The implementation of intranuclear cascade

Tomasz Golan (on behalf of NuWro Collaboration)

03-05.12.2017, NuWro Workshop 2017

Cascade by Metropolis

Intranuclear cascade

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

Total cross section

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 \(\pi^+n\)

Total cross section (low energies)

Nucleons below \(335\) MeV

\(\beta\) - velocity of incoming nucleon

Pions below \(51\) MeV

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

Interaction parameters (nucleons)

\(f_{inel}\) - the fraction of pion production
\(f_{\pi}\) - the fraction of single pion production
angular distribution in CMS

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

Interaction parameters (pions)

\(f_{inel}\) - 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

Cascade algorithm

The main loop

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

Free path

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)

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

N-N interactions

\(\pi\)-N interactions

Improvements of cascade model in NuWro (nucleons)

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

N-N inelastic

based on experimental data

proton-Carbon scattering

N-N nuclear correction

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

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

N-N nuclear correction on/off

proton (E = 1 GeV) on Carbon

ArgoNeut data

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

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

\[T_k < V(r)\]

Neutron / Proton

At this point protons and neutrons are treat the same way
Work in progress

Improvements of cascade model in NuWro (pions)

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

Low-energy pions

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

\(\Delta\) 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 \(\rho\))

Comparison with Oset et al.

High-energy pions

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

Charge fragmentation

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

Angular distributions

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)

Pion-Carbon scattering

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

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

no elastic hadron-nucleus!

Pion-Carbon scattering

Formation time and \(\Delta\) lifetime

Formation time

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

\(\Delta\) lifetime

in primary vertex \(\Delta\) decays instantly
its lifetime is included in cascade

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

Summary

Improvements in progress / planned:
- off-shell propagation
- reweighting
Kaon cascade?
Alternatives to intranuclear cascade?