Nonperturbative determination of anisotropy coefficients in lattice gauge theories

We propose a new nonperturbative method to compute the derivatives of gauge coupling constants with respect to anisotropic lattice spacings (anisotropy coefficients), which are required in an evaluation of thermodynamic quantities from numerical simulations on the lattice. Our method is based on a precise measurement of the finite temperature deconfining transition curve in the lattice coupling parameter space extended to anisotropic lattices by applying the spectral density method. We test the method for the cases of SU(2) and SU(3) gauge theories at the deconfining transition point on lattices with the lattice size in the time direction $N_t=4\mathrm{}–6.$ In both cases, there is a clear discrepancy between our results and perturbative values. A long standing problem, when one uses the perturbative anisotropy coefficients, is a nonvanishing pressure gap at the deconfining transition point in the SU(3) gauge theory. Using our nonperturbative anisotropy coefficients, we find that this problem is completely resolved: we obtain $\Delta {p/T}^4=0.001\mathrm{}\left(15\right)\mathrm{}$ and $-0.003\left(17\right)\mathrm{}$ on $N_t=4\mathrm{}$ and 6 lattices, respectively.