## Thursday, February 20, 2014

### What Mechanism Causes Neutrinos To Oscillate?

Background

Neutrino Oscillation

There is compelling empirical evidence that the neutrinos oscillate from one of the three active neutrino flavors to another of the three active neutrino flavors according to a quite simple formula, and there is some (increasingly weak) evidence that a better fit to the data is secured by extending the model to include a very small probability of oscillation to a fourth sterile neutrino flavor with a mass greater than that of the heaviest neutrino mass, but less than 1 eV.*

The consensus formula that is used to describe, for example, the probability P that a muon neutrino (neutrino generation 2) with a given (kinetic) energy E (in GeV/c^2) will oscillate into a tau neutrino (neutrino generation 3) over a given distance L (in kilometers) is assumed, on a consensus basis that produces quite good fits to the data, to follow the formula below (from page 138 of the 2012 dissertation of Michael R. Dziomba):

P(v2-->v3)=sin2(2Θ23)*sin2(1.27*Δm232*L/E)

where Θ23 is the empirically determined mixing angle between the second and third generation neutrinos, and Δm23 is the empirically determined difference in mass between the second and third generation neutrinos.  The current empirical best fit for sin2(2Θ23) is about .99 which implies that Θ23 is about 42 degrees, and the empirical best fit for Δm23 is about 0.048 eV/c2.

Thus, the probability of an oscillation to another flavor oscillates periodically as a function of distance for a neutrino of any given energy, and the higher the neutrino's energy, the more slowly this probability oscillates as a function of distance.

This general formulation is derived in depth, for example, in a 1998 review article on the proper calculations for particle oscillation in general (including neutral kaons and neutrinos). But, like the equation above, it essentially just assumes that oscillation happens in a black box without much concern for why a particle is actually able to display this behavior. The equation tells you how to predict what happens, but not why it does what it does at the critical moment when the oscillation takes place.

The formulas for the probability that neutrinos of one state will oscillate to another, in general, follow this form with the exception of the subscripts on the Θ mixing angle and Δm term. A full treatment of an individual source neutrino's probability of ending up in a particular end state would require consideration of all possible oscillations, rather than just one as in the example above.

Flavor Changing W Boson Emissions

There is only one other process in the Standard Model by which fermions change flavor involves weak force interactions.

In that process, up-type quarks transform themselves into down-type quarks (of a type determined by probabilities empirically measured in the pertinent constants of the CKM matrix) simultaneously with their emission of W+ bosons, down-type quarks transform themselves into up-type quarks simultaneously with their emission of W- bosons, charged leptons transform themselves into neutrinos with their emission of W- bosons, and neutrinos transform themselves into charged leptons with their emission of W+ bosons.  The probability of a particle emitting a W boson in a given time period is largely a function of the weak force coupling constant.

(Note in contrast, that strong force interactions are not flavor changing, although they do routinely create quark-antiquark pairs that permit the creation of new hadrons with more quarks than the source particle, a process which does not violate baryon number conservation, so long as this is permitted by mass-energy conservation.)

How does neutrino oscillation happen?

Neutrino Flavor Changes Via Two Step Virtual W Boson Emissions

If W boson emissions were involved in neutrino oscillation, this would involve, at least, a two step process.  First, a neutrino would emits a virtual W+ boson and becoming a virtual charged lepton.  Then, the virtual charged lepton would emit a virtual W- boson that would turn it into a neutrino of random flavor again.  In this virtual two step process, the virtual W+ and W- boson would annihilate each other and thus cancel out before either of these bosons can decay.

In the typical neutrino oscillation scenario, the probability of the intermediate charged lepton coming into existence, or of the virtual W bosons coming into existence and producing decay products, would be essentially nil because these possibilities would violate mass-energy conservation.

Of course, there could be additional contributions to the overall probabilities from four, six, etc. steps processes.

