New optimization methods for converging perturbative series with a field cutoff
Abstract
We take advantage of the fact that, in λφ^{4} problems, a large field cutoff φ_{max} makes a perturbative series converge toward values exponentially close to the exact values to make optimal choices of φ_{max}. For a perturbative series terminated at even order, it is in principle possible to adjust φ_{max} in order to obtain the exact result. For a perturbative series terminated at odd order, the error can only be minimized. It is, however, possible to introduce a mass shift m^{2}→m^{2}(1+η) in order to obtain the exact result. We discuss weak and strong coupling methods to determine φ_{max} and η. The numerical calculations in this article are performed with a simple integral with one variable. We give arguments indicating that the qualitative features observed should extend to quantum mechanics and quantum field theory. We find that optimization at even order is more efficient than optimization at odd order. We compare our methods with the linear δ expansion (LDE) (combined with the principle of minimal sensitivity), which provides an upper envelope for the accuracy curves of various Padé and PadéBorel approximants. Our optimization method performs better than the LDE at strong and intermediate coupling, but not at weak coupling, where it appears less robust and subject to further improvements. We also show that it is possible to fix the arbitrary parameter appearing in the LDE using the strong coupling expansion, in order to get accuracies comparable to ours.
 Publication:

Physical Review D
 Pub Date:
 February 2004
 DOI:
 10.1103/PhysRevD.69.045014
 arXiv:
 arXiv:hepth/0309022
 Bibcode:
 2004PhRvD..69d5014K
 Keywords:

 11.10.Ef;
 11.15.Bt;
 12.38.Cy;
 31.15.Md;
 Lagrangian and Hamiltonian approach;
 General properties of perturbation theory;
 Summation of perturbation theory;
 Perturbation theory;
 High Energy Physics  Theory;
 Condensed Matter  Statistical Mechanics;
 High Energy Physics  Lattice;
 High Energy Physics  Phenomenology;
 Quantum Physics
 EPrint:
 10 pages, 16 figures, uses revtex