论文标题

来自$ \ bar {q} $方程的两环NMHV振幅的符号和字母

The symbol and alphabet of two-loop NMHV amplitudes from $\bar{Q}$ equations

论文作者

He, Song, Li, Zhenjie, Zhang, Chi

论文摘要

我们研究了平面$ {\ cal n} = 4 $ super-yang-mills的两环NMHV振幅的符号和字母,该符号来自$ \ bar {q} $方程,该方程提供了计算多环振幅的第一原理方法。从一环n $ {}^2 $ MHV比率函数开始,我们详细说明了如何使用$ \ bar {q} $方程来获取两回合$ n $ n $ n $ nmhv振幅的总差异,其符号包含kinematics of Kinematics for $ N \ geq geq 8 $ 8 $ 8 $的字母。我们为符号的一部分提供明确的公式,其中涉及各种多重性的代数字母,我们发现$ 17-2M $乘型独立的字母,用于革兰氏clot量的给定平方根,$ 0 \ leq m \ leq m \ leq 4 $,取决于涉及正方形根部的粒子的数量。我们还观察到,这些代数字母可以作为具有MHV或NMHV树的单层四质量领先奇异性的极点。作为我们代数结果的副产品,我们发现了一大批两环NMHV的组成部分,可以将其写入两个双乘形成式积分的差异,尤其是简单且没有方形根。例如,我们提供了$ n = 9 $的完整符号,其字母包含$ 59 \ times 9 $有理信件,除了$ 11 \ times 9 $独立代数。我们还为所有多重性提供了全环NMHV持久条件。

We study the symbol and the alphabet for two-loop NMHV amplitudes in planar ${\cal N}=4$ super-Yang-Mills from the $\bar{Q}$ equations, which provide a first-principle method for computing multi-loop amplitudes. Starting from one-loop N${}^2$MHV ratio functions, we explain in detail how to use $\bar{Q}$ equations to obtain the total differential of two-loop $n$-point NMHV amplitudes, whose symbol contains letters that are algebraic functions of kinematics for $n\geq 8$. We present explicit formula with nice patterns for the part of the symbol involving algebraic letters for all multiplicities, and we find $17-2m$ multiplicative-independent letters for a given square root of Gram determinant, with $0\leq m\leq 4$ depending on the number of particles involved in the square root. We also observe that these algebraic letters can be found as poles of one-loop four-mass leading singularities with MHV or NMHV trees. As a byproduct of our algebraic results, we find a large class of components of two-loop NMHV, which can be written as differences of two double-pentagon integrals, particularly simple and absent of square roots. As an example, we present the complete symbol for $n=9$ whose alphabet contains $59\times 9$ rational letters, in addition to the $11 \times 9$ independent algebraic ones. We also give all-loop NMHV last-entry conditions for all multiplicities.

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