# Estimate mass of exchange boson by decay time

I have made a rough estimate that the minimum lifetime $\tau$ of the proton must be $10^{23} \, \mathrm{s}$. From this I would like to estimate the mass of the X boson which would mediate this decay in some GUT. With the lifetime I could already estimate the decay width via $\Gamma \approx 1/\tau$. This uses the Heisenberg uncertainty relation at its core, I believe.

In general, a higher exchange boson mass will lead to a longer lifetime. One can see this with electromagnetic and weak decays in neutral and charged pions. Is there any order-of-magnitude method to obtain the exchange boson mass?

The mass of the exchange boson enters in the propagator of the matrix element into the decay width. In principle, one should be able to find a dependency $M_\mathrm X(\Gamma)$ by setting up the Feynman diagram, using some rules and computing the phase space integrals. But isn't there an easier way to obtain a first estimate?

• The currently accepted experimental limits (PDG) seem to be $>2.1\times 10^{29}years\approx 6.6\times 10^{36}s$. How did you come up with your estimate? – CuriousOne Apr 24 '16 at 17:36
• I noticed that I am alive. Therefore the protons in my body do not produce more than one Gray per year. From this I derived a minimal lifetime. – Martin Ueding Apr 24 '16 at 18:27
• OK, that's a pretty good estimate that requires zero experimental effort. Someone should put that on a t-shirt! :-) – CuriousOne Apr 24 '16 at 18:37
• The professor supervising my talk about this topic suggested this as a neat introduction. I did not think about this obvious estimation myself, that is probably one of the downs of becoming a theorist :-). – Martin Ueding Apr 24 '16 at 20:14

The current limits on proton lifetime are $\tau_\mathrm p^\text {exp}\gtrsim 10^{34}$ years. In unified models, we can naively estimate the proton lifetime with this relation:
where $M_\mathrm{GUT}$ is the mass scale of your proton-decay mediating gauge boson (what you call $X$), $M_\mathrm p\approx 1\,\mathrm{GeV}$ is the proton mass, and finally $\alpha_U$ (typically, $\alpha_U\approx 1/40 \div 1/20$ ) is the coupling strength at the GUT scale.
• Is the $\alpha_U$ supposed to be in the range of $1/40$ and $1/20$? If I take the $\div$ symbol to mean division, that would be just $1/2$ in the end … – Martin Ueding Apr 24 '16 at 20:06
• yes, it's a typical range. I used $\div$ to express a range indeed. – xi45 Apr 24 '16 at 23:05