The fractal self-similar-Borel algorithm for the effective potential of the scalar field theory in one time plus one space dimensions

We develop a new resummation algorithm that can give accurate results for the resummation of a divergent series. First, we use a Borel resummation algorithm, originally due to Kleinert, Thoms and Janke, which has the advantage of using the strong coupling as well as the large order behaviors rather than the conventional resummation techniques which use only the large order behavior. The obtained series is supplemented by the fractal self-similar method. We applied the new algorithm to the effective potential of the (g4)(ϕ4)1+1 scalar field theory. We were able to predict the critical reduced temperature 1Gc≈φ=5-12, which reflects the fractal structure of the magnetization at the critical point. Also, our prediction is in complete agreement with lattice calculations in spite of the use of a relatively short input perturbation series which is up to g3 order.