### The Fat Lady Sings

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.

**Mohammad:** Hmmmm….

**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

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.

## 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.