动量空间中共形积分的解析延拓
Analytic Continuation of Conformal Integrals in Momentum Space
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中文总结 AI 辅助
本研究针对动量空间中无法在全物理运动学区域收敛的多重-K积分，利用括号方法推导了适配该区域的级数表示，得到了全区域收敛的三重-K积分双分支级数及一般多重-K积分的迭代级数。
中文摘要 AI 辅助
共形对称性对关联函数有强约束。在动量空间中，共形Ward恒等式通过三个贝塞尔函数的乘积积分（“三重-K积分”）求解三点函数；更一般地，这类积分（“多重-K积分”）是一大类更高点共形关联函数的构建块。这类积分属于广义超几何函数范畴，虽原则上已知其级数表示，但 none 在动量守恒选定的运动学空间整个物理区域内收敛。本研究中，我们构建了适配该物理域的级数表示。利用将定积分求值转化为求解线性代数方程组的括号方法，我们推导了多重-K积分的多种Lauricella型级数表示。对于通常以Appell F₄函数表示（其级数仅在三角不等式区域外收敛）的三重-K积分，我们发现了一种紧凑的双分支级数，其在整个物理区域及任意标度维度下均收敛。随后我们将该构建推广至一般多重-K积分：通过引入一组新的运动学变量，我们构建了一种对所有物理运动学构型均收敛的迭代级数表示。
英文摘要
Conformal symmetry strongly constrains correlation functions. In momentum space, the conformal Ward identities are solved for three-point functions by integrals of a product of three Bessel functions ("triple-$K$ integrals"); more generally, integrals of this type ("multiple-$K$ integrals"), serve as the building blocks of a wide class of higher-point conformal correlators. These integrals belong to the class of generalised hypergeometric functions, and while their series representations are known in principle, none converges throughout the entire physical region of kinematic space selected by momentum conservation. In this work, we construct series representations adapted to exactly this physical domain. Using the method of brackets, which turns the evaluation of definite integrals into solving a linear system of algebraic equations, we derive various Lauricella-type series representations for multiple-$K$ integrals. For triple-$K$ integrals, conventionally expressed in terms of the Appell $F_4$ function whose series converges only outside the triangle-inequality region, we find instead a compact, two-branch series that converges throughout the entire physical region and for arbitrary scaling dimensions. We then extend this construction to general multiple-$K$ integrals: by introducing a new set of kinematic variables, we build an iterative series representation that converges for all physical kinematic configurations.
