## October 4, 2011

I was chatting with our new postdoc, and the conversation went something like this:

Mohammad: So, why aren’t you blogging any more?
Jacques: If I were blogging, I would probably be writing posts about superluminal neutrinos. Surely, that’s not what you want.
Jacques: On the other hand, Cohen and Glashow wrote a very nice paper laying that whole miserable subject to rest.

The paper is based on the old Coleman-Glashow analysis of Lorentz-violation in the Standard Model. In the case at hand, super-luminal neutrinos would lose energy via the neutral-current process

(1)$\nu \to \nu + e^+ + e^-$

This process has a threshold energy, $\delta E = \frac{2 m_e}{\sqrt{v_\nu^2 - 1}}\sim 140 \text{MeV}$ for OPERA’s purported value of $v_\nu-1 \sim 2.48\times {10}^{-5}$.

In the Coleman-Glashow analysis, $v_\nu = \tfrac{d E}{d k}$ is constant for $E\gg m_\nu$. That’s already excluded by Supernova 1987a, which constrains $v_\nu-1\lt {10}^{-9}$ for $E$ in the range of a few MeV. OPERA already requires some more complicated dispersion relation, $E(k)$.

Regardless of the details, it’s clear that superluminal neutrinos rapidly lose energy, due to (1). Assuming $v_\nu$ is approximately constant over the relevant range of energies, it’s possible to integrate $\frac{d E}{d x} = - \frac{25}{448}\frac{G_F^2}{192\pi^3} E^6 {(v_\nu^2 - 1)}^3$ to obtain $E^{-5} - E_0^{-5} = \frac{125}{448}\frac{G_F^2}{192\pi^3} {(v_\nu^2 - 1)}^3 L \equiv E_T^{-5}$ To a very good approximation, the arrival energy of the neutrinos at the OPERA detector is independent of the initial energy, and is given by $E_T=12.5$ GeV, for OPERA’s $L=730$km.

Allowing $v_\nu$ to vary, over the energy range of interest, changes the behaviour quantitatively, but probably not qualitatively.

Unfortunately, OPERA sees neutrinos with a mean energy of 17.5 GeV (ranging up to 50 GeV), which rules out the possibility that they could be superluminal.

Posted by distler at October 4, 2011 5:27 PM

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### Re: The Fat Lady Sings

So how about Aref’eva and Volovich’s idea that you can get around this if the tachyon is a right-handed neutrino that’s a standard model singlet and shows up only by mass mixing? The left-handed neutrino feels the weak interaction but it’s subluminal, the right-handed neutrino is superluminal but doesn’t feel the weak interaction, and we never have the Cohen-Glashow case of a superluminal neutrino that feels the weak force.

Posted by: Mitchell Porter on October 5, 2011 4:00 AM | Permalink | Reply to this

### Re: The Fat Lady Sings

…the right-handed neutrino is superluminal but doesn’t feel the weak interaction, and we never have the Cohen-Glashow case of a superluminal neutrino that feels the weak force.

But they mix, via the off-diagonal mass term. (If they didn’t mix, the superluminal neutrino species would be completely decoupled from the physics at either end of the experiment.)

Coleman-Glashow already analyzed the implications, from neutrino oscillation experiments, of having the neutrino mass matrix being non-diagonal in the basis which diagonalizes the kinetic terms. They found a much-more-stringent bound on the velocity differences (on the order of ${10}^{-20}$).

Seriously, if you’re going to play this game, a minimal requirement would be to incorporate the known limits on this sort of Lorentz-violation.

And papers, like the Aref’eva-Volovich paper, which say “We do X,Y,Z” and then … don’t, aren’t worth paying attention-to. (My most charitable interpretation of the Aref’eva-Volovich paper is that they uploaded the introduction to the arXiv, intending to do the research later.)

Posted by: Jacques Distler on October 5, 2011 8:28 AM | Permalink | PGP Sig | Reply to this

### Re: The Fat Lady Sings

They found a much-more-stringent bound on the velocity differences (on the order of ${10}^{-20}$).

Sorry. I need to partially-retract that.

