Almost no intrinsic dispersion in OPERA’s 20 new neutrinos!

With OPERA adding details of their 20 single-neutrino measurements from the past few weeks to their preprint (which they have now submitted for publication), we now get a good idea of how sharp the 60-nanosecond advance (ahead of the speed of light) is for different neutrinos.

The results are absolutely astounding. Figure 18 of their updated paper shows that they are contained within a 50-nanosecond bin, from 40 to 90 nanoseconds (with the average being 62 nanoseconds). The reason that this is remarkable is that their synchronization of the OPERA master clock to the GPS timing is itself quantized within a 50-nanosecond-wide bin. Individual neutrino events could fall anywhere within this bin (i.e. the error in timing could be anywhere between -25 and +25 nanoseconds).

Their result implies that, over time, if they collected enough events (if they were still running the single-neutrino experiment), the distribution of received times may look very much like the distribution caused by the clock calibration itself. But if the neutrino travel time itself had any substantial distribution — say, if higher-energy neutrinos were slower than lower-energy ones — then this distribution would be convolved with the clock jitter.

But this is not seen.

This confirms what seemed evident in their September results: there is almost no intrinsic dispersion in travel times for their neutrinos.

It is difficult to see how this result could be reconciled with any version of Einsteinian or non-Einsteinian mechanics, if it is assumed that it is the neutrinos traveling a little bit faster than light over the 730 km to Italy.

The only remaining alternatives are that there is indeed a 60-nanosecond systematic uncertainty plaguing the experiment, or that there is an almost-constant 60-nanosecond advance occurring near the production point at CERN.

I have put forward the latter conjecture in my two previous blog posts.

OPERA really needs to perform a timing detection of muons at the far end of the hadron stop — at CERN — to confirm or rule out this possibility.

 

Could the OPERA tachyon be the unbroken Higgs?

In yesterday’s blog I looked at some of the numbers from the OPERA neutrino experiment, to see whether (if correct) they made any sense without invoking too many radical changes to our understanding of the Universe (other than a loss of causality, which is unavoidable once you have things travelling faster than the speed of light).

At first sight, it’s difficult to see how it can. The OPERA result would imply an “instantaneous momentum” (the tachyonic analogue of the “rest mass” of a normal bradyonic particle; see yesterday) on the order of 100 MeV. Not only doesn’t this fit at all well into the neutrino sector as we know it, it would also imply a dispersion of arrival times not seen in the experiment itself.

I speculated that the only way I could see such a “clean” (uniform) 60 ns advance in arrival times across such a spectrum of energies would be if there were some sort of “intermediate tachyon” that somehow managed to get 60 ns ahead of the speed of light, which then decayed back into the regular neutrinos that we know and love — with m^2 on the order of eV^2 or less — which would effectively travel the rest of the way to Italy at the speed of light.

Last night I was playing around with the Standard Model to see how it would react to the introduction of an exotic tachyonic intermediate particle, when it dawned on me that it already has one — in a way.

Let me roll back a little. The Higgs mechanism introduces a scalar field with a mass term that has a negative mass-squared. A tachyon? Not quite. Viewed as a pseudo-classical field theory, the wrong sign of the mass term yields a “potential energy” that is unbounded from below. So the Higgs mechanism postulates an additional fourth-order interaction term, with positive sign, so that the potential energy surface looks like the bottom of a beer bottle. The potential energy is then bounded from below, in the degenerate “ring” around the bottom of the bottle.

The vacuum is then taken to be that state in which the Higgs field is in its lowest-energy state, which has a value around 246 GeV. Since this is actually a ring of degenerate states with the same energy, the Universe arbitrarily “chooses” one such state; this is the “spontaneous symmetry breaking” of the theory.

For Feynman-diagram type calculations, we like to use perturbation theory around the vacuum state. When you crunch it through with the right gauge interactions and the right gauge chosen to remove unphysical fields, you end up with a set of particles and interactions that agrees extremely well with what we actually find in Nature, as well as a remnant “broken” Higgs boson, with a normal (bradyonic) mass (that is not, unfortunately, predicted by the theory). Moreover, the Higgs mechanism allows the Standard Model to remain renormalisable, despite having features which would otherwise make it a computational basket-case.

All that is elementary particle physics. What’s interesting is that the original, “unbroken” Higgs, with its tachyonic mass term, is still really there in the Lagrangian, albeit 246 GeV away from the vacuum state.  Of course, if we were to solve the equations of motion exactly, then either viewpoint would be equivalent — it’s merely shifting to a different set of dynamical coordinates. But we’re so accustomed to using perturbative theory around the vacuum that it’s easy to forget where it all came from.

