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?