Models of neutrinos consistent with OPERA's results I guess by now most people have heard about the new paper (arXiv:1109.4897) by the OPERA collaboration which claims to have observed superluminal neutrinos with 6$\sigma$ significance. Obviously this has been greeted with a great deal of skepticism, and there will no doubt be debate over systematic errors for a long time to come (and frankly I expect some unaccounted for systematic error to be the case here).
Obviously theorists abhor superluminal travel, and I am well aware of many of the reasons for this.
However, the paper has me wondering whether there have been any toy models put forward which would be both consistent with the OPERA paper, and with earlier bounds on neutrino velocity.
In particular, if taken with other previous papers (from MINOS and from observations of the 1987 supernova) you have the following bounds on neutrino velocity in various average energy regimes:

$>$30 GeV: $~\frac{|v-c|}{c} < 4\times 10^{-5}$
17 GeV: $~~~~\frac{v-c}{c} = (2.48 \pm 0.28 (stat) \pm 0.30 (sys))\times 10^{-5}$
3 GeV: $~~~~~\,\frac{v-c}{c} = (5.1 \pm 2.9) \times 10^{-5}$
10 MeV: $~~~~\frac{|v-c|}{c} < 2\times 10^{-9}$

Is there any proposed model which is actually consistent with such results? It seems that there has been a lot of pointing to the supernova bound (the 10MeV scale) as being inconsistent with the reported findings. However if there was a mechanism whereby the velocity were a monotonic function of energy (or depended on flavor), this argument would be negated. Do there exist any such proposed mechanisms?
 A: Luboš Motl and Matt Reece have given excellent answers to this question. I just wanted to add an answer about a relatively recent proposal.
Stephen Gubser wrote a very recent prerint in the style of a 'no-go theorem' that discusses the major issues with superluminal travel in extra-dimensions; Link. His main point is that it is "hard" to find a compact spacetime manifold that satisfies the null-energy condition (in extra-dimensions) and satisfies Einstein's equations. However, there are a few loopholes that Gubser points out:


*

*A spacetime dimension of $D=6$ may allow superluminal travel

*One likely cannot have a smooth manifold as a model of spacetime; we need singularities. This likely rules out other 'common' possibilities for spacetime such as the conifold (since it resolves to a $CY_3$ as per the famous construction by Candelas and de la Ossa)

A: It's very hard to imagine that there is any sensible model consistent with OPERA's results. (Aside from models of unaccounted-for systematic uncertainties in the experiment.) We know that we live in a world described to very high precision by Lorentz-invariant quantum field theory, so the most sensible way to look for Lorentz violation is to start with such a theory and add small Lorentz-breaking terms to the Lagrangian. Because we assume that they're small, the usual dimensional analysis that tells you if operators are relevant, marginal, or irrelevant in effective field theory still applies. So, for instance, instead of a particle having a kinetic term $\eta^{\mu\nu} \partial_\mu \phi \partial_\nu \phi$, we can add a piece $\epsilon \partial_t \phi \partial_t \phi$ that picks out a preferred time direction. This continues to be dimension 4, so it's a marginal operator -- you should expect such operators to be present with order-1 coefficients, and they should run logarithmically just as in usual field theory, so if you try to set all of them to zero except the ones for neutrinos, say, the others will be generated nonetheless. This is already a bad sign for the viability of such theories.
The Lorentz-violating extension of the SM, defined along these lines, has been studied by Coleman and Glashow and by Colladay and Kostelecky. (As Lubos mentions, Kostelecky has done a lot of careful and serious work on understanding exactly how well Lorentz invariance has been tested.) It turns out that there are not just marginal operators, but also relevant ones, which makes the whole picture look even worse. However, the relevant operators all violate CPT, so if you decide to consider only theories that violate Lorentz invariance while preserving CPT, you're on slightly safer ground, although marginal operators are not by any means safe. The bounds are extremely strong: a 2008 review by Kostelecky and Russell has convenient tables of various operators. From there you can find references to other papers like this one of Altschul that derives a bound of about $10^{-11}$ on the analogue of "$\epsilon$" for muons. If you want a theory where muon neutrinos see a violation of the speed of light, you'll inevitably generate such a thing for muons -- even if you cleverly try to arrange it to depend on electroweak symmetry breaking to evade the fact that they live in the same doublet, loops will inevitably generate the term for muons, at a level larger than $10^{-11}$. Electrons are even more strongly constrained, at the $10^{-15}$ level, and because neutrinos oscillate into different flavors, you can't avoid confronting that bound as well.
That's not to mention the supernova 1987A constraints on neutrinos, which tell you that if you want all this to work you need to go to even more exotic theories and look at energy dependence, as you noted in the question. You can try to do that with some higher-dimension operators, but again, effective field theory tells you that you can never do that sort of thing in isolation. Break the symmetry somewhere, and all allowed terms are generated. And they're all very well bounded by data. (The size of the effect tells you that these higher-dimension operators would be suppressed by relatively low scales, which is also not encouraging.)
I won't claim that I've given you a completely airtight argument here, but anyone who really wants to claim to have an explanation of how a particle can go faster than $c$ has to confront these effective field theory issues. Anyone who claims to be able to just sidestep them completely is either trying to fool you, or fooling themselves.
Edited to add: Because I think Moshe is right that this point isn't widely enough appreciated, while I'm on my soapbox I might as well point out for readers not well-versed in effective field theory that precisely the same argument should be deployed against the idea that physics is fundamentally discrete, or that we might be living inside a condensed matter system that flows near a Lorentz-invariant fixed point, or all sorts of other new kinds of science that people might try to sell you.
A: I am afraid that one has to go to a "very unusual segment" of theoretical literature if he wants any papers about superluminal neutrinos. Guang-jiong Ni has been authoring many papers about superluminal neutrinos a decade ago:

