Because you can’t do local updates. For the gluon action, you’re locally updating the gauge field. So at any point, you only need to know the nearest neighbours. Whereas with the quarks, you’ve “done” the path integral, so it cares about every gluon field, at every point.

Put more simply, you’re not updateing the “quark field at point x” you’re computing the entire effect of the quark field at once.

Also, it seems to me if you get good precision in the higher mass calculation, the scaling would naively appear to be linear in the eigenvalues.

The compute time scals as the inverse. And the physical chiral limit is determined by Chiral PT, which shows that the extrapolation is not linear.

Which again, computing power seems to have vastly superceded since I first heard about Lattice QCD (Moore’s law et al). So, obviously it cannot be linear, what exactly big O() is it?

For a fixed volume compute time scales roughly as the inverse 6th power of the lattice spacing, times the inverse of the lightest pion mass. 4 powers of the lattice spacing are just the number of points on the grid. One is critical slowing of the gluon update algorithm, and the remaing mass*spacing is the slowing of the quark matrix inversion.

In “real life” it’s even worse than this. But this sums it up pretty well.

Yea I never quite understood this. Why is the matrix so large in the first place, here as opposed to elsewhere (where other lqcd calculations seem to be so reliable). Is it merely b/c of the smallness of the lattice spacing necessary that leads to such huge matrices? Also, it seems to me if you get good precision in the higher mass calculation, the scaling would naively appear to be linear in the eigenvalues. Which again, computing power seems to have vastly superceded since I first heard about Lattice QCD (Moore’s law et al). So, obviously it cannot be linear, what exactly big O() is it?

Okay, in lattice QCD you write the fermion action as

S = \bar{\psi} M \psi

where M is some huge (but finite and totally well defined) matrix that depends on the gauge fields. Now you can’t really do grassman variables on the computer, so instead you perform the path integral over the fermions exactly, which gives you

det(M)

So you end up having to compute the determinant of a very large matrix. The major cost of doing this is computing the inverse of M. The cost of the algorithim you use to compute the inverse (conjugate gradiant) goes like the inverse of the smallest eigenvalue. And the smallest eigenvalue of the matrix M is the quark mass. So as you take the quark mass smaller and smaller, the cost of your simulation goes through the roof.

If one is calculating the meson and/or baryon spectrum with heavier quarks (ie. charm, bottom, etc …), is it reasonable to treat them non-relativisticly?

For the bottom quarks the answer is yes. A non-relativistic effective theory is what we use. For the charm it’s somewhat less clear. The charm is “heavy” but it’s not quite “heavy enough” to let you trust non-relativistic expansions in the same way you trust them for the b quark.

It still worries me if the calculation is about mass splittings instead of absolute.

That is entirely due to the fact that the chiral perturbation theory for the absolute masses still needs doing. The simulations don’t care one way or the other. The omega mass was computed, see the paper by Davies and Bernard.

Indeed it is already a bit regrettable the need to recourse to chiral extrapolation there. In some sense, we are not testing QCD anymore but QCD plus the effective theory we are expected to fit to.

I disagree. Chiral perturbation theory, just like heavy quark effective theory, is a consquence of QCD. Using fits based on chiral perturbation theory is a test of full QCD.

Or is the neutral pion one of these problematic low mass objects?

That I don’t know actually. One rarely makes a distinction between the charged and neutral pions.

The \eta’ is hard though, I know that.

Hmm I have taken a look to the paper; the small error in 5 experimental numbers (plus the four ones “exact” from tuning) are impressive, really more impressive than quenched approximations. It still worries me if the calculation is about mass splittings instead of absolute.

The light quark masses are the key problem in the modern simulations.

Indeed it is already a bit regrettable the need to recourse to chiral extrapolation there. In some sense, we are not testing QCD anymore but QCD plus the effective theory we are expected to fit to.

Acutally calculating meson masses is easy.

In surprises me that also the width for QCD-stable objects seems to be easy to calculate. In particular for the pseudoscalar, this means that the chiral anomaly is handled correctly, does it? Or is the neutral pion one of these problematic low mass objects?

What exactly is the nature of the problem in getting the up/down mass light enough?

If one is calculating the meson and/or baryon spectrum with heavier quarks (ie. charm, bottom, etc …), is it reasonable to treat them non-relativisticly? One would guess that mesons and/or baryons with light quarks would have to be treated relativistically.

Modern large scale simulations are unquenched. The paper I mentioned is an unquenched calculation.

Hmm but, besides a fundamental mass, how many free parameters do you need

You tune the 4 bare quark masses (up and down are degenerate, and you ignore the top) and the bare lattice spacing to reproduce 5 experimental numbers. That’s it, there are no “free parameters” beyond that. This is not a model.

Are the tuned values published elshewhere?

Yes, you’d have to dig a bit through the references. The papers by the MILC people (Bernard et. al.) are the relevent ones for light mesons.

could you inform us of which other groups are obtaining sucessful predictions for the hadronic spectrum?

Well, every group does the meson spectrum to some degree, if only to fix parameters. Large scale calculations are being done by the CP-PACS group, see hep-lat/0409124 for example. It’s not clear if they will be able to get to light enough quark masses though.

The light quark masses are the key problem in the modern simulations. That’s where the difficulty is, getting the up/down mass light enough. Acutally calculating meson masses is easy.

3 * mass_cascade – mass_nucleon

agrees with experiment at the few percent level. The paper hep-lat/0304004 has some other things. We’ve computed a few other light hadron quantities since then, the \Omega mass, for example.

Hmm but, besides a fundamental mass, how many free parameters do you need, and how sensitive the results are to veriations of these parameters? In 0304004, all the quark masses are tuned to reproduce the most important mass predictions (Are the tuned values published elshewhere? ).

And now you are here, I wonder… could you inform us of which other groups are obtaining sucessful predictions for the hadronic spectrum?

The CP-PACS group reproducde the light hadron spectrum to within around 10% using quenched (i.e. not entirely physical) lattice QCD in the mid-nineties. There’s a plot of this, which is what you probably saw in Wilczek’s talk. For modern full QCD simulations we’re not quite at the full hadron spectrum, but some things are done. In order to reduce systematic errors one often computes mass differences, for example, the combination

3 * mass_cascade – mass_nucleon

agrees with experiment at the few percent level. The paper hep-lat/0304004 has some other things. We’ve computed a few other light hadron quantities since then, the \Omega mass, for example.

It’s better in the heavy quark sector. Apart from a couple of lingering problems, the charm and bottom meson spectrum has been totally computed, and agrees with experiments. There are also calculations of various decay constants and form factors, which are a bit harder than masses.

My opinion, I think it is fair to say that the meson spectra has not been calculated.

You’re wrong, as even a cursory glance at the lattice liturature would show.

Note also we are speaking of unstable particles, so the “pole mass” spectra includes both mass and decay width

Montecarlo errors are high, finite size effects etc

Actually, those errors are fairly well under control for spectrum calculations. What really bites you is discretization errors, and errors in the chiral extrapolations.

Of course, when you try to do harder things (glueball masses, K \to (2,3)\pi) statistics and finite volume are much worse. But for meson masses, it’s pretty well under control.

I believe the calculation Robert mentions for K a Calabi-Yau gives a simple result: the low mass particles are massless and supersymmetry is unbroken. For obvious reasons this result is not heavily promoted.

A problem he doesn’t mention is how to give dynamics to the moduli parameters for the compactification manifold. Naively the effective action doesn’t depend on them so you end up with massless scalars, less naively you can believe in fixing them a la KKLT, then you have the landscape to deal with.