Relationship between the on-shell and BPHZ renormalization schemes 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!
 A: I believe that this may be what you’re looking for. I don’t claim to be an expert in renormalization, so all members of the Physics Stack Exchange Community are welcome to correct my answer.
To make the relationship between the two more precise, let’s call the on-shell renormalization scheme the ‘on-shell subtraction scheme’ and the BPHZ renormalization scheme the ‘BPHZ algorithm’.
The on-shell subtraction scheme tells us how to subtract the divergent parts of a Feynman diagram, while the BPHZ algorithm tells us how to apply the subtractions to a multi-loop Feynman diagram in a sensible manner so that the problem of overlapping divergences is overcome.
Hence, the relationship between the two concepts is precisely this:

One uses the on-shell subtraction scheme in conjunction with the BPHZ algorithm in order to cure the divergences of the $ \phi^{4} $-theory to all orders of perturbation theory.

Without a subtraction scheme that yields a subtraction operator, the BPHZ algorithm is useless, because we have to tell the algorithm how to subtract!
Without the BPHZ algorithm, the on-shell subtraction scheme is very limited in scope, because it doesn’t tell us how to apply subtractions properly to address the technical difficulties associated with overlapping divergences.
The BPHZ algorithm also works with other subtraction schemes, like the minimal subtraction scheme. It can also be used with the momentum subtraction scheme, of which the on-shell subtraction scheme is just a single instance.
Now, Anthony Duncan explains what I’ve just mentioned in awesome detail in his book The Conceptual Framework of Quantum Field Theory. You may also consult John Collins’ book Renormalization.
