One loop Fermi Constant running

[tomperson]

Hi

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

M_w²=(πα/G_F√2)/sin²θ_w(1-Δr)

where (Δr)_top=(3G_FM_t²)/8√2π²tan²θ_w

using values:-

α=α(M_Z)=(127.916)^-1
G_F=1.16634×10^-5
sin²θw=0.23116 => tan²θw=sin²θw/cos²θw=sin²θw/(1-sin²θw)=0.300661
M_t=172.9

This gives the result M_w=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.

 

[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.

 

[tomperson]

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