Spectral function in NuWro

Tomasz Golan (on behalf of NuWro Collaboration)

03-05.12.2017, NuWro Workshop 2017

Spectral Function



The probability of removing a nucleon with momentum \(\vec p\) and leaving residual nucleus with excitation energy \(E\).

\begin{eqnarray} & & P(\vec p, E) = \\ & & \sum\limits_n \left|\left<\psi_n^{A-1}|a_p|\psi_0^A\right>\right|^2\delta(E + E_0 - E_n) \end{eqnarray}

O. Benhar et al. PRD72 (2005) 053005

Spectral Function Begins


  • Artur Ankowski PhD Thesis (in Polish) - collaboration with O. Benhar

  • in NuWro spectral function for:

    • carbon
    • oxygen
    • iron
    • (approximated) calcium
    • (approximated) argon

Implementation

Starting point


  • differential cross section (in impulse approximation)

    \[d\sigma = x \int dE d^3p P(\vec p, E) \int d^3k' \delta(E_{k'} + E_{p'} - y)\frac{L_{\mu\nu}H^{\mu\nu}}{E_p E_{k'} E_{p'}}\]

    • \(x = \frac{G_F^2\cos^2\theta_C}{8\pi^2E_k}\)
    • \(y = E_k + M - E\) (constant for fixed \(E\) and \(\vec p\))

Momentum and removal energy




  • momentum (\(\vec p\)) and removal energy (\(E\)) are chosen randomly according to \(P(\vec p, E)\)

  • integral over \(d^3k'\) is performed for fixed \(\vec p\) and \(E\)

src: A. Ankowski thesis

Integral over final lepton kinematics


  • having \(\vec p\) and \(E\) fixed final lepton kinematics is calculated in CMS frame

  • this approach is much faster than the original implementation

  • with good numerical stability

  • and the result are the same

Spectator


  • if an interaction happens on correlated nucleon we assume a spectator to have momentum \(-\vec p\)

  • educated guess method is used to determine if this is a case

Pauli blocking


  • PB1: using avg. Fermi momentum \(\tilde p_F\)

    \[P(\vec p, E) \rightarrow P(\vec p, E)\cdot\Theta(\tilde p_F - |\vec p_{final}|)\]
  • PB2: using local Fermi momentum \(\tilde p_F\)

    \[P(\vec p, E) \rightarrow P(\vec p, E)\cdot\Theta(p_F(r) - |\vec p_{final}|)\]


  • PB3: using momentum distribution \(n(\vec p\)) = dE P(p, E)$ given by spectral function:

    \[P(\vec p, E) \rightarrow P(\vec p, E)\cdot\Theta(n(\vec p_{final}) - \text{random[0,1]})\]

Pauli blocking


Coulomb corrections

Coulomb correction to SF


  • protons repel each other (smaller binding energy)

  • protons energy levels are measured

  • for neutrons one need to apply Coulomb correction

    \[P_{hole}^n (\vec p, E) = P_{hole}^{p}(\vec p, E - \Delta)\]

  • \(\Delta = 2.8\) MeV (for Carbon, src: A. Ankowski, PRC86 (2012) 024616)

  • affects NC and neutrino CC

Charged lepton and Coulomb potential


  • Coulomb potential decrease (increase) energy of negative (positive) charged particles (leaving nucleus)

  • assuming constant shift \(\Delta = 3.5\) MeV (for Carbon, src: A. Ankowski et al., PRD91 (2015) 03305)

  • affects CC (opposite sign for neutrino and anti-neutrino)

Final state interactions

From IA to FSI


  • knock-out nucleon interacts with spectators, which modifies energy transfer

    \[\frac{d\sigma^{FSI}}{d\omega d\Omega} = \int d\omega' f_{\vec q}(\omega - \omega') \frac{d\sigma^{IA}}{d\omega' d\Omega}\]

  • note: fixed angle

  • \(f_{\vec q}\) - folding function

Folding function




\[f_{\vec q} (\omega) = \delta(\omega)\sqrt{T_A} + (1 - \sqrt{T_A})F_{\vec q}(\omega)\]

  • \(T_A\) - nuclear transparency
  • \(F_{\vec q}(\omega) = \frac{1}{\pi}\frac{U_W}{U_W^2 + \omega^2}\)
  • \(U_W\) is the imaginary part of the potential (\(U = U_V + iU_W\))
  • Real part of the potential shifts the energy transfer

\[f_{\vec q}(\omega - \omega') \rightarrow f_{\vec q}(\omega - \omega' - U_V)\]

Nuclear transparency




  • note: \(\sqrt{T_A} > 0.75\)

    \[f_{\vec q} (\omega) = \delta(\omega)\sqrt{T_A} + (1 - \sqrt{T_A})F_{\vec q}(\omega)\]

  • imaginary part of the potential does not have much impact

O. Benhar et al., PRD72 (2005) 053005

Folding function




  • calculated for fixed \(\vec q = 1\) GeV

  • it smears energy transfer distribution

  • in NuWro implementation Gauss fit is used

O. Benhar, PRC87 (2013) 024606

Real part of the potential



  • \(U_V\) shifts the energy transfer distribution

  • \(U_W\) smears the energy transfer distribution with the probability \((1 - \sqrt{T_A})\)

  • both \(U_V\) and \(T\) depends on the final nucleon kinetic energy

A. Ankowski et al., PRD91 (2015) 033005

Final nucleon energy




\[\frac{d\sigma^{FSI}}{d\omega d\Omega} = \int d\omega' f_{\vec q}(\omega - \omega') \frac{d\sigma^{IA}}{d\omega' d\Omega}\]

  • to calculate this integral for fixed scattering angle the following approximation is used for final nucleon kinetic energy

\[T_k = \frac{E_k^2 \cdot (1 - \cos\theta)}{M + E_k \cdot (1 - \cos\theta)}\]

MC Procedure



\[\frac{d\sigma^{FSI}}{d\omega d\Omega} = \int d\omega' f_{\vec q}(\omega - \omega') \frac{d\sigma^{IA}}{d\omega' d\Omega}\]

  • calculate \(\frac{d\sigma^{IA}}{d\omega' d\Omega}\)

  • calculate \(T_k\) of final nucleon, and then \(U_V\) and \(T_A\)

  • if rand[0,1] > \(\sqrt{T_A}\): \(\omega = \omega' + U_V\)

  • else \(\omega = \omega' + U_V + x\), where \(x\) is random from \(F_{\vec q}(\omega)\)

Reconstructed energy

Procedure


  • original results from A. Ankowski et al., PRD91 (2015) 033005 (include FSI and Coulomb corrections)

  • reconstructed energy formula (nucleon at rest)

    \[E_\nu = \frac{2E_l\tilde M - m^2 + \tilde M^2 - M^2}{2(\tilde M - E_l + |\vec k_l|\cos\theta)}\]

  • \(\tilde M = M - \varepsilon\)

  • \(\varepsilon = 19\) MeV for neutrino, \(\varepsilon = 6\) MeV for anti-neutrino

Comparison


Coulomb on/off


Summary


  • The implementation of spectral function in NuWro

    • is super fast

    • includes spectator knock-out

    • includes Coulomb correction (for Carbon)

    • includes FSI effects (for Carbon)

  • Approximated SF for Argon can be used (before the "real one" becomes available)