC}.  For the non-relativistic quark model, the quark 
exchange technique was developed as an alternative to the Feynman diagram 
technique at short distances \cite{QUEX}.

The CHIPS event generator models quark exchange processes, taking into
account kinematic and combinatorial factors for asymptotically free
quark-partons.  In the naive picture of the quark-exchange mechanism,
one quark-parton tunnels from the asymptotically free region of one hadron 
to the asymptotically free region of another hadron.  To conserve color, 
another quark-parton from the neighboring hadron must replace the first 
quark-parton in the quasmon.  This makes the tunneling mutual, and the 
resulting process is quark exchange.

The experimental data available for multihadron production at high energies 
show regularities in the secondary particle spectra that can be related to 
the simple kinematic, combinatorial, and phase space rules of such quark 
exchange and fusion mechanisms.  The CHIPS model combines these mechanisms
consistently.

\begin{figure}[tbp]
% \centerline{\epsfig{file=hadronic/theory_driven/ChiralInvariantPhaseSpace/plots/diagram.eps, height=2.5in, width=2.5in}}
%\resizebox{1.00\textwidth}{!}
%{
\includegraphics[angle=0,scale=0.6]{hadronic/theory_driven/ChiralInvariantPhaseSpace/plots/diagram.eps}
%\includegraphics[angle=0,scale=0.6]{plots/diagram.eps}
%}
\caption{Diagram of the quark exchange mechanism. }
\label{diagram}
\end{figure}

Fig.~\ref{diagram} shows a quark exchange diagram which helps to keep track 
of the kinematics of the process.  It was shown in Section \ref{annil} that
a quasmon, $Q$ is kinematically determined by a few asymptotically free 
quark-partons homogeneously distributed over the invariant phase space.  The 
quasmon mass $M$ is related to the number of quark-partons $N$ through
\begin{equation}
<M^{2}>=4N(N-1)\cdot T^{2},  \label{temperatureII}
\end{equation}
where $T$ is the temperature of the system.

The spectrum of quark partons can be calculated as
\begin{equation}
\frac{dW}{k^{\ast }dk^{\ast}}\propto 
 \left(1-\frac{2k^{\ast}}{M} \right)^{N-3},  \label{spectrum_1II}
\end{equation}
where $k^{\ast}$ is the energy of the primary quark-parton in the 
center-of-mass system of the quasmon.  After the primary quark-parton 
is randomized and the neighboring cluster, or parent cluster, $PC$, with 
bound mass $\tilde{\mu}$\ is selected, the quark exchange process begins.
To follow the process kinematically one should imagine a colored, compound 
system consisting of a stationary, bound parent cluster and the primary quark.
The primary quark has energy $k$ in the lab system,
\begin{equation}
k=k^{\ast }\cdot \frac{E_{N}+p_{N}\cdot \cos (\theta _{k})}{M_{N}},
\end{equation}
where $M_N$, $E_N$ and $p_N$ are the mass, energy, and momentum of the 
quasmon in the lab frame.  The mass of the compound system, $CF$, is
$\mu _{c}=\sqrt{(\tilde{\mu}+k)^{2}}$, where $\tilde{\mu}$ and $k$ are the
corresponding four-vectors.  This colored compound system decays into a
free outgoing nuclear fragment, $F$, with mass $\mu$ and a recoiling quark 
with energy $q$.  $q$ is measured in the CMS of $\tilde{\mu}$, which 
coincides with the lab frame in the present version of the model because no 
cluster motion is considered.  At this point one should recall that a colored
residual quasmon, $CRQ$, with mass $M_{N-1}$ remains after the radiation of
$k$.  $CRQ$ is finally fused with the recoil quark $q$ to form the residual 
quasmon $RQ$.  The minimum mass of $RQ$ should be greater than $M_{\min}$, 
which is determined by the minimum mass of a hadron (or Chipolino 
double-hadron as defined in Section \ref{annil}) with the same quark content.

All quark-antiquark pairs with the same flavor should be canceled in the
minimum mass calculations.  This imposes a restriction, which in the 
center-of-mass system of $\mu_{c}$, can be written as
\begin{equation}
2q\cdot (E-p\cdot \cos \theta_{qCQ})+M_{N-1}^{2}>M_{\min }^{2},
\label{min_mass}
\end{equation}
where $E$ is the energy and $p$ is the momentum of the colored residual
quasmon with mass $M_{N-1}$ in the CMS of $\mu _{c}$.  The restriction for 
$\cos\theta_{qCQ}$ then becomes
\begin{equation}
\cos \theta _{qCQ}<\frac{2qE-M_{\min }^{2}+M_{N-1}^{2}}{2qp},
\label{cost_restriction}
\end{equation}
which implies
\begin{equation}
q>\frac{M_{N-1}^{2}-M_{\min }^{2}}{2\cdot (E+p)}.  \label{resid_rest}
\end{equation}

A second restriction comes from the nuclear Coulomb barrier for charged
particles.  The Coulomb barrier can be calculated in the simple form:
\begin{equation}
E_{CB}=\frac{Z_{F}\cdot
Z_{R}}{A_{F}^{\frac{1}{3}}+A_{R}^{\frac{1}{3}}}\ (\rm{MeV}),
\label{CoulBar}
\end{equation}
where $Z_F$ and $A_F$ are the charge and atomic weight of the fragment, and 
$Z_R$ and $A_R$ are the charge and atomic weight of the residual nucleus. 
The obvious restriction is
\begin{equation}
   q<k+\tilde{\mu}-\mu -E_{CB}.  \label{cb_rest}
\end{equation}

In addition to \ref{resid_rest} and \ref{cb_rest}, the quark 
exchange mechanism imposes restrictions which are calculated below.  The 
spectrum of recoiling quarks is similar to the $k^{\ast}$ spectrum in the
quasmon (\ref{spectrum_1II}):
\begin{equation}
  \frac{dW}{q\ dq\ d\cos \theta }\propto 
  \left(1-\frac{2q}{\tilde{\mu}}\right)^{n-3}, \label{spectrum_2II}
\end{equation}
where $n$ is the number of quark-partons in the nucleon cluster.  It is 
assumed that $n = 3 A_C$, where $A_C$ is the atomic weight of the parent 
cluster.  The tunneling of quarks from one nucleon to another provides a
common phase space for all quark-partons in the cluster.

