Dynamical QCD simulations on anisotropic lattices

The simulation of QCD on dynamical ($N_f=2$) anisotropic lattices is described. A method for nonperturbative renormalisation of the parameters in the anisotropic gauge and quark actions is presented. The precision with which this tuning can be carried out is determined in dynamical simulations on $8^3\times 48$ and $8^3\times 80$ lattices.

Figure 1

A typical $t_{\min }$ plot, showing the energy for momentum $n^2=1$ on run 1, $8^3\times 80$ lattices from fits to time ranges $t_{\max }=40$ for various $t_{\min }$. A stable ground state energy determination, with a good ${\chi }^2$, is achieved for $22\le t_{\min }\le 30$.

Figure 2

A comparison of the dispersion relations determined from an $8^3\times 48$ lattice and an $8^3\times 80$ lattice. The solid lines are the best fits and the dotted lines are the 68% confidence levels. The figure also shows a comparison of all-to-all propagators and point propagators on the same (longer) lattice. The plot shows that the ground state energies have not reached a plateau on the shorter lattice. On the longer lattice the all-to-all and point data agree, while higher precision is achieved with all-to-all propagators.

Figure 3

Dispersion relations from runs 1, 4 and 5 on $8^3\times 80$ lattices using all-to-all propagators. The solid line is a fit to the four points and the dotted lines are the 68% confidence levels. The quality of all three fits is very good with ${\chi }^2/N_{\mathrm{d}.\mathrm{f}.}=2.0/2$, $1.9/2$, $2.0/2$ for runs 1,4 and 5, respectively.

Figure 4

Tuned values of input parameters $({\xi }_g^0,{\xi }_q^0)$ determined from the plane fit procedure on the $8^3\times 80$ lattice. The plots show the results for different values of $R$ and $T$ used to determine the gluon anisotropy. Each point corresponds to one bootstrap sample.

Figure 5

As in Fig. 4. The figure shows the results from a plane fit using parameters from runs 1, 4, 5 and 6 (marked with an $\times$). The big red (gray) cross at $({\xi }_g^0,{\xi }_q^0)=(8.42,7.43)$ indicates the result of the best fit.

Figure 6

The potential between fundamental static color sources for run 6, measured from static propagation in a coarse direction. Lines show fits to the Cornell potential, and are used in a crude determination of the lattice spacing.