Effective field theory in Srednicki's book

[brushguy]

Hi!

I'm currently learning for my QFT exam with the book from srednicki (here as pdf: http://web.physics.ucsb.edu/~mark/ms-qft-DRAFT.pdf) and I am trying to understand the chapter "Effective field theory" (p. 185 in the pdf above)

He first introduces an ultraviolet cutoff Λ and then computes the path integral over all fields with momentum above Λ.
Then he defines the "Wilsonian effective action" and writes down the corresponding lagrangian
( (29.11) on p. 186 ).

-) How can one derive eq (29.11) (the lagrangian density of the Wilsonian effective action) ?

-) Why can one compute the new coefficients m²(Λ) \alpha, λ(Λ), ... the way it is shown in eq (29.13) - (29.23) ?
I am not confused about how exactly the computation works, but with how he comes up with the ansatz itself. For example, why the new mass m(Λ) can be computed with eq (29.20)

Thanks in advance!

 

[eljose79]

Hi there,

Thank you for your question about the Wilsonian effective action in QFT. The derivation of equation (29.11) can be found in many textbooks on QFT, such as Peskin and Schroeder's "An Introduction to Quantum Field Theory" or Weinberg's "The Quantum Theory of Fields". Essentially, the Wilsonian effective action is derived by integrating out the high momentum modes of the fields, which are above the cutoff Λ. This is done by performing a series of momentum shell integrals, and then using the renormalization group equations to obtain the effective action at the scale Λ.

As for the ansatz for the new coefficients m2(Λ), α, λ(Λ), etc., this is a result of the renormalization group equations and the requirement that the effective action must reproduce the same physical predictions as the original action. The specific form of the ansatz is determined by the symmetries present in the theory. For example, in a theory with a scalar field, the new mass term m(Λ) is determined by the renormalization group equation for the mass parameter and the requirement that the effective action must be invariant under the symmetries of the theory.

I hope this helps to clarify your understanding of the Wilsonian effective action. Good luck with your QFT exam!