An additional equation follows from the mass shell condition for the
outgoing fragment,
\begin{equation}
\mu ^{2}=\tilde{\mu}^{2}+2\tilde{\mu}\cdot k-2\tilde{\mu}\cdot q-2k\cdot
q\cdot (1-\cos \theta _{kq}),  \label{hadronII}
\end{equation}
where $\theta _{kq}$ is the angle between quark-parton momenta in the lab 
frame.  From this equation $q$ can be calculated as 
\begin{equation}
q=\frac{\tilde{\mu}\cdot (k-\Delta )}{\tilde{\mu}+k\cdot 
(1-\cos \mathit{\theta }_{\mathit{kq}})},  \label{q-cos}
\end{equation}
where $\Delta $ is the covariant binding energy of the cluster 
$\Delta =\frac{\mu ^{2}-\tilde{\mu}^{2}}{2\tilde{\mu}}$. 
The quark exchange probability integral can be then written in the form: 
\begin{eqnarray}
&&P(k,\tilde{\mu},\mu )=  \nonumber \\
&&\int \delta \left[ \mu ^{2}-\tilde{\mu}^{2}-2\tilde{\mu}\cdot k+2\tilde{\mu
}\cdot q+2k\cdot {q}\cdot (1-\cos \theta _{kq})\right]   \nonumber \\
&&\ \ \ \ \ \ \ \ \times \ \left( 1-\frac{2{q}}{\tilde{\mu}}\right) ^{n-3}{q}
d{q\cdot }d\cos \theta _{kq}.
\end{eqnarray}
Using the $\delta$-function to perform the integration over $q$ one obtains 
\begin{eqnarray}
P(k,\tilde{\mu},\mu ) &=&\int \left( 1-\frac{2(k-\Delta )}{\tilde{\mu}
+k(1-\cos \theta _{\mathit{kq}})}\right) ^{n-3}  \nonumber \\
&&\times \ \frac{\tilde{\mu}(k-\Delta )}{2[\tilde{\mu}+k(1-\cos \mathit{
\theta }_{\mathit{kq}})]^{2}}d\mathit{\cos \theta }_{\mathit{kq}}
\end{eqnarray}
or 
\begin{eqnarray}
P(k,\tilde{\mu},\mu ) &=&\int \left( 1-\frac{2(k-\Delta )}{\tilde{\mu}
+k(1-\cos \theta _{\mathit{kq}})}\right) ^{n-3}  \nonumber \\
&&\times \ \left( \frac{\tilde{\mu}(k-\Delta )}{\tilde{\mu}+k(1-\cos 
\mathit{\theta }_{\mathit{kq}})}\right) ^{2}  \nonumber \\
&&\times \ d \left( \frac{\tilde{\mu}+k(1-\cos 
\mathit{\theta }_{\mathit{kq}})}{\tilde{\mu}(k-\Delta )}\right).
\end{eqnarray}
The result of the integration is 
\begin{eqnarray}
&&P(k,\tilde{\mu},\mu )=\frac{\tilde{\mu}}{4k(n-2)}  \nonumber \\
&&\times \ \left[ \left( 1-\frac{2(k-\Delta )}{\tilde{\mu}+2k}\right)
_{\rm{high}}^{n-2}-\left( 1-\frac{2(k-\Delta )}{\tilde{\mu}}\right)
_{\rm{low}}^{n-2}%
\right] .  \label{QUEX_Int}
\end{eqnarray}
For randomization it is convenient to make $z$ a random parameter 
\begin{equation}
z=1-\frac{2(k-\Delta )}{\tilde{\mu}+k(1-\cos 
\theta_{\mathit{kq}})}=1-\frac{2{q}}{\tilde{\mu}}.  \label{z(q)}
\end{equation}
From (\ref{QUEX_Int}) one can find the high and the low limits of the
randomization. The first limit is for $k$: $k>\Delta$. It is
similar to the restriction for quasmon fragmentation in vacuum:
$k^{\ast}>\frac{\mu^{2}}{2M}$. The second limit is
$k=\frac{\mu^{2}}{2\tilde{\mu}}$, when the low limit of randomization
becomes equal to zero. If $k<\frac{\mu^{2}}{2\tilde{\mu}}$, then
$-1<\cos\theta_{kq}<1$\ and
$z_{\rm{low}}=1-\frac{2(k-\Delta)}{\tilde{\mu}}$. If
$k>\frac{\mu^{2}}{2\tilde{\mu}}$, then the range of $\cos\theta
_{kq}$\ is $-1<\cos\theta_{kq}<\frac{\mu^{2}}{k\tilde{\mu}}-1$\ and
$z_{\rm{low}}=0$.  This value of $z_{\rm{low}}$\ should be corrected
using the Coulomb barrier restriction (\ref{cb_rest}), and the value of
$z_{\rm{high}}$ should be corrected using the minimum residual quasmon
restriction (\ref{resid_rest}). In the case of a quasmon with momentum much
less than $k$ it is possible to impose tighter restrictions than
(\ref{resid_rest}) because the direction of motion of the CRQ is
opposite to $k$. So
$\cos\theta_{qCQ}=-\cos\mathit{\theta}_{\mathit{kq}}$, and from
(\ref{q-cos}) one can find that
\begin{equation}
\cos \theta_{qCQ} =1-\frac{\tilde{\mu}\cdot (k-\Delta -q)}{k\cdot q}.
\label{cos_q}
\end{equation}
So in this case the equation (\ref{resid_rest})\ can be replaced by
the more stringent one: 
\begin{equation}
q>\frac{M_{N-1}^{2}-M_{\min }^{2}+2\frac{p\cdot 
\tilde{\mu}}{k}(k-\Delta )}{2\cdot (E+p+\frac{p\cdot \tilde{\mu}}{k})}.
\end{equation}

The integrated kinematical quark exchange probability (in the range
from $z_{\rm{low}}$ to $z_{\rm{high}}$) is 
\begin{equation}
\frac{\tilde{\mu}}{4k(n-2)}\cdot z^{n-2},  \label{z_probabII}
\end{equation}
and the total kinematic probability of hadronization of the quark-parton
with energy $k$ into a nuclear fragment with mass\ $\mu $ is 
\begin{equation}
\frac{\tilde{\mu}}{4k(n-2)}\cdot 
\left( z_{\rm{high}}^{n-2}-z_{\rm{low}}^{n-2}\right).
\label{tot_kin_probabII}
\end{equation}
This can be compared with the vacuum probability of the quark fusion mechanism 
from Section \ref{annil}: 
\begin{equation}
\frac{M-2k}{4k(N-3)}z_{\max }^{N-3}.
\end{equation}
The similarity is very important, as the absolute probabilities
define the competition between vacuum and nuclear channels.

Equations (\ref{z_probabII})\ and (\ref{tot_kin_probabII})\ can be used for
randomization of $z$: 
\begin{equation}
z=z_{\rm{low}}+\sqrt[n-2]{R}\cdot (z_{\rm{high}}-z_{\rm{low}}), 
 \label{z_randomII}
\end{equation}
where $R$\ is a random number, uniformly distributed in the interval (0,1).

Eq. (\ref{tot_kin_probabII})\ can be used to control the competition
between different nuclear fragments and hadrons in the hadronization
process, but in contrast to the case of ``in vacuum'' hadronization 
it is not enough to take into account only the quark combinatorics of the 
quasmon and the outgoing hadron. In the case of hadronization in nuclear 
matter, different parent bound clusters should be taken into account as well.
For example, tritium can be radiated as a result of quark exchange with a 
bound tritium cluster or as a result of quark exchange with a bound $^3$He 
cluster.

To calculate the yield of fragments it is necessary to calculate the
probability to find a cluster with certain proton and neutron content
in a nucleus. One could consider any particular probability as an
independent parameter, but in such a case the process of tuning the model
would be difficult. We proposed the following scenario of
clusterization. A gas of quasi-free nucleons is close to the phase
transition to a liquid bound by strong quark exchange forces. Precursors of 
the liquid phase are nuclear clusters, which may
be considered as ``drops'' of the liquid phase within the nucleus. Any
cluster can meet another nucleon and absorb it (making it bigger),
or it can release one of the nucleons (making it smaller).  The
first parameter $\varepsilon_{1}$\ is the percentage of quasi-free
nucleons not involved in the clusterization process. The rest of the
nucleons ($1-\varepsilon_{1}$) clusterize. 
We assume that since on the periphery of the nucleus the density 
is lower, one can consider only dibaryon clusters, and neglect 
triple-baryon clusters.  Still we denote the number of nucleons
clusterized in dibaryons on the periphery by the parameter
$\varepsilon_{2}$.  In the dense part of the nucleus, strong quark
exchange forces make clusters out of quasi-free nucleons with high
probability.  To characterize the distribution of clusters the
clusterization probability parameter $\omega$ was used.

If the number of nucleons involved in clusterization is
$a=(1-\varepsilon_{1}-\varepsilon _{2})\cdot A$, then the probability
to find a cluster consisting of $\nu$\ nucleons is defined by the 
distribution 
\begin{equation}
P_{\nu }\propto C_{\nu }^{a}\cdot \omega ^{\nu -1},
\end{equation}
where $C_{\nu }^{a}$ is the corresponding binomial coefficient. 
The coefficient of proportionality can be found from the equation 
\begin{equation}
a=b\cdot \sum\limits_{\nu =1}^{a}\nu \cdot C_{\nu }^{a}\cdot \omega ^{\nu
-1}=b\cdot a\cdot (1+\omega )^{a-1}.
\end{equation}
Thus, the number of clusters consisting of $\nu$\ nucleons is 
\begin{equation}
P_{\nu }=\frac{C_{\nu }^{a}\cdot \omega ^{\nu -1}}{(1+\omega )^{a-1}}.
\end{equation}
For clusters with an even number of nucleons we used only isotopically
symmetric configurations ($\nu=2n$, $n$\ protons and $n$\ neutrons) and
for odd clusters ($\nu =2n+1$) we used only two configurations: $n$\
neutrons with $n+1$\ protons and $n+1$\ neutrons with $n$\ protons. This 
restriction, which we call ``isotopic focusing'', can be considered an 
empirical rule of the CHIPS model which helps to describe data. It is 
applied in the case of nuclear
clusterization (isotopically symmetric clusters) and in the case of
hadronization in nuclear matter.  In the hadronization process the
quasmon is shifted from the isotopic symmetric state (e.g., by capturing
a negative pion) and transfers excess charge to the outgoing nuclear
cluster. This tendency is symmetric with respect to the quasmon and
the parent cluster.

The temperature parameter used to calculate the number of
quark-partons in a quasmon (see equation~\ref{temperatureII}) was chosen 
to be $T=180$ MeV, which is the same as in Section \ref{annil}.

