平面N=4超杨-米尔斯理论中的两环散射积分解析解

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本文档主要探讨了在平面$\mathcal{N}=4$超杨-米尔斯理论中的自举方法应用于两环（双圈）Feynman积分的符号解析计算。作者们针对七点和八点的两个环散射幅度，对特定的两环Feynman积分进行了深入研究。他们采用了启发自“自举”策略（bootstrapping），这是一种在高能物理计算中常用的数值和符号求解技术，旨在通过系统性的递归关系或约束条件来推测出复杂的数学结构。文中提到的方法结合了一组二级偏微分方程，这些方程为符号ansatz（符号假设）提供了强大的限制，有助于确定积分的符号表达式。符号ansatz是解析计算中的一种关键概念，它是一种假设的函数形式，用于表示Feynman积分的复变量表示。通过这些方程，研究者能够有效地缩小可能的符号形式，从而找到其精确解析。当完整符号字母集不可用时，作者们采取了混合方法，这可能是结合了符号推导和数值分析的手段，以弥补数据不完整时的不足。这种方法的灵活性使得他们能够在面对复杂问题时仍能推进计算，并得到有意义的结果。该研究发表于《Journal of High Energy Physics》（JHEP）的10期，2018年，是开放获取资源，表明其研究成果可供学术界广泛查阅和讨论。论文的作者来自多个知名机构，包括马克斯·普朗克物理研究所、约翰内斯·古腾堡大学和加州大学洛杉矶分校，他们的合作展示了在现代量子场论和计算物理领域中的跨学科协作。这篇论文的重要性在于它不仅推进了$\mathcal{N}=4$超杨-米尔斯理论中的精确计算，还展示了如何利用自举策略和高级数学工具解决高阶Feynman积分的符号解析问题，这对于理解基本粒子相互作用和量子场论的深层次性质具有重要意义。

JHEP10(2018)059

Caron-Huot [53] and extract the independent symbol alphabet or focus on the existing al-

phabet to describe the heptagon symbols relevant for MHV and NMHV scattering [37, 40].

In both cases we obtain equivalent 42-letter alphabets. The dimension of the alphabet coin-

cides with the dimension of the cluster algebra associated to the Grassmannian Gr(4, 7) [37].

A priori it is unclear if the symbol alphabet that is relevant for the full amplitudes is

suﬃcient to describe individual integrals. It is logically possible that Feynman integrals

depend on additional letters that drop out in the amplitude. As we will show below, this

is not the case for the seven point examples we discuss in this section but becomes relevant

starting at eight points. In the opposite direction, individual Feynman integrals might

live in lower-dimensional subspaces of the full heptagon-space, but for uniformity of the

discussion, it will be advantageous to embed the integrals into the more general setting

of the six dimensional heptagon space characterized by the 42 heptagon letters introduced

in [37, 40]. Here we follow the conventions of [37, 40] and denote the letters by a

i,j

1,1

h1234ih1567ih2367i

h1237ih1267ih3456i

, a

2,1

h1234ih2567i

h1267ih2345i

3,1

h1567ih2347i

h1237ih4567i

h1234i

4,1

h2457ih3456i

h2345ih4567i

5,1

h1245ih1367i − h1267ih1345i

h1234ih1567i

h1(23)(45)(67)i

h1234ih1567i

6,1

h1356ih1472i − h1372ih1456i

h1234ih1567i

h1(34)(56)(72)i

h1234ih1567i

(3.1)

The remaining letters are given by the cyclic permutation of the particle labels,

= a



7→Z

k+j−1

. (3.2)

Parity transformations act nontrivially on these letters, it leaves a

and a

invariant and

maps the other letters to cyclic images according to the rule [37, 40],

↔ a

, a

↔ a

, a

↔ a

, a

↔ a

, (3.3)

and cyclic images thereof.

First entry condition. Scattering amplitudes for massless particles as well as Feynman

integrals with massless internal propagators can have branch cuts only when Mandelstam

invariants s

or s

,...,i

go to zero or inﬁnity as dictated by locality and unitarity [41]. In

the bootstrap literature, this is known as the branch-cut condition and imposes constraints

on the allowed ﬁrst entries of a symbol where only Mandelstam invariants can appear. In

the example of the heptagon alphabet this branch cut condition imposes that only the

letters a

1,j

can appear as valid ﬁrst entries [37, 40],

1,1

h1234ih1567ih2367i

h1237ih1267ih3456i

712

, (3.4)

as a

1,1

and its cyclic images are the only letters which exclusively involve Mandelstam

invariants.

– 7 –

JHEP10(2018)059

Steinmann conditions. Feynman integrals in a local Lorentz-invariant quantum ﬁeld

theory are subject to analyticity constraints derived from locality. The Steinmann con-

ditions [62–64], which were originally derived in the context of axiomatic quantum ﬁeld

theory, are some of the key constraints that recently experienced a revived interest con-

cerning scattering amplitudes for massless particles [39, 40] (see also e.g. [65, 66] for the

use of Steinmann conditions in the multi-Regge-limit). At a practical level, the Steinmann

conditions state that scattering amplitudes (and Feynman integrals) can not have double-

discontinuities in overlapping kinematic channels. For a more detailed discussion we refer

the reader directly to [39, 40]. For the bootstrap program, disallowed double-discontinuities

are associated with certain exclusions on consecutive letters in the symbol.

At the level of multi-particle invariants in the heptagon case, this means that symbols

of Feynman integrals can not involve adjacent letters that are associated with overlapping

three-particle channels. Since the structure of the 42-letter heptagon alphabet (3.1) is such

that each of the 7 three-particle invariants is associated to a unique letter, the Steinmann

conditions are easily imposed at the symbol level. For example from (3.4) we see that

1,1

↔ s

712

which indicates that a

1,1

can not be adjacent to a

1,2

, a

1,3

, and a

1,6

(and cyclic

images thereof). For further details, we refer directly to [37, 40].

Integrability. Besides these two more detailed constraints that rely on the analytic struc-

ture of Feynman integrals, there is a general constraint imposed on symbols coming from

the fact that partial diﬀerentials ought to commute for well deﬁned functions. This con-

straint is known as the integrability condition. Starting from the symbol of a weight n

function F written in a factorized form

S(F ) =

i,j

S(F

i,j

) ⊗ log α

, ⊗log α

, (3.5)

where the sum i, j is over all letters of the alphabet α

i,j

and F

i,j

are weight n−2 functions.

Equipped with these deﬁnitions, we can write the integrability conditions as follows,

(∂

∂

− ∂

∂

)F = 0 ⇔

i,j



S(F

i,j

) − S(F

j,i

)





∂ log α

∂x



∂ log α

∂x



= 0 , (3.6)

for any combination of partial derivatives with respect to the underlying variables x

, x

These integrability equations give constraints on the antisymmetric combinations, S(F

i,j

)−

S(F

j,i

), which can then be imposed iteratively in weight to construct integrable symbols.

Further details can be found e.g. in [37, 40].

Following the discussion above, one can count the number of integrable symbols that

satisfy,

• First entry condition (only a

can appear in ﬁrst entry)

• Steinmann constraints (no overlapping discontinuities in incompatible channels)

• Integrability

– 8 –

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