Monday, May 29, 2006

Non-Relativistic QCD

This is another installment in our series about fermions on the lattice. In the previous posts in this series we had looked at various lattice discretisations of the continuum Dirac action, and how they dealt with the problem of doublers posed by the Nielsen-Ninomiya theorem. As it turned out, one of the main difficulties in this was maintaining chiral symmetry, which is important in the limit of vanishing quark mass. But what about the opposite limit -- the limit of infinite quark mass?

As it turns out, that limit is also difficult to handle, but for entirely different reasons: The correlation functions, from which the properties of bound states are extracted, show an exponential decay of the form $$C(T,0)\sim e^{-maT}$$, where $$t$$ is the number of timesteps, and $$ma$$ is the product of the state's mass and the lattice spacing. Now for a heavy quark, e.g. a bottom, and the lattice spacings that are feasible with the biggest and fastest computers in existence today, $$ma\approx 2$$, which means that the correlation functions for an $$\Upsilon$$ will decay like $$e^{-4T}$$, which is way too fast to extract a meaningful signal. (Making the lattice spacing smaller is so hard because in order to fill the same physical volume you need to increase the number of lattice points accordingly, which requires a large increase in computing power.)

Fortunately, in the case of heavy quark systems the kinetic energies of the heavy quarks are small compared to their rest masses, as evidenced by the relatively small splittings between the ground and excited states of heavy $$Q\bar{Q}$$ mesons. This means that the heavy quarks are moving at non-relativistic velocities $$v<<c$$ and can hence be well described by a Schrödinger equation instead of the full Dirac equation after integrating out the modes with energies of the order of $$E\gesim M$$. The corresponding effective field theory is known as Non-Relativistic QCD (NRQCD) and can be schematically written using the Lagrangian
$$\mathcal{L} = \psi^\dag \left(\Delta_4 - H\right)\psi$$
where $\psi$ is a non-relativistic two-component Pauli spinor and the Hamiltonian is
$$H = - \frac{\bm{\Delta}^2}{2M} + \textrm{(relativistic and other<br />corrections)}$$
In actual practice, this is not a useful way to write things, since it is numerically unstable for $$Ma<3$$; instead one uses an action that looks like
$$\mathcal{L} = \psi^\dag\psi - \psi^dag\left( 1 - \frac{a\delta H}{2} \right) \left( 1 - \frac{aH_0}{2n} \right)^n U_4^\dagger\left( 1 - \frac{aH_0}{2n} \right)^n \left( 1 - \frac{a \delta H}{2} \right)\psi$$
where
$$H_0 = - \frac{\bm{\Delta}^2}{2M}$$
whereas $$\delta H$$ incorporates the relativistic and other corrections, and $$n\ge 1$$ is a numerical stability parameter that makes the system stable for $$Ma>3/(2n)$$.

This complicated form makes NRQCD rather formidable to work with, but it can be and has been successfully used in the description of the $$\Upsilon$$ system and in other contexts. In fact, some of the most precise predictions from lattice QCD rely on NRQCD for the description of heavy quarks.

It should be noted that the covariant derivatives in NRQCD are nearest-neighbours differences -- the reasons for having to take symmetric derivatives don't apply in the non-relativistic case; hence there are no doublers in NRQCD.

Wednesday, May 24, 2006

Calling lattice bloggers!

As pointed out in Matthew's last post so far, this is the world's first (and only, hence best) group blog devoted to Lattice QCD. Unfortunately, Matthew's new job does not leave him too much time for blogging; therefore I'm running this blog all alone at the moment, which leads to the relatively low activity seen in recent weeks.

So I was wondering if there are any other lattice people out there who would like to join this blog and post here about their research work, the most recent papers on the arXiv or in the journals, interesting developments in the field, science in the news, and any other matters appropriate for a physics blog (as opposed to a mere physicists' blog). It would be great if this blog saw a little more activity!

Tuesday, May 23, 2006

Physics fun with sunglasses

Yesterday was a public holiday (Victoria Day) in Canada, and (as opposed to last year, when I went to my windowless office at the university in blissful ignorance of the Canadian holiday schedule, wondered why it was so empty and the lights on the corridors were off, and only figured it out when I was unable to obtain any lunch in the food court) I got to enjoy the sunshine on a lovely day.

I had completely forgotten how much fun sunglasses could be: I have these sunglass things (I don't really know what the technical term for them is) that can be clipped to my glasses to turn them into sunglasses; what makes them so much fun is that they really are nothing but polarisation filters! Of course polarisation filters make great sunglasses because the sunlight is unpolarised, and because the polarisation filter does not introduce a colour bias like an old-fashioned green or brown filter would. But as everybody remembers from their undergraduate optics course, light reflected from surfaces is partially polarised, and the same is true for scattered light. Therefore, when wearing your polarisation filters/sunglasses, the brightness of the road surface and of the blue sky will vary as you tilt your head towards the right or the left, which is quite fascinating. Unfortunately, other people will probably consider you to be crazy if they see you tilting your head from side to side while stepping forward and backward trying to determine the Brewster angle, or turning around your own axis trying to precisely locate the spot of maximal polarisation in the sky (which is how bees detect the direction towards the sun even if the sun itself is behind a cloud) -- but a real physicist shouldn't mind, right?