CHIPS is mostly a model of fragmentation, conserving energy, momentum, and 
charge. But to compare it with experimental data one needs to model also the 
first interaction of the projectile with the
nucleus.  For proton-antiproton annihilation this was easy, as we
assumed that in the interaction at rest, a proton and antiproton always
create a quasmon. In the case of pion capture the pion can be captured by
different clusters. We assumed that the probability of capture is
proportional to the number of nucleons in a cluster.  After the
capture the quasmon is formed, and the CHIPS generator produces
fragments consecutively and recursively, choosing at each step the
quark-parton four-momentum $k$, the type of parent and outgoing fragment,
and the four-momentum of the exchange quark-parton $q$, to produce
a final state hadron and the new quasmon with less energy.

In the CHIPS model we consider this process as a chaotic process
with large number of degrees of freedom and do not take into account
any final state interactions of outgoing hadrons. Nevertheless, when
the excitation energy dissipates, and in some step the quasmon mass
drops below the mass shell, the quark-parton mechanism of hadronization
fails. To model the event exclusively, it becomes necessary to
continue fragmentation at the hadron level. Such a fragmentation process
is known as nuclear evaporation. It is modeled using the
non-relativistic phase space approach.  In the non-relativistic case the
phase space of nucleons can be integrated as well as in the
ultra-relativistic case of quark-partons.

The general formula for the non-relativistic phase space can be found starting
with the phase space for two particles $\tilde{\Phi}_{2}$. It is
proportional to the center-of-mass momentum:
\begin{equation}
\tilde{\Phi}_2(W_2) \propto \sqrt{W_2},  \label{F2}
\end{equation}
where $W_2$\ is a total kinetic energy of the two non-relativistic
particles.  If the phase space integral is known for $n-1$\ hadrons
then it is possible to calculate the phase space integral for $n$\
hadrons:
\begin{eqnarray}
\tilde{\Phi}_{n}(W_n) &=&\int \tilde{\Phi}_{n-1}(W_{n-1}) \cdot
\delta (W_{n}-W_{n-1}-E_{\rm{kin}})  \nonumber \\
&&\times \sqrt{E_{\rm{kin}}}dE_{\rm{kin}} dW_{n-1}.  \label{Fn} 
\end{eqnarray}
Using (\ref{F2})\ and (\ref{Fn})\ one can find that 
\begin{equation}
\tilde{\Phi}_{n}(W_n)\propto W_{n}^{\frac{3}{2}n-\frac{5}{2}}
\end{equation}
and the spectrum of hadrons, defined by the phase space of residual
$n-1$ nucleons, can be written as 
\begin{equation}
\frac{dN}{\sqrt{E_{\rm{kin}}}dE_{\rm{kin}}} \propto 
\left(1-\frac{E_{\rm{kin}}}{W_{n}}\right)^{\frac{3}{2}n-4}.
\label{evap_spectr}
\end{equation}
This spectrum can be randomized. The only problem is from which level one
should measure the thermal kinetic energy when most nucleons in nuclei
are filling nuclear levels with zero temperature. To model the evaporation
process we used this unknown level as a parameter $U$\ of the evaporation
process. Comparison with experimental data gives $U=1.7$ MeV. Thus, the
total kinetic energy of $A$\ nucleons is 
\begin{equation}
W_{A}=U\cdot A+E_{\rm{ex}},
\end{equation}
where $E_{\rm{ex}}$ is the excitation energy of the nucleus.

To\  be\  radiated,\ \  the nucleon\ \  should\ \  overcome\ \  the threshold 
\begin{equation}
U_{\rm{thresh}}=U+U_{\rm{bind}}+E_{CB},
\end{equation}
where $U_{\rm{bind}}$\ is the separation energy of the nucleon, and
$E_{CB}$\ is the Coulomb barrier energy which is non-zero only for
positive particles and can be calculated using formula
(\ref{CoulBar}).

From several experimental investigations of nuclear pion capture at 
rest, four published results have been selected here, which
constitute, in our opinion, a representative data set covering a wide
range of target nuclei, types of produced hadrons and nuclear
fragments, and their energy range. In the first publication
\cite{MIPHI}\ the spectra of charged fragments (protons, deuterons,
tritons, $^{3}$He, $^{4}$He) in pion capture were measured on
17 nuclei within one experimental setup. To verify the spectra we
compared them for a carbon target with detailed measurements of the
spectra of charged fragments given in Ref.~\cite{Mechtersheimer}. In
addition, we took $^{6}$Li spectra for a carbon
target from the same paper.

The neutron spectra were added from Ref.~\cite{Cernigoi} and
Ref.~\cite{Madey}. We present data and Monte Carlo distributions as 
the invariant phase space function
$f=\frac{d\sigma}{pdE}$\ depending on the variable
$k=\frac{p+E_{\rm{kin}}}{2}$\ as defined in equation~(\ref{k}).

Spectra on $^{9}$Be, $^{12}$C, $^{28}$Si ($^{27}$Al for secondary
neutrons), $^{59}$Co ($^{64}$Cu for secondary neutrons), and
$^{181}$Ta\ are shown in Figs.~\ref{be0405}\ through~\ref{ta73108}.
The data are well-described, including the total energy spent in the
reaction to yield the particular type of fragments.

The evaporation process for nucleons is also well-described.  It is
exponential in $k$, and looks especially impressive for Si/Al and
Co/Cu data, where the Coulomb barrier is low, and one can see proton
evaporation as a continuation of the evaporation spectra from
secondary neutrons. This way the exponential behavior of the
evaporation process can be followed over 3 orders of
magnitude. Clearly seen is\ the\ transition region at\ \ $k \approx
90$\ MeV\ \ (kinetic energy $15-20$\ MeV)\ \ between the quark-level
hadronization process and the hadron-level evaporation process. For
light target nuclei the evaporation process becomes much less
prominent.

The $^{6}$Li spectrum on a carbon target exhibits an interesting regularity
when plotted as a function of $k$: it practically coincides with the
spectrum of $^{4}$He fragments, and shows exponential behavior in a
wide range of $k$, corresponding to a few orders of magnitude in the
invariant cross section.  To keep the figure readable, the $^{6}$Li 
spectrum generated by CHIPS was not plotted. It coincides with the 
$^{4}$He spectrum at $k > 200$\ MeV, and under-estimates lithium
emission at lower energies, similarly to the $^{3}$He and tritium data.

Between the region where hadron-level processes dominate and the
kinematic limit, all hadronic spectrum slopes become similar when plotted
as a function of $k$. In addition to this general behavior there is 
the effect of strong proton-neutron splitting. For protons and neutrons
it reaches almost an order of magnitude. To model such splitting in
the CHIPS generator, the mechanism of ``isotopic focusing'' was used,
which locally transfers the negative charge from the pion to the first
radiated nuclear fragment.

\begin{table}
\caption{Clusterization parameters}
\label{tab:1}
\begin{tabular}{llllll}
\hline\noalign{\smallskip}
& $^{9}$Be & $^{12}$C & $^{28}$Si & $^{59}$Co & $^{181}$Ta \\
\noalign{\smallskip}\hline\noalign{\smallskip}
$\varepsilon_{1}$ & 0.45 & 0.40 & 0.35 & 0.33 & 0.33 \\
$\varepsilon_{2}$ & 0.15 & 0.15 & 0.05 & 0.03 & 0.02 \\
$\omega $ & 5.00 & 5.00 & 5.00 & 5.00 & 5.00 \\
\noalign{\smallskip}\hline
\end{tabular}
\end{table}

Thus, the model qualitatively describes all typical features of the
pion capture process. The question is what can be extracted from the
experimental data with this tool. The clusterization parameters are
listed in Table~\ref{tab:1}. No formal fitting procedure has been
performed. A balanced qualitative agreement with all data was used to
tune the parameters. The difference between the $\frac{\varepsilon
_{2}}{\varepsilon _{1}}$\ ratio and the parameter $\omega$\ (which is
the same for all nuclei) is an indication that there is a
phase transition between the gas phase and the liquid phase of the
nucleus. The large value of the parameter $\omega$, determining the
average size of a nuclear cluster, is critical in describing 
the model spectra
at large $k$, where the fragment spectra approach the kinematic
limits.

