# I Sin2thw running

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1. Aug 5, 2016

### colubra

Hi, everybody!

I can not really understand the particular behavior of the sin2thw with respect to the scale (like on the plot below).

Clearly there are three regimes from the left to the right: a plato, a degradation and a steep rise. But why exactly does the curve behave like this? The W-boson mass scale is clearly one pole here, where weak interaction starts playing role, but this is not enough for understanding this.

The runnings of the EM and weak couplings must be causing this, but somehow they do not add up to the sin2thw curve in my head.

Would be grateful for any explanation or a link to a good source describing this!

2. Aug 5, 2016

### RGevo

Hi Colubra,

As you may know, the running is something like s2w(mu) = 1/ (1 + alpha_2(mu)/alpha_1(mu)), where these are the couplings of the SU2_L and U_Y. To one-loop, these individual couplings will run like:
alpha_i(mu) = alpha_i(mu0)/ (1 - alpha_i(mu0)/(2 Pi) b_i Log(mu/mu0) )

The coefficients b_i depend on the number of active particles. So, for b_1 it depends on the hypercharges of the active fermions and scalars within the theory. If I read this plot correctly, its probably the one-loop running of s2w and the little changes in direction refer to the points where the author has changed the number of active particles running in the loop. Above 200 MeV there is strange loops, then c-loops at ~1.5 GeV and b-loops at ~5 GeV. These contributions can be hard to evaluate, and might be included by including some alpha(hadronic) contributions... maybe. Depending on how/when these contributions are included, the running will change as these particles have different charges under SU(2)L and U(1)Y. Maybe you can try and check, the general formula for the b_i coefficients are giving in eq.3 of http://arxiv.org/pdf/hep-ph/0412163v2.pdf
Hope this helps a bit.