I Neutrino flavour eigenstates and expansion of the universe

[kimbyd]

The particles nu_e, nu_mu and nu_tau do not exist as physical particles. Talking about them as if they are can only lead to confusion. The physical particles are the mass eigenstates, nu_1, nu_2 and nu_3. Decays like $\nu_3 \rightarrow \nu_1 + \overline{\nu_2} + \nu_2$ can happen if they are kinematically allowed, i.e. in this case $m(\nu_3) > 2m(\nu_2) + m(\nu_1)$.

I think you'll find that the flavor numbers don't match on the left and right sides of that equation, making it questionable at best. This is why such interactions are always written in terms of the flavor eigenstates.

 

[Vanadium 50]

I think you'll find that the flavor numbers don't match on the left and right sides of that equation

Neither the left side nor the right side is in a flavor eigenstate. So how can that possibly be true?

Furthermore, the weak interaction does not conserve flavor.

 

[snorkack]

Furthermore, the weak interaction does not conserve flavor.

What precisely requires a muon to emit two neutrinos to get rid of its muonic flavour? Like an alternative
μ^-→e^-+e^-+e⁺
Spin is conserved, lepton number is conserved... what´s not conserved is lepton flavour. What´s wrong with neutrinoless muon decay?

 

[mfb]

It can happen. It is just extremely unlikely due to the tiny neutrino masses. See post 30. Something like $\displaystyle \frac{m_\nu^4}{m_W^4} = 10^{-50}$ branching fraction, give or take a few orders of magnitude.

Mu3e wants to set 10^-16 as upper limit in the next years. Well, they would prefer finding the decay, but then we need new physics.

 

[kimbyd]

It can happen. It is just extremely unlikely due to the tiny neutrino masses. See post 30. Something like $\displaystyle \frac{m_\nu^4}{m_W^4} = 10^{-50}$ branching fraction, give or take a few orders of magnitude.

I don't think that makes sense as an explanation, because in that instance this decay is also possible:
\mu \rightarrow e + \gamma

Since the photon has zero mass, it is kinematically favored over the neutrino decays.

The Wikipedia article points out that this decay is possible through neutrino oscillation of a virtual neutrino, but this is highly unlikely, probably due to the short amount of time involved for the decay interaction.

 

[Carlos L. Janer]

I don't think that makes sense as an explanation, because in that instance this decay is also possible:
\mu \rightarrow e + \gamma

Since the photon has zero mass, it is kinematically favored over the neutrino decays.

The Wikipedia article points out that this decay is possible through neutrino oscillation of a virtual neutrino, but this is highly unlikely, probably due to the short amount of time involved for the decay interaction.

What makes no sense to me is to consider the propagation of the quantum superposition of three particles in a classical curved space-time. Unfortunately, I am not aware that we have such a theory. Using QFT formalism in a curved space-time is contradictory (there's no Poincaré invariance). I don't know how it could possibly work.

 

[kimbyd]

What makes no sense to me is to consider the propagation of the quantum superposition of three particles in a classical curved space-time. Unfortunately, I am not aware that we have such a theory. Using QFT formalism in a curved space-time is contradictory (there's no Poincaré invariance). I don't know how it possibly work.

For the most part, the curvature of space-time is low enough that the fact that it isn't flat space-time is irrelevant. Just do the quantum examination for the case where space-time is flat and you'll get close to correct.

This breaks down if you're close to a black hole, but works fine in most other cases.

 

[mfb]

The Wikipedia article points out that this decay is possible through neutrino oscillation of a virtual neutrino, but this is highly unlikely, probably due to the short amount of time involved for the decay interaction.

It is not really an amount of time involved in internal lines in a Feynman diagram.

You say the same thing in different words. All the flavor violating decays are extremely unlikely because the neutrinos are so light.

 

[Carlos L. Janer]

For the most part, the curvature of space-time is low enough that the fact that it isn't flat space-time is irrelevant. Just do the quantum examination for the case where space-time is flat and you'll get close to correct.

