From: tjroberts137@sbcglobal.net   
      
   On 3/29/20 3:43 PM, Michael Moroney wrote:   
   > Re SN 1987A, what is believed to be the reaction(s) which produce the   
   > majority of its neutrinos?  What is the energy of those neutrinos?   
      
   The usual suspects: nuclear interactions giving neutrinos of a few MeV,   
   and particle decays that can be either lower or higher energy.   
      
   > Assuming the SN1987A progenitor is relatively stationary wrt us, and   
   > from the time between neutrino detection and visible evidence of the   
   > supernova, we know the neutrinos must be going at very close to c.   
   > However, there should be an estimate of their minimum speed (still   
   > nearly c, of course, and from that we should be able to determine the   
   > minimum gamma of the neutrinos from our reference. From that and the   
   > estimated energy of the neutrinos, we should be able to put an upper   
   > limit on their masses, and it would have to be very small for such a   
   > large gamma.  I assume this has been done, correct? Or does neutrino   
   > oscillation throw everything off?   
      
   The difficulty is understanding the astrophysical processes that emit   
   the neutrinos,and those that emit the light. The neutrinos were detected   
   BEFORE the light from SN1987A, so the astrophysical processes dominate   
   the time difference, not slower propagation due to nonzero neutrino mass.   
      
   By applying a common astrophysical model, Arnett and Rosner calculate an   
   upper bound on neutrino mass of 12 eV [Arnett and Rosner, PRL _58_   
   (1987), p1906]. That is significantly higher than other experiments'   
   upper limits. And the uncertainty in the astrophysics involved is large.   
      
   > Speaking of which, I don't see how a neutrino could convert to a   
   > different type with a different mass without violating conservation of   
   > energy, momentum or both. What is happening with this?   
      
   Neutrino oscillation is not a "conversion". Rather it is the usual   
   quantum mechanical process of projecting a wavefunction onto different   
   basis eigenstates. When the weak interaction creates a neutrino, it is   
   created in a "flavor eigenstate" -- either nu_e, nu_mu, or nu_tau   
   (electron, muon, and tau neutrino). But those are not the same as the   
   "mass eigenstates" (which are better called "eigenstates of the   
   propagation Hamiltonian"). These latter eigenstates describe how the   
   wavefunction (amplitude) propagates. Each mass eigenstate has nonzero   
   overlap with each flavor eigenstate, but with different factors, so each   
   flavor eigenstate has different fractions of the mass eigenstates. Since   
   the mass eigenstates have different masses, their amplitudes vary in   
   space along the propagation direction with different wavelengths (this   
   is the usual rotation of their complex phase). This would be   
   unobservable by itself, but when the neutrino interacts in the detector,   
   it does so as a flavor eigenstate, and the different mass eigenstates in   
   the wavefunction interfere, giving a different amplitude for the flavor   
   eigenstates than when it was created. That difference is a function of   
   which flavors are involved, the distance between creation and detection,   
   and the energy of the neutrino -- measurements on earth over several   
   hundred kilometers can provide considerable detail of the process   
   because the neutrino beam has a range of energies.   
      
   As for conservation of 4-momentum, remember that in QM no wavefunction   
   is perfectly sharp. The creation interaction expressed as a flavor   
   eigenstate has sufficient indeterminacy in 4-momentum to have nonzero   
   overlaps with each of the mass eigenstates. Each mass eigenstate   
   propagates independently, conserving 4-momentum. Since their amplitudes   
   vary differently in space, and the flavor eigenstates have different   
   fractions to the mass eigenstates, at the detector the overlap with   
   flavor eigenstates is different from what it was at creation. So at   
   creation the neutrino had a definite flavor, but at the detector it has   
   nonzero amplitude to be each of the three flavors; for a given neutrino   
   the detector will see only one of them, but for the beam there will be a   
   distributions of all three flavors, even if the beam was all created as   
   a single flavor.   
      
   Tom Roberts   
      
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