Using the same parameters of clusterization, the $\gamma$\ absorption 
data \cite{Ryckbosch} on Al and Ca nuclei were compared in 
Fig.~\ref{gam62}) to the CHIPS results. One can see that the
spectra of secondary protons and deuterons are qualitatively described
by the CHIPS model.

\begin{figure}[tbp]
% \centerline{\epsfig{file=hadronic/theory_driven/ChiralInvariantPhaseSpace/plots/be0405k.eps, height=4.5in, width=4.5in}}
%\resizebox{1.00\textwidth}{!}
%{
\includegraphics[angle=0,scale=0.6]{hadronic/theory_driven/ChiralInvariantPhaseSpace/plots/be0405k.eps}
%\includegraphics[angle=0,scale=0.6]{plots/be0405k.eps}
%}
\caption{\protect{Comparison of the CHIPS model results with
experimental data on proton, neutron, and nuclear fragment production
in the capture of negative pions on $^9$Be.
Proton~\cite{MIPHI} and neutron~\cite{Cernigoi}\ experimental spectra
are shown in the upper left panel by open circles and open squares,
respectively. The model calculations are shown by the two
corresponding solid lines. The same arrangement
is used to present $^{3}$He~\cite{MIPHI}
and tritium~\cite{MIPHI}
spectra in the lower left panel. Deuterium~\cite{MIPHI} 
and $^{4}$He~\cite{MIPHI} spectra are
shown in the right panels of the figure by open squares
and lines (CHIPS model). The average kinetic
energy carried away by each nuclear fragment is shown in the panels
by the two numbers: first is the average calculated using the
experimental data shown; second is the model result.}}
\label{be0405}
\end{figure}

\begin{figure}[tbp]
% \centerline{\epsfig{file=hadronic/theory_driven/ChiralInvariantPhaseSpace/plots/c0606k.eps, height=4.5in, width=4.5in}}
%\resizebox{1.00\textwidth}{!}
%{
\includegraphics[angle=0,scale=0.6]{hadronic/theory_driven/ChiralInvariantPhaseSpace/plots/c0606k.eps}
%\includegraphics[angle=0,scale=0.6]{plots/c0606k.eps}
%}
\caption{\protect{Same as in Figure~\ref{be0405}, for 
pion capture on $^{12}$C. The experimental neutron spectrum
is taken from \cite{Madey}. In addition, the detailed data on
charged particle production, including the $^{6}$Li spectrum, taken from
Ref.~\cite{Mechtersheimer}, are superimposed on the plots as a series of
dots.}}
\label{c0606}
\end{figure}

\begin{figure}[tbp]
% \centerline{\epsfig{file=hadronic/theory_driven/ChiralInvariantPhaseSpace/plots/si1414k.eps, height=4.5in, width=4.5in}}
%\resizebox{1.00\textwidth}{!}
%{
\includegraphics[angle=0,scale=0.6]{hadronic/theory_driven/ChiralInvariantPhaseSpace/plots/si1414k.eps}
%\includegraphics[angle=0,scale=0.6]{plots/si1414k.eps}
%}
\caption{\protect{Same as in Figure~\ref{be0405}, for
pion capture on $^{28}$Si nucleus. The experimental neutron spectrum
is taken from~\cite{Madey}, for the reaction on $^{27}$Al.}}
\label{si1414}
\end{figure}

\begin{figure}[tbp]
% \centerline{\epsfig{file=hadronic/theory_driven/ChiralInvariantPhaseSpace/plots/co2732k.eps, height=4.5in, width=4.5in}}
%\resizebox{1.00\textwidth}{!}
%{
\includegraphics[angle=0,scale=0.6]{hadronic/theory_driven/ChiralInvariantPhaseSpace/plots/co2732k.eps}
%\includegraphics[angle=0,scale=0.6]{plots/co2732k.eps}
%}
\caption{\protect{Same as in Figure~\ref{be0405}, for 
pion capture on $^{59}$Co. The experimental neutron spectrum
is taken from~\cite{Madey}, for the reaction on $^{64}$Cu.}}
\label{co2732}
\end{figure}

\begin{figure}[tbp]
% \centerline{\epsfig{file=hadronic/theory_driven/ChiralInvariantPhaseSpace/plots/ta73108k.eps, height=4.5in, width=4.5in}}
%\resizebox{1.00\textwidth}{!}
%{
\includegraphics[angle=0,scale=0.6]{hadronic/theory_driven/ChiralInvariantPhaseSpace/plots/ta73108k.eps}
%\includegraphics[angle=0,scale=0.6]{plots/ta73108k.eps}
%}
\caption{\protect{Same as in Figure~\ref{be0405}, for 
pion capture on $^{181}$Ta. The experimental neutron
spectrum is taken from~\cite{Madey}.}}
\label{ta73108}
\end{figure}

\begin{figure}[tbp]
% \centerline{\epsfig{file=hadronic/theory_driven/ChiralInvariantPhaseSpace/plots/gam62.eps, height=4.5in, width=4.5in}}
%\resizebox{0.70\textwidth}{!}
%{
\includegraphics[angle=0,scale=0.75]{hadronic/theory_driven/ChiralInvariantPhaseSpace/plots/gam62.eps}
%\includegraphics[angle=0,scale=0.75]{plots/gam62.eps}
%}
\caption{\protect{Comparison of CHIPS model with 
experimental data~\cite{Ryckbosch} on
proton and deuteron production at $90^{\circ}$ 
in photonuclear reactions on $^{27}$Al
and $^{40}$Ca at 59 -- 65 MeV. Open circles and solid squares represent the
experimental proton and deuteron spectra, 
respectively. Solid 
and dashed lines show the results of the corresponding CHIPS model
calculation. Statistical errors in the CHIPS results are not shown and
can be judged by the point-to-point variations in the lines. The
comparison is absolute, using the values of total
photonuclear cross section 3.6 mb for Al and 5.4 mb for Ca, 
as given in Ref.~\cite{Ahrens}.
}}
\label{gam62}
\end{figure}

The CHIPS model covers a wide spectrum of hadronic reactions with a
large number of degrees of freedom. In the case of nuclear reactions 
the CHIPS generator helps to understand phenomena such as the 
order-of-magnitude splitting of neutron and proton spectra, the high 
yield of energetic nuclear fragments, and the emission of nucleons 
which kinematically can be produced only if seven or more nucleons 
are involved in the reaction.

The CHIPS generator allows the extraction of collective parameters of 
a nucleus such as clusterization. The qualitative conclusion based on 
the fit to the experimental data is that most of the nucleons are 
clusterized, at least in heavy nuclei.  The nuclear clusters can be 
considered as drops of a liquid nuclear phase. The quark exchange 
makes the phase space of quark-partons of each cluster common, 
stretching the kinematic limits for particle production.

The hypothetical quark exchange process is important not only for
nuclear clusterization, but also for the nuclear hadronization 
process.  The quark exchange between the excited cluster (quasmon) 
and a neighboring nuclear cluster, even at low excitation level, 
operates with quark-partons at energies comparable with the nucleon 
mass. As a result it easily reaches the kinematic limits of the 
reaction, revealing the multi-nucleon nature of the process.

Up to now the most under-developed part of the model has been the 
initial interaction between projectile and target. That is why we 
started with proton-antiproton annihilation and pion capture on 
nuclei at rest, because the interaction cross section is not involved. 
The further development of the model will require a better 
understanding of the mechanism of the first interaction.  However, 
we believe that even the basic model will be useful in the 
understanding the nature of multi-hadron fragmentation.  Because 
of the model's features, it is a suitable candidate for the hadron 
production and hadron cascade parts of the newly developed event 
generation and detector simulation Monte Carlo computer codes.

\section[Modeling of real and virtual photon interactions]{Modeling of real and virtual photon interactions with nuclei below pion production threshold}

In the example of
the photonuclear reaction discussed in the Appendix D, namely
the description of $90^{\circ}$ proton and deuteron spectra in
$A({\gamma},X)$ reactions at $E_{\gamma} = 59-65$ MeV, the assumption
on the initial Quasmon excitation mechanism was the same. The
description of the $90^{\circ}$ data was satisfactory, but the
generated data showed very little angular dependence, because the 
velocity of the quasmons produced in the initial state was small, 
and the fragmentation process was almost isotropic.  Experimentally, 
the angular dependence of secondary protons in photo-nuclear reactions 
is quite strong even at low energies (see, for example,
Ref.~\cite{Ryckebusch}). This is a challenging experimental fact which
is difficult to explain in any model. It's enough to say that if the
angular dependence of secondary protons in the $\gamma ^{40}$Ca
interaction at 60 MeV is analyzed in terms of relativistic boost, then
the velocity of the source should reach $0.33 c$; hence the mass
of the source should be less than pion mass. The main point of this 
discussion is to show that the quark-exchange mechanism used in the
CHIPS model can not only model the clusterization of nucleons in nuclei
and hadronization of intranuclear excitations into nuclear fragments, 
but it can also model complicated mechanisms of the interaction of 
photons and hadrons in nuclear matter.

