Quantum Gravity in Tritium Decay?

If you've been working in a field for some while there comes the moment when you feel like you've heard it all before. So I was surprised when the other day I came across an idea to test the phenomenology of quantum gravity that I had not heard about before - and the paper is three years old already:

Hypersharp Resonant Capture of Neutrinos as a Laboratory Probe of the Planck Length
R. S. Raghavan
Phys. Rev. Lett. 102:091804 (2009).
arxiv:0903.0787

Time-Energy Uncertainty in Neutrino Resonance: Quest for the Limit of Validity of Quantum Mechanics
R. S. Raghavan
arXiv:0907.0878

In a nutshell, the idea is as follows. We previously discussed that the Planck length is expected to play the role of a minimal length. It takes high energies to resolve short distances, and once you reach the Planck scale, this creates large fluctuations of space-time which prevents a better resolution. This idea can be captured in what has become known as the "generalized uncertainty principle", a research direction that has recently become quite popular.

This all goes back, essentially, to Mead's idea which we discussed in the earlier post. Mead however had more to say about this: He wrote another paper, "Observable Consequences of Fundamental-Length Hypotheses" Phys. Rev. 143, 990–1005 (1966), in which he argued that such a Planck scale limit should, in principle, lead to a lower limit on the width of atomic spectral lines; it should create a fundamental blurring that can't be removed by better measurement. Raghavan in his paper now wants to test this limit. Rather than using photon emission though, he suggests to use tritium decay.

Tritium makes a β-decay to Helium and in this emits an electron and anti-electron neutrino. Normally the electron flies off and the energy spread of the outgoing neutrino is quite large, but Raghavan lays out some techniques by which this spread can be dramatically reduced. The starting point is that some fraction of the tritium the electrons doesn't fly off but is instead captured in a bound orbit around the Helium. Now if the tritium, normally a gas at room temperature, can be embedded in a solid, then the recoil energy can be very small; this is essentially the Mössbauer effect, just with neutrino emission, and this gives a hypersharp neutrino line. The first some slides of this pdf are a useful summary of the recoilless bound-state β-decay.

Raghavan estimates ΔE/E to be as small as 10^-29. The average lifetime of tritium is about 12 years. There are a lot of techniques involved in this estimate that I don't know much about, so I can't tell how feasible the experiment he proposes is. It sounds plausible to me though, give or take some orders of magnitude.

He then speaks in his paper about the energy-time uncertainty relation and its Planck scale modifications. Now it is true that if you have a generalized uncertainty for spatial spread Δx and momentum spread Δp, you expect there to be also one for ΔEΔt. Yet, normally the deviations from the usual Heisenberg uncertainty scale with the energy over the Planck mass. And for the emitted neutrinos with an average energy of some keV this a ridiculously small correction term.

So here then comes the input from Mead's paper. Mead argues that the ratio ΔE/E is, in the most conservative model, actually proportional to the Planck length over the size of the system l_Pl/R, which he takes to be the size of the nuclei. This is quite puzzling, because if you take the Planck length bound on a wavelength and make an error propagation to the frequency, what you'd get is actually ΔE/E larger or equal to l_PlE which is about 4 orders of magnitude smaller in the case at hand. The reason for this mismatch is that Mead in his argument speaks about the displacement of elementary particles in a potential. Now if the wavelength of the particles is larger than the typical extension of the potential this doesn't make much sense.

That having been said, one can of course consider the proposed parameterization as a model that is to be constrained, but this leaves the question how plausible it is that there be such a modification from quantum gravity. At first sight, I'd have said a low-energetic system like an atom is a hopeless place to look for quantum gravity, but then the precision of the suggested measurement would be amazing indeed. If it works that is. I'll have to do some more thinking to see if I can make sense of the argument for the scaling of the effect. Either way however, an experiment like the one Raghavan discusses, watching the decay of tritium under suitable conditions, would test a new range of parameters, which is always a good thing to do.