By Georg von Hippel | February 28th 2007 12:45 AM | Track Comments
The gauge theory known as Quantum Chromodynamics (QCD) has been enormously successful at describing all known phenomena of the strong interactions that bind quarks into hadrons. However, most of this success has been via numerical simulations of lattice QCD; very little is known about how to treat strongly interacting gauge theories like QCD analytically.
So it is quite exciting that in this paper, Leigh, Minic and Yelnikov present an analytical result for the glueball spectrum in (2+1) dimensions. They employ a Hamiltonian formalism pioneeered in a series of papers by Karabali, Kim and Nair. The main result is that the glueball spectrum of (2+1)-dimensional pure Yang-Mills theory can be expressed in terms of the zeros of the Bessel function $J_2(z)$. In particular, the masses of $0^{++}$ states can be written as the sum of two Bessel zeros:
\[
m(0^{++*^r}) = (j_{2,n_1}+j_{2,n_2})\frac{g^2 N}{4\pi}
\]
where $n_1$ and $n_2$ can be determined from $r$, and it is to be noted that the gauge coupling in (2+1) dimensions has the dimension of $\sqrt{Mass}$. Similarly, the masses of $0^{--}$ states can be written as the sum of three Bessel zeros:
\[
m(0^{--*^r}) = (j_{2,n_1}+j_{2,n_2}+j_{2,n_3})\frac{g^2 N}{4\pi}
\]
Their results agree reasonably well with lattice simulations of (2+1)-dimensional pure Yang-Mills theory.
There are some interesting implications of their results which are not discussed in their first paper, but which they discussed in a later paper. In particular, since for large m the Bessel zeros go like
\[
j_{m,n}\simeq\left(n+\frac{m}{2}+\frac{1}{4}\right)\pi
\]
for large excitation numbers, there will be almost degenerate states separated by gaps of $g^2N/4$, with the (almost) degeneracy of the r-th state given by the number of ways to partition (r+1), or (r+2), into two or three integers, respectively.
Another interesting implication of their results is that the mass difference between successive states of even parity and that of successive states of odd parity should be the same. This does not quite agree with what is found on the lattice, where the mass difference for the ++ states is about 1.6 times that for the -- states (which is similar to the difference in results obtained for the gluonic mass in (2+1) dimensions using self-consistent resummation methods with parity-even and parity-odd mass terms, respectively). From the analytical results this parity-dependence of the mass gap would appear to be some sort of artifact.
The later paper also gives a lot more details on their procedures: They start with the functional Schrödinger picture analysis of (2+1)d pure Yang-Mills theory performed by Karabali, Kim and Nair to re-express the theory in terms of new variables, and then make a generalised Gaussian ansatz for the vacuum wave functional containing an undetermined kernel $K(\Delta/m^2)$. The Schrödinger equation is then turned into an ordinary differential equation for $K(L)$, which can be solved in terms of Bessel functions, leading to the formulae given above for the masses of the glueball states.
These are very interesting results and their work may be considered a major breakthrough. 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: $[g_3^2]=[Mass]$, 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.











