# Neutron decay discrepancy

I have been reading the article Neutron Lifetime Puzzle Deepens, but No Dark Matter Seen on the present methods of measuring the life span of neutrons The bottle method measures a mean lifespan which is seconds shorter than the other the beam method. One explanation is that neutrons decay into dark matter, but it seems to me that this should be expected. We know that a faster particle like a neutron in a beam should, by special relativity, experience time slower than a particle that is moving slowly or standing still.

Is it possible that the difference between "bottle" and "beam" measurements of the neutron lifetime is due to relativity, since the "beam" neutrons are moving faster than the "bottle" neutrons? Or do these discrepancies persist when Relativity is taken into account and thus require another explanation?

## 1 Answer

In "beam" neutron decay experiments, the neutrons pass through some volume where their decay products can be captured by a detector and counted. The faster the neutrons are traveling, the less time the spend in the decay volume, and the less likely they are to decay. That reduces the count rate in the experiment, which reduces its statistical sensitivity. So beam-based neutron decay experiments generally use "thermal" or "cold" neutrons.

An ensemble of "thermalized" neutrons has come into near-equilibrium with some moderator at some temperature, and each neutron has $$\frac12 mv^2 \approx kT$$. For room temperature, $$T=300\rm\,K$$, the typical neutron velocity is about $$2000\rm\,m/s$$. The relativistic factor for such neutrons, which sets the scale for time-dilation effects, is

\begin{align} \gamma &= \left(1-\frac{v^2}{c^2}\right)^{-1/2} \\ &\approx 1 + \frac12 \frac{v^2}{c^2} \\ & = 1 + \frac12\left(\frac{2\times10^3\rm\,m/s}{3\times10^8\rm\,m/s}\right)^2 \\ & \sim 1 + 10^{-10} \end{align}

So for a thermal-neutron experiment, we expect time dilation effects to start in roughly the tenth significant figure. The puzzle in neutron lifetimes is in the fourth significant figure, a million times bigger. It's not an ignored relativistic correction. The best beam-based result is the NIST experiment, which used a cold beam, for which the relativistic correction is even smaller.