A summation rule for the four-coupling beta function of four-dimensional nonplanar Euclidean scalar φ4 theory under factorial coefficient bounds, and a domain on which the summed beta function is well defined
Abstract
Let c=(λ,α,μ,ν)∈ℝ4 denote the four running couplings of four-dimensional nonplanar Euclidean scalar φ4 theory, and let B(c)=∑p≥2∑|m|=pβ(m)cm be its formal beta-function series. Assuming the factorial coefficient bounds |β(m)|≤ (p−1)!Cp−1 for |m|=p, we construct an explicit summation rule 𝒮, given by a cut Borel--Laplace transform whose cut is placed by a scale-covariant real-analytic gauge. We prove that it assigns a well-defined real value to B on an explicit nonempty open domain, in fact on all of ℝ4. The summed function 𝒮B is linear and real, real-analytic on ℝ4∖{0}, C∞ on ℝ4 with Taylor series B at the origin, and Gevrey-$1$ asymptotic to B to all orders with an explicit remainder bound. At optimal truncation the remainder is O(e−η/(C|c|∞)) for any η<1. We show that 𝒮B reproduces the classical Borel sum and the ordinary sum of a convergent series up to errors of the same exponentially small order. Exact reproduction is impossible for any linear rule defined on the whole admissible class: regularity forces all weights to equal $1$, leading to divergence. Finally, we prove optimality under the stated hypothesis. Admissible series may diverge at every c≠0, and their Borel transforms may have the circle |τ|=1/(C|c|∞) as a natural boundary, so classical Borel-type summation need not be available on the class. Moreover, there exist two functions indistinguishable at the level of the coefficient bounds yet differing by exactly (2π/C)e−1/(C|c|∞). All constants are explicit and all arguments are self-contained.
Related
- Coefficient duality and interpolation of Euclidean flag Besov spaces
- The Levin Method for the Summation of One-dimensional and Multidimensional Infinite Series
- Cops and Robber game in higher-dimensional manifolds with spherical and Euclidean metric
- Pseudoconvex domains in complex Euclidean space with finite-dimensional Bergman spaces
- Two-Dimensional Euclidean Free Fermi Fields from Grassmannian Stochastic Analysis
BibTeX
@misc{zeng2026summation,
title = {{A summation rule for the four-coupling beta function of four-dimensional nonplanar Euclidean scalar $φ^4$ theory under factorial coefficient bounds, and a domain on which the summed beta function is well defined}},
author = {Puyu Zeng},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.29605},
eprint = {2609.29605},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.29605},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A summation rule for the four-coupling beta function of four-dimensional nonplanar Euclidean scalar φ^4 theory under factorial coefficient bounds, and a domain on which the summed beta function is well defined
AU - Puyu Zeng
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.29605
UR - https://arxiv.org/abs/2609.29605
AB - Let c=(λ,α,μ,ν)∈ℝ^4 denote the four running couplings of four-dimensional nonplanar Euclidean scalar φ^4 theory, and let B(c)=∑_p≥2∑_|m|=pβ(m)c^m be its formal beta-function series. Assuming the factorial coefficient bounds |β(m)|≤ (p−1)!C^p−1 for |m|=p, we construct an explicit summation rule 𝒮, given by a cut Borel--Laplace transform whose cut is placed by a scale-covariant real-analytic gauge. We prove that it assigns a well-defined real value to B on an explicit nonempty open domain, in fact on all of ℝ^4. The summed function 𝒮B is linear and real, real-analytic on ℝ^4∖{0}, C^∞ on ℝ^4 with Taylor series B at the origin, and Gevrey-$1$ asymptotic to B to all orders with an explicit remainder bound. At optimal truncation the remainder is O(e^−η/(C|c|_∞)) for any η<1. We show that 𝒮B reproduces the classical Borel sum and the ordinary sum of a convergent series up to errors of the same exponentially small order. Exact reproduction is impossible for any linear rule defined on the whole admissible class: regularity forces all weights to equal $1$, leading to divergence. Finally, we prove optimality under the stated hypothesis. Admissible series may diverge at every c≠0, and their Borel transforms may have the circle |τ|=1/(C|c|_∞) as a natural boundary, so classical Borel-type summation need not be available on the class. Moreover, there exist two functions indistinguishable at the level of the coefficient bounds yet differing by exactly (2π/C)e^−1/(C|c|_∞). All constants are explicit and all arguments are self-contained.
ER -