Wilsonian RG and Effective Field Theory

Question

I'm having trouble reconciling the discussions of the Wilsonian RG that appear in the texts of Peskin and Schroeder and Zee on the one hand, and those of Schwartz, Srednicki, and Weinberg on the other.

In the former, they seem to say that as one scales down to lower momentum, the couplings with negative mass dimension ("irrelevant couplings") scale to smaller and smaller values as one integrates out more high-momentum modes. Hence, at energy scales much smaller than the initial cutoff, the theory will look like a renormalizable QFT since the irrelevant couplings become small under the RG flow.

In contrast, the books of Schwartz, Srednicki, and Weinberg state that the Wilsonian RG analysis does NOT imply the irrelevant couplings scale to small values as one integrates out high-momentum modes, but merely that they become calculable functions of the relevant and marginal couplings. I.e., they become insensitive to the values of the irrelevant couplings of the initial large-cutoff Lagrangian.

My question is, how do I reconcile these two views?

My first exposure to the subject was Peskin and Schroeder, and I thought it all made perfect sense at the time. Now that I've read the more recent books of Schwartz, et al., I'm wondering if either

1. I've misinterpreted what P&S and Zee are saying when they discuss Wilsonian RG and effective field theories, or

2. they've made some simplifying assumptions that the treatments of Schwartz et. al. don't make.

Regarding the 2nd point, when discussing how the couplings scale under the RG, P&S largely ignore the "dynamical part" that comes from evaluating loop diagrams, in which case the scaling of the couplings boils down to simple dimensional analysis. In this case there's no accounting for operator mixing (i.e., that relevant and marginal couplings can feed into the flow of irrelevant couplings). This seems to be different from Schwartz's treatment, where he keeps information from the beta functions that encode information from the loop diagrams and allow for operator mixing. Could this be the reason why they seem to say different things about the size of irrelevant couplings as you lower the cutoff?

Answer 1

One of the main (but usually not explicitly given) assumption of the perturbative RG is that even in presence of irrelevant couplings, the RG flow starts close to the Gaussian Fixed Point (FP). That way, the negative mass operators flow toward zero, making the Gaussian FP a better and better approximation, until the relevant couplings kick in.

In that case, one ends up with a "renormalizable" theory, and one can just take care of the one or two relevant couplings, thus going back to the old school QFT RG.

However, Wilson does not assume that the irrelevant (with respect to the Gaussian FP) couplings have to be small. In fact, in most stat-phys applicatiosn, all couplings are of the same order! (For instance, in the Ising model, there is only one parameter $K=J/T$, so the corresponding field theory has all couplings of the same order.) But that does not prevent one to do some RG calculation in principle. In fact, in these models, the flow never goes close to the Gaussian FP, and the flow is non-perturbative right from the beginning.

One should however keep in mind that if one is only interested in the critical behavior of the system, thanks to universality close to a (Wilson-Fisher like) FP, one can also study a simpler theory (say a $\phi^4$ QFT) which is enough to describe the fixed point structure (usually). This is what saves the perturbative RG from oblivion.

Answer 2

That's a great question. OP has a point.

1. On one hand, the Wilsonian effective action (WEA) is defined via a 2-step procedure, cf. Refs. 1-4:

   • 1st step: WEA is the generator $W_c[J^H,\phi_L]$ of connected Feynman diagrams of heavy/high modes $\phi_H$ with wavevectors $\Lambda_L\leq |k|\leq \Lambda_H$ in a background of light/low modes $\phi_L$ with wavevectors $ |k|\leq \Lambda_L$ and heavy sources $J^H$, $$\begin{align} \exp&\left\{-\frac{1}{\hbar}W_c[J^H,\phi_L] \right\} \cr ~:=~& \int_{\Lambda_L\leq |k|\leq \Lambda_H} \! {\cal D}\frac{\phi_H}{\sqrt{\hbar}}~\exp\left\{ \frac{1}{\hbar} \left(-S[\phi_L+\phi_H]+J^H\phi_H\right)\right\}.\end{align}\tag{W1}$$ (The heavy sources $J^H$ are mainly introduced so that we can use generating techniques. They are usually put to zero in the end.) Here $\Lambda=\Lambda_L$ is a scale much larger than the experimental scale, and $\Lambda_H$ is a UV cut-off/regularization. (Say, $\Lambda_H\sim 1/a$ on a lattice with lattice constant $a$.)

