  For a long time, the predictive limits of perturbative quantum field theory
have been limited by our inability to carry out loop calculations to
arbitrarily high order, which become increasingly complex as the order of
perturbation theory is increased. This problem is exacerbated by the fact that
perturbation series derived from loop diagram (Feynman diagram) calculations
represent asymptotic (divergent) series which limits the predictive power of
perturbative quantum field theory. Here, we discuss an ansatz which could
overcome these limits, based on the observations that (i) for many
phenomenologically relevant field theories, one can derive dispersion relations
which relate the large-order growth (the asymptotic limit of "infinite loop
order") with the imaginary part of arbitrary correlation functions, for
negative coupling ("unstable vacuum"), and (ii) one can analyze the imaginary
part for negative coupling in terms of classical field configurations
(instantons). Unfortunately, the perturbation theory around instantons, which
could lead to much more accurate predictions for the large-order behavior of
Feynman diagrams, poses a number of technical as well as computational
difficulties. Here, we study, to further the above mentioned ansatz,
correlation functions in a one-dimensional (1D) field theory with a quartic
self-interaction and an O(N) internal symmetry group, otherwise known as the 1D
N-vector model. Our focus is on corrections to the large-order growth of
perturbative coefficients, i.e., the limit of a large number of loops in the
Feynman diagram expansion. We evaluate, in momentum space, the two-loop
corrections for the two-point correlation function, and its derivative with
respect to the momentum, as well as the two-point correlation function with a
wigglet insertion. Also, we study the four-point function.
