Running coupling of the weak interactions

[evilcman]

Can someone point me to a reference that explains how the (effective) running coupling behaves in the weak interactions(at 1-loop order). I couldn't find it...

If I understand correctly, than the coupling is g = \frac{e}{sin(\theta_W)} where e is the QED coupling, which increases with energy scale, and cos \theta_W = M_W / M_Z is the Weinberg angle. The vector boson masses are in turn determined by their couplings with the Higgs, and the renormalization group equations will determine the running of these (and other) couplings. Than from that we can get the running of this effective coupling. Can someone tell me where to find the results for such a computation? Is it approximately constant until the electroweak unification or not? If not, how much does it change, etc...

 

[blechman]

You can calculate the running of g the same way you calculate the running of g_s. The beta function is always the same:

\frac{dg}{d\log\mu}=-\frac{b_0g^3}{16\pi^2}+\mathcal{O}(g^5)\quad{\rm where}\quad b_0=\frac{11N-n_f-n_s}{3}

with N=2,3 for g,g_s, and n_f is the number of CHIRAL fermions (so for QCD which has VECTOR fermions, this would be 2n_f) and n_s is the number of scalars (Higgs bosons).

The g' running (hypercharge) is given by a similar formula, with N=0 (no gauge boson loops) and n_f\rightarrow 2\sum_f Y_f^2 where you sum over all fermions, and Y_f is the hypercharge of the fermion (in units of g'). This is the same formula in QED: there are TWO fermions (electron and positron), both with Q^2=1, so that b_0(QED)=-4/3. Plugging that into the formula gives

\frac{de}{d\log\mu}=+\frac{e^3}{12\pi^2}+\mathcal{O}(e^5)

which is correct.

Hope that helps!

 

[evilcman]

Well I am sure that would be the case if the weak interactions was a pure SU(2) gauge theory, but the full SM is more complicated, and I am not sure this still applies.

 

[blechman]

sure it does, you just have to be careful to count the fields correctly. For example, when counting quarks in SU(2), remember that you count each quark color as an extra (Weyl) field, so \Delta n_f=3~({\rm color})\times 3~({\rm generations})=9 in my formula above (you also have to add leptons). Besides that, it's exactly the same.

 

[evilcman]

What about above the W threshold?

 

[blechman]

You can only talk about g,g' couplings above the W threshold. Below that, the couplings no longer run (just like below the electron mass, the QED coupling does not run). Below the W threshold, the correct thing to do is to match onto QED + Fermi theory, and then you can talk about the running of G_F and e.