A 'Twistor String' Inspired Formula For Tree-Level Scattering Amplitudes in N=8 SUGRA
A 'Twistor String' Inspired Formula For Tree-Level Scattering Amplitudes in N=8 SUGRA
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发表时间：
2012-06
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通讯作者：
F. Cachazo;Yvonne Geyer
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作者：
F. Cachazo;Yvonne Geyer
我们提出了N = 8超重力的完全树级s矩阵的一个新公式。在k - r电荷扇区中n个粒子的新公式是格拉斯曼G（2,n）上的积分，并使用维罗内塞映射到G（k,n）。在G（2,n）中的点的像需要在2 b| 8平面的“补”中，从而使SU(8) r -对称得以体现。被积函数是两个行列式之比。分子是Hodges最近的MHV振幅行列式公式的类比。分母是2（n+k−2）×2（n+k）阶2（n+k−2）矩阵的次元×2（n+k−2）。正如霍奇斯公式对MHV振幅所做的那样，我们的被积函数使Sn下的完全不变性在所有部门都表现出来。新公式的正确性来自两个令人惊讶的事实。一个是在允许运动不变量以一种新颖的方式离壳时，Hodges的MHV公式与kawai - lewellen - type （KLT）公式的等价性。对于任意数量的粒子，我们给出了证明。第二个事实是定义维罗内塞嵌入的多项式方程的解的正交性。明确证明正交性的所有振幅在所有的r电荷扇区与八个或更少的粒子，从而为我们的建议提供非平凡的证据。
We propose a new formulation of the complete tree-level S-matrix of N = 8 supergravity. The new formula for n particles in the k R-charge sector is an integral over the Grassmannian G(2,n) and uses the Veronese map into G(k,n). The image of a point in G(2,n) is required to be in the “complement” of a 2|8-plane thus making the SU(8) R-symmetry manifest. The integrand is the ratio of two determinants. The numerator is an analog of Hodges’ recent determinant formula for MHV amplitudes. The denominator is a 2(n+k −2)×2(n+k −2) minor of a 2(n+k)×2(n+k) matrix of rank 2(n+k −2). Just as Hodges’ formula does for MHV amplitudes, our integrand makes the complete invariance under Sn manifest for all sectors. The validity of the new formula follows from two surprising facts. One is the equivalence of Hodges’ MHV formula and the Kawai-Lewellen-Tye (KLT) formula when kinematic invariants are allowed to be off-shell in a novel way. We give a proof of this for any number of particles. The second fact is an orthogonality property of the solutions to the polynomial equations defining the Veronese embedding. Explicit proof of the orthogonality is given for all amplitudes in all R-charge sectors with eight or less particles thus providing non-trivial evidence for our proposal.