Deriving 4-Point Fermi Interaction from Standard Model

[RedX]

Can someone explain how one derives the 4-point Fermi interaction from the full Standard Model?

I understand you set up a cutoff \Lambda, and the cost of this is that the coefficients of all terms in your Lagrangian become functions of this cutoff, and you also have an infinite number of new, non-renormalizeable terms. So your new propagator should be of the form 1/(Z(\Lambda)k^2-M^2), where Z(\Lambda) is the coefficient of your kinetic term, and since integrations are now only to \Lambda, this propagator can be replaced by (-1/M^2) since Z is at most log(\Lambda)=log(M)?

But after doing all this, your Lagrangian still involves the heavy boson terms. The book says to solve their equations of motion, and substitute this back into the Lagrangian. What is the justification for this?

 

[blechman]

As a first step, compute 2-2 scattering amplitude of quarks or leptons (at tree level) with the W-boson present.

You get an answer that is proportional to g^2/(p^2 - M_w^2) where g is the SU(2)_L coupling, and p^2 is related to the CoM energy.

Now imagine that you are at energies much lower than M_w. Then you can factor out an overall

-g^2/M_w^2

and write the answer as a power series in p^2/M_w^2.

Notice that the overall factor is just G_F (up to factors of 2 which also work out)!

Notice that the leading-order result (the 1 in the geometric series expansion) is EXACTLY what you would have gotten if you used Fermi theory!

Notice that all the other terms in the expansion are EXACTLY what you would get if you include power-corrections from the renormalization of Fermi theory!

So you see that this is certainly "justified".

I'm confused by your post, since what you call "Lambda" is actually M_w ! That's your cutoff.

For a very beautiful and excellent introduction to the ideas of Effective Field Theory, there are so many great review articles I couldn't even begin to include them all. Go to spires and look for title "effective field theory" and you'll get a bunch of them.

Personally, one of my most favorite, although there are many others as well, is the one by David B. Kaplan, titled "Effective Field Theory" (very original title!). That was one of the first EFT reviews I tackled when I was a student and I like it very much.

 

[RedX]

Thanks. I will look at the articles soon.

I was confused more by the generic way that you derive an effective field theory from a full-scale theory by integrating out the heavy field from the path integral. I've got most of it understood now.

But certainly if you calculate the tree-level scattering amplitude with the exchange of a W and make approximations, you should get the Fermi 4-point interaction.

 

[blechman]

check out those articles. they will help you greatly to understand this. if you're still at a loss, feel free to ask more questions.