Conventional renormalization to order in the coupling or loops?

[center o bass]

In conventional renormalization one is first supposed to compute a scattering amplitude of interest in terms of bare quantities $\lambda_0, m_0...$, then compute these bare quantities in terms of physical quantities, i.e. $m(m_0,\lambda_0,...), \lambda(m_0, \lambda_0,...)$ and substitute these back into the scattering amplitude. The amplitude should now be finite.

That is all nice, but suppose one computes the scattering amplitude to $O(\lambda_0^2 )$ and to first order in loops. To be consistent, should one now compute the mass and coupling to second order in the coupling or first order in loop corrections?

And why?

 

[DarMM]

At each order in perturbation theory you make the bare/physical replacement to the order previously computed. These should leave diveregences at the current order that can be absorbed by invoking the renormalization conditions.

It works because of the graph theoretic structure of Feynman integrals. Look for explanations of Zimmerman's Forest formula.

Part 1 of these lecture notes gives an nice intro and some references: https://arxiv.org/abs/1004.3462