This breaks down if you're close to a black hole, but works fine in most other cases.

Are you sure about that? I's a quotation I've been given several times. However, your tracing the trajectory and behaviour of relic neutrinos, that decoupled a long time ago from barionic mass and have been contribuiting to the universe expansion ever since. That does not sound right to me.

 

[Carlos L. Janer]

For the most part, the curvature of space-time is low enough that the fact that it isn't flat space-time is irrelevant. Just do the quantum examination for the case where space-time is flat and you'll get close to correct.

Could you give me some references where I could check under what assumptions that aproximation is valid?

 

[kimbyd]

It is not really an amount of time involved in internal lines in a Feynman diagram.

You say the same thing in different words. All the flavor violating decays are extremely unlikely because the neutrinos are so light.

That doesn't make sense to me. It could potentially explain why the $e^- + e^+ + e^-$ decay is disfavored, because the neutrinos are so much lighter than the electrons the decay to neutrinos will be far more common. That doesn't explain why the $e^- + \gamma$ decay is disfavored, as the photon is even less massive.

Where do you get the statement that it's the small neutrino mass that is important in these decay rates?

 

[kimbyd]

Could you give me some references where I could check under what assumptions that aproximation is valid?

Unfortunately I'm not aware of any references offhand. You'd likely be able to search for them about as well as I can.

But the picture is pretty simple: quantum superpositions only really matter in two general cases:
1. There's some kind of discontinuity, such as an event horizon. What happens if one part of the superposition crosses, but the other does not? Thus there are likely issues with the flat-space approximation near black holes.
2. The quantum system produces significant space-time curvature. In this case, because we can't properly describe how a superposition of states impacts space-time curvature, we can't say how such a system gravitates. This isn't really relevant for neutrinos because their density isn't high enough for any significant gravitational effect. It would have been high enough in the early universe, but back then the neutrinos were thermalized enough that they could be treated classically.

 

[Carlos L. Janer]

Unfortunately I'm not aware of any references offhand. You'd likely be able to search for them about as well as I can.

But the picture is pretty simple: quantum superpositions only really matter in two general cases:
1. There's some kind of discontinuity, such as an event horizon. What happens if one part of the superposition crosses, but the other does not? Thus there are likely issues near black holes with the flat-space approximation.
2. The quantum system produces significant space-time curvature. In this case, because we can't properly describe how a superposition of states impacts space-time curvature, we can't say how such a system gravitates. This isn't really relevant for neutrinos because their density isn't high enough for any significant gravitational effect. It would have been high enough in the early universe, but back then the neutrinos were thermalized enough that they could be treated classically.

Ok, I get what you say and I buy it.

However, you're ruling out yourself the possibility for the more massive neutrinos to quickly decay (by an interaction that is beyond the Standard Model) to the lightest one and I thought it was the idea that you were trying to explore with mfb and the reason you were having this conversation with Vanadium50 and mfb.

 

[mfb]

That doesn't make sense to me. It could potentially explain why the $e^- + e^+ + e^-$ decay is disfavored, because the neutrinos are so much lighter than the electrons the decay to neutrinos will be far more common. That doesn't explain why the $e^- + \gamma$ decay is disfavored, as the photon is even less massive.

Where do you get the statement that it's the small neutrino mass that is important in these decay rates?

Draw the Feynman diagram for $\mu \to e \gamma$. It has a neutrino oscillating to a different neutrino type.
The photon mass doesn't matter ($\mu \to Z e$ would be rare as well if it would be possible kinematically). The process is rare due to the flavor violation.

 

[kimbyd]

However, you're ruling out yourself the possibility for the more massive neutrinos to quickly decay (by an interaction that is beyond the Standard Model) to the lightest one and I thought it was the idea that you were trying to explore with mfb and the reason you were having this conversation with Vanadium50 and mfb.

