The fractal self-similar-Borel algorithm with applications to the specific heat exponent of XY model and the effective potential of
$(\\frac{g}{4}\\phi^{4})_{1+1}$ Scalar Field Theory
Abouzeid Shalaby
Abstract
We develop a novel algorithm for the resummation of a divergent series. The algorithm uses the ideas of Borel resummation as well as self similar non-perturbarive methods. The precision of that algorithm has been tested through the calculation of the critical exponents of the XY model as well as the calculation of the critical coupling of the of $(\\frac{g}{4}\\phi^{4})_{1+1}$ Scalar Field Theory. The result has been compared with lattice calculations for the critical coupling and a very good agreement has been found. For the critical exponents of the XY model our algorithm is consistent with the best experimental result obtained so far for the α exponent of the specific-heat peak in superfluid helium, found in a satellite experiment with a temperature resolution of nanoKelvin.