# Calculating Cross Section in terms of barns (help with converting from natural units)

I am trying to calculate the Glashow resonance to get something in terms of barns but I am getting confused switching between natural units and non-natural units. The Glashow Resonance is given by the following equation:

$$\sigma=\frac{4}{3}\frac{G_F^2 m_e E_{\nu}}{2\pi}\frac{M_{W}^{4}}{(M_{W}^{2}-2m_{e}E_{})^2+M_{W}^2\Gamma_{W}^2}$$ One thing right off the bat that is confusing me is that in the term in the denominator $$(M_{W}^{2}-2m_{e}E_{})^2+M_{W}^2\Gamma_{W}^2$$, the first term will have units of mass^4 (so $$\frac{\mathrm{GeV}^{4}}{c^8}$$), however the last term will have units of mass squared times energy squared (so $$\frac{\mathrm{GeV}^4}{c^4}$$). How can these possibly be combined in non-natural units where you dont have the luxury of $$c=1$$?

My other confusion is what the actual value of $$G_F$$ should be? I see here that $$G_F/(\hbar c)^3=1.66\cdot10^{-5}$$ GeV$$^{-2}$$, but does this imply I need to multiply by $$(\hbar c)^3$$ in the equation above, or do I just plug in $$1.66\cdot10^{-5}$$ GeV$$^{-2}$$? I believe there is a conversion of $$(\hbar c)=0.389$$GeV$$^{2}$$mb that should be plugged in, but when I do that I am getting units of mb$$^{3}$$ and it isn't immediately clear where that would cancel out to give me just mb. Perhaps answering the first question will help solve the second.

Thanks!

• How can these possibly be combined in non-natural units where you dont have the luxury of c=1? If you really wanted to work in, say, SI units (not recommended!), you would just use dimensional analysis to restore the missing factors of $c$ and $\hbar$. Commented Aug 16, 2023 at 5:44
• ah okay, so I am guessing there are factors of $\hbar$ on the decay width that I am not accounting for, is that right? Commented Aug 16, 2023 at 14:01
• Yes. See this question. Commented Aug 17, 2023 at 4:15

You only need the conversion factor 1 GeV$$^{-2}$$= 0.389379 mb for the very end. All masses, energies, and widths are in GeV, while you have the Fermi constant in natural units.
Deleted post correction: (Now, in these units, the formula you have is dimensionally inconsistent: it is GeV$$^{-2}$$ on the left, and GeV$$^0$$ on the right. Fix it. Do you understand the Glashow resonance? Why only one power of the Fermi constant in a cross section?).
• That was a typo on my part. It should have been $G_{F}^{2}$ and I have since edited it. Commented Aug 16, 2023 at 14:02