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## The difference between the weak and mass eigenstates in the PNMS matrix

Hi,

I am hoping someone could clear up a few things about neutrinos oscillations for me.

For the sake of this dicussion let's set up the neutrino mixing equations in such a way that the flavor eigenstates are a superposition of mass eigenstates.

So now for example we have

$|\nu_e>=\Sigma U_{1j}|\nu_j>$

Now here is where I am getting confused, the line above reads that the electron neutrino is a superposition of mass eigenstates, SO is it theoretically possible to measure the mass of one electron neutrino and find it is say $m_j$ and then measure the mass of a different electron neutrino and find its mass to be $m_k$ where the two measured masses are not equal. I.E find a pair of eletron neutrinos that have different masses?

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Leaving aside the practical consideration that no-one has yet been able to specifically measure the mass of any kind of neutrino, I think the mass and flavour of neutrinos are complementary measurements under the uncertainty principle - you cannot measure both similtaneously.  Think of the neutral kaons as an analogy - after a certain interval of time, a K0L that started life as a d$\bar{s}$ has a non-zero probability of being a $\bar{d}$s!

But surely if that was the case then it would be nonsensical for experiments such as supernemo to attempt measurements of mass as it would not be possible to assign the mass to a particle..

## The difference between the weak and mass eigenstates in the PNMS matrix

Sorry, my previous post was a bit opaque (and also slightly confused). What I was really trying to say was that neutrinos propagate as mass eigenstates that don't have a definite 'flavour' in the sense of being a sepcific $\nu$e, $\nu$μ or $\nu$τ.

Suppose in an experiment we detect neutrinos by getting them to interact with some heavy particles (eg atomic nuclei) in a detector, and observing the Cerenkov radiation produced by the e-/e+ particles into which they are converted. For simplicity we will ignore reactions that produce μs or τs. Let's also assume we can somehow very accurately measure the energy and momenta (pμ) of both the electron/positron and the recoiling nucleus. From these we could then calculate the energy and momentum of each incoming neutrino and hence its rest mass (m0 = pμpμ).

What we should then find is that the incoming neutrinos may be in different mass eigenstates ($\nu$1/$\nu$2/$\nu$3). A neutrino in any of these states can be "wearing the clothes of" an electron neutrino as it arrives into our detector, so may thus get caught and detected. So if we plotted the measured neutrino masses on the X axis of a graph and the number of neutrinos detected on the Y axis, we should expect to see a separate peak around the mass of each mass eigenstate.

Incidentally, SuperNEMO is an experiment designed to detect neutrinoless double beta decay, which is a somewhat different (though related) thing. It will be a very significant result if they find this, but that's a story for another day.

Mentor  If you took a bucket of stopped pions, and could very, very accurately measure the electron energy in the decay π→e+v, you would see three peaks, one for each of the mass eigenstates.   Essentially, there are three different decays:  π→e+v1,  π→e+v2,  π→e+v3.   Why is this a problem?

This isn't a problem I was confirming whether it is possible to find two electron (or mu or tau) neutrinos that have different masses.  I figured that is what the maths was saying, however I had spoken to someone who insisted this was not the case, which confused me when the maths is basically staring right at you.   EDIT  Yes that is the experimental aim of supernemo, but from that you can set the absolute mass scale, (not the mass squared difference). Yes it also would be significant as it would confirm the majarana'ness (spelt wrong) of neutrinos in general.   FURTHER EDIT  what made my confusion worse was after reading some papers in which they gave upperbounds for the masses of an electron, muon and tau neutrino, which seemed non-sensical.  My question is basically trivial in regards to the maths, however a lot of papers seem to imply opposite answers, spuring confusion.   Im not actually a moron guys, funnily enough I have just joined the T2K experiment and I realised there was an obvious hole in my knowledge of neutrino theory.

Mentor  This is ordinary QM.  If something is in a flavor eigenstate, it's not in a mass eigenstate.  "I was confirming whether it is possible to find two electron (or mu or tau) neutrinos that have different masses" is like saying "is it possible to find two photons that are polarized along x that have different polarizations along z?"

