The question.
Does there exist a mathematicaly precise, commonly accepted definition of the term "regularization procedure" in perturbative quantum field theory? If so, what is it?
Motivation and background.
As pointed out by user drake in his nice answer to my previous question Regulator-scheme-independence in QFT , it is often said that in a renormalizable quantum field theory, results for physical quantities (such as scattering amplitudes) when written in terms of other physical quantities (like physical masses, physical couplings, etc.) do not depend on the regularization procedure one chooses to use. In fact, user drake takes this desirable property as part of the definition of the term "renormalizable."
In my mind, in order for such a statement or definition to be meaningful and useful, it helps to have a precise, mathematically explicit notion of what constitutes a regularization procedure. That way, when one wants to compute something physical, one can use any procedure one wishes provided it satisfies some general properties.
My current understanding in a (small) nutshell.
When we perturbatively compute, say, correlation functions for some theory pre-regularization/renormalization, we obtain formal power series' in the bare parameters that characterize the theory. Such power series' contain expressions for loop integrals that generically diverge, often due to high-momentum (UV) effects, so we "regularize", namely we implement some procedure by which these integrals are made to depend on some parameter, call it $\Lambda$, and are rendered finite provided $\Lambda$ doesn't take on a certain limiting value $\Lambda_*$ (that could be $\infty$) corresponding to the physical regime (like the UV) that led to the original divergence. We then renormalize and find (in renormalizable theories) that $\Lambda$ drops out of physical results.
What sort of an answer am I looking for?
I am looking for something like this.
A regularization procedure is a prescription by which all divergent integrals encountered in perturbation theory are made to depend on a parameter $\Lambda$ and which satisfies the following properties: (1) All divergent integrals are rendered finite for all but a certain value of $\Lambda$. (2)…
I know that there are other properties, but I don't know what constitutes a sufficiently complete list of such properties such that if you were to show me some procedure, I could say "oh yes, that counts as a valid regularization procedure, good job!" Surely, for example, the putative regularization procedure cannot be too destructive; I cannot, for instance, simply replace every loop integral with $3\Lambda$ and call it a day because that would completely destroy all information about how many loops the corresponding diagrams contained. How much of the "formal structure" of the integrals does the procedure need to preserve?
As far as I can tell, there is no discussion of this in any of the standard QFT texts who simply adopt tried-and-true procedures like a hard UV cutoff, dim reg, Pauli-Villars, etc. without commenting on general conditions sufficient to guarantee that these particular procedures count as "good" ones. There is, of course, a lot of discussion of whether certain regulators preserve certain symmetries, but that's a distinct issue.
Edit. (January 8, 2014)
Discussions with fellow graduate students have led me to believe that the appropriate definition proceeds by appealing to the idea of effective field theory. In particular, if we view our theory as an effective low-energy description of some more complete theory that works at higher energy scales, then imposing a high momentum cutoff has a conceptually privileged position among regulators; it is the natural way of encoding the idea that the putative theory only works below a certain scale.
This can then be used to define a regularization procedure that, in some sense, reproduces the same structure encoded in regularizing with a cutoff. Unfortunately, I'm still not entirely sure if this is the correct way to think about this, and I'm also not sure how to formalize the notion of preserving the structure the comes out of cutoff regularization. My inclination is that the most important structure to preserve is the singular behavior of regularized integrals as the cutoff $\Lambda$ is taken to infinity.
Best Answer
The definite answer to your question is: There is no mathematicaly precise, commonly accepted definition of the term "regularization procedure" in perturbative quantum field theory.
Instead, there are various regularization schemes with their advantages and disatvantages.
Maybe you'll find Chapter B5: Divergences and renormalization of my theoretical physics FAQ at http://arnold-neumaier.at/physfaq/physics-faq.html illuminating. There I try to abstract the common features and explain in general terms what is needed to make renormalization work. The general belief is that the details of the regularization scheme don't matter, though in fact it is known that sometimes some regularization schemes give apparently incorrect results.
This is to be expected since the unregularized theory is ill-defined, and can be made well-defined in different ways, just as a divergent infinite series can be given infinitely many different meanings depending how you group the terms to sum them up.
If at any time in the future there will be a positive answer to your question, it will be most likely only when someone found a logically sound nonperturbative definition of the class of renormalizable quantum field theories.
On the other hand, if you want to have a mathematically rigorous treatment of some particular regularization schemes for some particular theories, you should read the books by (i) Salmhofer, Renormalization: an introduction, Springer 1999, and (ii) Scharf, Finite quantum electrodynamics: the causal approach, Springer 1995.