# Wilsonian RG and Effective Field Theories

• A

## Main Question or Discussion Point

Years ago after reading Ch. 12 of Peskin and Schroeder (and the analogous discussion in Zee), I thought I fully understood the modern Wilsonian view of renormalization, and how it explains why non-renormalizable field theories still have meaning/predictive power at energies well below the intrinsic cutoff or breakdown scale $\Lambda$. To recap the argument, P&S start with $\phi^4$ theory in Euclidean space with a sharp UV cutoff $\Lambda$. They then follow the 3-step Wilsonian RG procedure:

1. Integrate out modes in the momentum shell $b\Lambda < k < \Lambda$ where $b<1$, thereby shifting the starting parameters $m$ and $\Lambda$, but also generating higher-dimension operators like $\phi^6$ terms, etc.
2. Rescale momentum $k'=k/b$ and lengths $x' = bx$ so that the momentum integrals in the effective Lagrangian from Step 1 span the same range (i.e., $[0,\Lambda]$) as the starting Lagrangian.
3. Rescale fields to keep the quadratic term unchanged
Then one observes how the coefficients of the different operators in $L_{eff}$ scale as the 3-step transformation is iterated many times until $b^n\Lambda$ is near the scale of some low-energy process you want to compute. For weak coupling, (i.e., near the gaussian fixed point), this basically boils down to dimensional analysis, with the couplings with negative mass dimension decaying as you
scale towards low momenta (i.e., the so-called irrelevant operators), and those with positive mass dimension growing ("relevant operators"). Dimensionless ("marginal") couplings don't change-- more precisely, you need to consider the "dynamic" part of the RG transformation coming from loop integrals in step 1 to see if it decays or grows. In this way, you see that even if you started with some crazy non-renormalizable theory at the scale of $\Lambda$, at energies much less than this all the non-renormalizable terms become unimportant since their coefficients decay under the 3-step RG procedure.

Anyway, the above all seemed "obvious" until I encountered the analogous discussions in the more recent QFT books of Srednicki (Ch. 29) and Schwartz (Ch. 23). Weinberg also has a similar discussion in a section called "the floating cutoff". All of these, best as I can tell, are based on the 1984 Nucl. Phys. B paper of Polchinski (see http://www.sciencedirect.com/science/article/pii/0550321384902876). There, they also follow the Wilsonian approach of integrating out high-momentum modes, but they all make the point that the coefficients of the irrelevant operators aren't necessarily flowing to zero in the infrared, but rather they become insensitive to the values of the irrelevant couplings at the large cutoff scales. I.e., the coefficients with negative mass dimension aren't necessarily small as you integrate down to low momentum scales, but rather they become computeable functions of just the marginal and relevant couplings.

My question is, have I horrendously misunderstood the discussions in Peskin and Schroeder (and also Zee), which seemed so intuitive and trivial at the time? Are they not saying the irrelevant couplings are decaying to zero as you iterate the RG transformation? How do I square this with the discussions in Srednicki, Schwartz, Weinberg, and Polchinski?

Related High Energy, Nuclear, Particle Physics News on Phys.org
atyy
https://arxiv.org/abs/hep-th/9210046

Polchinski gives the definitions of marginal, relevant, and irrelevant operators on p3.

I think your understanding of Weinberg is correct. I believe Srednicki (eg. p193) says the same thing.

Last edited:
TensorAndTensor
I'm grateful for this thread because I'm deep into quantum gravity, which is non-renormalizable and as far as I know, generally accepted as being a good candidate for an effective field theory. I am also deep into Srednicki chapter 29.

As a relative newcomer to these techniques, this thread prompts some basic questions:
1) Which non-renormalizable theories serve as the best basic illustration of these concepts (strong interaction, etc)? There must have been some successes along the way since effective field theory continues to be pursued. Basically I'd like to come up to speed on cases where effective field techniques lifted a problem out of a calculational morass and added additional higher-order physical information.
2) Is it necessary to get proficient with path integration in order to do calculations in this framework, or do 'Feynman diagrams + added features' do the trick?

I have Weinberg, Srednicki, Peskin and Schroeder right at hand, as well as Zee (first edition), if that matters.