Zero Mass Particles: Neutrinos & Photons

[edpell]

Now that Neutrinos have mass is it true the only zero mass particle is the photon?

 

[diazona]

Gluons are also massless. So are gravitons, if they really exist.

 

[bcrowell]

We only know for sure that there are differences in mass between types of neutrinos are nonzero. It's still possible that one type of neutrino is massless.

 

[edpell]

Thank you.

 

[blechman]

We only know for sure that there are differences in mass between types of neutrinos are nonzero. It's still possible that one type of neutrino is massless.

Actually, I'm not sure that's true. If any neutrino is massless, then it cannot oscillate. And that would show up in measurements of the mixing angles.

I'm posting this off the top of my head so my logic might be wrong, but I think it's true. In that case, we know that all three neutrinos have mass (no matter how small). This leaves the photon and the gluon as the only massless particles (and the graviton, if you include it).

And even the gluon is tricky, since gluons are not "particles" in the usual sense. That is: you cannot talk about a single, isolated gluon, which is what you need to identify it with a "particle".

Hope that helps!

 

[edpell]

Now you have opened a can of worms. Why can't we talk about a single gluon? Does this have something to do with color conservation?

 

[blechman]

Oh, boy!

Well, without getting into too much detail, the problem is that gluons (and quarks) carry color charge. Our experience tells us that such particles are never seen in nature at low energies. This is sometimes called "color confinement". Thus gluons (as free particles) only exist at very high energies. At low energies (where we live) we only see hadrons (protons, pions, etc). All of these particles are massive.

As of this moment, there is no mathematical proof that ALL hadrons must be massive, but it is generally believed by physicists (and mathematicians) to be the case. This is called the "Yang Mills Mass Gap" and there is a $1,000,000 prize if you can prove it!

 

[edpell]

I just read Jaffe and Witten's statement of the problem. I am not even going to try to solve it.

 

[humanino]

If any neutrino is massless, then it cannot oscillate. And that would show up in measurements of the mixing angles.

All oscillation formula involve mass differences. The absolute measurements are even more challenging than the oscillation measurements. There is a dedicated chapter 14 in "Fundamentals of Neutrino Physics and Astrophysics" by C. Giunti and C. W. Kim (Oxford University Press 2007) which begins with

[...] the results of neutrino oscillation experiments have recently proved that neutrinos are massive. Since these experiments give only information on the neutrino squared-mass differences, we currently know that there are at least two massive neutrinos [...]

Look for instance at the effect of neutrino masses on the end point of electron energy spectrum in beta decay of tritium. For a mass of 10 eV the end point will be shifted by ... 10 eV out of about 20 keV. This is so challenging that systematic uncertainties give us negative estimates. So at present, the best we get is that the electron neutrino mass is less than about 2 eV.

Neither pion and tau decays or neutrinoless double beta decay are more sensitive. It remains a major challenge to provide an absolute scale (declare lightest non-zero) in the neutrino mass spectrum.

 

[blechman]

All oscillation formula involve mass differences. The absolute measurements are even more challenging than the oscillation measurements. There is a dedicated chapter 14 in "Fundamentals of Neutrino Physics and Astrophysics" by C. Giunti and C. W. Kim (Oxford University Press 2007) which begins with


Look for instance at the effect of neutrino masses on the end point of electron energy spectrum in beta decay of tritium. For a mass of 10 eV the end point will be shifted by ... 10 eV out of about 20 keV. This is so challenging that systematic uncertainties give us negative estimates. So at present, the best we get is that the electron neutrino mass is less than about 2 eV.

Neither pion and tau decays or neutrinoless double beta decay are more sensitive. It remains a major challenge to provide an absolute scale (declare lightest non-zero) in the neutrino mass spectrum.

I agree with everything you say, but I also claim that it does not disprove anything I said!

My logic is simply that a massless state cannot oscillate. This follows from nothing more than special relativity! Given that fact, if one of the neutrino states is massless, then that state must decouple from the oscillation, and this would have a very noticeable effect on the MNS matrix. This is inconsistent with the data on the mixing angles as it stands today.

You are quite right that oscillation measurements are only sensitive to the mass difference (mass^2 difference to be precise), and so they cannot tell us anything about this. But that does not invalidate my argument.

As an anecdotal aside: my first project ever involved measuring V_{tb} from top quark decay data at D0 in the mid 1990s. It was found that V_{tb} > 1 was completely consistent with data, although certainly not consistent with unitarity! However, when we publish measurements like "0.95 +- 0.1" no one is seriously suggesting that V_tb is larger than one! It's up to you, the careful reader, to interpret this statistic correctly.

