# Calculating proton decay lifetime correctly?

I'm trying to calculate the SU(5) model's prediction for the lower limit of a proton decay lifetime, for the channel $p \rightarrow \pi^{0} e^{+}$. I'm following this paper:

arXiv:hep-ph/0504276v1

It contains the following equation: As far as I can tell this actually contains a prediction based also on SO(10), but the paper states that setting $k_{2} = 0$ means that it reduces to just SU(5).

Anyway, I have the following inputs for this equation and having tried to calculate it in Excel, I end up way away from what I expect to find, which is a proton decaying in $10^{29} - 10^{31}$ years. Here's what I am using:

$$k_{1} = \frac{g_{GUT}}{4M_{XY}}, \hspace{0.5 cm} m_{p} = 938.3 \hspace{1 mm} MeV, \hspace{0.5 cm} f_{\pi} = 130 \hspace{1 mm} MeV, \hspace{0.5 cm} A_L = 1.43, \hspace{0.5 cm} \alpha = 0.003 \hspace{1 mm} GeV^{3}, \hspace{0.5 cm} D+F = 1.276, \hspace{0.5 cm} g_{GUT} = 4\pi(\frac{1}{40}), \hspace{0.5 cm} M_{XY} = 10^{14} \hspace{1 mm} GeV$$

The latter two I have taken from the left plot shown here where I've said $\alpha = g / 4\pi$: Anyway, when I compute this, what I find is a number of the order $\Gamma \approx 10^{-64}$. I assume one then gets the proton lifetime $\tau_{p}$. by taking $\tau_{p} = 1 / \Gamma$, but clearly I end up very far away from $10^{29} - 10^{31}$ years.

I have attached an Excel file where the calculation is broken up into a few steps and they all look fine to me. The final answer I get is shown in bold. Does anyone know what I'm doing wrong? I've tried messing with factors of 1/1000 to make MeV's into GeV's where possible, and also using a factor of (60 * 60 * 24 * 365) seconds per year, but that isn't enough to get me anywhere near the right order of magnitude.

I notice that this equation is different from the rough estimate (according to arXiv:hep-ph/0601023v3) for the proton decay rate:
$$\Gamma_{p} \approx \alpha_{GUT}^{2}\frac{m_{p}^{5}}{M_{GUT}^{4}}$$
where $\alpha_{GUT} = g_{GUT} / 4\pi$ and $M_{GUT} \equiv M_{XY}$ from above if I'm not mistaken. That equation uses the proton mass to the 5th power whereas the long equation above only has $m_p$ to the first power. Overall I'm confused... help?!

Thanks.

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mfb
Mentor
Probably an error with units somewhere. Did you do the calculation with units?

Probably an error with units somewhere. Did you do the calculation with units?
Well that's the thing, the paper I'm going from doesn't seem to indicate the units but I wondered if there's some factor that the author assumes everyone reading knows is supposed to be applied, in the same way that it might be assumed with out pointing it out that readers know your equations are in terms of $c = \hbar = 1$ (eg. in this case I thought maybe multiply whatever you get by 10^32 or something).

mfb
Mentor
You'll see that if you work with units, that is the point.

You'll see that if you work with units, that is the point.
View attachment 86840
So calculating with units...
$$\Gamma_{p} = \frac{0.938 GeV}{16\pi(0.130 GeV)^2} \times (1.43)^{2} \times (0.003 GeV^{3})^{2} \times (1 + 1.276)^{2} \times \frac{5\times (0.3142)^{4}}{ (4\times 10^{14} GeV)^{4} } \\ = 2.004 \times 10^{-64} \times (\frac{GeV}{GeV^{2}} \times GeV^{6} \times GeV^{-4} ) \\ = 2.004 \times 10^{-64} GeV$$
So that is at least dimensionally correct seeing as using natural units means this decay rate has units of energy, so decay time has units of $GeV^{-1}$... oh and then I can say $\tau_{p} = 1/\Gamma_{p} = 4.99 \times 10^{63} GeV^{-1}$ and
$$1 \hspace{1 mm} eV^{-1} = 6.58 \times 10^{-16} s \\ \rightarrow 1 \hspace{1 mm} eV = 1.520 \times 10^{15} \hspace{1 mm} s^{-1} \\ \rightarrow 1 GeV = 1.520 \times 10^{24} \hspace{1 mm} s^{-1} \\ \rightarrow \hspace{1 mm} 1 \hspace{1 mm} GeV^{-1} = 6.58 \times 10^{-25} \hspace{1 mm} s \\ 1 \hspace{1 mm} year = 60 \times 60 \times 24 \times 365 \hspace{1 mm} s = 31536000 \hspace{1 mm} s \\ \rightarrow 1 \hspace{1 mm} GeV^{-1} = 6.58 \times 10^{-25} \hspace{1 mm} s / 31536000 \hspace{1 mm} s \hspace{1 mm} yr^{-1} = 2.087 \times 10^{-32} \hspace{1 mm} yr$$
so $\tau_{p} = 4.99 \times 10^{63} \hspace{1 mm} GeV^{-1} = (4.99 \times 10^{63}) \times (2.087 \times 10^{-32}) \hspace{1 mm} yr = 1.04 \times 10^{32} \hspace{1 mm} yr$
ahhhh a sensible number. Still just one order of magnitude above the $10^{28} - 10^{31}$ range I expected to hit but... that must be easily fixed when I work out which number I have wrong.

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Actually something is STILL going wrong and I don't know why. The paper arXiv:hep-ph/0504276v1 calculates the proton lifetime to be $3\times 10^{33}$ years. I have attached another Excel file with the calculation set out clearly in an attempt to replicate this number and I'm way off.

Two things I notice they do:

1) is they use $M_{V} \equiv M_{XY} = 10^{16}$ GeV. For a non-SUSY calculation I thought this should be $M_{XY} = 10^{14}$ but whatever, if I use their number my calculation results in a proton lifetime of $\tau_{p} \approx 4.7 \times 10^{36}$ years. Off by a factor of 1000.

2) The only input I choose myself is that of $g_{GUT}$, this is the only variable they do not state a value for. Is it correct to have set $g_{GUT} = \sqrt{ 4\pi\alpha_{GUT} } = \sqrt{ 4\pi(\frac{1}{40}) } \approx 0.561$ ? Actually if they have used $M_{XY} = 10^{16}$ GeV (ie. SUSY GUT scale) then I should use the corresponding $\alpha_{GUT}^{-1} = 25$ as that is the SUSY unification value... If I adjust $g_{GUT} \approx 3.5$ I can reproduce their $\tau_{p} = 3\times 10^{33}$ years...

The only other possibility as far as I can tell is that I've done my unit conversion from GeV^-1 to years wrong...

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