In Ref. Appendix D a quark-exchange diagram was defined which
helps to keep track of the kinematics of the quark-exchange process
(see Fig.~1 in Apendix D). To apply the same diagram to
the first interaction of a photon with a nucleus, it is necessary to
assume that the quark-exchange process takes place in nuclei
continuously, even
without any external interaction. Nucleons with high momenta do not
leave the nucleus because of the lack of excess energy. The
hypothesis of the CHIPS model is that the quark-exchange forces
between nucleons \cite{NN QEX}\ continuously create clusters in normal
nuclei. Since a low-energy photon (below the pion production threshold)
cannot be absorbed by a free nucleon, other absorption mechanisms
involving more than one nucleon have to be used.

The simplest scenario is photon absorption by a quark-parton in
the nucleon. At low energies and in vacuum this does not work because 
there is no corresponding excited baryonic state. But in nuclear matter
it is possible to exchange this quark with a neighboring nucleon
or a nuclear cluster. The diagram for the process is shown in
Fig.~\ref{diagram1}. In this case the photon is absorbed by a
quark-parton from the parent cluster $\rm{PC}_1$, and then 
the secondary nucleon or cluster $\rm{PC}_2$
absorbs the entire momentum of the quark and photon. The exchange
quark-parton $q$ restores the balance of color, producing the 
final-state hadron F and the residual Quasmon RQ. The process looks like a 
knockout of a quasi-free nucleon or cluster out of the nucleus. It should be
emphasized that in this scenario the CHIPS event generator
produces not only ``quasi-free'' nucleons but ``quasi-free'' fragments
as well. The yield of these quasi-free nucleons or fragments is
concentrated in the forward direction.

The second scenario which provides for an angular dependence is the absorption
of the photon by a colored fragment ($\rm{CF}_2$ 
in Fig.~\ref{diagram2}). In this
scenario, both the primary quark-parton with momentum $k$ and the photon
with momentum $q_{\gamma}$ are absorbed by a parent cluster ($\rm{PC}_2$ in
Fig.~\ref{diagram2}), and the recoil quark-parton with momentum $q$
cannot fully compensate the momentum $k+q_{\gamma}$. 
As a result the radiation of the
secondary fragment in the forward direction becomes more probable.

In both cases the angular dependence is defined by the first act of
hadronization. Further fragmentation of the residual quasmon is
almost isotropic.

\begin{figure}[tbp]
% \centerline{\epsfig{file=hadronic/theory_driven/ChiralInvariantPhaseSpace/plots/diagram1.eps, height=2.5in, width=2.5in}}
%\resizebox{0.70\textwidth}{!}
%{
\includegraphics[angle=0,scale=0.6]{hadronic/theory_driven/ChiralInvariantPhaseSpace/plots/diagram1.eps}
%\includegraphics[angle=0,scale=0.6]{plots/diagram1.eps}
%}
\caption{\protect{Diagram of photon absorption in the quark
exchange mechanism. $\rm{PC}_{1,2}$ stand for parent clusters 
with bound masses
$\tilde{\mu}_{1,2}$, participating in the quark-exchange. $\rm{CF}_{1,2}$
stand for the colored nuclear fragments in the process of quark
exchange. F($\mu$) denotes the outgoing hadron with mass $\mu$ in the
final state. RQ is the residual Quasmon which carries the rest of the 
excitation energy and momentum. $M_{\min}$ characterizes 
its minimum mass defined by its quark content. Dashed lines indicate
colored objects. The photon is absorbed by a
quark-parton $k$ from the parent cluster $\rm{PC}_1$.
}}
\label{diagram1}
\end{figure}

\begin{figure}[tbp]
%  \centerline{\epsfig{file=hadronic/theory_driven/ChiralInvariantPhaseSpace/plots/diagram2.eps, height=2.5in, width=2.5in}}
%\resizebox{0.70\textwidth}{!}
%{
\includegraphics[angle=0,scale=0.6]{hadronic/theory_driven/ChiralInvariantPhaseSpace/plots/diagram2.eps}
%\includegraphics[angle=0,scale=0.6]{plots/diagram2.eps}
%}
\caption{\protect{Diagram of photon absorption in the 
quark-exchange mechanism. The notation is the same as in
Fig.~\ref{diagram1}. The photon is absorbed by the colored fragment
$\rm{CF}_2$.
}}
\label{diagram2}
\end{figure}

It was shown in Section \ref{annil} that the energy spectrum of quark
partons in a quasmon can be calculated as
\begin{equation}
\frac{dW}{k^{\ast }dk^{\ast }}\propto 
\left(1-\frac{2k^{\ast }}{M} \right)^{N-3},  \label{spectrum_1III}
\end{equation}
where $k^{\ast }$ is the energy of the primary quark-parton in the 
center-of-mass system of the quasmon, $M$\ is the mass of the quasmon. 
The number $N$ of quark-partons in the quasmon can be calculated 
from the equation
\begin{equation}
<M^{2}>=4\cdot N\cdot (N-1)\cdot T^{2}.  \label{temperatureIII}
\end{equation}
Here $T$ is the temperature of the system.

In the first scenario of the $\gamma A$ interaction
(Fig.~\ref{diagram1}), because both interacting particles are massless,
we assumed that the cross section for the interaction of a photon with 
a particular quark-parton is proportional to the charge of the 
quark-parton squared, and inversely proportional to the mass of the 
photon-parton system $s$, which can be calculated as
\begin{equation}
s=2\omega k(1-\cos (\theta _{k})).  \label{s}
\end{equation}
Here $\omega $\ is the energy of the photon, and $k$ is the energy of
the quark-parton in the laboratory system (LS):
\begin{equation}
k=k^{\ast }\cdot \frac{E_{N}+p_{N}\cdot \cos (\theta _{k})}{M_{N}}.
\end{equation}
For a virtual photon, equation~(\ref{s}) can be written as
\begin{equation}
s=2k(\omega -q_{\gamma}\cdot \cos (\theta _{k})),
\end{equation}
where $q_{\gamma}$ is the momentum of the virtual photon. In both cases 
equation~(\ref{spectrum_1III}) transforms into
\begin{equation}
\frac{dW}{dk^{\ast }}\propto \left(1-\frac{2k^{\ast }}{M} \right)^{N-3},
\end{equation}
and the angular distribution in $\cos (\theta _{k})$\ converges to a
$\delta $-function. In the case of a real photon 
$\cos (\theta _{k})=1$, and in the case of a virtual photon 
$\cos (\theta _{k})=\frac{\omega }{q_{\gamma}}$.

In the second scenario for the photon interaction 
(Fig.~\ref{diagram2}) we assumed that both the photon and the primary 
quark-parton, randomized according to 
Eq.~(\ref{spectrum_1III}), enter the parent cluster $\rm{PC}_2$,
and after that the normal procedure of quark exchange
continues, in which the recoiling quark-parton $q$ returns 
to the first cluster.

An additional parameter in the model is the relative contribution of
both mechanisms. As a first approximation we assumed equal
probability, but in the future, when more detailed data are obtained,
this parameter can be adjusted.

\begin{figure}[tbp]
% \centerline{\epsfig{file=hadronic/theory_driven/ChiralInvariantPhaseSpace/plots/gam62.eps, height=4.5in, width=4.5in}}
%\resizebox{0.80\textwidth}{!}
%{
\includegraphics[angle=0,scale=0.75]{hadronic/theory_driven/ChiralInvariantPhaseSpace/plots/gam62.eps}
%\includegraphics[angle=0,scale=0.75]{plots/gam62.eps}
%}
\caption{\protect{Comparison of the CHIPS model results (lines) with the 
experimental data~\cite{Ryckbosch} on proton spectra at $90^{\circ}$ 
in the photonuclear reactions on $^{40}$Ca at 59--65 MeV (open
circles),
and proton spectra at $60^{\circ}$ (triangles) and $150^{\circ}$ 
(diamonds).
Statistical errors in the CHIPS results are not shown but
can be judged by the point-to-point variations in the lines. The
comparison is absolute, using the value of the total
photonuclear cross section of 5.4 mb for Ca, as given in Ref.~\cite{Ahrens}.
} }
\label{gam62III}
\end{figure}

\begin{figure}[tbp]
% \centerline{\epsfig{file=hadronic/theory_driven/ChiralInvariantPhaseSpace/plots/gamm_c0606_e123.eps, height=4.5in, width=4.5in}}
%\resizebox{0.80\textwidth}{!}
%{
\includegraphics[angle=0,scale=0.75]{hadronic/theory_driven/ChiralInvariantPhaseSpace/plots/gamm_c0606_e123.eps}
%\includegraphics[angle=0,scale=0.75]{plots/gamm_c0606_e123.eps}
%}
\caption{\protect{Comparison of the CHIPS model results (lines) with the 
experimental data~\cite{Harty} on
proton spectra at $57^{\circ}$, $77^{\circ}$, $97^{\circ}$, 
$117^{\circ}$, and $127^{\circ}$
in the photonuclear reactions on $^{12}$C at 123 MeV (open
circles).  The value of the total photonuclear cross section was set to 1.8 mb.
}  }
\label{gam_123}
\end{figure}

