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Triviality vs perturbation theory: an analysis for mean-field $\varphi^4$-theory in four dimensions

Christoph Kopper, Pierre Wang

TL;DR

The work addresses the interplay between triviality and perturbation theory in the four-dimensional mean-field limit of the $\varphi^4$ theory. By employing renormalization-group flow equations with a UV cutoff, the authors define a renormalized coupling $\tilde{g}$ and derive perturbative expansions of the mean-field CAS functions that remain locally Borel summable. They establish compatibility of renormalization conditions (including BPHZ) and obtain explicit bounds on perturbative coefficients and remainders, showing that the perturbative series is asymptotic to the nonperturbative mean-field solution. The results provide a rigorous link between the perturbative and nonperturbative pictures in the mean-field truncation and confirm local Borel summability in the presence of a finite UV regulator, highlighting the continued relevance of perturbative methods within a trivially renormalized setting.

Abstract

We have constructed the mean-field trivial solution of the $\varphi^4$ theory $O(N)$ model in four dimensions in two previous papers using the flow equations of the renormalization group. Here we establish a relation between the trivial solutions we constructed and perturbation theory. We show that if an UV-cutoff is maintained, we can define a renormalized coupling constant $g$ and obtain the perturbative solutions of the mean-field flow equations at each order in perturbation theory. We prove the local Borel-summability of the renormalized mean-field perturbation theory in the presence of an UV cutoff and show that it is asymptotic to the non-perturbative solution.

Triviality vs perturbation theory: an analysis for mean-field $\varphi^4$-theory in four dimensions

TL;DR

The work addresses the interplay between triviality and perturbation theory in the four-dimensional mean-field limit of the theory. By employing renormalization-group flow equations with a UV cutoff, the authors define a renormalized coupling and derive perturbative expansions of the mean-field CAS functions that remain locally Borel summable. They establish compatibility of renormalization conditions (including BPHZ) and obtain explicit bounds on perturbative coefficients and remainders, showing that the perturbative series is asymptotic to the nonperturbative mean-field solution. The results provide a rigorous link between the perturbative and nonperturbative pictures in the mean-field truncation and confirm local Borel summability in the presence of a finite UV regulator, highlighting the continued relevance of perturbative methods within a trivially renormalized setting.

Abstract

We have constructed the mean-field trivial solution of the theory model in four dimensions in two previous papers using the flow equations of the renormalization group. Here we establish a relation between the trivial solutions we constructed and perturbation theory. We show that if an UV-cutoff is maintained, we can define a renormalized coupling constant and obtain the perturbative solutions of the mean-field flow equations at each order in perturbation theory. We prove the local Borel-summability of the renormalized mean-field perturbation theory in the presence of an UV cutoff and show that it is asymptotic to the non-perturbative solution.

Paper Structure

This paper contains 25 sections, 38 theorems, 217 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The flow equations in the mean-field approximation
  3. The non-perturbative mean-field flow equations
  4. Perturbative Flow equations in the mean-field approximation
  5. Mean-field flow equations for $\mathcal{A}_{n,j}^{\alpha_0,\alpha}$
  6. Mean-field flow equations for the remainder $\Delta f_{n}^{K+1}$
  7. The perturbative expansion of the trivial solution
  8. The perturbative expansion of the trivial solution
  9. Compatibility of the renormalization conditions
  10. The perturbative expansion of the regularized renormalized mean-field two-point function
  11. Local Borel summability of the mean-field regularized renormalized perturbation theory
  12. Local existence of the Borel transform for a real coupling and estimates on $\Delta f_n^{K+1}(\mu)$
  13. Local Borel summability of the regularized renormalized mean-field perturbation theory
  14. Generalities
  15. Properties of Gaussian measures
  16. ...and 10 more sections

Key Result

Proposition 2.1

There exists $\tilde{C}\equiv \tilde{C}(c_{0,2},c_{0,4})>1$ and $r<1$ such that

Figures (2)

  • Figure 1: The region of analyticity of the Borel-summable function
  • Figure 2: The region of analyticity of the Borel transform of a function satisfying the assumptions of Nevanlinna-Sokal theorem.

Theorems & Definitions (73)

  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • Proposition 2.4
  • proof
  • Proposition 2.5
  • proof
  • Proposition 2.6
  • ...and 63 more