Proof of a modular relation between 1-, 2- and 3-loop Feynman diagrams on a torus
Proof of a modular relation between 1-, 2- and 3-loop Feynman diagrams on a torus
复制标题
环面上 1、2 和 3 环费曼图之间模关系的证明
DOI：
10.1016/j.jnt.2017.07.022
复制
发表时间：
2015
影响因子：
0.7
通讯作者：
P. Vanhove
中科院分区：
文献类型：
--
作者：
E. D'hoker;M. Green;P. Vanhove
通过对环面复结构模上的非全纯模函数和的积分，确定了一类引力子II型超弦散射振幅的低能展开中高导数项的系数。在四重子振幅的情况下，这些模函数中的每一个都是与环面上自由无质量标量场的费曼图相关联的多重和。每个图中的线连接了对顶点插入点，线的数量定义了它的权值w，这对应于它在低能扩展中的顺序。先前关于一类四引力子振幅的低能量膨胀的结果导致了一些模函数之间的猜想关系，但是不同数量的环≤w−1。在本文中，我们将证明这些猜想关系中最简单的一个，即在权值w= 4时出现的关系，它用一环和二环模函数来表示三环模函数d4。作为一个副产品，我们证明了三个有趣的全纯模恒等式。
The coefficients of the higher-derivative terms in the low energy expansion of genus-one graviton Type II superstring scattering amplitudes are determined by integrating sums of non-holomorphic modular functions over the complex structure modulus of a torus. In the case of the four-graviton amplitude, each of these modular functions is a multiple sum associated with a Feynman diagram for a free massless scalar field on the torus. The lines in each diagram join pairs of vertex insertion points and the number of lines defines its weight w, which corresponds to its order in the low energy expansion. Previous results concerning the low energy expansion of the genus-one four-graviton amplitude led to a number of conjectured relations between modular functions of a given w, but different numbers of loops≤ w− 1. In this paper we shall prove the simplest of these conjectured relations, namely the one that arises at weight w= 4 and expresses the three-loop modular function D 4 in terms of modular functions with one and two loops. As a byproduct, we prove three intriguing new holomorphic modular identities.