Lattice gauge theories

There is a large number of properties of non-Abelian gauge theories that are nonperturbative and have not been checked directly by analytic calculations. All low energy phenomena fall in this class. The reason being that the coupling constant in non-abelian gauge theories runs with the energy scale. The solution of the leading order renormatlization group equation provides the following form of the running of the strong gauge coupling, $g$

$$g^2=\frac{g_0^2}{1+2\beta_Gg_0^2\log(\mu/\mu_0)}.$$

where the constant

$$\beta_g=\frac{11N-N_s-2N_f}{48\pi^2},$$

where $N$ is the Casimir operator of the gauge group ($N$ for $SU(N)$), and for future reference $N_s$ is the number of adjoint representation scalars and $N_f$ is the number of fundamental representation fermions. The scales $\mu$ and $\mu_0$ correspond to the couplings $g$ and $g_0$.

Now, while it is clear that the coupling goes to zero when the scale goes to infinity, which is nothing else then asymptotic freedom, it is also clear that it increases for towards smaller scales. Of, course at small $\mu$ the expression of the coupling becomes unreliable as it has been derived by perturbative methods. The net result is that perturbative QCD cannot be applied at low energies, usually unreliable below 1GeV, and certainly nonsensical at or below the QCD scale, $\mu_0\simeq 200$MeV.

At the same time, not speaking about masse, a host of phenomena are decidedly controlled by low energies and are non-perturbative. Examples are, confinement, spontaneous breaking of chiral symmetry, matrix elements of of currents appearing in weak and electromagnetic interaction, high temperature deconfining phase transition, nuclear physics, diffraction phenomena in high energy interactions, just to mention a few.

Until one is able to verify that the predictions of QCD are in agreement with observations in the above listed phemomena one cannot state that QCD is the correct theory describing strong interactions. Therefore it is of great importance to devise non-perturbative methods that probve the low energy region of QCD.

The only really successful method in this respect is lattice gauge theory. Lattice methods can be applied to study any field theory in the Euclidean form, but we will concentrate on QCD and first of all on pure QCD, consisting of gluons alone. As you all know, unlike QED, QCD is a full fledged interacting theory without matter fields and should show many of the properties of a theory of quarks and gluons, including confinement, physical bound states (glue balls), deconfinement transition. Quite amazingly, one can even study decidedly fermionic processes in pure gluonic QCD such as chiral symmetry breaking, and even masses of elementary particles.