This is a repost from Matthew's old blog.

Well, the whole “work” thing is not really going well today, so I guess I’ll continue my little story.

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.

That’s where I come it. The trouble is doing perturbation theory on a lattice is rather hard. The vertex rules in lattice gauge theory are miserably complicated, so even *deriving* them is hard. Then you have to compute something, which is hard because the lattice cutoff violates Poincare symmetry, so you can’t use all the textbook tricks.

Our group is developing tools to do these calculations to one, two and (for a special operator) three loop order. It’s a daunting task, my rule of thumb is that lattice perturbation theory is “one-loop” more complicated than continuum. That is a two loop lattice calculation is roughly as much effort as a three loop continuum one. One reason for this is that the toolkit for continuum perturbation theory has a lot more tools in it.

These calculations are absolutely crucial for getting high precision results from lattice simulations. The reason is the coupling constant is around 0.2 at current lattice spacings. So a one loop calculation corrects a 20% error, and a two loop calculation corrects a 4% error. If you want to produce 5% accurate results, you must have two loop perturbation theory.

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