Wilsonian effective field theories and renormalization due to marginal and irrelevant operators

Question

In the top-down approach, e.g., Fermi interaction obtained from EW Lagrangian, the loop corrections (using dimensional regularization) and renormalization of $G_F$ are done using the full EW Lagrangian. What happens when we don't have a UV complete theory (which we will never have anyway, in Wilsonian picture, since all QFTs are effective) especially in a bottom-up approach? How do we renormalize the theory with contribution from the irrelevant operators?

In particular, I seem to completely miss the point of relevant and marginal operators. Even the irrelevant operators are confusing to me. If I understand it correctly, all results are to be interpreted as the cutoff $\Lambda\rightarrow\infty$ limit. In the Fermi model, there should be a renormalization of fermion mass running with the dim. reg. parameter $\mu$ coming from the $\frac{G_F}{\Lambda^2}(\bar\psi\psi)^2$ proportional to $\frac{m^3}{\Lambda^2}\log{(\frac{m^2}{\mu^2})}$. This seems to vanish in the limit. There should also be a renormalization of $G_F$ proportional to $\frac{1}{\Lambda^4}$. Which also vanishes.

Should we only consider loop diagrams coming from relevant and marginal operators leaving out the irrelevant operators for tree diagrams only? How can SM, QED etc. be effective, since it has no cutoff scale $\Lambda$ attached to it? If all theories are only effective, shouldn't they only have irrelevant operators since contributions from relevant and marginal operator parts will be there no matter how large $\Lambda$ is?

Answer 1

One of the main confusions of the OP is about the status of operators in renormalizable and non-renomalizable theories when viewed as effective field theories. I will only address this point there.

As noted by the OP, the modern view of QFT is that all theories are effective (valid up to some physical high-energy scale $\Lambda_{eff}$), and whether a theory is (perturbatively) renormalizable or not is not a good criterion to decide which theory is good or bad. Of course, a theory is said to be perturbatively renormalizable if one can absorb in a finite number of couplings order by order in perturbation theory the infinities that appear in the calculations.

To simplify the discussion, I will assume that the theory, renormalizable or not, is regularized by a momentum cut-off $\Lambda$, which is here just thought of as a mathematical device to make the integrals finite. It has a priori nothing to do with the energy scale $\Lambda_{eff}$. This is probably the origin of the OP's confusion.

How do we reconcile the two visions (EFT vs renormalizability)?

Renormalizability, i.e. being allowed to send $\Lambda\to\infty$, does not imply that $\Lambda$ has to be infinite. Maybe (most probably in HEP) your renormalizable theory is valid only up to some $\Lambda_{eff}$. Saying that your renormalizable theory is an EFT means that in principle you could add irrelevant terms (aka non-renormalizable terms) to your theory to describe high-energy processes. But if it is renormalizable, then this means that as long as you are probing energies must smaller than $\Lambda_{eff}$, you could as well send it to infinity while tuning only a few renormalizable couplings (aka relevant and marginal).

Then $\Lambda_{eff}$ and $\Lambda$ indeed play here a similar role: knowing that the theory is valid up to a scale $\Lambda_{eff}$, we cut all integrals with a UV-cutoff $\Lambda\equiv \Lambda_{eff}$. As long as we are probing low energies, we can forget about all irrelevant/non-renormalizable couplings, and do our calculations with the renormalizable ones. Because the theory is renormalizable, we see a pattern in the calculation: all low-energy observables depend on $\Lambda\equiv \Lambda_{eff}$ in a way that allows us to "send $\Lambda\to\infty$''.

