双伴随振幅与Trop G(3,7)、X(3,7)方程的1272个解解析

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本文是一篇关于双伴随振幅、Trop G（3，7）和X（3，7）散射方程的深入分析文章，发表在《Journal of High Energy Physics》(JHEP)第4期（2020年）。作者Freddy Cachazo和Jairo M. Rojas探讨了热带格拉斯曼面(Tropical Grassmannians)与菲恩曼图(Feynman diagrams)的最新联系，这些在量子场论中起着关键作用。他们聚焦于n=7和k=3的特殊情况，即所谓的"biadjoint amplitudes"，这是一种在多粒子相互作用中的重要数学结构。在研究中，作者利用这个新发现的关系，对七点在复-projective空间CP2上的X(3,7)配置空间进行了详尽的探讨。X(3,7)的散射方程是描述物理系统中粒子碰撞过程的关键方程。通过两步法求解，他们首先实现了高效精确地获取1162个显式解决方案，这在计算上是一项关键成就。在第二步中，他们利用Trop G（3，7）的小面技术计算了360×360的双伴随振幅矩阵，并通过减去已知的1162个解的结果，得到了一个矩阵。通过对矩阵的秩进行计算，他们得出的结果是110，这表明除了明确的1162个解外，还有额外的110个解存在。这一发现对于理解和验证理论物理中的粒子交互模型具有重要意义，因为它揭示了更深层次的对称性和数学结构。这篇论文不仅提供了计算方法，还揭示了新的理论洞察，有助于推动量子场论和高能物理学领域的发展。其开放存取的特性使得研究成果能够广泛传播和进一步研究。整体而言，这篇文章为相关领域的研究人员提供了一个宝贵的工具和新视角，促进了未来在双伴随振幅和量子散射理论方面的深入探索。

JHEP04(2020)176

Having a higher k version of the scattering equations, one can immediately generalize

the Cachazo-He-Yuan (CHY) formula for biadjoint amplitudes [7]. Such amplitudes can

then be expanded in terms of higher k Feynman diagrams as deﬁned by facets of the tropical

Grassmanian [5]. Even though a direct quantum ﬁeld theory derivation of such generalized

Feynman diagrams is not yet known, it is important to mention that the known physically

relevant case of k = 2 implies via the duality X(2, n) ∼ X(n − 2, n) that k = n − 2 is also

physically relevant. It is therefore very natural to complete the spectrum of possibilities

between the two ends, i.e., for 2 < k < n − 2. At the very least one can expect that a good

understanding of such cases would deepen that of the k = 2 and k = n − 2 cases.

Given two orderings α and β one deﬁnes [5]

(3)

(α|β) :=

vol(SL(3, C))

a=1



∂S

∂x





∂S

∂y



(3)

(α)PT

(3)

(β) (1.4)

where k = 3 Parke-Taylor functions [8] are naturally generalized to

(3)

(1, 2, . . . , n) :=

|123| |234| · · · |n12|

. (1.5)

CHY integrals are localized to the solutions of the scattering equations [4]. For k = 2 it

is known that the number of solutions is (n − 3)!. For k = 3 it is known that there are 2

and 26 solutions for n = 5 and n = 6 respectively [5]. Also in [5] two lower bounds for the

number of solutions for n = 7 were obtained using soft limit arguments. The stronger of

the two is 1152. In this note we prove that the number of solutions is 1272. This number

is consistent with a computation done by T. Lam using uniform matriods over ﬁnite ﬁelds

which also reproduces all other known results [9].

In [5] the authors provided evidence that CHY integrals deﬁned on X(k, n) have a deep

and surprising connection to the tropical Grassmannian Trop G(k, n). The connection for

k = 2 is fairly direct. It maps all possible 2

n−1

− n − 1 propagators to all possible vertices

(or rays) of Trop G(2, n). In addition, there is also a bijection between every one of the

(2n − 5)!! tree-level Feynman diagrams in a cubic scalar theory and facets of Trop G(2, n).

In [5] the case of n = 6 and k = 3 was studied in detail. In this work we extend the

same analysis to n = 7, k = 3 biadjoint amplitudes and Trop G(3, 7). Luckily, the structure

of Trop G(3, 7) has been carefully studied by Herrmann, Jensen, Joswig, and Sturmfels [10]

and we make use of their results to carry out all our Feynman diagram computations which

lead to the explicit form of all 360 × 360 biadjoint amplitudes m

(3)

(α|β).

The strategy for determining the number of solutions to the X(3, 7) scattering equa-

tions is the following. We start with the dual version, i.e., the X(4, 7) scattering equations

near a soft limit. The strict soft limit gives equations on X(4, 6) w X(2, 6) and produces

6 solutions. Each solution is then used to ﬁnd a polynomial of degree 192 from the equa-

tions for the soft particle. This produces 6 × 192 = 1152 seeds of solutions from which

a numerical search for the true solutions can be launched. In the kinematics we use this

search returned 1162 solutions which we obtained with more than 2000 digits of precision.

Having the explicit form of all 360 × 360 biadjoint amplitudes m

(3)

(α|β) and their

corresponding approximation m

(3):approx

(α|β) from using (1.4) with 1162 solutions, we

– 2 –

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双伴随振幅与Trop G(3,7)、X(3,7)方程的1272个解解析

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