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

Abstract

Let $\mathbf{c}=(λ,α,μ,ν)\in\mathbb{R}^{4}$ denote the four running couplings of four-dimensional nonplanar Euclidean scalar $φ^{4}$ theory, and let $B(\mathbf{c})=\sum_{p\ge2}\sum_{|\mathbf{m}|=p}β(\mathbf{m})\mathbf{c}^{\mathbf{m}}$ be its formal beta-function series. Assuming the factorial coefficient bounds $|β(\mathbf{m})|\le (p-1)!C^{p-1}$ for $|\mathbf{m}|=p$, we construct an explicit summation rule $\mathcal{S}$, 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 $\mathbb{R}^{4}$. The summed function $\mathcal{S}B$ is linear and real, real-analytic on $\mathbb{R}^{4}\setminus\{0\}$, $C^\infty$ on $\mathbb{R}^{4}$ 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 $O(e^{-η/(C\|\mathbf{c}\|_\infty)})$ for any $η<1$. We show that $\mathcal{S}B$ 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 $\mathbf{c}\ne0$, and their Borel transforms may have the circle $|τ|=1/(C\|\mathbf{c}\|_\infty)$ 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 $(2π/C)e^{-1/(C\|\mathbf{c}\|_\infty)}$. All constants are explicit and all arguments are self-contained.