四维非平面欧几里得标量 $\varphi^{4}$ 理论在阶乘系数界下的四耦合 beta 函数求和规则，以及求和后 beta 函数良定义的定义域
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
- College of Cryptology and Cyber Science, Nankai University（南开大学密码学与网络科学学院）
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AI总结：
针对四维非平面欧几里得标量 $\varphi^{4}$ 理论，在阶乘系数界下构造显式求和规则 $\mathcal{S}$，证明其在 $\mathbb{R}^{4}$ 上良定义且复现经典 Borel 和，并证明最优性与不可精确复现性。
AI中文摘要：
设 $\mathbf{c}=(\lambda,\alpha,\mu,\nu)\in\mathbb{R}^{4}$ 表示四维非平面欧几里得标量 $\varphi^{4}$ 理论的四个运行耦合，并设 $B(\mathbf{c})=\sum_{p\ge2}\sum_{|\mathbf{m}|=p}\beta(\mathbf{m})\mathbf{c}^{\mathbf{m}}$ 为其形式 beta 函数级数。假设阶乘系数界 $|\beta(\mathbf{m})|\le (p-1)!C^{p-1}$ 对 $|\mathbf{m}|=p$ 成立，我们构造一个显式求和规则 $\mathcal{S}$，由带割线的 Borel--Laplace 变换给出，其割线由尺度协变的实解析规范放置。我们证明该规则在显式非空开域上，实际上在全部 $\mathbb{R}^{4}$ 上，为 $B$ 赋予良定义的实数值。求和函数 $\mathcal{S}B$ 是线性的且为实的，在 $\mathbb{R}^{4}\setminus\{0\}$ 上实解析，在 $\mathbb{R}^{4}$ 上为 $C^\infty$，在原点具有 Taylor 级数 $B$，并且对所有阶具有显式余项界的 Gevrey-$1$ 渐近到 $B$。在最优截断处，对任意 $\eta<1$，余项为 $O(e^{-\eta/(C\\|\mathbf{c}\\|_\infty)})$。我们证明 $\mathcal{S}B$ 复现了经典 Borel 和以及收敛级数的普通和，误差为相同指数小阶。对于定义在整个可容许类上的任何线性规则，精确复现是不可能的：正则性迫使所有权重等于 $1$，导致发散。最后，我们在所述假设下证明最优性。可容许级数可能在每个 $\mathbf{c}\ne0$ 处发散，且其 Borel 变换可能以圆 $|\tau|=1/(C\\|\mathbf{c}\\|_\infty)$ 为自然边界，因此经典 Borel 型求和在类上不一定可用。此外，存在两个函数在系数界层面不可区分，但恰好相差 $(2\pi/C)e^{-1/(C\\|\mathbf{c}\\|_\infty)}$。所有常数都是显式的，所有论证都是自洽的。
英文摘要：
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.
