At the end of the last post we saw that the bare coupling in lattice gauge theory is a bad coupling to do a perturbative expansion in. If we use a good coupling, like $\alpha_{V}$ we will find a much more convergent perturbative series. This is crucially important if we want to determine things like improvement coefficients to good accuracy. An open question at the end of the last post was *why* the bare lattice coupling is bad, that's the subject of this post.

To understand this, we need to understand how gauge fields are represented in the lattice. In lattice QCD, the matter fields are assigned to the sites of the lattice, and the gauge fields "live" on the links. This is very much in keeping with the geometric picture of gauge theory, where you have a parallel transporter ($U$) which allows you to compare matter fields at two different points. That is,

$$

\psi(x+\hat{\mu}) = U(x+\mu,x) \psi(x)

$$

In continuum gauge theory we have

$$

U(x+\mu,x) = P exp(ig \int_{x}^{x+\mu} dz^{\nu} A_{\nu}(z))

$$

Here P stands for path ordering.

This is actually much simpler on the lattice. On the lattice, there is a "shortest path" namely one link. So the parallel transporter simplifies to

$$

U(x+\mu,x) = U_{\mu}(x) = exp(i g a A_{\mu}(x))

$$

These parallel transporters (or link fields) are what one builds lattice actions out of. For example, a simple forward covariant derivative on a fermion field might be

$$

D_{\mu} \psi(x) = U_{\mu}(x)\psi(x+\mu) - \psi(x).

$$

Notice that the connection between the link field $U$ and the gauge field $A$ is non-linear. This is where the problems come in. Consider the expectation value of a single link field $(U)$. Expanding the expression for $U$ we find

$$

(U) = 1 - 2 \pi \alpha(\pi/a) a^{2} (A_{\mu}(x) A_{\mu}(x)) + ...

$$

I've used (A) = 0, and evaluated the strong coupling at the cutoff $\pi/a$. This looks fine, in the formal continuum limit (a->0) the lattice corrections vanish. But all is not well! There's a dangerous beast lurking in the loop integral (AA). Let's look at what it is.

The expectation value (A A) can be written as the integral over the momentum space gluon propagator

$$

(A A) = \int_{0}^{2\pi/a} d^{4}q / qhat^{2}

$$

where

$$

qhat^{2} = 4/a^2 \sum_{1}^{4} sin^2 (q_{\mu} a/2).

$$

In the limit where $a$ is small, we can replace $qhat^{2}$ by $q^{2}$, and we find that the integral is

$$

(A A) ~ 1/a^{2} + O(1)

$$

This is the disaster, because when we substitute this back into our expression for the expectation value of $U$ we find

$$

(U) = 1 - 2\pi \alpha(\pi/a) a^{2} (1/a^{2} + O(1)) + ...

$$

The $1/a^{2}$ from the loop integral cancels the $a^{2}$ from the expansion of the exponential leaving a lattice artifact suppressed only by the coupling constant, rather than the lattice spacing. The coupling constant is small (0.2) but not really small (like $\alpha a^{2} ~ 0.01).

It is factors like these (called gluon tadpoles) which spoil lattice perturbation theory in the bare coupling. The $\alpha_{V}$ coupling doesn't suffer from these corrections because tadpole effects come in an 2nd order in $\alpha$. It would be nice to have a way to systematically account for these effects. This method, called tadpole improvement (or mean-field improvement) was proposed by Lepage and Mackenzie, and is now in wide use.

The idea is to use link fields that have the tadpole effects largely canceled out of them. To do this you take all the link fields you have $U$ and multiple and divide by some average link field $u$. That is

$$

U \to u U/u = u \tilde{U}

$$

The new fields $\tidle{U}$ are much closer to their continuum values, since the average link field division cancels out most of the tadpole effects. The factors in the numerator $u$ can be absorbed into couplings and masses. When combined with a good coupling (such as $\alpha_{V}$) the agreement between perturbation theory and non-perturbative simulations (of short distance quantities) is much better.

There is one last thing one can do to improve the perturbative series, namely set the scale of the coupling in a more sensible way. That will be the subject of my next post.

## 4 comments:

Great blog!

Some general stupid questions:

* Can one do lattice QCD calculations at all in modern PCs (say current Pentium with 2 Gig RAM or Opterons with more than 4 Gig RAM)?

* If so, what level of precision can one get for lattice QCD calculations? And how slow ?

* Are there any basic open source lattice qcd software available that can be downloaded and run?

Many thanks!

Great blog!

Thanks! It's a little "physics heavy" at the moment. I hope to find time to post things at a little lower level at some point.Can one do lattice QCD calculations at all in modern PCs (say current Pentium with 2 Gig RAM or Opterons with more than 4 Gig RAM)?

That largely depends on what one wants to compute. In general, the answer is no, sort of :) Generating gauge configurations with unquenched fermions is supercomputer territory. Even computing fermionic observables on top of other peoples configurations is costly, though a dual opteron could make a go at it in some cases.There are things that can be done on workstations. One nice example is non-relativistic QCD, which doesn't require large matrix inversions. The perturbation theory that I do could also be done on a high horsepower workstation, though it'd be slow compared to the large cluster I currently use.

* If so, what level of precision can one get for lattice QCD calculations? And how slow ?

On a workstation, I wouldn't expect too much :) You could compute things like the quarkonium spectrum to a reasonable (10%) degree of accuracy.* Are there any basic open source lattice qcd software available that can be downloaded and run?

The MILC code is publically availible, you can find the links from their website http://physics.indiana.edu/~sg/milc.html. But that's heavy stuff, which would require massive computing. Some other major lattice collaborations are listed on the lattice web http://www.lqcd.org/ they might have some code.Usually though, for the stuff done on workstations, you write your own code. Lattice QCD programs tend to be really simple in theri basic structure. Perhaps I'll do a post on basic algorithms at some point.

One can probably do simple glueball spectroscopy studies on your PC. I can't imagine doing anything much more than this, hybrids, etc.

-Steven

There is a free open source lattice qcd code for arbitrary SU(n) gauge groups and object oriented design called fermiqcd

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