Of course, a neutrino cannot oscillate into a heavier neutrino flavor if its kinetic energy is less than the mass difference between the neutrino flavors of the end states, since mass-energy conservation forbids this possibility.  My assumption is that mass gained through neutrino oscillation results in a neutrino losing an equal amount of kinetic energy without altering the direction of its momentum, and that mass lost through neutrino oscillation results in a neutrino gaining an equal amount of kinetic energy without altering the direction of its momentum, on mass-energy and momentum conservation grounds.

Indeed, in the Standard Model, this process must exist and make some contribution to observed neutrino oscillations.

Does neutrino oscillation take place via the same mechanism as flavor changing in quarks?

As I understand the matter, however, the neutrino oscillation formula noted above is independent of the mechanism by which neutrino oscillation occurs.  So far as I know, this formula does not assume that the process involves this two step weak force process involving a virtual charged lepton and the emission of a virtual W+ and W- boson, at rates dictated by the strength of the coupling of the weak force to ordinary left handed neutrinos and ordinary right handed anti-neutrinos.

The PMNS matrix that governs the transition probabilities of oscillating neutrinos, however, is closely analogous to the CKM matrix that governs the transition probabilities of flavor changing quarks that emit W bosons.

But, while the CKM matrix is relevant only in the event that a quark emits a W boson, which happens at a rate that is a function of the strength of the quark's coupling to the weak force, nothing about the PMNS matrix implies that the oscillation event that triggers its application involves an emission of a boson by the neutrino or is related to the weak force in any way.

I assume that if the probability of a neutrino flavor change happened matched the prediction from first principles of the weak force, that this aspect of the neutrino oscillation process would be widely discussed, when, in fact, I've never seen anyone assert that this is the case.  So, my assumption without actually doing the calculations myself (which I don't have the expertise to do) is that virtual pairs of weak force interactions contribute only minimally to the overall neutrino oscillation rates that are observed.

When are particles prone to state changing and mixing?

Looking at the matter from a forest level view, as opposed to a detailed Standard Model explanation of each individual case, a pattern emerges.  Certain kinds of physical systems that share key properties with neutrinos seem to be particularly lightly tied to a particular state and prone to easily take on new or mixed states.

As particles (fundamental and composite alike) become charged and gain higher mass, their particle character tends to be very well defined and if they are transformed into something else with equal or lesser mass-energy, the transformation tends to take places decisively in a single direction (e.g. from higher rest mass to lower rest mass, rather than visa versa).

In contrast, particles with neutral electric charge, particularly when they are less massive, tend to be less well defined as distinct particle states and become more prone to oscillating or mixing different particle states that are similar in mass.

1.  As I noted in my previous post at this blog, many neutral pseudo-scalar mesons that have no charm or bottom quarks are proposed to be linear combinations of electric and color charge and flavor neutral pseudoscalar quarkonium pairs of up, down and/or strange quarks (each of which is much lighter than mesons containing charm or bottom quarks), to the point where the only observable state is the blended linear combination.

In particular, it appears that scalar mesons and axial vector mesons (aka pseudovector mesons) also arise from the mixing of pseudoscalar quarkonium pairs involving lighter quark types (up, down and strange), but on a much more extreme scale.  In contrast, less neutral vector quarkonium mesons (with spin and isospin 1) made of lighter quarks tend to be well defined and distinct particle states that aren't as prone to mixing. Similarly, mixing may explain why free glueballs are not observed; they mix with similar mesons.

2.  Electroweak theory supposes that the four electroweak bosons (the photon, W+, W- and Z boson) have their source in four massless bosons (the W+, neutral W, W- and neutral B bosons) which acquire mass and experience spontaneous symmetry breaks that causes the neutral W and neutral B bosons to coalesce into the massless photon (γ) and Z boson through the Higgs mechanism.  The photon and Z boson in this model are a linear combination of the neutral W and neutral B bosons that is a function of the trigonometric functions of the weak mixing angle (the cosine of which is equal to the mass of the real world W boson divided by the mass of the Z boson).  The W and Z bosons then "eat" three of the four Goldstone bosons leaving the Higgs boson as the sole remaining Goldstone boson of the weak force.

