Evaluating critical exponents in the optimized perturbation theory

We use the optimized perturbation theory, or linear δ expansion, to evaluate the critical exponents in the critical 3d O(N) invariant scalar field model. Regarding the implementation procedure, this is the first successful attempt to use the method in this type of evaluation. We present and discuss all the associated subtleties producing a prescription which can, in principle, be extended to higher orders in a consistent way. Numerically, our approach, taken at the lowest nontrivial order (second order) in the δ expansion produces a modest improvement in comparison to mean field values for the anomalous dimension η and correlation length ν critical exponents. However, it nevertheless points to the right direction of the values obtained with other methods, like the e-expansion. We discuss the possibilities of improving over our lowest-order results and on the convergence to the known values when extending the method to higher orders.