If we where to assume that protons decay and we know the half-life of protons, would it be possible to determine the half-life of deuterium?
If $^{1}$H (a single proton) has a half-life of, say, $10^{35}$ years, what estimations could be made for the half-life of $^{2}$H (a bound proton-neutron pair) or other “stable” nuclei?
The lifetime of deuterium would be very close to that of protium. The reason is that the energy released (the $Q$ value) in a proton decay would be most of the mass energy of the proton itself, which dwarfs the binding energy of the deuteron. If proton decay occurs, the dominant mode is likely to be something like $$p^{+}\rightarrow e^{+}+\pi^{0},$$ which conserves charge and $B-L$. This would be the primary decay mode in a $SU(5)$ grand unified theory (GUT), and although the $SU(5)$ version of this decay is ruled out experimentally, the decay modes in more elaborate theories—like $SO(10)$ GUTs—are similar. The key fact is that there is no baryon (made up of three quarks) with a mass lighter than the proton, and the next heaviest particles lighter than the proton are mesons with substantially smaller masses. The pion masses are 135 and 140 MeV, compared with the 938 MeV for the proton. Possible leptonic decay products are as light (106 MeV for a muon) or much lighter (for the positron or neutrinos). Even the spin-1 $\rho$ meson (which is an unlikely decay product because of its angular momentum) has mass over a hundred MeV less than that of the proton.
So the $Q$ value for a proton decay is going to be in the range of hundreds of MeV. Compare this with the binding energy $\Delta E=(\Delta m)c^{2}$ of the deuteron, which is only 2.2 MeV. The large energy difference means that the the proton decay is almost independent of whether or not it is bound. One way to think of this is considering the energy-time uncertainty relation. The proton decay process, once it starts, lasts a time something like $\hbar/Q$, which is of the order of $10^{-21}$ s. This is much less than the time $\hbar/\Delta m$, which is approximately the time it takes for the bound proton and neutron to orbit one-another. On the time scale of the proton the decay, the effect of the nearby neutron is practically negligible.
This is very different from the case for neutron decay. Since the neutron decays into a proton (plus other particles), whose mass is only slightly less than that of the parent neutron, the $Q$ value is small, about 0.78 MeV. This is small enough that the neutron decay becomes completely forbidden in the deuteron; the nuclear binding energy stabilizes the neutron. Even for larger, neutron-rich nuclei that do undergo $\beta$-decay, the nuclear binding has a huge impact on the lifetimes, since single-particle binding energies and the free neutron $Q$ value are similar in size.
