The negative binomial distribution (NBD) has been theorized to express a scale-invariant property of many-body systems and has been consistently shown to outperform other statistical models in both describing the multiplicity of quantum-scale events in particle collision experiments and predicting the prevalence of cosmological observables, such as the number of galaxies in a region of space. Despite its widespread applicability and empirical success in these contexts, a theoretical justification for the NBD from first principles has remained elusive for fifty years. The accuracy of the NBD in modeling hadronic, leptonic, and semileptonic processes is suggestive of a highly general principle, which is yet to be understood. This study demonstrates that a statistical event of the NBD can in fact be derived in a general context via the dynamical equations of a canonical ensemble of particles in Minkowski space. These results describe a fundamental feature of many-body systems that is consistent with data from the ALICE and ATLAS experiments and provides an explanation for the emergence of the NBD in these multiplicity observations. Two methods are used to derive this correspondence: the Feynman path integral and a hypersurface parametrization of a propagating ensemble.

翻译：负二项分布(NBD)理论上能够表达多体系统的尺度不变性质,并且在描述粒子碰撞实验中量子尺度事件的多重数以及预测宇宙学观测量的普遍性(如空间区域中的星系数量)方面,始终优于其他统计模型。尽管在这些背景下具有广泛适用性和经验成功,但关于NBD的第一性原理理论论证在五十年间仍悬而未决。NBD在描述强子、轻子和半轻子过程中的准确性暗示着一个尚未被理解的高度普适性原理。本研究证明,通过闵可夫斯基空间中正则系综粒子的动力学方程,实际上可以在一般背景下推导出NBD的统计事件。这些结果描述了多体系统的一个基本特征,与ALICE和ATLAS实验的数据一致,并为NBD在这些多重数观测中的出现提供了解释。本研究采用两种方法推导这一对应关系:费曼路径积分和传播系综的超曲面参数化。