Well in this case the flavor eigenstates are superpositions of mass eigenstates, and vice versa, each mass eigenstate is a superposition of flavor, so mathematically it is.  You take two neutrinos produced in two separate mass eigenstates (call them 1 and 2) and allow them to propagate somewhere, I then attempt to 'measure' their flavor, as both mass eigenstates have a non-zero probability of being in the electron eigenstates then it is possible to find two electron neutrinos with differing masses

Quote by Vanadium 50 					 				 				If you took a bucket of stopped pions, and could very, very accurately measure the electron energy in the decay π→e+v, you would see three peaks, one for each of the mass eigenstates.   Essentially, there are three different decays:  π→e+v1,  π→e+v2,  π→e+v3.   Why is this a problem?
Isn't electronness conserved? So only the decay π→e+v1 exists. It could also decay π→μ+v2. Anyways, neutrinos are created in a flavor eigenstate but propagate in a mass eigenstate.

Mentor

Quote by thedemon13666 					 				 				then it is possible to find two electron neutrinos with differing masses
An electron neutrino does not have a definite mass. It is in a mix of three mass eigenstates.

Likewise, a neutrino of definite mass is not in a flavor eigenstate.

So "the mass of an electron neutrino" is not a well-defined quantity.

Exactly, but if you made a mass measurement the wavefunction would collapse and it would have one of three masses.  (Im talking from a theoretical point of view with the mass measurement)

Mentor

Quote by Vanadium 50 					 				 				If you took a bucket of stopped pions, and could very, very accurately measure the electron energy in the decay π→e+v, you would see three peaks, one for each of the mass eigenstates.   Essentially, there are three different decays:  π→e+v1,  π→e+v2,  π→e+v3.

Quote by robert2734 					 				 				Isn't electronness conserved?
No. We know this because neutrino (flavor) oscillations exist. An antineutrino that was created together with an electron, e.g. in nuclear beta decay, can be absorbed in a reaction that produces a muon.

My interpretation of the mathematics is that if you could select a neutrino with a particular mass by measuring the electron energy precisely in the decay that V50 describes, then that neutrino would be a mixture of e, μ and τ neutrino states, at least from the moment that you measure the electron energy.

Furthermore, it would be a non-oscillating mixture because it would be a pure mass state. The probabilities of getting an e, μ or τ when the neutrino interacts, would be constant, and not vary with position or time as with neutrino oscillations.

Mentor

Quote by jtbell 					 				 				My interpretation of the mathematics is that if you could select a neutrino with a particular mass by measuring the electron energy precisely in the decay that V50 describes, then that neutrino would be a mixture of e, μ and τ neutrino states, at least from the moment that you measure the electron energy.
Because energy is conserved, this happens immediately, and it happens whether or not you actually measure the energy. You could have, and that's enough to collapse the wavefunction.

So, why don't neutrino beams exist in a non-oscillating set of pure mass eigenstates? Because the pion is not a free particle when it decays. It's constrained to be within the volume of the decay pipe, and by the Heisenberg Uncertainty Principle, that localization causes an uncertainty in the momentum too large to determine the neutrino mass from the electron energy. Essentially, it's quantum mechanics on a scale of tens or hundreds of meters doing this.

Ok I am a little confused.  Mathematically the flavor oscillates because of the way you form the pmns matrix. i.e mass = PMNS X flavor  But you could just form it the other way round so:   flavor = PMNS X mass  so why can you not think of the mass as oscillating?   Also could you ellaborate on why the pion is constrained to the volume of the pipe?

Mentor  Well, you can't just move matrices around like that.  You would need to make it the inverse.  And yes, you could write it like that, but why?  It's a little like F = am.   The pion is restricted to the decay pipe just like the air is restricted to a room.  It's in a box - or a pipe, in this case.

That wasnt my point, someone further up stated that the neutrino in the flavor eigenstate wouldnt oscillate between masses, it would be a superposition but fixed in time.... By moving them around I didnt simply mean swapping them, to one part in 10^42 the PNMS matrix is unitary, so where I have written PMNS, I mean PMNS-dagger.  Oh I see, I thought you were describing something deeply mathematical.... not that it is physically trapped in there, derp...

A quick calculation shows that the 10^(-3)*ev^(2) mass^(2) difference between neutrino mass eigenstates can even be "swallowed" into the pion width, its mass uncertainty due to its finite lifetime. the uncertainty on the pion mass squared is of order ev^(2), which is large compared to the neutrino mass squared  contribution to the invariant mass.

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