Note that here, "send $\Lambda\to\infty$'' does not really mean anything, as we have a physical interpretation of $\Lambda$ (since it is our physical scale $\Lambda_{eff}$). Instead, we should think in terms of a ratio: that of the energy scale that we probe, $E$, and the cut-off. Indeed, the correct limit is $E/\Lambda\to 0$, which can be thought of as $\Lambda\to\infty$ when the UV cut-off is a mathematical device (as people did before EFT). But in an EFT, $\Lambda\equiv \Lambda_{eff}$ is fixed, and it is instead $E$ that is send to $0$ (compared to $\Lambda_{eff}$), i.e. we probe low-energies (compared to $\Lambda_{eff}$).

Answer 2

OP asks many questions. In this answer we will focus on the statements in OP's linked references.

1. Let us consider a Wilsonian effective action (WEA) which is the generator $W[\phi_L]$ 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$. Here $\Lambda=\Lambda_L$ is a scale and $\Lambda_H$ is a UV cut-off/regularization. Let us assume that there is only a finite number of field species, and that the spacetime dimension $d>2$.

2. A typical interaction term in the WEA could in principle be a monomial in several species of fields with possible spacetime derivatives. Let us use an oversimplified notation $$ \int\!d^dx \frac{g_n}{n!}\phi_L^n~=~\int\!d^dx \frac{\lambda_n \Lambda^{[g_n]}}{n!}\phi_L^n\tag{A}$$ to represent a typical term, i.e. different species and derivatives are implicitly implied in the notation. Here we have introduced a dimensionless coupling constant $\lambda_n$ $$ g_n~=~ \lambda_n \Lambda^{[g_n]}, \qquad [\lambda_n]~=~0.\tag{B}$$

3. A coupling constant $g_n$ is relevant/marginal/irrelevant depending on whether the mass dimension $[g_n]$ is positive/zero/negative, respectively.

4. An irrelevant coupling constant $g_n$ lead to a non-renormalizable theory in the old Dyson sense, i.e. infinitely many superficially (UV) divergent Feynman diagrams with superficial degree of divergence (SDOD) $\geq 0$, cf. e.g. this Phys.SE post. It is important to realize that the UV-divergences are caused by sending the UV cutoff $\Lambda_H\to \infty$. However, Refs. 1-2 argue that it still makes sense as an effective field theory.

5. Note that there are only finitely many relevant/marginal coupling constants. We will for simplicity assume that all the dimensionless coupling constants $|\lambda_n|\ll 1$ are ripe for perturbation theory.

6. Let us assume that all relevant/marginal 2-vertices belong to the free (rather than the interaction) part of the action, i.e. contributes to the free propagator. One may then show that there is at most a finite number of superficially (UV) divergent Feynman diagrams contributing to the term $(A)$ at each $\hbar$/loop-order. In other words, at a certain order of mass-dimension and loops only finitely many counterterms are needed.

7. The leading Feynman diagram for the coupling constant $g_n$ has as few vertices as possible due to the perturbative ansatz. Typically the leading Feynman diagram has SDOD $=[g_n]$, i.e. an irrelevant coupling constant $g_n$ may depend on relevant/marginal couplings but is independent of the UV cut-off $\Lambda_H$.

8. We will now use the dimensionfull (dimensionless) coupling constants $g_n$ ($\lambda_n$) for the relevant (irrelevant) couplings, respectively. The WEA becomes a Taylor series $$W[\phi_L]~=~\sum_{m=0}^{\infty}\Lambda^{-m/2} W_{-m/2}[\phi_L]$$ in the explicit $1/\sqrt{\Lambda}$ dependence from the irrelevant interactions. Each term $W_{-m/2}$ only contains a finite number of monomial terms $(A)$. Refs. 1-2 argue that each term $(A)$ receive small contributions from (possible UV-divergent) diagrams when integrating over heavy modes, suppressed by the scale $\Lambda$.

References:

1. David B. Kaplan, Effective Field Theories, arXiv:nucl-th/9506035; p. 17.

2. Sourendu Gupta, Wilsonian Renormalization and Effective Field Theories, lecture 2; p. 15-18.

3. A. Falkowski, Saclay Lectures on EFT, 2017.