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.
Best Answer
Excellent question. I've long had this same confusion about the meaning of irrelevant couplings in the Wilsonian RG. After reading Schwartz and similar treatments in Weinberg, Srednicki, and the 1984 paper by Polchinski on effective Lagrangians, I think the point is that irrelevant couplings are driven to zero in the IR only to zeroth order in the marginal/relevant couplings-- i.e., what you see in a linearized treatment of the RG flow about a fixed point.
For example, imagine you start the RG flow at large cutoff at a point slightly off the critical surface. (It helps if you've seen pictures of RG flows as found in typical condensed matter treatments of Wilsonian RG like in Altand and Simons or the book of Cardy.) As you flow to lower scales, the irrelevant couplings decay as the trajectory passes close to the fixed point. However, the relevant couplings eventually push you away from the fixed point, and feed back into the flow of the irrelevant couplings to drive them to their final-- not necessarily small-- values in the IR focused along the so-called renormalized trajectory emanating from the fixed point in the relevant direction. Any RG trajectory with initial conditions tuned to pass near the fixed point loses its memory of the initial values of the irrelevant couplings it started with, as all such trajectories get focused along the renormalized trajectory as you flow to the IR. (Now that I think about it, I wonder, is this "focusing effect" of different trajectories with very different starting points as you flow to the IR why you can't run the Wilsonian RG equations backward, kind of like how you can't integrate the heat equation backwards in time?)
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., $\bar{c}\sim\mathcal{O}(1)$)?