# Regularization scheme independence in QFT

I know there are a few similar questions on the topic (1,2) , but I still feel they do not fully answer my questions (correct me if I am wrong!). What I am asking is a clarification on the commonly assumed fact that all physical observables - once expressed in terms of other observables - must not depend on the regularization procedure used. As pointed out in (2), it is difficult to define what a regularization procedure is in general, so I am wondering whether any of these apply:

• Results have been proven to coincide for the different schemes we know (dimensional regularization, Pauli-Villars, cutoff..etc)

• Calculations in different schemes give relations between different physical observables, so it's not always possible to compare them

The only example of regulator independence I have come across is in Matthew Schwartz's "Quantum field theory and the Standard Model", where he shows that a very reasonable class of regulators gives the same results in the calculation of the Casimir force.

In my opinion, sadly the best book on renormalisation is hardly known. It can be shown that a change of renormalisation prescription - reflected in a change of variables of the parameters, even by a function of the regulator - will leave us with the same theory, which you can read in Collins'.

Moreover, we are also free to arbitrarily change the scale of fields, e.g. $\phi \to \zeta\phi$. This is because, as elucidated by the LSZ formula, in stripping off factors for the external lines we eliminate any factors of $\zeta$ and thus the S-matrix is left invariant.

In section 7.1.2, it is proven that it holds,

$$G_N(p_1,\dots,p_N;g,m,\mu) = \zeta^{-N}G_N(p_1,\dots,p_N;g',m',\mu').$$

Doing this order by order in perturbation is comparatively easy: you simply compute and see that the above holds. To generalise to all orders, the key is Zimmermann approach's to renormalisation which provides a systematic means to handle any combination of subdivergences of a graph and in theory can be applied for any regulator.

In a nutshell: the S-matrix is invariant under rescalings of the fields, which is the only result of a change of renormalisation prescription. The proof of this uses an approach to renormalisation that is regulator independent.$^\dagger$

$\dagger$ While the physical predictions are independent of the regulator, like decay widths or cross sections, not everything is. In particular, the failure of commutativity of taking the trace of an operator and renormalising it is due to a lack of scale invariance which crops up with dimensional regularisation, but will not with BPHZ renormalisation.