Renormalization scheme dependence and robustness

Question

If a theory is perturbatively renormalizable (using old fashioned renormalization methods, not RG methods), will using different renormalization schemes give us the same answers? That is, will using different renormalization schemes give us the same empirically meaningful parameters?

Answer 1

The answer is either yes, or else physicists have essentially pulled the greatest heist of the century. The key to why the particular scheme you choose doesn't matter really comes down to the idea of a renormalization point, that is, the point in momentum space at which we fix the values of the correlators and compute the necessary counter terms. It's also interesting to note that, while RG methods come much later, the ideas are really inexorably tied to how renormalization works, so the only sense in which you can do renormalization without the renormalization group is if you put blind folds on and go out of your way to ignore the fact that you have some freedom in how you go about setting up your renormalization procedure.

Edit: Let me expound upon this last point some more. In any renormalization scheme you like, there are always some choices you need to make. In dimensional regularization you need to choose a mass scale $\mu$, in Pauli-Villars you need to choose a mass $M$, in cut-off you need to choose the actual cut-off $\Lambda$, the list goes on. There's always an ambiguity in how the actual procedure goes. In a very real sense, the renormalization group is the expression of the ambiguity. We derive the renormalization group equations, in fact, by explicitly demanding that physical quantities (correlators) should be independent of this ambiguity. So sure, you can fix this ambiguity by explicitly choosing your mass scale, mass or cutoff length, but that doesn't mean the ambiguity isn't there. That would be like saying that, because you've elected to work in Coulomb gauge in electrodynamics, gauge symmetry isn't there. It is there, you've just chosen to blind yourself to it.

Now, not all renormalization schemes make the selection of the renormalization point our choice, and not all of them make it clear what point is chosen...essentially, by losing explicit control of the renormalization point we gain computational simplicity and power.

Physicists are, of course, completely aware of these facts and if you go to the Particle Data Group website (which has many very helpful notes and all up-to-date data in particle physics), essentially all measurements are listed together with the method (renormalization scheme) used to compute them. This information tends to get left out of tables that appear on other sites, like Wikipedia (though not all pages ignore this information).

Edit: By the way, there's a nice discussion of some of these things in Nair's QFT book in detail, though that book tends to lean more heavily upon mathematics than some might like during a first pass at QFT. There is also a QFT book by Banks which contains essentially no detail (it's all left as exercises to the reader), but which does give a fairly nice discussion of a number of things (if I remember correctly). So it could be good if you're just looking to get at the big ideas of what's going on.

Edit 2: As Andrew rightly points out below, there is often ambiguity in the calculations we make, and the key is whether or not keeping said ambiguity actually buys us anything or not. So the counter point to the argument I have sort of been pushing for in the above would be: sure, that ambiguity is there, but do we actually gain anything by not simply fixing the ambiguity immediately as one might fix a gauge.

After all, one can play the game of electrodynamics without ever touching a vector potential (which suffers gauge ambiguity). You would very quickly run into problems which are exceptionally difficult to handle working only with the electric and magnetic fields, but which are quite solvable if you allow yourself to work with a vector potential. This is essentially the attitude that is taken in most first introductions to electrodynamics.

So what we see is that it can sometimes be advantageous in one way or another to work with objects that suffer from some ambiguity. The key fact we need to keep in mind for physics to ground ourselves is which objects are have no ambiguity in them (or at least, what kind of ambiguity we have where...all dimensionful quantities in physics are immediately ambiguous up to an overall scale -- the unit we choose, yet this is so familiar it causes us no concern). In quantum field theory, this if often the correlators since these are the objects which we can use to construct cross-sections, which are things we can then go out into the world and measure and hence which better have no ambiguity to them (well, units again).