揭示QCD高能极限中的胶子对散射关系：一环与两环的有趣关联

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本文探讨了量子色动力学(QCD)中胶子-胶子散射幅度在高能极限下的性质，主要关注的是在一环和两环级别的计算结果。作者Vittorio Del Duca，来自瑞士苏黎世ETH Zurich的理论物理研究所，研究了这些散射过程中的一个关键概念——胶子的单圈（one-loop）影响因子（gluon impact factor），这是量子场论中用于描述粒子相互作用强度的重要工具。在高能过程中，散射振幅的分析通常涉及对洛伦兹不变量的解析结构进行研究，尤其是对于Regge轨迹的理解，这在粒子物理学中是描述散射过程在能量增大的情况下的行为的一种模型。Regge轨迹是一种描述在高能下散射粒子交换的虚拟粒子质量与平方中心对撞能量之间关系的曲线。在这里，Del Duca发现了一个有趣的现象：在QCD胶子-胶子散射的高能极限下，单圈胶子影响因子的有限部分与两圈Regge轨迹的有限部分之间存在着意想不到的关联。文章的核心内容包括深入解析高能极限下的QCD算符维度和相互作用，以及探讨这种关联如何影响理论预测的精度和一致性。作者还特别提到了这篇论文是献给已故的物理学家Lev Nikolaevich Lipatov，他在高能物理领域有着重要贡献，特别是对于强相互作用的研究。此外，值得注意的是，该论文是在开放获取平台上发表的，由Springer出版，并通过SCOAP3项目资助，这意味着读者可以免费访问这篇重要的研究成果。全文讨论了理论方法、具体计算结果及其背后的物理意义，以及可能的后续研究方向，例如对多圈阶的胶子-胶子散射幅度的扩展分析和对Regge轨迹的进一步理解。这篇研究深入剖析了QCD中的高能极限行为，揭示了单圈与多圈散射过程的内在联系，不仅为理论物理提供了新的见解，也为实验数据的解读提供了潜在的理论依据。对于理解量子色动力学的强相互作用本质，以及在粒子碰撞实验如LHC中预测和解释高能现象具有重要意义。

JHEP02(2018)112

loops [31–36]. The one-loop impact factors [37–42] and the two-loop Regge trajectory [43–

47], which are building blocks to the BFKL equation at NLL accuracy, have poles in the

dimensional regulator ε in d = 4−2ε dimensions which can be understood, for the one-loop

impact factors, in terms of the one-loop cusp anomalous dimension and the one-loop quark

and gluon collinear anomalous dimensions [27, 28], and for the two-loop Regge trajectory

in terms of the two-loop cusp anomalous dimension [23].

Further, we know that the picture of high-energy factorisation based on one-reggeised-

gluon exchange breaks down at NNLO accuracy [47]. The violation can be explained

through infrared factorisation by showing that the real part of the amplitudes becomes non-

diagonal in the t-channel-exchange basis [24, 25]. Accordingly, it can be predicted how the

violation propagates to higher loops, and the three-loop prediction for the violation [27, 28],

which has NNLL accuracy, has been conﬁrmed by the explicit computation of the three-

loop four-point function of N = 4 SYM [51]. In the high-energy factorisation picture, that

violation is due to the contribution of the three-Reggeised-gluon exchange [29, 52]. Thus,

although a study of the BFKL ladder and ﬁrstly of its building blocks at NNLL accuracy

is yet to be undertaken, we already have a clear picture of how the factorisation violations

occur at NNLL accuracy.

While we have a precise knowledge of how the infrared poles occur and what they

mean in the loop corrections to impact factors and Regge trajectory, the ﬁnite parts of

those corrections are treated as free parameters, which neither infrared nor high-energy

factorisation can constrain. At NLL accuracy, they are ﬁxed either by direct computa-

tion [37, 38, 40, 43–46, 48, 49] or by extracting them from the knowledge of the scattering

amplitudes in the exact kinematics [41, 42, 47, 50]. The goal of this short note is to alert

the reader to the possibility that there might be some organised structure in the ﬁnite

parts of impact factors and Regge trajectory as well. We shall discuss amplitudes at NLL

accuracy, so we shall not be concerned with factorisation violations here.

In section 2, we focus on the ﬁnite part of the one-loop gluon impact factor, having

the ﬁnite part of the two-loop Regge trajectory as a target. In section 3, we discuss

exponentiation patterns that might help in understanding the ﬁndings of section 2. In

appendix A, we recall the coeﬃcients of the cusp and collinear anomalous dimensions,

used throughout the note. In appendix B, we provide for completeness also the one-loop

quark impact factor.

2 Playing with ﬁnite terms

In order to illustrate the issue, we shall consider the tree-level amplitude for gluon-gluon

scattering g

→ g

, which may be written as [3],

(0)aa

gg→gg

= 2s

)

i(0)

, p

)

j(0)

, p

)

, (2.1)

where a, a

, b, b

represent the colours of the scattering gluons, and G represents the adjoint

representation of SU(N), with (T

)

= if

acb

, with the colour matrices normalised in the

The one-loop central emission vertex [42, 48–50] has not been analysed yet in this fashion.

– 2 –

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