So I got to feel like an experimentalist for a little while, while also taking a pleasant walk in the park, sipping lime juice on a terrace above Wascana Lake and generally enjoying myself, all thanks to great weather and Her Majesty's official birthday in Canada: God save the Queen!

Oh, and of course those sunglasses are real fun to use with LCDs, too...

Thursday, May 18, 2006

Analytical (3+1)d Yang-Mills and ontology

A little while ago, there were two papers by Leigh, Minic and Yelnikov, in which they expanded on the previous work done by Karabali, Kim and Nair towards an analytical solution for (2+1)-dimensional pure Yang-Mills theory. By re-expressing the theory in terms of appropriate variables, they were able to find an ansatz for the vacuum wavefunctional in the Schrödinger picture which they could solve analytically, enabling them to find the spectrum of glueball masses. But can the same be done for the physical case of (3+1) dimensions?

In this paper, Freidel, Leigh and Minic seem to say "probably". Their generalisation to (3+1) dimensions is based on the idea of "corner variables", which are essentially untraced Wilson loops lying within the coordinate planes which go through the point at infinity. If the theory is expressed in terms of these, there are a lot of formal algebraic analogies with the (2+1)-dimensional case, which renders them hopeful that it may be possible to treat the (3+1)-dimensional theory in an analogous fashion. In this case the only problem left to solve would be to determine the kernel appearing in the ansatz for the wavefunctional.

There seems, however, to be a very important difference between the (2+1)d and (3+1)d cases, which they also mention but appear to consider as a relatively minor inconvenience that will be worked out: in (2+1) dimensions, the gauge coupling has a positive mass dimension: [g32]=[Mass], so the generation of a mass gap is expected on dimensional grounds just from looking at the Lagrangian, and it is even possible to compute the mass gap semi-perturbatively using self-consistent approximations. In (3+1) dimensions, there is no dimensionfull parameter in the Yang-Mills Lagrangian, so the existence of a mass gap is really an unexpected surprise. Of course an arbitrary mass scale will be introduced by regularisation, but even if this mass scale cancels from all mass ratios (as Freidel et.al. appear to assert it will), its arbitrariness still means that the overall mass scale of the theory will remain completely undetermined by the kind of analysis they propose. I am not sure if this can be a consistent situation.

The corner variables they use reminded me of a talk by the philosopher Holger Lyre given at a physics conference in Berlin in 2005. He discussed the Aharonov-Bohm effect and exhibited three possible ways of interpreting electrodynamics ontologically, which he called the A-, B- and C-interpretations. In the A-interpretation, the gauge potential A is assumed to be a real physical field: that is probably what most working physicists would reply when asked for the first time, and it has the advantage of making the locality of the interaction explicit; on the other hand, how can a quantity that depends on an arbitrary gauge choice be physically real? In the B-interpretation, the field strength B (and E) is considered to be physically real; this means physical reality is gauge-invariant, as it should be, but the interaction with matter becomes maximally nonlocal, which is very bad. In the C-interpretation, finally, the holonomies (C is for curves) of the gauge connection are taken to be the only physically real part of the theory: this leads to gauge-invariance and a form of locality (not a point interaction, but a Nahewirkungsprinzip). Ultimately, the C-interpretation would therefore appear to be the most palatable ontology of gauge theories. Finding a quantum formulation of gauge theories in the continuum that contains only Wilson loops as variables would be very desirable from this philosophical point of view alone, even if it does not lead to an analytical solution.

Friday, May 05, 2006

Language-dependent spectra

I just noticed that the sequence of colours in the visible electromagnetic spectrum seems to be named differently in English and German. In English, it generally appears to be red/orange/yellow/green/blue/indigo/violet (as in the mnemonic "Richard of York gave battle in vain."), whereas in German, it appears to be red/orange/yellow/light-green/dark-green/blue/violet (at least that is how I remember learning it in elementary school).

Now I wonder what the basic colours of the visible spectrum are called in other languages. In particular, I suppose they are rather different in languages that divide parts of the colour space differently anyway (such as, I believe, Gaelic and Russian, and probably lots of non-Indoeuropean languages). Does anybody have examples of how the spectrum is "different" in other languages?

Update: As a clarification: German-speakers don't call blue "dark-green" -- it's just that the conventional rendition of the colour spectrum in German splits the green part into two colour bands, whereas the English one does the same to the blue part. And a Franco-Canadian told me that in (Canadian) French it just is red/orange/yellow/green/blue/violet (six colours only).