No, I don't think we can rule out a BSM decay. I thought I had been explicit that such a decay, if it exists, would be beyond the standard model, because it would require a lack of conservation of flavor. My understanding is that the ways in which flavor is not conserved in the standard model just don't apply to these sorts of interactions.

Your statement that this could be recovered by looking at the reaction in terms of mass eigenstates rather than flavor eigenstates does nothing but obfuscate the difficulty. Mass isn't a conserved quantity, so it doesn't even make sense to write down an interaction equation in terms of mass eigenstates. We write such interactions in terms of flavor eigenstates because flavor is mostly conserved.

 

[kimbyd]

Draw the Feynman diagram for $\mu \to e \gamma$. It has a neutrino oscillating to a different neutrino type. The photon mass doesn't matter. The process is rare due to the flavor violation.

That's what I'm trying to say: the low neutrino mass isn't the dominant factor in suppressing such interactions.

 

[mfb]

That's what I'm trying to say: the low neutrino mass isn't the dominant factor in suppressing such interactions.

But... it is!

Flavor violation is rare because the neutrinos are light.

 

[PeterDonis]

There's some kind of discontinuity, such as an event horizon. What happens if one part of the superposition crosses, but the other does not?

An event horizon is not a discontinuity locally; it's just a lightlike surface, which locally looks like any other lightlike surface. Quantum field theory works fine locally across lightlike surfaces.

 

[kimbyd]

But... it is!

Flavor violation is rare because the neutrinos are light.

Why is that?

You could argue that the flavor violation is rare because neutrino oscillation happens slowly due to the small differences in masses between neutrinos, but that only explains part of it. The more important question, to me, is why the neutrino oscillation is necessary at all to produce the flavor violation.

 

[Vanadium 50]

This is a B level thread. I don't think any question whose answer starts "write down the Feynman diagram" can be answered at the B level.

 

[mfb]

Looks like a problem of the label, not a problem of the thread. Easy to fix.

One of OP's posts:

What makes no sense to me is to consider the propagation of the quantum superposition of three particles in a classical curved space-time. Unfortunately, I am not aware that we have such a theory. Using QFT formalism in a curved space-time is contradictory (there's no Poincaré invariance). I don't know how it could possibly work.

 

[Carlos L. Janer]

What is (or, better, could be) the renormalization (semi) group flow of the neutrino mass matrix? What particles could be involved? I suppose they're not the only parameters that do (could) flow. What are (could be) the others? Is it conceivable that neutrinos could carry a tiny electrical charge?

I'm asking out of sheer ignorance.

 

[kimbyd]

What is (or, better, could be) the renormalization (semi) group flow of the neutrino mass matrix? What particles could be involved? I suppose they're not the only parameters that do (could) flow. What are (could be) the others?

I'm not sure what you're asking here. Could you please clarify?

Is it conceivable that neutrinos could carry a tiny electrical charge?

I don't think there's any chance of this. Even a very small electric charge would overwhelm the weak force for most interactions, making neutrinos visible in a number of experiments where they are currently invisible. Here's one paper I found that went into these limits back in 1999:
http://wwwth.mpp.mpg.de/members/raffelt/mypapers/199906.pdf

 

[PeterDonis]

Using QFT formalism in a curved space-time is contradictory (there's no Poincaré invariance). I don't know how it could possibly work.

You should be very careful about extrapolating "I don't know" into "it's contradictory". QFT in curved spacetime is perfectly possible. See, for example, Wald's 1993 monograph "Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics". The basic idea is to first reformulate QFT in flat spacetime in a way that does not rely on global Poincare invariance, and then extend the reformulated version to curved spacetime. But it's not a simple exercise.

 

[Carlos L. Janer]

I'm not sure what you're asking here. Could you please clarify?

What I have in mind is rather vague.

If there's a unitary matrix relating flavor eigenstates to mass eigenstates, I suppose that this matrix must "flow" as the interaction energy of the flavor neutrinos changes (after all, the SM is a renormalizable gauge theory).

