FINITE VOLUME EFFECTS AND ANISOTROPIES

In the following, we are particularly interested in the deviation of the gluon propagator from the tree level form. We will therefore factor out the tree level behaviour and plot q²D(q²) rather than D(q²) itself.

  

Figure 1: The gluon propagator multiplied by q² as a function of q for the small lattice. The filled triangles denote momenta directed along the time axis, while the filled squares denote momenta directed along one of the spatial axes.

  

Figure 2: The gluon propagator multiplied by q² as a function of qfor the large lattice. The symbols are as in fig. 1.

Fig. 1 shows the gluon propagator on the small lattice as a function of qa. For low momentum values on the small lattice, there are large discrepancies due to finite volume effects between points representing momenta along the time axis and those representing momenta along the spatial axes. These discrepancies are absent from the data from the large lattice, shown in fig. 2. This indicates that finite volume effects here are under control.

However, at higher momenta, there are anisotropies which remain for the large lattice data, and which are of approximately the same magnitude for the two lattices. These anisotropies are considerably reduced on the fine lattice, indicating that they arise from finite lattice spacing errors. In order to eliminate these anisotropies, we select momenta lying within a cylinder of radius $\Delta$$\hat{q}$a = 2 x 2$\pi$/32 along the 4-dimensional diagonals. The result of this cut on the large lattice is shown in fig. 3. A more detailed discussion of these cuts can be found in [7].

  

Figure 3: The gluon propagator multiplied by q² as a function of qfor the large lattice, after the cylindrical cut.