Is it possible that OPERA is creating an “unbroken Higgs” particle, i.e., a tachyonic excitation around the unbroken Higgs field at zero, rather than around its vacuum expectation value? It certainly couldn’t do that in a vacuum, with only tens of GeV available to play with. But our condensed matter colleagues never fail to admonish us for treating everything as if it were perturbations around the vacuum. What if it were possible to excite the unbroken Higgs field within the confines of condensed matter?

A possible scenario for the OPERA experiment could then be the following. As the CNGS mesons travel down the 1000 metre vacuum tunnel, they decay into muons and muon-neutrinos, which are still essentially travelling together at almost the speed of light. They enter the hadron stop. Inside that material, they convert back into an unbroken Higgs (remember that the unbroken Higgs has different quantum numbers from the normal “broken” Higgs, so this is not forbidden).

This unbroken Higgs excitation (particle) travels through the hadron stop with kinematics corresponding to its mass term — which is tachyonic. Assuming that the unbroken Higgs has a ‘w’ value (see yesterday’s post) that is much larger than the tens of GeV of the beam energy, then the unbroken Higgs will be in its “low energy regime”, regardless of whether the recombining muon and neutrino happen to be from the same original decay or not. Now, for a tachyon, a low-energy particle moves at almost infinite speed. The unbroken Higgs gets to the other side of the hadron stop in close to no time at all.

At that point, it re-emerges into the vacuum, and the condensed matter dynamical environment is gone. Unstable in this vacuum environment, it decays back into a muon and a neutrino, which continue to travel at essentially the speed of light. The muons are detected or otherwise swept away. The neutrinos continue on into the rock beyond CERN, on their way to the OPERA detector in Italy. Without their corresponding muons, the cannot convert into unbroken Higgs particles for the rest of their trip.

Do the numbers add up? The OPERA paper states that the hadron stop is 18 metres long. Anything travelling at essentially the speed of light would take 60 nanoseconds to traverse it. The unbroken Higgs, on the other hand, gets across it almost instantaneously. The neutrinos would therefore be 60 nanoseconds ahead of where they were supposed to be. For the rest of the trip to Italy, they travel at their expected speed of essentially the speed of light. The arrival time distribution is advanced by 60 nanoseconds, for all energies.

This is a seductive calculation, but it has one fatal flaw: If it were true, then the muons detected after the hadron stop would also be 60 nanoseconds ahead of where they were supposed to be. OPERA reported no such effect, and since the nice juicy charged muons right there at CERN are much easier to detect than the poltergeist neutrinos 730 kilometres away, one would have to hazard the guess that such an effect was not present.

Do OPERA’s tachyonic neutrinos make sense?

Having been suitably humbled on the statistics of the OPERA experiment in recent days, I’ve been having a bit of play with the numbers to see whether their measurements would make sense, at even an elementary level. (I agree that the systematic errors look precarious — but they have had hundreds of sensible people looking at them for months. Any important discovery will always be on that precarious edge. Most are looking into that already, so I’ll leave it to them for now.)

So let’s assume, for the sake of this blog entry, that the OPERA measurement is sound. What would that mean for physics?

For starters, it would not mean that Einstein would be turning in his grave. I’ve been sorely disappointed in the past few days by physicists, who should know better, claiming that tachyonic neutrinos (i.e. travelling faster than the speed of light) would utterly destroy the special and general theories of relativity; it indicates that there are some in the current generation of physicists who haven’t been given a thorough enough grounding in the basics. For sure, if someone had managed to accelerate a regular (bradyonic) particle through the “light barrier”, then relativity would be in trouble: that’s simply not possible in relativity as we know it, with a finite amount of energy. But Lorentz kinematics has nothing to say about particles that always go faster than the speed of light. Again, such tachyons would allow signals to be sent backwards in time, and so causality would take a battering; but there is nothing in Einstein’s kinematics that prevents it.

After all, we already know that antiparticles are effectively regular particles travelling backwards in time, which is something that was not in the realm of physics at the time that Einstein did his seminal work. (My colleagues and I wrote up a short paper for the pedagogical American Journal of Physics, explaining how this can be understood and made sense of, even at the classical level, some time ago.) Just as relativistic kinematics handles antiparticles with relative ease (once you figure which equations extend seamlessly to that domain), so too can it handle tachyons.

There’s almost a century of work in tachyons available for us to contemplate, but there are some simple calculations that we can do with little more than Einstein’s original equations. Consider this one:

m^2 = E^2 ( 1 – v^2 ),

where (like all theoretical particle physicists) I’m using “natural” units in which the speed of light, c, has the numerical value of 1. This equation isn’t generally used much in particle physics — it’s something you only see in an elementary textbook on special relativity — but it is useful for considering the OPERA experiment. Factorising the parentheses, we have

m^2 = E^2 ( 1 – v ) ( 1 + v ).