Notably, for these purposes, in electroweak theory the neutral W and B bosons that are linearly combined are both electrically neutral and massless prior to becoming a photon and Z boson via spontaneous symmetry breaking.

3. I have previously noted the manner in which the Higgs boson appears in some ways to resemble some sort of composite of the four electroweak bosons (the W+, W-, Z and photon), with the same aggregate electric charge, their aggregate mass divided by the square root of four (the number of particles involved), and so on.

4.  CP violation tends to be seen in the decays of relatively light particles with neutral electric charge such as the neutral kaon, although it is also observed in the much heavier neutral B meson and neutral D meson.  There are early experimental indications that CP violation exists in the mixing of the highly chiral active neutrinos as well.

5. In theory, a neutral fermion that is its own anti-particle (something that is impossible for a charged fermion), which some physicists argue could include the neutrino (something I think is highly unlikely because of the important of distinction between particle v. antiparticle character for neutrinos), can acquire what is called Majorana mass in a process akin to state mixing of similar particles.  Quasi-particles in superconducting systems seem to behave in this manner.

6. The search is on for any evidence of neutron and anti-neutron oscillation (although I suspect that this simply doesn't happen absent neutron annihilation class energies).

7. Up and down quarks, which have nearly degenerate mass, are particularly prone to mutate into each other via flavor changing weak force process.  Indeed, these transitions are the only flavor changing processes routinely observed in nature today.

8. Supersymmetry theories, generically, have a sparticle spectrum that does not match one to one with the Standard Model particle spectrum as a result of mixing that is predicted to occur in the sparticle spectrum.

Towards Oscillation As A Highly Suppressed Default Rule, Rather Than An Exception

Conceptually, it isn't entirely unfair to think of particles with neutral electric charge (and even more so, those with true scalar or otherwise even parity spins) as being less firmly tied to the direction of time than charged particles (particularly fermions) with odd parity.

Put another way, perhaps it is useful to think about everything that changes between one particle (be it composite or fundamental) state and another as a barrier to that transition that requires energy to change.  So, the less a particle has to change to reach a new state, the more likely it is to happen.

In the case of neutrinos, the absolute scale of the mass differences between neutrino mass states is so tiny relative to the differences between any other two particles with distinct masses (just dozens of meV or less, in each case), and there is never any change in electric charge or weak coupling strength or color charge or parity between neutrino mass states, so the barriers to oscillation are lowest for these particles and are so low that their mixings with each other are particularly great.

It might make sense to think of neutrino oscillation as the norm and default rule, and of all of the other particles as exceptions to the general principle that particles freely oscillate into each other, with their oscillations suppressed by their electric charge, color charge and much greater mass differences.

Or, perhaps one might go so far as to say that each of these features makes them more wave-like and less point-like than other particles in this context, and that these aspects involve a continuum between extremes rather than a discrete either or choice (something that is indeed true in general in quantum mechanics). Since they are fermions rather than bosons, neutrinos can not react the absolute extreme of literally mixing with each other at the same time and place, as bosons do, but they can do the next best thing, by freely and frequently oscillating from one flavor state into another.

In this view, hypothetical massless fermions (none of which have ever been observed outside theoretical physics models) would all maximally oscillate with each other infinitely rapidly to the point that all possible kinds of massless electric charge, color charge, parity neutral fermions would be indistinguishable from each other as distinct particles entirely, and neutrinos escape that extreme circumstance only as a result of their mass which is intimately connected to the fact that they have, at least, a weak force charge.

A corollary of this hypothesis would be that hypothetical sterile right handed neutrinos that do not mix with left handed neutrinos by some newly postulated force, would necessarily have zero mass as well. Hence, they would also not interact even via gravity. And, if they had no interactions via any force with anything, they would, by definition, not exist. In this view, massless photons and gluons can exist only because they have electromagnetic and strong force interactions, respectively, with particles that have electric and color charge, respectively, which connect them to the world and endow them with energy.

The "Standard Model" Line

What I have articulated isn't the same as the more technically minded "technical" Standard Model description of what is going on from a mechanism point of view, although it is not equivalent and is also more poetic, less mechanical, and gives goes a bit more deeply than the not terribly introspective typical description of how the Standard Model following the lead of Pontecorvo of PMNS matrix fame describes the process.

