One of the most interesting calculations in Lattice QCD is the determination of the strong coupling constant. This calculation also nicely demonstrates the need for perturbation theory in lattice QCD in a context outside of improving actions, which I talked about previous postings.

Okay, so what do we want to do? Well, we want to determine the strong coupling constant \alpha renormalized in the MS-bar scheme and at the scale set by the Z mass (M_Z ~ 90 GeV). So we fire up our computers, and get to work. The first thing we want to do is a non-perturbative simulation of QCD, using lattice Monte-Carlo methods. In order to do this, we need to tune 5 input parameters in our simulation.

1) The light quark mass (we take the up and down quarks to have the same mass

2) The strange quark mass

3) The charm quark mass

4) The bottom quark mass

5) The lattice spacing

To do this we proceed in exactly the same way as any other field theory. We pick 5 measurements, and tune our inputs (the “bare” quantities) until the 5 results we get agree with the 5 measurements. In our case, we’ll use the pion mass to fix the light quark mass and the K mass for the strange. For the other three we use some combination of spin splittings in heavy quark bound states. We don’t use meson masses, as they are more sensitive to lattice errors. The important point here is that we have 5 measured quantities, and we tune the inputs to agree with them. After that, no other input from experiment is needed. That is, this is a first principles QCD calculation.

With our parameters tuned, we run the simulation, go away for some period of time (a long time if we are using dynamical fermions), and finally get our result. The result of a Monte-Carlo simulation is a set of gauge fields {A} which we can measure quantities. There are lots of quantities we can measure, for example, the self energy of a static quark, sitting at the spatial origin boils down to computing

U(x=0,t=0)U(0,1)U(0,2)...U(0,L)

where

U = exp(iA)

and L is the length of the lattice. Obviously if we had a bunch of different sets of gauge fields we could average over them. It’s this average which would be the result we want. The more sets of gauge fields we have the more accurate our results.

Now the static quark self energy is interesting, but it is not quite what we want to measure for our purposes. What we want to do is measure some short distance quantity. Why? Well, recall that QCD is perturbative at short distances, so if we measure a short distance quantity *and* have a perturbative expansion for it, we can solve for the coupling. At one loop it looks like this: we measure the average <Om> on the lattice, then we compute in perturbation theory <O> = 1 + \alpha O1 + ... Set them equal, and solve

\alpha = [<Om> - 1 ] / O1

We have now determined the strong coupling on the lattice!

In practice, the short distance quantity of choice has been the average plaquette, which is the product of gauge fields on a 1x1 square on the lattice. Often one also uses large squares and rectangles in order to get a few different determinations, as a check. As well, one loop is not enough, you need at least second order calculations. My supervisor Howard Trottier, and a colleague Quentin Mason, have calculated these short distance quantities out to three loops, which greatly improves the accuracy.

We now have in hand a value for the strong coupling. The value we have depends on what we used in our perturbative expansion. Normally, one uses a definition based on the perturbative expression for the static quark potential (this will be the subject of another posting). The coupling has been evaluated at a scale that is close to the lattice cutoff pi/a but not quite. We’ll call that scale q* (more about this in a later posting as well, for now, think of q* as being within 10% of pi/a). So what we have extracted is

\alpha_{V}(q*)

and what we want is \alpha_{MS-bar}(M_Z).

We go from point a to point b in two steps. First we convert

\alpha_{V}(q*) to \alpha_{MS-bar}(q*), then we run that number up to M_Z using the known three loop running. The latter step is well known (and not lattice specific) so I’ll just say how to do the former.

To convert from \alpha_{V} to \alpha_{MS-bar} means computing some quantity perturbativly in both schemes, and matching them. In practice, the quantity that is used is the two point function for background field gluons. This quantity is used because the combination of the background field and the coupling is not renormalized. So you can compute this in the MS-bar scheme and with a lattice regulator, and equate the two, solving for \alpha_{V} as a series in \alpha_{MS-bar}.

And that’s how it’s done! The results are pretty good, using a modern, unquenched simulation, a preliminary number is

\alpha_{MS-bar}(M_Z) = 0.1181(15)

which agrees with, and has lower errors than, the PDG average.