UV divergences in perturbation theory

Question

I've seen people claim that UV divergences appear in perturbation theory are due to taking the continuum limit "too soon" or "sloppily". What does this mean?

Answer 1

Let me take the point of view of Wilsonian renormalization. Then we do not allow momenta larger than some cutoff $\Lambda$ in the theory, and in particular momenta larger than $\Lambda$ do not appear in loop diagrams. The Wilson coefficients (the coefficients of operators in the Lagrangian) depend on $\Lambda$, in such a way that physical observables do not depend on the cutoff. After fixing the Wilson coefficients, we can take the continuum limit $\Lambda\rightarrow\infty$. Since physical quantities do not depend on the cutoff by construction, there is no problem with this limit. (Well, I'm simplifying, since there are subtle issues like Landau poles that can arise, but no low energy observables will depend on the limit).

If we do not introduce a cutoff before cutting loop diagrams, then divergent loop integrals will be infinite, instead of cutoff-dependent. Then we lose control of the calculation, and it is not clear how to fix the Wilson coefficients to cancel the divergences. This is as opposed to cancelling cutoff dependence as in the previous paragraph, which is calculable.