标题： 散射振幅的正性性质 显示英文标题

标题： Positivity properties of scattering amplitudes

摘要： 我们研究量子场论(QFT)中的正性性质。 我们提供了证据，并在某些情况下给出了证明，表明散射振幅的许多基本组成部分，以及在某些情况下完整的振幅，满足无限多个正性条件：在指定的运动学区域内，这些函数及其所有带符号的导数都是非负的。 这类函数在数学文献中被称为完全单调(CM)。 通过积分表示来验证完全单调性是一种强大的方法。 因此，我们证明了这种方法适用于具有欧几里得区域的平面和非平面费曼积分，以及与宇宙学关联函数和弦论积分相关的某些欧拉积分。 这意味着，例如，微扰理论中的许多基本组成部分，如主积分，可以选择为完全单调的。 我们还讨论了两种展示完整振幅完全单调性的途径。 一种与解析S矩阵的性质有关。 另一种是CM性质与正几何之间的密切联系。 受此启发，我们研究了平面最大超对称杨-米尔斯理论中的正性性质。 我们基于已知的解析多环结果提供了证据，表明CM性质扩展到了该理论中的几个物理量。 这包括六粒子最大螺旋性破坏(MHV)振幅的适当归一化的有限余项函数、库仑分支上的四点散射振幅、四点关联函数，以及角度依赖的尖点异常维数。 然而，我们的发现不仅限于超对称理论。 结果显示，CM性质分别在QCD和QED的尖点异常维数中成立，分别为三环和四环。 我们评论了一些开放性问题，并讨论了完全单调性的可能数值应用。 显示英文摘要

摘要： We investigate positivity properties in quantum field theory (QFT). We provide evidence,and in some case proofs, that many building blocks of scattering amplitudes, and in some cases the full amplitudes, satisfy an infinite number of positivity conditions: the functions, as well as all their signed derivatives, are non-negative in a specified kinematic region. Such functions are known as completely monotonic(CM) in the mathematics literature. A powerful way to certify complete monotonicity is via integral representations. We thus show that it applies to planar and non-planar Feynman integrals possessing a Euclidean region,as well as to certain Euler integrals relevant to cosmological correlators and stringy integrals. This implies that in particular that many basic building blocks of perturbation theory, such as master integrals, can be chosen to be completely monotone. We also discuss two pathways for showing complete monotonicity for full amplitudes. One is related to properties of the analytic S-matrix. The other one is a close connection between the CM property and Positive Geometry. Motivated by this, we investigate positivity properties in planar maximally supersymmetric Yang-Mills theory. We present evidence, based on known analytic multi-loop results, that the CM property extends to several physical quantities in this theory. This includes the (suitably normalized) finite remainder function of the six-particle maximally-helicity-violating (MHV) amplitude, four-point scattering amplitudes on the Coulomb branch,four-point correlation functions, as well as the angle-dependent cusp anomalous dimension. Our findings are however not limited to supersymmetric theories. It is shown that the CM property holds for the QCD and QED cusp anomalous dimensions, to three and four loops, respectively. We comment on open questions, and on possible numerical applications of complete monotonicity.