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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).

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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.

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  • $\begingroup$ Let us continue this discussion in chat. $\endgroup$
    – SAMCRO
    May 11, 2020 at 20:17
  • $\begingroup$ 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. $\endgroup$
    – G. Smith
    May 11, 2020 at 21:53
  • $\begingroup$ So what is the mean lifetime of a proton, according to this calculation? $\endgroup$
    – PM 2Ring
    May 11, 2020 at 23:57
  • $\begingroup$ 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). $\endgroup$
    – SAMCRO
    May 12, 2020 at 0:54

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