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.

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