# One loop Fermi Constant running

1. Apr 20, 2012

### tomperson

Hi

I am attempting to calculate the mass of the W boson according to one loop energies using the equation,

Mw2=(πα/GF√2)/sin2θw(1-Δr)

where (Δr)top=(3GFMt2)/8√2π2tan2θw

using values:-

α=α(MZ)=(127.916)-1
GF=1.16634×10-5
sin2θw=0.23116 => tan2θw=sin2θw/cos2θw=sin2θw/(1-sin2θw)=0.300661
Mt=172.9

This gives the result Mw=72.2922, which is very wrong.

I suspect at least part of the discrepancy comes from the fact that the Tree Level Fermi Constant has been used, yet many hours of scouring the Internet has not revealed any running values of it. Is my assumption correct, or is there something more fundamentally wrong with what I have done?

I appreciate any assistance you can lend.

2. Apr 20, 2012

### fzero

You probably made a mistake plugging numbers in (that formula is ugly). I find 81.6, which is a bit better (WolframAlpha).

I haven't seen much about the definition of a running $G_F$, but apparently the reported value is measured from the muon lifetime using eq 10.4 in http://pdg.lbl.gov/2011/reviews/rpp2011-rev-standard-model.pdf Looking at that, it looks like any running of the weak coupling is accounted for by the radiative corrections there. In other words, perhaps

$$G_F = \frac{\sqrt{2} g_2^2(M_Z)}{8M_W^2}$$

is the reported value.

If that's not the case, it doesn't seem like $G_F$ runs too much. With

$$g_2^2 = \frac{4\pi \alpha}{\sin^2\theta_W},$$

I find $G_F(M_Z) = 1.1622\cdot 10^{-5} ~\mathrm{GeV}^{-2}$, which is very close to the reported value.

3. Apr 22, 2012

### tomperson

Thank you. Maths is great until numbers get involved :3