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- Script: N. Metropolis
- Director: J. Sobczyk
- Cast: C. Juszczak, T. Golan, K. Niewczas

*src: N. Metropolis et al., Phys. Rev. 110 (1958) 204*

Nucleons

- \(\sigma_{ii}\) - same isospin
- \(\sigma_{ij}\) - different isospin

Pions

- \(\sigma_{ii}\) - \(\pi^+p\) or \(\pi^-n\)
- \(\sigma_{ij}\) - \(\pi^-p\) or \(\pi^+n\)

*Nucleons below \(335\) MeV*

- \(\beta\) - velocity of incoming nucleon

*Pions below \(51\) MeV*

\(\gamma\) - total energy in \(m_{\pi^0c^2}\)

\(\eta\) - momentum in \(m_{\pi^0c}\)

\(f_{inel}\) - the fraction of pion production

\(f_{\pi}\) - the fraction of single pion production

angular distribution in CMS

\[\frac{d\sigma}{d\Omega} = A\cos^4\theta + B\cos^3\theta + 1\]

\(f_{inel}\) - the fraction of pion production

\(f_{\pi}\) - the fraction of single pion production

angular distribution in CMS

\[\frac{d\sigma}{d\Omega} = A\cos^4\theta + B\cos^3\theta + 1\]

- \(f_{CE}\) - the fraction (of inelastic events) that is charge exchange

General idea

```
until there are particles to propagate
until there are nucleons in nucleus
take a particle from the queue
calculate free path
move particle
if there is no interaction
put the particle back to the queue
otherwise
generate interaction
put all created particles
into the queue
```

- The probability of passing \(\lambda\) without any interactions

\[ P(\lambda) = e^{-\lambda / \tilde\lambda}\] - Mean free path

\[\tilde\lambda = \left[\sigma_p\rho_p(r) + \sigma_n\rho_n(r)\right]^{-1}\] - Free path (an interaction happens if \(\lambda < 0.2\) fm)

\[\lambda = - \tilde\lambda\cdot\ln(\text{rand[0,1]})\]

*all changes are done in a way to keep the structure the same*

*based on experimental data*

*src: V.R. Pandharipande and S.C. Pieper, PRC45 (1992) 791*

- effective mass calculated using potential form
*R.B. Wiringa, PRC38 (1988) 2967*

*proton (E = 1 GeV) on Carbon*

*src: K. Partyka, “Exclusive 1mu+np topologies in ArgoNeuT”, NuInt12, 2012 O. Palamara, “QE or not QE, that is the question”, INT workshop, Seattle, 2013*

binding energy is subtracted from nucleon energy in the primary vertex

the value is stored and use later in the cascade

nuclear potential is defined as

\[V(r) = E_F(r) + E_B\]nucleon is jailed in a nucleus if

\[T_k < V(r)\]

At this point protons and neutrons are treat the same way

Work in progress

*all changes are done in a way to keep the structure the same*

for low-energy pions (\(T_k < 350\) MeV) E. Oset et al (

*Phys. Lett. B165 (1985) 13–18*) is used (as in NEUT)\(\Delta\) width modification in nuclear matter

\[\frac{1}{2}\tilde\Gamma \rightarrow \frac{1}{2}\tilde\Gamma - \text{Im}\Sigma_\Delta\]- \(\tilde\Gamma\) - reduced \(\Delta\) width (due to Pauli blocking)
- \(\Sigma_\Delta\) - \(\Delta\) self-energy

the parametrization of \(\Delta\) self-energy is taken from

*E. Oset et al., Nucl. Phys. A468 (1987) 631–652*

\[\text{Im}\Sigma_\Delta(E_\pi) = -\left[C_Q(\rho/\rho_0)^\alpha + C_{A2}(\rho/\rho_0)^\beta + C_{A3}(\rho/\rho_0)^\gamma\right]\]\(C_Q\), \(C_{A2}\), \(C_{A3}\), \(\alpha\), \(\beta\), \(\gamma\) - functions of pion energy

\(C_{A}\) - pion absorption

implementation: cross sections 2D tables (\(T_k\) and \(\rho\))

Metropolis-like tables based on data

new parameter \(f_{2\pi}\) gives the fraction of double pion production among all non-single pion production processes

for single pion production see a table on the right

for double pion production \(ii\): half is assumed to be with neutal pion

all other cases - equally likely

for QEL and CEX \(\pi\)-N scattering (in CMS)

\[\frac{d\sigma}{d\Omega} \sim \sum\limits_{i=0}^{7}a_i\cos^i\theta\]with \(a_i\) being extracted from SAID model

separately for each channel (

*ii*,*ij*,*0*, and*CEX*)

\[\sigma = \frac{N_i}{N}\pi R^2\]

\(R\) - density \(10^5\) smaller than in the center

*no elastic hadron-nucleus!*

formation time for DIS (Ranft)

\[t_f = \tau_0\frac{E\cdot M}{\mu_T^2}\]\(E\), \(M\) - hadron energy and mass

\(\mu_T^2 = M^2 + p_T^2\) - transverse mass

in primary vertex \(\Delta\) decays instantly

its lifetime is included in cascade

\[t_\Delta = \frac{E_\Delta}{M\Gamma}\]\(\Gamma = 120\) MeV

Improvements in progress / planned:

- off-shell propagation
- reweighting

Kaon cascade?

Alternatives to intranuclear cascade?