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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!

Schoppe
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  • 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 $\hbar$. – Ghoster Aug 16 '23 at 05:44
  • ah okay, so I am guessing there are factors of $\hbar$ on the decay width that I am not accounting for, is that right? – Schoppe Aug 16 '23 at 14:01
  • Yes. See this question. – Ghoster Aug 17 '23 at 04:15

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You seem to be throwing the baby out with the bathwater. The whole point of natural units is to work all expressions out in powers of GeV, always setting c=1, ℏ=1. Bid them farewell forever and ever! Take advantage of the "luxury": it is the lifeline in this field.

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?).

Cosmas Zachos
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