Their really stringent bound comes from when the kinetic terms are non-diagonal in the flavour-eigenstate basis. Here, we’re taking them to be diagonal (only the SM singlet neutrino has a Lorentz-violating kinetic term).

That means that the Aref’eva-Volovich scenario isn’t immediately excluded. But it ain’t out of the woods, either.

Posted by: Jacques Distler on October 5, 2011 2:02 PM | Permalink | PGP Sig | Reply to this

### Re: The Fat Lady Sings

Honestly,
that paper doesn’t seem to me very nice.
Although I agree that it’s very very, very difficult to find an appealing model that can incorporate the OPERA result, I also believe that Cohen and Glashow treat the anomaly in a very superficial way.
They take one of the most surprising measurements of the last 50 years and apply a method that is the least conservative I can think of.
Which is: “let’s just say that if the speed of neutrinos is greater than c, then I can take a fast neutrino and make it decay to itself plus two more particles… you know, you have to be able to do that if the Lorentz symmetry is not what we think it is…”
To me, this kind of argument not only is not putting the discussion about the result to rest, but it’s also very silly to say that if neutrinos are faster than light then they necessarily don’t obey anymore to certain decay laws that normally prevent their energy from depleting.
It’s like saying that the cosmological constant must be zero otherwise a stable-in-size universe would not be likely to exist.
Papers like Mattingly’s http://arxiv.org/abs/1110.0783 Moffat’s http://arxiv.org/abs/1110.1330 or Amelino-Camelia’s http://arxiv.org/abs/1110.0521 and http://arxiv.org/abs/1109.5172 talk about it in a much more conservative way.

Posted by: Ruggero on October 10, 2011 4:57 AM | Permalink | Reply to this

### Re: The Fat Lady Sings

Yes, there have been lots of really crappy papers, submitted to the arXivs, on this subject.

The Mattingly et al paper looks serious. It’s a nice elaboration of the ideas presented here.

As to the others? Well … the less said the better.

Posted by: Jacques Distler on October 10, 2011 9:51 AM | Permalink | PGP Sig | Reply to this

### Re: The Fat Lady Sings

Oh, I didn’t mean to endorse any of those papers.
I agree with you that the Mattingly et al one looks more solid, but what I meant was just that *even* those papers, regardless their implications, treat the matter in a more conservative ways.

In my opinion, if you want to write a paper saying that some result is incompatible with the limits we have, you really have to be conservative. Because a faster than light neutrino would necessarily mean that there is something we got wrong about the Lorentz symmetry in our universe, it would be exactly in that domain that we would need to try to understand what’s possible and what’s not. We should not be just taking the easiest possible model and say that that one rejects the measurement.

This said, I don’t think the anomaly will last. Mine was just a comment about the method.

Posted by: Ruggero on October 10, 2011 10:51 AM | Permalink | Reply to this

### Cohen-Glashow paper

Err, isn’t the Cohen-Glashow paper is based on the beyond-relativity theory developed by Coleman and Glashow himself some time ago? That theory presumes that the Lorentz symmetry is broken by the presence of the preferred reference frame, and it has not been confirmed by experiment yet. Moreover, it is not the only theory available at this moment hence the Cohen-Glashow arguments are not universally applicable at most (actually, they merely indicate that the Coleman-Glashow theory couldnt explain the neutrino anomaly if it existed). The paper 1110.0521 gives the example where the Cohen-Glashow arguments fail, namely, when the Lorentz is deformed rather than broken.

Posted by: Quest on October 24, 2011 5:17 AM | Permalink | Reply to this

### Re: Cohen-Glashow paper

The paper 1110.0521 gives the example where the Cohen-Glashow arguments fail, namely, when the Lorentz is deformed rather than broken.

That paper is garbage (on many levels).

What is relevant here is that that Lorentz-invariance (of the undeformed variety) is exquisitely well-tested in a wide variety of circumstances, including the propagation of neutrinos at lower energies (10s of MeV, rather than 10s of GeV).

So deforming the Lorentz algebra (whatever its other problems) is irrelevant to the problem at hand.

Posted by: Jacques Distler on October 24, 2011 7:46 AM | Permalink | PGP Sig | Reply to this

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