\begin{figure}[tbp]
% \centerline{\epsfig{file=hadronic/theory_driven/ChiralInvariantPhaseSpace/plots/gamm_c0606_e151.eps, height=4.5in, width=4.5in}}
%\resizebox{0.80\textwidth}{!}
%{
\includegraphics[angle=0,scale=0.75]{hadronic/theory_driven/ChiralInvariantPhaseSpace/plots/gamm_c0606_e151.eps}
%\includegraphics[angle=0,scale=0.75]{plots/gamm_c0606_e151.eps}
%}
\caption{\protect{Same as in Fig.~\ref{gam_123}, for the photon energy 151 MeV.}
}
\label{gam_151}
\end{figure}

\begin{figure}[tbp]
% \centerline{\epsfig{file=hadronic/theory_driven/ChiralInvariantPhaseSpace/plots/vgam_c0606k.eps, height=4.5in, width=4.5in}}
%\resizebox{0.80\textwidth}{!}
%{
\includegraphics[angle=0,scale=0.75]{hadronic/theory_driven/ChiralInvariantPhaseSpace/plots/vgam_c0606k.eps}
%\includegraphics[angle=0,scale=0.75]{plots/vgam_c0606k.eps}
%}
\caption{\protect{Comparison of the CHIPS model results (line) with 
the experimental data~\cite{Bates} (open circles) on the
proton spectrum measured in parallel kinematics in the
$^{12}$C(e,e$^{\prime}$p)\ reaction at an energy transfer equal to 210
MeV and momentum transfer equal to 585 MeV/$c$.  Statistical errors in
the CHIPS result are not shown but can be judged by the point-to-point
variations in the line.  The relative normalization is arbitrary.
}  }
\label{vgam}
\end{figure}

We begin the comparison with the data on proton production in the 
$^{40}$Ca$(\gamma,X)$\ reaction at $90^{\circ}$\ and 59--65 MeV
\cite{Ryckbosch}, and at $60^{\circ}$\ and $150^{\circ}$\ and 60 MeV
\cite{Abeele}.  We analyzed these data together to compare the angular
dependence generated by CHIPS with experimental data. The data are
presented as a function of the invariant inclusive cross section 
$f=\frac{d\sigma }{p_{p}dE_{p}}$\ depending on the variable 
$k=\frac{T_{p}+p_{p}}{2}$, 
where $T_{p}$\ and $p_{p}$\ are the kinetic energy and momentum of the
secondary proton. As one can see from Fig.~\ref{gam62III}, the angular
dependence of the proton yield in photoproduction on $^{40}$Ca at 
$60$ MeV is reproduced quite well by the CHIPS event generator.

The second set of measurements that we use for the benchmark
comparison deals with the secondary proton yields in
$^{12}$C$(\gamma,X)$ reactions at 123 and 151 MeV \cite{Harty}, 
which is still below the pion production threshold on
a free nucleon. Inclusive spectra of protons have been measured in 
$\gamma ^{12}$C reactions at $57^{\circ}$, $77^{\circ}$, $97^{\circ}$, 
$117^{\circ}$, and $127^{\circ}$. 
Originally, these data were presented as a function of 
the missing energy. We present the data in Figs.~\ref{gam_123} 
and \ref{gam_151} together with CHIPS calculations in
the form of the invariant inclusive cross section dependent on $k$. 
All parameters of the model such as temperature $T$ and parameters
of clusterization for the particular nucleus were the same as in 
Appendix D, where pion capture spectra were fitted.
The agreement between the experimental data and the CHIPS model results 
is quite remarkable. Both data and calculations show significant strength
in the proton yield cross section up to the kinematic limits of the
reaction. The angular distribution in the model is not as prominent as
in the experimental data, but agrees well qualitatively.

Using the same parameters, we applied the CHIPS event generator to the
$^{12}$C(e,e$^{\prime }$p) reaction measured in Ref.\cite{Bates}. The
proton spectra were measured in parallel kinematics in the interaction
of virtual photons with energy $\omega = 210$ MeV and momentum
$q_{\gamma} = 585$ MeV/$c$. To account for the experimental conditions
in the CHIPS event generator, we have selected protons generated in
the forward direction with respect to the direction of the virtual
photon, with the relative angle $\Theta_{qp} < 6^{\circ}$.  The CHIPS
generated distribution and the experimental data are shown in
Fig.~\ref{vgam} in the form of the invariant inclusive cross section as a
function of $k$.  The CHIPS event generator works only with ground
states of nuclei so we did not expect any narrow peaks for
$^{1}p_{3/2}$-shell knockout or for other shells. Nevertheless we
found that the CHIPS event generator fills in the so-called
``$^{1}s_{1/2}$-shell knockout'' region, which is usually artificially
smeared by a Lorentzian~\cite{Lorentzian}.  In the regular
fragmentation scenario the spectrum of protons below $k = 300$ MeV is
normal; it falls down to the kinematic limit. The additional yield at
$k > 300$ MeV is a reflection of the specific first act of
hadronization with the quark exchange kinematics. The slope increase
with momentum is approximated well by the model, but it is obvious
that the yield close to the kinematic limit of the $2 \rightarrow 2$
reaction can only be described in detail if the excited states of the
residual nucleus are taken into account.

The angular dependence of the proton yield in low-energy photo-nuclear
reactions is described in the CHIPS model and event generator. The
most important assumption in the description is the hypothesis of a
direct interaction of the photon with an asymptotically free quark in
the nucleus, even at low energies. This means that asymptotic freedom of
QCD and dispersion sum rules~\cite{sum_rules} can in some way be
generalized for low energies.  The knockout of a proton from a nuclear
shell or the homogeneous distributions of nuclear evaporation cannot
explain significant angular dependences at low energies.

The same mechanism appears to be capable of modeling proton yields in
such reactions as the $^{16}$C(e,e$^{\prime }$p) reaction measured at MIT
Bates \cite{Bates}, where it was shown that the region of missing
energy above 50 MeV reflects ``two-or-more-particle knockout'' (or the
``continuum'' in terms of the shell model). The CHIPS model may help
to understand and model such phenomena.

\section[Chiral invariant phase-space decay]{Chiral invariant phase-space decay in high energy hadron nuclear reactions}

\noindent \qquad Chiral invariant phase-space decay can be used to
de-excite an excited hadronic system. This possibility can be exploited
to replace the intra-nuclear cascading after a high energy primary
interaction takes place. The basic assumption in this is that the energy
loss of the high energy hadron in nuclear matter is approximately 
constant per unit path length (about 1 GeV/fm). This energy is extracted 
from the soft part of the particle spectrum of the primary interaction, 
and from particles with formation times that place them within the 
nuclear boundaries.

Several approaches of transfering this energy into quasmons were studied, 
and comparisons with energy spectra of particles emitted in the backward 
hemisphere were made for a range of materials. Best results were achieved 
with a model that creates one quasmon per particle absorbed in the nucleus.


