## Large-N gauge theories on the lattice

Today I have found on arXiv a very nice review about large-N gauge theories on the lattice (see here). The authors, Biagio Lucini and Marco Panero, are well-known experts on lattice gauge theories being this their main area of investigation. This review, to appear on Physics Report, gives a nice introduction to this approach to manage non-perturbative regimes in gauge theories. This is essential to understand the behavior of QCD, both at zero and finite temperatures, to catch the behavior of bound states commonly observed. Besides this, the question of confinement is an open problem yet. Indeed, a theoretical understanding is lacking and lattice computations, especially in the very simplifying limit of large number of colors N as devised in the ’70s by ‘t Hooft, can make the scenario clearer favoring a better analysis.

What is seen is that confinement is fully preserved, as one gets an exact linear increasing potential in the limit of N going to infinity, and also higher order corrections are obtained diminishing as N increases. They are able to estimate the string tension obtaining (Fig. 7 in their paper):

$\centering{\frac{\Lambda_{\bar{MS}}}{\sigma^\frac{1}{2}}\approx a+\frac{b}{N^2}}.$

This is a reference result for whoever aims to get a solution to the mass gap problem for a Yang-Mills theory as the string tension must be an output of such a result. The interquark potential has the form

$m(L)=\sigma L-\frac{\pi}{3L}+\ldots$

This ansatz agrees with numerical data to distances $3/\sqrt{\sigma}$! Two other fundamental results these authors cite for the four dimensional case is the glueball spectrum:

$\frac{m_{0^{++}}}{\sqrt{\sigma}}=3.28(8)+\frac{2.1(1.1)}{N^2},$
$\frac{m_{0^{++*}}}{\sqrt{\sigma}}=5.93(17)-\frac{2.7(2.0)}{N^2},$
$\frac{m_{2^{++}}}{\sqrt{\sigma}}=4.78(14)+\frac{0.3(1.7)}{N^2}.$

Again, these are reference values for the mass gap problem in a Yang-Mills theory. As my readers know, I was able to get them out from my computations (see here). More recently, I have also obtained higher order corrections and the linear rising potential (see here) with the string tension in a closed form very similar to the three-dimensional case. Finally, they give the critical temperature for the breaking of chiral symmetry. The result is

$\frac{T_c}{\sqrt{\sigma}}=0.5949(17)+\frac{0.458(18)}{N^2}.$

This result is rather interesting because the constant is about $\sqrt{3/\pi^2}$. This result has been obtained initially by Norberto Scoccola and Daniel Gómez Dumm (see here) and confirmed by me (see here). This result pertains a finite temperature theory and a mass gap analysis of Yang-Mills theory should recover it but here the question is somewhat more complex. I would add to these lattice results also the studies of propagators for a pure Yang-Mills theory in the Landau gauge, both at zero and finite temperatures. The scenario has reached a really significant level of maturity and it is time that some of the theoretical proposals put forward so far compare with it. I have just cited some of these works but the literature is now becoming increasingly vast with other really meaningful techniques beside the cited one.

As usual, I conclude this post on such a nice paper with the hope that maybe time is come to increase the level of awareness of the community about the theoretical achievements on the question of the mass gap in quantum field theories.

Biagio Lucini, & Marco Panero (2012). SU(N) gauge theories at large N arXiv arXiv: 1210.4997v1

Marco Frasca (2008). Yang-Mills Propagators and QCD Nuclear Physics B (Proc. Suppl.) 186 (2009) 260-263 arXiv: 0807.4299v2

Marco Frasca (2011). Beyond one-gluon exchange in the infrared limit of Yang-Mills theory arXiv arXiv: 1110.2297v4

D. Gomez Dumm, & N. N. Scoccola (2004). Characteristics of the chiral phase transition in nonlocal quark models Phys.Rev. C72 (2005) 014909 arXiv: hep-ph/0410262v2

Marco Frasca (2011). Chiral symmetry in the low-energy limit of QCD at finite temperature Phys. Rev. C 84, 055208 (2011) arXiv: 1105.5274v4

### 4 Responses to Large-N gauge theories on the lattice

1. ohwilleke says:

“to distances 3/root lowercase sigma” – what does this correspond to in SI units (or in lattice units, if it does not have a direct real world analogy)?

Likewise, what are the units for the glueball spectrum and m(L) numbers?

(When I was studying chemistry and physics as an exchange student in New Zealand, we used to receive a 50% grade reduction for any answer or even intermediate step towards an answer without units expressly shown (even when it is obvious or part of a series of same unit problems) and this has colored by stylistic tendencies when discussing physics and chemistry ever since).

2. mfrasca says:

Hi ohwilleke,

Well $\sqrt{\sigma}=420\ MeV$ then you need a unit conversion from MeV to fm that is $1\ MeV^{-1}=197\ fm$. As you should know one fermi is $10^{-15}\ m$.

As $m(L)$ is a mass and $\sqrt{\sigma}$ has the same dimension, this ratio is just a pure number and must be recovered by any successful theory aimed to understand the mass gap in order to get a Millenium prize.

Marco

3. ohwilleke says:

Thanks again. Much appreciated.

• mfrasca says:

You are welcome.