# What exactly is regularization in QFT?

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.

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Just as a quick remark, it is not true that all quantities are independent on the regularization scheme, only universal quantities are. See for instance my answer (and the reference given at the end, which might also answer part of your questions) : physics.stackexchange.com/q/73403 –  Adam Jan 5 at 23:11
@Adam I should add that if you think you'd be able to add anything specific beyond your answer to that question, even if it doesn't constitute what you might consider a complete and/or exhaustive answer, then I'd encourage you to contribute an answer (and would appreciate any such insight). –  joshphysics Jan 7 at 7:42
I think you're barking up the right tree by focusing on effective field theory. But my inner mathematical pedant wonders: what about dimensional regularization? –  user1504 Jan 10 at 15:41
@drake: Not all QFT need to be Minkovskian. QFT is not all about HEP. And some regularizations are perfectly physical, such as lattice regularizations, in the case of a model defined on the lattice. And this does not prevent to recover a Lorentz (euclian) invariance at long-distance. –  Adam Feb 4 at 21:41

I'm going to give a silly answer, but I think this is the best we can do. A regulator is any prescription for defining the path integral such that after adding a sum of local counterterms to the action and allowing the physical couplings to depend on the renormalization scale $\mu$, the correlation functions are equal to those obtained by taking a continuum limit of a lattice theory. This is similar in motivation to your most recent edit, except that we really need to use a lattice instead of a Euclidean momentum space cutoff, because the latter breaks gauge invariance.

I'm pessimistic about the existence of a better answer, just because some regulators satisfy some nice properties (i.e. unitarity, symmetries, etc.), while others don't.

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+1: I increasingly feel that this is basically right, and I'm similarly pessimistic. –  joshphysics Feb 5 at 3:41

Regularization is a rewriting of your integral so that you can deal its divergences using other tricks.

For example in QFT you calculate some amplitude to a certain order in perturbation theory. The integrals that represent loop diagrams diverge. The most common regularization procedure is called dimensional regularization where you parametrize the dimension of your loop integral to, for example, d=4-c.

It turns out that your integral was divergent for large momentum, therefore having a branch cut. Now after dimensional regularization the branch cut divergence became a simple pole when c=0. As you know, it is easier to deal with a simple pole than a branch cut.

After making your integral more manageable you can renormalize and then separate the divergent and finite parts of your result. You eventually use other tricks to remove the divergent part, for example add counterterms to your lagrangian.

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I appreciate the response, but I'd ask you to read the question, the other answer, and comments more carefully. I unfortunately am aware of what regularization is at the level you suggest; I'm looking for something more mathematically precise and descriptive. –  joshphysics Feb 4 at 21:27