Reduction of theN-component scalar model at the two-loop level

Dimensional reduction of high temperature field theories improves IR features of their perturbative treatment. A crucial question is the following: What three-dimensional theory is representing the full system the most faithful way? A careful investigation of the induced three-dimensional counterterm structure of the finite temperature 4D $\mathrm{O}\mathrm{}\left(N\right)\mathrm{}$ symmetric scalar theory at the two-loop level leads to proposing the presence of nonlocal operators in the effective theory. A three-dimensional matching process is applied for the construction of the optimal local, superrenormalizable approximation.