超越对数精度：八粒子三环振幅在N=4超杨-米尔斯理论中的多Regge极限计算

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本文主要探讨了在平面N=4超级杨-米尔斯理论的多Regge极限中，对于八粒子振幅的计算，特别是在超越了传统的主导对数精度方面的重要进展。作者Robin Marzucca和Bram Verbeek来自UCLouvain的CP3宇宙学、粒子物理和现象学中心，他们从八粒子的全环路色散积分形式出发，对该问题进行了深入研究。首先，他们针对八粒子情况分析了色散积分，将它与一个明确定义的傅立叶-梅林变换联系起来。傅立叶-梅林变换是解析数论中的一个重要工具，它在处理复变函数的频域表示时非常有效。利用这种变换的性质，尤其是其卷积积结构，作者能够以接近对数精度计算出三环的八粒子MHV振幅，这是一种特殊的量子色动力学（QCD）中的高阶计算，对于理解粒子相互作用具有关键意义。接下来，他们通过MHV（最大哈密顿量）结果，进一步推导出了三环八粒子振幅在多Regge运动学中的表达。在多Regge极限下，粒子之间的散射行为遵循一种指数增长的规律，这是粒子在高能情况下相互作用的重要特性。他们的计算结果显示，三环八粒子振幅由权重相等的单值多项式对数组合而成，这些多项式对数反映了振幅的奇异行为，这些奇异点结构与之前对数精度下的分类是一致的。值得注意的是，这项工作不仅提升了计算精度，还可能揭示了更深层次的物理原理，比如可能与超对称性和重整化群流动有关的结构。此外，该成果对于检验理论模型、验证广义的QCD软因子以及进一步探索强相互作用的固定点行为都有重要的实践价值。这篇发表在《Journal of High Energy Physics》(JHEP) 2019年7月号上的论文，通过对八粒子振幅的细致分析和高级计算技术的应用，为深化我们对N=4超级杨-米尔斯理论在高能物理领域的理解提供了重要突破。

JHEP07(2019)039

Amplitudes in planar N = 4 SYM obey dual conformal invariance, which implies that the

kinematical dependence can be expressed in terms of conformal cross-ratios

≡

i+1j

ij+1

i+1j+1

, (2.6)

where x

= x

− x

and indices are cyclically identiﬁed, i + N ' i. Of these, only 3N − 15

are algebraically independent in four dimensions.

In the multi-Regge-limit, amplitudes only depend on the transverse momenta p

equivalently the transverse dual coordinates x

deﬁned by

i+3

≡ x

i+2

− x

i+1

, 0 ≤ i ≤ N − 4, (2.7)

Since the kinematics is determined entirely by the x

which obey dual conformal symmetry,

we can identify the conﬁguration space as N − 2 points in CP

, or equivalently the moduli

space of genus zero curves with N − 2 marked points M

0,N−2

. Upon ﬁxing the residual

SL(2, C) symmetry of M

0,N−2

, we can ﬁnd various coordinate systems corresponding to

N − 5 cross ratios formed out of the dual coordinates x

, such as

≡

− x

i+3

) (x

i+2

− x

i+1

)

− x

i+1

) (x

i+2

− x

i+3

)

. (2.8)

These coordinates are well-suited to describe the Fourier-Mellin transforms in amplitudes

in MRK, and we will hence refer to them as Fourier-Mellin coordinates. A set of local

coordinates in which the functional dependence of MRK amplitudes is particularly simple

are the simplicial coordinates, obtained by ﬁxing three of the N − 2 points to 0, 1, and ∞.

A special set of simplicial coordinates named simplicial MRK coordinates is given by

, . . . , x

N−2

) → (1, 0, ρ

, . . . , ρ

N−5

, ∞) . (2.9)

They are related to the Fourier-Mellin coordinates by

(1 − ρ

i+1

)(ρ

− ρ

i−1

)

(1 − ρ

i−1

)(ρ

− ρ

i+1

)

, ρ

= 0 , ρ

N−4

= ∞ . (2.10)

This set of coordinates is particularly interesting, as it was shown ﬁrst for the two loop

symbol to NLLA in [25] and later to all orders at LLA at function level in [28] that multi-

leg amplitudes in the multi-Regge limit of N = 4 SYM factorize into a ﬁnite number of

building blocks with fewer legs when expressed in these coordinates. This property was

used to compute the scattering of N particles up to ﬁve loops at LLA for MHV amplitudes.

2.2 Single-valued polylogarithms

As a consequence of the kinematics being described entirely by M

0,N−2

, amplitudes in MRK

correspond to a family of iterated integrals called multiple polylogarithms (MPLs) [31],

deﬁned by the recursion

G(a

, . . . , a

; z) =

t − a

G(a

, . . . , a

; t) , (2.11)

– 4 –

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