\section{Neutrino-nuclear interactions}
\label{numunuc}

The simulation of DIS reactions includes reactions with high $Q^2$. The 
first approximation of the $Q^2$-dependent photonuclear cross-sections 
at high $Q^2$ was made in \cite{photNuc}, where the modified photonuclear 
cross sections of virtual photons \cite{Electronuc} were used.  The 
structure functions of protons and deuterons have been approximated in 
CHIPS by the sum of
non-perturbative multiperipheral and non-perturbative direct
interactions of virtual photons with hadronic partons:
\begin{figure}[tbp]
% \centerline{\epsfig{file=hadronic/theory_driven/ChiralInvariantPhaseSpace/plots/vgam_c0606k.eps, height=4.5in, width=4.5in}}
%\resizebox{0.80\textwidth}{!}
%{
\includegraphics[angle=0,scale=0.60]{hadronic/theory_driven/ChiralInvariantPhaseSpace/plots/gabsa.eps}
%\includegraphics[angle=0,scale=0.60]{plots/gabsa.eps}
%}
\caption{
Fit of $\gamma A$ cross sections with different $H$ values. Data are
from \cite{photNuc}.
}
\label{gamC}
\end{figure}
\begin{equation}
F_2(x,Q^2)=[A(Q^2)\cdot x^{-\Delta(Q^2)}+B(Q^2)\cdot
x]\cdot(1-x)^{N(Q^2)-2},
\label{DIS}
\end{equation}
where $A(Q^2)=\bar{e^2_S}\cdot D\cdot U$, $B(Q^2)=\bar{e^2_V}\cdot(1-D)\cdot V$,
$\bar{e^2}_{V(p)}=\frac{1}{3}$, $\bar{e^2}_{V(d)}=\frac{5}{18}$,
$\bar{e^2_S}=\frac{1}{3}-\frac{\frac{1}{3}-\frac{5}{18}}{1+m^2_\phi/Q^2}
+\frac{\frac{1}{3}-\frac{5}{18}}{1+m^2_{J/\psi}/Q^2}-
\frac{\frac{1}{3}-\frac{19}{63}}{1+m^2_{\Upsilon}/Q^2}$,
$N=3+\frac{0.5}{\alpha_s(Q^2)}$,
$\alpha_s(Q^2)=\frac{4\pi}{\beta_0 ln(1+\frac{Q^2}{\Lambda^2})}$,
$\beta_0^{(n_f=3)}=9$, $\Lambda=200~MeV$,
$U=\frac{(3~C(Q^2)+N-3)\cdot\Gamma(N-\Delta)}
{N\cdot\Gamma(N-1)\cdot\Gamma(1-\Delta)}$, $V=3(N-1)$,
$D(Q^2)=H\cdot  S(Q^2)\left(1-\frac{1}{2}S(Q^2)\frac{\bar{e^2_V}}{\bar{e^2_S}}
\right)$,
$S={\left(1+\frac{m^2_\rho}{Q^2}\right)^{-\alpha_P(Q^2)}}$,
$\alpha_P=1+\Delta(Q^2)$, $\Delta=\frac{1+r}{12.5+2r}$,
$r=\left(\frac{Q^2}{1.66}\right)^{1/2}$, $C=\frac{1+f}{g\cdot (1+f/.24)}$,
$f=\left(\frac{Q^2}{0.08}\right)^2$, $g=1+\frac{Q^2}{21.6}$.
The parton distributions are normalized to the unit total momentum
fraction.

\begin{figure}[tbp]
% \centerline{\epsfig{file=hadronic/theory_driven/ChiralInvariantPhaseSpace/plots/vgam_c0606k.eps, height=4.5in, width=4.5in}}
%\resizebox{0.80\textwidth}{!}
%{
\includegraphics[angle=0,scale=0.60]{hadronic/theory_driven/ChiralInvariantPhaseSpace/plots/f23nud.eps}
%\includegraphics[angle=0,scale=0.60]{plots/f23nud.eps}
%}
\caption{
  Fit of $f_{2d}(x,Q^2)$ (filled circles, solid lines) and
  $f_{3d}(x,Q^2)$ (open circles, dashed lines) structure functions
  measured by the WA25 experiment \cite{WA25}.
}
\label{nuD}
\end{figure}

The photonuclear cross sections are calculated by the eikonal formula:
\begin{equation}
\sigma_\gamma^{tot}=\left[\frac{4\pi\alpha}{Q^2}F_2\left(\frac{Q^2}
{2M\nu},Q^2\right)\right]^{\nu=E}_{Q^2=0},
\label{eikonal}
\end{equation}
An example of the approximation is shown in Fig.~\ref{gamC}. One can
see that the hadronic resonances are ``melted'' in nuclear matter and
the multi-peripheral part of the cross section (high energy) is 
shadowed.

The differential cross section of the $(\nu,\mu)$ reaction was
approximated as
\begin{equation}
\frac{yd^2\sigma^{\nu,\bar\nu}}{dydQ^2}=\frac{G^2_F\cdot        M^4_W}{4\pi\cdot
(Q^2+M^2_W)^2}\left[c_1(y)\cdot f_2(x,Q^2)\pm c_2(y)\cdot xf_3(x,Q^2)\right],
\label{difsec}
\end{equation}
where $c_1(y)=2-2y+\frac{y^2}{1+R}$, $R=\frac{\sigma_L}{\sigma_T}$,
$c_2(y)=y(2-y)$. As $\bar{e^2_V}=\bar{e^2_S}=1$ in
Eq.\ref{DIS}, hence $f_2(x,Q^2)=\left[D\cdot U\cdot
x^{-\Delta}+(1-D)\cdot V\cdot x\right]\cdot(1-x)^{N-2}$,
$xf_3(x,Q^2)=\left[ D\cdot U_{f3}\cdot x^{-\Delta}
+(1-D)\cdot V\cdot x\right]\cdot(1-x)^{N-2}$, with
$D=H\cdot S(Q^2)\cdot\left(1-\frac{1}{2}S(Q^2)\right)$ and
$U_{f3}=\frac{3\cdot    C(Q^2)\cdot\Gamma(N-\Delta)}
{N\cdot\Gamma(N-1)\Gamma(1-\Delta)}$. The approximation is compared
with data in Fig.\ref{nuD} for deuterium \cite{WA25} and in
Fig.\ref{nuFe} for iron \cite{CDHSW,CCFR}. It must be emphasized
that the CHIPS parton distributions are the same as for
electromagnetic reactions.

\begin{figure}[tbp]
% \centerline{\epsfig{file=hadronic/theory_driven/ChiralInvariantPhaseSpace/plots/vgam_c0606k.eps, height=4.5in, width=4.5in}}
%\resizebox{0.80\textwidth}{!}
%{
\includegraphics[angle=0,scale=0.6]{hadronic/theory_driven/ChiralInvariantPhaseSpace/plots/f23nufe.eps}
%\includegraphics[angle=0,scale=0.6]{plots/f23nufe.eps}
%}
\caption{
  Fit of $f_{2Fe}(x,Q^2)$ (filled markers, solid lines) and
  $f_{3Fe}(x,Q^2)$ (open markers, dashed lines) structure functions
  measured by the CDHSW \cite{CDHSW} (circles) and CCFR \cite{CCFR}
  (squares) experiments.
}
\label{nuFe}
\end{figure}

For the $(\nu,\mu)$ amplitudes one can not apply the optical theorem, 
To calculate the total cross sections, it is therefore necessary to 
integrate the differential cross sections first over $x$ and then over 
$Q^2$. For the $(\nu,\mu)$ reactions the differential cross section 
can be integrated with good accuracy even for low energies because it 
does not have the $\frac{1}{Q^4}$ factor of the boson propagator. The 
quasi-elastic part of the total cross-section can be calculated for 
$W<m_N+m_\pi$. The total $(\nu,\mu)$ cross-sections are shown in
Fig.\ref{totqe}(a,b). The dashed curve corresponds to the GRV \cite{GRV}
approximation of parton distributions and the dash-dotted curves
correspond to the KMRS \cite{KMRS} approximation. Neither approximation
fits low energies, because the perturbative calculations
give parton distributions only for $Q^2 > 1~GeV^2$. In \cite{Comby} an
attempt was made to freeze the DIS parton distributions at $Q^2=1$ and
to use them at low $Q^2$. The $W<1.4~GeV$ part of DIS was replaced by
the quasi-elastic and one pion production contributions, calculated on
the basis of the low energy models. The results of \cite{Comby} are
shown by the dotted lines. The nonperturbative CHIPS approximation
(solid curves) fits both total and quasi-elastic cross sections even at 
low energies.

\begin{figure}[tbp]
% \centerline{\epsfig{file=hadronic/theory_driven/ChiralInvariantPhaseSpace/plots/vgam_c0606k.eps, height=4.5in, width=4.5in}}
%\resizebox{0.80\textwidth}{!}
%{
\includegraphics[angle=0,scale=0.60]{hadronic/theory_driven/ChiralInvariantPhaseSpace/plots/numu_cs.eps}
%\includegraphics[angle=0,scale=0.60]{plots/numu_cs.eps}
%}
\caption{
 Fit of total (a,b) and quasi-elastic (c,d) cross-sections of
 $(\nu,\mu)$ reactions (Geant4 database). The solid line
 is the CHIPS approximation (for other lines see text).
}
\label{totqe}
\end{figure}

The quasi-elastic $(\nu,\mu)$ cross sections are shown in
Fig.\ref{totqe}(c,d). The CHIPS approximation (solid line) is compared 
with calculations made in \cite{Comby} (the dotted line) and the best 
fit of the $V-A$ theory was made in \cite{VMA} (the dashed lines). One 
can see that CHIPS gives reasonable agreement.

