We derive a recursion relation for loop-level scattering amplitudes of Lagrangian field theories that generalises the tree-level Berends-Giele recursion relation in Yang-Mills theory. The origin of this recursion relation is the homological perturbation lemma, which allows us to compute scattering amplitudes from minimal models of quantum homotopy algebras in a recursive way. As an application of our techniques, we give an alternative proof of the relation between non-planar and planar colour-stripped scattering amplitudes.

中文翻译：

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环路振幅和量子同伦代数

我们推导出拉格朗日场理论的循环级散射幅度的递归关系，该关系概括了杨-米尔斯理论中的树级 Berends-Giele 递归关系。这种递归关系的起源是同调微扰引理，它使我们能够以递归方式从量子同伦代数的最小模型中计算散射幅度。作为我们技术的应用，我们给出了非平面和平面彩色剥离散射幅度之间关系的替代证明。