http://arxiv.org/abs/hep-ph/0103051
http://arxiv.org/abs/hep-th/0201077
http://arxiv.org/abs/hep-ph/0203060
http://arxiv.org/abs/hep-ph/0306028

and probably others. They are pretty much cited by the same author only so you may become the second person in the world who has read them. For somewhat more well-known papers on tachyonic neutrinos, see

http://arxiv.org/abs/hep-ph/9810355
http://arxiv.org/abs/hep-ph/9607477
http://arxiv.org/abs/hep-th/9411230

which were raised by the observations of apparently superluminal neutrinos in the decay of the tritium atoms. Well, the older ones were written before the tritium atom decay anomaly. An even older paper is

http://www.sciencedirect.com/science/article/pii/0370269385904605

which reviewed the experimental situation of tachyonic neutrinos as of 1985. You may want to check many more papers by Alan Kostelecky because he's been working on similar possible ways how the Lorentz symmetry could be broken for decades and he is a rather serious researcher. See also 

http://www.sciencedirect.com/science/article/pii/0370269386904806

A paper that actually claimed to have a model of superluminal neutrinos is

http://arxiv.org/abs/hep-ph/0009291

where two Weyl equations were joined into a twisted Dirac equation of a sort. Not sure whether it made any sense. On Sunday, I will post an article on my blog about a vague way to get different speeds of light from noncommutative geometry (in string theory or otherwise):

http://motls.blogspot.com/2011/09/superluminal-neutrinos-from.html

As you noted as well, the functional dependence of the velocity on the neutrino energy would have to be an extremely unusual function which de facto invalidates the Opera results without any loopholes. However, there could be a loophole: the neutrino could become highly tachyonic only while it moves through the rocks. "Index refraction for neutrinos" could be smaller than one for common materials such as rocks. It sounds of course as incompatible with relativity as the tachyonic neutrinos in the vacuum but by splitting the experimental data into the vacuum data and rocks data, you could get more sensible velocity dependence on energy in both cases.
A: A Modified Newtonian Model Accurately Predicts the
Neutrino OPERA Detected Velocity
Ramzi Suleiman†
University of Haifa
Abstract
Motivated by a recent neutrino OPERA experimental finding, which showed that neutrino particles travel faster than light, I here propose a modified Newtonian model, which adopts a relativistic perspective. Assuming that there is no absolute frame of reference, I use Newton’s laws of motion to derive a novel result for relative time. The derived result yields a precise prediction of the velocity of neutrino reported in the aforementioned neutrino OPERA experiment. 
For the complete paper in PDF check: http://ssrn.com/abstract=1995121
