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\begin{document}


The nuclear incoherent (NI) $\pi ^{0}$ photoproduction cross section is
calculated using an extended version of the Monte Carlo Multicollisional
intranuclear cascade model (MCMC). The model uses the Impulse Approximation
(IA) for the $\pi ^{0}$ photoproduction and assumes that the incoming photon
interacts directly with a single nucleon. In this PWIA, the NI cross section
is proportional to the single nucleon cross section times a factor that
accounts for the short range correlations\cite{Eng64}. The $\pi ^{0}-$
nucleus FSI and the shadowing of the incoming photon distort the PWIA due to
the $\pi ^{0}$ absorption and re-scattering. Furthermore, additional neutral
pions can also be produced at forward angles in secondary scatterings. Such
complicated scenario can be described semi-classically as two correlated
intranuclear cascade mechanisms triggered by the produced $\pi ^{0}$ and
struck nucleon. In this approach, the dynamics of the nuclear ensemble is
taken into account using the concept of interaction probability in a\
relativistic and time-dependent Monte Carlo algorithm.

The basic features and improvements of the MCMC model \cite%
{Rod04,Rod05,Rod06,Rod07} are: i) the inclusion of the $\pi ^{0}$
photoproduction mechanism within 4.0 to 6.0 GeV in terms of $\rho $ and $%
\omega $ exchange (see Fig.1), constrained by the available data\cite%
{Bra70,And71}, ii) the use of two different nuclear densities for Carbon and
Lead\cite{Mey72}, iii) the incorporation of an accurate momentum
distribution for $^{12}$C based on the global $1s$ and $1p$ proton knock-out
data\cite{Lap00}, iv) a rigorous non-stochastic Pauli-blocking both for the
photoproduction and multiple $\pi ^{0}N$ scatterings, v) the implementation
of the shadowing effect using a VMD model with formation time constraint\cite%
{Bau78}, and vi) a consistent analysis of the full $\pi ^{0}-$ nucleus FSI,
as well as the use of realistic (diffractive) angular distributions for the $%
\pi ^{0}N\rightarrow \pi ^{0}N$ elastic channel.

Most of the $\pi ^{0}N\rightarrow X$ channels relevant for the PrimEx were
incorporated in the routine combining the properties of isospin invariance
and time reversal of the strong interactions with the experimental results
of charged pions. Collisions of the type $\pi N,\pi N^{\ast },\pi \Delta
,NN,NN^{\ast }$ and $N\Delta $ are taken into account, but collisions
between pairs of pions are not considered due to a much lower pion density.
The energy dependence of the cross sections for the entrance channels $\pi
^{0}p$ and $\pi ^{0}n$ are presented in Figs. 2 and 3, respectively.

The results of the single differential cross sections at 5.2 GeV from Carbon
and Lead taking into account an elasticity cut of $0.92$ are presented in
Figs. 4 and 5, respectively. The double differential cross sections within
the same kinematics are shown in Figs. 6 and 7 for Carbon and Lead,
respectively. Fig. 8 presents the total $\pi ^{0}$ energy spectra due to
incoherent production.

\begin{thebibliography}{99}
\bibitem{Eng64} C. A. Engelbrecht, Phys. Rev. \textbf{133}, B988 (1964).

\bibitem{Rod04} T. E. Rodrigues, J. D. T. Arruda-Neto, A. Deppman, V. P.
Likhachev, J. Mesa, C. Garcia, K. Shtejer, G. Silva, S. B. Duarte and O. A.
P. Tavares, Phys. Rev. C \textbf{69}, 064611 (2004).

\bibitem{Rod05} T. E. Rodrigues, J. D. T. Arruda-Neto, J. Mesa, C. Garcia,
K. Shtejer, D. Dale and I. Nakagawa, Phys. Rev. C \textbf{71}, 051603(%
\textbf{R}) (2005).

\bibitem{Rod06} T. E. Rodrigues, J. D. T. Arruda-Neto, J. Mesa, C. Garcia,
K. Shtejer, D. Dale and I. Nakagawa, Braz. J. of Phys., vol. \textbf{36},
no. 4b, 1366 (2006).

\bibitem{Rod07} T. E. Rodrigues, \textit{Incoherent }$\pi ^{0}$\textit{\
photoproduction in the PrimEx kinematics via the MCMC intranuclear cascade
model}, PrimEx Note \# 52 (2007).

\bibitem{Bra70} M. Braunschweig, W. Braunschweig, D. Husmann, K. L\"{u}%
belsmeyer and D. Schmitz, Nucl. Phys. B \textbf{20}, 191 (1970).

\bibitem{And71} R. L. Anderson \textit{et al.}, Phys. Rev. D \textbf{4},
1937 (1971).

\bibitem{Mey72} W. T. Meyer \textit{et al.}, Phys. Rev. Lett. \textbf{28},
1344 (1972).

\bibitem{Lap00} L. Lapik\'{a}s \textit{et al.}, Phys. Rev. C \textbf{61},
064325 (2000).

\bibitem{Bau78} T. H. Bauer, R. D. Spital, and D. R. Yennie, Rev. Mod. Phys. 
\textbf{50}, 261 (1978).
\end{thebibliography}

\bigskip \newpage

Figure captions:

Fig.1: Differential cross section for $\pi ^{0}$\ photoproduction from the
proton. The blue lines include the strong (red) and the Coulomb
contributions to the cross section. The solid histograms are the cascade
inputs, which take into account only the strong part. The data points were
taken from refs.\cite{Bra70} (squares) and\cite{And71} (circles).

Fig. 2: $\pi ^{0}p\rightarrow X$ channels implemented in the MCMC model.

Fig. 3: $\pi ^{0}n\rightarrow X$ channels implemented in the MCMC model.

Fig. 4: Nuclear Incoherent (NI) cross section $\left( \frac{d\sigma }{%
d\Omega }\right) $ for $\pi ^{{\footnotesize 0}}$\ photoproduction from
Carbon. The black histogram is the PWIA, which is $A$ times the single
nucleon cross section. The red histogram includes the Pauli-principle during
the photoproduction, while the magenta represents the full calculation after
including the $\pi ^{{\footnotesize 0}}-nucleus$ FSI. The blue/olive line is
a polynomial fitting that reproduces the results without/with shadowing. For
details see ref.\cite{Rod07}.

Fig. 5: Nuclear Incoherent (NI) cross section $\left( \frac{d\sigma }{%
d\Omega }\right) $ for $\pi ^{{\footnotesize 0}}$\ photoproduction from
Lead. The notation is the same as in Fig. 4.

Fig. 6: Double differential cross section $\left( \frac{d^{2}\sigma }{%
d\theta dE_{\pi ^{0}}}\right) $ for incoherent $\pi ^{{\footnotesize 0}}$\
photoproduction from Carbon without shadowing. To include shadowing just
multiply the non-shadowed results by 0.76.

Fig. 7: Double differential cross section $\left( \frac{d^{2}\sigma }{%
d\theta dE_{\pi ^{0}}}\right) $ for incoherent $\pi ^{{\footnotesize 0}}$\
photoproduction from Lead without shadowing. To include shadowing just
multiply the non-shadowed results by 0.75.

Fig. 8: $\pi ^{{\footnotesize 0}}$\ energy spectra due to incoherent
production from Carbon (black) and Lead (red) including full FSI of the
produced pions. The results are for pion elasticity grater than 0.92 and do
not include shadowing. To include shadowing just multiply the non-shadowed
results by 0.76 for Carbon and 0.75 for Lead.

\end{document}