For the case of particles travelling at approximately the speed of light (as we have in OPERA), namely v being approximately 1, the last factor 1 + v is well approximated by 2, and so we have

m^2  \approx  -2 E^2 ( v – 1 ).

Note that tachyonic particles simply have a negative mass-squared. This might sound strange — and indeed this property leads some to describe a tachyon, misleadingly, as having an “imaginary mass” — but think about what “mass” really is: Go back to elementary special relativity, where it is generally called “rest mass” (until you get familiar with it): the “mass” of a particle is indeed just its energy in a frame in which it is at rest. But just as a bradyon can never be accelerated through the light barrier, nor can a tachyon be decelerated through the light barrier; it can never travel slower than the speed of light, and most certainly can never be at rest. So saying that a tachyon has an “imaginary (rest) mass” is not really useful.

So what does m^2 mean for a tachyon? Well, for any free tachyon there is a frame of reference (up to a spatial rotation, as always) in which its velocity v is infinite; in other words, the tachyon travels from its creation point in space to its destruction point in space (and all points in space on a straight line between those points) instantaneously. Let’s call this the “instantaneous frame” for a given free tachyon, by analogy with the “rest frame” for a free bradyon. By the above, if m^2 is a physical constant for the tachyonic particle, then the energy E must be zero in such a frame. A particle with zero energy? Sounds strange, but possible: the tachyon would still have a finite (three-)momentum. And in fact, the magnitude of its momentum in its instantaneous frame is just the square-root of -m^2. So you can think of \sqrt{-m^2} as its “instantaneous momentum” (for want of a better term); the magnitude of its three-momentum in any other frame is always greater than this value (just as, for a regular bradyonic particle, the magnitude of its energy in any frame other than its rest frame is always greater than its energy in its rest frame, namely, its mass).

OK, so let’s go back to the equation above. Now, although the OPERA neutrinos have a distribution of energies E, this equation can at least give us an order-of-magnitude estimate for -m^2. Overall, the average neutrino energy in their experiment is 17 GeV, and v – 1 = ( 2.48 +/- 0.28 +/- 0.30 ) x 10^{-5}. Pretending, for the sake of this order-of-magnitude calculation, that all of the neutrinos had an energy of exactly 17 GeV, then the above equation would give

-m^2 = ( 14300 +/- 1600 +/- 1700 ) MeV^2.

To make things simple, let’s denote the “instantaneous momentum” by w:

w \id \sqrt{ -m^2 }.

Then for this approximate calculation, we would have

w = ( 120 +/- 7 +/- 7 ) MeV.

These uncertainties aren’t correct, because I have approximated the energy distribution by a delta function at E = 17 GeV. But it at least tells us that, if correct, the OPERA experiment is pointing to some sort of tachyonic particle with a w value around 100 MeV.

We can do one better than this, even without having access to the entire OPERA data set. They divided the charged current subset of their data into two nearly equal halves, by energy, by cutting at 20 GeV. The mean energy of the lower-energy half was 13.9 GeV, and the mean energy of the upper-energy half was 42.9 GeV. Converting their nanosecond results into v – 1 values, and again making the (wrong!) approximation that the energy distribution of each half is a delta function at the mean energy value, I get

w = ( 92 +/- 17 +/- 7 ) MeV  around  13.9 GeV,

w = ( 320 +/- 50 +/- 20 ) MeV  around  42.9 GeV.

As noted by many people already, the lack of any significant energy dependence in the OPERA time shift results seems to suggest that there is a systematic error at play here. For the above calculation, this is reflected in the fact that the estimated w value increases as the energy increases. (Of course, these uncertainties are also not correct, because we have not convolved in the energy distribution, so it is not proof of a problem.)

The fundamental problem is that, even if we were to believe that one or more of the neutrino mass eigenstates were tachyonic (i.e. that one or more of the m^2 eigenvalues were negative), and even if only those particular mass eigenstates were in play for the OPERA experiment (more on this below), then there should be an energy dependence in the time shift: lower-energy neutrinos should get to Italy faster than the higher-energy neutrinos. (This sounds like it’s the wrong way around, but that’s the way it works for tachyons; remember, zero-energy tachyons have infinite speed!) This should have caused dispersion in the arrival time PDF, and the resultant widening of the PDF (in particular, the two “end regions”) should have been measurable at the precision of the experiment. For example, if we take the central estimate of w = 120 MeV, then a 13.9 GeV particle should get to Italy 91 ns early, a 17 GeV particle should get there 61 ns early, and a 42.9 GeV particle should get there just 10 ns early.

