Abstract
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.
Highlights
The predictive limits of perturbative quantum field theory have been limited by our inability to carry out loop calculations to an 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 series which limits the predictive power of perturbative quantum field theory
Even though the problem of the calculation of the functional determinant has been outlined in Ref. [18], we here revisit the derivation, with an emphasis on those aspects of the path integral Jacobian that are important for the calculation of correlation functions
Summary
We use a formulation of the functional determinant [18], which allows us to separate the path integral around the nontrivial (instanton) saddle point into integrals over the collective coordinates (the start point of the instanton and the variables characterizing the internal space of the theory), as well as integrals over the transverse fluctuations around the nontrivial saddle point, in the internal symmetry group. In one dimension, the field fluctuations are not strong enough to induce a phase transition, we clarify the connection of our calculations to the quantities entering the renormalizationgroup (RG) equations in Appendix A. III, we continue with the calculation of the two-point and four-point functions as well as the derivative of the two-point function at zero momentum transfer and the wigglet insertion All of these functions enter the Callan-Symanzik [24,25] RG equations.
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