   A priori the various terms in the WEA are not normalized. A typical kinetic term is of the form $\int\!d^dx\frac{Z_{\phi}}{2}(\partial\phi_L)^2$, while a typical interaction term is of the form $\int\!d^dx\frac{Z_gg_n}{n!}\phi_L^n$, adorned with $Z$-factors.

   • 2nd step: The definition (W1) of WEA implicitly assumes coarse-graining/blocking:

     • We now rescale the integration variables $$\begin{align} k^{\prime}~=~&k/b, \cr x^{\prime}~=~&xb, \cr b~:=~&\Lambda_L/\Lambda_H~<~1,\end{align}\tag{W2}$$ in the action $W_c[J^H,\phi_L]$, so that the light/low modes $\phi_L$ has wavevectors $ |k^{\prime}|\leq \Lambda_H$.

     • We also rescale the light fields $$\phi_L^{\prime}~:=~Z_{\phi}^{1/2}\phi_L/ b^{[\phi]},\tag{W3}$$ so that the kinetic term $$\int\!d^dx\frac{Z_{\phi}}{2}(\partial\phi_L)^2 ~=~\int\!d^dx^{\prime}\frac{1}{2}(\partial^{\prime}\phi^{\prime}_L)^2\tag{W4}$$ is canonically normalized.

   Similarly, a typical interaction term becomes of the form $$\int\!d^dx\frac{Z_gg_n}{n!}\phi_L^n~=~\int\!d^dx^{\prime}\frac{g_n^{\prime}}{n!}\phi_L^{\prime n},\tag{W5}$$ so that the new coupling constant becomes $$ g_n^{\prime}~=~\frac{Z_g}{Z_{\phi}^{n/2}}g_n/b^{[g_n]}.\tag{W6} $$ Here the irrelevant couplings (with $[g_n]<0$) die out in the IR if (and that's a big if) we can neglect the $Z$-factors.

2. On the other hand, the Wilson-Polchinski effective action (WPEA) is defined as $$\exp\left\{ \frac{1}{\hbar} \left(-\frac{1}{2}\phi_L G_L^{-1}\phi_L -W_{\rm int}[J,\phi_L]\right)\right\},\tag{WP1} $$ where $$\begin{align} \exp&\left\{-\frac{1}{\hbar}W_{\rm int}[J,\phi_L] \right\}\cr ~:=~&\int_{\Lambda_L\leq |k|\leq \Lambda_H} \! {\cal D}\frac{\phi_H}{\sqrt{\hbar}}~\cr &\exp\left\{ \frac{1}{\hbar} \left( -\frac{1}{2}\phi_H G_H^{-1}\phi_H -S_{\rm int}[\phi_L\!+\!\phi_H]+J (\phi_L\!+\!\phi_H)\right)\right\},\end{align}\tag{WP2}$$ cf. Refs. 5-7. Here the Greens functions $G_{H/L}$ for high and low modes are multiplied with a smooth regulator/filter that depends on $\Lambda=\Lambda_L$.

   Note that $W_{\rm int}[J,\phi_L]$ does not include the free term $\frac{1}{2}\phi_L G_L^{-1}\phi_L$, only the corresponding counterterm. Here we do not perform the 2nd step, but we might make the couplings dimensionless $$\lambda_n~:=~g_n/\Lambda^{[g_n]}.\tag{WP3} $$ In the IR limit $\Lambda\to 0$, the irrelevant couplings depend on marginal and relevant couplings, while they become independent of the UV cut-off $\Lambda_H$. Deceivingly, eq. (WP3) may naively suggest that the irrelevant couplings $\lambda_n$ die out in the IR, but in practice $\lambda_n$ typically flow to a finite fixed-point value, while it's $g_n=\lambda_n\Lambda^{[g_n]}\to \infty$ that blow up for $\Lambda\to 0$.

References:

1. M.E. Peskin & D.V. Schroeder, An Intro to QFT; section 12.1.

2. A. Zee, QFT in a nutshell, 2010; section VI.8.

3. D. Tong, Statistical Field Theory; chapter 3, p. 55-58.

4. M. Srednicki, QFT, 2007; chapter 29, p. 181-182. Canonical normalization (W4) and rescaling (W3) are discussed on p. 184. A prepublication draft PDF file is available here.

5. M.D. Schwartz, QFT & the standard model, 2014; section 23.6.

6. S. Weinberg, Quantum Theory of Fields, Vol. 1, 1995; section 12.4.

7. J. Polchinski, Renormalization and effective lagrangians, Nucl. Phys. B231 (1984) 269.