色对偶切割分块的一项练习：从正性出发的$\u003cmathcal{N}\u003e=8$超引力
An exercise in Color-Dual Cut Tiling: $\mathcal{N}=8$ Supergravity from Positivity
AI总结：
本文提出色对偶切割分块算法，证明树级BCJ关系可减少圈级散射振幅计算的显式切割数，用平面切割生成三圈最大超对称规范理论的非平面扇区，并通过广义双复制得到最大超对称引力表示。
摘要：
色与运动学之间的BCJ对偶为多圈散射振幅的计算带来两大优势。首先，为将被积函数固定为规范理论形式所需进行的有序切割数量极少——在所有已知案例中，这将呈阶乘增长的费曼图数量缩减至个位数的基。其次，通过双复制方法可从规范理论振幅简单生成相关的引力振幅。越来越多的证据表明，即便半经典理论满足色对偶性，仍存在一些不存在局域色对偶表示的情况。那么我们能否不在被积函数层面显式体现对偶性的前提下，仍然简化计算呢？本文在此方向上迈出了重要一步，证明了树级BCJ关系的满足足以大幅减少必须执行的显式切割数量，即便圈级关系并未被显式满足。我们提出了一种凝聚算法——色对偶切割分块（color-dual cut tiling），该算法通过对离壳被积函数信息的种子施加最简单的在壳条件，识别并构建出完整的被积函数。具体而言，我们证明对于最大超对称规范理论中的三圈2对2散射，完全由正性约束决定的平面切割中包含的信息足以生成所有非平面扇区。此外，我们利用广义双复制方法，生成了最大超对称引力作为平面SYM输入泛函的一种表示。我们讨论了该过程可能的推广方式，最后评述了该方法在其他显式幺正切割计算成本高昂的相关感兴趣案例中的适用性。
英文摘要：
The BCJ duality between color and kinematics brings two advantages to calculating multi-loop scattering amplitudes. First the number of ordered cuts that need to be performed to fix the integrand to a gauge theory is minimal -- reducing the factorially-growing number of diagrams to a single-digit basis in all known cases. Second is the trivial generation of related gravitational amplitudes from gauge theory amplitudes via double-copy. Mounting evidence suggests there are some cases where no local color-dual representations exist, even when the semi-classical theory is color-dual. Can we still simplify the calculations without making the duality manifest at the level of the integrand? Here we take a non-trivial step in this direction by showing that the satisfaction of tree-level BCJ relations is sufficient to dramatically reduce the number of explicit cuts that must be performed, even when the loop-level relations are not explicitly satisfied. We introduce an agglomerative algorithm, color-dual cut tiling, that identifies and builds the entire integrand from the simplest on-shell conditions applied to a seed of off-shell integrand information. Specifically, we demonstrate that for two-to-two scattering at three loops in the maximally supersymmetric gauge theory there is sufficient information contained in planar cuts -- completely determined by positivity constraints -- to generate all of the non-planar sector. Additionally, we make use of the generalized double copy to generate a representation of maximally supersymmetric gravity as a functional of the planar SYM input. We discuss how the process might generalize, and then close by commenting on the applicability of this method for additional cases of interest where performing explicit unitarity cuts is expensive.
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