Beta function of electroweak theory

[Sleuth]

Hi everyone,

I have another (probably too) simple question for particle physicists on this forum, but I often realize that my understanding of QFT is still rather poor.

Do you know where I can find the electroweak beta function explicitly written down (at one-loop, of course)?
I would like to have a look at the explicit expression for it. The reason is that, as it's well known, non-abelian theories can be asymptotically free. But then, what about SU(2)xU(1) with SSB, i.e. the ew sector of the standard model? Is there a choice of parameters for which the electroweak interactions would become asymptotically free? Or does the SSB mechanism make this impossible?

Thanks!
Sleuth

 

[Sleuth]

I wonder, was the question below everyones' dignity ? xD

 

[Vanadium 50]

I wonder if anyone will answer the question now. Most people do not react well to "My servants! Attend me!"

 

[Sleuth]

Ehm... people do take themselves too seriously sometimes :) i was just making a joke. I thought the question would really be too stupid. Take it easy; ) i guess asking physics questions over Christmas is not the best thing someone can do :D

 

[Hepth]

I think http://arxiv.org/abs/hep-ph/9807548 might help?

 

[Sleuth]

Thanks Hepth! The paper looks like it goes in the right direction. Going through it fast I haven't found a precise quantitative statement on the running of the ew couplings, but maybe I should take some time to read it more carefully!

Thanks

 

[RGevo]

Hi Sleuth,

I hope you find the following interesting:

Please look at http://arxiv.org/pdf/hep-ph/0412163v2.pdf.

Equation 2 is the general equation for the running coupling, and the relevant parameter is the b_i piece given in equation 3.

If you compute this for the SM(at 173 GeV) you find
b1=41/6 and b2 = -19/6.

b2 is for example coming from 12 Weyl fermions, 1 scalar, C_2(G) = 2 for SU(2), and T_R(I) = 1/2. So, if you want s_w(mu)^2 or something, you can compute 1/(1+alpha(b1,mu)/alpha(b2,mu)).

On the other hand, if you want to flip the signs etc. of the running beyond some high energy scales etc. where new particles might be living, you have to change the particle content (consistently) and compute the running of the beta functions under these conditions.

Hope this helps.

Edit, added scale at which the SM content gives those values. I believe it includes the top quark.