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Rigorous Science

# Can pion production effectively shorten the lifetime of neutrons?

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The known charge-conserving decay modes of free neutrons all involve the production of a proton, an electron and an electron antineutrino: $$n\to p^++e^-+\bar{\nu}_{e}$$ This beta decay is why, outside an atom, a neutron is unstable, with a lifetime of 15 minutes. This is actually quite a long time compared to the decays of other particles mediated by the weak nuclear force.

I recently learned that there are other ways for a neutron to (any I'm definitely abusing terminology here) "decay" into a proton and a charged pion, if it can interact with a photon: $$\gamma+n\to p^++\pi^-$$ and similarly, a proton interacting with a photon can lead to a neutron and a positively charged pion. Both of these processes are known as charged pion photoproduction.

Imagine an environment with a large quantity of free neutrons - say, several million. Normally, assuming they avoided interacting with any other particles, they would undergo beta decay after, on average, 15 minutes. But let's say that as soon as the neutrons are released, they're bathed in a sea of high-energy photons, energetic enough to enable pion photoproduction. Handwaving away the precise details of how these photons are produced, could the photons trigger photoproduction to the extent that it would be the predominant "decay" mode over beta decay, or is the reaction simply too unlikely for it to ever dominate?

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Yeah sure photoproduction can dominate, no problem.

A neutron's proper lifetime is a fixed property of the particle, but the mean free path length that a neutron can travel without interacting with a photon depends on the medium. If your medium is jam-packed with six zillion photons per cubic femtometer, your neutrons are very likely to absorb a photon almost instantly. The exact number density of photons that you need for "insanely fast nearly instant absorption" depends on the cross section for the photon-neutron interaction... or the photon-neutron coupling strength or the interaction strength if you think in those terms... but you can always pack in more photons to compensate for a small interaction probability. The photons all need to have at least enough energy to mediate the photoproduction interaction, and there might be some limit where like I dunno the energy density is so high that the quantum vacuum breaks down and it's no longer meaningful to talk about objects like "neutrons" and "photons," but I'm pretty sure that an electromagnetic process like photoproduction will be able to outpace a weak decay long before then, even though the neutron carries no net electric charge.

If you want to go into slightly more detail, maybe assume that the neutrons are moving very very slowly in the lab frame and you use 300 MeV photons[1] because you want your photon-neutron center-of-mass energy to land under the $\Delta$ (1232) mass peak. The interaction cross section shoots way up if you put yourself under a resonance peak... that's why they're called peaks. For anyone not from particle physics, check out Figure 2 from Sam Ting's (in)famous Phys. Rev. 33 paper about the discovery of the $J$ particle... or was it the $\psi$?? Anyway: the other nice thing about excited states like the $\Delta$ is that they decay to $p\pi$ via the strong force, which is super fast so you don't need to worry that it will impair your ability to outpace neutron decay. Aside from the $\Delta$ there are also plenty of excited isospin-1/2 nucleon states to choose from, but I never hear anybody talk about those for some reason. Go figure.

As the $\Delta$'s decay, photons from the perfectly-prepared 300 MeV bath will scatter off the newly-created charged particles. This will tend to thermalize the photon bath, so the photon-neutron center-of-mass energy will in general no longer lie under the $\Delta$ peak. The electromagnetic photon-neutron interaction rate will slow down, and strongly-mediated proton-neutron and pion-neutron interactions will start to become important. These strongly-mediated scatterings will generate a few excited hadronic states that decay back to protons and neutrons and pions, and the main effect will be to include the neutrons and protons and pions in the thermalization process. If the medium manages to really mellow out in less than something similar to 15 minutes, you might even see some of the new protons and neutrons bind into light nuclei, but if the temperature of the photon bath is still anywhere above like 30 MeV... that's the binding energy of helium-4... those photons will break any hopeful new nuclei apart.

The temperature after 15 minutes will depend on things like the number of energetic photons and the number of cold neutrons that you had in the first place, plus the actual photon-neutron inellastic cross section, and also the photon-proton and photon-pion elastic cross sections for the thermalization process. You can also include light-on-light scatterings if you want to get real precise... those might be non-negligible at the densities we need to overpower natural neutron decay, but I sort of doubt it. The point is that you can definitely use a bath of photons to not only interrupt the natural decay of a bunch of neutrons, but also to turn particle physics problems into statistical mechanics problems.

1. 1232 MeV/c² is pretty close to the neutron mass of 940 MeV/c², so the neutron probably won't need much of a boost to get to the photon-neutron center-of-mass frame... if I give the photon 300 MeV/c of momentum, that's 1/3rd of the neutron mass, so the neutron needs a boost to about 1/3rd of the speed of light to cancel the photon's momentum... yeah 1/3rd of lightspeed is not particularly relativistic, so I can approximate that the neutron's mass accounts for most of its energy and use this formula... $$v/c=pc/E\approx pc/mc^2\approx (300~\mathrm{MeV})/(940~\mathrm{MeV})\approx 1/3 .$$ Great everything is self-consistent so I can in fact give the photons roundabout 300 MeV/c of energy. The threshold for pion photoproduction is going to be closer to the pion mass of 150 MeV, if you prefer that number, and there might even be some sort of resonant structure just above threshold but I only know about the $\Delta$ so I'm sticking with 300 MeV. You can probably use that number to calculate the photon number density needed to reach Planck-scale physics, but I am not sure how to do that and I really just wanted a concrete photon energy to throw in the text. ↩︎

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