## Archive for October 1, 2011

### Theories about Superluminal Neutrinos

Predictably, and happily, several theorists have posted calculations and speculations to the archives, spurred by the observation by the OPERA Collaboration that neutrinos appear to travel faster than the speed of light.

I say “happily” because a result like this one provokes new ideas in the heads of theorists and we all learn something in the process. Of course, experimenters have ideas, too, and I’ll bet that people in MINOS and T2K are hard at work, hoping to confirm or refute the OPERA result.

The theoretical ideas are wildly diverse and most of them are tentative. Although I am not a theorist and I am not capable of judging any of these ideas, I’ll try to convey the essence of the papers I liked the best. No offense to papers that I don’t cite here.

The first paper by Giacomo Cacciapaglia, Aldo Deandrea, Luca Panizzi (arXiv:1109.4980 23-Sep) describes a rather careful quantitative analysis of the available neutrino data – mainly SN1987a, MINOS and OPERA. As I explained

earlier this week, the authors find it difficult to reconcile the lack-of-spread of the MINOS data with a quadratic energy dependence and indicate that some kind of step function at E_{ν} around 1 GeV is required.

The second paper by Jean Alexandre, John Ellis and Nick E. Mavromatos (arXiv:1109.6296 28-Sep) also look at the question of how strong the E_{ν} dependence has to be in order to satisfy SN1987a and OPERA. They point out that conventional tachyons don’t fit because the effect should *decrease* with energy – i.e., the effect for SN1987a would be huge. Despite the difficulties with the data, the authors forge ahead to construct two example theories that allow for superluminal neutrinos. Their first speculation involves a Lifshitz-type Field Theory of gravity, leading to an interesting dispersion relation ω^{2} = m^{6} + M^{4}p^{2} + 2M^{2}p^{4} + p^{6}, where m is a dynamical mass, M is a mass parameter characterizing Lorentz Violation and p is the momentum. Clearly this rises faster than p^{2}, but it still does not fit the data all that well. Their second speculation is more interesting and is based on a Lorentz-violating gauge theory, involving a fermion coupling to a new U(1) gauge field. In this scenario, the light cone seen by fermions differs from the one seen by photons. An interesting point is that this theory generates fermion masses dynamically, and a Higgs mechanism is not needed. A Lorentz-violating term is included in the Lagrangian which alters the fermion propagator. In general, subluminal velocities result from this term, unless one adds a constant background gauge field B_{μ}. If the couplings are weak but the field is strong, superluminal velocities can occur. An important point is that neutrinos and anti-neutrinos would have different velocities. Also, the velocity would depend on the angle with respect to the background field, B_{μ} – so some experiments would see a positive TOF with respect to light, others a negative, and some no effect at all. This field B_{μ} could vary in strength on galactic distance scales, and have little effect on SN1987a while a major effect on earth-based neutrino sources. In any case, a like one of the closing statements by the authors: **Superluminal neutrinos should not be discarded as a phenomenological impossibility, but rather be regarded as a scenario to be probed and constrained by experiment.** Right on.

The third paper could not be farther in spirit form the second. Written by Andrew G. Cohen, Sheldon L. Glashow (arXiv:1109.6562 29-Sep), the abstract contains the sentence **Thus we refute the superluminal interpretation of the OPERA results**! The authors have noticed that superluminal particles can slow down by emitting radiation including pairs of fermions, and they completed the lowest-order calculation for neutrinos. The dominant mechanism is ν→ν+e^{+}e^{–}. Although the prefactor is roughtly 10^{-6}G_{F}^{2}, the rate of emission of the e^{+}e^{–} pairs by the neutrino is proportional to E^{6} and the energy loss is proportional to E^{5}. For the 730 km baseline of the OPERA experiment, the terminal energy is 12.5 GeV which is much lower than observed by OPERA – the neutrinos have radiated away the rest. According to Cohen and Glashow, the OPERA energy distribution directly contradicts the notion that the neutrinos are traveling faster than the speed of light. They go on to say that observations by the IceCube Collaboration of multi-TeV neutrinos passing through the earth is an extremely strong constraint on the order of δv/c < 1.7×10^{-11}. This is a formidable result and its hard to think that the calculation is wrong.

Nonetheless, let me put aside the objection from Cohen and Glashow to point to two interesting papers.

The fourth paper is by A. Nicolaidis (arXiv:1109.6354 28-Sep) who sketches how [sterile] neutrinos could take a “short-cut” through the bulk in an extra-dimensional scenario. You’ll recall that these theories attempt to solve the hierarchy problem by extending Gauss’s Law to extra dimensions in which gravity can propagate but the gauge interactions of the standard model cannot. In essence, the neutrino can follow a geodesic in a many-dimensional space while light and other standard model particles are constrained to move along a curved surface. He writes down a toy model and concludes that one can match the OPERA result with the size of one extra dimension of about 2.7μm and a curvature parameter Ak on the order of 10^{-2}. He does not discuss the facts from SN1987a or MINOS, however, and hopes that an experiment can be done with NESTOR which is 1676 km away from CERN.

The last paper I’ll mention today (-again- sorry that I do not try to discuss all of the papers that have appeared) was written by Marco Matone (arXiv:1109.6631 29-Sep). This curious paper begins with Hamilton-Jacobi theory in quantum mechanics. I don’t understand the calculations at all but the upshot seems to be that some extra constants of integration appear when one addresses the time evolution of the particle. In a particular rather generic case, the usual result for the velocity of a quantum particle is modified by a function of these extra constants, and this function can have a value greater than one, thereby leading to superluminal solutions. While Matone’s calculation is relativistic, it ignores field theory so it probably cannot be taken at face value. That said, I think it is great that the OPERA result can spur thoughts about fundamental quantum mechanics!

So here’s what we have – ad hoc parametrization of the dispersion relation, two or more parametrizations pulled out of Lorentz-violating field theory, extra dimensions and unconventional quantum mechanics – plus an observation based on “simple” physics that the OPERA results are simply impossible. You might look upon this as theoretical bedlam, but I am happy to see so many creative ideas emerging, which can help shape the experiments *we need to do* in order to verify or refute the OPERA results.