Why do I tell this story? The data cannot tell us anything about the absolute masses of the neutrinos, as such there is room in the oscillation data for one of them to be massless. But when you also include the fact that the MNS matrix is not block-diagonal, then a massless neutrino is not feasible. This is beyond the scope of the experiment, but not too far beyond (I've only assumed Special Relativity!).

Again: I might be wrong about this, and am ready to be convinced otherwise if anyone can tell me why a massless neutrino can participate in oscillations!

BTW, on slightly different topic: you seem upset that the neutrino mass can be negative, but remember that these are fermions, and there is nothing wrong with a negative-mass fermion; only scalars have a problem with negative mass (squared).

 

[bcrowell]

My logic is simply that a massless state cannot oscillate.

But isn't the physical state a superposition of the electron-, mu-, and tau-neutrino states? So isn't the physical state not an eigenstate of mass?

 

[blechman]

But isn't the physical state a superposition of the electron-, mu-, and tau-neutrino states? So isn't the physical state not an eigenstate of mass?

Be careful! The "physical state" is always the "mass eigenstate", since it is these states that propagate in time. When you say "electron neutrino" you really mean a special linear combination of the three "physical states" that interacts with the electron and W boson.

 

[blechman]

Rereading your post, bcrowell, there might be some confusion. The "electron neutrino" is a "flavor eigenstate" that is a linear combination of "mass eigenstates".

The "electron/muon/tau" label are NOT mass labels.

 

[hamster143]

I don't see how it follows from special relativity that massless states can't oscillate. If it's something to do with bad behavior of energy and momentum (E=p for a massless state, but E>p for a massive state), four-momentum is already non-conserved (at least on the face of it) even in normal oscillations.

 

[blechman]

I don't see how it follows from special relativity that massless states can't oscillate. If it's something to do with bad behavior of energy and momentum (E=p for a massless state, but E>p for a massive state), four-momentum is already non-conserved (at least on the face of it) even in normal oscillations.

Neutrino oscillation does not violate conservation of energy/momentum! Come on, now!

No, this logic is the same logic that says that massless particles cannot decay: a massless particle is moving at the speed of light. Such particles have no rest frame, and hence you cannot define clocks and rulers for them. Therefore they cannot decay/oscillate since they don't know when to do it!

Put in a slightly more rigorous language: the spacetime interval for a massless particle moving at the speed of light is exactly zero. Therefore length contraction is complete, and the universe shrinks to a point! From the point of view of the massless particle: it can travel from one end of the universe to the other instantaneously! So there is no "time" for it to oscillate.

One can thus imagine a thought experiment where you set up the massless neutrino as an initial state. Then you would have to find that this neutrino never oscillates, and this would put constraints on the MNS matrix. These constraints are not realized to my understanding.

As humanino correctly pointed out, this is all thought experiments, but even so, the logic seems sound to me. Unless you can point out the flaw...

 

[Vanadium 50]

four-momentum is already non-conserved (at least on the face of it) even in normal oscillations.

Why do you say that? In normal oscillations the particle propagates in a mass eigenstate, so 4-momentum is still conserved.

A simple way to see it is that a massless particle travels at c, so it is infinitely time dilated, so never has time to oscillate.

 

[blechman]

You know: the more I think about it, the more I don't believe my own arguments!

I have to think a bit more about it. But perhaps a massless neutrino is not a problem...

Grrr... Quantum Mechanics can give me a headache sometimes!

 

[blechman]

Why do you say that? In normal oscillations the particle propagates in a mass eigenstate, so 4-momentum is still conserved.

A simple way to see it is that a massless particle travels at c, so it is infinitely time dilated, so never has time to oscillate.

AH! so you agree with me?! Before I renegged...

It's a subtle question. It might be that because we are measuring FLAVOR states and not mass states, nothing rules out a massless $\nu_1$. My thought experiment might not be valid...

>>HEADACHE!<<

 

[blechman]

The counter to my counter that confuses me so is "rho-gamma" mixing in hadronic physics. There the photon can mix with a spin-1 combination of quarks, the (off shell) rho meson, and this is a real and measured effect. In fact, it is even a strong constraint in more complicated models such as technicolor, with rho replaced with "techni-rho"; or KK gauge bosons in extra dimension theories; etc.

So I'm confused about this mixing. Anyone have any thoughts on why this can happen?

 

[hamster143]

Why do you say that? In normal oscillations the particle propagates in a mass eigenstate, so 4-momentum is still conserved.

A simple way to see it is that a massless particle travels at c, so it is infinitely time dilated, so never has time to oscillate.

If time dilation were a factor, we'd find that oscillation probability depends not only on $\Delta m^2$, but also on the absolute value of the mass, because, as mass approaches zero, proper time between the moment our neutrino is emitted in the center of the Sun and the moment it's detected by our detector goes to zero as well.

