Thursday, March 17, 2005

The Strong Coupling Constant

First off, before I get to the substance of the post, let me welcome Mark Trodden to the physics blog world. He's part of the even more elite "Central New York physics" blog world.

This post is going to be an outline of how one computes a value for the strong coupling constant ($\alpha_{s}$) using lattice QCD. Now the strong coupling constant isn't really constant at all. It depends on two things, the regulation scheme you use, and the energy you determine it at. The Particle Data Group always quotes the values in the MSbar scheme (a particular type of dimensional regulation, don't worry if you don't know what it is) at the energy scale $\mu=M_{Z}=91$ GeV. So that's our goal, to get a number for $\alpha$ from lattice QCD. The world average, which includes an older lattice calculation, is $\alpha_{s}(M_{Z}) = 0.1187(20)$, where the number in brackets is the error on the last two digits.

How do we get this number from lattice QCD? Well, let's think about the inputs to a lattice QCD simulation, they are 5 quark masses, and the lattice spacing (the finite volume doesn't matter in this calculation). Now, in the language of quantum field theory, these are bare parameters, quantum effects will renormalize them, so the actual physical values that the simulation predicts will be different. For example, you might put 75 MeV in for the strange quark mass, but if you turn the crank, and then look at a physical quantity which tells you m_{s} you might find that it's 72.3 GeV. The same is true of the lattice spacing $a$. The spacing you put in, say $a=0.09$ fm ($1$ fm = $10^{-15}$m), will get renormalized, we'll call the renormalized spacing $a'$.

Unlike quark masses, which are very hard to extract, it's pretty easy to extract the renormalized spacing. What you do is compute some mass differences of heavy mesons. The ideal case, that our group uses, is the bound state of a b and an anti-b quark, called an Upsilon. Like the hydrogen atom (or better, positronium) the Upsilon has a ground state, and a whole spectrum of excited states. In addition, because the b quark is so heavy, the dynamics of the system is basically non-relativistic. The latter fact means that we can compute masses of all the excited states fairly cheaply. Even better, we can compute the mass differences between the excited states. This is better because many systematic errors cancel in the differences.

The general procedure then is to start with gauge configurations generated by the MILC collaboration. These are configurations of gauge fields which have the non-perturbative effects of the light quarks "folded in" (or in lattice QCD jargon, they're unquenched). With these configurations we can determine the mass splittings in the Upsilon system, using our non-relativistic quark formalism. Of course, we don't get the actual mass differences out, we're working on a computer, which will only give us dimensionless numbers, so what we get is DM * a', that is the mass difference, multiplied by the renormalized lattice spacing.

Hopefully, you can see where this is going, since we know DM * a', and Upsilon mesons are things you can actually make in the lab, we can use our calculation, and a measurement of DM, to extract a'. We're really fortunate, because several groups (including CLEO here at Cornell) have measured DM very precisely. With that, we can get very precise values for a'.

Now what does this have to do with the strong coupling constant? Remember, I said that the coupling constant depends on the scale at which we measure it, well a' sets the scale for everything on the lattice. And now we know the scale.

The next step is to actually get a value for \alpha, which takes some doing. The first thing you do is take your non-perturbative gauge field configuurations and compute something that you expect to be perturbative. A popular choice is the average value of the gauge fields around 1x1 square on the lattice (which we'll call P). This is a very short distance thing, around 1 fm per side. Now remember, in QCD if something is short distance, we ought to be be able to compute in in perturbation theory. So we fire up our perturbation theory codes and compute P.

My part in this calculation was computing P (and some other quantities) to second order, that is computing P1 and P2 in the expansion,
$$P = 1 + P1 \alpha_{V} + P2 \alpha_{V}^{2} + P3 \alpha_{V}^{3} + ...$$
In this expression $\alpha_{V}$ is the QCD coupling constant, evaluated in a lattice regulated theory (not the MS bar thing we want, but close) and at the scale a' (which we know). Unfortunately, for really high precision second order calculations are not enough, and through heroic efforts two coworkers (Howard Trottier and Quentin Mason) computed the third order coefficient P3.

Now we have everything we need, we've determined P from our simulation, and we also know what it is in perturbation theory. With that we can solve for $\alpha_{V}$ at the scale a'! There are a couple of extra things we have to do after than, first we convert from the lattice regulated coupling to the MSbar one we want, and then we run the scale up to M_{Z}. These steps require more perturbation theory at third order (Howard and Quentin had to do the former calculation, the latter (scale running) was already known), so they're also non-trivial.

What's more, if you just do this using one lattice spacing a', and one short distance quantity (P) you don't get a very precise answer. One reason is that the perturbation theory is not very convergent. That is, contributions from the P4 term (which we haven't computed) are not very small. There are two things we did to get around this problem. One is to run at three different lattice spacings. This allows one to estimate P4 (and P5) from fits, which helps control the error. The other is to use more than one short distance quantity. In the paper there are 28 different short distance quantities used. It turns out that the first trick, using multiple lattice spacings, is the one that really cuts down the error.

So what's the final result? We find
$$\alpha_{s}(M_{Z}) = 0.1177(13)$$
which is more accurate than the world average, and is the single most accurate determination of $\alpha_{s}$.

Of all lattice QCD calculations, I think this one is the most elegant. It nicely mixes perturbative and non-perturbative physics in a non-trivial way. To quote the paper, this calculation "demonstrates that the QCD of confinement is the same theory as the QCD of jets; lattice QCD is full QCD, encompassing both its perturbative and nonperturbative aspects."