Since there are different known sources of dense neutrino fluxes: solar neutrinos, nuclear reactor and supernova neutrinos. What do the measurements on these sources tell us about these flows? What are the particles involved in this renormalization? W+- and Z vectorial and Higgs bosons? Are they not a bit too heavy?

I cannot make much sense of what I have in mind.

 

[Carlos L. Janer]

You should be very careful about extrapolating "I don't know" into "it's contradictory". QFT in curved spacetime is perfectly possible. See, for example, Wald's 1993 monograph "Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics". The basic idea is to first reformulate QFT in flat spacetime in a way that does not rely on global Poincare invariance, and then extend the reformulated version to curved spacetime. But it's not a simple exercise.

I've been told (by some other staff members) that this only works when you study Hawking's radiation and becomes useless when you try apply these ideas to the universe. But you're right, I never tried to work out the involved maths and maybe I should try (at least to find out why it does not work).

 

[PeterDonis]

I've been told (by some other staff members) that this only works when you study Hawking's radiation and becomes useless when you try apply these ideas to the universe.

I don't think it's just limited to Hawking radiation. The key limitation is that the curved spacetime geometry has to be assumed and held fixed, and the QFT is done in that fixed background spacetime. This can be done in any curved geometry; when studying Hawking radiation it's done in the Schwarzschild geometry, but it could be done in the FRW geometry that cosmologists use to describe the universe. But if there is significant stress-energy associated with the quantum fields (as one would expect, for example, in trying to model the universe as a whole), there is no way to dynamically solve, within the QFT, for the spacetime geometry that is determined by that stress-energy. You have to assume a spacetime geometry, do the QFT in that geometry, compute the expectation value of the stress-energy tensor from the QFT, and check to see if it is consistent with the spacetime geometry you assumed. If it isn't, you have to go back and start over again.

(Even this, of course, is not a full theory of quantum gravity, because you're still treating the spacetime geometry as classical.)

 

[Carlos L. Janer]

I suppose that my post #55 does not make any sense at all.

However, the idea that the parameter values involved in elementary particle interactions (mass, "charge" or interaction strength and field normalization constant) depended on the interaction energy scale was deeply engraved in my mind.

Could someone care to explain me what is it that I'm not getting right?

The post was about the renormalization of the unitary matrix that relates neutrino flavor eigenstates to neutrino mass eigenstates and any clarification is welcomed.

 

[Carlos L. Janer]

We cannot rule out neutrino/photon interactions, and at loop-level we have them even in the SM, but they have to be extremely weak. And that is elastic scattering - I still don't see how you would get a decay.

I don't really see why the scattering MUST be elastic. The mass difference between the neutrino mass eigenstates is so tiny, that the incoming and outgoing neutrinos may be different. Instead of an EM elastic scattering (at one loop level) of a neutrino you would have the decay (at one loop level) of a neutrino into a lighter one and the emission of a soft photon. Wouldn't you?

In the RF of the incoming neutrino, the outgoing neutrino and the outgoing photon would propagate in opposite directions. Four momentum would not be conserved exactly but, my guess is (I would actually have to make the calculations) that the difference would be unmeasurable. Moreover, we're talking here about an interaction that is taking place in an expanding universe.

 

[PeterDonis]

the idea that the parameter values involved in elementary particle interactions (mass, "charge" or interaction strength and field normalization constant) depended on the interaction energy scale was deeply engraved in my mind.

the renormalization of the unitary matrix that relates neutrino flavor eigenstates to neutrino mass eigenstates

These questions really belong in a separate thread in the quantum physics forum. Please start one if you would like to pursue them. Also, before starting a new thread, it might help to look through this Wikipedia article and the references it gives:

https://en.wikipedia.org/wiki/Neutrino_oscillation

The original question in this thread has been answered, so it is now closed.