 

[Vanadium 50]

It's a subtle question. It might be that because we are measuring FLAVOR states and not mass states, nothing rules out a massless $\nu_1$.

I'm not so sure. I'd want to repeat the derivation separately for the two helicity states and see what happens. The problem with a massless $\nu_1$ is that it has only one helicity state, but the particles it is oscillating into have two.

 

[Vanadium 50]

rho-omega mixing with the photon, or more generally vector meson dominance is a different phenomenon from oscillations. What you have in the former case is a vertex where you can place several different particles with the quantum numbers of the photon: including (unsurprisingly) the photon. What it does not mean is that a freely propagating rho will turn into a photon and back again.

 

[blechman]

What it does not mean is that a freely propagating rho will turn into a photon and back again.

as that would undoubtably violate energy/momentum conservation! ;-) Yes, the off-shellness is important.

I'll think more about the neutrino question. Thanks for answering this!

 

[humanino]

rho-omega is just what I had in mind. We already need Heisenberg for oscillations to respect energy momentum anyway, so I doubt you can enforce no-mixing with massless without appealing to some superselection rule.

 

[blechman]

So as I drove 700 miles across the US, I had plenty of time to think to myself about this, and I now think I understand what I did wrong.

Short answer: I withdraw my previous statement - there is still the possibility of a single massless neutrino, as was originally stated by bcrowell.

Long answer: Ironically, I should have been tipped off by hamster143's comment about violations of energy/momentum! It is easiest to see what is happening (at least to me) in the wave-picture of quantum mechanics:

When you create an electron neutrino, say, what you have really created is a linear combination of three "physical states" (that is, mass eigenstates) that each have a different mass, therefore energy, therefore momentum! So as this "electron neutrino" state propogates through time, its components are traveling at different speeds, and the wavefunction "modulates" into a new state, say, the muon neutrino. Then when we observe it again, we see oscillation.

This argument has NOTHING to do with the value of the mass of the neutrinos! All it required was that the neutrinos have DIFFERENT masses, so that the wavefunction components move at different speeds (since, after all, momentum IS conserved!). This is the same effect, in the wavefunction picture, that you get in nonlinear optics where the light of different frequencies moves at different speeds in the medium, creating a "shimmering effect".

My thought experiment is bogus: while it is true that a massless state will not oscillate, it is also true that a MASSIVE state will not oscillate either! This is because these states are energy eigenstates and therefore only pick up an overall phase after time evolution. There's nothing to oscillate into!

My argument works for DECAYS of particles, but is irrelevant for oscillations of this form. The only thing that matters is that there must be a mass DIFFERENCE so we can get this modulation effect, and this is clearly shown by the probability vanishing as two masses approach each other.

OK, hope that's clear! Sorry for the confusion.

 

[blechman]

I'm not so sure. I'd want to repeat the derivation separately for the two helicity states and see what happens. The problem with a massless $\nu_1$ is that it has only one helicity state, but the particles it is oscillating into have two.

That is not necessarily true. If the masses are Majorana, you still have a single-chirality state (notice I say "chirality" and not "helicity", since you have to make the distinction for massive states, and it is "chirality" that matters!).

 

[vitruvianman]

Now that Neutrinos have mass is it true the only zero mass particle is the photon?

Yes, neutrinos have mass and its measurement is made in 1980's. I don't think that there may be particles without mass, but "particles" how we understand.

But tachyons (actually all theoritical particles which travel "backwards" :) ) has a strange theoritical mass, which is a complex number, like (I'm just making up) $m=3^{-16}+2i$

Looks frightening doesn't it?

 

[humanino]

Fairly short letter :
Playing with Neutrino Masses
Sheldon L. Glashow

Most of what is known about neutrino masses and mixings results from studies of oscillation phenomena. We focus on those neutrino properties that are not amenable to such studies: $\Sigma$, the sum of the absolute values of the neutrino masses; $m_\beta$, the effective mass of the electron neutrino; and $m_{\beta\beta}$, the parameter governing neutrinoless double beta decay. Each of these is the subject of ongoing experimental or observational studies. Here we deduce constraints on these observables resulting from anyone of six ad hoc hypotheses that involve the three complex mass parameters $m_i$:

• ($1$) Their product or
• ($2$) sum vanishes;
• ($3$) Their absolute values, like those of charged leptons or quarks of either charge, do not form a triangle;
• ($4$) The $e$-$e$ entry of the neutrino mass matrix vanishes;
• ($5$) Both the $\mu$-$\mu$ and $\tau$-$\tau$ entries vanish;
• ($6$) All three diagonal entries are equal in magnitude. The title of this note reflects the lack of any theoretical basis for any of these simple assertions.

It seems, if you really have an argument against one massless neutrino, Glashow would support publication