# Why we can know what flavor the neutrino is created but not their mass? [duplicate]

All neutrino knows which of the 3 flavours they must be upon their creation, however their mass becomes uncertain... why is that? isn't mass supposed to tie to which ever flavours like for example an electron neutrino has a mass of say x electron volt and tau neutrino y electron volt but clearly this is not the case. So my question is what is it that are oscillating their flavour or mass?

## marked as duplicate by John Rennie, Kyle Kanos, Jon Custer, user191954, BuzzDec 14 '18 at 5:27

• @JohnRennie: actually I don't understand what the answer meant but can I safely say that until it is observed both its mass and flavour are in superposition state? – user6760 Dec 13 '18 at 7:26
• A neutrino is produced (e.g. in beta decay) in a flavour eigenstate. However this is not a mass eigenstate. – John Rennie Dec 13 '18 at 7:37
• @JohnRennie: just to be clear we cannot know what each flavour's mass really is or these 3 flavours do not have a fixed mass? I think I'm getting close to the answer. – user6760 Dec 13 '18 at 8:02
• A flavour eigenstate is in a superposition of mass eigenstates. – John Rennie Dec 13 '18 at 8:16

isn't mass supposed to tie to which ever flavours

Intuitively, yes, it's "supposed to"—but it doesn't. It's strange, but that's how it seems to work.

Here's a rough overview: Neutrinos are created in weak interactions. In these interactions, a charged lepton (like an electron) can emit a W boson and "turn into" a neutrino; likewise, a W boson can produce a neutrino/antilepton (or lepton/antineutrino) pair. These interactions conserve flavor: an electron will always appear with an electron-(anti)neutrino, a muon with a muon-(anti)neutrino, etc. These states, of a single, well-defined flavor, are called "flavor eigenstates".

Likewise, a neutrino with a single, well-defined mass is in a "mass eigenstate". If you've only studied non-relativistic QM, this might seem strange—isn't mass just a constant of the theory? Doesn't "mass eigenstate" imply a "mass operator"? What would that operator look like? To make sense of it, you need to move to a relativistic mindset. In relativity, a particle's energy and momentum are closely related; they form a four-vector, and this vector's length is a scalar quantity: $$mc^2$$. That suggests you can define a sort of "mass operator" using energy and momentum operators. Similar to how a particle's flavor state determines how it participates in weak interactions, a particle's mass state determines how its free states propagate through space.

It turns out, though, that for neutrinos, the flavor eigenstates are not, in fact, mass eigenstates. A flavor eigenstate is a superposition of mass eigenstates, and vice-versa. This means that the flavors of neutrinos don't actually have individual, well-defined masses. There simply are "three neutrino flavors" and "three neutrino masses", but we can't assign a one-to-one correspondence between them.

The reason this causes neutrino oscillations is because, as I mentioned, a particle's mass state determines how it propagates through space. A neutrino created by an electron interaction is necessarily an electron neutrino, so it must be in a superposition of three mass states. These three mass states then propagate (as plane waves) at different speeds, and as they do, their respective waves get out-of-phase and interfere with each other. The result ends up being that, if you measure the neutrino at some later time, it'll still be in a mixture of mass states, but won't necessarily be the same mixture. The new mixture, generally, won't correspond to the same flavor eigenstate the neutrino was created in, but will be a superposition of flavor states. When you measure it, then, there's some chance you'll get a different flavor than you started with. This is the source of "neutrino oscillations".

• They don't "come from" anything any more than the flavors come from anything. They're just experimentally measured quantities. Physically, they describe how different components of a neutrino state propagate through free space, so we measure them by looking at how neutrinos propagate; in practice, we perform that measurement by looking at neutrino oscillations. – TheMac Dec 13 '18 at 8:14
• If we wanted to write out all the particles in the Standard Model, we could just as well say it includes three neutrinos, $\nu_1$, $\nu_2$, and $\nu_3$, with masses $m_1$, $m_2$, and $m_3$. Those three neutrinos wouldn't interact with other particles in straightforward way, though, since they're don't have well defined flavors anymore, so from a practical standpoint it makes more sense to think the three flavor eigenstates as the more "fundamental" particles of our theory, and accept that those flavors don't have well-defined masses as a peculiarity that we'll just have to deal with. – TheMac Dec 13 '18 at 8:14
• so you're saying until observation is made, the neutrino first start off as a specific flavour then as it moves it's mass is in a superposition of all 3 diff mass value... upon observation we can tell its flavour by its mass am I close? – user6760 Dec 13 '18 at 8:15
• Pretty much! It doesn't move into a mass superposition though, it's always in a mass superposition—it's just which superposition it's in that varies. – TheMac Dec 13 '18 at 8:16