The beta function for dimensional reguarlization in MS scheme looks like: β(ε,λ)=-ελ+f(λ) But does this mean that the total derivative of β with respect to ε is nonzero, namely equal to -λ ? λ should be invariant to total changes in ε? But doesn't β also have to be invariant with respect to changes in either λ0 or ε? If you were using a cutoff scheme, then I think β has to be independent of the cutoff (or I guess what happens is the cutoff changes, but the bare couplings change, but the renormalized coupling should not be affected by any of that). The analogy should be the same, except now ε is your bare cutoff and λ0= λ0(ε) should adjust as you change ε so that β doesn't depend on ε at all.