# Does neutrino oscillation from electron neutrino to muon neutrino conserve energy?

when an electron neutrino is produced in the sun, it have the total energy of E. the total energy E have the relation with mass and momentum as E2=p2+m2. during the journey of its propagation from sun to earth, the electron neutrino change their flavor to muon neutrino which have mass m', momentum p', and energy E'.

E2-p2=m2 (initial)

E'2-p'2=m'2 where m'>m (final)

then the four vector momentum is not conserved ie m'>m.

if we say the energy and momentum is conserved, E=E', p=p', then m'=m. but in reality we know than m'>m

I am confusing...can anybody helps?

Demystifier
Gold Member

The neutrino is in the state which is a SUPERPOSITION of neutrinos with two different masses and energies. Thus the energy and the mass are uncertain, but the average energy is conserved with time. What changes (oscillates) with time is the probability that the neutrino will have this or that mass. When you do a measurement, you "collapse" the state into a state with a definite mass.

blechman

I made this mistake (both here and in my own research, I'm sad to say!) more than once.

I've recently come to the conclusion that it's easiest (although by no means necessary!) to think of it in terms of wave mechanics. In particular, compare it to CLASSICAL E&M (light):

Let's say you have a beam of (polychromatic) light traveling through a (nonlinear) medium. Then you know that the different frequencies of light that make up the beam will be traveling at different speeds, since the index of refraction depends on the frequency of the light. This is the phenomenon of "dispersion" from optics.

Now the frequency of the light is its energy. But energy is ALWAYS conserved! It's not that energy is not conserved here, but that the components of the light that make the beam all have different energies.

Now return to the neutrino problem: think of an "electron neutrino" as a beam of "polychromatic light" (since the electron neutrino is composed of various mass eigenstates, it's like the light being composed of various wavelengths). Now the analogy is virtually exact.

Hope that helps!

If you have access, questions such as "how a precise measurement of energy and momentum prevents coherence and thus oscillations" are dealt with in details in PRD vol 48 n 9 (1993)