Optimized $\delta $ - Expansion for Lattice U(1) and SU(2) with Interpolating Continuum Action

Embedding the lattice gauge theory into a continuum theory allows to use the continuum action as trial action in the variational calculation. Only originally divergent graphs contribute. This leads to a very simple scheme which makes it possible to write down explicit expressions for the plaquette energy $E$ for U(1) in arbitrary space time dimension for the first three orders of the expansion. For dimensions three and four one can even go up to fourth order. This allows a rather thorough empirical investigation of the convergence properties of the $\delta$-expansion, in particular near the phase transition or the transition region, respectively. As already found in previous work, the principle of minimal sensitivity can be only applied for $\beta$ above a certain value, because otherwise no extremum with respect to the variational parameter exists. One can, however, extend the range of applicability down to small $\beta$, by calculating instead of $E$ some power $E^\kappa$, or by performing an appropriate Pad\'e transformation. We find excellent agreement with the data for $\beta$ above the transition region for the second and higher orders. Below the transition region the agreement is rather poor in low orders, but quite impressive in fourth order. For SU(2) we performed the calculation up to second order. The agreement with the data is somewhat worse than in the abelian case.