Musings

Small Pond

I hope he doesn’t mind my embarrassing him further, but Matthew Nobes has the first in (what I hope is) a series of posts in which he discusses some of the analytical tools of modern lattice gauge theory. A lot of the improvement in lattice gauge theory has come about not so much by “brute-force” going to finer lattices as from being smarter about our computations. The number of lattice sites scales like $a^{-4}$ so, even tremendous strides in the computer power available lets you cut the lattice spacing by only a relatively modest amount.

Instead, the big improvements have come from being “smarter” about what to compute. By comparing lattice perturbation theory to the continuum, and using the results to fine-tune the lattice action, one can minimize the effects of the discretization and get vastly improved results on the same “coarse” lattice.

1. In comparing lattice perturbation theory to the continuum, we are interested in comparing physical quantities at some distance scale, $L$, long compared to the lattice spacing, but still much shorter than the size of the box. We should take care to use a “renormalization group-improved” lattice perturbation theory, rather than expressing our answers in terms of “bare” coupling(s) in the lattice action. The RG-improved perturbation series is generally much more accurate than the naive one in the bare lattice coupling.
2. Moreover, we can make the RG flow converge faster to the continuum by working, not with the “naive” Wilson action, but with an “improved” lattice action, containing “higher derivative” interactions, with cleverly-chosen coefficients.

Choose the coefficients correctly, and the errors scale to zero much faster as $a/L\to 0$. Matthew’s one of the guys who figures out how to do this in practice.

Recall that we were worried about finite spacing errors in lattice field theory. As an example we were using a scalar field coupled to gluons. The basic action was $$\phi D^{2} \phi$$ and this has $a^2$ errors. I said that we could use $$\phi (D^2 + C a^2 D^4) \phi$$ to reduce these errors. Clearly this involves picking some value for $C$, but how do we do that?

It pays to remember what the lattice is doing for us. It’s cutting the theory off at the small distance $a$, or in momentum space at high energy/momentum. So the spacing errors are reflecting a problem with the high energy (short distance) part of the theory. Now way back at the start of the first post we noted that QCD is perturbative at high energy. So we ought to be able to correct for the spacing errors perturbativly, by matching our lattice theory to the continuum theory to some order in perturbation theory. We pick some scattering amplitude, and fiddle with $C$, order by order. Done properly, this lowers the spacing errors, at a modest performance cost.