Monday, December 12, 2005

Analytical results for the glueball spectrum

In a recent 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 n1 and n2 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 paper (they say they are going to publish another, more detailed, one). 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.

It will be interesting to see what is in Leigh, Minic and Yelnikov's detailed paper, in particular how the higher-spin glueballs turn out.

4 comments:

Georg said...

If anybody can explain to me why the fourth image won't show, I would be extremely grateful.

Unfortunately, the Blogger people seem to have no interest in actually being helpful with this (I haven't received a reply to my request about how to automatically upload images yet), so this blog may well have to migrate elsewhere in the not-too-distant future.

Arun said...

It seems that your link is to a .png file, while the file on the server is a .jpg file. This URL worked for me:

http://photos1.blogger.com/blogger/1971/1881/320/analtical_results_for_glueball_spectrum_4.jpg

Georg said...

That's pretty bizarre, because I definitely uploaded a PNG file -- apparently the Blogger software converted it to JPEG instead. Weird.

Georg said...

Thanks for the hint, Arun.