A quite comprehensive treatment of a variety of neutrino physics and neutrino oscillation issues from the issues at the foundation of trying to integrate massive neutrinos into the Standard Model from 1987 articulates essentially the same conceptual framework for neutrino oscillation at page 24 that is the consensus view today (by getting into the mud of the material I stated in an earlier footnote that I would disregard):
The oscillations of neutrinos are analogous in their quantum-mechanical nature to K°<=±K° oscillations. Suppose that the state vectors of the neutrinos taking part in the weak interactions (ve,vM,vT,. . . ) are superpositions of the state vectors of neutrinos (Dirac or Majorana) with different masses.
What would be the behavior of a neutrino beam in this case?
It is clear that at some distance from the source of neutrinos of a given type, the state vectors of neutrinos with different masses (because of the difference in the masses) would acquire different phase factors. The state vector of a neutrino would then be a superposition of the state vectors of neutrinos of different (all possible, in principle) types. It is obvious that the probability of finding a neutrino of a given type would be a periodic function of the distance between the source and the detector. This phenomenon was called neutrino oscillations (Pontecorvo, 1957, 1958).
In order for oscillations of neutral kaons, neutrinos, etc. to be possible, the following conditions have to be realized: (i) the particle interaction Lagrangian should contain terms that preserve some quantum numbers (strangeness in the case of kaons, lepton numbers in the case of leptons, etc.); (ii) the total Lagrangian (and the mass term) should not be diagonal with respect to these quantum numbers, and the relevant quantum-number non-conserving couplings should be much weaker than those preserving the quantum numbers. The states with definite mass (and width) would then be superpositions of states possessing definite strangeness in the case of neutral kaons, definite lepton numbers in the case of neutrinos, etc.
In other words, neutrino oscillation happens because neutrino flavors for purposes of the weak force and neutrino mass states are two separate things with separate sources in the Standard Model equations that will get out of synch with each other in an oscillating fashion in a way that leads neutrinos to change flavor now and then in a predictable way over time.

Oscillation is a fundamental property of neutrinos because the three weak interaction states of a neutrino at its origin are fundamentally constructed in the Standard Model as a mix of the three different mass states that are inherently indeterminate.

This oscillation is conceptually more like the process of the wave function of probability amplitudes regarding other properties of a quantum particle (such as its location) collapsing when a particle is measured that produces random outcomes regarding its final destination which are distributed proportionately to the square of the probability amplitudes of a particle being located at a particular place, as opposed to the kind of flavor changing process seen in flavor changing W bosons interactions. It is a part of the equation of how a neutrino particle propagates rather that an interaction.

Later in the same treatment from 1987, at footnote 22, the paper also notes that in that case of higher generation neutrinos with masses of under 100 eV, which could experience "radiative decay" just as the the charged leptons do into a lighter lepton of the same electric charge and lepton number together with a photon, but that they are metastable because they have predicted mean lifetimes longer than the age of the universe (i.e. 13.6 billion years or so). So, the neutrino oscillation process is distinct in source from the flavor changing decays of heavy charged leptons into lighter charged leptons.

This analysis is also partially stymied by the same uncertainty regarding the origin of neutrino mass (Dirac, Majorana or Pseudo-Dirac) that still remains an open question today, 27 years later, despite all of the experimental advances that we have made since then.  Each leading approach has its own problems and is an awkward fit to the overall Standard Model framework, although the Dirac mass fix is somewhat more minimalist.

A New (or Old) Neutrino Oscillation Boson?

Another possibility is that neutrino oscillation actually does involve flavor changing interactions triggered by the emission of a boson much like that of flavor changing charged W boson interactions, with some boson other than the W boson. Perhaps a flavor changing neutral current is mediated by virtual Z bosons in a manner that is somehow suppressed completely or nearly completely in charged particles, but not in the electrically neutral fundamental particles that are neutrinos.

