# Understanding irrelevant operators in Wilsonian RG

I had always understood irrelevant operators as operators whose coefficients got smaller at lower energy scales, but there's a passage from Schwartz's Quantum Field Theory and the Standard Model which confuses me. In his discussion of the Wilsonian renormalization group equations (section 23.6), he appears to suggest that the important feature of irrelevant operators is not that they get small, but rather the fact that the low energy physics is insensitive to the UV values of them.

Values of couplings when the cutoff is low are insensitive to the boundary conditions associated with irrelevant operators when the cutoff is high.

Specifically, he uses this in an example with QED

To relate all this rather abstract manipulation to physics, recall the calculation of the electron magnetic moment from Chapter 17. We found that the moment was $$g=2$$ at tree-level and $$g = 2 + \frac{\alpha}{\pi}$$ at 1-loop. If we had added to the QED Lagrangian an operator of the form $$\mathcal{O}_\sigma = \frac{e}{4} \bar{\psi} \sigma^{\mu \nu} \psi F_{\mu \nu}$$ with some coefficient $$C_\sigma$$, this would have given $$g = 2 + \frac{\alpha}{\pi} + C_\sigma$$. Since the measured value of $$g$$ is in excellent agreement with the calculation ignoring $$C_\sigma$$, we need an explanation of why $$\mathcal{O}_\sigma$$ should be absent or have a small coeffiicient. [...] Say we do add $$\mathcal{O}_\sigma$$ to the QED Lagrangian with even a very large coefficient, but with the cutoff set to some very high scale $$\Lambda_H \sim M_{PI} \sim 10^{19}$$ GeV. Then, when the cutoff is lowered,, even a little bit, whatever you set your coefficient to at $$M_{PI}$$ would be totally irrelevant: the coefficient of $$\mathcal{O}_\sigma$$ would now be determined completely in terms of $$\alpha$$. Hence $$g$$ becomes a calculable function of $$\alpha$$.

I do not see the equivalence of

$$g$$ becomes a calculable function of $$\alpha$$

with

the measured value of $$g$$ is in excellent agreement with the calculation ignoring $$C_\sigma$$

Given Schwartz's explanation of Wilsonian RG, it seems the most we could conclude is $$g = 2 + \frac{\alpha}{\pi} + C_\sigma(\alpha)$$, where $$C_\sigma(\alpha)$$ is determined by the RGE's. This would seem analogous to the example he works through explicitly, where the coefficient of the irrelevant operator does not become small, but rather becomes a function of only the relevant coefficients at low energy. More generally, I do not see how the notion of irrelevant operators in this framework justifies disregarding them at low energies.

• In physics.stackexchange.com/q/372306 I gave a long answer (from scratch) to a few things regarding renormalization and in particular that it amounts to parametrizing the low energy irrelevant couplings in terms of the low energy relevant couplings. Have a look. Feb 18, 2019 at 17:53
• Mar 1, 2022 at 21:46

As to the meat of your question, I think he's saying the RG tells us at some $$\Lambda << M_{PL}$$ (but still much larger than the energy scales we're interested in) we have, $$C_{\sigma}(\Lambda) \sim \frac{\bar{c}(\alpha(\Lambda))}{\Lambda}$$,
So therefore the anomalous magnetic moment would be $$\frac{\alpha}{2\pi}\biggl(\frac{1}{2m_e} + \frac{\bar{c}}{2\Lambda}\biggr)$$ and the excellent agreement with experiment with just the first term therefore puts a bound on the $$\Lambda$$. Though I guess one has to assume $$\bar{c}$$ is of natural size (i.e., $$\bar{c}\sim\mathcal{O}(1)$$)?