The $Q^2$ spectra for each energy are known as an intermediate result
of the calculation of total or quasi-elastic cross sections. For the
quasi-elastic interactions ($W<m_N+m_\pi$) one can use $x=1$ and
simulate a binary reaction. In the final state the recoil nucleon has
some probability of interacting with the nucleus. If $W>m_N+m_\pi$ the
$Q^2$ value is randomized and therefore the $Q^2$ dependent
coefficients (the number of partons in non-perturbative phase space
$N$, the Pomeron intercept $\alpha_P$, the fraction of the direct
interactions, etc.) can be calculated. Then for fixed energy and 
$Q^2$ the neutrino interaction with quark-partons (directly or through 
the Pomeron ladder) can be randomized and the secondary parton
distribution can be calculated. In vacuum or in nuclear matter the
secondary partons are creating quasmons \cite{CHIPS1,CHIPS2} which
decay to secondary hadrons.

\section{Conclusion.}

\noindent \qquad For users who would like to improve the
interaction part of the CHIPS event generator for their own
specific reactions, some advice concerning data presentation 
is useful.

It is a good idea to use a normalized invariant function $\rho (k)$%
\[
\rho =\frac{2E\cdot d^{3}\sigma }{\sigma _{tot}\cdot d^{3}p}\propto \frac{%
d\sigma }{\sigma _{tot}\cdot pdE}, 
\]
where $\sigma _{tot}$\ is the total cross section of the reaction.
The simple rule, then, is to divide the distribution over the hadron 
energy $E$ by the momentum and by the reaction cross section. The argument 
$k$ can be calculated for any outgoing hadron or fragment as 
\[
k=\frac{E+p-B\cdot m_{N}}{2}, 
\]
which is the energy of the primary quark-parton. Because the spectrum 
of the quark-partons is universal for all the secondary hadrons or
fragments, the distributions over this parameter have a similar shape 
for all the secondaries. They should differ only when the kinematic 
limits are approached or in the evaporation region. This feature is 
useful for any analysis of experimental data, independent of the CHIPS 
model.

% The released version of the CHIPS event generator is not perfect yet, 
% so in case of an error it is necessary to distinguish between the error 
% of the test program ({\bf CHIPStest.cc}) and the error in the body of 
% the generator.  Usually the error printing contains the address of the 
% routine, but sometimes the name is abbreviated so that instead of 
% {\bf G4QEnvironment}, {\bf G4Quasmon}, or {\bf G4QNucleus}, one will 
% find {\bf G4QE}, {\bf G4Q}, or {\bf G4QN}. The errors in
% {\bf CHIPStest.cc} can be easily analyzed. Even if sometimes energy or
% charge is not conserved, this check can be excluded in order to keep 
% going. On the other hand, if the error is in the body it is difficult 
% to fix. The normal procedure is to uncomment the flags of the debugging 
% prints in the corresponding part of the source code and try to find out 
% the reason. Anyway inform authors about the error. Do not forget to attach the
% {\bf CHIPStest.cc} and the {\bf chipstest.in} files.

Some concluding remarks should be made about the parameters of the model. 
The main parameter, the critical temperature T$_{c}$, should not be varied.
A large set of data confirms the value {\bf 180 MeV} while from the mass 
spectrum of hadrons it can be found more precisely as 182 MeV. The 
clusterization parameter is {\bf 4.} which is just about 4$\pi /3.$
If the quark exchange starts at the mean distance between baryons in the
dense part of the nucleus, then the radius of the clusterization sphere is
twice the ''the radius of the space occupied by the baryon''.
It gives 8 for the parameter, but the space occupied by the baryon can not
be spherical; only cubic subdivision of space is possible so the factor 
$\pi/6 $ appears. But this is a rough estimate, so {\bf 4} or even {\bf 5} 
can be tried.  The surface parameter $fD$ varies slightly with $A$,
growing from 0 to 0.04. For the present CHIPS version the recommended
parameters for low energies are:

\begin{tabular}{llllllllll}
{\bf A} & {\bf T} & {\bf s/u} & {\bf eta} & {\bf noP} & {\bf fN} & {\bf fD}
& {\bf Cp} & {\bf rM} & {\bf sA} \\ 
{\bf Li} & 180. & 0.1 & 0.3 & 223 & .4 & .00 & 4. & 1.0 & 0.4 \\ 
{\bf Be} & 180. & 0.1 & 0.3 & 223 & .4 & .00 & 4. & 1.0 & 0.4 \\ 
{\bf  C} & 180. & 0.1 & 0.3 & 223 & .4 & .00 & 4. & 1.0 & 0.4 \\ 
{\bf  O} & 180. & 0.1 & 0.3 & 223 & .4 & .02 & 4. & 1.0 & 0.4 \\ 
{\bf  F} & 180. & 0.1 & 0.3 & 223 & .4 & .03 & 4. & 1.0 & 0.4 \\ 
{\bf Al} & 180. & 0.1 & 0.3 & 223 & .4 & .04 & 4. & 1.0 & 0.4 \\ 
{\bf Ca} & 180. & 0.1 & 0.3 & 223 & .4 & .04 & 4. & 1.0 & 0.4 \\ 
{\bf Cu} & 180. & 0.1 & 0.3 & 223 & .4 & .04 & 4. & 1.0 & 0.4 \\ 
{\bf Ta} & 180. & 0.1 & 0.3 & 223 & .4 & .04 & 4. & 1.0 & 0.4 \\ 
{\bf  U} & 180. & 0.1 & 0.3 & 223 & .4 & .04 & 4. & 1.0 & 0.4
\end{tabular}

The vacuum hadronization weight parameter can be bigger for light
nuclei and smaller for heavy nuclei, but $1.0$ is a good guess. The 
s/u parameter is not yet tuned, as it demands strange particle 
production data. A guess is that if there are as many $u\bar{u}$ 
and $d\bar{d}$ pairs in the reaction as in the $p\bar{p}$ 
interaction, the parameter can be 0.1. In other cases it is closer 
to 0.3 as in other event generators. But it is bestnot to touch any 
parameters for the first experience with the CHIPS event generator.
Only the incident momentum, the PDG code of the projectile, and the 
CHIPS style PDG code of the target need be changed.


\section{Status of this document}

02.12.05 neutrino interactions section and figures added by M.V. Kossov \\
26.04.03 first four sections re-written by D.H. Wright \\
01.01.01 created by M.V. Kossov and H.P. Wellisch \\

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%****************************

\begin{latexonly}

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\bibitem{Cernigoi}  C.~Cernigoi et al., Nucl. Phys. \textbf{A456} (1986) 599.

\bibitem{Madey}  R.~Madey et al., Phys. Rev. C \textbf{25} (1982) 3050.

\bibitem{Ryckbosch}  D.~Ryckbosch et al., Phys. Rev. C \textbf{42} (1990) 444.

\bibitem{Ahrens}  J.~Ahrens et al.,  Nucl. Phys. \textbf{A446} (1985) 229c.

\bibitem{Ryckebusch} Jan~Ryckebusch et al., Phys. Rev. C \textbf{49} (1994)
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\bibitem{Abeele} C.~Van~den~Abeele; private communication cited
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\bibitem{Harty} P.D.~Harty et al. (unpublished); 
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\bibitem{Bates}  L.B.~Weinstein et al., Phys. Rev. Lett. \textbf{64} (1990)
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\bibitem{Lorentzian}  J.P.~Jeukenne and C.~Mahaux, Nucl. Phys. A \textbf{394}
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\end{thebibliography}

\end{latexonly}

\begin{htmlonly}

\section{Bibliography}

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% event generator,High Energy Accelerator Research Organization (KEK)
% Internal 2000-17, February 2001, H/R

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%%%%%%%%%%%%%%%

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Moscow Physics and Engineering Institute Preprint No. 034-90, Moscow, 1990.

\item G.~Mechtersheimer et al., Nucl. Phys. 
                          \textbf{A324} (1979) 379.

\item C.~Cernigoi et al., Nucl. Phys. \textbf{A456} (1986) 599.

\item R.~Madey et al., Phys. Rev. C \textbf{25} (1982) 3050.

\item D.~Ryckbosch et al., Phys. Rev. C \textbf{42} (1990) 444.

\item J.~Ahrens et al.,  Nucl. Phys. \textbf{A446} (1985) 229c.

\item Jan~Ryckebusch et al., Phys. Rev. C \textbf{49} (1994)
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\item C.~Van~den~Abeele; private communication cited
in the reference: Jan~Ryckebusch et al., Phys. Rev. C \textbf{49} (1994)
2704.

\item P.D.~Harty et al. (unpublished); 
private communication cited
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2704.

\item L.B.~Weinstein et al., Phys. Rev. Lett. \textbf{64} (1990)
1646.

\item J.P.~Jeukenne and C.~Mahaux, Nucl. Phys. A \textbf{394}
(1983) 445.

\end{enumerate}

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