Testing Dimensional Reduction in SU(2) Gauge Theory

At high temperature, every $(d+1)$-dimensional theory can be reformulated as an effective theory in $d$ dimensions. We test the numerical accuracy of this Dimensional Reduction for (3+1)-dimensional SU(2) by comparing perturbatively determined effective couplings with lattice results as the temperature is progressively lowered. We observe an increasing disagreement between numerical and perturbative values from $T=4 T_c$ downwards, which may however be due to somewhat different implementations of dimensional reduction in the two cases.