Let us denote the half-life of the proton by $Y_p$. (There is, of course, no experimental evidence that $Y_p<\infty$, but there are theories that assert it, so this is really a question about those theories).

The question is: what, in that case, is $Y_C$, the half-life of a carbon-12 nucleus?

A naive answer would be that since $^{12}C$ contains six protons, $Y_C=\frac{1}{6}Y_p$.

However, protons do not decay. Quarks do. Since neutrons are made of as many quarks as protons are, they should decay into non-baryons just like protons. Since $^{12}C$ contains twelve nucleons, that makes $Y_C=\frac{1}{12}Y_p$.

  1. Which is it? $\frac{1}{6}Y_p$ or $\frac{1}{12}Y_p$?

There is a hidden assumption in all this: that the half-life of a quark is unaffected by the baryon or meson in which it finds itself. On the other hand, the half-life of a neutron is strongly affected by the nucleus in which it finds itself (or doesn't).

  1. Is the assumption of environment-independence correct for the decay of quarks into leptons?

There is one more assumption. Protons and neutrons are each made of two different kinds of quarks.

  1. Do the theories that make quarks decay into leptons assign identical half-lives for this process for both up quarks and down quarks?

Half-lives of bound systems don't usually have any simple scaling laws of the kind you have in mind. The half-life would depend in part on the phase space available to the particles that were produced, as well as factors having to do with nuclear structure. However, this decay is a rather high-energy process. Decay of a proton into a neutral pion and a positron has a $Q$ value of 802.8 MeV. Because a 12C nucleus has a different binding energy than a 11B nucleus, the $Q$ value of your decay would be different, probably lower by on the order of a few MeV. But this is pretty small compared to eight hundred MeV, so it would probably have a small effect. So I would guess that in this example, because of the disparate energy scales, the fact that the proton was bound inside a nucleus would have little effect on the half-life.

For similar reasons, I would expect there not to be much difference between the contribution of the neutrons and that of the protons. So 1/12 of the proton's half-life is probably a pretty reasonable estimate.


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