For a long time, 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 (divergent) series which limits the predictive power of perturbative quantum field theory. Here, we discuss an ansatz that 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. These quantities, computed at zero momentum transfer, enter the renormalization-group functions (Callan-Symanzik equation) of the model. Our calculations pave the way for further development of related methods in field theory and for a better understanding of field-theoretical expansions at large order.
Two-loop corrections to the large-order behavior of correlation functions in the one-dimensional N -vector model
Malatesta E. M.;
2020
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 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 (divergent) series which limits the predictive power of perturbative quantum field theory. Here, we discuss an ansatz that 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. These quantities, computed at zero momentum transfer, enter the renormalization-group functions (Callan-Symanzik equation) of the model. Our calculations pave the way for further development of related methods in field theory and for a better understanding of field-theoretical expansions at large order.
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