Created attachment 129 [details] Mu+ decay distributions in muon rest frame, compared to V-A distributions from Barr et al. The current description of muon decay (p10) says: "The momenta of the two neutrinos are not sampled from their V − A distributions. Instead they are generated back-to-back and isotropically in the neutrinos’ center-of-mass frame, with the magnitude of the neutrino momentum chosen to conserve energy in the decay. The two neutrinos are then boosted opposite to the momentum of the decay electron. This approximation is sufficient for most simulations because the neutrino is usually not observed in any detector." The situation is very much better than that would lead one to believe. Based on my analysis, I suggest adding the following: The approximation used to generate the two neutrinos has the following properties in the muon rest frame: 1. The total energy of the 3 decay products is correct within 0.5 MeV. 2. The total 3-momentum of the 3 decay products is correct (zero) within 0.5 MeV/c. 3. The individual neutrinos have energy distributions that are quite close to their V-A distributions. 4. The individual neutrinos have isotropic angular distributions. The result is that 4-momentum is only approximately conserved in the muon decay process, and the correlations among the decay products are incorrect. But if one generates a neutrino beam from many muon decays, the resulting distributions for each neutrino type are statistically indistinguishable from the correct distributions for unpolarized muon decays. The V-A distributions in the attached plots are from Barr et al, Phys. Rev. D39, p3532 (1989). Their heights were manually varied to visually fit the data.
I just realized that the current description must be wrong. It clearly implies that the distributions of the two neutrinos are identical. They manifestly are not identical (see plots attached to original report).
I was just browsing the bug reports when I saw this. Looking at the comments in the code: G4MuonDecayChannel.hh - This class describes muon decay kinemtics. This version neglects muon polarization assumes the pure V-A coupling gives incorrect energy spectrum for neutrinos G4MuonDecayChannel.cc - M. Melissas ( melissas AT cppm.in2p3.fr), J. Brunner ( brunner AT cppm.in2p3.fr) Adding V-A fluxes for neutrinos using a new algortithm : and the History file contains this note: - May 05, 2005 M.Asai (particles-V07-00-03) - Add neutrino energy spectrum in MuonDecayChannel (Jurgen) So, it seems the documentation is outdated. The original logic for generating the neutrinos is still found in G4MuonDecayChannelWithSpin. // daughter 1 ,2 (neutrinos) // create neutrinos in the C.M frame of two neutrinos G4double energy2 = parentmass*(1.0 - x/2.0); G4double vmass = std::sqrt((energy2-daughtermomentum[0])*(energy2+daughtermomentum[0])); G4double beta = -1.0*daughtermomentum[0]/energy2; G4double costhetan = 2.*G4UniformRand()-1.0; G4double sinthetan = std::sqrt((1.0-costhetan)*(1.0+costhetan)); G4double phin = twopi*G4UniformRand()*rad; G4double sinphin = std::sin(phin); G4double cosphin = std::cos(phin); As for G4MuonDecayChannel this code was replaced by the present since version 7.1.
The section on muon decay has now been updated to reflect the code in G4MuonDecayChannel. The text on neutrino distributions now reads: The magnitudes of the two neutrino momenta are also sampled from the $V-A$ distribution and constrained by energy conservation. The direction of the electron neutrino is sampled using \begin{eqnarray*} cos(\theta) = 1 - 2/E_e - 2/E_{\nu e} + 2/E_e/E_{\nu e} \end{eqnarray*} and the muon anti-neutrino momentum is chosen to conserve momentum. This will appear in the next version of the Physics Reference Manual which is expected soon.
