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It is known that isolated neutrons are unstable, but that neutrons inside nuclei can be stable.

I know there are candidate theories that allow proton decay but I wonder whether in these theories protons might be stable inside nuclei. Is it so?

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Neutron decay is the conversion of a down quark to an up quark with emission of a (virtual) $W^-$ that decays to an electron and electron anti-neutrino. The energy scales involved are comparable to the energy scales involved in binding nucleons together into a nucleus, so it's not too surprising that the decay can be prevented by binding the neutron to a proton.

By contrast proton decay involves the conversion of a down quark into a positron with emission of an X boson. The mass of the X boson is not known but is likely to be around $10^{15}$ times the mass of a proton or neutron. So this interaction occurs on a completely different energy scale, and it's exceedingly unlikely that whether the proton is bound into a nucleus or not would make any difference to the decay rate.

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