论文标题
动量内核的谎言括号
A Lie bracket for the momentum kernel
论文作者
论文摘要
我们开发了新的数学工具，用于研究使用Lie多项式属性的树级散射幅度的双拷贝和颜色界二元性。我们表明，定义的$ S $地图是简化超阳的 - 米尔斯多颗粒超级场所实际上是谎言多项式双重空间上的一个新的谎言括号。我们基于Berends-Giele递归递归的双节标量表树振幅，介绍{\它是多项式电流}，该振幅占据了多项式。野外理论幅度幅度是从分子振幅中获得的，该分子是从自由谎言代数到运动学数据的同态的分子振幅。示出了针对BIADJOINT标量，Yang-Mills理论和非线性Sigma模型的示例。这些理论满足了伯恩 - 卡拉斯科 - 约翰逊的振幅关系，遵循了我们证明的谎言多项式振幅和BCJ分子的存在所证明的身份。 通过嵌套S-Map Lie支架获得了从谎言多项式到其双重的KLT图。该地图的矩阵元素产生了最近提出的“广义klt矩阵”，当该矩阵的条目仅限于基础时，这将减少为通常的KLT矩阵。使用此功能，我们给出了代数证明，以取消KLT公式中的双极用于重力振幅。从这个角度来看，我们在分子和颜色基因二元方面发表了一些评论。
We develop new mathematical tools for the study of the double copy and colour-kinematics duality for tree-level scattering amplitudes using the properties of Lie polynomials. We show that the $S$-map that was defined to simplify super-Yang--Mills multiparticle superfields is in fact a new Lie bracket on the dual space of Lie polynomials. We introduce {\it Lie polynomial currents} based on Berends-Giele recursion for biadjoint scalar tree amplitudes that take values in Lie polynomials. Field theory amplitudes are obtained from the Lie polynomial amplitudes by numerators characterized as homomorphisms from the free Lie algebra to kinematic data. Examples are presented for the biadjoint scalar, Yang--Mills theory and the nonlinear sigma model. That these theories satisfy the Bern-Carrasco-Johansson amplitude relations follows from the identities we prove for the Lie polynomial amplitudes and the existence of BCJ numerators. A KLT map from Lie polynomials to their dual is obtained by nesting the S-map Lie bracket; the matrix elements of this map yield a recently proposed `generalized KLT matrix', and this reduces to the usual KLT matrix when its entries are restricted to a basis. Using this, we give an algebraic proof for the cancellation of double poles in the KLT formula for gravity amplitudes. We finish with some remarks on numerators and colour-kinematics duality from this perspective.