On top of this, one would expect, from the raft of experimental evidence already available, that there is at least one neutrino mass eigenstate whose m^2 value is within eV^2 of zero, not MeV^2. (The Review of Particle Physics would tend to indicate establishment of all of the mass eigenstates to satisfy this bound, but it’s not clear to me that the “physical boundary” mentioned on page 556 — presumably that m^2 >= 0 — might not allow some wiggle room; in any case, I doubt that many of the analyses would have been taking into account the possibility of tachyonic mass eigenstates with w on the order of 100 MeV.) With the mixing angles in the neutrino sector being so large, and with (postulated) m^2 values as large as these, it’s difficult to see how the small m^2 eigenstate(s) wouldn’t be coupled in to the OPERA measurements. In that case, not only would you expect to see the dispersion in arrival times described above, you would also expect to see another “copy” of the source PDF with an anomalous shift of zero (i.e. travelling at exactly the speed of light, to within the precision of this experiment). Again, that was not seen by OPERA.

The catch-all response to all this, of course, is that, if the OPERA result is correct, then there must be “new physics” in the neutrino sector. But what sort of “new physics” would yield a shift in the arrival times that is so energy-independent?

There’s only one simple explanation that comes to my mind: Leave the neutrinos with the m^2 values that we believe them to have (on the order of or less than eV^2), but have them not created directly (as with the Standard Model) but rather via some sort of intermediate tachyonic particle (that gets 60 ns or so ahead of the light-cone in the OPERA experiment) before decaying into neutrinos. That would explain the clean shape of the OPERA distribution, and would be consistent with SN1987A and all the other experimental evidence for small m^2.

This sort of model may also explain why so many neutrino experiments have tended to give an estimate of m^2 that is negative, albeit small (including Fermilab’s MINOS). If there is some sort of intermediate tachyon that comes into play wherever the Standard Model would currently assume to be a neutrino creation vertex, which decays soon after into a neutrino, then every calculation will be skewed that little bit towards a negative m^2; but without explicitly including the intermediate tachyon, each calculation would come up with a different skewed estimate of this negative m^2 (which will be smaller, the longer the “baseline”, i.e. it depends on how much of the measurement is of the neutrino, and how much is of the intermediate tachyon).

While unconventional, this sort of model would also leave us with a Universe that is only a “little bit acausal”, just as it is only a “little bit CP-violating”. It wouldn’t make it impossible to make a tachyonic anti-telephone using this intermediate tachyon, but it would certainly make it a difficult engineering problem.

All this sounds far-fetched, of course, and by far the best guess by all at the moment is that the OPERA result has an unfound systematic error. But it’s important to do the “what-if” analysis, just in case.

You just never know.

Prime Minister Gillard?

It may seem fanciful to believe that the Labor Caucus could “do a Hawkie” on Kevin Rudd during his first term in office, but it is less difficult to believe that Julia Gillard’s low public profile in recent weeks may be a brilliant Labor tactic to insulate her (no pun intended) from the messes that the Rudd Government has found itself in. That Labor may be keeping her in reserve as “Plan B” should not be such a stretch of the imagination.

Naïve extrapolation of current polling trends would suggest that Rudd is still a shoo-in to win this year’s election; but no one (other than a hard-line climate alarmist, perhaps) believes that naïve extrapolations are useful for anything significant in the real world. Labor optimists point out that the last one-term federal government was nearly eighty years ago; but the parallels are chilling: Scullin and Rudd won power from the only two Australian Prime Ministers to ever lose their seats. It has been reasonably argued that the surreal “out of sight, out of mind” conditions of the ensuing Parliaments created environments much more conducive to Australians rapidly forgetting the Prime Minister they voted out.

Labor desperately needs to rid itself of its ministerial dead wood — Garrett, Wong and Conroy — before the calling of the federal election; but for Kevin Rudd to do so this late in the game would destroy his credibility. Dumping Rudd in favour of Gillard, on the other hand, not only switches out a leader that Australians are increasingly unimpressed with, but moreover carries with it the implied prerogative of the new Prime Minister to determine her own Cabinet.

Whether such benefits would outweigh the embarrassment of dumping their own Prime Minister is a question that Labor’s power-brokers are undoubtedly assessing on a daily basis. But Julia Gillard has performed remarkably well while those around her have stumbled. Going slightly too far in industrial relations will not harm her; all she needs to do is fine-tune the system at the edges to restore clarity, certainty and fairness, and Australian employers and employees will quickly adapt. Tony Abbott, on the other hand, has surprised many by his treading into the dangerous territory of the ghost of John Howard — but at least he has shown that he has the guts that Australians expect of their leaders.

A Gillard versus Abbott election — with fresh faces on both sides — would be closer to the political battle that Australians have come to expect.