Monday, October 25, 2004

I'm still getting used to doing this, and I think I prefer the blogger interface/experiance to livejournal. It's still not a mathML enabled blog, but I really don't have the time to set up a proper server, etc, yet. So for now, I'm sticking to this.

In the last post I talked about determining the strong coupling constant from lattice QCD. You'll recall that I talked about having to use a special coupling, determined from the perturbative static quark potential. I also mentioned that the perturbative scale for the average plaquette was not $\pi/a$, but was instead some scale $q^{*}$. The next four posts will explain those cryptic comments. They will basically be a summery of a famous 1992 paper by Lepage and Mackenzie, in which they saved lattice perturbation theory.

We'll start by introducing the problem.

The procedure I outlined, for determining the strong coupling constant, could also be run in reverse. That is, I'll start out knowing the coupling constant at $M_{Z}$ run it down to the lattice scale $\pi/a$ and convert it to the bare lattice coupling. Then I use that to compute the second order value of the average plaquette. I get some number from this, X. Now I fire up my Monte-Carlo simulation, and I compute the average plaquette that way, and I get some number Y. The trouble is X and Y are not equal. Even if I allow for an error coming from third order perturbation theory, it's way, way off.

This is a disaster. It means that peturbation theory on the lattice is suspect. You'll recall that the coefficients in improved actions were to be determined perturbativly, and using improved actions was crucial in error reduction. If lattice perturbation theory fails, we can no longer do this, and we'll be forced to wait many years until computers get faster. We need to figure out what is wrong with lattice perturbation theory, and, if possible, fix it.

The next post will talk about problem #1, a bad choice of coupling constant.