Scaling and Asymptotic Scaling in the SU(2) Gauge Theory

We determine the critical couplings for the deconfinement phase transition in $SU(2)$ gauge theory on $N_{\tau }\times N_{\sigma }^3$ lattices with $N_{\tau }=8$ and 16 and $N_{\sigma }$ varying between 16 and 48. A comparison with string tension data shows scaling of the ratio $T_c\mathrm{/}\sqrt{\sigma }$ in the entire coupling regime $\beta =2.30-2.75$, while the individual quantities still exhibit large scaling violations. We find $T_c\mathrm{/}\sqrt{\sigma }=0.69(2)$. We also discuss in detail the extrapolation of $T_c / Lambda_{\rm{\bar{M} \bar{S}}}$and$\sqrt{\sigma} / Lambda_{\rm{\bar{M}\bar{S}}}$ to the continuum limit. Our result, which is consistent with the above ratio, is $T_c / Lambda_{\rm{\bar{M}\bar{S}}} = 1.23(11)$and$\sqrt{\sigma} / Lambda_{\rm{\bar{M}\bar{S}}} = 1.79(12)$. We also comment upon corresponding results for $SU(3)$ gauge theory and four flavour QCD.