Special thanks go to Rajan Gupta in collaboration with whom all the numerical work was done and from whom I learned many tricks of the trade. The properties of the SU(2) and SU(3) lattice gauge theories are investigated using the Real Space Monte-Carlo Renormalization Group method. The only candidate field theory of strong interactions that is both renormalizable and consistent with the known symmetry properties of the hadrons is Quantum Chromo-Dynamics (QCD).
The equations of the renormalization group show that all dimensional quantities have a non-analytic coupling dependence of the form exp ( -g-2). In particular, the phenomenon of restriction and chiral symmetry breaking can be easily explained in the strong coupling limit. However, the connection to continuum field theory is only possible in the limit of the lattice spacing going to zero.
This happens in the weak coupling boundary, and therefore one has to take a smooth boundary from the strong coupling region to the weak coupling region. Phenomena such as confinement are expected to be present even in this truncated version of the theory, and therefore the study of the pure gauge theory is a first step towards understanding the non-perturbative physics.
LATTICE GAUGE THEORIES
This is more easily understood by turning to the Euclidean time formulation of field theory. The different action choices differ in the contribution of the higher order terms to the lattice distance. Therefore, as one approaches the continuum limit, the correlation length diverges in terms of the lattice spacing.
A handy technique is strong coupling expansion, which is discussed in the next section. This effect manifests itself as a strong dependence of the value of the observable on the number of terms preserved in the expansion (see Figure 3). This is necessary before we can claim that QCD is a theory of strong interactions.
In the first case, one loses all the effects of the anomaly, while in the second case the approach to the continuum boundary becomes more difficult. 2.3]Comparison of the strong coupling results for the string tension with the Monte-Carlo data taken from the second reference in [13], for the SU(3) gauge theory.
MONTE-CARLO RENORMALISATION GROUP
Block Transformations
For SU(2) theory, the link variable of a block is defined as the normalized mean of the eight link variables on the original grid that connect the block y V to the block y +f.L (see Figure 3.6) [5]. We need one gauge degree of freedom that remains for the position of the block as a whole; therefore profiling can only be done at 15 places within each block. Here, the block link variable consists of nine path-ordered products of the link variables on the original grid [~0].
The central path is topologically different from the rest, and should therefore be considered with a different weight in the construction of the block connection variable. Also of the available ~9 degrees of freedom, only 36 are used (9 paths per direction). No meter confirmation is required in the construction of the block variables. iv) Matter fields (complex scalars) defined at the locations can also be renormalized simultaneously, while preserving the exact local gauge invariance of the theory.
Now let's look at the construction of the "block link" variables between the block sites in this transformation. This does not affect the calculation of the {3 function, since grids of the same size (i.e. the same boundary conditions) are compared there. To ensure that every link is used in the construction of the block link, we need to associate the 36 positively oriented links coming from the 9 sites with the 4 positively oriented links from the block site.
Eight out of the 36 links can be equated to the identity by making a meter transformation at the 8 nearest neighbor sites. In practice, the gauge determination is taken care of by constructing each of the 7 links as the path-ordered product of the 3 links joining the sites on the renormalized grid. The dimensional freedom left at the block sites allows this procedure to be repeated for successive applications of the RG transformation.
Having enumerated the paths and shown that their computation is simple, we now describe the construction of the block link from these paths. Therefore, some information about normalization is absorbed into the definition of renormalized joins. This detail is not important in the calculation of the {3- function since the same approximation is used for both meshes.
APPENDIX
The first term on the right side of the above equation is maximized when. Since Y and 2 are determined as soon as X is, to complete the solution we must find the phase that characterizes X. 3.2]The integration of a high momentum contribution in the A.rp4 theory leads to six-point vertices in the effective theory.
The lines represent the links on the original grid used in the construction of the block link.
ATTRACTIVE IRFP
REPELLENT 0 PHASE
STRUCTURE
UNSTABLE UVFP
UVFP
THE {1-FUNCTION
Results for SU(3)
As expected, the phase structure in the extended coupling constant space causes the {)'-function to fall below the perturbed result on the v,rea.l{couplLT'lg side of the crossover region. Variables on grids 14 and (v3)4 could be matched simultaneously and even variables on grids 34 showed a rough match. Crosses are based on matches on the 14 grid, and circles are based on matches on the (v3)4 grid.
The expected values of the absolute magnitude of the Wilson line are also shown as squares. Also sh as squares are the expected values of the absolute magnitude of the Wilson line.
Chapter 5
IMPROVING THE ACTION
Results for SU(2)
The validity of the results v.ill therefore be tested by repeating the calculation with the improved action determined here. This constitutes the largest set of operators that we could use in the improved action analysis with the current data. In the central region of the couplings we have studied, the currents from both the Wilson axis and the M-K line, when projected in the !KF,KAl plane, converge to an approximate path given by KA KF.
The improved performance projection is now very close to the result found using the M-K approximation [6]. When gba:re was smaller, the K6P I KF ratio increased and appeared to approach the enhanced tree-level value, while the effect of higher representations was reduced. The fact that the Wilson and M-K actions lie on opposite sides of the KA coupling RT and yet both converge to roughly the same trajectory after the first iteration means that the fKA1l points near the RT are in the bounded coupling space we explored.
We found that for a given start fKA L the renormalized coupling KP was much smaller than the value of Kf for which the long-range behavior of the theory differs from Kfo by the scale factor of the RG transformation. The calculation of the improved action is very sensitive to the number of operators included. To improve the statistics and check the stability of the improved action achieved here, we are currently running the program on a.
The small contribution of higher representative operators means that the effect of the non-trivial phase structure is smaller here than in the SU(2) case. The grating size N v'3 is chosen after taking into account all finite size effects; i.e. the maximum correlation length must be a factor of rv3 smaller than it. Based on the autocorrelation leP.gth and the update criterion tL>ne, co:r~figurations separated by a certain nlL>nber of updates are blocked from dov.-n to N4 lattices. .
The ratio of the lattice correlation length to the lattice size remains unchanged, as both are reduced by the same v'3 scale factor. The termination of the MCRG procedure provides both the necessary {3 function to control scaling and an improved estimate of the renormalized trajectory. 5.4]The projection of the trajectories of the Migdal-Kadanoff approximate renormalization scheme onto the [Kp,KA] space of the SU(2) gauge theory.
