# Converting width in $\rm GeV$ to lifetime in years

I am trying to reproduce the results of this paper1. In particular Eqs. (25) and (26). I would like to convert the proton decay width (given in units of $$\rm GeV$$) to a lifetime in years. The formula is below:

$$\Gamma (p\to \pi^0 \mu^+)_{\rm flipped}= \frac{g_5^4m_p |V_{ud}|^2 |(U_l)_{21}|^2 }{32\pi M_X^4}(1-\frac{m_\pi^2}{m_p^2})^2 A_L^2 A_{S_1}^2 \left(\langle \pi^0\vert (ud)_Ru_L\vert p\rangle_{\mu}\right)^2.$$

Here $$A_{L}=1.2$$, $$A_{S}=1.5$$, $$m_{\pi}=0.139$$ GeV, $$m_{p}=0.938$$ GeV, $$\left(\langle \pi^0\vert (ud)_Ru_L\vert p\rangle_{\mu}\right) = -0.118 \rm{GeV}^2$$, $$|V_{ud}|^2\approx 0.97$$. The lifetime ($$\tau=1/\Gamma$$ in natural units) formula derived from the one above is

$$\tau (p\to \pi^0 \mu^+)_{\rm flipped}\simeq 9.7 \times 10^{35} \times |(U_l)_{21}|^{-2} \biggl(\frac{M_X}{10^{16}~{\rm GeV}}\biggr)^4 \biggl(\frac{0.0378}{\alpha_5}\biggr)^2 ~{\rm yrs}~.$$

where $$\alpha_{5}=\frac{g^2_{5}}{4\pi}$$. I have tried plugging in these numbers and then converting from GeV to years $$(1\rm{GeV}=4.79347\times10^{31} \rm{year}^{-1})$$. Not sure what I am doing wrong as I get far too small a lifetime. Help is appreciated.

1 Reference: J. Ellis, M. A. G. Garcia, N. Nagata, D. V. Nanopoulos & K. A. Olive, "Proton decay: flipped vs. unflipped $$\rm SU(5)$$", J. High Energ. Phys. 2020, 21 (2020).

Using $$(1-\frac{m_\pi^2}{m_p^2})^2\sim 0.96\quad A_L^2A_S^2\sim 3.24 \quad |V_{ud}|^2\sim 0.94\quad \left(\langle \pi^0\vert (ud)_Ru_L\vert p\rangle_{\mu}\right)^2\sim 0.013924 \rm{GeV}^4$$. Inputting the numerical constants one finds

$$\Gamma \approx 8.59\times10^{-69}\left(\frac{\alpha_5}{0.0378}\right)^2 \left(\frac{10^{16} \rm{GeV}}{M_X}\right)^4 [\rm{GeV}] |(U_l)_{21}|^2$$

Inverting this we find

$$\tau \approx 1.16687\times 10^{68}\left(\frac{0.0378}{\alpha_5}\right)^{2} \left(\frac{M_X}{10^{16} \rm{GeV}}\right)^4 |(U_l)_{21}|^{-2}[\rm{GeV}]^{-1}$$

and multiplying by the conversion factor $$\rm{GeV}^{-1} = 2.08768\times10^{-32}$$ years

$$\tau \approx 2\times10^{36} \left(\frac{0.0378}{\alpha_5}\right)^{2} \left(\frac{M_X}{10^{16} \rm{GeV}}\right)^4 |(U_l)_{21}|^{-2}$$ years.

• May 11, 2020 at 20:17
• This looks correct now. Three final tiny quibbles: There is no reason to put brackets around “GeV”, just like you don’t put them around “years”. You wrote $\alpha_S$ instead of $\alpha_5$. You have some intermediate values with an unjustified number of significant digits. But all the serious problems have been fixed. May 11, 2020 at 21:53
• So what is the mean lifetime of a proton, according to this calculation? May 11, 2020 at 23:57
• I fixed the alpha's G. Smith. So the lifetime of the proton (assuming a 100%) branching ratio i.e. proton decays to a neutral pion and muon all the time, is limited to be >7.7 10^{33} years. From this calculation you can tune the mass of the gauge boson associated to the GUT (M_X) to be sufficiently heavy such that this constraint can be easily satisfied. So the lifetime depends on the mass of this M_X. Also on the mixing maxing entries of U (PMNS matrix). May 12, 2020 at 0:54