ANL-Osaka DCC model
neutrino
Structure function
The hadron tensor of meson production reaction on nucleon $(N(P) + q \rightarrow f(P') )$ defined as
\( W^{\alpha,\mu \nu} = \sum_{\bar{i},f}(2\pi)^3\delta^4(P+q-P')\frac{E_N}{m_N} < f |J^{\mu}(q)|i>< f |J^{\nu}(q)|i>^* \)
is parametrized in a conventional form as
\(W^{\alpha,\mu \nu} = - g^{\mu \nu} W_1^\alpha + \frac{P^\mu P^\nu}{m_N^2} W_2^\alpha +i \frac{\epsilon^{\mu\nu\rho\sigma}P_\rho q_\sigma}{2m_N^2}W_3^\alpha + \frac{q^\mu q^\nu}{m_N^2}W_4^\alpha + \frac{P^\mu q^\nu + q^\mu P^\nu}{m_N^2} W_5^\alpha\)
$\alpha = EM, CC, NC$ for electromagnetic, charged current and neutral current reactions. The currents are expressed in terms of vector current $V^\mu$ and axial vector current $A^\mu$ as described in [ref],
\( J^{EM}_\mu = V^{3}_\mu + V^{Iso Scalar}_\mu \)
\( J^{CC}_\mu = V_\mu^{\pm} - A_\mu^{\pm} \)
\( J^{NC}_\mu = (1 - 2 \sin^2\theta_W)J^{EM}_\mu - V^{Iso Scalar}_\mu - A^{3}_\mu \)
It is noticed that the weak and electromagnetic coupling constants are not included our $W^\alpha_i$.
Cross section formula
The inclusive cross sections(charm or strangeness changing reactions are not included) for EM,CC and NC reactions are given as
\( \frac{d\sigma^{EM}}{dE_{l'}d\Omega_{l'} } = \frac{4 \alpha^2}{Q^4} \frac{|\vec{p}_{l^\prime}|} {|\vec{p}_l|} E_lE_{l'}[2W_1^{EM} (\sin^2\frac{\chi}{2} - \frac{m_l^2}{E_l E_{l'}}) + W_2^{EM} (\cos^2 \frac{\chi}{2}+ \frac{m_l^2}{E_l E_{l'}})] \)
\( \frac{d\sigma^{CC}}{dE_{l}d\Omega_{l} } = \frac{G_F^2 V_{ud}^2 |\vec{p}_{l}|E_{l}}{2\pi^2} [2W_1^{CC} \sin^2\frac{\chi}{2} + W_2^{CC} \cos^2 \frac{\chi}{2} \pm \frac{W_3^{CC}}{m_N}( (E_\nu + E_{l})\sin^2\frac{\chi}{2} - \frac{m_{l}^2}{2E_{l}} ) \)
\( + \frac{m_{l}^2}{m_N^2}W_4^{CC} \sin^2 \frac{\chi}{2} - \frac{m_{l}^2}{m_N E_{l}}W_5^{CC}] \)
\( \frac{d\sigma^{NC}}{dE_{\nu'}d\Omega_{\nu'} } = \frac{G_F^2 E_{\nu'}^2}{2\pi^2} [2W_1^{NC} \sin^2\frac{\theta_{\nu'}}{2} + W_2^{NC} \cos^2 \frac{\theta_{\nu'}}{2} \pm \frac{W_3^{NC}}{m_N} (E_\nu + E_{\nu'})\sin^2\frac{\theta}{2}] \)
Here $\pm$ is for neutrino and anti-neutrino reactions. $\displaystyle \cos\chi = \frac{p_{l'}}{E_{l'}}\cos \theta_{l'}$. $E_{l,l'},\theta$ are energy and scattering angle of lepton in the target rest frame.
Download stucture functions
Sample code
Fortran Code
1. structure functions at working directory, for example, tar xzf wcc.tar.gz
2. run code
Example to calculate cross sections
Output of the code is
\( \frac{d\sigma^\alpha}{dWdQ^2} = \frac{\pi W}{m_N|\vec{p}_l||\vec{p}_{l'}|} \frac{d\sigma^\alpha}{dE_{l'}d\Omega_{l'}} \)
References
The structure functions $W^\alpha$ are calculated based on
1. Dynamical coupled-channels model for neutrino-induced meson productions in resonance region
S. X. Nakamura, H. Kamano, T. Sato, Phys. Rev. D 92, 074024 (2015)
More references on DCC model, amplitudes and cross sections of meson electromagnetic production reaction are found at ANL-Osaka Partial-Wave Amplitudes (PWA)
2. Neutrino-induced forward meson-production reactions in nucleon resonance region
H. Kamano, S. X. Nakamura, T.-S. H. Lee, T. Sato, Phys. Rev. D 86 , 097503 (2012)
3. Nucleon resonances within a dyamical coupled-channels model of $\pi N$ and $\gamma N$ reactions
H. Kamano, S. X. Nakamura, T.-S. H. Lee and T. Sato, Phys. Rev. C 88, 035209 (2013)
4. Isospin decomposition of $\gamma N \rightarrow N^*$ transitions within a dynamical coupled-channels model
H. Kamano, S.X. Nakamura, T.-S. H. Lee, T. Sato, Phys. Rev. C 94, 015201 (2016)
5. Angular distribution in electroweak pion production off nucleons: Odd parity hadron terms, strong relative phases and model dependence
J. E. Sobczyk, E. Hernandez, S. X. Nakamura, J. Nieves, T. Sato, Phys. Rev. D 98, 073001 (2018)
SL model(reaction model for the $\Delta$ resonance region.)
6. Meson-exchange model for $\pi N$ scattering and $\gamma N \rightarrow \pi N$ reaction
T. Sato, T. -S. H. Lee, Phys. Rev. C 54, 2660 (1996)
7. Dynamical study of the $\Delta$ excitation in $N(e,e'\pi)$ reactions
T. Sato and T. -S. H. Lee, Phys. Rev. C 63, 055201 (2001)
8. Dynamical model of weak pion production reactions
T. Sato, D. Uno, T.-S. H. Lee, Phys. Rev. C, 065201 (2003)
9. Quark-hadron duality and parity violating asymmetry of electroweak reactions in the $\Delta$ region
K. Matsui, T. Sato, T.-S. H. Lee, Phys. Rev. C 72, 025204 (2005)
Application of the model for nuclear reactions.
10. Neutrino-nucleus reactions in the delta resonance region
B. Szczerbinska, T. Sato, K. Kubodera, T.-S. H. Lee, Phys. Lett. B 649, 132 (2007)
11. Dynamical model of coherent pion production in neutrino-nucleus scattering
S. X. Nakamura, T. Satom T.-S. H. Lee, B. Szczerbinska, K. Kubodera, Phys. Rev. C 81, 035502 (2010)
12. Incoherent pion production in neutrino-deuteron interactions
Jia-Jun Wu, T. Sato, T.-S. H. Lee, Phys. Rev. C 91, 035203 (2015)
13. Impact of final state interactions on neutrino-nucleon pion production cross sections extracted from neutrino-deuteron reaction data
S. X. Nakamura, H. Kamano, T. Sato, Phys. Rev. D 99, 031301(R) (2019)