Or, perhaps there might be some undiscovered, possibly charge-phobic, Z' boson, either massive or without mass, that mediates this interaction in a way analogous to the W boson, which might or might not be part of the electroweak forces. Perhaps this neutrino oscillation Z' boson, if one existed, would have a mass relative to the regular Z boson similar to the mass of the charged leptons relative to their neutrino partners. Thus, the Z' boson might have a mass on the order of 45 eV as opposed to the approximately 90 GeV of the Z boson.

Perhaps the charge phobic Z' boson could also be the boson that mediates a force that operates between neutral dark matter particles.  A charge-phobic boson would also not be impaired in its ability to mediate interactions between electrically neutral dark matter particles.  The sweet spot for a "dark photon" mass is about 1 MeV to 100 MeV in the kind of sophisticated self-interacting dark matter model that comes closest to reproducing the kind of dark matter halo shapes inferred from astronomy observations of galaxies.  This is well within the workable range for a Z' particle that would mediate neutrino oscillations as well, however. (A recent study by Matt Strassler, et al. has established that any such Z' boson cannot produce significant numbers of charged leptons if produced in Higgs boson decays, but this wouldn't be expected in the charge-phobic Z' case.)

More radically, in the case of a massless Z', one could imagine the Z' boson as a neutral W boson that somehow escaped becoming a part of the weak mixing angle linear combination that creates the photon and Z boson of electroweak unification.  This would not, however, be a good fit to a dark photon in a sophisticated self-interacting dark matter model of the kind necessary to come close to reproducing the dark matter halo shapes that are inferred from astronomy observations.

At 45 eV, it would ordinarily be prevented by mass-energy conservation from decaying into anything other than pairs of neutrinos and anti-neutrinos, but at anywhere from about 10 to 100 MeV, it would be massive enough to also decay to pairs of first generation quarks or charged leptons, and even at 1.2 MeV it would be massive enough to decay to pairs of electrons.  But, if the boson was truly charge-phobic, it would not do either of these things because it does not interact with charged particles since it is charge phobic so it would create no quark pairs or charged lepton pairs even if had enough mass-energy to do so.

Experimental Tests For Boson Mediated Neutrino Oscillations

There would be tell tale sign of neutrino oscillation mediated by any form of boson, as opposed to neutrino oscillations that is not mediated by another particle. For example, in the unmediated scenario, all changes in rest mass due to neutrino oscillations would be converted into or out of kinetic energy while maintaining momentum in the same direction. But, in the case of boson facilitated neutrino oscillations, the aggregate changes in neutrino beam mass due to neutrino oscillations would differ from the aggregate changes in neutrino beam velocity due to that adjustment in its kinetic energy.

Another experimental signature of boson mediated neutrino oscillations, particular in the light Z' boson case, would be a bump in the number of neutrinos and antineutrinos that is equal in magnitude that have combined mass-energy equal to half of the Z' boson mass.

A third experimental signature of boson mediated neutrino oscillations that would be more model specific would be a suppression of neutrino oscillations within strong electromagnetic fields, if the Z' boson was charge-phobic. This is more of a long shot and not very general, but has the virtue of being quite easy to test experimentally by placing a powerful electromagnetic field along the path of the neutrino beam leaving a nuclear reactor for which there is already a baseline level of neutrino events in place. This could even be scheduled to be turned off and on so that the experiment would use the reactor's own electricity during times when the demand for electricity from the grid was low and turn off during periods of high electrical grid demand, providing data sets that were naturally robust to effects associated with seasonal change during the year or differences in neutrino production from year to year for a variety of hard to determine reasons (e.g. a natural aging cycle of the nuclear fuel used at the reactors).

Footnotes

* There is a technical distinction between the three "weak force" neutrino flavors and the three neutrino mass eigenstates. This footnote acknowledges that I am deliberately ignoring that distinction in this discussion for those who understand the distinction, in the interest of clarity for a lay reader. I do not believe that this intentional imprecision does great injustice to the gist of the idea discussed in this post, except in setting for the "Standard Model" explanation as I do later on.