Perturbative expansions with a large field cutoff

• For two non-trivial problems (anharmonic oscillator and hierarchical model) improved perturbative series can be obtained by cutting off the large field contributions.
• The modified series converge to values exponentially close to the exact ones.
• For larger than some critical value, the method outperforms Padé's approximants and Borel summations.
• The method can also be used for series which are not Borel summable such as the double-well potential series.

    
  Figure 2: Number of significant digits for the double-well at order 3 to 6 for regular perturbation (black) compared to series obtained with and (blue) or (green). As the order increases, the black curves reach the one-instanton contribution (red) over wider regions to the left while the two other sets reach the accuracy level obtained numerically for (purple) or (brown).
  

• Semi-classical methods can be used to calculate the modified Feynman rules, estimate the error and optimize the field cutoff.
• More details in: Y. Meurice, Simple Method to Make Asymptotic Series of Feynman Diagrams Converge, Phys. Rev. Lett. 88, 141601 (2002).

Work in Progress, Plans:

• We are working on various methods to calculate the coefficients of the modified perturbative series.
• For one dimensional problems, we found a perturbative version of the accurate numerical methods which motivated this approach.
• We were able to reproduce these results using Monte Carlo methods. This is illustrated below for the first perturbative coefficient.

   
  Figure 3: First perturbative coefficient for different field cutoffs
  

• We were able to calculate the first six perturbative coefficients with this method.
• We are planning to use similar in higher dimension provided that our proposal for 16 node cluster submitted to the University gets funded.
• Our goal will be to improve the calculation of the critical exponents obtained with conventional field theory methods.
• Brian Kessler has been working on the comparison between the field cutoff method and variational methods based on the -expansion. He found that the optimal field cutoff and the optimal variational parameter obey simple scaling relation in the limits of strong and weak couplings.