Relationship between the on-shell and BPHZ renormalization schemes

Question

In his book Quantum Field Theory - A Tourist Guide for Mathematicians, Gerald Folland introduces the on-shell renormalization scheme for the $ \phi^{4} $-scalar field theory. According to my understanding, this is a scheme by which counter-terms are determined, one order at a time, by fixing the pole and residue of the sum $ \Pi(p) $ of all non-trivial $ 1\text{PI} $-diagram insertions into a propagator with external momentum $ p $.

I know how to make calculations with this scheme up to second loop order, so naïvely, one would think that this process can be continued, without difficulty, to higher orders. However, I also know that this kind of thinking is wrong (because this is supposed to be a non-trivial problem!), but I do not see how it can be wrong. I am aware of the problems posed by sub-divergences, but given the algorithmic nature of the on-shell renormalization scheme, I do not see why calculating higher-order corrections might interfere with the calculations for lower-order ones.

The BPHZ renormalization scheme gives a rigorous proof of renormalization to all orders, and it is rather elegant in the sense that it tells us how to renormalize individual Feynman integrals. Hence, I am trying to see how the on-shell scheme can be shown to be equivalent to the BPHZ procedure.

Thank you very much!

This post imported from StackExchange Physics at 2014-11-21 06:30 (UTC), posted by SE-user Transcendental